View source for Richtmyer-Meshkov instability and re-accelerated inhomogeneous flows

<strong><nowiki>Richtmyer–Meshkov instability</nowiki></strong> 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  <figref>RM_shockinterface.png</figref>.
[[Image:RM_shockinterface.png|thumb|300px|right|RM_shockinterface|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.
[[Image:vorticitydeposition.png|thumb|300px|right|vorticitydeposition|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 <math>\mathbf{\omega}=\mathbf{\nabla}\times\mathbf{v}\ .</math> 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
:<math>\label{vorticity equation}
\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}\,,
</math>

<!--
+\nu \,\nabla^{2}\mathbf{\omega }-
\frac{\nu }{\rho }\,\mathbf{\nabla \!}\rho \times \left( \nabla^{2}\mathbf{v}+
\frac{1}{3}\,\mathbf{\nabla \nabla \cdot v}\right),
</math>
-->
<!-- (Mügler & Gauthier 2000) -->
where <math>\mathrm{D}/\mathrm{D}t=\partial/\partial t+\mathbf{v}\cdot{\mathbf \nabla}\ .</math> 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  <figref>Vorticitydeposition.png</figref>. This distribution would then result in increasing amplitude in time, as well as the formation of the mushroom structures shown in  <figref>RM_shockinterface.png</figref>. 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 <math>M</math> 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
[[Image:2D_reshock.jpg|thumb|360px|right|2D_reshock|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.]]


 

:<math>\label{Atwood number}
A = \frac{\rho_1 - \rho_2}{\rho_1 + \rho_2}
</math>

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  <figref>2D_reshock.jpg</figref>) by a second shock reflected from the endwall of a shock tube, disordered small-scale structure forms (see  <figref>Hybrid2d_density.jpg</figref>).


===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 <math>a_{b},a_{s}</math> in the single-mode sinusoidally-perturbed interface case (with post-shock amplitude <math>a_{0}^{+}\ ,</math> perturbation wavelength <math>\lambda\ ,</math> and perturbation wavenumber <math>k=2\pi/\lambda</math>), and of the mixing layer widths <math>h_{b},h_{s}</math> 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).
<!--
Shock polar analysis was also used to derive an analytic expression for the circulation deposited per unit length of the unshocked planar interface (inclined at an angle to the shock) both within and beyond the regular refraction regime, for fast-slow interactions (Samtaney & Zabusky 1993, 1994).
-->

===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,
:<math>\label{mass}
\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\,,
</math>
 <!-- this comment be removed after conversion to new site -->
:<math>\label{momentum}
\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}} 
</math>
 <!-- this comment be removed after conversion to new site -->
:<math>\label{energy}
\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) 
</math>
 <!-- this comment be removed after conversion to new site -->
where <math>\rho</math> is the density, <math>m_{r}</math> is the mass fraction of fluid <math>r=1,2\ ,</math> <math>u_{i}</math> is the velocity, <math>J_{j}^{r}=\rho D_{r}\partial m_{r}/\partial x_{j}</math> is the diffusional mass flux, <math>p</math> is the pressure, <math>\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]</math> is the viscous stress tensor, and
<math>e = u^2/2 + p/\!\left[\rho\left(\gamma-1\right)\right]</math> 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 <math>\gamma</math>) 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., <math>\sigma_{ij} = \chi = 0</math>). Equations \eqref{mass}– \eqref{energy} are formally the same equations describing Rayleigh–Taylor instability, except that there are no body force terms in Eqs. \eqref{momentum} and \eqref{energy}. Note that summing Eq. \eqref{mass} over <math>r</math> using <math>\sum_{r}m_{r}=1</math> and <math>\sum_{r}J_{j}^{r}=0</math> 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 <math>\delta</math>-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 Richtmyer model==== 
* The Richtmyer model -->
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 <math>\mathrm{d}^{2}a/\mathrm{d}t^{2}=gAk</math> with <math>a=\left(a_{b}+a_{s}\right)\!/2</math> the amplitude, <math>A</math> the Atwood number, and <math>g</math> the acceleration], by replacing the constant gravitational acceleration with an impulsive acceleration <math>\Delta u\delta (t)</math> (where <math>\Delta u</math> is the velocity jump across the interface imparted by the shock):
:<math>\label{Richtmyer model}
\frac{\mathrm{d}a}{\mathrm{d}t}=k\,\Delta u\,A^{+}\,a_{0}^{+}\equiv v_{0}\,,
</math>

where <math>A^{+}</math> is the post-shock Atwood number.
<!-- ====The Meyer–Blewett model==== 
* The Meyer–Blewett model
-->
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. 
\eqref{Richtmyer model} was obtained for the light-to-heavy case; however, better agreement in the heavy-to-light case than that given by \eqref{Richtmyer model} was obtained by averaging the pre- and post-shock amplitudes, <math>\mathrm{d}a/\mathrm{d}t=k\,\Delta u\,A^{+}\left(a_{0}^{+}+a_{0}^{-}\right)\!/2.</math>
<!--
:<math>
\frac{\mathrm{d}a}{\mathrm{d}t}=k\,\Delta u\,A^{+}\left(a_{0}^{+}+a_{0}^{-}\right)/2\,.
</math>
-->

