Richtmyer-Meshkov Mixing Research Articles

Simulations of three-layer Richtmyer–Meshkov mixing in a shock tube

The Richtmyer–Meshkov instability causes perturbations to grow after a shock traverses a fluid density interface. This increases the mixing rate between fluid from either side of the interface. We use the Flash Eulerian hydrodynamic code to investigate alterations when a thin third layer of intermediate density is placed along the interface, effectively creating two adjacent unstable interfaces. This is a common occurrence in engineering applications where a thin barrier initially separates two materials. We find that the width of the mixing layer is similar or slightly reduced; however, the total mass of mixed material can actually increase. The mixing layer becomes more compact and efficient. However, the normalized mixed mass decreases, meaning that finger entrainment becomes more important than in the simple two-layer case. The effect of adding the central layer appears to decrease when the Atwood number is decreased. The Flash results are also benchmarked against two-layer experimental data from a shock tube at the University of Arizona.

Self-similar Reynolds-averaged mechanical–scalar turbulence models for reshocked Richtmyer–Meshkov instability-induced mixing in the small Atwood number limit

Analytical self-similar solutions to two-, three-, and four-equation Reynolds-averaged mechanical–scalar turbulence models describing incompressible turbulent Richtmyer–Meshkov instability-induced mixing in planar geometry derived in the small Atwood number limit [O. Schilling, “Self-similar Reynolds-averaged mechanical–scalar turbulence models for Rayleigh–Taylor, Richtmyer–Meshkov, and Kelvin–Helmholtz instability-induced mixing in the small Atwood number limit,” Phys. Fluids 33, 085129 (2021)] are extended to construct models for reshocked Richtmyer–Meshkov mixing. The models are based on the turbulent kinetic energy K and its dissipation rate ε, together with the scalar variance S and its dissipation rate χ modeled either differentially or algebraically. The three- and four-equation models allow for a simultaneous description of mechanical and scalar mixing, i.e., mixing layer growth and molecular mixing. Mixing layer growth parameters and other physical observables were obtained explicitly as functions of the model coefficients and were used to calibrate the model coefficients. Here, the solutions for the singly shocked Richtmyer–Meshkov case for the mixing layer width and the turbulent fields are used to construct piecewise-continuous generalizations of these quantities for times after reshock. For generality, the post-reshock mixing layer width is not assumed to grow with the same power-law as the pre-reshock width, and an impulsive approximation applied to Rayleigh–Taylor instability growth is used to establish the expression for the post-reshock width. A four-equation model is then used to illustrate the spatiotemporal behavior of the mean and turbulent fields and late-time turbulent equation budgets across the mixing layer. The reference solutions derived here can provide systematic calibrations and better understanding of mechanical–scalar turbulence models and their predictions for reshocked Richtmyer–Meshkov instability-induced turbulent mixing in the very large Reynolds number limit.

Advection versus diffusion in Richtmyer-Meshkov mixing

The Richtmyer-Meshkov (RM) instability is one of the most severe degradation mechanisms for inertial confinement fusion (ICF), and mitigating it has been a priority for the global ICF effort. In this Letter, the instability's ability to atomically mix is linked to its background decay of residual turbulent energy. We show how recently derived inequalities from the mathematical theory of PDEs constrain the evolution. A model RM process at leading order may diffusively mix or retain imprints of its initial structures indefinitely, depending on initial conditions, and there exists a theoretical range of zero-mixing for certain values of parameters. The results may apply to other systems resembling scalar transport in decaying turbulence.

