Small-amplitude perturbations in the three-dimensional cylindrical Richtmyer–Meshkov instability

We first study the linear stability of an interface between two fluids following the passage of an imploding or exploding shock wave. Assuming incompressible flow between the refracted waves following shock impact, we derive an expression for the asymptotic growth rate for a three-dimensional combination of azimuthal and axial perturbations as a function of the Atwood ratio, the axial and azimuthal wave numbers, the initial radial position and perturbation amplitude of the interface, and the interface velocity gain due to the shock interaction. From the linearized theory, a unified expression for the impulsive asymptotic growth rate in plane, cylindrical, and spherical geometries is obtained which clearly delineates the effects of perturbation growth due to both geometry and baroclinic vorticity deposition. Several different limit cases are investigated, allowing recovery of Mikaelian's purely azimuthal theory and Richtmyer's plane model. We discuss the existence of three-dimensional perturbations with zero growth, typical of curvilinear geometries, as first observed by Mikaelian. The effect of shock proximity on the interface growth rate is studied in the case of a reflected shock. Analytical predictions of the effect of the incident shock strength and the perturbation wave numbers are then compared with results obtained from highly resolved numerical simulations of cylindrical imploding Richtmyer–Meshkov instability for ideal gases. A parallel is made with the instability growth in spherical and plane geometry. In particular, we propose a representation of the perturbation growth by considering the volume of the perturbed layer. This volume is found to grow faster in the plane case than in the imploding cylindrical geometry, among other results

Growth of shocked gaseous interfaces in a conical geometry

The results of experiments on Richtmyer-Meshkov instability growth of multimode initial perturbations on an air-sulfur hexafluoride (SF6) interface in a conical geometry are presented. The experiments are done in a relatively larger shock tube. A nominally planar interface is formed by sandwiching a polymeric membrane between wire-mesh frames. A single incident shock wave ruptures the membrane resulting in multimode perturbations. The instability develops from the action of baroclinically deposited vorticity at the interface. The visual thickness delta of the interface is measured from schlieren photographs obtained in each run. Data are presented for delta at times when the interface has become turbulent. The data are compared with the experiments of Vetter [Shock Waves 4, 247 (1995)] which were done in a straight test section geometry, to illustrate the effects of area convergence. It is found from schlieren images that the interface thickness grows about 40% to 50% more rapidly than in Vetter's experiments. Laser induced scattering is used to capture the air-helium interface at late times. Image processing of pictures is also used to determine the interface thickness in cases where it was not clear from the pictures and to obtain the dominant eddy-blob sizes in the mixing zone. Some computational studies are also presented to show the global geometry changes of the interface when it implodes into a conical geometry in both light-heavy and heavy-light cases

The magnetised Richtmyer–Meshkov instability in two-fluid plasmas

We investigate the effects of magnetisation on the two-fluid plasma Richtmyer–Meshkov instability of a single-mode thermal interface using a computational approach. The initial magnetic field is normal to the mean interface location. Results are presented for a magnetic interaction parameter of 0.1 and plasma skin depths ranging from 0.1 to 10 perturbation wavelengths. These are compared to initially unmagnetised and neutral fluid cases. The electron flow is found to be constrained to lie along the magnetic field lines resulting in significant longitudinal flow features that interact strongly with the ion fluid. The presence of an initial magnetic field is shown to suppress the growth of the initial interface perturbation with effectiveness determined by plasma length scale. Suppression of the instability is attributed to the magnetic field's contribution to the Lorentz force. This acts to rotate the vorticity vector in each fluid about the local magnetic-field vector leading to cyclic inversion and transport of the out-of-plane vorticity that drives perturbation growth. The transport of vorticity along field lines increases with decreasing plasma length scales and the wave packets responsible for vorticity transport begin to coalesce. In general, the two-fluid plasma Richtmyer–Meshkov instability is found to be suppressed through the action of the imposed magnetic field with increasing effectiveness as plasma length scale is decreased. For the conditions investigated, a critical skin depth for instability suppression is estimated

Effects of magnetic fields on magnetohydrodynamic cylindrical and spherical Richtmyer-Meshkov instability

The effects of seed magnetic fields on the Richtmyer-Meshkov instability driven by converging cylindrical and spherical implosions in ideal magnetohydrodynamics are investigated. Two different seed field configurations at various strengths are applied over a cylindrical or spherical density interface which has a single-dominant-mode perturbation. The shocks that excite the instability are generated with appropriate Riemann problems in a numerical formulation and the effect of the seed field on the growth rate and symmetry of the perturbations on the density interface is examined. We find reduced perturbation growth for both field configurations and all tested strengths. The extent of growth suppression increases with seed field strength but varies with the angle of the field to interface. The seed field configuration does not significantly affect extent of suppression of the instability, allowing it to be chosen to minimize its effect on implosion distortion. However, stronger seed fields are required in three dimensions to suppress the instability effectively

Turbulent mixing driven by spherical implosions. Part 1. Flow description and mixing-layer growth

