A high-order discontinuous Galerkin method for compressible interfacial flows with consistent and conservative Phase Fields

Abstract

Excessive and non-uniform numerical diffusion poses challenges for accurate simulations of compressible interfacial flows with shocks using diffuse interface methods. Even with high-order accurate methods, material interfaces continually diffuse, thus making material regions ambiguous and deleteriously impacting wave propagation across interfaces. Simulations with low-order methods are particularly affected by this issue, requiring a significant number of degrees of freedom to resolve interfaces. At the same time, Phase-Field methods are able to maintain interfaces with constant and uniform thickness over time. However, while implemented into second-order accurate solvers, a general fully conservative, bounds-preserving, and high-order accurate treatment for high-speed flows is lacking. To address these issues we adapt the general consistent and conservative Phase-Field model and the associated theoretical and numerical analysis to high-order accurate discontinuous Galerkin schemes with non-conformal subcell finite volume discontinuity capturing. In the proposed approach, discontinuities are detected a priori by physical sensors and captured by robust finite volume schemes on uniform subcell grids. By computing the Phase-Field mechanism using subcell data and reconstructing high-order representations in each cell, the calculation of the Phase-Field fluxes is robust, efficient, and can be directly incorporated into the high-order discontinuous Galerkin framework. Further, because of the linearity of the chosen projection and reconstruction operators, the properties of the general Phase-Field scheme are retained in the high-order framework, including consistency of reduction, Galilean invariance, maintenance of appropriate interfacial conditions, and conservation. We verify the proposed scheme with several test cases that demonstrate its accuracy in smooth regions, the numerically exact reduction consistency of the solver, its ability to simultaneously represent smooth and sharp features accurately, the advantage of the discontinuous Galerkin discretization with non-conformal subcell finite volume over conventional finite volume schemes and high-order limiters, and the capability of the scheme to robustly simulate flows with large density, pressure, velocity, and material property gradients. Validation against experimental data is also included.