<!-- ====The Vandenboomgaerde–Mügler–Gauthier model====
* The Vandenboomgaerde–Mügler–Gauthier model -->
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.
<!--
, obtaining
:<math>
\frac{\mathrm{d}a}{\mathrm{d}t}=k\,\Delta u\left[ \frac{A^{+}\,a_{0}^{+}+A^{-}\,a_{0}^{-}}{2}
-\frac{\left( A^{+}-A^{-}\right) \!\left(a_{0}^{+}-a_{0}^{-}\right) }{6}\right] .
</math>
-->
<!--
<math>\mathrm{d}a/\mathrm{d}t=k\,\Delta u\left[ \left(A^{+}\,a_{0}^{+}+A^{-}\,a_{0}^{-}\right)\!/2
-\left( A^{+}-A^{-}\right) \!\left(a_{0}^{+}-a_{0}^{-}\right)\!/6\right].</math>
-->

===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.

<!-- ====The Zhang–Sohn model====
* The Zhang–Sohn model  -->
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 
:<math>\label{Zhang-Sohn model}
\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}\,, 
</math>

where <math>v_{0}</math> is the Richtmyer velocity \eqref{Richtmyer model}. Padé approximants (Zhang & Sohn 1997b) were subsequently applied to extend the range of validity of the growth rates in time.
<!-- ====The Vandenboomgaerde–Gauthier–Mügler model====
* The Vandenboomgaerde–Gauthier–Mügler model -->
Vandenboomgaerde et al. (2002) proposed a simplified version of the Zhang–Sohn perturbation expansion, in which the growth rate is given by a <math>2N</math>th-degree polynomial.
<!--
<math>\mathrm{d}a/\mathrm{d}t=a_{0}^{+}\sigma \sum_{n=0}^{N}\left(2\,n+1\right)\!
\left( k\,a_{0}^{+}\,\sigma \,t\right)^{2n}\,P_{2n+1}(A^{+})\ ,</math>
:<math>
\frac{\mathrm{d}a}{\mathrm{d}t}=a_{0}^{+}\sigma \sum_{n=0}^{N}\left(2\,n+1\right)
\left( k\,a_{0}^{+}\,\sigma \,t\right)^{2n}\,P_{2n+1}(A^{+})\,, 
</math>
where <math>\sigma \equiv \frac{k\,\Delta u}{2}\left( A^{+}+\frac{A^{-}}{1-\Delta u/u_{\mathrm{\mathrm{shock}}}}\right)</math> and <math>P_{2n+1}</math> are a series approximation in <math>A^{+}\ .</math>
-->

===Potential flow models===

Potential flow models describe the amplitude evolution through the late-time, nonlinear regime by the bubble and spike velocity (<math>v_{b,s}=\mathrm{d}a_{b,s}/\mathrm{d}t</math>) 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.

<!-- ====The Jacobs–Sheeley model==== 
* The Jacobs–Sheeley model -->
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 <math>1/\!\left(kt\right)</math> dependence of the velocity at late times.
<!--
:<math>
\frac{\mathrm{d}a}{\mathrm{d}t}=\frac{v_{0}}{\sqrt{\left[ k\,v_{0}\,t+\frac{\pi }{2}\sinh \left( k\,v_{0}\right) \right] ^{2}+\frac{\pi ^{2}}{4}}}.
</math>
-->

<!-- ====The Goncharov model====
* The Goncharov model -->
Goncharov (2002) extended the two-dimensional Layzer model to the <math>A^{+}\neq 1</math> 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 <math>t\rightarrow \infty</math> limit of the result: 
:<math>\label{Goncharov model}
v_{b}(t)=\frac{3+A^{+}}{3\left( 1+A^{+}\right) k\,t}\,. 
</math>


