Three-Dimensional Simulations of Subaerial Landslide-Generated Waves: Comparing OpenFOAM and FLOW-3D HYDRO Models

지표 산사태로 발생한 파랑의 3차원 시뮬레이션: OpenFOAM과 FLOW-3D HYDRO 모델 비교

Ramtin Sabeti, Mohammad Heidarzadeh, Alessandro Romano, Gabriel Barajas Ojeda & Javier L. Lara

Abstract

The recent destructive landslide tsunamis, such as the 2018 Anak Krakatau event, were fresh reminders for developing validated three-dimensional numerical tools to accurately model landslide tsunamis and to predict their hazards. In this study, we perform Three-dimensional physical modelling of waves generated by subaerial solid-block landslides, and use the data to validate two numerical models: the commercial software FLOW-3D HYDRO and the open-source OpenFOAM package. These models are key representatives of the primary types of modelling tools—commercial and open-source—utilized by scientists and engineers in the field. This research is among a few studies on 3D physical and numerical models for landslide-generated waves, and it is the first time that the aforementioned two models are systematically compared. We show that the two models accurately reproduce the physical experiments and give similar performances in modelling landslide-generated waves. However, they apply different approaches, mechanisms and calibrations to deliver the tasks. It is found that the results of the two models are deviated by approximately 10% from one another. This guide helps engineers and scientists implement, calibrate, and validate these models for landslide-generated waves. The validity of this research is confined to solid-block subaerial landslides and their impact in the near-field zone.

1 Introduction and Literature Review

Subaerial landslide-generated waves represent major threats to coastal areas and have resulted in destruction and casualties in several locations worldwide (Heller et al., 2016; Paris et al., 2021). Interest in landslide-generated tsunamis has risen in the last decade due to a number of devastating events, especially after the December 2018 Anak Krakatau tsunami which left a death toll of more than 450 people (Grilli et al., 2021; Heidarzadeh et al., 2020a). Another significant subaerial landslide tsunami occurred on 16 October 1963 in Vajont dam reservoir (Northern Italy), when an impulsive landslide-generated wave overtopped the dam, killing more than 2000 people (Heller & Spinneken, 2013; Panizzo et al., 2005). The largest tsunami run-up (524 m) was recorded in Lituya Bay landslide tsunami event in 1958 where it killed five people (Fritz et al., 2009).

To achieve a better understanding of subaerial landslide tsunamis, laboratory experiments have been performed using two- and three-dimensional (2D, 3D) set-ups (Bellotti & Romano, 2017; Di Risio et al., 2009; Fritz et al., 2004; Romano et al., 2013; Sabeti & Heidarzadeh, 2022a). Results of physical models are essential to shed light on the nonlinear physical phenomena involved. Furthermore, they can be used to validate numerical models (Fritz et al., 2009; Grilli & Watts, 2005; Liu et al., 2005; Takabatake et al., 2022). However, the complementary development of numerical tools for modelling of landslide-generated waves is inevitable, as these models could be employed to accelerate understanding the nature of the processes involved and predict the detailed outcomes in specific areas (Cremonesi et al. 2011). Due to the high flexibility of numerical models and their low costs in comparison to physical models, validated numerical models can be used to replicate actual events at a fair cost and time (e.g., Cecioni et al., 2011; Grilli et al., 2017; Heidarzadeh et al., 2020b, 2022; Horrillo et al., 2013; Liu et al., 2005; Løvholt et al., 2005; Lynett & Liu, 2005).

Table 1 lists some of the existing numerical models for landslide tsunamis although the list is not exhaustive. Traditionally, Boussinesq-type models, and Shallow water equations have been used to simulate landslide tsunamis, among which are TWO-LAYER (Imamura and Imteaz,1995), LS3D (Ataie-Ashtiani & Najafi Jilani, 2007), GLOBOUSS (Løvholt et al., 2017), and BOUSSCLAW (Kim et al., 2017). Numerical models that solve Navier–Stokes equations showed good capability and reliability to simulate subaerial landslide-generated waves (Biscarini, 2010). Considering the high computational cost of solving the full version of Navier–Stokes equations, a set of methods such as RANS (Reynolds-averaged Navier–Stokes equations) are employed by some existing numerical models (Table 1), which provide an approximate averaged solution to the Navier–Stokes equations in combination with turbulent models (e.g., k–ε, k–ω). Multiphase flow models were used to simulate the complex dynamics of landslide-generated waves, including scenarios where the landslide mass is treated as granular material, as in the work by Lee and Huang (2021), or as a solid block (Abadie et al., 2010). Among the models listed in Table 1, FLOW-3D HYDRO and OpenFOAM solve Navier–Stokes equations with different approaches (e.g., solving the RANS by IHFOAM) (Paris et al., 2021; Rauter et al., 2022). They both offer a wide range of turbulent models (e.g., Large Eddy Simulation—LES, k–ε, k–ω model with Renormalization Group—RNG), and they both use the VOF (Volume of Fluid) method to track the water surface elevation. These similarities are one of the motivations of this study to compare the performance of these two models. Details of governing equations and numerical schemes are discussed in the following.

Numerical models	Approach	Developer
FLOW-3D HYDRO	This CFD package solves Navier–Stokes equations using finite-difference and finite volume approximations, along with Volume of Fluid (VOF) method for tracking the free surface	Flow Science, Inc. (https://www.flow3d.com/)
MIKE 21	This model is based on the numerical solution of 2D and 3D incompressible RANS equations subject to the assumptions of Boussinesq and hydrostatic pressure	Danish Hydraulic Institute (DHI) (https://www.mikepoweredbydhi.com/products/mike-21-3)
OpenFOAM (IHFOAM solver)	IHFOAM is a newly developed 3D numerical two-phase flow solver. Its core is based on OpenFOAM^®. IHFOAM can also solve two-phase flow within porous media using RANS/VARANS equations	IHCantabria research institute (https://ihfoam.ihcantabria.com/)
NHWAVE	NHWAVE is a 3D shock-capturing non-Hydrostatic model which solves the incompressible Navier–Stokes equations in terrain and surface-following sigma coordinates	Kirby et al. (2022) (https://sites.google.com/site/gangfma/nhwave, https://github.com/JimKirby/NHWAVE)
GLOBOUSS	GloBouss is a depth-averaged model based on the standard Boussinesq equations including higher order dispersion terms, Coriolis terms, and numerical hydrostatic correction terms	Løvholt et al. (2022) (https://www.duo.uio.no/handle/10852/10184)
BOUSSCLAW	BoussClaw is a new hybrid Boussinesq type model which is an extension of the GeoClaw model. It employs a hybrid of finite volume and finite difference methods to solve Boussinesq equations	Clawpack Development Team (http://www.clawpack.org/)Kim et al. (2017)
THETIS-MUI	THETIS is a multi-fluid Navier–Stokes solver which can be considered a one-fluid model as only one velocity is defined at each point of the mesh and there is no mixing between the three considered fluids (water, air, and slide). It applies VOF method	TREFLE department of the I2M Laboratory at Bordeaux, France (https://www.i2m.u-bordeaux.fr/en)
LS3D	A 2D depth-integrated numerical model which applies a fourth-order Boussinesq approximation for an arbitrary time-variable bottom boundary	Ataie-Ashtiani and Najafi Jilani (2007)
LYNETT- Mild-Slope Equation (MSE)	MSE is a depth-integrated version of the Laplace equation operating under the assumption of inviscid flow and mildly varying bottom slopes	Lynett and Martinez (2012)
Tsunami 3D	A simplified 3D Navier–Stokes model for two fluids (water and landslide material) using VOF for tracking of water surface	Horrillo et al. (2013)Kim et al. (2020)
(Cornell Multi-grid Coupled Tsunami Mode (COMCOT)	COMCOT adopts explicit staggered leap-frog finite difference schemes to solve Shallow Water Equations in both Spherical and Cartesian Coordinates	Liu et al. (1998); Wang and Liu (2006)
TWO-LAYER	A mathematical model for a two-layer flow along a non-horizontal bottom. Conservation of mass and momentum equations are depth integrated in each layer, and nonlinear kinematic and dynamic conditions are specified at the free surface and at the interface between fluids	Imamura and Imteaz (1995)

Table 1 Some of the existing numerical models for simulating landslide-generated waves

In this work, we apply two Computational Fluid Dynamic (CFD) frameworks, FLOW-3D HYDRO, and OpenFOAM to simulate waves generated by solid-block subaerial landslides in a 3D set-up. We calibrate and validate both numerical models using our physical experiments in a 3D wave tank and compare the performances of these models systematically. These two numerical models are selected among the existing CFD solvers because they have been reported to provide valuable insights into landslide-generated waves (Kim et al., 2020; Romano et al., 2020a, b ; Sabeti & Heidarzadeh, 2022a). As there is no study to compare the performances of these two models (FLOW-3D HYDRO and OpenFOAM) with each other in reproducing landslide-generated waves, this study is conducted to offer such a comparison, which can be helpful for model selection in future research studies or industrial projects. In the realm of tsunami generation by subaerial landslides, the solid-block approach serves as an effective representative for scenarios where the landslide mass is more cohesive and rigid, rather than granular. This methodology is particularly relevant in cases such as the 2018 Anak Krakatau or 1963 Vajont landslides, where the landslide’s nature aligns closely with the characteristics simulated by a solid-block model (Zaniboni & Tinti, 2014; Heidarzadeh et al., 2020a, 2020b).

The objectives of this research are: (i) To provide a detailed implementation and calibration for simulating solid-block subaerial landslide-generated waves using FLOW-3D HYDRO and OpenFOAM, and (ii) To compare the performance of these two numerical models based on three criteria: free surface elevation of the landslide-generated waves, capabilities of the models in simulating 3D features of the waves in the near-field, velocity fields, and velocity variations at different locations. The innovations of this study are twofold: firstly, it is a 3D study involving physical and numerical modelling and thus the data can be useful for other studies, and secondly, it compares the performance of two popular CFD models in modelling landslide-generated waves for the first time. The validated models such as those reported in this study and comparison of their performances can be useful for engineers and scientists addressing landslide tsunami hazards worldwide.

2 Data and Methods

2.1 Physical Modelling

To validate our numerical models, a series of three-dimensional physical experiments were carried out at the Hydraulic Laboratory of the Brunel University London (UK) in a 3D wave tank 2.40 m long, 2.60 m wide, and 0.60 m high (Figs. 1 and 2). To mitigate experimental errors and enhance the reliability of our results, each physical experiment was conducted three times. The reported data in the manuscript reflects the average of these three trials, assuming no anomalous outliers, thus ensuring an accurate reflection of the experimental tests. One experiment was used for validation of our numerical models. The slope angle (α) and water depth (h) were 45° and 0.246 m, respectively for this experiment. The movement of the sliding mass was recorded by a digital camera with a sampling frequency of 120 frames per second, which was used to calculate the slide impact velocity (vs). The travel distance (D), defined as the distance from the toe of the sliding mass to the water surface, was D=0.045 m. The material of the solid block used in our study was concrete with a density of 2600 kg/m³. Table 2 provides detailed information on the dimensions and kinematics of this solid block used in our physical experiments.

Figure 1. The geometrical and kinematic parameters of a subaerial landslide tsunami. Parameters are: h, water depth; aM, maximum wave amplitude; α, slope angle;vs, slide velocity; ls, length of landslide; bs, width of landslide; s, thickness of landslide; SWL, still water level; D, travel distance (the distance from the toe of the sliding mass to the water surface); L, length of the wave tank; and W, width of the wave tank and H, is the hight of the wave tank

Figure 2. a Wave tank setup of the physical experiments of this study. b Numerical simulation setup for the FLOW-3D HYDRO Model. c The numerical set-up for the OpenFOAM model. The location of the physical wave gauge (represented by numerical gauge WG-3 in the numerical simulations) is at X = 1.03 m, Y = 1.21 m, and Z = 0.046 m. d Top view showing the locations of numerical wave gauges (WG-1, WG-2, WG-3, WG-4, WG-5)

Parameter, unit	Value/type
Slide width (bs), m	0.26
Slide length (ls), m	0.20
Slide thickness (s), m	0.10
Slide volume (V), m³	2.60 × 10^–3
Specific gravity, (γs)	2.60
Slide weight (ms), kg	6.86
Slide impact velocity (vs), m/s	1.84
Slide Froude number (Fr)	1.18
Material	Concrete

Table 2 Geometrical and kinematic information of the sliding mass used for physical experiments in this study

We took scale effects into account during physical experiments by considering the study by Heller et al. (2008) who proposed a criterion for avoiding scale effects. Heller et al. (2008) stated that the scale effects can be negligible as long as the Weber number (W=ρgh2/σ; where σ is surface tension coefficient) is greater than 5.0 × 10³ and the Reynolds number (R=g0.5h1.5/ν; where ν is kinematic viscosity) is greater than 3.0 × 10⁵ or water depth (h) is approximately above 0.20 m. Considering the water temperature of approximately 20 °C during our experiments, the kinematic viscosity (ν) and surface tension coefficient (σ) of water become 1.01 × 10^–6 m²/s and 0.073 N/m, respectively. Therefore, the Reynolds and Weber numbers were as R= 3.8 × 10⁵ and W= 8.1 × 10⁵, indicating that the scale effect can be insignificant in our experiments. To record the waves, we used a twin wire wave gauge provided by HR Wallingford (https://equipit.hrwallingford.com). This wave gauge was placed at X = 1.03 m, Y = 1.21 m based on the coordinate system shown in Fig. 2a.

2.2 Numerical Simulations

The numerical simulations in this work were performed employing two CFD packages FLOW-3D HYDRO, and OpenFOAM which have been widely used in industry and academia (e.g., Bayon et al., 2016; Jasak, 2009; Rauter et al., 2021; Romano et al., 2020a, b; Yin et al., 2015).

2.2.1 Governing Equations and Turbulent Models

2.2.1.1 FLOW-3D HYDRO

The FLOW-3D HYDRO solver is based on the fundamental law of mass, momentum and energy conservation. To estimate the influence of turbulent fluctuations on the flow quantities, it is expressed by adding the diffusion terms in the following mass continuity and momentum transport equations:

quation (1) is the general mass continuity equation, where u is fluid velocity in the Cartesian coordinate directions (x), Ax is the fractional area open to flow in the x direction, VF is the fractional volume open to flow, ρ is the fluid density, R and ξ are coefficients that depend on the choice of the coordinate system. When Cartesian coordinates are used, R is set to unity and ξ is set to zero. RDIF and RSOR are the turbulent diffusion and density source terms, respectively. Uρ=Scμ∗/ρ, in which Sc is the turbulent Schmidt number, μ∗ is the dynamic viscosity, and ρ is fluid density. RSOR is applied to model mass injection through porous obstacle surfaces.

The 3D equations of motion are solved with the following Navier–Stokes equations with some additional terms:

where t is time, Gx is accelerations due to gravity, fx is viscous accelerations, and bx is the flow losses in porous media.

According to Flow Science (2022), FLOW-3D HYDRO’s turbulence models differ slightly from other formulations by generalizing the turbulence production with buoyancy forces at non-inertial accelerations and by including the influence of fractional areas/volumes of the FAVOR method (Fractional Area-Volume Obstacle Representation) method. Here we use k–ω model for turbulence modelling. The k–ω model demonstrates enhanced performance over the k-ε and Renormalization-Group (RNG) methods in simulating flows near wall boundaries. Also, for scenarios involving pressure changes that align with the flow direction, the k–ω model provides more accurate simulations, effectively capturing the effects of these pressure variations on the flow (Flow Science, 2022). The equations for turbulence kinetic energy are formulated as below based on Wilcox’s k–ω model (Flow Science, 2022):

where kT is turbulent kinetic energy, PT is the turbulent kinetic energy production, DiffKT is diffusion of turbulent kinetic energy, GT is buoyancy production, β∗=0.09 is closure coefficient, and ω is turbulent frequency.

2.2.1.2 OpenFOAM

For the simulations conducted in this study, OpenFOAM utilizes the Volume-Averaged RANS equations (VARANS) to enable the representation of flow within porous material, treated as a continuous medium. The momentum equation incorporates supplementary terms to accommodate frictional forces from the porous media. The mass and momentum conservation equations are linked to the VOF equation (Jesus et al., 2012) and are expressed as follows:

where the gravitational acceleration components are denoted bygj. The term u¯i=1Vf∫Vf0ujdV represents the volume averaged ensemble averaged velocity (or Darcy velocity) component, Vf is the fluid volume contained in the average volumeV,τ is the surface tension constant (assumed to be 1 for the water phase and 0 for the air phase), and fσi is surface tension, defined as fσi=σκ∂α∂xi, where σ (N/m) is the surface tension constant and κ (1/m) is the curvature (Brackbill et al., 1992). μeff is the effective dynamic viscosity that is defined as μeff=μ+ρνt and takes into account the dynamic molecular (μ) and the turbulent viscosity effects (ρνt). νt is eddy viscosity, which is provided by the turbulence closure model. n is the porosity, defined as the volume of voids over total volume, and P∗=1Vf∫∂Vf0P∗dS is the ensemble averaged pressure in excess of hydrostatic pressure. The coefficient A accounts for the frictional force induced by laminar Darcy-type flow, B considers the frictional force under turbulent flow conditions, and c accounts for the added mass. These coefficients (A,B, and c) are defined based on the work of Engelund (1953) and later modified by Van Gent (1995) as given below:

where D50 is the mean nominal diameter of the porous material, KC is the Keulegan–Carpenter number, a and b are empirical nondimensional coefficients (see Lara et al., 2011; Losada et al., 2016) and γ = 0.34 is a nondimensional parameter as proposed by Van Gent (1995). The k-ω Shear Stress Transport (SST) turbulence is employed to capture the effect of turbulent flow conditions (Zhang & Zhang, 2023) with the enhancement proposed by Larsen and Fuhrman (2018) for the over-production of turbulence beneath surface waves. Boundary layers are modelled with wall functions. The reader is referred to Larsen and Fuhrman (2018) for descriptions, validations, and discussions of the stabilized turbulence models.

2.2.2 FLOW-3D HYDRO Simulation Procedure

In our specific case in this study, FLOW-3D HYDRO utilizes the finite-volume method to numerically solve the equations described in the previous Sect. 2.2.1.1, ensuring a high level of accuracy in the computational modelling. The use of structured rectangular grids in FLOW-3D HYDRO offers the advantages of easier development of numerical methods, greater transparency in their relation to physical problems, and enhanced accuracy and stability of numerical solutions. (Flow Science, 2022). Curved obstacles, wall boundaries, or other geometric features are embedded in the mesh by defining the fractional face areas and fractional volumes of the cells that are open to flow (the FAVOR method). The VOF method is employed in FLOW-3D HYDRO for accurate capturing of the free-surface dynamics (Hirt and Nichols 1981). This approach then is upgraded to method of the TruVOF which is a split Lagrangian method that typically produces lower cumulative volume error than the alternative methods (Flow Science, 2022).

For numerical simulation using FLOW-3D HYDRO, the entire flow domain was 2.60 m wide, 0.60 m deep and 2.50 m long (Fig. 2b). The specific gravity (γs) for solid blocks was set to 2.60 in our model, aligning closely with the density of the actual sliding mass, which was approximately determined in our physical experiments. The fluid medium was modelled as water with a density of 1000 kg/m³ at 20 °C. A uniform grid comprising of one single mesh plane was applied with a grid size of 0.005 m. The top, front and back of the mesh areas were defined as symmetry, and the other surfaces were of wall type with no-slip conditions around the walls.

To simulate turbulent flows, k-ω model was used because of its accuracy in modelling turbulent flows (Menter 1992). Landslide movement was replicated in simulations using coupled motion objects, which implies that the movement of landslides is based on gravity and the friction between surfaces rather than a specified motion in which the model should be provided by force and torques. The time intervals of the numerical model outputs were set to 0.02 s to be consistent with the actual sampling rates of our wave gauges in the laboratory. In order to calibrate the FLOW-3D HYDRO model, the friction coefficient is set to 0.45, which is consistent with the Coulombic friction measurements in the laboratory. The Courant Number (C=UΔtΔx) is considered as the criterion for the stability of numerical simulations which gives the maximum time step (Δt) for a prespecified mesh size (Δx) and flow speed (U). The Courant number was always kept below one.

2.2.3 OpenFOAM Simulation Procedure

OpenFOAM is an open-source platform containing several C⁺⁺ libraries which solves both 3D Reynolds-Averaged Navier–Stokes equations (RANS) and Volume-Averaged RANS equations (VARANS) for two-phase flows (https://www.openfoam.com/documentation/user-guide). Its implementation is based on a tensorial approach using object-oriented programming techniques and the Finite Volume Method (McDonald 1971). In order to simulate the subaerial landslide-generated waves, the IHFOAM solver based on interFoam (Higuera et al., 2013a, 2013b), and the overset mesh framework method are employed. The implementation of the overset mesh method for porous mediums in OpenFOAM is described in Romano et al. (2020a, b) for submerged rigid and impermeable landslides.

The overset mesh technique, as outlined by Romano et al. (2020a, b), uses two distinct domains: a moving domain that captures the dynamics of the rigid landslide and a static background domain to characterize the numerical wave tank. The overlapping of these domains results in a composite mesh that accurately depicts complex geometrical transformations while preserving mesh quality. A porous media with a very low permeability (n = 0.001) was used to simulate the impermeable sliding surfaces. RANS equations were solved within the porous media. The Multidimensional Universal Limiter with Explicit Solution (MULES) algorithm is employed for solving the (VOF) equation, ensuring precision in tracking fluid interfaces. Simultaneously, the PIMPLE algorithm is employed for the effective resolution of velocity–pressure coupling in the Eqs. 7 and 8. A background domain was created to reproduce the subaerial landslide waves with dimensions 2.50 m (x-direction) × 2.60 m (y-direction) × 0.6 m (z-direction) (Fig. 2c). The grid size is set to 0.005 m for the background mesh. A moving domain was applied in an area of 0.35 m (x-direction) × 0.46 m (y-direction) × 0.32 m (z-direction) with a grid spacing of 0.005 m and applying a body-fitted mesh approach, which contains the rigid and impermeable wedges. Wall condition with No-slip is defined as the boundary for the four side walls (left, right, front and back, in Fig. 1). Also, a non-slip boundary condition is specified to the bottom, whereas the top boundary is defined as open. The experimental slide movement time series is used to model the landslide motion in OpenFOAM. The applied equation is based on the analytical solution by Pelinovsky and Poplavsky (1996) which was later elaborated by Watts (1998). The motion of a sliding rigid body is governed by the following equation:

where, m represents the mass of the landslide, s is the displacement of the landslide down the slope, t is time elapsed, g stands for the acceleration due to gravity, θ is the slope angle, Cf is the Coulomb friction coefficient, Cm is the added mass coefficient, m0 denotes the mass of the water displaced by the moving landslide, A is the cross-sectional area of the landslide perpendicular to the direction of motion, ρ is the water density, and Cd is the drag coefficient.

2.2.4 Mesh Sensitivity Analysis

In order to find the most efficient mesh size, mesh sensitivity analyses were conducted for both numerical models (Fig. 3). We considered the influence of mesh density on simulated waveforms by considering three mesh sizes (Δx) of 0.0025 m, 0.005 m and 0.010 m. The results of FLOW-3D HYDRO revealed that the largest mesh deviates 9% (Fig. 3a, Δx = 0.0100 m) from two other finer meshes. Since the simulations by FLOW-3D HYDRO for the finest mesh (Δx = 0.0025 m) do not show any improvements in comparison with the 0.005 m mesh, therefore the mesh with the size of Δx = 0.0050 m is used for simulations (Fig. 3a). A similar approach was followed for mesh sensitivity of OpenFOAM mesh grids. The mesh with the grid spacing of Δx = 0.0050 m was selected for further simulations since a satisfactory independence was observed in comparison with the half size mesh (Δx = 0.0025 m). However, results showed that the mesh size with the double size of the selected mesh (Δx = 0.0100 m) was not sufficiently fine to minimize the errors (Fig. 3b).

Figure 3. a, b Sensitivity of numerical simulations to the sizes of the mesh (Δx) for FLOW-3D HYDRO, and OpenFOAM, respectively. The location of the wave gauge 3 (WG-3) is at X = 1.03 m, Y = 1.21 m, and Z = -0.55 m (see Fig. 2d)

In terms of computational cost, the time required for 2 s simulations by FLOW-3D HYDRO is approximately 4.0 h on a PC Intel^® Core™ i7-8700 CPU with a frequency of 3.20 GHz equipped with a 32 GB RAM. OpenFOAM requires 20 h to run 2 s of numerical simulation on 2 processors on a PC Intel^® Core™ i9-9900KF CPU with a frequency of 3.60 GHz equipped with a 364 GB RAM. Differences in computational time for simulations run with FLOW-3D HYDRO and OpenFOAM reflect the distinct characteristics of each numerical methods, and the specific hardware setups.

2.2.5 Validation

We validated both numerical models based on our laboratory experimental data (Fig. 4). The following criterion was used to assess the level of agreement between numerical simulations and laboratory observations:

where ε is the mismatch error, Obsi is the laboratory observation values, Simi is the simulation values, and the mathematical expression |X| represents the absolute value of X. The slope angle (α), water depth (h) and travel distance (D) were: α = 45°, h = 0.246 m and D = 0.045 m in both numerical models, consistent with the physical model. We find the percentage error between each simulated data point and its corresponding observed value, and subsequently average these errors to assess the overall accuracy of the simulation against the observed time series. Our results revealed that the mismatch errors between physical experiments and numerical models for the FLOW-3D HYDRO and OpenFOAM are 8% and 18%, respectively, indicating that our models reproduce the measured waveforms satisfactorily (Fig. 4). The simulated waveform by OpenFOAM shows a minor mismatch at t = 0.76 s which resulted from a droplet immediately after the slide hits the water surface in the splash zone. In term of the maximum negative amplitude, the simulated waves by OpenFOAM indicates a relatively better performance than FLOW-3D HYDRO, whereas the maximum positive amplitude (aM) simulated by FLOW-3D HYDRO is closer to the experimental value. The recorded maximum positive amplitude in physical experiment is 0.022 m, whereas it is 0.020 m for FLOW-3D HYDRO and 0.017 m for OpenFOAM simulations. In acknowledging the deviations observed, it is pertinent to highlight that while numerical models offer robust insights, the difference in meshing techniques and the distinct computational methods to resolve the governing equations in FLOW-3D HYDRO and OpenFOAM have contributed to the variance. Moreover, the intrinsic uncertainties associated with the physical experimentation process, including the precision of wave gauges and laboratory conditions, are non-negligible factors influencing the results.

Figure 4. Validation of the simulated waves (brown line for FLOW-3D HYDRO and green line for OpenFOAM) using the laboratory-measured waves (black solid diamonds). This physical experiment was conducted for wave gauge 3 (WG-3) located at X = 1.03 m, Y = 1.21 m, and Z = -0.55 m (see Fig. 2d). Here,
ε shows the errors between simulations and actual physical measurements using Eq. (13)

3 Results

Following the validations of the two numerical models (FLOW-3D HYDRO and OpenFOAM), a series of simulations were performed to compare the performances of these two CFD solvers. The generation process of landslide waves, waveforms, and velocity fields are considered as the basis for comparing the performance of the two models (Figs. 5, 6, 7 and 8).

Figure 5.Comparison between the simulated waveforms by FLOW-3D HYDRO (black) and OpenFOAM (red) at four different locations in the near-field zone (WG-1,2,4 and 5). WG is the abbreviation for wave gauge. The mismatch (Δ) between the two models at each wave gauge is calculated using Eq. (14)

**Figure 6**. Comparison of water surface elevations produced by solid-block subaerial landslides for the two numerical models FLOW-3D-HYDRO (a–c) and OpenFOAM (**e–g**) at different times

**Figure 7**. Snapshots of the simulations at different times for FLOW-3D HYDRO (a–c) and OpenFOAM (e–g) showing velocity fields (colour maps and arrows). The colormaps indicate water particle velocity in m/s, and the lines indicate the velocities of water particles

**Figure 8**. Comparison of velocity variations at (WG-3) for FLOW-3D HYDRO (light blue) and OpenFOAM (brown)

3.1 Comparison of Waveforms

Five numerical wave gauges were placed in our numerical models to measure water surface oscillations in the near-field zone (Fig. 5). These gauges offer an azimuthal coverage of 60° (Fig. 2d). Figure 5 reveals that the simulated waveforms from two models (FLOW-3D HYDRO and OpenFOAM) are similar. The highest wave amplitude (aM) is recorded at WG-3 for both models, whereas the lowest amplitude is recorded at WG-5 and WG-1 which can be attributed to the longer distances of these gauges from the source region as well as their lateral offsets, resulting in higher wave energy dissipation at these gauges. The sharp peaks observed in the simulated waveforms, such as the red peak between 0.8–1.0 s in Fig. 5a from OpenFOAM, the red peak between 0.6–0.8 s in Fig. 5b also from OpenFOAM, and the black peak between 1.4–1.6 s in Fig. 5d from FLOW-3D HYDRO, are due to the models’ spatial and temporal discretization. They reflect the sensitivity of the models to capturing transient phenomena, where the chosen mesh and time-stepping intervals are key factors in the models’ ability to track rapid changes in the flow field. To quantify the deviations of the two models from one another, we apply the following equation for mismatch calculation:

where Δ is the mismatch error, Sim1 is the simulation values from FLOW-3D HYDRO, Sim2 is the simulation values from OpenFOAM, and the mathematical expression |X| implies the absolute value of X. We calculate the percentage difference for each corresponding pair of simulation results, then take the mean of these percentage differences to determine the average deviation between the two simulation time series. Using Eq. (14), we found a deviation range from 9 to 11% between the two models at various numerical gauges (Fig. 5), further confirming that the two models give similar simulation results.

3.2 Three-Dimensional Vision of Landslide Generation Process by Numerical Models

A sequence of four water surface elevation snapshots at different times is shown in Fig. 6 for both numerical modes. In both simulations, the sliding mass travels a constant distance of 0.045 m before hitting the water surface at t = 0.270 s which induces an initial change in water surface elevation (Figs. 6a and e). At t = 0.420 s, the mass is fully immersed for both simulations and an initial dipole wave is generated (Figs. 6b and f). Based on both numerical models, the maximum positive amplitude (0.020 m for FLOW-3D HYDRO, and 0.017 m for OpenFOAM) is observed at this stage (Fig. 6). The maximum propagation of landslide-simulated waves along with more droplets in the splash zone could be seen at t = 0.670 s for both models (Fig. 6c and g). The observed distinctions in water surface elevation simulations as illustrated in Fig. 6 are rooted in the unique computational methodologies intrinsic to each model. In the OpenFOAM simulations, a more diffused water surface elevation profile is evident. Such diffusion is an outcome of the simulation’s intrinsic treatment of turbulent kinetic energy dissipation, aligning with the solver’s numerical dissipation characteristics. These traits are influenced by the selected turbulence models and the numerical advection schemes, which prioritize computational stability, possibly at the expense of interface sharpness. The diffusion in the wave pattern as rendered by OpenFOAM reflects the application of a turbulence model with higher dissipative qualities, which serves to moderate the energy retained during wave propagation. This approach can provide insights into the potential overestimation of energy loss under specific simulation conditions. In contrast, the simulations from FLOW-3D HYDRO depict a more localized wave pattern, indicative of a different approach to turbulent dissipation. This coherence in wave fronts is a function of the model’s specific handling of the air–water interface and its targeted representation of the energy dynamics resulting from the landslide’s interaction with the water body. They each have specific attributes that cater to different aspects of wave simulation fidelity, thereby contributing to a more comprehensive understanding of the phenomena under study.

3.3 Wave Velocity Analysis

We show four velocity fields at different times during landslide motion in Fig. 7 and one time series of velocity (Fig. 8) for both numerical models. The velocity varies in the range of 0–1.9 m/s for both models, and the spatial distribution of water particle velocity appears to be similar in both. The models successfully reproduce the complex wavefield around the landslide generation area, which is responsible for splashing water and mixing with air around the source zone (Fig. 7). The first snapshot at t = 0.270 s (Fig. 7a and e) shows the initial contact of the sliding mass with water surface for both numerical models which generates a small elevation wave in front of the mass exhibiting a water velocity of approximately 1.2 m/s. The slide fully immerses for the first time at t = 0.420 s producing a water velocity of approximately 1.5 m/s at this time (Fig. 7b and f). The last snapshot (t = 0.670 s) shows 1.20 s after the slide hits the bottom of the wave tank. Both models show similar patterns for the propagation of the waves towards the right side of the wave tank. The differences in water surface profiles close to the slope and solid block at t = 0.67 s, observed in the FLOW-3D HYDRO and OpenFOAM simulations (Figs. 6 and 7), are due to the distinct turbulence models employed by each (RNG and k-ω SST, respectively) which handle the complex interactions of the landslide-induced waves with the structures differently. Additionally, the methods of simulating landslide movement further contribute to this discrepancy, with FLOW-3D HYDRO’s coupled motion objects possibly affecting the waves’ initiation and propagation unlike OpenFOAM’s prescribed motion from experimental data. In addition to the turbulence models, the variations in VOF methodologies between the two models also contribute to the observed discrepancies.

For the simulated time series of velocity, both models give similar patterns and close maximum velocities (Fig. 8). For both models the WG-3 located at X = 1.03 m, Y = 1.21 m, and Z = − 0.55 m (Fig. 2d) were used to record the time series. WG-3 is positioned 5 mm above the wave tank bottom, ensuring that the measurements taken reflect velocities very close to the bottom of the wave tank. The maximum velocity calculated by FLOW-3D HYDRO is 0.162 m/s while it is 0.132 m/s for OpenFOAM, implying a deviation of approximately 19% from one another. Some oscillations in velocity records are observed for both models, but these oscillations are clearer and sharper for OpenFOAM. Although it is hard to see velocity oscillations in the FLOW-3D HYDRO record, a close look may reveal some small oscillations (around t = 0.55 s and 0.9 s in Fig. 8). In fact, velocity oscillations are expected due to the variations in velocity of the sliding mass during the travel as well as due to the interferences of the initial waves with the reflected wave from the beach. In general, it appears that the velocity time series of the two models show similar patterns and similar maximum values although they have some differences in the amplitudes of the velocity oscillations. The differences between the two curves are attributed to factors such as difference in meshing between the two models, turbulence models, as well as the way that two models record the outputs.

4 Discussions

An important step for CFD modelling in academic or industrial projects is the selection of an appropriate numerical model that can deliver the task with satisfactory performance and at a reasonable computational cost. Obviously, the major drivers when choosing a CFD model are cost, capability, flexibility, and accessibility. In this sense, the existing options are of two types as follows:

Commercial models, such as FLOW-3D HYDRO, which are optimised to solve free-surface flow problems, with customer support and an intuitive Graphical User Interface (GUI) that significantly facilitates meshing, setup, simulation monitoring, visualization, and post-processing. They usually offer high-quality customer support. Although these models show high capabilities and flexibilities for numerical modelling, they are costly, and thus less accessible.
Open-source models, such as OpenFOAM, which come without a GUI but with coded tools for meshing, setup, parallel running, monitoring, post-processing, and visualization. Although these models offer no customer support, they have a big community support and online resources. Open-source models are free and widely accessible, but they may not be necessarily always flexible and capable.

OpenFOAM provides freedom for experimenting and diving through the code and formulating the problem for a user whereas FLOW-3D HYDRO comes with high-level customer supports, tutorial videos and access to an extensive set of example simulations (https://www.flow3d.com/). While FLOW-3D-HYDRO applies a semi-automatic meshing process where users only need to input the 3D model of the structure, OpenFOAM provides meshing options for simple cases, and in many advanced cases, users need to create the mesh in other software (e.g., ANSYS) (Ariza et al., 2018) and then convert it to OpenFOAM format. Auspiciously, there are numerous online resources (https://www.openfoam.com/trainings/about-trainings), and published examples for OpenFOAM (Rauter et al., 2021; Romano et al., 2020a, b; Zhang & Zhang, 2023).

The capabilities of both FLOW-3D HYDRO and OpenFOAM to simulate actual, complex landslide-generated wave events have been showcased in significant case studies. The study by Ersoy et al. (2022) applied FLOW-3D HYDRO to simulate impulse waves originating from landslides near an active fault at the Çetin Dam Reservoir, highlighting the model’s capacity for detailed, site-specific modelling. Concurrently, the work by Alexandre Paris (2021) applied OpenFOAM to model the 2017 Karrat Fjord landslide tsunami events, providing a robust validation of OpenFOAM’s utility in capturing the dynamics of real-world geophysical phenomena. Both instances exemplify the sophisticated computational approaches of these models in aiding the prediction and analysis of natural hazards from landslides.

As for limitations of this study, we acknowledge that our numerical models are validated by one real-world measured wave time series. However, it is believed that this one actual measurement was sufficient for validation of this study because it was out of the scope of this research to fully validate the FLOW-3D HYDRO and OpenFOAM models. These two models have been fully validated by more actual measurements by other researchers in the past (e.g., Sabeti & Heidarzadeh, 2022b). It is also noted that some of the comparisons made in this research were qualitative, such as the 3D wave propagation snapshots, as it was challenging to develop quantitative comparisons for snapshots. Another limitation of this study concerns the number of tests conducted here. We fixed properties such as water depth, slope angle, and travel distance throughout this study because it was out of the scope of this research to perform sensitivity analyses.

5 Conclusions

We configured, calibrated, validated and compared two numerical models, FLOW-3D HYDRO, and OpenFOAM, using physical experiments in a 3D wave tank. These validated models were used to simulate subaerial solid-block landslides in the near-field zone. Our results showed that both models are fully compatible with investigating waves generated by subaerial landslides, although they use different approaches to simulate the phenomenon. The properties of solid-block, water depth, slope angle, and travel distance were kept constant in this study as we focused on comparing the performance of the two models rather than conducting a full sensitivity analysis. The findings are as follows:

Different settings were used in the two models for modelling landslide-generated waves. In terms of turbulent flow modelling, we used the Renormalization Group (RNG) turbulence model in FLOW-3D HYDRO, and k-ω (SST) turbulence model in OpenFOAM. Regarding meshing techniques, the overset mesh method was used in OpenFOAM, whereas the structured cartesian mesh was applied in FLOW-3D HYDRO. As for simulation of landslide movement, the coupled motion objects method was used in FLOW-3D HYDRO, and the experimental slide movement time series were prescribed in OpenFOAM.
Our modelling revealed that both models successfully reproduced the physical experiments. The two models deviated 8% (FLOW-3D HYDRO) and 18% (OpenFOAM) from the physical experiments, indicating satisfactory performances. The maximum water particle velocity was approximately 1.9 m/s for both numerical models. When the simulated waveforms from the two numerical models are compared with each other, a deviation of 10% was achieved indicating that the two models perform approximately equally. Comparing the 3D snapshots of the two models showed that there are some minor differences in reproducing the details of the water splash in the near field.
Regarding computational costs, FLOW-3D HYDRO was able to complete the same simulations in 4 h as compared to nearly 20 h by OpenFOAM. However, the hardware that were used for modelling were not the same; the computer used for the OpenFOAM model was stronger than the one used for running FLOW-3D HYDRO. Therefore, it is challenging to provide a fair comparison for computational time costs.
Overall, we conclude that the two models give approximately similar performances, and they are both capable of accurately modelling landslide-generated waves. The choice of a model for research or industrial projects may depend on several factors such as availability of local knowledge of the models, computational costs, accessibility and flexibilities of the model, and the affordability of the cost of a license (either a commercial or an open-source model).

Reference

Abadie, S., Morichon, D., Grilli, S., & Glockner, S. (2010). Numerical simulation of waves generated by landslides using a multiple-fluid Navier–Stokes model. Coastal Engineering, 57(9), 779–794. https://doi.org/10.1016/j.coastaleng.2010.04.002
Ariza, C., Casado, C., Wang, R.-Q., Adams, E., & Marugán, J. (2018). Comparative evaluation of OpenFOAM® and ANSYS® Fluent for the modeling of annular reactors. Chemical Engineering & Technology, 41(7), 1473–1483. https://doi.org/10.1002/ceat.201700455
Ataie-Ashtiani, B., & Najafi Jilani, A. (2007). A higher-order Boussinesq-type model with moving bottom boundary: Applications to submarine landslide tsunami waves. Pure and Applied Geophysics, 164(6), 1019–1048. https://doi.org/10.1002/fld.1354
Bayon, A., Valero, D., García-Bartual, R., & López-Jiménez, P. A. (2016). Performance assessment of OpenFOAM and FLOW-3D in the numerical modeling of a low Reynolds number hydraulic jump. Environmental Modelling & Software, 80, 322–335. https://doi.org/10.1016/j.envsoft.2016.02.018
Bellotti, G., & Romano, A. (2017). Wavenumber-frequency analysis of landslide-generated tsunamis at a conical island. Part II: EOF and modal analysis. Coastal Engineering, 128, 84–91. https://doi.org/10.1016/j.coastaleng.2017.07.008
Biscarini, C. (2010). Computational fluid dynamics modelling of landslide generated water waves. Landslides, 7(2), 117–124. https://doi.org/10.1007/s10346-009-0194-z
Brackbill, J. U., Kothe, D. B., & Zemach, C. (1992). A continuum method for modeling surface tension. Journal of computational physics, 100(2), 335–354.
Cecioni, C., Romano, A., Bellotti, G., Di Risio, M., & De Girolamo, P. (2011). Real-time inversion of tsunamis generated by landslides. Natural Hazards & Earth System Sciences, 11(9), 2511–2520. https://doi.org/10.5194/nhess-11-2511-2011
Cremonesi, M., Frangi, A., & Perego, U. (2011). A Lagrangian finite element approaches the simulation of water-waves induced by landslides. Computers & Structures, 89(11–12), 1086–1093.
Del Jesus, M., Lara, J. L., & Losada, I. J. (2012). Three-dimensional interaction of waves and porous coastal structures: Part I: Numerical model formulation. Coastal Engineering, 64, 57–72. https://doi.org/10.1016/J.COASTALENG.2012.01.008
Di Risio, M., De Girolamo, P., Bellotti, G., Panizzo, A., Aristodemo, F., Molfetta, M. G., & Petrillo, A. F. (2009). Landslide-generated tsunamis runup at the coast of a conical island: New physical model experiments. Journal of Geophysical Research: Oceans. https://doi.org/10.1029/2008JC004858
Engelund, F., & Munch-Petersen, J. (1953). Steady flow in contracted and expanded rectangular channels. La Houille Blanche, (4), 464–481.
Ersoy, H., Oğuz Sünnetci, M., Karahan, M., & Perinçek, D. (2022). Three-dimensional simulations of impulse waves originating from concurrent landslides near an active fault using FLOW-3D software: A case study of Çetin Dam Reservoir (Southeast Turkey). Bulletin of Engineering Geology and the Environment, 81(7), 267. https://doi.org/10.1007/s10064-022-02675-8
Flow Science. (2022). FLOW-3D HYDRO version 12.0 user’s manual. Santa Fe, NM, USA. Retrieved from https://www.flow3d.com/. 6 Aug 2023.
Fritz, H. M., Hager, W. H., & Minor, H. E. (2004). Near field characteristics of landslide generated impulse waves. Journal of waterway, port, coastal, and ocean engineering, 130(6), 287–302.
Fritz, H. M., Mohammed, F., & Yoo, J. (2009). Lituya bay landslide impact generated mega-tsunami 50th anniversary. In: Tsunami science four years after the 2004 Indian Ocean Tsunami (pp. 153–175). Birkhäuser Basel, Switzerland.
Grilli, S. T., Shelby, M., Kimmoun, O., Dupont, G., Nicolsky, D., Ma, G., Kirby, J. T., & Shi, F. (2017). Modelling coastal tsunami hazard from submarine mass failures: Effect of slide rheology, experimental validation, and case studies off the US East Coast. Natural Hazards, 86(1), 353–391. https://doi.org/10.1007/s11069-016-2692-3
Grilli, S. T., & Watts, P. (2005). Tsunami generation by submarine mass failure. I: Modelling, experimental validation, and sensitivity analyses. Journal of Waterway, Port, Coastal, and Ocean Engineering, 131(6), 283–297. https://doi.org/10.1061/(ASCE)0733-950X
Grilli, S. T., Zhang, C., Kirby, J. T., Grilli, A. R., Tappin, D. R., Watt, S. F. L., et al. (2021). Modeling of the Dec. 22nd, 2018, Anak Krakatau volcano lateral collapse and tsunami based on recent field surveys: Comparison with observed tsunami impact. Marine Geology. https://doi.org/10.1016/j.margeo.2021.106566
Heidarzadeh, M., Gusman, A., Ishibe, T., Sabeti, R., & Šepić, J. (2022). Estimating the eruption-induced water displacement source of the 15 January 2022 Tonga volcanic tsunami from tsunami spectra and numerical modelling. Ocean Engineering, 261, 112165. https://doi.org/10.1016/j.oceaneng.2022.112165
Heidarzadeh, M., Ishibe, T., Sandanbata, O., Muhari, A., & Wijanarto, A. B. (2020a). Numerical modeling of the subaerial landslide source of the 22 December 2018 Anak Krakatoa volcanic tsunami, Indonesia. Ocean Engineering, 195, 106733. https://doi.org/10.1016/j.oceaneng.2019.106733
Heidarzadeh, M., Putra, P. S., Nugroho, H. S., & Rashid, D. B. Z. (2020b). Field survey of tsunami heights and runups following the 22 December 2018 Anak Krakatau volcano tsunami, Indonesia. Pure and Applied Geophysics, 177, 4577–4595. https://doi.org/10.1007/s00024-020-02587-w
Heller, V., Bruggemann, M., Spinneken, J., & Rogers, B. D. (2016). Composite modelling of subaerial landslide–tsunamis in different water body geometries and novel insight into slide and wave kinematics. Coastal Engineering, 109, 20–41. https://doi.org/10.1016/j.coastaleng.2015.12.004
Heller, V., Hager, W. H., & Minor, H. E. (2008). Scale effects in subaerial landslide generated impulse waves. Experiments in Fluids, 44(5), 691–703. https://doi.org/10.1007/s00348-007-0427-7
Heller, V., & Spinneken, J. (2013). Improved landslide-tsunami prediction: Effects of block model parameters and slide model. Journal of Geophysical Research: Oceans, 118(3), 1489–1507. https://doi.org/10.1002/jgrc.20099
Higuera, P., Lara, J. L., & Losada, I. J. (2013a). Realistic wave generation and active wave absorption for Navier–Stokes models: Application to OpenFOAM®. Coastal Engineering, 71, 102–118. https://doi.org/10.1016/j.coastaleng.2012.07.002
Higuera, P., Lara, J. L., & Losada, I. J. (2013b). Simulating coastal engineering processes with OpenFOAM®. Coastal Engineering, 71, 119–134. https://doi.org/10.1016/j.coastaleng.2012.06.002
Hirt, C.W. and Nichols, B.D., 1981. Volume of fluid (VOF) method for the dynamics of free boundaries. Journal of computational physics, 39(1), 201–225.
Horrillo, J., Wood, A., Kim, G.-B., & Parambath, A. (2013). A simplified 3-D Navier–Stokes numerical model for landslide-tsunami: Application to the Gulf of Mexico. Journal of Geophysical Research, 118(12), 6934–6950. https://doi.org/10.1002/2012JC008689
Imamura, F., & Imteaz, M. A. (1995). Long waves in two-layers: Governing equations and numerical model. Science of Tsunami Hazards, 13(1), 3–24.
Jasak, H. (2009). OpenFOAM: Open source CFD in research and industry. International Journal of Naval Architecture and Ocean Engineering, 1(2), 89–94. https://doi.org/10.2478/IJNAOE-2013-0011
Kim, G. B., Cheng, W., Sunny, R. C., Horrillo, J. J., McFall, B. C., Mohammed, F., Fritz, H. M., Beget, J., & Kowalik, Z. (2020). Three-dimensional landslide generated tsunamis: Numerical and physical model comparisons. Landslides, 17(5), 1145–1161. https://doi.org/10.1007/s10346-019-01308-2
Kim, J., Pedersen, G. K., Løvholt, F., & LeVeque, R. J. (2017). A Boussinesq type extension of the GeoClaw model-a study of wave breaking phenomena applying dispersive long wave models. Coastal Engineering, 122, 75–86. https://doi.org/10.1016/j.coastaleng.2017.01.005
Kirby, J. T., Grilli, S. T., Horrillo, J., Liu, P. L. F., Nicolsky, D., Abadie, S., Ataie-Ashtiani, B., Castro, M. J., Clous, L., Escalante, C., Fine, I., et al. (2022). Validation and inter-comparison of models for landslide tsunami generation. Ocean Modelling, 170, 101943. https://doi.org/10.1016/j.ocemod.2021.101943
Lara, J. L., Ruju, A., & Losada, I. J. (2011). Reynolds averaged Navier–Stokes modelling of long waves induced by a transient wave group on a beach. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 467(2129), 1215–1242.
Larsen, B. E., & Fuhrman, D. R. (2018). On the over-production of turbulence beneath surface waves in Reynolds-averaged Navier–Stokes models. Journal of Fluid Mechanics, 853, 419–460.
Lee, C. H., & Huang, Z. (2021). Multi-phase flow simulation of impulsive waves generated by a sub-aerial granular landslide on an erodible slope. Landslides, 18(3), 881–895. https://doi.org/10.1007/s10346-020-01560-z
Liu, P. L. F., Woo, S. B., & Cho, Y. S. (1998). Computer programs for tsunami propagation and inundation. Technical Report. Cornell University, Ithaca, New York.
Liu, F., Wu, T.-R., Raichlen, F., Synolakis, C. E., & Borrero, J. C. (2005). Runup and rundown generated by three-dimensional sliding masses. Journal of Fluid Mechanics, 536(1), 107–144. https://doi.org/10.1017/S0022112005004799
Losada, I. J., Lara, J. L., & del Jesus, M. (2016). Modeling the interaction of water waves with porous coastal structures. Journal of Waterway, Port, Coastal, and Ocean Engineering, 142(6), 03116003
Løvholt, F., Harbitz, C. B., & Haugen, K. (2005). A parametric study of tsunamis generated by submarine slides in the Ormen Lange/Storegga area off western Norway. In: Ormen Lange—An integrated study for safe field development in the Storegga submarine area (pp. 219–231). Elsevier. https://doi.org/10.1016/B978-0-08-044694-3.50023-8
Løvholt, F., Bondevik, S., Laberg, J. S., Kim, J., & Boylan, N. (2017). Some giant submarine landslides do not produce large tsunamis. Geophysical Research Letters, 44(16), 8463–8472
Løvholt, F. J. M. R., Griffin, J., & Salgado-Gálvez, M. A. (2022). Tsunami hazard and risk assessment on the global scale. Complexity in Tsunamis, Volcanoes, and their Hazards, 213–246
Lynett, P., & Liu, P. L. F. (2005). A numerical study of the run-up generated by three-dimensional landslides. Journal of Geophysical Research: Oceans. https://doi.org/10.1029/2004JC002443
Lynett, P. J., & Martinez, A. J. (2012). A probabilistic approach for the waves generated by a submarine landslide. Coastal Engineering Proceedings, 33, 15–15. https://doi.org/10.9753/icce.v33.currents.15
McDonald, P. W. (1971). The computation of transonic flow through two-dimensional gas turbine cascades (Vol. 79825, p. V001T01A089). American Society of Mechanical Engineers
Menter, F. R. (1992). Improved two-equation k-omega turbulence models for aerodynamic flows (No. A-92183). https://ntrs.nasa.gov/citations/19930013620
Panizzo, A., De Girolamo, P., Di Risio, M., Maistri, A., & Petaccia, A. (2005). Great landslide events in Italian artificial reservoirs. Natural Hazards and Earth System Sciences, 5(5), 733–740. https://doi.org/10.5194/nhess-5-733-2005
Paris, A. (2021). Comparison of landslide tsunami models and exploration of fields of application. Doctoral dissertation, Université de Pau et des Pays de l’Adour.
Paris, A., Heinrich, P., & Abadie, S. (2021). Landslide tsunamis: Comparison between depth-averaged and Navier–Stokes models. Coastal Engineering, 170, 104022. https://doi.org/10.1016/j.coastaleng.2021.104022
Pelinovsky, E., & Poplavsky, A. (1996). Simplified model of tsunami generation by submarine landslides. Physics and Chemistry of the Earth, 21(1–2), 13–17. https://doi.org/10.1016/S0079-1946(97)00003-7
Rauter, M., Hoße, L., Mulligan, R. P., Take, W. A., & Løvholt, F. (2021). Numerical simulation of impulse wave generation by idealized landslides with OpenFOAM. Coastal Engineering, 165, 103815. https://doi.org/10.1016/j.coastaleng.2020.103815
Rauter, M., Viroulet, S., Gylfadóttir, S. S., Fellin, W., & Løvholt, F. (2022). Granular porous landslide tsunami modelling–the 2014 Lake Askja flank collapse. Nature Communications, 13(1), 678. https://doi.org/10.1038/s41467-022-28356-2
Romano, A., Bellotti, G., & Di Risio, M. (2013). Wavenumber–frequency analysis of the landslide-generated tsunamis at a conical island. Coastal Engineering, 81, 32–43. https://doi.org/10.1016/j.coastaleng.2013.06.007
Romano, A., Lara, J., Barajas, G., Di Paolo, B., Bellotti, G., Di Risio, M., Losada, I., & De Girolamo, P. (2020a). Tsunamis generated by submerged landslides: Numerical analysis of the near-field wave characteristics. Journal of Geophysical Research: Oceans, 125(7), e2020JC016157. https://doi.org/10.1029/2020JC016157
Romano, M., Ruggiero, A., Squeglia, F., Maga, G., & Berisio, R. (2020b). A structural view of SARS-CoV-2 RNA replication machinery: RNA synthesis, proofreading and final capping. Cells, 9(5), 1267.
Sabeti, R., & Heidarzadeh, M. (2022a). Numerical simulations of water waves generated by subaerial granular and solid-block landslides: Validation, comparison, and predictive equations. Ocean Engineering, 266, 112853. https://doi.org/10.1016/j.oceaneng.2022.112853
Sabeti, R., & Heidarzadeh, M. (2022b). Numerical simulations of tsunami wave generation by submarine landslides: Validation and sensitivity analysis to landslide parameters. Journal of Waterway, Port, Coastal, and Ocean Engineering, 148(2), 05021016. https://doi.org/10.1061/(ASCE)WW.1943-5460.0000694
Takabatake, T., Han, D. C., Valdez, J. J., Inagaki, N., Mäll, M., Esteban, M., & Shibayama, T. (2022). Three-dimensional physical modeling of tsunamis generated by partially submerged landslides. Journal of Geophysical Research: Oceans, 127(1), e2021JC017826. https://doi.org/10.1029/2021JC017826
Van Gent, M. R. A. (1995). Porous flows through rubble-mound material. Journal of waterway, port, coastal, and ocean engineering, 121(3), 176–181.
Wang, X., & Liu, P. L. F. (2006). An analysis of 2004 Sumatra earthquake fault plane mechanisms and Indian Ocean tsunami. Journal of Hydraulic Research, 44, 147–154. https://doi.org/10.1080/00221686.2006.9521671
Watts, P. (1998). Wavemaker curves for tsunamis generated by underwater landslides. Journal of Waterway, Port, Coastal, and Ocean Engineering, 124(3), 127–137. https://doi.org/10.1061/(ASCE)0733-950X(1998)124:3(127)
Yin, Y., Zhang, C., Imamura, F., Harris, J. C., & Li, Z. (2015). Numerical analysis on wave generated by the Qianjiangping landslide in Three Gorges Reservoir. China. Landslides, 12(2), 355–364. https://doi.org/10.1007/s10346-015-0564-7
Zaniboni, F., & Tinti, S. (2014). Numerical simulations of the 1963 Vajont landslide, Italy: application of 1D Lagrangian modelling. Natural hazards, 70, 567–592.
Zhang, C., & Zhang, M. (2023). Numerical investigation of solitary wave attenuation and mitigation caused by vegetation using OpenFOAM. Coastal Engineering Journal, 65(2), 198–216. https://doi.org/10.1080/21664250.2022.2163844

Discharge Formula and Hydraulics of Rectangular Side Weirs in the Small Channel and Field Inlet

소규모 수로 및 유입구에서의 직사각형 측면 위어의 유량 공식 및 수리학

Yingying Wang, Mouchao Lv, Wen’e Wang, Ming Meng

Abstract

In this study, experimental investigations were conducted on rectangular side weirs with different widths and heights. Corresponding simulations were also performed to analyze hydraulic characteristics including the water surface profile, flow velocity, and pressure. The relationship between the discharge coefficient and the Froude number, as well as the ratios of the side weir height and width to upstream water depth, was determined. A discharge formula was derived based on a dimensional analysis. The results demonstrated good agreement between simulated and experimental data, indicating the reliability of numerical simulations using FLOW-3D software (version 11.1). Notably, significant fluctuations in water surface profiles near the side weir were observed compared to those along the center line or away from the side weir in the main channel, suggesting that the entrance effect of the side weir did not propagate towards the center line of the main channel. The proposed discharge formula exhibited relative errors within 10%, thereby satisfying the flow measurement requirements for small channels and field inlets.

1. Introduction

Sharp crested weirs are used to obtain discharge in open channels by solely measuring the water head upstream of the water. Side weirs, as a kind of sharp-crested weir, are extensively used for flow measurement, flow diversion, and flow regulation in open channels. Side weirs can be placed directly in the channel direction or field inlet, without changing the original structure of the channel. Thus, side weirs have certain advantages in the promotion and application of flow measurement facilities in small channels and field inlets. The rectangular sharp-crested weir is the most commonly available, and many scholars have conducted research on it.
Research on side weirs started in 1934. De Marchi studied the side weir in the rectangular channel and derived the theoretical formula based on the assumption that the specific energy of the main flow section of the rectangular channel in the side weir section was constant [1]. Ackers discussed the existing formulas for the prediction of the side weir discharge coefficient [2]. Chen concluded that the momentum theorem was more suitable for the analytical calculation of the side weir based on the experimental data [3]. Based on previous theoretical research, more and more scholars began to carry out experimental research on side weirs. Uyumaz and Muslu conducted experiments under subcritical and supercritical flow regimes and derived expressions for the side weir discharge and water surface profiles for these regimes by comparing them with experimental results [4]. Borghei et al. developed a discharge coefficient equation for rectangular side weirs in subcritical flow [5]. Ghodsian [6] and Durga and Pillai [7] developed a discharge coefficient equation of rectangular side weirs in supercritical flow. Mohamed proposed a new approach based on the video monitoring concept to measure the free surface of flow over rectangular side weirs [8]. Durga conducted experiments on rectangular side weirs of different lengths and sill heights and discussed the application of momentum and energy principles to the analysis of spatially varied flow under supercritical conditions. The results showed that the momentum principle was fitting better [7]. Omer et al. obtained sharp-crested rectangular side weirs discharge coefficients in the straight channel by using an artificial neural network model for a total of 843 experiments [9]. Emiroglu et al. studied water surface profile and surface velocity streamlines, and developed a discharge coefficient formula of the upstream Froude number, the ratios of weir length to channel width, weir length to flow depth, and weir height to flow depth [10]. Other investigators [11,12,13,14] have conducted experiments to study flow over rectangular side weirs in different flow conditions.
Numerous studies have been conducted in laboratories to this day. Compared to experimental methods, the numerical simulation method has many attractive advantages. We can easily obtain a wide range of hydraulic parameters of side weirs using numerical simulation methods, without investing a lot of manpower and resources. In addition, we can conduct small changes in inlet condition, outlet condition, and geometric parameters, and study their impact on the flow characteristics of side weirs. Therefore, with the development and improvement of computational fluid dynamics, the numerical simulation method has begun to be widely applied on side weirs. Salimi et al. studied the free surface changes and the velocity field along a side weir located on a circular channel in the supercritical regime by numerical simulation [15]. Samadi et al. conducted a three-dimensional simulation on rectangular sharp-crested weirs with side contraction and without side contraction and verified the accuracy of numerical simulation compared with the experimental results [16]. Aydin investigated the effect of the sill on rectangular side weir flow by using a three-dimensional computational fluid dynamics model [17]. Azimi et al. studied the discharge coefficient of rectangular side weirs on circular channels in a supercritical flow regime using numerical simulation and experiments [18]. The discharge coefficient over the two compound side weirs (Rectangular and Semi-Circle) was modeled by using the FLOW-3D software to describe the flow characteristics in subcritical flow conditions [19]. Safarzadeh and Noroozi compared the hydraulics and 3D flow features of the ordinary rectangular and trapezoidal plan view piano key weirs (PKWs) using two-phase RANS numerical simulations [20]. Tarek et al. investigated the discharge performance, flow characteristics, and energy dissipation over PK and TL weirs under free-flow conditions using the FLOW-3D software [21].
As evident from the aforementioned, the majority of studies have primarily focused on determining the discharge coefficient, while comparatively less attention has been devoted to investigating the hydraulic characteristics of rectangular side weirs. Numerical simulations were conducted on different types of side weirs, including compound side weirs and piano key weirs, in different cross-section channels under different flow regimes. It is imperative to derive the discharge formula and investigate other crucial flow parameters such as depth, velocity, and pressure near side weirs for their effective implementation in water measurement. In this study, a combination of experimental and numerical simulation methods was employed to examine the relationship between the discharge coefficient and its influencing factors; furthermore, a dimensionless analysis was utilized to derive the discharge formula. Additionally, water surface profiles near side weirs and pressure distribution at the bottom of the side channel were analyzed to assess safety operation issues associated with installing side weirs.

2. Principle of Flow Measurement

Flow discharge over side weirs is a function of different dominant physical and geometrical quantities, which is defined as

where Q is flow discharge over the side weir, b is the side weir width, B is the channel width, P is the side weir height, v is the mean velocity, h₁ is water depth upstream the side weir in the main channel, g is the gravitational acceleration, μ is the dynamic viscosity of fluid, ρ is fluid density, and i is the channel slope (Figure 1).

**Figure 1.** Definition sketch of parameters of rectangular side weir under subcritical flow. Note: h₁ and h₂ represent water depth upstream and downstream of the side weir in the main channel, respectively; y₁ and y₂ represent weir head upstream and downstream of the side weir in the main channel, respectively.

In experiments when the upstream weir head was over 30 mm, the effects of surface tension on discharge were found to be minor [22]. The viscosity effect was far less than the gravity effect in a turbulent flow. Hence μ and σ were excluded from the analysis [23,24]. In addition, the channel width, the channel slope, and the fluid density were all constant, so the discharge formula can be simplified as:

According to the Buckingham π theorem, the following relationship among the dimensionless parameters is established:

Selected h₁ and g as basic fundamental quantities, and the remaining physical quantities were represented in terms of these fundamental quantities as follows:

In which

Based on dimensional analysis, the following equations were derived.

Namely

So the discharge formula can be simplified as:

In a sharp-crested weir, discharge over the weir is proportional to 𝐻1.51H11.5 (H₁ is the upstream total head above the crest, namely H₁ = y₁ + v²/2 g), so Equation (6) can be transformed as follows:

Consequently, the discharge formula over rectangular side weirs is defined as follows, in which 𝑚=𝑓(𝑏ℎ1m=f(bh1,𝑃ℎ1,𝐹𝑟1)Ph1,Fr1). Parameter m represents the dimensionless discharge coefficient. Parameter Fr₁ represents the Froude number at the upstream end of the side weir in the main channel.

3. Experiment Setup

The experimental setup contained a storage reservoir, a pumping station, an electromagnetic flow meter, a control valve, a stabilization pond, rectangular channels, a side weir, and a sluice gate. The layout of the experimental setup is shown in Figure 2. Water was supplied from the storage reservoir using a pump. The flow discharge was measured with an electromagnetic flow meter with precision of ±3‰. Water depth was measured with a point gauge with an accuracy of ±0.1 mm. The flow velocity was measured with a 3D Acoustic Doppler Velocimeter (Nortek Vectrino, manufactured by Nortek AS in Rud, Norway). In order to eliminate accidental and human error, multiple measurements of the water depth and flow velocity at the same point were performed and the average values were used as the actual water depth and flow velocity of the point. The main and side channels were both rectangular open channels measuring 47 cm in width and 60 cm in height. The geometrical parameters of rectangular side weirs are shown in Table 1.

**Figure 2.** Layout of the test system.

**Table 1.** The geometrical parameters of rectangular side weirs.

When water passes through a side weir, its quality point is affected not only by gravity but also by centrifugal inertia force, leading to an inclined water surface within that particular cross-section before reaching the weir. In order to examine water profiles adjacent to side weirs, cross-sectional measurements were conducted at regular intervals of 12 cm both upstream and downstream of each side weir, denoted as sections ① to ⑩, respectively. Measuring points were positioned near the side weir (referred to as “Side I”), along the center line of the main channel (referred to as “Side II”), and far away from the side weir (referred to as “Side III”) for each cross-section. The schematic diagram illustrating these measuring points is presented in Figure 3.

**Figure 3.** Schematic diagram of measurement points.

4. Numerical Simulation Settings

4.1. Mathematical Model

4.1.1. Governing Equations

Establishing the controlling equations is a prerequisite for solving any problem. For the flow analysis problem of water flowing over a side weir in a rectangular channel, assuming that no heat exchange occurs, the continuity equation (Equation (9)) and momentum equation (Equation (10)) can be used as the controlling equations as follows:

The continuity equation:

Momentum equation:

where: ρ is the fluid density, kg/m³; t is time, s; u_i, u_j are average flow velocities, u₁, u₂, u₃ represent average flow velocity components in Cartesian coordinates x, y, and z, respectively, m/s; μ is dynamic viscosity of fluid, N·s/m²; p is the pressure, pa; S_i is the body force, S₁ = 0, S₂ = 0, S₃ = −ρg, N [24].

4.1.2. RNG k-ε Model

The water flow in the main channel is subcritical flow. When the water flows through the side weir, the flow line deviates sharply, the cross section suddenly decreases, and due to the blocking effect of the side weir, the water reflects and diffracts, resulting in strong changes in the water surface and obvious three-dimensional characteristics of the water flow [25]. Therefore the RNG k–ε model is selected. The model can better handle flows with greater streamline curvature, and its corresponding k and ε equation is, respectively, as follows:

where: k is the turbulent kinetic energy, m²/s²; μ_eff is the effective hydrodynamic viscous coefficient; G_k is the generation item of turbulent kinetic energy k due to gradient of the average flow velocity; C∗1εC1ε*, C_2ε are empirical constants of 1.42 and 1.68, respectively; ε is turbulence dissipation rate, kg·m²/s².

4.1.3. TruVOF Model

Because the shape of the free surface is very complex and the overall position is constantly changing, the fluid flow phenomenon with a free surface is a typical flow phenomenon that is difficult to simulate. The current methods used to simulate free surfaces mainly include elevation function method, the MAC method [26] and the VOF (Volume of Fluid) method [27]. The VOF method is a method proposed by Hirt and Nichols to deal with the complex motion of the free surface of a fluid, which can describe all the complexities of the free surface with only one function. The basic idea of the method is to define functions α_w and α_a, which represent the volume percentage of the calculation area occupied by water and air, respectively. In each unit cell, the sum of the volume fractions of water and air is equal to 1, i.e.,

The TruVOF calculation method can accurately track the change of free liquid level and accurately simulate the flow problems with free interface. Its equation is:

where: u_¯m is the average velocity of the mixture; t is the time; F is the volume fraction of the required fluid.

4.2. Parameter Setting and Boundary Conditions

To streamline the iterative calculation and minimize simulation time, we selected a main channel measuring 7.5 m in length and a side channel measuring 2.5 m in length for simulation. Three-dimensional geometrical models were developed using the software AutoCAD (version 2016-Simplified Chinese). The spatial domain was meshed using a constructed rectangular hexahedral mesh and each cell size was 2 cm. A volume flow rate was set in the channel inlet with an auto-adjusted fluid height. An outflow–outlet condition was positioned at the end of the side channel. A symmetry boundary condition was set in the air inlet at the top of the model, which represented that no fluid flows through the boundary. The lower Z (Zmin) and both of the side boundaries were treated as a rigid wall (W). No-slip conditions were applied at the wall boundaries. Figure 4 illustrates these boundary conditions.

**Figure 4.** Diagram of boundary conditions.

5. Results

5.1. Water Surface Profiles

Water surface profiles were crucial parameters for selecting water-measuring devices. Upon analyzing the consistent patterns observed in different conditions, one specific condition was chosen for further analysis. To validate the reliability of numerical simulation, measured and simulated water depths of rectangular side weirs with different widths and heights at a discharge rate of 25 L/s were extracted for comparison (Table 2 and Figure 5). The results in Table 2 and Figure 5 indicate a maximum absolute relative error value of 9.97% and all absolute relative error values within 10%, demonstrating satisfactory agreement between experimental and simulated results.

**Figure 5.** Comparison between measured and simulated flow depth.

P/cm	Section Position	b = 20 cm			b = 30 cm			b = 40 cm			b = 47 cm
P/cm	Section Position	h_m/cm	h_s/cm	R/%	h_m/cm	h_s/cm	R/%	h_m/cm	h_s/cm	R/%	h_m/cm	h_s/cm	R/%
7	④	21.49	19.4	9.73	17.74	16.9	4.74	16.07	14.51	9.71	13.79	12.50	9.35
	④′	20.48	19.05	6.98	17.78	16.14	9.22	15.69	14.31	8.80
	⑥	20.71	19.02	8.16	17.82	16.31	8.47	15.92	14.53	8.73	15.23	13.80	9.39
	⑧′	22.00	20.22	8.09	18.27	16.74	8.37	16.59	14.96	9.83
	⑧	22.37	20.17	9.83	17.73	16.80	5.25	16.27	15.08	7.31	15.36	14.36	6.51
10	④	24.15	22.6	6.42	19.96	18.84	5.61	19.03	18.58	2.36	16.83	15.85	5.82
	④′	24.21	22.05	8.92	19.49	18.19	6.67	18.75	18.35	2.13
	⑥	24.01	21.78	9.29	19.65	18.34	6.67	18.95	18.63	1.69	17.52	16.09	8.16
	⑧′	24.88	22.4	9.97	20.65	19.21	6.97	20.12	19.29	4.13
	⑧	24.03	22.96	4.45	21.16	19.34	8.60	19.71	19.43	1.42	18.39	17.36	5.60
15	④	28.85	27.56	4.47	25.86	24.09	6.84	24.05	21.89	8.98	22.73	20.80	8.49
	④′	28.49	26.97	5.34	25.19	23.84	5.36	23.42	21.46	8.37
	⑥	28.85	26.98	6.48	25.72	23.99	6.73	23.23	21.82	6.07	23.10	21.05	8.87
	⑧′	28.96	27.30	5.73	26.38	24.19	8.30	24.18	22.27	7.90
	⑧	29.18	27.96	4.18	26.57	24.54	7.64	24.57	22.33	9.12	23.20	21.10	9.05
20	④	33.29	32.34	2.85	30.63	29.02	5.26	28.49	26.87	5.69	26.99	25.81	4.37
	④′	33.14	31.95	3.59	29.75	28.62	3.80	28.11	26.79	4.70
	⑥	33.32	31.79	4.59	30.04	28.45	5.29	28.99	26.86	7.35	27.42	26.72	2.55
	⑧′	34.02	32.39	4.79	30.69	28.95	5.67	29.59	27.25	7.91
	⑧	34.62	32.84	5.14	31.44	29.29	6.84	29.51	27.31	7.46	28.21	27.00	4.29

Table 2. Comparison of measured and simulated water depths on Side I of each side weir at a discharge of 25 L/s

Due to the diversion caused by the side weir, there was a rapid variation in flow near the side weir in the main channel. In order to investigate the impact of the side weir on water flow in the main channel, water surface profiles on Side I, Side II, and Side III were plotted with a side weir width and height both set at 20 cm at a discharge rate of 25 L/s (Figure 6). As depicted in Figure 6, within a certain range of the upstream end of the main channel, water depths on Side I, Side II, and Side III were nearly equal with almost horizontal profiles. As the distance between the location of water flow and the upstream end of the weir crest decreased gradually, there was a gradual decrease in water depth on Side I along with an inclined trend in its corresponding profile; however, both Side II and Side III still maintained almost horizontal profiles. When approaching closer to the side weir area with flowing water, there was an evident reduction in water depth on Side I accompanied by a significant downward trend visible across an expanded decline range. The minimum point occurred near the upstream end of the weir crest before gradually increasing again towards downstream sections. At the crest section of the side weir, there is an upward trend observed in the water surface. The water surface tended to stabilize downstream of the main channel within a certain range from the downstream end of the weir crest. There was no significant change in the water surface profiles of Side Ⅱ and Side Ⅲ in the crest section. It can be inferred that the side weir entrance effect occurred only between Side Ⅰ and Side Ⅱ. M. Emin reported the same pattern [10].

**Figure 6.** Water surface profiles on Side I, Side II, and Side III with a side weir width of 20 cm and height of 15 cm at a discharge of 25 L/s.

For a more accurate study on the entrance effect of the side weir on the Water Surface Profile (WSP) for Side I; a comparative analysis conducted using different widths but the same height (15 cm) at a discharge rate of 25 L/s is presented through Figure 7, Figure 8, Figure 9 and Figure 10.

