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

TL;DR

This paper introduces an immersed boundary method for the discrete velocity model of the Boltzmann equation, enabling accurate and stable fluid-structure interaction simulations on Cartesian grids with complex geometries.

Contribution

It develops a novel upwind-weighted interpolation and a cut-cell correction in velocity space, ensuring second-order accuracy and robustness for 2D and 3D problems without dimension-specific modifications.

Findings

Achieves second-order accuracy in physical and velocity space.

Demonstrates stability and robustness for arbitrary geometries.

Provides predictions comparable to body-conformal solvers.

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.