View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector ADVANCES IN APPLIED MATHEMATICS 10,201-227 (1989) The Interaction of Shock Waves with Fluid Interfaces JOHN GROVE* Courant Institute of Mathematical Sciences, New York University, New York, New York 10012 Enhanced resolution for the computation of the interaction of shock waves with fluid interfaces is achieved through a detailed mathematical analysis of 2-dimen- sional wave interactions produced during the collision of the waves. This computa- tion is carried to late times, which are characterized by interface instability and chaotic mixing processes. Algorithms for incorporating the wave interaction analysis and the resulting bifurcation of front topology give an important extension of the front tracking method and are presented here. The mathematical analysis shows that the customary theory for oblique 2-dimensional wave interactions is equivalent to a l-dimensional Riemann problem for steady (supersonic) flow. This analysis, known for polytropic gases, is extended here to a general equation of state. Moreover,the asymptotic limit of a small incident angle is analyzed to obtain a well-conditioned numerical algorithm. This limit is found to define a l-dimensional unsteady Riemann problem. Q 1989 Academic Press, Inc. 1. INTRODUCTION There are two distinct time periods for the analysis of the interaction of a shock wave with a fluid interface. In the short time periods 2dimensional wave interactions and bifurcations of front topology dominate the problem. Over a longer time period, interface instability, entrainment, and chaotic mixing are the major phenomena. The main thrust of the paper is to use a detailed mathematical analysis of the short term 2-dimensional wave interactions to give enhanced resolution computations of long term mixing processes. Enhanced resolution is ob- tained through the front tracking algorithm [8, 12-141. The enhanced *Supported in part by the Army Research Office Grant DAAG29-84-K-0130, 201 0196-8858/89 $7.50 Copyright 0 1989 by Academic Press. Inc. All rights of reproduction in any form reserved 202 JOHN GROVE resolution appears to be at least a factor of 4.4 per linear dimension, or 85 per space time grid block, compared to other methods. An important extension of this method is required for the problems considered here. The shock-contact interaction problem presents a level of complexity in the fluid flow not previously handled by the front tracking method. The required extension of the method to deal with such problems is among the principal new results reported in this paper. These results are achieved by carrying out a mathematical analysis of the 2-dimensional interaction of shock waves with fluid interfaces and incorporating this information di- rectly into the computational process. The application of this procedure to numerical studies of shock-contact interactions, as reported here, consti- tutes a proof of principle for the front tracking method in the present context. The mathematical analysis begins with the observation that the cumber- some shock polar analysis for oblique 2-dimensional wave interactions can be interpreted as a l-dimensional Riemann problem for steady (supersonic) flow. The customary analysis of this problem, valid for polytropic gases, is extended to a general equation of state. This extension, while not needed for the computational examples presented here, will be required for a subsequent comparison of computation with experiment. It is also shown that for large flow speeds, the steady state Riemann problem has a l-dimen- sional Riemann problem as an asymptotic limit. The analysis of this asymptotic limit was necessary to obtain a well-conditioned algorithm in the limit of small incident angles. A further contribution of this paper is to describe the algorithms for incorporating bifurcations of front topology into the front tracking compu- tation. These algorithms generalize .previous bifurcation algorithms ob- tained for the case of scalar waves [16]. Computational examples are presented. Chaotic mixing caused by the acceleration of a fluid interface by a shock wave is known as the Richtmyer-Meshkov instability [6, 261. Recent calculations of D. L. Youngs [34] use a volume in cell Eulerian method [17, 24, 25, 331 with the monotonic advection method of Van Leer [19, 201 to enhance the interface resolution and to minimize the numerical diffusion. His problems are not exactly comparable to the ones presented here, but it appears that the front tracking method gives an increased resolution by a factor of at least 4.4 per linear dimension. Comparisons of front tracking in the context of other problems, with the solutions obtained using different numerical methods, show increased resolutions by factors of up to 3-5 per linear dimension, or 27-125 per space time grid block. One