**Figure 7.** Water surface profile on Side Ⅰ with a side weir width of 20 cm and height of 15 cm at a discharge of 25 L/s.

**Figure 8.** Water surface profile on Side Ⅰ with a side weir width of 30 cm and height of 15 cm at a discharge of 25 L/s.

**Figure 9.** Water surface profile on Side Ⅰ with a side weir width of 40 cm and height of 15 cm at a discharge of 25 L/s.

**Figure 10.** Water surface profile on Side Ⅰ with a side weir width of 47 cm and height of 15 cm at a discharge of 25 L/s.

According to Figure 7, Figure 8, Figure 9 and Figure 10, the water depth upstream of the main channel started to decrease as it approached the upstream end of the weir crest and then gradually increased at the weir crest section. In other words, the water surface profile exhibited a backwater curve along the length of the weir crest. The water depth remained relatively stable downstream of the main channel within a certain range from the downstream end of the weir crest. Additionally, there was a higher water depth downstream of the main channel compared to that upstream of the main channel. Furthermore, an increase in the width of the side weir led to a gradual reduction in fluctuations on its water surface.

5.2. Velocity Distribution

The law of flow velocity distribution near the side weir is the focus of research and analysis, so the simulated and measured values of flow velocity near the side weir were compared and analyzed. Take the discharge of 25 L/s, the height of 15 cm, and the width of 30 cm of the side weir as an example to illustrate. Figure 11 shows the measured and simulated velocity distribution in the x-direction of cross-section ④. As can be seen from Figure 11, the diagrams of the measured and simulated velocity distribution were relatively consistent, and the maximum absolute relative error between the measured and simulated values at the same measurement point was 9.37%, and the average absolute relative error was 3.97%, which indicated a satisfactory agreement between the experimental and simulated results.

**Figure 11.** Velocity distribution in the x-direction of section ④: when the discharge is 25 L/s, the height of the side weir is 15 cm and the width of the side weir is 30 cm. (a) Measured velocity distribution; (b) Simulated velocity distribution.

From Figure 11, it can be seen that the flow velocity gradually increased from the bottom of the channel towards the water surface in the Z-direction, and the flow velocity gradually increased from Side Ⅲ to Side Ⅰ in the Y-direction. The maximum flow velocity occurred near the weir crest.

Figure 12 shows the distribution of flow velocity at different depths (z/P = 0.3, z/P = 0.8, z/P = 1.6) with a side weir width of 30 cm and height of 15 cm at a discharge of 25 L/s. The water flow line began to bend at a certain point upstream of the main channel, and the closer it was to the upstream end of the weir crest, the greater the curvature. The maximum curvature occurred at the downstream end of the weir crest. The flow patterns at the bottom, near the side weir crest, and above the side weir crest were significantly different. There was a reverse flow at the bottom of the main channel, where the forward and reverse flows intersect, resulting in a detention zone. The maximum flow velocity at the bottom layer occurred at the upstream end of the side weir crest. When the location of water flow approached the weir crest, the maximum flow velocity occurred at the upstream end of the weir crest. The maximum flow velocity on the water surface occurred at the downstream end of the weir crest. As the water depth decreased, the position of the maximum flow velocity gradually moved from the upstream end of the side weir to the downstream end of the side weir.

**Figure 12.** Distribution of flow velocity at different depths with a side weir width of 30 cm and height of 15 cm at a discharge of 25 L/s. (a) z/P = 0.3; (b) z/P = 0.8; (c) z/P = 1.6.

5.3. Side Channel Pressure Distribution

When water flowed through the side weir, an upstream water level was formed, resulting in a pressure zone at the junction with the side channel. This pressure zone led to increased water pressure on the floor of the side channel, which affected its stability and durability. In small channels or fields where erosion resistance is weak, excessive pressure can cause scour holes. Therefore, analyzing the pressure distribution in the side channel is necessary to select an appropriate height and width for the side weir that effectively reduces its impact on the bottom plate.

To investigate the impact of side weir width on hydraulic characteristics, pressure data was collected at a discharge rate of 25 L/s for side weirs with heights of 20 cm and widths ranging from 20 cm to 47 cm. The pressure distribution map was drawn, as shown in Figure 13.

**Figure 13.** Comparison of pressure distribution on the bottom plate of the side channel with different widths of side weirs when the discharge is 25 L/s and the height of side weirs is 20 cm. (a) P = 20 cm, b = 20 cm; (b) P = 20 cm, b = 30 cm; (c) P = 20 cm, b = 40 cm; (d) P = 20 cm, b = 47 cm.

As can be seen from Figure 13, the pressure at the bottom of the side channel decreased as the width of the side weir increased. This uneven distribution of water flow on the weir was caused by the sharp bending of water flow lines and the influence of centrifugal inertia force over a short period. After passing through the side weir, the water flow became symmetrically distributed with respect to the axis of the side channel, leaning towards the right bank at a certain distance. As we increased the width of the side weir, we noticed that its position gradually approached the side weir and maximum pressure decreased at this location where the water tongue formed due to flowing through it (Figure 13). For a constant height (20 cm) but varying widths (20 cm, 30 cm, 40 cm, and 47 cm), we measured maximum pressures at these positions as follows: 103,713 Pa, 103,558 Pa, 103,324 Pa, and 103,280 Pa, respectively. Consequently, increasing width reduced the impact on the side channel from water flowing through it while changing pressure distribution from concentration to dispersion in a vertical direction. In the practical application of side weirs, appropriate height should be selected based on the bottom plate’s capacity to withstand the pressure exerted by flowing water within channels.

To investigate how height affects the hydraulic characteristics of rectangular side weirs further (Figure 14), we extracted pressures on bottom plates when discharge was fixed at 25 L/s while varying heights were set as follows: 7 cm, 10 cm, 15 cm, and 20 cm, respectively.

**Figure 14.** Comparison of pressure distribution on the bottom plate of the side channel with different heights of side weirs when discharge is 25 L/s and the width of side weirs is 20 cm. (a) P = 7 cm, b = 20 cm; (b) P = 10 cm, b = 20 cm; (c) P = 15 cm, b = 20 cm; (d) P = 20 cm, b = 20 cm.

As shown in Figure 14, when the width of the side weir was constant, the pressure at the bottom of the side channel increased with the height of the side weir. As the height of the side weir increased, the water tongue formed by flow through the side weir gradually moved away from it in a downstream direction. In terms of vertical water flow, as the height of the side weir increased, the position of maximum pressure at which the water tongue falls shifted closer to the axis of the side channel from its right bank. Moreover, an increase in height resulted in higher maximum pressure at this falling point. For a constant width (20 cm) and varying heights (7 cm, 10 cm, 15 cm, and 20 cm), corresponding maximum pressures at this landing point were measured as 102,422 Pa, 102,700 Pa, 103,375 Pa, and 103,766 Pa, respectively. Consequently, increasing width led to a greater impact on both flow through and pressure distribution within the side channel; transforming it from scattered to concentrated along its lengthwise direction. Therefore, when applying such weirs practically one should select an appropriate width based on what pressure can be sustained by their respective channel bottom plates.

5.4. Discharge Coefficient

Based on dimensionless analysis, the influencing parameters of the discharge coefficient were obtained. To study the effect of parameters Fr₁, b/h₁, and P/h₁, discharge coefficient values were plotted against Fr₁, b/h₁, and P/h₁, shown in Figure 15, Figure 16 and Figure 17. The discharge coefficient decreased as parameters Fr₁ and b/h₁ increased. The discharge coefficient increased as parameter P/h₁ increased. As Uyumaz and Muslu reported in a previous study, the variation of the discharge coefficient with respect to the Froude number showed a second-degree curve for a subcritical regime [4].

**Figure 15.** Variation of discharge coefficient values against Froude number.

**Figure 16.** Variation of discharge coefficient values against the percentage of the side weir width to the upstream flow depth over the side weir.

**Figure 17.** Variation of discharge coefficient values against the percentage of the side weir height to the upstream flow depth over the side weir.

Quantitative analysis between discharge coefficient values and parameters Fr₁, b/h_1, and P/h₁ was conducted using data analysis software (IBM SPSS Statistics 19). The various coefficients obtained are shown in Table 3.

Model	Unstandardized Coefficients		Standardized Coefficients	t	Sig
Model	B	Std. Error	Beta (β)	t	Sig
Constant	−1.294	0.155		−8.369	0.000
Fr₁	3.430	0.286	3.401	12.013	0.000
b/h₁	−0.004	0.004	−0.045	−0.944	0.348
P/h₁	2.401	0.167	4.064	14.394	0.000

Table 3. Coefficient.

The value of t and Sig are the significance results of the independent variable, and the value of Sig corresponding to the value of t is less than 0.05, indicating that the independent variable has a significant impact on the dependent variable. Therefore, the values of Sig corresponding to the parameters Fr₁ and P/h₁ were less than 0.05, indicating that the parameters Fr₁ and P/h₁ have a significant impact on the discharge coefficient. On the contrary, the parameter b/h₁ has less impact on the discharge coefficient. Therefore, quantitative analysis between discharge coefficient values and parameters Fr₁, and P/h₁ was conducted using data analysis software by removing factor b/h₁. The model summary, ANOVA, and coefficient obtained are shown respectively in Table 4, Table 5 and Table 6. R and adjusted R square in Table 4 were approaching 1, which indicated the goodness of fit of the regression model was high. The value of Sig corresponding to the value of F in Table 5 was less than 0.05, which indicated that the regression equation was useful. The values of Sig corresponding to the parameters Fr₁ and P/h₁ in Table 6 were less than 0.05, indicating that the parameters Fr₁ and P/h₁ have a significant impact on the discharge coefficient.

Model	R	R Square	Adjusted R Square	Std. Error of the Estimate
1	0.913 ^a	0.833	0.829	0.03232

Table 4. Model Summary ^b. Note: ^a. Predictors:(Constant), Fr₁, P/h₁; ^b. Discharge coefficient.

Model		Sum of Squares	df	Mean Square	F	Sig
1	Regression	0.402	2	0.201	192.545	0.000 ^a
	Residual	0.080	77	0.001
	Total	0.483	79

Table 5. ANOVA ^b. Note: ^a. Predictors:(Constant), Fr₁, P/h₁; ^b. Discharge coefficient.

Model	Unstandardized Coefficients		Standardized Coefficients	t	Sig
Model	B	Std. Error	Beta (β)	t	Sig
Constant	−1.326	0.151		−8.796	0.000
Fr₁	3.479	0.281	3.449	12.396	0.000
P/h₁	2.427	0.164	4.108	14.765	0.000

Table 6. Coefficient ^a. Note: ^a. Predictors:(Constant), Fr₁, P/h₁.

Based on the above analysis, the flow coefficient formula has been obtained, shown as follows:

Discharge formula were obtained by substituting Equation (15) into Equation (12), as shown in Equation (16).

where Q ∈ [0.006, 0.030], m³/s; b ∈ [0.20, 0.47], m; P ∈ [0.07, 0.20], m.

Figure 18 showed the measured discharge coefficient values with those calculated from discharge formulas in Table 3. The scatter of the data with respect to perfect line was limited to ±10%.

**Figure 18.** Comparison of the measured discharge coefficient values with those calculated from discharge formulas in Table 3.

6. Discussions

Determining water surface profile near the side weir in the main channel is one of the tasks of hydraulic calculation for side weirs. As the water flows through the side weir, discharge in the main channel is gradually decreasing, namely dQ/ds<0. According to the Equation (17) derived from Qimo Chen [3], it can be inferred that the value of 𝑑ℎ/𝑑𝑠 is greater than zero in subcritical flow (Fr < 1), that is, the water surface profile near the side weir in the main channel is a backwater curve. Due to the side weir entrance effect at the upstream end, water surface profiles drop slightly at the upstream end of the side weir crest, as EI-Khashab [28] and Emiroglu et al. [29] reported in previous experimental studies.

In this study, the water surface profile exhibited a backwater curve along the length of the weir crest. Therefore, during side weir application, it is crucial to ensure that downstream water levels do not exceed the highest water level of the channel.

The head on the weir is one of the important factors that flow over side weirs depends on. At the same time, the head depends on the water surface profile near the side weir in the main channel. Therefore, further research on the quantitative analysis of water surface profile needs to be conducted. Mohamed Khorchani proposed a new approach based on the video monitoring concept to measure the free surface of flow over side weirs. It points out a new direction for future research [8].

The maximum flow velocity, a key parameter in assessing the efficiency of a weir, occurs at the upstream end of the weir crest, typically near the crest. This is attributed to the convergence of the flow as it approaches the crest, resulting in a significant increase in velocity. It was found that in this study the minimum flow velocity occurred at the bottom of the main channel away from the side weir. Under such conditions, the accumulation of sediments could lead to siltation, which in turn can affect the accuracy of flow measurement through side weirs. This is because the presence of sediments can alter the flow patterns and cause errors in the measurement. Therefore, it becomes crucial to explore methods to optimize the selection of side weirs in order to minimize or eliminate the effects of sedimentation on flow measurement.

Pressure distribution plays a crucial role in ensuring structural safety for side weirs since small channels and field inlets have relatively limited pressure-bearing capacities. Therefore, it is important to select an appropriate geometrical parameter of rectangular side weirs based on their ability to withstand the pressure exerted on their bottom combined with pressure distribution data at the bottom of the side channel we have obtained in this study.

The discharge coefficient formula (Equation (15)), which incorporates Fr₁ and P/h₁, was derived based on dimensional analysis. However, it is worth noting that previous research has contradicted this formula by suggesting that the discharge coefficient solely depends on the Froude number. This conclusion can be observed in this study such as in Equations (18)–(23) in Table 7 of the manuscript [30,31,32,33,34,35], which clearly demonstrate the dependency of the discharge coefficient on the Froude number. In contrast, our derived discharge coefficient formula (Equation (15)) offers a more streamlined and simplified approach compared to Equation (25) [36] and Equation (29) [10]—making it easier to comprehend and apply—an advantageous feature particularly valuable in fluid dynamics where intricate calculations can be time-consuming. Furthermore, our derived discharge coefficient formula (Equation (15)) exhibits a broader application scope than that of Equation (24) [37] as shown in Table 8. Equation (26) [38] and Equation (27) [5] are specifically applicable under high flow discharge conditions. Conversely, our derived discharge coefficient formula (Equation (15)) is better suited for low-flow discharge conditions.

**Table 7.** Discharge coefficient formulas of rectangular side weirs presented in previous studies.

Discharge/(L·s⁻¹)	Width of Side Weir/cm	Height of Side Weir/cm	Number of Formula
10~14	10~20	6~12	(24)
35–100	20~75	1~19	(26), (27)
6~30	20~47	7~20	(15)

Table 8. Application scope of discharge coefficient formulas.

In addition to the factors studied in the paper, factors such as the sediment content in the flow, the bottom slope, and the cross-section shape of the channel also have a certain impact on the hydraulic characteristics of the side weir. Further numerical simulation methods can be used to study the hydraulic characteristics and the influencing factors of the side weir. Water measurement facilities generally require high accuracy of water measurement, the flow of sharp-crested side weirs is complex, and the water surface fluctuates greatly. While conducting numerical simulations, experimental research on prototype channels is necessary to ensure the reliability of the results and provide reference for the body design and optimization of side weirs in small channels and field inlets.

7. Conclusions

This paper presents a comprehensive study that encompasses both experimental and numerical simulation research on rectangular side weirs of varying heights and widths within rectangular channels. A thorough analysis of the experimental and numerical simulation results has been conducted, leading to the derivation of several notable conclusions:

A comparative analysis was conducted on the measured and simulated values of water depth and flow velocity. Both of the maximum absolute relative errors were within 10%, which indicated that the numerical simulation of the side weir was feasible and effective.
The water surface profile exhibited a backwater curve along the length of the weir crest. The side weir entrance effect occurred only between Side Ⅰ and Side Ⅱ. This indicates that flow patterns and associated hydraulic forces at the weir entrance play a crucial role in determining water level distribution along the weir crest.
The maximum flow velocity of the cross-section at the upstream end of the weir crest occurred near the weir crest, while the minimum flow velocity occurred at the bottom of the main channel away from the side weir. As the water depth decreased, the position of the maximum flow velocity gradually moved from the upstream end of the side weir to the downstream end of the side weir.
When the height of the side weir remains constant, an increase in the width of the side weir leads to a decrease in pressure at the bottom of the side channel. Conversely, when the width of the side weir is kept constant, an increase in its height results in an increase in pressure at the bottom of the side channel. Therefore, during practical applications involving side weirs, it is crucial to select an appropriate weir width based on the maximum pressure that can be sustained by the channel’s bottom plate.
The discharge coefficient was found to depend on the upstream Froude number Fr₁ and the percentage of the side weir height to the upstream flow depth over the side weir P/h₁. The relationship between the discharge coefficient and parameters Fr₁ and P/h₁ was obtained using multiple regression analysis, which was of linear form and provided an easy means to estimate the discharge coefficient. The discharge formula is of high accuracy with relative errors within 10%, which met the water measurement accuracy requirements of small channels in irrigation areas.

Reference

De Marchi, G. Essay on the performance of lateral weirs. L’Energ Electr. 1934, 11, 849.
Ackers, P. A theoretical consideration of side weirs as storm water overflows. Proc. Inst. Civ. Eng. 1957, 6, 250–269.
Chen, Q.M.; Xie, P.Z.; Chen, Q.R. Experiment on hydraulic characteristics of side weir. J. Fuzhou Univ. 1979, 19, 26–29.
Uyumaz, A.; Muslu, Y. Flow over side weir in circular channels. ASCE J. Hydraul. Eng. 1985, 111, 144–160.
Borghei, M.; Jalili, M.R.; Ghodsian, M. Discharge coefficient for sharp-crested side weir in subcritical flow. ASCE J. Hydraul. Eng. 1999, 125, 1051–1056.
Ghodsian, M. Supercritical flow over rectangular side weir. Can. J. Civ. Eng. 2003, 30, 596–600.
Durga Rao, K.H.V.; Pillai, C.R.S. Study of Flow Over Side Weirs Under Supercritical Conditions. Water Resour Manag. 2008, 22, 131–143.
Khorchani, M.; Blanpain, O. Free surface measurement of flow over side weirs using the video monitoring concept. Flow Meas. Instrum. 2004, 15, 111–117.
Bilhan, O.; Emiroglu, M.E.; Kisi, O. Application of two different neural network techniques to lateral outflow over rectangular side weirs located on a straight channel. Adv. Eng. Softw. 2010, 41, 831–837.
Emiroglu, M.E.; Agaccioglu, H.; Kaya, N. Discharging capacity of rectangular side weirs in straight open channels. Flow Meas. Instrum. 2011, 22, 319–330.
Azza, N.; Al-Talib, A.N. Flow over oblique side weir. J. Damascus Univ. 2012, 28, 15–22.
Bagheri, S.; Kabiri-Samani, A.R.; Heidarpour, M. Discharge coefficient of rectangular sharp-crested side weirs part i: Traditional weir equation, Flow Measure. Instrumentation 2014, 35, 109–115.
Shariq, A.; Hussain, A.; Ansari, M.A. Lateral flow through the sharp crested side rectangular weirs in open channels. Flow Measure. Instrumentation 2018, 59, 8–17.
Li, G.D.; Shen, G.Y.; Li, S.S.; Lu, Q.N. Prediction Model of Side Weir Discharge Capacity Based on LS-SVM. J. Basic Sci. Eng. 2023, 4, 843–851.
Shabanlou, S.; Salimi, M.S. Free surface and velocity field in a circular channel along the side weir in supercritical flow conditions. Flow Meas. Instrum. 2014, 38, 108–115.
Samadi, A.; Arvanaghi, H.; Abbaspour, A. Three-Dimensional Simulation of Free Surface Flow over Rectangular Sharp crested Weirs. Int. J. Agric. Biosci. 2015, 4, 83–86.
Aydin, M.C. Investigation of a Sill Effect on Rectangular Side-Weir Flow by Using CFD. J. Irrig. Drain. Eng. 2016, 142.
Azimi, H.; Shabanlou, S.; Ebtehaj, I.; Bonakdari, H. Discharge Coefficient of Rectangular Side Weirs on Circular Channels. Int. J. Nonlinear Sci. Numer. Simul. 2016, 17, 391–399.
Khassaf, S.I.; Attiyah, A.N.; Al-Yousify, H.A. Experimental investigation of compound side weir with modeling using computational fluid dynamic. Energy Environ. 2018, 7, 169–178.
Safarzadeh, A.; Noroozi, B. 3D Hydrodynamics of Trapezoidal Piano Key Spillways. Int. J. Civ. Eng. 2017, 15, 89–101.
Selim, T.; Hamed, A.K.; Elkiki, M.; Eltarabily, M.G. Numerical investigation of flow characteristics and energy dissipation over piano key and trapezoidal labyrinth weirs under free-flow conditions. Model. Earth Syst. Environ. 2023, 10, 1253–1272.
Novak, P.; Cabelka, J. Models in Hydraulic Engineering; Pitman: London, UK, 1981.
Henderson, F.M. Open Channel Flow; Prentice-Hall: Englewood Cliffs, NJ, USA, 1966.
Wang, F.J. Computational Fluid Dynamics Analysis-Theory and Application of CFD; Tsinghua University Press: Beijing, China, 2004.
Zhu, Y.L.; Ma, X.Y.; Zhan, G.L.; Lv, J.W. Numerical simulation of flow in flat V-weir. Yellow River 2010, 32, 99–100.
Harlow, F.H.; Welch, J.E. Numberical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys. Fluids 1965, 8, 2182–2189.
Hirt, C.W.; Nichols, B.D. Volume of fluid (VOF) method for dynamics of free boundaries. Phys. Fluids 1981, 39, 201–221.
El-Khashab, A.M.M. Hydraulics of Flow over Side Weirs. Ph.D. Thesis, University of Southampton, Southampton, UK, 1975.
Emiroglu, M.E.; Kaya, N.; Agaccioglu, H. Discharge capacity of labyrinth side-weir located on a straight channel. ASCE J. Irrig. Drain. Eng. 2010, 136, 37–46.
Subramanya, K.; Awasthy, S.C. Spatially varied flow over side weirs. ASCE J. Hydraul. Div. 1972, 98, 1–10.
Nandesamoorthy, T.; Thomson, A. Discussion of spatially varied flow over side weir. ASCE J. Hydraul. Eng. 1972, 98, 2234–2235.
Yu-Tech, L. Discussion of spatially varied flow over side weir. ASCE J. Hydraul. Div. 1972, 98, 2046–2048.
Ranga Raju, K.G.; Prasad, B.; Grupta, S.K. Side weir in rectangular channel. ASCE J. Hydraul. Div. 1979, 105, 547–554.
Hager, W.H. Lateral outflow over side weirs. ASCE J. Hydraul. Eng. 1987, 113, 491–504.
Cheong, H.F. Discharge coefficient oflateral diversion from trapezoidal channel. ASCE J. Irrig. Drain. Eng. 1991, 117, 321–333.
Swamee, P.K.; Santosh, K.P.; Masoud, S.A. Side weir analysis using elementary discharge coefficient. ASCE J. Irrig. Drain. Eng. 1994, 120, 742–755.
Singh, R.; Manivannan, D.; Satyanarayana, T. Discharge coefficient of rectangular side weirs. ASCE J. Irrig. Drain. Eng. 1994, 120, 814–819.
Jalili, M.R.; Borghei, S.M. Discussion of Discharge coefficient of rectangular side weir. ASCE J. Irrig. Drain. Eng. 1996, 122, 132.

Three-dimensional flow structure in a confluence-bifurcation unit

합류 분기 유닛의 3차원 유동 구조

Di Wang, Xiaoyong Cheng, Zhixuan Cao, Jinyun Deng

Abstract

Enhanced understanding of flow structure in braided rivers is essential for river regulation, flood control, and infrastructure safety across the river. It has been revealed that the basic morphological element of braided rivers is confluence-bifurcation units. However, flow structure in these units has so far remained poorly understood with previous studies having focused mainly on single confluences/bifurcations. Here, the flow structure in a laboratory-scale confluence-bifurcation unit is numerically investigated based on the FLOW–3D® software platform. Two discharges are considered, with the central bars submerged or exposed respectively when the discharge is high or low. The results show that flow convergence and divergence in the confluence-bifurcation unit are relatively weak when the central bars are submerged. Based on comparisons with a single confluence/bifurcation, it is found that the effects of the upstream central bar on the flow structure in the confluence-bifurcation unit reign over those of the downstream central bar. Concurrently, the high-velocity zone in the confluence-bifurcation unit is less concentrated than that in a single confluence while being more concentrated than that observed in a single bifurcation. The present work unravels the flow structure in a confluence-bifurcation unit and provides a unique basis for further investigating morphodynamics in braided rivers.

1 Introduction

Confluences and bifurcations commonly exist in alluvial rivers and usually are important nodes of riverbed planform (Szupiany et al., 2012; Hackney et al., 2018). Flow convergence and divergence in these junctions result in highly three-dimensional (3D) flow characteristics, which greatly influence sediment transport, and hence riverbed evolution and channel formation (Le et al., 2019; Xie et al., 2020). Braided rivers, characterized by unstable networks of channels separated by central bars (Ashmore, 2013), have confluence-bifurcation units as their basic morphological elements (Ashmore, 1982; 1991; 2013; Federici & Paola, 2003; Jang & Shimizu, 2005). In particular, confluence-bifurcation units exhibit a distinct morphology from single confluences/bifurcations and bifurcation-confluence regions because two adjacent central bars are included. Within a confluence-bifurcation unit, two tributaries converge at the upstream bar tail and soon diverge to two anabranches again at the downstream bar head. Therefore, the flow structure in the unit may be significantly influenced by both the two central bars, and thus considerably different from that in single confluences, single bifurcations, and bifurcation-confluence regions, where the flow is affected by only one central bar. Enhanced understanding of flow structure in confluence-bifurcation units is urgently needed, which is essential for water resources management, river regulation, flood control, protection of river ecosystems and the safety of infrastructures across the rivers such as bridges, oil pipelines and communication cables (Redolfi et al., 2019; Ragno et al., 2021).

The flow dynamics, turbulent coherent structures, and turbulent characteristics in single confluences have been widely studied since the 1980s (Yuan et al., 2022). Flow dynamics at river channel confluences have been systematically and completely analyzed, which can be characterized by six major regions of flow stagnation, flow deflection, flow separation, maximum velocity, flow recovery and distinct shear layers (Best, 1987). For example, the field observation of Roy et al. (1988) and Roy and Bergeron (1990) highlighted the flow separation zones and recirculation at downstream natural confluence corners. Ashmore et al. (1992) measured the flow field in a natural confluence and found flow accelerates suddenly at the confluence junction with two separated high-velocity cores merging into one single core at the channel centre. De Serres et al. (1999) investigated the three-dimensional flow structure at a river confluence and identified the existence of the mixing layer, stagnation zones, separation zones and recovery zones. Sharifipour et al. (2015) numerically studied the flow structure in a 90° single confluence and found that the size of the separation zone decreases with the width ratio between the tributary and the main channel. Recently, three main classes of large-scale turbulent coherent structures (Duguay et al., 2022) have been presented, i.e. vertical-orientated vortices or Kelvin-Helmholtz instabilities (Rhoads & Sukhodolov, 2001; Constantinescu et al., 2011; 2016; Biron et al., 2019), channel-scale ‘back-to-back’ helical cells, (Mosley, 1976; Ashmore, 1982; Ashmore et al., 1992; Ashworth, 1996; Best, 1987; Rhoads & Kenworthy, 1995; Bradbrook et al., 1998; Lane et al., 2000), and smaller, strongly coherent streamwise-orientated vortices (Constantinescu et al., 2011; Sukhodolov & Sukhodolova, 2019; Duguay et al., 2022). However, no consensus on a universal turbulent coherent structure mode has been reached so far (Duguay et al., 2022). In addition, some studies (Ashworth, 1996; Constantinescu et al., 2011; Sukhodolov et al., 2017; Le et al., 2019; Yuan et al., 2023) have focused on turbulent characteristics, e.g. turbulent kinetic energy, turbulent dissipation rate and Reynolds stress, which can be critical parameters to further explaining the diversity of these turbulent coherent structure modes.

Investigations on the flow structure in single bifurcations have mainly focused on hydrodynamics in anabranches (Hua et al., 2009; van der Mark & Mosselman, 2013; Iwantoro et al., 2022) and around bifurcation bars (McLelland et al., 1999; Bertoldi & Tubino, 2005; 2007; Marra et al., 2014), whereas few studies have considered the effects of bifurcations on the upstream flow structure. Thomas et al. (2011) found that the velocity core upstream of the bifurcation is located near the water surface and towards the channel center in experimental investigations of a Y-shaped bifurcation. Miori et al. (2012) simulated flow in a Y-shaped bifurcation and found two circulation cells upstream of the bifurcation with flow converging at the water surface and diverging near the bed. Szupiany et al. (2012) reported velocity decreasing and back-to-back circulation cells upstream of the bifurcation junction in the field observation of a bifurcation of the Rio Parana River. These investigations provide insight into how bifurcations affect the flow patterns upstream, yet there is a need for further research on the dynamics of flow occurring immediately before the bifurcation junction.

Generally, the findings of studies on bifurcation-confluence regions are similar to those concerning single confluences and bifurcations. Hackney et al. (2018) measured the hydrodynamic characteristics in a bifurcation-confluence of the Mekong River and found the velocity cores located at the channel centre and strong secondary current occurring under low discharges. Le et al. (2019) reported a high-turbulent-kinetic-energy (high-TKE) zone located near the bed in their numerical simulation of flow in a natural bifurcation-confluence region. Moreover, a stagnation zone was found upstream of the confluence and back-to-back secondary current cells were detected at the confluence according to Xie et al. (2020) and Xu et al. (2022). Overall, these studies have further unraveled the flow patterns in river confluences and bifurcations.

Unfortunately, limited attention has been paid to the flow structure in confluence-bifurcation units. Parsons et al. (2007) investigated a large confluence-bifurcation unit in Rio Parana, Argentina, and no classical back-to-back secondary current cells were observed under a discharge of 12000 m3·s−1. To date, the differences in flow structure between confluence-bifurcation units and single confluences/bifurcations have remained far from clear. In addition, although the effects of discharge on flow structure have been investigated in several studies on single confluences/bifurcations, (Hua et al., 2009; Le et al., 2019; Luz et al., 2020; Xie et al., 2020; Xu et al., 2022), cases with fully submerged central bars were not considered, which is typical in braided rivers during floods. In-depth studies concerning these issues are urgently needed to gain better insight into the flow structure in confluence-bifurcation units of braided rivers.

This paper aims to (1) reveal the 3D flow structure in a confluence-bifurcation unit under different discharges and (2) elucidate the differences in the flow structure between confluence-bifurcation units and single confluence/bifurcation cases. Using the commercial computational fluid dynamics software FLOW-3D® (Version 11.2; https://www.flow3d.com; Flow Science, Inc.), fixed-bed simulations of a laboratory-scale confluence-bifurcation unit are conducted, and cases of a single confluence/bifurcation are also included for comparison. Two discharges are considered, with the central bars fully submerged or exposed respectively when the discharge is high or low. Based on the computational results, the 3D flow structure in the confluence-bifurcation unit conditions is analyzed from various aspects including free surface elevation, time-averaged flow velocity distribution, recirculation vortex structure, secondary current, and turbulent kinetic energy and dissipation rate. In particular, the flow structure in the confluence-bifurcation unit is compared with that in the single confluence/bifurcation cases to unravel the differences.h

2. Conceptual flume and computational cases

2.1. Conceptual flume

In this paper, a laboratory-scale conceptual flume is designed and used in numerical simulations. Figure 1(a–d) shows the morphological characteristics of the flume. To ensure that the conceptual flume reflects morphology features of natural braided channels, key parameters governing the flume morphology, e.g. unit length, width, and channel width-depth ratio, are determined according to studies on morphological characteristics of natural confluence-bifurcation units (Hundey & Ashmore, 2009; Ashworth, 1996; Orfeo et al., 2006; Parsons et al., 2007; Sambrook Smith et al., 2005; Kelly, 2006; Ashmore, 2013; Egozi & Ashmore, 2009; Redolfi et al., 2016; Ettema & Armstrong, 2019).

Figure 1. The sketch of the conceptual flume: (a) the original flume, (b) the central bar: (c) the sketch of cross-section C-C, (d) the sketch of cross-section D-D, (e) the modified part for the single confluence, (f) the modified part for the single bifurcation, (g) the position of different cross-sections. The red dashed boxes denote the regions of primary concern.

2.1.1. Length and width scales of the confluence-bifurcation unit

The length and width scales of the flume are first determined. The inner relation among the length LCB and average width B of a confluence-bifurcation unit and the average width Bi of a single branch was statistically studied by Hundey and Ashmore (2009), which indicates the following relations:
𝐿CB =(4∼5)⁢𝐵 (1)
𝐵 =1.41⁢𝐵𝑖 (2)
In addition, Ashworth (1996) gave B = 2Bi in his experimental research on mid-bar formation downstream of a confluence, while the confluence-bifurcation unit of Rio Parana, Argentina has a relation of B≈1.71Bi (Orfeo et al., 2006; Parsons et al., 2007). Accordingly, the following relations are used in the present paper:
𝐿CB =4⁢𝐵 (3)
𝐵 =1.88⁢𝐵𝑖 (4)
where LCB = 6 m, B = 1.5 m and Bi = 0.8 m.

2.1.2. Central bar morphology

The idealized plane pattern of central bars in braided rivers is a slightly fusiform leaf shape with a short upstream side and a long downstream side (Ashworth, 1996; Sambrook Smith et al., 2005; Kelly, 2006; Ashmore, 2013). To simplify the design, the bar is approximated as a combination of two different semi-ellipses (Figure 1(b)). The major axis Lb is two to ten times longer than the minor axis Bb according to the statistical data in Kelly’s study, and the regression equation is given as (Kelly, 2006):
𝐿𝑏=4.62⁢𝐵0.96𝑏 (5)
In this study, the bar width Bb is set as 0.8 m, whilst the lengths of downstream (LT1) and upstream sides (LT2) are 2 and 1.5 m, respectively (Figure 1(b)). Thus, the relation of Lb and Bb is given as:
𝐿𝑏=(𝐿𝑇⁢1+𝐿𝑇⁢2)=4.375⁢𝐵𝑏 (6)
The lengths of the inlet and outlet parts are determined as Lin = Lout = 8 m, which ensures negligible effects of boundary conditions without exceptional computational cost.

2.1.3. Width-depth ratio

Channel flow capacity can be significantly affected by cross-section shapes. For natural rivers, cross-section shapes can be generalized into three sorts based on the following width-depth curve (Redolfi et al., 2016):
𝐵=𝜓⁢𝐻𝜑(7)
Braided rivers usually have ψ = 5∼50 and φ>1, which indicates a rather wide and shallow cross-section. The central bar form should also be taken into account, so a parabolic cross-section shape is used here with ψ = 8 and φ>1 (Figure 1(c,d)).

2.1.4. Bed slope

In addition, natural braided rivers are usually located in mountainous areas and thus have a relatively large bed slope. According to flume experiments and field observations, the bed slope Sb is mostly in the range of 0.01∼0.02, and a few are below 0.01 (Ashworth, 1996; Egozi & Ashmore, 2009; Ashmore, 2013; Redolfi et al., 2016; Ettema & Armstrong, 2019). In this study, Sb takes 0.005.

2.1.5. Complete sketch of the conceptual flume

In summary, the flume is 29 m long, 2.4 m wide, and 0.6 m high. The plane coordinates (x-direction and y-direction) used in the calculation process are shown in Figure 1
(a). Note that the inlet corresponds to x = 0 m, and the centreline of the flume is located at y = 1.3 m. Besides, the thalweg elevation of the outlet is set as z = 0 m.

2.2. Computational cases

As stated before, the first aim of this paper is to reveal the flow structure in the confluence-bifurcation unit under different discharges. Therefore, two basic cases are set first: (1) case 1a under a low discharge (0.05 m³·s⁻¹) with exposed central bars and (2) case 2a under a high discharge (0.30 m³·s⁻¹) with fully submerged central bars. A total of 22 cross-sections are identified to examine the results (Figure 1(g)).

Further, cases of a single confluence/bifurcation are generated by splitting the original confluence-bifurcation unit into two parts. Part 1 only includes the upstream central bar and focuses on the flow convergence downstream of CS04 (Figure 1(e)), while Part 2 only includes the downstream central bar and focuses on the flow divergence upstream of CS19 (Figure 1(f)). Notably, the numbers of corresponding cross-sections in the original flume are reserved to facilitate comparison. The outlet section of the single confluence as well as the inlet section of the single bifurcation is extended to make the total length equivalent to the original flume (29 m). Also, two discharge conditions (0.05 and 0.30 m³·s⁻¹), which correspond to exposed and fully submerged central bars, are considered for the single confluence/bifurcation. In total, six computational cases are conducted, as listed in Table 1. As the conceptual flume is designed to be symmetrical about the centreline, the momentum flux ratio (M_r) of the two branches should be 1 in all six cases. This is confirmed by further examining the computational results.

Case	Configuration	Q_in (m³·s⁻¹)	Z_out (m)	M_r	Condition of bars
1a	CBU	0.05	0.15	1	Exposed
1b	SC	0.05	0.15	1	Exposed
1c	SB	0.05	0.15	1	Exposed
2a	CBU	0.30	0.34	1	Submerged
2b	SC	0.30	0.34	1	Submerged
2c	SB	0.30	0.34	1	Submerged

Table 1. Computational cases with inlet and outlet boundary conditions.

3. Numerical method

In this section, the 3D Large Eddy Simulation (LES) model integrated in the FLOW-3D® (Version 11.2; https://www.flow3d.com; Flow Science, Inc.) software platform is introduced, including governing equations and boundary conditions. Information on computational meshes with mesh independence test can be found in the Supplementary material.

3.1. Governing equations

The LES model was applied in the present paper to simulate flow in the laboratory-scale confluence-bifurcation unit. The LES model has been proven to be effective in simulating turbulent flow in river confluences and bifurcations (Constantinescu et al., 2011; Le et al., 2019). The basic idea of the LES model is that one should directly compute all turbulent flow structures that can be resolved by the computational meshes and only approximate those features that are too small to be resolved (Smagorinsky, 1963). Therefore, a filtering operation is applied to the original Navier-Stokes (NS) equations for incompressible fluids to distinguish the large-scale eddies and small-scale eddies (Liu et al., 2018). The filtered NS equations are then generated, which can be expressed in the form of a Cartesian tensor as (Liu, 2012):

(10) where ¯𝑢𝑖 is the resolved velocity component in the i – direction (i goes from 1 to 3, denoting the x-, y – and z-directions, respectively); t is the flow time; ρ is the density of the fluid; ¯𝑝 is the pressure; ν is the kinematic viscosity; τ_ij is the sub-grid scale (SGS) stress; ¯𝐺𝑖 is the body acceleration. In FLOW–3D®, the full NS equations are discretized and solved using the finite-volume/finite-difference method (Bombardelli et al., 2011; Lu et al., 2023).

Due to the filtering process, the velocity can be divided into a resolved part (¯𝑢⁡(𝑥,𝑡)) and an approximate part (𝑢′⁡(𝑥,𝑡)) which is also known as the SGS part (Liu, 2012). To achieve model closure, the standard Smagorinsky SGS stress model is introduced here (Smagorinsky, 1963):
𝜏ij−13⁢𝜏kk⁢𝛿ij=−2⁢𝜈SGS⁢¯𝑆ij(11)
where ν_SGS is the SGS turbulent viscosity, and ¯𝑆ij is the resolved rate-of-strain tensor for the resolved scale defined by (Smagorinsky, 1963):
¯𝑆ij=12⁢(∂¯𝑢𝑖∂𝑥𝑗+∂¯𝑢𝑗∂𝑥𝑖)(12)
In the standard Smagorinsky SGS stress model, the eddy viscosity is modelled by (Smagorinsky, 1963):
𝜈SGS=(𝐶𝑠⁢¯𝛥)2⁢∣¯𝑆∣,∣¯𝑆∣=√2⁢¯𝑆ij⁢¯𝑆ij(13)
¯𝛥=(ΔxΔyΔz⁢)1/3(14)
where C_s is the Smagorinsky constant, Δx, Δy, and Δz are mesh scales. In FLOW–3D®, C_s is between 0.1 to 0.2 (Smagorinsky, 1963).
One of the key problems in simulating 3D open channel flow is the calculation of free surface. FLOW–3D® uses the Volume of Fluid (VOF) method (Hirt & Nichols, 1981) to track the change of free surface. The VOF method introduces a fluid phase fraction function f to characterize the proportion of a certain fluid in each mesh cell. In that case, the surface position can be precisely located if the mesh cell is fine enough. To monitor the change of f with time and space, the following convection equation is added:

For open channel flow, only two kinds of fluids are involved: water and air. If f is the fraction of water, the state of the fluid in each mesh cell can be defined as:

In FLOW–3D®, the interface between water and air is assumed to be shear-free, which means that the drag force on the water from the air is negligible. Moreover, in most cases, the details of the gas motion are not crucial for the heavier water motion so the computational processes will be more efficient.

3.2. Boundary conditions

Six boundary conditions need to be preset in the 3D numerical simulation process. Discharge boundary conditions are used for the inlet of the flume, where the free surface elevation is automatically calculated based on the free surface elevation boundary conditions set for the outlet. The specific information on the inlet and outlet boundary conditions for all computational cases is shown in Table 1. Moreover, because the free surface moves temporally, the free surface boundary conditions are just set as no shear stress and having a normal pressure, and the position of the free surface will be automatically adjusted over time by the VOF method in FLOW–3D®. Furthermore, the bed and two side walls are all set to be no-slip for fixed bed conditions, and a standard wall function is employed at the wall boundaries for wall treatment.

The inlet turbulent boundary conditions also need to be considered. They are set by default here. The turbulent velocity fluctuations V^’ are assumed to be 10% of the mean flow velocity with the turbulent kinetic energy (TKE) (per unit mass) equaling 0.5V^’2. The maximum turbulent mixing length is assumed to be 7% of the minimum computational domain scale, and the turbulent dissipation rate is evaluated automatically from the TKE.

4. Results and discussion

4.1. Flow structure in the confluence-bifurcation unit

4.1.1. Free surface elevation

Figure 2 shows the free surface elevation at five different longitudinal profiles (i.e. α = 0.2, 0.4, 0.5, 0.6, 0.8) for cases 1a and 2a. The parameter α was defined as follows:𝛼=𝑠𝐵(17) where s is the transverse distance between a certain profile and the left boundary of the flume. In general, the longitudinal change of free surface in the two cases is very similar despite different discharge levels. The free surface elevation decreases as the channel narrows from the upstream bifurcation to the front of the confluence-bifurcation unit. On the contrary, when the flow diverges again at the end of the confluence-bifurcation unit, the free surface elevation increases with channel widening. However, whether the fall or rise of free surface elevation in case 1a is much sharper than that in case 2a, especially at profiles with α = 0.2 and 0.8 (Figure 2(a)), which indicates there may be distinct flow states between the two cases. To further illustrate this finding, the Froude number F_r at different cross-sections (CS08∼CS15) is examined. In case 2a, the flow remains subcritical within the confluence-bifurcation unit. By contrast, in case 1a, a local supercritical flow is observed near the side banks of CS09 (i.e. α = 0.2 and 0.8), with F_r being about 1.2. This local supercritical flow can lead to a hydraulic drop followed by a hydraulic jump, which accounts for the sharp change of the free surface. The foregoing reveals that when central bars are exposed under relatively low discharge, supercritical flow is more likely to occur near the side banks of the confluence junction due to flow convergence.

**Figure 2. Five time-averaged free surface elevation profiles in the confluence-bifurcation unit, in which** α **denotes the lateral position of the certain profile. Note that the black dashed line denotes the position of CS09, where** Fr is about 1.2 near the side banks (α **= 0.2 and 0.8) in case 1a.** Z’ = z/h₂, X’ = x/B, h₂ **is the maximum flow depth at the outlet boundary of cases 2a, 2b and 2c,** h₂= **0.34 m.**

Moreover, in both cases 1a and 2a, the free surface is higher at the channel centre than near the side banks, whether at the front or the end of the confluence-bifurcation unit. Thus, lateral free surface slopes from the centre to the side banks are generated. For example, the lateral free surface slopes at CS09 are 0.022 and 0.016 respectively for cases 1a and 2a. These lateral slopes can lead to lateral pressure gradient force whose direction is from the channel centreline to the side banks. Notably, the lateral surface slope in case 1a is steeper than that in case 2a, which may also result from the effect of the supercritical flow.

4.1.2. Time-averaged streamwise flow velocity

Figure 3. Time-averaged flow velocity distribution at three different slices over z-direction in the confluence-bifurcation unit: (a)∼(c) case 1a, (d)∼(f) case 2a. The flow direction is from the left to the right. StZ = Stagnation Zones, MiL = Mixing Layer. X’ = x/B, Y’ = y/B, Ui’ = Ui/Uti, Ui denotes the time-averaged streamwise flow velocity in case series i (i = 1,2), Uti denotes the cross-section-averaged streamwise flow velocity in case series i, Ut1 = 0.385 m/s, for case 2a Ut2 = 0.714 m/s.

**Figure 4. Time-averaged flow velocity contours at eight different cross-sections in the confluence-bifurcation unit: (a) case 1a, (b) case 2a.**

Besides the shared features described above, some differences between the two cases are also identified. First, flow stagnation zones at the upstream bar tail are found exclusively in case 1a as the central bars are exposed (Figure 3
(a–c)). Second, in case 1a the mixing layer is obvious in both the lower or upper flows (Figure 3
(a–c)), while in case 2a the mixing layer can be inconspicuous in the upper flow (Figure 3
(f)). Third, in case 1a, two high-velocity cores gradually transform into one single core downstream of the confluence [Figure 4
(a), CS08∼CS11] and are divided into two cores again at the downstream bar head [Figure 4
(a), CS15]. By contrast, in case 2a, the two cores merge much more rapidly [Figure 4
(a), CS08∼CS09], and no obvious reseparation of the merged core is found at the downstream bar head (Figure 3
(d–f)). The latter two differences between cases 1a and 2a indicate that the flow convergence and divergence are relatively weak when the central bars are fully submerged. It is noticed that when the central bars are exposed, the flow in branches needs to steer around the central bar, which can cause a large angle between the two flow directions at the confluence, and thus relatively strong flow convergence and divergence may occur. By contrast, when the central bars are fully submerged, the flow behavior resembles that of a straight channel, with flow predominantly moving straight along the main axis of the central bars. Therefore, a small angle between two tributary flow forms, and thus flow convergence and divergence are relatively mild.

4.1.3. Recirculation vortex

A recirculation vortex with a vertical axis is a typical structure usually found where flow steers sharply, and is generated from flow separation (Lu et al., 2023). This vortex structure is found in the confluence-bifurcation unit in the present study, marking several significant flow separation zones. Figure 5 shows the recirculation vortex structure at the bifurcation junction of the confluence-bifurcation unit. In both cases 1a and 2a, two recirculation vortices BV1 and BV2 are found at the bifurcation junction corner. Moreover, BV1 and BV2 seem well-established near the bed but tend to transform into premature ones in the upper flow, and there is also a tendency for the cores of BV1 and BV2 to shift downstream as they transition from the lower to the upper flow (Figure 5(a–c,d–f)). This finding indicates that flow separation zones exist at the bifurcation junction corner, and the vortex structure is similar in the separation zones under low and high discharges. These flow separation zones are generated due to the inertia effect as flow suddenly diverges and steers towards the curved side banks of the channel (Xie et al., 2020). Notably, two additional vortices BV3 and BV4 are found at both sides of the downstream bar in case 1a (Figure 5(a–c)), but no such vortices exist in case 2a. This difference shows that flow separation zones at both sides of the downstream bar are hard to form when the bars are completely submerged under the high discharge.

Figure 5. Recirculation vortices at the bifurcation junction (streamline view at three different slices over z-direction): (a)∼(c) case 1a, (d)∼(f) case 2a. The red solid line marked out the position of these vortices (BV1∼BV4).

Similarly, Figure 6 shows the recirculation vortex structure at the confluence junction of the confluence-bifurcation unit. No noteworthy similarities but a key difference between the two cases are observed at this site. Two vortices CV1 and CV2 are found downstream of the confluence junction corner in case 1a (Figure 6(c)), which mark two separation zones. Conversely, no such separation zones are found in case 2a. In fact, separation zones were reported at similar sites under relatively low discharges in some previous studies (Ashmore et al., 1992, Luz et al., 2020, Sukhodolov & Sukhodolova, 2019; Xie et al., 2020). Nevertheless, the flow separation zones at the confluence corner are very restricted in the present study (Figure 6(c)). Ashmore et al. (1992) also reported that no, or very restricted flow separation zones occur downstream of natural river confluence corners, primarily because of the relatively slow change in bank orientation compared with the sharp corners of laboratory confluences where separation is pronounced (Best & Reid, 1984; Best, 1988). In the present study, the bank orientation also changes slowly, which may explain why flow separation zones are inconspicuous at the confluence corner.

Figure 6. Recirculation vortices at the confluence junction (streamline view at three different slices over z-direction): (a)∼(c) case 1a, (d)∼(f) case 2a. The red solid line marked out the position of these vortices (CV1 & CV2).

The differences in the distribution of recirculation vortices discussed above may be mainly attributed to the difference in the angle between the tributary flows under different discharges. Some previous studies have reported that the confluence/bifurcation angle can significantly influence the flow structure at confluences/bifurcations (Best & Roy, 1991; Ashmore et al., 1992; Miori et al., 2012). Although the confluence/bifurcation angle is fixed due to the determined central bar shape in the present study, the angle between two tributary flows is affected by the varying discharge. When the central bars are exposed under the low discharge, the flow is characterized by a more pronounced curvature of the streamlines, and a large angle between the two tributary flows is noted (Figure 6(b)), causing strong flow convergence and divergence. By contrast, a small angle forms as the central bars are submerged, thereby leading to relatively weak flow convergence/divergence (Figure 6(e)). Overall, the differences mentioned above can be attributed to the differences in the intensity of flow convergence and divergence under different discharges.

It should be noted that some previous studies (Constantinescu et al., 2011; Sukhodolov & Sukhodolova, 2019) presented that there is a wake mode in the mixing layer of two streams at the confluence junction. The wake mode means that in the mixing layer, multiple streamwise coherent vortices moving downstream will form, which is similar to the flow structure around a bluffing body (Constantinescu et al., 2011). However, no such structure has been found within the confluence-bifurcation unit in this study. According to the numerical simulations of Constantinescu et al. (2011), a wake mode was found at a river confluence with a concordant bed and a momentum flux ratio of about 1. The confluence has a much larger angle (∼60°) between the two streams when compared to the confluence junction of the confluence-bifurcation unit in the present study where the angle is about 25°. As flow mechanics at river confluences may include several dominant mechanisms depending on confluence morphology, momentum ratio, the angle between the tributaries and the main channel, and other factors (Constantinescu et al., 2011), the relatively small confluence angle in the present study may explain why the wake mode is absent. The possible effects of the confluence/bifurcation angle are reserved for future study. Additionally, flow separation can lead to reduced local sediment transport capacity, thus causing considerable sediment deposition under natural conditions. Hence, the bank may migrate towards the inner side of the channel at the positions of CV1, CV2, BV1, and BV2, while the bar may expand laterally at the positions of BV3 and BV4.

4.1.4. Secondary current

Secondary current is the flow perpendicular to the mainstream axis (Thorne et al., 1985) and can be categorized into two primary types based on its origin: (1) Secondary current generated by the interaction between centrifugal force and pressure gradient force; (2) Secondary current resulting from turbulence heterogeneity and anisotropy (Lane et al., 2000). There are some widely recognized definitions of secondary current strength (SCS) (Lane et al., 2000). In this paper, the secondary current cells are identified by visible vortex with a streamwise axis, and the definition of SCS proposed by Shukry (1950) is used:

where u_x, u_y, and u_z are flow velocities in x, y, and z directions and u_x represents the mainstream flow velocity.

Figure 7 presents contour plots of SCS and the secondary current structure at key cross-sections of the study area. When the central bars are exposed, at the upstream bar tail (CS08), intense transverse flow occurs with flow converging to the centreline, but no secondary current cell is formed (Figure 7(a)). This is consistent with the findings of Hackney et al. (2018). At the confluence junction (CS09), transverse flow still plays a major role in the secondary current structure, with flow converging to the centreline at the surface and diverging to side banks near the bed (Figure 7(b)). Moreover, ‘back-to-back’ helical cells, which are two vortices rotating reversely, tend to generate at CS09 with their cores located near the side banks (Figure 7(b)) (Mosley, 1976; Ashmore, 1982; Ashmore et al., 1992), yet their forms are rather premature. As the flow goes downstream, the cores of the helical cells gradually rise to the upper flow and approach towards the centreline, and the helical cells become well-established (Figure 7(c–e)). When the flow diverges again at the downstream bar head (CS15), the helical cells attenuate rapidly, and the secondary current structure is once again characterized predominantly by transverse flow (Figure 7(f)).

Figure 7. Distribution of secondary current strength and secondary current cells at six different cross-sections: (a)∼(f) case 1a, (g)∼(l) case 2a. The secondary current cells are identified by visible lateral vortices (streamline view). The zero distance of each cross-section is located on the right bank.

When the central bars are fully submerged under the high discharge, the secondary current structure at the upstream bar tail and the confluence junction exhibits a resemblance to that under the low discharge (Figure 7(g,h)). However, at CS09, two pairs of cells with different scales tend to form under the high discharge (Figure 7(h)). The large and premature helical cells are similar to those under the low discharge, whereas the small helical cells are located near side banks possibly due to wall effects. As the flow moves downstream, the large helical cells tend to diminish rapidly and merge with the small ones near both side walls (Figure 7(i–k)). Moreover, the secondary current structure is once again characterized predominantly by transverse flow at CS14 under the high discharge, which occurs earlier than that under the low discharge (Figure 7(k)). At the downstream bar head, transverse flow still takes a dominant place, while the helical cells seem to become premature with increased scale (Figure 7(l)).

In general, in both cases 1a and 2a, the lateral distribution of SCS at all cross-sections is symmetrical about the channel centreline, where SCS is relatively small. A relatively high SCS is detected at both the upstream bar tail and the downstream bar head due to the effects of centrifugal force caused by flow steering. SCS decreases rapidly from the upstream bar tail (CS08) to the entrance of the downstream bifurcation junction (CS14), followed by a sudden increase at the downstream bar head (CS15) (Figure 7
(a–e, g–k)). However, the distribution of high-SCS zones is different between the two discharges. Under the low discharge, high-SCS zones appear along the bottom near the centerline and at the free surface on both sides of the centreline. Although similar high-SCS zones are found along the bottom near the centerline under the high discharge, the high-SCS zones are not found at the free surface. Furthermore, it is noticed that more obvious high-SCS zones appear under the low discharge compared with the high discharge, especially at CS09. This may be attributed to the differences in the intensity of flow convergence and divergence under different submerging conditions of the central bars. When the central bars are exposed, flow convergence and divergence are strong and sharp flow steering occurs, thereby causing large SCS. By contrast, when the central bars are fully submerged, flow convergence and divergence are relatively weak, and thus small SCS is observed.

4.1.5. Turbulent characteristics

Turbulent characteristics reflect the performance of energy and momentum transfer activities in flow (Sukhodolov et al., 2017). Comprehensive analysis of turbulent characteristics is crucial as they greatly impact the incipient motion, settling behavior, diffusion pattern, and transport process of sediment. Here, the TKE and turbulent dissipation rate (TDR) of flow in the confluence-bifurcation unit are analyzed.

Figure 8 shows the distribution of TKE on various cross-sections in cases 1a and 2a. In the same way, Figure 10 shows the distribution of TDR. The values of TKE and TDR are nondimensionalized with mid-values of TKE = 0.005 m2·s−2 and TDR = 0.007 m3·s−2. In both cases 1a and 2a, the distributions of TKE and TDR show symmetrical patterns concerning the channel centreline. High-TKE and high-TDR zones exhibit a belt distribution near the channel bottom (McLelland et al., 1999; Ashworth, 1996; Constantinescu et al., 2011), indicating that turbulence primarily originates at the channel bottom due to the influence of bed shear stress. A sudden increase of TKE (Weber et al., 2001) and TDR occurs near the channel bottom at the confluence junction [Figure 8 and 9, CS08∼CS09] and from the entrance of the bifurcation junction (CS14) to the downstream bar head (CS15) (Figures 8 and 9).

Figure 8. Turbulent kinetic energy contours at eight different cross-sections in the confluence-bifurcation unit: (a) case 1a, (b) case 2a. TKE = turbulent kinetic energy. TKE’ = dimensionless value of TKE, with regard to a mid-value of TKE = 0.005 m²·s⁻².

Figure 9. Turbulent dissipation rate contours at eight different cross-sections in the confluence-bifurcation unit: (a) case 1a, (b) case 2a. TDR = turbulent dissipation rate. TDR’ = dimensionless value of TDR, with regard to a mid-value of TDR = 0.007 m³·s⁻².

Figure 10. Comparison of the distribution of time-averaged streamwise flow velocity along the flow depth at different cross-sections between the confluence-bifurcation unit and the single confluence. (a)∼(f) 1a vs. 1b, (g)∼(l) 2a vs. 2b.

Despite the common turbulent characteristics between cases 1a and 2a, additional high-TKE zones are found in the upper flow at the upstream bar tail (CS08), the confluence junction (CS09) and the downstream bar head (CS15) (Figure 8) when the central bars are fully submerged. The formation mechanism of these high-TKE zones near the water surface is more complicated, which may result from interactions of velocity gradient, secondary current structure and wall shear stress (Engel & Rhoads, 2017; Lu et al., 2023).

4.2. Comparison with single confluence/bifurcation cases

In this section, the results of a single confluence (cases 1b and 2b) and a single bifurcation (cases 1c and 2c) are compared with those of the confluence-bifurcation unit (cases 1a and 2a) under two discharges. Flow structure at CS08∼CS15 is mainly concerned below.

4.2.1. Comparison with single confluence cases

First, the patterns of time-averaged streamwise velocity, TKE and TDR within the single confluence (presented by contour plots in the supplementary materials) are assessed and then compared with those within the confluence-bifurcation unit (Figures 4, 8, and 9). It is found that distributions of these parameters are similar in the confluence-bifurcation unit and the single confluence from the upstream bar tail (CS08) to the entrance of the bifurcation junction (CS14), despite varying discharges. As the existence of the downstream central bar is the main difference between the single confluence and the confluence-bifurcation unit, this finding indicates that the downstream bar may have limited influence on the flow structure in the confluence-bifurcation unit. In other words, the flow structure in the confluence-bifurcation unit appears to be mainly shaped by the presence of the upstream bar, with its impact potentially reaching as far as the entrance of the bifurcation (CS14). Moreover, under the low discharge, the two high-velocity cores seem to merge later (at CS11) in the single confluence than in the confluence-bifurcation unit (at CS10), which indicates the convergence of two tributary flows may achieve a steady state faster in the confluence-bifurcation unit. To further elucidate the differences, results on the distribution of time-averaged streamwise velocity and TKE along the flow depth are discussed below.

4.2.1.1. Time-averaged streamwise velocity

Figure 10 shows the distribution of time-averaged streamwise flow velocity along the flow depth at different cross-sections. Note that α = 0.5 denotes the channel centreline and α = 0.7 denotes a position near the side banks. As only marginal differences are found at α = 0.3 and 0.7, only profiles at α = 0.7 are displayed for clarity.

Under the low discharge, no obvious difference in the distribution of time-averaged streamwise flow velocity is observed at the upstream bar tail (Figure 10(a)). At the confluence junction (Figure 10(b)), the velocities near the side banks (α = 0.7) are larger than those at the centre (α = 0.5) in both the confluence-bifurcation unit and the single confluence, which suggests that the two tributary flows have not sufficiently merged. The two tributary flows achieve convergence at CS11 in both the confluence-bifurcation unit and the single confluence (Figure 10(c)), with the velocity at the centre (α = 0.5) is larger than that near the side banks. Nevertheless, the velocities at the centre (α = 0.5) and near the side banks (α = 0.7) are closer to each other in the confluence-bifurcation unit than those in the single confluence, which represents less sufficient flow convergence in the confluence-bifurcation unit than in the single confluence. Therefore, it can be inferred that the convergence of two tributary flows may achieve a steady state faster in the confluence-bifurcation unit. After reaching the steady state, the velocity near the side banks (α = 0.7) is smaller in the single confluence than in the confluence-bifurcation unit despite close values at the centre (α = 0.5) (Figure 10(d,e)). This leads to a more pronounced disparity between velocities at the centre and near the side banks in the single confluence than that observed in the confluence-bifurcation unit. In other words, the high-velocity zone is more concentrated on the channel centreline in the single confluence, while the lateral distribution of flow velocity tends to be more uniform in the confluence-bifurcation unit. This may be attributed to the influence of the downstream central bar, which is further proved by comparing the velocity profiles at CS15 (Figure 10(e)).

As for the high discharge condition, from CS08 to CS14, the quantitative differences in velocity distribution between the confluence-bifurcation unit and the single confluence seem small. This indicates that the effect of morphology appears to be subdued when the central bars are fully submerged under the high discharge. It should be also noted that under both the low and high discharge, velocity profiles at the corresponding location exhibit the same shapes in the confluence-bifurcation unit and the single confluence, which indicates that the upstream confluence may dominate the flow structure in the confluence-bifurcation unit.

4.2.1.2. Secondary current

Figure 11 shows contour plots of SCS and the secondary current structure for single confluence cases. Compared with Figure 7, under both low and high discharge conditions, the distribution of SCS and the structure of helical cells in the confluence-bifurcation unit and the single confluence are very similar from CS08 to CS12 (Figure 7(a–d, g–j) and Figure 11(a–d, g–j)]. This indicates that the secondary current structure in the confluence-bifurcation unit exhibits certain consistent features when compared to those in the single confluence, thus proving that the effects of the upstream central bar may dominate the flow structure in the confluence-bifurcation unit. However, the secondary current structure at CS14 and CS15 is different between the confluence-bifurcation unit and the single confluence (Figure 7 and 11(e, f, k,l)). Under the low discharge, transverse flow is from the side banks to the centre and relatively high SCS is found near the side banks at CS14 in the single confluence, while the transverse flow is always from the centre to the side banks and SCS is relatively low at the corresponding sites in the confluence-bifurcation unit (Figure 11(e)). Under the high discharge, the helical cells near the side walls almost diminish in the single confluence, while they still exist in the confluence-bifurcation unit at CS14 (Figure 11(k)). Under both low and high discharges, the secondary current pattern at CS15 is similar to that at CS14 in the single confluence, while they are different in the confluence-bifurcation unit due to the existence of the downstream central bar. This comparison indicates that the existence of the downstream central bar can influence the upstream secondary current structure, nevertheless, the effects are fairly limited.

Figure 11. Secondary current at different cross-sections in the single confluence condition: (a)∼(f) case 1b, (g)∼(l) case 2b. The zero distance of each cross-section is located on the right bank.

4.2.1.3. Turbulent kinetic energy

Figure 12 shows TKE distribution along the flow depth at different cross-sections. Under the low discharge, in general, the maximum TKE tends to appear near the channel bottom in both the confluence-bifurcation unit and the single confluence. No obvious difference is observed at the upstream bar tail (CS08) (Figure 12(a)). Downstream this site (at CS09), the maximum TKE near the side banks (α = 0.7) is larger than that at the channel centre in the single confluence, while they are close to each other in the confluence-bifurcation unit (Figure 12(b)). This can also be attributed to the insufficient convergence of the two tributary flows. At CS11, flow convergence achieves a steady state in the confluence-bifurcation unit, while it remains insufficient in the single confluence. As flow convergence reaches a steady state at CS12, the maximum TKE in the single confluence exhibits a more concentrated distribution on the channel centre than that in the confluence-bifurcation unit (Figure 12(d)). This effect becomes more obvious downstream at CS14 (Figure 12(e)). The less-concentrated distribution of the maximum TKE in the confluence-bifurcation unit can be owing to the effects of the downstream central bar as well, which appears analogous to that mentioned in 4.2.1.1.

Figure 12. Comparison of the distribution of TKE along the flow depth at different cross-sections between the confluence-bifurcation unit and the single confluence. (a)∼(f) 1a vs. 1b, (g)∼(l) 2a vs. 2b.

Under the high discharge condition, two peaks of TKE appear in both the confluence-bifurcation unit and the single confluence (Figure 12(g–l)). Moreover, in both the confluence-bifurcation unit and the single confluence, from the upstream bar tail to the downstream bar head, the peak of TKE in the upper flow is larger at the channel centre (α = 0.5), while the peak of TKE in the lower flow is larger near the side banks (α = 0.7). However, the disparity between the TKE near the side banks and at the channel centre seems to be larger in the single confluence, while the TKE in the confluence-bifurcation unit takes a more uniform distribution. Even though, TKE profiles at the corresponding location exhibit highly similar shapes in the confluence-bifurcation unit and the single confluence, suggesting that the effects of channel morphology seem to be inhibited when the central bars are submerged under the high discharge.

4.2.2. Comparison with single bifurcation cases

Distributions of time-averaged streamwise velocity, TKE and TDR at corresponding cross-sections are also compared between the single bifurcation (see the Supplementary material) and the confluence-bifurcation unit (Figures 4, 8 and 9). Unlike the high similarity in flow characteristics exhibited between the confluence-bifurcation unit and the single confluence, significant differences are found between the confluence-bifurcation unit and the single bifurcation, especially at CS08∼CS14. On the one hand, the high-velocity zones are broader and asymmetrical concerning the channel centreline in the single bifurcation, with a belt-like and an approximately elliptic-like distribution respectively under the low and high discharges. By contrast, the high-velocity zone is a core that concentrates on the channel centre in the confluence-bifurcation unit. Moreover, the maximum velocity seems smaller in the single bifurcation than that in the confluence-bifurcation unit. On the other hand, the high-TKE belt near the channel bottom appears to be narrower in the single bifurcation than in the confluence-bifurcation unit, especially at CS08∼CS14 under the low discharge. Furthermore, additional high-TKE zones are found near the side walls at CS08∼CS11 in the single bifurcation, of which the scale is obviously smaller than those in the confluence-bifurcation unit. In addition, TKE at the channel centre is smaller near the free surface in the single bifurcation than that in the confluence-bifurcation unit. Nevertheless, the distributions of velocity, TKE and TDR seem to be similar in the confluence-bifurcation unit and the single bifurcation at CS15. As the existence of the upstream central bar is the main difference between the single confluence and the confluence-bifurcation unit, all the above findings indicate that the upstream central bar greatly influences the flow structure in the confluence-bifurcation unit. On the other hand, the downstream central bar may have a restricted influence on the flow structure in the confluence-bifurcation unit, whose impact may be limited to a range between the entrance of the bifurcation (CS14) and the downstream bar head (CS15). To further elucidate the differences, results on the distribution of time-averaged streamwise velocity and TKE along the flow depth are discussed below.

4.2.2.1. Time-averaged streamwise velocity

Figure 13 shows the distribution of time-averaged streamwise velocity along the flow depth at different cross-sections. Under the low discharge, distinct distribution patterns of flow velocity between the confluence-bifurcation unit and the single bifurcation are found at CS08, CS09 and CS11, which can be attributed to the effects of upstream flow convergence (Figure 13(a–c)). However, when the flow convergence reaches a steady state in the confluence-bifurcation unit (Figure 13(d–f)), the high-velocity zone is more concentrated in the confluence-bifurcation unit than in the single bifurcation due to to the significant influence of the upstream central bar on the flow structure. The velocity profiles at the downstream bar head can be a shred of evidence as well, with the maximum velocity larger at the channel centre but smaller near the side banks in the confluence-bifurcation unit than in the single bifurcation.

Figure 13. Comparison of the distribution of time-averaged streamwise flow velocity along the flow depth at different cross-sections between the confluence-bifurcation unit and the single bifurcation. (a)∼(f) 1a vs. 1c, (g)∼(l) 2a vs. 2c.

Under the high discharge, the distribution of velocity seems to exhibit limited differences between the two kinds of morphology, which indicates that the effects of channel morphology may be less noticeable when the central bars are fully submerged under the high discharge. Nevertheless, the velocity in the lower flow (below a relative depth of 0.45) shows a uniform lateral distribution in the single bifurcation, as the velocity profile at the channel centreline (α = 0.5) is in line with that near the side banks (α = 0.7) (Figure 13(g–l)). However, in the confluence-bifurcation unit, different velocity distributions in the lower flow can be observed at the channel centreline (α = 0.5) and near the side banks (α = 0.7). The foregoing results indicate that when the central bars are fully submerged, the high-velocity zones are more concentrated on the channel centreline in the confluence-bifurcation unit, while the lateral distribution of flow velocity within the single bifurcation tends to be more uniform, especially near the bifurcation junction (Figure 13(j,k)). This can also be attributed to the dominant influence of the upstream central bar over the downstream central bar.

It is also noted that the flow velocity distribution along the flow depth in the confluence-bifurcation unit is of a similar pattern despite varying discharges. As a critical point, the maximum velocity appears in the upper flow. The distribution above the critical point is approximately linear whereas it appears logarithmic below. By contrast, despite the similarity observed under the low discharge, the flow velocity distribution along the flow depth within the single bifurcation exhibits a distinct pattern under the high discharge, especially near the side banks (Figure 13(e–h)). On the one hand, the critical point in the upper flow no longer corresponds to the maximum velocity. On the other hand, the velocity distribution deviates from logarithmic below the critical point, with the maximum velocity appearing at a relative depth of 0.45. Succinctly, the distribution of streamwise velocity along the flow depth may retain the same pattern regardless of discharge levels in the confluence-bifurcation unit, while it may exhibit distinct patterns under different discharge levels in the single bifurcation.

4.2.2.2. Secondary current

Figure 14 shows contour plots of SCS and the distribution of secondary current for single bifurcation cases. In general, the value of SCS near the side banks at CS08∼CS14 (Figure 14(a–d, g–j)) in the single bifurcation seems smaller than that in the confluence-bifurcation unit (Figure 7(a–d, g–j)), especially under the low discharge. SCS distribution at CS14 is similar in the confluence-bifurcation unit and the single bifurcation under both low and high discharges. This difference in SCS distribution between the confluence-bifurcation unit and the single bifurcation indicates that the downstream bifurcation may have a restricted influence on the flow structure in the confluence-bifurcation unit. This influence is limited to a range between the entrance of the bifurcation (CS14) and the downstream bar head (CS15).

Figure 14. Secondary current at different cross-sections in the single bifurcation condition: (a)∼(f) case 1c, (g)∼(l) case 2c. The zero distance of each cross-section is located on the right bank.

In addition, the secondary current structure may also present different patterns in response to varying channel morphologies and discharge conditions. Under the low discharge condition, multiple unstable helical cells with asymmetrical distribution are formed from CS08 to CS12 in the single bifurcation (Figure 14(a–d)), while no obvious helical cells are found at CS14 and CS15 (Figure 14(d,e)). These findings are quite different from the stable and symmetrical helical cells at all cross-sections shown in the confluence-bifurcation unit (Figure 7). This difference may be attributed to the significant influence of the upstream central bar and the limited influence of the downstream central bar. Under the high discharge condition, only one pair of premature helical cells are found from CS08 to CS12 in the single bifurcation with their cores located near the side banks (Figure 14(e,f)). As the flow moves downstream, the helical cells gradually develop and become well-established (Figure 14(g,h)). These helical cells in the single bifurcation show symmetric cross-sectional distribution and a similar longitudinal development as in the confluence-bifurcation unit. However, in the confluence-bifurcation unit, two pairs of helical cells appear upstream of CS12 and CS14 and gradually fuse to one pair under the high discharge. As the ‘two-pairs’ structure in the confluence-bifurcation unit origins from the upstream confluence, the differences in the secondary current structure between the single bifurcation and the confluence-bifurcation unit under the high discharge can also be owing to the effects of the upstream central bar in excess of those of the downstream central bar.

4.2.2.3. Turbulent kinetic energy

Figure 15 shows the TKE distribution along the flow depth at different cross-sections. Under the low discharge, when the two tributary flows have not achieved sufficient convergence in the confluence-bifurcation unit, the maximum TKE is more concentrated in the single bifurcation (Figure 15(a–c)). As flow convergence achieves a steady state, more concentrated high-TKE zones appear at the channel centre within the confluence-bifurcation unit, confirming the finding that the effects of the upstream central bar reign over those of the downstream central bar in the confluence-bifurcation unit. However, things can be very complicated under the high discharge. For TKE distribution at the channel centreline, two peaks appear in the confluence-bifurcation unit with one close to the free surface and the other near the bed (Figure 15(g–l)). By contrast, only one peak near the bed is present in the single bifurcation. Therefore, a larger TKE can be found in the upper flow of the channel centreline in the confluence-bifurcation unit. For TKE distribution near the side banks, two peaks appear in both the confluence-bifurcation unit and the single bifurcation at CS09∼CS14 (Figure 15(h–l)). The upper peak is larger but the lower peak is smaller within the single bifurcation than those within the confluence-bifurcation unit. These significant discordances in TKE distribution under the high discharge further prove that the effects of the upstream bar on the flow structure in the confluence-bifurcation unit are more prominent than those of the downstream central bar.

