In diverse areas of science and technology, including inertial confinement fusion (ICF), astrophysics, geophysics, and engineering processes, turbulent mixing induced by hydrodynamic instabilities is of scientific interest as well as practical significance. Because of the fundamental roles they often play in ICF and other applications, three classes of hydrodynamic instability-induced turbulent flows—those arising from the Rayleigh-Taylor, Richtmyer-Meshkov, and Kelvin-Helmholtz instabilities—have attracted much attention. ICF implosions, supernova explosions, and other applications illustrate that these phases of instability growth do not occur in isolation, but instead are connected so that growth in one phase feeds through to initiate growth in a later phase. Essentially, a description of these flows must encompass both the temporal and spatial evolution of the flows from their inception. Hydrodynamic instability will usually start from potentially infinitesimal spatial perturbations, will eventually transition to a turbulent flow, and then will reach a final state of a true multiscale problem. Indeed, this change in the spatial scales can be vast, with hydrodynamic instability evolving from just a few microns to thousands of kilometers in geophysical or astrophysical problems. These instabilities will evolve through different stages before transitioning to turbulence, experiencing linear, weakly, and highly nonlinear states. The challenges confronted by researchers are enormous. The inherent difficulties include characterizing the initial conditions of such flows and accurately predicting the transitional flows. Of course, fully developed turbulence, a focus of many studies because of its major impact on the mixing process, is a notoriously difficult problem in its own right. In this pedagogical review, we will survey challenges and progress, and also discuss outstanding issues and future directions.

Turbulent mixing and transition criteria of flows induced by hydrodynamic instabilities

A theoretical framework to study linear and nonlinear Richtmyer-Meshkov instability (RMI) is presented. This instability typically develops when an incident shock crosses a corrugated material interface separating two fluids with different thermodynamic properties. Because the contact surface is rippled, the transmitted and reflected wavefronts are also corrugated, and some circulation is generated at the material boundary. The velocity circulation is progressively modified by the sound wave field radiated by the wavefronts, and ripple growth at the contact surface reaches a constant asymptotic normal velocity when the shocks/rarefactions are distant enough. The instability growth is driven by two effects: an initial deposition of velocity circulation at the material interface by the corrugated shock fronts and its subsequent variation in time due to the sonic field of pressure perturbations radiated by the deformed shocks. First, an exact analytical model to determine the asymptotic linear growth rate is presented and its dependence on the governing parameters is briefly discussed. Instabilities referred to as RM-like, driven by localized non-uniform vorticity, also exist; they are either initially deposited or supplied by external sources. Ablative RMI and its stabilization mechanisms are discussed as an example. When the ripple amplitude increases and becomes comparable to the perturbation wavelength, the instability enters the nonlinear phase and the perturbation velocity starts to decrease. An analytical model to describe this second stage of instability evolution is presented within the limit of incompressible and irrotational fluids, based on the dynamics of the contact surface circulation. RMI in solids and liquids is also presented via molecular dynamics simulations for planar and cylindrical geometries, where we show the generation of vorticity even in viscid materials.

Richtmyer-Meshkov instability: theory of linear and nonlinear evolution

Theoretical and heuristic modelling methods are reviewed for studying turbulence phenomena of fluids and plasmas. Emphasis is placed on understanding of effects on turbulence characteristics due to inhomogeneities of field and plasma parameters. The similarity and dissimilarity between the methods for fluids and plasmas are sought in order to shed light on the properties that are shared or not by fluid and plasma turbulence.

/pdf/turbulence-theories-and-modelling-of-fluids-and-plasmas-2aianpjwlj.pdf

Turbulence theories and modelling of fluids and plasmas

In diverse areas of science and technology, including inertial confinement fusion (ICF), astrophysics, geophysics, and engineering processes, turbulent mixing induced by hydrodynamic instabilities is of scientific interest as well as practical significance. Because of the fundamental roles they often play in ICF and other applications, three classes of hydrodynamic instability-induced turbulent flows—those arising from the Rayleigh-Taylor, Richtmyer-Meshkov, and Kelvin-Helmholtz instabilities—have attracted much attention. ICF implosions, supernova explosions, and other applications illustrate that these phases of instability growth do not occur in isolation, but instead are connected so that growth in one phase feeds through to initiate growth in a later phase. Essentially, a description of these flows must encompass both the temporal and spatial evolution of the flows from their inception. Hydrodynamic instability will usually start from potentially infinitesimal spatial perturbations, will eventually transition to a turbulent flow, and then will reach a final state of a true multiscale problem. Indeed, this change in the spatial scales can be vast, with hydrodynamic instability evolving from just a few microns to thousands of kilometers in geophysical or astrophysical problems. These instabilities will evolve through different stages before transitioning to turbulence, experiencing linear, weakly, and highly nonlinear states. The challenges confronted by researchers are enormous. The inherent difficulties include characterizing the initial conditions of such flows and accurately predicting the transitional flows. Of course, fully developed turbulence, a focus of many studies because of its major impact on the mixing process, is a notoriously difficult problem in its own right. In this pedagogical review, we will survey challenges and progress, and also discuss outstanding issues and future directions.In diverse areas of science and technology, including inertial confinement fusion (ICF), astrophysics, geophysics, and engineering processes, turbulent mixing induced by hydrodynamic instabilities is of scientific interest as well as practical significance. Because of the fundamental roles they often play in ICF and other applications, three classes of hydrodynamic instability-induced turbulent flows—those arising from the Rayleigh-Taylor, Richtmyer-Meshkov, and Kelvin-Helmholtz instabilities—have attracted much attention. ICF implosions, supernova explosions, and other applications illustrate that these phases of instability growth do not occur in isolation, but instead are connected so that growth in one phase feeds through to initiate growth in a later phase. Essentially, a description of these flows must encompass both the temporal and spatial evolution of the flows from their inception. Hydrodynamic instability will usually start from potentially infinitesimal spatial perturbations, will eventually t...

