Numerical simulation of compressible multifluid flows using an adaptive positivity‐preserving RKDG‐GFM approach

Abstract

Summary The Runge‐Kutta discontinuous Galerkin method together with a refined real‐ghost fluid method is incorporated into an adaptive mesh refinement environment for solving compressible multifluid flows, where the level set method is used to capture the moving material interface. To ensure that the Riemann problem is exactly along the normal direction of the material interface, a simple and efficient modification is introduced into the original real‐ghost fluid method for constructing the interfacial Riemann problem, and the initial conditions of the Riemann problem are obtained directly from the solution polynomials of the discontinuous Galerkin finite element space. In addition, a positivity‐preserving limiter is introduced into the Runge‐Kutta discontinuous Galerkin method to suppress the failure of preserving positivity of density or pressure for the problems involving strong shock wave or shock interaction with material interface. For interfacial cells in adaptive mesh refinement, the data transfer between different grid levels is achieved by using a L2 projection approach along with the least squares fitting. Various numerical cases, including multifluid shock tubes, underwater explosions, and shock‐induced collapse of a underwater air bubble, are computed to assess the capability of the present adaptive positivity‐preserving RKDG‐GFM approach, and the simulated results show that the present approach is quite robust and can provide relatively reasonable results across a wide variety of flow regimes, even for problems involving strong shock wave or shock wave impacting high acoustic impedance mismatch material interface.

An arbitrary Lagrangian–Eulerian method for fluid–structure interactions due to underwater explosions

“…The interactions between shock wave stage and bubble motion stage is studied in a continuous process (Liu et al , 2018). Ge et al give a Runge–Kutta discontinuous Galerkin (RKDG) method with adaptive mesh refinement for compressible multifluid flows in Ge et al (2019), where the level set method is used to capture the moving material interface. A homogeneous mixture model is given to study the dynamics of an underwater explosion bubble in the work of Phan et al (2019).…”

Positivity‐preserving scheme for the flux reconstruction framework with a novel implementation

“…Remarkable works such as References 5‐7 report great improvement of robustness owing to its application. The positivity‐preserving property may be realized through a posteriori correction or reconstruction, 8 and extended to more problems 9‐12 . In References 13‐15, a popular positivity‐preserving scheme is developed for solving different equations on various types of meshes.…”

“…The positivity-preserving property may be realized through a posteriori correction or reconstruction, 8 and extended to more problems. [9][10][11][12] In References 13-15, a popular positivity-preserving scheme is developed for solving different equations on various types of meshes. In this scheme, a positivity-preserving flux in the local Lax-Friedrichs form is adopted.…”