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TURBULENCE MODEL NUMERICAL ANALYSIS

2D NACA 0012 Airfoil Validation for Turbulence Model Numerical Analysis

SA Model Results for M = 0.15, Re_c = 6 million, alpha = 10 deg, T_ref = 540 R

Please refer to AIAA Journal, Vol. 54, No. 9, 2016, pp. 2563-2588, https://doi.org/10.2514/1.J054555 and AIAA Paper 2015-1746, https://doi.org/10.2514/6.2015-1746.

Results are shown for the NACA 0012 at M=0.15, Re=6 million based on chord, and alpha=10 deg. Two different CFD codes (FUN3D and CFL3D) and 3 different grid families have been employed in an effort to try to discern the grid-converged result for this case. (For the case without farfield point vortex correction BC, TAU results are also included.) Results here are for the "standard" SA model. However, note that FUN3D and TAU make use of the SA-neg variant, which was designed for improved numerical behavior. SA-neg is passive to the original (SA) model in well-resolved flowfields, and hence is expected to yield essentially identical results for the cases here. Furthermore, FUN3D and TAU both used clipping method "c" for $\hat S$ , while CFL3D used clipping method "a" (see Note 1 on the Spalart-Allmaras equation page). For both codes, the farfield value of the Spalart turbulence variable is $\tilde \nu_{farfield} \geq 3 \nu_{\infty}$ . In both codes the Prandtl number Pr is taken to be constant at 0.72, and turbulent Prandtl number Pr_t is taken to be constant at 0.9. The dynamic viscosity is computed using Sutherland's Law (See White, F. M., "Viscous Fluid Flow," McGraw Hill, New York, 1974, p. 28). In Sutherland's Law, the local value of dynamic viscosity is determined by plugging the local value of temperature (T) into the following formula:

$\mu = \mu_0 \left( \frac{T}{T_0} \right)^{3/2} \left( \frac{T_0 + S}{T+S} \right)$

where $\mu_0 = 1.716 \times 10^{-5} kg/(ms)$ , $T_0 = 491.6 R$ , and $S = 198.6 R$ . The same formula can be found online (with temperature constants given in degrees K and some small conversion differences). Note that in terms of the reference quantities for this particular case, Sutherland's Law can equivalently be written:

$\left( \frac{\mu}{\mu_{ref}} \right) = \left( \frac{T}{T_{ref}} \right)^{3/2} \left( \frac{T_{ref} + S}{T+S} \right)$

where $\mu_{ref}$ is the reference dynamic viscosity that corresponds to the freestream in this case, and freestream $T_{ref}$ (as defined on the previous page) is 540R. This latter form may be more convenient for nondimensional codes. (Specific details regarding an implementation of Sutherland's Law in nondimensional codes can be found in handwritten notes describing Sutherland's Law in CFL3D and FUN3D.)

The difference between the three grid families is in their trailing edge streamwise (TES) spacing: the grid family I has the coarsest TES spacing (similar to the spacing used for the 2D NACA 0012 Airfoil Validation Case), the grid family II has the finest TES spacing, and the grid family III has an intermediate TES spacing. All grids have a farfield extent of about 500c.

Results are given both without and with a farfield Point Vortex correction BC:

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Recent significant updates:
03/10/2015 - added description of clipping method for $\hat S$ used

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Last Updated: 11/10/2021