Single-mode Richtmyer-Meshkov Instability Research Articles

Analysis of single-mode Richtmyer–Meshkov instability using high-order incompressible vorticity—streamfunction and shock-capturing simulations

Two- and three-dimensional simulation results obtained using a new high-order incompressible, variable-density vorticity–streamfunction (VS) method and data from previous ninth-order weighted essentially nonoscillatory (WENO) shock-capturing simulations [M. Latini and O. Schilling, “A comparison of two- and three-dimensional single-mode reshocked Richtmyer-Meshkov instability growth,” Physica D 401, 132201 (2020)] are used to investigate the nonlinear dynamics of single-mode Richtmyer–Meshkov instability using a model of a Mach 1.3 air(acetone)/SF6 shock tube experiment [J. W. Jacobs and V. V. Krivets, “Experiments on the late-time development of single-mode Richtmyer–Meshkov instability,” Phys. Fluids 17, 034105 (2005)]. A comparison of the density fields from both simulations with the experimental images demonstrates very good agreement in the large-scale structure with both methods but differences in the small-scale structure. The WENO method captures the small-scale disordered structure observed in the experiment, while the VS method partially captures such structure and yields a strong rotating core. The perturbation amplitude growth from the simulations generally agrees well with the experiment. The simulation bubble and spike amplitudes agree well at early times. At later times, the WENO bubble amplitude is smaller than the VS amplitude and vice versa for the spike amplitude. The predictions of nonlinear single-mode instability growth models are shown to agree with the simulation amplitudes at early-to-intermediate times but underpredict the amplitudes at later times in the nonlinear regime. Visualizations of the mass fraction and enstrophy isosurfaces, velocity and vorticity fields, and baroclinic vorticity production and vortex stretching terms from the three-dimensional simulations indicate that, with the exception of the small-scale structure within the rollups, the VS and WENO results are in good agreement.

Experiments on the single-mode Richtmyer–Meshkov instability with reshock at high energy densities

The hydrodynamic instability growth of a reshocked single-mode interface between high energy density fluids is studied. A laser-driven shock wave is used to drive an initially solid, sinusoidal interface between a dense plastic (1.43 g/cc) and a light foam (≈ 0.110 g/cc). After the interface has grown to a nonlinear state where the amplitude is of order of the wavelength, it is reshocked. The reshock compresses the nonlinear perturbation, which then grows at about twice the rate. While the pre-reshock growth rate is sensitive to the initial amplitude and wavelength of the perturbation, the post-reshock growth rate is comparatively insensitive to the initial condition. Qualitatively, we observe that the perturbations are less coherent after reshock, consistent with the idea that having a reshock accelerates the transition to turbulence. We find that some memory of the initial condition remains, even after reshock at late time: it appears if the initial perturbations have large enough wavelengths, and the flow structure of size comparable to the initial wavelength persists through reshock. Our results agree with design simulations and are consistent with the phenomenology of reshock studies in conventional gaseous shock tubes.

Ion kinetic effects on the evolution of Richtmyer–Meshkov instability and interfacial mix

Plasma effects, such as the multi-component kinetic diffusion and self-generated electromagnetic fields, are recognized as a pivotal key to understanding the physics of interface evolution in inertial confinement fusion and supernova remnants. In this work, a two-dimensional hybrid fluid-PIC code is used to investigate the ion kinetic effects of the single-mode Richtmyer–Meshkov instability (RMI) at the interface between hydrogen plasma and carbon plasma. After an electrostatic shockwave passing through the perturbed interface, the RMI, which reshapes the interface, grows via the vorticity depositing as well as the self-generated magnetic field. After scaling the growth of the interfacial mix region with time, the density transition layer has been found to exceed the disturbance wavelength and lead to a suppression of the instability evolution finally.

2D single-mode Richtmyer–Meshkov instability

We study the evolution of the single-mode Richtmyer–Meshkov instability for a wide range of Atwood numbers, shock strengths and perturbation amplitudes using Youngs’ hydrodynamical simulation code TURMOIL. We compare our results to previously published analytic models for the impulsively-driven growth rate, and propose a modification to them to treat the reduction of growth found at high initial perturbation amplitudes and high Mach numbers. It is known that the overall asymmetry between bubbles and spikes and their eventual deceleration can be interpreted as the result of nonlinear coupling to higher harmonic modes. However, we find that for light-to-heavy interfaces at moderate to high impinging shock Mach numbers, the shape of the growing bubbles varies in time, with the initial curved bubble surfaces flattening and inverting to generate a second low velocity jet. For high shock Mach numbers and low initial surface amplitudes, the process of inversion can recur on numerous occasions. We interpret this as being the result of vorticity deposited by the transmitted shock in the bulk of the heavy material, away from the initial interface.

