High-resolution simulations and modeling of reshocked single-mode Richtmyer-Meshkov instability: Comparison to experimental data and to amplitude growth model predictions

The reshocked single-mode Richtmyer-Meshkov instability is simulated in two spatial dimensions using the fifth- and ninth-order weighted essentially nonoscillatory shock-capturing method with uniform spatial resolution of 256 points per initial perturbation wavelength. The initial conditions and computational domain are modeled after the single-mode, Mach 1.21 air(acetone)/SF6 shock tube experiment of Collins and Jacobs [J. Fluid Mech. 464, 113 (2002)]. The simulation densities are shown to be in very good agreement with the corrected experimental planar laser-induced fluorescence images at selected times before reshock of the evolving interface. Analytical, semianalytical, and phenomenological linear and nonlinear, impulsive, perturbation, and potential flow models for single-mode Richtmyer-Meshkov unstable perturbation growth are summarized. The simulation amplitudes are shown to be in very good agreement with the experimental data and with the predictions of linear amplitude growth models for small times, and with those of nonlinear amplitude growth models at later times up to the time at which the driver-based expansion in the experiment (but not present in the simulations or models) expands the layer before reshock. The qualitative and quantitative differences between the fifth- and ninth-order simulation results are discussed. Using a local and global quantitative metric, the prediction of the Zhang and Sohn [Phys. Fluids 9, 1106 (1997)] nonlinear Padé model is shown to be in best overall agreement with the simulation amplitudes before reshock. The sensitivity of the amplitude growth model predictions to the initial growth rate from linear instability theory, the post-shock Atwood number and amplitude, and the velocity jump due to the passage of the shock through the interface is also investigated numerically

Physics of reshock and mixing in single-mode Richtmyer-Meshkov instability

The ninth-order weighted essentially nonoscillatory (WENO) shock-capturing method is used to investigate the physics of reshock and mixing in two-dimensional single-mode Richtmyer-Meshkov instability to late times. The initial conditions and computational domain were adapted from the Mach 1.21 air (acetone)/SF6 shock tube experiment of Collins and Jacobs [J. Fluid Mech. 464, 113 (2002)]: the growth of the bubble and spike amplitudes from fifth- and ninth-order WENO simulations of this experiment were compared to the predictions of linear and nonlinear amplitude growth models, and were shown to be in very good agreement with the experimental data prior to reshock by Latini, Schilling, and Don [Phys. Fluids 19, 024104 (2007)]. In the present investigation, the density, vorticity, baroclinic vorticity production, and simulated density Schlieren fields are first presented to qualitatively describe the reshock process. The baroclinic circulation deposition on the interface is shown to agree with the predictions of the Samtaney-Zabusky model and with linear instability theory. The time evolution of the positive and negative circulation on the interface is considered before and after reshock: it is shown that the magnitudes of the circulations are equal before as well as after reshock, until the interaction of the reflected rarefaction with the layer induces flow symmetry breaking and different evolutions of the magnitude of the positive and negative circulation. The post-reshock mixing layer growth is shown to be in generally good agreement with three models predicting linear growth for a short time following reshock. Next, a comprehensive investigation of local and global mixing properties as a function of time is performed. The distribution and amount of mixed fluid along the shock propagation direction is characterized using averaged mole fraction profiles, a fast kinetic reaction model, and mixing fractions. The modal distribution of energy in the mixing layer is quantified using the spectra of the fluctuating kinetic energy, fluctuating enstrophy, pressure variance, density variance, and baroclinic vorticity production variance. It is shown that a broad range of scales already exists prior to reshock, indicating that the single-mode Richtmyer-Meshkov instability develops nontrivial spectral content from its inception. The comparison of the spectra to the predictions of classical inertial subrange scalings in two-dimensional turbulence shows that the post-reshock spectra may be consistent with many of these scalings over wave number ranges less than a decade. At reshock, fluctuations in all fields (except for the density) are amplified across all scales. Reshock strongly amplifies the circulation, profiles, and mixing fractions, as well as the energy spectra and statistics, leading to enhanced mixing followed by a decay. The mole and mixing fraction profiles become nearly self-similar at late times following reshock; the mixing fraction exhibits an approach toward unity across the layer at the latest time, signifying nearly complete mixing of the gases. To directly quantify the amplification of fluctuations by reshock, the previously considered quantities are compared immediately after and before reshock. Finally, to investigate the decay of fluctuations in the absence of additional waves interacting with the mixing layer following reshock, the boundary condition at the end of the computational domain is changed from reflecting to outflow to allow the reflected rarefaction wave to exit the domain. It is demonstrated that the reflected rarefaction has an important role in breaking symmetry and achieving late-time statistical isotropy of the velocity field

Large-eddy simulation and multiscale modelling of a Richtmyer–Meshkov instability with reshock

Large-eddy simulations of the Richtmyer–Meshkov instability with reshock are pre- sented and the results are compared with experiments. Several configurations of shocks initially travelling from light (air) to heavy (sulfur hexafluoride, SF6) have been simulated to match previous experiments and good agreement is found in the growth rates of the turbulent mixing zone (TMZ). The stretched-vortex subgrid model used in this study allows for subgrid continuation modelling, where statistics of the unresolved scales of the flow are estimated. In particular, this multiscale modelling allows the anisotropy of the flow to be extended to the dissipation scale, eta, and estimates to be formed for the subgrid probability density function of the mixture fraction of air/SF6 based on the subgrid variance, including the effect of Schmidt number

