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Thomas P. Abraham September 6, 1999 09:30

A Simple Question
Hello Everyone:

I have a question for you all. The fluid is incompressible. Both the flow and the energy equations are solved. The standard approach is to apply velocity at the inlet and a constant static pressure at the outlet. We get the static pressure at the inlet as part of the solution.

Now, to my question. Does the actual value of static pressure at the outlet effect the solution? Let's say the static pressure at the inlet is Pinlet and the static pressure at the outlet is Poutlet. I would think (Pinlet - Poutlet) would be the same independent of the value of Poutlet. Since temperature depends on the absolute value of static pressure, I would think that actual value of Poutlet effects the thermal field. I will be trying this out soon and let you all know.

In the meantime, all your feedbacks are welcome and appreciated.

Special note to John: You are most welcome to add your philosophical view points because I really enjoy reading all that.



Dr. Hrvoje Jasak September 6, 1999 10:58

Re: A Simple Question
If the fluid is incompressible (and you have no buoyancy), the energy equation is decoupled. The only term in the Navier-Stokes equations containing the pressure is grad p: the pressure level is therefore irrelevant. If you've got problems ask yourself where in the code you're using the pressure level (is energy really decoupled, does pressure equation residual normalisation depend on p etc.).

John C. Chien September 6, 1999 14:11

Re: A Simple Question
(1). It is holiday today. Have a nice holiday! (2). You are doing just fine. (3). As discussed before, for incompressible flow, the governing equations can be expressed in term of the velocity equations and the vorticity equations (vector equation) because the pressure terms in the momentum equations can be eliminated by the cross vector operation. This can be found in most fluid dynamics books. (4). Some people use this vorticity-velocity formulation to solve the incompressible flow, along with the energy equation if required. In this formulation, there is no pressure term or the pressure to worry about. When the problem is 2-D, the formulation becomes the classical stream-function-vorticity-equation formulation, which has been used for a long time. ( you get the same answer whether using stream function or velocity equations in 2-D). This is a preferred method for liquid type application. (5). For the gas type application, some people like to use the primitive variable formulation approach, that is the use of the density, u,v,w,and e as the variables in the governing equations even for the very low Mach number flows. ( In the low Mach number range (M<0.3), the density effect is small.) (6). In the primitive variable formulation, there is a so-called pressure based approach in addition to the transient compressible density based approach, to solve the equations. In the pressure based approach, the momentum equations are solved with the pressure gradient terms known. This known pressure gradient must be computed from somewhere, and as long as it is computed it will have some errors in it. And also, since this pressure gradient term is of first order type, it will control only the pressure from both side of a cell or grid point. Thus, the pressure at the center of the cell of the mesh point is never involved directly in the calculation of the momentum equation,if the second-order accurate difference scheme is used for the pressure term. That means, at the time when the velocity is computed with a given pressure gradient term, the pressure level at the mesh point (or the center of the cell) can be anything. (7). At this point, it becomes clear that the central issue is " how to obtain this pressure gradient term". So, following the same thinking logic of the vorticity-velocity formulation, a pressure equation can also be derived for the pressure. This second-order pressure always works with the continuity equation. After this, it is then up to the user to find out how to solve the pressure equation. Some methods solve the pressure directly, some others solve the pressure difference through iterative correction procedure. (8). Regardless of which approach taken, sooner or later, one has to rely on the physics of fluid dynamics, if the incompressible flow is to be solved. One person will tell you that you don't need to know the pressure level or pressure field at all. On the other hand, in the primitive approach, pressure-based formulation, one is always facing this pressure coupling problem. (9). In practice, when the pressure is specified at one point and solved for the whole flow field, you will have some errors in the computed pressure. At least for non-mesh-independent pressure solution, there are errors. If you don't have the accurate pressure field solution, the pressure gradient terms in the momentum equation will be in error. And thus, the computed velocity field also will be in error. (10). On the other hand, in the vorticity-velocity formulation, this coupling effect does not exist. The error is not coupled between the velocity field and the pressure field there. But they are related. (I am not trying to sell you this vorticity idea, but rather to indicate to you that by looking at the vorticity approach you can get a better reference point when you are confused.)(11). What I am trying to point out is, even though the physics is there, it will be distorted through the errors acquired ,depending on the solution procedure taken. It is always very confusing when using the pressure based correction method for the incompressible flow calculation. (In some commercial codes, you don't even know where this pressure point is specified in the flow field,at inlet or exit, not mentioning the pressure output in terms of the pressure correction instead of the real pressure itself.) So, outside the strictly incompressible formulation, all you can do is try to reduce the error and get the solution closer to the physics of the flow. When confused, look at the vorticity side as a reference.

Thomas P. Abraham September 6, 1999 18:19

Re: A Simple Question
Hello Hrvoje and John,

Thanks to both of you for your insights,


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