<!-- ====The Sohn models==== 
* The Sohn models -->
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 <math> v_{b}(t)=2/\!\left[\left( 2+A^{+}\right) k\,t\right]\ .</math>
<!--
:<math>
v_{b}(t)=\frac{2}{\left( 2+A^{+}\right) k\,t}\,.
</math>
Sohn (2004) also extended the Zufiria (1988) model to arbitrary Atwood numbers.
the bubble velocity is 
:<math>
v_{b}(t)=\left[ \frac{3+A^{+}}{3\left( 1+A^{+}\right) }-\frac{1}{q}+\frac{2\,A^{+}}{3\left( 1+A^{+}\right) q^{2}}\right]
\frac{1}{k\,t}\,,
</math>
where <math>q=q(A^{+})</math> is the root of the cubic polynomial
<math>\left(3-A^{+}\right)q^{3}-\left( 21+9A^{+}\right)q^{2}+\left( 3+15A^{+}\right)q-4A^{+}=0
\ .</math>
-->

===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 <math>M=1.3</math>–<math>3.5</math> in air and SF<math>_{6}\ .</math> The model 
captures the initial linear growth, and the later nonlinear growth of the bubble and spike, 
:<math>\label{Sadot model}
\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}}  
</math>

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

===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 <math>h(t)\propto t^{\theta}</math> with <math>\theta\in \left(1/3,1\right)</math> (Mikaelian 1989; Gauthier & Bonnet 1990; Alon et al. 1994, 1995). Buoyancy-drag models have been formulated to describe <math>h(t)</math> (Dimonte 2000). Two-equation turbulence models can also be reduced to ordinary differential equations via the similarity transformation <math>\eta\propto x/t^{\theta}\ ,</math> the solutions of which give the exponent <math>\theta</math> as a function of the model parameters.

==Experiments==

===Shock tube experiments===

[[Image:Montage4.jpg|thumb|500px|right|montage|A sequence of PLIF images from the shock tube experiments described in Jacobs and Krivets (2005) in which a <math>M = 1.3</math> shock wave accelerates an air/SF<math>_6</math> 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.
<!-- The presence of the membrane fragments in the flow also impedes visualization. -->
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.
<!-- In addition to these single mode experiments, attempts have been made to study broadband multimode perturbations by performing experiments initiated with an apparently flat interface: it is believed that the instability that evolves is seeded with a multi-mode random initial perturbation. However, because this perturbation is too small to be measured, this assumption remains unverified. Nevertheless, these experiments are thought to be more applicable to naturally occurring phenomena. -->
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.
<!-- 
Richtmyer–Meshkov instability as defined above involves the interaction of a planar shock wave with a perturbed planar interface. However, interactions of nonplanar shocks and/or nonplanar interfaces are often included in the general topic. For example, many of the applications for Richtmyer–Meshkov instability involve cylindrical or spherical geometries in which a converging shock impinges upon a cylindrical or spherical interface. As a result there have been a number of experimental efforts devoted to these geometries (Haas & Sturtevant 1987; Jacobs 1993). In addition, there has been a large amount of work done at the Los Alamos National Laboratory on the study of the Richtmyer–Meshkov instability of two closely spaced interfaces using a 
‘’gas curtain’’ (Jacobs et al. 1993, 1995; Rightley et al. 1997; Prestridge et al. 2000). The interaction of planar shocks with cylindrical or spherical inhomogeneities has also been classified as a variation of Richtmyer–Meshkov instability.
-->

===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).
<!-- The second animation shows a 2D simulation of this experiment (Mueschke et al. 2005). -->
<!-- [[Media:IRMI.avi]] -->
[[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==

[[Image:hybrid2d_density.jpg|thumb|500px|right|hybrid2d_density|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.
[[Image:WENO3D_isosurface.jpg|thumb|500px|right|WENO3D_isosurface|Mass fraction (top) and enstrophy (bottom) isosurfaces from a 3D ninth-order WENO simulation of the <math>M = 1.5</math> 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, <math>M=1.2</math> air(acetone)/SF<math>_{6}</math> Richtmyer–Meshkov instability experiment (Latini et al. 2007a,b) and the <math>M = 1.3</math> 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===

<!-- [[Image:Hybrid2D_density.png|thumb|250px|right|Figlabel|Density field from a 2D Navier–Stokes simulation of the Jacobs and Krivets (2005) experiment using a hybrid fifth-order WENO/sixth-order central difference method.]]
-->
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<math>_{6}</math> in a shock tube experiment with <math>M=1.26</math> (Bates et al. 2007).


