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K-kL-MEAH2015m Expected Results - 2D Bump-in-channel

Previously on this page the results were reported as k-kL-MEAH2015 solutions, but more properly they should be referred to as k-kL-MEAH2015m.

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 bump-in-channel flow with the K-kL-MEAH2015m model from Menter/Egorov and Abdol-Hamid - see full description on k-kL 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. 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).

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 k-kL-MEAH2015 solutions, but more properly they should be referred to as k-kL-MEAH2015m 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 k-kL-MEAH2015m were the following:

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

$(kL)_{\infty} = 1.5589 \times 10^{-6} \mu_{\infty}a_{\infty}/\rho_{\infty}$

The above two equations represent the "standard" k-kL-MEAH2015m 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 V6.6:

Typical Input File for 2D Bump

FUN3D:

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	0.44	0.052%	0.145%	0.271%
FUN3D	8.32	< 0.001%	< 0.001%	0.153%
x=0.6321975
CFL3D	0.09	0.100%	1.510%	0.258%
FUN3D	oscillatory convergence	0.081%	N/A	N/A
x=0.8678025
CFL3D	0.78	0.693%	0.975%	1.207%
FUN3D	1.06	0.833%	0.777%	0.963%

The data file that generated the above plot is given here: cf_convergence_kkl.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	3.22	0.061%	0.007%	0.784%
CFL3D	C_d,p	oscillatory convergence	0.214%	N/A	N/A
CFL3D	C_d,v	1.23	0.098%	0.073%	0.091%
CFL3D	C_L	0.65	0.353%	0.619%	0.779%
FUN3D	C_d	oscillatory convergence	0.011%	N/A	N/A
FUN3D	C_d,p	oscillatory convergence	0.039%	N/A	N/A
FUN3D	C_d,v	oscillatory convergence	0.007%	N/A	N/A
FUN3D	C_L	0.71	0.095%	0.149%	0.187%

The data file that generated the above plot is given here: force_convergence_kkl.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. Both codes indicate turbulence "activation" downstream of the leading edge, 0.025 < x < 0.05 ("activation" is where the turbulence model transitions on its own from laminar to turbulent). 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_kklmeah2015.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_kklmeah2015.dat.

The nondimensionalized eddy viscosity contours, k contours, and (kL) contours from the two codes on the finest 1409 x 641 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.) (The CFL3D contour plots have blank spaces because only cell centers values were output and multiple zones were used.)
eddy viscosity contours for CFL3D eddy viscosity contours for FUN3D

k contours for CFL3D k contours for FUN3D

kL contours for CFL3D kL contours for FUN3D

The CFL3D data files that generated the above plots are given here: mut_contours_cfl3d_kklmeah2015.dat.gz (9.5 MB), k_contours_cfl3d_kklmeah2015.dat.gz (9.9 MB), kl_contours_cfl3d_kklmeah2015.dat.gz (9.9 MB) (structured, multiblock, at cell centers). 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. (FUN3D files are not given in order to save space.)

Using the finest 1409 x 641 grid, an extracted nondimensional eddy viscosity profile at x=0.75 is shown below.
eddy viscosity at x=0.75

The nondimensional k and (kL) profiles at x=0.75 from the 1409 x 641 grid are shown in the following plots.
nondimensional k profile at x=0.75 nondimensional (kL) profile at x=0.75

The data file that generated the turbulence profiles at x=0.75 is given here: k-kl-mut-kklmeah2015.dat.

U-Velocity profiles are shown at x=0.75 for the finest grid in the following plot.
standard velocity profile at x=0.75

The data file that generated the above plot is given in u-kklmeah2015.dat.

The k-kL-MEAH2015 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

K-kL-MEAH2015 results from TAU are shown alongside the CFL3D and FUN3D results below. All three codes are consistent. Note that TAU used different freestream turbulence intensity (Tu) and freestream eddy viscosity: Tu=0.1% and $\mu_t / \mu = 0.1$ . This shows up in plots of k and kL (in the freestream only), but does not influence the mean flow results.
convergence of Cf at x=0.75 vs h, incl TAU convergence of Cf at x=0.632 vs h, incl TAU
convergence of Cf at x=0.868 vs h, incl TAU convergence of bump drag coefficient vs h, incl TAU
convergence of bump pressure drag coefficient vs h, incl TAU convergence of bump viscous drag coefficient vs h, incl TAU
convergence of bump lift coefficient vs h, incl TAU skin friction coefficient over the bump, incl TAU
pressure coefficient over the bump, incl TAU eddy viscosity at x=0.75, incl TAU
nondimensional k profile at x=0.75, incl TAU nondimensional kL profile at x=0.75, incl TAU
standard velocity profiles at x=0.75, incl TAU