Wilcox f_k and f_w modifiers

Wilcox k-omega variants scale dissipation coefficients with bounded functions of cross-diffusion and vortex stretching.

\[x_k = (\nabla k\cdot\nabla\omega)/\omega^3\]

\[\begin{split}f_k = \begin{cases} 1 & x_k \le 0 \\ (1+680 x_k^2)/(1+400 x_k^2) & x_k > 0 \end{cases}\end{split}\]

\[f_w^{1998}=(1+70 x_w)/(1+80 x_w), \qquad f_w^{2006}=(1+85 x_w)/(1+100 x_w)\]

Intermediate checks: (f_k(0)=1), (f_wto 7/8) as (x_wtoinfty) (1998).

References

• Wilcox, D. C. Turbulence Modeling for CFD.

• NASA TMR — Wilcox k-omega variants.

• See verification_layers.rst and references.rst.

Pure APIs: WilcoxXkFromCrossTerm, WilcoxFkFromXk, WilcoxFw1998, WilcoxFw2006, WilcoxVortexStretchingInvariant.

Intermediate verification

Closed-form checks of \(f_k(x_k)\) live in turb_wilcox_fbeta_test and tools/verification/data/wilcox_fk_xk_theory.csv (Contract Workbench).

The beta modifiers \(f_w^{1998}=(1+70 x_w)/(1+80 x_w)\) and \(f_w^{2006}=(1+85 x_w)/(1+100 x_w)\) are locked in turb_wilcox_fbeta_test with theory tables under tools/verification/data/wilcox_fw*_theory.csv.