Adaptive solution techniques for simulating underwater explosions and implosions
Introduction
In this paper, we present adaptive solution techniques for simulating underwater explosion and implosion problems. These are multi-phase flow problems in which we solve a liquid component (water obeying Tait equation of state) and a gas component (gas is assumed to be spatially uniform, and adiabatic). A multi-phase flow calculation can be based on either solving a two-fluid system, i.e., both gas and liquid (compressible or incompressible) equations are solved, or based on solving a single fluid system, i.e., only liquid (compressible or incompressible) equations are solved, and the gas contents are treated as spatially uniform. For instance Fedkiw et al. [10], [7] used compressible (gas)–compressible (water) and compressible (gas)–incompressible (water) models. Compressible (gas)–compressible (water) models are also used by [23], [27], [14]. Our method solves the compressible liquid equations together with a compressible, spatially uniform, gas model. Multi-phase flow methods, in which the liquid is treated as compressible, are often formulated as an explicit method (i.e. [23], [10], [27], [14]). The time step for an explicit method is constrained by both the magnitude of the underlying velocity field and the magnitude of the sound speed of water. Alternatively, the time step for a semi-implicit method is not constrained by the sound speed of water. The following semi-implicit approaches have been developed for treating water as a compressible fluid [31], [30], [9], [32]. In the context of a multi-phase flow problem, very little attention has been given to semi-implicit approaches for discretizing the equations for the compressible flow of water [33].
Here we use semi-implicit discretizations to solve underwater explosion and implosion problems. The semi-implicit discretization approach enables us to solve many problems involving underwater explosions/implosions with a much larger time step than using an explicit discretization approach. An efficiency comparison between our semi-implicit methods and an explicit method due to Wardlaw [27] is made in the results section.
Adaptive Mesh Refinement (AMR) is another core feature of our algorithms. Adding adaptivity to our calculations makes our semi-implicit methods even more efficient. The need for the employment of the AMR technology in our computations is as follows. When solving underwater explosion and implosion problems, it is important to resolve the flow only around the high gradient regions such as shocks or material discontinuities in order to obtain efficient and accurate solution representations and save CPU time. Automating dynamic Adaptive Mesh Refinement techniques are customized to serve these purposes. The idea behind the Adaptive Mesh Refinement technique [5], [4] is to overlay successively finer resolution grid patches on top of underlying coarse grids, and to introduce a recursive time integration algorithm, then to synchronize the data between different grid levels. The time integration algorithm can be applied in two ways. One way is that all grid levels are advanced with the finest grid level time step. This is called the no-time-subcycling procedure. Another way is that a coarser grid level is advanced with a coarse time step and a finer grid level is advanced with multiple fine time steps until the finer level time reaches to the coarser level time. This is called the time-subcycling procedure.
Early Adaptive Mesh Refinement (AMR) techniques [5], [4], [3] are developed for solving hyperbolic conservation laws. Later, they are extended to solve the compressible Navier–Stokes equations [18], [17]. There have been significant efforts to solve incompressible or weakly compressible flows adaptively [1], [15], [11], [13], [2], [8], [19].
Our method shares some common features with [1] in the sense that we both subcycle in time and we both provide velocity continuity across the coarse/fine grids. Almgren et al. [1] method includes a synchronization projection step to provide velocity and pressure continuity across the coarse/fine grids. In [1], firstly, the velocity field is advanced on each level separately allowing velocity mismatch across the coarse/fine levels, then a multi-level composite projection step is applied at the end of each coarse level to correct the velocity differences. As a result their correction procedure modifies the solution on both coarse and fine levels. On the other hand, during our time advancing step, we solve the pressure equation on the current level and all levels above simultaneously. This produces accurate pressure boundary conditions for the next finer level. Also, since we solve on all levels l′ ⩾ l simultaneously, the velocity mismatch error, that gets corrected at the end of the ensuing fine time level integration steps, is significantly reduced. During the synchronization step, we solve the synchronization equations in the underlying coarse regions with Neumann boundary conditions. In this way, we maintain the velocity continuity without modifying the solution at the fine levels, i.e., we assume that the fine level velocity is correct and should not be changed.
The contents of the present paper is as follows. In Section 2, we describe the governing dynamics equations. In Section 3, we review the AMR grid hierarchy and define the components of our adaptive semi-implicit algorithms. In Section 4, we present the numerical results from the computations of underwater explosion/implosion test cases. Section 5 includes some concluding remarks.
Section snippets
Governing equations
Here we are interested in studying non-linear bubble dynamics by simulating underwater explosions and implosions. An underwater explosion can be modeled as a high pressure gas bubble which generates a shock wave. An underwater implosion can be modeled as a collapse of a low pressure gas bubble. We make the following assumptions when modeling the flow in water: the bubble growth and collapse are assumed to be axisymmetric. The system is assumed to be adiabatic, i.e., we ignore the heat
Components of the new adaptive solution techniques
In this section, we will describe the components of the new Adaptive Mesh Refinement (AMR) techniques. In the first subsection, we will describe the basic structure of the multi-level grids and give detail information about the grid management. In the second subsection, we will describe the time integration algorithms and point out some algorithmic facts. The third subsection will be dedicated to the synchronization procedure.
Spherical explosion bubble growth and collapse
A region of high pressure JWL gas, with a radius of 16 cm, fills a small bubble at the lower left corner of a two-dimensional domain as shown in Fig. 6. The initial material states for this problem are:
The high pressure bubble initiates a spherically propagating shock wave. While the shock front rapidly leaves the region, the bubble continues to grow. Fig. 7, produced by using our non-conservative semi-implicit method,
Conclusions
We presented adaptive solution techniques for simulating underwater explosions and implosions. We solved several test problems to show the performance of our methods. We validated our results by performing grid refinement studies in which we added additional levels of adaptivity. Also we validated our results by comparing shock speed, shock amplitude, and material interface speed, with benchmark results produced by Wardlaw [27]. Also, we compared our two semi-implicit formulations (conservative
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The authors were supported in part by Office of Naval Research (ONR) STTR under Contract N00014-02-C-0543, and by Weidlinger Associates Inc.