The effect of initial amplitude and convergence ratio on instability development and deposited fluctuating kinetic energy in the single-mode Richtmyer–Meshkov instability in spherical implosions

This paper investigates the growth of perturbations on the inner surface of a dense imploding spherical shell due to hydrodynamic instabilities. The perturbations change in amplitude due to Richtmyer–Meshkov instability and Rayleigh-Taylor instability, geometric convergence, and compressibility. Two mode numbers (ℓ=5,50) and three different perturbation amplitudes (a0=0.1λ,0.01λ,0.001λ) are applied to the surface of the dense inner shell. Two independent codes were used to perform simulations with these six perturbation profiles at four convergence ratios, ranging from 3.6 to 7.3, for a total of 48 cases. The mixing layer amplitudes show good agreement between the two simulation codes across the range of convergence ratios, mode numbers, and initial amplitudes. The growth of the mixing layer is employed to validate an extension of a recently proposed Bell-Plesset model, showing good agreement across the range of convergence ratios. Persistent and substantial shock-deposited fluctuating kinetic energy is observed within the light core, away from the perturbed interface. Temporal evolution of fluctuating kinetic energy indicates a time delay between the peak radial and theta directions, consistent with a build-up of vortical motion. A “jet” like phenomena is observed at the peaks and troughs of the initial perturbations in both simulation codes across a variety of cases. It is postulated that these occur due to the shape singularity shown to develop in spatially periodic perturbed planar shock waves in ideal gas dynamics. Significant anisotropy of kinetic energy components is present at all times.

A lattice Boltzmann method for simulation of multi-species shock accelerated flows

ABSTRACTA numerical scheme for simulating multi-species shock accelerated flows using lattice Boltzmann approach has been proposed. It uses the moment conservation approach of Yang, Shu, and Wu and extends it to multi-species fluid problems. The multi-species method of Wang et al. has been modified by use of a predictor–corrector approach. This has helped in preventing the pressure oscillations while handling multi-species. Simulation of 2D shock cylinder interaction with this solver has shown good agreement with the experimental data and could capture material discontinuity and unsteady shocks. The simulation of a single mode Richtmyer–Meshkov instability showed that the solver is able to capture the development of spike and bubble as per the experimental findings of Aure and Jacobs. The dissipation in the proposed scheme was further reduced by the use of fifth-order weighted essentially non-oscillatory (WENO). Validated with multiple problems, this method has been found to capture shock instability with good accuracy with a check on pressure oscillations.

A comparative study of the single-mode Richtmyer–Meshkov instability

In this work, the single-mode Richtmyer–Meshkov instability is studied numerically to find a reasonable nonlinear theoretical model which can be applied to predict the interface evolution from the linear stage to the early nonlinear stage. The cut-cell-based sharp-interface methods $$\hbox {MuSiC}^{+}$$ (Chang et al. in J Comput Phys 242:946–990, 2013) and CCGF (Bai and Deng in Adv Appl Math Mech 9(5):1052–1075, 2017) are applied to generate numerical results for comparisons. Classical Air–SF $$_{6}$$ and Air–Helium conditions are applied in this study, and initial amplitude and Atwood number are varied for comparison. Comparisons to the simulation results from the literature show the applicability of $$\hbox {MuSiC}^{+}$$ and CCGF. Comparisons to the nonlinear theoretical models show that ZS (Zhang and Sohn in Phys Lett A 212:149–155, 1996; Phys Fluids 9:1106–1124, 1997), SEA (Sadot et al. in Phys Rev Lett 80:1654–1657, 1998), and DR (Dimonte and Ramaprabhu in Phys Fluids 22:014104, 2010) models are valid for both spike and bubble growth rates, and MIK (Mikaelian in Phys Rev E 67:026319, 2003) and ZG (Zhang and Guo in J Fluid Mech 786:47–61, 2016) models are valid for bubble growth rate, when the initial perturbation is small and the Atwood number is low, but only the DR model is applicable for both spike and bubble growth rates when the initial perturbation amplitude and the Atwood number are large. A new term of non-dimensional initial perturbation amplitude is presented and multiplied to the DR model to get a unified fitted DR model, which gives consistent results to the simulation ones for small and large initial amplitudes.

Reynolds number effects on the single-mode Richtmyer-Meshkov instability.

