Wave-Appropriate Reconstruction of Compressible Multiphase and Multicomponent Flows: Fully Conservative and Semi-Conservative Eigenstructures

TL;DR

This paper derives the eigenstructure of models for compressible multiphase flows, enabling more accurate and oscillation-free interface reconstruction by leveraging wave-decoupling and characteristic decomposition.

Contribution

It introduces complete eigenstructures for conservative and semi-conservative formulations, improving interface accuracy and stability in multiphase flow simulations.

Findings

Eigenvectors contain a thermodynamic jump term enforcing pressure equilibrium.

Reconstruction in characteristic space satisfies equilibrium conditions.

Test cases confirm oscillation-free, accurate results.

Abstract

Compressible multiphase and multicomponent solvers require accurate interface representation without spurious pressure oscillations. At material interfaces, pressure and velocity are continuous while density and the equation of state exhibit abrupt discontinuities. Standard approaches reconstruct primitive or characteristic variables to capture these properties, but do not clarify the failure mechanisms of conservative reconstruction or fully leverage the wave-decoupling advantages of characteristic decomposition. This work derives the complete eigenstructure of the Allaire five-equation model for two variable sets. In the fully conservative~(FC) formulation, $\mathbf{U} = [\alpha_1\rho_1,\,\alpha_2\rho_2,\,\rho u,\,\rho v,\,\rho E,\,\alpha_1]^T$, eigenvectors contain a thermodynamic jump term~$\Psi$ that enforces $dp=0$ and $du=0$ at material contacts by compensating for compressibility mismatches. In the semi-conservative~(SC) formulation, $\mathbf{V} = [\alpha_1\rho_1,\,\alpha_2\rho_2,\,\rho u,\,\rho v,\,p,\,\alpha_1]^T$, the volume-fraction eigenvector carries a structural zero in the pressure slot, enforcing equilibrium without thermodynamic correction. Explicit left and right eigenvectors are derived for one- and two-dimensional stiffened-gas flows. Both formulations satisfy Abgrall's equilibrium condition when reconstruction is performed in characteristic space; reconstruction in physical space yields $\mathcal{O}(1)$ pressure and velocity errors at interfaces regardless of the variable set. The eigenvector structure further reveals that the shear wave is decoupled from all thermodynamic and interface fields in both formulations, extending this result from single-species to compressible multiphase flows including gas-liquid configurations. One- and two-dimensional gas-gas and gas-liquid test cases confirm oscillation-free, accurate results.