Predicting different turbulent mixing problems with the same k–ɛ model and model coefficients

In the foreseeable future, the Reynolds-averaged Navier–Stokes (RANS) method will remain the most viable approach for simulating practical flows. The RANS method mainly comprises two parts: a physical model and model coefficients. Hitherto, although researchers have made great effort to improve them, there is still much room for improving the satisfaction of RANS prediction. Given an available but possibly non-accurate physical model, is it possible to achieve a satisfactory RANS prediction? Our recent work on determination of k–L mixing model coefficients [Zhang et al., J. Fluid Mech. 905, A26 (2020)] sheds light on this problem. In that study, a satisfactory RANS prediction is achieved by systematically setting the values of model coefficients to match the given model. In this paper, we further extend this methodology to the widely used k–ɛ model. Consequently, with the same k–ɛ model and the same model coefficients, we successfully predict different turbulent mixing problems induced by various interface instabilities, which broadly occur in both nature and engineering. The investigated problems include both basic mixing problems, e.g., classical Rayleigh–Taylor (RT), Richtmyer–Meshkov (RM), and Kelvin–Helmholtz mixing, and complex mixing problems, e.g., reshocked RM mixing, RT mixing with initial tilted interface, and spherical implosion mixing. The successful predictions of the k–ɛ model, as well as those of the k–L model, indicate that the idea mentioned above may be applied to other RANS models and bring hope for predicting engineering problems with the same physical model and the same model coefficients.

Growth mechanism of interfacial fluid-mixing width induced by successive nonlinear wave interactions.

Interfacial fluid mixing induced by successive waves, such as shock, rarefaction, and compression waves, plays a fundamental role in engineering applications, e.g., inertial confinement fusion, and in natural phenomena, e.g., supernova explosion. These waves bring nonuniform, unsteady external forces into the mixing zone, which leads to a complex mixing process. The growth rate of the mixing width is analyzed by decomposing the turbulent flow field into the averaged field and the fluctuating counterpart. The growth rate is thus divided into three parts: (i) the stretching or compression (S(C)) effect induced by the averaged-velocity difference between two ends of the mixing zone, (ii) the penetration effect induced by the fluctuations which represent the penetration of the two species into each other, and (iii) the diffusive effect, which is induced by the molecular diffusion and is negligible in high-Reynolds-number flows at Schmidt number of order unity. The penetration effect is further divided into the Richtmyer-Meshkov (RM) effect, which is induced by fluctuations that were deposited by earlier wave interactions, and the Rayleigh-Taylor (RT) effect, which is caused by the fluctuations that arise in an overall acceleration of the mixing zone. During the passage of the rarefaction waves, the mixing zone is stretched, while during the passage of the compression waves or shock waves, the mixing zone is compressed. To illustrate these effects, a physical model of RM mixing with reshock is used. By combining the S(C), RM, and RT effects, the entire evolution of mixing width is restructured, which agrees well with numerical simulations for problems with a wide range of density ratios.

A new idea to predict reshocked Richtmyer–Meshkov mixing: constrained large-eddy simulation

Abstract

Numerical study and buoyancy–drag modeling of bubble and spike distances in three-dimensional spherical implosions

High-resolution three-dimensional implicit large eddy simulations of implosion in spherical geometries are presented. The growth of perturbations is due to Rayleigh–Taylor (RT) and Richtmyer–Meshkov (RM) instabilities and also to geometric convergence and compression effects. RM and RT instabilities have been studied extensively in planar configurations, but there are comparatively few studies on spherical geometries. Planar geometries lack the effect of convergence that changes the morphology and growth of perturbations in spherical geometries. This paper presents a study of turbulent mixing in spherical geometries considering different narrowband (NB) and broadband multimode initial perturbations and examines several quantities including the evolution of the integral mixing layer width and integral bubble and spike heights using novel integral definitions. The growth of the bubble and spike is modeled using a Buoyancy–Drag (BD) approach that is based on simple ordinary differential equations to model the growth of the turbulent mixing layer. In a recent study, Youngs and Thornber [“Buoyancy-drag modelling of bubble and spike distances for single-shock Richtmyer-Meshkov mixing,” Physica D 410, 132517 (2020)] constructed modifications to the BD equations to take into account the early stages of the mixing process that are dependent on the initial conditions. Those modifications are shown to be important to obtain correct results. The current study adopted the same modifications and adapted the BD equations to the spherical implosion case. The results of the BD model are compared with those of different initial NB cases that include different initial amplitudes and wavelengths of the perturbations, for validation purposes. The predictions from the new BD model are in very good agreement with the numerical results; however, there exist some limitations in the accuracy of the model, in particular the use of the interface position and fluid velocity from one-dimensional data.

