# Richtmyer-Meshkov instability and re-accelerated inhomogeneous flows

Post-publication activity

Curator: Jeffrey W. Jacobs

Richtmyer–Meshkov instability is the fluid flow phenomenon that occurs when a shock wave impinges (in the normal direction) upon an interface separating two gases having different densities (Richtmyer 1960; Meshkov 1969; see Brouillette 2002). The instability causes perturbations on the interface to grow, producing vortical structures which potentially result in a turbulent mixing layer, as illustrated in Figure 1.

Figure 1: Richtmyer–Meshkov instability is produced by the interaction of a planar shock wave and a perturbed interface and often results in the “mushroom” structures shown below.

## Introduction

### History

The Richtmyer–Meshkov instability derives its name from the pioneering work of Robert D. Richtmyer at Los Alamos National Laboratory (LANL) who performed the first instability analysis, and Evgeny E. Meshkov at the Russian Federal Nuclear Center-All-Russian Research Institute of Experimental Physics (RFNC-VNIIEF) who performed the first experiments to study this instability.

Figure 2: The misalignment of the pressure gradient from the shock wave with the density gradient at the interface produces the vorticity distribution shown below.

### Description of the instability

Although the shock interaction process is inherently compressible, Richtmyer–Meshkov instability can also be produced by the impulsive acceleration of two incompressible fluids. In this sense, Richtmyer–Meshkov instability is related to the better known Rayleigh–Taylor instability, and is often referred to as impulsive Rayleigh–Taylor instability. The physical mechanism producing the instability is most easily described using the dynamics of the vorticity field $$\mathbf{\omega}=\mathbf{\nabla}\times\mathbf{v}\ .$$ Vorticity is deposited on the perturbed interface by the misalignment of the pressure gradient of the shock wave and the density gradient at the fluid interface, as governed by the baroclinic production (first term on the right side) in the inviscid 3D vorticity equation $\tag{1} \frac{\mathrm{D}\mathbf{\omega }}{\mathrm{D}t}=\frac{\mathbf{\nabla \!}\rho \times \mathbf{\nabla \!}p}{\rho ^{2}}+\mathbf{\omega \cdot \nabla v}- \mathbf{\omega \,\nabla \cdot v}\,,$

where $$\mathrm{D}/\mathrm{D}t=\partial/\partial t+\mathbf{v}\cdot{\mathbf \nabla}\ .$$ If the interaction of a planar shock wave with a sinusoidally-perturbed interface is considered, the baroclinic term would result in a vorticity distribution that is maximum near the points of minimum deflection and decays to zero at points of maximum deflection yielding a distribution similar to that of a row of vortices having alternating signs as shown in Figure 2. This distribution would then result in increasing amplitude in time, as well as the formation of the mushroom structures shown in Figure 1. The configuration depicted illustrates the interaction with a light-to-heavy system. If this configuration is reversed (i.e., interaction with a heavy-to-light system) the sign of the vorticity would be reversed causing a flattening of the interface followed by the growth of the perturbation, resulting in mushroom structures 180° out of phase from those of the light/heavy system. Following shock interaction, the baroclinic production mechanism continues to act via the pressure gradient produced by vortices and the interface to continually alter the vorticity. The most important parameters governing the instability are the shock wave Mach number and the Atwood number. The shock Mach number $$M$$ determines the effects of compressibility in the flow, i.e., for weak shocks the flow following shock interaction may be adequately described assuming incompressible flow. The Atwood number

Figure 3: Density (left), vorticity (middle), and pressure (right) fields from a 2D fifth-order WENO simulation of the Collins and Jacobs (2002) experiment at 6.38 ms. Reshock compresses the evolving interface, redepositing vorticity on the interface. The pressure field shows the transmitted shock and the reflected rarefaction wave.

