SST blending function F1

Menter SST switches between inner (Wilcox) and outer (k-epsilon-like) constants with a continuous blending function \(F_1\).

Definitions

\[CD_{k\omega} = 2\rho\sigma_{\omega 2} (\nabla k\cdot\nabla\omega)/\omega\]

\[\mathrm{arg}_1 = \min\Big( \max\Big( \frac{\sqrt{k}}{\beta^* d\omega},\, \frac{500\nu}{\omega d^2 Re} \Big),\, \frac{4\rho\sigma_{\omega 2} k}{CD_{k\omega} d^2} \Big)\]

\[F_1 = \tanh(\mathrm{arg}_1^4)\]

Intermediate verification

• Algebra: pure APIs for \(CD\), \(\mathrm{arg}_1\), \(F_1\).

• Curve: \(F_1(\mathrm{arg}_1)\to 0\) as \(\mathrm{arg}_1\to 0\) and \(F_1\to 1\) for large \(\mathrm{arg}_1\) (gtest).

• Later: wall-normal \(F_1(y)\) profiles vs NASA TMR / reference codes (CSV/plot), still driven by these APIs.

References

• Menter, F. R. Two-equation eddy-viscosity turbulence models (SST).

• NASA TMR — SST (https://turbmodels.larc.nasa.gov/sst.html).

• See sst_k_source.rst, sst_omega_source.rst, references.rst.

Pure APIs: SstCrossDiffScalar, SstCdkwWithFloor, SstBlendingArg1, SstF1FromArg1, SstGradientDotProduct.

Intermediate verification

Closed-form checks of \(F_1(\mathrm{arg}_1)=\tanh(\mathrm{arg}_1^4)\) and the convex blend \(\phi=F_1\phi_1+(1-F_1)\phi_2\) live in turb_sst_policy_test and theory table tools/verification/data/sst_f1_arg1_theory.csv (Contract Workbench Plot tab).