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2DZPH: 2D Zero Pressure Gradient High Mach Number Flat Plate Validation Case

The purpose here is to provide a validation case for turbulence models. Unlike verification, which seeks to establish that a model has been implemented correctly, validation compares CFD results against data in an effort to establish a model's ability to reproduce physics. A large sequence of nested grids of the same family are provided here if desired. Data are also provided for comparison. For this particular flat plate case, the data are from theory.

For the grids given below, running at a Reynolds number per unit length of Re = 15 million is sufficient to achieve desired Re_theta levels, and to allow the turbulence models to fully activate far enough upstream. The following plot shows the layout of the simple flat plate grids, along with typical boundary conditions for supersonic flow. (Note that particular variations of the BCs at the inflow, top wall, and outflow may also work and yield similar results for this problem.)

Note: at high Mach numbers (e.g., M > 2), CFD transition to fully turbulent wall-bounded flow can be delayed compared with flow at lower M. Also, the delayed transition location can vary significantly with the freestream levels set. See, for example, Journal of Spacecraft and Rockets, Vol. 47, No. 1, 2010, pp. 11-20, https://doi.org/10.2514/1.45350 and AIAA Journal, Vol. 47, No. 4, 2009, pp. 982-993, https://doi.org/10.2514/1.39947 (which is based on AIAA Paper 2008-4403). For this test case, results are only plotted for regions of the flow that were fully turbulent.
supersonic flat plate grid layout & BCs

GRIDS (same as those used for Flat Plate Verification case)

The following supersonic flat plate cases are under consideration:

M_freestream=2, T_w/T_freestream=1.712 (T_w/T_aw approx =1.0)
M_freestream=5, T_w/T_freestream=1.090 (T_w/T_aw approx =0.2)
M_freestream=5, T_w/T_freestream=2.725 (T_w/T_aw approx =0.5)
M_freestream=5, T_w/T_freestream=5.450 (T_w/T_aw approx =1.0)

Two quantities of interest are desired for comparison:

Wall skin friction coefficient vs. Re_theta (for 4000 < Re_theta < 13,000)
u+ vs. y+ (at Re_theta=10,000)

(It is worth noting that the second case - M=5, T_w/T_aw approx 0.2 - tends to exhibit greater grid sensitivity in its CFD results than the other cases for wall skin friction coefficient.)

Definitions are given here for the relevant quantities, including Re_theta, Cf, u+, and y+

$Re_{\theta} = \frac{\rho_{\infty} U_{\infty} \theta}{\mu_{\infty}}$

$\theta = \int_0^{\infty} \frac{\rho}{\rho_{\infty}} \frac{u}{U_{\infty}} \left( 1 - \frac{u}{U_{\infty}} \right) dy$

$c_f = \frac{\tau_w}{\frac{1}{2} \rho_{\infty} U_{\infty}^2}$

$\tau_w = \mu_w \left( \frac{\partial u}{\partial y} \right)_w$

$u^+ = \frac{u}{v^*} = \frac{u}{\sqrt{\tau_w / \rho_w}}$

$y^+ = \frac{y v^*}{\nu_w} = \frac{y \sqrt{\tau_w / \rho_w}}{\nu_w}$

It should be noted that computing Re_theta typically involves an additional post-processing step for many CFD codes (numerically integrating to obtain theta). Although this step is relatively straightforward for the flat plate, nonetheless some numerical errors are unavoidably introduced which may vary depending on the postprocessing program employed. When viewing comparisons, this additonal potential source of error should be taken into consideration.

The so-called van Driest transformations (see, e.g., White, F. M., "Viscous Fluid Flow," McGraw-Hill, 1974, New York, pp. 632 ff) are used to obtain theoretical correlations for the compressible wall skin friction coefficient and wall variables.

For incompressible wall skin friction, the Karman-Schoenherr (K-S) relation (Schoenherr, K. E., Trans. SNAME. 40:279-313, 1932) is:

$c_{f,incomp}(Re_{\theta}) = \left( 17.08 \left({\rm log}_{10}Re_{\theta} \right)^2 + 25.11 \left( {\rm log}_{10}Re_{\theta} \right) + 6.012 \right)^{-1}$

The van Driest II transformation can be used to transform the incompressible levels to levels valid for supersonic flows with various wall temperatures. Details are as follows:

The approximate adiabatic wall temperature is:

$T_{aw}=T_{\infty}\left(1+r\frac{\gamma-1}{2}M^2 \right)$

where r=0.89 is the recovery factor (an empirical factor introduced because in practice energy recovery is not perfect). The compressible skin friction is related to the incompressible value in van Driest II via:

$C_f = \frac{1}{F_c} {c_{f,incomp}} \left(Re_{\theta}^{'}\right)$

where:

${Re}_{\theta}^{'}=Re_{\theta}F_{Re\theta}$

$F_{Re\theta}=\frac{\mu_\infty}{\mu_w}$

Using Sutherland's law, this becomes:

