Lattice Boltzmann model for non-ideal compressible fluid dynamics

TL;DR

This work develops a lattice Boltzmann method for non-ideal compressible flows using two distribution functions on a first-neighbor lattice to recover the Euler and Navier–Stokes equations with thermodynamic consistency. By introducing shifted equilibria and correction terms for both the momentum and energy populations, the model achieves positive-definite dissipation and independent control over bulk viscosity while incorporating Korteweg-type interfacial stresses and Fourier heat flux under a van der Waals equation of state. Multiscale analysis confirms the hydrodynamic limit, and extensive numerical validations demonstrate accurate dispersion/dissipation of hydrodynamic modes, correct liquid–vapour coexistence and interface dynamics, and faithful shock-tube and shock–liquid interactions, including non-classical wave phenomena. The approach extends the applicability of lattice Boltzmann methods to high-speed, non-ideal compressible flows with a minimal kinetic stencil, enabling efficient simulations of complex multiphase and near-critical regimes.

Abstract

We present a lattice Boltzmann formulation for the simulation of compressible, non-ideal fluid flows. The method employs first-neighbor lattices and introduces a consistent set of correction terms through quasi-equilibrium attractors, ensuring positive-definite and Galilean-invariant Navier-Stokes dissipation rates. This construction circumvents the need for extended stencils or ad hoc regularization, while maintaining numerical stability and thermodynamic consistency across a broad range of flow regimes. The resulting model accurately reproduces both Euler- and Navier-Stokes-level hydrodynamics. As a stringent validation, we demonstrate, for the first time within a lattice Boltzmann framework, quantitatively accurate simulations of drop-shock interactions at Mach numbers up to 1.47. The proposed approach thus extends the applicability of lattice Boltzmann methods to high-speed, non-ideal compressible flows with a minimal kinetic stencil.

Figures (12)

• Figure 1: Pressure-temperature diagram for $\ce{CO2}$. The color-bar indicates the compressibility factor $Z=P/\rho RT$. Figure reproduced from guardone2024nonideal.
• Figure 2: Speed of sound for nitrogen $\ce{N2}$ on the saturated liquid and vapor branches. Line: analytical solution from Eq. \ref{['eq:vdW_speed_of_sound']}, Markers: simulations.
• Figure 3: Kinematic viscosity as measured from shear wave decay simulations at different Mach numbers. Plain black line: analytical viscosity, square markers: viscosity measured from simulations.
• Figure 4: Left panel: Temperature and density distribution along channel for thermal Couette flow considering different Prandtl numbers. Triangle, square and circular markers are analytical results for ${\rm Pr}\in\{0.6, 1.2, 4.9\}$ respectively. Plain and dashed lines are temperature and density profiles from simulations. Here Ma=0.8 for all cases. Right panel: Temperature and density distribution for different Ma numbers. Triangle, square and circular markers are analytical results for ${\rm Ma}\in\{0.8, 1.2, 1.6\}$ respectively. Plain and dashed lines are temperature and density profiles from simulations. Here Pr=1.2 for all cases.
• Figure 5: Normal dissipation rate $\alpha$ as measured from normal wave decay simulations at different Mach numbers. Plain black line: analytical dissipation rate, square markers: dissipation rate measured from simulations.