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Wilcox2006-klim-m Expected Results - 2D Zero Pressure Gradient Flat Plate

Previously on this page the results were reported as Wilcox2006-klim solutions, but more properly they should be referred to as Wilcox2006-klim-m.

Results are shown here from 2 compressible codes so that the user may compare their own compressible code results. Multiple grids were used so the user can see trends with grid refinement. Different codes will behave differently with grid refinement depending on many factors (including code order of accuracy and other numerics), but it would be expected that as the grid is refined the results will tend toward an "infinite grid" solution that is the same. Be careful when comparing details: any differences in boundary conditions or flow conditions may affect results.

Two independent compressible RANS codes, CFL3D and FUN3D, were used to compute this zero-pressure-gradient flat plate flow with the Wilcox2006-klim-m model (see full description on Wilcox k-omega page). The full series of 5 grids were used. CFL3D is a cell-centered structured-grid code, and FUN3D is a node-centered unstructured-grid code (FUN3D can solve on mixed element grids, so this case was computed on the same hexahedral grid used by CFL3D). Both codes used Roe's Flux Difference Splitting and a UMUSCL upwind approach. In CFL3D its standard UMUSCL (kappa=0.33333) scheme was used, whereas in FUN3D the option UMUSCL 0.5 was used. Both codes were run with full Navier-Stokes (as opposed to thin-layer, which is CFL3D's default mode of operation), and both codes used first-order upwinding for the advective terms of the turbulence model. Details about the codes can be found on their respective websites, the links for which are given on this site's home page. The codes were not run to machine-zero iterative convergence, but an attempt was made to converge sufficiently so that results of interest were well within normal engineering tolerance and plotting accuracy. It should be kept in mind that many of the files given below contain computed values directly from the codes, using a precision greater than the convergence tolerance (i.e., the values in the files are not necessarily as precise as the number of digits given).

Note that in both CFL3D and FUN3D, the production term

$P = \tau_{ij}\frac{\partial u_i}{\partial x_j}$

is approximated by

$P = 2 \mu_t S_{ij}S_{ij}$

which is exact for incompressible flow. For this particular low-speed flow, the approximation is very accurate. (Previously on this page the results were reported as Wilcox2006-klim solutions, but more properly they should be referred to as Wilcox2006-klim-m because of this approximation and the fact that the $(2/3) \overline \rho k \delta_{ij}$ term is ignored in tau_ij in the momentum and energy equations.)

For the CFL3D and FUN3D tests reported below, the turbulent inflow boundary conditions used for Wilcox2006-klim-m were the following:

$k_{farfield} = 9 \times 10^{-9} a_{\infty}^2$

$\omega_{farfield} = 1 \times 10^{-6} \frac{\rho_{\infty}a_{\infty}^2}{\mu_{\infty}}$

The above two equations represent the "standard" Wilcox2006-klim-m boundary condition values used by both CFL3D and FUN3D, chosen to achieve a not-too-low level of freestream turbulent kinetic energy, a not-too-severe rate of freestream turbulence decay, and a reasonable level of freestream turbulent eddy viscosity of $\mu_t/\mu_{\infty} = 0.009$ .

For the interested reader, typical input files for this problem are given here:

CFL3D V6.5:

Typical Input File for 2D Flat Plate

FUN3D:

The following plot shows the convergence of the wall skin friction coefficient at x=0.97008 with grid size for the two codes. In the plot the x-axis is plotting 1/N^1/2, which is proportional to grid spacing (h). At the left of the plot, h=0 represents an infinitely fine grid. As can be seen, both codes go toward approximately the same result on an infinitely refined grid.
convergence of Cf at x=0.97
vs h

Using the uncertainty estimation procedure from the Fluids Engineering Division of the ASME (Celik, I. B., Ghia, U., Roache, P. J., Freitas, C. J., Coleman, H., Raad, P. E., "Procedure for Estimation and Reporting of Uncertainty Due to Discretization in CFD Applications," Journal of Fluids Engineering, Vol. 130, July 2008, 078001, https://doi.org/10.1115/1.2960953), described in Summary of Uncertainty Procedure, the finest 3 grids yield the following for skin friction coefficient at x=0.97:

Code	Computed apparent order, p	Approx rel fine-grid error, e_a²¹	Extrap rel fine-grid error, e_ext²¹	Fine-grid convergence index, GCI_fine²¹
CFL3D	1.08	0.279%	0.248%	0.311%
FUN3D	1.25	0.258%	0.187%	0.234%

The data file that generated the above plot is given here: cf_convergence w06.dat.

