6 2D Turbulent Flat-Plate Boundary Layer, M=2 and M=5 — NASA TMR SA Verification

Skin friction versus Re_θ and the law of the wall for the Spalart–Allmaras model on a cooled supersonic/hypersonic flat plate (against van Driest theory and CFL3D)

6.1 Run Provenance and Artifacts

Item	Value
Wake CFD baseline version	`0.23.0`
Solver	DBNS
Compute device	CPU
Parallel execution	8 MPI ranks
Arithmetic precision	Mixed
Run date	2026-10-18

Note

Results depend on the solver build that produced them. Read every figure in this report together with the version and date above — re-running the same input on a later build may give different numbers.

All twenty runs (four conditions × five meshes) used the same input apart from the mesh, the freestream and wall-temperature values, and the iteration cap.

6.1.1 Downloadable Files

Not available: standalone input decks and solver result files have not yet been published for download.

6.2 Verification Objective

This case simulates a zero-pressure-gradient turbulent boundary layer on an isothermal flat plate at supersonic and hypersonic Mach numbers with the Spalart–Allmaras (SA) model. It follows the high-Mach flat-plate case of the NASA Turbulence Modeling Resource (TMR), which uses the same mesh family as the low-speed flat-plate case (case 02) but adds compressibility, a real Sutherland viscosity, and a cooled wall. It verifies Wake CFD for:

the skin-friction coefficient Cf as a function of the momentum-thickness Reynolds number Re_θ, against van Driest II theory and the CFL3D SA result published by TMR
the direction and size of the change in Cf as the mesh is refined, at four Mach number / wall-temperature conditions
the near-wall velocity profile in wall coordinates (u+ versus y+)

The TMR page treats this case as a comparison with theory rather than with other codes, and compares only the fully turbulent region of the plate. This report follows that: the primary quantity is the error in Cf against theory at Re_θ = 10,000, and the CFL3D SA result is shown as a second reference.

6.3 Reference Solution and Data

6.3.1 Analytical Solution

The reference curves are those published by TMR: the skin friction from van Driest II theory (Cf versus Re_θ) and the law of the wall from van Driest I theory (u+ versus y+), each given per condition. TMR states that the theory is applicable to the fully turbulent region only, and compares Cf over 4000 < Re_θ < 13000 and u+ at Re_θ = 10,000.

6.3.2 Numerical Reference

TMR also publishes a CFL3D SA result on the 545×385 mesh for each condition. The CFL3D data cover only Re_θ ≈ 4,000 to 14,000, as distributed. TMR describes such code results as representative rather than definitive, so they are shown for context and the theory is the reference for the error figures. CFL3D and FUN3D use a freestream nu_tilde/nu = 3 for SA; the same value was used here.

6.4 Simulation Conditions

Four conditions were run. The freestream temperature is T∞ = 300 K and the unit Reynolds number is Re = 1.5×10^7 /m for all four.

Condition (label in figures)	Mach	`Tw/T∞`	`Tw`	`ρ∞` [kg/m³]	`p∞` [Pa]	`U∞` [m/s]
M=2, Tw/Tinf=1.712	2	1.712	513.6 K	0.398722	34,335.92	694.44
M=5, Tw/Tinf=1.090	5	1.090	327.0 K	0.159489	13,734.37	1736.09
M=5, Tw/Tinf=2.725	5	2.725	817.5 K	0.159489	13,734.37	1736.09
M=5, Tw/Tinf=5.45	5	5.45	1635.0 K	0.159489	13,734.37	1736.09

Item	Setting
Dimension	3D (one-cell spanwise extrusion; effectively 2D flow)
Geometry	Flat plate, `x = [0, 2] m`, leading edge at `x = 0`
Analysis	Steady, compressible DBNS, RANS (Spalart–Allmaras)
Gas model	Ideal gas (`R = 287.05 J/(kg·K)`, `γ = 1.4`), Sutherland viscosity (`μ_ref = 1.716×10⁻⁵ Pa·s`, `T_ref = 273.15 K`, `S = 110.4 K`), constant Prandtl number `0.72`
Wall	No-slip, isothermal at `Tw` above

For M = 5, the adiabatic wall temperature is about 5.45 T∞, so the three M = 5 conditions have Tw/Taw ≈ 0.2, 0.5, and 1.0. TMR notes that the strongly cooled Tw/T∞ = 1.09 condition is more sensitive to the mesh than the others.

6.5 Mesh

The meshes are the grids distributed by TMR for the flat-plate cases (the same family as case 02), used as distributed, with one cell in the spanwise direction. The y+ values are the wall values reported by the solver at the end of the run.

