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Simulations and analysis of two- and three-dimensional single-mode Richtmyer-Meshkov instability using weighted essentially non-oscillatory and vortex methods

Citation

Latini, Marco (2007) Simulations and analysis of two- and three-dimensional single-mode Richtmyer-Meshkov instability using weighted essentially non-oscillatory and vortex methods. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd-12082006-124547

Abstract

An incompressible vorticity-streamfunction (VS) method is developed to investigate the single-mode Richtmyer-Meshkov instability in two and three dimensions. The initial vortex sheet (representing the initial shocked interface) is thickened to regularize the limit of classical Lagrangian vortex methods. In the limit of smaller thickness, the initial velocity converges to the velocity of a vortex sheet. The vorticity on the Cartesian grid follows the vorticity evolution equation augmented by the baroclinic vorticity production term (to capture the effects of the instability on the layer) and a viscous dissipation term. The equations are discretized using a fourth-order in space and third-order in time semi-implicit Adams-Bashforth backward differentiation scheme. The convergence properties of the method with respect to varying the diffuse interface thickness and viscosity are investigated. It is shown that the small-scale structures within the roll-up are more sensitive to the diffuse interface thickness than to the viscosity. By contrast, the large-scale quantities, including the perturbation, bubble, and spike amplitudes are less sensitive. Fourth-order point-wise convergence is achieved, provided that a sufficiently fine grid is used.

In two dimensions, the VS method is applied to investigate late-time nonlinear effects of the single-mode Mach 1.3 air(acetone)/SF_6 shock tube experiment of Jacobs and Krivets. The results are also compared to those from compressible ninth-order weighted essentially non-oscillatory (WENO) simulations. The density fields from the WENO and VS methods agree with the experimental PLIF images in the large-scale structures but differ in the small-scale structures. The WENO method exhibits small-scale disordered structure similar to that in the experiment, while the VS method does not capture such structure, but shows a strong rotating core. The perturbation amplitudes from the two methods are in good agreement and match the experimental data points well. The WENO bubble amplitude is smaller than the VS amplitude and vice versa for the spike amplitude. Comparing amplitudes from simulations with varying Mach number shows that as the Mach number increases, the differences in the bubble and spike amplitudes increase due to intensifying pressure perturbations not present in the incompressible VS method. The perturbation amplitude from the WENO and VS methods is also compared to the predictions of nonlinear amplitude growth models in which the growth rate was reduced to account for the diffuse initial interface. In general, the model predictions agree with the simulation amplitudes at early-to-intermediate times and underpredict at later times, corresponding to the late nonlinear regime.

The WENO simulation is used to investigate reshock, which occurs when the transmitted shock reflects from the end wall of the test section and interacts with the evolving layer. The post-reshock mixing layer width agrees well with the predictions of reshock models for short times until the interaction of the reflected rarefaction with the layer.

The VS simulation was also compared to classical Lagrangian and vortex-in-cell simulations as the Atwood number was varied. For low Atwood numbers, all three simulations agree. As the Atwood number increases, the VS simulation shows differences in the bubble and spike amplitudes compared to the Lagrangian and VIC simulations, as the baroclinic vorticity production for a diffuse layer is different from that of a thin layer. The simulation amplitudes agree with the predictions of nonlinear amplitude growth models at early times. The growth models underpredict the amplitudes at later times.

The investigation is extended to three dimensions, where the initial perturbation is a product of sinusoids and the initial vorticity deposition is given by linear instability analysis. The instability evolution and dynamics of vorticity are visualized using the mass fraction and enstrophy isosurface, respectively. For the WENO and VS methods, two roll-ups corresponding to the bubble and spike regions form, and the vorticity shows the formation of a ring-like structure. The perturbation amplitudes from the WENO and VS methods are in excellent agreement. The bubble and spike amplitude are in good agreement at early times. At later times, the WENO bubble amplitude is smaller than the VS amplitude and vice versa for the spike. The nonlinear three-dimensional Zhang-Sohn model agrees with the simulation amplitudes at early times, and underpredicts later. In three dimensions, the enstrophy iso-surface after reshock shows significant fragmentation and the formation of small, short, tubular structures. Simulations with different initial amplitudes show that the mixing layer width after reshock does not depend on the pre-shock amplitude. Finally, the effects of Atwood number are investigated using the VS method and the amplitudes are compared to the predictions of the Zhang-Sohn model. The simulation and the models are in agreement at early times, while the models underpredict later.

The VS method constitutes a useful numerical approach to investigate the Richtmyer-Meshkov instability in two and three dimensions. The VS method and, more generally, vortex methods are valid tools for predicting the large-scale instability features, including the perturbation amplitudes, into the late nonlinear regime.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:convergence properties; reshock; Richtmyer-Meshkov instability; vortex methods; vorticity-streamfunction method; WENO methods
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Applied And Computational Mathematics
Thesis Availability:Restricted to Caltech community only
Research Advisor(s):
  • Meiron, Daniel I. (advisor)
  • Schilling, Oleg (advisor)
Thesis Committee:
  • Meiron, Daniel I. (chair)
  • Pullin, Dale Ian
  • Pierce, Niles A.
  • Schilling, Oleg
  • Hou, Thomas Y.
Defense Date:22 September 2006
Author Email:mlatini (AT) acm.caltech.edu
Record Number:CaltechETD:etd-12082006-124547
Persistent URL:http://resolver.caltech.edu/CaltechETD:etd-12082006-124547
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:4868
Collection:CaltechTHESIS
Deposited By: Imported from ETD-db
Deposited On:11 Jan 2007
Last Modified:26 Dec 2012 03:12

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