Phenomenological energy exchange of diatomic gases: Comparison of Pullin and Borgnakke-Larsen models in direct simulation Monte Carlo method

TL;DR

The paper addresses thermodynamic nonequilibrium in hypersonic rarefied diatomic flows by comparing the Borgnakke–Larsen (BL) model with Pullin’s kinetically consistent energy-exchange scheme implemented in DSMC for VHS molecules. It introduces a parameterization that links Pullin’s partition parameters ${ m \\phi}$ and ${\rm \\psi}$ to the rotational collision number $Z$, enabling physically grounded energy exchange across all degrees of freedom. Across 0‑D, 1‑D, 2‑D, and 3‑D test cases, the Pullin formulations (full and simplified) show excellent agreement with BL and experimental data, with Pullin offering a rigorous foundation and comparable performance in highly rarefied regimes. While full Pullin incurs modest extra cost in near-continuum flows, the simplified variant provides similar accuracy at reduced overhead, making it a robust alternative for hypersonic aerothermodynamics and high-enthalpy nonequilibrium simulations.

Abstract

In hypersonic rarefied flows, insufficient intermolecular collisions cause significant deviations between translational and rotational temperatures, leading to strong thermal nonequilibrium. For diatomic gases such as nitrogen and oxygen, the direct simulation Monte Carlo (DSMC) method commonly employs the Borgnakke-Larsen (BL) model to simulate translational-rotational energy exchange (relaxation) processes. Although widely used, the BL model lacks a rigorous theoretical foundation and assumes that only a fraction of collisions lead to rotational relaxation. To address these shortcomings, Pullin introduced a kinetically consistent relaxation model into the gas kinetic theory. By employing the Beta function for energy partitioning, a concrete collision cross section that satisfies the detailed balance condition is constructed. In this study, a comparative investigation of the BL and Pullin models is performed within the DSMC framework, where both original and simplified equations are considered and parameterized by physical accommodated coefficient in the Beta function. A series of test cases--including zero-dimensional rotational relaxation of nitrogen, one-dimensional planar Couette flow and normal shock wave, two-dimensional hypersonic flow past a cylinder, and three-dimensional hypersonic flow around an X38-like vehicle--are performed to assess the accuracy and efficiency of these models. The results confirm the consistency between the Pullin and BL models. Owing to its rigorous theoretical foundation and accurate physical representation, the Pullin model is expected to provide substantial support for the extension of subsequent theoretical studies and numerical simulations. Moreover, in the highly rarefied flow regime (Knudsen number greater than 1, or altitudes above 100 km), the simplified Pullin model exhibits performance comparable to that of the BL model.

Figures (18)

• Figure 1: Comparison of the implementation procedures of the BL and Pullin models within the DSMC framework. Top: BL model; bottom: Pullin model.
• Figure 2: Rotational relaxation in nitrogen gas. Squares: BL model; triangles: modified BL model; gradients: Pullin model; circles: simplified Pullin model; solid line: analytical solution.
• Figure 3: Equilibrium distributions of nitrogen molecules: (a) molecular speeds and (b) rotational energies. Squares: BL model; triangles: modified BL model; gradients: Pullin model; circles: simplified Pullin model; solid line: theoretical solution.
• Figure 4: Couette flow profiles at $\mathrm{Kn}=0.1$: (a) density; (b) velocity; (c) temperature; (d) heat flux. Triangles: Pullin model; gradients: simplified Pullin model; solid line: BL model.
• Figure 5: Temperature profiles of shock waves at different Mach numbers: (a) $\mathrm{Ma}=1.53$; (b) $\mathrm{Ma}=2.0$; (c) $\mathrm{Ma}=6.1$; (d) $\mathrm{Ma}=10.0$. Solid line: Pullin model; dashed line: simplified Pullin model; triangles: BL model.