$\tag{2} A = \frac{\rho_1 - \rho_2}{\rho_1 + \rho_2}$

is a dimensionless measure of the density ratio. The instability develops symmetrically for small values of the Atwood number. Moderate to large (approaching unity) values produce asymmetry resulting in the broadening of regions where light fluid penetrates the heavy one (bubbles) and narrowing in regions where the heavy fluid penetrates the lighter one (spikes). When the evolving structure is reshocked (see Figure 3) by a second shock reflected from the endwall of a shock tube, disordered small-scale structure forms (see Figure 5).

### Applications

The Richtmyer–Meshkov instability occurs in physical applications ranging from astrophysics to supersonic combustion. For example, this instability is believed to occur when the outward propagating shock wave generated by the collapsing core of a dying star passes through the helium–hydrogen interface. Observations of the optical output of supernova 1987A appear to indicate that the outer regions were much more uniformly mixed than expected, suggesting that significant shock-induced mixing had occurred (Arnett et al. 1989; Arnett 2000). Richtmyer–Meshkov instability is also of critical importance to inertial confinement fusion (ICF). In this case, the shell encapsulating the deuterium–tritium fuel becomes Richtmyer–Meshkov unstable as it is accelerated inward by the ablation of its outer surface by x-ray radiation. The degree of compression attainable in laser fusion experiments is ultimately limited by the Rayleigh–Taylor and Richtmyer–Meshkov instabilities. Thus, these instabilities represent important barriers to attaining gain (i.e., positive net energy yield) in laser fusion (Takabe et al. 1988; Dittrich et al. 1994; Li et al. 2002; Wilson et al. 2003; Rygg 2006). Richtmyer–Meshkov instability is also important in high-speed combustion applications. For example, the interaction of a shock wave with a flame front in a scramjet engine produces this instability, which can greatly enhance the fuel–oxidizer mixing rate (Waitz et al. 1993).

## Theory and modeling

Most of the modeling of the development of the Richtmyer–Meshkov instability has been focused on the growth of the bubble and spike amplitudes $$a_{b},a_{s}$$ in the single-mode sinusoidally-perturbed interface case (with post-shock amplitude $$a_{0}^{+}\ ,$$ perturbation wavelength $$\lambda\ ,$$ and perturbation wavenumber $$k=2\pi/\lambda$$), and of the mixing layer widths $$h_{b},h_{s}$$ in the multi-mode case. Widely used linear, weakly-nonlinear, potential flow, and phenomenological Richtmyer–Meshkov instability growth models were recently reviewed (Latini et al. 2007b). The evolution of shock-accelerated density-stratified interfaces has also been modeled using a vorticity deposition-evolution and coherent structure paradigm (Hawley & Zabusky 1989; Zabusky 1999).

### Governing equations

The evolution of the Richtmyer–Meshkov instability is described by the compressible, multi-species fluid dynamics equations expressed by mass, momentum, and energy conservation, $\tag{3} \frac{\partial}{\partial t}\!\left(\rho\, m_{r}\right) + \frac{\partial}{\partial x_{j}} \!\left(\rho\, m_{r}\, u_{j} + J_{j}^{r}\right) = 0\,,$ $\tag{4} \frac{\partial}{\partial t}\!\left(\rho\, u_{i}\right) + \frac{\partial}{\partial x_{j}}\!\left(\rho\, u_{i}\, u_{j}\right) = - \frac{\partial p}{\partial x_{i}} + \frac{\partial \sigma_{ij}}{\partial x_{j}}$ $\tag{5} \frac{\partial}{\partial t}\!\left(\rho\, e\right) + \frac{\partial}{\partial x_{j}}\!\left[\left(\rho\, e + p\right)u_{j}\right] = \frac{\partial}{\partial x_{j}}\!\left(\chi\,\frac{\partial T}{\partial x_{j}}\right) + \frac{\partial}{\partial x_{j}}\!\left(\sigma_{ij}\,u_{i}\right)$ where $$\rho$$ is the density, $$m_{r}$$ is the mass fraction of fluid $$r=1,2\ ,$$ $$u_{i}$$ is the velocity, $$J_{j}^{r}=\rho D_{r}\partial m_{r}/\partial x_{j}$$ is the diffusional mass flux, $$p$$ is the pressure, $$\sigma_{ij}=\mu\left[\partial u_{i}/\partial x_{j} + \partial u_{j}/\partial x_{i}-(2/3)\delta_{ij}\partial u_{k}/\partial x_{k}\right]$$ is the viscous stress tensor, and $$e = u^2/2 + p/\!\left[\rho\left(\gamma-1\right)\right]$$ is the total (kinetic plus internal) energy per unit mass (for an ideal gas). An ideal gas equation of state is typically assumed for both fluids. Averaged values of the thermodynamic parameters (such as $$\gamma$$) are often used, or some type of blending of the parameters. Most descriptions of this instability are based on the non-dissipative version of the above equations (i.e., $$\sigma_{ij} = \chi = 0$$). Equations (3)– (5) are formally the same equations describing Rayleigh–Taylor instability, except that there are no body force terms in Eqs. (4) and (5). Note that summing Eq. (3) over $$r$$ using $$\sum_{r}m_{r}=1$$ and $$\sum_{r}J_{j}^{r}=0$$ gives the usual continuity equation. In the Richtmyer–Meshkov instability, shock passage through the perturbed interface separating the two fluids initiates the instability growth, whereas in the Rayleigh–Taylor instability, a continuous acceleration initiates the growth.

