Connection with Validation Databases

While the tasks of Verification and Validation (V&V) address separate issues—the demonstration of grid convergence and the accurate reproduction of actual physical processes, respectively—

ideally these efforts ought to be combined and applied jointly. Comparison with test data by itself is insufficient as a means of establishing confidence in a given CFD code or physical model.

Similarly, physical models cannot be fully evaluated without first ruling out or at least understanding the magnitude of discretization errors. As an extreme example, it is possible to perform solution verification for an inviscid solution to a backward-facing step. One can demonstrate grid convergence, even though the selected physical model is completely inappropriate and incorrect. Once this verification is accomplished, one may readily identify where the physical modeling employed falls short. Verification is thus an equally important aspect of CFD code and model development. With the development of a single-grid error prediction method, the ETE, the task of solution verification has become much less labor and time intensive.

Figure 11 depicts how the error quantification capabilities in CRISP CFD may be combined with the CRAFT Tech Automated Validation Environment (CRAVE) [67]. After implementing a new physical model, the analyst selects a suitable validation test case from the CRAVE database to assess the model. The existing package for the grid and input deck is extracted and the flow solver is run using the new model. Previously, one would then run automated scripts from CRAVE to extract the new results and compare with prior baseline simulations and stored test data. Instead, the analyst now has the option to run an error analysis for the solution, and obtain numerical error bars to include in the resulting plots. If the experimental data falls outside the predicted error bars, this indicates a discrepancy in physical modeling, not in the solution of the equations. Such insight is both useful and valuable. Time permitting, users may employ the AMR capabilities in CRISP CFD to subsequently reduce these errors and re-evaluate the solution along with test data.

Figure 11. Use of CRISP CFD with CRAVE.

7. Example Applications 7.1. Low Speed Airfoil

Turbulent low speed flow past a two-dimensional airfoil is presented as a realistic wall bounded flow example case. The airfoil, analyzed in previous work by the author is at zero angle-of-attack in a Mach 0.3 freestream. Experimental data containing surface pressure coefficients and detailed velocity measurements is given in the paper by Nakayama [68]. Note that no experimental data for section lift and drag coefficient is available in the original paper. Figure 12 illustrates the mesh sequence used in the current study, while Figure 13a depicts the flowfield through nondimensional velocity contours on the fine grid, along with selected stations in the wake where velocity profile data is available for comparison.

Figure 12. Mesh sequence for turbulent airfoil test case of Nakayama [68].

Figure 13b through Figure 13d present the results from the error transport equation for each of the meshes employed. The streamwise (U) component of velocity is compared with experimental data for each of the three wake stations shown in Figure 13a. As one might expect, the magnitude of the errors reduces as mesh resolution is increased and the solution accuracy is improved.

Examining the errors for the last station, it is observed that the predicted error increases from coarse to the medium mesh, and then reduces from the medium to the fine mesh. This is an instance where the initial mesh is too coarse to accurately capture the error. The ETE is solved on

a) Coarse b) Medium c) Fine

FLOW SOLVER

CRISP CFD ERROR ANALYSIS

TO SIMULTANEOUS VERIFICATION AND VALIDATION FROM VALIDATION DATABASE…

FLOW SOLVER

CRISP CFD ERROR ANALYSIS

TO SIMULTANEOUS VERIFICATION AND VALIDATION FROM VALIDATION DATABASE…

the same grid used to solve the Navier-Stokes equations, and thus is subject to the same numerical discretization issues. Figure 14 compares the predicted lift and drag coefficients for the three-grid sequence. The error bars derived from the inviscid ETE solution provide confirmation of the level of accuracy achieved, at least for . In contrast, the predicted errors in are considerably larger than those provided by Richardson extrapolation. The primary contributor to this is the computed error in wall shear stress as discussed in prior work by the author [49, 50].

Using the viscous ETE formulation, the error predictions are not affected, however the error bars are greatly improved and more in line with the results of Richardson extrapolation.

Figure 13. ETE predictions for low speed airfoil.

CL C_D

b) Wake profiles on coarse grid

c) Wake profiles on medium grid d) Wake profiles on fine grid

Figure 14. Error predictions for aerodynamic coefficients for a low speed airfoil. height H = 1.27 cm is given as a reference length. The tunnel span has a length of 12H, and the tunnel height prior to the step is 8H. The boundary layer is well developed prior to the step with a thickness of approximately 1.5H. The inlet was modeled far upstream to achieve this boundary layer growth. The approach flow itself has a centerline velocity of 44.2 m/s, at standard atmospheric pressure and temperature.

A series of four grids were employed, as summarized in Table 5, with the medium grid shown in Figure 16 for reference. All grids were hexahedral, with the medium grid in the region of interest containing 27,200 cells. The cell spacing for the coarse grid was halved in both the x and y directions, while the cell spacing was doubled successively in both directions for the fine and very fine grids. Contours of velocity magnitude are presented in Figure 17, in which the boundary layer thickness and reattachment point are clearly evident. We present error predictions for two sets of simulations. The node-centered solver CRUNCH CFD was used along with the standard k-ε turbulence model. We also present results for this case using the cell-centered solver AVUS [71] and the k-ω turbulence model.

Table 5. Resolution in the region of interest for backward facing step study.

Grid Resolution Number of Cells

1 Coarse 6,800

2 Medium 27,200

3 Fine 108,800

4 Very Fine 435,200

Figure 15. Geometry of backward facing step.

Grid