Early Time Modifications to the Buoyancy-Drag Model for Richtmyer–Meshkov Mixing

Abstract The Buoyancy-Drag model is a simple model, based on ordinary differential equations, for estimating the growth in the width of a turbulent mixing zone at an interface between fluids of different densities due to Richtmyer–Meshkov and Rayleigh–Taylor instabilities. The model is calibrated to give the required self-similar behavior for mixing in simple situations. However, the early stages of the mixing process are very dependent on the initial conditions and modifications to the Buoyancy-Drag model are then needed to obtain correct results. In a recent paper, Thornber et al. (2017, “Late-Time Growth Rate, Mixing, and Anisotropy in the Multimode Narrowband Richtmyer–Meshkov Instability: The θ-Group Collaboration,” Phys. Fluids, 29, p. 105107), a range of three-dimensional simulation techniques was used to calculate the evolution of the mixing zone integral width due to single-shock Richtmyer–Meshkov mixing from narrowband initial random perturbations. Further analysis of the results of these simulations gives greater insight into the transition from the initial linear behavior to late-time self-similar mixing and provides a way of modifying the Buoyancy-Drag model to treat the initial conditions accurately. Higher-resolution simulations are used to calculate the early time behavior more accurately and compare with a multimode model based on the impulsive linear theory. The analysis of the iLES data also gives a new method for estimating the growth exponent, θ (mixing zone width ∼ tθ), which is suitable for simulations which do not fully reach the self-similar state. The estimates of θ are consistent with the theoretical model of Elbaz and Shvarts (2018, “Modal Model Mean Field Self-Similar Solutions to the Asymptotic Evolution of Rayleigh-Taylor and Richtmyer-Meshkov Instabilities and Its Dependence on the Initial Conditions,” Phys. Plasmas, 25, p. 062126).

Modeling of turbulent mixing with an improved K–L model

Turbulent mixing, induced by Rayleigh–Taylor (RT), Richtmyer–Meshkov (RM), and Kelvin–Helmholtz (KH) instabilities, broadly occurs in both natural phenomena, such as supernova explosions, and engineering applications, such as inertial confinement fusion (ICF). These three instabilities usually simultaneously exist and are highly coupled to drive and affect turbulent mixing, which raises a great challenge for turbulence modeling. In this study, an improved version of the K–L model is proposed. The modifications include that: (i) the deviatoric shear stress is considered to describe the KH instability; (ii) the concept of characteristic acceleration is introduced to better distinguish RT and RM instabilities; and (iii) an enthalpy diffusion is directly derived from the internal energy equation to model the turbulent diffusion term. Then, a unified set of model coefficients is systematically derived based on the self-similar analysis and physical observations. This model is validated by canonical RT, RM, and KH mixings and further investigated for more complex cases, including the RM mixing with multiple reshocks, the two-dimensional RT mixing called “tilted-rig,” and the simple spherical implosion, a much simplified version of an ICF implosion. Good agreement with the corresponding experimental and numerical data is achieved, revealing the ability of the present model to describe combined buoyancy, shock, and shear effects, which will contribute to a further application in real problems.

Two-equation and multi-fluid turbulence models for Richtmyer–Meshkov mixing

This paper concerns an investigation of two different approaches in modeling the turbulent mixing induced by the Richtmyer–Meshkov instability (RMI): A two-equation K-L multi-component Reynolds-averaged Navier–Stokes model and a two-fluid model. We have improved the accuracy of the K-L model by implementing new modifications, including a realizability condition for the Reynolds stress tensor and a threshold in the production of the turbulence kinetic energy. We examine the models in the one-dimensional (1D) form in the (re)-shocked mixing of a double-planar air and sulfur-hexafluoride (SF6) interface of the Atwood number |At| ≃ 0.6853. Furthermore, we investigated the models’ accuracy to RMI-induced mixing of a (re)-shocked planar-inverse chevron air–SF6 interface. Relevant integral quantities in time, as well as instantaneous profiles and contour plots, are used to assess the models’ accuracy against high-resolution implicit large eddy simulations. The proposed modifications improve the efficiency of the K-L model. The model is designed as a simple model capable of capturing the self-similar growth of Rayleigh–Taylor and Richtmyer–Meshkov flows. The two-fluid model remains more accurate but is also computationally more expensive.