The Reynolds number effects on the nonlinear growth rates of the Richtmyer-Meshkov instability are investigated using two-dimensional numerical simulations. A decrease in Reynolds number gives an increased time to reach nonlinear saturation, with Reynolds number effects only significant in the range Re<256. Within this range there is a sharp change in instability properties. The bubble and spike amplitudes move towards equal size at lower Reynolds numbers and the bubble velocities decay faster than predicted by Sohn's model [S.-I. Sohn, Phys. Rev. E 80, 055302 (2009)PLEEE81539-375510.1103/PhysRevE.80.055302]. Predicted amplitudes show reasonable agreement with the existing theory of Carles and Popinet [P. Carles and S. Popinet, Phys. Fluids Lett. 13, 1833 (2001)10.1063/1.1377863; Eur. J. Mech. B 21, 511 (2002)EJBFEV0997-754610.1016/S0997-7546(02)01199-8] and Mikaelian [K. O. Mikaelian, Phys. Rev. E 47, 375 (1993)1063-651X10.1103/PhysRevE.47.375; K. O. Mikaelian, Phys. Rev. E 87, 031003 (2013)PLEEE81539-375510.1103/PhysRevE.87.031003], with the former being the closest match to the current computations.

Numerical Investigation of a Shock Accelerated Heavy Gas Cylinder in the Self-Similar Regime

A detailed numerical simulation of a shock accelerated heavy gas (SF6) cylinder surrounded by air gas is presented. It is a simplified configuration of the more general shock-accelerated inhomogeneous flows which occur in a wide variety of astrophysical systems. From the snapshots of the time evolution of the gas cylinder, we find that the evolution of the shock accelerated gas cylinder is in some ways similar to the roll-ups of a vortex sheet for both roll up into a spiral and fall into a self-similar behavior. The systemic and meaningful analyses of the negative circulation, the center of vorticity and the vortex spacing are in a good agreement with results obtained from the prediction of vorticity dynamics. Unlike the mixing zone width in single-mode or multi-mode Richtmyer-Meshkov instability which doesn’t exist, a single power law of time owing to the bubble and spike fronts follow a power law of tθ with different power exponents, the normalized length of the shock accelerated gas cylinder follows a single power law with θ = 0.43 in its self-similar regime obtained from the numerical results.

An Investigation Into Nonlinear Growth Rate of Two-Dimensional and Three-Dimensional Single-Mode Richtmyer–Meshkov Instability Using an Arbitrary-Lagrangian–Eulerian Algorithm

This paper presents an investigation into the use of a moving mesh algorithm for solving unsteady turbulent mixing problems. The growth of a shock induced mixing zone following reshock, using an initial setup comparable to that of existing experimental work, is used to evaluate the behavior of the numerical scheme for single-mode Richtmyer–Meshkov instability (SM-RMI). Subsequently the code is used to evaluate the growth rate for a range of different initial conditions. The initial growth rate for three-dimensional (3D) SM Richtmyer–Meshkov is also presented for a number of different initial conditions. This numerical study details the development of the mixing layer width both prior to and after reshock. The numerical scheme used includes an arbitrary Lagrangian–Eulerian grid motion which is successfully used to reduce the mesh size and computational time while retaining the accuracy of the simulation results. Varying initial conditions shows that the growth rate after reshock is independent of the initial conditions for a SM provided that the initial growth remains in the linear regime.

A Hybrid Compressible–Incompressible Computational Fluid Dynamics Method for Richtmyer–Meshkov Mixing

This paper presents a hybrid compressible–incompressible approach for simulating the Richtmyer–Meshkov instability (RMI) and associated mixing. The proposed numerical approach aims to circumvent the numerical deficiencies of compressible methods at low Mach (LM) numbers, when the flow has become essentially incompressible. A compressible flow solver is used at the initial stage of the interaction of the shock wave with the fluids interface and the development of the RMI. When the flow becomes sufficiently incompressible, based on a Mach number prescribed threshold, the simulation is carried out using an incompressible flow solver. Both the compressible and incompressible solvers use Godunov-type methods and high-resolution numerical reconstruction schemes for computing the fluxes at the cell interfaces. The accuracy of the model is assessed by using results for a two-dimensional (2D) single-mode RMI.

Numerical simulation of a Richtmyer–Meshkov instability with an adaptive central-upwind sixth-order WENO scheme

Two-dimensional simulations of the single-mode Richtmyer–Meshkov instability (RMI) are conducted and compared to experimental results of Jacobs and Krivets (2005 Phys. Fluids 17 034105). The employed adaptive central-upwind sixth-order weighted essentially non-oscillatory (WENO) scheme (Hu et al 2010 J. Comput. Phys. 229 8952–65) introduces only very small numerical dissipation while preserving the good shock-capturing properties of other standard WENO schemes. Hence, it is well suited for simulations with both small-scale features and strong gradients. A generalized Roe average is proposed to make the multicomponent model of Shyue (1998 J. Comput. Phys. 142 208–42) suitable for high-order accurate reconstruction schemes. A first sequence of single-fluid simulations is conducted and compared to the experiment. We find that the WENO-CU6 method better resolves small-scale structures, leading to earlier symmetry breaking and increased mixing. The first simulation, however, fails to correctly predict the global characteristic structures of the RMI. This is due to a mismatch of the post-shock parameters in single-fluid simulations when the pre-shock states are matched with the experiment. When the post-shock parameters are matched, much better agreement with the experimental data is achieved. In a sequence of multifluid simulations, the uncertainty in the density gradient associated with transition between the fluids is assessed. Thereby the multifluid simulations show a considerable improvement over the single-fluid simulations.