$F_{Re\theta}={\left(\frac{T_\infty}{T_w}}\right)^{3/2}\frac{T_w+110.4}{T_\infty+110.4}$

where temperatures are in degrees K. Other correlation parameters are:

$F_c=\frac{T_{aw}/T_\infty-1}{\left(sin^{-1}A+sin^{-1}B)\right^2}$

$A=\frac{2a^{2}-b}{\left(b^{2}+4a^{2})\right^{1/2}$

$B=\frac{b}{\left(b^{2}+4a^{2})\right^{1/2}$

$a=\left(r\frac{\gamma-1}{2}M^{2}\frac{T_\infty}{T_w}\right)^{1/2}=\left[\left(\frac{T_{aw}}{T_\infty}-1\right)\frac{T_\infty}{T_w}\right]^{1/2}$

$b=\frac{T_{aw}}{T_w}-1$

For u+ vs. y+, the incompressible law-of-the-wall based on Coles' mean velocity profile (Coles, D., J. Fluid Mech. 1(2):191-226, 1956, https://doi.org/10.1017/S0022112056000135 Coles, D., RAND Corp Rept. R-403-PR, 1962, https://www.rand.org/pubs/reports/R403.html), is:

$u^+_{incomp} = \frac{1}{\kappa} {\rm ln} \left( y^+ \right) + C + \frac{2 \Pi}{\kappa} \left[ {\rm sin} \left( \frac{\pi y}{2 \delta} \right) \right]^2$

along with a van Driest type damping near the wall. A program provided by P. G. Huang was used to compute these mean incompressible profiles. See Bardina et al, NASA TM 110446, April 1997, https://ntrs.nasa.gov/citations/19970017828 for more details.

The van Driest I transformation can be used to transform the incompressible law-of-the-wall to one valid for supersonic flows with various wall temperatures. Details are as follows. The transformation uses:

$u_{eqiv}^{+}=\frac{u_{eqiv}}{u_\tau}=\frac{u_\infty}{au_\tau}\left(sin^{-1}\frac{2a^{2}\frac{u}{u_\infty}-b}{Q}+sin^{-1}\frac{b}{Q}\right)$

so that:

$u^{+}=\frac{u}{u_\tau}=\frac{u_\infty}{2u_{\tau}a^2}\left[Q*sin\left(\frac{au_\tau}{u_\infty}u_{incomp}^{+}-sin^{-1}\frac{b}{Q}\right)+b\right]$

where:

$Q=\left(b^{2}+4a^{2}\right)^{1/2}$

$u_\tau=\sqrt{\frac{\tau_w}{\rho_w}}$

$\tau_w=\frac{1}{2}\rho_{\infty}u_{\infty}^{2}c_f$

$\rho_\infty=\frac{p_\infty}{R_{gas}T_\infty}$

$\rho_w=\frac{p_\infty}{R_{gas}T_w}$

$u_\infty=\sqrt{{\gamma}R_{gas}T_\infty}M_\infty$

Note that the theoretical correlations are imperfect. Not only is there some uncertainty in the incompressible correlations themselves (see discussion on the Subsonic Flat Plate Validation Case webpage), but there is further uncertainty about the best transformations for producing compressible correlations (see, e.g., White, F. M., "Viscous Fluid Flow," McGraw-Hill, New York, 1974). For example, the van Driest II transformation may yield too-high skin friction for strongly-cooled walls (Huang, P. G., Bradshaw, P., and Coakley, T. J., AIAA Journal, Vol. 31, No. 9, Sept. 1993, pp. 1600-1604, https://doi.org/10.2514/3.11820). Results from the van Driest transformations are plotted in the following figures, using freestream temperature of 300K and standard air conditions:
other skin friction correlations
other similarity laws

The data files for these correlations are provided here:

What to Expect:

RESULTS
LINK TO EQUATIONS
MRR Level

SA
SA eqns
4

SST-Vm
SST-Vm eqns
3

Wilcox2006-klim-m
Wilcox2006-klim-m eqns
2

EASMko2003-S
EASMko2003-S eqns
1

K-e-Rt
K-e-Rt eqns
1

(Other turbulence model results may be added in the future.)

Note that the OVERFLOW code has documented its results for this validation case (for the SA-noft2 and SST-V turbulence models) in NAS Technical Paper 2016-01 (pdf file) (18.3 MB) by Jespersen, Pulliam, and Childs.

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Recent significant updates:
03/21/2023 - mentioned possible delay in transition to fully turbulent flow at higher M
08/28/2020 - changed SST-V naming to SST-Vm

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Responsible NASA Official: Ethan Vogel
Page Curator: Clark Pederson
Last Updated: 03/12/2025

RESULTS	LINK TO EQUATIONS	MRR Level
SA	SA eqns	4
SST-Vm	SST-Vm eqns	3
Wilcox2006-klim-m	Wilcox2006-klim-m eqns	2
EASMko2003-S	EASMko2003-S eqns	1
K-e-Rt	K-e-Rt eqns	1