Consistent kinetic modeling of compressible flows with variable Prandtl numbers: Double-distribution quasi-equilibrium approach

TL;DR

The paper addresses the challenge of consistently modeling compressible flows across all Prandtl numbers and energy splits within a kinetic framework. It develops a QE-based double-distribution approach (f and g) with two relaxation times, enabling accurate recovery of the Navier–Stokes–Fourier equations via Chapman–Enskog analysis, while maintaining conservation and Galilean invariance. The method employs high-order Hermite-based Grad expansions and discrete quadratures (e.g., D2Q16, D2Q25) to faithfully reproduce NSF fluxes and dissipation, validated through conservation tests, dispersion/dissipation studies, and a viscous shock–vortex benchmark. The results demonstrate robust performance across wide ranges of $Ma$, $\theta$, $\gamma$, and $Pr$, offering an efficient, scalable framework suitable for moderate-to-high-Mach compressible flows and paving the way for extensions to high-Mach and hypersonic regimes via reference-frame corrections and adaptive refinement.

Abstract

A consistent kinetic modeling and discretization strategy for compressible flows across all Prandtl numbers and specific heat ratios is developed using the quasi-equilibrium approach within two of the most widely used double-distribution frameworks. The methodology ensures accurate recovery of the Navier-Stokes-Fourier equations, including all macroscopic moments and dissipation rates, through detailed hydrodynamic limit analysis and careful construction of equilibrium and quasi-equilibrium attractors. Discretization is performed using high-order velocity lattices with a static reference frame in a discrete velocity Boltzmann context to isolate key modeling aspects such as the necessary requirements on expansion and quadrature orders. The proposed models demonstrate high accuracy, numerical stability and Galilean invariance across a wide range of Mach numbers and temperature ratios. Separate tests for strict conservation and measurements of all dissipation rates confirm these insights for all Prandtl numbers and specific heat ratios. Simulations on a sensitive two-dimensional shock-vortex interaction excellently reproduce viscous Navier-Stokes-Fourier-level physics. The proposed models establish an accurate, efficient and scalable framework for kinetic simulations of compressible flows with moderate supersonic speeds and discontinuities at arbitrary Prandtl numbers and specific heat ratios, offering a valuable tool for studying complex problems in fluid dynamics and paving the way for future extensions to the lattice Boltzmann context, by application of correction terms, as well as high-Mach and hypersonic regimes, employing target-designed reference frames.

Figures (7)

• Figure 1: Results of the shock tube problem with $\mathrm{Pr} = \{0.5,1,2\}$ for the total energy split in the left column and internal-non-translational in the right column. The top row depicts a comparison of the density with the reference (Riemann solution for inviscid problem), whereas the bottom row shows the relative error in conservation of mass and total energy, where the machine epsilon for double precision is depicted as a reference.
• Figure 2: Results of the dispersion tests with $\gamma = \{5/3, 8/6\}$ and $\mathrm{Pr} = \{0.5,1,2\}$ for the total energy split in the left column and internal-non-translational in the right column. The top row depicts a comparison of the measured normalized speed of sound $c_s^* = \sqrt{\gamma R \theta}$ at various normalized temperatures $\theta = T/T_{ref}$ for $\mathrm{Ma} = 0$, i.e. $u_x = u_0 = 0$, whereas the bottom row shows $c_s^*$ at various Mach numbers of the mean flow at $\theta = 1$, i.e. $T = T_0 = T_{ref}$. The gray circles indicate corresponding measurement points between the two rows.
• Figure 3: Results of the dissipation tests for the shear mode with $\mathrm{Pr} = \{0.5,1,2\}$ for the total energy split in the left column and internal-non-translational in the right column. The measured versus imposed values of kinematic shear viscosity is depicted in the top row at $\mathrm{Ma} = 0$, whereas the bottom row shows the measured values at various Mach numbers. The gray circles indicate corresponding measurement points between the two rows.
• Figure 4: Results of the dissipation tests for the bulk mode with $\gamma = \{5/3, 8/6\}$ and $\mathrm{Pr} = \{0.5,1,2\}$ for the total energy split in the left column and internal-non-translational in the right column. The measured versus imposed values of kinematic bulk viscosity is depicted in the top row at $\mathrm{Ma} = 0$, whereas the bottom row shows the measured values at various Mach numbers. The gray circles indicate corresponding measurement points between the two rows.
• Figure 5: Results of the dissipation tests for the entopic mode with $\mathrm{Pr} = \{0.5,1,2\}$ for the total energy split in the left column and internal-non-translational in the right column. The measured versus imposed values of thermal diffusivity is depicted in the top row at $\mathrm{Ma} = 0$, whereas the bottom row shows the measured values at various Mach numbers. The gray circles indicate corresponding measurement points between the two rows.