I want to confirm my opinion about the pressure correction method. I have seen a very interesting defference in the calculation of flow separation in a 2D sysmmetric diffuser: With the pressure correction method one gets a symmetric flow separation, but with the time marching method one gets a non-symmetric flow separation. The latter agrees well with the reality. I would say that the pressure correction method stablizes the flow strongly and hence is not suitable for many flows. Maybe I am wrong. Does anybody have the same experience?

X. Ye

The pressure correction method is a procedure for solving the discrete form of the incompressible continuity and momentum equations. If has no influence at all on the predicted answer which is determined by the discretization of the continuity and momentum equations. For the same set of algebraic equations but solved by two alternative methods one must get the same answer (or set of answers since attaching to either wall are solutions and the symmetric solution exists but may be unstable depending on things like the exit boundary condition/treatment).

A simple check is to take the converged solution using method A and evaluate the residuals using method B. If it is not fully converged then the discretization schemes are different and the residuals will show you where.

If you do have two different discretization schemes then you can pin down the real culprit. Pressure smoothing and exit treatments would be high on my list of suspects. The former is often worked into the presentation of the solution procedure and not considered as a separate entity which can be a source of confusion.

I am not quite sure which pressure correction method and what time marching method do you use. I simply share my thought and hope it make sense to you.

In the pressure correction method, there is no direct link for the pressure between the continuity and momentum eqs. The pressure and the velocity are numerically linked by a third equation. So the solution converges to a quasi-static relation between the pressure and the velocity. In the time marching method, the continuity and momentum equations are solved simutanuously and there is a direct link for the pressure between these eqs. So the solution will converge to a dynamic relation between the pressure and the velocity. Back to your 2-D diffuser problem, the separated flow is physicallly unsteady due to the dynamic shedding of vortex. This makes the flow asymmetric even if the boundary conditions are symmetric. The time marching method is able to catch this behavior more or less so you may neiver get fully steady state solution. If you stop the solution at different time steps, you may see difference is the separation. However, with the pressure correction method, when the boundary conditions are steady and symmetric, the dynamic varition of pressure/velocity is stablized to quasi-static relation. It will nicely converge to a false steady state solution, and natually, the flow must be symmetric.

HL.

How a stalled diffuser behaves is largely determined by what is going on downstream. Stalled diffusers are certainly not always unstable. In fact, I would say the opposite is the more usual case and they switch from attached diffusing flow to stably stalled flow which can be a bugger to unstick if they go during start up (both experiment and LES simulation). However, it is obviously varies depending on the diffuser and installation.

Steady state (and transient) pressure correction methods have no problems predicting stably stalled diffusers so long as the exit conditions are properly specified (and you have not specified a plane of symmetry down the middle!). In fact, a few years ago I used it several times as an undergraduate CFD class exercise.

I am struggling to understand your answer to point out a flaw in the reasoning. In a converged solution, I cannot see how the pressure and velocity can be numerically linked other than via the discrete momentum equation (in which they both appear) since this would introduce an additional equation to the continuity and momentum equation and you would run out of unknowns?

I think the difference comes from the program technique. To get a non-symemetric solution, one may make some difference in the symmetric flows. Check your program, confirm whether the spatial integral is completely symmetric. For example, in your time marching method, when the pressure Poisson equation is solved, if the spatial integral is always done in the same direction (I=1,IMAX) and the newest information is used, the solution at the large node (I) is newer than the one at the small node (I) so that the solution becomes non-symmetric. But if you swith the integral direction between I=1,IMAX and I=IMAX,1,-1 in every other iteration step, you will find the solution remains symmetric. Similarly, the symmetric solution with the pressure correction method will be broken down if you remain the spatial integral in the same direction and use the newest information in the iteration. I have obtained Karman vortex street by integrating always in the same direction, without any other disturbence.

hi,

I want to know the discribtion of the time marching method. Can somebody tell me some references about time marching method? Thanks.

Wei

You can find this method in a large amount of literature. First of all, the book of Charls Hirsch "Numerical Computation of Internal and external Flows" is not bad.

