Richtmyer–Meshkov instability for elastic–plastic solids in converging geometries☆
Introduction
The instability caused by the passage of a shock wave on a perturbed interface separating two fluids of different properties was first described by and named after Richtmyer (1960) and Meshkov (1970). The former constructed an approximate analytical solution for the growth rate of a slightly perturbed contact discontinuity using an impulsive model that replicates the passage of a shock parallel to the main position of the interface. The latter qualitatively confirmed the predicted results by means of shock-tube experiments. Baroclinic vorticity, generated by the misalignment of the density gradient (normal to the interface) and the pressure gradient (normal to the shock) causes the interface perturbations to grow unless there exists a mechanism that could dissipate or advect such vorticity. This problem was originally regarded as the impulsive limit of the Rayleigh–Taylor instability (RTI), in which a perturbed density interface evolves under the effect of a continuous or sustained acceleration (Taylor, 1950). However, in recent years, the Richtmyer–Meshkov instability (RMI) has gained in relevance and independence from RTI studies, as indicated by several annual reviews in the subject (e.g., Zabusky, 1999, Brouillette, 2002) and its relevance to multiple fields, including compressible turbulent mixing (Mikaelian, 1989, Youngs, 1994, Vetter and Sturtevant, 1995, Zhou et al., 2003, Lombardini, 2008), combustion (Khokhlov et al., 1999), magnetohydrodynamics (Wheatley et al., 2005), plasma physics (Lindl et al., 1992), inertial confinement fusion (Lindl et al., 1992), astrophysical phenomena (Arnett, 2000, Fryxell et al., 1991, Maran et al., 2000, Stone et al., 1995), and solid mechanics (Plohr and Plohr, 2005, Piriz et al., 2006).
Most efforts to describe the Richtmyer–Meshkov instability by means of analytical models, numerical simulations, or experiments have been performed for fluid–fluid interfaces and mainly for planar geometries with the aim of understanding the start-up process (Lombardini and Pullin, 2009b, Griffond, 2006, Vandenboomgaerde et al., 1998, Wouchuk and Nishihara, 1996) or the subsequent nonlinear regime (Samtaney and Zabusky, 1993). The RMI in converging geometries mostly appears in experiments aiming to achieve inertial confinement fusion (ICF) (Lindl, 1998), plasma generation (Lanier et al., 2003), and in astrophysical phenomena, such as supernova formation (Fryxell et al., 1991). This geometrical configuration represents a more complex problem owing to the acceleration of the shock and interface as they approach the axis/origin, the second shock–interface interaction (re-shock) after the transmitted shock reaches the axis, and the presence of secondary instabilities such as Rayleigh–Taylor (Yu and Livescu, 2008, Mikaelian, 2004) (interface acceleration) and Kelvin–Helmholtz (discontinuity in tangential velocities across the interface). Studies of RMI in curved or converging geometries are scarce even in the fluid case. Mikaelian (2004) developed a model for the start-up process in converging geometries that applies to both RMI and RTI. Zhang and Graham (1998) performed numerical simulations to qualitatively explain the differences between the four possible problem configurations: shock moving from heavy to light or light to heavy material across the interface and imploding or exploding direction of the shock. Finally, Lombardini (2008) and Lombardini and Pullin (2009a) studied in detail the influence of initial parameters (shock strength and wave number of the interface perturbation) of heavy–light and light–heavy systems in the imploding configuration.
In the field of solid mechanics, there exist multiple features of interest that make the flow intrinsically different from the fluid case. Shear waves are able to carry vorticity away from the interface, reducing and even canceling the growth of the interface amplitude. The elastic regime of the Richtmyer–Meshkov solid flow was analytically investigated in López Ortega et al. (2010) using a linearized, incompressible, and impulsive-driven model for the time evolution of the interface that revealed distinct regions of stability depending on the shear modulus and density of the materials. Plohr and Plohr (2005) developed a semi-numerical model for the shock-driven motion of an interface separating two elastic and compressible materials. The results obtained were limited to a narrow range of initial conditions and material parameters for which the interface separating two solids was oscillatory and stable after the shock wave–interface interaction. Piriz et al. (2006) constructed a semi-analytical model restricted to oscillatory motion of the interface and conducted simulations to derive an approximate result for the long-term behavior of the perturbations.
