# Weighted Essentially Non-Oscillatory Simulations and Modeling of Complex Hydrodynamic Flows. Part 2. Single-Mode Richtmyer-Meshkov Instability with Reshock

## Abstract

The Richtmyer-Meshkov instability is a fundamental fluid instability that occurs when perturbations on an interface separating gases with different properties grow following the passage of a shock. This instability is typically studied in shock tube experiments, and constitutes a fundamental example of a complex hydrodynamic flow. Numerical simulations and models for the instability growth and evolution have also been used to further understand the physics of the Richtmyer-Meshkov instability. In the present work, the formally high-order accurate weighted essentially non-oscillatory (WENO) shock-capturing method using a third-order total-variation diminishing (TVD) Runge-Kutta time-evolution scheme (as implemented in the HOPE code [57]) is applied to simulate the single-mode Richtmyer-Meshkov instability with reshock in two spatial dimensions. The initial conditions and computational domain for the simulations are modeled after the Collins and Jacobs [23] single-mode, Mach 1.21 air(acetone)/SF6 shock tube experiment. The following boundary conditions are used: (1) periodic in the spanwise direction corresponding to the cross-section of the test section; (2) outflow at the entrance of the test section in the streamwise direction, and; (3) reflecting at the end wall of the test section in the streamwise direction. The present investigation has three principal motivations: (1) to provide additional validation of the HOPEmore »

- Authors:

- Publication Date:

- Research Org.:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 15014825

- Report Number(s):
- UCRL-TR-207593

TRN: US200802%%1316

- DOE Contract Number:
- W-7405-ENG-48

- Resource Type:
- Technical Report

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUMM MECHANICS, GENERAL PHYSICS; BOUNDARY CONDITIONS; ENERGY SPECTRA; HYDRODYNAMICS; INSTABILITY; POTENTIAL FLOW; RESOLUTION; SHOCK TUBES; SIMULATION; SPATIAL RESOLUTION; TIME DEPENDENCE

### Citation Formats

```
Schilling, O, and Latini, M.
```*Weighted Essentially Non-Oscillatory Simulations and Modeling of Complex Hydrodynamic Flows. Part 2. Single-Mode Richtmyer-Meshkov Instability with Reshock*. United States: N. p., 2004.
Web. doi:10.2172/15014825.

```
Schilling, O, & Latini, M.
```*Weighted Essentially Non-Oscillatory Simulations and Modeling of Complex Hydrodynamic Flows. Part 2. Single-Mode Richtmyer-Meshkov Instability with Reshock*. United States. https://doi.org/10.2172/15014825

```
Schilling, O, and Latini, M. Wed .
"Weighted Essentially Non-Oscillatory Simulations and Modeling of Complex Hydrodynamic Flows. Part 2. Single-Mode Richtmyer-Meshkov Instability with Reshock". United States. https://doi.org/10.2172/15014825. https://www.osti.gov/servlets/purl/15014825.
```

```
@article{osti_15014825,
```

title = {Weighted Essentially Non-Oscillatory Simulations and Modeling of Complex Hydrodynamic Flows. Part 2. Single-Mode Richtmyer-Meshkov Instability with Reshock},

author = {Schilling, O and Latini, M},

abstractNote = {The Richtmyer-Meshkov instability is a fundamental fluid instability that occurs when perturbations on an interface separating gases with different properties grow following the passage of a shock. This instability is typically studied in shock tube experiments, and constitutes a fundamental example of a complex hydrodynamic flow. Numerical simulations and models for the instability growth and evolution have also been used to further understand the physics of the Richtmyer-Meshkov instability. In the present work, the formally high-order accurate weighted essentially non-oscillatory (WENO) shock-capturing method using a third-order total-variation diminishing (TVD) Runge-Kutta time-evolution scheme (as implemented in the HOPE code [57]) is applied to simulate the single-mode Richtmyer-Meshkov instability with reshock in two spatial dimensions. The initial conditions and computational domain for the simulations are modeled after the Collins and Jacobs [23] single-mode, Mach 1.21 air(acetone)/SF6 shock tube experiment. The following boundary conditions are used: (1) periodic in the spanwise direction corresponding to the cross-section of the test section; (2) outflow at the entrance of the test section in the streamwise direction, and; (3) reflecting at the end wall of the test section in the streamwise direction. The present investigation has three principal motivations: (1) to provide additional validation of the HOPE code against available experimental data; (2) to provide numerical simulation data for detailed analysis of mixing induced by the Richtmyer-Meshkov instability with reshock, and; (3) to systematically investigate the dependence of mixing properties on both the order of WENO reconstruction and spatial resolution. The present study constitutes the first comprehensive application of the high-resolution WENO method to the Richtmyer-Meshkov instability with reshock, as well as analysis of the resulting mixing. First, analytical, semi-analytical, and phenomenological models for the growth of a single- and multi-mode perturbation are reviewed (impulsive, vortex, perturbation, potential flow, and asymptotic power-law growth models), including models for diffuse and reshocked interfaces. A model for baroclinic circulation deposition is also reviewed. Numerical simulations are performed using the third-, fifth-, and ninth-order WENO method with spatial resolutions corresponding to a uniform grid with 128, 256, and 512 points per initial perturbation wavelength. The density from the fifth- and ninth-order simulation is compared to the corrected experimental PLIF images of Collins and Jacobs at selected times. The amplitude obtained from the fifth-order simulation at a resolution of 256 points per initial perturbation wavelength is compared to the experimental data of Collins and Jacobs and to the predictions of linear and nonlinear amplitude growth models before and after reshock. The prediction of the Zhang-Sohn nonlinear amplitude growth model is in best agreement with the simulation data prior to reshock. The simulation data is also in excellent agreement with the experimentally-measured amplitude prior to reshock. The absence of the initial rarefaction wave (resulting from the rupture of the membrane that generates the first shock in the experiment) in the numerical simulations results in a time lag between the numerical and experimental interface evolution following reshock. The results of this component of the present investigation also serve as an additional validation of the HOPE code as applied to a shock-induced hydrodynamic instability. Second, local and global properties of the mixing during the linear, nonlinear, pre- and post-reshock, and late-time phases are investigated and discussed, including a quantitative investigation of the time-dependence and structure of various related mixing parameters defined in terms of the mole fraction and one-dimensional energy spectra. Spatial averaging of quantities along the spanwise (periodic) flow direction yields streamwise profiles, and is used to define instantaneous Reynolds and Favre averages and fluctuations. The fluctuations are Fourier-transformed along the spanwise direction to define time-dependent energy (abstract truncated)},

doi = {10.2172/15014825},

url = {https://www.osti.gov/biblio/15014825},
journal = {},

number = ,

volume = ,

place = {United States},

year = {2004},

month = {10}

}