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A High-resolution Spatiotemporal Coupling Ghost Fluid Method for Two-Dimensional Compressible Multimedium Flows with Source Terms

Zhixin Huo

TL;DR

The paper tackles thermodynamic inconsistency and inadequate multidimensional treatment in Riemann-based ghost fluid methods for compressible multimedium flow with source terms. It introduces a generalized Riemann problem (GRP) based spatiotemporal coupling ghost fluid method that uses nonlinear geometric optics to embed entropy transport and the Lax–Wendroff/Cauchy–Kowalevski approach to include tangential fluxes and source terms, providing linearly distributed ghost-fluid states. The authors develop RP- and GRP-based ghost fluid definitions, derive GRP fluxes, and demonstrate through axisymmetric tests (spherical bubble–shock interactions and bubble collapse in water) that the GRP-based method offers superior accuracy and robustness over conventional RP-based approaches. This work advances reliable high-resolution simulations of multidimensional, thermodynamically active multimedium flows, with potential impact on reactive-flow and interface-dynamics problems in engineering and physics.

Abstract

While exact and approximate Riemann solvers are widely used, they exhibit two fundamental limitations: 1) Fail to represent continuous entropy transport processes, resulting in thermodynamic incompatibility that limits their applicability to compressible flows. 2) Consider only the effects of normal components at interfaces while neglecting the effects of tangential flux and source term, making them unsuitable for multidimensional problems and cases involving source terms. These limitations persist in Riemann problem-based ghost fluid methods. To address these challenges, we developed a novel spatiotemporal coupling high-resolution ghost fluid method featuring two key advancements: 1) Integration of nonlinear geometrical optics to properly account for thermodynamic entropy evolution. 2) Implementation of the Lax-Wendroff/Cauchy-Kowalevski approach to incorporate tangential fluxes and source term effects. These enhancements have been systematically applied to Riemann problem-based ghost fluid methods. Comprehensive numerical experiments demonstrate significant improvements in simulation accuracy and robustness compared to conventional approaches.

A High-resolution Spatiotemporal Coupling Ghost Fluid Method for Two-Dimensional Compressible Multimedium Flows with Source Terms

TL;DR

The paper tackles thermodynamic inconsistency and inadequate multidimensional treatment in Riemann-based ghost fluid methods for compressible multimedium flow with source terms. It introduces a generalized Riemann problem (GRP) based spatiotemporal coupling ghost fluid method that uses nonlinear geometric optics to embed entropy transport and the Lax–Wendroff/Cauchy–Kowalevski approach to include tangential fluxes and source terms, providing linearly distributed ghost-fluid states. The authors develop RP- and GRP-based ghost fluid definitions, derive GRP fluxes, and demonstrate through axisymmetric tests (spherical bubble–shock interactions and bubble collapse in water) that the GRP-based method offers superior accuracy and robustness over conventional RP-based approaches. This work advances reliable high-resolution simulations of multidimensional, thermodynamically active multimedium flows, with potential impact on reactive-flow and interface-dynamics problems in engineering and physics.

Abstract

While exact and approximate Riemann solvers are widely used, they exhibit two fundamental limitations: 1) Fail to represent continuous entropy transport processes, resulting in thermodynamic incompatibility that limits their applicability to compressible flows. 2) Consider only the effects of normal components at interfaces while neglecting the effects of tangential flux and source term, making them unsuitable for multidimensional problems and cases involving source terms. These limitations persist in Riemann problem-based ghost fluid methods. To address these challenges, we developed a novel spatiotemporal coupling high-resolution ghost fluid method featuring two key advancements: 1) Integration of nonlinear geometrical optics to properly account for thermodynamic entropy evolution. 2) Implementation of the Lax-Wendroff/Cauchy-Kowalevski approach to incorporate tangential fluxes and source term effects. These enhancements have been systematically applied to Riemann problem-based ghost fluid methods. Comprehensive numerical experiments demonstrate significant improvements in simulation accuracy and robustness compared to conventional approaches.
Paper Structure (17 sections, 55 equations, 13 figures)

This paper contains 17 sections, 55 equations, 13 figures.

Table of Contents

  1. Introduction
  2. The two-dimensional multi-medium compressible flow problems with source term
  3. The ghost fluid method for multi-medium flow problems
  4. The finite volume method for single-medium flow problems
  5. The approximation of the numerical fluxes
  6. The numerical fluxes with Riemann solver
  7. The numerical fluxes with GRP solver
  8. Quasi 1-D acoustic case
  9. Quasi 1-D genuinely nonlinear case
  10. The definition of the ghost fluid states
  11. The Riemann problem based ghost fluid method
  12. The GRP based ghost fluid method
  13. Numerical tests
  14. Spherical bubble shock interaction problem
  15. The spherical shock induced collapse of an air bubble in water
  16. ...and 2 more sections

Figures (13)

  • Figure 1: The mathematical model of the two-dimensional compressible multi-medium flow.
  • Figure 2: The schedule of the ghost fluid method for double medium in 2D.
  • Figure 3: The schematic of an axisymmetric object and its cross section.
  • Figure 4: Schematic of initial flow configuration for 'spherical bubble shock interaction problem'.
  • Figure 5: Shadow-photographs of the interaction of an $M_s=1.25$ shock wave moving from right to left over a spherical helium volume (4.5 cm diameter) at $t_4=223\mu s$.
  • ...and 8 more figures