Mesh	Streamwise × wall-normal	Node count (including spanwise extrusion)
Coarsest	35 × 25	1,750
Coarse	69 × 49	6,762
Medium	137 × 97	26,578
Fine	273 × 193	105,378
Finest	545 × 385	419,650

First-cell wall y+ (average over the plate):

Condition	35×25	545×385
M=2, Tw/Tinf=1.712	2.13	0.13
M=5, Tw/Tinf=1.090	2.87	0.20
M=5, Tw/Tinf=2.725	1.02	0.06
M=5, Tw/Tinf=5.45	0.45	0.03

The `35×25` mesh in three dimensions — one cell in the spanwise direction; the plate is the bottom face, the upstream inlet is on the left, and the colors distinguish boundary groups

6.6 Boundary and Initial Conditions

Boundary	Segment	Type	Setting
Wall	Plate, `x∈[0,2]`	`wall`	no-slip, isothermal at `Tw` (see conditions)
Symmetry plane	Upstream slip segment + spanwise front/rear faces	`symmetry`	—
Inlet	Upstream	`farfield`	freestream Mach, `p∞`, `T∞`, flow in `+x`, freestream `nu_tilde/nu = 3`
Outlet	Downstream	`pressure_outlet`	supersonic outlet, back pressure `p∞`
Farfield	Top	`farfield`	same as inlet

Initial field: uniform freestream (Mach, pressure, temperature, flow in +x). The initial turbulence value is the solver default.

6.7 Numerical Methods

Item	Setting
Turbulence model	SA-neg variant with the `ft2` term, `Pr_t = 0.9`
Convective scheme	Roe, second-order reconstruction with no limiter, U-MUSCL `kappa = 0`, entropy fix off (all solver defaults; the reconstruction gradient is the same Green–Gauss gradient)
Gradient method	Green–Gauss
Time integration	Euler implicit; FGMRES (error `0.01`, at most 15 iterations) with SGS preconditioning for the flow equations (2 sweeps, ω = 0.9)
CFL	Constant, 50 on all runs. The three `545×385` `M = 5` runs were continued from the iteration-8,000 state with CFL 100 up to iteration 16,000
Convergence criterion	Absolute flow residual `−12` (log10), or a 12-order reduction from the initial residual
Iteration cap	`M = 2`: 8,000 (`35×25`, `69×49`), 40,000 (`137×97`, `273×193`), 60,000 (`545×385`); `M = 5`: 8,000 on all meshes, 16,000 for the `545×385` runs as described above
Wall treatment	Solver default (`automatic`); no alternative was compared

Note

Assumptions and differences from the references.

The TMR reference is the standard SA model; the SA-neg variant used here is the same where nu_tilde is non-negative. The effect of the variant was not isolated.
CFL3D and FUN3D advect the turbulence variable with first-order upwinding, as stated by TMR. Wake CFD advects the turbulence variable with a first-order scheme in the build used (0.21.5), so this matches.
The CFL value was chosen to make the runs converge in a practical number of iterations; at CFL 5 the M = 5 runs stalled on the fine meshes. Converged solutions did not depend on the CFL value in a check on the 137×97 mesh (CFL 20 and 50 gave the same Cf at Re_θ = 10,000).

6.8 Results

6.8.1 Convergence History

Iterations to termination, and the final absolute flow residual (log10) for the runs that ended on the iteration cap:

Condition	35×25	69×49	137×97	273×193	545×385
M=2, Tw/Tinf=1.712	718	718	2,214	8,008	28,530
M=5, Tw/Tinf=1.090	418	718	2,619	8,000 (−10.59)	16,000 (−9.72)
M=5, Tw/Tinf=2.725	432	620	2,225	8,000 (−11.82)	16,000 (−10.67)
M=5, Tw/Tinf=5.45	626	619	1,917	6,357	16,000 (−11.69)

Fifteen of the twenty runs reached the residual criterion of −12. Five ended on the iteration cap with a residual between −9.7 and −11.8, still decreasing: the 273×193 runs of M=5, Tw/Tinf=1.090 and 2.725, and the three 545×385 M = 5 runs. The criterion was not met in these five, and they are treated as converged for this comparison: on 545×385 the Cf at Re_θ = 10,000 differs from the 273×193 value by 0.6 percentage points of the theory value or less (table below), so the mesh trend is not changed by these runs ending short of the criterion, as far as this comparison shows. A longer run was not made to confirm that the Cf values no longer change.

Residual history of the flow and `nu_tilde` residuals on the five meshes — M=2, Tw/Tinf=1.712

6.8.2 Skin Friction

Cf was taken from the wall shear stress, and Re_θ from the momentum thickness computed from the density and velocity profile at each streamwise station, integrated up to the first point where u/U∞ ≥ 0.99. The figures show Cf over Re_θ ≥ 1,000; closer to the leading edge Cf rises steeply near the first points and Re_θ is not monotonic in x, so that region is not plotted. All errors below are at Re_θ = 10,000.