Buoyancy–Drag modelling of bubble and spike distances for single-shock Richtmyer–Meshkov mixing

The Buoyancy–Drag model is a simple model, based on ordinary differential equations, for estimating the growth of a turbulent mixing zone at an interface between fluids of different density due to Richtmyer–Meshkov and Rayleigh–Taylor instabilities. The early stages of the mixing process are very dependent on the initial conditions and modifications to the Buoyancy–Drag model are needed to obtain correct results. In a recent paper, Youngs & Thornber (2019), the results of three-dimensional turbulent mixing simulations were used to construct the modifications required to represent the evolution of the overall width of the mixing zone due to single-shock Richtmyer–Meshkov mixing evolving from narrowband initial random perturbations. The present paper extends this analysis to give separate equations for the bubble and spike distances (the depths to which the mixing zone penetrates the dense and light fluids). The data analysis depends on novel integral definitions of the bubble and spike distances which vary smoothly with time. Results are presented for two pre-shock density ratios, ρ1∕ρ2=3and20. New insights are given for the variation of asymmetry of the mixing zone with time. At early time, values of the spike-to-bubble distance are very high. The asymmetry greatly reduces as mixing proceeds towards a self-similar state. For the overall (integral) mixing width, W, the Buoyancy–Drag model gives satisfactory results at both density ratios using the same parameters. However, for the bubble and spike distances the behaviour is very different at the two density ratios. The method used to analyse the data provides a new way of estimating the self-similar growth exponent θ (W∼tθ). The values obtained are approximate because of the difficulty in running the three dimensional simulations far enough into the self-similar regime. Estimates of θ are consistent with the theoretical value of 1/3 given by the model of Elbaz & Shvarts (2018). The corrected form of the Buoyancy–Drag model gives accurate fits to the data for W,hbandhsover the whole time range with θ =1/3 for ρ1∕ρ2=3 and θ=0.35 for ρ1∕ρ2=20.

Unified prediction of reshocked Richtmyer–Meshkov mixing with K-L model

Hydrodynamic instabilities, including Rayleigh–Taylor, Richtmyer–Meshkov (RM), and Kelvin–Helmholtz, induced turbulent mixing broadly occur in both natural phenomena, such as supernova explosions, and high-energy-density applications, such as inertial confinement fusion. Reshocked RM mixing is the most fundamental physical process that is closely related to practical problems, as it involves three classical instabilities. In complex applications, the Reynolds-averaged Navier–Stokes model analysis continues to play a major role. However, there are very few turbulence models that facilitate unified predictions of the outcome of reshocked RM mixing experiments under different physical conditions. Thus, we aim to achieve this objective using the K-L model based on three considerations: deviatoric shear stress is considered when constructing Reynolds stress tensor; the model coefficients used are derived based on a new systematic procedure; the performance of different numerical schemes are studied to ensure high resolution but basically no numerical oscillation. Consequently, a unified prediction is obtained for the first time for a series of reshocked RM mixing experiments under incident shock Mach numbers Ma = 1.2–1.98, Atwood numbers At = ±0.67, and test section lengths 8 cm ≤ δ ≤ 110 cm. The results reveal the feasibility of demonstrating different reshocked RM processes using a single model, without adjusting the model coefficients, which sheds light on the further application of the present model to practical engineering, such as inertial confinement fusion.

