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Direct numerical simulation of interfacial instabilities: A consistent, conservative, all-speed, sharp-interface method. (English) Zbl 1299.76097
Summary: We present a conservative and consistent numerical method for solving the Navier-Stokes equations in flow domains that may be separated by any number of material interfaces, at arbitrarily-high density/viscosity ratios and acoustic-impedance mismatches, subjected to strong shock waves and flow speeds that can range from highly supersonic to near-zero Mach numbers. A principal aim is prediction of interfacial instabilities under superposition of multiple potentially-active modes (Rayleigh-Taylor, Kelvin-Helmholtz, Richtmyer-Meshkov) as found for example with shock-driven, immersed fluid bodies (locally oblique shocks) – accordingly we emphasize fidelity supported by physics-based validation, including experiments. Consistency is achieved by satisfying the jump discontinuities at the interface within a conservative 2nd-order scheme that is coupled, in a conservative manner, to the bulk-fluid motions. The jump conditions are embedded into a Riemann problem, solved exactly to provide the pressures and velocities along the interface, which is tracked by a level set function to accuracy of $$O(\Delta x^5,\Delta t^4)$$. Subgrid representation of the interface is achieved by allowing curvature of its constituent interfacial elements to obtain $$O(\Delta x^3)$$ accuracy in cut-cell volume, with attendant benefits in calculating cell- geometric features and interface curvature $$(O(\Delta x^3))$$. Overall the computation converges at near-theoretical $$O(\Delta x^2)$$. Spurious-currents are down to machine error and there is no time-step restriction due to surface tension. Our method is built upon a quadtree-like adaptive mesh refinement infrastructure. When necessary, this is supplemented by body-fitted grids to enhance resolution of the gas dynamics, including flow separation, shear layers, slip lines, and critical layers. Comprehensive comparisons with exact solutions for the linearized Rayleigh-Taylor and Kelvin-Helmholtz problems demonstrate excellent performance. Sample simulations of liquid drops subjected to shock waves demonstrate for the first time ab initio numerical prediction of the key interfacial features and phenomena found in recent experimental and theoretical studies of this class of problems [T.G. Theofanous, Ann. Rev. Fluid Mech. 43, 661–690 (2011; Zbl 1299.76217)].

##### MSC:
 76F65 Direct numerical and large eddy simulation of turbulence 76M25 Other numerical methods (fluid mechanics) (MSC2010)
Gerris; AUSM
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