Fundamental Statistical Descriptions of Plasma Turbulence in Magnetic Fields

The self-similar evolution to turbulence of a multi-mode Rayleigh-Taylor mix at small density differences (A t ∼ 7.5 x 10 -4 ), is investigated through particle image velocimetry (PIV), and high-resolution thermocouple measurements. The density difference has been achieved through a temperature difference in the fluid. Cold fluid enters above the hot in a closed channel to form an unstable interface. This buoyancy-driven mixing experiment allows for long data collection times, short transients, and is statistically steady. First-, second-, and third-order statistics with spectra of velocity and temperature fields are presented. Analysis of the measurements has shed light on the structure of mixing as it develops to a self-similar regime in this flow. The onset of self-similarity is marked by the development of a self-preserving form of the temperature spectra, and the collapse of velocity profiles expressed in self-similar units

Experimental investigation of Rayleigh–Taylor mixing at small Atwood numbers

We review the theoretical developments in the field of Rayleigh-Taylor instabilities and turbulent mixing, discuss what is known and what is not known about the phenomenon, and outline the features of similarity of the turbulent mixing process. Based on the physical intuition and on the results of rigorous theoretical studies, we put forward some new ideas on how to grasp the essentials of the mixing process and consider the influence of momentum transport on the invariants and on scaling and statistical properties of the unsteady turbulent mixing.

Review of theoretical modelling approaches of Rayleigh-Taylor instabilities and turbulent mixing.

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 turbulence model is developed to described the self-similar growth of the Rayleigh-Taylor (RT) and Richtmyer-Meshkov (RM) instabilities. The model describes the dominant eddies in the mixing zone with evolutionary equations for their characteristic dimension L and energy per unit mass K≡V2∕2. The equations are based on the successful buoyancy-drag models for RT and RM flows, but constructed only with local parameters so that it can be applied to multidimensional flows with multiple shells of materials. The model has several unknown coefficients that are determined by comparing analytical and numerical solutions with RT and RM experiments.

K-L turbulence model for the self-similar growth of the Rayleigh-Taylor and Richtmyer-Meshkov instabilities

This study gives an updated account of our current ability to describe multimaterial compressible turbulent flows by means of a one-point transport model. Evolution equations are developed for a number of second-order correlations of turbulent data, and approximations of the gradient type are applied to additional correlations to close the system of equations. The principal fields of interest are the one- point Reynolds tensor for variable-density flow, the turbulent energy dissipation rate, and correlations for density-velocity and density- density fluctuations. This single-field description of turbulent flows is compared in some detail to two-field flow equations for nonturbulent, highly dispersed flow with separate variables for each field. This comparison suggests means for improved modeling of some correlations not subjected to evolution equations.

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Turbulence transport equations for variable-density turbulence and their relationship to two-field models

The spectrum of inhomogeneous turbulence is modeled by an approach that is not limited to regimes of large Reynolds numbers or small mean-flow strain rates. In its simplest form and applied to incompressible flow, the model depends on five phenomenological constants defining the strength of turbulence coupling to mean flow, turbulence transport in physical and wave-number space, and mixing of stress-tensor components. The implications for homogeneous isotropic turbulence are investigated in detail and found to correspond well to the conclusions from more fundamental theories. Under appropriate limiting conditions, a turbulent system described by the model will relax over time into a state of approximate spectral equilibrium permitting a reduction to a “one-point” model for the system that is substantially like the familiar K-e model. This yields preliminary estimates of the present model's parameters and points to the way to improved modeling of flows beyond the applicability of the K-e method.

Spectral transport model for turbulence

In this study, a turbulence model is developed for flows containing large density gradients but moving with small Mach number In such cases, the turbulence initiates in regions with the steepest spatial variations in mean density This model generalizes standard turbulence models and is comprised of transport equations for second-order correlations constructed from fluctuating components of density, velocity, and internal energy From these, instability-driven source terms for turbulence arise naturally We also exhibit the effect of nonhomogeneities in the flow on the level of turbulence Many applications of this model can be envisioned, including meteorology, the study of stratified flow, and investigation of volcanic plumes formed during eruptions 21 refs, 3 figs

Conservation and transport properties of turbulence with large density variations

Crenulative turbulence is a nonlinear extension of the Bell-Plesset instability, usually observed in a converging system in which there is a nonhomogeneous response of stress to strain and/or strain rate. In general, crenelation occurs in any circumstance in which the mean flow streamlines converge the material more strongly than the compressibility can accommodate. Elements of the material slip past each other, resulting in local fluctuations in velocity from that of the mean flow, producing a type of turbulence that is more kinematic than inertial. For a homogeneous material, crenelation occurs at the atomic or molecular scale. With nonhomogeneous stress response at larger scales, the crenulative process can also occur at those larger scales. The results are manifested by a decrease in the rate of dissipation to heat, and by the configurationally-irreversible mixing of nonhomogeneities across any mean-flow-transported interface. We obtain a mathematical description of the crenulative process by means of Reynolds decomposition of the appropriate variables, and the derivation of transport equations for the second-order moments that arise in the mean-flow momentum and energy equations. The theory is illustrated by application to the spherical convergence of an incompressible fluid with nonhomogeneous distribution of kinematic viscosity.

https://digital.library.unt.edu/ark:/67531/metadc685325/m2/1/high_res_d/3545.pdf

R. M. Rauenzahn

Papers

Turbulence transport equations for variable-density turbulence and their relationship to two-field models

Spectral transport model for turbulence

Conservation and transport properties of turbulence with large density variations

Spectral transport model for turbulence

Crenulative Turbulence in a Converging Nonhomogeneous Material