An immersed boundary method for the discrete velocity model of the Boltzmann equation
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.
Summary
This paper introduces a novel ghost-cell immersed boundary method (GCIBM) for simulating fluid-structure interactions in rarefied gas dynamics, specifically addressing the Boltzmann equation's discrete velocity model (DVM). The key challenges tackled are the accurate representation of velocity slip and temperature jump phenomena at the fluid-structure interface within a Cartesian grid framework. The method incorporates the Maxwell gas-surface interaction model by constructing ghost-cell particle distribution functions. The two main advancements are an upwind-weighted compact interpolation strategy in physical space for numerical stability and a cut-cell correction methodology in velocity space to mitigate quadrature accuracy degradation caused by surface discontinuities. The GCIBM framework is designed to be equally applicable to both 2D and 3D problems without dimension-specific modifications. The authors provide rigorous analysis demonstrating that the approach maintains second-order accuracy in both physical and velocity space while ensuring robust numerical stability. Comprehensive numerical experiments validate the designed accuracy, showing predictions comparable to body-conformal solvers while retaining the advantages of Cartesian grid methods (simplicity, flexibility, scalability). This unified and physically consistent IBM framework is valuable for simulating dynamic interactions between non-equilibrium flows and structural components across a wide range of flow regimes.
Key Insights
- •Novel Upwind-Weighted Compact Interpolation: Developed an upwind-weighted compact interpolation strategy in physical space for constructing ghost cell values, ensuring numerical stability and robustness without relying on large stencils or normal-direction projections.
- •Cut-Cell Correction in Velocity Space: Proposed a cut-cell correction methodology in velocity space to address the degradation of quadrature accuracy caused by surface discontinuities. This approach accounts for the mixed nature of cells intersected by the discontinuity in velocity space.
- •Second-Order Accuracy: Rigorous analysis proves that the approach maintains second-order accuracy across both physical and velocity space.
- •Stability for Arbitrary Geometries: The upwind weighting and compact stencil design ensure numerical stability and robustness for arbitrary geometries.
- •General Applicability: The method is equally applicable to both two- and three-dimensional problems without requiring any dimension-specific modifications.
- •Performance Comparable to Body-Conformal Solvers: Numerical experiments demonstrate that the solution algorithm delivers precise predictions comparable to body-conformal solvers.
- •Limitations: The analysis of surface discontinuity effects focuses on boundaries with moderate curvature and doesn't consider complex scenarios with both convex and concave segments with small characteristic scales of variation.
Practical Implications
- •Aerospace Engineering: The method is directly applicable to aerospace engineering endeavors involving fluid-structure interactions, particularly in rarefied gas dynamics scenarios like very-low-earth-orbit flight.
- •Micro-Electro-Mechanical Systems (MEMS): The framework can be used to simulate flows in MEMS devices where rarefied gas effects are significant.
- •Practitioners in CFD: Engineers and researchers can leverage the developed GCIBM framework for simulating non-equilibrium flows around complex geometries, especially when Cartesian grid methods offer advantages over body-conformal approaches.
- •Future Research: The work opens up avenues for further research, including extending the method to handle more complex geometries with varying curvature and investigating its performance with different gas-surface interaction models. The algorithm can also be extended to include heat transfer between the fluid and the immersed boundary.