==References==

* Aglitskiy, Y., Metzler, N., Karasik, M., Serlin, V., Velikovich, A. L., Obenschain, S. P., Mostovych, A. N., Schmitt, A. J., Weaver, J., Gardner, J. H. & Walsh, T. 2006 Perturbation evolution started by Richtmyer–Meshkov instability in planar laser targets. ''Physics of Plasmas'' '''13''', 080703-1–080703-4.
* Aleshin, A. N., Gamalli, E. G., Zaitsev, S. G., Lazareva, E. V., Lebo, I. G. & Rozanov, V. B. 1988 Nonlinear and transitional stages in the onset of the Richtmyer–Meshkov instability. ''Soviet Technical Physics Letters'' '''14''', 466–468.
* Alon, U., Hecht, J., Mukamel, D. & Shvarts, D. 1994 Scale invariant mixing rates of hydrodynamically unstable interfaces. ''Physical Review Letters'' '''72''', 2867–2870.
* Alon, U., Hecht, J., Ofer, D. & Shvarts, D. 1995 Power laws and similarity of Rayleigh–Taylor and Richtmyer–Meshkov mixing fronts at all density ratios. ''Physical Review Letters'' '''74''', 534–537.
* Arnett, D. 2000 The role of mixing in astrophysics. ''Astrophysical Journal Supplement'' '''127''', 213–217.
* Arnett, W. D., Fryxell, B. A. & Müller, E. 1989 Instabilities and nonradial motion in SN-1987A. ''Astrophysical Journal'' '''341''', L63–L66.
* Baltrusaitis, R. M., Gittings, M. L., Weaver, R. P., Benjamin, R. F. & Budzinski, J. M. 1996 Simulation of shock-generated instabilities. ''Physics of Fluids'' '''8''', 2471–2483.
* Bates, K. R., Nikiforakis, N. & Holder, D. 2007 Richtmyer–Meshkov instability induced by the interaction of a shock wave with a rectangular block of SF<math>_{6}\ .</math> ''Physics of Fluids'' '''19''', 036101-1–036101-16.
* Benjamin, R. F. 1992 Experimental Observations of Shock Stability and Shock-Induced Turbulence. ''Advances in Compressible Turbulent Mixing'', edited by W. P. Dannevik, A. C. Buckingham, and C. E. Leith, 341–348, Lawrence Livermore National Laboratory Report Conf-8810234.
* Bonazza, R. & Sturtevant, B. 1996 X-ray measurements of growth rates at a gas interface accelerated by shock waves. ''Physics of Fluids'' '''8''', 2496–2512.
* Brouillette, M. 2002 The Richtmyer–Meshkov instability. ''Annual Review of Fluid Mechanics'' '''34''', 445–468.
* Brouillette, M. & Sturtevant, B. 1994 Experiments on the Richtmyer–Meshkov instability: single-scale perturbations on a continuous interface. ''Journal of Fluid Mechanics'' '''263''', 271–292.
* Castilla, R. & Redondo, J. M. 1994 Mixing Front Growth in RT and RM Instabilities. ''Proceedings of the Fourth International Workshop on the Physics of Compressible Turbulent Mixing'', Cambridge, United Kingdom, edited by P. F. Linden, D. L. Youngs, and S. B. Dalziel, 11–31.
* Cavailler, C., Mercier, P., Rodriguez, G. & Haas, J. F. 1990 A new vertical shock tube for Rayleigh–Taylor instability measurements. ''Proceedings of the Seventeenth ISSWST'', edited by Y. Kim, 564, American Institute of Physics.
* Cohen, R. H., Dannevik, W. P., Dimits, A. M., Eliason, D. E., Mirin, A. A., Zhou, Y., Porter, D. H. & Woodward, P. R. 2002 Three-dimensional simulation of a Richtmyer–Meshkov instability with a two-scale initial perturbation. ''Physics of Fluids'' '''14''', 3692–3709.
* Collins, B. D. & Jacobs, J. W. 2002 PLIF flow visualization and measurements of the Richtmyer–Meshkov instability of an air/SF<math>_{6}</math> interface. ''Journal of Fluid Mechanics'' '''464''', 113–136.
* Costa, B. & Don, W. S. 2007 High order hybrid central–WENO finite difference scheme for conservation laws. ''Journal of Computational and Applied Mathematics'' '''204''', 209–218.
* Dimonte, G. 1999 Nonlinear evolution of the Rayleigh–Taylor and Richtmyer–Meshkov instabilities. ''Physics of Plasmas'' '''6''', 2009–2015.
* Dimonte, G. 2000 Spanwise homogeneous buoyancy-drag model for Rayleigh–Taylor mixing and experimental evaluation. ''Physics of Plasmas'' '''7''', 2255–2269.
* Dimonte, G., Frerking, C. E. & Schneider, M. 1995 Richtmyer–Meshkov instability in the turbulent regime. ''Physical Review Letters'' '''74''', 4855–4858.
* Dimonte, G., Frerking, C. E., Schneider, M. & Remington, B. 1996 Richtmyer–Meshkov instability with strong radiatively driven shocks. ''Physics of Plasmas'' '''3''', 614–630.
* Dimonte, G. & Remington, B. 1993 Richtmyer–Meshkov experiments on the Nova laser at high compression. ''Physical Review Letters'' '''70''', 1806–1809.
