Hi,

In laminar flows in complex geometries (with recirculations, moving walls etc.) without any modelling requirement (neither combustion, phase change or multiphase), is it possible – by means of doing the mesh more and more fine - to achieve any desired level of accuracy of the results?

In other words, can we do the overall numerical error as small as we want in the aforementioned case?

Thanks in advance,

Gorka

There exists no error estimates for the Navier-Stokes equations, hence, it is not possible to in general say something about the accuracy. You can do grid refinenment studies etc. to make it likely that your results have a certain degree of accuracy. However, you can not prove this mathematically.

Thanks for your response Jonas,

Actually I am looking for a more pragmatic answer. Lets suppose that we have to calculate an isothermal laminar flow in a very complex geometry with a good quality mesh and using a tested code that works with well-posed (consistent, conservative etc.) numerical schemes. Doing the mesh more and more fine, can we aspire to reduce the numerical error to the computer round off error, and then affirm that the results would be realistic without the need to compare with experimental data???

Thanks in advance,

Gorka

Let us assume that the NS equations are exact (excluding, e.g., breakdown of continuity for small Knudsen No.). Then, if the discretization is CONSISTENT, it is converging to the differential equations in the limit of infinitely small cell size. Therefore, to my understanding, refining the grid will bring you closer to the exact solution (assuming, of course, that the BCs are correct, that the iterative solution is converged, etc).

I agree that the error estimates are unavailable, but performing refinement study gives some insight on the grid dependence and the rate of convergence wrt to the grid size.

Thanks for your response Rami,

Of course, you are right. To obtain realistic results the Boundary Conditions should be realistic too.

Gorka

You can never be 100% sure of how accurate your results are since there is no mathematical proof to rely on.

However, for simple laminar flows experience says that you can be quite sure that a good CFD code will produce good results, especially if you perform a grid-refinement study. However, if the flow is complex with separations, unsteadiness etc. you are more likely to produce incorrect results. If the flow is nicely attached and steady I'd trust the results.

Jonas and Rami are both right. A numerical scheme is one approach to proving existence of a solution. Mathematicians sometimes use a finite-difference scheme to provide a constructive proof of existence of a solution to simpler equations than the NS. In those cases, they are able to prove certain things about how the difference scheme will behave as one refines the mesh. For the NS equations, such proofs are not available. In general, one should not expect such proofs for steady-state solutions, except perhaps for trivial cases. However, numerical experiments such as Gorka contemplates verge on being heuristic "proofs" for any particular geometry and boundary conditions. One may not be able to mathematically prove in advance that the procedure will not blow up on the next finer mesh, or that the sequence of "steady-state" numerical solutions will converge with respect to mesh refinement. However, numerical experimentation with a consistent scheme on a sequence of increasingly fine meshes, gives one confidence that a solution exists, and that one is approaching it as the mesh is refined. Engineers must press on with a Platonic attitude, while mathematicians compete for one of the Clay Institute prizes!