A discrete Boltzmann model with state-dependent power-law relaxation time for nonequilibrium transport in compressible flows

Abstract

Thermodynamic nonequilibrium effects play a central role in momentum and energy transport in compressible flows. In conventional BGK kinetic models, the relaxation time $\tau$ is taken as a constant, which neglects the dependence of the relaxation process on local macroscopic states. To overcome this limitation, we develop a discrete Boltzmann model with a density- and temperature-dependent power-law relaxation time, termed DTRT-DBM, in which $\tau=\tau_0(\rho/\rho_0)^a(T/T_0)^b$. This formulation extends the discrete Boltzmann framework to flows with spatially varying nonequilibrium intensity. The model is validated by the Sod shock tube and by analytical solutions for viscous stress and heat flux, demonstrating accurate recovery of both macroscopic wave structures and nonequilibrium quantities across shock waves, rarefaction waves, and contact discontinuities. On this basis, phase diagrams of viscous stress and heat flux are constructed to examine how these quantities depend on the power-law exponents $a$ and $b$. The extrema of these quantities depend exponentially on the model parameters and exhibit regime-dependent behaviour. The roles of $a$ and $b$ are not symmetric: the nonequilibrium response is more sensitive to $a$ when density gradients dominate, but more sensitive to $b$ when temperature gradients dominate. Within the parameter range and flow configurations examined here, higher-order viscous stress increases the growth rate of the total viscous-stress extremum, whereas higher-order heat flux reduces the growth rate of the total heat-flux extremum. These results show that the proposed model can capture different higher-order nonequilibrium responses in compressible flows and provides a framework for the modelling and analysis of multiscale nonequilibrium processes.