The Simulation of Shock- and Impact-Driven Flows with Mie-Grüneisen Equations of State

Citation

Ward, Geoffrey M. (2011) The Simulation of Shock- and Impact-Driven Flows with Mie-Grüneisen Equations of State. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/8Q2Q-GT29. https://resolver.caltech.edu/CaltechTHESIS:12162010-115725941

Abstract

An investigation of shock- and impact-driven flows with Mie-Grüneisen equation of state derived from a linear shock-particle speed Hugoniot relationship is presented. Cartesian mesh methods using structured adaptive refinement are applied to simulate several flows of interest in an Eulerian frame of reference. The flows central to the investigation include planar Richtmyer-Meshkov instability, the impact of a sphere with a plate, and an impact-driven Mach stem.

First, for multicomponent shock-driven flows, a dimensionally unsplit, spatially high-order, hybrid, center-difference, limiter methodology is developed. Effective switching between center-difference and upwinding schemes is achieved by a set of robust tolerance and Lax-entropy-based criteria [49]. Oscillations that result from such a mixed stencil scheme are minimized by requiring that the upwinding method approaches the center-difference method in smooth regions. To attain this property a blending limiter is introduced based on the norm of the deviation of WENO reconstruction weights from ideal. The scheme is first demonstrated successfully for the linear advection equation in spatially fourth- and sixth-order forms. A spatially fourth-order version of the method that combines a skew-symmetric kinetic-energy preserving center-difference scheme with a Roe-Riemann solver is then developed and implemented in Caltech's Adaptive Mesh Refinement, Object-oriented C++ (AMROC) [16,17] framework for Euler flows.

The solver is then applied to investigate planar Richtmyer-Meshkov instability in the context of an equation of state comparison. Comparisons of simulations with materials modeled by isotropic stress Mie-Grüneisen equations of state derived from a linear shock-particle speed Hugoniot relationship [36,52] to those of perfect gases are made with the intention of exposing the role of the equation of state. First, results for single- and triple-mode planar Richtmyer-Meshkov instability between mid-ocean ridge basalt (MORB) and molybdenum modeled by Mie-Grüneisen equations of state are presented for the case of a reflected shock. The single-mode case is explored for incident shock Mach numbers of 1.5 and 2.5. For the planar triple-mode case a single incident Mach number of 2.5 is examined with the initial corrugation wave numbers related by k₁=k₂+k₃. A comparison is drawn to Richtmyer-Meshkov instability in fluids with perfect gas equations of state utilizing matching of a nondimensional pressure jump across the incident shock, the post-shock Atwood ratio, post-shock amplitude-to-wavelength ratio, and time nondimensionalized by the Rcithmyer linear-growth rate time constant prediction. Result comparison demonstrates difference in start-up time and growth rate oscillations. Growth rate oscillation frequency is seen to correlate directly to the expected oscillation frequency of the transmitted and reflected shocks. For the single-mode cases, further comparison is given for vorticity distribution and corrugation centerline shortly after shock interaction that demonstrates only minor differences.

Additionally, examined is single-mode Richtmyer-Meshkov instability when a reflected expansion wave is present for incident Mach numbers of 1.5 and 2.5. Comparison to perfect gas solutions in such cases yields a higher degree of similarity in start-up time and growth rate oscillations. Vorticity distribution and corrugation centerline shortly after shock interaction is also examined. The formation of incipient weak shock waves in the heavy fluid driven by waves emanating from the perturbed transmitted shock is observed when an expansion wave is reflected.

Next, the ghost fluid method [83] is explored for application to impact-driven flows with Mie-Grüneisen equations of state in a vacuum. Free surfaces are defined utilizing a level-set approach. The level-set is reinitialized to the signed distance function periodically by solution to a Hamilton-Jacobi differential equation in artificial time. Flux reconstruction along each Cartesian direction of the domain is performed by subdividing in a way that allows for robust treatment of grid-scale sized voids. Ghost cells in voided regions near the material-vacuum interface are determined from surface-normal Riemann problem solution. The method is then applied to several impact problems of interest. First, a one-dimensional impact problem is examined in Mie-Grüneisen aluminum with simple point erosion used to model separation by spallation under high tension. A similar three-dimensional axisymmetric simulation of two rods impacting is then performed without a model for spallation. Further results for three-dimensional axisymmetric simulation of a sphere hitting a plate are then presented.

Finally, a brief investigation of the assumptions utilized in modeling solids as isotropic fluids is undertaken. An Eulerian solver approach to handling elastic and elastic-plastic solids is utilized for comparison to the simple fluid model assumption. First, in one dimension an impact problem is examined for elastic, elastic-plastic, and fluid equations of state for aluminum. The results demonstrate that in one dimension the fluid models the plastic shock structure of the flow well. Further investigation is made using a three-dimensional axisymmetric simulation of an impact problem involving a copper cylinder surrounded by aluminum. An aluminum slab impact drives a faster shock in the outer aluminum region yielding a Mach reflection in the copper. The results demonstrate similar plastic shock structures. Several differences are also notable that include a lack of roll-up instability at the material interface and slip-line emanating from the Mach stem's triple point.

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