Figure 15. Comparison of the distribution of TKE along the flow depth at different cross-sections between the confluence-bifurcation unit and the single bifurcation. (a)∼(f) 1a vs. 1c, (g)∼(l) 2a vs. 2c.

4.2.3. Further discussion of the comparisons

The above subsections have revealed significant differences in flow structure within the confluence-bifurcation unit and the single confluence and bifurcation, which directly result from the distinct channel morphologies and vary with the discharge conditions as well. These differences are summarized and further discussed below.

The distinctive morphology of a confluence-bifurcation unit plays a pivotal role in governing streamwise flow velocity distribution, secondary current structure, and turbulent kinetic energy distribution within the channel. Generally, from the upstream bar tail (CS08) to the entrance of the bifurcation (CS14), the flow structure in the confluence-bifurcation unit is highly similar to that in the single confluence, while it exhibits great differences (as shown in 4.2.2) between the confluence-bifurcation unit and the single bifurcation. This indicates that the upstream central bar greatly influences the flow structure in the confluence-bifurcation unit, with the effects spreading to the entrance of the bifurcation. At the downstream bar head (CS15), the flow structure (e.g. the transverse flow patterns) in the confluence-bifurcation unit exhibits high similarity to that in the single bifurcation. However, these similarities do not spread to upstream cross-sections, suggesting that the influence of the downstream central bar is limited at the bifurcation junction. In a word, the effects of the upstream central bar on the flow structure in the confluence-bifurcation unit are in excess of those of the downstream central bar.

However, despite the influence of channel morphology, discharge may also have some important effects on the streamwise flow velocity distribution. On the one hand, when the central bars are exposed under the low discharge, the high-velocity zone is less concentrated in the confluence-bifurcation unit than in the single confluence, while it is more concentrated in the confluence-bifurcation unit than in the single bifurcation. On the other hand, it is noticed that when the central bars are fully submerged under the high discharge, reduced differences in flow structure between the confluence-bifurcation unit and the single confluence/bifurcation are witnessed, and thus the morphology effect seems to be subdued.

4.3. Implications

The present work unravels the flow structure in a laboratory-scale confluence-bifurcation unit and takes the first step to further investigating morphodynamics in such channel morphology. Based on the comparison with a single confluence/bifurcation, the findings provide insight into the complex 3D interactions between water flow and channel morphology. The distinct flow structure in the laboratory-scale confluence-bifurcation unit may appreciably alter sediment transport and morphological evolution, of which research is underway. As the basic morphological element of braided river planform is confluence-bifurcation units, the present work should have direct implications for flow structure in natural braided rivers. This is pivotal for the sustainable management of braided rivers which deals with water and land resources planning, eco-hydrological well-being, and infrastructure safety such as cross-river bridges and oil pipelines (Redolfi et al., 2019; Ragno et al., 2021).

Notably, braided rivers worldwide (e.g. in the Himalayas, North America, and New Zealand) have undergone increased pressures and will continue to evolve due to forces of global climate change and intensified anthropogenic activities (Caruso et al., 2017; Hicks et al., 2021; Lu et al., 2022). In particular, channel aggradation caused by increased sediment supply as well as exploitation of braidplain compromise space for flood conveyance, making the rivers prone to flooding. In this sense, an enhanced understanding of the flow structure under high discharge when central bars are fully submerged is essential for mitigating flooding hazards.

5. Conclusions

This study has numerically investigated the 3D flow structure in a laboratory-scale confluence-bifurcation unit based on the LES model integrated in the FLOW–3D® software platform. Two different discharges are considered with the central bars fully submerged or exposed respectively when the discharge is high or low. Cases of a single confluence/bifurcation are included for comparison. The key findings of this paper are as follows:

Several differences are highlighted in the comparison of the flow structure in the confluence-bifurcation unit between the two discharges. When the central bars are fully submerged under the high discharge, the mixing layer of two tributary flows is less obvious, and two high-velocity cores merge more rapidly as compared with those under the low discharge. Besides, flow separation zones are found neither at the confluence corner nor on both sides of the downstream bar when the central bars are fully submerged. Moreover, SCS seems to be smaller near the side banks under the high discharge than under the low discharge. Therefore, it is suggested that flow convergence/divergence is relatively weak in the confluence-bifurcation unit when central bars are fully submerged under the high discharge.
From the upstream bar tail to the entrance of the bifurcation, the flow structure in the confluence-bifurcation unit is highly similar to that in the single confluence, while it exhibits great differences from that in the single bifurcation. Only at the downstream bar head does the flow structure in the confluence-bifurcation unit exhibit high similarity to that in the single bifurcation. Consequently, the effects of the upstream central bar on the flow structure in the confluence-bifurcation unit reign over those of the downstream central bar.
Despite the influence of channel morphology, discharge may also have significant effects on the distribution of streamwise flow velocity. On the one hand, when the central bars are exposed under the low discharge, the high-velocity zone is less concentrated in the confluence-bifurcation unit than in the single confluence, while it is more concentrated in the confluence-bifurcation unit than in the single bifurcation. On the other hand, when the central bars are fully submerged under the high discharge, reduced differences in flow structure between the confluence-bifurcation unit and the single confluence/bifurcation are witnessed, and thus the morphology effect seems to be subdued.

It is noticed that the effects of other factors (e.g. confluence and bifurcation angles, bed discordance) on the flow structure in the confluence-bifurcation unit are not discussed here. Studies on these issues are warranted and reserved for future work.

Reference

Ashmore, P. E. (1982). Laboratory modelling of gravel braided stream morphology. Earth Surface Processes and Landforms, 7(3), 201–225. https://doi.org/10.1002/esp.3290070301
Ashmore, P. E. (1991). How do gravel-bed rivers braid? Canadian Journal of Earth Sciences, 28(3), 326–341. https://doi.org/10.1139/e91-030
Ashmore, P. E. (2013). Morphology and dynamics of braided rivers. In J. Shroder, & (Editor in Chief) E. Wohl (Eds.), Treatise on geomorphology (Vol. 9, pp. 289–312). https://doi.org/10.1016/B978-0-12-374739-6.00242-6
Ashmore, P. E., Ferguson, R. I., Prestegaard, K. L., Ashworth, P. J., & Paola, C. (1992). Secondary flow in anabranch confluences of a braided, gravel-bed stream. Earth Surface Processes and Landforms, 17(3), 299–311. https://doi.org/10.1002/esp.3290170308
Ashworth, P. J. (1996). Mid channel bar growth and its relationship to local flow strength and direction. Earth Surface Processes and Landforms, 21(2), 103–123.
Bertoldi, W., & Tubino, M. (2005). Bed and bank evolution of bifurcating channels. Water Resources Research, 41(7), W07001. https://doi.org/10.1029/2004WR003333
Bertoldi, W., & Tubino, M. (2007). River bifurcations: Experimental observations on equilibrium configurations. Water Resources Research, 43(10), W10437. https://doi.org/10.1029/2007WR005907
Best, J. L. (1987). Flow dynamics at river channel confluences: Implications for sediment transport and bed morphology. In F. G. Ethridge, R. M. Flores, & M. D. Harvey (Eds.), Recent developments in fluvial sedimentology (pp. 27–35).
Best, J. L. (1988). Sediment transport and bed morphology at river channel confluences. Sedimentology, 35(3), 481–498. https://doi.org/10.1111/j.1365-3091.1988.tb00999.x
Best, J. L., & Reid, I. (1984). Separation zone at open-channel junctions. Journal of Hydraulic Engineering, 110(11), 1588–1594. https://doi.org/10.1061/(ASCE)0733-9429(1984)110:11(1588)
Best, J. L., & Roy, A. G. (1991). Mixing-layer distortion at the confluence of channels of different depth. Nature, 350(6317), 411–413. https://doi.org/10.1038/350411a0
Biron, P. M., Buffin-Bélanger, T., & Martel, N. (2019). Three-dimensional turbulent structures at a medium-sized confluence with and without an ice cover. Earth Surface Processes and Landforms, 44(15), 3042–3056. https://doi.org/10.1002/esp.4718
Bombardelli, F. A., Meireles, I., & Matos, J. (2011). Laboratory measurements and multi-block numerical simulations of the mean flow and turbulence in the nonaerated skimming flow region of steep stepped spillways. Environmental Fluid Mechanics, 11(3), 263–288. https://doi.org/10.1007/s10652-010-9188-6
Bradbrook, K. F., Biron, P. M., Lane, S. N., Richards, K. S., & Roy, A. G. (1998). Investigation of controls on secondary circulation in a simple confluence geometry using a three-dimensional numerical model. Hydrological Processes, 12(8), 1371–1396. https://doi.org/10.1002/(SICI)1099-1085(19980630)12:8<1371::AID-HYP620>3.0.CO;2-C
Caruso, B., Newton, S., King, R., & Zammit, C. (2017). Modelling climate change impacts on hydropower lake inflows and braided rivers in a mountain basin. Hydrological Sciences Journal, 62(6), 928–946. https://doi.org/10.1080/02626667.2016.1267860
Constantinescu, G., Miyawaki, S., Rhoads, B., & Sukhodolov, A. (2016). Influence of planform geometry and momentum ratio on thermal mixing at a stream confluence with a concordant bed. Environmental Fluid Mechanics, 16(4), 845–873. https://doi.org/10.1007/s10652-016-9457-0
Constantinescu, G., Miyawaki, S., Rhoads, B., Sukhodolov, A., & Kirkil, G. (2011). Structure of turbulent flow at a river confluence with momentum and velocity ratios close to 1: Insight provided by an eddy-resolving numerical simulation. Water Resources Research, 47(5), W05507. https://doi.org/10.1029/2010WR010018
De Serres, B., Roy, A. G., Biron, M. P., & Best, J. L. (1999). Three-dimensional structure of flow at a confluence of river channels with discordant beds. Geomorphology, 26(4), 313–335. https://doi.org/10.1016/S0169-555X(98)00064-6
Duguay, J., Biron, P., & Buffin-Bélanger, T. (2022). Large-scale turbulent mixing at a mesoscale confluence assessed through drone imagery and eddy-resolved modelling. Earth Surface Processes and Landforms, 47(1), 345–363. https://doi.org/10.1002/esp.5251
Egozi, R., & Ashmore, P. E. (2009). Experimental analysis of braided channel pattern response to increased discharge. Journal of Geophysical Research: Earth Surface, 114, F02012. https://doi.org/10.1029/2008JF001099
Engel, F. L., & Rhoads, B. L. (2017). Velocity profiles and the structure of turbulence at the outer bank of a compound meander bend. Geomorphology, 295, 191–201. https://doi.org/10.1016/j.geomorph.2017.06.018
Ettema, R., & Armstrong, D. L. (2019). Bedload and channel morphology along a braided, sand-bed channel: Insights from a large flume. Journal of Hydraulic Research, 57(6), 822–835. https://doi.org/10.1080/00221686.2018.1555557
Federici, B., & Paola, C. (2003). Dynamics of channel bifurcations in noncohesive sediments. Water Resources Research, 39(6), 1162. https://doi.org/10.1029/2002WR001434
Hackney, C. R., Darby, S. E., Parsons, D. R., Leyland, J., Aalto, R., Nicholas, A. P., & Best, J. L. (2018). The influence of flow discharge variations on the morphodynamics of a diffluence-confluence unit on a large river. Earth Surface Processes and Landforms, 43(2), 349–362. https://doi.org/10.1002/esp.4204
Hicks, D. M., Baynes, E. R. C., Measures, R., Stecca, G., Tunnicliffe, J., & Fredrich, H. (2021). Morphodynamic research challenges for braided river environments: Lessons from the iconic case of New Zealand. Earth Surface Processes and Landforms, 46(1), 188–204. https://doi.org/10.1002/esp.5014
Hirt, C. W., & Nichols, B. D. (1981). Volume of fluid (VOF) method for the dynamics of free boundaries. Journal of Computational Physics, 39(1), 201–225. https://doi.org/10.1016/0021-9991(81)90145-5
Hua, Z. L., Gu, L., & Chu, K. J. (2009). Experiments of three-dimensional flow structure in braided rivers. Journal of Hydrodynamics, 21(2), 228–237. https://doi.org/10.1016/S1001-6058(08)60140-7
Hundey, E. J., & Ashmore, P. E. (2009). Length scale of braided river morphology. Water Resources Research, 45(8), W08409. https://doi.org/10.1029/2008WR007521
Iwantoro, A. P., van der Vegt, M., & Kleinhans, M. G. (2022). Stability and asymmetry of tide-influenced river bifurcations. Journal of Geophysical Research: Earth Surface, 127(6), e2021JF006282. https://doi.org/10.1029/2021JF006282
Jang, C. L., & Shimizu, Y. (2005). Numerical simulation of relatively wide, shallow channels with erodible banks. Journal of Hydraulic Engineering, 131, 565–575.
Kelly, S. (2006). Scaling and hierarchy in braided rivers and their deposits: Examples and implications for reservoir modelling. In G. H. Smith, J. L. Best, C. S. Bristow, & G. E. Petts (Eds.), Braided rivers: Process, deposits, ecology and management (pp. 75–106).
Lane, S. N., Bradbrook, K. F., Richards, K. S., Biron, P. M., & Roy, A. G. (2000). Secondary circulation cells in river channel confluences: Measurement artefacts or coherent flow structures? Hydrological Processes, 14(11-12), 2047–2071. https://doi.org/10.1002/1099-1085(20000815/30)14:11/12<2047::AID-HYP54>3.0.CO;2-4
Le, T. B., Khosronejad, A., Sotiropoulos, F., Bartelt, N., Woldeamlak, S., & Dewall, P. (2019). Large-eddy simulation of the Mississippi River under base-flow condition: Hydrodynamics of a natural diffluence-confluence region. Journal of Hydraulic Research, 57(6), 836–851. https://doi.org/10.1080/00221686.2018.1534282
Liu, C. B., Li, J., Bu, W. Y., Ma, W. X., Shen, G., & Yuan, Z. (2018). Large eddy simulation for improvement of performance estimation and turbulent flow analysis in a hydrodynamic torque converter. Engineering Applications of Computational Fluid Mechanics, 12(1), 635–651. https://doi.org/10.1080/19942060.2018.1489896
Liu, Z. (2012). Investigation of flow characteristics around square cylinder with inflow turbulence. Engineering Applications of Computational Fluid Mechanics, 6(3), 426–446. https://doi.org/10.1080/19942060.2012.11015433
Lu, G. W., Liu, J. X., Cao, Z. X., Li, Y. W., Lei, X. T., & Li, Y. (2023). A computational study of 3D flow structure in two consecutive bends subject to the influence of tributary inflow in the middle Yangtze River. Engineering Applications of Computational Fluid Mechanics, 17(1), 2183901. https://doi.org/10.1080/19942060.2023.2183901
Lu, H. Y., Li, Z. W., Hu, X. Y., Chen, B., & You, Y. C. (2022). Morphodynamic processes in a large gravel–bed braided channel in response to runof change: A case study in the Source Region of Yangtze River. Arabian Journal of Geosciences, 15(5), 377. https://doi.org/10.1007/s12517-022-09641-y
Luz, L. D., Szupiany, R. N., Parolin, M., Silva, A., & Stevaux, J. C. (2020). Obtuse-angle vs. confluent sharp meander bends: Insights from the Paraguay-Cuiabá confluence in the tropical Pantanal wetlands, Brazil. Geomorphology, 348, 106907. https://doi.org/10.1016/j.geomorph.2019.106907
Marra, W. A., Parsons, D. R., Kleinhans, M. G., Keevil, G. M., & Thomas, R. E. (2014). Near-bed and surface flow division patterns in experimental river bifurcations. Water Resources Research, 50(2), 1506–1530. https://doi.org/10.1002/2013WR014215
McLelland, S. J., Ashworth, P. J., Best, J. L., Roden, J., & Klaassen, G. J. (1999). Flow structure and transport of sand-grade suspended sediment around an evolving braid bar, Jamuna River, Bangladesh. Fluvial Sedimentology VI, 28, 43–57. https://doi.org/10.1002/9781444304213.ch4
Miori, S., Hardy, R. J., & Lane, S. N. (2012). Topographic forcing of flow partition and flow structures at river bifurcations. Earth Surface Processes and Landforms, 37(6), 666–679. https://doi.org/10.1002/esp.3204
Mosley, M. P. (1976). An experimental study of channel confluences. The Journal of Geology, 84(5), 535–562. https://doi.org/10.1086/628230
Orfeo, O., Parsons, D. R., Best, J. L., Lane, S. N., Hardy, R. J., Kostaschuk, R., Szupiany, R. N., & Amsler, M. L. (2006). Morphology and flow structures in a large confluence-diffluence: Rio Parana, Argentina. In R. M. L. Ferreira, C. T. L. Alves, G. A. B. Leal, & A. H. Cardoso (Eds.), River Flow 2006 (pp. 1277–1282).
Parsons, D. R., Best, J. L., Lane, S. N., Orfeo, O., Hardy, R. J., & Kostaschuk, R. (2007). Form roughness and the absence of secondary flow in a large confluence–diffluence, Rio Paraná, Argentina. Earth Surface Processes and Landforms, 32(1), 155–162. https://doi.org/10.1002/esp.1457
Ragno, N., Redolfi, M., & Tubino, M. (2021). Coupled morphodynamics of river bifurcations and confluences. Water Resources Research, 57(1), e2020WR028515. https://doi.org/10.1029/2020WR028515
Redolfi, M., Tubino, M., Bertoldi, W., & Brasington, J. (2016). Analysis of reach-scale elevation distribution in braided rivers: Definition of a new morphologic indicator and estimation of mean quantities. Water Resources Research, 52(8), 5951–5970. https://doi.org/10.1002/2015WR017918
Redolfi, M., Zolezzi, G., & Tubino, M. (2019). Free and forced morphodynamics of river bifurcations. Earth Surface Processes and Landforms, 44(4), 973–987. https://doi.org/10.1002/esp.4561
Rhoads, B. L., & Kenworthy, S. T. (1995). Flow structure at an asymmetrical stream confluence. Geomorphology, 11(4), 273–293. https://doi.org/10.1016/0169-555X(94)00069-4
Rhoads, B. L., & Sukhodolov, A. N. (2001). Field investigation of three-dimensional flow structure at stream confluences: 1. Thermal mixing and time-averaged velocities. Water Resources Research, 37(9), 2393–2410. https://doi.org/10.1029/2001WR000316
Roy, A. G., & Bergeron, N. (1990). Flow and particle paths at a natural river confluence with coarse bed material. Geomorphology, 3(2), 99–112. https://doi.org/10.1016/0169-555X(90)90039-S
Roy, A. G., Roy, R., & Bergeron, N. (1988). Hydraulic geometry and changes in flow velocity at a river confluence with coarse bed material. Earth Surface Processes and Landforms, 13(7), 583–598. https://doi.org/10.1002/esp.3290130704
Sambrook Smith, G. H., Ashworth, P. J., Best, J. L., Woodward, J., & Simpson, C. J. (2005). The morphology and facies of sandy braided rivers: Some considerations of scale invariance. In M. D. Blum, S. B. Marriott, & S. F. Leclair (Eds.), Fluvial sedimentology VII. International association of sedimentologists. Special Publication No. 35 (pp. 145–158). Blackwell.
Sharifipour, M., Bonakdari, H., Zaji, A. H., & Shamshirband, S. (2015). Numerical investigation of flow field and flowmeter accuracy in open-channel junctions. Engineering Applications of Computational Fluid Mechanics, 9(1), 280–290. https://doi.org/10.1080/19942060.2015.1008963
Shukry, A. (1950). Flow around bends in an open flume. Transactions of the American Society of Civil Engineers, 115(1), 751–778. https://doi.org/10.1061/TACEAT.0006426
Smagorinsky, J. (1963). General circulation experiments with the primitive equations. Monthly Weather Review, 91(3), 99–164. https://doi.org/10.1175/1520-0493(1963)091<0099:GCEWTP>2.3.CO;2
Sukhodolov, A. N., Krick, J., Sukhodolova, T. A., Cheng, Z. Y., Rhoads, B. L., & Constantinescu, G. S. (2017). Turbulent flow structure at a discordant river confluence: Asymmetric jet dynamics with implications for channel morphology. Journal of Geophysical Research: Earth Surface, 122(6), 1278–1293. https://doi.org/10.1002/2016JF004126
Sukhodolov, A. N., & Sukhodolova, T. A. (2019). Dynamics of flow at concordant gravel bed river confluences: Effects of junction angle and momentum flux ratio. Journal of Geophysical Research: Earth Surface, 124(2), 588–615. https://doi.org/10.1029/2018JF004648
Szupiany, R. N., Amsler, M. L., Hernandez, J., Parsons, D. R., Best, J. L., Fornari, E., & Trento, A. (2012). Flow fields, bed shear stresses, and suspended bed sediment dynamics in bifurcations of a large river. Water Resources Research, 48(11), W11515. https://doi.org/10.1029/2011WR011677.
Thomas, R. E., Parsons, D. R., Sandbach, S. D., Keevil, G. M., Marra, W. A., Hardy, R. J., Best, J. L., Lane, S. N., & Ross, J. A. (2011). An experimental study of discharge partitioning and flow structure at symmetrical bifurcations. Earth Surface Processes and Landforms, 36(15), 2069–2082. https://doi.org/10.1002/esp.2231
Thorne, C. R., Zevenbergen, L. W., Pitlick, J. C., Rais, S., Bradley, J. B., & Julien, P. Y. (1985). Direct measurements of secondary currents in a meandering sand-bed river. Nature, 315, 746–747. https://doi.org/10.1038/315746a0.
van der Mark, C. F., & Mosselman, E. (2013). Effects of helical flow in one-dimensional modelling of sediment distribution at river bifurcations. Earth Surface Processes and Landforms, 38(5), 502–511. https://doi.org/10.1002/esp.3335
Wang, X. G., Yan, Z. M., & Guo, W. D. (2007). Three-dimensional simulation for effects of bed discordance on flow dynamics at Y-shaped open channel confluences. Journal of Hydrodynamics, 19(5), 587–593. https://doi.org/10.1016/S1001-6058(07)60157-7
Weber, L. J., Schumate, E. D., & Mawer, N. (2001). Experiments on flow at a 90° open-channel junction. Journal of Hydraulic Engineering, 127(5), 340–350. https://doi.org/10.1061/(ASCE)0733-9429(2001)127:5(340)
Xie, Q. C., Yang, J., & Lundström, T. S. (2020). Flow and sediment behaviours and morpho-dynamics of a diffluence−Confluence unit. River Research and Applications, 36(8), 1515–1528. https://doi.org/10.1002/rra.3697
Xu, L., Yuan, S. Y., Tang, H. W., Qiu, J. J., Whittaker, C., & Gualtieri, C. (2022). Mixing dynamics at the large confluence between the Yangtze River and Poyang Lake. Water Resources Research, 58(11), e2022WR032195. https://doi.org/10.1029/2022WR032195
Yuan, S. Y., Xu, L., Tang, H. W., Xiao, Y., & Gualtieri, C. (2022). The dynamics of river confluences and their effects on the ecology of aquatic environment: A review. Journal of Hydrodynamics, 34(1), 1–14. https://doi.org/10.1007/s42241-022-0001-z
Yuan, S. Y., Yan, G. H., Tang, H. W., Xiao, Y., Rahimi, H., Aye, M. N., & Gualtieri, C. (2023). Effects of tributary floodplain on confluence hydrodynamics. Journal of Hydraulic Research, 61(4), 552–572. https://doi.org/10.1080/00221686.2023.2231413

Download PDF

Consumer Products | 소비자 제품의 설계 및 제조

자유 표면 흐름은 가정과 사무실 환경 모두에서 사용되는 소비자 제품의 설계 및 제조에서 일반적입니다. 예를 들어, 병 채우기는 매일 대규모로 이루어지는 프로세스입니다. 생산 속도를 극대화하면서 낭비를 최소화하도록 이러한 프로세스를 설계하면 시간이 지남에 따라 상당한 비용 절감으로 이어질 수 있습니다. FLOW-3D는 또한 스프레이 노즐을 설계하고 다공성 재료 및 기타 소비재 구성 요소의 흡수 기능을 모델링하는 데 사용할 수 있습니다. FLOW-3D 의 공기 유입, 다공성 매체 및 표면 장력을 포함한 고급 다중 물리 모델을 사용하면 소비자 제품 설계를 정확하게 시뮬레이션하고 최적화하는 것이 쉽습니다.

충전재

유입된 공기는 생산 라인에서 용기를 채울 때 액체의 부피를 늘릴 수 있습니다. 아래 왼쪽 이미지는 높이가 약 20cm인 병을 1.2초 동안 채우는 것을 보여줍니다. 색상 음영은 액체에 있는 공기의 부피 분율을 나타냅니다. 병에서 혼합 시간이 짧고 혼합 정도가 높기 때문에 공기가 표면으로 올라가 빠져나갈 시간이 없었습니다. 그러나 오른쪽 이미지에서 볼 수 있듯이 약 1.7초의 추가 시간이 지나면 공기가 표면으로 올라가면서 발생하는 액체 부피 감소가 명확하게 보입니다. FLOW-3D 의 드리프트 플럭스 모델을 사용하면 액체에 있는 기포와 같은 구성 요소를 분리하여 분리할 수 있습니다.

Tide® 병 충전의 빠른 평가

이 기사에서는 FLOW-3D를 사용하여 새로운 타이드 병 디자인의 충전을 모델링하는 방법을 설명하며, Procter and Gamble Company의 기술 섹션 책임자인 John McKibben이 기고했습니다 .

지금 오전 9시인데 긴급 이메일을 받았다고 상상해보세요.

방금 새로운 Tide® 병 디자인 중 하나가 손잡이에 채워지고 충전 장비에 문제가 생길 수 있다는 것을 깨달았습니다. 우리는 프로토타입 병이 없으며 몇 주 동안 없을 것입니다. 디자이너와 소비자는 디자인의 모습을 좋아하지만, 채우는 방식이 생산 시설에 쇼스토퍼가 될 수 있습니다.

이런 상황이 제게 주어졌을 때, 저는 3D 지오메트리(그림 1)의 스테레오 리소그래피(.stl) 파일을 요청하여 응답을 시작했고, 제가 무엇을 할 수 있는지 알아보고자 했습니다. 저는 FLOW-3D가 .stl 파일을 사용하여 지오메트리를 입력하고 충전을 위한 자유 표면 문제를 해결할 수 있을 것이라는 것을 알고 있었습니다. 저는 이것이 잠재적인 문제에 대한 좋은 정성적 이해를 제공할 것으로 기대했지만, 이 애플리케이션에 얼마나 정확할지에 대해 약간 불확실했습니다.

시뮬레이션 설정 및 실행

오후 1시경에 저는 지오메트리 파일, 유량, 유체 특성을 받았습니다. 몇 시간 이내에 시뮬레이션이 실행되어 예비 결과가 나왔습니다. 저는 제 고객을 초대하여 결과를 잠깐 살펴보게 했고 그는 “사장의 상사”를 데려와서 살펴보게 했습니다. 그래서 저녁 5시경에 예비 결과를 살펴보고 원래 우려했던 것이 문제가 아니라는 것을 확인했습니다.

하지만 결과는 몇 가지 다른 의문을 제기했습니다. 손잡이에 채우면 유입 유체 제트가 많이 깨졌습니다. 이렇게 하면 유입 공기와 거품의 양이 늘어날 것이라는 걸 알았습니다(결국 세탁 세제를 채우고 있으니까요). FLOW-3D 공기 유입 모델을 테스트하기로 했습니다. 이 모델은 원래 난류 제트용으로 개발되었고, 이 층류 문제를 살펴보면 얼마나 잘 수행될지 확신할 수 없었습니다.

그림 2는 공기 유입 모델이 있는 경우와 없는 경우 병 충전 모델의 결과를 보여줍니다. 유입 공기가 포함되면 충전 레벨이 상당히 증가한다는 점에 유의하십시오. 유입 공기가 병 상단에서 유체를 강제로 밀어내지는 않지만 공기 유입 정확도를 확인해야 할 만큼 충분히 가깝습니다. 그림 3은 공기 유입 레벨을 몇 주 후에 실행한 실험 이미지와 비교합니다(시제품 병이 출시된 후). 제트 분리 및 충전 레벨의 질적 일치는 우수하며 시뮬레이션이 병 설계를 선별하기에 충분히 정확하다는 것을 확인했습니다.

홍조

변기가 어떻게 작동하는지 궁금한 적이 있나요? 사실 꽤 복잡합니다. 손잡이를 밀면 물이 변기 그릇을 채우기 시작합니다. 변기 그릇의 유체 수위가 트랩 상단(변기 그릇 뒤) 위로 올라가면 웨어 유형의 흐름이 시작됩니다. 흐름이 충분히 빠르면 변기 그릇에 거품이 형성되어 사이펀이 생성됩니다. 그 지점에서 사이펀이 변기 그릇에서 물을 끌어내고 변기가 물을 흘립니다. 많은 지역에서 물 절약은 중요한 문제이며, 저유량 변기는 가정과 상업용 모두에 필요합니다. 하지만 변기가 첫 번째 시도에서 제 역할을 하지 못하면 물 절약 목표는 달성되지 않습니다. FLOW-3D를 사용하면 다양한 설계를 모델링하여 최적의 결과를 얻을 수 있습니다.

식품 가공

식품 가공 산업은 복잡한 유체, 일반적으로 비뉴턴 유체, 슬러리, 고체와 유체의 혼합물을 관리하여 분배 장비를 최적으로 설계하고 제조하기 위한 다양한 요구 사항이 있습니다. 이는 상업용 장비의 일관성과 내구성 및 품질에 필수적입니다. 또한 포장 디자인의 혁신을 통해 한 제품을 다른 제품과 명확히 구별할 수 있습니다. 예를 들어, 꿀, 케첩 또는 크리머를 깨끗하고 정확하게 분배하는 것은 소비자가 매장에서 내리는 선택일 수 있습니다. 운송 및 보관 요구 사항에는 더 나은 모양 엔지니어링과 더 많은 용기 재료 선택이 필요합니다. 1.5리터 물병이나 세탁 세제를 움직이거나 떨어뜨리는 동안의 유체 하중은 상류 설계의 중요한 부분이 될 수 있습니다.

꿀, 옥수수 시럽, 치약과 같은 점성 유체는 일반적으로 고체 표면에 닿으면 코일을 형성하는 경향이 있습니다. 이 효과는 관찰하기에 흥미롭고 재미있지만, 공기가 제품에 끌려들어 포장이 어려워질 수 있는 포장 공정에서는 환영받지 못할 수 있습니다. 코일링이 발생하는 조건은 유체의 점도, 유체가 떨어지는 거리, 유체의 속도에 따라 달라집니다. FLOW-3D는 다양한 물리적 공정 매개변수를 연구하여 효율적인 공정을 설계하는 데 도움이 되는 정확한 도구를 제공합니다.

혼입

지난 수십 년 동안 컴퓨터화된 측정 및 시뮬레이션 기술의 발전으로 인해 혼합에 대한 이해가 크게 진전되었습니다. 유동 모델링 기술의 지속적인 발전 덕분에 혼합 장비의 유동 의존적 프로세스에 대한 자세한 통찰력을 CFD 소프트웨어를 사용하여 쉽게 시뮬레이션하고 이해할 수 있습니다. 오늘날 블렌딩에서 고체 현탁액, 재킷 반응기의 열 전달에서 발효에 이르기까지 광범위한 응용 분야가 FLOW-3D 의 혼합 기술을 사용하여 모델링됩니다. FLOW-3D 시뮬레이션은 임펠러의 모든 구성과 모든 용기 형상의 혼합 조건에서 블렌딩 시간, 순환 및 전력 수와 같은 주요 혼합 매개변수를 평가하는 데 도움이 될 수 있습니다. 이러한 시뮬레이션은 실험적 방법을 사용하여 보완합니다. 이러한 장비의 유동 의존적 프로세스를 예측하고 이해하기 위해 CFD 소프트웨어를 사용하면 제품 품질을 향상시키고 많은 제품의 비용과 출시 시간을 모두 줄일 수 있습니다.

비뉴턴 유체

혈액, 케첩, 치약, 샴푸, 페인트, 로션과 같은 비뉴턴 유체는 다양한 점도를 가진 복잡한 유동학을 가지고 있습니다. FLOW-3D 는 변형 및/또는 온도에 따라 달라지는 비뉴턴 점도를 가진 이러한 유체를 모델링합니다. 전단 및 온도에 따른 점도는 Carreau, 거듭제곱 법칙 함수 또는 단순히 표 형식의 입력을 통해 설명됩니다. 일부 폴리머, 세라믹 및 반고체 금속의 특징인 시간 종속 또는 틱소트로피 거동도 시뮬레이션할 수 있습니다.

핸드 로션 펌프는 종종 여러 가지 설계 문제와 관련이 있습니다. 펌프가 공기 공극을 가두지 않고 효과적으로 작동하고 로션의 연속적인 흐름을 생성하는 것이 중요합니다. 좋은 설계는 노력이 덜 필요하고 이상적으로는 로션을 원하는 곳으로 향하게 합니다. FLOW-3D 의 이동 객체 모델은 노즐이 아래로 눌리는 것을 시뮬레이션하여 저장소의 로션을 가압하는 데 사용됩니다. 로션의 압력과 로션을 추출하는 데 필요한 힘을 연구할 수 있습니다. 여러 설계 변수는 동일한 고정 구조 메시 내에서 쉽게 분석할 수 있습니다.

다공성 재료

다공성 매체에서 유체의 이동에 대한 수치 모델링은 어려울 수 있지만 FLOW-3D 에는 다공성 재료와 관련된 문제를 해결하는 데 유용한 기능이 많이 포함되어 있습니다. FAVOR™ 기술에는 사용자가 연속적인 다공성 매체를 표현할 수 있도록 하는 데 필요한 다공성 변수가 포함되어 있습니다. FLOW-3D를 사용하면 사용자가 포화 및 불포화 흐름 조건을 모두 시뮬레이션할 수 있습니다. 거듭제곱 법칙 관계를 사용하면 불포화 흐름 조건에서 모세관 압력 과 포화 사이의 비선형 관계를 모델링 할 수 있습니다. 별도의 충전 및 배수 곡선을 사용하여 히스테리시스 현상을 모델링할 수 있습니다. 서로 직접 접촉하는 경우에도 서로 다른 다공성, 투과성 및 습윤성 속성을 서로 다른 장애물에 할당할 수 있습니다. 투과성은 흐름 방향에 따라 지정할 수 있으므로 사용자가 다공성 매체의 이방성 동작을 모델링할 수 있습니다. 유체와 다공성 매체 간의 열 전달을 고려할 수 있습니다.

분무

소용돌이 분무 노즐은 화학 세정제, 의약품 및 연료에서 액체를 분사하는 일반적인 방법입니다. 액체를 성공적으로 분무하려면 일반적으로 노즐로 침투하는 공기 코어를 형성해야 합니다. CFD는 최적의 분무 콘에 대한 기하학, 소용돌이 속도 및 유체 특성의 영향을 탐색하는 효과적인 방법입니다.

이 예에서 2차원 축대칭 소용돌이 흐름이 시뮬레이션되었습니다. 대칭 축을 따라 공기 코어가 노즐의 전체 길이를 거의 관통했습니다. 왼쪽 플롯은 평면에서 속도 분포를 나타내는 벡터가 있는 압력 분포입니다. 오른쪽 플롯은 속도의 소용돌이 구성 요소로 채색되어 있으며 빨간색은 더 높은 값을 나타냅니다.

분무 콘의 규모와 입자 크기가 너무 광범위하기 때문에 분무의 완전한 분무를 직접 계산하는 것은 불가능합니다. 또한 분무는 외부 교란, 노즐의 미세한 결함 및 기타 영향과 밀접하게 관련된 혼란스러운 프로세스입니다. 그러나 노즐을 떠날 때 분무 콘의 특성(예: 벽 두께, 콘 각도, 축 및 방위 속도)을 예측할 수 있다면 이러한 유형의 흐름 장치를 최적화하는 데 큰 도움이 됩니다.

소용돌이 스프레이 노즐 — 소용돌이 분무 노즐의 FLOW-3D 시뮬레이션

Products

자유 표면 흐름은 가정과 사무실 환경 모두에서 사용되는 소비자 제품의 설계 및 제조에서 일반적입니다.

예를 들어, 병 채우기는 매일 대규모로 진행되는 프로세스입니다. 생산 속도를 최대화하면서 낭비를 최소화하도록 이러한 프로세스를 설계하면 시간이 지남에 따라 상당한 비용 절감으로 이어질 수 있습니다. FLOW-3D는 또한 스프레이 노즐을 설계하고 다공성 재료 및 기타 소비재 구성 요소의 흡수 기능을 모델링하는 데 사용할 수 있습니다.

공기 혼입, 다공성 매질 및 표면 장력을 포함한 FLOW-3D의 고급 다중 물리 모델을 사용하면 소비자 제품 설계를 정확하게 시뮬레이션하고 최적화 할 수 있습니다.

Top 20 Fastest Desktops for 2024

본 자료는 Computer에 대한 전문적인 지식보다는 수치해석을 주 목적으로 FLOW-3D 를 이용하기 위한 해석용 컴퓨터를 선택할 때 도움을 주기 위한 자료입니다.

흔히 고성능 컴퓨터는 표준 데스크톱 컴퓨터와 어떻게 다른지 궁금하게 생각하는데, 보통 HPC(high performance computing)는 더욱 강력한 프로세서, 더 큰 메모리, 뛰어난 그래픽 성능을 갖추고 있어 일반적인 표준 데스크톱보다 훨씬 빠르게 여러 가지 복잡한 작업을 동시에 처리할 수 있습니다. 따라서 시중에서 판매하는 고사양의 컴퓨터에 CPU, Memory, Graphic Card 등을 보완하거나 고사양으로 만들어진 컴퓨터를 구매하게 되면, 단일 노드의 HPC(high performance computing)와 유사한 성능을 확보할 수 있습니다.

하지만 개인이 여러 컴퓨터를 대상으로 테스트를 수행하기 어렵기 때문에, 전문가들이 테스트를 수행하여 공개하는 보고서를 참조하여 도움을 얻는 것이 효율적입니다.

아래 전문 성능비교 테스트 보고서가 시스템 선택에 도움이 될 것으로 생각합니다. 참고로, 당사는 기사를 제공하는 기관과 전혀 관련이 없음을 알려드립니다.

Top 20 Fastest Desktops for 2024

Edit: 2024-11-28

원문 출처: https://www.pcbenchmarks.net/fastest-desktop.html


Position	Score	BL#	CPU Type	CPU speed (MHz)	#Phys. CPUs	OS	Motherboard	RAM	Video card	Date uploaded
1	26054.3	2223537	Intel Core i9-14900KS	3188	1	Windows 11 Pro for Workstations build 26100 (64-bit)	ASUSTeK COMPUTER INC. ROG MAXIMUS Z790 APEX ENCORE	49.0 GB	GeForce RTX 4090	2024-10-19 05:20:40
2	25140.3	2102766	Intel Core i9-14900KS	3188	1	Windows 11 Pro for Workstations build 26100 (64-bit)	ASUSTeK COMPUTER INC. ROG MAXIMUS Z790 APEX ENCORE	49.0 GB	GeForce RTX 4090	2024-05-16 19:37:40
3	25130.7	2229143	Intel Core i9-14900KS	3188	1	Windows 11 Pro for Workstations build 26100 (64-bit)	ASUSTeK COMPUTER INC. ROG MAXIMUS Z790 FORMULA	32.5 GB	GeForce RTX 4090	2024-10-25 16:08:42
4	25022.6	2096097	Intel Core i9-14900KF	3188	1	Windows 11 Pro for Workstations build 26100 (64-bit)	ASUSTeK COMPUTER INC. ROG MAXIMUS Z790 APEX ENCORE	49.0 GB	GeForce RTX 4090	2024-05-08 01:22:49
5	24977.1	2093965	Intel Core i9-14900KF	3187	1	Windows 11 Pro for Workstations build 22631 (64-bit)	ASUSTeK COMPUTER INC. ROG MAXIMUS Z790 APEX ENCORE	49.0 GB	GeForce RTX 4090	2024-05-04 21:23:54
6	24550.7	1756060	Intel Core i9-13900KS	3188	1	Windows 10 Home build 19045 (64-bit)	Micro-Star International Co., Ltd. MAG Z790 TOMAHAWK WIFI DDR4(MS-7D91)	32.5 GB	GeForce RTX 4090	2023-02-27 01:36:21
7	24124.9	2010540	Intel Core i9-14900KF	3188	1	Windows 11 Pro for Workstations build 22631 (64-bit)	ASUSTeK COMPUTER INC. ROG MAXIMUS Z790 APEX ENCORE	32.6 GB	GeForce RTX 4090	2024-01-30 06:25:31
8	23924.4	1989560	Intel Core i9-13900KS	3188	1	Windows 11 Pro for Workstations build 22631 (64-bit)	ASUSTeK COMPUTER INC. ROG MAXIMUS Z790 APEX ENCORE	32.6 GB	GeForce RTX 4090	2024-01-06 11:51:42
9	23117.0	1986111	Intel Core i9-14900K	3187	1	Windows 11 Pro for Workstations build 22631 (64-bit)	ASUSTeK COMPUTER INC. ROG MAXIMUS Z790 APEX ENCORE	32.6 GB	GeForce RTX 4090	2024-01-02 23:37:24
10	23035.8	2196479	Intel Core i9-14900K	3188	1	Windows 11 Professional Edition build 26120 (64-bit)	ASUSTeK COMPUTER INC. ROG MAXIMUS Z790 FORMULA	32.5 GB	GeForce RTX 4090	2024-09-16 06:54:48
11	23011.3	2219063	AMD Ryzen 9 9950X	4301	1	Windows 11 Pro for Workstations build 26100 (64-bit)	ASUSTeK COMPUTER INC. ROG CROSSHAIR X870E HERO	48.8 GB	GeForce RTX 4090	2024-10-14 01:33:56
12	23002.9	2219004	AMD Ryzen 9 9950X	4301	1	Windows 11 Pro for Workstations build 26100 (64-bit)	ASUSTeK COMPUTER INC. ROG CROSSHAIR X870E HERO	48.8 GB	GeForce RTX 4090	2024-10-13 22:45:27
13	22696.6	2216075	Intel Core i9-14900KS	3188	1	Windows 11 Professional Edition build 26100 (64-bit)	ASUSTeK COMPUTER INC. ROG MAXIMUS Z790 APEX ENCORE	49.0 GB	GeForce RTX 4090	2024-10-10 10:13:14
14	22676.2	1936617	Intel Core i9-14900K	3187	1	Windows 11 Professional Edition build 22631 (64-bit)	ASUSTeK COMPUTER INC. ROG MAXIMUS Z790 APEX ENCORE	32.6 GB	GeForce RTX 4090	2023-10-31 18:47:25
15	22626.0	2214448	AMD Ryzen 9 9950X	4300	1	Windows 11 Pro for Workstations build 26100 (64-bit)	ASUSTeK COMPUTER INC. ROG CROSSHAIR X870E HERO	32.4 GB	GeForce RTX 4090	2024-10-08 12:43:52
16	22561.7	1947982	Intel Core i9-14900K	3188	1	Windows 11 Professional Edition build 22631 (64-bit)	ASUSTeK COMPUTER INC. ROG MAXIMUS Z790 FORMULA	32.5 GB	GeForce RTX 4090	2023-11-16 12:20:10
17	22561.5	2214771	AMD Ryzen 9 9950X	4300	1	Windows 11 Pro for Workstations build 26100 (64-bit)	ASUSTeK COMPUTER INC. ROG CROSSHAIR X870E HERO	32.4 GB	GeForce RTX 4090	2024-10-08 21:38:56
18	22423.6	2014883	Intel Core i9-14900K	3188	1	Windows 11 Professional Edition build 22635 (64-bit)	ASUSTeK COMPUTER INC. ROG MAXIMUS Z790 FORMULA	32.5 GB	GeForce RTX 4090	2024-02-03 17:43:54
19	22378.5	2079225	Intel Core i9-14900K	3188	1	Windows 11 Professional Edition build 26100 (64-bit)	ASUSTeK COMPUTER INC. ROG MAXIMUS Z790 FORMULA	32.5 GB	GeForce RTX 4090	2024-04-16 04:53:25
20	22368.4	1947167	Intel Core i9-14900K	3187	1	Windows 11 Professional Edition build 22631 (64-bit)	ASUSTeK COMPUTER INC. ROG MAXIMUS Z790 FORMULA	32.5 GB	GeForce RTX 4090	2023-11-15 11:19:39

CPU 벤치마크

아래는 차트에 나타나는 모든 단일 및 다중 소켓 CPU 유형의 목록입니다. 열특정 프로세서 이름을 클릭하면 해당 프로세서가 나타나는 차트로 이동하여 강조 표시됩니다.

https://www.cpubenchmark.net/CPU_mega_page.html


CPU Name	Cores	CPU Mark	Thread Mark	TDP (W)	Socket	Category
AMD Ryzen Threadripper PRO 7995WX	96	153,598	3,964	350	sTR5	Desktop
AMD Ryze n Threadripper 7980X	64	133,902	3,956	350	sTR5	Desktop
AMD Ryzen Threadripper PRO 7985WX	64	133,194	3,890	350	sTR5	Desktop
[Dual CPU] AMD Ryzen Threadripper PRO 3995WX	64	113,693	2,559	280	sWRX8	Desktop, Server
[Dual CPU] AMD Ryzen Threadripper PRO 3975WX	32	98,811	2,676	280	sWRX8	Desktop, Server
AMD Ryzen Threadripper 7970X	32	98,685	4,137	350	sTR5	Desktop
AMD Ryzen Threadripper PRO 7975WX	32	95,623	4,065	350	sTR5	Desktop
AMD Ryzen Threadripper PRO 5995WX	64	93,192	3,207	280	sWRX8	Desktop, Server
AMD Ryzen Threadripper PRO 3995WX	64	83,697	2,598	280	sWRX8	Desktop, Server
AMD Ryzen Threadripper 7960X	24	83,624	4,123	350	sTR5	Desktop
AMD Ryzen Threadripper PRO 7965WX	24	80,920	3,945	350	sTR5	Desktop
AMD Ryzen Threadripper 3990X	64	80,659	2,565	280	sTRX4	Desktop
AMD Ryzen Threadripper PRO 5975WX	32	75,654	3,315	280	sWRX8	Desktop, Server
Intel Core Ultra 9 285K	24	68,768	5,087	125	FCLGA1851	Desktop
AMD Ryzen Threadripper PRO 5965WX	24	66,680	3,359	280	sWRX8	Desktop, Server
AMD Ryzen 9 9950X	16	66,358	4,731	170	AM5	Desktop
[Dual CPU] AMD Ryzen Threadripper PRO 3955WX	16	63,885	2,439	280	sWRX8	Desktop, Server
AMD Ryzen Threadripper 3970X	32	63,115	2,665	280	sTRX4	Desktop
AMD Ryzen 9 7950X	16	62,704	4,275	170	AM5	Desktop
AMD Ryzen 9 7950X3D	16	62,511	4,149	120	AM5	Desktop
AMD Ryzen Threadripper PRO 3975WX	32	62,477	2,656	280	sWRX8	Desktop, Server
Intel Core i9-14900KS	24	62,374	4,871	150	FCLGA1700	Desktop
Intel Core Ultra 7 265KF	20	61,964	4,956	125	FCLGA1851	Desktop
Intel Core i9-13900KS	24	61,536	4,746	150	FCLGA1700	Desktop
Intel Core i9-14900K	24	60,116	4,735	125	FCLGA1700	Desktop
AMD Ryzen Threadripper PRO 7955WX	16	59,968	4,096	350	sTR5	Desktop
Intel Core i9-14900KF	24	59,563	4,710	125	FCLGA1700	Desktop
Intel Core Ultra 7 265K	20	59,162	4,784	125	FCLGA1851	Desktop
Intel Cor e i9-13900K	24	58,996	4,620	125	FCLGA1700	Desktop
Intel Core i9-13900KF	24	58,304	4,609	125	FCLGA1700	Desktop
AMD Ryzen Threadripper 3960X	24	54,891	2,682	280	sTRX4	Desktop
AMD Ryzen 9 9900X	12	54,695	4,684	120	AM5	Desktop

Hardware Selection for FLOW-3D Products – FLOW-3D

부분 업데이트 / ㈜에스티아이씨앤디 솔루션사업부

In this blog, Flow Science’s IT Manager Matthew Taylor breaks down the different hardware components and suggests some ideal configurations for getting the most out of your FLOW-3D products.

개요

본 자료는 Flow Science의 IT 매니저 Matthew Taylor가 작성한 자료를 기반으로 STI C&D에서 일부 자료를 보완한 자료입니다. 본 자료를 통해 FLOW-3D 사용자는 최상의 해석용 컴퓨터를 선택할 때 도움을 받을 수 있을 것으로 기대합니다.

수치해석을 하는 엔지니어들은 사용하는 컴퓨터의 성능에 무척 민감합니다. 그 이유는 수치해석을 하기 위해 여러 준비단계와 분석 시간들이 필요하지만 당연히 압도적으로 시간을 소모하는 것이 계산 시간이기 때문일 것입니다.

따라서 수치해석용 컴퓨터의 선정을 위해서 단위 시간당 시스템이 처리하는 작업의 수나 처리량, 응답시간, 평균 대기 시간 등의 요소를 복합적으로 검토하여 결정하게 됩니다.

또한 수치해석에 적합한 성능을 가진 컴퓨터를 선별하는 방법으로 CPU 계산 처리속도인 Flops/sec 성능도 중요하지만 수치해석을 수행할 때 방대한 계산 결과를 디스크에 저장하고, 해석결과를 분석할 때는 그래픽 성능도 크게 좌우하기 때문에 SSD 디스크와 그래픽카드에도 관심을 가져야 합니다.

FLOW SCIENCE, INC. 에서는 일반적인 FLOW-3D를 지원하는 최소 컴퓨터 사양과 O/S 플랫폼 가이드를 제시하지만, 도입 담당자의 경우, 최상의 조건에서 해석 업무를 수행해야 하기 때문에 가능하면 최고의 성능을 제공하는 해석용 장비 도입이 필요합니다. 이 자료는 2022년 현재 FLOW-3D 제품을 효과적으로 사용하기 위한 하드웨어 선택에 대해 사전에 검토되어야 할 내용들에 대해 자세히 설명합니다. 그리고 실행 중인 시뮬레이션 유형에 따라 다양한 구성에 대한 몇 가지 아이디어를 제공합니다.

CPU 최신 뉴스

2024년 04월 01일 기준

CPU Benchmarks — 이미지 출처 : https://www.cpubenchmark.net/high_end_cpus.html

CPU의 선택

CPU는 전반적인 성능에 큰 영향을 미치며, 대부분의 경우 컴퓨터의 가장 중요한 구성 요소입니다. 그러나 데스크탑 프로세서를 구입할 때가 되면 Intel 과 AMD의 모델 번호와 사양을 이해하는 것이 어려워 보일 것입니다.
그리고, CPU 성능을 평가하는 방법에 의해 가장 좋은 CPU를 고른다고 해도 보드와, 메모리, 주변 Chip 등 여러가지 조건에 의해 성능이 달라질 수 있기 때문에 성능평가 결과를 기준으로 시스템을 구입할 경우, 단일 CPU나 부품으로 순위가 정해진 자료보다는 시스템 전체를 대상으로 평가한 순위표를 보고 선정하는 지혜가 필요합니다.

PassMark - CPU Mark
High End CPUs
Updated 31st of March 2024 — PassMark – CPU Mark High End CPUs Updated 31st of March 2024

<출처>https://www.cpubenchmark.net/high_end_cpus.html

수치해석을 수행하는 CPU의 경우 예산에 따라 Core가 많지 않은 CPU를 구매해야 하는 경우도 있을 수 있습니다. 보통 Core가 많다고 해석 속도가 선형으로 증가하지는 않으며, 해석 케이스에 따라 적정 Core수가 있습니다. 이 경우 예산에 맞는 성능 대비 최상의 코어 수가 있을 수 있기 때문에 Single thread Performance 도 매우 중요합니다. 아래 성능 도표를 참조하여 예산에 맞는 최적 CPU를 찾는데 도움을 받을 수 있습니다.

CPU 성능 분석 방법

부동소수점 계산을 하는 수치해석과 밀접한 Computer의 연산 성능 벤치마크 방법은 대표적으로 널리 사용되는 아래와 같은 방법이 있습니다.

SPECfp 벤치 마크
https://en.wikipedia.org/wiki/SPECfp참고로 “SPEC CPU2017” 성능평가 기준(현재까지 개발된 가장 최신 평가기준임)으로 평가된 2018년 2분기 성능평가 목록은 아래 사이트에서 확인할 수 있습니다.
https://www.spec.org/cpu2017/results/res2018q2/
LINPACK 벤치 마크
https://en.wikipedia.org/wiki/LINPACK_benchmarks

FLOW-3D의 CFD 솔버 성능은 CPU의 부동 소수점 성능에 전적으로 좌우되기 때문에 계산 집약적인 프로그램입니다. 현재 출시된 사용 가능한 모든 CPU를 벤치마킹할 수는 없지만 상대적인 성능을 합리적으로 비교할 수는 있습니다.

특히, 수치해석 분야에서 주어진 CPU에 대해 FLOW-3D 성능을 추정하거나 여러 CPU 옵션 간의 성능을 비교하기 위한 최상의 옵션은 Standard Performance Evaluation Corporation의 SPEC CPU2017 벤치마크(현재까지 개발된 가장 최신 평가기준임)이며, 특히 SPECspeed 2017 Floating Point 결과가 CFD Solver 성능을 매우 잘 예측합니다.

이는 유료 벤치마크이므로 제공된 결과는 모든 CPU 테스트 결과를 제공하지 않습니다. 보통 제조사가 ASUS, Dell, Lenovo, HP, Huawei 정도의 제품에 대해 RAM이 많은 멀티 소켓 Intel Xeon 기계와 같은 값비싼 구성으로 된 장비 결과들을 제공합니다.

CPU 비교를 위한 또 다른 옵션은 Passmark Software의 CPU 벤치마크입니다. PerformanceTest 제품군은 유료 소프트웨어이지만 무료 평가판을 사용할 수 있습니다. 대부분의 CPU는 저렴한 옵션을 포함하여 나열됩니다. 부동 소수점 성능은 전체 벤치마크의 한 측면에 불과하지만 다양한 워크로드에서 전반적인 성능을 제대로 테스트합니다.

예산을 결정하고 해당 예산에 해당하는 CPU를 선택한 후에는 벤치마크를 사용하여 가격에 가장 적합한 성능을 결정할 수 있습니다.

<참고>

SPEC의 벤치 마크 ( https://www.spec.org/benchmarks.html#cpu )

SPEC CPU 2017 (현재까지 가장 최근에 개발된 CPU 성능측정 기준)

다른 컴퓨터 시스템에서 컴퓨팅 계산에 대한 집약적인 워크로드를 비교하는데 사용할 수 있는 성능 측정을 제공하도록 설계된 SPEC CPU 2017에는 SPECspeed 2017 정수, SPECspeed 2017 부동 소수점, SPECrate 2017 정수 및 SPECrate 2017 부동 소수점의 4 가지 제품군으로 구성된 43 개의 벤치 마크가 포함되어 있습니다. SPEC CPU 2017에는 에너지 소비 측정을 위한 선택적 메트릭도 포함되어 있습니다.

벤치마크 결과보고서는 제조사별, 모델별로 테스트한 결과를 아래 사이트에 가면 볼 수 있습니다.

https://www.spec.org/cgi-bin/osgresults

<보고서 샘플>

SPEC CPU 2017

Designed to provide performance measurements that can be used to compare compute-intensive workloads on different computer systems, SPEC CPU 2017 contains 43 benchmarks organized into four suites: SPECspeed 2017 Integer, SPECspeed 2017 Floating Point, SPECrate 2017 Integer, and SPECrate 2017 Floating Point. SPEC CPU 2017 also includes an optional metric for measuring energy consumption.

SPEC CPU 2006
[Retired]
SPEC CPU 2000
[Retired]
SPEC CPU 95
[Retired]
SPEC CPU 92
[Retired]

클럭 대 코어

일반적으로 클럭 속도가 높은 칩은 CPU 코어를 더 적게 포함합니다. FLOW-3D는 병렬화가 잘되어 있지만, 디스크 쓰기와 같이 일부 작업은 기본적으로 단일 스레드 방식으로 수행됩니다. 따라서 데이터 출력이 빈번하거나 큰 시뮬레이션은 종종 더 많은 코어가 아닌, 더 높은 클럭 속도를 활용합니다. 마찬가지로 코어 및 소켓의 다중 스레딩은 오버헤드를 발생시키므로 작은 문제의 해석일 경우 사용되는 코어 수를 제한하면 성능이 향상될 수 있습니다.

CPU 아키텍처

CPU 아키텍처는 중요합니다. 최신 CPU는 일반적으로 사이클당 더 많은 기능을 제공합니다. 즉, 현재 세대의 CPU는 일반적으로 동일한 클럭 속도에서 이전 CPU보다 성능이 우수합니다. 또한 전력 효율이 높아져 와트당 성능이 향상될 수 있습니다. Flow Science에는 구형 멀티 소켓 12, 16, 24 코어 Xeon보다 성능이 뛰어난 최근 세대 10~12 Core i9 CPU 시스템을 보유하고 있습니다.

오버클럭

해석용 장비에서는 CPU를 오버클럭 하지 않는 것이 좋습니다. 하드웨어를 다년간의 투자라고 생각한다면, 오버클럭화는 발열을 증가시켜 수명을 단축시킵니다. CPU에 따라 안정성도 저하될 수 있습니다. CPU를 오버클럭 할 때는 세심한 열 관리가 권장됩니다.

하이퍼스레딩

<이미지출처:https://gameabout.com/krum3/4586040>

하이퍼스레딩은 물리적으로 1개의 CPU를 가상으로 2개의 CPU처럼 작동하게 하는 기술로 파이프라인의 단계수가 많고 각 단계의 길이가 짧을때 유리합니다. 다만 수치해석 처럼 모든 코어의 CPU를 100% 사용중인 장시간 수행 시뮬레이션은 일반적으로 Hyper Threading이 비활성화 된 상태에서 더 잘 수행됩니다. FLOW-3D는 100% CPU 사용률이 일반적이므로 새 하드웨어를 구성할 때 Hyper Threading을 비활성화하는 것이 좋습니다. 설정은 시스템의 BIOS 설정에서 수행합니다.

몇 가지 워크로드의 경우에는 Hyper Threading을 사용하여 약간 더 나은 성능을 보이는 경우가 있습니다. 따라서, 최상의 런타임을 위해서는 두 가지 구성중에서 어느 구성이 더 적합한지 시뮬레이션 유형을 테스트하는 것이 좋습니다.

스케일링

여러 코어를 사용할 때 성능은 선형적이지 않습니다. 예를 들어 12 코어 CPU에서 24 코어 CPU로 업그레이드해도 시뮬레이션 런타임이 절반으로 줄어들지 않습니다. 시뮬레이션 유형에 따라 16~32개 이상의 CPU 코어를 선택할 때는 FLOW-3D 및 FLOW-3D CAST의 HPC 버전을 사용하거나 FLOW-3D CLOUD로 이동하는 것을 고려하여야 합니다.

AMD Ryzen 또는 Epyc CPU

AMD는 일부 CPU로 벤치마크 차트를 석권하고 있으며 그 가격은 매우 경쟁력이 있습니다. FLOW SCIENCE, INC. 에서는 소수의 AMD CPU로 FLOW-3D를 테스트했습니다. 현재 Epyc CPU는 이상적이지 않고 Ryzen은 성능이 상당히 우수합니다. 발열은 여전히 신중하게 다뤄져야 할 문제입니다.

Graphics 고려 사항

FLOW-3D는 OpenGL 드라이버가 만족스럽게 수행되는 최신 그래픽 카드가 필요합니다. 최소한 OpenGL 3.0을 지원하는 것이 좋습니다. 권장 옵션은 엔비디아의 쿼드로 K 시리즈와 AMD의 파이어 프로 W 시리즈입니다.

특히 엔비디아 쿼드로(NVIDIA Quadro)는 엔비디아가 개발한 전문가 용도(워크스테이션)의 그래픽 카드입니다. 일반적으로 지포스 그래픽 카드가 게이밍에 초점이 맞춰져 있지만, 쿼드로는 다양한 산업 분야의 전문가가 필요로 하는 영역에 광범위한 용도로 사용되고 있습니다. 주로 산업계의 그래픽 디자인 분야, 영상 콘텐츠 제작 분야, 엔지니어링 설계 분야, 과학 분야, 의료 분석 분야 등의 전문가 작업용으로 사용되고 있습니다. 따라서 일반적인 소비자를 대상으로 하는 지포스 그래픽 카드와는 다르계 산업계에 포커스 되어 있으며 가격이 매우 비싸서 도입시 예산을 고려해야 합니다.

유의할 점은 엔비디아의 GTX 게이밍 하드웨어는 볼륨 렌더링의 속도가 느리거나 오동작 등 몇 가지 제한 사항이 있습니다. 일반적으로 노트북에 내장된 통합 그래픽 카드보다는 개별 그래픽 카드를 강력하게 추천합니다. 최소한 그래픽 메모리는 512MB 이상을 권장합니다.

PassMark - G3D Mark
High End Videocards — PassMark – G3D Mark High End Videocards

출처 : https://www.videocardbenchmark.net/high_end_gpus.html

원격데스크탑 사용시 고려 사항

Flow Science는 nVidia 드라이버 버전이 341.05 이상인 nVidia Quadro K, M 또는 P 시리즈 그래픽 하드웨어를 권장합니다. 이 카드와 드라이버 조합을 사용하면 원격 데스크톱 연결이 완전한 3D 가속 기능을 갖춘 기본 하드웨어에서 자동으로 실행됩니다.

원격 데스크톱 세션에 연결할 때 nVidia Quadro 그래픽 카드가 설치되어 있지 않으면 Windows는 소프트웨어 렌더링을 사용합니다. FLOW-3D 가 소프트웨어 렌더링을 사용하고 있는지 확인하려면 FLOW-3D 도움말 메뉴에서 정보를 선택하십시오. GDI Generic을 소프트웨어 렌더링으로 사용하는 경우 GL_RENDERER 항목에 표시됩니다.

하드웨어 렌더링을 활성화하는 몇 가지 옵션이 있습니다. 쉬운 방법 중 하나는 실제 콘솔에서 FLOW-3D를 시작한 다음 원격 데스크톱 세션을 연결하는 것입니다. Nice Software DCV 와 같은 일부 VNC 소프트웨어는 기본적으로 하드웨어 렌더링을 사용합니다.

RAM 고려 사항

프로세서 코어당 최소 4GB의 RAM은 FLOW-3D의 좋은 출발입니다. POST Processor를 사용하여 후처리 작업을 할 경우 충분한 양의 RAM을 사용하는 것이 좋습니다.

현재 주력제품인 DDR4보다 2배 빠른 DDR5가 곧 출시된다는 소식도 있습니다.

일반적으로 FLOW-3D를 이용하여 해석을 할 경우 격자(Mesh)수에 따라 소요되는 적정 메모리 크기는 아래와 같습니다.페이지 보기

초대형 (2억개 이상의 셀) : 최소 128GB
대형 (60 ~ 1억 5천만 셀) : 64 ~ 128GB
중간 (30-60백만 셀) : 32-64GB
작음 (3 천만 셀 이하) : 최소 32GB

HDD 고려 사항

수치해석은 해석결과 파일의 데이터 양이 매우 크기 때문에 읽고 쓰는데, 속도면에서 매우 빠른 SSD를 적용하면 성능면에서 큰 도움이 됩니다. 다만 SSD 가격이 비싸서 가성비 측면을 고려하여 적정수준에서 결정이 필요합니다.

CPU와 저장장치 간 데이터가 오고 가는 통로가 그림과 같이 3가지 방식이 있습니다. 이를 인터페이스라 부르며 SSD는 흔히 PCI-Express 와 SATA 통로를 이용합니다.

흔히 말하는 NVMe는 PCI-Express3.0 지원 SSD의 경우 SSD에 최적화된 NVMe (NonVolatile Memory Express) 전송 프로토콜을 사용합니다. 주의할 점은 MVMe중에서 SATA3 방식도 있기 때문에 잘 구별하여 구입하시기 바랍니다.

그리고 SSD를 선택할 경우에도 SSD 종류 중에서 PCI Express 타입은 매우 빠르고 가격이 고가였지만 최근에는 많이 저렴해졌습니다. 따라서 예산 범위내에서 NVMe SSD등 가장 효과적인 선택을 하는 것이 좋습니다.
( 참고 : 해석용 컴퓨터 SSD 고르기 참조 )

기존의 물리적인 하드 디스크의 경우, 디스크에 기록된 데이터를 읽기 위해서는 데이터를 읽어내는 헤드(바늘)가 물리적으로 데이터가 기록된 위치까지 이동해야 하므로 이동에 일정한 시간이 소요됩니다. (이러한 시간을 지연시간, 혹은 레이턴시 등으로 부름) 따라서 하드 디스크의 경우 데이터를 읽기 위한 요청이 주어진 뒤에 데이터를 실제로 읽기까지 일정한 시간이 소요되는데, 이 시간을 일정한 한계(약 10ms)이하로 줄이는 것이 불가능에 가까우며, 데이터가 플래터에 실제 기록된 위치에 따라서 이러한 데이터에의 접근시간 역시 차이가 나게 됩니다.

하지만 HDD의 최대 강점은 가격대비 용량입니다. 현재 상용화되어 판매하는 대용량 HDD는 12TB ~ 15TB가 공급되고 있으며, 이는 데이터 저장이나 백업용으로 가장 좋은 선택이 됩니다.
결론적으로 데이터를 직접 읽고 쓰는 드라이브는 SSD를 사용하고 보관하는 용도의 드라이브는 기존의 HDD를 사용하는 방법이 효과적인 선택이 될 수 있습니다.

PassMark – Disk Rating High End Drives

출처 : https://www.harddrivebenchmark.net/high_end_drives.html

상기 벤치마크 테스트는 테스트 조건에 따라 그 성능 곡선이 달라질 수 있기 때문에 조건을 확인할 필요가 있습니다. 예를 들어 Windows7, windows8, windows10 , windows11 모두에서 테스트한 결과를 평균한 점수와 자신이 사용할 컴퓨터 O/S에서 테스트한 결과는 다를 수 있습니다. 상기 결과에 대한 테스트 환경에 대한 내용은 아래 사이트를 참고하시기 바랍니다.

참고 : 테스트 환경

페이지 보기

Sacrificial Piles as Scour Countermeasures in River Bridges A Numerical Study using FLOW-3D

하천 교량의 파괴 대책으로서 희생파일에 대한 FLOW-3D를 이용한 수치 연구

Mohammad Nazari-Sharabian, Aliasghar Nazari-Sharabian, Moses Karakouzian, Mehrdad Karami

Abstract

Scour is defined as the erosive action of flowing water, as well as the excavating and carrying away materials from beds and banks of streams, and from the vicinity of bridge foundations, which is one of the main causes of river bridge failures. In the present study, implementing a numerical approach, and using the FLOW-3D model that works based on the finite volume method (FVM), the applicability of using sacrificial piles in different configurations in front of a bridge pier as countermeasures against scouring is investigated. In this regard, the numerical model was calibrated based on an experimental study on scouring around an unprotected circular river bridge pier. In simulations, the bridge pier and sacrificial piles were circular, and the riverbed was sandy. In all scenarios, the flow rate was constant and equal to 45 L/s. Furthermore, one to five sacrificial piles were placed in front of the pier in different locations for each scenario. Implementation of the sacrificial piles proved to be effective in substantially reducing the scour depths. The results showed that although scouring occurred in the entire area around the pier, the maximum and minimum scour depths were observed on the sides (using three sacrificial piles located upstream, at three and five times the pier diameter) and in the back (using five sacrificial piles located upstream, at four, six, and eight times the pier diameter) of the pier. Moreover, among scenarios where single piles were installed in front of the pier, installing them at a distance of five times the pier diameter was more effective in reducing scour depths. For other scenarios, in which three piles and five piles were installed, distances of six and four times the pier diameter for the three piles scenario, and four, six, and eight times the pier diameter for the five piles scenario were most effective.

Keywords

Scouring; River Bridges; Sacrificial Piles; Finite Volume Method (FVM); FLOW-3D.

References

Karakouzian, Chavez, Hayes, and Nazari-Sharabian. “Bulbous Pier: An Alternative to Bridge Pier Extensions as a Countermeasure Against Bridge Deck Splashing.” Fluids 4, no. 3 (July 24, 2019): 140. doi:10.3390/fluids4030140.

Karami, Mehrdad, Abdorreza Kabiri-Samani, Mohammad Nazari-Sharabian, and Moses Karakouzian. “Investigating the Effects of Transient Flow in Concrete-Lined Pressure Tunnels, and Developing a New Analytical Formula for Pressure Wave Velocity.” Tunnelling and Underground Space Technology 91 (September 2019): 102992. doi:10.1016/j.tust.2019.102992.

Karakouzian, Moses, Mohammad Nazari-Sharabian, and Mehrdad Karami. “Effect of Overburden Height on Hydraulic Fracturing of Concrete-Lined Pressure Tunnels Excavated in Intact Rock: A Numerical Study.” Fluids 4, no. 2 (June 19, 2019): 112. doi:10.3390/fluids4020112.

Chiew, Yee-Meng. “Scour protection at bridge piers.” Journal of Hydraulic Engineering 118, no. 9 (1992): 1260-1269. doi:10.1061/(ASCE)0733-9429(1992)118:9(1260).

Shen, Hsieh Wen, Verne R. Schneider, and Susumu Karaki. “Local scour around bridge piers.” Journal of the Hydraulics Division (1969): 1919-1940.

Richardson, E.V., and Davis, S.R. “Evaluating Scour at Bridges”. Hydraulic Engineering Circular. (2001), 18 (HEC-18), Report no. FHWA NHI 01–001, U.S. Department of Transportation, Federal Highway Administration, Washington, DC, USA.

Elsaeed, Gamal, Hossam Elsersawy, and Mohammad Ibrahim. “Scour Evaluation for the Nile River Bends on Rosetta Branch.” Advances in Research 5, no. 2 (January 10, 2015): 1–15. doi:10.9734/air/2015/17380.

Chang, Wen-Yi, Jihn-Sung Lai, and Chin-Lien Yen. “Evolution of scour depth at circular bridge piers.” Journal of Hydraulic Engineering 130, no. 9 (2004): 905-913. doi:10.1061/(ASCE)0733-9429(2004)130:9(905).

Unger, Jens, and Willi H. Hager. “Riprap failure at circular bridge piers.” Journal of Hydraulic Engineering 132, no. 4 (2006): 354-362. doi:10.1061/(ASCE)0733-9429(2006)132:4(354).

Abdeldayem, Ahmed W., Gamal H. Elsaeed, and Ahmed A. Ghareeb. “The effect of pile group arrangements on local scour using numerical models.” Advances in Natural and Applied Sciences 5, no. 2 (2011): 141-146.

Sheppard, D. M., B. Melville, and H. Demir. “Evaluation of Existing Equations for Local Scour at Bridge Piers.” Journal of Hydraulic Engineering 140, no. 1 (January 2014): 14–23. doi:10.1061/(asce)hy.1943-7900.0000800.

Melville, Bruce W., and Anna C. Hadfield. “Use of sacrificial piles as pier scour countermeasures.” Journal of Hydraulic Engineering 125, no. 11 (1999): 1221-1224. doi:10.1061/(ASCE)0733-9429(1999)125:11(1221).

Yao, Weidong, Hongwei An, Scott Draper, Liang Cheng, and John M. Harris. “Experimental Investigation of Local Scour Around Submerged Piles in Steady Current.” Coastal Engineering 142 (December 2018): 27–41. doi:10.1016/j.coastaleng.2018.08.015.

Link, Oscar, Marcelo García, Alonso Pizarro, Hernán Alcayaga, and Sebastián Palma. “Local Scour and Sediment Deposition at Bridge Piers During Floods.” Journal of Hydraulic Engineering 146, no. 3 (March 2020): 04020003. doi:10.1061/(asce)hy.1943-7900.0001696.

Khan, Mujahid, Mohammad Tufail, Muhammad Fahad, Hazi Muhammad Azmathullah, Muhammad Sagheer Aslam, Fayaz Ahmad Khan, and Asif Khan. “Experimental analysis of bridge pier scour pattern.” Journal of Engineering and Applied Sciences 36, no. 1 (2017): 1-12.

Yang, Yifan, Bruce W. Melville, D. M. Sheppard, and Asaad Y. Shamseldin. “Clear-Water Local Scour at Skewed Complex Bridge Piers.” Journal of Hydraulic Engineering 144, no. 6 (June 2018): 04018019. doi:10.1061/(asce)hy.1943-7900.0001458.