Supernova, nuclear synthesis, fluid instabilities, and interfacial mixing

Supernovae and their remnants are a central problem in astrophysics due to their role in the stellar evolution and nuclear synthesis. A supernova’s explosion is driven by a blast wave causing the development of Rayleigh–Taylor and Richtmyer–Meshkov instabilities and leading to intensive interfacial mixing of materials of a progenitor star. Rayleigh–Taylor and Richtmyer–Meshkov mixing breaks spherical symmetry of a star and provides conditions for synthesis of heavy mass elements in addition to light mass elements synthesized in the star before its explosion. By focusing on hydrodynamic aspects of the problem, we apply group theory analysis to identify the properties of Rayleigh–Taylor and Richtmyer–Meshkov dynamics with variable acceleration, discover subdiffusive character of the blast wave-induced interfacial mixing, and reveal the mechanism of energy accumulation and transport at small scales in supernovae.

Single-Interface Richtmyer–Meshkov Turbulent Mixing at the Los Alamos Vertical Shock Tube

Mach number and initial conditions effects on Richtmyer–Meshkov (RM) mixing are studied by the vertical shock tube (VST) at Los Alamos National Laboratory (LANL). At the VST, a perturbed stable light-to-heavy (air–SF6, A = 0.64) interface is impulsively accelerated with a shock wave to induce RM mixing. We investigate changes to both large and small scales of mixing caused by changing the incident Mach number (Ma = 1.3 and 1.45) and the three-dimensional (3D) perturbations on the interface. Simultaneous density (quantitative planar laser-induced fluorescence (PLIF)) and velocity (particle image velocimetry (PIV)) measurements are used to characterize preshock initial conditions and the dynamic shocked interface. Initial conditions and fluid properties are characterized before shock. Using two types of dynamic measurements, time series (N = 5 realizations at ten locations) and statistics (N = 100 realizations at a single location) of the density and velocity fields, we calculate several mixing quantities. Mix width, density-specific volume correlations, density–vorticity correlations, vorticity, enstrophy, strain, and instantaneous dissipation rate are examined at one downstream location. Results indicate that large-scale mixing, such as the mix width, is strongly dependent on Mach number, whereas small scales are strongly influenced by initial conditions. The enstrophy and strain show focused mixing activity in the spike regions.

Special Issue on the 13th International Workshop on the Physics of Compressible Turbulent Mixing

The study of compressible turbulent mixing associated with Richtmyer-Meshkov (RM), Rayleigh-Taylor (RT), and Kelvin-Helmholtz (KH) instabilities is motivated by diverse applications in science and engineering, including supersonic combustion, detonation, instability of collapsing gas bubbles, stratified flows in geophysical applications, chemical engineering, inertial confinement fusion (ICF), supernovae, and molecular clouds. Further, the interaction of shock waves with materials is also of interest in biomedical applications, such as fragmentation of cancer cells during shockwave chemotherapy and cavitation damage to human tissues during lithotripsy. In many of these applications, the Reynolds number is very high and the instabilities rapidly lead to turbulent mixing. In the case of ICF, which is regarded as a promising approach to controlled thermonuclear fusion: (1) these instabilities lead to the growth of perturbations on the interfaces within the capsules; (2) perturbations grow into the nonlinear regime by mode coupling and eventually cause mixing of materials; and (3) material mixing inhibits thermonuclear burning of the fuel.There are ongoing research efforts at centers in the USA, UK, France, Russia, and China to design improved shock tube experiments in order to further elucidate RM instability and better understand turbulent mixing across material interfaces. Novel experiments are also being performed to further investigate RT and KH instabilities, as well as combinations of these instabilities. There are, however, a number of difficulties associated with experiments, including separating the initial gas regions, the interpretation of visualization images and challenges with diagnostics, and the inability to measure certain quantities of interest either because of their intrinsic nature (e.g., vorticity) or because of practical limitations in the experimental configurations. Computational, theoretical, and modeling studies have been widely conducted in parallel with experiments to complement the data obtained from the experiments, help design and interpret results from experiments, develop reduced model descriptions (such as Reynolds-averaged and buoyancy-drag models), and validate numerical methods and codes by comparisons to experimental data. Numerical simulation techniques applied to these instability and mixing problems include direct numerical simulation (DNS), large eddy simulation (LES), and monotone or implicit LES (ILES).The 13th International Workshop on the Physics of Compressible Turbulent Mixing (IWPCTM13) was organized in the UK by Cranfield University at Woburn Abbey (July 15–20). The conference was cohosted by the Atomic Weapons Establishment, with additional sponsorship from the Commissariat à l’Énergie Atomique et aux Énergies Alternatives, Los Alamos National Laboratory, and Lawrence Livermore National Laboratory. The conference was attended by more than 70 scientists from the UK, USA, Russia, France, China, India, and Spain. Experimental, theoretical, and computational studies of mixing processes applied to a range of problems were discussed at the workshop. The IWPCTM13 offered an excellent opportunity to an international group of delegates from national laboratories, institutes, and universities to review the state-of-the-art research on compressible turbulent mixing and to stimulate the discussion of future research in this field (for more information on this series of workshops and for proceedings from previous workshops, see http://www.iwpctm.org). This special issue of the Journal of Fluids Engineering publishes peer-reviewed papers originally presented at the IWPCTM13.Both experimental and computational papers are included. Experimental studies concern laser-driven hydrodynamic experiments; comparisons of tilted rocket rig experiments with DNS, ILES, and turbulence model predictions; and shock tube experiments. The computational studies focus on LES and Reynolds-averaged turbulence modeling of RT and RM mixing in fluids and solids. The results showed that significant progress has been made with regard to simulation and modeling techniques for turbulent mixing. However, further research is required to assess the accuracy and efficiency of different turbulence models against numerical simulation and experimental data in order to reduce the modeling uncertainty in applications such as ICF. A review of phase separation, microchannel devices is also presented. Finally, the papers discussed at the conference included applications of advanced computational methods and models in astrophysics.The guest editors of this special issue would like to thank all of the authors for their efforts in preparing their contributions as well as all of the reviewers for their assistance in reviewing these papers. Finally, we would like to thank the Journal of Fluids Engineering Editor, Malcolm Andrews, the Editorial Office, Amber Grady-Fuller, and the Production Coordinator at the American Society of Mechanical Engineers for their encouragement and assistance during the course of this special issue publication.

