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SSG/LRR-RSM-w2012 Expected Results - 2D Zero Pressure Gradient Flat Plate

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 TAU, were used to compute this zero-pressure-gradient flat plate flow with the SSG/LRR-RSM-w2012 second-moment Reynolds stress transport model (see full description on SSG/LRR Full Reynolds Stress Model page). The full series of 5 grids were used. CFL3D is a cell-centered structured-grid code (NASA Langley), and TAU is a node-centered unstructured-grid code (DLR). CFL3D used Roe's Flux Difference Splitting, whereas TAU was run using central discretization with artificial matrix dissipation for the mean flow equations and upwinding for the turbulence equations. Additionally in TAU, low Mach number preconditioning was applied. Both codes were run with full Navier-Stokes, and both codes used first-order upwinding for the advective terms of the turbulence model. (However, TAU was also run with second-order upwinding of the turbulence model, and it made only minor differences; see near bottom of this page.) Details about the codes can be found on their respective websites (CFL3D, TAU). The codes were not necessarily 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. For example, for CFL3D the density residual was typically driven down below 10^-13. 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).

For the CFL3D tests reported below, the turbulent inflow boundary conditions used for SSG/LRR-RSM-w2012 were the following:

$\hat R_{11, farfield} = \hat R_{22, farfield} = \hat R_{33, farfield} = 6 \times 10^{-9} a_{\infty}^2$

(meaning that $k_{farfield} = (\overline{u_1''u_1''} + \overline{u_2''u_2''} + \overline{u_3''u_3''})/2 = 9 \times 10^{-9} a_{\infty}^2$ ),

$\hat R_{12, farfield} = \hat R_{13, farfield} = \hat R_{23, farfield} = 0$

and

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

The above equations represent the "standard" SSG/LRR-RSM-w2012 boundary condition values used by CFL3D. In terms of freestream turbulence intensity (Tu) and freestream eddy viscosity, these boundary conditions for this particular problem (with M=0.2) correspond to: Tu=0.039% and $\mu_t / \mu = 0.009$ . The freestream values used by TAU were Tu=0.1% and $\mu_t / \mu = 0.1$ . For freestream BCs, both codes assume isotropic turbulence conditions (identical normal stresses, zero diagonal stresses).

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

CFL3D:

Typical Input File for 2D Flat Plate

TAU:

Typical Input File for 2D Flat Plate

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	0.99	0.202%	0.205%	0.257%
TAU	1.19	0.260%	0.202%	0.253%

The data file that generated the above plot is given here: cf_convergence_ssglrrrsm.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. TAU, 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	1.24	0.173%	0.127%	0.158%
TAU	1.23	0.286%	0.212%	0.266%

The data file that generated the above plot is given here: drag_convergence_ssglrrrsm.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 ("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_ssglrrrsm.dat.

Contours of the nondimensional Reynolds stress variables ( $\hat R_{ij}$ ) as well as nondimensional omega from the two codes on the finest 545 x 385 grid are shown in the following plots (z-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. Note also that in both codes for this case, the "z"-direction is up. Therefore, for a 2-D computation the 12 and 23 components of the Reynolds stress are identically zero. The results from TAU shown here are not in their native nondimensional form, but have been re-nondimensionalized to match the native form in CFL3D.
R11 contours for CFL3D R11 contours for TAU

R22 contours for CFL3D R22 contours for TAU

R33 contours for CFL3D R33 contours for TAU

R13 contours for CFL3D R13 contours for TAU

omega contours for CFL3D omega contours for TAU

The data files that generated the above plots are given here: turb_contours_cfl3d_ssglrrrsm.dat.gz (5.8 MB), (structured, at cell centers) and turb_contours_tau_ssglrrrsm.dat.gz (11.0 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.

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+_rsme.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_rsme.dat (extracted only for CFL3D).

TAU was also run using 2nd order turbulence advection (as opposed to 1st order). The change made only small differences, as shown in the following plots.
effect of turb advection spatial order
on convergence of Cf at x=0.97 vs h

The codes were also run with the LRR/SSG-RSM-w2012-SD variant. Results were slightly different from LRR/SSG-RSM-w2012, but the two codes CFL3D and TAU were again consistent with each other as the grid was refined, as shown in the following plots.
effect of simple diffusion variant
on convergence of Cf at x=0.97 vs h

Results from FUN3D are shown alongside the CFL3D and TAU results below. All three codes are consistent.
skin friction coefficient over the plate, incl FUN3D
convergence of Cf at x=0.86 vs h, incl FUN3D convergence of plate drag coefficient vs h, incl FUN3D

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

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Recent significant updates:
09/05/2014 - added some FUN3D results

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Last Updated: 12/01/2014