Moussa, Yasser Abdallah Mohamed, Tarek Hemdan Nasr-Allah, and Amera Abd-Elhasseb. “Studying the Effect of Partial Blockage on Multi-Vents Bridge Pier Scour Experimentally and Numerically.” Ain Shams Engineering Journal 9, no. 4 (December 2018): 1439–1450. doi:10.1016/j.asej.2016.09.010.

Guan, Dawei, Yee-Meng Chiew, Maoxing Wei, and Shih-Chun Hsieh. “Characterization of Horseshoe Vortex in a Developing Scour Hole at a Cylindrical Bridge Pier.” International Journal of Sediment Research 34, no. 2 (April 2019): 118–124. doi:10.1016/j.ijsrc.2018.07.001.

Dougherty, E.M. “CFD Analysis of Bridge Pier Geometry on Local Scour Potential” (2019). LSU Master’s Theses. 5031.

Vijayasree, B. A., T. I. Eldho, B. S. Mazumder, and N. Ahmad. “Influence of Bridge Pier Shape on Flow Field and Scour Geometry.” International Journal of River Basin Management 17, no. 1 (November 10, 2017): 109–129. doi:10.1080/15715124.2017.1394315.

Farooq, Rashid, and Abdul Razzaq Ghumman. “Impact Assessment of Pier Shape and Modifications on Scouring Around Bridge Pier.” Water 11, no. 9 (August 23, 2019): 1761. doi:10.3390/w11091761.

Link, Oscar, Cristian Castillo, Alonso Pizarro, Alejandro Rojas, Bernd Ettmer, Cristián Escauriaza, and Salvatore Manfreda. “A Model of Bridge Pier Scour During Flood Waves.” Journal of Hydraulic Research 55, no. 3 (November 18, 2016): 310–323. doi:10.1080/00221686.2016.1252802.

Karakouzian, Moses, Mehrdad Karami, Mohammad Nazari-Sharabian, and Sajjad Ahmad. “Flow-Induced Stresses and Displacements in Jointed Concrete Pipes Installed by Pipe Jacking Method.” Fluids 4, no. 1 (February 21, 2019): 34. doi:10.3390/fluids4010034.

Flow Science, Inc. FLOW-3D User’s Manual, Flow Science (2018).

Brethour, J. Modeling Sediment Scour. Flow Science, Santa Fe, NM. (2003).

Brethour, James, and Jeff Burnham. “Modeling sediment erosion and deposition with the FLOW-3D sedimentation & scour model.” Flow Science Technical Note, FSI-10-TN85 (2010): 1-22.

Balouchi, M., and Chamani, M.R. “Investigating the Effect of using a Collar around a Bridge Pier, on the Shape of the Scour Hole”. Proceedings of the First International Conference on Dams and Hydropower (2012) (In Persian).

Bayon, Arnau, Daniel Valero, Rafael García-Bartual, Francisco José Vallés-Morán, and P. Amparo López-Jiménez. “Performance Assessment of OpenFOAM and FLOW-3D in the Numerical Modeling of a Low Reynolds Number Hydraulic Jump.” Environmental Modelling & Software 80 (June 2016): 322–335. doi:10.1016/j.envsoft.2016.02.018.

Aminoroayaie Yamini, O., S. Hooman Mousavi, M. R. Kavianpour, and Azin Movahedi. “Numerical Modeling of Sediment Scouring Phenomenon Around the Offshore Wind Turbine Pile in Marine Environment.” Environmental Earth Sciences 77, no. 23 (November 24, 2018). doi:10.1007/s12665-018-7967-4.

Nazari-Sharabian, Mohammad, Masoud Taheriyoun, Sajjad Ahmad, Moses Karakouzian, and Azadeh Ahmadi. “Water Quality Modeling of Mahabad Dam Watershed–Reservoir System under Climate Change Conditions, Using SWAT and System Dynamics.” Water 11, no. 2 (February 24, 2019): 394. doi:10.3390/w11020394.

DOI: 10.28991/cej-2020-03091531

Download PDF

Computational Fluid Dynamics Study of Perforated Monopiles

Mary Kathryn Walker
Florida Institute of Technology, mwalker2022@my.fit.edu

Robert J. Weaver, Ph.D.
Associate Professor
Ocean Engineering and Marine Sciences
Major Advisor

Chungkuk Jin, Ph.D.
Assistant Professor
Ocean Engineering and Marine Sciences

Kelli Z. Hunsucker, Ph.D.
Assistant Professor
Ocean Engineering and Marine Sciences

Richard B. Aronson, Ph.D.
Professor and Department Head
Ocean Engineering and Marine Sciences

Abstract

모노파일은 해상 풍력 터빈 건설에 사용되며 일반적으로 설계 수명은 25~50년입니다. 모노파일은 수명 주기 동안 부식성 염수 환경에 노출되어 구조물을 빠르게 분해하는 전기화학적 산화 공정을 용이하게 합니다. 이 공정은 모노파일을 보호 장벽으로 코팅하고 음극 보호 기술을 구현하여 완화할 수 있습니다.

역사적으로 모노파일 설계자는 파일 내부가 완전히 밀봉되고 전기화학적 부식 공정이 결국 사용 가능한 모든 산소를 소모하여 반응을 중단시킬 것이라고 가정했습니다. 그러나 도관을 위해 파일 벽에 만든 관통부는 종종 누출되어 신선하고 산소화된 물이 내부 공간으로 유입되었습니다.

표준 부식 방지 기술을 보다 효과적으로 적용할 수 있는 산소화된 환경으로 내부 공간을 재고하는 새로운 모노파일 설계가 연구되고 있습니다. 이러한 새로운 모노파일은 간조대 또는 조간대 수준에서 벽에 천공이 있어 신선하고 산소화된 물이 구조물을 통해 흐를 수 있습니다.

이러한 천공은 또한 구조물의 파도 하중을 줄일 수 있습니다. 유체 역학적 하중 감소의 크기는 천공의 크기와 방향에 따라 달라집니다. 이 연구에서는 천공의 크기에 따른 모노파일의 힘 감소 분석에서 전산 유체 역학(CFD)의 적용 가능성을 연구하고 주어진 파도의 접근 각도 변화의 효과를 분석했습니다.

모노파일의 힘 감소를 결정하기 위해 이론적 3D 모델을 제작하여 FLOW-3D® HYDRO를 사용하여 테스트했으며, 천공되지 않은 모노파일을 제어로 사용했습니다. 이론적 데이터를 수집한 후, 동일한 종류의 천공이 있는 물리적 스케일 모델을 파도 탱크를 사용하여 테스트하여 이론적 모델의 타당성을 확인했습니다.

CFD 시뮬레이션은 물리적 모델의 10% 이내, 이전 연구의 5% 이내에 있는 것으로 나타났습니다. 물리적 모델과 시뮬레이션 모델을 검증한 후, 천공의 크기가 파도 하중 감소에 뚜렷한 영향을 미치고 주어진 파도의 접근 각도에 대한 테스트를 수행할 수 있음을 발견했습니다.

접근 각도의 변화는 모노파일을 15°씩 회전하여 시뮬레이션했습니다. 이 논문에 제시된 데이터는 모노파일의 방향이 통계적으로 유의하지 않으며 천공 모노파일의 설계 고려 사항이 되어서는 안 된다는 것을 시사합니다.

또한 파도 하중 감소와 구조적 안정성 사이의 균형을 찾기 위해 천공의 크기와 모양에 대한 연구를 계속하는 것이 좋습니다.

Monopiles are used in the construction of offshore wind turbines and typically have a design life of 25 to 50 years. Over their lifecycle, monopiles are exposed to a corrosive saltwater environment, facilitating a galvanic oxidation process that quickly degrades the structure. This process can be mitigated by coating the monopile in a protective barrier and implementing cathodic protection techniques. Historically, monopile designers assumed the interior of the pile would be completely sealed and the galvanic corrosion process would eventually consume all the available oxygen, halting the reaction. However, penetrations made in the pile wall for conduit often leaked and allowed fresh, oxygenated water to enter the interior space. New monopile designs are being researched that reconsider the interior space as an oxygenated environment where standard corrosion protection techniques can be more effectively applied. These new monopiles have perforations through the wall at intertidal or subtidal levels to allow fresh, oxygenated water to flow through the structure. These perforations can also reduce wave loads on the structure. The magnitude of the hydrodynamic load reduction depends on the size and orientation of the perforations. This research studied the applicability of computational fluid dynamics (CFD) in analysis of force reduction on monopiles in relation to size of a perforation and to analyze the effect of variation in approach angle of a given wave. To determine the force reduction on the monopile, theoretical 3D models were produced and tested using FLOW-3D® HYDRO with an unperforated monopile used as the control. After the theoretical data was collected, physical scale models with the same variety of perforations were tested using a wave tank to determine the validity of the theoretical models. The CFD simulations were found to be within 10% of the physical models and within 5% of previous research. After the physical and simulated models were validated, it was found that the size of the perforations has a distinct impact on the wave load reduction and testing for differing approach angles of a given wave could be conducted. The variation in approach angle was simulated by rotating the monopile in 15° increments. The data presented in this paper suggests that the orientation of the monopile is not statistically significant and should not be a design consideration for perforated monopiles. It is also suggested to continue the study on the size and shape of the perforations to find the balance between wave load reduction and structural stability.

Figure 1: Overview sketch of typical monopile (MP) foundation and transition piece (TP) design with an internal j-tube (Hilbert et al., 2011) — Figure 1: Overview sketch of typical monopile (MP) foundation and transition
piece (TP) design with an internal j-tube (Hilbert et al., 2011)

Computational-Fluid-Dynamics-Study-of-Perforated-Monopiles-1 Download

References
Andersen, J., Abrahamsen, R., Andersen, T., Andersen, M., Baun, T., & Neubauer,
J. (2020). Wave Load Mitigation by Perforation of Monopiles. Journal of
Marine Science and Engineering, 8(5), 352.
https://doi.org/10.3390/jmse8050352
Bakker A. (2008) Lectures on Applied Computational Fluid Dynamics.
www.bakker.org.
Bustamante, A., Vera-Tudela, L., & Kühn, M. (2015). Evaluation of wind farm
effects on fatigue loads of an individual wind turbine at the EnBW baltic 1
offshore wind farm. Journal of Physics: Conference Series, 625, 012020.
https://doi.org/10.1088/1742-6596/625/1/012020
Chakrabarti SK. Hydrodynamics of offshore structures. Springer Verlag;1987.
Christiansen, R. (2020). Living Docks: Structural Implications and Determination
of Force Coefficients of Oyster Mats on Dock Pilings in the Indian River
Lagoon [Master’s Thesis, Florida Institute of Technology].
Clauss, G. (1992). Offshore Structures, Volume 1, Conceptual Design and
Hydromechanics. Springer, London, UK.
COMSOL Multiphysics® v. 6.1. www.comsol.com. COMSOL AB, Stockholm,
Sweden.
Delwiche, A. & Tavares, I. (2017). Retrofit Strategy using Aluminum Anodes for
the Internal section of Windturbine Monopiles. NACE Internation
Corrosion Conference & Expo, Paper no. 8955.
Det Norske Veritas (2014) Fatigue design of offshore steel structures. Norway.
70
Det Norske Veritas (1989). Rules for the Classification of Fixed Offshore
Installations. Technical report, DNV, Hovik, Norway.
DNV. (2011). DNV-RP-C203 Fatigue Design of Offshore Steel Structures (tech.
rep.). http://www.dnv.com
Elger, D. F., LeBret, B. A., Crowe, C. T., & Roberson, J. A. (2022). Engineering
fluid mechanics. John Wiley & Sons, Inc.
FLOW-3D® Version 12.0 Users Manual (2018). FLOW-3D [Computer software].
Santa Fe, NM: Flow Science, Inc. https://www.flow3d.com
Gaertner, Evan, Jennifer Rinker, Latha Sethuraman, Frederik Zahle, Benjamin
Andersen, Garrett Barter, Nikhar Abbas, Fanzhong Meng, Pietro Bortolotti,
Witold Skrzypinski, George Scott, Roland Feil, Henrik Bredmose,
Katherine Dykes, Matt Shields, Christopher Allen, and Anthony Viselli.
(2020). Definition of the IEA 15-Megawatt Offshore Reference Wind.
Golden, CO: National Renewable Energy Laboratory. NREL/TP-5000-

https://www.nrel.gov/docs/fy20osti/75698.pdf
Goodisman, Jerry (2001). “Observations on Lemon Cells”. Journal of Chemical
Education. 78 (4): 516–518. Bibcode:2001JChEd..78..516G.
doi:10.1021/ed078p516. Goodisman notes that many chemistry textbooks
use an incorrect model for a cell with zinc and copper electrodes in an
acidic electrolyte
Hilbert, L.R. & Black, Anders & Andersen, F. & Mathiesen, Troels. (2011).
Inspection and monitoring of corrosion inside monopile foundations for
offshore wind turbines. European Corrosion Congress 2011, EUROCORR
3. 2187-2201.
H. J. Landau, “Sampling, data transmission, and the Nyquist rate,” in Proceedings
of the IEEE, vol. 55, no. 10, pp. 1701-1706, Oct. 1967, doi:
10.1109/PROC.1967.5962.
71
Journee, J. M., and W. W. Massie. Offshore Hydrodynamics, First Edition.
Delft University of Technology, 2001.
Keulegan, G. H., and L. H. Carpenter. “Forces on Cylinders and Plates in an
Oscillating Fluid.” Journal of Research of the National Bureau of
Standards, vol. 60, no. 5, 1958, pp. 423–40.
Lahlou, O. (2019). Experimental and Numerical Analysis of the Drag Force on
Surfboards with Different Shapes (thesis).
L. H. Holthuijsen. Waves in Oceanic and Coastal Waters. Cam-bridge University
Press, 2007. doi:10.1017/cbo9780511618536.
MacCamy, R.C., Fuchs, R.A.: Wave Forces on Piles: a Diffraction Theory. Corps
of Engineers Washington DC Beach Erosion Board (1954)
M. M. Maher and G. Swain, “The Corrosion and Biofouling Characteristics of
Sealed vs. Perforated Offshore Monopile Interiors Experiment Design
Comparing Corrosion and Environment Inside Steel Pipe,” OCEANS 2018
MTS/IEEE Charleston, Charleston, SC, USA, 2018, pp. 1-4, doi:
10.1109/OCEANS.2018.8604522.
Morison, J. R.; O’Brien, M. P.; Johnson, J. W.; Schaaf, S. A. (1950), “The force
exerted by surface waves on piles”, Petroleum Transactions, American
Institute of Mining Engineers, 189 (5): 149–154, doi:10.2118/950149-G
Paluzzi, Alexander John, “Effects of Perforations on Internal Cathodic Protection
and Recruitment of Marine Organisms to Steel Pipes” (2023). Theses and
Dissertations. 1403. https://repository.fit.edu/etd/1403
Ploeg, J.V.D. (2021). Perforation of monopiles to reduce hydrodynamic loads and
enable use in deep waters [Master’s Thesis, Delft University of
Technology] Institutional Repository at Delft University of Technology.
http://resolver.tudelft.nl/uuid:91eada6f-4f2b-4ae6-be59-2b5ff0590c6f.
72
Shi, W., Zhang, S., Michailides, C., Zhang, L., Zhang, P., & Li, X. (2023).
Experimental investigation of the hydrodynamic effects of breaking waves
on monopiles in model scale. Journal of Marine Science and Technology,
28(1), 314–325. https://doi.org/10.1007/s00773-023-00926-9
Santamaria Gonzalez, G.A. (2023) Advantages and Challenges of Perforated
Monopiles in Deep Water Sites [Master’s Thesis, Delft University of
Technology] Institutional Repository at Delft University of Technology.
http://resolver.tudelft.nl/uuid:490791b6-a912-4bac-a007-f77012c01107
Sarpkaya, T. and Isaacson, M. (1981). Mechanics of Wave Forces on Offshore
Structures. Number ISBN 0-442-25402-4. Van Nostrand Reinhold
Company Inc., New York.
Tang, Y., Shi, W., Ning, D., You, J., & Michailides, C. (2020). Effects of spilling
and plunging type breaking waves acting on large monopile offshore wind
turbines. Frontiers in Marine Science, 7.
https://doi.org/10.3389/fmars.2020.00427
Teja, R. (2021, June 25). Wheatstone bridge: Working, examples, applications.
ElectronicsHub. https://www.electronicshub.org/wheatstone-bridge/
The MathWorks Inc. (2022). MATLAB version: 9.13.0 (R2022b), Natick,
Massachusetts: The MathWorks Inc. https://www.mathworks.com
Wave gauges. Edinburgh Designs. (2016).
http://www4.edesign.co.uk/product/wavegauges/
Wilberts, F. (2017). MEASUREMENT DRIVEN FATIGUE ASSESSMENT OF
OFFSHORE WIND TURBINE FOUNDATIONS (Master’s Thesis,
Uppsala University).

PDF 전체 원본 보기

Numerical Investigation of the Local Scour for Tripod Pile Foundation

Waqed H. Hassan | Zahraa Mohammad Fadhe^* | Rifqa F. Thiab | Karrar Mahdi
Civil Engineering Department, Faculty of Engineering, University of Warith Al-Anbiyaa, Kerbala 56001, Iraq
Civil Engineering Department, Faculty of Engineering, University of Kerbala, Kerbala 56001, Iraq
Corresponding Author Email: Waqed.hammed@uowa.edu.iq

OPEN ACCESS

Abstract:

This work investigates numerically a local scour moves in irregular waves around tripods. It is constructed and proven to use the numerical model of the seabed-tripod-fluid with an RNG k turbulence model. The present numerical model then examines the flow velocity distribution and scour characteristics. After that, the suggested computational model Flow-3D is a useful tool for analyzing and forecasting the maximum scour development and the flow field in random waves around tripods. The scour values affecting the foundations of the tripod must be studied and calculated, as this phenomenon directly and negatively affects the structure of the structure and its design life. The lower diagonal braces and the main column act as blockages, increasing the flow accelerations underneath them. This increases the number of particles that are moved, which in turn creates strong scouring in the area. The numerical model has a good agreement with the experimental model, with a maximum percentage of error of 10% between the experimental and numerical models. In addition, Based on dimensional analysis parameters, an empirical equation has been devised to forecast scour depth with flow depth, median size ratio, Keulegan-Carpenter (Kc), Froud number flow, and wave velocity that the results obtained in this research at various flow velocities and flow depths demonstrated that the maximum scour depth rate depended on wave height with rising velocities and decreasing particle sizes (d₅₀) and the scour depth attains its steady-current value for V_w < 0.75. As the Froude number rises, the maximum scour depth will be large.

Keywords:

local scour, tripod foundation, Flow-3D, waves

1. Introduction

New energy sources have been used by mankind since they become industrialized. The main energy sources have traditionally been timber, coal, oil, and gas, but advances in the science of new energies, such as nuclear energy, have emerged [1, 2]. Clean and renewable energy such as offshore wind has grown significantly during the past few decades. There are numerous different types of foundations regarding offshore wind turbines (OWTs), comprising the tripod, jacket, gravity foundation, suction anchor (or bucket), and monopile [3, 4]. When the water depth is less than 30 meters, Offshore wind farms usually employ the monopile type [4]. Engineers must deal with the wind’s scouring phenomenon turbine foundations when planning and designing wind turbines for an offshore environment [5]. Waves and currents generate scour, this is the erosion of soil near a submerged foundation and at its location [6]. To predict the regional scour depth at a bridge pier, Jalal et al. [7-10] developed an original gene expression algorithm using artificial neural networks. Three monopiles, one main column, and several diagonal braces connecting the monopiles to the main column make up the tripod foundation, which has more complicated shapes than a single pile. The design of the foundation may have an impact on scour depth and scour development since the foundation’s form affects the flow field [11, 12]. Stahlmann [4] conducted several field investigations. He discovered that the main column is where the greatest scour depth occurred. Under the main column is where the maximum scour depth occurs in all experiments. The estimated findings show that higher wave heights correspond to higher flow velocities, indicating that a deeper scour depth is correlated with finer silt granularity [13] recommends as the design value for a single pile. These findings support the assertion that a tripod may cause the seabed to scour more severely than a single pile. The geography of the scour is significantly more influenced by the KC value (Keulegan–Carpenter number)

The capability of computer hardware and software has made computational fluid dynamics (CFD) quite popular to predict the behavior of fluid flow in industrial and environmental applications has increased significantly in recent years [14].

Finding an acceptable piece of land for the turbine’s construction and designing the turbine pile precisely for the local conditions are the biggest challenges. Another concern related to working in a marine environment is the effect of sea waves and currents on turbine piles and foundations. The earth surrounding the turbine’s pile is scoured by the waves, which also render the pile unstable.

In this research, the main objective is to investigate numerically a local scour around tripods in random waves. It is constructed and proven to use the tripod numerical model. The present numerical model is then used to examine the flow velocity distribution and scour characteristics.

2. Numerical Model

To simulate the scouring process around the tripod foundation, the CFD code Flow-3D was employed. By using the fractional area/volume method, it may highlight the intricate boundaries of the solution domain (FAVOR).

This model was tested and validated utilizing data derived experimentally from Schendel et al. [15] and Sumer and Fredsøe [6]. 200 runs were performed at different values of parameters.

2.1 Momentum equations

The incompressible viscous fluid motion is described by the three RANS equations listed below [16]:

(1)

\frac{\partial u}{\partial t}+\frac{1}{{{V}_{F}}}\left( u{{A}_{x}}\frac{\partial u}{\partial x}+v{{A}_{y}}\frac{\partial u}{\partial y}+w{{A}_{z}}\frac{\partial u}{\partial z} \right)=-\frac{1}{\rho }\frac{\partial p}{\partial x}+{{G}_{x}}+fx

(2)

\frac{\partial v}{\partial t}+\frac{1}{{{V}_{F}}}\left( u{{A}_{x}}\frac{\partial v}{\partial x}+v{{A}_{y}}\frac{\partial v}{\partial y}+w{{A}_{z}}\frac{\partial v}{\partial z} \right)=-\frac{1}{\rho }\frac{\partial p}{\partial y}+{{G}_{y}}+\text{f}y

(3)

\frac{\partial w}{\partial t}+\frac{1}{{{V}_{F}}}\left( u{{A}_{x}}\frac{\partial w}{\partial x}+v{{A}_{y}}\frac{\partial w}{\partial y}+w{{A}_{z}}\frac{\partial w}{\partial z} \right)=-\frac{1}{\rho }\frac{\partial p}{\partial z}+{{G}_{z}}+\text{fz}

where, respectively, u, v, and w represent the x, y, and z flow velocity components; volume fraction (VF), area fraction (A_i; I=x, y, z), water density (f), viscous force (f_i), and body force (G_i) are all used in the formula.

2.2 Model of turbulence

Several turbulence models would be combined to solve the momentum equations. A two-equation model of turbulence is the RNG k-model, which has a high efficiency and accuracy in computing the near-wall flow field. Therefore, the flow field surrounding tripods was captured using the RNG k-model.

2.3 Model of sediment scour

2.3.1 Induction and deposition

Eq. (4) can be used to determine the particle entrainment lift velocity [17].

(4)

{{u}_{lift,i}}={{\alpha }_{i}}{{n}_{s}}d_{*}^{0.3}{{\left( \theta -{{\theta }_{cr}} \right)}^{1.5}}\sqrt{\frac{\parallel g\parallel {{d}_{i}}\left( {{\rho }_{i}}-{{\rho }_{f}} \right)}{{{\rho }_{f}}}}

α𝛼_i is the Induction parameter, n_s the normal vector is parallel to the seafloor, and for the present numerical model, n_s=(0,0,1), θ𝜃_cr is the essential Shields variable, g is the accelerated by gravity, d_i is the size of the particles, ρ_i is species density in beds, and d_∗ The diameter of particles without dimensions; these values can be obtained in Eq. (5).

(5)

{{d}_{*}}={{d}_{i}}{{\left( \frac{\parallel g\parallel {{\rho }_{f}}\left( {{\rho }_{i}}-{{\rho }_{f}} \right)}{\mu _{f}^{2}} \right)}^{1/3}}

μ𝜇_f is this equation a dynamic viscosity of the fluid. c_r was determined from an equation based on Soulsby [18].

(6)

{{\theta }_{cr}}=\frac{0.3}{1+1.2{{d}_{*}}}+0.055\left[ 1-\text{exp}\left( -0.02{{d}_{*}} \right) \right]

The equation was used to determine how quickly sand particles set Eq. (7):

(7)

{{\mathbf{u}}_{\text{nsettling},i}}=\frac{{{v}_{f}}}{{{d}_{i}}}\left[ {{\left( {{10.36}^{2}}+1.049d_{*}^{3} \right)}^{0.5}}-10.36 \right]

v_f stands for fluid kinematic viscosity.

2.3.2 Transportation for bed loads

Van Rijn [19] states that the speed of bed load conveyance was determined as:

(8)

{{~}_{\text{bedload},i}}=\frac{{{q}_{b,i}}}{{{\delta }_{i}}{{c}_{b,i}}{{f}_{b}}}

f_b is the essential particle packing percentage, q_b, i is the bed load transportation rate, and c_b, I the percentage of sand by volume i. These variables can be found in Eq. (9), Eq. (10), f_b, δ𝛿_i the bed load thickness.

(9)

{{q}_{b,i}}=8{{\left[ \parallel g\parallel \left( \frac{{{\rho }_{i}}-{{\rho }_{f}}}{{{\rho }_{f}}} \right)d_{i}^{3} \right]}^{\frac{1}{2}}}

(10)

{{\delta }_{i}}=0.3d_{*}^{0.7}{{\left( \frac{\theta }{{{\theta }_{cr}}}-1 \right)}^{0.5}}{{d}_{i}}

In this paper, after the calibration of numerous trials, the selection of parameters for sediment scour is crucial. Maximum packing fraction is 0.64 with a shields number of 0.05, entrainment coefficient of 0.018, the mass density of 2650, bed load coefficient of 12, and entrainment coefficient of 0.01.

3. Model Setup

To investigate the scour characteristics near tripods in random waves, the seabed-tripod-fluid numerical model was created as shown in Figure 1. The tripod basis, a seabed, and fluid and porous medium were all components of the model. The seabed was 240 meters long, 40 meters wide, and three meters high. It had a median diameter of d₅₀ and was composed of uniformly fine sand. The 2.5-meter main column diameter D. The base of the main column was three dimensions above the original seabed. The center of the seafloor was where the tripod was, 130 meters from the offshore and 110 meters from the onshore. To prevent wave reflection, the porous media were positioned above the seabed on the onshore side.

image013.png

Figure 1. An illustration of the numerical model for the seabed-tripod-fluid

3.1 Generation of meshes

Figure 2 displays the model’s mesh for the Flow-3D software grid. The current model made use of two different mesh types: global mesh grid and nested mesh grid. A mesh grid with the following measurements was created by the global hexahedra mesh grid: 240m length, 40m width, and 32m height. Around the tripod, a finer nested mesh grid was made, with dimensions of 0 to 32m on the z-axis, 10 to 30 m on the x-axis, and 25 to 15 m on the y-axis. This improved the calculation’s precision and mesh quality.

image014.png

Figure 2. The mesh block sketch

3.2 Conditional boundaries

To increase calculation efficiency, the top side, The model’s two x-z plane sides, as well as the symmetry boundaries, were all specified. For u, v, w=0, the bottom boundary wall was picked. The offshore end of the wave boundary was put upstream. For the wave border, random waves were generated using the wave spectrum from the Joint North Sea Wave Project (JONSWAP). Boundary conditions are shown in Figure 3.

image015.png

Figure 3. Boundary conditions of the typical problem

The wave spectrum peak enhancement factor (=3.3 for this work) and can be used to express the unidirectional JONSWAP frequency spectrum.

3.3 Mesh sensitivity

Before doing additional research into scour traits and scour depth forecasting, mesh sensitivity analysis is essential. Three different mesh grid sizes were selected for this section: Mesh 1 has a 0.45 by 0.45 nested fine mesh and a 0.6 by 0.6 global mesh size. Mesh 2 has a 0.4 global mesh size and a 0.35 nested fine mesh size, while Mesh 3 has a 0.25 global mesh size and a nested fine mesh size of 0.15. Comparing the relative fine mesh size (such as Mesh 2 or Mesh 3) to the relatively coarse mesh size (such as Mesh 1), a larger scour depth was seen; this shows that a finer mesh size can more precisely represent the scouring and flow field action around a tripod. Significantly, a lower mesh size necessitates a time commitment and a more difficult computer configuration. Depending on the sensitivity of the mesh guideline utilized by Pang et al., when Mesh 2 is applied, the findings converge and the mesh size is independent [20]. In the next sections, scouring the area surrounding the tripod was calculated using Mesh 2 to ensure accuracy and reduce computation time. The working segment generates a total of 14, 800,324 cells.

3.4 Model validation

Comparisons between the predicted outcomes from the current model and to confirm that the current numerical model is accurate and suitably modified, experimental data from Sumer and Fredsøe [6] and Schendel et al. [15] were used. For the experimental results of Run 05, Run 15, and Run 22 from Sumer and Fredsøe [6], the experimental A9, A13, A17, A25, A26, and A27 results from Schendel et al. [15], and the numerical results from the current model are shown in Figure 4. The present model had d₅₀=0.051cm, the height of the water wave(h)=10m, and wave velocity=0.854 m.s^-1.

image016.png

Figure 4. Cell size effect

image017.png

Figure 5. Comparison of the present study’s maximum scour depth with that authored by Sumer and Fredsøe [6] and Schendel et al. [15]

According to Figure 5, the highest discrepancy between the numerical results and experimental data is about 10%, showing that overall, there is good agreement between them. The ability of the current numerical model to accurately depict the scour process and forecast the maximum scour depth (S) near foundations is demonstrated by this. Errors in the simulation were reduced by using the calibrated values of the parameter. Considering these results, a suggested simulated scouring utilizing a Flow-3D numerical model is confirmed as a superior way for precisely forecasting the maximum scour depth near a tripod foundation in random waves.

3.5 Dimensional analysis

The variables found in this study as having the greatest impacts, variables related to flow, fluid, bed sediment, flume shape, and duration all had an impact on local scouring depth (t). Hence, scour depth (S) can be seen as a function of these factors, shown as:

(11)

S=f\left(\rho, v, V, h, g, \rho s, d_{50}, \sigma g, V_w, D, d, T_v, t\right)

With the aid of dimensional analysis, the 14-dimensional parameters in Eq. (11) were reduced to 6 dimensionless variables using Buckingham’s -theorem. D, V, and were therefore set as repetition parameters and others as constants, allowing for the ignoring of their influence. Eq. (12) thus illustrates the relationship between the effect of the non-dimensional components on the depth of scour surrounding a tripod base.

(12)

\frac{S}{D}=f\left(\frac{h}{D}, \frac{d 50}{D}, \frac{V}{V W}, F r, K c\right)

where, SD𝑆𝐷 are scoured depth ratio, VVw𝑉𝑉𝑤 is flow wave velocity, d50D𝑑50𝐷 median size ratio, $Fr representstheFroudnumber,and𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑠𝑡ℎ𝑒𝐹𝑟𝑜𝑢𝑑𝑛𝑢𝑚𝑏𝑒𝑟,𝑎𝑛𝑑Kc$ is the Keulegan-Carpenter.

4. Result and Discussion

4.1 Development of scour

Similar to how the physical model was used, this numerical model was also used. The numerical model’s boundary conditions and other crucial variables that directly influence the outcomes were applied (flow depth, median particle size (d₅₀), and wave velocity). After the initial 0-300 s, the scour rate reduced as the scour holes grew quickly. The scour depths steadied for about 1800 seconds before reaching an asymptotic value. The findings of scour depth with time are displayed in Figure 6.

4.2 Features of scour

Early on (t=400s), the scour hole began to appear beneath the main column and then began to extend along the diagonal bracing connecting to the wall-facing pile. Gradually, the geography of the scour; of these results is similar to the experimental observations of Stahlmann [4] and Aminoroayaie Yamini et al. [1]. As the waves reached the tripod, there was an enhanced flow acceleration underneath the main column and the lower diagonal braces as a result of the obstructing effects of the structural elements. More particles are mobilized and transported due to the enhanced near-bed flow velocity, it also increases bed shear stress, turbulence, and scour at the site. In comparison to a single pile, the main column and structural components of the tripod have a significant impact on the flow velocity distribution and, consequently, the scour process and morphology. The main column and seabed are separated by a gap, therefore the flow across the gap may aid in scouring. The scour hole first emerged beneath the main column and subsequently expanded along the lower structural components, both Aminoroayaie Yamini et al. [1] and Stahlmann [4] made this claim. Around the tripod, there are several different scour morphologies and the flow velocity distribution as shown in Figures 7 and 8.

image023.png

Figure 6. Results of scour depth with time

image024.png

image025.png

image026.png

image027.png

Figure 7. The sequence results of scour depth around tripod development (reached to steady state) simulation time

image028.png

image029.png

image030.png

image031.png

Figure 8. Random waves of flow velocity distribution around a tripod

4.3 Wave velocity’s (V_w) impact on scour depth

In this study’s section, we looked at how variations in wave current velocity affected the scouring depth. Bed scour pattern modification could result from an increase or decrease in waves. As a result, the backflow area produced within the pile would become stronger, which would increase the depth of the sediment scour. The quantity of current turbulence is the primary cause of the relationship between wave height and bed scour value. The current velocity has increased the extent to which the turbulence energy has changed and increased in strength now present. It should be mentioned that in this instance, the Jon swap spectrum random waves are chosen. The scour depth attains its steady-current value for Vw<0.75, Figure 9 (a) shows that effect. When (V) represents the mean velocity=0.5 m.s^-1.

image032.png

(a)

image033.png

(b)

image034.png

(c)

image035.png

(d)

Figure 9. Main effects on maximum scour depth (S_max) as a function of column diameter (D)

4.4 Impact of a median particle (d₅₀) on scour depth

In this section of the study, we looked into how variations in particle size affected how the bed profile changed. The values of various particle diameters are defined in the numerical model for each run numerical modeling, and the conditions under which changes in particle diameter have an impact on the bed scour profile are derived. Based on Figure 9 (b), the findings of the numerical modeling show that as particle diameter increases the maximum scour depth caused by wave contact decreases. When (d₅₀) is the diameter of Sediment (d₅₀). The Shatt Al-Arab soil near Basra, Iraq, was used to produce a variety of varied diameters.

4.5 Impact of wave height and flow depth (h) on scour depth

One of the main elements affecting the scour profile brought on by the interaction of the wave and current with the piles of the wind turbines is the height of the wave surrounding the turbine pile causing more turbulence to develop there. The velocity towards the bottom and the bed both vary as the turbulence around the pile is increased, modifying the scour profile close to the pile. According to the results of the numerical modeling, the depth of scour will increase as water depth and wave height in random waves increase as shown in Figure 9 (c).

4.6 Froude number’s (Fr) impact on scour depth

No matter what the spacing ratio, the Figure 9 shows that the Froude number rises, and the maximum scour depth often rises as well increases in Figure 9 (d). Additionally, it is crucial to keep in mind that only a small portion of the findings regarding the spacing ratios with the smallest values. Due to the velocity acceleration in the presence of a larger Froude number, the range of edge scour downstream is greater than that of upstream. Moreover, the scouring phenomena occur in the region farthest from the tripod, perhaps as a result of the turbulence brought on by the collision of the tripod’s pile. Generally, as the Froude number rises, so does the deposition height and scour depth.

4.7 Keulegan-Carpenter (KC) number

The geography of the scour is significantly more influenced by the KC value. Greater KC causes a deeper equilibrium scour because an increase in KC lengthens the horseshoe vortex’s duration and intensifies it as shown in Figure 10.

The result can be attributed to the fact that wave superposition reduced the crucial KC for the initiation of the scour, particularly under small KC conditions. The primary variable in the equation used to calculate This is the depth of the scouring hole at the bed. The following expression is used to calculate the Keulegan-Carpenter number:

Kc=Vw∗TpD𝐾𝑐=𝑉𝑤∗𝑇𝑝𝐷 (13)

where, the wave period is T_p and the wave velocity is shown by V_w_.

image037.png

Figure 10. Relationship between the relative maximum scour depth and KC

5. Conclusion

(1) The existing seabed-tripod-fluid numerical model is capable of faithfully reproducing the scour process and the flow field around tripods, suggesting that it may be used to predict the scour around tripods in random waves.

(2) Their results obtained in this research at various flow velocities and flow depths demonstrated that the maximum scour depth rate depended on wave height with rising velocities and decreasing particle sizes (d₅₀).

(3) A diagonal brace and the main column act as blockages, increasing the flow accelerations underneath them. This raises the magnitude of the disturbance and the shear stress on the seafloor, which in turn causes a greater number of particles to be mobilized and conveyed, as a result, causes more severe scour at the location.

(4) The Froude number and the scouring process are closely related. In general, as the Froude number rises, so does the maximum scour depth and scour range. The highest maximum scour depth always coincides with the bigger Froude number with the shortest spacing ratio.

Since the issue is that there aren’t many experiments or studies that are relevant to this subject, therefore we had to rely on the monopile criteria. Therefore, to gain a deeper knowledge of the scouring effect surrounding the tripod in random waves, further numerical research exploring numerous soil, foundation, and construction elements as well as upcoming physical model tests will be beneficial.

Nomenclature

CFD	Computational fluid dynamics
FAVOR	Fractional Area/Volume Obstacle Representation
VOF	Volume of Fluid
RNG	Renormalized Group
OWTs	Offshore wind turbines
Greek Symbols
ε, ω	Dissipation rate of the turbulent kinetic energy, m²s^-3
Subscripts
d₅₀	Median particle size
V_f	Volume fraction
G_T	Turbulent energy of buoyancy
K_T	Turbulent velocity
P_T	Kinetic energy of the turbulence
Α_i	Induction parameter
n_s	Induction parameter
ΘΘ_cr	The essential Shields variable
D_i	Diameter of sediment
d_∗	The diameter of particles without dimensions
µ_f	Dynamic viscosity of the fluid
q_b,_i	The bed load transportation rate
C_s,i	Sand particle’s concentration of mass
D	Diameter of pile
D_f	Diffusivity
D	Diameter of main column
Fr	Froud number
Kc	Keulegan–Carpenter number
G	Acceleration of gravity g
H	Flow depth
Vw	Wave Velocity
V	Mean Velocity
Tp	Wave Period
S	Scour depth

References

[1] Aminoroayaie Yamini, O., Mousavi, S.H., Kavianpour, M.R., Movahedi, A. (2018). Numerical modeling of sediment scouring phenomenon around the offshore wind turbine pile in marine environment. Environmental Earth Sciences, 77: 1-15. https://doi.org/10.1007/s12665-018-7967-4

[2] Hassan, W.H., Hashim, F.S. (2020). The effect of climate change on the maximum temperature in Southwest Iraq using HadCM3 and CanESM2 modelling. SN Applied Sciences, 2(9): 1494. https://doi.org/10.1007/s42452-020-03302-z

[3] Fazeres-Ferradosa, T., Rosa-Santos, P., Taveira-Pinto, F., Pavlou, D., Gao, F.P., Carvalho, H., Oliveira-Pinto, S. (2020). Preface: Advanced research on offshore structures and foundation design part 2. In Proceedings of the Institution of Civil Engineers-Maritime Engineering. Thomas Telford Ltd, 173(4): 96-99. https://doi.org/10.1680/jmaen.2020.173.4.96

[4] Stahlmann, A. (2013). Numerical and experimental modeling of scour at foundation structures for offshore wind turbines. In ISOPE International Ocean and Polar Engineering Conference. ISOPE, pp. ISOPE-I.

[5] Petersen, T.U., Sumer, B.M., Fredsøe, J. (2014). Edge scour at scour protections around offshore wind turbine foundations. In 7th International Conference on Scour and Erosion. CRC Press, pp. 587-592.

[6] Sumer, B.M., Fredsøe, J. (2001). Scour around pile in combined waves and current. Journal of Hydraulic Engineering, 127(5): 403-411. https://doi.org/10.1061/(ASCE)0733-9429(2001)127:5(403)

[7] Jalal, H.K., Hassan, W.H. (2020). Effect of bridge pier shape on depth of scour. In IOP Conference Series: Materials Science and Engineering. IOP Publishing, 671(1): 012001. https://doi.org/10.1088/1757-899X/671/1/012001

[8] Hassan, W.H., Jalal, H.K. (2021). Prediction of the depth of local scouring at a bridge pier using a gene expression programming method. SN Applied Sciences, 3(2): 159. https://doi.org/10.1007/s42452-020-04124-9

[9] Jalal, H.K., Hassan, W.H. (2020). Three-dimensional numerical simulation of local scour around circular bridge pier using Flow-3D software. In IOP Conference Series: Materials Science and Engineering. IOP Publishing, 745(1): 012150. https://doi.org/10.1088/1757-899X/745/1/012150

[10] Hassan, W.H., Attea, Z.H., Mohammed, S.S. (2020). Optimum layout design of sewer networks by hybrid genetic algorithm. Journal of Applied Water Engineering and Research, 8(2): 108-124. https://doi.org/10.1080/23249676.2020.1761897

[11] Hassan, W.H., Hussein, H.H., Alshammari, M.H., Jalal, H.K., Rasheed, S.E. (2022). Evaluation of gene expression programming and artificial neural networks in PyTorch for the prediction of local scour depth around a bridge pier. Results in Engineering, 13: 100353. https://doi.org/10.1016/j.rineng.2022.100353

[12] Hassan, W.H., Hh, H., Mohammed, S.S., Jalal, H.K., Nile, B.K. (2021). Evaluation of gene expression programming to predict the local scour depth around a bridge pier. Journal of Engineering Science and Technology, 16(2): 1232-1243. https://doi.org/10.1016/j.rineng.2022.100353

[13] Nerland, C. (2010). Offshore wind energy: Balancing risk and reward. In Proceedings of the Canadian Wind Energy Association’s 2010 Annual Conference and Exhibition, Canada, p. 2000.

[14] Hassan, W.H., Nile, B.K., Mahdi, K., Wesseling, J., Ritsema, C. (2021). A feasibility assessment of potential artificial recharge for increasing agricultural areas in the kerbala desert in Iraq using numerical groundwater modeling. Water, 13(22): 3167. https://doi.org/10.3390/w13223167

[15] Schendel, A., Welzel, M., Schlurmann, T., Hsu, T.W. (2020). Scour around a monopile induced by directionally spread irregular waves in combination with oblique currents. Coastal Engineering, 161: 103751. https://doi.org/10.1016/j.coastaleng.2020.103751

[16] Yakhot, V., Orszag, S.A. (1986). Renormalization group analysis of turbulence. I. Basic theory. Journal of Scientific Computing, 1(1): 3-51. https://doi.org/10.1007/BF01061452

[17] Mastbergen, D.R., Van Den Berg, J.H. (2003). Breaching in fine sands and the generation of sustained turbidity currents in submarine canyons. Sedimentology, 50(4): 625-637. https://doi.org/10.1046/j.1365-3091.2003.00554.x

[18] Soulsby, R. (1997). Dynamics of marine sands. https://doi.org/10.1680/doms.25844

[19] Van Rijn, L.C. (1984). Sediment transport, part I: Bed load transport. Journal of Hydraulic Engineering, 110(10): 1431-1456. https://doi.org/10.1061/(ASCE)0733-9429(1984)110:10(1431)

[20] Pang, A.L.J., Skote, M., Lim, S.Y., Gullman-Strand, J., Morgan, N. (2016). A numerical approach for determining equilibrium scour depth around a mono-pile due to steady currents. Applied Ocean Research, 57: 114-124. https://doi.org/10.1016/j.apor.2016.02.010

Estimating maximum initial wave amplitude of subaerial landslide tsunamis: A three-dimensional modelling approach

해저 산사태 쓰나미의 최대 초기 파동 진폭 추정: 3차원 모델링 접근법

Ramtin Sabeti ^a, Mohammad Heidarzadeh ^ab

^aDepartment of Architecture and Civil Engineering, University of Bath, Bath BA27AY, UK
^bHydroCoast Consulting Engineers Ltd, Bath, UK

https://doi.org/10.1016/j.ocemod.2024.102360

Highlights

•Landslide travel distance is considered for the first time in a predictive equation.
•Predictive equation derived from databases using 3D physical and numerical modeling.
•The equation was successfully tested on the 2018 Anak Krakatau tsunami event.
•The developed equation using three-dimensional data exhibits a 91 % fitting quality.

Abstract

Landslide tsunamis, responsible for thousands of deaths and significant damage in recent years, necessitate the allocation of sufficient time and resources for studying these extreme natural hazards. This study offers a step change in the field by conducting a large number of three-dimensional numerical experiments, validated by physical tests, to develop a predictive equation for the maximum initial amplitude of tsunamis generated by subaerial landslides. We first conducted a few 3D physical experiments in a wave basin which were then applied for the validation of a 3D numerical model based on the Flow3D-HYDRO package. Consequently, we delivered 100 simulations using the validated model by varying parameters such as landslide volume, water depth, slope angle and travel distance. This large database was subsequently employed to develop a predictive equation for the maximum initial tsunami amplitude. For the first time, we considered travel distance as an independent parameter for developing the predictive equation, which can significantly improve the predication accuracy. The predictive equation was tested for the case of the 2018 Anak Krakatau subaerial landslide tsunami and produced satisfactory results.

Keywords

Tsunami, Subaerial landslide, Physical modelling, Numerical simulation, FLOW-3D HYDRO

1. Introduction and literature review

The Anak Krakatau landslide tsunami on 22nd December 2018 was a stark reminder of the dangers posed by subaerial landslide tsunamis (Ren et al., 2020; Mulia et al. 2020a; Borrero et al., 2020; Heidarzadeh et al., 2020; Grilli et al., 2021). The collapse of the volcano’s southwest side into the ocean triggered a tsunami that struck the Sunda Strait, leading to approximately 450 fatalities (Syamsidik et al., 2020; Mulia et al., 2020b) (Fig. 1). As shown in Fig. 1, landslide tsunamis (both submarine and subaerial) have been responsible for thousands of deaths and significant damage to coastal communities worldwide. These incidents underscored the critical need for advanced research into landslide-generated waves to aid in hazard prediction and mitigation. This is further emphasized by recent events such as the 28th of November 2020 landslide tsunami in the southern coast mountains of British Columbia (Canada), where an 18 million m³ rockslide generated a massive tsunami, with over 100 m wave run-up, causing significant environmental and infrastructural damage (Geertsema et al., 2022).

Physical modelling and numerical simulation are crucial tools in the study of landslide-induced waves due to their ability to replicate and analyse the complex dynamics of landslide events (Kim et al., 2020). In two-dimensional (2D) modelling, the discrepancy between dimensions can lead to an artificial overestimation of wave amplification (e.g., Heller and Spinneken, 2015). This limitation is overcome with 3D modelling, which enables the scaled-down representation of landslide-generated waves while avoiding the simplifications inherent in 2D approaches (Erosi et al., 2019). Another advantage of 3D modelling in studying landslide-generated waves is its ability to accurately depict the complex dynamics of wave propagation, including lateral and radial spreading from the slide impact zone, a feature unattainable with 2D models (Heller and Spinneken, 2015).

Physical experiments in tsunami research, as presented by authors such as Romano et al. (2020), McFall and Fritz (2016), and Heller and Spinneken (2015), have supported 3D modelling works through validation and calibration of the numerical models to capture the complexities of wave generation and propagation. Numerical modelling has increasingly complemented experimental approach in tsunami research due to the latter’s time and resource-intensive nature, particularly for 3D models (Li et al., 2019; Kim et al., 2021). Various numerical approaches have been employed, from Eulerian and Lagrangian frameworks to depth-averaged and Navier–Stokes models, enhancing our understanding of tsunami dynamics (Si et al., 2018; Grilli et al., 2019; Heidarzadeh et al., 2017, 2020; Iorio et al., 2021; Zhang et al., 2021; Kirby et al., 2022; Wang et al., 2021, 2022; Hu et al., 2022). The sophisticated numerical techniques, including the Particle Finite Element Method and the Immersed Boundary Method, have also shown promising results in modelling highly dynamic landslide scenarios (Mulligan et al., 2020; Chen et al., 2020). Among these methods and techniques, FLOW-3D HYDRO stands out in simulating landslide-generated tsunami waves due to its sophisticated technical features such as offering Tru Volume of Fluid (VOF) method for precise free surface tracking (e.g., Sabeti and Heidarzadeh 2022a). TruVOF distinguishes itself through a split Lagrangian approach, adeptly reducing cumulative volume errors in wave simulations by dynamically updating cell volume fractions and areas with each time step. Its intelligent adaptation of time step size ensures precise capture of evolving free surfaces, offering unparalleled accuracy in modelling complex fluid interfaces and behaviour (Flow Science, 2023).

Predictive equations play a crucial role in assessing the potential hazards associated with landslide-generated tsunami waves due to their ability to provide risk assessment and warnings. These equations can offer swift and reasonable evaluations of potential tsunami impacts in the absence of detailed numerical simulations, which can be time-consuming and expensive to produce. Among multiple factors and parameters within a landslide tsunami generation, the initial maximum wave amplitude (Fig. 1) stands out due to its critical role. While it is most likely that the initial wave generated by a landslide will have the highest amplitude, it is crucial to clarify that the term “initial maximum wave amplitude” refers to the highest amplitude within the first set of impulse waves. This parameter is essential in determining the tsunami’s impact severity, with higher amplitudes signalling a greater destructive potential (Sabeti and Heidarzadeh 2022a). Additionally, it plays a significant role in tsunami modelling, aiding in the prediction of wave propagation and the assessment of potential impacts.

In this study, we initially validate the FLOW-3D HYDRO model through a series of physical experiments conducted in a 3D wave tank at University of Bath (UK). Upon confirmation of the model’s accuracy, we use it to systematically vary parameters namely landslide volume, water depth, slope angle, and travel distance, creating an extensive database. Alongside this, we perform a sensitivity analysis on these variables to discern their impacts on the initial maximum wave amplitude. The generated database was consequently applied to derive a non-dimensional predictive equation aimed at estimating the initial maximum wave amplitude in real-world landslide tsunami events.

Two innovations of this study are: (i) The predictive equation of this study is based on a large number of 3D experiments whereas most of the previous equations were based on 2D results, and (ii) For the first time, the travel distance is included in the predictive equation as an independent parameter. To evaluate the performance of our predictive equation, we applied it to a previous real-world subaerial landslide tsunami, i.e., the Anak Krakatau 2018 event. Furthermore, we compare the performance of our predictive equation with other existing equations.

2. Data and methods

The methodology applied in this research is a combination of physical and numerical modelling. Limited physical modelling was performed in a 3D wave basin at the University of Bath (UK) to provide data for calibration and validation of the numerical model. After calibration and validation, the numerical model was employed to model a large number of landslide tsunami scenarios which allowed us to develop a database for deriving a predictive equation.

2.1. Physical experiments

To validate our numerical model, we conducted a series of physical experiments including two sets in a 3D wave basin at University of Bath, measuring 2.50 m in length (W_L), 2.60 m in width (W_W), and 0.60 m in height (W_H) (Fig. 2a). Conducting two distinct sets of experiments (Table 1), each with different setups (travel distance, location, and water depth), provided a robust framework for validation of the numerical model. For wave measurement, we employed a twin wire wave gauge from HR Wallingford (https://equipit.hrwallingford.com). In these experiments, we used a concrete prism solid block, the dimensions of which are outlined in Table 2. In our experiments, we employed a concrete prism solid block with a density of 2600 kg/m³, chosen for its similarity to the natural density of landslides, akin to those observed with the 2018 Anak Krakatau tsunami, where the landslide composition is predominantly solid rather than granular. The block’s form has also been endorsed in prior studies (Watts, 1998; Najafi-Jilani and Ataie-Ashtiani, 2008) as a suitable surrogate for modelling landslide-induced waves. A key aspect of our methodology was addressing scale effects, following the guidelines proposed by Heller et al. (2008) as it is described in Table 1. To enhance the reliability and accuracy of our experimental data, we conducted each physical experiment three times which revealed all three experimental waveforms were identical. This repetition was aimed at minimizing potential errors and inconsistencies in laboratory measurements.

Table 1. The locations and other information of the laboratory setups for making landslide-generated waves in the physical wave basin. This table details the specific parameters for each setup, including slope range (α), slide volume (V), kinematic viscosity (ν), water depth (h), travel distance (D), surface tension coefficient of water (σ), Reynolds number (R), Weber number (W), and the precise coordinates of the wave gauges (WG).

Lab	α(°)	V (m³)	h (m)	D (m)		(ν) (m²/s)	(σ) (N/m)	Acceptable range for avoiding scale effects*	Observed values of W and R ^⁎⁎
Lab 1	45	2.60 × 10⁻³	0.247	0.070	X₁=1.090 m	1.01 × 10⁻⁶	0.073	R > 3.0 × 10⁵	R₁ = 3.80 × 10⁵
					Y₁=1.210 m				R₁ = 3.80 × 10⁵
					Y₁=1.210 m				W₁ = 8.19 × 10⁵
					Z₁=0.050m			W >5.0 × 10³	W₁ = 8.19 × 10⁵
Lab 2	45	2.60 × 10⁻³	0.246	0.045	X₂=1.030 m	1.01 × 10⁻⁶	0.073		R₂ = 3.78 × 10⁵
					Y₂=1.210 m				W₂ = 8.13 × 10⁵
					Z₂=0.050 m				W₂ = 8.13 × 10⁵

⁎

The acceptable ranges for avoiding scale effects are based on the study by Heller et al. (2008).⁎⁎

The Reynolds number (R) is given by g^0.5h^1.5/ν, with ν denoting the kinematic viscosity. The Weber number (W) is W = ρgh²/σ, where σ represents surface tension coefficient and ρ = 1000kg/m³ is the density of water. In our experiments, conducted at a water temperature of approximately 20 °C, the kinematic viscosity (ν) and the surface tension coefficient of water (σ) are 1.01 × 10⁻⁶ m²/s and 0.073 N/m, respectively (Kestin et al., 1978).

Table 2. Specifications of the solid block used in physical experiments for generating subaerial landslides in the laboratory.

Solid-block attributes	Property metrics	Geometric shape
Slide width (b_s)	0.26 m
Slide length (l_s)	0.20 m
Slide thickness (s)	0.10 m
Slide volume (V)	2.60 × 10⁻³ m³
Specific gravity, (γ_s)	2.60
Slide weight (m_s)	6.86 kg

2.2. Numerical simulations applying FLOW-3D hydro

The detailed theoretical framework encompassing the governing equations, the computational methodologies employed, and the specific techniques used for tracking the water surface in these simulations are thoroughly detailed in the study by Sabeti et al. (2024). Here, we briefly explain some of the numerical details. We defined a uniform mesh for our flow domain, carefully crafted with a fine spatial resolution of 0.005 m (i.e., grid size). The dimensions of the numerical model directly matched those of our wave basin used in the physical experiment, being 2.60 m wide, 0.60 m deep, and 2.50 m long (Fig. 2). This design ensures comprehensive coverage of the study area. The output intervals of the numerical model are set at 0.02 s. This timing is consistent with the sampling rates of wave gauges used in laboratory settings. The friction coefficient in the FLOW-3D HYDRO is designated as 0.45. This value corresponds to the Coulombic friction measurements obtained in the laboratory, ensuring that the simulation accurately reflects real-world physical interactions.

In order to simulate the landslide motion, we applied coupled motion objects in FLOW-3D-HYDRO where the dynamics are predominantly driven by gravity and surface friction. This methodology stands in contrast to other models that necessitate explicit inputs of force and torque. This approach ensures that the simulation more accurately reflects the natural movement of landslides, which is heavily reliant on gravitational force and the interaction between sliding surfaces. The stability of the numerical simulations is governed by the Courant Number criterion (Courant et al., 1928), which dictates the maximum time step (Δt) for a given mesh size (Δx) and flow speed (U). According to Courant et al. (1928), this number is required to stay below one to ensure stability of numerical simulations. In our simulations, the Courant number is always maintained below one.

In alignment with the parameters of physical experiments, we set the fluid within the mesh to water, characterized by a density of 1000 kg/m³ at a temperature of 20 °C. Furthermore, we defined the top, front, and back surfaces of the mesh as symmetry planes. The remaining surfaces are designated as wall types, incorporating no-slip conditions to accurately simulate the interaction between the fluid and the boundaries. In terms of selection of an appropriate turbulence model, we selected the k–ω model that showed a better performance than other turbulence methods (e.g., Renormalization-Group) in a previous study (Sabeti et al., 2024). The simulations are conducted using a PC Intel® Core™ i7-10510U CPU with a frequency of 1.80 GHz, and a 16 GB RAM. On this PC, completion of a 3-s simulation required approximately 12.5 h.

2.3. Validation

The FLOW-3D HYDRO numerical model was validated using the two physical experiments (Fig. 3) outlined in Table 1. The level of agreement between observations (O_i) and simulations (S_i) is examined using the following equation:(1)�=|��−��|×100where ε represents the mismatch error, O_i denotes the observed laboratory values, and S_i represents the simulated values from the FLOW-3D HYDRO model. The results of this validation process revealed that our model could replicate the waves generated in the physical experiments with a reasonable degree of mismatch (ε): 14 % for Lab 1 and 8 % for Lab 2 experiments, respectively (Fig. 3). These values indicate that while the model is not perfect, it provides a sufficiently close approximation of the real-world phenomena.

In terms of mesh efficiency, we varied the mesh size to study sensitivity of the numerical results to mesh size. First, by halving the mesh size and then by doubling it, we repeated the modelling by keeping other parameters unchanged. This analysis guided that a mesh size of ∆x = 0.005 m is the most effective for the setup of this study. The total number of computational cells applying mesh size of 0.005 m is 9.269 × 10⁶.

2.4. The dataset

The validated numerical model was employed to conduct 100 simulations, incorporating variations in four key landslide parameters namely water depth, slope angle, slide volume, and travel distance. This methodical approach was essential for a thorough sensitivity analysis of these variables, and for the creation of a detailed database to develop a predictive equation for maximum initial tsunami amplitude. Within the model, 15 distinct slide volumes were established, ranging from 0.10 × 10⁻³ m³ to 6.25 × 10⁻³ m³ (Table 3). The slope angle varied between 35° and 55°, and water depth ranged from 0.24 m to 0.27 m. The travel distance of the landslides was varied, spanning from 0.04 m to 0.07 m. Detailed configurations of each simulation, along with the maximum initial wave amplitudes and dominant wave periods are provided in Table 4.

Table 3. Geometrical information of the 15 solid blocks used in numerical modelling for generating landslide tsunamis. Parameters are: l_s, slide length; b_s, slide width; s, slide thickness; γ_s, specific gravity; and V, slide volume.

Solid block	l_s (m)	b_s (m)	s (m)	V (m³)	γ_s
Block-1	0.310	0.260	0.155	6.25 × 10⁻³	2.60
Block-2	0.300	0.260	0.150	5.85 × 10⁻³	2.60
Block-3	0.280	0.260	0.140	5.10 × 10⁻³	2.60
Block-4	0.260	0.260	0.130	4.39 × 10⁻³	2.60
Block-5	0.240	0.260	0.120	3.74 × 10⁻³	2.60
Block-6	0.220	0.260	0.110	3.15 × 10⁻³	2.60
Block-7	0.200	0.260	0.100	2.60 × 10⁻³	2.60
Block-8	0.180	0.260	0.090	2.11 × 10⁻³	2.60
Block-9	0.160	0.260	0.080	1.66 × 10⁻³	2.60
Block-10	0.140	0.260	0.070	1.27 × 10⁻³	2.60
Block-11	0.120	0.260	0.060	0.93 × 10⁻³	2.60
Block-12	0.100	0.260	0.050	0.65 × 10⁻³	2.60
Block-13	0.080	0.260	0.040	0.41 × 10⁻³	2.60
Block-14	0.060	0.260	0.030	0.23 × 10⁻³	2.60
Block-15	0.040	0.260	0.020	0.10 × 10⁻³	2.60

Table 4. The numerical simulation for the 100 tests performed in this study for subaerial solid-block landslide-generated waves. Parameters are a_M, maximum wave amplitude; α, slope angle; h, water depth; D, travel distance; and T, dominant wave period. The location of the wave gauge is X=1.030 m, Y=1.210 m, and Z=0.050 m. The properties of various solid blocks are presented in Table 3.

Test-	Block No	α (°)	h (m)	D (m)	T(s)	a_M (m)
1	Block-7	45	0.246	0.029	0.510	0.0153
2	Block-7	45	0.246	0.030	0.505	0.0154
3	Block-7	45	0.246	0.031	0.505	0.0156
4	Block-7	45	0.246	0.032	0.505	0.0158
5	Block-7	45	0.246	0.033	0.505	0.0159
6	Block-7	45	0.246	0.034	0.505	0.0160
7	Block-7	45	0.246	0.035	0.505	0.0162
8	Block-7	45	0.246	0.036	0.505	0.0166
9	Block-7	45	0.246	0.037	0.505	0.0167
10	Block-7	45	0.246	0.038	0.505	0.0172
11	Block-7	45	0.246	0.039	0.505	0.0178
12	Block-7	45	0.246	0.040	0.505	0.0179
13	Block-7	45	0.246	0.041	0.505	0.0181
14	Block-7	45	0.246	0.042	0.505	0.0183
15	Block-7	45	0.246	0.043	0.505	0.0190
16	Block-7	45	0.246	0.044	0.505	0.0197
17	Block-7	45	0.246	0.045	0.505	0.0199
18	Block-7	45	0.246	0.046	0.505	0.0201
19	Block-7	45	0.246	0.047	0.505	0.0191
20	Block-7	45	0.246	0.048	0.505	0.0217
21	Block-7	45	0.246	0.049	0.505	0.0220
22	Block-7	45	0.246	0.050	0.505	0.0226
23	Block-7	45	0.246	0.051	0.505	0.0236
24	Block-7	45	0.246	0.052	0.505	0.0239
25	Block-7	45	0.246	0.053	0.510	0.0240
26	Block-7	45	0.246	0.054	0.505	0.0241
27	Block-7	45	0.246	0.055	0.505	0.0246
28	Block-7	45	0.246	0.056	0.505	0.0247
29	Block-7	45	0.246	0.057	0.505	0.0248
30	Block-7	45	0.246	0.058	0.505	0.0249
31	Block-7	45	0.246	0.059	0.505	0.0251
32	Block-7	45	0.246	0.060	0.505	0.0257
33	Block-1	45	0.246	0.045	0.505	0.0319
34	Block-2	45	0.246	0.045	0.505	0.0294
35	Block-3	45	0.246	0.045	0.505	0.0282
36	Block-4	45	0.246	0.045	0.505	0.0262
37	Block-5	45	0.246	0.045	0.505	0.0243
38	Block-6	45	0.246	0.045	0.505	0.0223
39	Block-7	45	0.246	0.045	0.505	0.0196
40	Block-8	45	0.246	0.045	0.505	0.0197
41	Block-9	45	0.246	0.045	0.505	0.0198
42	Block-10	45	0.246	0.045	0.505	0.0184
43	Block-11	45	0.246	0.045	0.505	0.0173
44	Block-12	45	0.246	0.045	0.505	0.0165
45	Block-13	45	0.246	0.045	0.404	0.0153
46	Block-14	45	0.246	0.045	0.404	0.0124
47	Block-15	45	0.246	0.045	0.505	0.0066
48	Block-7	45	0.202	0.045	0.404	0.0220
49	Block-7	45	0.204	0.045	0.404	0.0219
50	Block-7	45	0.206	0.045	0.404	0.0218
51	Block-7	45	0.208	0.045	0.404	0.0217
52	Block-7	45	0.210	0.045	0.404	0.0216
53	Block-7	45	0.212	0.045	0.404	0.0215
54	Block-7	45	0.214	0.045	0.505	0.0214
55	Block-7	45	0.216	0.045	0.505	0.0214
56	Block-7	45	0.218	0.045	0.505	0.0213
57	Block-7	45	0.220	0.045	0.505	0.0212
58	Block-7	45	0.222	0.045	0.505	0.0211
59	Block-7	45	0.224	0.045	0.505	0.0208
60	Block-7	45	0.226	0.045	0.505	0.0203
61	Block-7	45	0.228	0.045	0.505	0.0202
62	Block-7	45	0.230	0.045	0.505	0.0201
63	Block-7	45	0.232	0.045	0.505	0.0201
64	Block-7	45	0.234	0.045	0.505	0.0200
65	Block-7	45	0.236	0.045	0.505	0.0199
66	Block-7	45	0.238	0.045	0.404	0.0196
67	Block-7	45	0.240	0.045	0.404	0.0194
68	Block-7	45	0.242	0.045	0.404	0.0193
69	Block-7	45	0.244	0.045	0.404	0.0192
70	Block-7	45	0.246	0.045	0.505	0.0190
71	Block-7	45	0.248	0.045	0.505	0.0189
72	Block-7	45	0.250	0.045	0.505	0.0187
73	Block-7	45	0.252	0.045	0.505	0.0187
74	Block-7	45	0.254	0.045	0.505	0.0186
75	Block-7	45	0.256	0.045	0.505	0.0184
76	Block-7	45	0.258	0.045	0.505	0.0182
77	Block-7	45	0.259	0.045	0.505	0.0183
78	Block-7	45	0.260	0.045	0.505	0.0191
79	Block-7	45	0.261	0.045	0.505	0.0192
80	Block-7	45	0.262	0.045	0.505	0.0194
81	Block-7	45	0.263	0.045	0.505	0.0195
82	Block-7	45	0.264	0.045	0.505	0.0195
83	Block-7	45	0.265	0.045	0.505	0.0197
84	Block-7	45	0.266	0.045	0.505	0.0197
85	Block-7	45	0.267	0.045	0.505	0.0198
86	Block-7	45	0.270	0.045	0.505	0.0199
87	Block-7	30	0.246	0.045	0.505	0.0101
88	Block-7	35	0.246	0.045	0.505	0.0107
89	Block-7	36	0.246	0.045	0.505	0.0111
90	Block-7	37	0.246	0.045	0.505	0.0116
91	Block-7	38	0.246	0.045	0.505	0.0117
92	Block-7	39	0.246	0.045	0.505	0.0119
93	Block-7	40	0.246	0.045	0.505	0.0121
94	Block-7	41	0.246	0.045	0.505	0.0127
95	Block-7	42	0.246	0.045	0.404	0.0154
96	Block-7	43	0.246	0.045	0.404	0.0157
97	Block-7	44	0.246	0.045	0.404	0.0162
98	Block-7	45	0.246	0.045	0.505	0.0197
99	Block-7	50	0.246	0.045	0.505	0.0221
100	Block-7	55	0.246	0.045	0.505	0.0233

In all these 100 simulations, the wave gauge was consistently positioned at coordinates X=1.09 m, Y=1.21 m, and Z=0.05 m. The dominant wave period for each simulation was determined using the Fast Fourier Transform (FFT) function in MATLAB (MathWorks, 2023). Furthermore, the classification of wave types was carried out using a wave categorization graph according to Sorensen (2010), as shown in Fig. 4a. The results indicate that the majority of the simulated waves are on the border between intermediate and deep-water waves, and they are categorized as Stokes waves (Fig. 4a). Four sample waveforms from our 100 numerical experiments are provided in Fig. 4b.

The dataset in Table 4 was used to derive a new predictive equation that incorporates travel distance for the first time to estimate the initial maximum tsunami amplitude. In developing this equation, a genetic algorithm optimization technique was implemented using MATLAB (MathWorks 2023). This advanced approach entailed the use of genetic algorithms (GAs), an evolutionary algorithm type inspired by natural selection processes (MathWorks, 2023). This technique is iterative, involving selection, crossover, and mutation processes to evolve solutions over several generations. The goal was to identify the optimal coefficients and powers for each landslide parameter in the predictive equation, ensuring a robust and reliable model for estimating maximum wave amplitudes. Genetic Algorithms excel at optimizing complex models by navigating through extensive combinations of coefficients and exponents. GAs effectively identify highly suitable solutions for the non-linear and complex relationships between inputs (e.g., slide volume, slope angle, travel distance, water depth) and the output (i.e., maximum initial wave amplitude, a_M). MATLAB’s computational environment enhances this process, providing robust tools for GA to adapt and evolve solutions iteratively, ensuring the precision of the predictive model (Onnen et al., 1997). This approach leverages MATLAB’s capabilities to fine-tune parameters dynamically, achieving an optimal equation that accurately estimates a_M. It is important to highlight that the nondimensionalized version of this dataset is employed to develop a predictive equation which enables the equation to reproduce the maximum initial wave amplitude (a_M) for various subaerial landslide cases, independent of their dimensional differences (e.g., Heler and Hager 2014; Heller and Spinneken 2015; Sabeti and Heidarzadeh 2022b). For this nondimensionalization, we employed the water depth (h) to nondimensionalize the slide volume (V/h³) and travel distance (D/h). The slide thickness (s) was applied to nondimensionalize the water depth (h/s).

2.5. Landslide velocity

In discussing the critical role of landslide velocity for simulating landslide-generated waves, we focus on the mechanisms of landslide motion and the techniques used to record landslide velocity in our simulations (Fig. 5). Also, we examine how these methods were applied in two distinct scenarios: Lab 1 and Lab 2 (see Table 1 for their details). Regarding the process of landslide movement, a slide starts from a stationary state, gaining momentum under the influence of gravity and this acceleration continues until the landslide collides with water, leading to a significant reduction in its speed before eventually coming to a stop (Fig. 5) (e.g., Panizzo et al. 2005).

To measure the landslide’s velocity in our simulations, we attached a probe at the centre of the slide, which supplied a time series of the velocity data. The slide’s velocity (v_s) peaks at the moment it enters the water (Fig. 5), a point referred to as the impact time (t_Imp). Following this initial impact, the slides continue their underwater movement, eventually coming to a complete halt (t_Stop). Given the results in Fig. 5, it can be seen that Lab 1, with its longer travel distance (0.070 m), exhibits a higher peak velocity of 1.89 m/s. This increase in velocity is attributed to the extended travel distance allowing more time for the slide to accelerate under gravity. Whereas Lab 2, featuring a shorter travel distance (0.045 m), records a lower peak velocity of 1.78 m/s. This difference underscores how travel distance significantly influences the dynamics of landslide motion. After reaching the peak, both profiles show a sharp decrease in velocity, marking the transition to submarine motion until the slides come to a complete stop (t_Stop). There are noticeable differences observable in Fig. 5 between the Lab-1 and Lab-2 simulations, including the peaks at 0.3 s . These variations might stem from the placement of the wave gauge, which differs slightly in each scenario, as well as the water depth’s minor discrepancies and, the travel distance.

2.6. Effect of air entrainment

In this section we examine whether it is required to consider air entrainment for our modelling or not as the FLOW-3D HYDRO package is capable of modelling air entrainment. The process of air entrainment in water during a landslide tsunami and its subsequent transport involve two key components: the quantification of air entrainment at the water surface, and the simulation of the air’s transport within the fluid (Hirt, 2003). FLOW-3D HYDRO employs the air entrainment model to compute the volume of air entrained at the water’s surface utilizing three approaches: a constant density model, a variable density model accounting for bulking, and a buoyancy model that adds the Drift-FLUX mechanism to variable density conditions (Flow Science, 2023). The calculation of the entrainment rate is based on the following equation:(2)��=��[2(��−��−2�/��)]1/2where parameters are: V_air, volume of air; C_air, entrainment rate coefficient; A_s, surface area of fluid; ρ, fluid density; k, turbulent kinetic energy; g_n, gravity normal to surface; L_t, turbulent length scale; and σ, surface tension coefficient. The value of k is directly computed from the Reynolds-averaged Navier-Stokes (RANS) (k–w) calculations in our model.

In this study, we selected the variable density + Drift-FLUX model, which effectively captures the dynamics of phase separation and automatically activates the constant density and variable density models. This method simplifies the air-water mixture, treating it as a single, homogeneous fluid within each computational cell. For the phase volume fractions f₁and f₂, the velocities are expressed in terms of the mixture and relative velocities, denoted as u and u_r, respectively, as follows:(3)��1��+�.(�1�)=��1��+�.(�1�)−�.(�1�2��)=0(4)��2��+�.(�2�)=��2��+�.(�2�)−�.(�1�2��)=0

The outcomes from this simulation are displayed in Fig. 6, which indicates that the influence of air entrainment on the generated wave amplitude is approximately 2 %. A value of 0.02 for the entrained air volume fraction means that, in the simulated fluid, approximately 2 % of the volume is composed of entrained air. In other words, for every unit volume of the fluid-air mixture at that location, 2 % is air and the remaining 98 % is water. The configuration of Test-17 (Table 4) was employed for this simulation. While the effect of air entrainment is anticipated to be more significant in models of granular landslide-generated waves (Fritz, 2002), in our simulations we opted not to incorporate this module due to its negligible impact on the results.

3. Results

In this section, we begin by presenting a sequence of our 3D simulations capturing different time steps to illustrate the generation process of landslide-generated waves. Subsequently, we derive a new predictive equation to estimate the maximum initial wave amplitude of landslide-generated waves and assess its performance.

