We introduce a new numerical method for solving the compressible Navier-Stokes equations in flow domains that may be separated by any number of interfaces, fluid-to-fluid or fluid-to-solid, at arbitrarily high density ratios and acoustic-impedance mismatch, and subjected to pressure waves of arbitrarily high amplitudes and steepness. A principal aim is the ab initio prediction of interfacial instabilities under superposition of multiple active modes (Rayleigh-Taylor, Kelvin-Helmholtz, Richtmeyer-Meshkov) along with mean flow development, as found for example in rapidly-accelerated immersed-fluid bodies. We insist on a sharp-interface treatment (fluid-property and normal-stress jumps), and a unique feature of our approach is solving for motion of each element of the piecewise-curved interface as an exact Riemann problem within an overall conservative scheme that couples consistently the bulk-fluid motions to that of the interface. Sample simulations of liquid drops subjected to shock waves demonstrate for the first time the numerical prediction of the key phenomena found in recent experimental and theoretical studies of this class of problems [Theofanous, Ann. Rev. Fluid Mech. 43:661-90,2011].
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