We present large-eddy simulations (LES) of turbulent mixing at a perturbed, spherical interface separating two fluids of differing densities and subsequently impacted by a spherically imploding shock wave. This paper focuses on the differences between two fundamental configurations, keeping fixed the initial shock Mach number ≈1.2, the density ratio (precisely |A_0|≈0.67) and the perturbation shape (dominant spherical wavenumber ℓ_0=40 and amplitude-to-initial radius of 3%): the incident shock travels from the lighter fluid to the heavy fluid or, inversely, from the heavy to the light fluid. After describing the computational problem we present results on the radially symmetric flow, the mean flow, and the growth of the mixing layer. Turbulent statistics are developed in Part 2 (Lombardini, M., Pullin, D. I. & Meiron, D. I. J. Fluid Mech., vol. 748, 2014, pp. 113–142). A wave-diagram analysis of the radially symmetric flow highlights that the light–heavy mixing layer is processed by consecutive reshocks, and not by reverberating rarefaction waves as is usually observed in planar geometry. Less surprisingly, reshocks process the heavy–light mixing layer as in the planar case. In both configurations, the incident imploding shock and the reshocks induce Richtmyer–Meshkov (RM) instabilities at the density layer. However, we observe differences in the mixing-layer growth because the RM instability occurrences, Rayleigh–Taylor (RT) unstable scenarios (due to the radially accelerated motion of the layer) and phase inversion events are different. A small-amplitude stability analysis along the lines of Bell (Los Alamos Scientific Laboratory Report, LA-1321, 1951) and Plesset (J. Appl. Phys., vol. 25, 1954, pp. 96–98) helps quantify the effects of the mean flow on the mixing-layer growth by decoupling the effects of RT/RM instabilities from Bell–Plesset effects associated with geometric convergence and compressibility for arbitrary convergence ratios. The analysis indicates that baroclinic instabilities are the dominant effect, considering the low convergence ratio (≈2) and rather high (ℓ>10) mode numbers considered

Linear simulations of the cylindrical Richtmyer-Meshkov instability in magnetohydrodynamics

Numerical simulations and analysis indicate that the Richtmyer-Meshkov instability (RMI) is suppressed in ideal magnetohydrodynamics (MHD) in Cartesian slab geometry. Motivated by the presence of hydrodynamic instabilities in inertial confinement fusion and suppression by means of a magnetic field, we investigate the RMI via linear MHD simulations in cylindrical geometry. The physical setup is that of a Chisnell-type converging shock interacting with a density interface with either axial or azimuthal (2D) perturbations. The linear stability is examined in the context of an initial value problem (with a time-varying base state) wherein the linearized ideal MHD equations are solved with an upwind numerical method. Linear simulations in the absence of a magnetic field indicate that RMI growth rate during the early time period is similar to that observed in Cartesian geometry. However, this RMI phase is short-lived and followed by a Rayleigh-Taylor instability phase with an accompanied exponential increase in the perturbation amplitude. We examine several strengths of the magnetic field (characterized by β = 2p/B ) and observe a significant suppression of the instability for β ≤ 4. The suppression of the instability is attributed to the transport of vorticity away from the interface by Alfvén fronts

Magnetohydrodynamic implosion symmetry and suppression of Richtmyer-Meshkov instability in an octahedrally symmetric field

We present numerical simulations of ideal magnetohydrodynamics showing suppression of the Richtmyer-Meshkov instability in spherical implosions in the presence of an octahedrally symmetric magnetic field. This field configuration is of interest owing to its high degree of spherical symmetry in comparison with previously considered dihedrally symmetric fields. The simulations indicate that the octahedral field suppresses the instability comparably to the other previously considered candidate fields for light-heavy interface accelerations while retaining a highly symmetric underlying flow even at high field strengths. With this field, there is a reduction in the root-mean-square perturbation amplitude of up to approximately 50% at representative time under the strongest field tested while maintaining a homogeneous suppression pattern compared to the other candidate fields

Covering Shock Wave Induced Interfacial Mixing: Numerical Study and a Control Primer

This document is aiming toward deepening the understanding of the phenomena of mixing and the effect of the initial conditions in the cylindrical & spherical Richtmyer-Meshkov and Rayleigh-Taylor Instabilities. This work is focused on identifying the most energetic structures of the ow in order to define a reduced order model intended for modeling the evolution of the mixing layer after reshocking the density interface. Initially, Simulations are implemented for the two dimensional case of a cylindrical shock wave convergently approaching an initially wave-like perturbed density discontinuity formed by a target of Sulfur Hexauoride immersed into unshocked air with Atwood number of 0.67. The perturbation is varied by setting different values for the wave amplitude and wave-number; the amplitude and wave-number effects on late-time mixing are studied separately and then such perturbation features are coupled together in the analysis of single- and multi-mode well-defined cylindrical perturbations. The simulation data is then utilized as a mechanism for obtaining a model equation intended to predict the mixing layer evolution using a Proper Orthogonal Decomposition. The ultimate goal of the POD is to model the evolution after reshock which has been the main issue to be tackled since available models fail to predict the extent of the mixing layer after reshocking the interface. Considering three-dimensional effects as in spherical shock-interface interaction gives a better depiction of the small-scale interactions but spherical cases are only quickly addressed. The main effect is the vortex stretching affectation on the vorticity evolution. Furthermore, mixing layers in 3D spherical simulations are found to be wider than its 2D simplified framework. Nonetheless, useful insight is gained by reducing the problen under study to a cylindrical two-dimensional symmetrical system