3.1. Wave generation and propagation

To demonstrate the wave generation process in our simulation, we reference Test-17 from Table 4, where we employed Block-7 (Tables 3, 4). In this configuration, the slope angle was set to 45°, with a water depth of 0.246 m and a travel distance at 0.045 m (Fig. 7). At 0.220 s, the initial impact of the moving slide on the water is depicted, marking the onset of the wave generation process (Fig. 7a). Disturbances are localized to the immediate area of impact, with the rest of the water surface remaining undisturbed. At this time, a maximum water particle velocity of 1.0 m/s – 1.2 m/s is seen around the impact zone (Fig. 7d). Moving to 0.320 s, the development of the wave becomes apparent as energy transfer from the landslide to the water creates outwardly radiating waves with maximum water particle velocity of up to around 1.6 m/s – 1.8 m/s (Fig. 7b, e). By the time 0.670 s, the wave has fully developed and is propagating away from the impact point exhibiting maximum water particle velocity of up to 2.0 m/s – 2.1 m/s. Concentric wave fronts are visible, moving outwards in all directions, with a colour gradient signifying the highest wave amplitude near the point of landslide entry, diminishing with distance (Fig. 7c, f).

3.2. Influence of landslide parameters on tsunami amplitude

In this section, we investigate the effects of various landslide parameters namely slide volume (V), water depth (h), slipe angle (α) and travel distance (D) on the maximum initial wave amplitude (a_M). Fig. 8 presents the outcome of these analyses. According to Fig. 8, the slide volume, slope angle, and travel distance exhibit a direct relationship with the wave amplitude, meaning that as these parameters increase, so does the amplitude. Conversely, water depth is inversely related to the maximum initial wave amplitude, suggesting that the deeper the water depth, the smaller the maximum wave amplitude will be (Fig. 8b).

Fig. 8a highlights the pronounced impact of slide volume on the a_M, demonstrating a direct correlation between the two variables. For instance, in the range of slide volumes we modelled (Fig. 8a), The smallest slide volume tested, measuring 0.10 × 10⁻³ m³, generated a low initial wave amplitude (a_M= 0.0066 m) (Table 4). In contrast, the largest volume tested, 6.25 × 10⁻³ m³, resulted in a significantly higher initial wave amplitude (a_M= 0.0319 m) (Table 4). The extremities of these results emphasize the slide volume’s paramount impact on wave amplitude, further elucidated by their positions as the smallest and largest a_M values across all conducted tests (Table 4). This is corroborated by findings from the literature (e.g., Murty, 2003), which align with the observed trend in our simulations.

The slope angle’s influence on a_M was smooth. A steady increase of wave amplitude was observed as the slope angle increased (Fig. 8c). In examining travel distance, an anomaly was identified. At a travel distance of 0.047 m, there was an unexpected dip in a_M, which deviates from the general increasing trend associated with longer travel distances. This singular instance could potentially be attributed to a numerical error. Beyond this point, the expected pattern of increasing a_M with longer travel distances resumes, suggesting that the anomaly at 0.047 m is an outlier in an otherwise consistent trend, and thus this single data point was overlooked while deriving the predictive equation. Regarding the inverse relationship between water depth and wave amplitude, our result (Fig. 8b) is consistent with previous reports by Fritz et al. (2003), (2004), and Watts et al. (2005).

The insights from Fig. 8 informed the architecture of the predictive equation in the next Section, with slide volume, travel distance, and slope angle being multiplicatively linked to wave amplitude underscoring their direct correlations with wave amplitude. Conversely, water depth is incorporated as a divisor, representing its inverse relationship with wave amplitude. This structure encapsulates the dynamics between the landslide parameters and their influence on the maximum initial wave amplitude as discussed in more detail in the next Section.

3.3. Predictive equation

Building on our sensitivity analysis of landslide parameters, as detailed in Section 3.2, and utilizing our nondimensional dataset, we have derived a new predictive equation as follows:(5)��/ℎ=0.015(tan�)0.10(�ℎ3)0.90(�ℎ)0.10(ℎ�)−0.11where, V is sliding volume, h is water depth, α is slope angle, and s is landslide thickness. It is important to note that this equation is valid only for subaerial solid-block landslide tsunamis as all our experiments were for this type of waves. The performance of this equation in predicting simulation data is demonstrated by the satisfactory alignment of data points around a 45° line, indicating its accuracy and reliability with regard to the experimental dataset (Fig. 9). The quality of fit between the dataset and Eq. (5) is 91 % indicating that Eq. (5) represents the dataset very well. Table 5 presents Eq. (5) alongside four other similar equations previously published. Two significant distinctions between our Eq. (5) and these others are: (i) Eq. (5) is derived from 3D experiments, whereas the other four equations are based on 2D experiments. (ii) Unlike the other equations, our Eq. (5) incorporates travel distance as an independent parameter.

Table 5. Performance comparison among our newly-developed equation and existing equations for estimating the maximum initial amplitude (a_M) of the 2018 Anak Krakatau subaerial landslide tsunami. Parameters: a_M, initial maximum wave amplitude; h, water depth; v_s, landslide velocity; V, slide volume; b_s, slide width; l_s, slide length; s, slide thickness; α, slope angle; and ��, volume of the final immersed landslide. We considered ��= V as the slide volume.

Event	Predictive equations	Author (year)	Observed a_M (m) ^⁎⁎	Calculated a_M (m)	Error, ε (%) ^⁎⁎⁎⁎
2018 Anak Krakatau tsunami (Subaerial landslide) *	��/ℎ=1.32��ℎ	Noda (1970)	134	134	0
	��/ℎ=0.667(0.5(��ℎ)2)0.334(��)0.754(��)0.506(�ℎ)1.631	Bolin et al. (2014) ^⁎⁎⁎	134	5942	4334
	��/ℎ=0.25(��ℎ2)0.8	Robbe-Saule et al. (2021)	134	31	77
	��/ℎ=0.4545(tan�)0.062(�ℎ3)0.296(ℎ�)−0.235	Sabeti and Heidarzadeh (2022b)	134	126	6
	��/ℎ=0.015(tan�)0.10(�ℎ3)0.911(�ℎ)0.10(ℎ�)−0.11	This study	134	130	2.9

⁎

Geometrical and kinematic parameters of the 2018 Anak Krakatau subaerial landslide based on Heidarzadeh et al. (2020), Grilli et al. (2019) and Grilli et al. (2021): V=2.11 × 10⁷ m³, h= 50 m; s= 114 m; α= 45°; l_s=1250 m; b_s= 2700 m; v_s=44.9 m/s; D= 2500 m; a_M= 100 m −150 m.⁎⁎

a_M= An average value of a_M = 134 m is considered in this study.⁎⁎⁎

The equation of Bolin et al. (2014) is based on the reformatted one reported by Lindstrøm (2016).⁎⁎⁎⁎

Error is calculated using Eq. (1), where the calculated a_M is assumed as the simulated value.

Additionally, we evaluated the performance of this equation using the real-world data from the 2018 Anak Krakatau subaerial landslide tsunami. Based on previous studies (Heidarzadeh et al., 2020; Grilli et al., 2019, 2021), we were able to provide a list of parameters for the subaerial landslide and associated tsunami for the 2018 Anak Krakatau event (see footnote of Table 5). We note that the data of the 2018 Anak Krakatau event was not used while deriving Eq. (5). The results indicate that Eq. (5) predicts the initial amplitude of the 2018 Anak Krakatau tsunami as being 130 m indicating an error of 2.9 % compared to the reported average amplitude of 134 m for this event. This performance indicates an improvement compared to the previous equation reported by Sabeti and Heidarzadeh (2022a) (Table 5). In contrast, the equations from Robbe-Saule et al. (2021) and Bolin et al. (2014) demonstrate higher discrepancies of 4200 % and 77 %, respectively (Table 5). Although Noda’s (1970) equation reproduces the tsunami amplitude of 134 m accurately (Table 5), it is crucial to consider its limitations, notably not accounting for parameters such as slope angle and travel distance.

It is essential to recognize that both travel distance and slope angle significantly affect wave amplitude. In our model, captured in Eq. (5), we integrate the slope angle (α) through the tangent function, i.e., tan α. This choice diverges from traditional physical interpretations that often employ the cosine or sine function (e.g., Heller and Hager, 2014; Watts et al., 2003). We opted for the tangent function because it more effectively reflects the direct impact of slope steepness on wave generation, yielding superior estimations compared to conventional methods.

The significance of this study lies in its application of both physical and numerical 3D experiments and the derivation of a predictive equation based on 3D results. Prior research, e.g. Heller et al. (2016), has reported notable discrepancies between 2D and 3D wave amplitudes, highlighting the important role of 3D experiments. It is worth noting that the suitability of applying an equation derived from either 2D or 3D data depends on the specific geometry and characteristics inherent in the problem being addressed. For instance, in the case of a long, narrow dam reservoir, an equation derived from 2D data would likely be more suitable. In such contexts, the primary dynamics of interest such as flow patterns and potential wave propagation are predominantly two-dimensional, occurring along the length and depth of the reservoir. This simplification to 2D for narrow dam reservoirs allows for more accurate modelling of these dynamics.

This study specifically investigates waves initiated by landslides, focusing on those characterized as solid blocks instead of granular flows, with slope angles confined to a range of 25° to 60°. We acknowledge the additional complexities encountered in real-world scenarios, such as dynamic density and velocity of landslides, which could affect the estimations. The developed equation in this study is specifically designed to predict the maximum initial amplitude of tsunamis for the aforementioned specified ranges and types of landslides.

4. Conclusions

Both physical and numerical experiments were undertaken in a 3D wave basin to study solid-block landslide-generated waves and to formulate a predictive equation for their maximum initial wave amplitude. At the beginning, two physical experiments were performed to validate and calibrate a 3D numerical model, which was subsequently utilized to generate 100 experiments by varying different landslide parameters. The generated database was then used to derive a predictive equation for the maximum initial wave amplitude of landslide tsunamis. The main features and outcomes are:

•The predictive equation of this study is exclusively derived from 3D data and exhibits a fitting quality of 91 % when applied to the database.
•For the first time, landslide travel distance was considered in the predictive equation. This inclusion provides more accuracy and flexibility for applying the equation.
•To further evaluate the performance of the predictive equation, it was applied to a real-world subaerial landslide tsunami (i.e., the 2018 Anak Krakatau event) and delivered satisfactory performance.

CRediT authorship contribution statement

Ramtin Sabeti: Conceptualization, Methodology, Validation, Software, Visualization, Writing – review & editing. Mohammad Heidarzadeh: Methodology, Data curation, Software, Writing – review & editing.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Funding

RS is supported by the Leverhulme Trust Grant No. RPG-2022-306. MH is funded by open funding of State Key Lab of Hydraulics and Mountain River Engineering, Sichuan University, grant number SKHL2101. We acknowledge University of Bath Institutional Open Access Fund. MH is also funded by the Great Britain Sasakawa Foundation grant no. 6217 (awarded in 2023).

Acknowledgements

Authors are sincerely grateful to the laboratory technician team, particularly Mr William Bazeley, at the Faculty of Engineering, University of Bath for their support during the laboratory physical modelling of this research. We appreciate the valuable insights provided by Mr. Brian Fox (Senior CFD Engineer at Flow Science, Inc.) regarding air entrainment modelling in FLOW-3D HYDRO. We acknowledge University of Bath Institutional Open Access Fund.

Data availability

All data used in this study are given in the body of the article.

References

Baptista et al., 2020M.A. Baptista, J.M. Miranda, R. Omira, I. El-HussainStudy of the 24 September 2013 Oman Sea tsunami using linear shallow water inversionArab. J. Geosci., 13 (14) (2020), p. 606View in Scopus Google Scholar
Bolin et al., 2014H. Bolin, Y. Yueping, C. Xiaoting, L. Guangning, W. Sichang, J. ZhibingExperimental modeling of tsunamis generated by subaerial landslides: two case studies of the Three Gorges Reservoir, ChinaEnviron. Earth Sci., 71 (2014), pp. 3813-3825View at publisher CrossRef View in Scopus Google Scholar
Borrero et al., 2020J.C. Borrero, T. Solihuddin, H.M. Fritz, P.J. Lynett, G.S. Prasetya, V. Skanavis, S. Husrin, Kushendratno, W. Kongko, D.C. Istiyanto, A. DaulatField survey and numerical modelling of the December 22, 2018, Anak Krakatau TsunamiPure Appl. Geophys, 177 (2020), pp. 2457-2475View at publisher CrossRef View in Scopus Google Scholar
Chen et al., 2020F. Chen, V. Heller, R. BrigantiNumerical modelling of tsunamis generated by iceberg calving validated with large-scale laboratory experimentsAdv. Water Resour., 142 (2020), Article 103647View PDF View article View in Scopus Google Scholar
Courant et al., 1928R. Courant, K. Friedrichs, H. LewyÜber die partiellen Differenzengleichungen der mathematischen PhysikMath. Ann., 100 (1) (1928), pp. 32-74View in Scopus Google Scholar
Ersoy et al., 2019H. Ersoy, M. Karahan, K. Gelişli, A. Akgün, T. Anılan, M.O. Sünnetci, B.K. YahşiModelling of the landslide-induced impulse waves in the Artvin Dam reservoir by empirical approach and 3D numerical simulationEng. Geol., 249 (2019), pp. 112-128View PDF View article View in Scopus Google Scholar
Flow Science 2023Flow Science. (2023). Flow-3D Hydro version 12.0 user’s manual. Santa Fe, NM, USA. Retrieved from https://www.flow3d.com/. 6 Aug 2023.Google Scholar
Fritz, 2002H.M. FritzDoctoral DissertationETH Zurich (2002)Google Scholar
Fritz et al., 2003H.M. Fritz, W.H. Hager, H.E. MinorLandslide generated impulse waves. 1. Instantaneous flow fieldsExp. Fluids, 35 (2003), pp. 505-519View in Scopus Google Scholar
Fritz et al., 2004H.M. Fritz, W.H. Hager, H.E. MinorNear field characteristics of landslide generated impulse wavesJ. Waterw. Port Coastal Ocean Eng., 130 (6) (2004), pp. 287-302View in Scopus Google Scholar
Geertsema et al., 2022M. Geertsema, B. Menounos, G. Bullard, J.L. Carrivick, J.J. Clague, C. Dai, D. Donati, G. Ekstrom, J.M. Jackson, P. Lynett, M. PichierriThe 28 Nov 2020 landslide, tsunami, and outburst flood – a hazard cascade associated with rapid deglaciation at Elliot Creek, BC, CanadaGeophys. Res. Lett., 49 (6) (2022)Google Scholar
Grilli et al., 2021S.T. Grilli, C. Zhang, J.T. Kirby, A.R. Grilli, D.R. Tappin, S.F.L. Watt, J.E. Hunt, A. Novellino, S. Engwell, M.E.M. Nurshal, M. AbdurrachmanModeling of the Dec. 22nd, 2018, Anak Krakatau volcano lateral collapse and tsunami based on recent field surveys: comparison with observed tsunami impactMar. Geol., 440 (2021), Article 106566View PDF View article View in Scopus Google Scholar
Grilli et al., 2019S.T. Grilli, D.R. Tappin, S. Carey, S.F. Watt, S.N. Ward, A.R. Grilli, S.L. Engwell, C. Zhang, J.T. Kirby, L. Schambach, M. MuinModelling of the tsunami from the Dec. 22, 2018, lateral collapse of Anak Krakatau volcano in the Sunda Straits, IndonesiaSci. Rep., 9 (1) (2019), p. 11946 View at publisher This article is free to access.View in Scopus Google Scholar
Heidarzadeh et al., 2023M. Heidarzadeh, A.R. Gusman, I.E. MuliaThe landslide source of the eastern Mediterranean tsunami on 6 Feb 2023 following the Mw 7.8 Kahramanmaraş (Türkiye) inland earthquakeGeosci. Lett., 10 (1) (2023), p. 50 View at publisher This article is free to access.View in Scopus Google Scholar
Heidarzadeh et al., 2022M. Heidarzadeh, A.R. Gusman, A. Patria, B.T. WidyantoroPotential landslide origin of the Seram Island tsunami in Eastern Indonesia on 16 June 2021 following an Mw 5.9 earthquakeBull. Seismol. Soc. Am., 112 (5) (2022), pp. 2487-2498View at publisher CrossRef View in Scopus Google Scholar
Heidarzadeh et al., 2020M. Heidarzadeh, T. Ishibe, O. Sandanbata, A. Muhari, A.B. WijanartoNumerical modeling of the subaerial landslide source of the 22 Dec 2018 Anak Krakatoa volcanic tsunami, IndonesiaOcean. Eng., 195 (2020), Article 106733View PDF View article View in Scopus Google Scholar
Heidarzadeh et al., 2017M. Heidarzadeh, T. Harada, K. Satake, T. Ishibe, T. TakagawaTsunamis from strike-slip earthquakes in the Wharton Basin, northeast Indian Ocean: March 2016 M w7. 8 event and its relationship with the April 2012 M w 8.6 eventGeophys. J. Int., 211 (3) (2017), pp. 1601-1612, 10.1093/gji/ggx395 View at publisher This article is free to access.View in Scopus Google Scholar
Heller et al., 2008V. Heller, W.H. Hager, H.E. MinorScale effects in subaerial landslide generated impulse wavesExp. Fluids, 44 (2008), pp. 691-703View at publisher CrossRef View in Scopus Google Scholar
Heller and Hager, 2014V. Heller, W.H. HagerA universal parameter to predict subaerial landslide tsunamis?J. Mar. Sci. Eng., 2 (2) (2014), pp. 400-412View at publisher CrossRef View in Scopus Google Scholar
Heller and Spinneken, 2015V. Heller, J. SpinnekenOn the effect of the water body geometry on landslide–tsunamis: physical insight from laboratory tests and 2D to 3D wave parameter transformationCoastal Eng., 104 (2015), pp. 113-134View PDF View article View in Scopus Google Scholar
Heller et al., 2016V. Heller, M. Bruggemann, J. Spinneken, B.D. RogersComposite modelling of subaerial landslide–tsunamis in different water body geometries and novel insight into slide and wave kinematicsCoastal Eng., 109 (2016), pp. 20-41View PDF View article View in Scopus Google Scholar
Hirt, 2003C.W. HirtModeling Turbulent Entrainment of Air at a Free SurfaceFlow Science, Inc (2003)Google Scholar
Hu et al., 2023G. Hu, K. Satake, L. Li, P. DuOrigins of the tsunami following the 2023 Turkey–Syria earthquakeGeophys. Res. Lett., 50 (18) (2023)Google Scholar
Hu et al., 2022G. Hu, W. Feng, Y. Wang, L. Li, X. He, Ç. Karakaş, Y. TianSource characteristics and exacerbated tsunami hazard of the 2020 Mw 6.9 Samos earthquake in Eastern Aegean SeaJ. Geophys. Res., 127 (5) (2022)e2022JB023961Google Scholar
Iorio et al., 2021V. Iorio, G. Bellotti, C. Cecioni, S.T. GrilliA numerical model for the efficient simulation of multiple landslide-induced tsunamis scenariosOcean Model, 168 (2021), Article 101899View PDF View article View in Scopus Google Scholar
Kestin et al., 1978J. Kestin, M. Sokolov, W.A. WakehamViscosity of liquid water in the range− 8 C to 150 CJPCRD, 7 (3) (1978), pp. 941-948View at publisher CrossRef View in Scopus Google Scholar
Kim et al., 2020G.B. Kim, W. Cheng, R.C. Sunny, J.J. Horrillo, B.C. McFall, F. Mohammed, H.M. Fritz, J. Beget, Z. KowalikThree-dimensional landslide generated tsunamis: numerical and physical model comparisonsLandslides, 17 (2020), pp. 1145-1161View at publisher CrossRef View in Scopus Google Scholar
Kirby et al., 2022J.T. Kirby, S.T. Grilli, J. Horrillo, P.L.F. Liu, D. Nicolsky, S. Abadie, B. Ataie-Ashtiani, M.J. Castro, L. Clous, C. Escalante, I. Fine, J.M. González-Vida, F. Løvholt, P. Lynett, G. Ma, J. Macías, S. Ortega, F. Shi, S. Yavari-Ramshe, C. ZhangValidation and inter-comparison of models for landslide tsunami generationOcean Model., 170 (2022), Article 101943View PDF View article View in Scopus Google Scholar
Li et al., 2019G. Li, G. Chen, P. Li, H. JingEfficient and accurate 3-D numerical modelling of landslide tsunamiWater, 11 (10) (2019), p. 2033View at publisher CrossRef View in Scopus Google Scholar
MathWorks 2023MathWorksMATLAB User ManualThe MathWorks Inc., MA, USA (2023), p. 282Google Scholar
McFall and Fritz, 2016B.C. McFall, H.M. FritzPhysical modelling of tsunamis generated by three-dimensional deformable granular landslides on planar and conical island slopesProc. R. Soc. A. Math. Phys. Eng. Sci., 472 (2188) (2016), Article 20160052View at publisher CrossRef Google Scholar
Mulia et al., 2020aI.E. Mulia, S. Watada, T.C. Ho, K. Satake, Y. Wang, A. AditiyaSimulation of the 2018 tsunami due to the flank failure of Anak Krakatau volcano and implication for future observing systemsGeophys. Res. Lett., 47 (14) (2020), Article e2020GL087334 View at publisher This article is free to access.View in Scopus Google Scholar
Murty, 2003T.S. MurtyTsunami wave height dependence on landslide volumePure Appl. Geophys., 160 (2003), pp. 2147-2153View in Scopus Google Scholar
Mulia et al., 2020bI.E. Mulia, S. Watada, T.C. Ho, K. Satake, Y. Wang, A. AditiyaSimulation of the 2018 tsunami due to the flank failure of Anak Krakatau volcano and implication for future observing systemsGeophys. Res. Lett., 47 (14) (2020)Google Scholar
Mulligan et al., 2020R.P. Mulligan, A. Franci, M.A. Celigueta, W.A. TakeSimulations of landslide wave generation and propagation using the particle finite element methodJ. Geophys. Res. Oceans, 125 (6) (2020)Google Scholar
Najafi-Jilani and Ataie-Ashtiani, 2008A. Najafi-Jilani, B. Ataie-AshtianiEstimation of near-field characteristics of tsunami generation by submarine landslideOcean Eng., 35 (5) (2008), pp. 545-557View PDF View article View in Scopus Google Scholar
Onnen et al., 1997C. Onnen, R. Babuška, U. Kaymak, J.M. Sousa, H.B. Verbruggen, R. IsermannGenetic algorithms for optimization in predictive controlControl Eng. Pract, 5 (10) (1997), pp. 1363-1372View PDF View article View in Scopus Google Scholar
Panizzo et al. 2005A. Panizzo, P. De Girolamo, M. Di Risio, A. Maistri, A. PetacciaGreat landslid event in Italian artificial reservoirsNat. Hazards Earth Syst. Sci., 5 (5) (2005), pp. 733-740View at publisher CrossRef View in Scopus Google Scholar
Ren et al., 2020Z. Ren, Y. Wang, P. Wang, J. Hou, Y. Gao, L. ZhaoNumerical study of the triggering mechanism of the 2018 Anak Krakatau tsunami: eruption or collapsed landslide?Nat. Hazards, 102 (2020), pp. 1-13View in Scopus Google Scholar
Robbe-Saule et al., 2021M. Robbe-Saule, C. Morize, Y. Bertho, A. Sauret, A. Hildenbrand, P. GondretFrom laboratory experiments to geophysical tsunamis generated by subaerial landslidesSci. Rep., 11 (1) (2021), pp. 1-9Google Scholar
Sabeti and Heidarzadeh, 2022aR. Sabeti, M. HeidarzadehNumerical simulations of water waves generated by subaerial granular and solid-block landslides: validation, comparison, and predictive equationsOcean Eng., 266 (2022), Article 112853View PDF View article View in Scopus Google Scholar
Sabeti and Heidarzadeh, 2022bR. Sabeti, M. HeidarzadehA new empirical equation for predicting the maximum initial amplitude of submarine landslide-generated wavesLandslides, 19 (2) (2022), pp. 491-503 View at publisher This article is free to access.CrossRef View in Scopus Google Scholar
Sabeti et al. 2024R. Sabeti, M. Heidarzadeh, A. Romano, G. Barajas Ojeda, J.L. LaraThree-Dimensional Simulations of Subaerial Landslide-Generated Waves: Comparing OpenFOAM and FLOW-3D HYDRO ModelsPure Appl. Geophys. (2024), 10.1007/s00024-024-03443-x View at publisher This article is free to access.Google Scholar
Si et al., 2018P. Si, H. Shi, X. YuA general numerical model for surface waves generated by granular material intruding into a water bodyCoastal Eng., 142 (2018), pp. 42-51View PDF View article View in Scopus Google Scholar
Sorensen, 2010R.M. SorensenBasic Coastal Engineering(3rd edition), Springer Science & Business Media (2010), p. 324Google Scholar
Syamsidik et al., 2020Benazir Syamsidik, M. Luthfi, A. Suppasri, L.K. ComfortThe 22 December 2018 Mount Anak Krakatau volcanogenic tsunami on Sunda Strait coasts, Indonesia: tsunami and damage characteristicsNat. Hazards Earth Syst. Sci., 20 (2) (2020), pp. 549-565View in Scopus Google Scholar
Synolakis et al., 2002C.E. Synolakis, J.P. Bardet, J.C. Borrero, H.L. Davies, E.A. Okal, E.A. Silver, D.R. TappinThe slump origin of the 1998 Papua New Guinea tsunamiProc. R. Soc. Lond. A Math. Phys. Eng. Sci., 45 (2002), pp. 763-789View in Scopus Google Scholar
Wang et al., 2022Y. Wang, H.Y. Su, Z. Ren, Y. MaSource properties and resonance characteristics of the tsunami generated by the 2021 M 8.2 Alaska earthquakeJ. Geophys. Res. Oceans, 127 (3) (2022), Article e2021JC018308 View at publisher This article is free to access.View in Scopus Google Scholar
Wang et al., 2021Y. Wang, M. Heidarzadeh, K. Satake, G. HuCharacteristics of two tsunamis generated by successive M w 7.4 and M w 8.1 earthquakes in Kermadec Islands on March 4, 2021Nat. Haz. Earth Sys. Sci., 22 (2021), pp. 1073-1082, 10.5194/nhess-22-1073-2022 View at publisher View in Scopus Google Scholar
Watts et al., 2005P. Watts, S.T. Grilli, D.R. Tappin, G.J. FryerTsunami generation by submarine mass failure. II: predictive equations and case studiesJ. Waterw. Port Coast. Ocean Eng., 131 (6) (2005), pp. 298-310View in Scopus Google Scholar
Watts et al., 2003P. Watts, S.T. Grilli, J.T. Kirby, G.J. Fryer, D.R. TappinLandslide tsunami case studies using a Boussinesq model and a fully nonlinear tsunami generation modelNat. Hazards Earth Sys. Sci., 3 (5) (2003), pp. 391-402View at publisher CrossRef View in Scopus Google Scholar
Watts, 1998P. WattsWavemaker curves for tsunamis generated by underwater landslidesJ. Waterw. Port. Coast. Ocean. Eng., 124 (3) (1998), pp. 127-137Google Scholar
Zhang et al., 2021C. Zhang, J.T. Kirby, F. Shi, G. Ma, S.T. GrilliA two-layer non-hydrostatic landslide model for tsunami generation on irregular bathymetry. 2. Numerical discretization and model validationOcean Model., 160 (2021), Article 101769View PDF View article View in Scopus Google Scholar

PDF 원문 보기

Figure 3 – Free surface views. Bottom left: k-ε RNG model. Bottom right: LES.

Physical Modeling and CFD Comparison: Case Study of a HydroCombined Power Station in Spillway Mode

물리적 모델링 및 CFD 비교: 방수로 모드의 HydroCombined 발전소 사례 연구

Gonzalo Duró, Mariano De Dios, Alfredo López, Sergio O. Liscia

ABSTRACT

This study presents comparisons between the results of a commercial CFD code and physical model measurements. The case study is a hydro-combined power station operating in spillway mode for a given scenario. Two turbulence models and two scales are implemented to identify the capabilities and limitations of each approach and to determine the selection criteria for CFD modeling for this kind of structure. The main flow characteristics are considered for analysis, but the focus is on a fluctuating frequency phenomenon for accurate quantitative comparisons. Acceptable representations of the general hydraulic functioning are found in all approaches, according to physical modeling. The k-ε RNG, and LES models give good representation of the discharge flow, mean water depths, and mean pressures for engineering purposes. The k-ε RNG is not able to characterize fluctuating phenomena at a model scale but does at a prototype scale. The LES is capable of identifying the dominant frequency at both prototype and model scales. A prototype-scale approach is recommended for the numerical modeling to obtain a better representation of fluctuating pressures for both turbulence models, with the complement of physical modeling for the ultimate design of the hydraulic structures.

본 연구에서는 상용 CFD 코드 결과와 물리적 모델 측정 결과를 비교합니다. 사례 연구는 주어진 시나리오에 대해 배수로 모드에서 작동하는 수력 복합 발전소입니다.

각 접근 방식의 기능과 한계를 식별하고 이러한 종류의 구조에 대한 CFD 모델링의 선택 기준을 결정하기 위해 두 개의 난류 모델과 두 개의 스케일이 구현되었습니다. 주요 흐름 특성을 고려하여 분석하지만 정확한 정량적 비교를 위해 변동하는 주파수 현상에 중점을 둡니다.

일반적인 수리학적 기능에 대한 허용 가능한 표현은 물리적 모델링에 따라 모든 접근 방식에서 발견됩니다. k-ε RNG 및 LES 모델은 엔지니어링 목적을 위한 배출 유량, 평균 수심 및 평균 압력을 잘 표현합니다.

k-ε RNG는 모델 규모에서는 변동 현상을 특성화할 수 없지만 프로토타입 규모에서는 특성을 파악합니다. LES는 프로토타입과 모델 규모 모두에서 주요 주파수를 식별할 수 있습니다.

수력학적 구조의 궁극적인 설계를 위한 물리적 모델링을 보완하여 두 난류 모델에 대한 변동하는 압력을 더 잘 표현하기 위해 수치 모델링에 프로토타입 규모 접근 방식이 권장됩니다.

Figure 1 – Physical scale model (left). Upstream flume and point gauge (right)

Figure 4 – Water levels: physical model (maximum values) and CFD results (mean values)

Figure 5 – Instantaneous pressures [Pa] and velocities [m/s] at model scale (bay center)

Keywords

CFD validation, hydro-combined, k-ε RNG, LES, pressure spectrum

PDF Download

REFERENCES

ADRIAN R. J. (2007). “Hairpin vortex organization in wall turbulence.” Phys. Fluids 19(4), 041301.
DEWALS B., ARCHAMBEAU P., RULOT F., PIROTTON M. and ERPICUM S. (2013). “Physical and
Numerical Modelling in Low-Head Structures Design.” Proc. International Workshop on Hydraulic
Design of Low-Head Structures, Aachen, Germany, Bundesanstalt für Wasserbau Publ., D.B. BUNG
and S. PAGLIARA Editors, pp.11-30.
GRENANDER, U. (1959). Probability and Statistics: The Harald Cramér Volume. Wiley.
HIRT, C. W. and NICHOLS B. D. (1981). “Volume of fluid (VOF) method for the dynamics of free
boundaries.” Journal of Computational Physics 39(1): 201-225.
JOHNSON M. C. and SAVAGE B. M. (2006). “Physical and numerical comparison of flow over ogee
spillway in the presence of tailwater.” J. Hydraulic Eng. 132(12): 1353–1357.
KHAN L.A., WICKLEIN E.A., RASHID M., EBNER L.L. and RICHARDS N.A. (2004).
“Computational fluid dynamics modeling of turbine intake hydraulics at a hydropower plant.” Journal
of Hydraulic Research, 42:1, 61-69
LAROCQUE L.A., IMRAN J. and CHAUDHRY M. (2013). “3D numerical simulation of partial breach
dam-break flow using the LES and k–ϵ turbulence models.” Jl of Hydraulic Research, 51:2, 145-157
LI S., LAI Y., WEBER L., MATOS SILVA J. and PATEL V.C. (2004). “Validation of a threedimensional numerical model for water-pump intakes.” Journal of Hydraulic Research, 42:3, 282-292
NOVAK P., GUINOT V., JEFFREY A. and REEVE D.E. (2010). “Hydraulic modelling – An
introduction.” Spon Press, London and New York, ISBN 978-0-419-25010-4, 616 pp.

PDF Download

Figure (17): Stream Lines Indicating Average Flow Speed in the Model with Various Nose shapes, Measured at Mid-Depth and at the Flow Surface Level, at a Flow Rate of 78 Liters per Second.

Conducting experimental and numerical studies to analyze theimpact of the base nose shape on flow hydraulics in PKW weirusing FLOW-3D

FLOW-3D를 사용하여 PKW 둑의 흐름 수력학에 대한 베이스 노즈 모양의 영향을 분석하기 위한 실험 및 수치 연구 수행

Behshad Mardasi ¹
Rasoul Ilkhanipour Zeynali ²
Majid Heydari ³

Abstract

Weirs are essential structures used to manage excess water flow from behind dams to downstream areas. Enhancing discharge efficiency often involves extending the effective length of Piano Key Weirs (PKW) in dams or regulating flow within irrigation and drainage networks. This study employed both numerical and laboratory investigations to assess the impact of different base nose shapes installed beneath the outlet keys and varying Input to output key width ratios (Wi/Wo) on discharges ranging from 5 to 80 liters per second. Furthermore, the study aimed to achieve research objectives and compare the performance of Piano Key Weirs with Ogee Weir. For numerical simulation, the optimal number of cells for meshing was determined, and an appropriate turbulence model was selected. The results indicated that the numerical model accurately simulated the laboratory sample with a high degree of precision. Moreover, the numerical model closely approximated PKW for all parameters Q, H, and Cd compared to the laboratory sample. The findings revealed that in laboratory models with a maximum discharge area of 80 liters per second, the weir with Wi/Wo=1.2 and a flow head value of 285 mm exhibited the lowest value, whereas the weir with Wi/Wo=0.71 and a flow head value of 305 mm showed the highest, attributed to the higher discharge in the input-output ratio. Additionally, as the ratio of flow head to weir height H/P increased, the discharge coefficient Cd decreased. Comparing the flow conditions in weirs with different base nose shapes, it was observed that the weir with a spindle nose shape (PKW1.2S) outperformed the PKW with a flat (PKW1.2), semi-cylindrical (PKW1.2CL) and triangular base nose (PKW1.2TR). The results emphasized that models featuring semi-cylindrical and flat noses exhibited notable flow deviation and abrupt disruption upon impact with the nose. However, this effect was significantly reduced in models equipped with triangular and spindle-shaped noses. Also, the coefficient of discharge in PKW1.2S and PKW1.2TR weirs, compared to the PKW1.20 weir, increased by 27% and 20%, respectively.

웨어는 댐 뒤에서 하류 지역으로의 과도한 물 흐름을 관리하는 데 사용되는 필수 구조물입니다. 배출 효율을 높이는 데에는 댐의 피아노 키 위어(PKW) 유효 길이를 연장하거나 관개 및 배수 네트워크 내 흐름을 조절하는 것이 포함됩니다.

이 연구에서는 콘센트 키 아래에 설치된 다양한 베이스 노즈 모양과 초당 5~80리터 범위의 배출에 대한 다양한 입력 대 출력 키 너비 비율(Wi/Wo)의 영향을 평가하기 위해 수치 및 실험실 조사를 모두 사용했습니다. 또한 본 연구에서는 연구 목적을 달성하고 Piano Key Weir와 Ogee Weir의 성능을 비교하는 것을 목표로 했습니다.

수치 시뮬레이션을 위해 메시 생성을 위한 최적의 셀 수를 결정하고 적절한 난류 모델을 선택했습니다. 결과는 수치 모델이 높은 정밀도로 실험실 샘플을 정확하게 시뮬레이션했음을 나타냅니다. 더욱이, 수치 모델은 실험실 샘플과 비교하여 모든 매개변수 Q, H 및 Cd에 대해 PKW에 매우 근접했습니다.

연구 결과, 최대 배출 면적이 초당 80리터인 실험실 모델에서는 Wi/Wo=1.2, 플로우 헤드 값이 285mm인 웨어가 가장 낮은 값을 나타냈고, Wi/Wo=0.71 및 a인 웨어는 가장 낮은 값을 나타냈습니다. 플로우 헤드 값은 305mm로 가장 높은 것으로 나타났는데, 이는 입출력 비율의 높은 토출량에 기인합니다. 또한, 웨어 높이에 대한 유수두 비율 H/P가 증가함에 따라 유출계수 Cd는 감소하였다.

베이스 노즈 모양이 다른 웨어의 흐름 조건을 비교해 보면, 스핀들 노즈 모양(PKW1.2S)의 웨어가 평면(PKW1.2), 반원통형(PKW1.2CL) 및 삼각형 모양의 PKW보다 성능이 우수한 것으로 관찰되었습니다. 베이스 노즈(PKW1.2TR) 결과는 반원통형 및 편평한 노즈를 특징으로 하는 모델이 노즈에 충격을 가할 때 눈에 띄는 흐름 편차와 급격한 중단을 나타냄을 강조했습니다.

그러나 삼각형 및 방추형 노즈를 장착한 모델에서는 이러한 효과가 크게 감소했습니다. 또한 PKW1.20보에 비해 PKW1.2S보와 PKW1.2TR보의 유출계수는 각각 27%, 20% 증가하였다.

Keywords

Piano Key Weir, Base Nose Shape, Flow Hydraulics, Numerical Model, Triangular
Nose Shape, Flat Nose Shape, Semi-Cylindrical Nose Shape, Spindle Nose Shape

Reference

Chow, V.T. (1959). “Open channel hydraulics.” McGraw-Hill Book Company, New York,
NY.
Ouamane, A., and Lempérière, F. (2006). “Design of a new economic shape of weir.” Proc.,
Intl. Symp. on Dams in the Societies of the 21st Century, 463-470, Barcelona, Spain.
Crookston, B. M., Anderson, A., Shearin-Feimster, L., and Tullis, B. P. (2014). “Mitigation
investigation of flow-induced vibrations at a rehabilitated spillway.” Proc., 5th IAHR Intl.
Symp. on Hydraulic Structures, Univ. of Queensland Brisbane, Brisbane, Australia.
Machiels, O. (2012). “Experimental study of the hydraulic behaviour of Piano Key Weirs.”
Ph.D. Dissertation, Faculty of Applied Science, University of Liège, Liège, Belgium.
Blanc, P., and Lempérière, F. (2001). “Labyrinth spillways have a promising future.” Intl. J.
of Hydropower and Dams, 8(4), 129-131.
Muslu, Y. (2001). “Numerical analysis for lateral weir flow.” J. of Irrigation and Drainage
Eng., ASCE, 127, 246.
Erpicum, S., Machiels, O., Dewals, B., Pirotton, M., and Archambeau, P. (2012).
“Numerical and physical hydraulic modeling of Piano Key Weirs.” Proc., ASIA 2012 – 4th
Intl. Conf. on Water Resources and Renewable Energy Development in Asia, Chiang Mai,
Thailand.
Tullis, J.P., Amanian, N., and Waldron, D. (1995). “Design of Labyrinth Spillways.” J. of
Hydraulic Eng., ASCE, 121.
Lux, F.L., and Hinchcliff, D. (1985). “Design and construction of labyrinth spillways.”
Proc., 15th Intl. Congress on Large Dams, ICOLD, Vol. 4, 249-274, Paris, France.
Erpicum, S., Laugier, F., Ho to Khanh, M., & Pfister, M. (2017). Labyrinth and Piano Key
Weirs III–PKW 2017. CRC Press, Boca Raton, FL.
Kabiri-Samani, A., and Javaheri, A. (2012). “Discharge coefficient for free and submerged flow over Piano Key weirs.” Hydraulic Research J., 50(1), 114-120.
Hien, T.C., Son, H.T., and Khanh, M.H.T. (2006). “Results of some piano Key weirs
hydraulic model tests in Vietnam.” Proc., 22nd ICOLD Congress, CIGB/ICOLD,
Barcelona, Spain.
Laugier, F., Lochu, A., Gille, C., Leite Ribeiro, M., and Boillat, J-L. (2009). “Design and
construction of a labyrinth PKW spillway at St-Marc Dam.” Hydropower and Dams J.,
15(5), 100-107.
Cicero, G.M., Menon, J.M., Luck, M., and Pinchard, T. (2011). “Experimental study of side
and scale effects on hydraulic performances of a Piano Key Weir.” In: Erpicum, S., Laugier,
F., Boillat, J-L, Pirotton, M., Reverchon, B., and Schleiss, A-J (Eds.), Labyrinth and Piano
Key Weirs, 167-172, CRC Press, London.
Pralong, J., Vermeulen, J., Blancher, B., Laugier, F., Erpicum, S., Machiels, O., Pirotton,
M., Boillat, J.L, Leite Ribeiro, M., and Schleiss, A.J. (2011). “A naming convention for the
piano key weirs geometrical parameters.” In: Erpicum, S., Laugier, F., Boillat, J-L, Pirotton,
M., Reverchon, B., and Schleiss, A-J (Eds.), Labyrinth and Piano Key Weirs, 271-278,
CRC Press, London.
Denys, F. J. M., and Basson, G. R. (2018). “Transient hydrodynamics of Piano Key Weirs.”
Proc., 7th IAHR Intl. Symp. on Hydraulic Structures, ISHS2018, 518-527,
DigitalCommons@USU, Logan, UT.
Anderson, A., and Tullis, B. P. (2018). “Finite crest length weir nappe oscillation.” J. of
Hydraulic Eng., ASCE, 144(6), 04018020. https://doi.org/10.1061/(ASCE)HY.1943-
7900.0001461
Erpicum, S., Laugier, F., Boillat, J.-L., Pirotton, M., Reverchon, B., and Schleiss, A. J.
(2011). “Labyrinth and Piano Key Weirs–PKW 2011.” CRC Press, Boca Raton, FL.
Aydin, C.M., and Emiroglu, M.E. (2011). “Determination of capacity of labyrinth side weir
by CFD.” Flow Measurement and Instrumentation, 29, 1-8.
Cicero, G.M., Delisle, J.R., Lefebvre, V., and Vermeulen, J. (2013). “Experimental and
numerical study of the hydraulic performance of a trapezoidal PKW.” Proc., Intl. Workshop
on Labyrinths and Piano Key Weirs PKW II 2013, 265-272, CRC Press.
Anderson, R. M. (2011). “Piano Key Weir Head Discharge Relationships.” Master’s Thesis,
Utah State University, Logan, Utah.
Crookston, B.M., Anderson, R.M., and Tullis, B.P. (2018). “Free-flow discharge estimation
method for Piano Key weir geometries.” J. of Hydro-environment Research, 19, 160-167

PDF 원문 보기

Coupled CFD-DEM simulation of interfacial fluid–particle interaction during binder jet 3D printing

바인더 제트 3D 프린팅 중 계면 유체-입자 상호 작용에 대한 CFD-DEM 결합 시뮬레이션

Joshua J. Wagner, C. Fred Higgs III

https://doi.org/10.1016/j.cma.2024.116747

Abstract

The coupled dynamics of interfacial fluid phases and unconstrained solid particles during the binder jet 3D printing process govern the final quality and performance of the resulting components. The present work proposes a computational fluid dynamics (CFD) and discrete element method (DEM) framework capable of simulating the complex interfacial fluid–particle interaction that occurs when binder microdroplets are deposited into a powder bed. The CFD solver uses a volume-of-fluid (VOF) method for capturing liquid–gas multifluid flows and relies on block-structured adaptive mesh refinement (AMR) to localize grid refinement around evolving fluid–fluid interfaces. The DEM module resolves six degrees of freedom particle motion and accounts for particle contact, cohesion, and rolling resistance. Fully-resolved CFD-DEM coupling is achieved through a fictitious domain immersed boundary (IB) approach. An improved method for enforcing three-phase contact lines with a VOF-IB extension technique is introduced. We present several simulations of binder jet primitive formation using realistic process parameters and material properties. The DEM particle systems are experimentally calibrated to reproduce the cohesion behavior of physical nickel alloy powder feedstocks. We demonstrate the proposed model’s ability to resolve the interdependent fluid and particle dynamics underlying the process by directly comparing simulated primitive granules with one-to-one experimental counterparts obtained from an in-house validation apparatus. This computational framework provides unprecedented insight into the fundamental mechanisms of binder jet 3D printing and presents a versatile new approach for process parameter optimization and defect mitigation that avoids the inherent challenges of experiments.

바인더 젯 3D 프린팅 공정 중 계면 유체 상과 구속되지 않은 고체 입자의 결합 역학이 결과 구성 요소의 최종 품질과 성능을 좌우합니다. 본 연구는 바인더 미세액적이 분말층에 증착될 때 발생하는 복잡한 계면 유체-입자 상호작용을 시뮬레이션할 수 있는 전산유체역학(CFD) 및 이산요소법(DEM) 프레임워크를 제안합니다.

CFD 솔버는 액체-가스 다중유체 흐름을 포착하기 위해 VOF(유체량) 방법을 사용하고 블록 구조 적응형 메쉬 세분화(AMR)를 사용하여 진화하는 유체-유체 인터페이스 주위의 그리드 세분화를 국지화합니다. DEM 모듈은 6개의 자유도 입자 운동을 해결하고 입자 접촉, 응집력 및 구름 저항을 설명합니다.

완전 분해된 CFD-DEM 결합은 가상 도메인 침지 경계(IB) 접근 방식을 통해 달성됩니다. VOF-IB 확장 기술을 사용하여 3상 접촉 라인을 강화하는 향상된 방법이 도입되었습니다. 현실적인 공정 매개변수와 재료 특성을 사용하여 바인더 제트 기본 형성에 대한 여러 시뮬레이션을 제시합니다.

DEM 입자 시스템은 물리적 니켈 합금 분말 공급원료의 응집 거동을 재현하기 위해 실험적으로 보정되었습니다. 우리는 시뮬레이션된 기본 과립과 내부 검증 장치에서 얻은 일대일 실험 대응물을 직접 비교하여 프로세스의 기본이 되는 상호 의존적인 유체 및 입자 역학을 해결하는 제안된 모델의 능력을 보여줍니다.

이 계산 프레임워크는 바인더 제트 3D 프린팅의 기본 메커니즘에 대한 전례 없는 통찰력을 제공하고 실험에 내재된 문제를 피하는 공정 매개변수 최적화 및 결함 완화를 위한 다용도의 새로운 접근 방식을 제시합니다.

Introduction

Binder jet 3D printing (BJ3DP) is a powder bed additive manufacturing (AM) technology capable of fabricating geometrically complex components from advanced engineering materials, such as metallic superalloys and ultra-high temperature ceramics [1], [2]. As illustrated in Fig. 1(a), the process is comprised of many repetitive print cycles, each contributing a new cross-sectional layer on top of a preceding one to form a 3D CAD-specified geometry. The feedstock material is first delivered from a hopper to a build plate and then spread into a thin layer by a counter-rotating roller. After powder spreading, a print head containing many individual inkjet nozzles traverses over the powder bed while precisely jetting binder microdroplets onto select regions of the spread layer. Following binder deposition, the build plate lowers by a specified layer thickness, leaving a thin void space at the top of the job box that the subsequent powder layer will occupy. This cycle repeats until the full geometries are formed layer by layer. Powder bed fusion (PBF) methods follow a similar procedure, except they instead use a laser or electron beam to selectively melt and fuse the powder material. Compared to PBF, binder jetting offers several distinct advantages, including faster build rates, enhanced scalability for large production volumes, reduced machine and operational costs, and a wider selection of suitable feedstock materials [2]. However, binder jetted parts generally possess inferior mechanical properties and reduced dimensional accuracy [3]. As a result, widescale adoption of BJ3DP to fabricate high-performance, mission-critical components, such as those common to the aerospace and defense sectors, is contingent on novel process improvements and innovations [4].

A major obstacle hindering the advancement of BJ3DP is our limited understanding of how various printing parameters and material properties collectively influence the underlying physical mechanisms of the process and their effect on the resulting components. To date, the vast majority of research efforts to uncover these relationships have relied mainly on experimental approaches [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], which are often expensive and time-consuming and have inherent physical restrictions on what can be measured and observed. For these reasons, there is a rapidly growing interest in using computational models to circumvent the challenges of experimental investigations and facilitate a deeper understanding of the process’s fundamental phenomena. While significant progress has been made in developing and deploying numerical frameworks aimed at powder spreading [20], [21], [22], [23], [24], [25], [26], [27] and sintering [28], [29], [30], [31], [32], simulating the interfacial fluid–particle interaction (IFPI) in the binder deposition stage is still in its infancy. In their exhaustive review, Mostafaei et al. [2] point out the lack of computational models capable of resolving the coupled fluid and particle dynamics associated with binder jetting and suggest that the development of such tools is critical to further improving the process and enhancing the quality of its end-use components.

We define IFPI as a multiphase flow regime characterized by immiscible fluid phases separated by dynamic interfaces that intersect the surfaces of moving solid particles. As illustrated in Fig. 1(b), an elaborate IFPI occurs when a binder droplet impacts the powder bed in BJ3DP. The momentum transferred from the impacting droplet may cause powder compaction, cratering, and particle ejection. These ballistic disturbances can have deleterious effects on surface texture and lead to the formation of large void spaces inside the part [5], [13]. After impact, the droplet spreads laterally on the bed surface and vertically into the pore network, driven initially by inertial impact forces and then solely by capillary action [33]. Attractive capillary forces exerted on mutually wetted particles tend to draw them inward towards each other, forming a packed cluster of bound particles referred to as a primitive [34]. A single-drop primitive is the most fundamental building element of a BJ3DP part, and the interaction leading to its formation has important implications on the final part characteristics, such as its mechanical properties, resolution, and dimensional accuracy. Generally, binder droplets are deposited successively as the print head traverses over the powder bed. The traversal speed and jetting frequency are set such that consecutive droplets coalesce in the bed, creating a multi-drop primitive line instead of a single-drop primitive granule. The binder must be jetted with sufficient velocity to penetrate the powder bed deep enough to provide adequate interlayer binding; however, a higher impact velocity leads to more pronounced ballistic effects.

A computational framework equipped to simulate the interdependent fluid and particle dynamics in BJ3DP would allow for unprecedented observational and measurement capability at temporal and spatial resolutions not currently achievable by state-of-the-art imaging technology, namely synchrotron X-ray imaging [13], [14], [18], [19]. Unfortunately, BJ3DP presents significant numerical challenges that have slowed the development of suitable modeling frameworks; the most significant of which are as follows:

1.Incorporating dynamic fluid–fluid interfaces with complex topological features remains a nontrivial task for standard mesh-based CFD codes. There are two broad categories encompassing the methods used to handle interfacial flows: interface tracking and interface capturing [35]. Interface capturing techniques, such as the popular volume-of-fluid (VOF) [36] and level-set methods [37], [38], are better suited for problems with interfaces that become heavily distorted or when coalescence and fragmentation occur frequently; however, they are less accurate in resolving surface tension and boundary layer effects compared to interface tracking methods like front-tracking [39], arbitrary Lagrangian–Eulerian [40], and space–time finite element formulations [41]. Since interfacial forces become increasingly dominant at decreasing length scales, inaccurate surface tension calculations can significantly deteriorate the fidelity of IFPI simulations involving <100 μm droplets and particles.
2.Dynamic powder systems are often modeled using the discrete element method (DEM) introduced by Cundall and Strack [42]. For IFPI problems, a CFD-DEM coupling scheme is required to exchange information between the fluid and particle solvers. Fully-resolved CFD-DEM coupling suggests that the flow field around individual particle surfaces is resolved on the CFD mesh [43], [44]. In contrast, unresolved coupling volume averages the effect of the dispersed solid phase on the continuous fluid phases [45], [46], [47], [48]. Comparatively, the former is computationally expensive but provides detailed information about the IFPI in question and is more appropriate when contact line dynamics are significant. However, since the pore structure of a powder bed is convoluted and evolves with time, resolving such solid–fluid interfaces on a computational mesh presents similar challenges as fluid–fluid interfaces discussed in the previous point. Although various algorithms have been developed to deform unstructured meshes to accommodate moving solid surfaces (see Bazilevs et al. [49] for an overview of such methods), they can be prohibitively expensive when frequent topology changes require mesh regeneration rather than just modification through nodal displacement. The pore network in a powder bed undergoes many topology changes as particles come in and out of contact with each other, constantly closing and opening new flow channels. Non-body-conforming structured grid approaches that rely on immersed boundary (IB) methods to embed the particles in the flow field can be better suited for such cases [50]. Nevertheless, accurately representing these complex pore geometries on Cartesian grids requires extremely high mesh resolutions, which can impose significant computational costs.
3.Capillary effects depend on the contact angle at solid–liquid–gas intersections. Since mesh nodes do not coincide with a particle surface when using an IB method on structured grids, imposing contact angle boundary conditions at three-phase contact lines is not straightforward.

While these issues also pertain to PBF process modeling, resolving particle motion is generally less crucial for analyzing melt pool dynamics compared to primitive formation in BJ3DP. Therefore, at present, the vast majority of computational process models of PBF assume static powder beds and avoid many of the complications described above, see, e.g., [51], [52], [53], [54], [55], [56], [57], [58], [59]. Li et al. [60] presented the first 2D fully-resolved CFD-DEM simulations of the interaction between the melt pool, powder particles, surrounding gas, and metal vapor in PBF. Following this work, Yu and Zhao [61], [62] published similar melt pool IFPI simulations in 3D; however, contact line dynamics and capillary forces were not considered. Compared to PBF, relatively little work has been published regarding the computational modeling of binder deposition in BJ3DP. Employing the open-source VOF code Gerris [63], Tan [33] first simulated droplet impact on a powder bed with appropriate binder jet parameters, namely droplet size and impact velocity. However, similar to most PBF melt pool simulations described in the current literature, the powder bed was fixed in place and not allowed to respond to the interacting fluid phases. Furthermore, a simple face-centered cubic packing of non-contacting, monosized particles was considered, which does not provide a realistic pore structure for AM powder beds. Building upon this approach, we presented a framework to simulate droplet impact on static powder beds with more practical particle size distributions and packing arrangements [64]. In a study similar to [33], [64], Deng et al. [65] used the VOF capability in Ansys Fluent to examine the lateral and vertical spreading of a binder droplet impacting a fixed bimodal powder bed with body-centered packing. Li et al. [66] also adopted Fluent to conduct 2D simulations of a 100 μm diameter droplet impacting substrates with spherical roughness patterns meant to represent the surface of a simplified powder bed with monosized particles. The commercial VOF-based software FLOW-3D offers an AM module centered on process modeling of various AM technologies, including BJ3DP. However, like the above studies, particle motion is still not considered in this codebase. Ur Rehman et al. [67] employed FLOW-3D to examine microdroplet impact on a fixed stainless steel powder bed. Using OpenFOAM, Erhard et al. [68] presented simulations of different droplet impact spacings and patterns on static sand particles.

Recently, Fuchs et al. [69] introduced an impressive multipurpose smoothed particle hydrodynamics (SPH) framework capable of resolving IFPI in various AM methods, including both PBF and BJ3DP. In contrast to a combined CFD-DEM approach, this model relies entirely on SPH meshfree discretization of both the fluid and solid governing equations. The authors performed several prototype simulations demonstrating an 80 μm diameter droplet impacting an unconstrained powder bed at different speeds. While the powder bed responds to the hydrodynamic forces imparted by the impacting droplet, the particle motion is inconsistent with experimental time-resolved observations of the process [13]. Specifically, the ballistic effects, such as particle ejection and bed deformation, were drastically subdued, even in simulations using a droplet velocity ∼ 5× that of typical jetting conditions. This behavior could be caused by excessive damping in the inter-particle contact force computations within their SPH framework. Moreover, the wetted particles did not appear to be significantly influenced by the strong capillary forces exerted by the binder as no primitive agglomeration occurred. The authors mention that the objective of these simulations was to demonstrate their codebase’s broad capabilities and that some unrealistic process parameters were used to improve computational efficiency and stability, which could explain the deviations from experimental observations.

In the present paper, we develop a novel 3D CFD-DEM numerical framework for simulating fully-resolved IFPI during binder jetting with realistic material properties and process parameters. The CFD module is based on the VOF method for capturing binder–air interfaces. Surface tension effects are realized through the continuum surface force (CSF) method with height function calculations of interface curvature. Central to our fluid solver is a proprietary block-structured AMR library with hierarchical octree grid nesting to focus enhanced grid resolution near fluid–fluid interfaces. The GPU-accelerated DEM module considers six degrees of freedom particle motion and includes models based on Hertz-Mindlin contact, van der Waals cohesion, and viscoelastic rolling resistance. The CFD and DEM modules are coupled to achieve fully-resolved IFPI using an IB approach in which Lagrangian solid particles are mapped to the underlying Eulerian fluid mesh through a solid volume fraction field. An improved VOF-IB extension algorithm is introduced to enforce the contact angle at three-phase intersections. This provides robust capillary flow behavior and accurate computations of the fluid-induced forces and torques acting on individual wetted particles in densely packed powder beds.

We deploy our integrated codebase for direct numerical simulations of single-drop primitive formation with powder beds whose particle size distributions are generated from corresponding laboratory samples. These simulations use jetting parameters similar to those employed in current BJ3DP machines, fluid properties that match commonly used aqueous polymeric binders, and powder properties specific to nickel alloy feedstocks. The cohesion behavior of the DEM powder is calibrated based on the angle of repose of the laboratory powder systems. The resulting primitive granules are compared with those obtained from one-to-one experiments conducted using a dedicated in-house test apparatus. Finally, we demonstrate how the proposed framework can simulate more complex and realistic printing operations involving multi-drop primitive lines.

Section snippets

Mathematical description of interfacial fluid–particle interaction

This section briefly describes the governing equations of fluid and particle dynamics underlying the CFD and DEM solvers. Our unified framework follows an Eulerian–Lagrangian approach, wherein the Navier–Stokes equations of incompressible flow are discretized on an Eulerian grid to describe the motion of the binder liquid and surrounding gas, and the Newton–Euler equations account for the positions and orientations of the Lagrangian powder particles. The mathematical foundation for

CFD solver for incompressible flow with multifluid interfaces

This section details the numerical methodology used in our CFD module to solve the Navier–Stokes equations of incompressible flow. First, we introduce the VOF method for capturing the interfaces between the binder and air phases. This approach allows us to solve the fluid dynamics equations considering only a single continuum field with spatial and temporal variations in fluid properties. Next, we describe the time integration procedure using a fractional-step projection algorithm for

DEM solver for solid particle dynamics

This section covers the numerical procedure for tracking the motion of individual powder particles with DEM. The Newton–Euler equations (Eqs. (10), (11)) are ordinary differential equations (ODEs) for which many established numerical integrators are available. In general, the most challenging aspects of DEM involve processing particle collisions in a computationally efficient manner and dealing with small time step constraints that result from stiff materials, such as metallic AM powders. The

Unified CFD-DEM solver

The preceding sections have introduced the CFD and DEM solution algorithms separately. Here, we discuss the integrated CFD-DEM solution algorithm and related details.

Binder jet process modeling and validation experiments

In this section, we deploy our CFD-DEM framework to simulate the IFPI occurring during the binder droplet deposition stage of the BJ3DP process. The first simulations attempt to reproduce experimental single-drop primitive granules extracted from four nickel alloy powder samples with varying particle size distributions. The experiments are conducted with a dedicated in-house test apparatus that allows for the precision deposition of individual binder microdroplets into a powder bed sample. The

Conclusions

This paper introduces a coupled CFD-DEM framework capable of fully-resolved simulation of the interfacial fluid–particle interaction occurring in the binder jet 3D printing process. The interfacial flow of binder and surrounding air is captured with the VOF method and surface tension effects are incorporated using the CSF technique augmented by height function curvature calculations. Block-structured AMR is employed to provide localized grid refinement around the evolving liquid–gas interface.

CRediT authorship contribution statement

Joshua J. Wagner: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Software, Visualization, Writing – original draft, Writing – review & editing. C. Fred Higgs III: Conceptualization, Funding acquisition, Investigation, Methodology, Project administration, Resources, Supervision, Writing – original draft, Writing – review & editing.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported by a NASA Space Technology Research Fellowship, United States of America, Grant No. 80NSSC19K1171. Partial support was also provided through an AIAA Foundation Orville, USA and Wilbur Wright Graduate Award, USA . The authors would like to gratefully acknowledge Dr. Craig Smith of NASA Glenn Research Center for the valuable input he provided on this project.

References (155)

ZiaeeM. et al.Binder jetting: A review of process, materials, and methodsAddit. Manuf.(2019)
MostafaeiA. et al.Binder jet 3D printing—process parameters, materials, properties, modeling, and challengesProg. Mater. Sci.(2021)
MiyanajiH. et al.Effect of printing speed on quality of printed parts in binder jetting processAddit. Manuf.(2018)
CraneN.B.Impact of part thickness and drying conditions on saturation limits in binder jet additive manufacturingAddit. Manuf.(2020)
ColtonT. et al.Influence of droplet velocity, spacing, and inter-arrival time on line formation and saturation in binder jet additive manufacturingAddit. Manuf.(2021)
InkleyC.G. et al.Impact of controlled prewetting on part formation in binder jet additive manufacturingAddit. Manuf.(2023)
ParteliE.J. et al.Particle-based simulation of powder application in additive manufacturingPowder Technol.(2016)
HaeriS. et al.Discrete element simulation and experimental study of powder spreading process in additive manufacturingPowder Technol.(2017)
MeierC. et al.Critical influences of particle size and adhesion on the powder layer uniformity in metal additive manufacturingJ. Mater Process. Technol.(2019)
HeY. et al.Combined effect of particle size and surface cohesiveness on powder spreadability for additive manufacturingPowder Technol.(2021)

그림 12: 시간 경과에 따른 속도 카운터: 30초 그림 13: 시간 경과에 따른 속도 카운터: 20초

Gemelo digital del puente de Kalix: cargas estructurales de futuros eventos climáticos extremos

Kalix Bridge 디지털 트윈: 미래 극한 기후 현상으로 인한 구조적 부하

Este documento está relacionado con un proyecto en curso para el cual se está desarrollando e implementando un gemelo digital estructural del puente de Kalix en Suecia.
이 문서는 스웨덴 Kalix 교량의 구조적 디지털 트윈이 개발 및 구현되고 있는 진행 중인 프로젝트와 관련이 있습니다.

Autores: Mahyar Kazemian¹, Sajad Nikdel², Mehrnaz MohammadEsmaeili³, Vahid Nik⁴, Kamyab Zandi^*5

RESUMEN Las cargas ambientales, como el viento y el caudal de los ríos, juegan un papel esencial en el diseño y evaluación estructural de puentes de grandes luces. El cambio climático y los eventos climáticos extremos son amenazas para la confiabilidad y seguridad de la red de transporte.

Esto ha llevado a una creciente demanda de modelos de gemelos digitales para investigar la resistencia de los puentes en condiciones climáticas extremas. El puente de Kalix, construido sobre el río Kalix en Suecia en 1956, se utiliza como banco de pruebas en este contexto.

La estructura del puente, realizada en hormigón postensado, consta de cinco vanos, siendo el más largo de 94 m. En este estudio, las características aerodinámicas y los valores extremos de la simulación numérica del viento, como la presión en la superficie, se obtienen utilizando la simulación de remolinos desprendidos retardados (DDES) de Spalart-Allmaras como un enfoque de turbulencia RANS-LES híbrido que es práctico y computacionalmente eficiente para cerca de la pared densidad de malla impuesta por el método LES.

La presión del viento en la superficie se obtiene para tres escenarios climáticos extremos, que incluyen un clima con mucho viento, un clima extremadamente frío y el valor de cálculo para un período de retorno de 3000 años. El resultado indica diferencias significativas en la presión del viento en la superficie debido a las capas de tiempo que provienen de la simulación del flujo de viento transitorio. Para evaluar el comportamiento estructural en el escenario de viento crítico, se considera el valor más alto de presión en la superficie para cada escenario.

Además, se realiza un estudio hidrodinámico en los pilares del puente, en el que se simula el flujo del río por el método VOF, y se examina el proceso de movimiento del agua alrededor de los pilares de forma transitoria y en diferentes momentos. En cada una de las superficies del pilar se calcula la presión superficial aplicada por el caudal del río con el caudal volumétrico más alto registrado.

Para simular el flujo del río, se ha utilizado la información y las condiciones meteorológicas registradas en períodos anteriores. Los resultados muestran que la presión en la superficie en el momento en que el flujo del río golpea los pilares es mucho mayor que en los momentos posteriores. Esta cantidad de presión se puede usar como carga crítica en los cálculos de interacción fluido-estructura (FSI).

Finalmente, para ambas secciones, la presión en la superficie del viento, el campo de velocidades con respecto a las líneas de sondas auxiliares, los contornos del movimiento circunferencial del agua alrededor de los pilares y el diagrama de presión en ellos se informan en diferentes intervalos de tiempo.

요약 바람, 강의 흐름과 같은 환경 하중은 장대 교량의 설계 및 구조 평가에 필수적인 역할을 합니다. 기후 변화와 기상 이변은 교통 네트워크의 신뢰성과 보안에 위협이 됩니다.

이로 인해 극한 기상 조건에서 교량의 복원력을 조사하기 위한 디지털 트윈 모델에 대한 수요가 증가했습니다. 1956년 스웨덴 칼릭스 강 위에 건설된 칼릭스 다리는 이러한 맥락에서 테스트베드로 사용됩니다.

포스트텐션 콘크리트로 만들어진 교량 구조는 5개 경간으로 구성되며 가장 긴 길이는 94m입니다. 본 연구에서는 하이브리드 RANS-LES 난류 접근 방식인 Spalart-Allmaras 지연 분리 와류 시뮬레이션(DDES)을 사용하여 수치적 바람 시뮬레이션의 공기역학적 특성과 표면압 등 극한값을 얻습니다. LES 방법으로 부과된 벽 근처 메쉬 밀도.

바람이 많이 부는 기후, 극도로 추운 기후, 그리고 3000년의 반환 기간에 대해 계산된 값을 포함한 세 가지 극한 기후 시나리오에 대해 표면 풍압을 얻습니다. 결과는 과도 풍류 시뮬레이션에서 나오는 시간 레이어로 인해 표면 풍압에 상당한 차이가 있음을 나타냅니다. 임계 바람 시나리오에서 구조적 거동을 평가하기 위해 각 시나리오에 대해 가장 높은 표면 압력 값이 고려됩니다.

또한 교량 기둥에 대한 유체 역학 연구를 수행하여 하천의 흐름을 VOF 방법으로 시뮬레이션하고 기둥 주변의 물 이동 과정을 일시적이고 다른 시간에 조사합니다. 각 기둥 표면에서 기록된 체적 유량이 가장 높은 강의 흐름에 의해 적용되는 표면 압력이 계산됩니다.

강의 흐름을 시뮬레이션하기 위해 이전 기간에 기록된 정보와 기상 조건이 사용되었습니다. 결과는 강의 흐름이 기둥에 닿는 순간의 표면 압력이 나중에 순간보다 훨씬 높다는 것을 보여줍니다. 이 압력의 양은 유체-구조 상호작용(FSI) 계산에서 임계 하중으로 사용될 수 있습니다.

마지막으로 두 섹션 모두 바람 표면의 압력, 보조 프로브 라인에 대한 속도장, 기둥 주위 물의 원주 운동 윤곽 및 압력 다이어그램이 서로 다른 시간 간격으로 보고됩니다.

키워드: 디지털 트윈 , 풍력 공학, 콘크리트 교량, 유체역학, CFD 시뮬레이션, DDES 난류 모델, Kalix 교량

Palabras clave: Gemelo digital , Ingeniería eólica, Puente de hormigón, Hidrodinámica, Simulación CFD, Modelo de turbulencia DDES, Puente Kalix

1. Introducción

Las infraestructuras de transporte son la columna vertebral de nuestra sociedad y los puentes son el cuello de botella de la red de transporte [1]. Además, el cambio climático que da como resultado tasas de deterioro más altas y los eventos climáticos extremos son amenazas importantes para la confiabilidad y seguridad de las redes de transporte. Durante la última década, muchos puentes se han dañado o fallado por condiciones climáticas extremas como tifones e inundaciones.

Wang et al. analizó los impactos del cambio climático y mostró que se espera que el deterioro de los puentes de hormigón sea aún peor que en la actualidad, y se prevé que los eventos climáticos extremos sean más frecuentes y con mayor gravedad [2].

Además, la demanda de capacidad de carga a menudo aumenta con el tiempo, por ejemplo, debido al uso de camiones más pesados para el transporte de madera en el norte de Europa y América del Norte. Por lo tanto, existe una necesidad creciente de métodos confiables para evaluar la resistencia estructural de la red de transporte en condiciones climáticas extremas que tengan en cuenta los escenarios futuros de cambio climático.

Los activos de transporte por carretera se diseñan, construyen y explotan basándose en numerosas fuentes de datos y varios modelos. Por lo tanto, los ingenieros de diseño usan modelos establecidos proporcionados por las normas; ingenieros de construccion
documentar los datos en el material real y proporcionar planos según lo construido; los operadores recopilan datos sobre el tráfico, realizan inspecciones y planifican el mantenimiento; los científicos del clima combinan datos y modelos climáticos para
predecir eventos climáticos futuros, y los ingenieros de evaluación calculan el impacto de la carga climática extrema en la estructura.

Dadas las fuentes abrumadoras y la complejidad de los datos y modelos, es posible que la información y los cálculos actualizados no estén disponibles para decisiones cruciales, por ejemplo, con respecto a la seguridad estructural y la operabilidad de la infraestructura durante episodios de eventos extremos. La falta de una integración perfecta entre los datos de la infraestructura, los modelos estructurales y la toma de decisiones a nivel del sistema es una limitación importante de las soluciones actuales, lo que conduce a la inadaptación e incertidumbre y crea costos e ineficiencias.

El gemelo digital estructural de la infraestructura es una simulación estructural viva que reúne todos los datos y modelos y se actualiza desde múltiples fuentes para representar su contraparte física. El Digital Twin estructural, mantenido durante todo el ciclo de vida de un activo y fácilmente accesible en cualquier momento, proporciona al propietario/usuarios de la infraestructura una idea temprana de los riesgos potenciales para la movilidad inducidos por eventos climáticos, cargas de vehículos pesados e incluso el envejecimiento de un infraestructura de transporte.

En un proyecto en curso, estamos desarrollando e implementando un gemelo digital estructural para el puente de Kalix en Suecia. El objetivo general del presente artículo es presentar un método y estudiar los resultados de la cuantificación de las cargas estructurales resultantes de eventos climáticos extremos basados en escenarios climáticos futuros para el puente de Kalix. El puente de Kalix, construido sobre el río Kalix en Suecia en 1956, está hecho de una viga cajón de hormigón postensado. El puente se utiliza como banco de pruebas para la demostración de métodos de evaluación y control de la salud estructural (SHM) de última generación.

El objetivo específico de la investigación actual es dar cuenta de parámetros climáticos como el viento y el flujo de agua, que imponen cargas estáticas y dinámicas en las estructuras. Nuestro método, en el primer paso, consiste en simulaciones de flujo de viento y simulaciones de flujo de agua utilizando un modelado CFD transitorio basado en el modelo de turbulencia LES/DES para cuantificar las cargas de viento e hidráulicas; esto constituye el punto focal principal de este artículo.

En el siguiente paso, se estudiará la respuesta estructural del puente mediante la transformación de los perfiles de carga eólica e hidráulica en cargas estructurales en el análisis de EF estructural no lineal. Por último, el modelo estructural se actualizará incorporando sin problemas los datos del SHM y, por lo tanto, creando un gemelo digital estructural que refleje la verdadera respuesta de la estructura. Los dos primeros enfoques de investigación permanecen fuera del alcance inmediato del presente artículo.

2. Descripción del puente de Kalix

El puente de Kalix consta de 5 vanos largos de los cuales el más largo tiene unos 94 metros y el más corto 43,85 m. El puente es de hormigón postensado, el cual se cuela in situ de forma segmentaria y una viga cajón no prismática como se muestra en la Fig. 1. El puente es simétrico en geometría y hay una bisagra en el punto medio. El ancho del tablero del puente en la losa superior e inferior es de aproximadamente 13 my 7,5 m, respectivamente. El espesor del muro es de 45 cm y el espesor de la losa inferior varía de 20 cm a
50 cm.

Fig. 1. Geometría y secciones del puente

3. Simulación de viento

Las pruebas en túnel de viento solían ser la única forma de examinar la reacción de los puentes a las cargas de viento Consulte [3]; sin embargo, estos experimentos requieren mucho tiempo y son costosos. Se requieren cerca de 6 a 8 semanas para realizar una prueba típica en un túnel de viento Consulte [4]. Los últimos logros en la capacidad computacional de las computadoras brindan oportunidades para la simulación práctica del viento alrededor de puentes utilizando la dinámica de fluidos computacional (CFD).

