An immersed boundary method for the discrete velocity model of the Boltzmann equation

TL;DR

This work introduces a ghost-cell immersed boundary method (GCIBM) for the discrete velocity model of the Boltzmann equation on Cartesian grids to simulate non-equilibrium gas–structure interactions. It couples Maxwell gas-surface boundary conditions with a velocity-space cut-cell correction and a particle-velocity-based upwind interpolation to capture velocity slip and temperature jump across both $2$-D and $3$-D geometries. The approach is shown to be second-order accurate in both physical and velocity space, stable, and capable of delivering predictions in near-wall regions that are competitive with body-conformal solvers across hypersonic, supersonic, and three-dimensional problems. This unified framework preserves Cartesian-grid advantages while providing a physically consistent treatment of rarefied flows around immersed structures, with potential extensions to moving or deformable bodies.

Abstract

Computational modeling and simulation of fluid-structure interactions constitute a fundamental cornerstone for advancing aerospace engineering endeavors. This paper addresses the notion and implementation of the immersed boundary method for the discrete velocity model of the Boltzmann equation. The method incorporates the Maxwell gas-surface interaction model into the construction of ghost-cell particle distribution functions, facilitating meticulous characterization of velocity slip and temperature jump effects within a Cartesian grid framework, which ultimately achieves accurate prediction of aerodynamic parameters. This study presents two principal advancements. First, an upwind-weighted compact interpolation strategy is developed in physical space, which ensures numerical stability and robustness for arbitrary geometries without relying on large stencils or normal-direction projections. Second, a cut-cell correction methodology is proposed in velocity space to address the degradation of quadrature accuracy caused by surface discontinuities. The resulting framework is equally applicable to both two- and three-dimensional problems without requiring any dimension-specific modifications. Rigorous analysis is provided to prove that the approach maintains second-order accuracy across both physical and velocity space, while ensuring robust numerical stability. Comprehensive numerical experiments demonstrate that the solution algorithm achieves the designed accuracy and delivers precise predictions comparable to body-conformal solvers, while retaining the simplicity, flexibility, and scalability of the Cartesian grid method. The proposed approach provides a unified and physically consistent immersed boundary framework for simulating dynamic interactions between non-equilibrium flows and structural components across a wide range of flow regimes.

Figures (25)

• Figure 1: Schematic of the computational module near the gas-solid surface.
• Figure 2: Schematic of the solution stencil when $N_S$ reaches its maximum and minimum.
• Figure 3: Schematic of the discontinuity in velocity space at the intersect point.
• Figure 4: A single cut cell in velocity space.
• Figure 5: Flow chart of the solution. The boxes labeled with blue refer to the immersed boundary implementation, while those in red represent the outside solution procedure.