Viscous face spectral radius (model 2)

Purpose

Model 2 estimates (lambda_{mathrm{vis}}) using an explicit length scale \(d\) (cell size or wall distance). Family overview: Viscous face spectral radius (overview).

Formula

\[c_1=\frac{2\mu_{\mathrm{tot}}}{\rho\,d\,\mathrm{Re}},\qquad \lambda_{\mathrm{vis}}=\frac{1}{2} c_1 A =\frac{\mu_{\mathrm{tot}} A}{\rho\,d\,\mathrm{Re}}\]

Derivation (sketch)

Diffusion rate \(\nu/d^{2}\) with \(\nu\sim\mu/(\rho\,\mathrm{Re})\) yields an inverse-time scale. Mapping onto a face contribution linear in \(A\) produces the form above (engineering factors \(2\) and \(1/2\)).

Unlike model 1, there is no separate energy Prandtl channel; a single total viscosity is used.

Source map

  • ViscousFaceSpectralRadiusModel2 in TimeStepPolicy

  • Tests: tests/solver/timestep_policy_test.cpp (when intermediate curve tests are present)

  • Contract Workbench id: viscous_spectral_radius_model2 (when wired)

What this proves / does not prove

Proves (when tests present): algebra and \(1/d\) scaling versus the pure API.

Does not prove: optimal geometric definition of \(d\) on a mesh.

References

  • Diffusion time-scale argument (standard CFD notes; Blazek discusses viscous spectral limits in boundary layers in the same chapters as model 1).

  • Length-based viscous CFL estimates appear in many unstructured codes; SU2-style (mu S^{2}/rho) is a related but not identical form (area-squared without explicit (d)).

  • Family overview: Viscous face spectral radius (overview).