Es beneficioso investigar la presión del viento en los componentes del puente utilizando una simulación por computadora. Es necesario determinar los parámetros de simulación del puente y el campo de viento a su alrededor; por lo tanto, se pueden evaluar con precisión sus impactos en las fuerzas aplicadas en el puente.

Las demandas de diseño de las estructuras de puentes requieren una investigación rigurosa de la acción del viento, especialmente en condiciones climáticas extremas. Garantizar la estabilidad de los puentes de grandes luces, ya que sus características y formaciones son más propensas a la carga de viento, se encuentra entre las principales consideraciones de diseño [3].

3.1. Parámetros de simulación

La velocidad básica del viento se elige 22 m/s según el mapa de viento de Suecia y la ubicación del puente de Kalix según EN 1991-1-4 [5] y el código sueco BFS 2019: 1 EKS 11; ver figura 1. La superficie libre sobre el agua se considera un área expuesta a la carga de viento. La dirección del ataque del viento dominante se considera perpendicular al tablero del puente.

Las simulaciones actuales se basan en tres escenarios que incluyen: viento extremo, frío extremo y valor de diseño para un período de retorno de 3000 años. Cada condición tiene diferentes valores de temperatura, viento básico
velocidad, viscosidad cinemática y densidad del aire, como se muestra en la Tabla 1. Los conjuntos de datos meteorológicos se sintetizaron para dos semanas meteorológicas extremas durante el período de 30 años de 2040-2069, considerando 13 escenarios climáticos futuros diferentes con diferentes modelos climáticos globales (GCM) y rutas de concentración representativas (RCP).

Se seleccionaron una semana de frío extremo y una semana de viento extremo utilizando el enfoque desarrollado
de Nik [7]. El planteamiento se adaptó a las necesidades de este trabajo, considerando el horario semanal en lugar de mensual. Se ha verificado la aplicación del enfoque para simulaciones complejas, incluidos los sistemas de energía Consulte [7] Consulte [8], hidrotermal Consulte [ 9] y simulaciones de microclimas Consulte [10].

Para considerar las condiciones climáticas extremas de una infraestructura muy importante, el valor de la velocidad básica del viento debe transferirse del período de retorno de 50 años a 3000 años como se indica en la ecuación 1 [6]. El perfil de velocidad y turbulencia se crea en base a EN 1991-1-4 [5] para la categoría de terreno 0 (Z₀ = 0,003 my Z_mín = 1 m), donde Z₀ y Z_mín son la longitud de rugosidad y la altura mínima, respectivamente. La variación de la velocidad del viento con la altura se define en la ecuación 2, donde c_o (z) es el factor de orografía tomado como 1, v_m (z) es la velocidad media del viento a la altura z, k_r es el factor del terreno que depende de la longitud de la rugosidad , e I_v (z) es la intensidad de la turbulencia; ver ecuación 3.��50=[0.36+0.1ln12�] 1��=��·ln��0·�� [2]��=��=�1�0�·ln�/�0 �� ≤�≤�� [3]��=�� <�� [4]

Velocidad del viento, variación de la velocidad del viento con la altura, intensidad de la turbulencia

Se calcula que el valor de la velocidad del viento para T = período de retorno de 3000 años es de 31 m/s; por lo tanto, los diagramas de velocidad del viento e intensidad de turbulencia se obtienen como se muestra en la figura 2.

Tabla. 1. Información meteorológica para tres escenarios

Fig. 2. Valor de cálculo para la información del periodo de retorno de 3000 años: (a) Velocidad del viento y (b) Perfil de intensidad de turbulencia, y (c) Especificaciones del modelo

Fig. 2. Valor de cálculo para la información del periodo de retorno de 3000 años: (a) Velocidad del viento y (b) Intensidad de la turbulencia perfil, y (c) Especificaciones del modelo

3.2. Modelo de turbulencia

Para que las investigaciones sean precisas en el flujo alrededor de estructuras importantes como puentes, se aplica un enfoque híbrido que incluye simulaciones de remolinos desprendidos retardados (DDES) y es computacionalmente eficiente [11] [12]. Este modelo de turbulencia usa un método RANS cerca de las capas límite y el método LES lejos de las capas límite y en el área del flujo de la región separada ‘.

En el primer paso, el enfoque de simulación de remolinos separados se ha ampliado para adquirir predicciones de fuerza fiables en los modelos con un gran impacto del flujo separado. Hay varios ejemplos en la parte de revisión de Spalart Consulte [11] para varios casos que usan la aplicación del modelo de turbulencia de simulación de remolino separado (DES).

La formulación DES inicial [13] se desarrolla utilizando el enfoque de Spalart-Allmaras. Con respecto a la transición del enfoque RANS al LES, se revisa el término de destrucción en la ecuación de transporte de viscosidad modificada: la distancia entre un punto en el dominio y la superficie sólida más cercana (d) se sustituye por el factor introducido por:�~=��(�.��·∆)

Factor que sustituye la distancia entre el punto en el dominio y la superficie sólida más cercana (d)

donde C_DES es un coeficiente, se considera como 0,65 y Δ es una escala de longitud asociada con el espaciado de la rejilla local:�=��(��.��.��)

Escala de longitud asociada con el espaciado de rejilla local

Se ha empleado un enfoque modificado de DES, conocido como simulación de remolinos desprendidos retardados (DDES), para dominar el probable problema de la “separación inducida por la rejilla” (GIS) que está relacionado con la geometría de la rejilla. El objetivo de este nuevo enfoque es confirmar que el modelado de turbulencia se mantiene en modo RANS en todas las capas de contorno [14]. Por lo tanto, la definición del parámetro se modifica como se define:�~=�-��(0. �-��·�) 6

Modificación del parámetro d

donde f_d es una función de filtro que considera un valor de 0 en las capas límite cercanas al muro (zona RANS) y un valor de 1 en las áreas donde se realizó la separación del flujo (zona LES).

3.3. Rejilla computacional y resultados

RWIND 2.01 Pro se emplea para la simulación de viento CFD, que usa el código CFD externo OpenFOAM® versión 17.10. La simulación CFD tridimensional se realiza como una simulación de viento transitorio para flujo turbulento incompresible utilizando el algoritmo SIMPLE (Método semi-implícito para ecuaciones vinculadas a presión).

En la simulación actual, el solucionador de estado estacionario se considera como la condición inicial, lo que significa que cuando se está calculando el flujo transitorio, el cálculo del estado estacionario de la condición inicial comienza en la primera parte de la simulación y tan pronto como se calcula. completado, el cálculo de transitorios se iniciará automáticamente.

Fig. 3. Dominio del túnel de viento y rejilla computacional de referencia (8.057.279 celdas)

La cuadrícula computacional se realiza mediante 8.057.279 celdas tridimensionales y 8.820.901 nudos, también se consideran las dimensiones del dominio del túnel de viento 2000 m * 1000 m * 100 m (largo, ancho, alto) como se muestra en la figura 3. El volumen mínimo de la celda es de 6,34 * 10-5 m3, el volumen máximo es de 812,30 m3 y la desviación máxima es de 1,80.

La presión residual final se considera 5 * 10-5. El proceso de generación de mallas e independencia de la rejilla se ha realizado utilizando los cuatro tamaños de malla que se muestran en la figura 4 para la malla de referencia, y finalmente se ha conseguido la independencia de la rejilla.

Fig. 4. Estudio de rejilla de cuatro tamaños de malla computacional a través de la línea de sondeo.

Se han realizado tres simulaciones para obtener el valor de la presión del viento para condiciones climáticas extremas y el valor de cálculo del viento que se muestra en la Fig. 5. Para cada escenario, el resultado de la presión del viento se obtiene utilizando el modelo de turbulencia transitoria DDES con respecto a 30 (s) de duración que incluye 60 capas de tiempo (Δt = 0,5 s).

Se puede observar que el área frontal del puente está expuesta a la presión del viento positiva y la cantidad de presión aumenta en la altura cerca del borde del tablero para todos los escenarios. Además, la Fig. 5. ilustra los valores negativos de la presión del viento en su totalidad en la superficie de la cubierta. El valor de pertenencia para el período de 3000 años es mucho más alto que los otros escenarios.

Es importante tener en cuenta que el intervalo de la velocidad del viento de entrada tiene un gran impacto en el valor de la presión en la superficie más que en los otros parámetros. Además, para cada escenario, el intervalo más alto de presión del viento y succión durante el tiempo total debe considerarse como una carga de viento crítica impuesta a la estructura. El valor más bajo de la presión en la superficie se obtiene en el escenario de condiciones de frío extremo, mientras que en condiciones de mucho viento, el valor de la presión se vuelve un orden de magnitud más alto.

Fig. 5. Contorno de presión superficial y diagrama para 60 capas de tiempo (Δt = 0.5 s) a través de una línea de sondeo para tres escenarios.

Además, es importante tener en cuenta que el comportamiento del puente sería completamente diferente debido a las diferentes temperaturas del aire, y puede ocurrir un posible caso crítico en el escenario que experimente una presión menor. Con respecto al valor de entrada de cada escenario, el rango más alto de presión del viento pertenece al nivel de diseño debido al período de retorno de 3000 años, que ha recibido la velocidad del viento más alta como velocidad de entrada.

4. Simulación hidráulica

Los pilares de los puentes a través del río pueden bloquear el flujo al reducir la sección transversal del río, crear corrientes parásitas locales y cambiar la velocidad del flujo, lo que puede ejercer presión en las superficies de los pilares. Cuando el río fluye hacia los pilares del puente, el proceso del flujo de agua alrededor de la base se puede dividir en dos partes: aplicando presión en el momento en que el agua golpea el pilar del puente y después de la presión inicial cuando el agua fluye alrededor de los pilares [15].

Cuando el agua alcanza los pilares del puente a una cierta velocidad, el efecto de la presión sobre los pilares es mucho mayor que la presión del fluido que queda a su alrededor. Debido a los desarrollos de la ciencia de la computación, así como al desarrollo cada vez mayor de los códigos dinámicos de fluidos computacionales, se han utilizado ampliamente varias simulaciones numéricas y se ha demostrado que los resultados de muchas simulaciones son consistentes con los resultados experimentales [16].

Por ello, en esta investigación se ha utilizado el método de la dinámica de fluidos computacional para simular los fenómenos que gobiernan el comportamiento del flujo de los ríos. Para este estudio se ha seleccionado una solución tridimensional basada en cálculos numéricos utilizando el modelo de turbulencia LES. La simulación tridimensional del flujo del río en diferentes direcciones y velocidades nos permite calcular y analizar todas las presiones en la superficie de los pilares del puente en diferentes intervalos de tiempo.

4.1. Parámetros de simulación

El flujo del río se puede definir como un flujo de dos fases, que incluye agua y aire, en un canal abierto. El flujo de canal abierto es un flujo de fluido con una superficie libre en la que la presión atmosférica se distribuye uniformemente y se crea por el peso del fluido. Para simular este tipo de flujo se utiliza el método multifase VOF.

El programa Flow3D, disponible en el mercado, utiliza los métodos de fracciones volumétricas VOF y FAVOF. En el método VOF, el dominio de modelado se divide primero en celdas de elementos o volúmenes de controles más pequeños. Para los elementos que contienen fluidos, se mantienen valores numéricos para cada una de las variables de flujo dentro de ellos.

Estos valores representan la media volumétrica de los valores en cada elemento. En las corrientes superficiales libres, no todas las celdas están llenas de líquido; algunas celdas en la superficie de flujo están medio llenas. En este caso, se define una cantidad llamada volumen de fluido, F, que representa la parte de la celda que se llena con el fluido.

Después de determinar la posición y el ángulo de la superficie del flujo, será posible aplicar las condiciones de contorno apropiadas en la superficie del flujo para calcular el movimiento del fluido. A medida que se mueve el fluido, el valor de F también cambia con él. Las superficies libres son monitoreadas automáticamente por el movimiento de fluido dentro de una red fija. El método FAVOR se usa para determinar la geometría.

También se puede usar otra cantidad de fracción volumétrica para determinar el nivel de un cuerpo rígido desocupado ( V_f ). Cuando se conoce el volumen que ocupa el cuerpo rígido en cada celda, el límite del fluido dentro de la red fija se puede determinar como VOF. Este límite se usa para determinar las condiciones de contorno del muro que sigue el arroyo. En general, la ecuación de continuidad de masa es la siguiente:��𝜕�𝜕�+𝜕𝜕�(��)+�𝜕𝜕�(��)+𝜕𝜕�(��)+��=�� 10

Ecuación de continuidad de masa

Las ecuaciones de movimiento para los componentes de la velocidad de un fluido en coordenadas 3D, o en otras palabras, las ecuaciones de Navier-Stokes, son las siguientes:𝜕�𝜕�+1��𝜕�𝜕�+��𝜕�𝜕�+��𝜕�𝜕�+��2��=-1�𝜕�𝜕�+��+��-��-��-��-�� 11𝜕�𝜕�+1��𝜕�𝜕�+��𝜕�𝜕�+��𝜕�𝜕�+��=-�1�𝜕�𝜕�+��+��-��-��-��-�� 12𝜕�𝜕�+1��𝜕�𝜕�+��𝜕�𝜕�+��𝜕�𝜕�=-1�𝜕�𝜕�+��+��-��-��-��-�� 13

Ecuaciones de Navier-Stokes

Donde V_F es la relación del volumen abierto al flujo, ρ es la densidad del fluido, (u, v, w) son las componentes de la velocidad en las direcciones x, y y z, respectivamente, R _SOR es la función de la fuente, (A_x, A_y, A_z ) son las áreas fraccionales, (G_x, G_y, G_z ) son las fuerzas gravitacionales, (f_x, f_y, f_z ) son las aceleraciones de la viscosidad y (b_x, b_y, b_z ) son las pérdidas de flujo en medios porosos en las direcciones x, y, z, respectivamente [17].

La zona de captación del río Kalix es grande y amplia, por lo que tiene un clima subpolar con inviernos fríos y largos y veranos suaves y cortos. Aproximadamente el 50% de las precipitaciones en esta zona es nieve. En mayo, por lo general, el deshielo provoca un aumento significativo en el caudal del río. Las condiciones climáticas del río se resumen en la Tabla 2, [18].

Contrariamente a la tendencia general de este estudio, la previsión de las condiciones meteorológicas mencionadas está utilizando la información meteorológica registrada en los períodos pasados. En función de la información meteorológica disponible, definimos las condiciones de contorno al realizar los cálculos.

Tabla 2: Parámetros del modelo y tabla 3:Condiciones de contorno del modelo

4.2 Cuadrícula computacional y resultados

Primero, según las dimensiones de los pilares en tres direcciones X, Y, Z, y según la dimensión longitudinal de los pilares (D = 8,5 m; véase la figura 7), el dominio se extiende 10D aguas arriba y 20D aguas abajo. Se ha utilizado el método de mallado estructurado (cartesiano) y el software Flow3D para resolver este problema. Para una cuadrícula correcta, el dominio se debe dividir en diferentes secciones.

Esta división se basa en lugares con fuertes pendientes. Usando la creación de una nueva superficie, el dominio se puede dividir en varias secciones para crear una malla regular con las dimensiones correctas y apropiadas, se puede especificar el número de celdas en cada superficie.

Fig. 6: Estudio de rejilla para el dominio hídrico

Esto aumenta el volumen final de las células. Por esta razón, hemos dividido este dominio en tres niveles: Grueso, medio y fino. Los resultados de los estudios de independencia de la red se muestran en la figura 6. Para comprobar los resultados calculados, primero debemos asegurarnos de que la corriente de entrada sea la correcta. Para hacer esto, el caudal de entrada se mide en el dominio de la solución y se compara con el valor base. Las dimensiones del dominio de la solución se especifican en la figura 7. Esta figura también contribuye al reconocimiento de los pilares del puente y su denominación de superficies.

Como se muestra en la Fig. 8, el caudal del río se encuentra dentro del intervalo admisible durante el 90% del tiempo de simulación y el caudal de entrada se ha simulado correctamente. Además, en la Fig. 9, la velocidad media del río se calcula en función del caudal y del área de la sección transversal del río.

Para extraer la cantidad de presión aplicada a los diferentes lados de las columnas, hemos seleccionado el intervalo de tiempo de simulación de 10 a 25 segundos (tiempo de estabilización de descarga en la cantidad de 1800 metros cúbicos por segundo). Los resultados calculados para cada lado se muestran en la Fig. 10 y 11. Los contornos de velocidad también se muestran en las Figuras 12 y 13. Estos contornos se ajustan en función de la velocidad del fluido en un momento dado.

Debido a las dimensiones del dominio de la solución y al caudal del río, el flujo de agua llega a los pilares del puente en el décimo segundo y la presión inicial del flujo del río afecta las superficies de los pilares del puente. Esta presión inicial decrece con el tiempo y se estabiliza en un rango determinado para cada lado según el área y el porcentaje de interacción con el flujo. Para los cálculos de interacción fluido-estructura (FSI), se puede usar la presión crítica calculada en el momento en que la corriente golpea los pilares.

Fig. 7: Dibujo del dominio hidrostático

Fig. 8: caudal del río; La figura 9: Caudal de la velocidad del río; La figura 10: Presión en la pila del puente – I; La figura 11: Presión en la pila del puente – II

Fig. 12: Contador de velocidad en el tiempo: 30s Fig. 13: Contador de velocidad en el tiempo: 20 s

5. Conclusión

Los efectos de las condiciones meteorológicas extremas, incluido el viento dinámico y el flujo de agua, se investigaron numéricamente para el puente de Kalix. Se definieron tres escenarios para las simulaciones dinámicas de viento, incluido el clima con mucho viento, el clima extremadamente frío y el valor de diseño para un período de retorno de 3.000 años. Aprovechando las simulaciones CFD, se determinaron las presiones del viento en pasos de 60 tiempos (30 segundos) utilizando el modelo de turbulencia transitoria DDES.

Los resultados indican diferencias significativas entre los escenarios, lo que implica la importancia de los datos de entrada, especialmente el diagrama de velocidades del viento. Se observó que el valor de diseño para el período de devolución de 3000 años tiene un impacto mucho mayor que los otros escenarios. Además, se mostró la importancia de considerar el rango más alto de presión del viento en la superficie a través de los pasos de tiempo para evaluar el comportamiento estructural del puente en la condición más crítica.

Además, se consideró el caudal máximo del río para una simulación transitoria según las condiciones meteorológicas registradas, y los pilares del puente se sometieron al caudal máximo del río durante 30 segundos. Por lo tanto, además de las condiciones físicas del flujo del río y cómo cambia la dirección del flujo aguas abajo, se cuantificaron las presiones máximas del agua en el momento en que el flujo golpea los pilares.

En el trabajo futuro, el rendimiento estructural del puente de Kalix será evaluado por
imposición de la carga del viento, la presión del agua y la carga del tráfico, creando así un gemelo digital estructural que refleja la verdadera respuesta de la estructura.

6. Reconocimiento

Los autores agradecen enormemente el apoyo de Dlubal Software por proporcionar la licencia de RWIND Simulation, así como de Flow Sciences Inc. por proporcionar la licencia de FLOW-3D.

Autores: Mahyar Kazemian¹, Sajad Nikdel², Mehrnaz MohammadEsmaeili³, Vahid Nik⁴, Kamyab Zandi^*5

¹Candidato a doctorado, becario en el Departamento de Ingeniería de Timezyx Inc., Canadá.

²M.Sc. estudiante, pasante en el Departamento de Ingeniería, Timezyx Inc., Canadá.

³Estudiante de licenciatura, pasante en el Departamento de Ingeniería, Timezyx Inc., Canadá.

⁴ Profesor adjunto en la división de Física de la construcción de la Universidad de Lund y la Universidad Tecnológica de Chalmers, Suecia.

^{* 5} Director, Timezyx Inc., Vancouver, BC V6N 2R2, Canadá. E-mail: kamyab.zandi@timezyx.com

Referencias

Jančula, M., Jošt, J., & Gocál, J. (2021). Influencia de las acciones ambientales agresivas en las estructuras de los puentes. Transportation Research Procedia, 55 , 1229–1235. https://doi.org/10.1016/j.trpro.2021.07.104
Wang, X., Nguyen, M., Stewart, MG, Syme, M. y Leitch, A. (2010). Análisis de los impactos del cambio climático en el deterioro de la infraestructura de hormigón – Informe de síntesis. CSIRO, Canberra.
Kemayou, BTM (2016). Análisis de secciones de tableros de puentes por el método de la pseudocompresibilidad basado en FDM y LES: Mejora del rendimiento mediante la implementación de la computación en paralelo (tesis). Universidad de Arkansas.
Larsen, A. y Walther, JH (1997). Análisis aeroelástico de secciones de vigas de puentes basado en simulaciones discretas de vórtices. Journal of Wind Engineering and Industrial Aerodynamics, 67–68 , 253–265. https://doi.org/10.1016/s0167-6105(97)00077-9
Eurocódigo 1: Acciones en estructuras. (2006). Instituto Británico de Normas.
ASCE. Cargas mínimas de cálculo para edificios y otras estructuras. (2013). Sociedad Estadounidense de Ingenieros Civiles.
Nik, VM (2016). Facilitación de la simulación energética para el clima futuro: síntesis de conjuntos de datos meteorológicos típicos y extremos a partir de modelos climáticos regionales (RCM). Applied Energy, 177 , 204–226. https://doi.org/10.1016/j.apenergy.2016.05.107
Perera, AT, Nik, VM, Chen, D., Scartezzini, J.‑L. y Hong, T. (2020). Cuantificación de los impactos del cambio climático y los eventos climáticos extremos en los sistemas energéticos. Nature Energy, 5 (2), 150–159. https://doi.org/10.1038/s41560-020-0558-0
Nik, VM (2017). Aplicación de conjuntos de datos meteorológicos típicos y extremos en la simulación higrotérmica de componentes de construcción para el clima futuro: un estudio de caso para un muro de entramado de madera. Energy and Buildings, 154 , 30–45. https://doi.org/10.1016/j.enbuild.2017.08.042
Hosseini, M., Javanroodi, K. y Nik, VM (2022). Evaluación de impacto de alta resolución del cambio climático en el rendimiento energético de los edificios considerando los eventos meteorológicos extremos y el microclima – Investigando las variaciones en el confort térmico interior y los grados-día. Ciudades sostenibles y sociedad, 78 , 103634. https://doi.org/10.1016/j.scs.2021.103634
Spalart, P. R. (2009). Simulación de remolinos separados. Revisión anual de mecánica de fluidos, 41 , 181–202. https://doi.org/10.1146/annurev.fluid.010908.165130
Spalart, PR, et al. (2006) Una nueva versión de simulación de remolinos separados, resistente a densidades de rejilla ambiguas. Dinámica de fluidos teórica y computacional, 2006. 20 (3), 181-195. https://doi.org/10.1007/s00162-006-0015-0
Spalart, PR (1997). Comentarios sobre la viabilidad de LES para alas y sobre una aproximación híbrida RANS/LES. En Actas de la Primera Conferencia Internacional de AFOSR sobre DNS/LES. Prensa de Greyden.
Boudreau, M., Dumas, G. y Veilleux, J.-C. (2017). Evaluación de la capacidad del enfoque de modelado de turbulencia DDES para simular la estela de un cuerpo de farol. Aeroespacial, 4 (3), 41. https://doi.org/10.3390/aerospace4030041
Wang, Y., Zou, Y., Xu, L. y Luo, Z. (2015). Análisis de la presión del flujo de agua en pilas de puentes considerando el efecto del impacto. Problemas matemáticos en ingeniería, 2015 , 1–8. https://doi.org/10.1155/2015/687535
Qi, H., Zheng, J. y Zhang, C. (2020). Simulación numérica del campo de velocidades alrededor de dos pilares de pilas en tándem del puente longitudinal. Fluidos, 5 (1), 32. https://doi.org/10.3390/fluids5010032
Jalal, H. K. y Hassan, W. H (2020). Simulación numérica tridimensional de la socavación local alrededor de la pila de un puente circular utilizando el software flow-3d. Ciclo de conferencias de IOP: Ciencia e ingeniería de materiales, 745 , 012150. https://doi.org/10.1088/1757-899x/745/1/012150
Herzog, S. D., Conrad, S., Ingri, J., Persson, P. y Kritzberg, E. S (2019). Cambios inducidos por crecidas de primavera en la especiación y destino del Fe a mayor salinidad. Geoquímica aplicada, 109 , 104385. https://doi.org/10.1016/j.apgeochem.2019.104385

원본 보기

NUMERICAL ANALYSIS OF THE HYDRODYNAMICS CHARACTERISTICS OF TORPEDO ANCHOR INSTALLATION UNDER THE INFLUENCE OF OCEAN CURRENTS

魚雷錨擲錨過程受海流擲下之運移特性數值分析

번역된 기고 제목: 해류의 영향에 따른 어뢰 앵커 설치의 유체 역학 특성에 대한 수치 분석

Translated title of the contribution: NUMERICAL ANALYSIS OF THE HYDRODYNAMICS CHARACTERISTICS OF TORPEDO ANCHOR INSTALLATION UNDER THE INFLUENCE OF OCEAN CURRENTS

L. Y. Chen, R. Y. Yang

Department of Hydraulic and Ocean Engineering

Abstract

The gravity-installed anchor (GIA) is a type of the anchor foundation that is installed by penetrating the seabed using the weight of the anchor body. It has the advantages of high installation efficiency, low cost, and no requirement of additional installation facilities. The GIA type used in this study is the torpedo anchor, which has been ap-plied in practical cases widely. The purpose of this study is to investigate the numerical analysis of the anchor trans-porting during the installation of the torpedo anchor under the action of ocean currents. Therefore, this article con-siders external environmental conditions and the different forms of torpedo anchors by using computational fluid dynamics (CFD) software, FLOW-3D, to simulate the fluid-solid interaction effect on the torpedo anchor. The falling time, impact velocity, displaced angle, and horizontal displacement of the torpedo anchor were observed at an installation height (i.e., the distance between the seabed and the anchor release height) of 85 meters. The obtained results show that when the current velocity is greater, the torpedo anchor will have a larger displaced angle, which will affect the impact velocity of the anchor on the seabed and may cause insufficient penetration depth, leading to installation failure.

중력설치형 앵커(GIA)는 앵커 본체의 무게를 이용하여 해저를 관통하여 설치하는 앵커 기초의 일종이다. 설치 효율성이 높고, 비용이 저렴하며, 추가 설치 시설이 필요하지 않다는 장점이 있습니다. 본 연구에서 사용된 GIA 유형은 어뢰앵커로 실제 사례에 널리 적용되어 왔다.

본 연구의 목적은 해류의 작용에 따라 어뢰앵커 설치 시 앵커 이송에 대한 수치해석을 연구하는 것이다. 따라서 이 기사에서는 어뢰 앵커에 대한 유체-고체 상호 작용 효과를 시뮬레이션하기 위해 전산유체역학(CFD) 소프트웨어인 FLOW-3D를 사용하여 외부 환경 조건과 다양한 형태의 어뢰 앵커를 고려합니다.

어뢰앵커의 낙하시간, 충격속도, 변위각, 수평변위 등은 설치높이(즉, 해저와 앵커 해제 높이 사이의 거리) 85m에서 관찰되었다. 얻은 결과는 현재 속도가 더 높을 때 어뢰 앵커의 변위 각도가 더 커져 해저에 대한 앵커의 충격 속도에 영향을 미치고 침투 깊이가 부족하여 설치 실패로 이어질 수 있음을 보여줍니다.

Ocean currentsEngineering & Materials Science100%
AnchorsEngineering & Materials Science74%
Numerical analysisEngineering & Materials Science63%
HydrodynamicsEngineering & Materials Science62%
GravitationEngineering & Materials Science9%
Computational fluid dynamicsEngineering & Materials Science4%
FluidsEngineering & Materials Science3%
CostsEngineering & Materials Science
해류
앵커
수치해석
유체 역학
중력
전산유체역학

원본 바로가기

Influences of the Powder Size and Process Parameters on the Quasi-Stability of Molten Pool Shape in Powder Bed Fusion-Laser Beam of Molybdenum

몰리브덴 분말층 융합-레이저 빔의 용융 풀 형태의 준안정성에 대한 분말 크기 및 공정 매개변수의 영향

Abstract

Formation of a quasi-steady molten pool is one of the necessary conditions for achieving excellent quality in many laser processes. The influences of distribution characteristics of powder sizes on quasi-stability of the molten pool shape during single-track powder bed fusion-laser beam (PBF-LB) of molybdenum and the underlying mechanism were investigated.

The feasibility of improving quasi-stability of the molten pool shape by increasing the laser energy conduction effect and preheating was explored. Results show that an increase in the range of powder sizes does not significantly influence the average laser energy conduction effect in PBF-LB process. Whereas, it intensifies fluctuations of the transient laser energy conduction effect.

It also leads to fluctuations of the replenishment rate of metals, difficulty in formation of the quasi-steady molten pool, and increased probability of incomplete fusion and pores defects. As the laser power rises, the laser energy conduction effect increases, which improves the quasi-stability of the molten pool shape. When increasing the laser scanning speed, the laser energy conduction effect grows.

However, because the molten pool size reduces due to the decreased heat input, the replenishment rate of metals of the molten pool fluctuates more obviously and the quasi-stability of the molten pool shape gets worse. On the whole, the laser energy conduction effect in the PBF-LB process of Mo is low (20-40%). The main factor that affects quasi-stability of the molten pool shape is the amount of energy input per unit length of the scanning path, rather than the laser energy conduction effect.

Moreover, substrate preheating can not only enlarge the molten pool size, particularly the length, but also reduce non-uniformity and discontinuity of surface morphologies of clad metals and inhibit incomplete fusion and pores defects.

준안정 용융 풀의 형성은 많은 레이저 공정에서 우수한 품질을 달성하는 데 필요한 조건 중 하나입니다. 몰리브덴의 단일 트랙 분말층 융합 레이저 빔(PBF-LB) 동안 용융 풀 형태의 준안정성에 대한 분말 크기 분포 특성의 영향과 그 기본 메커니즘을 조사했습니다.

레이저 에너지 전도 효과와 예열을 증가시켜 용융 풀 형태의 준안정성을 향상시키는 타당성을 조사했습니다. 결과는 분말 크기 범위의 증가가 PBF-LB 공정의 평균 레이저 에너지 전도 효과에 큰 영향을 미치지 않음을 보여줍니다. 반면, 과도 레이저 에너지 전도 효과의 변동이 강화됩니다.

이는 또한 금속 보충 속도의 변동, 준안정 용융 풀 형성의 어려움, 불완전 융합 및 기공 결함 가능성 증가로 이어집니다. 레이저 출력이 증가함에 따라 레이저 에너지 전도 효과가 증가하여 용융 풀 모양의 준 안정성이 향상됩니다. 레이저 스캐닝 속도를 높이면 레이저 에너지 전도 효과가 커집니다.

그러나 열 입력 감소로 인해 용융 풀 크기가 줄어들기 때문에 용융 풀의 금속 보충 속도의 변동이 더욱 뚜렷해지고 용융 풀 형태의 준안정성이 악화됩니다.

전체적으로 Mo의 PBF-LB 공정에서 레이저 에너지 전도 효과는 낮다(20~40%). 용융 풀 형상의 준안정성에 영향을 미치는 주요 요인은 레이저 에너지 전도 효과보다는 스캐닝 경로의 단위 길이당 입력되는 에너지의 양입니다.

또한 기판 예열은 용융 풀 크기, 특히 길이를 확대할 수 있을 뿐만 아니라 클래드 금속 표면 형태의 불균일성과 불연속성을 줄이고 불완전한 융합 및 기공 결함을 억제합니다.

Fig. 1View in article
Fig. 2View in article
Fig. 3View in article
Fig. 4View in article
Fig. 5View in article
Fig. 6View in article
Fig. 7View in article
Fig. 8View in article
Fig. 9View in article
Fig. 10View in article
Fig. 11View in article
Fig. 12View in article
Fig. 13View in article
Fig. 14View in article
Fig. 15View in article
Fig. 16View in article
Fig. 17View in article
Fig. 18View in article
Fig. 19View in article
Fig. 20View in article

References

M. Sharifitabar, F.O. Sadeq, and M.S. Afarani, Synthesis and Kinetic Study of Mo (Si, Al)2 Coatings on the Surface of Molybdenum Through Hot Dipping into a Commercial Al-12 wt.% Si Alloy Melt, Surf. Interfaces, 2021, 24, p 101044.Article CAS Google Scholar
Z. Zhang, X. Li, and H. Dong, Response of a Molybdenum Alloy to Plasma Nitriding, Int. J. Refract. Met. Hard Mater., 2018, 72, p 388–395.Article CAS Google Scholar
C. Tan, K. Zhou, M. Kuang, W. Ma, and T. Kuang, Microstructural Characterization and Properties of Selective Laser Melted Maraging Steel with Different Build Directions, Sci. Technol. Adv. Mater., 2018, 19(1), p 746–758.Article CAS Google Scholar
C. Tan, F. Weng, S. Sui, Y. Chew, and G. Bi, Progress and Perspectives in Laser Additive Manufacturing of Key Aeroengine Materials, Int. J. Mach. Tools Manuf, 2021, 170, p 103804.Article Google Scholar
S.A. Khairallah and A. Anderson, Mesoscopic Simulation Model of Selective Laser Melting of Stainless Steel Powder, J. Mater. Process. Technol., 2014, 214(11), p 2627–2636.Article CAS Google Scholar
S.A. Khairallah, A.T. Anderson, A. Rubenchik, and W.E. King, Laser Powder-Bed Fusion Additive Manufacturing: Physics of Complex Melt Flow and Formation Mechanisms of Pores, Spatter, and Denudation Zones, Acta Mater., 2016, 108, p 36–45.Article CAS ADS Google Scholar
K.Q. Le, C. Tang, and C.H. Wong, On the Study of Keyhole-Mode Melting in Selective Laser Melting Process, Int. J. Therm. Sci., 2019, 145, p 105992.Article Google Scholar
M. Bayat, A. Thanki, S. Mohanty, A. Witvrouw, S. Yang, J. Thorborg, N.S. Tiedje, and J.H. Hattel, Keyhole-Induced Porosities in Laser-Based Powder Bed Fusion (L-PBF) of Ti6Al4V: High-Fidelity Modelling and Experimental Validation, Addit. Manuf., 2019, 30, p 100835.CAS Google Scholar
B. Liu, G. Fang, L. Lei, and X. Yan, Predicting the Porosity Defects in Selective Laser Melting (SLM) by Molten Pool Geometry, Int. J. Mech. Sci., 2022, 228, p 107478.Article Google Scholar
W. Ge, J.Y.H. Fuh, and S.J. Na, Numerical Modelling of Keyhole Formation in Selective Laser Melting of Ti6Al4V, J. Manuf. Process., 2021, 62, p 646–654.Article Google Scholar
W. Ge, S. Han, S.J. Na, and J.Y.H. Fuh, Numerical Modelling of Surface Morphology in Selective Laser Melting, Comput. Mater. Sci., 2021, 186, p 110062.Article Google Scholar
Y.-C. Wu, C.-H. San, C.-H. Chang, H.-J. Lin, R. Marwan, S. Baba, and W.-S. Hwang, Numerical Modeling of Melt-Pool Behavior In Selective Laser Melting with Random Powder Distribution and Experimental Validation, J. Mater. Process. Technol., 2018, 254, p 72–78.Article Google Scholar
C. Tang, J.L. Tan, and C.H. Wong, A Numerical Investigation on the Physical Mechanisms of Single Track Defects in Selective Laser Melting, Int. J. Heat Mass Transf., 2018, 126, p 957–968.Article CAS Google Scholar
X. Zhou, X. Liu, D. Zhang, Z. Shen, and W. Liu, Balling Phenomena in Selective Laser Melted Tungsten, J. Mater. Process. Technol., 2015, 222, p 33–42.Article CAS Google Scholar
J.D.K. Monroy and J. Ciurana, Study of the Pore Formation on CoCrMo Alloys by Selective Laser Melting Manufacturing Process, Procedia Eng., 2013, 63, p 361–369.Article CAS Google Scholar
L. Kaserer, J. Braun, J. Stajkovic, K.H. Leitz, B. Tabernig, P. Singer, I. Letofsky-Papst, H. Kestler, and G. Leichtfried, Fully Dense and Crack Free Molybdenum Manufactured by Selective Laser Melting Through Alloying with Carbon, Int. J. Refract. Met. Hard Mater., 2019, 84, p 105000.Article CAS Google Scholar
T.B.T. Majumdar, E.M.C. Ribeiro, J.E. Frith, and N. Birbilis, Understanding the Effects of PBF Process Parameter Interplay on Ti-6Al-4V Surface Properties, PLoS ONE, 2019, 14, p e0221198.Article CAS PubMed PubMed Central Google Scholar
A.K.J.-R. Poulin, P. Terriault, and V. Brailovski, Long Fatigue Crack Propagation Behavior of Laser Powder Bed-Fused Inconel 625 with Intentionally- Seeded Porosity, Int. J. Fatigue, 2019, 127, p 144–156.Article CAS Google Scholar
P. Rebesan, M. Ballan, M. Bonesso, A. Campagnolo, S. Corradetti, R. Dima, C. Gennari, G.A. Longo, S. Mancin, M. Manzolaro, G. Meneghetti, A. Pepato, E. Visconti, and M. Vedani, Pure Molybdenum Manufactured by Laser Powder Bed Fusion: Thermal and Mechanical Characterization at Room and High Temperature, Addit. Manuf., 2021, 47, p 102277.CAS Google Scholar
D. Wang, C. Yu, J. Ma, W. Liu, and Z. Shen, Densification and Crack Suppression in Selective Laser Melting of Pure Molybdenum, Mater. Des., 2017, 129, p 44–52.Article CAS Google Scholar
K.-H. Leitz, P. Singer, A. Plankensteiner, B. Tabernig, H. Kestler, and L.S. Sigl, Multi-physical Simulation of Selective Laser Melting, Met. Powder Rep., 2017, 72, p 331–338.Article Google Scholar
D.G.J. Zhang, Y. Yang, H. Zhang, H. Chen, D. Dai, and K. Lin, Influence of Particle Size on Laser Absorption and Scanning Track Formation Mechanisms of Pure Tungsten Powder During Selective Laser Melting, Engineering, 2019, 5, p 736–745.Article CAS Google Scholar
L. Caprio, A.G. Demir, and B. Previtali, Influence of Pulsed and Continuous Wave Emission on Melting Efficiency in Selective Laser Melting, J. Mater. Process. Technol., 2019, 266, p 429–441.Article CAS Google Scholar
D. Gu, M. Xia, and D. Dai, On the Role of Powder Flow Behavior in Fluid Thermodynamics and Laser Processability of Ni-based Composites by Selective Laser Melting, Int. J. Mach. Tools Manuf, 2018, 137, p 67–78.Article Google Scholar
W.-I. Cho, S.-J. Na, C. Thomy, and F. Vollertsen, Numerical Simulation of Molten Pool Dynamics in High Power Disk Laser Welding, J. Mater. Process. Technol., 2012, 212(1), p 262–275.Article CAS Google Scholar
S.W. Han, J. Ahn, and S.J. Na, A Study on Ray Tracing Method for CFD Simulations of Laser Keyhole Welding: Progressive Search Method, Weld. World, 2016, 60, p 247–258.Article CAS Google Scholar
W. Ge, S. Han, Y. Fang, J. Cheon, and S.J. Na, Mechanism of Surface Morphology in Electron Beam Melting of Ti6Al4V Based on Computational Flow Patterns, Appl. Surf. Sci., 2017, 419, p 150–158.Article CAS ADS Google Scholar
W.-I. Cho, S.-J. Na, C. Thomy, and F. Vollertsen, Numerical Simulation of Molten Pool Dynamics in High Power Disk Laser Welding, J. Mater. Process. Technol., 2012, 212, p 262–275.Article CAS Google Scholar
W. Ma, J. Ning, L.-J. Zhang, and S.-J. Na, Regulation of Microstructures and Properties of Molybdenum-Silicon-Boron Alloy Subjected to Selective Laser Melting, J. Manuf. Process., 2021, 69, p 593–601.Article Google Scholar
S. Haeri, Y. Wang, O. Ghita, and J. Sun, Discrete Element Simulation and Experimental Study of Powder Spreading Process in Additive Manufacturing, Powder Technol., 2016, 306, p 45–54.Article Google Scholar
D. Yao, X. Liu, J. Wang, W. Fan, M. Li, H. Fu, H. Zhang, X. Yang, Q. Zou, and X. An, Numerical Insights on the Spreading of Practical 316 L Stainless Steel Powder in SLM Additive Manufacturing, Powder Technol., 2021, 390, p 197–208.Article CAS Google Scholar
S. Vock, B. Klöden, A. Kirchner, T. Weißgärber, and B. Kieback, Powders for Powder Bed Fusion: A Review, Prog. Addit. Manuf., 2019, 4, p 383–397.Article Google Scholar
X. Luo, C. Yang, Z.Q. Fu, L.H. Liu, H.Z. Lu, H.W. Ma, Z. Wang, D.D. Li, L.C. Zhang, and Y.Y. Li, Achieving Ultrahigh-Strength in Beta-Type Titanium Alloy by Controlling the Melt Pool Mode in Selective Laser Melting, Mater. Sci. Eng. A, 2021, 823, p 141731.Article CAS Google Scholar
J. Braun, L. Kaserer, J. Stajkovic, K.-H. Leitz, B. Tabernig, P. Singer, P. Leibenguth, C. Gspan, H. Kestler, and G. Leichtfried, Molybdenum and Tungsten Manufactured by Selective Laser Melting: Analysis of Defect Structure and Solidification Mechanisms, Int. J. Refract. Met. Hard Mater., 2019, 84, p 104999.Article CAS Google Scholar
L. Kaserera, J. Brauna, J. Stajkovica, K.-H. Leitzb, B. Tabernigb, P. Singerb, I. Letofsky-Papstc, H. Kestlerb, and G. Leichtfried, Fully Dense and Crack Free Molybdenum Manufactured by Selective Laser Melting Through Alloying with Carbon, Int. J. Refract Metal Hard Mater., 2019, 84, p 105000.Article Google Scholar

Fig. 3. Free surface and substrate profiles in all Sp and Ls cases at t = 1 s, t = 3 s, and t = 5 s, arranged left to right (note: the colour contours correspond to the horizontal component of the flow velocity (u), expressed in m/s).

Numerical investigation of dam break flow over erodible beds with diverse substrate level variations

다양한 기질 수준 변화를 갖는 침식성 층 위의 댐 파손 흐름에 대한 수치 조사

Alireza Khoshkonesh¹, Blaise Nsom², Saeid Okhravi³*, Fariba Ahmadi Dehrashid⁴, Payam Heidarian⁵,
Silvia DiFrancesco⁶
¹Department of Geography, School of Social Sciences, History, and Philosophy, Birkbeck University of London, London, UK.
² Université de Bretagne Occidentale. IRDL/UBO UMR CNRS 6027. Rue de Kergoat, 29285 Brest, France.
³ Institute of Hydrology, Slovak Academy of Sciences, Dúbravská cesta 9, 84104, Bratislava, Slovak Republic.
⁴Department of Water Science and Engineering, Faculty of Agriculture, Bu-Ali Sina University, 65178-38695, Hamedan, Iran.
⁵Department of Civil, Environmental, Architectural Engineering and Mathematics, University of Brescia, 25123 Brescia, Italy.
⁶Niccol`o Cusano University, via Don C. Gnocchi 3, 00166 Rome, Italy. * Corresponding author. Tel.: +421-944624921. E-mail: saeid.okhravi@savba.sk

Abstract

This study aimed to comprehensively investigate the influence of substrate level difference and material composition on dam break wave evolution over two different erodible beds. Utilizing the Volume of Fluid (VOF) method, we tracked free surface advection and reproduced wave evolution using experimental data from the literature. For model validation, a comprehensive sensitivity analysis encompassed mesh resolution, turbulence simulation methods, and bed load transport equations. The implementation of Large Eddy Simulation (LES), non-equilibrium sediment flux, and van Rijn’s (1984) bed load formula yielded higher accuracy compared to alternative approaches. The findings emphasize the significant effect of substrate level difference and material composition on dam break morphodynamic characteristics. Decreasing substrate level disparity led to reduced flow velocity, wavefront progression, free surface height, substrate erosion, and other pertinent parameters. Initial air entrapment proved substantial at the wavefront, illustrating pronounced air-water interaction along the bottom interface. The Shields parameter experienced a one-third reduction as substrate level difference quadrupled, with the highest near-bed concentration observed at the wavefront. This research provides fresh insights into the complex interplay of factors governing dam break wave propagation and morphological changes, advancing our comprehension of this intricate phenomenon.

이 연구는 두 개의 서로 다른 침식층에 대한 댐 파괴파 진화에 대한 기질 수준 차이와 재료 구성의 영향을 종합적으로 조사하는 것을 목표로 했습니다. VOF(유체량) 방법을 활용하여 자유 표면 이류를 추적하고 문헌의 실험 데이터를 사용하여 파동 진화를 재현했습니다.

모델 검증을 위해 메쉬 해상도, 난류 시뮬레이션 방법 및 침대 하중 전달 방정식을 포함하는 포괄적인 민감도 분석을 수행했습니다. LES(Large Eddy Simulation), 비평형 퇴적물 플럭스 및 van Rijn(1984)의 하상 부하 공식의 구현은 대체 접근 방식에 비해 더 높은 정확도를 산출했습니다.

연구 결과는 댐 붕괴 형태역학적 특성에 대한 기질 수준 차이와 재료 구성의 중요한 영향을 강조합니다. 기판 수준 차이가 감소하면 유속, 파면 진행, 자유 표면 높이, 기판 침식 및 기타 관련 매개변수가 감소했습니다.

초기 공기 포집은 파면에서 상당한 것으로 입증되었으며, 이는 바닥 경계면을 따라 뚜렷한 공기-물 상호 작용을 보여줍니다. 기판 레벨 차이가 4배로 증가함에 따라 Shields 매개변수는 1/3로 감소했으며, 파면에서 가장 높은 베드 근처 농도가 관찰되었습니다.

이 연구는 댐 파괴파 전파와 형태학적 변화를 지배하는 요인들의 복잡한 상호 작용에 대한 새로운 통찰력을 제공하여 이 복잡한 현상에 대한 이해를 향상시킵니다.

Keywords

Dam break; Substrate level difference; Erodible bed; Sediment transport; Computational fluid dynamics CFD.

PDF Download

REFERENCES

Aleixo, R., Soares-Frazão, S., Zech, Y., 2010. Velocity profiles in
dam-break flows: water and sediment layers. In: Proc. Int. Conf.
on Fluvial Hydraulics “River Flow 2010”, pp. 533–540.
An, S., Ku, H., Julien, P.Y., 2015. Numerical modelling of local
scour caused by submerged jets. Maejo Int. J. Sci. Technol., 9, 3,
328–343.
Bahmanpouri, F., Daliri, M., Khoshkonesh, A., Namin, M.M.,
Buccino, M., 2021. Bed compaction effect on dam break flow over
erodible bed; experimental and numerical modeling. J. Hydrol.,
594, 125645. https://doi.org/10.1016/j.jhydrol.2020.125645
Baklanov, A., 2007. Environmental risk and assessment modelling
– scientific needs and expected advancements. In: Ebel, A.,
Davitashvili, T. (Eds.): Air, Water and Soil Quality Modelling
for Risk and Impact Assessment Springer, Dordrecht, pp. 29–44.
Biscarini, C., Di Francesco, S., Nardi, F., Manciola, P., 2013.
Detailed simulation of complex hydraulic problems with
macroscopic and mesoscopic mathematical methods. Math.
Probl. Eng., 928309. https://doi.org/10.1155/2013/928309
Cao, Z., Pender, G., Wallis, S., Carling, P., 2004. Computational
dam-break hydraulics over erodible sediment bed. J. Hydraul.
Eng., 130, 7, 689–703.
Catucci, D., Briganti, R., Heller, V., 2021. Numerical validation of novel
scaling laws for air entrainment in water. Proc. R. Soc. A, 477, 2255,20210339. https://doi.org/10.1098/rspa.2021.0339
Dehrashid, F.A., Heidari, M., Rahimi, H., Khoshkonesh, A., Yuan,
S., Tang, X., Lu, C., Wang, X., 2023. CFD modeling the flow
dynamics in an open channel with double-layered vegetation.
Model. Earth Syst. Environ., 9, 1, 543–555.
Desombre, J., Morichon, D., Mory, M., 2013. RANS v2-f simulation
of a swash event: Detailed flow structure. Coastal Eng., 71, 1–12.
Dodangeh, E., Afzalimehr, H., 2022. Incipient motion of sediment
particles in the presence of bed forms under decelerating and
accelerating flows. J. Hydrol. Hydromech., 70, 1, 89–102.
Dong, Z., Wang, J., Vetsch, D.F., Boes, R.M., Tan, G., 2019.
Numerical simulation of air entrainment on stepped
spillways. In: E-proceedings of the 38th IAHR World Congress
(pp. 1494). September 1–6, 2019, Panama City, Panama. DOI:
10.3850/38WC092019-0755
Flow3D [computer software]. 2023. Santa Fe, NM: Flow Science,
Inc.
Fraccarollo, L., Capart, H., 2002. Riemann wave description of
erosional dam-break flows. J. Fluid Mech., 461, 183–228.
Gu, Z., Wang, T., Meng, W., Yu, C.H., An, R., 2023. Numerical
investigation of silted-up dam-break flow with different silted-up
sediment heights. Water Supply, 23, 2, 599–614.
Gualtieri, P., De Felice, S., Pasquino, V., Doria, G.P., 2018. Use of
conventional flow resistance equations and a model for the
Nikuradse roughness in vegetated flows at high submergence. J.
Hydrol. Hydromech., 66, 1, 107–120.
Heller, V., 2011. Scale effects in physical hydraulic engineering
models. J. Hydraul. Res., 49, 3, 293–306.
Hirt, C.W., 2003. Modeling turbulent entrainment of air at a free
surface. Flow Science, Inc.
Hirt, C.W., Nichols, B.D., 1981. Volume of fluid (VOF) method for
the dynamics of free boundaries. J. Comput. Phys., 39, 1, 201–
225.
Issakhov, A., Zhandaulet, Y., Nogaeva, A., 2018. Numerical
simulation of dam break flow for various forms of the obstacle
by VOF method. Int. J. Multiphase Flow, 109, 191–206.
Khayyer, A., Gotoh, H., 2010. On particle-based simulation of a dam
break over a wet bed. J. Hydraul. Res., 48, 2, 238–249.
Khoshkonesh, A., Daliri, M., Riaz, K., Dehrashid, F.A.,
Bahmanpouri, F., Di Francesco, S., 2022. Dam-break flow
dynamics over a stepped channel with vegetation. J. Hydrol., 613,128395. https://doi.org/10.1016/j.jhydrol.2022.128395
Khoshkonesh, A., Nsom, B., Gohari, S., Banejad, H., 2019.
A comprehensive study on dam-break flow over dry and wet
beds. Ocean Eng., 188, 106279.
https://doi.org/10.1016/j.oceaneng.2019.106279
Khoshkonesh, A., Sadeghi, S.H., Gohari, S., Karimpour, S., Oodi,
S., Di Francesco, S., 2023. Study of dam-break flow over a
vegetated channel with and without a drop. Water Resour.
Manage., 37, 5, 2107–2123.
Khosravi, K., Chegini, A.H.N., Cooper, J., Mao, L., Habibnejad, M.,
Shahedi, K., Binns, A., 2021. A laboratory investigation of bedload transport of gravel sediments under dam break flow. Int. J.
Sediment Res., 36, 2, 229–234.
Kim, Y., Zhou, Z., Hsu, T.J., Puleo, J.A., 2017. Large eddy
simulation of dam‐break‐driven swash on a rough‐planar beach.
J. Geophys. Res.: Oceans, 122, 2, 1274–1296.
Kocaman, S., Ozmen-Cagatay, H., 2012. The effect of lateral
channel contraction on dam break flows: Laboratory experiment.
J. Hydrol., 432, 145–153.
Leal, J.G., Ferreira, R.M., Cardoso, A.H., 2006. Dam-break wavefront celerity. J. Hydraul. Eng., 132, 1, 69–76.
Leal, J.G.A.B., Ferreira, R.M., Cardoso, A.H., 2003. Dam-break
wave propagation over a cohesionless erodible bed. In: Proc.
30rd IAHR Congress, 100, 261–268.
Li, Y. L., Ma, Y., Deng, R., Jiang, D.P., Hu, Z., 2019. Research on
dam-break induced tsunami bore acting on the triangular
breakwater based on high order 3D CLSVOF-THINC/WLICIBM approaching. Ocean Eng., 182, 645–659.
Li, Y.L., Yu, C.H., 2019. Research on dam-break flow induced front
wave impacting a vertical wall based on the CLSVOF and level
set methods. Ocean Eng., 178, 442–462.
Mei, S., Chen, S., Zhong, Q., Shan, Y., 2022. Detailed numerical
modeling for breach hydrograph and morphology evolution
during landslide dam breaching. Landslides, 19, 12, 2925–2949.
Meng, W., Yu, C.H., Li, J., An, R., 2022. Three-dimensional simulation
of silted-up dam-break flow striking a rigid structure. Ocean Eng.,
261, 112042. https://doi.org/10.1016/j.oceaneng.2022.112042
Meyer-Peter, E., Müller, R., 1948. Formulas for bed-load transport.
In: IAHSR 2nd meeting, Stockholm, appendix 2. IAHR.
Nielsen, P., 1984. Field measurements of time-averaged suspended
sediment concentrations under waves. Coastal Eng., 8, 1, 51–72.
Nielsen, P., 2018. Bed shear stress, surface shape and velocity field
near the tips of dam-breaks, tsunami and wave runup. Coastal
Eng., 138, 126–131.
Nsom, B., Latrache, N., Ramifidisoa, L., Khoshkonesh, A., 2019.
Analytical solution to the stability of gravity-driven stratified
flow of two liquids over an inclined plane. In: 24th French
Mechanics Congress in Brest. Brest, p. 244178.
Nsom, B., Ravelo, B., Ndong, W., 2008. Flow regimes in horizontal
viscous dam-break flow of Cayous mud. Appl. Rheol., 18, 4,
43577-1. https://doi.org/10.1515/arh-2008-0012
Oguzhan, S., Aksoy, A.O., 2020. Experimental investigation of the
effect of vegetation on dam break flood waves. J. Hydrol.
Hydromech., 68, 3, 231–241.
Okhravi, S., Gohari, S., Alemi, M., Maia, R., 2022. Effects of bedmaterial gradation on clear water scour at single and group of
piles. J. Hydrol. Hydromech., 70, 1, 114–127.
Okhravi, S., Gohari, S., Alemi, M., Maia, R., 2023. Numerical
modeling of local scour of non-uniform graded sediment for two
arrangements of pile groups. Int. J. Sediment Res., 38, 4, 597–614.
Parambath, A., 2010. Impact of tsunamis on near shore wind power
units. Master’s Thesis. Texas A&M University. Available
electronically from https://hdl.handle.net/1969.1/ETD-TAMU2010-12-8919
Pintado-Patiño, J.C., Puleo, J.A., Krafft, D., Torres-Freyermuth, A.,

Hydrodynamics and sediment transport under a dambreak-driven swash: An experimental study. Coastal Eng., 170,
https://doi.org/10.1016/j.coastaleng.2021.103986
Riaz, K., Aslam, H.M.S., Yaseen, M.W., Ahmad, H.H.,
Khoshkonesh, A., Noshin, S., 2022. Flood frequency analysis
and hydraulic design of bridge at Mashan on river Kunhar. Arch.
Hydroengineering Environ. Mech., 69, 1, 1–12.
Ritter, A., 1892. Die Fortpflanzung der Wasserwellen. Zeitschrift
des Vereines Deutscher Ingenieure, 36, 33, 947–954. (In
German.)
Smagorinsky, J., 1963. General circulation experiments with the
primitive equations: I. The basic experiment. Mon. Weather
Rev., 91, 3, 99–164.
Soulsby, R.L., 1997. Dynamics of marine sands: a manual for
practical applications. Oceanogr. Lit. Rev., 9, 44, 947.
Spinewine, B., Capart, H., 2013. Intense bed-load due to a sudden
dam-break. J. Fluid Mech., 731, 579–614.
Van Rijn, L.C., 1984. Sediment transport, part I: bed load transport.
J. Hydraul. Eng., 110, 10, 1431–1456.
Vosoughi, F., Rakhshandehroo, G., Nikoo, M.R., Sadegh, M.,
Experimental study and numerical verification of
silted-up dam break. J. Hydrol., 590, 125267.
https://doi.org/10.1016/j.jhydrol.2020.125267
Wu, W., Wang, S.S., 2008. One-dimensional explicit finite-volume
model for sediment transport. J. Hydraul. Res., 46, 1, 87–98.
Xu, T., Huai, W., Liu, H., 2023. MPS-based simulation of
dam-break wave propagation over wet beds with a
sediment layer. Ocean Eng., 281, 115035.
https://doi.org/10.1016/j.oceaneng.2023.115035
Yang, S., Yang, W., Qin, S., Li, Q., Yang, B., 2018. Numerical study
on characteristics of dam-break wave. Ocean Eng., 159, 358–371.
Yao, G.F., 2004. Development of new pressure-velocity solvers in
FLOW-3D. Flow Science, Inc., USA.

PDF Download

Schematic diagram of HP-LPBF melting process.

Modeling and numerical studies of high-precision laser powder bed fusion

Yi Wei ;Genyu Chen;Nengru Tao;Wei Zhou
https://doi.org/10.1063/5.0191504

In order to comprehensively reveal the evolutionary dynamics of the molten pool and the state of motion of the fluid during the high-precision laser powder bed fusion (HP-LPBF) process, this study aims to deeply investigate the specific manifestations of the multiphase flow, solidification phenomena, and heat transfer during the process by means of numerical simulation methods. Numerical simulation models of SS316L single-layer HP-LPBF formation with single and double tracks were constructed using the discrete element method and the computational fluid dynamics method. The effects of various factors such as Marangoni convection, surface tension, vapor recoil, gravity, thermal convection, thermal radiation, and evaporative heat dissipation on the heat and mass transfer in the molten pool have been paid attention to during the model construction process. The results show that the molten pool exhibits a “comet” shape, in which the temperature gradient at the front end of the pool is significantly larger than that at the tail end, with the highest temperature gradient up to 1.69 × 10⁸ K/s. It is also found that the depth of the second track is larger than that of the first one, and the process parameter window has been determined preliminarily. In addition, the application of HP-LPBF technology helps to reduce the surface roughness and minimize the forming size.

Topics

Heat transfer, Nonequilibrium thermodynamics, Solidification process, Computer simulation, Discrete element method, Lasers, Mass transfer, Fluid mechanics, Computational fluid dynamics, Multiphase flows

PDF Download

I. INTRODUCTION

Laser powder bed fusion (LPBF) has become a research hotspot in the field of additive manufacturing of metals due to its advantages of high-dimensional accuracy, good surface quality, high density, and high material utilization.^1,2 With the rapid development of electronics, medical, automotive, biotechnology, energy, communication, and optics, the demand for microfabrication technology is increasing day by day.³ High-precision laser powder bed fusion (HP-LPBF) is one of the key manufacturing technologies for tiny parts in the fields of electronics, medical, automotive, biotechnology, energy, communication, and optics because of its process characteristics such as small focal spot diameter, small powder particle size, and thin powder layup layer thickness.^4–13 Compared with LPBF, HP-LPBF has the significant advantages of smaller focal spot diameter, smaller powder particle size, and thinner layer thickness. These advantages make HP-LPBF perform better in producing micro-fine parts, high surface quality, and parts with excellent mechanical properties.

HP-LPBF is in the exploratory stage, and researchers have already done some exploratory studies on the focal spot diameter, the amount of defocusing, and the powder particle size. In order to explore the influence of changing the laser focal spot diameter on the LPBF process characteristics of the law, Wildman et al.¹⁴ studied five groups of different focal spot diameter LPBF forming 316L stainless steel (SS316L) processing effect, the smallest focal spot diameter of 26 μm, and the results confirm that changing the focal spot diameter can be achieved to achieve the energy control, so as to control the quality of forming. Subsequently, Mclouth et al.¹⁵ proposed the laser out-of-focus amount (focal spot diameter) parameter, which characterizes the distance between the forming plane and the laser focal plane. The laser energy density was controlled by varying the defocusing amount while keeping the laser parameters constant. Sample preparation at different focal positions was investigated, and their microstructures were characterized. The results show that the samples at the focal plane have finer microstructure than those away from the focal plane, which is the effect of higher power density and smaller focal spot diameter. In order to explore the influence of changing the powder particle size on the characteristics of the LPBF process, Qian et al.¹⁶ carried out single-track scanning simulations on powder beds with average powder particle sizes of 70 and 40 μm, respectively, and the results showed that the melt tracks sizes were close to each other under the same process parameters for the two particle-size distributions and that the molten pool of powder beds with small particles was more elongated and the edges of the melt tracks were relatively flat. In order to explore the superiority of HP-LPBF technology, Xu et al.¹⁷ conducted a comparative analysis of HP-LPBF and conventional LPBF of SS316L. The results showed that the average surface roughness of the top surface after forming by HP-LPBF could reach 3.40 μm. Once again, it was verified that HP-LPBF had higher forming quality than conventional LPBF. On this basis, Wei et al.⁶ comparatively analyzed the effects of different laser focal spot diameters on different powder particle sizes formed by LPBF. The results showed that the smaller the laser focal spot diameter, the fewer the defects on the top and side surfaces. The above research results confirm that reducing the laser focal spot diameter can obtain higher energy density and thus better forming quality.

LPBF involves a variety of complex systems and mechanisms, and the final quality of the part is influenced by a large number of process parameters.^18–24 Some research results have shown that there are more than 50 factors affecting the quality of the specimen. The influencing factors are mainly categorized into three main groups: (1) laser parameters, (2) powder parameters, and (3) equipment parameters, which interact with each other to determine the final specimen quality. With the continuous development of technologies such as computational materials science and computational fluid dynamics (CFD), the method of studying the influence of different factors on the forming quality of LPBF forming process has been shifted from time-consuming and laborious experimental characterization to the use of numerical simulation methods. As a result, more and more researchers are adopting this approach for their studies. Currently, numerical simulation studies on LPBF are mainly focused on the exploration of molten pool, temperature distribution, and residual stresses.

Finite element simulation based on continuum mechanics and free surface fluid flow modeling based on fluid dynamics are two common approaches to study the behavior of LPBF molten pool.^25–28 Finite element simulation focuses on the temperature and thermal stress fields, treats the powder bed as a continuum, and determines the molten pool size by plotting the elemental temperature above the melting point. In contrast, fluid dynamics modeling can simulate the 2D or 3D morphology of the metal powder pile and obtain the powder size and distribution by certain algorithms.²⁹ The flow in the molten pool is mainly affected by recoil pressure and the Marangoni effect. By simulating the molten pool formation, it is possible to predict defects, molten pool shape, and flow characteristics, as well as the effect of process parameters on the molten pool geometry.^30–34 In addition, other researchers have been conducted to optimize the laser processing parameters through different simulation methods and experimental data.^35–46 Crystal growth during solidification is studied to further understand the effect of laser parameters on dendritic morphology and solute segregation.^47–54 A multi-scale system has been developed to describe the fused deposition process during 3D printing, which is combined with the conductive heat transfer model and the dendritic solidification model.^55,56
Relevant scholars have adopted various different methods for simulation, such as sequential coupling theory,⁵⁷ Lagrangian and Eulerian thermal models,⁵⁸ birth–death element method,²⁵ and finite element method,⁵⁹ in order to reveal the physical phenomena of the laser melting process and optimize the process parameters. Luo et al.⁶⁰ compared the LPBF temperature field and molten pool under double ellipsoidal and Gaussian heat sources by ANSYS APDL and found that the diffusion of the laser energy in the powder significantly affects the molten pool size and the temperature field.
The thermal stresses obtained from the simulation correlate with the actual cracks,⁶¹ and local preheating can effectively reduce the residual stresses.⁶² A three-dimensional thermodynamic finite element model investigated the temperature and stress variations during laser-assisted fabrication and found that powder-to-solid conversion increases the temperature gradient, stresses, and warpage.⁶³ Other scholars have predicted residual stresses and part deflection for LPBF specimens and investigated the effects of deposition pattern, heat, laser power, and scanning strategy on residual stresses, noting that high-temperature gradients lead to higher residual stresses.^64–67

In short, the process of LPBF forming SS316L is extremely complex and usually involves drastic multi-scale physicochemical changes that will only take place on a very small scale. Existing literature employs DEM-based mesoscopic-scale numerical simulations to investigate the effects of process parameters on the molten pool dynamics of LPBF-formed SS316L. However, a few studies have been reported on the key mechanisms of heating and solidification, spatter, and convective behavior of the molten pool of HP-LPBF-formed SS316L with small laser focal spot diameters. In this paper, the geometrical properties of coarse and fine powder particles under three-dimensional conditions were first calculated using DEM. Then, numerical simulation models for single-track and double-track cases in the single-layer HP-LPBF forming SS316L process were developed at mesoscopic scale using the CFD method. The flow genesis of the melt in the single-track and double-track molten pools is discussed, and their 3D morphology and dimensional characteristics are discussed. In addition, the effects of laser process parameters, powder particle size, and laser focal spot diameter on the temperature field, characterization information, and defects in the molten pool are discussed.

II. MODELING

A. 3D powder bed modeling

HP-LPBF is an advanced processing technique for preparing target parts layer by layer stacking, the process of which involves repetitive spreading and melting of powders. In this process, both the powder spreading and the morphology of the powder bed are closely related to the results of the subsequent melting process, while the melted surface also affects the uniform distribution of the next layer of powder. For this reason, this chapter focuses on the modeling of the physical action during the powder spreading process and the theory of DEM to establish the numerical model of the powder bed, so as to lay a solid foundation for the accuracy of volume of fluid (VOF) and CFD.

1. DEM

DEM is a numerical technique for calculating the interaction of a large number of particles, which calculates the forces and motions of the spheres by considering each powder sphere as an independent unit. The motion of the powder particles follows the laws of classical Newtonian mechanics, including translational and rotational,^38,68–70 which are expressed as follows:��¨=��+∑��ij,

(1)��¨=∑�(�ij×�ij),

(2)

where �� is the mass of unit particle i in kg, ��¨ is the advective acceleration in m/s², And g is the gravitational acceleration in m/s². �ij is the force in contact with the neighboring particle � in N. �� is the rotational inertia of the unit particle � in kg · m². ��¨ is the unit particle � angular acceleration in rad/s². �ij is the vector pointing from unit particle � to the contact point of neighboring particle �⁠.

Equations (1) and (2) can be used to calculate the velocity and angular velocity variations of powder particles to determine their positions and velocities. A three-dimensional powder bed model of SS316L was developed using DEM. The powder particles are assumed to be perfect spheres, and the substrate and walls are assumed to be rigid. To describe the contact between the powder particles and between the particles and the substrate, a non-slip Hertz–Mindlin nonlinear spring-damping model⁷¹ was used with the following expression:�hz=��+��[(��ij−�eff��)−(��+�eff��)],

(3)

where �hz is the force calculated using the Hertzian in M. �� and �� are the radius of unit particles � and � in m, respectively. �� is the overlap size of the two powder particles in m. ��⁠, �� are the elastic constants in the normal and tangential directions, respectively. �ij is the unit vector connecting the centerlines of the two powder particles. �eff is the effective mass of the two powder particles in kg. �� and �� are the viscoelastic damping constants in the normal and tangential directions, respectively. �� and �� are the components of the relative velocities of the two powder particles. �� is the displacement vector between two spherical particles. The schematic diagram of overlapping powder particles is shown in Fig. 1.

FIG. 1.

VIEW LARGE DOWNLOAD SLIDE

Schematic diagram of overlapping powder particles.

Because the particle size of the powder used for HP-LPBF is much smaller than 100 μm, the effect of van der Waals forces must be considered. Therefore, the cohesive force �jkr of the Hertz–Mindlin model was used instead of van der Waals forces,⁷² with the following expression:�jkr=−4��0�*�1.5+4�*3�*�3,

(4)1�*=(1−��2)��+(1−��2)��,

(5)1�*=1��+1��,

(6)

where �* is the equivalent Young’s modulus in GPa; �* is the equivalent particle radius in m; �0 is the surface energy of the powder particles in J/m²; α is the contact radius in m; �� and �� are the Young’s modulus of the unit particles � and �⁠, respectively, in GPa; and �� and �� are the Poisson’s ratio of the unit particles � and �⁠, respectively.

2. Model building

Figure 2 shows a 3D powder bed model generated using DEM with a coarse powder geometry of 1000 × 400 × 30 μm³. The powder layer thickness is 30 μm, and the powder bed porosity is 40%. The average particle size of this spherical powder is 31.7 μm and is normally distributed in the range of 15–53 μm. The geometry of the fine powder was 1000 × 400 × 20 μm³, with a layer thickness of 20 μm, and the powder bed porosity of 40%. The average particle size of this spherical powder is 11.5 μm and is normally distributed in the range of 5–25 μm. After the 3D powder bed model is generated, it needs to be imported into the CFD simulation software for calculation, and the imported geometric model is shown in Fig. 3. This geometric model is mainly composed of three parts: protective gas, powder bed, and substrate. Under the premise of ensuring the accuracy of the calculation, the mesh size is set to 3 μm, and the total number of coarse powder meshes is 1 704 940. The total number of fine powder meshes is 3 982 250.

FIG. 2.

VIEW LARGE DOWNLOAD SLIDE

Three-dimensional powder bed model: (a) coarse powder, (b) fine powder.

FIG. 3.

VIEW LARGE DOWNLOAD SLIDE

Geometric modeling of the powder bed computational domain: (a) coarse powder, (b) fine powder.

B. Modeling of fluid mechanics simulation

In order to solve the flow, melting, and solidification problems involved in HP-LPBF molten pool, the study must follow the three governing equations of conservation of mass, conservation of energy, and conservation of momentum.⁷³ The VOF method, which is the most widely used in fluid dynamics, is used to solve the molten pool dynamics model.

1. VOF

VOF is a method for tracking the free interface between the gas and liquid phases on the molten pool surface. The core idea of the method is to define a volume fraction function F within each grid, indicating the proportion of the grid space occupied by the material, 0 ≤ F ≤ 1 in Fig. 4. Specifically, when F = 0, the grid is empty and belongs to the gas-phase region; when F = 1, the grid is completely filled with material and belongs to the liquid-phase region; and when 0 < F < 1, the grid contains free surfaces and belongs to the mixed region. The direction normal to the free surface is the direction of the fastest change in the volume fraction F (the direction of the gradient of the volume fraction), and the direction of the gradient of the volume fraction can be calculated from the values of the volume fractions in the neighboring grids.⁷⁴ The equations controlling the VOF are expressed as follows:𝛻��+�⋅(��→)=0,

(7)

where t is the time in s and �→ is the liquid velocity in m/s.

FIG. 4.

VIEW LARGE DOWNLOAD SLIDE

Schematic diagram of VOF.

The material parameters of the mixing zone are altered due to the inclusion of both the gas and liquid phases. Therefore, in order to represent the density of the mixing zone, the average density �¯ is used, which is expressed as follows:⁷²�¯=(1−�1)�gas+�1�metal,

(8)

where �1 is the proportion of liquid phase, �gas is the density of protective gas in kg/m³, and �metal is the density of metal in kg/m³.

2. Control equations and boundary conditions

Figure 5 is a schematic diagram of the HP-LPBF melting process. First, the laser light strikes a localized area of the material and rapidly heats up the area. Next, the energy absorbed in the region is diffused through a variety of pathways (heat conduction, heat convection, and surface radiation), and this process triggers complex phase transition phenomena (melting, evaporation, and solidification). In metals undergoing melting, the driving forces include surface tension and the Marangoni effect, recoil due to evaporation, and buoyancy due to gravity and uneven density. The above physical phenomena interact with each other and do not occur independently.

FIG. 5.

VIEW LARGE DOWNLOAD SLIDE

Schematic diagram of HP-LPBF melting process.