Note that in this particular flat plate case, when looking at the total integrated drag coefficient on the plate, formal order-property convergence may not be generally achievable. This is because the skin friction (which is the only contributor to the drag in this case) is singular (tends toward infinity) at the leading edge. The finer the grid, the more nearly singular the local behavior on a finite grid. FUN3D, which is a node-centered code, solves for flow variables at the leading edge, so it may be more sensitive to the singular behavior than CFL3D, which is a cell-centered code. There also appears to be some locally minor anomalous behavior at the aft end of the plate, which is likely a function of how each code handles the interaction of the solid wall boundary condition with the outflow pressure boundary condition near the bottom right corner of the grid. Nonetheless, both codes are tending toward a similar integrated drag coefficient value as the grid is refined.
convergence of plate drag
coefficient vs h

Code	Computed apparent order, p	Approx rel fine-grid error, e_a²¹	Extrap rel fine-grid error, e_ext²¹	Fine-grid convergence index, GCI_fine²¹
CFL3D	1.71	0.242%	0.107%	0.133%
FUN3D	1.26	0.530%	0.378%	0.474%

The data file that generated the above plot is given here: drag_convergence_w06.dat.

The surface skin friction coefficient from both codes on the finest 545 x 385 grid over the entire plate is shown in the next plot. Again, local anomalous behavior exists near the leading edge (x=0) due to singular behavior of the solution, and near the trailing edge (x=2) most likely due to boundary condition interaction. These behaviors differ for the two codes, and result in small local deviations that can be seen when zoomed into the two locations. In addition, both codes indicate turbulence "activation" at slightly different locations very near the leading edge, 0 < x < 0.01 ("activation" is where the turbulence model transitions on its own from laminar to turbulent). But both codes are seen to yield nearly identical results over most of the plate.
skin friction coefficient over the plate

The data file that generated the above plot is given here: cf_plate_w06.dat.

The nondimensionalized eddy viscosity contours, k contours, and omega contours from the two codes on the finest 545 x 385 grid are shown in the following plots (y-scale expanded for clarity). Results from the two codes on this grid are essentially indistinguishable. (Note legends do not necessarily reflect min and max values.)
eddy viscosity contours for CFL3D eddy viscosity contours for FUN3D

k contours for CFL3D k contours for FUN3D

omega contours for CFL3D omega contours for FUN3D

The data files that generated the above plots are given here: mut_contours_cfl3d_w06.dat.gz (1.5 MB), k_contours_cfl3d_w06.dat.gz (1.5 MB), omega_contours_cfl3d_w06.dat.gz (1.5 MB) (structured, at cell centers) and mut_contours_fun3d_w06.dat.gz (2.6 MB), k_contours_fun3d_w06.dat.gz (2.6 MB), omega_contours_fun3d_w06.dat.gz (2.6 MB) (unstructured, at grid points). Note that these are all gzipped Tecplot formatted files, so you must either have Tecplot or know how to read their format in order to use these files.

Using the finest 545 x 385 grid, an extracted nondimensional eddy viscosity profile at x=0.97 is shown below.
eddy viscosity at x=0.97

The data file that generated the eddy viscosity profile at x=0.97 is given here: mut_0.97_w06.dat.

The nondimensional k and omega profiles at x=0.97008 from the 545 x 385 grid are shown in the following plots.
nondimensional k profile at x=0.97 nondimensional omega profile at x=0.97

The data file that generated the eddy viscosity profile at x=0.97 is given here: w06_omega_k.dat.

In terms of inner wall variables, u+ and y+, the finest grid yields the following results, which are shown at two x-locations of x=0.97008 and x=1.90334. The law-of-the-wall theory with kappa=0.41 and B=5.0 is also shown (see White, F. M., Viscous Fluid Flow, McGraw-Hill Book Company, New York, 1974, p. 472).
velocity profiles in inner wall variables

The data file that generated the above plot is given in flatplate_u+y+_w06.dat for the CFD (extracted only for CFL3D), and in u+y+theory.dat for the theory (other theoretical curves, not shown in the plot, are also included in this latter file).

Standard velocity profiles are shown at the same two x-locations of x=0.97008 and x=1.90334 for the finest grid in the following plot.

The data file that generated the above plot is given in flatplate_u_w06.dat (extracted only for CFL3D).

Note for users of OpenFOAM.

Return to: 2D Zero Pressure Gradient Flat Plate Verification Case Intro Page

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Recent significant updates:
9/18/2014 - added link to note for users of OpenFOAM
09/20/2013 - modified FUN3D results to use same UMUSCL as other pages

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Last Updated: 03/24/2021