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 Spalart-Allmaras turbulence model
(version SA - see full description on
Spalart-Allmaras 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).
For the CFL3D and FUN3D tests reported below, the turbulent inflow boundary condition used for SA was:
.
In both CFL3D and FUN3D, this was not the default setting, so special keywords needed to be
set in both codes.
For the interested reader, typical input files for this problem are given here:
CFL3D V6.5:
FUN3D (original):
FUN3D (new - should yield nearly the same converged results as original):
All FUN3D results below are from the original run (rev 32421).
Note: prior to April 2, 2010, the drag coefficient numbers posted on this page were scaled too high by a
factor of 2, because an incorrect reference area (of 1 instead of 2) was used. This has been corrected here.
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/N1/2, which is proportional to
grid spacing (h).
At the left of the plot, h=0 represents an infinitely fine grid.
As can be seen, CFL3D approaches from above and FUN3D approaches from
below, but both go toward approximately the same result on an infinitely refined grid.
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, ea21 |
Extrap rel fine-grid error, eext21 |
Fine-grid convergence index, GCIfine21 |
| CFL3D |
1.98 |
0.041% |
0.014% |
0.017% |
| FUN3D |
1.34 |
0.034% |
0.022% |
0.028% |
The data file that generated the above plot is given here:
cf_convergence.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.
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 drag coefficient:
| Code |
Computed apparent order, p |
Approx rel fine-grid error, ea21 |
Extrap rel fine-grid error, eext21 |
Fine-grid convergence index, GCIfine21 |
| CFL3D |
1.75 |
0.051% |
0.022% |
0.027% |
| FUN3D |
0.80 |
0.159% |
0.215% |
0.269% |
The data file that generated the above plot is given here:
drag_convergence.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. But both codes are seen to yield nearly identical results
over most of the plate.
The data file that generated the above plot is given here:
cf_plate.dat.
The eddy viscosity contours (nondimensionalized by freestream laminar viscosity)
from the two codes on the finest 545 x 385 grid are shown
in the following plots (y-scale expanded for clarity). They are essentially indistinguishable.
(Note legends do not necessarily reflect min and max values.)
The data files that generated the above plots are given here:
mut_contours_cfl3d.dat.gz (1.3 MB) (structured, at cell centers) and
mut_contours_fun3d.dat.gz (2.6 MB) (unstructured, at grid points). Note
that these are both 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, along with a plot of the maximum nondimensional
eddy viscosity as a function of x.
The data file that generated the eddy viscosity profile at x=0.97 is given here:
mut_0.97.dat.
The data file that generated the max eddy viscosity plot is given here:
find_peak_mut.dat (extracted only for CFL3D).
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).
The data file that generated the above plot is given in
flatplate_u+y+.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.dat (extracted only for CFL3D).
FUN3D was also run for this case using the same grids cut into triangles.
The following plot shows the convergence of the wall skin friction coefficient
at x=0.97008, compared to the earlier results on quad grids. The solution is
approaching the same result as the grid is refined.
The uncertainty estimation procedure applied to the "FUN3D, triangles" results yields:
| Code |
Computed apparent order, p |
Approx rel fine-grid error, ea21 |
Extrap rel fine-grid error, eext21 |
Fine-grid convergence index, GCIfine21 |
| FUN3D, triangles |
0.85 |
0.380% |
0.471% |
0.592% |
The data file that generated the "FUN3D, triangles" results in the above plot
is given in
cf_convergence_fun3d_tri.dat.
Results for total integrated drag coefficient using the triangular grids
are shown in the following plot, alongside the quad grid results.
The uncertainty estimation procedure applied to the "FUN3D, triangles" results yields:
| Code |
Computed apparent order, p |
Approx rel fine-grid error, ea21 |
Extrap rel fine-grid error, eext21 |
Fine-grid convergence index, GCIfine21 |
| FUN3D, triangles |
0.78 |
0.490% |
0.679% |
0.854% |
The data file that generated the "FUN3D, triangles" results in the above plot
is given in
drag_convergence_fun3d_tri.dat.
The surface skin friction coefficient on the finest 545 x 385 grid cut into
triangles is shown in the next plot. To plotting accuracy,
triangle-grid results are almost the same as the results
on the finest quad grid.
The data file that generated the "FUN3D, triangles" results in the
above plot is given here:
cf_plate_fun3d_tri.dat.
Results from VULCAN, OVERFLOW, USM3D, and REFRESCO are shown alongside the CFL3D and FUN3D results below
(note that SA-noft2 yields essentially identical results for this case as SA).
NAS Technical Paper 2014-03 (pdf file)
(16.2 MB) by Childs, Pulliam, and Jespersen provides details of the OVERFLOW results for this case.
Recall that all results above (and elsewhere on this website) are from compressible CFD codes.
To give an indication of the magnitude of the difference between compressible (M=0.2) and incompressible results using SA
for this case, sample skin friction convergence at x=0.97 using different compressible/incompressible codes are plotted below:
The data file of incompressible results from the above plot is given here:
cf_incomp_results_sa.dat.
Note for users of OpenFOAM.
Return to: 2D Zero Pressure Gradient Flat Plate Verification Case Intro Page
Return to: Turbulence Modeling Resource Home Page
Recent significant updates:
11/27/2024 - added REFRESCO results to plots of Cf
12/05/2014 - added link to NAS Technical paper of OVERFLOW results for the verification cases
9/18/2014 - added link to note for users of OpenFOAM
12/19/2013 - added USM3D results
7/11/2013 - added comparison with incompressible results
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Last Updated: 11/27/2024