Laser heat sourceThe Gaussian surface heat source model is used as the laser heat source model with the following expression:�=2�0��2exp(−2�12��2),(9)where � is the heat flow density in W/m², �0 is the absorption rate of SS316L, �� is the radius of the laser focal spot in m, and �1 is the radial distance from the center of the laser focal spot in m. The laser focal spot can be used for a wide range of applications.
Energy absorptionThe formula for calculating the laser absorption �0 of SS316L is as follows:�0=0.365(�0[1+�0(�−20)]/�)0.5,(10)where �0 is the direct current resistivity of SS316L at 20 °C in Ω m, �0 is the resistance temperature coefficient in ppm/°C, � is the temperature in °C, and � is the laser wavelength in m.
Heat transferThe basic principle of heat transfer is conservation of energy, which is expressed as follows:𝛻𝛻𝛻�(��)��+�·(��→�)=�·(�0��)+��,(11)where � is the density of liquid phase SS316L in kg/m³, �� is the specific heat capacity of SS316L in J/(kg K), 𝛻� is the gradient operator, t is the time in s, T is the temperature in K, 𝛻�� is the temperature gradient, �→ is the velocity vector, �0 is the coefficient of thermal conduction of SS316L in W/(m K), and �� is the thermal energy dissipation term in the molten pool.
Molten pool flowThe following three conditions need to be satisfied for the molten pool to flow:
- Conservation of mass with the following expression:𝛻�·(��→)=0.(12)
- Conservation of momentum (Navier–Stokes equation) with the following expression:𝛻𝛻𝛻𝛻��→��+�(�→·�)�→=�·[−pI+�(��→+(��→)�)]+�,(13)where � is the pressure in Pa exerted on the liquid phase SS316L microelement, � is the unit matrix, � is the fluid viscosity in N s/m², and � is the volumetric force (gravity, atmospheric pressure, surface tension, vapor recoil, and the Marangoni effect).
- Conservation of energy, see Eq. (11)
Surface tension and the Marangoni effectThe effect of temperature on the surface tension coefficient is considered and set as a linear relationship with the following expression:�=�0−��dT(�−��),(14)where � is the surface tension of the molten pool at temperature T in N/m, �� is the melting temperature of SS316L in K, �0 is the surface tension of the molten pool at temperature �� in Pa, and σdσ/ dT is the surface tension temperature coefficient in N/(m K).In general, surface tension decreases with increasing temperature. A temperature gradient causes a gradient in surface tension that drives the liquid to flow, known as the Marangoni effect.
Metal vapor recoilAt higher input energy densities, the maximum temperature of the molten pool surface reaches the evaporation temperature of the material, and a gasification recoil pressure occurs vertically downward toward the molten pool surface, which will be the dominant driving force for the molten pool flow.⁷⁵ The expression is as follows:��=0.54�� exp ��−��0��,(15)where �� is the gasification recoil pressure in Pa, �� is the ambient pressure in kPa, �� is the latent heat of evaporation in J/kg, �0 is the gas constant in J/(mol K), T is the surface temperature of the molten pool in K, and T_e is the evaporation temperature in K.
Solid–liquid–gas phase transitionWhen the laser hits the powder layer, the powder goes through three stages: heating, melting, and solidification. During the solidification phase, mutual transformations between solid, liquid, and gaseous states occur. At this point, the latent heat of phase transition absorbed or released during the phase transition needs to be considered.⁶⁸ The phase transition is represented based on the relationship between energy and temperature with the following expression:�=��,(�<��),�(��)+�−��−��,(��<�<��)�(��)+(�−��)��,(��<�),,(16)where �� and �� are solid and liquid phase density, respectively, of SS316L in kg/m³. �� and �� unit volume of solid and liquid phase-specific heat capacity, respectively, of SS316L in J/(kg K). �� and ��⁠, respectively, are the solidification temperature and melting temperature of SS316L in K. �� is the latent heat of the phase transition of SS316L melting in J/kg.

3. Assumptions

The CFD model was computed using the commercial software package FLOW-3D.⁷⁶ In order to simplify the calculation and solution process while ensuring the accuracy of the results, the model makes the following assumptions:

It is assumed that the effects of thermal stress and material solid-phase thermal expansion on the calculation results are negligible.
The molten pool flow is assumed to be a Newtonian incompressible laminar flow, while the effects of liquid thermal expansion and density on the results are neglected.
It is assumed that the surface tension can be simplified to an equivalent pressure acting on the free surface of the molten pool, and the effect of chemical composition on the results is negligible.
Neglecting the effect of the gas flow field on the molten pool.
The mass loss due to evaporation of the liquid metal is not considered.
The influence of the plasma effect of the molten metal on the calculation results is neglected.

It is worth noting that the formulation of assumptions requires a trade-off between accuracy and computational efficiency. In the above models, some physical phenomena that have a small effect or high difficulty on the calculation results are simplified or ignored. Such simplifications make numerical simulations more efficient and computationally tractable, while still yielding accurate results.

4. Initial conditions

The preheating temperature of the substrate was set to 393 K, at which time all materials were in the solid state and the flow rate was zero.

5. Material parameters

The material used is SS316L and the relevant parameters required for numerical simulations are shown in Table I.^46,77,78

TABLE I.

SS316L-related parameters.

Property	Symbol	Value
Density of solid metal (kg/m³)	�metal	7980
Solid phase line temperature (K)	��	1658
Liquid phase line temperature (K)	��	1723
Vaporization temperature (K)	��	3090
Latent heat of melting (⁠ J/kg⁠)	��	2.60×105
Latent heat of evaporation (⁠ J/kg⁠)	��	7.45×106
Surface tension of liquid phase (N /m⁠)	�	1.60
Liquid metal viscosity (kg/m s)	��	6×10−3
Gaseous metal viscosity (kg/m s)	�gas	1.85×10−5
Temperature coefficient of surface tension (N/m K)	��/�T	0.80×10−3
Molar mass (⁠ kg/mol⁠)	M	0.05 593
Emissivity	�	0.26
Laser absorption	�0	0.35
Ambient pressure (kPa)	��	101 325
Ambient temperature (K)	�0	300
Stefan–Boltzmann constant (W/m² K⁴)	�	5.67×10−8
Thermal conductivity of metals (⁠ W/m K⁠)	�	24.55
Density of protective gas (kg/m³)	�gas	1.25
Coefficient of thermal expansion (/K)	��	16×10−6
Generalized gas constant (⁠ J/mol K⁠)	R	8.314

III. RESULTS AND DISCUSSION

With the objective of studying in depth the evolutionary patterns of single-track and double-track molten pool development, detailed observations were made for certain specific locations in the model, as shown in Fig. 6. In this figure, P1 and P2 represent the longitudinal tangents to the centers of the two melt tracks in the XZ plane, while L1 is the transverse profile in the YZ plane. The scanning direction is positive and negative along the X axis. Points A and B are the locations of the centers of the molten pool of the first and second melt tracks, respectively (x = 1.995 × 10⁻⁴, y = 5 × 10⁻⁷, and z = −4.85 × 10⁻⁵).

FIG. 6.

VIEW LARGE DOWNLOAD SLIDE

Schematic diagram of observation position.

A. Single-track simulation

A series of single-track molten pool simulation experiments were carried out in order to investigate the influence law of laser power as well as scanning speed on the HP-LPBF process. Figure 7 demonstrates the evolution of the 3D morphology and temperature field of the single-track molten pool in the time period of 50–500 μs under a laser power of 100 W and a scanning speed of 800 mm/s. The powder bed is in the natural cooling state. When t = 50 μs, the powder is heated by the laser heat and rapidly melts and settles to form the initial molten pool. This process is accompanied by partial melting of the substrate and solidification together with the melted powder. The molten pool rapidly expands with increasing width, depth, length, and temperature, as shown in Fig. 7(a). When t = 150 μs, the molten pool expands more obviously, and the temperature starts to transfer to the surrounding area, forming a heat-affected zone. At this point, the width of the molten pool tends to stabilize, and the temperature in the center of the molten pool has reached its peak and remains largely stable. However, the phenomenon of molten pool spatter was also observed in this process, as shown in Fig. 7(b). As time advances, when t = 300 μs, solidification begins to occur at the tail of the molten pool, and tiny ripples are produced on the solidified surface. This is due to the fact that the melt flows toward the region with large temperature gradient under the influence of Marangoni convection and solidifies together with the melt at the end of the bath. At this point, the temperature gradient at the front of the bath is significantly larger than at the end. While the width of the molten pool was gradually reduced, the shape of the molten pool was gradually changed to a “comet” shape. In addition, a slight depression was observed at the top of the bath because the peak temperature at the surface of the bath reached the evaporation temperature, which resulted in a recoil pressure perpendicular to the surface of the bath downward, creating a depressed region. As the laser focal spot moves and is paired with the Marangoni convection of the melt, these recessed areas will be filled in as shown in Fig. 7(c). It has been shown that the depressed regions are the result of the coupled effect of Marangoni convection, recoil pressure, and surface tension.⁷⁹ By t = 500 μs, the width and height of the molten pool stabilize and show a “comet” shape in Fig. 7(d).

FIG. 7.

VIEW LARGE DOWNLOAD SLIDE

Single-track molten pool process: (a) t = 50 ��⁠, (b) t = 150 ��⁠, (c) t = 300 ��⁠, (d) t = 500 ��⁠.

Figure 8 depicts the velocity vector diagram of the P1 profile in a single-track molten pool, the length of the arrows represents the magnitude of the velocity, and the maximum velocity is about 2.36 m/s. When t = 50 μs, the molten pool takes shape, and the velocities at the two ends of the pool are the largest. The variation of the velocities at the front end is especially more significant in Fig. 8(a). As the time advances to t = 150 μs, the molten pool expands rapidly, in which the velocity at the tail increases and changes more significantly, while the velocity at the front is relatively small. At this stage, the melt moves backward from the center of the molten pool, which in turn expands the molten pool area. The melt at the back end of the molten pool center flows backward along the edge of the molten pool surface and then converges along the edge of the molten pool to the bottom center, rising to form a closed loop. Similarly, a similar closed loop is formed at the front end of the center of the bath, but with a shorter path. However, a large portion of the melt in the center of the closed loop formed at the front end of the bath is in a nearly stationary state. The main cause of this melt flow phenomenon is the effect of temperature gradient and surface tension (the Marangoni effect), as shown in Figs. 8(b) and 8(e). This dynamic behavior of the melt tends to form an “elliptical” pool. At t = 300 μs, the tendency of the above two melt flows to close the loop is more prominent and faster in Fig. 8(c). When t = 500 μs, the velocity vector of the molten pool shows a stable trend, and the closed loop of melt flow also remains stable. With the gradual laser focal spot movement, the melt is gradually solidified at its tail, and finally, a continuous and stable single track is formed in Fig. 8(d).

FIG. 8.

VIEW LARGE DOWNLOAD SLIDE

Vector plot of single-track molten pool velocity in XZ longitudinal section: (a) t = 50 ��⁠, (b) t = 150 ��⁠, (c) t = 300 ��⁠, (d) t = 500 ��⁠, (e) molten pool flow.

In order to explore in depth the transient evolution of the molten pool, the evolution of the single-track temperature field and the melt flow was monitored in the YZ cross section. Figure 9(a) shows the state of the powder bed at the initial moment. When t = 250 μs, the laser focal spot acts on the powder bed and the powder starts to melt and gradually collects in the molten pool. At this time, the substrate will also start to melt, and the melt flow mainly moves in the downward and outward directions and the velocity is maximum at the edges in Fig. 9(b). When t = 300 μs, the width and depth of the molten pool increase due to the recoil pressure. At this time, the melt flows more slowly at the center, but the direction of motion is still downward in Fig. 9(c). When t = 350 μs, the width and depth of the molten pool further increase, at which time the intensity of the melt flow reaches its peak and the direction of motion remains the same in Fig. 9(d). When t = 400 μs, the melt starts to move upward, and the surrounding powder or molten material gradually fills up, causing the surface of the molten pool to begin to flatten. At this time, the maximum velocity of the melt is at the center of the bath, while the velocity at the edge is close to zero, and the edge of the melt starts to solidify in Fig. 9(e). When t = 450 μs, the melt continues to move upward, forming a convex surface of the melt track. However, the melt movement slows down, as shown in Fig. 9(f). When t = 500 μs, the melt further moves upward and its speed gradually becomes smaller. At the same time, the melt solidifies further, as shown in Fig. 9(g). When t = 550 μs, the melt track is basically formed into a single track with a similar “mountain” shape. At this stage, the velocity is close to zero only at the center of the molten pool, and the flow behavior of the melt is poor in Fig. 9(h). At t = 600 μs, the melt stops moving and solidification is rapidly completed. Up to this point, a single track is formed in Fig. 9(i). During the laser action on the powder bed, the substrate melts and combines with the molten state powder. The powder-to-powder fusion is like the convergence of water droplets, which are rapidly fused by surface tension. However, the fusion between the molten state powder and the substrate occurs driven by surface tension, and the molten powder around the molten pool is pulled toward the substrate (a wetting effect occurs), which ultimately results in the formation of a monolithic whole.^38,80,81

FIG. 9.

VIEW LARGE DOWNLOAD SLIDE

Evolution of single-track molten pool temperature and melt flow in the YZ cross section: (a) t = 0 ��⁠, (b) t = 250 ��⁠, (c) t = 300 ��⁠, (d) t = 350 ��⁠, (e) t = 400 ��⁠, (f) t = 450 ��⁠, (g) t = 500 ��⁠, (h) t = 550 ��⁠, (i) t = 600 ��⁠.

The wetting ability between the liquid metal and the solid substrate in the molten pool directly affects the degree of balling of the melt,^82,83 and the wetting ability can be measured by the contact angle of a single track in Fig. 10. A smaller value of contact angle represents better wettability. The contact angle α can be calculated by�=�1−�22,

(17)

where �1 and �2 are the contact angles of the left and right regions, respectively.

FIG. 10.

VIEW LARGE DOWNLOAD SLIDE

Schematic of contact angle.

Relevant studies have confirmed that the wettability is better at a contact angle α around or below 40°.⁸⁴ After measurement, a single-track contact angle α of about 33° was obtained under this process parameter, which further confirms the good wettability.

B. Double-track simulation

In order to deeply investigate the influence of hatch spacing on the characteristics of the HP-LPBF process, a series of double-track molten pool simulation experiments were systematically carried out. Figure 11 shows in detail the dynamic changes of the 3D morphology and temperature field of the double-track molten pool in the time period of 2050–2500 μs under the conditions of laser power of 100 W, scanning speed of 800 mm/s, and hatch spacing of 0.06 mm. By comparing the study with Fig. 7, it is observed that the basic characteristics of the 3D morphology and temperature field of the second track are similar to those of the first track. However, there are subtle differences between them. The first track exhibits a basically symmetric shape, but the second track morphology shows a slight deviation influenced by the difference in thermal diffusion rate between the solidified metal and the powder. Otherwise, the other characteristic information is almost the same as that of the first track. Figure 12 shows the velocity vector plot of the P2 profile in the double-track molten pool, with a maximum velocity of about 2.63 m/s. The melt dynamics at both ends of the pool are more stable at t = 2050 μs, where the maximum rate of the second track is only 1/3 of that of the first one. Other than that, the rest of the information is almost no significant difference from the characteristic information of the first track. Figure 13 demonstrates a detailed observation of the double-track temperature field and melts flow in the YZ cross section, and a comparative study with Fig. 9 reveals that the width of the second track is slightly wider. In addition, after the melt direction shifts from bottom to top, the first track undergoes four time periods (50 μs) to reach full solidification, while the second track takes five time periods. This is due to the presence of significant heat buildup in the powder bed after the forming of the first track, resulting in a longer dynamic time of the melt and an increased molten pool lifetime. In conclusion, the level of specimen forming can be significantly optimized by adjusting the laser power and hatch spacing.

FIG. 11.

VIEW LARGE DOWNLOAD SLIDE

Double-track molten pool process: (a) t = 2050 ��⁠, (b) t = 2150 ��⁠, (c) t = 2300 ��⁠, (d) t = 2500 ��⁠.

FIG. 12.

VIEW LARGE DOWNLOAD SLIDE

Vector plot of double-track molten pool velocity in XZ longitudinal section: (a) t = 2050 ��⁠, (b) t = 2150 ��⁠, (c) t = 2300 ��⁠, (d) t = 2500 ��⁠.

FIG. 13.

VIEW LARGE DOWNLOAD SLIDE

Evolution of double-track molten pool temperature and melt flow in the YZ cross section: (a) t = 2250 ��⁠, (b) t = 2300 ��⁠, (c) t = 2350 ��⁠, (d) t = 2400 ��⁠, (e) t = 2450 ��⁠, (f) t = 2500 ��⁠, (g) t = 2550 ��⁠, (h) t = 2600 ��⁠, (i) t = 2650 ��⁠.

In order to quantitatively detect the molten pool dimensions as well as the remolten region dimensions, the molten pool characterization information in Fig. 14 is constructed by drawing the boundary on the YZ cross section based on the isothermal surface of the liquid phase line. It can be observed that the heights of the first track and second track are basically the same, but the depth of the second track increases relative to the first track. The molten pool width is mainly positively correlated with the laser power as well as the scanning speed (the laser line energy density �⁠). However, the remelted zone width is negatively correlated with the hatch spacing (the overlapping ratio). Overall, the forming quality of the specimens can be directly influenced by adjusting the laser power, scanning speed, and hatch spacing.

FIG. 14.

VIEW LARGE DOWNLOAD SLIDE

Double-track molten pool characterization information on YZ cross section.

In order to study the variation rule of the temperature in the center of the molten pool with time, Fig. 15 demonstrates the temperature variation curves with time for two reference points, A and B. Among them, the red dotted line indicates the liquid phase line temperature of SS316L. From the figure, it can be seen that the maximum temperature at the center of the molten pool in the first track is lower than that in the second track, which is mainly due to the heat accumulation generated after passing through the first track. The maximum temperature gradient was calculated to be 1.69 × 10⁸ K/s. When the laser scanned the first track, the temperature in the center of the molten pool of the second track increased slightly. Similarly, when the laser scanned the second track, a similar situation existed in the first track. Since the temperature gradient in the second track is larger than that in the first track, the residence time of the liquid phase in the molten pool of the first track is longer than that of the second track.

FIG. 15.

VIEW LARGE DOWNLOAD SLIDE

Temperature profiles as a function of time for two reference points A and B.

C. Simulation analysis of molten pool under different process parameters

In order to deeply investigate the effects of various process parameters on the mesoscopic-scale temperature field, molten pool characteristic information and defects of HP-LPBF, numerical simulation experiments on mesoscopic-scale laser power, scanning speed, and hatch spacing of double-track molten pools were carried out.

1. Laser power

Figure 16 shows the effects of different laser power on the morphology and temperature field of the double-track molten pool at a scanning speed of 800 mm/s and a hatch spacing of 0.06 mm. When P = 50 W, a smaller molten pool is formed due to the lower heat generated by the Gaussian light source per unit time. This leads to a smaller track width, which results in adjacent track not lapping properly and the presence of a large number of unmelted powder particles, resulting in an increase in the number of defects, such as pores in the specimen. The surface of the track is relatively flat, and the depth is small. In addition, the temperature gradient before and after the molten pool was large, and the depression location appeared at the biased front end in Fig. 16(a). When P = 100 W, the surface of the track is flat and smooth with excellent lap. Due to the Marangoni effect, the velocity field of the molten pool is in the form of “vortex,” and the melt has good fluidity, and the maximum velocity reaches 2.15 m/s in Fig. 16(b). When P = 200 W, the heat generated by the Gaussian light source per unit time is too large, resulting in the melt rapidly reaching the evaporation temperature, generating a huge recoil pressure, forming a large molten pool, and the surface of the track is obviously raised. The melt movement is intense, especially the closed loop at the center end of the molten pool. At this time, the depth and width of the molten pool are large, leading to the expansion of the remolten region and the increased chance of the appearance of porosity defects in Fig. 16(c). The results show that at low laser power, the surface tension in the molten pool is dominant. At high laser power, recoil pressure is its main role.

FIG. 16.

VIEW LARGE DOWNLOAD SLIDE

Simulation results of double-track molten pool under different laser powers: (a) P = 50 W, (b) P = 100 W, (c) P = 200 W.

Table II shows the effect of different laser powers on the characteristic information of the double-track molten pool at a scanning speed of 800 mm/s and a hatch spacing of 0.06 mm. The negative overlapping ratio in the table indicates that the melt tracks are not lapped, and 26/29 indicates the melt depth of the first track/second track. It can be seen that with the increase in laser power, the melt depth, melt width, melt height, and remelted zone show a gradual increase. At the same time, the overlapping ratio also increases. Especially in the process of laser power from 50 to 200 W, the melting depth and melting width increased the most, which increased nearly 2 and 1.5 times, respectively. Meanwhile, the overlapping ratio also increases with the increase in laser power, which indicates that the melting and fusion of materials are better at high laser power. On the other hand, the dimensions of the molten pool did not change uniformly with the change of laser power. Specifically, the depth-to-width ratio of the molten pool increased from about 0.30 to 0.39 during the increase from 50 to 120 W, which further indicates that the effective heat transfer in the vertical direction is greater than that in the horizontal direction with the increase in laser power. This dimensional response to laser power is mainly affected by the recoil pressure and also by the difference in the densification degree between the powder layer and the metal substrate. In addition, according to the experimental results, the contact angle shows a tendency to increase and then decrease during the process of laser power increase, and always stays within the range of less than 33°. Therefore, in practical applications, it is necessary to select the appropriate laser power according to the specific needs in order to achieve the best processing results.

TABLE II.

Double-track molten pool characterization information at different laser powers.

Laser power (W)	Depth (μm)	Width (μm)	Height (μm)	Remolten region (μm)	Overlapping ratio (%)	Contact angle (°)
50	16	54	11	/	−10	23
100	26/29	74	14	18	23.33	33
200	37/45	116	21	52	93.33	28

2. Scanning speed

Figure 17 demonstrates the effect of different scanning speeds on the morphology and temperature field of the double-track molten pool at a laser power of 100 W and a hatch spacing of 0.06 mm. With the gradual increase in scanning speed, the surface morphology of the molten pool evolves from circular to elliptical. When � = 200 mm/s, the slow scanning speed causes the material to absorb too much heat, which is very easy to trigger the overburning phenomenon. At this point, the molten pool is larger and the surface morphology is uneven. This situation is consistent with the previously discussed scenario with high laser power in Fig. 17(a). However, when � = 1600 mm/s, the scanning speed is too fast, resulting in the material not being able to absorb sufficient heat, which triggers the powder particles that fail to melt completely to have a direct effect on the bonding of the melt to the substrate. At this time, the molten pool volume is relatively small and the neighboring melt track cannot lap properly. This result is consistent with the previously discussed case of low laser power in Fig. 17(b). Overall, the ratio of the laser power to the scanning speed (the line energy density �⁠) has a direct effect on the temperature field and surface morphology of the molten pool.

FIG. 17.

VIEW LARGE DOWNLOAD SLIDE

Simulation results of double-track molten pool under different scanning speed: (a) � = 200 mm/s, (b) � = 1600 mm/s.

Table III shows the effects of different scanning speed on the characteristic information of the double-track molten pool under the condition of laser power of 100 W and hatch spacing of 0.06 mm. It can be seen that the scanning speed has a significant effect on the melt depth, melt width, melt height, remolten region, and overlapping ratio. With the increase in scanning speed, the melt depth, melt width, melt height, remelted zone, and overlapping ratio show a gradual decreasing trend. Among them, the melt depth and melt width decreased faster, while the melt height and remolten region decreased relatively slowly. In addition, when the scanning speed was increased from 200 to 800 mm/s, the decreasing speeds of melt depth and melt width were significantly accelerated, while the decreasing speeds of overlapping ratio were relatively slow. When the scanning speed was further increased to 1600 mm/s, the decreasing speeds of melt depth and melt width were further accelerated, and the un-lapped condition of the melt channel also appeared. In addition, the contact angle increases and then decreases with the scanning speed, and both are lower than 33°. Therefore, when selecting the scanning speed, it is necessary to make reasonable trade-offs according to the specific situation, and take into account the factors of melt depth, melt width, melt height, remolten region, and overlapping ratio, in order to achieve the best processing results.

TABLE III.

Double-track molten pool characterization information at different scanning speeds.

Scanning speed (mm/s)	Depth (μm)	Width (μm)	Height (μm)	Remolten region (μm)	Overlapping ratio (%)	Contact angle (°)
200	55/68	182	19/32	124	203.33	22
1600	13	50	11	/	−16.67	31

3. Hatch spacing

Figure 18 shows the effect of different hatch spacing on the morphology and temperature field of the double-track molten pool under the condition of laser power of 100 W and scanning speed of 800 mm/s. The surface morphology and temperature field of the first track and second track are basically the same, but slightly different. The first track shows a basically symmetric morphology along the scanning direction, while the second track shows a slight offset due to the difference in the heat transfer rate between the solidified material and the powder particles. When the hatch spacing is too small, the overlapping ratio increases and the probability of defects caused by remelting phenomenon grows. When the hatch spacing is too large, the neighboring melt track cannot overlap properly, and the powder particles are not completely melted, leading to an increase in the number of holes. In conclusion, the ratio of the line energy density � to the hatch spacing (the volume energy density E) has a significant effect on the temperature field and surface morphology of the molten pool.

FIG. 18.

VIEW LARGE DOWNLOAD SLIDE

Simulation results of double-track molten pool under different hatch spacings: (a) H = 0.03 mm, (b) H = 0.12 mm.

Table IV shows the effects of different hatch spacing on the characteristic information of the double-track molten pool under the condition of laser power of 100 W and scanning speed of 800 mm/s. It can be seen that the hatch spacing has little effect on the melt depth, melt width, and melt height, but has some effect on the remolten region. With the gradual expansion of hatch spacing, the remolten region shows a gradual decrease. At the same time, the overlapping ratio also decreased with the increase in hatch spacing. In addition, it is observed that the contact angle shows a tendency to increase and then remain stable when the hatch spacing increases, which has a more limited effect on it. Therefore, trade-offs and decisions need to be made on a case-by-case basis when selecting the hatch spacing.

TABLE IV.

Double-track molten pool characterization information at different hatch spacings.

Hatch spacing (mm)	Depth (μm)	Width (μm)	Height (μm)	Remolten region (μm)	Overlapping ratio (%)	Contact angle (°)
0.03	25/27	82	14	59	173.33	30
0.12	26	78	14	/	−35	33

In summary, the laser power, scanning speed, and hatch spacing have a significant effect on the formation of the molten pool, and the correct selection of these three process parameters is crucial to ensure the forming quality. In addition, the melt depth of the second track is slightly larger than that of the first track at higher line energy density � and volume energy density E. This is mainly due to the fact that a large amount of heat accumulation is generated after the first track, forming a larger molten pool volume, which leads to an increase in the melt depth.

D. Simulation analysis of molten pool with powder particle size and laser focal spot diameter

Figure 19 demonstrates the effect of different powder particle sizes and laser focal spot diameters on the morphology and temperature field of the double-track molten pool under a laser power of 100 W, a scanning speed of 800 mm/s, and a hatch spacing of 0.06 mm. In the process of melting coarse powder with small laser focal spot diameter, the laser energy cannot completely melt the larger powder particles, resulting in their partial melting and further generating excessive pore defects. The larger powder particles tend to generate zigzag molten pool edges, which cause an increase in the roughness of the melt track surface. In addition, the molten pool is also prone to generate the present spatter phenomenon, which can directly affect the quality of forming. The volume of the formed molten pool is relatively small, while the melt depth, melt width, and melt height are all smaller relative to the fine powder in Fig. 19(a). In the process of melting fine powders with a large laser focal spot diameter, the laser energy is able to melt the fine powder particles sufficiently, even to the point of overmelting. This results in a large number of fine spatters being generated at the edge of the molten pool, which causes porosity defects in the melt track in Fig. 19(b). In addition, the maximum velocity of the molten pool is larger for large powder particle sizes compared to small powder particle sizes, which indicates that the temperature gradient in the molten pool is larger for large powder particle sizes and the melt motion is more intense. However, the size of the laser focal spot diameter has a relatively small effect on the melt motion. However, a larger focal spot diameter induces a larger melt volume with greater depth, width, and height. In conclusion, a small powder size helps to reduce the surface roughness of the specimen, and a small laser spot diameter reduces the minimum forming size of a single track.

FIG. 19.

VIEW LARGE DOWNLOAD SLIDE

Simulation results of double-track molten pool with different powder particle size and laser focal spot diameter: (a) focal spot = 25 μm, coarse powder, (b) focal spot = 80 μm, fine powder.

Table V shows the maximum temperature gradient at the reference point for different powder sizes and laser focal spot diameters. As can be seen from the table, the maximum temperature gradient is lower than that of HP-LPBF for both coarse powders with a small laser spot diameter and fine powders with a large spot diameter, a phenomenon that leads to an increase in the heat transfer rate of HP-LPBF, which in turn leads to a corresponding increase in the cooling rate and, ultimately, to the formation of finer microstructures.

TABLE V.

Maximum temperature gradient at the reference point for different powder particle sizes and laser focal spot diameters.

Laser power (W)	Scanning speed (mm/s)	Hatch spacing (mm)	Average powder size (μm)	Laser focal spot diameter (μm)	Maximum temperature gradient (×10⁷ K/s)
100	800	0.06	31.7	25	7.89
100	800	0.06	11.5	80	7.11

IV. CONCLUSIONS

In this study, the geometrical characteristics of 3D coarse and fine powder particles were first calculated using DEM and then numerical simulations of single track and double track in the process of forming SS316L from monolayer HP-LPBF at mesoscopic scale were developed using CFD method. The effects of Marangoni convection, surface tension, recoil pressure, gravity, thermal convection, thermal radiation, and evaporative heat dissipation on the heat and mass transfer in the molten pool were considered in this model. The effects of laser power, scanning speed, and hatch spacing on the dynamics of the single-track and double-track molten pools, as well as on other characteristic information, were investigated. The effects of the powder particle size on the molten pool were investigated comparatively with the laser focal spot diameter. The main conclusions are as follows:

The results show that the temperature gradient at the front of the molten pool is significantly larger than that at the tail, and the molten pool exhibits a “comet” morphology. At the top of the molten pool, there is a slightly concave region, which is the result of the coupling of Marangoni convection, recoil pressure, and surface tension. The melt flow forms two closed loops, which are mainly influenced by temperature gradients and surface tension. This special dynamic behavior of the melt tends to form an “elliptical” molten pool and an almost “mountain” shape in single-track forming.
The basic characteristics of the three-dimensional morphology and temperature field of the second track are similar to those of the first track, but there are subtle differences. The first track exhibits a basically symmetrical shape; however, due to the difference in thermal diffusion rates between the solidified metal and the powder, a slight asymmetry in the molten pool morphology of the second track occurs. After forming through the first track, there is a significant heat buildup in the powder bed, resulting in a longer dynamic time of the melt, which increases the life of the molten pool. The heights of the first track and second track remained essentially the same, but the depth of the second track was greater relative to the first track. In addition, the maximum temperature gradient was 1.69 × 10⁸ K/s during HP-LPBF forming.
At low laser power, the surface tension in the molten pool plays a dominant role. At high laser power, recoil pressure becomes the main influencing factor. With the increase of laser power, the effective heat transfer in the vertical direction is superior to that in the horizontal direction. With the gradual increase of scanning speed, the surface morphology of the molten pool evolves from circular to elliptical. In addition, the scanning speed has a significant effect on the melt depth, melt width, melt height, remolten region, and overlapping ratio. Too large or too small hatch spacing will lead to remelting or non-lap phenomenon, which in turn causes the formation of defects.
When using a small laser focal spot diameter, it is difficult to completely melt large powder particle sizes, resulting in partial melting and excessive porosity generation. At the same time, large powder particles produce curved edges of the molten pool, resulting in increased surface roughness of the melt track. In addition, spatter occurs, which directly affects the forming quality. At small focal spot diameters, the molten pool volume is relatively small, and the melt depth, the melt width, and the melt height are correspondingly small. Taken together, the small powder particle size helps to reduce surface roughness, while the small spot diameter reduces the forming size.

REFERENCES

S. L. Sing and W. Y. Yeong , “ Laser powder bed fusion for metal additive manufacturing: Perspectives on recent developments,” Virtual Phys. Prototyping. 15, 359–370 (2020).https://doi.org/10.1080/17452759.2020.1779999
Google ScholarCrossref
A. M. Khorasani , I. G. Jithin , J. K. Veetil , and A. H. Ghasemi , “ A review of technological improvements in laser-based powder bed fusion of metal printers,” Int. J. Adv. Manuf. Technol. 108, 191–209 (2020).https://doi.org/10.1007/s00170-020-05361-3
Google ScholarCrossref
Y. Qin , A. Brockett , Y. Ma , A. Razali , J. Zhao , C. Harrison , W. Pan , X. Dai , and D. Loziak , “ Micro-manufacturing: Research, technology outcomes and development issues,” Int. J. Adv. Manuf. Technol. 47, 821–837 (2010).https://doi.org/10.1007/s00170-009-2411-2
Google ScholarCrossref
B. Nagarajan , Z. Hu , X. Song , W. Zhai , and J. Wei , “ Development of micro selective laser melting: The state of the art and future perspectives,” Engineering. 5, 702–720 (2019).https://doi.org/10.1016/j.eng.2019.07.002
Google ScholarCrossref
Y. Wei , G. Chen , W. Li , Y. Zhou , Z. Nie , J. Xu , and W. Zhou , “ Micro selective laser melting of SS316L: Single tracks, defects, microstructures and thermal/mechanical properties,” Opt. Laser Technol. 145, 107469 (2022).https://doi.org/10.1016/j.optlastec.2021.107469
Google ScholarCrossref
Y. Wei , G. Chen , W. Li , M. Li , Y. Zhou , Z. Nie , and J. Xu , “ Process optimization of micro selective laser melting and comparison of different laser diameter for forming different powder,” Opt. Laser Technol. 150, 107953 (2022).https://doi.org/10.1016/j.optlastec.2022.107953
Google ScholarCrossref
H. Zhiheng , B. Nagarajan , X. Song , R. Huang , W. Zhai , and J. Wei , “ Formation of SS316L single tracks in micro selective laser melting: Surface, geometry, and defects,” Adv. Mater. Sci. Eng. 2019, 9451406.https://doi.org/10.1155/2019/9451406
Crossref
B. Nagarajan , Z. Hu , S. Gao , X. Song , R. Huang , M. Seita , and J. Wei , “ Effect of in-situ laser remelting on the microstructure of SS316L fabricated by micro selective laser melting,” in Advanced Surface Enhancement, edited by Sho Itoh and Shashwat Shukla , Lecture Notes in Mechanical Engineering ( Springer Singapore, Singapore, 2020), pp. 330–336.
Google ScholarCrossref
H. Zhiheng , B. Nagarajan , X. Song , R. Huang , W. Zhai , and J. Wei , “ Tailoring surface roughness of micro selective laser melted SS316L by in-situ laser remelting,” in Advanced Surface Enhancement, edited by Sho Itoh and Shashwat Shukla , Lecture Notes in Mechanical Engineering ( Springer Singapore, Singapore, 2020), pp. 337–343.
Google Scholar
J. Fu , Z. Hu , X. Song , W. Zhai , Y. Long , H. Li , and M. Fu , “ Micro selective laser melting of NiTi shape memory alloy: Defects, microstructures and thermal/mechanical properties,” Opt. Laser Technol. 131, 106374 (2020).https://doi.org/10.1016/j.optlastec.2020.106374
Google ScholarCrossref
E. Abele and M. Kniepkamp , “ Analysis and optimisation of vertical surface roughness in micro selective laser melting,” Surf. Topogr.: Metrol. Prop. 3, 034007 (2015).https://doi.org/10.1088/2051-672X/3/3/034007
Google ScholarCrossref
S. Qu , J. Ding , J. Fu , M. Fu , B. Zhang , and X. Song , “ High-precision laser powder bed fusion processing of pure copper,” Addit. Manuf. 48, 102417 (2021).https://doi.org/10.1016/j.addma.2021.102417
Google ScholarCrossref
Y. Wei , G. Chen , M. Li , W. Li , Y. Zhou , J. Xu , and Z. wei , “ High-precision laser powder bed fusion of 18Ni300 maraging steel and its SiC reinforcement composite materials,” J. Manuf. Process. 84, 750–763 (2022).https://doi.org/10.1016/j.jmapro.2022.10.049
Google ScholarCrossref
B. Liu , R. Wildman , T. Christopher , I. Ashcroft , and H. Richard , “ Investigation the effect of particle size distribution on processing parameters optimisation in selective laser melting process,” in 2011 International Solid Freeform Fabrication Symposium ( University of Texas at Austin, 2011).
Google Scholar
T. D. McLouth , G. E. Bean , D. B. Witkin , S. D. Sitzman , P. M. Adams , D. N. Patel , W. Park , J.-M. Yang , and R. J. Zaldivar , “ The effect of laser focus shift on microstructural variation of Inconel 718 produced by selective laser melting,” Mater. Des. 149, 205–213 (2018).https://doi.org/10.1016/j.matdes.2018.04.019
Google ScholarCrossref
Y. Qian , Y. Wentao , and L. Feng , “ Mesoscopic simulations of powder bed fusion: Research progresses and conditions,” Electromachining Mould 06, 46–52 (2017).https://doi.org/10.3969/j.issn.1009-279X.2017.06.012
Google Scholar
J. Fu , S. Qu , J. Ding , X. Song , and M. W. Fu , “ Comparison of the microstructure, mechanical properties and distortion of stainless Steel 316L fabricated by micro and conventional laser powder bed fusion,” Addit. Manuf. 44, 102067 (2021).https://doi.org/10.1016/j.addma.2021.102067
Google ScholarCrossref
N. T. Aboulkhair , I. Maskery , C. Tuck , I. Ashcroft , and N. M. Everitt , “ The microstructure and mechanical properties of selectively laser Melted AlSi10Mg: The effect of a conventional T6-like heat treatment,” Mater. Sci. Eng. A 667, 139–146 (2016).https://doi.org/10.1016/j.msea.2016.04.092
Google ScholarCrossref
S. Y. Chen , J. C. Huang , C. T. Pan , C. H. Lin , T. L. Yang , Y. S. Huang , C. H. Ou , L. Y. Chen , D. Y. Lin , H. K. Lin , T. H. Li , J. S. C. Jang , and C. C. Yang , “ Microstructure and mechanical properties of open-cell porous Ti-6Al-4V fabricated by selective laser melting,” J. Alloys Compd. 713, 248–254 (2017).https://doi.org/10.1016/j.jallcom.2017.04.190
Google ScholarCrossref
Y. Bai , Y. Yang , D. Wang , and M. Zhang , “ Influence mechanism of parameters process and mechanical properties evolution mechanism of Maraging steel 300 by selective laser melting,” Mater. Sci. Eng. A 703, 116–123 (2017).https://doi.org/10.1016/j.msea.2017.06.033
Google ScholarCrossref
Y. Bai , Y. Yang , Z. Xiao , M. Zhang , and D. Wang , “ Process optimization and mechanical property evolution of AlSiMg0.75 by selective laser melting,” Mater. Des. 140, 257–266 (2018).https://doi.org/10.1016/j.matdes.2017.11.045
Google ScholarCrossref
Y. Liu , M. Zhang , W. Shi , Y. Ma , and J. Yang , “ Study on performance optimization of 316L stainless steel parts by high-efficiency selective laser melting,” Opt. Laser Technol. 138, 106872 (2021).https://doi.org/10.1016/j.optlastec.2020.106872
Google ScholarCrossref
D. Gu , Y.-C. Hagedorn , W. Meiners , G. Meng , R. J. S. Batista , K. Wissenbach , and R. Poprawe , “ Densification behavior, microstructure evolution, and wear performance of selective laser melting processed commercially pure titanium,” Acta Mater. 60, 3849–3860 (2012).https://doi.org/10.1016/j.actamat.2012.04.006
Google ScholarCrossref
N. Read , W. Wang , K. Essa , and M. M. Attallah , “ Selective laser melting of AlSi10Mg alloy: Process optimisation and mechanical properties development,” Mater. Des. 65, 417–424 (2015).https://doi.org/10.1016/j.matdes.2014.09.044
Google ScholarCrossref
I. A. Roberts , C. J. Wang , R. Esterlein , M. Stanford , and D. J. Mynors , “ A three-dimensional finite element analysis of the temperature field during laser melting of metal powders in additive layer manufacturing,” Int. J. Mach. Tools Manuf. 49(12–13), 916–923 (2009).https://doi.org/10.1016/j.ijmachtools.2009.07.004
Google ScholarCrossref
K. Dai and L. Shaw , “ Finite element analysis of the effect of volume shrinkage during laser densification,” Acta Mater. 53(18), 4743–4754 (2005).https://doi.org/10.1016/j.actamat.2005.06.014
Google ScholarCrossref
K. Carolin , E. Attar , and P. Heinl , “ Mesoscopic simulation of selective beam melting processes,” J. Mater. Process. Technol. 211(6), 978–987 (2011).https://doi.org/10.1016/j.jmatprotec.2010.12.016
Google ScholarCrossref
F.-J. Gürtler , M. Karg , K.-H. Leitz , and M. Schmidt , “ Simulation of laser beam melting of steel powders using the three-dimensional volume of fluid method,” Phys. Procedia 41, 881–886 (2013).https://doi.org/10.1016/j.phpro.2013.03.162
Google ScholarCrossref
P. Meakin and R. Jullien , “ Restructuring effects in the rain model for random deposition,” J. Phys. France 48(10), 1651–1662 (1987).https://doi.org/10.1051/jphys:0198700480100165100
Google ScholarCrossref
J-m Wang , G-h Liu , Y-l Fang , and W-k Li , “ Marangoni effect in nonequilibrium multiphase system of material processing,” Rev. Chem. Eng. 32(5), 551–585 (2016).https://doi.org/10.1515/revce-2015-0067
Google ScholarCrossref
W. Ye , S. Zhang , L. L. Mendez , M. Farias , J. Li , B. Xu , P. Li , and Y. Zhang , “ Numerical simulation of the melting and alloying processes of elemental titanium and boron powders using selective laser alloying,” J. Manuf. Process. 64, 1235–1247 (2021).https://doi.org/10.1016/j.jmapro.2021.02.044
Google ScholarCrossref
U. S. Bertoli , A. J. Wolfer , M. J. Matthews , J.-P. R. Delplanque , and J. M. Schoenung , “ On the limitations of volumetric energy density as a design parameter for selective laser melting,” Mater. Des. 113, 331–340 (2017).https://doi.org/10.1016/j.matdes.2016.10.037
Google ScholarCrossref
W. E. King , H. D. Barth , V. M. Castillo , G. F. Gallegos , J. W. Gibbs , D. E. Hahn , C. Kamath , and A. M. Rubenchik , “ Observation of keyhole-mode laser melting in laser powder-bed fusion additive manufacturing,” J. Mater. Process. Technol. 214(12), 2915–2925 (2014).https://doi.org/10.1016/j.jmatprotec.2014.06.005
Google ScholarCrossref
L. Cao , “ Numerical simulation of the impact of laying powder on selective laser melting single-pass formation,” Int. J. Heat Mass Transfer 141, 1036–1048 (2019).https://doi.org/10.1016/j.ijheatmasstransfer.2019.07.053
Google ScholarCrossref
L. Huang , X. Hua , D. Wu , and F. Li , “ Numerical study of keyhole instability and porosity formation mechanism in laser welding of aluminum alloy and steel,” J. Mater. Process. Technol. 252, 421–431 (2018).https://doi.org/10.1016/j.jmatprotec.2017.10.011
Google ScholarCrossref
K. Q. Le , C. Tang , and C. H. Wong , “ On the study of keyhole-mode melting in selective laser melting process,” Int. J. Therm. Sci. 145, 105992 (2019).https://doi.org/10.1016/j.ijthermalsci.2019.105992
Google ScholarCrossref
J.-H. Cho and S.-J. Na , “ Theoretical analysis of keyhole dynamics in polarized laser drilling,” J. Phys. D: Appl. Phys. 40(24), 7638 (2007).https://doi.org/10.1088/0022-3727/40/24/007
Google ScholarCrossref
W. Ye , “ Mechanism analysis of selective laser melting and metallurgy process based on base element powder of titanium and boron,” Ph.D. dissertation ( Nanchang University, 2021).
Google Scholar
R. Ammer , M. Markl , U. Ljungblad , C. Körner , and U. Rüde , “ Simulating fast electron beam melting with a parallel thermal free surface lattice Boltzmann method,” Comput. Math. Appl. 67(2), 318–330 (2014).https://doi.org/10.1016/j.camwa.2013.10.001
Google ScholarCrossref
H. Chen , Q. Wei , S. Wen , Z. Li , and Y. Shi , “ Flow behavior of powder particles in layering process of selective laser melting: Numerical modeling and experimental verification based on discrete element method,” Int. J. Mach. Tools Manuf. 123, 146–159 (2017).https://doi.org/10.1016/j.ijmachtools.2017.08.004
Google ScholarCrossref
F. Verhaeghe , T. Craeghs , J. Heulens , and L. Pandelaers , “ A pragmatic model for selective laser melting with evaporation,” Acta Mater. 57(20), 6006–6012 (2009).https://doi.org/10.1016/j.actamat.2009.08.027
Google ScholarCrossref
C. H. Fu and Y. B. Guo , “ Three-dimensional temperature gradient mechanism in selective laser melting of Ti-6Al-4V,” J. Manuf. Sci. Eng. 136(6), 061004 (2014).https://doi.org/10.1115/1.4028539
Google ScholarCrossref
Y. Xiang , Z. Shuzhe , L. Junfeng , W. Zhengying , Y. Lixiang , and J. Lihao , “ Numerical simulation and experimental verification for selective laser single track melting forming of Ti6Al4V,” J. Zhejiang Univ. (Eng. Sci.) 53(11), 2102–2109 + 2117 (2019).https://doi.org/10.3785/j.issn.1008-973X.2019.11.007
Google Scholar
Q. He , H. Xia , J. Liu , X. Ao , and S. Lin , “ Modeling and numerical studies of selective laser melting: Multiphase flow, solidification and heat transfer,” Mater. Des. 196, 109115 (2020).https://doi.org/10.1016/j.matdes.2020.109115
Google ScholarCrossref
L. Cao , “ Mesoscopic-scale numerical simulation including the influence of process parameters on SLM single-layer multi-pass formation,” Metall. Mater. Trans. A 51, 4130–4145 (2020).https://doi.org/10.1007/s11661-020-05831-z
Google ScholarCrossref
L. Cao , “ Mesoscopic-scale numerical investigation including the influence of process parameters on LPBF multi-layer multi-path formation,” Comput. Model. Eng. Sci. 126(1), 5–23 (2021).https://doi.org/10.32604/cmes.2021.014693
Google ScholarCrossref
H. Yin and S. D. Felicelli , “ Dendrite growth simulation during solidification in the LENS process,” Acta Mater. 58(4), 1455–1465 (2010).https://doi.org/10.1016/j.actamat.2009.10.053
Google ScholarCrossref
P. Nie , O. A. Ojo , and Z. Li , “ Numerical modeling of microstructure evolution during laser additive manufacturing of a nickel-based superalloy,” Acta Mater. 77, 85–95 (2014).https://doi.org/10.1016/j.actamat.2014.05.039
Google ScholarCrossref
Z. Liu and H. Qi , “ Effects of substrate crystallographic orientations on crystal growth and microstructure formation in laser powder deposition of nickel-based superalloy,” Acta Mater. 87, 248–258 (2015).https://doi.org/10.1016/j.actamat.2014.12.046
Google ScholarCrossref
L. Wei , L. Xin , W. Meng , and H. Weidong , “ Cellular automaton simulation of the molten pool of laser solid forming process,” Acta Phys. Sin. 64(01), 018103–018363 (2015).https://doi.org/10.7498/aps.64.018103
Google ScholarCrossref
R. Acharya , J. A. Sharon , and A. Staroselsky , “ Prediction of microstructure in laser powder bed fusion process,” Acta Mater. 124, 360–371 (2017).https://doi.org/10.1016/j.actamat.2016.11.018
Google ScholarCrossref
M. R. Rolchigo and R. LeSar , “ Modeling of binary alloy solidification under conditions representative of additive manufacturing,” Comput. Mater. Sci. 150, 535–545 (2018).https://doi.org/10.1016/j.commatsci.2018.04.004
Google ScholarCrossref
S. Geng , P. Jiang , L. Guo , X. Gao , and G. Mi , “ Multi-scale simulation of grain/sub-grain structure evolution during solidification in laser welding of aluminum alloys,” Int. J. Heat Mass Transfer 149, 119252 (2020).https://doi.org/10.1016/j.ijheatmasstransfer.2019.119252
Google ScholarCrossref
W. L. Wang , W. Q. Liu , X. Yang , R. R. Xu , and Q. Y. Dai , “ Multi-scale simulation of columnar-to-equiaxed transition during laser selective melting of rare earth magnesium alloy,” J. Mater. Sci. Technol. 119, 11–24 (2022).https://doi.org/10.1016/j.jmst.2021.12.029
Google ScholarCrossref
Q. Xia , J. Yang , and Y. Li , “ On the conservative phase-field method with the N-component incompressible flows,” Phys. Fluids 35, 012120 (2023).https://doi.org/10.1063/5.0135490
Google ScholarCrossref
Q. Xia , G. Sun , J. Kim , and Y. Li , “ Multi-scale modeling and simulation of additive manufacturing based on fused deposition technique,” Phys. Fluids 35, 034116 (2023).https://doi.org/10.1063/5.0141316
Google ScholarCrossref
A. Hussein , L. Hao , C. Yan , and R. Everson , “ Finite element simulation of the temperature and stress fields in single layers built without-support in selective laser melting,” Mater. Des. 52, 638–647 (2013).https://doi.org/10.1016/j.matdes.2013.05.070
Google ScholarCrossref
J. Ding , P. Colegrove , J. Mehnen , S. Ganguly , P. M. Sequeira Almeida , F. Wang , and S. Williams , “ Thermo-mechanical analysis of wire and arc additive layer manufacturing process on large multi-layer parts,” Comput. Mater. Sci. 50(12), 3315–3322 (2011).https://doi.org/10.1016/j.commatsci.2011.06.023
Google ScholarCrossref
Y. Du , X. You , F. Qiao , L. Guo , and Z. Liu , “ A model for predicting the temperature field during selective laser melting,” Results Phys. 12, 52–60 (2019).https://doi.org/10.1016/j.rinp.2018.11.031
Google ScholarCrossref
X. Luo , M. Liu , L. Zhenhua , H. Li , and J. Shen , “ Effect of different heat-source models on calculated temperature field of selective laser melted 18Ni300,” Chin. J. Lasers 48(14), 1402005–1402062 (2021).https://doi.org/10.3788/CJL202148.1402005
Google ScholarCrossref
J. F. Li , L. Li , and F. H. Stott , “ Thermal stresses and their implication on cracking during laser melting of ceramic materials,” Acta Mater. 52(14), 4385–4398 (2004).https://doi.org/10.1016/j.actamat.2004.06.005
Google ScholarCrossref
P. Aggarangsi and J. L. Beuth , “ Localized preheating approaches for reducing residual stress in additive manufacturing,” paper presented at the 2006 International Solid Freeform Fabrication Symposium, The University of Texas in Austin on August 14–16, 2006.
K. Dai and L. Shaw , “ Thermal and mechanical finite element modeling of laser forming from metal and ceramic powders,” Acta Mater. 52(1), 69–80 (2004).https://doi.org/10.1016/j.actamat.2003.08.028
Google ScholarCrossref
A. H. Nickel , D. M. Barnett , and F. B. Prinz , “ Thermal stresses and deposition patterns in layered manufacturing,” Mater. Sci. Eng. A 317(1–2), 59–64 (2001).https://doi.org/10.1016/S0921-5093(01)01179-0
Google ScholarCrossref
M. F. Zaeh and G. Branner , “ Investigations on residual stresses and deformations in selective laser melting,” Prod. Eng. 4(1), 35–45 (2010).https://doi.org/10.1007/s11740-009-0192-y
Google ScholarCrossref
P. Bian , J. Shi , Y. Liu , and Y. Xie , “ Influence of laser power and scanning strategy on residual stress distribution in additively manufactured 316L steel,” Opt. Laser Technol. 132, 106477 (2020).https://doi.org/10.1016/j.optlastec.2020.106477
Google ScholarCrossref
B. M. Marques , C. M. Andrade , D. M. Neto , M. C. Oliveira , J. L. Alves , and L. F. Menezes , “ Numerical analysis of residual stresses in parts produced by selective laser melting process,” Procedia Manuf. 47, 1170–1177 (2020).https://doi.org/10.1016/j.promfg.2020.04.167
Google ScholarCrossref
W. Mu , “ Numerical simulation of SLM forming process and research and prediction of forming properties,” MA thesis ( Anhui Jianzhu University, 2022).
Google Scholar
Y. Zhang , “ Multi-scale multi-physics modeling of laser powder bed fusion process of metallic materials with experiment validation,” Ph.D. dissertation ( Purdue University, 2018).
Google Scholar
Y. Qian , “ Mesoscopic simulation studies of key processing issues for powder bed fusion technology,” Ph.D. dissertation ( Tsinghua University, 2019).
Google Scholar
N. V. Brilliantov , S. Frank , J.-M. Hertzsch , and T. Pöschel , “ Model for collisions in granular gases,” Phys. Rev. E 53(5), 5382–5392 (1996).https://doi.org/10.1103/PhysRevE.53.5382
Google ScholarCrossref
Z. Xiao , “ Research on microscale selective laser melting process of high strength pure copper specimens,” MA thesis ( Hunan University, 2022).
Google Scholar
Z. Li , K. Mukai , M. Zeze , and K. C. Mills , “ Determination of the surface tension of liquid stainless steel,” J. Mater. Sci. 40(9–10), 2191–2195 (2005).https://doi.org/10.1007/s10853-005-1931-x
Google ScholarCrossref
R. Scardovelli and S. Zaleski , “ Analytical relations connecting linear interfaces and volume fractions in rectangular grids,” J. Comput. Phys. 164(1), 228–237 (2000).https://doi.org/10.1006/jcph.2000.6567
Google ScholarCrossref
D.-W. Cho , W.-I. Cho , and S.-J. Na , “ Modeling and simulation of arc: Laser and hybrid welding process,” J. Manuf. Process. 16(1), 26–55 (2014).https://doi.org/10.1016/j.jmapro.2013.06.012
Google ScholarCrossref
76.Flow3D. Version 11.1.0: User Manual ( FlowScience, Santa Fe, NM, USA, 2015).
Y. Tian , L. Yang , D. Zhao , Y. Huang , and J. Pan , “ Numerical analysis of powder bed generation and single track forming for selective laser melting of ss316l stainless steel,” J. Manuf. Process. 58, 964–974 (2020).https://doi.org/10.1016/j.jmapro.2020.09.002
Google ScholarCrossref
C. Tang , K. Q. Le , and C. H. Wong , “ Physics of humping formation in laser powder bed fusion,” Int. J. Heat Mass Transfer 149, 119172 (2020).https://doi.org/10.1016/j.ijheatmasstransfer.2019.119172
Google ScholarCrossref
L. Cao , “ Mesoscopic-scale simulation of pore evolution during laser powder bed fusion process,” Comput. Mater. Sci. 179, 109686 (2020).https://doi.org/10.1016/j.commatsci.2020.109686
Google ScholarCrossref
R. Li , J. Liu , Y. Shi , W. Li , and W. Jiang , “ Balling behavior of stainless steel and nickel powder during selective laser melting process,” Int. J. Adv. Manuf. Technol. 59(9–12), 1025–1035 (2012).https://doi.org/10.1007/s00170-011-3566-1
Google ScholarCrossref
S. A. Khairallah and A. Anderson , “ Mesoscopic simulation model of selective laser melting of stainless steel powder,” J. Mater. Process. Technol. 214(11), 2627–2636 (2014).https://doi.org/10.1016/j.jmatprotec.2014.06.001
Google ScholarCrossref
J. Liu , D. Gu , H. Chen , D. Dai , and H. Zhang , “ Influence of substrate surface morphology on wetting behavior of tracks during selective laser melting of aluminum-based alloys,” J. Zhejiang Univ. Sci. A 19(2), 111–121 (2018).https://doi.org/10.1631/jzus.A1700599
Google ScholarCrossref
L. Li , J. Li , and T. Fan , “ Phase-field modeling of wetting and balling dynamics in powder bed fusion process,” Phys. Fluids 33, 042116 (2021).https://doi.org/10.1063/5.0046771
Google ScholarCrossref
X. Nie , Z. Hu , H. Zhu , Z. Hu , L. Ke , and X. Zeng , “ Analysis of processing parameters and characteristics of selective laser melted high strength Al-Cu-Mg alloys: from single tracks to cubic samples,” J. Mater. Process. Technol. 256, 69–77 (2018).https://doi.org/10.1016/j.jmatprotec.2018.01.030
Google ScholarCrossref

원본 바로가기

Figure 3. Computed contour of velocity magnitude (m/s) for Run 1 to Run 15.

FLOW-3D 소프트웨어를 이용한 유입구 및 배플 위치가 침전조 제거 효율에 미치는 영향

Ali Poorkarimi¹
Khaled Mafakheri²
Shahrzad Maleki²

Journal of Hydraulic Structures
J. Hydraul. Struct., 2023; 9(4): 76-87
DOI: 10.22055/jhs.2024.44817.1265

Abstract

중력에 의한 침전은 부유 물질을 제거하기 위해 물과 폐수 처리 공정에 널리 적용됩니다. 이 연구에서는 침전조의 제거 효율에 대한 입구 및 배플 위치의 영향을 간략하게 설명합니다. 실험은 CCD(중심복합설계) 방법론을 기반으로 수행되었습니다. 전산유체역학(CFD)은 유압 설계, 미래 발전소에 대한 계획 연구, 토목 유지 관리 및 공급 효율성과 관련된 복잡한 문제를 모델링하고 분석하는 데 광범위하게 사용됩니다. 본 연구에서는 입구 높이, 입구로부터 배플까지의 거리, 배플 높이의 다양한 조건에 따른 영향을 조사하였다. CCD 접근 방식을 사용하여 얻은 데이터를 분석하면 축소된 2차 모델이 R² = 0.77의 결정 계수로 부유 물질 제거를 예측할 수 있음이 나타났습니다. 연구 결과, 유입구와 배플의 부적절한 위치는 침전조의 효율에 부정적인 영향을 미칠 수 있음을 보여주었습니다. 입구 높이, 배플 거리, 배플 높이의 최적 값은 각각 0.87m, 0.77m, 0.56m였으며 제거 효율은 80.6%였습니다.

Sedimentation due to gravitation is applied widely in water and wastewater treatment processes to remove suspended solids. This study outlines the effect of the inlet and baffle position on the removal efficiency of sedimentation tanks. Experiments were carried out based on the central composite design (CCD) methodology. Computational fluid dynamics (CFD) is used extensively to model and analyze complex issues related to hydraulic design, planning studies for future generating stations, civil maintenance, and supply efficiency. In this study, the effect of different conditions of inlet elevation, baffle’s distance from the inlet, and baffle height were investigated. Analysis of the obtained data with a CCD approach illustrated that the reduced quadratic model can predict the suspended solids removal with a coefficient of determination of R² = 0.77. The results showed that the inappropriate position of the inlet and the baffle can have a negative effect on the efficiency of the sedimentation tank. The optimal values of inlet elevation, baffle distance, and baffle height were 0.87 m, 0.77 m, and 0.56 m respectively with 80.6% removal efficiency.

Keywords

Sedimentation tank, Particle removal, Central Composite Design, Computational
Fluid Dynamics, Flow-3D

References

Shahrokhi, M., F. Rostami, M.A. Md. Said, S.R.S. Yazdi, and Syafalni, (2013). Experimental
investigation of the influence of baffle position on the flow field, sediment concentration, and
efficiency of rectangular primary sedimentation tanks. Journal of Hydraulic Engineering,.
139(1): p. 88-94.
Shahrokhi, M., F. Rostami, M.A.M. Said, and S.R.S. Yazdi, (2012). The effect of number of
baffles on the improvement efficiency of primary sedimentation tanks. Applied Mathematical
Modelling,. 36(8): p. 3725-3735.
Borna, M., A. Janfeshan, E. Merufinia, and A. Asnaashari, (2014). Numerical simulations of
distribution and sediment transmission in pre-settled pools using Finite Volume Method and
comparison with experimental results. Journal of Civil Engineering and Urbanism,. 4(3): p.
287-292.
Rad, M.J., P.E. Firoozabadi, and F. Rostami, (2022). Numerical Investigation of the Effect
Dimensions of Rectangular Sedimentation Tanks on Its Hydraulic Efficiency Using Flow-3D
Software. Acta Technica Jaurinensis,. 15(4): p. 207-220.
Hirom, K. and T.T. Devi, (2022). Application of computational fluid dynamics in sedimentation
tank design and its recent developments: A review. Water, Air, & Soil Pollution,. 233: p. 1-
26.
Shahrokhi, M., F. Rostami, M.A.M. Said, S.-R. Sabbagh-Yazdi, S. Syafalni, and R. Abdullah,
(2012). The effect of baffle angle on primary sedimentation tank efficiency. Canadian Journal
of Civil Engineering,. 39(3): p. 293-303.
Tarpagkou, R. and A. Pantokratoras, (2014). The influence of lamellar settler in sedimentation
tanks for potable water treatment—A computational fluid dynamic study. Powder
Technology,. 268: p. 139-149.
Ekama, G. and P. Marais, (2004). Assessing the applicability of the 1D flux theory to full-scale
secondary settling tank design with a 2D hydrodynamic model. Water research,. 38(3): p. 495-
506.
Gharagozian, A., (1998). Circular secondary clarifier investigations using a numerical model.,
University of California, Los Angeles.
Shahrokhi, M., F. Rostami, and M.A.M. Said, (2013). Numerical modeling of baffle location
effects on the flow pattern of primary sedimentation tanks. Applied Mathematical Modelling,.
37(6): p. 4486-4496.
Razmi, A.M., R. Bakhtyar, B. Firoozabadi, and D.A. Barry, (2013). Experiments and
numerical modeling of baffle configuration effects on the performance of sedimentation tanks.
Canadian Journal of Civil Engineering,. 40(2): p. 140-150.
Liu, Y., P. Zhang, and W. Wei, (2016). Simulation of effect of a baffle on the flow patterns
and hydraulic efficiency in a sedimentation tank. Desalination and Water Treatment,. 57(54):
p. 25950-25959.
Saeedi, E., E. Behnamtalab, and S. Salehi Neyshabouri, (2020). Numerical simulation of baffle
effect on the performance of sedimentation basin. Water and environment journal,. 34(2): p.
212-222.
Miri, J.K., B. Aminnejad, and A. Zahiri, (2023). Numerical Study of Flow Pattern, Sediment
Field and Effect of the Arrangement of Guiding Blades (Baffles) on Sedimentation in PreSedimentation Basins by Numerical Models. Water Resources,. 50(1): p. 68-81.
Heydari, M.M., M. Rahbani, and S.M.M. Jamal, (2014). Experimental and numerical
investigations of baffle effect on the removal efficiency of sedimentation basin. Advances in
Environmental Biology,: p. 1015-1022.
Guo, H., S.J. Ki, S. Oh, Y.M. Kim, S. Wang, and J.H. Kim, (2017). Numerical simulation of
separation process for enhancing fine particle removal in tertiary sedimentation tank mounting
adjustable baffle. Chemical engineering science,. 158: p. 21-29.

PDF View

Lab-on-a-Chip 시스템의 혈류 역학에 대한 검토: 엔지니어링 관점

Review on Blood Flow Dynamics in Lab-on-a-Chip Systems: An Engineering Perspective

Bin-Jie Lai
,
Li-Tao Zhu
,
Zhe Chen*
,
Bo Ouyang*
, and
Zheng-Hong Luo*

Abstract

다양한 수송 메커니즘 하에서, “LOC(lab-on-a-chip)” 시스템에서 유동 전단 속도 조건과 밀접한 관련이 있는 혈류 역학은 다양한 수송 현상을 초래하는 것으로 밝혀졌습니다.

본 연구는 적혈구의 동적 혈액 점도 및 탄성 거동과 같은 점탄성 특성의 역할을 통해 LOC 시스템의 혈류 패턴을 조사합니다. 모세관 및 전기삼투압의 주요 매개변수를 통해 LOC 시스템의 혈액 수송 현상에 대한 연구는 실험적, 이론적 및 수많은 수치적 접근 방식을 통해 제공됩니다.

전기 삼투압 점탄성 흐름에 의해 유발되는 교란은 특히 향후 연구 기회를 위해 혈액 및 기타 점탄성 유체를 취급하는 LOC 장치의 혼합 및 분리 기능 향상에 논의되고 적용됩니다. 또한, 본 연구는 보다 정확하고 단순화된 혈류 모델에 대한 요구와 전기역학 효과 하에서 점탄성 유체 흐름에 대한 수치 연구에 대한 강조와 같은 LOC 시스템 하에서 혈류 역학의 수치 모델링의 문제를 식별합니다.

전기역학 현상을 연구하는 동안 제타 전위 조건에 대한 보다 실용적인 가정도 강조됩니다. 본 연구는 모세관 및 전기삼투압에 의해 구동되는 미세유체 시스템의 혈류 역학에 대한 포괄적이고 학제적인 관점을 제공하는 것을 목표로 한다.

원문 PDF 다운로드

KEYWORDS:

1. Introduction

1.1. Microfluidic Flow in Lab-on-a-Chip (LOC) Systems

Over the past several decades, the ability to control and utilize fluid flow patterns at microscales has gained considerable interest across a myriad of scientific and engineering disciplines, leading to growing interest in scientific research of microfluidics.

(1) Microfluidics, an interdisciplinary field that straddles physics, engineering, and biotechnology, is dedicated to the behavior, precise control, and manipulation of fluids geometrically constrained to a small, typically submillimeter, scale.

(2) The engineering community has increasingly focused on microfluidics, exploring different driving forces to enhance working fluid transport, with the aim of accurately and efficiently describing, controlling, designing, and applying microfluidic flow principles and transport phenomena, particularly for miniaturized applications.

(3) This attention has chiefly been fueled by the potential to revolutionize diagnostic and therapeutic techniques in the biomedical and pharmaceutical sectorsUnder various driving forces in microfluidic flows, intriguing transport phenomena have bolstered confidence in sustainable and efficient applications in fields such as pharmaceutical, biochemical, and environmental science. The “lab-on-a-chip” (LOC) system harnesses microfluidic flow to enable fluid processing and the execution of laboratory tasks on a chip-sized scale. LOC systems have played a vital role in the miniaturization of laboratory operations such as mixing, chemical reaction, separation, flow control, and detection on small devices, where a wide variety of fluids is adapted. Biological fluid flow like blood and other viscoelastic fluids are notably studied among the many working fluids commonly utilized by LOC systems, owing to the optimization in small fluid sample volumed, rapid response times, precise control, and easy manipulation of flow patterns offered by the system under various driving forces.

(4)The driving forces in blood flow can be categorized as passive or active transport mechanisms and, in some cases, both. Under various transport mechanisms, the unique design of microchannels enables different functionalities in driving, mixing, separating, and diagnosing blood and drug delivery in the blood.

(5) Understanding and manipulating these driving forces are crucial for optimizing the performance of a LOC system. Such knowledge presents the opportunity to achieve higher efficiency and reliability in addressing cellular level challenges in medical diagnostics, forensic studies, cancer detection, and other fundamental research areas, for applications of point-of-care (POC) devices.

(6)

1.2. Engineering Approach of Microfluidic Transport Phenomena in LOC Systems

Different transport mechanisms exhibit unique properties at submillimeter length scales in microfluidic devices, leading to significant transport phenomena that differ from those of macroscale flows. An in-depth understanding of these unique transport phenomena under microfluidic systems is often required in fluidic mechanics to fully harness the potential functionality of a LOC system to obtain systematically designed and precisely controlled transport of microfluids under their respective driving force. Fluid mechanics is considered a vital component in chemical engineering, enabling the analysis of fluid behaviors in various unit designs, ranging from large-scale reactors to separation units. Transport phenomena in fluid mechanics provide a conceptual framework for analytically and descriptively explaining why and how experimental results and physiological phenomena occur. The Navier–Stokes (N–S) equation, along with other governing equations, is often adapted to accurately describe fluid dynamics by accounting for pressure, surface properties, velocity, and temperature variations over space and time. In addition, limiting factors and nonidealities for these governing equations should be considered to impose corrections for empirical consistency before physical models are assembled for more accurate controls and efficiency. Microfluidic flow systems often deviate from ideal conditions, requiring adjustments to the standard governing equations. These deviations could arise from factors such as viscous effects, surface interactions, and non-Newtonian fluid properties from different microfluid types and geometrical layouts of microchannels. Addressing these nonidealities supports the refining of theoretical models and prediction accuracy for microfluidic flow behaviors.

The analytical calculation of coupled nonlinear governing equations, which describes the material and energy balances of systems under ideal conditions, often requires considerable computational efforts. However, advancements in computation capabilities, cost reduction, and improved accuracy have made numerical simulations using different numerical and modeling methods a powerful tool for effectively solving these complex coupled equations and modeling various transport phenomena. Computational fluid dynamics (CFD) is a numerical technique used to investigate the spatial and temporal distribution of various flow parameters. It serves as a critical approach to provide insights and reasoning for decision-making regarding the optimal designs involving fluid dynamics, even prior to complex physical model prototyping and experimental procedures. The integration of experimental data, theoretical analysis, and reliable numerical simulations from CFD enables systematic variation of analytical parameters through quantitative analysis, where adjustment to delivery of blood flow and other working fluids in LOC systems can be achieved.

Numerical methods such as the Finite-Difference Method (FDM), Finite-Element-Method (FEM), and Finite-Volume Method (FVM) are heavily employed in CFD and offer diverse approaches to achieve discretization of Eulerian flow equations through filling a mesh of the flow domain. A more in-depth review of numerical methods in CFD and its application for blood flow simulation is provided in Section 2.2.2.

1.3. Scope of the Review

In this Review, we explore and characterize the blood flow phenomena within the LOC systems, utilizing both physiological and engineering modeling approaches. Similar approaches will be taken to discuss capillary-driven flow and electric-osmotic flow (EOF) under electrokinetic phenomena as a passive and active transport scheme, respectively, for blood transport in LOC systems. Such an analysis aims to bridge the gap between physical (experimental) and engineering (analytical) perspectives in studying and manipulating blood flow delivery by different driving forces in LOC systems. Moreover, the Review hopes to benefit the interests of not only blood flow control in LOC devices but also the transport of viscoelastic fluids, which are less studied in the literature compared to that of Newtonian fluids, in LOC systems.

Section 2 examines the complex interplay between viscoelastic properties of blood and blood flow patterns under shear flow in LOC systems, while engineering numerical modeling approaches for blood flow are presented for assistance. Sections 3 and 4 look into the theoretical principles, numerical governing equations, and modeling methodologies for capillary driven flow and EOF in LOC systems as well as their impact on blood flow dynamics through the quantification of key parameters of the two driving forces. Section 5 concludes the characterized blood flow transport processes in LOC systems under these two forces. Additionally, prospective areas of research in improving the functionality of LOC devices employing blood and other viscoelastic fluids and potentially justifying mechanisms underlying microfluidic flow patterns outside of LOC systems are presented. Finally, the challenges encountered in the numerical studies of blood flow under LOC systems are acknowledged, paving the way for further research.

2. Blood Flow Phenomena

ARTICLE SECTIONS

Jump To

2.1. Physiological Blood Flow Behavior

Blood, an essential physiological fluid in the human body, serves the vital role of transporting oxygen and nutrients throughout the body. Additionally, blood is responsible for suspending various blood cells including erythrocytes (red blood cells or RBCs), leukocytes (white blood cells), and thrombocytes (blood platelets) in a plasma medium.Among the cells mentioned above, red blood cells (RBCs) comprise approximately 40–45% of the volume of healthy blood.

(7) An RBC possesses an inherent elastic property with a biconcave shape of an average diameter of 8 μm and a thickness of 2 μm. This biconcave shape maximizes the surface-to-volume ratio, allowing RBCs to endure significant distortion while maintaining their functionality.

(8,9) Additionally, the biconcave shape optimizes gas exchange, facilitating efficient uptake of oxygen due to the increased surface area. The inherent elasticity of RBCs allows them to undergo substantial distortion from their original biconcave shape and exhibits high flexibility, particularly in narrow channels.RBC deformability enables the cell to deform from a biconcave shape to a parachute-like configuration, despite minor differences in RBC shape dynamics under shear flow between initial cell locations. As shown in Figure 1(a), RBCs initiating with different resting shapes and orientations displaying display a similar deformation pattern

(10) in terms of its shape. Shear flow induces an inward bending of the cell at the rear position of the rim to the final bending position,

(11) resulting in an alignment toward the same position of the flow direction.

Figure 1. Images of varying deformation of RBCs and different dynamic blood flow behaviors. (a) The deforming shape behavior of RBCs at four different initiating positions under the same experimental conditions of a flow from left to right, (10) (b) RBC aggregation, (13) (c) CFL region. (18) Reproduced with permission from ref (10). Copyright 2011 Elsevier. Reproduced with permission from ref (13). Copyright 2022 The Authors, under the terms of the Creative Commons (CC BY 4.0) License https://creativecommons.org/licenses/by/4.0/. Reproduced with permission from ref (18). Copyright 2019 Elsevier.

The flexible property of RBCs enables them to navigate through narrow capillaries and traverse a complex network of blood vessels. The deformability of RBCs depends on various factors, including the channel geometry, RBC concentration, and the elastic properties of the RBC membrane.