* Dimonte, G. & Schneider, M. 1997 Turbulent Richtmyer–Meshkov instability experiments with strong radiatively driven shocks. ''Physics of Plasmas'' '''4''', 4347–4357.
* Dimonte, G. & Schneider, M. 2000 Density ratio dependence of Rayleigh–Taylor mixing for sustained and impulsive acceleration histories. ''Physics of Fluids'' '''12''', 304–321.
* Dittrich, T. R., Hammel, B. A., Keane, C. J., McEachern, R., Turner, R. E., Haan, S. W. & Suter, L. J. 1994 Diagnosis of pusher-fuel mix in indirectly driven Nova implosions. ''Physical Review Letters'' '''73''', 2324–2327.
* Farley, D. R., Peyser, T. A., Logory, L. M., Murray, S. D. & Burke, E. W. 1999 High Mach number mix instability experiments on an unstable density interface using a single-mode, nonlinear initial perturbation. ''Physics of Plasmas'' '''6''', 4304–4317.
* Fincke, J. R., Lanier, N. E., Batha, S. H., Hueckstaedt, R. M., Magelssen, G. R., Rothman, S. D., Parker, K. W. & Horsfield, C. J. 2004 Postponement of saturation of the Richtmyer–Meshkov instability in a convergent geometry. ''Physical Review Letters'' '''93''', 115003-1–115003-4.
* Fincke, J. R., Lanier, N. E., Batha, S. H., Hueckstaedt, R. M., Magelssen, G. R., Rothman, S. D., Parker, K. W. & Horsfield, C. J. 2005 Effect of convergence on growth of the Richtmyer–Meshkov instability. ''Laser and Particle Beams'' '''23''', 21–25.
* Gauthier, S. & Bonnet, M. 1990 A <math>k</math>-<math>\varepsilon</math> model for turbulent mixing in shock-tube flows induced by Rayleigh–Taylor instability. ''Physics of Fluids A'' '''2''', 1685–1694.
* Glimm, J., Grove, J. W., Zhang, Y. & Dutta, S. 2002 Numerical study of axisymmetric Richtmyer–Meshkov instability and azimuthal effect of spherical mixing. ''Journal of Statistical Physics'' '''107''', 241–260.
* Goncharov, V. N. 2002 Analytical model of nonlinear, single-mode, classical Rayleigh–Taylor instability at arbitrary Atwood numbers. ''Physical Review Letters'' '''88''', 134502-1–134502-4.
* Hawley, J. F. & Zabusky, N. J. 1989 Vortex paradigm for shock-accelerated density-stratified interfaces. ''Physical Review Letters'' '''63''', 1241–1244.
* Hill, D. J., Pantano, C. & Pullin, D. I. 2006 Large-eddy simulation and multiscale modelling of a Richtmyer–Meshkov instability with reshock. ''Journal of Fluid Mechanics'' '''557''', 29–61.
* Holmes, R. L., Dimonte, G., Fryxell, B., Gittings, M. L., Grove, J. W., Schneider, M., Sharp, D. H., Velikovich, A. L., Weaver, R. P. & Zhang, Q. 1999 Richtmyer–Meshkov instability growth: experiment, simulation and theory. ''Journal of Fluid Mechanics'' '''389''', 55–79.
* Holmes, R. L., Grove, J. W. & Sharp, D. H. 1995 Numerical investigation of Richtmyer–Meshkov instability using front-tracking. ''Journal of Fluid Mechanics'' '''301''', 51–64.
* Jacobs, J. W. & Krivets, V. V. 2005 Experiments on the late-time development of single-mode Richtmyer–Meshkov instability. ''Physics of Fluids'' '''17''', 034105-1–034105-10.
* Jacobs, J. W. & Sheeley, J. M. 1996 Experimental study of incompressible Richtmyer–Meshkov instability. ''Physics of Fluids'' '''8''', 405–415.
* Jones, M. A. & Jacobs, J. W. 1997 A membraneless experiment for the study of Richtmyer–Meshkov instability of a shock-accelerated gas interface. ''Physics of Fluids'' '''9''', 3078–3085.
* Kane, J., Arnett, D., Remington, B. A., Glendinning, S. G., Bazan, G., Drake, R. P., Fryxell, B. A., Teyssier, R. & Moore, K. 1999 Scaling supernova hydrodynamics to the laboratory. ''Physics of Plasmas'' '''6''', 2065–2071.
* Kane, J., Arnett, D., Remington, B. A., Glendinning, S. G., Castor, J., Wallace, R., Rubenchik, A. & Fryxell, B. A. 1997 Supernova-relevant hydrodynamic instability experiments on the NOVA laser. ''Astrophysical Journal'' '''478''', L75–L78.
* Kotelnikov, A. D., Ray, J. & Zabusky, N. J. 2000 Vortex morphologies on reaccelerated interfaces: Visualization, quantification and modeling of one- and two-mode compressible and incompressible environments. ''Physics of Fluids'' '''12''', 3245–3264.
* Kucherenko, Yu. A., Balabin, S. I., Cheret, R. & Haas, J.-F. 1997 Experimental investigation into inertial properties of Rayleigh–Taylor turbulence. ''Laser and Particle Beams'' '''15''', 25–31.
