A comparison of two- and three-dimensional single-mode reshocked Richtmyer–Meshkov instability growth

doi:10.1016/j.physd.2019.132201

Physica D: Nonlinear Phenomena

Volume 401, January 2020, 132201

https://doi.org/10.1016/j.physd.2019.132201 Get rights and content

Abstract

The growth dynamics of two- and three-dimensional single-mode reshocked Richtmyer–Meshkov instability are compared systematically using data from high-resolution implicit large-eddy simulations of a model of the Mach 1.3 air(acetone) and sulfur hexafluoride (Jacobs and Krivets, 2005) shock tube experiment. The vorticity deposition by the incident shock and the dynamics of interface evolution are examined quantitatively and qualitatively. The perturbation amplitudes from the two- and three-dimensional simulations are compared to the experimental data and to the predictions of several nonlinear instability growth models. It is shown that the perturbation amplitudes from the two- and three-dimensional simulations with matching initial Richtmyer velocity are in excellent agreement with the experimental data. In addition, the dynamics of reshock (not considered in the experiment) are described in detail, and the post-reshock mixing layer amplitude growth rate is compared to the predictions of several reshock models. It is shown that using two-dimensional simulations to understand three-dimensional dynamics is valid only at early-to-intermediate times before reshock; at intermediate-to-late times after reshock the three-dimensional growth is generally larger than the corresponding two-dimensional growth. The reshock dynamics are also different between two and three dimensions. The quantitative results, together with visualizations of the flow field, were also used to contrast the difference between two- and three-dimensional vorticity and enstrophy dynamics.

Introduction

Richtmyer–Meshkov instability [1], [2] develops when an interface separating a heavier and a lighter material is impulsively accelerated: perturbations grow into bubbles ‘penetrating’ into the heavier fluid and spikes ‘penetrating’ into the lighter fluid, eventually developing roll-ups and complex structures through secondary shear instabilities. This instability is of fundamental interest to fluid dynamics and turbulent mixing [3], [4], [5], and for its applications to inertial confinement fusion [6], supersonic combustion [7], and astrophysics [8] (see Refs. [9], [10] for a recent review). One of the challenges in better understanding Richtmyer–Meshkov instability is the accurate modeling of the growth of the mixing layer in the nonlinear phase, as well as predicting the statistical properties and dynamics of turbulent mixing induced by this instability [11], [12], [13], [14].

The single-mode dynamics of this instability are examined here in two and three dimensions using the formally high-order accurate weighted essentially nonoscillatory (WENO) method. The simulations are performed on a model of the Mach 1.3 reshocked Richtmyer–Meshkov instability experiment of Jacobs and Krivets [15] to both provide validation of the simulation results by comparison to experimental data, and to examine the evolution of the instability from the early linear stage, through reshock, and to late times following reshock. The WENO method is a shock-capturing method used for discretizing the compressible Euler equations of gas dynamics. As such, ab initio simulations are performed with a shock generated in an air(acetone) mixture interacting with a diffuse sinusoidal interface initially separating this mixture from sulfur hexafluoride.

Other numerical simulations have been performed to investigate and compare various aspects of the evolution of single- and multimode Richtmyer–Meshkov instability in two and three dimensions using a variety of numerical algorithms and methods [16], [17], [18], [19]. Also, the predictions of the van Leer method and fifth- and ninth-order WENO methods applied to the Jacobs–Krivets experiment were compared [20], with an emphasis on the effect of grid resolution on the instability using coarse two-dimensional grids. An adaptive central-upwind sixth-order WENO method was previously used to simulate the Jacobs–Krivets experiment in two dimensions [21], with an emphasis on the differences between single- and multifluid formulations.

This paper is organized as follows. A description of the present numerical simulations, including the numerical method, initial and boundary conditions, and characterization of the vorticity deposition by the shock is given in Section 2. A description of the dynamics of the interface evolution is presented in Section 3. A comparison of the numerical amplitude to the predictions of several nonlinear instability growth models is discussed in Section 4. The dynamics of reshock are discussed in Section 5, including a comparison of the numerical amplitude to the predictions of several reshock models. This study is extended to three dimensions in Section 6. Finally, a summary of the results and conclusions is given in Section 7.

Section snippets

Numerical method

The weighted essentially nonoscillatory (WENO) method is a widely used shock-capturing scheme that has previously been used in the investigation of Richtmyer–Meshkov instability [11], [12], [22], [23] and more generally of complex flows with shocks [24]. The numerical simulations of reshocked Richtmyer–Meshkov instability here were performed using the characteristic projection-based, finite-difference WENO method with ninth-order flux reconstruction [25], [26]. A methods-of-lines discretization

Dynamics of the instability evolution

A two-dimensional simulation of a model of the Jacobs and Krivets [15] $M a_{i} = 1.3$ air(acetone)/SF₆ shock tube experiment is performed using the ninth-order WENO method. The grid resolution and other numerical parameters are given in Table 6. Simulations for a similar two-dimensional, single-mode Richtmyer–Meshkov instability with reshock using the same code with different grid resolutions and comparison of results are discussed in detail elsewhere [22]. The density fields from the simulation are

Comparison of the perturbation, bubble, and spike amplitudes to experimental data and model predictions

Here, the amplitude and bubble and spike velocity are compared to the predictions of several nonlinear growth models. Nonlinear Padé models for the amplitude growth are discussed in Section 4.1, and models for the bubble and spike velocities are presented in Section 4.2. A comparison of the amplitudes from the numerical simulation to experimental results and to the predictions of the models is discussed in Section 4.3.

