Galilean-Invariant Characteristic-Based Volume Penalization Method for Supersonic Flows with Moving Boundaries

Nurlybek Kasimov,Giuliano De Stefano,Eric Dymkoski,Oleg V Vasilyev

doi:10.3390/fluids6080293

Abstract

This work extends the characteristic-based volume penalization method, originally developed and demonstrated for compressible subsonic viscous flows in (J. Comput. Phys. 262, 2014), to a hyperbolic system of partial differential equations involving complex domains with moving boundaries. The proposed methodology is shown to be Galilean-invariant and can be used to impose either homogeneous or inhomogeneous Dirichlet, Neumann, and Robin type boundary conditions on immersed boundaries. Both integrated and non-integrated variables can be treated in a systematic manner that parallels the prescription of exact boundary conditions with the approximation error rigorously controlled through an a priori penalization parameter. The proposed approach is well suited for use with adaptive mesh refinement, which allows adequate resolution of the geometry without over-resolving flow structures and minimizing the number of grid points inside the solid obstacle. The extended Galilean-invariant characteristic-based volume penalization method, while being generally applicable to both compressible Navier–Stokes and Euler equations across all speed regimes, is demonstrated for a number of supersonic benchmark flows around both stationary and moving obstacles of arbitrary shape.

Highlights

Numerical simulation of complex geometry flows in a computationally efficient manner represents a challenging problem, especially in the presence of moving/deformable boundaries
The present study focuses on evaluating the novel aspects of the extended method that is developed for solving systems of hyperbolic partial differential equations (PDEs) with discontinuous solutions, in the presence of stationary or moving obstacles of arbitrary complexity
The first test is represented by the one-dimensional shock tube problem, where the shock reflection from the wall is quantitively evaluated, with the wall being approximated by the GI-characteristic-based volume penalization (CBVP) method

Summary

Introduction

Numerical simulation of complex geometry flows in a computationally efficient manner represents a challenging problem, especially in the presence of moving/deformable boundaries. Solid bodies are introduced by imposing appropriate boundary conditions upon surfaces, and to that end, several approaches have been developed. These methods can be separated into two major groups: body-fitted mesh [1,2,3] and immersed boundary (IB) methods [4,5,6]. The process of mesh generation is highly dependent upon the obstacle geometry and can become computationally expensive, especially for complex surfaces. This issue is compounded for moving or deforming obstacles, which require continuous adaptation or re-meshing throughout computation of the solution [6]. The grid generation process may be very expensive: it is not an easy task to generate a good-quality grid, as even simple geometries and simulations for moving boundary problems become prohibitively expensive due to grid generation and solution interpolation to the new mesh at each time step

Methods

Results

Conclusion