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

Weighted essentially non-oscillatory (WENO) simulations of the reshocked two-dimensional single-mode Richtmyer-Meshkov instability using third-, fifth- and ninth-order spatial flux reconstruction and uniform grid resolutions corresponding to 128, 256 and 512 points per initial perturbation wavelength are presented. The dependence of the density, vorticity, simulated density Schlieren and baroclinic production fields, mixing layer width, circulation deposition, mixing profiles, production and mixing fractions, energy spectra, statistics, probability distribution functions, numerical turbulent kinetic energy and enstrophy production/dissipation rates, numerical Reynolds numbers, and numerical viscosity on the order and resolution is investigated to long evolution times. The results are interpreted using the implicit numerical dissipation in the characteristic projection-based, finite-difference WENO method. It is shown that higher order higher resolution simulations have lower numerical dissipation. The sensitivity of the quantities considered to the order and resolution is further amplified following reshock, when the energy deposition by the second shock-interface interaction induces the formation of small-scale structures. Lower-order lower-resolution simulations preserve large-scale structures and flow symmetry to late times, while higher-order higher-resolution simulations exhibit fragmentation of the structures, symmetry breaking and increased mixing. Similar flow features are qualitatively and quantitatively captured by either approximately doubling the order or the resolution. Additionally, the computational scaling showsmore » that increasing the order is more advantageous than increasing the resolution for the flow considered here. The present investigation suggests that the ninth-order WENO method is well-suited for the simulation and analysis of complex multi-scale flows and mixing generated by shock-induced hydrodynamic instabilities.« less

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Effects of WENO flux reconstruction order and spatial resolution on reshocked two-dimensional Richtmyer-Meshkov instability

Particle image velocimetry (PIV) has become the chief experimental technique for velocity field measurements in fluid flows. The technique yields quantitative visualizations of the instantaneous flow patterns, which are typically used to support the development of phenomenological models for complex flows or for validation of numerical simulations. However, due to the complex relationship between measurement errors and experimental parameters, the quantification of the PIV uncertainty is far from being a trivial task and has often relied upon subjective considerations. Recognizing the importance of methodologies for the objective and reliable uncertainty quantification (UQ) of experimental data, several PIV-UQ approaches have been proposed in recent years that aim at the determination of objective uncertainty bounds in PIV measurements. This topical review on PIV uncertainty quantification aims to provide the reader with an overview of error sources in PIV measurements and to inform them of the most up-to-date approaches for PIV uncertainty quantification and propagation. The paper first introduces the general definitions and classifications of measurement errors and uncertainties, following the guidelines of the International Organization for Standards (ISO) and of renowned books on the topic. Details on the main PIV error sources are given, considering the entire measurement chain from timing and synchronization of the data acquisition system, to illumination, mechanical properties of the tracer particles, imaging of those, analysis of the particle motion, data validation and reduction. The focus is on planar PIV experiments for the measurement of two- or three-component velocity fields. Approaches for the quantification of the uncertainty of PIV data are discussed. Those are divided into a-priori UQ approaches, which provide a general figure for the uncertainty of PIV measurements, and a-posteriori UQ approaches, which are data-based and aim at quantifying the uncertainty of specific sets of data. The findings of a-priori PIV-UQ based on theoretical modelling of the measurement chain as well as on numerical or experimental assessments are discussed. The most up-to-date approaches for a-posteriori PIV-UQ are introduced, highlighting their capabilities and limitations. As many PIV experiments aim at determining flow properties derived from the velocity fields (e.g. vorticity, time-average velocity, Reynolds stresses, pressure), the topic of PIV uncertainty propagation is tackled considering the recent investigations based on Taylor series and Monte Carlo methods. Finally, the uncertainty quantification of 3D velocity measurements by volumetric approaches (tomographic PIV and Lagrangian particle tracking) is discussed.

/pdf/uncertainty-quantification-in-particle-image-velocimetry-44buxu5nve.pdf

Uncertainty quantification in particle image velocimetry

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

A vertical shock tube is used to perform experiments on the Richtmyer–Meshkov instability with a three-dimensional random initial perturbation. A membraneless flat interface is formed by opposed gas flows in which the light and heavy gases enter the shock tube from the top and from the bottom of the shock tube driven section. An air/SF
 $$_{6}$$
 gas combination is used and a Mach number $$ M = 1.2$$
 incident shock wave impulsively accelerates the interface. Initial perturbations on the interface are created by vertically oscillating the gas column within the shock tube to produce Faraday waves on the interface resulting in a short wavelength, three-dimensional perturbation. Planar Mie scattering is used to visualize the flow in which light from a laser sheet is scattered by smoke seeded in the air, and image sequences are captured using three high-speed video cameras. Measurements of the integral penetration depth prior to reshock show two growth behaviors, both having power law growth with growth exponents in the range found in previous experiments and simulations. Following reshock, all experiments show very consistent linear growth with a growth rate in good agreement with those found in previous studies.