(12) Both flexibility and deformability are vital in the process of oxygen exchange among blood and tissues throughout the body, allowing cells to flow in vessels even smaller than the original cell size prior to deforming.As RBCs serve as major components in blood, their collective dynamics also hugely affect blood rheology. RBCs exhibit an aggregation phenomenon due to cell to cell interactions, such as adhesion forces, among populated cells, inducing unique blood flow patterns and rheological behaviors in microfluidic systems. For blood flow in large vessels between a diameter of 1 and 3 cm, where shear rates are not high, a constant viscosity and Newtonian behavior for blood can be assumed. However, under low shear rate conditions (0.1 s

^–1) in smaller vessels such as the arteries and venules, which are within a diameter of 0.2 mm to 1 cm, blood exhibits non-Newtonian properties, such as shear-thinning viscosity and viscoelasticity due to RBC aggregation and deformability. The nonlinear viscoelastic property of blood gives rise to a complex relationship between viscosity and shear rate, primarily influenced by the highly elastic behavior of RBCs. A wide range of research on the transient behavior of the RBC shape and aggregation characteristics under varied flow circumstances has been conducted, aiming to obtain a better understanding of the interaction between blood flow shear forces from confined flows.

For a better understanding of the unique blood flow structures and rheological behaviors in microfluidic systems, some blood flow patterns are introduced in the following section.

2.1.1. RBC Aggregation

RBC aggregation is a vital phenomenon to be considered when designing LOC devices due to its impact on the viscosity of the bulk flow. Under conditions of low shear rate, such as in stagnant or low flow rate regions, RBCs tend to aggregate, forming structures known as rouleaux, resembling stacks of coins as shown in Figure 1(b).

(13) The aggregation of RBCs increases the viscosity at the aggregated region,

(14) hence slowing down the overall blood flow. However, when exposed to high shear rates, RBC aggregates disaggregate. As shear rates continue to increase, RBCs tend to deform, elongating and aligning themselves with the direction of the flow.

(15) Such a dynamic shift in behavior from the cells in response to the shear rate forms the basis of the viscoelastic properties observed in whole blood. In essence, the viscosity of the blood varies according to the shear rate conditions, which are related to the velocity gradient of the system. It is significant to take the intricate relationship between shear rate conditions and the change of blood viscosity due to RBC aggregation into account since various flow driving conditions may induce varied effects on the degree of aggregation.

2.1.2. Fåhræus-Lindqvist Effect

The Fåhræus–Lindqvist (FL) effect describes the gradual decrease in the apparent viscosity of blood as the channel diameter decreases.

(16) This effect is attributed to the migration of RBCs toward the central region in the microchannel, where the flow rate is higher, due to the presence of higher pressure and asymmetric distribution of shear forces. This migration of RBCs, typically observed at blood vessels less than 0.3 mm, toward the higher flow rate region contributes to the change in blood viscosity, which becomes dependent on the channel size. Simultaneously, the increase of the RBC concentration in the central region of the microchannel results in the formation of a less viscous region close to the microchannel wall. This region called the Cell-Free Layer (CFL), is primarily composed of plasma.

(17) The combination of the FL effect and the following CFL formation provides a unique phenomenon that is often utilized in passive and active plasma separation mechanisms, involving branched and constriction channels for various applications in plasma separation using microfluidic systems.

2.1.3. Cell-Free Layer Formation

In microfluidic blood flow, RBCs form aggregates at the microchannel core and result in a region that is mostly devoid of RBCs near the microchannel walls, as shown in Figure 1(c).

(18) The region is known as the cell-free layer (CFL). The CFL region is often known to possess a lower viscosity compared to other regions within the blood flow due to the lower viscosity value of plasma when compared to that of the aggregated RBCs. Therefore, a thicker CFL region composed of plasma correlates to a reduced apparent whole blood viscosity.

(19) A thicker CFL region is often established following the RBC aggregation at the microchannel core under conditions of decreasing the tube diameter. Apart from the dependence on the RBC concentration in the microchannel core, the CFL thickness is also affected by the volume concentration of RBCs, or hematocrit, in whole blood, as well as the deformability of RBCs. Given the influence CFL thickness has on blood flow rheological parameters such as blood flow rate, which is strongly dependent on whole blood viscosity, investigating CFL thickness under shear flow is crucial for LOC systems accounting for blood flow.

2.1.4. Plasma Skimming in Bifurcation Networks

The uneven arrangement of RBCs in bifurcating microchannels, commonly termed skimming bifurcation, arises from the axial migration of RBCs within flowing streams. This uneven distribution contributes to variations in viscosity across differing sizes of bifurcating channels but offers a stabilizing effect. Notably, higher flow rates in microchannels are associated with increased hematocrit levels, resulting in higher viscosity compared with those with lower flow rates. Parametric investigations on bifurcation angle,

(20) thickness of the CFL,

(21) and RBC dynamics, including aggregation and deformation,

(22) may alter the varying viscosity of blood and its flow behavior within microchannels.

2.2. Modeling on Blood Flow Dynamics

2.2.1. Blood Properties and Mathematical Models of Blood Rheology

Under different shear rate conditions in blood flow, the elastic characteristics and dynamic changes of the RBC induce a complex velocity and stress relationship, resulting in the incompatibility of blood flow characterization through standard presumptions of constant viscosity used for Newtonian fluid flow. Blood flow is categorized as a viscoelastic non-Newtonian fluid flow where constitutive equations governing this type of flow take into consideration the nonlinear viscometric properties of blood. To mathematically characterize the evolving blood viscosity and the relationship between the elasticity of RBC and the shear blood flow, respectively, across space and time of the system, a stress tensor (τ) defined by constitutive models is often coupled in the Navier–Stokes equation to account for the collective impact of the constant dynamic viscosity (η) and the elasticity from RBCs on blood flow.The dynamic viscosity of blood is heavily dependent on the shear stress applied to the cell and various parameters from the blood such as hematocrit value, plasma viscosity, mechanical properties of the RBC membrane, and red blood cell aggregation rate. The apparent blood viscosity is considered convenient for the characterization of the relationship between the evolving blood viscosity and shear rate, which can be defined by Casson’s law, as shown in eq 1.

𝜇=𝜏0𝛾˙+2𝜂𝜏0𝛾˙⎯⎯⎯⎯⎯⎯⎯√+𝜂�=�0�˙+2��0�˙+�

(1)where τ

₀ is the yield stress–stress required to initiate blood flow motion, η is the Casson rheological constant, and γ̇ is the shear rate. The value of Casson’s law parameters under blood with normal hematocrit level can be defined as τ

₀ = 0.0056 Pa and η = 0.0035 Pa·s.

(23) With the known property of blood and Casson’s law parameters, an approximation can be made to the dynamic viscosity under various flow condition domains. The Power Law model is often employed to characterize the dynamic viscosity in relation to the shear rate, since precise solutions exist for specific geometries and flow circumstances, acting as a fundamental standard for definition. The Carreau and Carreau–Yasuda models can be advantageous over the Power Law model due to their ability to evaluate the dynamic viscosity at low to zero shear rate conditions. However, none of the above-mentioned models consider the memory or other elastic behavior of blood and its RBCs. Some other commonly used mathematical models and their constants for the non-Newtonian viscosity property characterization of blood are listed in Table 1 below.

(24−26)Table 1. Comparison of Various Non-Newtonian Models for Blood Viscosity

(24−26)

Model	Non-Newtonian Viscosity	Parameters
Power Law	(2)	n = 0.61, k = 0.42
Carreau	(3)	μ₀ = 0.056 Pa·s, μ_∞ = 0.00345 Pa·s, λ = 3.1736 s, m = 2.406, a = 0.254
Walburn–Schneck	(4)	C₁ = 0.000797 Pa·s, C₂ = 0.0608 Pa·s, C₃ = 0.00499, C₄ = 14.585 g^–1, TPMA = 25 g/L
Carreau–Yasuda	(5)	μ₀ = 0.056 Pa·s, μ_∞ = 0.00345 Pa·s, λ = 1.902 s, n = 0.22, a = 1.25
Quemada	(6)	μ_p = 0.0012 Pa·s, k_∞ = 2.07, k₀ = 4.33, γ̇_c = 1.88 s^–1

The blood rheology is commonly known to be influenced by two key physiological factors, namely, the hematocrit value (H

_t) and the fibrinogen concentration (c

_f), with an average value of 42% and 0.252 gd·L

^–1, respectively. Particularly in low shear conditions, the presence of varying fibrinogen concentrations affects the tendency for aggregation and rouleaux formation, while the occurrence of aggregation is contingent upon specific levels of hematocrit.

(27) The study from Apostolidis et al.

(28) modifies the Casson model through emphasizing its reliance on hematocrit and fibrinogen concentration parameter values, owing to the extensive knowledge of the two physiological blood parameters.The viscoelastic response of blood is heavily dependent on the elasticity of the RBC, which is defined by the relationship between the deformation and stress relaxation from RBCs under a specific location of shear flow as a function of the velocity field. The stress tensor is usually characterized by constitutive equations such as the Upper-Convected Maxwell Model

(29) and the Oldroyd-B model

(30) to track the molecule effects under shear from different driving forces. The prominent non-Newtonian features, such as shear thinning and yield stress, have played a vital role in the characterization of blood rheology, particularly with respect to the evaluation of yield stress under low shear conditions. The nature of stress measurement in blood, typically on the order of 1 mPa, is challenging due to its low magnitude. The occurrence of the CFL complicates the measurement further due to the significant decrease in apparent viscosity near the wall over time and a consequential disparity in viscosity compared to the bulk region.In addition to shear thinning viscosity and yield stress, the formation of aggregation (rouleaux) from RBCs under low shear rates also contributes to the viscoelasticity under transient flow

(31) and thixotropy

(32) of whole blood. Given the difficulty in evaluating viscoelastic behavior of blood under low strain magnitudes and limitations in generalized Newtonian models, the utilization of viscoelastic models is advocated to encompass elasticity and delineate non-shear components within the stress tensor. Extending from the Oldroyd-B model, Anand et al.

(33) developed a viscoelastic model framework for adapting elasticity within blood samples and predicting non-shear stress components. However, to also address the thixotropic effects, the model developed by Horner et al.

(34) serves as a more comprehensive approach than the viscoelastic model from Anand et al. Thixotropy

(32) typically occurs from the structural change of the rouleaux, where low shear rate conditions induce rouleaux formation. Correspondingly, elasticity increases, while elasticity is more representative of the isolated RBCs, under high shear rate conditions. The model of Horner et al.

(34) considers the contribution of rouleaux to shear stress, taking into account factors such as the characteristic time for Brownian aggregation, shear-induced aggregation, and shear-induced breakage. Subsequent advancements in the model from Horner et al. often revolve around refining the three aforementioned key terms for a more substantial characterization of rouleaux dynamics. Notably, this has led to the recently developed mHAWB model

(35) and other model iterations to enhance the accuracy of elastic and viscoelastic contributions to blood rheology, including the recently improved model suggested by Armstrong et al.

(36)

2.2.2. Numerical Methods (FDM, FEM, FVM)

Numerical simulation has become increasingly more significant in analyzing the geometry, boundary layers of flow, and nonlinearity of hyperbolic viscoelastic flow constitutive equations. CFD is a powerful and efficient tool utilizing numerical methods to solve the governing hydrodynamic equations, such as the Navier–Stokes (N–S) equation, continuity equation, and energy conservation equation, for qualitative evaluation of fluid motion dynamics under different parameters. CFD overcomes the challenge of analytically solving nonlinear forms of differential equations by employing numerical methods such as the Finite-Difference Method (FDM), Finite-Element Method (FEM), and Finite-Volume Method (FVM) to discretize and solve the partial differential equations (PDEs), allowing for qualitative reproduction of transport phenomena and experimental observations. Different numerical methods are chosen to cope with various transport systems for optimization of the accuracy of the result and control of error during the discretization process.FDM is a straightforward approach to discretizing PDEs, replacing the continuum representation of equations with a set of finite-difference equations, which is typically applied to structured grids for efficient implementation in CFD programs.

(37) However, FDM is often limited to simple geometries such as rectangular or block-shaped geometries and struggles with curved boundaries. In contrast, FEM divides the fluid domain into small finite grids or elements, approximating PDEs through a local description of physics.

(38) All elements contribute to a large, sparse matrix solver. However, FEM may not always provide accurate results for systems involving significant deformation and aggregation of particles like RBCs due to large distortion of grids.

(39) FVM evaluates PDEs following the conservation laws and discretizes the selected flow domain into small but finite size control volumes, with each grid at the center of a finite volume.

(40) The divergence theorem allows the conversion of volume integrals of PDEs with divergence terms into surface integrals of surface fluxes across cell boundaries. Due to its conservation property, FVM offers efficient outcomes when dealing with PDEs that embody mass, momentum, and energy conservation principles. Furthermore, widely accessible software packages like the OpenFOAM toolbox

(41) include a viscoelastic solver, making it an attractive option for viscoelastic fluid flow modeling.

(42)

2.2.3. Modeling Methods of Blood Flow Dynamics

The complexity in the blood flow simulation arises from deformability and aggregation that RBCs exhibit during their interaction with neighboring cells under different shear rate conditions induced by blood flow. Numerical models coupled with simulation programs have been applied as a groundbreaking method to predict such unique rheological behavior exhibited by RBCs and whole blood. The conventional approach of a single-phase flow simulation is often applied to blood flow simulations within large vessels possessing a moderate shear rate. However, such a method assumes the properties of plasma, RBCs and other cellular components to be evenly distributed as average density and viscosity in blood, resulting in the inability to simulate the mechanical dynamics, such as RBC aggregation under high-shear flow field, inherent in RBCs. To accurately describe the asymmetric distribution of RBC and blood flow, multiphase flow simulation, where numerical simulations of blood flows are often modeled as two immiscible phases, RBCs and blood plasma, is proposed. A common assumption is that RBCs exhibit non-Newtonian behavior while the plasma is treated as a continuous Newtonian phase.Numerous multiphase numerical models have been proposed to simulate the influence of RBCs on blood flow dynamics by different assumptions. In large-scale simulations (above the millimeter range), continuum-based methods are wildly used due to their lower computational demands.

(43) Eulerian multiphase flow simulations offer the solution of a set of conservation equations for each separate phase and couple the phases through common pressure and interphase exchange coefficients. Xu et al.

(44) utilized the combined finite-discrete element method (FDEM) to replicate the dynamic behavior and distortion of RBCs subjected to fluidic forces, utilizing the Johnson–Kendall–Roberts model

(45) to define the adhesive forces of cell-to-cell interactions. The iterative direct-forcing immersed boundary method (IBM) is commonly employed in simulations of the fluid–cell interface of blood. This method effectively captures the intricacies of the thin and flexible RBC membranes within various external flow fields.

(46) The study by Xu et al.

(44) also adopts this approach to bridge the fluid dynamics and RBC deformation through IBM. Yoon and You utilized the Maxwell model to define the viscosity of the RBC membrane.

(47) It was discovered that the Maxwell model could represent the stress relaxation and unloading processes of the cell. Furthermore, the reduced flexibility of an RBC under particular situations such as infection is specified, which was unattainable by the Kelvin–Voigt model

(48) when compared to the Maxwell model in the literature. The Yeoh hyperplastic material model was also adapted to predict the nonlinear elasticity property of RBCs with FEM employed to discretize the RBC membrane using shell-type elements. Gracka et al.

(49) developed a numerical CFD model with a finite-volume parallel solver for multiphase blood flow simulation, where an updated Maxwell viscoelasticity model and a Discrete Phase Model are adopted. In the study, the adapted IBM, based on unstructured grids, simulates the flow behavior and shape change of the RBCs through fluid-structure coupling. It was found that the hybrid Euler–Lagrange (E–L) approach

(50) for the development of the multiphase model offered better results in the simulated CFL region in the microchannels.To study the dynamics of individual behaviors of RBCs and the consequent non-Newtonian blood flow, cell-shape-resolved computational models are often adapted. The use of the boundary integral method has become prevalent in minimizing computational expenses, particularly in the exclusive determination of fluid velocity on the surfaces of RBCs, incorporating the option of employing IBM or particle-based techniques. The cell-shaped-resolved method has enabled an examination of cell to cell interactions within complex ambient or pulsatile flow conditions

(51) surrounding RBC membranes. Recently, Rydquist et al.

(52) have looked to integrate statistical information from macroscale simulations to obtain a comprehensive overview of RBC behavior within the immediate proximity of the flow through introduction of respective models characterizing membrane shape definition, tension, bending stresses of RBC membranes.At a macroscopic scale, continuum models have conventionally been adapted for assessing blood flow dynamics through the application of elasticity theory and fluid dynamics. However, particle-based methods are known for their simplicity and adaptability in modeling complex multiscale fluid structures. Meshless methods, such as the boundary element method (BEM), smoothed particle hydrodynamics (SPH), and dissipative particle dynamics (DPD), are often used in particle-based characterization of RBCs and the surrounding fluid. By representing the fluid as discrete particles, meshless methods provide insights into the status and movement of the multiphase fluid. These methods allow for the investigation of cellular structures and microscopic interactions that affect blood rheology. Non-confronting mesh methods like IBM can also be used to couple a fluid solver such as FEM, FVM, or the Lattice Boltzmann Method (LBM) through membrane representation of RBCs. In comparison to conventional CFD methods, LBM has been viewed as a favorable numerical approach for solving the N–S equations and the simulation of multiphase flows. LBM exhibits the notable advantage of being amenable to high-performance parallel computing environments due to its inherently local dynamics. In contrast to DPD and SPH where RBC membranes are modeled as physically interconnected particles, LBM employs the IBM to account for the deformation dynamics of RBCs

(53,54) under shear flows in complex channel geometries.

(54,55) However, it is essential to acknowledge that the utilization of LBM in simulating RBC flows often entails a significant computational overhead, being a primary challenge in this context. Krüger et al.

(56) proposed utilizing LBM as a fluid solver, IBM to couple the fluid and FEM to compute the response of membranes to deformation under immersed fluids. This approach decouples the fluid and membranes but necessitates significant computational effort due to the requirements of both meshes and particles.Despite the accuracy of current blood flow models, simulating complex conditions remains challenging because of the high computational load and cost. Balachandran Nair et al.

(57) suggested a reduced order model of RBC under the framework of DEM, where the RBC is represented by overlapping constituent rigid spheres. The Morse potential force is adapted to account for the RBC aggregation exhibited by cell to cell interactions among RBCs at different distances. Based upon the IBM, the reduced-order RBC model is adapted to simulate blood flow transport for validation under both single and multiple RBCs with a resolved CFD-DEM solver.

(58) In the resolved CFD-DEM model, particle sizes are larger than the grid size for a more accurate computation of the surrounding flow field. A continuous forcing approach is taken to describe the momentum source of the governing equation prior to discretization, which is different from a Direct Forcing Method (DFM).

(59) As no body-conforming moving mesh is required, the continuous forcing approach offers lower complexity and reduced cost when compared to the DFM. Piquet et al.

(60) highlighted the high complexity of the DFM due to its reliance on calculating an additional immersed boundary flux for the velocity field to ensure its divergence-free condition.The fluid–structure interaction (FSI) method has been advocated to connect the dynamic interplay of RBC membranes and fluid plasma within blood flow such as the coupling of continuum–particle interactions. However, such methodology is generally adapted for anatomical configurations such as arteries

(61,62) and capillaries,

(63) where both the structural components and the fluid domain undergo substantial deformation due to the moving boundaries. Due to the scope of the Review being blood flow simulation within microchannels of LOC devices without deformable boundaries, the Review of the FSI method will not be further carried out.In general, three numerical methods are broadly used: mesh-based, particle-based, and hybrid mesh–particle techniques, based on the spatial scale and the fundamental numerical approach, mesh-based methods tend to neglect the effects of individual particles, assuming a continuum and being efficient in terms of time and cost. However, the particle-based approach highlights more of the microscopic and mesoscopic level, where the influence of individual RBCs is considered. A review from Freund et al.

(64) addressed the three numerical methodologies and their respective modeling approaches of RBC dynamics. Given the complex mechanics and the diverse levels of study concerning numerical simulations of blood and cellular flow, a broad spectrum of numerical methods for blood has been subjected to extensive review.

(64−70) Ye at al.

(65) offered an extensive review of the application of the DPD, SPH, and LBM for numerical simulations of RBC, while Rathnayaka et al.

(67) conducted a review of the particle-based numerical modeling for liquid marbles through drawing parallels to the transport of RBCs in microchannels. A comparative analysis between conventional CFD methods and particle-based approaches for cellular and blood flow dynamic simulation can be found under the review by Arabghahestani et al.

(66) Literature by Li et al.

(68) and Beris et al.

(69) offer an overview of both continuum-based models at micro/macroscales and multiscale particle-based models encompassing various length and temporal dimensions. Furthermore, these reviews deliberate upon the potential of coupling continuum-particle methods for blood plasma and RBC modeling. Arciero et al.

(70) investigated various modeling approaches encompassing cellular interactions, such as cell to cell or plasma interactions and the individual cellular phases. A concise overview of the reviews is provided in Table 2 for reference.

Table 2. List of Reviews for Numerical Approaches Employed in Blood Flow Simulation

Reference	Numerical methods
Li et al. (2013) (68)	Continuum-based modeling (BIM), particle-based modeling (LBM, LB-FE, SPH, DPD)
Freund (2014) (64)	RBC dynamic modeling (continuum-based modeling, complementary discrete microstructure modeling), blood flow dynamic modeling (FDM, IBM, LBM, particle-mesh methods, coupled boundary integral and mesh-based methods, DPD)
Ye et al. (2016) (65)	DPD, SPH, LBM, coupled IBM-Smoothed DPD
Arciero et al. (2017) (70)	LBM, IBM, DPD, conventional CFD Methods (FDM, FVM, FEM)
Arabghahestani et al. (2019) (66)	Particle-based methods (LBM, DPD, direct simulation Monte Carlo, molecular dynamics), SPH, conventional CFD methods (FDM, FVM, FEM)
Beris et al. (2021) (69)	DPD, smoothed DPD, IBM, LBM, BIM
Rathnayaka (2022) (67)	SPH, CG, LBM

3. Capillary Driven Blood Flow in LOC Systems

ARTICLE SECTIONS

Jump To

3.1. Capillary Driven Flow Phenomena

Capillary driven (CD) flow is a pivotal mechanism in passive microfluidic flow systems

(9) such as the blood circulation system and LOC systems.

(71) CD flow is essentially the movement of a liquid to flow against drag forces, where the capillary effect exerts a force on the liquid at the borders, causing a liquid–air meniscus to flow despite gravity or other drag forces. A capillary pressure drops across the liquid–air interface with surface tension in the capillary radius and contact angle. The capillary effect depends heavily on the interaction between the different properties of surface materials. Different values of contact angles can be manipulated and obtained under varying levels of surface wettability treatments to manipulate the surface properties, resulting in different CD blood delivery rates for medical diagnostic device microchannels. CD flow techniques are appealing for many LOC devices, because they require no external energy. However, due to the passive property of liquid propulsion by capillary forces and the long-term instability of surface treatments on channel walls, the adaptability of CD flow in geometrically complex LOC devices may be limited.

3.2. Theoretical and Numerical Modeling of Capillary Driven Blood Flow

3.2.1. Theoretical Basis and Assumptions of Microfluidic Flow

The study of transport phenomena regarding either blood flow driven by capillary forces or externally applied forces under microfluid systems all demands a comprehensive recognition of the significant differences in flow dynamics between microscale and macroscale. The fundamental assumptions and principles behind fluid transport at the microscale are discussed in this section. Such a comprehension will lay the groundwork for the following analysis of the theoretical basis of capillary forces and their role in blood transport in LOC systems.

At the macroscale, fluid dynamics are often strongly influenced by gravity due to considerable fluid mass. However, the high surface to volume ratio at the microscale shifts the balance toward surface forces (e.g., surface tension and viscous forces), much larger than the inertial force. This difference gives rise to transport phenomena unique to microscale fluid transport, such as the prevalence of laminar flow due to a very low Reynolds number (generally lower than 1). Moreover, the fluid in a microfluidic system is often assumed to be incompressible due to the small flow velocity, indicating constant fluid density in both space and time.Microfluidic flow behaviors are governed by the fundamental principles of mass and momentum conservation, which are encapsulated in the continuity equation and the Navier–Stokes (N–S) equation. The continuity equation describes the conservation of mass, while the N–S equation captures the spatial and temporal variations in velocity, pressure, and other physical parameters. Under the assumption of the negligible influence of gravity in microfluidic systems, the continuity equation and the Eulerian representation of the incompressible N–S equation can be expressed as follows:

∇·𝐮⇀=0∇·�⇀=0

(7)

−∇𝑝+𝜇∇2𝐮⇀+∇·𝝉⇀−𝐅⇀=0−∇�+�∇2�⇀+∇·�⇀−�⇀=0

(8)Here, p is the pressure, u is the fluid viscosity,

𝝉⇀�⇀ represents the stress tensor, and F is the body force exerted by external forces if present.

3.2.2. Theoretical Basis and Modeling of Capillary Force in LOC Systems

The capillary force is often the major driving force to manipulate and transport blood without an externally applied force in LOC systems. Forces induced by the capillary effect impact the free surface of fluids and are represented not directly in the Navier–Stokes equations but through the pressure boundary conditions of the pressure term p. For hydrophilic surfaces, the liquid generally induces a contact angle between 0° and 30°, encouraging the spread and attraction of fluid under a positive cos θ condition. For this condition, the pressure drop becomes positive and generates a spontaneous flow forward. A hydrophobic solid surface repels the fluid, inducing minimal contact. Generally, hydrophobic solids exhibit a contact angle larger than 90°, inducing a negative value of cos θ. Such a value will result in a negative pressure drop and a flow in the opposite direction. The induced contact angle is often utilized to measure the wall exposure of various surface treatments on channel walls where different wettability gradients and surface tension effects for CD flows are established. Contact angles between different interfaces are obtainable through standard values or experimental methods for reference.

(72)For the characterization of the induced force by the capillary effect, the Young–Laplace (Y–L) equation

(73) is widely employed. In the equation, the capillary is considered a pressure boundary condition between the two interphases. Through the Y–L equation, the capillary pressure force can be determined, and subsequently, the continuity and momentum balance equations can be solved to obtain the blood filling rate. Kim et al.

(74) studied the effects of concentration and exposure time of a nonionic surfactant, Silwet L-77, on the performance of a polydimethylsiloxane (PDMS) microchannel in terms of plasma and blood self-separation. The study characterized the capillary pressure force by incorporating the Y–L equation and further evaluated the effects of the changing contact angle due to different levels of applied channel wall surface treatments. The expression of the Y–L equation utilized by Kim et al.

(74) is as follows:

𝑃=−𝜎(cos𝜃b+cos𝜃tℎ+cos𝜃l+cos𝜃r𝑤)�=−�(cos⁡�b+cos⁡�tℎ+cos⁡�l+cos⁡�r�)

(9)where σ is the surface tension of the liquid and θ

_b, θ

_t, θ

_l, and θ

_r are the contact angle values between the liquid and the bottom, top, left, and right walls, respectively. A numerical simulation through Coventor software is performed to evaluate the dynamic changes in the filling rate within the microchannel. The simulation results for the blood filling rate in the microchannel are expressed at a specific time stamp, shown in Figure 2. The results portray an increasing instantaneous filling rate of blood in the microchannel following the decrease in contact angle induced by a higher concentration of the nonionic surfactant treated to the microchannel wall.

Figure 2. Numerical simulation of filling rate of capillary driven blood flow under various contact angle conditions at a specific timestamp. (74) Reproduced with permission from ref (74). Copyright 2010 Elsevier.

When in contact with hydrophilic or hydrophobic surfaces, blood forms a meniscus with a contact angle due to surface tension. The Lucas–Washburn (L–W) equation

(75) is one of the pioneering theoretical definitions for the position of the meniscus over time. In addition, the L–W equation provides the possibility for research to obtain the velocity of the blood formed meniscus through the derivation of the meniscus position. The L–W equation

(75) can be shown below:

𝐿(𝑡)=𝑅𝜎cos(𝜃)𝑡2𝜇⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√�(�)=��⁡cos(�)�2�

(10)Here L(t) represents the distance of the liquid driven by the capillary forces. However, the generalized L–W equation solely assumes the constant physical properties from a Newtonian fluid rather than considering the non-Newtonian fluid behavior of blood. Cito et al.

(76) constructed an enhanced version of the L–W equation incorporating the power law to consider the RBC aggregation and the FL effect. The non-Newtonian fluid apparent viscosity under the Power Law model is defined as

𝜇=𝑘·(𝛾˙)𝑛−1�=�·(�˙)�−1

(11)where γ̇ is the strain rate tensor defined as

𝛾˙=12𝛾˙𝑖𝑗𝛾˙𝑗𝑖⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√�˙=12�˙��˙��. The stress tensor term τ is computed as τ = μγ̇

_ij. The updated L–W equation by Cito

(76) is expressed as

𝐿(𝑡)=𝑅[(𝑛+13𝑛+1)(𝜎cos(𝜃)𝑅𝑘)1/𝑛𝑡]𝑛/𝑛+1�(�)=�[(�+13�+1)(�⁡cos(�)��)1/��]�/�+1

(12)where k is the flow consistency index and n is the power law index, respectively. The power law index, from the Power Law model, characterizes the extent of the non-Newtonian behavior of blood. Both the consistency and power law index rely on blood properties such as hematocrit, the appearance of the FL effect, the formation of RBC aggregates, etc. The updated L–W equation computes the location and velocity of blood flow caused by capillary forces at specified time points within the LOC devices, taking into account the effects of blood flow characteristics such as RBC aggregation and the FL effect on dynamic blood viscosity.Apart from the blood flow behaviors triggered by inherent blood properties, unique flow conditions driven by capillary forces that are portrayed under different microchannel geometries also hold crucial implications for CD blood delivery. Berthier et al.

(77) studied the spontaneous Concus–Finn condition, the condition to initiate the spontaneous capillary flow within a V-groove microchannel, as shown in Figure 3(a) both experimentally and numerically. Through experimental studies, the spontaneous Concus–Finn filament development of capillary driven blood flow is observed, as shown in Figure 3(b), while the dynamic development of blood flow is numerically simulated through CFD simulation.

Figure 3. (a) Sketch of the cross-section of Berthier’s V-groove microchannel, (b) experimental view of blood in the V-groove microchannel, (78) (c) illustration of the dynamic change of the extension of filament from FLOW 3D under capillary flow at three increasing time intervals. (78) Reproduced with permission from ref (78). Copyright 2014 Elsevier.

Berthier et al.

(77) characterized the contact angle needed for the initiation of the capillary driving force at a zero-inlet pressure, through the half-angle (α) of the V-groove geometry layout, and its relation to the Concus–Finn filament as shown below:

𝜃<𝜋2−𝛼sin𝛼1+2(ℎ2/𝑤)sin𝛼<cos𝜃{�<�2−�sin⁡�1+2(ℎ2/�)⁡sin⁡�<cos⁡�

(13)Three possible regimes were concluded based on the contact angle value for the initiation of flow and development of Concus–Finn filament:

𝜃>𝜃1𝜃1>𝜃>𝜃0𝜃0no SCFSCF without a Concus−Finn filamentSCF without a Concus−Finn filament{�>�1no SCF�1>�>�0SCF without a Concus−Finn filament�0SCF without a Concus−Finn filament

(14)Under Newton’s Law, the force balance with low Reynolds and Capillary numbers results in the neglect of inertial terms. The force balance between the capillary forces and the viscous force induced by the channel wall is proposed to derive the analytical fluid velocity. This relation between the two forces offers insights into the average flow velocity and the penetration distance function dependent on time. The apparent blood viscosity is defined by Berthier et al.

(78) through Casson’s law,

(23) given in eq 1. The research used the FLOW-3D program from Flow Science Inc. software, which solves transient, free-surface problems using the FDM in multiple dimensions. The Volume of Fluid (VOF) method

(79) is utilized to locate and track the dynamic extension of filament throughout the advancing interface within the channel ahead of the main flow at three progressing time stamps, as depicted in Figure 3(c).

4. Electro-osmotic Flow (EOF) in LOC Systems

ARTICLE SECTIONS

Jump To

The utilization of external forces, such as electric fields, has significantly broadened the possibility of manipulating microfluidic flow in LOC systems.

(80) Externally applied electric field forces induce a fluid flow from the movement of ions in fluid terms as the “electro-osmotic flow” (EOF).Unique transport phenomena, such as enhanced flow velocity and flow instability, induced by non-Newtonian fluids, particularly viscoelastic fluids, under EOF, have sparked considerable interest in microfluidic devices with simple or complicated geometries within channels.

(81) However, compared to the study of Newtonian fluids and even other electro-osmotic viscoelastic fluid flows, the literature focusing on the theoretical and numerical modeling of electro-osmotic blood flow is limited due to the complexity of blood properties. Consequently, to obtain a more comprehensive understanding of the complex blood flow behavior under EOF, theoretical and numerical studies of the transport phenomena in the EOF section will be based on the studies of different viscoelastic fluids under EOF rather than that of blood specifically. Despite this limitation, we believe these studies offer valuable insights that can help understand the complex behavior of blood flow under EOF.

4.1. EOF Phenomena

Electro-osmotic flow occurs at the interface between the microchannel wall and bulk phase solution. When in contact with the bulk phase, solution ions are absorbed or dissociated at the solid–liquid interface, resulting in the formation of a charge layer, as shown in Figure 4. This charged channel surface wall interacts with both negative and positive ions in the bulk sample, causing repulsion and attraction forces to create a thin layer of immobilized counterions, known as the Stern layer. The induced electric potential from the wall gradually decreases with an increase in the distance from the wall. The Stern layer potential, commonly termed the zeta potential, controls the intensity of the electrostatic interactions between mobile counterions and, consequently, the drag force from the applied electric field. Next to the Stern layer is the diffuse mobile layer, mainly composed of a mobile counterion. These two layers constitute the “electrical double layer” (EDL), the thickness of which is directly proportional to the ionic strength (concentration) of the bulk fluid. The relationship between the two parameters is characterized by a Debye length (λ

_D), expressed as

𝜆𝐷=𝜖𝑘B𝑇2(𝑍𝑒)2𝑐0⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√��=��B�2(��)2�0

(15)where ϵ is the permittivity of the electrolyte solution, k

_B is the Boltzmann constant, T is the electron temperature, Z is the integer valence number, e is the elementary charge, and c

₀ is the ionic density.

Figure 4. Schematic diagram of an electro-osmotic flow in a microchannel with negative surface charge. (82) Reproduced with permission from ref (82). Copyright 2012 Woodhead Publishing.

When an electric field is applied perpendicular to the EDL, viscous drag is generated due to the movement of excess ions in the EDL. Electro-osmotic forces can be attributed to the externally applied electric potential (ϕ) and the zeta potential, the system wall induced potential by charged walls (ψ). As illustrated in Figure 4, the majority of ions in the bulk phase have a uniform velocity profile, except for a shear rate condition confined within an extremely thin Stern layer. Therefore, EOF displays a unique characteristic of a “near flat” or plug flow velocity profile, different from the parabolic flow typically induced by pressure-driven microfluidic flow (Hagen–Poiseuille flow). The plug-shaped velocity profile of the EOF possesses a high shear rate above the Stern layer.Overall, the EOF velocity magnitude is typically proportional to the Debye Length (λ

_D), zeta potential, and magnitude of the externally applied electric field, while a more viscous liquid reduces the EOF velocity.

4.2. Modeling on Electro-osmotic Viscoelastic Fluid Flow

4.2.1. Theoretical Basis of EOF Mechanisms

The EOF of an incompressible viscoelastic fluid is commonly governed by the continuity and incompressible N–S equations, as shown in eqs 7 and 8, where the stress tensor and the electrostatic force term are coupled. The electro-osmotic body force term F, representing the body force exerted by the externally applied electric force, is defined as

𝐹⇀=𝑝𝐸𝐸⇀�⇀=��⇀, where ρ

_E and

𝐸⇀�⇀ are the net electric charge density and the applied external electric field, respectively.Numerous models are established to theoretically study the externally applied electric potential and the system wall induced potential by charged walls. The following Laplace equation, expressed as eq 16, is generally adapted and solved to calculate the externally applied potential (ϕ).

∇2𝜙=0∇2�=0

(16)Ion diffusion under applied electric fields, together with mass transport resulting from convection and diffusion, transports ionic solutions in bulk flow under electrokinetic processes. The Nernst–Planck equation can describe these transport methods, including convection, diffusion, and electro-diffusion. Therefore, the Nernst–Planck equation is used to determine the distribution of the ions within the electrolyte. The electric potential induced by the charged channel walls follows the Poisson–Nernst–Plank (PNP) equation, which can be written as eq 17.

∇·[𝐷𝑖∇𝑛𝑖−𝑢⇀𝑛𝑖+𝑛𝑖𝐷𝑖𝑧𝑖𝑒𝑘𝑏𝑇∇(𝜙+𝜓)]=0∇·[��∇��−�⇀��+��∇(�+�)]=0

(17)where D

_i, n

_i, and z

_i are the diffusion coefficient, ionic concentration, and ionic valence of the ionic species I, respectively. However, due to the high nonlinearity and numerical stiffness introduced by different lengths and time scales from the PNP equations, the Poisson–Boltzmann (PB) model is often considered the major simplified method of the PNP equation to characterize the potential distribution of the EDL region in microchannels. In the PB model, it is assumed that the ionic species in the fluid follow the Boltzmann distribution. This model is typically valid for steady-state problems where charge transport can be considered negligible, the EDLs do not overlap with each other, and the intrinsic potentials are low. It provides a simplified representation of the potential distribution in the EDL region. The PB equation governing the EDL electric potential distribution is described as

∇2𝜓=(2𝑒𝑧𝑛0𝜀𝜀0)sinh(𝑧𝑒𝜓𝑘b𝑇)∇2�=(2��0��0)⁡sinh(��b�)

(18)where n

₀ is the ion bulk concentration, z is the ionic valence, and ε

₀ is the electric permittivity in the vacuum. Under low electric potential conditions, an even further simplified model to illustrate the EOF phenomena is the Debye–Hückel (DH) model. The DH model is derived by obtaining a charge density term by expanding the exponential term of the Boltzmann equation in a Taylor series.

4.2.2. EOF Modeling for Viscoelastic Fluids

Many studies through numerical modeling were performed to obtain a deeper understanding of the effect exhibited by externally applied electric fields on viscoelastic flow in microchannels under various geometrical designs. Bello et al.

(83) found that methylcellulose solution, a non-Newtonian polymer solution, resulted in stronger electro-osmotic mobility in experiments when compared to the predictions by the Helmholtz–Smoluchowski equation, which is commonly used to define the velocity of EOF of a Newtonian fluid. Being one of the pioneers to identify the discrepancies between the EOF of Newtonian and non-Newtonian fluids, Bello et al. attributed such discrepancies to the presence of a very high shear rate in the EDL, resulting in a change in the orientation of the polymer molecules. Park and Lee

(84) utilized the FVM to solve the PB equation for the characterization of the electric field induced force. In the study, the concept of fractional calculus for the Oldroyd-B model was adapted to illustrate the elastic and memory effects of viscoelastic fluids in a straight microchannel They observed that fluid elasticity and increased ratio of viscoelastic fluid contribution to overall fluid viscosity had a significant impact on the volumetric flow rate and sensitivity of velocity to electric field strength compared to Newtonian fluids. Afonso et al.

(85) derived an analytical expression for EOF of viscoelastic fluid between parallel plates using the DH model to account for a zeta potential condition below 25 mV. The study established the understanding of the electro-osmotic viscoelastic fluid flow under low zeta potential conditions. Apart from the electrokinetic forces, pressure forces can also be coupled with EOF to generate a unique fluid flow behavior within the microchannel. Sousa et al.

(86) analytically studied the flow of a standard viscoelastic solution by combining the pressure gradient force with an externally applied electric force. It was found that, at a near wall skimming layer and the outer layer away from the wall, macromolecules migrating away from surface walls in viscoelastic fluids are observed. In the study, the Phan-Thien Tanner (PTT) constitutive model is utilized to characterize the viscoelastic properties of the solution. The approach is found to be valid when the EDL is much thinner than the skimming layer under an enhanced flow rate. Zhao and Yang

(87) solved the PB equation and Carreau model for the characterization of the EOF mechanism and non-Newtonian fluid respectively through the FEM. The numerical results depict that, different from the EOF of Newtonian fluids, non-Newtonian fluids led to an increase of electro-osmotic mobility for shear thinning fluids but the opposite for shear thickening fluids.Like other fluid transport driving forces, EOF within unique geometrical layouts also portrays unique transport phenomena. Pimenta and Alves

(88) utilized the FVM to perform numerical simulations of the EOF of viscoelastic fluids considering the PB equation and the Oldroyd-B model, in a cross-slot and flow-focusing microdevices. It was found that electroelastic instabilities are formed due to the development of large stresses inside the EDL with streamlined curvature at geometry corners. Bezerra et al.

(89) used the FDM to numerically analyze the vortex formation and flow instability from an electro-osmotic non-Newtonian fluid flow in a microchannel with a nozzle geometry and parallel wall geometry setting. The PNP equation is utilized to characterize the charge motion in the EOF and the PTT model for non-Newtonian flow characterization. A constriction geometry is commonly utilized in blood flow adapted in LOC systems due to the change in blood flow behavior under narrow dimensions in a microchannel. Ji et al.

(90) recently studied the EOF of viscoelastic fluid in a constriction microchannel connected by two relatively big reservoirs on both ends (as seen in Figure 5) filled with the polyacrylamide polymer solution, a viscoelastic fluid, and an incompressible monovalent binary electrolyte solution KCl.

Figure 5. Schematic diagram of a negatively charged constriction microchannel connected to two reservoirs at both ends. An electro-osmotic flow is induced in the system by the induced potential difference between the anode and cathode. (90) Reproduced with permission from ref (90). Copyright 2021 The Authors, under the terms of the Creative Commons (CC BY 4.0) License https://creativecommons.org/licenses/by/4.0/.

In studying the EOF of viscoelastic fluids, the Oldroyd-B model is often utilized to characterize the polymeric stress tensor and the deformation rate of the fluid. The Oldroyd-B model is expressed as follows:

𝜏=𝜂p𝜆(𝐜−𝐈)�=�p�(�−�)

(19)where η

_p, λ, c, and I represent the polymer dynamic viscosity, polymer relaxation time, symmetric conformation tensor of the polymer molecules, and the identity matrix, respectively.A log-conformation tensor approach is taken to prevent convergence difficulty induced by the viscoelastic properties. The conformation tensor (c) in the polymeric stress tensor term is redefined by a new tensor (Θ) based on the natural logarithm of the c. The new tensor is defined as

Θ=ln(𝐜)=𝐑ln(𝚲)𝐑Θ=ln(�)=�⁡ln(�)�

(20)in which Λ is the diagonal matrix and R is the orthogonal matrix.Under the new conformation tensor, the induced EOF of a viscoelastic fluid is governed by the continuity and N–S equations adapting the Oldroyd-B model, which is expressed as

∂𝚯∂𝑡+𝐮·∇𝚯=𝛀Θ−ΘΩ+2𝐁+1𝜆(eΘ−𝐈)∂�∂�+�·∇�=�Θ−ΘΩ+2�+1�(eΘ−�)

(21)where Ω and B represent the anti-symmetric matrix and the symmetric traceless matrix of the decomposition of the velocity gradient tensor ∇u, respectively. The conformation tensor can be recovered by c = exp(Θ). The PB model and Laplace equation are utilized to characterize the charged channel wall induced potential and the externally applied potential.The governing equations are numerically solved through the FVM by RheoTool,

(42) an open-source viscoelastic EOF solver on the OpenFOAM platform. A SIMPLEC (Semi-Implicit Method for Pressure Linked Equations-Consistent) algorithm was applied to solve the velocity-pressure coupling. The pressure field and velocity field were computed by the PCG (Preconditioned Conjugate Gradient) solver and the PBiCG (Preconditioned Biconjugate Gradient) solver, respectively.Ranging magnitudes of an applied electric field or fluid concentration induce both different streamlines and velocity magnitudes at various locations and times of the microchannel. In the study performed by Ji et al.,

(90) notable fluctuation of streamlines and vortex formation is formed at the upper stream entrance of the constriction as shown in Figure 6(a) and (b), respectively, due to the increase of electrokinetic effect, which is seen as a result of the increase in polymeric stress (τ

_xx).

(90) The contraction geometry enhances the EOF velocity within the constriction channel under high E

_app condition (600 V/cm). Such phenomena can be attributed to the dependence of electro-osmotic viscoelastic fluid flow on the system wall surface and bulk fluid properties.

(91)

Figure 6. Schematic diagram of vortex formation and streamlines of EOF depicting flow instability at (a) 1.71 s and (b) 1.75 s. Spatial distribution of the elastic normal stress at (c) high E_app condition. Streamline of an electro-osmotic flow under E_app of 600 V/cm (90) for (d) non-Newtonian and (e) Newtonian fluid through a constriction geometry. Reproduced with permission from ref (90). Copyright 2021 The Authors, under the terms of the Creative Commons (CC BY 4.0) License https://creativecommons.org/licenses/by/4.0/.

As elastic normal stress exceeds the local shear stress, flow instability and vortex formation occur. The induced elastic stress under EOF not only enhances the instability of the flow but often generates an irregular secondary flow leading to strong disturbance.

(92) It is also vital to consider the effect of the constriction layout of microchannels on the alteration of the field strength within the system. The contraction geometry enhances a larger electric field strength compared with other locations of the channel outside the constriction region, resulting in a higher velocity gradient and stronger extension on the polymer within the viscoelastic solution. Following the high shear flow condition, a higher magnitude of stretch for polymer molecules in viscoelastic fluids exhibits larger elastic stresses and enhancement of vortex formation at the region.

(93)As shown in Figure 6(c), significant elastic normal stress occurs at the inlet of the constriction microchannel. Such occurrence of a polymeric flow can be attributed to the dominating elongational flow, giving rise to high deformation of the polymers within the viscoelastic fluid flow, resulting in higher elastic stress from the polymers. Such phenomena at the entrance result in the difference in velocity streamline as circled in Figure 6(d) compared to that of the Newtonian fluid at the constriction entrance in Figure 6(e).

(90) The difference between the Newtonian and polymer solution at the exit, as circled in Figure 6(d) and (e), can be attributed to the extrudate swell effect of polymers

(94) within the viscoelastic fluid flow. The extrudate swell effect illustrates that, as polymers emerge from the constriction exit, they tend to contract in the flow direction and grow in the normal direction, resulting in an extrudate diameter greater than the channel size. The deformation of polymers within the polymeric flow at both the entrance and exit of the contraction channel facilitates the change in shear stress conditions of the flow, leading to the alteration in streamlines of flows for each region.

4.3. EOF Applications in LOC Systems

4.3.1. Mixing in LOC Systems

Rather than relying on the micromixing controlled by molecular diffusion under low Reynolds number conditions, active mixers actively leverage convective instability and vortex formation induced by electro-osmotic flows from alternating current (AC) or direct current (DC) electric fields. Such adaptation is recognized as significant breakthroughs for promotion of fluid mixing in chemical and biological applications such as drug delivery, medical diagnostics, chemical synthesis, and so on.

(95)Many researchers proposed novel designs of electro-osmosis micromixers coupled with numerical simulations in conjunction with experimental findings to increase their understanding of the role of flow instability and vortex formation in the mixing process under electrokinetic phenomena. Matsubara and Narumi

(96) numerically modeled the mixing process in a microchannel with four electrodes on each side of the microchannel wall, which generated a disruption through unstable electro-osmotic vortices. It was found that particle mixing was sensitive to both the convection effect induced by the main and secondary vortex within the micromixer and the change in oscillation frequency caused by the supplied AC voltage when the Reynolds number was varied. Qaderi et al.

(97) adapted the PNP equation to numerically study the effect of the geometry and zeta potential configuration of the microchannel on the mixing process with a combined electro-osmotic pressure driven flow. It was reported that the application of heterogeneous zeta potential configuration enhances the mixing efficiency by around 23% while the height of the hurdles increases the mixing efficiency at most 48.1%. Cho et al.

(98) utilized the PB model and Laplace equation to numerically simulate the electro-osmotic non-Newtonian fluid mixing process within a wavy and block layout of microchannel walls. The Power Law model is adapted to describe the fluid rheological characteristic. It was found that shear-thinning fluids possess a higher volumetric flow rate, which could result in poorer mixing efficiency compared to that of Newtonian fluids. Numerous studies have revealed that flow instability and vortex generation, in particular secondary vortices produced by barriers or greater magnitudes of heterogeneous zeta potential distribution, enhance mixing by increasing bulk flow velocity and reducing flow distance.To better understand the mechanism of disturbance formed in the system due to externally applied forces, known as electrokinetic instability, literature often utilize the Rayleigh (Ra) number,

(1) as described below:

𝑅𝑎𝑣=𝑢ev𝑢eo=(𝛾−1𝛾+1)2𝑊𝛿2𝐸el2𝐻2𝜁𝛿Ra�=�ev�eo=(�−1�+1)2��2�el2�2��

(22)where γ is the conductivity ratio of the two streams and can be written as

𝛾=𝜎el,H𝜎el,L�=�el,H�el,L. The Ra number characterizes the ratio between electroviscous and electro-osmotic flow. A high Ra

_v value often results in good mixing. It is evident that fluid properties such as the conductivity (σ) of the two streams play a key role in the formation of disturbances to enhance mixing in microsystems. At the same time, electrokinetic parameters like the zeta potential (ζ) in the Ra number is critical in the characterization of electro-osmotic velocity and a slip boundary condition at the microchannel wall.To understand the mixing result along the channel, the concentration field can be defined and simulated under the assumption of steady state conditions and constant diffusion coefficient for each of the working fluid within the system through the convection–diffusion equation as below:

∂𝑐𝒊∂𝑡+∇⇀(𝑐𝑖𝑢⇀−𝐷𝑖∇⇀𝑐𝒊)=0∂��∂�+∇⇀(��⇀−��∇⇀��)=0

(23)where c

_i is the species concentration of species i and D

_i is the diffusion coefficient of the corresponding species.The standard deviation of concentration (σ

_sd) can be adapted to evaluate the mixing quality of the system.

(97) The standard deviation for concentration at a specific portion of the channel may be calculated using the equation below:

𝜎sd=∫10(𝐶∗(𝑦∗)−𝐶m)2d𝑦∗∫10d𝑦∗⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯�sd=∫01(�*(�*)−�m)2d�*∫01d�*

(24)where C*(y*) and C

_m are the non-dimensional concentration profile and the mean concentration at the portion, respectively. C* is the non-dimensional concentration and can be calculated as

𝐶∗=𝐶𝐶ref�*=��ref, where C

_ref is the reference concentration defined as the bulk solution concentration. The mean concentration profile can be calculated as

𝐶m=∫10(𝐶∗(𝑦∗)d𝑦∗∫10d𝑦∗�m=∫01(�*(�*)d�*∫01d�*. With the standard deviation of concentration, the mixing efficiency

(97) can then be calculated as below:

𝜀𝑥=1−𝜎sd𝜎sd,0��=1−�sd�sd,0

(25)where σ

_sd_,0 is the standard derivation of the case of no mixing. The value of the mixing efficiency is typically utilized in conjunction with the simulated flow field and concentration field to explore the effect of geometrical and electrokinetic parameters on the optimization of the mixing results.

5. Summary

ARTICLE SECTIONS

Jump To

5.1. Conclusion

Viscoelastic fluids such as blood flow in LOC systems are an essential topic to proceed with diagnostic analysis and research through microdevices in the biomedical and pharmaceutical industries. The complex blood flow behavior is tightly controlled by the viscoelastic characteristics of blood such as the dynamic viscosity and the elastic property of RBCs under various shear rate conditions. Furthermore, the flow behaviors under varied driving forces promote an array of microfluidic transport phenomena that are critical to the management of blood flow and other adapted viscoelastic fluids in LOC systems. This review addressed the blood flow phenomena, the complicated interplay between shear rate and blood flow behaviors, and their numerical modeling under LOC systems through the lens of the viscoelasticity characteristic. Furthermore, a theoretical understanding of capillary forces and externally applied electric forces leads to an in-depth investigation of the relationship between blood flow patterns and the key parameters of the two driving forces, the latter of which is introduced through the lens of viscoelastic fluids, coupling numerical modeling to improve the knowledge of blood flow manipulation in LOC systems. The flow disturbances triggered by the EOF of viscoelastic fluids and their impact on blood flow patterns have been deeply investigated due to their important role and applications in LOC devices. Continuous advancements of various numerical modeling methods with experimental findings through more efficient and less computationally heavy methods have served as an encouraging sign of establishing more accurate illustrations of the mechanisms for multiphase blood and other viscoelastic fluid flow transport phenomena driven by various forces. Such progress is fundamental for the manipulation of unique transport phenomena, such as the generated disturbances, to optimize functionalities offered by microdevices in LOC systems.

The following section will provide further insights into the employment of studied blood transport phenomena to improve the functionality of micro devices adapting LOC technology. A discussion of the novel roles that external driving forces play in microfluidic flow behaviors is also provided. Limitations in the computational modeling of blood flow and electrokinetic phenomena in LOC systems will also be emphasized, which may provide valuable insights for future research endeavors. These discussions aim to provide guidance and opportunities for new paths in the ongoing development of LOC devices that adapt blood flow.

5.2. Future Directions

5.2.1. Electro-osmosis Mixing in LOC Systems

Despite substantial research, mixing results through flow instability and vortex formation phenomena induced by electro-osmotic mixing still deviate from the effective mixing results offered by chaotic mixing results such as those seen in turbulent flows. However, recent discoveries of a mixing phenomenon that is generally observed under turbulent flows are found within electro-osmosis micromixers under low Reynolds number conditions. Zhao

(99) experimentally discovered a rapid mixing process in an AC applied micromixer, where the power spectrum of concentration under an applied voltage of 20 V

_p-p induces a −5/3 slope within a frequency range. This value of the slope is considered as the O–C spectrum in macroflows, which is often visible under relatively high Re conditions, such as the Taylor microscale Reynolds number Re > 500 in turbulent flows.

(100) However, the Re value in the studied system is less than 1 at the specific location and applied voltage. A secondary flow is also suggested to occur close to microchannel walls, being attributed to the increase of convective instability within the system.Despite the experimental phenomenon proposed by Zhao et al.,

(99) the range of effects induced by vital parameters of an EOF mixing system on the enhanced mixing results and mechanisms of disturbance generated by the turbulent-like flow instability is not further characterized. Such a gap in knowledge may hinder the adaptability and commercialization of the discovery of micromixers. One of the parameters for further evaluation is the conductivity gradient of the fluid flow. A relatively strong conductivity gradient (5000:1) was adopted in the system due to the conductive properties of the two fluids. The high conductivity gradients may contribute to the relatively large Rayleigh number and differences in EDL layer thickness, resulting in an unusual disturbance in laminar flow conditions and enhanced mixing results. However, high conductivity gradients are not always achievable by the working fluids due to diverse fluid properties. The reliance on turbulent-like phenomena and rapid mixing results in a large conductivity gradient should be established to prevent the limited application of fluids for the mixing system. In addition, the proposed system utilizes distinct zeta potential distributions at the top and bottom walls due to their difference in material choices, which may be attributed to the flow instability phenomena. Further studies should be made on varying zeta potential magnitude and distribution to evaluate their effect on the slip boundary conditions of the flow and the large shear rate condition close to the channel wall of EOF. Such a study can potentially offer an optimized condition in zeta potential magnitude through material choices and geometrical layout of the zeta potential for better mixing results and manipulation of mixing fluid dynamics. The two vital parameters mentioned above can be varied with the aid of numerical simulation to understand the effect of parameters on the interaction between electro-osmotic forces and electroviscous forces. At the same time, the relationship of developed streamlines of the simulated velocity and concentration field, following their relationship with the mixing results, under the impact of these key parameters can foster more insight into the range of impact that the two parameters have on the proposed phenomena and the microfluidic dynamic principles of disturbances.

In addition, many of the current investigations of electrokinetic mixers commonly emphasize the fluid dynamics of mixing for Newtonian fluids, while the utilization of biofluids, primarily viscoelastic fluids such as blood, and their distinctive response under shear forces in these novel mixing processes of LOC systems are significantly less studied. To develop more compatible microdevice designs and efficient mixing outcomes for the biomedical industry, it is necessary to fill the knowledge gaps in the literature on electro-osmotic mixing for biofluids, where properties of elasticity, dynamic viscosity, and intricate relationship with shear flow from the fluid are further considered.

5.2.2. Electro-osmosis Separation in LOC Systems

Particle separation in LOC devices, particularly in biological research and diagnostics, is another area where disturbances may play a significant role in optimization.

(101) Plasma analysis in LOC systems under precise control of blood flow phenomena and blood/plasma separation procedures can detect vital information about infectious diseases from particular antibodies and foreign nucleic acids for medical treatments, diagnostics, and research,

(102) offering more efficient results and simple operating procedures compared to that of the traditional centrifugation method for blood and plasma separation. However, the adaptability of LOC devices for blood and plasma separation is often hindered by microchannel clogging, where flow velocity and plasma yield from LOC devices is reduced due to occasional RBC migration and aggregation at the filtration entrance of microdevices.

(103)It is important to note that the EOF induces flow instability close to microchannel walls, which may provide further solutions to clogging for the separation process of the LOC systems. Mohammadi et al.

(104) offered an anti-clogging effect of RBCs at the blood and plasma separating device filtration entry, adjacent to the surface wall, through RBC disaggregation under high shear rate conditions generated by a forward and reverse EOF direction.

Further theoretical and numerical research can be conducted to characterize the effect of high shear rate conditions near microchannel walls toward the detachment of binding blood cells on surfaces and the reversibility of aggregation. Through numerical modeling with varying electrokinetic parameters to induce different degrees of disturbances or shear conditions at channel walls, it may be possible to optimize and better understand the process of disrupting the forces that bind cells to surface walls and aggregated cells at filtration pores. RBCs that migrate close to microchannel walls are often attracted by the adhesion force between the RBC and the solid surface originating from the van der Waals forces. Following RBC migration and attachment by adhesive forces adjacent to the microchannel walls as shown in Figure 7, the increase in viscosity at the region causes a lower shear condition and encourages RBC aggregation (cell–cell interaction), which clogs filtering pores or microchannels and reduces flow velocity at filtration region. Both the impact that shear forces and disturbances may induce on cell binding forces with surface walls and other cells leading to aggregation may suggest further characterization. Kinetic parameters such as activation energy and the rate-determining step for cell binding composition attachment and detachment should be considered for modeling the dynamics of RBCs and blood flows under external forces in LOC separation devices.

Figure 7. Schematic representations of clogging at a microchannel pore following the sequence of RBC migration, cell attachment to channel walls, and aggregation. (105) Reproduced with permission from ref (105). Copyright 2018 The Authors under the terms of the Creative Commons (CC BY 4.0) License https://creativecommons.org/licenses/by/4.0/.

5.2.3. Relationship between External Forces and Microfluidic Systems

In blood flow, a thicker CFL suggests a lower blood viscosity, suggesting a complex relationship between shear stress and shear rate, affecting the blood viscosity and blood flow. Despite some experimental and numerical studies on electro-osmotic non-Newtonian fluid flow, limited literature has performed an in-depth investigation of the role that applied electric forces and other external forces could play in the process of CFL formation. Additional studies on how shear rates from external forces affect CFL formation and microfluidic flow dynamics can shed light on the mechanism of the contribution induced by external driving forces to the development of a separate phase of layer, similar to CFL, close to the microchannel walls and distinct from the surrounding fluid within the system, then influencing microfluidic flow dynamics.One of the mechanisms of phenomena to be explored is the formation of the Exclusion Zone (EZ) region following a “Self-Induced Flow” (SIF) phenomenon discovered by Li and Pollack,

(106) as shown in Figure 8(a) and (b), respectively. A spontaneous sustained axial flow is observed when hydrophilic materials are immersed in water, resulting in the buildup of a negative layer of charges, defined as the EZ, after water molecules absorb infrared radiation (IR) energy and break down into H and OH

⁺^–.

Figure 8. Schematic representations of (a) the Exclusion Zone region and (b) the Self Induced Flow through visualization of microsphere movement within a microchannel. (106) Reproduced with permission from ref (106). Copyright 2020 The Authors under the terms of the Creative Commons (CC BY 4.0) License https://creativecommons.org/licenses/by/4.0/.

Despite the finding of such a phenomenon, the specific mechanism and role of IR energy have yet to be defined for the process of EZ development. To further develop an understanding of the role of IR energy in such phenomena, a feasible study may be seen through the lens of the relationships between external forces and microfluidic flow. In the phenomena, the increase of SIF velocity under a rise of IR radiation resonant characteristics is shown in the participation of the external electric field near the microchannel walls under electro-osmotic viscoelastic fluid flow systems. The buildup of negative charges at the hydrophilic surfaces in EZ is analogous to the mechanism of electrical double layer formation. Indeed, research has initiated the exploration of the core mechanisms for EZ formation through the lens of the electrokinetic phenomena.

(107) Such a similarity of the role of IR energy and the transport phenomena of SIF with electrokinetic phenomena paves the way for the definition of the unknown SIF phenomena and EZ formation. Furthermore, Li and Pollack

(106) suggest whether CFL formation might contribute to a SIF of blood using solely IR radiation, a commonly available source of energy in nature, as an external driving force. The proposition may be proven feasible with the presence of the CFL region next to the negatively charged hydrophilic endothelial glycocalyx layer, coating the luminal side of blood vessels.

(108) Further research can dive into the resonating characteristics between the formation of the CFL region next to the hydrophilic endothelial glycocalyx layer and that of the EZ formation close to hydrophilic microchannel walls. Indeed, an increase in IR energy is known to rapidly accelerate EZ formation and SIF velocity, depicting similarity to the increase in the magnitude of electric field forces and greater shear rates at microchannel walls affecting CFL formation and EOF velocity. Such correlation depicts a future direction in whether SIF blood flow can be observed and characterized theoretically further through the lens of the relationship between blood flow and shear forces exhibited by external energy.

The intricate link between the CFL and external forces, more specifically the externally applied electric field, can receive further attention to provide a more complete framework for the mechanisms between IR radiation and EZ formation. Such characterization may also contribute to a greater comprehension of the role IR can play in CFL formation next to the endothelial glycocalyx layer as well as its role as a driving force to propel blood flow, similar to the SIF, but without the commonly assumed pressure force from heart contraction as a source of driving force.

5.3. Challenges

Although there have been significant improvements in blood flow modeling under LOC systems over the past decade, there are still notable constraints that may require special attention for numerical simulation applications to benefit the adaptability of the designs and functionalities of LOC devices. Several points that require special attention are mentioned below:

1.	The majority of CFD models operate under the relationship between the viscoelasticity of blood and the shear rate conditions of flow. The relative effect exhibited by the presence of highly populated RBCs in whole blood and their forces amongst the cells themselves under complex flows often remains unclearly defined. Furthermore, the full range of cell populations in whole blood requires a much more computational load for numerical modeling. Therefore, a vital goal for future research is to evaluate a reduced modeling method where the impact of cell–cell interaction on the viscoelastic property of blood is considered.
2.	Current computational methods on hemodynamics rely on continuum models based upon non-Newtonian rheology at the macroscale rather than at molecular and cellular levels. Careful considerations should be made for the development of a constructive framework for the physical and temporal scales of micro/nanoscale systems to evaluate the intricate relationship between fluid driving forces, dynamic viscosity, and elasticity.
3.	Viscoelastic fluids under the impact of externally applied electric forces often deviate from the assumptions of no-slip boundary conditions due to the unique flow conditions induced by externally applied forces. Furthermore, the mechanism of vortex formation and viscoelastic flow instability at laminar flow conditions should be better defined through the lens of the microfluidic flow phenomenon to optimize the prediction of viscoelastic flow across different geometrical layouts. Mathematical models and numerical methods are needed to better predict such disturbance caused by external forces and the viscoelasticity of fluids at such a small scale.
4.	Under practical situations, zeta potential distribution at channel walls frequently deviates from the common assumption of a constant distribution because of manufacturing faults or inherent surface charges prior to the introduction of electrokinetic influence. These discrepancies frequently lead to inconsistent surface potential distribution, such as excess positive ions at relatively more negatively charged walls. Accordingly, unpredicted vortex formation and flow instability may occur. Therefore, careful consideration should be given to these discrepancies and how they could trigger the transport process and unexpected results of a microdevice.

Author Information

ARTICLE SECTIONS

Jump To

Corresponding Authors
- Zhe Chen – Department of Chemical Engineering, School of Chemistry and Chemical Engineering, State Key Laboratory of Metal Matrix Composites, Shanghai Jiao Tong University, Shanghai 200240, P. R. China; Email: zaccooky@sjtu.edu.cn
- Bo Ouyang – Department of Chemical Engineering, School of Chemistry and Chemical Engineering, State Key Laboratory of Metal Matrix Composites, Shanghai Jiao Tong University, Shanghai 200240, P. R. China; Email: bouy93@sjtu.edu.cn
- Zheng-Hong Luo – Department of Chemical Engineering, School of Chemistry and Chemical Engineering, State Key Laboratory of Metal Matrix Composites, Shanghai Jiao Tong University, Shanghai 200240, P. R. China; https://orcid.org/0000-0001-9011-6020; Email: luozh@sjtu.edu.cn
Authors
- Bin-Jie Lai – Department of Chemical Engineering, School of Chemistry and Chemical Engineering, State Key Laboratory of Metal Matrix Composites, Shanghai Jiao Tong University, Shanghai 200240, P. R. China; https://orcid.org/0009-0002-8133-5381
- Li-Tao Zhu – Department of Chemical Engineering, School of Chemistry and Chemical Engineering, State Key Laboratory of Metal Matrix Composites, Shanghai Jiao Tong University, Shanghai 200240, P. R. China; https://orcid.org/0000-0001-6514-8864
NotesThe authors declare no competing financial interest.

Acknowledgments

ARTICLE SECTIONS

Jump To

This work was supported by the National Natural Science Foundation of China (No. 22238005) and the Postdoctoral Research Foundation of China (No. GZC20231576).

Vocabulary

ARTICLE SECTIONS

Jump To

Microfluidics	the field of technological and scientific study that investigates fluid flow in channels with dimensions between 1 and 1000 μm
Lab-on-a-Chip Technology	the field of research and technological development aimed at integrating the micro/nanofluidic characteristics to conduct laboratory processes on handheld devices
Computational Fluid Dynamics (CFD)	the method utilizing computational abilities to predict physical fluid flow behaviors mathematically through solving the governing equations of corresponding fluid flows
Shear Rate	the rate of change in velocity where one layer of fluid moves past the adjacent layer
Viscoelasticity	the property holding both elasticity and viscosity characteristics relying on the magnitude of applied shear stress and time-dependent strain
Electro-osmosis	the flow of fluid under an applied electric field when charged solid surface is in contact with the bulk fluid
Vortex	the rotating motion of a fluid revolving an axis line

References

ARTICLE SECTIONS

Jump To

This article references 108 other publications.

1Neethirajan, S.; Kobayashi, I.; Nakajima, M.; Wu, D.; Nandagopal, S.; Lin, F. Microfluidics for food, agriculture and biosystems industries. Lab Chip 2011, 11 (9), 1574– 1586, DOI: 10.1039/c0lc00230eView Google Scholar
2Whitesides, G. M. The origins and the future of microfluidics. Nature 2006, 442 (7101), 368– 373, DOI: 10.1038/nature05058View Google Scholar
3Burklund, A.; Tadimety, A.; Nie, Y.; Hao, N.; Zhang, J. X. J. Chapter One – Advances in diagnostic microfluidics; Elsevier, 2020; DOI: DOI: 10.1016/bs.acc.2019.08.001 .View Google Scholar
4Abdulbari, H. A. Chapter 12 – Lab-on-a-chip for analysis of blood. In Nanotechnology for Hematology, Blood Transfusion, and Artificial Blood; Denizli, A., Nguyen, T. A., Rajan, M., Alam, M. F., Rahman, K., Eds.; Elsevier, 2022; pp 265– 283.View Google Scholar
5Vladisavljević, G. T.; Khalid, N.; Neves, M. A.; Kuroiwa, T.; Nakajima, M.; Uemura, K.; Ichikawa, S.; Kobayashi, I. Industrial lab-on-a-chip: Design, applications and scale-up for drug discovery and delivery. Advanced Drug Delivery Reviews 2013, 65 (11), 1626– 1663, DOI: 10.1016/j.addr.2013.07.017View Google Scholar
6Kersaudy-Kerhoas, M.; Dhariwal, R.; Desmulliez, M. P. Y.; Jouvet, L. Hydrodynamic blood plasma separation in microfluidic channels. Microfluid. Nanofluid. 2010, 8 (1), 105– 114, DOI: 10.1007/s10404-009-0450-5View Google Scholar
7Popel, A. S.; Johnson, P. C. Microcirculation and Hemorheology. Annu. Rev. Fluid Mech. 2005, 37 (1), 43– 69, DOI: 10.1146/annurev.fluid.37.042604.133933View Google Scholar
8Fedosov, D. A.; Peltomäki, M.; Gompper, G. Deformation and dynamics of red blood cells in flow through cylindrical microchannels. Soft Matter 2014, 10 (24), 4258– 4267, DOI: 10.1039/C4SM00248BView Google Scholar
9Chakraborty, S. Dynamics of capillary flow of blood into a microfluidic channel. Lab Chip 2005, 5 (4), 421– 430, DOI: 10.1039/b414566fView Google Scholar
10Tomaiuolo, G.; Guido, S. Start-up shape dynamics of red blood cells in microcapillary flow. Microvascular Research 2011, 82 (1), 35– 41, DOI: 10.1016/j.mvr.2011.03.004View Google Scholar
11Sherwood, J. M.; Dusting, J.; Kaliviotis, E.; Balabani, S. The effect of red blood cell aggregation on velocity and cell-depleted layer characteristics of blood in a bifurcating microchannel. Biomicrofluidics 2012, 6 (2), 24119, DOI: 10.1063/1.4717755View Google Scholar
12Nader, E.; Skinner, S.; Romana, M.; Fort, R.; Lemonne, N.; Guillot, N.; Gauthier, A.; Antoine-Jonville, S.; Renoux, C.; Hardy-Dessources, M.-D. Blood Rheology: Key Parameters, Impact on Blood Flow, Role in Sickle Cell Disease and Effects of Exercise. Frontiers in Physiology 2019, 10, 01329, DOI: 10.3389/fphys.2019.01329View Google Scholar
13Trejo-Soto, C.; Lázaro, G. R.; Pagonabarraga, I.; Hernández-Machado, A. Microfluidics Approach to the Mechanical Properties of Red Blood Cell Membrane and Their Effect on Blood Rheology. Membranes 2022, 12 (2), 217, DOI: 10.3390/membranes12020217View Google Scholar
14Wagner, C.; Steffen, P.; Svetina, S. Aggregation of red blood cells: From rouleaux to clot formation. Comptes Rendus Physique 2013, 14 (6), 459– 469, DOI: 10.1016/j.crhy.2013.04.004

Tag: CFD

Abstract

1 Introduction and Literature Review

2 Data and Methods

2.1 Physical Modelling

2.2 Numerical Simulations

2.2.1 Governing Equations and Turbulent Models

2.2.1.1 FLOW-3D HYDRO

2.2.1.2 OpenFOAM

2.2.2 FLOW-3D HYDRO Simulation Procedure

2.2.3 OpenFOAM Simulation Procedure

2.2.4 Mesh Sensitivity Analysis

2.2.5 Validation

3 Results

3.1 Comparison of Waveforms

3.2 Three-Dimensional Vision of Landslide Generation Process by Numerical Models

3.3 Wave Velocity Analysis

4 Discussions

5 Conclusions

Reference

Abstract

1. Introduction

2. Principle of Flow Measurement

3. Experiment Setup

4. Numerical Simulation Settings

4.1. Mathematical Model

4.1.1. Governing Equations

4.1.2. RNG k-ε Model

4.1.3. TruVOF Model

4.2. Parameter Setting and Boundary Conditions

5. Results

5.1. Water Surface Profiles

5.2. Velocity Distribution

5.3. Side Channel Pressure Distribution

5.4. Discharge Coefficient

6. Discussions

7. Conclusions

Reference

합류 분기 유닛의 3차원 유동 구조

Abstract

1 Introduction

2. Conceptual flume and computational cases

2.1. Conceptual flume

2.1.1. Length and width scales of the confluence-bifurcation unit

2.1.2. Central bar morphology

2.1.3. Width-depth ratio

2.1.4. Bed slope

2.1.5. Complete sketch of the conceptual flume

2.2. Computational cases

3. Numerical method

3.1. Governing equations

3.2. Boundary conditions

4. Results and discussion

4.1. Flow structure in the confluence-bifurcation unit

4.1.1. Free surface elevation

4.1.2. Time-averaged streamwise flow velocity

4.1.3. Recirculation vortex

4.1.4. Secondary current

4.1.5. Turbulent characteristics

4.2. Comparison with single confluence/bifurcation cases

4.2.1. Comparison with single confluence cases

4.2.1.2. Secondary current

4.2.1.3. Turbulent kinetic energy

4.2.2. Comparison with single bifurcation cases

4.2.3. Further discussion of the comparisons

4.3. Implications

5. Conclusions

Reference

충전재

시뮬레이션 설정 및 실행

홍조

식품 가공

혼입

비뉴턴 유체

다공성 재료

분무

Products

관련 기술자료

Top 20 Fastest Desktops for 2024

Top 20 Fastest Desktops for 2024