I met this phenomenon: in calculations of incompressible flows using zonal methods, pressure is oscillating near grid interfaces, although velocity is very smooth. This phenomenon becomes obvious as mesh size ratio is large across the interfaces. It seems that the oscillations cannot be eliminated. Besides, it may be an open question that how much the oscillations affect the accuracy of the whole solution.

Do you once had such experience or know some clues to overcome the problem?

Thanks a lot.

Hi, Hansong.

This reminds me something I met in my own practice. I was solving NSE in primitive variables. The scheme was strictly conservative, and to obtain a second order of accuracy I approximated mass fluxes through the cell borders also with second order. Provided that the approximation error varies smoothly this indeed ensures the second order. However, near a certain line, due to a sudden change in the grid structure the approximation error (for fluxes) changed suddenly, remaining of the second order. This, of course, resulted in a loss of approximation at this line: it reduced to the first order. (Well, O(h**2) change divided by the step size h gives O(h) error in continuity). However, it is known that often (in other PDE problems) this does not result in reducing accuracy of the solution. Indeed, we are in fact solving

\nabla u=h*E, h being the grid step,

with E being nonzero within a layer of thickness of order h, too. Therefore, the first derivatives of u will be obtained with O(h) error in this layer, but u itself with h*O(h)=O(h**2). Hence it seems OK.

The problem is that in the Laplace operator of the momentum equation this u is differentiated 2 times, and the pressure only once. As a result, the pressure has an error of order h. while the velocity can sometimes be obtained with h**2 accuracy. Depending on the problem, O(h) error in p may also lead to the error of order h in u.

My solution was to use third order approximation for mass fluxes in the continuity equation in the entire calculation domain.

Well, the recommendation: consider in detail the approximation error at the grid interface both in continuity and momentum equations, remembering that they are not independent.

Hope this may help.

Yours Sergei

Dear Sergei,

Thanks you much for your suggestions and I will consider them carefully.

Currently I use primitive variables and Dirichelt conditions at grid interfaces. Subzones overlap with each other a little. I also tried to use Neumann conditions.

I guess that the oscillations have something to do with interaction between waves with different wave numbers.

By the way, have you published your results?

Hansong Tang

>I guess that the oscillations have something to do with >interaction between waves
>with different wave numbers.

Waves in incompressible flow? Do you mean vorticity waves? Waves are rare a convenient notion when thinking about incompressible flow. Vorticity is convected as a small stick thrown into a stream, rotated and stretched by it, and it diffuses like a drop of ink in a pure water. To interpret something as a wave you need quite complicated constructs in incompressible case. Mathematically speaking, the equations are parabolic or elliptic, but certainly not hyperbolic.

You need to explain why only the pressure oscillates. One possible mechanism I discussed. Another is known. If a non-staggered rectangular grid is used (=all variables are sought for at the centers of the same grid cells), the pressure values in black and white cells (like on a chessboard) can differ by a constant. But in that case oscillations are not restricted to the vicinity of the interface. It hardly can be your case.

By the way, what is the length scale of the oscillations? If what I talked about was relevant then the wavelength of the oscillations is the wavelength of the jump in the approximation error. Now, if you have different grid steps along the interface on different sides of the interface, say, steps h_1=1/a and h_2=1/b, a>b, then one can expect oscillations with wavelength h_osc=1/(a+b) modulated with the wavelength 1/(a-b). (The idea is that this is similar to sin(x/a)-sin(x/b)). Is it so?

>By the way, have you published your results?

No. That was not a result. That was an error in the program. Generally, I publish theoretical results and use numerics for illustrations, or for proving that a particular problem has a solution with expected properties etc. With a few exceptions, in my publications numerical calculations are in the background, results are presented but numerical methods are not described because they are standard.

OK. Good luck. When you will find the true reason, share the experience.

Yours Sergei

My case is: mesh spacing jumps across grid interfaces, i.e., the spacing is discontinuous in the normal direction of the interfaces. I checked pressure contours and found that the oscillations mainly happen at the small spacing side and are confined within a few layers near the interfaces. The length scales of the oscillations are about the size of grid spacing.

I also tried cases that mesh size is dicontinuous in the direction of interfaces. THe results seemed to be similar.

What I meant by interaction between waves is that I feel that certain small structures from the fine grid cannot be resolved by the corse grid, and thus pressure adjusts itself and accumulates something around the interfaces.

I checked Shyy's papers. One of them says that the oscillations can be eliminated by enforcing mass conservation. But I am quite sure if this is true.

If I feel I get to something useful about this, I will share with you.

Hansong

You are probably not using the finite-difference methods. And even with the finite-difference method, smooth mesh stretching is essential to minimize the error. (1). try to bring the mesh size from both sides equal at the interface location first. (2). check out whether the problem is still there. ( you can let the mesh to grow bigger on the coarse mesh side.) (3). don't try to fix the problem, you need a sound mathematical foundation. (4). try a one dimensional model problem, and look directly at the final algebraic equation you are trying to solve, at the interior points and boundary points. (5). if possible, try to state your problem more clearly ( both the problem you are trying to solve and the problem you run into.)

Dear Chien,

Actually, I am using finite differencing, and the mesh jump at grid interfaces is desired (for applications such as local mesh refinement).

You are right. I am considering what is the true reason for the oscillations and try to formulate the problem if it is a real one. One question is that if you know some other people also once had such experience. It seems that the phemoneon (that oscillation) is not addressed much.

Thanks for you suggestion

Hansong Tang

Dear Hansong,

An afterthought:

I had oscillations of the length equal to the grid step when solving steady NSE using Newton iterations, with a direct solver for the linearized system. The third-order upstream differencing (Leonard scheme) was used, but the cells were not rectangular (strongly skewed, in fact). Naturally, the matrix was not diagonally dominant, so oscillations were a form of the well-known numerical instability. As could be expected, refining the grid removed the oscillations.

Therefore, a suggestion: check if refining the grid affect the oscillations. Normally, if the grid Reynolds number is less than something about unity, such an instability is not observed. If refining the grid removes the oscillations, but is too restrictive, it is worth to try to change the scheme/stencil in such a way that the main diagonal becomes more dominant even if it cannot be made really dominant.

Who knows, may be this can help. Good luck again.

Yours Sergei.