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LBM Viscosity (Chapman–Enskog)

\[\nu = c_s^2\left(\tau - \frac{1}{2}\right), \qquad c_s^2 = \frac{1}{3}\ \text{(D2Q9, lattice units)}\]

Variables

Symbol Meaning Units
\(\nu\) emergent kinematic viscosity lattice units (\(\Delta x^2/\Delta t\))
\(\tau\) BGK relaxation time time steps
\(c_s\) lattice sound speed \(\Delta x/\Delta t\)

Assumptions

  • BGK (single-relaxation-time) collision operator
  • A lattice satisfying the moment conditions through fourth order
  • Low Mach number, \(u/c_s \lesssim 0.1\) (truncated Hermite equilibrium)
  • \(\tau\) varying slowly in space, if at all

Derivation sketch

Chapman–Enskog expansion \(f_i = f_i^{(0)} + \varepsilon f_i^{(1)} + \dots\) with multiple time scales. The \(O(\varepsilon^1)\) momentum flux supplies the deviatoric stress, giving a viscosity \(c_s^2\tau\). Taylor-expanding the discrete update to second order in \(\Delta t\) contributes an additional negative numerical viscosity \(c_s^2\Delta t/2\); subtracting it gives the physical value. In lattice units \(\Delta t = 1\), hence the \(-\tfrac12\).

Notes

  • Stability floor: \(\nu > 0\) requires \(\tau > \tfrac12\). High Reynolds number pushes \(\tau\) toward the floor, which is the central practical difficulty of BGK LBM.
  • Verifiable: fit a Poiseuille profile and compare. Agreement to <1% is achievable across \(\tau \in [0.6, 1.2]\); see the worked example.
  • The thermal lattice obeys the same form, \(\alpha = c_s^2(\tau_T - \tfrac12)\), giving an independently tunable Prandtl number.