* Latini, M. 2007 Simulation and analysis of two- and three-dimensional single-mode Richtmyer–Meshkov instability using weighted essentially non-oscillatory and vortex methods. Ph.D. thesis, California Institute of Technology.
* Latini, M., Schilling, O. & Don, W. S. 2007a Effects of WENO flux reconstruction order and spatial resolution on reshocked two-dimensional Richtmyer–Meshkov instability. ''Journal of Computational Physics'' '''221''', 805–836.
* Latini, M., Schilling, O. & Don, W. S. 2007b High-resolution simulations and modeling of reshocked single-mode Richtmyer–Meshkov instability: comparison to experimental data and to amplitude growth model predictions. ''Physics of Fluids'' '''19''', 024104-1–024104-19.
* Latini, M., Schilling, O. & Don, W. S. 2007c Richtmyer–Meshkov Instability-Induced Mixing: Initial Conditions Modeling, Three-Dimensional Simulations and Comparisons to Experiment. ''Proceedings of the Tenth International Workshop on the Physics of Compressible Turbulent Mixing'', edited by M. Legrand and M. Vandenboomgaerde, 349–352.
* Layzer, D. 1955 On the instability of superposed fluids in a gravitational field. ''Astrophysical Journal'' '''122''', 1–12.
* Li, C. K., Séguin, F. H., Frenje, J. A., Kurebayashi, S., Petrasso, R. D., Meyerhofer, D. D., Soures, J. M., Delettrez, J. A., Glebov, V. Yu., Radha, P. B., Regan, S. P., Roberts, S., Sangster, T. C. & Stoeckl, C. 2002 Effects of fuel-shell mix upon direct-drive, spherical implosions on OMEGA. ''Physical Review Letters'' '''89''', 165002-1–165002-4.
* Li, X. L. & Zhang, Q. 1997 A comparative numerical study of the Richtmyer–Meshkov instability with nonlinear analysis in two and three dimensions. ''Physics of Fluids'' '''9''', 3069–3077.
* Meshkov, E. E. 1969 Instability of the interface of two gases accelerated by a shock wave. ''Fluid Dynamics'' '''4''', 101–104.
* Meyer, K. A. & Blewett, P. J. 1972 Numerical investigation of the stability of a shock-accelerated interface between two fluids. ''Physics of Fluids'' '''15''', 753–759.
* Mikaelian, K. O. 1989 Turbulent mixing generated by Rayleigh–Taylor and Richtmyer–Meshkov instabilities. ''Physica D'' '''36''', 343–357.
<!-- * Mueschke, N. J., Kraft, W. N., Dubois, O., Andrews, M. J. & Jacobs, J. W. 2005 Numerical Investigation of Single-Mode Richtmyer–Meshkov Instability. FEDSM2005-77189, ''Proceedings of the 2005 ASME Fluids Engineering Division Summer Meeting and Exhibition'', 19–23 June 2005, Houston, TX. -->
<!-- * Mügler, C. & Gauthier, S. 2000 Two-dimensional Navier–Stokes simulations of gaseous mixtures induced by Richtmyer–Meshkov instability. ''Physics of Fluids'' '''12''', 1783–1798. -->
* Niederhaus, C. E. & Jacobs, J. W. 2003 Experimental study of the Richtmyer–Meshkov instability of incompressible fluids. ''Journal of Fluid Mechanics'' '''485''', 243–277.
* Pantano, C., Deiterding, R., Hill, D. J. & Pullin, D. I. 2007 A low numerical dissipation patch-based adaptive mesh refinement method for large-eddy simulation of compressible flows. ''Journal of Computational Physics'' '''221''', 63–87.
* Peng, G., Zabusky, N. J. & Zhang, S. 2003 Vortex-accelerated secondary baroclinic vorticity deposition and late-intermediate time dynamics of a two-dimensional Richtmyer–Meshkov interface. ''Physics of Fluids'' '''15''', 3730–3744.
* Remington, B. A., Drake, R. P. & Ryutov, D. D. 2006 Experimental astrophysics with high power lasers and <math>Z</math> pinches. ''Reviews of Modern Physics'' '''78''', 755–807.
* Remington, B. A., Drake, R. P., Takabe, H. & Arnett, D. 2000 A review of astrophysics experiments on intense lasers. ''Physics of Plasmas'' '''7''', 1641–1652.
* Richtmyer, R. D. 1960 Taylor instability in shock acceleration of compressible fluids. ''Communications on Pure and Applied Mathematics'' '''13''', 297–319.
* Rygg, J. R. 2006 Shock convergence and mix dynamics in inertial confinement fusion. Ph.D. thesis, Massachusetts Institute of Technology.
* Sadot, O., Erez, L., Alon, U., Oron, D. & Levin, L. A. 1998 Study of nonlinear evolution of single-mode and two bubble interaction under Richtmyer–Meshkov instability. ''Physical Review Letters'' '''80''', 1654–1657.
<!-- * Samtaney, R. & Zabusky, N. J. 1993 On shock polar analysis and analytical expressions for vorticity deposition in shock-accelerated density-stratified interfaces. ''Physics of Fluids A'' '''5''', 1285–1287.