Analysis of two-dimensional reshock

Fig. 8 shows the $x$ – $t$ diagram from the WENO simulation. The locations of the bubble $ℓ_{b} (t)$ and of the spike $ℓ_{s} (t)$ are shown using the dash–dot and the dashed lines, respectively. The interface location is also shown using a solid line. The horizontal distance between the spike and bubble is the perturbation width $h (t)$ , and half of this distance is the perturbation amplitude $a (t)$ . Reshock occurs at $\approx 5.65 ms$ , when the shock wave refracts at the evolving interface, generating a transmitted shock in

Three-dimensional single-mode Richtmyer–Meshkov instability

Presented here are three-dimensional WENO simulations of single-mode Richtmyer–Meshkov instability using two values of the initial perturbation amplitude and analysis of the results.

Summary and conclusions

The ninth-order WENO shock-capturing method was applied to simulate a model of the Mach 1.3 air(acetone) and SF₆ Jacobs and Krivets single-mode Richtmyer–Meshkov instability experiment [15] extended numerically to include reshock. In this experiment, the larger Mach number compared to previous studies allowed the investigation of late-time effects prior to the arrival of the transmitted shock in the reshock phase. The present study analyzed the development of the instability prior to reshock,

Acknowledgments

This work is dedicated to honoring the exemplary scientific career of David L. Youngs. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

This document was prepared as an account of work sponsored by an agency of the United States government. Neither the United States government nor Lawrence Livermore National Security, LLC, nor any of their employees makes any warranty, expressed or implied, or

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Citation Excerpt :
See also, Refs. [56] and [57] for reshocked single- and multi-mode RMI induced flows, respectively.) Recently, ILES was used to elucidate the differences between two- and three-dimensional dynamics of single-mode reshocked RMI [58]. Finally, we remark that vorticity can also be generated on a shock structure itself, particularly from nonlinear triple-points that form on the shock geometry, and this can also be a source of turbulence [55].
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¹: Present address: Northrop Grumman Aerospace Systems, Palmdale, CA 93550, USA.

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A comparison of two- and three-dimensional single-mode reshocked Richtmyer–Meshkov instability growth

Abstract

Introduction

Section snippets

Numerical method

Dynamics of the instability evolution

Comparison of the perturbation, bubble, and spike amplitudes to experimental data and model predictions

Analysis of two-dimensional reshock

Three-dimensional single-mode Richtmyer–Meshkov instability

Summary and conclusions

Acknowledgments

Phys. Rep.

Phys. Rep.

J. Comput. Phys.

J. Comput. Phys.

Acta Math. Sci.

J. Comput. Phys.

J. Comput. Phys.

Prog. Aerosp. Sci.

J. Comput. Phys.

Physica D

Physica D

Physica D

Physica D

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

Taylor instability in shock acceleration of compressible fluids

Comm. Pure Appl. Math.

Instability of the interface of two gases accelerated by a shock wave

Sov. Fluid Dyn.

Numerical simulation of mixing by Rayleigh-Taylor and Richtmyer-Meshkov instabilities

Laser Part. Beams

Vortex paradigm for accelerated inhomogeneous flows: Visiometrics for the Rayleigh-Taylor and Richtmyer-Meshkov environments

Ann. Rev. Fluid Mech.

The Richtmyer-Meshkov instability

Ann. Rev. Fluid Mech.

Two Mechanisms of Vorticity Generation in Combusting Flow Fields

The role of mixing in astrophysics

Astrophys. J. Suppl.

High-resolution simulations and modeling of reshocked single-mode Richtmyer-Meshkov instability: comparison to experimental data and to amplitude growth model predictions

Phys. Fluids

Physics of reshock and mixing in single-mode Richtmyer-Meshkov instability

Phys. Rev. E

Late-time growth rate, mixing, and anisotropy in the multimode narrowband Richtmyer–Meshkov Instability: The θ-group collaboration

Phys. Fluids

Turbulent transport and mixing in the multimode narrowband Richtmyer-Meshkov instability

Phys. Fluids

Experiments on the late-time development of single-mode Richtmyer-Meshkov instability

Phys. Fluids

Numerical simulation of Richtmyer-Meshkov instabilities

Phys. Fluids A

Comparison of two- and three-dimensional simulations of miscible Richtmyer-Meshkov instability with multimode initial conditions

Phys. Fluids

Numerical investigation of 3D effects on a 2D-dominated shocked mixing layer

Phys. Fluids

Late-time growth rate, mixing, and anisotropy in the multimode narrowband Richtmyer–Meshkov Instability: The $θ$ -group collaboration