extreme case showed an increased resolution of 50 per linear dimension, or 503 = 1.25 X lo6 per space time grid block 1151. SHOCK WAVES WITH FLUID INTERFACES 203 2. THE EQUATIONS OF MOTION AND THERMODYNAMIC PRELIMINARIES The equations of motion are the Euler equations in two space dimensions for a non-viscous, non-heat conducting gas: J,P + v . (PI) = 0, (2.1.1) J,(wI) + v * (WI @ 9) + VP = 0, (2.1.2) d,(p@q + v * ((p&+ P)q) = 0. (2.1.3) Here q is the particle velocity, p is the mass density, P is the thermody- namic pressure, d = ]q] 2/2 + E is the specific total energy, and E is the specific internal energy. System (2.1) is closed by a caloric equation of state E = E(V, S) (2.2) expressing the specific internal energy as a function of the specific volume V = l/p, and the specific entropy S. The equation of state represents the response of the fluid to changes in temperature and volume and is restricted by the laws of physics. In particular, E(V, S) satisfies the first law of thermodynamics, TdS = dE + PdV, (2-3) where T is the absolute temperature. The assumption that the fluid remains in stable thermodynamic equilibrium implies that E(V, S) is convex [32]. This implies that the sound speed c2 = (h’P/Jp)l s is real, and P is monotone in V for constant S. Other restrictions on E(V, S) that govern the monotonicity and asymptotic properties of the wave curves for (2.1) are derived in [27, 231, and will be assumed when necessary. These added assumptions supply sufficient conditions for the existence and uniqueness of the solution to the one dimensional unsteady Riemann problem. 3. DIMENSIONAL REDUCTION FOR ~-DIMENSIONAL ELEMENTARY WAVE INTERACTIONS An elementary wave is by definition a steady, scale invariant solution of a system of hyperbolic conservation laws [lo, 11, 141. It consists of discrete waves moving at a constant velocity. According to the streamline directions, these waves are defined as either incoming or outgoing. The analysis presented in the next two sections will consider the case where the elementary wave consists of two incoming waves with supersonic states on their downstream edges. The task in studying such elementary 204 JOHN GROVE waves is to determine the outgoing waves, while the incoming waves are regarded as data. In hydrodynamics these waves include the crossing of two shock waves, the overtaking of one shock by another, and the diffraction of a shock at a fluid interface [14]. Other elementary waves include a Mach triple point and the interaction of several contact discontinuities. This section starts with the new observation that the outgoing waves are determined as solutions of a l-dimensional Riemann problem, namely that defined for steady state 2-dimensional flow (with the stream direction as a “time” axis). This point of view is equivalent to the more conventional description of such wave interactions by the intersection of shock polars, but has the advantage of greater mathematical clarity. It unifies the role of simple waves and shocks in the solution and stresses the analogy between the solution to such problems and the Riemann problem for l-dimensional unsteady flow. The main result here is built on this observation. The existence of a solution to the steady 2-dimensional Riemann problem for large flow speeds and a general equation of state is demonstrated below. This existence result carries over to the corresponding statement for 2- dimensional elementary waves. In general such solutions are not unique. However, it will be seen that the leading order asymptotic behavior for large flow speeds of the supersonic steady problem is the l-dimensional unsteady problem. The lowest pressure (highest Mach number) solution to the 2-dimensional steady Riemann problem is the asymptotically “correct” solution. This fact agrees with the experimental observation of Henderson [l-3, 51, that the lowest pressure solution to the shock polar analysis is the one observed experimentally. Each “supersonic” elementary wave can be viewed as the downstream scattering of the interacting incoming waves. The two incoming waves set up a pair of supersonic states that serve as upstream “initial” conditions for the downstream flow. It is in this sense that the downstream scattering of the incoming waves is the solution to a Riemann problem. 4. THE WAVE ANALYSIS FOR A GENERAL EQUATION OF STATE Before continuing with this discussion, it is necessary to summarize the wave analysis of steady planar flow. This discussion is based on the analysis of a general system of hyperbolic conservation laws [18]. In the following q and 8 are the polar coordinates of the stream velocity q. The restriction of (2.1) to steady planar flow yields a system of four conservation laws in the two space variables x and y. This system is non-strictly hyperbolic for supersonic flow. It has three families of charac- teristics.