Viscous face spectral radius (model 1)

Purpose

Model 1 estimates the viscous contribution to the local spectral radius for explicit or semi-implicit time-step control. It is a Blazek-style engineering bound, not an exact Jacobian eigenvalue.

Formula

With laminar viscosity \(\mu\), turbulent viscosity \(\mu_t\), face area \(A\), density \(\rho\), and Reynolds number \(\mathrm{Re}\):

\[ \begin{align}\begin{aligned}c_1=\frac{4}{3}(\mu+\mu_t),\qquad c_2=\gamma\left(\frac{\mu}{\mathrm{Pr}_l}+\frac{\mu_t}{\mathrm{Pr}_t}\right)\\\lambda_{\mathrm{vis}}=A^{2}\cdot\frac{2\max(c_1,c_2)}{\mathrm{Re}\,\rho}\end{aligned}\end{align} \]

OneFLOW stores inverse Prandtl numbers, so \(c_2=\gamma(\mu\cdot\mathrm{invPr}_l+\mu_t\cdot\mathrm{invPr}_t)\).

Derivation

Diffusion model

A one-dimensional diffusion equation \(u_t=\nu u_{xx}\) damps a Fourier mode with rate \(\nu k^{2}\). On a finite-volume cell, the mesh wave number is estimated from the face-to-volume scale:

\[k \sim \frac{A}{V} \quad\Rightarrow\quad \text{decay rate} \sim \nu\left(\frac{A}{V}\right)^{2}.\]

For time-step control the viscous spectral contribution is therefore taken proportional to \(A^{2}\nu_{\mathrm{eff}}\), with density and Reynolds scaling so that the result has the same dimensions as the inviscid face spectral radius and can enter

\[\Delta t \sim \frac{\mathrm{CFL}\cdot V}{\sum(\lambda_{\mathrm{inv}}+\lambda_{\mathrm{vis}})}.\]

Momentum channel

The effective dynamic viscosity is \(\mu+\mu_t\). A three-dimensional engineering factor \(4/3\) accounts for longitudinal / bulk-viscous effects in the spectral estimate:

\[c_1=\frac{4}{3}(\mu+\mu_t).\]

Energy channel

Heat conduction uses laminar and turbulent Prandtl numbers. With an ideal-gas factor \(\gamma\) the energy diffusivity scale is

\[c_2=\gamma\left(\frac{\mu}{\mathrm{Pr}_l}+\frac{\mu_t}{\mathrm{Pr}_t}\right).\]

Assembly

The tighter (larger) of the two channels is retained, scaled by \(2/(\mathrm{Re}\,\rho)\) and by \(A^{2}\):

\[\lambda_{\mathrm{vis}} =A^{2}\cdot\frac{2\max(c_1,c_2)}{\mathrm{Re}\,\rho}.\]

The factor \(2\) is an engineering constant of the estimate, not a unique eigenvalue coefficient from a single continuous operator.

Relation to inviscid spectrum

The inviscid face contribution remains

\[\lambda_{\mathrm{inv}}=\frac{1}{2}A\bigl(|v_n|+c\bigr)\]

(InviscidFaceSpectralRadius). Cell time steps may combine inviscid and viscous limits through CellTimeStepFromSpectrum and CombineInvAndVisTimeStep.

Source map

  • ViscousFaceSpectralRadiusModel1 in TimeStepPolicy

  • Related: InviscidFaceSpectralRadius, CellTimeStepFromSpectrum

  • Tests: tests/solver/timestep_policy_test.cpp (ViscousFaceSpectralRadiusModel1CurveMatchesClosedForm)

  • Theory table: tools/verification/data/viscous_spectral_radius_model1_theory.csv

  • Contract Workbench id: viscous_spectral_radius_model1

Viscous spectral radius model1 vs mut

What this proves / does not prove

Proves: model-1 algebra matches the pure API on a sample \(\mu_t\) grid; scaling with \(A^{2}\) holds in the unit test.

Does not prove: exact Jacobian spectral radius; optimality of CFL on a case mesh; models 2 and 3.

References

  • J. Blazek, Computational Fluid Dynamics: Principles and Applications, Elsevier (2nd/3rd ed.), temporal discretisation / maximum time step: viscous spectral radius with (maxigl( frac{4}{3}(mu_L+mu_T),gamma(mu_L/mathrm{Pr}_L+mu_T/mathrm{Pr}_T)igr)) and surface-element scaling. Primary structural source for model 1.

  • Overview and comparative notes: Viscous face spectral radius (overview).

  • SU2 / AIAA papers: edge-based (lambda_{mathrm{vis}}proptomu S^{2}/

ho)

for comparison of scaling only.

  • OneFLOW intermediate curve: tools/verification/data/viscous_spectral_radius_model1_theory.csv.