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TURBULENCE MODEL NUMERICAL ANALYSIS

3D Modified Supersonic Square Duct Validation

SA-QCR2000 Model Results

Link to SA-QCR2000 equations

Results are shown for the 3D supersonic square duct at M=3.9, Re=508,000 based on channel height, reference temperature of 520 R. Two different CFD codes (FUN3D and USM3D) have been employed. Results here are for the SA-QCR2000 variant of the SA model. 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 $T_{ref}$ is 520R. 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 following results were reported in the paper AIAA-2021-1552, https://doi.org/10.2514/6.2021-1552 and https://doi.org/10.2514/6.2021-1552.c1 (with the exception of the FUN3D results on Grid 1, which were run later). Note that FUN3D and USM3D both ran this case with first order spatial accuracy on the turbulence advection term.

Plots below show centerline velocity as a function of x. Results behave similarly between the two codes.
SA-QCR2000 - centerline velocity vs. x

Results that generated the above plots can be found in the following data files: fun3d_centerline_u.dat, usm3d_centerline_u.dat.

The first plot below shows grid convergence of centerline u-velocity at x/D=40, while the second plot shows the same thing at x/D=50. Here, h represents the number of degrees of freedom to the minus 1/3 power (N^(-1/3)). The two codes appear to be approaching similar values as the grid is refined (h approaches zero).
SA-QCR2000 - convergence of centerline u at x/D=40

Results that generated the above plots can be found in the following data files: fun3d_convergence_centerline_u.dat, usm3d_convergence_centerline_u.dat.

Next is shown drag coefficient convergence (left) and maximum eddy viscosity convergence (right). The drag coefficient appears to have a small mismatch between the codes as the grid is refined, on the order of one drag count. The max eddy viscosity trends of both codes agree very well.
SA-QCR2000 - convergence of drag coefficient SA-QCR2000 - convergence of maximum eddy viscosity

Results that generated the above plots can be found in the following data files: fun3d_convergence_cd_and_maxmut.dat, usm3d_convergence_cd_and_maxmut.dat.

The u-velocity profiles are shown below for a diagonal cut and a vertical cut at x/D=40. The two codes agree nearly perfectly, regardless of the grid level.
SA-QCR2000 - diagonal cut of u at x/D=40 SA-QCR2000 - vertical cut of u at x/D=40

Results that generated the above plots can be found in the following data files: fun3d_u_diag_40.dat, fun3d_u_vert_40.dat, usm3d_u_diag_40.dat, usm3d_u_vert_40.dat.

The u-velocity profiles are shown below for a diagonal cut and a vertical cut at x/D=50. Again, the two codes agree nearly perfectly, regardless of the grid level.
SA-QCR2000 - diagonal cut of u at x/D=50 SA-QCR2000 - vertical cut of u at x/D=50

Results that generated the above plots can be found in the following data files: fun3d_u_diag_50.dat, fun3d_u_vert_50.dat, usm3d_u_diag_50.dat, usm3d_u_vert_50.dat.

The skin friction coefficient along z at x/D=40 and 50 is shown below for FUN3D and USM3D. There is excellent agreement between the two codes as the grid is refined.
SA-QCR2000 - Cf at x/D=40 SA-QCR2000 - Cf at x/D=50

Results that generated the above plots can be found in the following data files: fun3d_cf_40_50.dat, usm3d_cf_40_50.dat.

An older set of results (on older, somewhat different grids) can be found here: 3D Modified Supersonic Square Duct Validation - SA-QCR2000 Model Results (old).

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Recent significant updates:
07/21/2021 - Updated results to include FUN3D on the finest grid

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Last Updated: 11/10/2021