Models of Richtmyer-Meshkov instability in continuously stratified fluids

Citation

Meloon, Mark Robert (1998) Models of Richtmyer-Meshkov instability in continuously stratified fluids. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/S4XY-H779. https://resolver.caltech.edu/CaltechETD:etd-01242008-153024

Abstract

The Richtmyer-Meshkov instability occurring when a planar shock wave passes through a sinusoidal region of continuous density gradient is studied numerically. Models are used to calculate the propagation of the shock through the inhomogeneity and to determine the late time behavior of the shocked fluid layer. The results from the models are compared with computations of the nonlinear Euler equations to determine ranges of validity. The models enjoy some success for problems involving weak incident shocks. As the Mach number is increased, however, the complex interactions between the transmitted and reflected fronts and the shocked density layer play an increasingly important role in the development of the flow and cause the models to fail. The popular impulse approximation is applied to the continuously stratified fluid configuration through the use of a model due to Saffman and Meiron. The predictions of the late time growth rate of the interface and interfacial circulation from the model are compared with calculations from the nonlinear Euler equations. It is shown that for weak incident shocks the model is a very accurate prediction of the asymptotic behavior of the interface for a wide range of problems including those with interfaces of finite amplitude and thickness. For stronger shocks, post-shock values for Atwood ratio, amplitude and layer thickness are used in the model to obtain accurate predictions of late time growth rate for high Atwood ratio configurations. Poor agreement is seen for low Atwood ratios. Comparisons between circulation calculations and pointwise values of vorticity between the model and Euler simulations reveal that the impulse model does not predict the correct vorticity distribution for high or low Atwood ratios. A numerical implementation of the Biot-Savart law is used to calculate the growth rate strictly from the vorticity field in the compressible Euler simulations. The good agreement between the compressible and incompressible growth rates, as well as direct measurement of the discrete divergence in the flow, indicates that compressible effects are only important in the initialization of the instability and that the subsequent evolution is determined from the vorticity distribution. The vorticity generated by subsequent oscillations of the transmitted and reflected shocks is shown to have a non-negligible effect on the interfacial growth rate. It is conjectured that the success of the impulse approximation in predicting the asymptotic growth rate for problems involving moderate to strong shocks and high Atwood ratios is simply the result of fortuitous cancellation between regions of vorticity not computed accurately by the model. The theory of Geometrical Shock Dynamics is used to propagate the shock through the region of density inhomogeneity and beyond. An important feature of the model is the neglect of interactions between the shock and the flow behind. A procedure for computing circulation in the entire flow from Geometrical Shock Dynamics is developed and implemented. For low to moderate Mach numbers, the initial circulation deposited on the layer is calculated with reasonable accuracy. At late times, however, the shape of the shock front does not agree with Euler calculations, resulting in incorrect calculation of the time evolution of total circulation of the flow. The agreement between the model and Euler simulations becomes poorer with increasing incident shock strength. By performing comparisons of local shock strength between the method and Euler simulation, it is shown that the method of Geometrical Shock Dynamics does not perform as well for problems involving nonuniform sound speed as had previously been believed. This suggests that the nonuniform flow conditions behind the shock, characterized by the vorticity baroclinically deposited on the interface during the shock refraction phase, plays a significant role in determining the evolution of the transmitted shock front.

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