*Samtaney, R. & Zabusky, N. J. 1994 Circulation deposition on shock-accelerated planar and curved density-stratified interfaces: models and scaling laws. ''Journal of Fluid Mechanics'' '''269''', 45–78.
-->
* Schilling, O., Latini, M. & Don, W. S. 2007 Physics of reshock and mixing in single-mode Richtmyer–Meshkov instability. ''Physical Review E'' '''76''', 026319-1–026319-28.
* Sohn, S.-I. 2003 Simple potential-flow model of Rayleigh–Taylor and Richtmyer–Meshkov instabilities for all density ratios. ''Physical Review E'' '''67''', 026301-1–026301-5.
<!-- * Sohn, S.-I. 2004 Density dependence of a Zufiria-type model for Rayleigh–Taylor bubble fronts. ''Physical Review E'' '''70''', 045301-1–045301-4.
-->
* Taccetti, J. M., Batha, S. H., Fincke, J. R., Delamater, N. D., Lanier, N. E., Magelssen, G. R., Hueckstaedt, R. M., Rothman, S. D., Horsfield, C. J. & Parker, K. W. 2005 Richtmyer–Meshkov instability reshock experiments using laser-driven double-cylinder implosions. ''Astrophysics and Space Science'' '''298''', 327–331.
* Takabe, H., Yamanaka, M., Mima, K., Yamanaka, C., Azechi, H., Miyanaga, N., Nakatsuka, M., Jitsuno, T., Norimatsu, T., Takagi, M., Nishimura, H., Nakai, M., Yabe, T., Sasaki, T., Yoshida, K., Nishihara, K., Kato, Y., Izawa, Y., Yamanaka, T. & Nakai, S. 1988 Scalings of implosion experiments for high neutron yield. ''Physics of Fluids'' '''31''', 2884–2893.
* Taylor, G. I. 1950 The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. ''Proceedings of the Royal Society of London A'' '''201''', 192–196.
* Thornber, B. 2007 Implicit large eddy simulation for unsteady multi-component compressible turbulent flows. Ph.D. thesis, Cranfield University.
* Vandenboomgaerde, M., Gauthier, S. & Mügler, C. 2002 Nonlinear regime of a multimode Richtmyer–Meshkov instability: A simplified perturbation theory. ''Physics of Fluids'' '''14''', 1111–1122.
* Vandenboomgaerde, M., Mügler, C. & Gauthier, S. 1998 Impulsive model for the Richtmyer–Meshkov instability. ''Physical Review E'' '''58''', 1874–1881.
* Vasilenko, A. M., Buryakov, O. V., Kuropatenko, V. F., Olkhovskaya, V. I., Ratnikov, V. P. & Jakovlev, V. G. 1992 Experimental Research of Gravitational Instability and Turbulization of Flow at the Noble Gases Interface. ''Advances in Compressible Turbulent Mixing'', edited by W. P. Dannevik, A. C. Buckingham, and C. E. Leith, 581–606, Lawrence Livermore National Laboratory Report Conf-8810234.
* Vetter, M. & Sturtevant, B. 1995 Experiments on the Richtmyer–Meshkov instability of an air/SF<math>_{6}</math> interface. ''Shock Waves'' '''4''', 247–252.
* Waitz, I. A., Marble, F. E. & Zukoski, E. E. 1993 Investigation of a contoured wall injector for hypervelocity mixing augmentation. ''AIAA Journal'' '''31''', 1014–1021.
* Wilson, D. C., Scannapieco, A. J., Cranfill, C. W., Clover, M. R., Hoffman, N. M. & Collins, J. 2003 Degradation of radiatively driven inertial confinement fusion capsule implosions by multifluid interpenetration mixing. ''Physics of Plasmas'' '''10''', 4427–4434.
* Youngs, D. L. 1994 Numerical simulation of mixing by Rayleigh–Taylor and Richtmyer–Meshkov instabilities. ''Laser and Particle Beams'' '''12''', 725–750.
* Youngs, D. L. 2007 Rayleigh–Taylor and Richtmyer–Meshkov mixing. ''Implicit Large Eddy Simulation: Computing Turbulent Fluid Dynamics'', edited by F. F. Grinstein, L. G. Margolin, and W. J. Rider, 392–409, Cambridge University Press.
* Zabusky, N. J. 1999 Vortex paradigm for accelerated inhomogeneous flows: visiometrics for the Rayleigh–Taylor and Richtmyer–Meshkov environments. ''Annual Review of Fluid Mechanics'' '''31''', 495–536.
* Zabusky, N. J., Kotelnikov, A. D., Gulak, Y. & Peng, G. 2003 Amplitude growth rate of a Richtmyer–Meshkov unstable two-dimensional interface to intermediate times. ''Journal of Fluid Mechanics'' '''475''', 147–162.
* Zhang, Q. & Sohn, S.-I. 1997a Nonlinear theory of unstable fluid mixing driven by shock wave. ''Physics of Fluids'' '''9''', 1106–1124.
* Zhang, Q. & Sohn, S.-I. 1997b Padé approximation to an interfacial fluid mixing problem. ''Applied Mathematics Letters'' '''10''', 121–127.
<!-- * Zufiria, J. A. 1988 Bubble competition in Rayleigh Taylor instability. ''Physics of Fluids'' '''31''', 440–446.
-->