### Impulsive models

Impulsive models based on representing the shock as an instantaneous $$\delta$$-function acceleration were developed by adapting existing models for the Rayleigh–Taylor instability (constant acceleration) to the case of an impulsive acceleration. These models predict linear growth in time of the perturbation amplitude that captures the early stages of the instability evolution before nonlinear saturation effects become important.

The first impulsive model proposed for the growth of a single-mode perturbation is due to Richtmyer (1960), who modified earlier work (Taylor 1950) on the growth of a small-amplitude single-mode perturbation when a light fluid is accelerated continuously into a heavier fluid [the Rayleigh–Taylor instability described by $$\mathrm{d}^{2}a/\mathrm{d}t^{2}=gAk$$ with $$a=\left(a_{b}+a_{s}\right)\!/2$$ the amplitude, $$A$$ the Atwood number, and $$g$$ the acceleration], by replacing the constant gravitational acceleration with an impulsive acceleration $$\Delta u\delta (t)$$ (where $$\Delta u$$ is the velocity jump across the interface imparted by the shock): $\tag{6} \frac{\mathrm{d}a}{\mathrm{d}t}=k\,\Delta u\,A^{+}\,a_{0}^{+}\equiv v_{0}\,,$

where $$A^{+}$$ is the post-shock Atwood number. Meyer and Blewett (1972) performed two-dimensional simulations of the single-mode Richtmyer–Meshkov instability and computed growth rates corresponding to a shock propagating from a light to a heavy gas and vice versa. Good agreement with Eq. (6) was obtained for the light-to-heavy case; however, better agreement in the heavy-to-light case than that given by (6) was obtained by averaging the pre- and post-shock amplitudes, $$\mathrm{d}a/\mathrm{d}t=k\,\Delta u\,A^{+}\left(a_{0}^{+}+a_{0}^{-}\right)\!/2.$$

Vandenboomgaerde et al. (1998) modified the Richtmyer model by replacing the impulsive acceleration, post-shock Atwood number, and post-shock amplitude with linearly time-varying values from pre- to post-shock quantities.

### Perturbation models

Perturbation models based on the asymptotic expansion of the linear perturbation equations generate asymptotic series with limited radii of convergence: the convergence can be improved using Padé approximants.