Incident shock Mach number effects on Richtmyer-Meshkov mixing in a heavy gas layer

Experiments were performed at the horizontal shock tube facility at Los Alamos National Laboratory to study the effect of incident shock Mach number (M) on the development of Richtmyer-Meshkov instability after a shock wave impulsively accelerates a varicose-perturbed, heavy-gas curtain. Three cases of incident shock strength were experimentally investigated: M = 1.21, 1.36, and 1.50. We discuss the state of the mixing and the mechanisms that drive the mixing at both large and small scales by examining the time evolution of 2D density fields derived from quantitative planar laser-induced fluorescence measurements. Several differences in qualitative flow features are identified as a result of Mach number variation, and differences in vortex interaction, observed using particle image velocimetry, play a critical role in the development of the flow field. Several quantities, including mixing layer width, mixing layer area, interface length, instantaneous mixing rate, the density self-correlation parameter, probability density functions of the density field, and mixing progress variables are examined as a function of time. These quantities are also examined versus time scaled with the convection velocity of the mixing layer. A higher incident Mach number yields greater mixing uniformity at a given downstream location, while a lower Mach number produces a greater amount of total mixing between the two gases, suggesting possible implications for optimization in applications with confined geometries.

Growth rate of a shocked mixing layer with known initial perturbations

AbstractWe derive a growth-rate model for the Richtmyer–Meshkov mixing layer, given arbitrary but known initial conditions. The initial growth rate is determined by the net mass flux through the centre plane of the perturbed interface immediately after shock passage. The net mass flux is determined by the correlation between the post-shock density and streamwise velocity. The post-shock density field is computed from the known initial perturbations and the shock jump conditions. The streamwise velocity is computed via Biot–Savart integration of the vorticity field. The vorticity deposited by the shock is obtained from the baroclinic torque with an impulsive acceleration. Using the initial growth rate and characteristic perturbation wavelength as scaling factors, the model collapses the growth-rate curves and, in most cases, predicts the peak growth rate over a range of Mach numbers ($1. 1\leq {M}_{i} \leq 1. 9$), Atwood numbers ($- 0. 73\leq A\leq - 0. 35$ and $0. 22\leq A\leq 0. 73$), adiabatic indices ($1. 40/ 1. 67\leq {\gamma }_{1} / {\gamma }_{2} \leq 1. 67/ 1. 09$) and narrow-band perturbation spectra. The mixing layer at late times exhibits a power-law growth with an average exponent of $\theta = 0. 24$.

