Revisiting the Lattice Boltzmann Method Through a Nonequilibrium Thermodynamics Perspective

Abstract

In this paper we incorporate a non-equilibrium thermodynamics perspective that is consistent with the Onsager reciprocity principle into the lattice Boltzmann framework to propose a novel regularized lattice Boltzmann formulation for modeling the Navier-Stokes-Fourier equations. The new method is applied to one-dimensional isothermal situations wherein the advantages of incorporating such a non-equilibrium perspective can be explicitly appreciated. In such situations, the non-equilibrium contribution of the lattice populations obtained by the new method completely vanish and the lattice update is reduced to evaluating the equilibrium distribution function only. Such a counter-intuitive one-dimensional mesoscopic description is not obtained in any other existing lattice Boltzmann scheme. We therefore numerically test the proposed formulation on two complex problems namely shockwave and nonlinear wave propagation and compare results with analytical results along with six existing lattice Boltzmann schemes; it is found that the new method indeed yields results that are more stable and accurate. These results highlight the potency of the non-equilibrium thermodynamics based approach for obtaining accurate and stable lattice Boltzmann computations while also providing new insights into established lattice Boltzmann simulation methods.

Derivation of extended-OBurnett and super-OBurnett equations and their analytical solution to plane Poiseuille flow at non-zero Knudsen number

“…Similarly, Xu (2003) solved the force-driven plane Poiseuille flow problem by employing the BGK–Burnett and the () BGK–super-Burnett schemes in the Boltzmann equations and demonstrated better results than those obtained using the N-S equations. An alternate way of solving the Boltzmann–BGK equation directly is through the lattice Boltzmann method (LBM) route; recent attempts to employ LBM schemes to solve high- flows have been detailed in Jonnalagadda, Sharma & Agrawal (2023). More recently, the ability of higher-order transport equations to address canonical wall-bounded flows in the slip and transition regimes has been investigated.…”