Zhang and Sohn (1997a) developed a model for the growth of a two-dimensional Richtmyer–Meshkov unstable interface from early to late times in the case of a reflected shock (light-to-heavy transition). The interface dynamics are modeled using the linear, compressible flow equations for early times and using the nonlinear, incompressible flow equations for later times. Perturbation expansions for the initial interface and velocity potentials are substituted into the equations, and differential equations are solved for terms of the same order, giving a series approximation that can be evaluated at the spike and bubble tips. The amplitude growth is given by $\tag{7} \frac{\mathrm{d}a}{\mathrm{d}t}=\left\{ 1-k^{2}\,v_{0}\,a_{0}^{+}\,t+\left[\left( A^{+}\right) ^{2}-\frac{1}{2}\right] \!\left( k\,v_{0}\,t\right)^{2}\right\} v_{0}\,,$

where $$v_{0}$$ is the Richtmyer velocity (6). Padé approximants (Zhang & Sohn 1997b) were subsequently applied to extend the range of validity of the growth rates in time. Vandenboomgaerde et al. (2002) proposed a simplified version of the Zhang–Sohn perturbation expansion, in which the growth rate is given by a $$2N$$th-degree polynomial.

### Potential flow models

Potential flow models describe the amplitude evolution through the late-time, nonlinear regime by the bubble and spike velocity ($$v_{b,s}=\mathrm{d}a_{b,s}/\mathrm{d}t$$) evolution. Layzer (1955) developed the first potential flow model to describe the Rayleigh–Taylor instability, which was subsequently extended to the Richtmyer–Meshkov instability by others. These models predict that the bubble velocity in a Richtmyer–Meshkov instability approaches zero asymptotically.

Jacobs and Sheeley (1996) developed a vortex model valid for very small Atwood numbers by assuming that the flow consists of an alternating array of line vortices placed at points midway between the crests and troughs, giving a logarithmic dependence of the amplitude and a $$1/\!\left(kt\right)$$ dependence of the velocity at late times.

Goncharov (2002) extended the two-dimensional Layzer model to the $$A^{+}\neq 1$$ case using a parabolic expansion of the perturbation near the bubble and spike tips to obtain five ordinary differential equations. The asymptotic bubble velocity is obtained by taking the $$t\rightarrow \infty$$ limit of the result: $\tag{8} v_{b}(t)=\frac{3+A^{+}}{3\left( 1+A^{+}\right) k\,t}\,.$

Sohn (2003) also extended the Layzer model to fluids with arbitrary density ratios using a simpler form for the initial potentials than Goncharov, leading to a simplified system requiring the solution of three ordinary differential equations. The asymptotic bubble velocity is given by $$v_{b}(t)=2/\!\left[\left( 2+A^{+}\right) k\,t\right]\ .$$

### Phenomenological models

The empirical Sadot et al. (1998) model is based on fits to experimental data and on asymptotic growth laws. This model was tested against experimental data and appears to be valid over $$M=1.3$$–$$3.5$$ in air and SF$$_{6}\ .$$ The model captures the initial linear growth, and the later nonlinear growth of the bubble and spike, $\tag{9} \frac{\mathrm{d}a_{b,s}}{\mathrm{d}t}=\frac{\left( 1+k\,v_{0}\,t\right) v_{0}}{1+\left( 1\pm A^{+}\right) k\,v_{0}\,t +\frac{1\pm A^{+}}{1+A^{+}}\,\frac{\left( k\,v_{0}\,t\right) ^{2}}{2\pi C}}$

[the upper (+) and lower (-) sign in $$\pm$$ denotes the bubble and spike, respectively] where $$C=1/(3\pi )$$ for $$A^{+}\geq 0.5\ .$$ In the limit $$A^{+}\rightarrow 0\ ,$$ $$C=1/(2\pi )\ .$$

### Self-similarity

There has been relatively little effort to theoretically model the late-time asymptotic growth of the Richtmyer–Meshkov mixing layer from multi-mode perturbations. The self-similar mixing layer in Richtmyer–Meshkov unstable flows is predicted to grow as $$h(t)\propto t^{\theta}$$ with $$\theta\in \left(1/3,1\right)$$ (Mikaelian 1989; Gauthier & Bonnet 1990; Alon et al. 1994, 1995). Buoyancy-drag models have been formulated to describe $$h(t)$$ (Dimonte 2000). Two-equation turbulence models can also be reduced to ordinary differential equations via the similarity transformation $$\eta\propto x/t^{\theta}\ ,$$ the solutions of which give the exponent $$\theta$$ as a function of the model parameters.