Experiments on the Richtmyer-Meshkov instability with an imposed, random initial perturbation

The effect of initial conditions on transition to turbulence is studied in a variable-density shock-driven flow. Richtmyer–Meshkov instability (RMI) evolution of fluid interfaces with two different imposed initial perturbations is observed before and after interaction with a second shock reflected from the end wall of a shock tube (reshock). The first perturbation is a predominantly single-mode long-wavelength interface which is formed by inclining the entire tube to 80 relative to the horizontal, yielding an amplitude-to-wavelength ratio, , and thus can be considered as half the wavelength of a triangular wave. The second interface is multi-mode, and contains additional shorter-wavelength perturbations due to the imposition of shear and buoyancy on the inclined perturbation of the first case. In both cases, the interface consists of a nitrogen-acetone mixture as the light gas over carbon dioxide as the heavy gas (Atwood number, ) and the shock Mach number is . The initial condition was characterized through Proper Orthogonal Decomposition and density energy spectra from a large set of initial condition images. The evolving density and velocity fields are measured simultaneously using planar laser-induced fluorescence (PLIF) and particle image velocimetry (PIV) techniques. Density, velocity, and density–velocity cross-statistics are calculated using ensemble averaging to investigate the effects of additional modes on the mixing and turbulence quantities. The density and velocity data show that a distinct memory of the initial conditions is maintained in the flow before interaction with reshock. After reshock, the influence of the long-wavelength inclined perturbation present in both initial conditions is still apparent, but the distinction between the two cases becomes less evident as smaller scales are present even in the single-mode case. Several methods are used to calculate the Reynolds number and turbulence length scales, which indicate a transition to a more turbulent state after reshock. Further evidence of transition to turbulence after reshock is observed in the velocity and density fluctuation spectra, where a scaling close to is observed for almost one decade, and in the enstrophy fluctuation spectra, where a scaling close to is observed for a similar range. Also, based on normalized cross correlation spectra, local isotropy is reached at lower wave numbers in the multi-mode case compared with the single-mode case before reshock. By breakdown of large scales to small scales after reshock, rapid decay can be observed in cross-correlation spectra in both cases.

Evaluation of turbulent mixing transition in a shock-driven variable-density flow

The effects of incident shock strength on the mixing transition in the Richtmyer–Meshkov instability (RMI) are experimentally investigated using simultaneous density–velocity measurements. This effort uses a shock with an incident Mach number of 1.9, in concert with previous work at Mach 1.55 (Mohaghar et al., J. Fluid Mech., vol. 831, 2017 pp. 779–825) where each case is followed by a reshock wave. Single- and multi-mode interfaces are used to quantify the effect of initial conditions on the evolution of the RMI. The interface between light and heavy gases ( for the higher Mach number case. This could possibly be related to the high anisotropy observed in this flow resulting from strong, large-scale streamwise fluctuations produced by large-scale shear.

The transition to turbulence in shock-driven mixing: effects of Mach number and initial conditions

Recent developments of burst-mode lasers and imaging systems have opened new realms of simultaneous diagnostics for velocity and density fields at a rate of 1 kHz–1 MHz. These enable the exploration of previously unimaginable shock-driven turbulent flow fields that are of significant importance to problems in high-energy density physics. The current work presents novel measurements using simultaneous measurements of velocity and scalar fields at 60 kHz to investigate Richtmyer-Meshkov instability (RMI) in a spatio-temporal approach. The evolution of scalar fields and the vorticity dynamics responsible for the same are shown, including the interaction of shock with the interface. This temporal information is used to validate two vorticity-deposition models commonly used for initiation of large scale simulations, and have been previously validated only via simulations or integral measures of circulation. Additionally, these measurements also enable tracking the evolution and mode merging of individual flow structures that were previously not possible owing to inherently random variations in the interface at the smallest scales. A temporal evolution of symmetric vortex merging and the induced mixing prevalent in these problems is presented, with implications for the vortex paradigms in accelerated inhomogenous flows.

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John Carter

Papers

Evaluation of turbulent mixing transition in a shock-driven variable-density flow

The transition to turbulence in shock-driven mixing: effects of Mach number and initial conditions

Time-Resolved Measurements of Turbulent Mixing in Shock-Driven Variable-Density Flows.