An exactly curl-free finite-volume scheme for a hyperbolic compressible barotropic two-phase model

TL;DR

This work develops a second-order structure-preserving finite volume method for the compressible barotropic two-phase Romenski model, ensuring the discrete preservation of the curl-free involution on the relative velocity $\boldsymbol{w}$ by employing a staggered grid and compatible gradient/curl operators. The method combines a path-conservative MUSCL-Hancock framework for the nonconservative terms with a curl-free discretization for $\boldsymbol{w}$, preserving the invariant $\nabla \times \boldsymbol{w}=0$ up to machine precision across multidimensional tests. Numerical experiments, including 1D Riemann problems, a stationary vortex, circular explosion, dambreak, and Kelvin-Helmholtz instability, demonstrate second-order convergence and the exact curl-free property, even with compatible wall boundaries. The approach provides a robust, physically consistent tool for hyperbolic two-phase flows and suggests avenues for higher-order, thermodynamically compatible or dissipative extensions within the SHTC framework.

Abstract

We present a new second order accurate structure-preserving finite volume scheme for the solution of the compressible barotropic two-phase model of Romenski et. al in multiple space dimensions. The governing equations fall into the wider class of symmetric hyperbolic and thermodynamically compatible (SHTC) systems and consist of a set of first-order hyperbolic partial differential equations (PDE). In the absence of algebraic source terms, the model is subject to a curl-free constraint for the relative velocity between the two phases. The main objective of this paper is, therefore, to preserve this structural property exactly also at the discrete level. The new numerical method is based on a staggered grid arrangement where the relative velocity field is stored in the cell vertexes while all the remaining variables are stored in the cell centers. This allows the definition of discretely compatible gradient and curl operators, which ensure that the discrete curl errors of the relative velocity field remain zero up to machine precision. A set of numerical results confirms this property also experimentally.

Figures (11)

• Figure 1: Sketch of the staggered grid. The blue elements are the cells of the main grid $\Omega_{p,q}$, and those delimited by dashed red lines are the cells of the dual grid. Left: the relative velocity $w_{}$ is defined at the vertices of the main grid, and the rest of variables, such as $u_{}$, are defined in the cell centers of the control volumes. Right: The scalar field $\phi$ is also defined in the cell centers.
• Figure 2: 1D Riemann problem solved with the structured preserving finite volume scheme on a Cartesian staggered mesh with 30000 cells at final time $t=0.25$. Top: densities of each phase, $\rho^{I}$ and $\rho^{I\!I}$. Center: mixture density $\rho$ and volume fraction $\alpha^{I}$. Bottom: mixture velocity $u_{}$ and relative velocity $w_{}=u^{I}_{}-u^{I\!I}_{}$. The zooms show the shock inside the rarefaction which are a very special feature of the exact solution of this Riemann problem, see Thein2022.
• Figure 3: Left: Solution of the stationary vortex with the curl-free scheme solved on a staggered mesh with $1024\times1024$ cells at time $t=1000$. Right: Comparison between the approximated solution (dashed red line) and the exact solution (solid black line) at time $t=1000$.
• Figure 4: Time-evolution of the $L^1$ norm of the discrete curl errors using the staggered compatible curl-free discretization (red line) and without the compatible curl-free discretization (black line) for the 2D stationary vortex problem, using a mesh with $1024\times 1024$ cells.
• Figure 5: Solution of the 2D circular explosion problem with the structured preserving finite volume scheme for the initial condition showed in Table \ref{['tab:CE']} solved on a Cartesian staggered mesh with $4800\times4800$ cells at time $t=0.1$, in comparison with the radial reference solution. Top: densities of each phase, $\rho^{I}$ and $\rho^{I\!I}$. Center: mixture density $\rho$ and volume fraction $\alpha^{I}$. Bottom: mixture velocity $u_{}$ and relative velocity $w_{}=u^{I}_{}-u^{I\!I}_{}$.