Preventing numerical errors generated by interface-capturing schemes in compressible multi-material flows

In the present work, errors generated in computations of compressible multi-material flows using shock-capturing schemes are examined, specifically pressure oscillations (when the specific heats ratio is variable), but also temperature spikes and species conservation errors. These numerical errors are generated at material discontinuities due to an inconsistent treatment of the convective terms. Though temperature errors are irrelevant to solutions to the Euler equations, it is shown that they have the potential to lead to problems when physical diffusion is included, i.e., for the Navier–Stokes equations. These errors are studied analytically and numerically by considering the one-dimensional advection of isolated material discontinuities. A methodology preventing such errors for weighted essentially non-oscillatory (WENO) schemes is presented, in which modified WENO weights are used to solve the transport equation for mass fraction in conservative form to prevent temperature and species conservation errors. Pressure errors are prevented by solving an additional transport equation for a given function of the ratio of specific heats. Several multi-dimensional problems with various discontinuities (shocks, material interfaces and contact discontinuities), including the single-mode Richtmyer–Meshkov instability, and turbulence are considered to test the method.

Erratum: Physics of reshock and mixing in single-mode Richtmyer-Meshkov instability [Phys. Rev. E76, 026319 (2007)

Erratum: Physics of reshock and mixing in single-mode Richtmyer-Meshkov instability [Phys. Rev. E<b>76</b>, 026319 (2007)

Shock-resolved Navier–Stokes simulation of the Richtmyer–Meshkov instability start-up at a light–heavy interface

The single-mode Richtmyer–Meshkov instability is investigated using a first-order perturbation of the two-dimensional Navier–Stokes equations about a one-dimensional unsteady shock-resolved base flow. A feature-tracking local refinement scheme is used to fully resolve the viscous internal structure of the shock. This method captures perturbations on the shocks and their influence on the interface growth throughout the simulation, to accurately examine the start-up and early linear growth phases of the instability. Results are compared to analytic models of the instability, showing some agreement with predicted asymptotic growth rates towards the inviscid limit, but significant discrepancies are noted in the transient growth phase. Viscous effects are found to be inadequately predicted by existing models.

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.

Assessment of Very High Order of Accuracy in Implicit LES models

This paper looks at the use of high-resolution and very high-order methods for implicit large-eddy simulation (ILES), with the specific example of simulating the multicomponent two-dimensional single-mode Richtmyer–Meshkov instability for which experimental data is available. The two gases are air and SF6, making stringent demands on the models used in the code. The interface between the two gases is initialized with a simple sinusoidal perturbation over a wavelength of 59mm, and a shock of strength Mach 1.3 is passed through this interface. The main comparison is between the second-order monotone upwind-centered scheme for conservation law methods of van Leer (1979, “Towards the Ultimate Conservative Difference Scheme,” J. Comput. Phys. 32, pp. 101–136) and the current state-of-the-art weighted essentially nonoscillatory interpolation, which is presented to ninth order, concentrating on the effect on resolution of the instability on coarse grids. The higher-order methods as expected provide better resolved and more physical features than the second-order methods on the same grid resolution. While it is not possible to make a definitive statement, the simulations have indicated that the extra time required for the higher-order reconstruction is less than the time saved by being able to obtain the same or better accuracy at lower computational cost (fewer grid points). It should also be noted that all simulations give a good representation of the growth rate of the instability, comparing very favorably to the experimental results, and as such far better than the currently existing theoretical models. This serves to further indicate that the ILES approach is capable of providing accurately physical information despite the lack of any formal subgrid model.

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.

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

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 shows 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.

Experiments on the late-time development of single-mode Richtmyer–Meshkov instability

The late-time development of Richtmyer–Meshkov instability is studied in shock tube experiments. This investigation makes use of the experimental apparatus and visualization methods utilized in the earlier study of Collins and Jacobs [J. Fluid Mech. 464, 113 (2002)] but employs stronger shocks and initial perturbations with shorter wavelengths to obtain much later-time (in the dimensionless sense) images of the single-mode instability. These modifications produce a very detailed look at the evolution of the late-time single-mode instability, revealing the transition and development of turbulence in the vortex cores that eventually results in the disintegration of the laminar vortex structures into small scale features. Amplitude measurements taken from these images are shown to be effectively collapsed when plotted in dimensionless variables defined using the wave number and the initial growth rate. The amplitude measurements are compared with several late-time nonlinear models and solutions. The best agreement is obtained with the model of Sadot et al. [Phys. Rev. Lett. 80, 1654 (1998)] which can be slightly improved by modifying the expression for the late-time asymptotic growth rate.