Analysis of Iterative Methods for the Steady and Unsteady Stokes Problem: Application to Spectral Element Discretizations

A new and detailed analysis of the basic Uzawa algorithm for decoupling of the pressure and the velocity in the steady and unsteady Stokes operator is presented. The paper focuses on the following new aspects: explicit construction of the Uzawa pressure-operator spectrum for a semiperiodic model problem; general relationship of the convergence rate of the Uzawa procedure to classical inf-sup discretization analysis; and application of the method to high-order variational discretization

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

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

Turbulent mixing driven by spherical implosions. Part 2. Turbulence statistics

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 one, or inversely, from the heavy to the light fluid. In Part 1 (Lombardini, M., Pullin, D. I. & Meiron, D. I., J. Fluid Mech., vol. 748, 2014, pp. 85-112), we described the computational problem and presented results on the radially symmetric flow, the mean flow, and the growth of the mixing layer. In particular, it was shown that both configurations reach similar convergence ratios ≈2. Here, turbulent mixing is studied through various turbulence statistics. The mixing activity is first measured through two mixing parameters, the mixing fraction parameter Theta and the effective Atwood ratio A(e), which reach similar late time values in both light-heavy and heavy-light configurations. The Taylor-scale Reynolds numbers attained at late times are estimated ≈2000 in the light-heavy case and 1000 in the heavy-light case. An analysis of the density self-correlation b, a fundamental quantity in the study of variable-density turbulence, shows asymmetries in the mixing layer and non-Boussinesq effects generally observed in high-Reynolds-number Rayleigh-Taylor (RT) turbulence. These traits are more pronounced in the light-heavy mixing layer, as a result of its flow history, in particular because of RT-unstable phases (see Part 1). Another measure distinguishing light-heavy from heavy-light mixing is the velocity-to-scalar Taylor microscales ratio. In particular, at late times, larger values of this ratio are reported in the heavy-light case. The late-time mixing displays the traits some of the traits of the decaying turbulence observed in planar Richtmyer-Meshkov (RM) flows. Only partial isotropization of the flow (in the sense of turbulent kinetic energy (TKE) and dissipation) is observed at late times, the Reynolds normal stresses (and, thus, the directional Taylor microscales) being anisotropic while the directional Kolmogorov microscales approach isotropy. A spectral analysis is developed for the general study of statistically isotropic turbulent fields on a spherical surface, and applied to the present flow. The resulting angular power spectra show the development of an inertial subrange approaching a Kolmogorov-like -5/3 power law at high wavenumbers, similarly to the scaling obtained in planar geometry. It confirms the findings of Thomas & Kares (Phys. Rev. Lett., vol. 109, 2012, 075004) at higher convergence ratios and indicates that the turbulent scales do not seem to feel the effect of the spherical mixing-layer curvature

Chaotic domains: A numerical investigation

We study the chaotic domain state in rotating convection using a model equation that allows for a continuous range of roll orientations as in the experimental system. Methods are developed for extracting the domain configuration from the resulting patterns that should be applicable to a wide range of domain states. Comparison with the truncated three mode amplitude equation description is made

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

Numerical simulations of the Richtmyer-Meshkov instability in solid-vacuum interfaces using calibrated plasticity laws

The Richtmyer-Meshkov instability of interfaces separating elastic-plastic materials from vacuum (heavy-light configuration) is studied by means of computational techniques. A fully Eulerian multimaterial algorithm that solves consistently the Euler equations and the time evolution of the deformations in the material is applied to three distinct materials (copper, aluminum, and stainless steel). If a perfectly plastic constitutive relation is considered, an empirical law is computed that relates the long-term perturbation amplitude of the interface, its maximum growth rate, the initial density, and the yield stress of the material. It is shown that this linear relation can be extended to materials that follow more complex plastic behavior which can account for rate dependency, hardening, and thermal softening, and to situations in which small-perturbation theory is no longer valid. In effect, the yield stress computed from measurements of the long-term amplitude and maximum growth rate closely matches the von Mises stress found at the interface of solid materials for a wide range of cases with different initial parameters

The 1998 Center for Simulation of Dynamic Response in Materials Annual Technical Report

Introduction: This annual report describes research accomplishments for FY 98 of the Center for Simulation of Dynamic Response of Materials. The Center is constructing a virtual shock physics facility in which the full three dimensional response of a variety of target materials can be computed for a wide range of compressive, tensional, and shear loadings, including those produced by detonation of energetic materials. The goals are to facilitate computation of a variety of experiments in which strong shock and detonation waves are made to impinge on targets consisting of various combinations of materials, compute the subsequent dynamic response of the target materials, and validate these computations against experimental data

Domain Coarsening in Systems Far from Equilibrium

The growth of domains of stripes evolving from random initial conditions is studied in numerical simulations of models of systems far from equilibrium such as Rayleigh-Benard convection. The scaling of the size of the domains deduced from the inverse width of the Fourier spectrum is studied for both potential and nonpotential models. The morphology of the domains and the defect structures are however quite different in the two cases, and evidence is presented for a second length scale in the nonpotential case.Comment: 11 pages, RevTeX; 3 uufiles encoded postscript figures appende