Collision Operators

The collision operator is the heart of the method: it is the single step where physics is injected, by relaxing the populations toward equilibrium. The kinetic primer introduced the simplest such operator, BGK. This chapter explains why BGK alone is not good enough for high-Reynolds turbulent flow, and builds up the family of regularized operators that Nassu uses instead.

Why plain BGK is not enough for LES

For a high-Reynolds LES the lattice viscosity must be tiny, which from the viscosity-relaxation relation (10) drives the relaxation time toward its limit, \(\tau \to 0.5\) (\(\omega \to 2\)). This is exactly the regime BGK handles worst.

The reason is a subtle defect in how BGK relaxes the populations. On a discrete velocity set there are more populations than there are physically meaningful moments. The extra degrees of freedom carry the so-called ghost moments: combinations of populations above second order that have no role in the Navier-Stokes dynamics. BGK relaxes all populations at the single rate \(\omega\), including these ghost moments. Near \(\omega = 2\) the ghost moments are barely damped, accumulate, and contaminate the very quantity an LES depends on, the rate-of-strain tensor read from the second-order non-equilibrium moment [1][2][3]. The result is numerical noise that destabilizes the simulation precisely where high Reynolds numbers require it to be robust.

The solution is regularization: before colliding, filter the non-equilibrium part of the populations so that it contains only the physically meaningful moments and the ghost content is discarded [1]. The populations are first split into equilibrium and non-equilibrium portions, \(f_{i}=f_{i}^{\mathrm{eq}}+f_{i}^{\mathrm{neq}}\), and the non-equilibrium part is rebuilt from its low-order moments alone. Carrying this idea to third order, in a way that reuses the Hermite structure of the equilibrium, gives the recursive regularized-BGK (RR-BGK) operator, which is the default operator in Nassu.

Why RR-BGK and not plain BGK

For low-Reynolds flows BGK is the obvious, cheapest choice. It fails exactly where computational wind engineering lives: at high Reynolds number the relaxation frequency sits near \(\omega = 2\), where BGK lets the unphysical ghost moments grow and pollute the rate-of-strain the LES model depends on. RR-BGK discards that ghost content every step, so it stays stable close to \(\omega = 2\) and keeps the stress clean for the subgrid model. The same regularization also makes the memory-efficient moment-only collision possible.

The pages below develop these operators in detail:

  1. RR-BGK derives the recursive regularized operator that Nassu uses in production, from the Hermite reconstruction of the equilibrium and non-equilibrium populations, and covers the hybrid variant (HRR-BGK) that builds on the same regularization.

  2. Moments Collision shows how regularization lets the solver store only the macroscopic moments per node, rebuilding the populations on the fly for streaming, which roughly halves the memory footprint and runtime in three dimensions.