Turbulent Mixing and Transition Criteria of Flows Induced by Hydrodynamic Instabilities

The classical Richtmyer‐Meshkov ~RM! instability develops when a planar shock wave interacts with a corrugated interface between two different fluids. A larger family of so-called RM-like hydrodynamic interfacial instabilities is discussed. All of these feature a perturbation growth at an interface, which is driven mainly by vorticity, either initially deposited at the interface or supplied by external sources. The inertial confinement fusion relevant physical conditions that give rise to the RM-like instabilities range from the early-time phase of conventional ablative laser acceleration to collisions of plasma shells ~like components of nested-wire-arrays, double-gas-puff Z-pinch loads, supernovae ejecta and interstellar gas!. In the laser ablation case, numerous additional factors are involved: the mass flow through the front, thermal conduction in the corona, and an external perturbation drive ~laser imprint!, which leads to a full stabilization of perturbation growth. In contrast with the classical RM case, mass perturbations can exhibit decaying oscillations rather than a linear growth. It is shown how the early-time perturbation behavior could be controlled by tailoring the density profile of a laser target or a Z-pinch load, to diminish the total mass perturbation seed for the Rayleigh‐Taylor instability development. © 2000 American Institute of Physics. @S1070-664X~00!93305-6#

Richtmyer–Meshkov-like instabilities and early-time perturbation growth in laser targets and Z-pinch loads

Stability of a uniform contact surface is investigated in the case when a nonuniform shock wave passes through the surface. The nonuniform shock is generated by a rippled piston that moves with constant velocity. The amplitude of the shock oscillates and decreases as it propagates. A uniform contact surface is found to be unstable after the nonuniform shock passes across it. The growth rate depends sensitively on the phase of the oscillating shock wave at the time when the shock hits the contact surface. The physical mechanism of the instability is qualitatively discussed. The linear and nonlinear evolutions of the instability are studied. In particular, the dependence of the linear case on the Atwood number for a weak shock is investigated. Properties of this stability are found to exhibit differences from those of the standard Richtmyer-Meshkov instability in both the linear and nonlinear cases. {copyright} {ital 1996 The American Physical Society.}

Instability of a contact surface driven by a nonuniform shock wave

The Lyapunov exponents and instantaneous expansion rates in a phase space of Coulomb many-body systems are measured with the use of a three-dimensional particle code SCOPE [K. Nishihara, Kakuyugo Kenkyu 66, 253 (1991)]. The code calculates particle dynamics determined by Coulomb forces among individual particles. The Lyapunov exponents normalized by plasma frequency are found to be proportional to ${\mathrm{\ensuremath{\Gamma}}}^{\mathrm{\ensuremath{-}}2\mathrm{/}5}$ in the range of 1\ensuremath{\leqslant}\ensuremath{\Gamma}\ensuremath{\leqslant}160, where \ensuremath{\Gamma} is the Coulomb coupling constant of the ion one-component plasma. There is a large jump of the Lyapunov exponent near \ensuremath{\Gamma}\ensuremath{\sim}170, which corresponds to the phase transition from the liquid to the solid state in the one-component plasma. In the solid state, the normalized Lyapunov exponents are proportional to ${\mathrm{\ensuremath{\Gamma}}}^{\mathrm{\ensuremath{-}}6\mathrm{/}5}$ for 170300. The observed dependence is discussed in analogy to a rigid-body particle system and a weakly nonlinear lattice system for liquid and solid states, respectively. Diffusion coefficients are found to be proportional to the third power of the Lyapunov exponent in the liquid state, that is, for 1\ensuremath{\leqslant}\ensuremath{\Gamma}\ensuremath{\leqslant}160. These results imply that the Lyapunov exponent is in close relation to the transport processes. The instantaneous expansion rate starts from a small value and increases rapidly to a large peak value before declining slowly towards an asymptotic value. This stage is called the Lyapunov transient stage. Products of the transient time and the Lyapunov exponent are found to be 1.5--2. Information of the initial state is lost after the transient time. The chaotic behavior of the instantaneous expansion rate is also shown.

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Y. Ueshima

Papers

Instability of a contact surface driven by a nonuniform shock wave

Particle simulation of lyapunov exponents in one-component strongly coupled plasmas