<!-- INSTRUCTIONS

Write section titles in "Sentence case", i.e., only the first word capitalized.

Throughout your article, ''use Italica to emphasize'' points. 

Use boldface '''to provide definitions'''. 

===Subsection b===

Use Latex for all your equations, e.g., <math>F=ma\ ,</math> or (on a separate line)
:<math>\label{mylabel}
F=ma
\ .</math>


===Subsection c===

Refer to figures and equations as <figref>Hybrid2D_density.png</figref> and Eq. \eqref{mylabel}.

==Section d==

   Avoid starting paragraphs with spaces, since they will be shown in a box with no line breaks. 
   This is useful though to provide code

===Subsection e===

[[Media:RTbw.avi]],

===Subsection f===

Provide the list of the most essential references. Keep it short. Cite references like (Author et al. 1999) in the article.

* Researcher1 A.B. and Researcher2 C.D. (1955) Title of the article. Journal, 111:123-145 

* Researcher3 A.B. and Researcher4 C.D. (1957) Title of the article. Journal, 112:231-251 


Provide a list of recent books that you recommend to read to learn more about your topic. It would help the editor if you could send him 
the ISBN of the books so that he could make direct links to them.

* Author F.N. (2007) Title of the book. Publisher

* Author2 F.N. (2007) Title of the book. Publisher
-->

==Recommended reading==

* Atzeni, S. & Meyer-ter-Vehn, J. 2004 ''The Physics of Inertial Fusion: Beam Plasma Interaction, Hydrodynamics, Hot Dense Matter'', International Series of Monographs on Physics Vol. 125, Oxford University Press.
* Eliezer, S. 2002 ''The Interaction of High Power Lasers with Plasmas'', Series in Plasma Physics Vol. 11, Institute of Physics Press.
* Nakai, S. & Takabe, H. 1996 Principles of inertial confinement fusion–physics of implosion and the concept of inertial fusion energy. ''Reports on Progress in Physics'' '''59''', 1071–1131.
* Pfalzner, S. 2006 ''An Introduction to Inertial Confinement Fusion'', Series in Plasma Physics, CRC Press.

==External links==

<!-- make links to your websites, one link per author. -->

[http://en.wikipedia.org/wiki/Richtmyer-Meshkov_instability Wikipedia article]

[http://www.ame.arizona.edu/research/fluidlab/RMISTE.htm Richtmyer–Meshkov instability experiments at the University of Arizona]

[http://www.ame.arizona.edu/research/fluidlab/IRMIE.htm Incompressible Richtmyer–Meshkov instability experiments at the University of Arizona]

[http://silver.neep.wisc.edu/~shock/ Wisconsin Shock Tube Laboratory at the University of Wisconsin]

[https://www.llnl.gov/str/April03/Miller.html Simulation at LLNL of an incompressible Richtmyer–Meshkov instability experiment]

<!-- Provide an alphabetical list of links to other relevant articles in Scholarpedia. -->

==See also==

[[Rayleigh-Taylor instability]], [[Inertial confinement fusion]], [[Turbulence]], [[Type Ia supernovae]] 


[[Category: Hydrodynamic instabilities]]
[[Category:Fluid Dynamics]]
[[Category:Multiple_Curators]]