Toward a Process for Quantifying Uncertainty in a CFD Solution: Analysis of Mesh Refinement

The objective of this study was to formulate a process to quantify the uncertainty associated with computational fluid dynamics (CFD) analysis. A generic, cone-nosed missile simulation using COBALT as the flow solver package was selected as a test case. The flow conditions were indicative of a missile traveling at Mach 3 at 30,000 feet above sea level. An unstructured meshing scheme was employed, using manual refinement and clustering to capture the flow characteristics. Three subsequent meshes were generated by doubling the spacing in every direction within the grid, creating four different mesh densities, with a constant, one-dimensional refinement ratio of 2. Two Grid Convergence Indices (GCl\´s) were then formulated, using the three finest meshes and the three coarsest meshes. The axial force value was the quantity of interest used to conduct the GCI analysis. The results of the first prediction, conducted with the three finest meshes, indicated that the "exact" solution fell within the range [590.527, 594.795], whereas the second study indicated that the "exact" value fell within the range [584.892, 612.992]. The "exact" solution is defined by the GCI study to be the Richardson Extrapolated value for a particular set of solutions. In other words, the "exact" value would be the predicted solution for a grid with grid spacing equal to zero. This process was then repeated on a simplified case over a blunt-body with a structured meshing scheme. The same flow conditions were used for this case as for the cone- nosed missile. The same trend was observed in the GCI results of the simplified case, showing that the GCI "exact" value range deduced from the denser meshes fell within the range of "exact" values produced from the coarser meshes. The results also indicated that the COBALT solver tended to produce second-order convergence values (p values) with respect to grid refinement. The values obtained for p in the missile tests were 2.03157 and 1.14883 for the fine and- coarse GCI studies, respectively. In the blunt-body cases, the p values were found to be 1.95777 and 2.28039 for the fine and coarse meshes, respectively. It seemed odd at first that two of the resulting p values were greater than two, considering the fact that it is more usual for nominally second order solvers to produce p values slightly less than two. This can possible be attributed to the fact that the study was conducted assuming inviscid flow, and therefore the solver was only manipulating the convective terms from the Navier-Stokes equation while neglecting the diffusion terms. Many flow solvers will use higher order techniques to solve the convective terms, while using nominally second order techniques to solve the diffusion terms. Though this conclusion may not give any quantifiable measurement of the overall uncertainty associated with results of the aforementioned studies, the GCI study performed does provide a good indication of the uncertainty due to mesh resolution associated with the COBALT flow solver.