Viscous face spectral radius (overview)

Purpose

For Navier-Stokes time-step control, the inviscid face spectral radius

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

must be supplemented by a viscous estimate (lambda_{mathrm{vis}}). Viscous limits dominate in boundary layers and can force a much smaller local (Delta t). OneFLOW provides three pure policies in TimeStepPolicy:

These are engineering bounds, not exact eigenvalues of the viscous flux Jacobian.

Theory basis

A diffusion mode (u_t=nu u_{xx}) decays at rate (nu k^{2}). On a finite-volume mesh, (k) is estimated either from face/volume geometry ((ksim A/V), leading to (A^{2}) forms) or from an explicit length (d) (leading to (1/d) forms). Non-dimensional NS equations place a factor (1/mathrm{Re}) on viscous fluxes, so (mathrm{Re}) appears in the spectral estimates.

Comparison

Model

Core formula

Inputs

Role

1

(A^{2}cdot 2max(c_1,c_2)/(mathrm{Re}rho))

(mu,mu_t,mathrm{Pr},gamma,A)

Full Blazek-type channel split

2

(mu_{mathrm{tot}} A/(rho,d,mathrm{Re}))

(mu_{mathrm{tot}},d,A)

Explicit length scale

3

(A^{2}/(mathrm{Re}rho))

(A,mathrm{Re},rho)

Cheapest bound

Source map

  • ViscousFaceSpectralRadiusModel1 / Model2 / Model3

  • InviscidFaceSpectralRadius, CellTimeStepFromSpectrum, CombineInvAndVisTimeStep

  • Header: codes/ns/include/TimeStepPolicy.h

References (primary and comparative)

Primary textbook (form of model 1)

  • J. Blazek, Computational Fluid Dynamics: Principles and Applications, Elsevier (2nd/3rd ed.). Chapter on temporal discretisation / determination of the maximum time step. Structured-grid viscous spectral radius (schematic):

    Lambda_v ~ max( (4/3)(mu_L+mu_T)/rho ,
                gamma (mu_L/Pr_L + mu_T/Pr_T)/rho ) * (Delta S)^2

    with a multiplier (C) (often 1–4) when combining with convective radii in (Delta t). Unstructured variants sum face contributions with the same (max) of momentum and energy diffusivities. ISBN / DOI vary by edition; see Elsevier monograph 9780080445069 (2nd) and later editions.

Open-source codes (comparative, not OneFLOW proofs)

  • SU2 — local time step from convective and viscous spectral radii; viscous contribution of the form (sum C,(mu/rho),S^{2}) along edges (see e.g. AIAA 2015-1949 and SU2 theory/solver documentation). Useful for terminology and scaling; coefficients differ.

  • Other FV RANS solvers — similar (lambda_{mathrm{vis}}proptomu S^{2}/rho) or (A^{2}) estimates appear widely; treat as implementation folklore unless tied to a specific published formula.

Commercial manuals

  • Public ANSYS Fluent / STAR-CCM+ theory guides discuss CFL and time-step limits at a high level; they do not publish OneFLOW-equivalent internal spectral-radius coefficients. Use only for terminology, not as a correctness proof for OneFLOW formulas.

What literature does and does not prove

  • Literature supports the structure of model 1 (momentum vs energy diffusivity, (A^{2}) or ((Delta S)^{2}) scaling, eddy viscosity).

  • Exact numerical prefactors ((2), (1/2), choice of (d)) are implementation choices locked by OneFLOW unit tests, not unique theorems.

  • Open/closed source manuals are for traceability and comparison, not permission to copy proprietary text or claim identical behavior.