## Experiments

### Shock tube experiments

Figure 4: A sequence of PLIF images from the shock tube experiments described in Jacobs and Krivets (2005) in which a $$M = 1.3$$ shock wave accelerates an air/SF$$_6$$ interface.

Richtmyer’s linear instability model was first verified experimentally by Meshkov, who performed shock tube experiments in which a perturbed interface between a variety of gases was produced using a sinusoidally-shaped thin membrane, and the flow was visualized using Schlieren photography. Others subsequently used similar techniques to investigate stronger shocks, three-dimensional perturbations, and reacceleration by the reflected shock wave (reshock) (Aleshin et al. 1988; Vasilenko et al. 1992; Benjamin 1992; Brouillette & Sturtevant 1994; Vetter & Sturtevant 1995; Sadot et al. 1998). While a thin membrane allows the formation of a sharp interface, its use does have limitations. As the membrane is shattered by the incident shock wave, pieces of the membrane become incorporated into the flow, potentially affecting the development of the instability. In attempts to eliminate the membrane, a number of techniques have been developed to form the interface using a thin plate to separate the gases (Brouillette & Sturtevant 1994; Cavailler et al. 1990; Bonazza & Sturtevant 1996), the wake of which (when extracted prior to shock tube firing) provides a pseudo-sinusoidal interfacial perturbation. These experiments are limited by the fact that the initial perturbation is uncontrolled, nonuniform, and often unreproducible. The interfaces created by this method are also very diffuse, having thicknesses equaling or exceeding the perturbation wavelength, which significantly slows the instability growth. More recently, a technique for generating a membraneless interface between two gases using opposed gas flows was developed (Jones & Jacobs 1997; Collins & Jacobs 2002; Jacobs & Krivets 2005): this produces repeatable 2D and 3D single-mode perturbations for shock tube experiments visualized using planar laser-induced fluorescence (PLIF) and particle-image velocimetry (PIV). Early work on Richtmyer–Meshkov instability used single-mode perturbations. Work has continued using both 2D and 3D single-mode perturbations to explore the late-time regime, as well as to test nonlinear growth models. Single-mode experiments have demonstrated good agreement with Richtmyer’s model for weak shocks (allowing the incompressible assumption) and small perturbation amplitudes as long as membrane effects can be neglected. In addition, late-time single-mode experiments have yielded good agreement with the Sadot et al. (1998) model. However, late-time multimode experiments have yielded inconclusive comparisons with existing self-similar growth models.

### Impulsive experiments

In addition to shock tube experiments, the incompressible Richtmyer–Meshkov instability has also been studied by impulsively accelerating containers of incompressible fluids. Castilla and Redondo (1993) first exploited this technique by dropping tanks containing a liquid and air onto a cushioned surface. This technique was improved upon by Jacobs and Sheeley (1996) and Niederhaus and Jacobs (2003) by mounting the tank onto a rail system and then allowing it to bounce off of a fixed spring; miscible liquids were used. More recently, Dimonte and Schneider (2000) and Kucherenko et al. (1997) have adapted their Rayleigh–Taylor facilities to produce impulsive accelerations. Incompressible experiments are useful since it is inherently easier to produce sharp interfaces between liquids than between gases. In addition, visualization is made simpler in liquid phase experiments by eliminating the difficulty of visualizing gas flows. The animation below shows a PLIF visualization of a single-mode incompressible Richtmyer–Meshkov instability experiment from the work described in Niederhaus and Jacobs (2003). Media:hrun292.mov