Turbulent mixing in a Richtmyer–Meshkov fluid layer after reshock: velocity and density statistics

AbstractThe properties of turbulent mixing in a Richtmyer–Meshkov (RM) unstable fluid layer are studied under the impact of a single shock followed by a reshock wave using simultaneous velocity–density measurements to provide new insights into the physics of RM mixing. The experiments were conducted on a varicose ${\mathrm{SF} }_{6} $ fluid layer (heavy fluid) interposed in air (light fluid) inside a horizontal shock tube at an incident Mach number of 1.21 and a reflected reshock Mach number of 1.14. The light–heavy–light fluid layer is observed to develop a nonlinear growth pattern, with no transition to turbulence upon impact by a single shock (up to $tU/ \lambda = 23. 4$). However, upon reshock, enhanced mixing between the heavy and light fluids along with a transition to a turbulent state characterized by the generation of significant turbulent velocity fluctuations (${\sigma }_{u} / U\ensuremath{\sim} 0. 3$) is observed. The streamwise and spanwise root-mean-squared velocity fluctuation statistics show similar trends across the fluid layer after reshock, with no observable preference for the direction of the shock wave motion. The measured streamwise mass flux ($ \overline{{\rho }^{\ensuremath{\prime} } {u}^{\ensuremath{\prime} } } $) shows opposing signs on either side of the density peak within the fluid layer, consistent with the turbulent material transport being driven along the direction of the density gradient. Measurements of three of the six independent components of the general Reynolds stress tensor (${R}_{ij} = \overline{\rho { u}_{i}^{\ensuremath{\prime} \ensuremath{\prime} } { u}_{j}^{\ensuremath{\prime} \ensuremath{\prime} } } $) show that the self-correlation terms ${R}_{11} $ and ${R}_{22} $ are similar in magnitude across much of the fluid layer, and much larger than the cross-correlation term ${R}_{12} $. Most importantly, the Reynolds stresses (${R}_{ij} $) are dominated by the mean density, cross-velocity product term ($ \overline{\rho } \hspace{0.167em} \overline{{ u}_{i}^{\ensuremath{\prime} } { u}_{j}^{\ensuremath{\prime} } } $), with the mass flux product and triple correlation terms being negligibly smaller in comparison. A lack of homogeneous mixing (and, possibly, a long-term imprint of the initial conditions) is observed in the spanwise turbulent mass flux measurements, with important implications for the simulation and modelling of RM mixing flows.

Solutions to Buoyancy—Drag Equation for Dynamical Evolution of Rayleigh—Taylor and Richtmyer—Meshkov Mixing Zone

With a self-similar parameter b(At) = Hi/λi, where At is the Atwood number, Hi and λi are the amplitude and wavelength of bubble (i = 1) and spike (i = 2) respectively, we derive analytically the solutions to the buoyancy—drag equation recently proposed for dynamical evolution of Rayleigh—Taylor and Richtmyer—Meshkov mixing zone. Numerical solutions are obtained with a simple form of b(At) = 1/(1 + At) and comparisons with recent LEM (linear electric motor) experiments are made, and an agreement is found with properly chosen initial conditions.

Quantitative modeling of bubble competition in Richtmyer-Meshkov instability

We present a quantitative model for the evolution of single and multiple bubbles in the Richtmyer-Meshkov (RM) instability. The higher-order solutions for a single-mode bubble are obtained, and distinctions between RM and Rayleigh-Taylor bubbles are investigated. The results for multiple-bubble competition from the model shows that the higher-order correction to the solution of the bubble curvature has a large influence on the growth rate of the RM bubble front. The model predicts that the bubble front of RM mixing grows as h approximately ttheta with theta approximately (0.3-0.35)+/-0.02 .