Grid Convergence Study ! - Is it Necessary?

Figure 1: Grid convergence study: SPICES-2013 Workshop, Flying Wing Configuration.

1220 words / About 6 mins

Grids play an equal and important role as the solver, in generating accurate results. The gridding methodology chosen (structured/unstructured/cartesian, etc) for computation is not the only choice that matters, but also the spatial resolution of the flow field. Though one might argue that, with the development of accurate algorithms, the demarcation between solutions obtained from structured /unstructured /cartesian is blurring, the necessity of conducting a grid convergence study (GCS), is something which is unconditionally agreed upon in the CFD community, before venturing into full-scale production runs.

The Eternal Dilemma!

Is the generated grid fine enough to get accurate results? In spite of addressing the basic needs of grid generation – capturing the geometry and respecting the physics, etc. one is not 100 percent sure, as to whether the CFD prediction is farther or nearer to the true solution. If at one end of the spectrum, the level of accuracy of the predicted results is of concern, at the other, the question lingers, is the level of accuracy actually worth the time and computational resource spent?

It’s a known fact that a fine grid would give more accurate results than a coarse grid. However, this comes at a cost, a) Larger grid generation and computing time, b) Increase in computational hardware requirements (RAM, hard disk space), and (c) Need to run the solver with more processors.

This brings us to the quintessential question, was the delta improvement in the solution obtained as we moved from the coarse grid to the finer grid of any worth?

How do we know the break-even point where the grid is sufficiently fine enough to be optimal in terms of the time and computational resources and still make CFD predictions which are reasonably accurate within a known level of tolerance?

Grid Convergence Study

The solution lies in conducting a systematic gridding exercise called the grid convergence study or mesh refinement study. For the concerned geometry, a series of grids are generated and CFD computations are performed and the variation in results with each grid level is analyzed. In general, at least 3 levels of grids – coarse, medium, and fine are generated for the study. In a more rigorous grid independent study, additional levels of grids – tiny, extra-fine, super-fine, and ultra-fine are also made use of, making the number of levels to 7. In international workshops like the AIAA DPW and AIAA HiLiftPW, at least 5 grid levels are expected for a transport aircraft configuration.

GCS is based on the fundamental principle that with the increase in fineness of grids, the spatial discretization errors will asymptotically approach to zero and thereby helping to achieve a grid-independent solution. What this means is that the solution is independent of the mesh resolution and any further refinement is not going to improve the solution.

**Figure 3:** Grid convergence study: Mesh at the leading edge.

How to Generate GCS Grids?

In mesh refinement study, systematic refinement of the grid in every nook and corner of the computational domain is carried out with each successive level in the grid family. The mesh density on the surface, in the viscous padding and flow field, is increased in an orderly manner. Conventionally, the number of surface elements increases by a factor of 2, while the volume cell count increases by a factor of 3 with each successive grid. In a structured grid, this translates to 1.5 times the growth in the number of points in each coordinate direction. Also, every grid in the grid family maintains the same stretching factors, same topology, etc. Even the first spacing is reduced in a systematic way, with the estimated y-plus decreasing from 1.0 to 0.1 with successively refined grids.

This is a tough challenge and even experts in grid generation have difficulty in generating a grid family adhering to the stringent guidelines. AIAA DPW and AIAA HiLiftPW gridding guidelines can serve as a good source for CFD practitioners to have clarity on what is expected.

The point to note is that these above-specified growth factors like the increment in total grid size by thrice, surface mesh by twice, etc., are not sacrosanct. One can have a different value say 1.5 times for volume and still generate a good grid family for GCS. What is critically important is the strict adherence of these fixed values for every level in the grid family. The finest of the grid generated should be sufficiently fine enough to declare the obtained CFD results as grid converged results.

This aspect is clearly seen in the last two Drag Prediction Workshop. In DPW5, the guidelines demanded to generate 6 grid levels for a wing-body configuration varying from 0.6 million to 140 million with a volume growth factor of 3. Interestingly, in DPW6, the guidelines requested the participants to generate 6 grids for the same configuration, with grid size varying from 20 million to 150 million, with a volume growth factor of 1.5.

**Figure 4:** Grid convergence study: Mesh at the wing tip region.

Wrong Notions About GCS

Many CFD engineers assume that GCS means, generating 2-3 grids with the total volume cell count increasing in some arbitrary manner — say a 2 million, 3 million, and 7 million for a configuration of interest, perform simulations and declare to achieve a grid-independent solution. In such assumptions the subtlety is missed out, GCS requires a systematic orderly variation in spatial resolution in every nook and corner of the domain. It is not some random refinement in the fluid domain. Increasing volume cells only or refining only surface mesh will not lead to building a GCS grid-family.

The grids so generated may still be qualitatively fine and the solver will produce results with some variation in results among the grids, but it doesn’t mean a grid-independent solution has been achieved. Unfortunately, this wrong understanding of GCS grids is strongly prevalent among many CFD practitioners, undermining the CFD results they generate.

GCS - Mesh at the trailing edge — **Figure 5:** Grid convergence study – Mesh at the trailing edge.

Interpreting the GCS Results

Once the family of grids is generated, simulations are done on each grid and the flow field parameters of interest say, CL, CD, etc. are plotted against the grid size. Usually, the change is the value from tiny to coarse to the medium is large. Starting from the fine grid, the delta change in solution becomes small, asymptoting to a grid-independent solution. From the plotted graph, one can pick the smallest of the grid, which gives a grid-independent solution for routine production runs. This is the optimal grid, which will provide the right solution with minimal solver run time.

GCS are done in CFD to make sure that the results obtained from simulations are due to the boundary conditions and the physics used and not because of mesh resolution. If the solution is invariant of the grid density, then we have achieved grid independence.

Grid convergence study - Mesh in the base region — **Figure 6:** Grid convergence study – Mesh in the base region.

Again, careful discretion of the difference between the asymptotic numerical value and true value is required. Attaining a grid independence solution does not mean getting true or exact value. Grid independent numerical value only means the spatial discretization errors have been eliminated and the asymptotic value may still have errors causing it to converge to a value different from the true value.

In general, it is a good practice to do GCS for a new configuration or a fluid flow problem, one can confidently apply the same mesh sizing the next time a similar geometry is used for CFD simulation. GCS helps in gaining more confidence in the generated results and also brings greater reliability to the CFD predicted data.