From time to time it has been observed that large velocities may appear in the vicinity of grid elements that are severely, but not completely, blocked by solid obstacles. On some occasions these regions even exhibit an instability in which these velocities continue to grow with time. In this note the reason for these problems, or at least for some of these problems, is discussed and a solution is given.
Problems Associated with Abrupt Area Changes
Instabilities that sometimes arise in a flow solution can often be traced to locations where there is a large change in flow area across a grid element. For example, consider an element having a small area fraction on its left side and a full open area on its right side. When flow is moving fluid from the left to the right there is a large reduction in velocity at the right in order to have a conserved volume of flow (i.e., uA should be constant in the direction of flow, where A is the fractional area open at the side of an element). At the right side the velocity receives a larger advective flux from the left side than it should because no mechanism is in place to allow for an expansion of the flow between the left and right sides. To recover the correct velocity at the right side the pressure in the element must decrease more than it otherwise should. In some circumstances this decrease may then pull in a cross–flow that grows with time.
It is well known that there should be significant flow losses at abrupt area expansions and contractions. Such losses were never considered in the original development of the FAVOR™ method, so it is not surprising that their absence could cause some distortions in the flow. In the next section a brief discussion of flow losses is given, which is then followed by a proposed addition to the advective flow that aims to account for these losses.