X. Ye

Thank you for the message. About your calculation of von Karman vortex street, I am very interested to know which numerical method did you use, pressure correction method or time marching method. Do your calculation results agree well qualitatively with the reality? As I know, the von Karman vortex street is unsteady and unsymmetric. I think, a unsymmetric flow in a symmetric geometry and boundary condition is caused always by a small disturbence, no matter it is a physical disturbence or a numerical disturbence described in you message. There are a lot of such unsymmetric flows around us. I think, now is the time to check whether the pressure correction method is suitable for such flows.

X. Ye

Many numerical methods fall into the category of pressure correction, either based on time-split, or Hodge decomposition, or projection, or fractional step. I agree with Andy that such methods generally have no problem predicting steady or unsteady flows. This is because of the biharmonic property of pressure as pointed out by Gresho, PM. I have tested several projection methods in 2D and 3D computations and they work well. I uploaded some results at my website

http://web.hku.hk/~chentong/research/research.htm

and hope you may find something interesting.

By the way, I don't like to name a method by "time-marching", since basically all unsteady methods involve time marching or time integration.

Thank you for your message. I have visited your CFD site, very impressive!

X. Ye

I was following Mr. Ye's statement that his non-symmetric solution was close to the reality. If the geometry and boundary conditions are symmetric and the flow is not, the only possible answer is that this particular flow is natually unsteady. It doesn't mean there is no stable stalled diffuser. Continue from there, if the flow is natually unsteady, steady state pressure correction methods will certainly gives you a steady state solution which must be symmetric. In the converged steady state solution, the pressure and velocity are natually linked by the momentum eqs. But if the flow is unsteady due to instability, it is possible for the PCM to give an false converged stable solution in which the unsteady terms in the momentum equation is numerically cancled by the pressure and velocity combination. But the time marching methods will not give your a converged solution so that the solution may appear asymmetric.

As I said in the first place, I don't know which PCM did Mr. Ye was using. There are certainly several unsteady pressure based methods and people still call them pressure correction methods. I used to call all unsteady solvers time marching methods simply because they need an initial condition and then let time go on, and they may converge to a steady state solution, or they may produce time-dependent solutions, depends on the nature reality of the flow.

Hope that will clear your confusion, or make you more confused . Eather way, thank you for your comments :|.

(1). Where can I find this " The latter agrees well with reality." document or paper? (2). Could you mention a couple of papers to support your observation? That is a 2-D transient calculation agree well with test results, especially for asymmetric separated diffuser flows. Thank you in advance.

At the risk of causing offence, I would like to disagree with a few of your statements (but promise to leave it there).

"I was following Mr. Ye's statement that his non-symmetric solution was close to the reality."

Yes. A stalled symmetric diffuser (one which is asked to diffuse the flow too much and is forced to separate) will do one of two things. Firstly, it will attach to one of the walls (irrelevant which one) and form a large reciculation on the other - this is a unsymmetric, stable, steady state solution which can be difficult to shift. Secondly, it may intermittently attach and separate from both walls. If you were to time average the flow over many shedding periods, the time-averaged flowfield is almost certain to be symmetric (although the possibility of asymmetry exists). This time averaged symmetry of vortex shedding was used in an ERCOFTAC exercise a few years ago to assess a number of LES codes for the flow over a rectangle (I think - it may have been a cylinder). A surprising number of codes failed this test.

"If the geometry and boundary conditions are symmetric and the flow is not, the only possible answer is that this particular flow is natually unsteady"

No. The non-linear Navier-Stokes equations quite happily admit asymmetric solutions to problems with symmetric boundary conditions. A stalled laminar diffuser is one but there are plenty of other cases as X. Ye observed.

"if the flow is natually unsteady, steady state pressure correction methods will certainly gives you a steady state solution which must be symmetric"

Although you may sometimes get a converged symmetric steady solution. You can also get a converged asymmetric solution. However, in my experience, the most likely outcome is that the solution procedure will (quite rightly) fail to converge.

"But if the flow is unsteady due to instability, it is possible for the PCM to give an false converged stable solution in which the unsteady terms in the momentum equation is numerically cancled by the pressure and velocity combination"

I am not sure I fully understand this but if the solution procedure has converged all time derivative terms are zero and there is nothing to cancel.

