Cannot get convergence with 2nd order scheme for velocity convection term
 

Hi everybody,
I know there exists a lot of posts like the one I am writing on the forum, but I still do not succeed in obtaining what I would want. I will try to be as clear as possible in order to help you giving me assistance.

The object of study is a swirl generator, constituted by a pipe with 4 axial inlets and 8 tangential inlets with an average Reynolds number of 7000. I found in literature that a suitable turbulence model may be the k-Omega SST and that the problem may be treated as globally steady, thus I am using simpleFoam solver. I have created the mesh by means of snappyHexMesh, with the introduction of surface layers in order to get at least y+ < 5. I report the checkMesh output.

Firstly, I ran the simulation with the attached fvSchemes, fvSolution and the following BCs (N.B.: "Big" and "Small" stay for the tangential inlets, "inlet" includes all the 4 axial inlets).

U
Inlets flowRateInletVelocity
Outlet zeroGradient
Walls noSlip
internalField potentialFoam

p
Inlets fixedFluxPressure
Outlet fixedValue 0
Walls fixedFluxPressure
internalField 0

k
Inlets fixedValue
Outlet inletOutlet
Walls kLowReWallFunction

k
Inlets fixedValue
Outlet inletOutlet
Walls omegaWallFunction

In this way, the numerical solution is stable without any doubt and I could get residuals as small as I wanted. Then, the idea was to switch to 2nd order scheme for the convection term by exploiting the "1st order solution" as the initial condition.

What I have done is just changing the upwind scheme for velocity into bounded Gauss vanLeerV, as suggested in literature. I also decided to start with quite low values of URFs (p 0.1, U 0.3, k/omega 0.2) and further increase them at some points of the run. Nevertheless, the residuals smoothly decrease up to a certain level (about 5e-4 for velocity components and k, 1e-3 for pressure, 5e-5 for omega) and then become subjected to "long wave-length oscillations" (they increase for about 100 iterations, then decrease for 100 iterations and so on but the mean value remains unchanged). This occurs even if I do not increase my URFs and keep them to the respective initial values.

What I have tried to do
First of all, I have tried to use bounded Gauss linearUpwindV grad(U) or bounded Gauss limitedLinear 1 for the convection scheme. However, I did not get any benefit with respect to the problem I have discussed above.

Then, although the mesh does not seem too ugly, I have tried to introduce 1, 2 or 3 nNonOrthogonalCorrectors. I have also modify the laplacian and surface-normal gradient schemes to limited 0.5 or even limited 0.333. I could not get better results.

I have also investigated the fvSolution dictionary to seek for stability. I have tried to use smoothSolver for velocity and turbulence quantities; I have also tried to switch to SIMPLEC algorithm. Even in this case, the situation did not get better.

Finally, I tried to initialize the "2nd order solution" just with potentialFoam. The problem remained unchanged.

Final question
I did no try to modify my BCs...could they represent the problem? Instead, do you think this could be just the outcome of some intrinisc transient behaviour of the flow and I have to start a transient simulation?

Thanks in advance,
Jacopo

Does your convergence become worse on a coarser mesh and better on a finer mesh?

Hi Domenico, thanks for your reply. Actually, I started with a different mesh, which had almost the same mesh spacing but was characterized by worse quality parameters (e.g. Max skewness 3.24 instead of 1.99, 25000 concave cells instead of 300, Average Non-Orthogonality 9 instead of 4). The problem was the same.

Right. My biased way to look into this, is to say that the mesh might not be appropriate to support a second order scheme. Before trying on a computationally more expensive finer mesh, it might make sense to consider a computationally cheaper mesh. The coarser mesh and the current mesh could then be compared. Alternatively, it might be of value of look into the residual field. Alternatively, it might make sense to look into more versatile mesh generators in case that your geometry is sufficiently complex.

Hi, I generated a coarser mesh. Quality parameters are almost the same, whereas the number of cells has been reduced from 600.000 to 100.000. Nevertheless, I have the same problem as for the finer mesh and it seems to take place in correspondence of the same point of the simulation.

In particular, the problem seems to arise from the treatment of pressure. Any hint?

A coarser mesh makes experimenting easier. Have you tried plotting the residual fields? Or lowering the inlet velocity? Can you share the plots of convergence with first and second order scheme for the convective term?