Atwood ratio dependence of Richtmyer-Meshkov flows under reshock conditions using large-eddy simulations

We study the shock-driven turbulent mixing that occurs when a perturbed planar density interface is impacted by a planar shock wave of moderate strength and subsequently reshocked. The present work is a systematic study of the influence of the relative molecular weights of the gases in the form of the initial Atwood ratio A. We investigate the cases A = ± 0.21, ±0.67 and ±0.87 that correspond to the realistic gas combinations air–CO_2, air–SF_6 and H_2–air. A canonical, three-dimensional numerical experiment, using the large-eddy simulation technique with an explicit subgrid model, reproduces the interaction within a shock tube with an endwall where the incident shock Mach number is ~1.5 and the initial interface perturbation has a fixed dominant wavelength and a fixed amplitude-to-wavelength ratio ~0.1. For positive Atwood configurations, the reshock is followed by secondary waves in the form of alternate expansion and compression waves travelling between the endwall and the mixing zone. These reverberations are shown to intensify turbulent kinetic energy and dissipation across the mixing zone. In contrast, negative Atwood number configurations produce multiple secondary reshocks following the primary reshock, and their effect on the mixing region is less pronounced. As the magnitude of A is increased, the mixing zone tends to evolve less symmetrically. The mixing zone growth rate following the primary reshock approaches a linear evolution prior to the secondary wave interactions. When considering the full range of examined Atwood numbers, measurements of this growth rate do not agree well with predictions of existing analytic reshock models such as the model by Mikaelian (Physica D, vol. 36, 1989, p. 343). Accordingly, we propose an empirical formula and also a semi-analytical, impulsive model based on a diffuse-interface approach to describe the A-dependence of the post-reshock growth rate

Growth of a Richtmyer-Meshkov turbulent layer after reshock

This paper presents a numerical study of a reshocked turbulent mixing layer using high-order accurate Implicit Large-Eddy-Simulations (ILES). Existing theoretical approaches are discussed, and the theory of Youngs (detailed in Ref. 1) is extended to predict the behaviour of a reshocked mixing layer formed initially from a shock interacting with a broadband instability. The theory of Mikaelian2 is also extended to account for molecular mixing in the single-shocked layer prior to reshock. Simulations are conducted for broadband and narrowband initial perturbations and results for the growth rate of the reshocked layer and the decay rate of turbulent kinetic energy show excellent agreement with the extended theoretical approach. Reshock causes a marginal decrease in mixing parameters for the narrowband layer, but a significant increase for the broadband initial perturbation. The layer properties are observed to be very similar post-reshock, however, the growth rate exponent for the mixing layer width is higher in the broadband case, indicating that the reshocked layer still has a dependence (although weakened) on the initial conditions. These results have important implications for Unsteady Reynolds Averaged Navier Stokes modelling of such instabilities

Multicomponent Reynolds-Averaged Navier--Stokes Modeling of Reshocked Richtmyer--Meshkov Instability-Induced Turbulent Mixing Using the Weighted Essentially Nonoscillatory Method

Elucidating reshocked Richtmyer--Meshkov instability is important for improving current techniques in predicting turbulence in complex flows and advancing many areas of high-energy-density physics. Shock-driven turbulent mixing induced by reshocked Richtmyer--Meshkov instability is investigated here using a multicomponent Reynolds-averaged Navier--Stokes (RANS) model including mixture molecular transport and thermodynamic coefficients closed with a two-equation K-epsilon turbulence model. The model is implemented in a hydrodynamics code using a third-order weighted essentially nonoscillatory (WENO) finite-difference method for the advection terms and a second-order centered WENO method for the gradients in the source and diffusion terms. Turbulent mixing generated by a shock accelerated perturbed air-sulfur hexafluoride interface is simulated for a variety of experiments with incident shock Mach numbers 1.20 < Mas < 1.98. Parametric studies are conducted to study the model sensitivity to variations in buoyancy production model coefficients, initial conditions, and incident shock Mach number. The time-evolution of the predicted mixing layer widths corresponding to different reshock times by variations in shock tube test section lengths is also considered. The RANS model results are compared with experimental data, previous large-eddy simulation (LES) and turbulence model predictions, and the early-time analytical self-similar mixing layer width. The model is also applied to reshocked Richtmyer--Meshkov instability for positive and negative Atwood numbers of 0.21, 0.67, and 0.87 with Mas=1.50, as experimental and numerical simulation data for Atwood numbers different from 0.67 are sparse. Shock-driven instabilities are also considered with larger incident shock Mach numbers 3.00 and 5.00, as limited turbulent mixing investigations with larger Mas values have been conducted. These studies are considered for cases with negative Atwood numbers, -0.21, -0.67, and -0.87. The budgets of the turbulent kinetic energy and turbulent kinetic energy dissipation rate transport equations are investigated to determine the key mechanisms in turbulent mixing. Results for convergence under grid refinement for mixing layer widths and the mean and turbulent fields are also presented. These investigations are considered for early-time and post-reshock mixing, as well as for changes in the mixing due to secondary expansion, rarefaction, and reshock waves.PHDNuclear Engineering & Radiological SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/100088/1/timoran_1.pd