`Cf` versus `Re_θ` — M=2, Tw/Tinf=1.712, five meshes, linear (top) and logarithmic (bottom) `Re_θ` axis

Difference of Wake CFD Cf relative to van Driest II theory at Re_θ = 10,000:

Condition	35×25	69×49	137×97	273×193	545×385
M=2, Tw/Tinf=1.712	−1.60%	−0.85%	−0.81%	−0.73%	−0.72%
M=5, Tw/Tinf=1.090	−17.14%	−13.13%	−11.05%	−9.87%	−9.28%
M=5, Tw/Tinf=2.725	−4.88%	−3.71%	−3.14%	−3.01%	−2.97%
M=5, Tw/Tinf=5.45	−0.73%	−0.54%	−0.38%	−0.17%	−0.22%

Difference of Wake CFD Cf relative to the CFL3D SA result at Re_θ = 10,000 (CFL3D is on the 545×385 mesh only):

Condition	CFL3D vs. theory	35×25	69×49	137×97	273×193	545×385
M=2, Tw/Tinf=1.712	−0.15%	−1.49%	−0.74%	−0.69%	−0.62%	−0.61%
M=5, Tw/Tinf=1.090	−8.08%	−9.86%	−5.50%	−3.23%	−1.94%	−1.30%
M=5, Tw/Tinf=2.725	−1.08%	−3.84%	−2.65%	−2.08%	−1.95%	−1.91%
M=5, Tw/Tinf=5.45	+2.44%	−3.09%	−2.91%	−2.75%	−2.55%	−2.60%

M = 2 and M = 5, Tw/T∞ = 5.45 (the warmest wall): the error against theory falls with the mesh and ends within 0.8% of theory on the three finest meshes for M = 2 and within 0.4% for Tw/T∞ = 5.45. On the finest mesh Wake CFD is 0.22% (Tw/T∞ = 5.45) and 0.72% (M = 2) below theory.
M = 5, Tw/T∞ = 2.725: the error against theory goes from −4.88% to −2.97%, changing little over the last two refinements (−3.01% to −2.97%).
M = 5, Tw/T∞ = 1.090 (the most strongly cooled wall): the error against theory remains large, −9.28% on the finest mesh, and is still moving with the mesh (−11.05%, −9.87%, −9.28% over the last three meshes). CFL3D SA is also 8.08% below theory at this Re_θ, and Wake CFD on the finest mesh is within 1.30% of CFL3D. For this condition the difference from theory therefore appears to be a property of the SA model rather than a discretization error of this run, but that attribution was not tested (for example with another turbulence model).
Offset from CFL3D at the warmer M = 5 walls. For Tw/T∞ = 2.725 and 5.45 Wake CFD stays about 1.9% and 2.6% below CFL3D on the two finest meshes, with almost no change from 273×193 to 545×385, so it does not decrease with refinement. It was not isolated whether this comes from differences in setting (for example the viscosity law or the Re_θ extraction) or from the models. The theory comparison is the primary result for this case, and Wake CFD is close to theory for these two conditions (−2.97% and −0.22%).

6.8.3 Law of the Wall

u+ = u/u_τ with u_τ = sqrt(τ_w/ρ_w) is plotted against y+ at the station nearest Re_θ = 10,000 on each mesh, together with the van Driest I curve and the CFL3D SA result from TMR. Compared visually, the curves of the three finest meshes lie on top of each other and on the CFL3D curve; the coarser meshes sit above them in the log region and the outer plateau. The theory curve lies below the computed profiles in the buffer and log regions; the gap is largest for Tw/T∞ = 1.09, where the Cf error against theory is also largest. Only the Cf values above were quantified.

`u+` versus `y+` at `Re_θ ≈ 10,000` — M=2, Tw/Tinf=1.712

6.9 Conclusions and Applicable Scope

On the finest mesh, Wake CFD’s SA result agrees with van Driest II theory at Re_θ = 10,000 within 0.72% (M=2, Tw/T∞=1.712), 0.22% (M=5, Tw/T∞=5.45) and 2.97% (M=5, Tw/T∞=2.725). For the strongly cooled M=5, Tw/T∞=1.09 wall it is 9.28% below theory, where the CFL3D SA result published by TMR is also 8.08% below theory and Wake CFD is within 1.30% of CFL3D.
Refining the mesh moves Cf toward its finest-mesh value in all four conditions, with one small exception: for Tw/T∞ = 5.45 the last step goes from −0.17% to −0.22% against theory. For Tw/T∞ = 1.09 the change from 273×193 to 545×385 is still 0.6 percentage points, so grid independence is not established for that condition.
Fifteen of the twenty runs met the residual criterion. Five (the two 273×193 and three 545×385 M = 5 runs listed above) ended on the iteration cap at residuals between −9.7 and −11.8 and are treated as converged for the comparison; they were not run longer to confirm the Cf values no longer change.
Against CFL3D, Wake CFD shows a difference of about 2% to 3% on the finest mesh at the two warmer M = 5 walls that does not decrease with refinement; its cause was not identified.
These results apply to a zero-pressure-gradient, isothermal flat plate at M = 2 and M = 5 with the SA model, over the fully turbulent range Re_θ ≈ 4,000 to 13,000. They cannot be generalized to pressure gradients, shock–boundary-layer interaction, transition, or other turbulence models. The effect of the wall treatment option and the leading-edge region were not examined.