### Laser-driven experiments

High-energy-density laser facilities (such as the NOVA laser at the Lawrence Livermore National Laboratory and the OMEGA laser at the Laboratory for Laser Energetics at the University of Rochester) have also been successfully utilized to study the Richtmyer–Meshkov instability produced by very strong shock waves in both planar (Dimonte & Remington 1993; Dimonte et al. 1995, 1996; Dimonte & Schneider 1997; Farley et al. 1999; Dimonte 1999; Aglitskiy et al. 2006) and convergent (Fincke et al. 2004, 2005; Taccetti et al. 2005) geometries. Strong shocks are possible but imaging of the instability evolution using face-on and side-on x-ray radiography is problematic. Many of the experiments performed on laser facilities have explored the role of Rayleigh–Taylor and Richtmyer–Meshkov instabilities in supernovae hydrodynamics (Kane et al. 1997, 1999) in an emerging field referred to as laboratory astrophysics (Remington et al. 2000, 2006).

## Simulations

Figure 5: Density and vorticity fields from a 2D hybrid fifth-order WENO/sixth-order central difference simulation of the Jacobs and Krivets (2005) experiment at 4 ms (early time), 5.6 ms (before reshock), and 6 ms (after reshock).

As in many other types of fluid flows, numerical simulations of the Richtmyer–Meshkov instability and its turbulent evolution have complemented experiments. In particular, simulations provide detailed data that otherwise cannot be (or is too difficult to be) measured experimentally due to diagnostic limitations. A variety of numerical methods have been applied to the singly- and multiply-shocked Richtmyer–Meshkov instability. The numerical method typically uses a shock-capturing algorithm. While the vast majority of simulations to date have solved the Euler equations, both large-eddy and Navier–Stokes simulations of the Richtmyer–Meshkov instability have also been recently performed.

Figure 6: Mass fraction (top) and enstrophy (bottom) isosurfaces from a 3D ninth-order WENO simulation of the $$M = 1.5$$ Vetter and Sturtevant (1995) experiment at 3 ms (before reshock), 3.25 ms (during reshock), 5 ms (during arrival of the reflected rarefaction), and 8 ms (late time) (Latini et al. 2007c). The enstrophy isosurfaces show the fragmentation of tubular structure into smaller, randomly-oriented structure following reshock.

### Monotone-integrated and implicit large-eddy simulations

Monotone-integrated and implicit large-eddy simulations solve the non-dissipative compressible fluid equations, relying on the discretization errors in the algorithm to provide a grid-resolution-dependent dissipation. Such methods include higher-order Godunov and total variation diminishing (TVD) methods (Holmes et al. 1995; Baltrusaitis et al. 1996; Li & Zhang 1997; Kotelnikov et al. 2000; Glimm et al. 2002), the Van Leer method (Youngs 1994, 2007; Thornber 2007), and the piecewise-parabolic method (Cohen et al. 2002; Zabusky et al. 2003; Peng et al. 2003). Comparisons of the predictions of various methods have also been presented (Holmes et al. 1999). More recently, formally higher order methods such as the fifth- and ninth-order weighted essentially non-oscillatory (WENO) methods were applied to simulate the Collins and Jacobs (2002) single-mode, $$M=1.2$$ air(acetone)/SF$$_{6}$$ Richtmyer–Meshkov instability experiment (Latini et al. 2007a,b) and the $$M = 1.3$$ Jacobs and Krivets (2005) experiment (Latini 2007). The physics of reshock was also investigated using this numerical method (Schilling et al. 2007). Hybrid methods combining upwinding in regions near discontinuities and less dissipative central difference methods in smooth flow regions using multiresolution analysis have been developed to reduce the numerical dissipation in pure upwinding methods (Costa & Don 2007).

### Large-eddy simulations

Large-eddy simulations solve the formally filtered large-scale fluid equations with explicit subgrid-scale models for the unclosed correlations. Hybrid upwind/central difference methods were extended to adaptive grid large-eddy simulations of the Richtmyer–Meshkov instability (Hill et al. 2006; Pantano et al. 2007).

### Navier–Stokes simulations

A multicomponent form of the dissipative compressible fluid dynamics equations was solved to simulate the interaction of a shock wave with a rectangular block of SF$$_{6}$$ in a shock tube experiment with $$M=1.26$$ (Bates et al. 2007).

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