"But the time marching methods will not give your a converged solution so that the solution may appear asymmetric."

Not sure I fully understand this but a "proper" (time accurate) time marching method (e.g. LES) is fully converged at every time step. Alternatively, if a time step is simply use to relax the solution procedure only a steady state solution is a "solution" and intermediate solution fields have no physical meaning.

As a final point, I would like repeat that stalled diffusers are greatly influenced by what happens downstream. You can maintain symmetry by restricting the downstream flow in a variety of ways.

read also John Anderson's book on cfd. it is easy to understand and explains the philosophy behind the time marching method/concept very well

I guess in your time-marching method, you are using a method that the time integral of the momentum equation is explicit Euler/Runger-Kutta method, and the spatial integral is done always in the same direction. While in your pressure correction method, both time integral and spatial integral are implicit. If this is true, this may be the reason why you obtained a non-symmetric solution in your time-marching computation and a symmetric one in your pressure correction computation. As I have said in the last message, in time-marching, if the spatial integral is always done in the same direction, a difference between the symmetric flows are numerically produced, and becomes larger and larger as the time integral marches forward. In your pressure correction computation, maybe the Re number is not large enough and the numerical disturbence is suppressed by the viscous effect. To get a non-symmetric solution, you may try to increase Re number, or add a artificial disturbence (for example a small shear flow U=(1+0.05*y)*U0 in the inflow) to make a non-symmetric inflow condition in the initial stage of computation and recover the disturbence to be zero (U=U0) when the flow becomes non-symmetric, then after converged, you will get a non-symmetric solution as similar as the one of the time-marching method.

I do not think that it is easy to get non-symmetric solution with a physical disturbence because it is very difficult to avoid numerical disturbence. The spectral method may be a good approach.

Try to get a non-symmetric solution in your pressure correction method by the method I tell you here, and tell us what happened.

Good luck.

Z. Lei

Thank you for your good tips. I'll try once with cormmercial codes such as CFX as you indicated. In fact, I work mostly on the explicit time marching method as you guessed. I gathered the experience of the pressure correction method six years ago on computing gas flow in gas turbine combustion chamber. At that time I used the SIMPLER method for steady flows and I believed that the pressure correction method can never simulate such asymmetric flows under symmetric geometry and boundary conditions correctly. But now I think there have been a lot of pressure correction methods, some of them maybe have been improved well for insteady flows as Tony Chen indicated. This is my conclusion from this very interesting discussion. Thank you again.

Dear John,

You can look into the paper of L.J. Lenke and H. Simon "Turbulence Modelling for Separated Flows", Finite Voulumes for Complex Applications II, editors R. Vilsmeiere, F. Benkhaldoun, D. Hänel, Hermes Science Publications, Paris, 1999. I think, the asymmetric separation of 2D symmetric diffuser is a well-known phenomenon of asymmetric flow under symmetric geometry and bounday conditions.

X. Ye

Your comments certainly did not cause any offence. But I have to say that you did not understand what I was talking about. Just look at your first paragragh about a stalled diffuser. You may see the first kind of separation in experiments in which you may never have perfect symmetric conditions, but you may not see this in numerical simulation if your conditions are perfect symmetric. At least in the past 10 years neither myself nor my students have seen such soltuions. Unless like someone said you switch the sweep directions to cause artificial numerical asymmetrice. It is a mathematic issue. For given equations and boundary conditions, the steady state solutions, if exist, must be unique, otherwise it is meaningless. If mathematically the solution is unique but numerically is not, you have consistency problems in your numerical algorithm.

But the second kind of separation, time averaged symmetric solutions, are very common. This is my whole point. In most time marching schemes, the time steps used in CFD calculations are smaller than the vortex shedding period. So what you see in your CFD solution is generally not time averaged, it is an instantaneous solution at certain time level. However, in schemes developed for steady state solutions, the time scales varies from place to place and the solutions are not at the same time frame at all. That smeared out all these small high frequency waves and the solutions just look like time-averaged solutions.

I used to be CFD algorithm developer before I came here as a CFD code development/application engineer. So maybe we have different view on some aspects. But I think this is not the right place to discuss this issue.