A Framework of Runge–Kutta, Discontinuous Galerkin, Level Set and Direct Ghost Fluid Methods for the Multi-Dimensional Simulation of Underwater Explosions

Nan Si,Alan Brown

doi:10.3390/fluids7010013

Abstract

This work describes the development of a hybrid framework of Runge–Kutta (RK), discontinuous Galerkin (DG), level set (LS) and direct ghost fluid (DGFM) methods for the simulation of near-field and early-time underwater explosions (UNDEX) in early-stage ship design. UNDEX problems provide a series of challenging issues to be solved. The multi-dimensional, multi-phase, compressible and inviscid fluid-governing equations must be solved numerically. The shock front in the solution field must be captured accurately while maintaining the total variation diminishing (TVD) properties. The interface between the explosive gas and water must be tracked without letting the numerical diffusion across the material interface lead to spurious pressure oscillations and thus the failure of the simulation. The non-reflecting boundary condition (NRBC) must effectively absorb the wave and prevent it from reflecting back into the fluid. Furthermore, the CFD solver must have the capability of dealing with fluid–structure interactions (FSI) where both the fluid and structural domains respond with significant deformation. These issues necessitate a hybrid model. In-house CFD solvers (UNDEXVT) are developed to test the applicability of this framework. In this development, code verification and validation are performed. Different methods of implementing non-reflecting boundary conditions (NRBCs) are compared. The simulation results of single and multi-dimensional cases that possess near-field and early-time UNDEX features—such as shock and rarefaction waves in the fluid, the explosion bubble, and the variation of its radius over time—are presented. Continuing research on two-way coupled FSI with large deformation is introduced, and together with a more complete description of the direct ghost fluid method (DGFM) in this framework will be described in subsequent papers.

Highlights

In an underwater explosion, a series of events occur sequentially, including ignition, the chemical reaction of explosives, the generation and spreading of a shock wave outwards and a rarefaction wave inwards, the interaction between the shock wave and near-by structures, local and bulk cavitation in the fluid, the explosive bubble expanding and contracting, multiple bubble pulses being generated due to it, the interaction between the bubble pulses and nearby structures, reflected waves from either the structures or the free surface and their interaction with bubble, and bubble jetting, etc
The structure experiences primary shock wave loading and deformation, the collapse of the local cavitation and reloading exerted by the fluid, loading caused by bubble pulses and bubble jetting which leads to further deformation, whipping, and other global and dynamic responses [1,2,3,4]
This paper summarizes the development of a computational fluid dynamics (CFD) solver based on a hybrid framework of algorithms for the simulation of near-field and early-time underwater explosions (UNDEX)

Summary

Introduction

A series of events occur sequentially, including ignition, the chemical reaction of explosives, the generation and spreading of a shock wave outwards and a rarefaction wave inwards, the interaction between the shock wave and near-by structures, local and bulk cavitation in the fluid, the explosive bubble expanding and contracting, multiple bubble pulses being generated due to it, the interaction between the bubble pulses and nearby structures, reflected waves from either the structures or the free surface and their interaction with bubble, and bubble jetting, etc. The NRBC must effectively absorb the shockwave and prevent it from reflecting back into the fluid, so that the computational domain does not need to be extremely large This is even more important as the simulation runs longer, for example, in order to capture the simulation of the bubble radius and its variation over time. The second paper will focus on the direct ghost fluid method (DGFM) In this hybrid framework, multi-dimensional, multi-phase, compressible and inviscid fluid-governing equations, i.e., the Euler equations, are solved numerically using the Runge–Kutta method as the time marching scheme, the discontinuous Galerkin method as the spatial discretization scheme, the level set method to track the material interface between explosive gas and water with the help of the direct ghost fluid method [5], and the embedded boundary method so that the fluid solver is capable of simulating the fluid with large deformation

Survey of Related Work

Modeling and Simulation

Order of Accuracy Verifications

Simulations of Single and Multi-Dimensional Configurations

Conclusions