Hello,

I'm using a 3D solver to compute the flow through a radial impeller passage. To make the computation, I have the choice between 2 numerical schemes: a second order central sheme with second and fourth order numerical dissipation and a first order upwind scheme.

The problem is that I have convergence problem when I use the central scheme. The massflow and all other values oscillate.

I find a solution with the upwind scheme but the massflow seems to be smaller than with the other scheme. I asked myself whether the massflow difference between the two shemes comes from the numerical viscosity of the first order scheme. What do you think about that?

Do the oscillations I found using the second order scheme come from a supersonic region?

I've checked and in certain regions the velocity is (M=0.8 1). Near impeller entry and in diffusor outlet.

Thanks for your answer

F.Ursenbacher

(1). I think you should be very happy with the code you use, because it can produce different answers using different options. (2). I would say that it is likely different user will also get different answers. (3). That is fairly consistent. (4). For the person who is interested in saving money, he can use the upwind option. (5). For the person who is interested in DNS, the transient nature of the second-order central difference solution would be very attractive. (6). So, there is no guarantee that the second-order method with a second-order and a fourth-order artificial viscosity terms is better than the first-order difference. (7). This is important, because when you run into the shock waves, it is likely that everything will be down graded to the first-order accuracy. (8). So, the conclusion is, they are not supposed to be the same, and you are all right. (9). Well, about the relationship between the oscillation and the supersonic region, you can easily find the answer by lowering the Mach number in the flow field. (by simply increasing the inlet temperature will lower the Mach number) (10). It is hard to know the answer unless you actually try it out. (11). The oscillation itself is not a bad thing though, because some people like to think that they are getting higher-order solutions. For them, a converged first-order upwind solution is just like venela icecream, it is not very exciting.

>For the person who is interested in DNS, the transient nature of the second-order central difference solution would be very attractive.

Huh? What does this mean?

As regards to losing mass when using upwind, I'm not sure what you mean by "losing" mass; however, if you have a (consistent) scheme that is conservative then you shouldn't be losing mass, no matter what order differencing you use. You may/will get a wrong (unconverged) velocity profile, but its integral (of course including density effects) should satisfy continuity. Theoretically, the upwind scheme introduces (numerical) diffusion. And all diffusion is supposed to do is to spread the diffused variable out. Period. It shouldn't create or destroy mass!

As for the oscillations with central differencing - these are quite famous. Without knowing the details of your code and simulation, I can guess that your CFL condition is not satisfied properly. That is, you are using too coarse a grid or too large a timestep. A simulation at a finer level should reduce the oscillations. If you (can) reduce your Mach number, you "may" see a correction to your problem as well. That's because you have effectively changed your CFL condition.

Adrin Gharakhani

(1). I shold have used the words "less attractive, relative to the upwind method". (2). And I am sure that people in DNS use much higher order methods, or spectral method. The solution is forced into a series of waves and oscillations as well, long before one even started the solution procedure.