The near-continuum mechanism for extended Boltzmann theory: the non-equilibrium relaxation

TL;DR

This work develops an analytically tractable extended Boltzmann framework for polyatomic gases by adopting the integrable Pullin equation, which enforces detailed balance in energy redistribution between translational and rotational modes. By combining a mixed Hermite–Laguerre expansion with moment integrals, it derives explicit relaxation rates for temperature, pressure deviator, and heat flux, and links these rates to transport coefficients via Chapman–Enskog expansion. The study then constructs a relaxation-rate–based kinetic model of Rykov type that recovers both shear/bulk viscosities and thermal conductivities while accurately capturing coupled translational-rotational heat-flux relaxation. Validation against DSMC and UGKS across zero-D, shock, Couette, lid-driven cavity, and hypersonic wake flows shows improved predictions of non-equilibrium features and heat-flux coupling, particularly under strong thermal non-equilibrium (T_t ≠ T_r). The results highlight the importance of heat-flux coupling and non-equilibrium temperature ratios in determining transport properties and demonstrate the model’s potential for reliable multiscale flow simulations and extensions to vibrational energy.

Abstract

The collision phenomenon of polyatomic gases is described by the collision operator of extended Boltzmann equation or the energy-exchange model in particle direct simulations, for example, the Borgnakke-Larsen model. However, as a collision kernel, it dose not guarantee the entrinsic detailed balance and is not integrable. In this work, the Pullin equation, which possesses an integrable collision kernel and satisfies the detailed balance constraint, is adopted as an extended Boltzmann equation for the theoretical analysis of near-continuum relaxation mechanisms. For clarity, only the translational and rotational degrees are considered in this work. Explicit analytical expressions for the temporal relaxation of macroscopic variables, including the stress force, (translational/rotational) temperature and heat flux, are obtained at the first time. This is achieved by approximating the distribution function in mixed Hermite and Laguerre for rotation and computing the collision operator moments, enabling a direct description of macroscopic non-equilibrium evolution. Base on the same elementary moment (integral) of collision operator, the macroscopic transport coefficients is found in Chapman-Enskog framework. The long-standing speculation, that thermal conduction coefficient should be depended on the degrees of thermal non-equilibrium, is rigorously confirmed and evaluated. When thermal equilibrium is enforced, the present thermal conduction coefficients can be degenerated to the famous results of Mason and Monchick. Given the correct relaxation rate, a Rykov-type novel relaxation model for Pullin equation is proposed. It can recover the interaction of transaltional and rotatioanl heat fluxes in relaxation process, which is ignored in the widely used Rykov equation. Finally, the precision of this new Rykov-type equation is examined using a series of benchmark test cases.

Figures (21)

• Figure 1: Mathematical derivation process of macroscopic quantity relaxation rate
• Figure 2: Variation of the Eucken factor with the degree of non-equilibrium
• Figure 3: Comparison of calculated macroscopic transport coefficients with the NIST database
• Figure 4: Time evolution of temperature and heat flux during zero‑dimensional homogeneous relaxation from a strongly non‑equilibrium initial state corresponding to a Ma=8 normal shock. (a)-(c) are the temperature profiles with HS, VHS and Maxwell molecules. (d)-(f) are the heat flux profiles with HS, VHS and Maxwell molecules.
• Figure 5: Profiles for nitrogen gas shock structure at Ma = 1.53, 4.0, 7.0. (a)-(c) are the density distribution at Ma = 1.53, 4.0, 7.0. (d)-(f) are the temperature distribution at Ma = 1.53, 4.0, 7.0.