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SSG/LRR-RSM-w2012 Expected Results - 2D Bump-in-channel

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 bump-in-channel 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. Both codes were run with full Navier-Stokes, 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 (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 Bump

TAU:

Typical Input File for 2D Bump

The following plots show the convergence of the wall skin friction coefficient at the bump peak (at x=0.75), in front of the bump peak (at x=0.6321975), and aft of the peak (at x=0.8678025) 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=0.75 vs h

convergence of Cf at x=0.6321975 vs h

convergence of Cf at x=0.8678025 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=0.75, x=0.6321975, and x=0.8678025:

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
x=0.75
CFL3D	1.11	0.352%	0.301%	0.378%
TAU	1.52	0.336%	0.180%	0.225%
x=0.6321975
CFL3D	0.47	0.360%	0.927%	1.072%
TAU	0.45	0.373%	1.018%	1.100%
x=0.8678025
CFL3D	1.18	0.491%	0.387%	0.482%
TAU	1.32	0.459%	0.309%	0.385%

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

The following plots show: (1) total drag coefficient, (2) pressure drag coefficient, (3) viscous drag coefficient, and (4) total lift coefficient for the bump. In this bump case the surface skin friction is singular (tends toward infinity) at the leading edge. The finer the grid, the more nearly singular the local behavior on a finite grid. There is also locally anomalous behavior in Cf at the back end of the bump wall (at x=1.5), as is often seen in CFD solutions near trailing edges (see, e.g., Swanson and Turkel, AIAA Paper 87-1107, 1987, https://doi.org/10.2514/6.1987-1107). Both of these behaviors may have some influence on the convergence/order-property of the integrated viscous component of the drag coefficient. As seen in the following plots, both codes are tending toward similar integrated force coefficient values as the grid is refined.
convergence of bump drag
coefficient vs h

convergence of bump pressure drag
coefficient vs h

convergence of bump viscous drag
coefficient vs h

convergence of bump lift
coefficient vs h

Code	Quantity	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	C_d	0.75	0.237%	0.350%	0.437%
CFL3D	C_d,p	oscillatory convergence	0.749%	N/A	N/A
CFL3D	C_d,v	0.65	0.382%	0.675%	1.875%
CFL3D	C_L	1.00	0.350%	0.350%	0.439%
TAU	C_d	0.07	0.056%	1.068%	0.144%
TAU	C_d,p	1.40	0.799%	0.487%	3.308%
TAU	C_d,v	oscillatory convergence	0.053%	N/A	N/A
TAU	C_L	1.06	0.094%	0.087%	0.109%

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

The surface skin friction coefficient from both codes on the finest 1409 x 641 grid over the entire bump 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=1.5) due to numerical influences. 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 < 0.025 ("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 bump wall.
skin friction coefficient over the bump

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

The surface pressure coefficient from both codes on the finest 1409 x 641 grid over the entire bump wall is shown in the next plot. Both codes yield nearly identical results.
pressure coefficient over the bump

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

Contours of the nondimensional Reynolds stress variables ( $\hat R_{ij}$ ) as well as nondimensional omega from the two codes on the finest 1409 x 641 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. (The CFL3D contour plots have blank spaces because only cell centers values were output and multiple zones were used.)
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 (27.3 MB), (structured, at cell centers) and turb_contours_tau_ssglrrrsm.dat.gz (52.4 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.

The SSG/LRR-RSM-w2012 model relies on the minimum distance to the nearest wall. For this case, contours of this function are shown in the following plot, for the grid 1 level down from the finest grid.
minimum distance function

The data file that generated the above plot is given in bump_1levdown.mindist.dat.gz (gzipped file, 3.9 MB, unstructured, at grid points). Note that this is a gzipped Tecplot formatted file, so you must either have Tecplot or know how to read their format in order to use it. It is important to note that computing minimum distance by searching along grid lines is incorrect, and is not the same as computing actual minimum distance to the nearest wall for this grid. Using the former method will yield some minor differences in the results. The following sketches demonstrate the concept of minimum distance. Improperly-calculated minimum distance functions will particularly produce incorrect results for cases in which the grid lines are not perfectly normal to the body surface. Note that when the nearest wall point is a sharp convex corner or edge (like an airfoil or wing trailing edge) then the correct minimum distance is the distance to that corner or edge, which is not a wall normal.
sketch 1 demonstrating the concept of minimum distance function sketch 2 demonstrating the concept of minimum distance function

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=0.868 vs h

SSG/LRR-RSM-w2012 results from FUN3D are shown alongside the CFL3D and TAU results below. All three codes are consistent. FUN3D used the same freestream turbulence intensity (Tu) and freestream eddy viscosity as CFL3D.
convergence of Cf at x=0.75 vs h, incl FUN3D convergence of Cf at x=0.632 vs h, incl FUN3D
convergence of Cf at x=0.868 vs h, incl FUN3D convergence of bump drag coefficient vs h, incl FUN3D
convergence of bump pressure drag coefficient vs h, incl FUN3D convergence of bump viscous drag coefficient vs h, incl FUN3D
convergence of bump lift coefficient vs h, incl FUN3D skin friction coefficient over the bump, incl FUN3D
pressure coefficient over the bump, incl FUN3D

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

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Last Updated: 03/01/2023