The problem of examining the Richtmyer–Meshkov flow in elastic–plastic solids is of special relevance since shocks often stress materials beyond their yield point. When the material surrounding the interface enters the plastic regime, its ability to flow without being submitted to large changes in stresses causes the interface to grow in an unstable manner, similar to a fluid-like behavior. In the long term, transitions between the plastic and elastic regimes at a material interface can occur due to relaxation of stresses, potentially modifying the time evolution of the perturbations of the interface from unstable to stable behavior. Unfortunately, plasticity theory is intrinsically non-linear due to yield criteria being based on norms (e.g., von Mises) or differences in absolute value (e.g., Tresca) of shear stresses. Purely analytical models are extremely difficult to formulate and the study of the problem using numerical techniques becomes necessary. Piriz et al. (2008) expanded their semi-analytical model to elastic–plastic solids in planar geometry, predicting an initial fluid-like constant growth rate driven by the solid exhibiting plastic behavior that eventually turns into stable behavior as stresses are relaxed below the yield stress. However, this model requires input from numerical simulations for closure. Nonetheless, results suggest the existence of a relation between the maximum amplitude achieved by the interface before it enters the elastic regime and the strength of the material. Dimonte et al. (2011) later performed a series of simulations using hypo-elastic continuum mechanics and molecular dynamics codes contrasted with experiments that drew the same conclusions with respect to the relation between the yield stress and the maximum amplitude achieved by perturbations. The numerical method applied in this paper, which makes use of hyper-elastic constitutive laws (Marsden and Hughes, 1993) instead, was used to confirm Dimonte et al.'s results and extend the range of validity of the analysis to different materials, plasticity laws, and shock conditions (López Ortega et al., 2014a).
This paper examines for the first time the Richtmyer–Meshkov instability that results from the motion of a shock-driven interface separating an elastic–plastic solid from a gas in a cylindrical geometry making use of numerical techniques. In the configuration considered here, the gas is occupying the inner region close to the axis with the solid material surrounding it. A shock is generated in the solid and propagates inwards towards the interface. The shock–interface interaction produces a transmitted shock in the fluid, a radially outward-moving expansion in the solid, and the growth of the interface perturbations. Upon reflecting from the axis, the shock in the fluid interacts with the interface for a second time (re-shock), slowing its motion in a Rayleigh–Taylor unstable manner and producing a secondary Richtmyer–Meshkov instability. Previously obtained planar results (Piriz et al., 2008, Dimonte et al., 2011, López Ortega et al., 2014a) are only valid in the start-up regime of the interface motion as the continuous acceleration of the interface and compression of the materials prevent relaxation of stresses and a stable behavior of the interface. We study the effect of diverse initial conditions and material parameters on the interface stability and behavior before and after the re-shock. Parameters of interest describing the initial geometry of the interface are the wave number n, the mean radius R0, and the amplitude η0 of the sinusoidal perturbations of the interface. The motion is also dependent on the initial solid–gas Atwood ratio and the incoming shock Mach number Ms. Finally, the effect of the ratio of yield stress to the shear modulus μ0 is considered. Table 1 summarizes the range of parameters employed for each of these variables.
The paper is structured as follows. Section 2 describes the equations of motion for solid materials in an Eulerian framework. In particular, equations that track the history of deformations are added to the classical system of Euler equations for mass, linear momentum, and total energy conservation. We also comment on the equations of state used in the modeling of solid and fluid media. A brief description of the discretization of these equations and the algorithm employed for carrying out multi-material simulations is also provided. Section 3 introduces the initial conditions and computational domain. A simulation that considers the parameters marked in bold in Table 1 is shown in Section 4. The purpose of this section is to describe the most important features of the flow before moving into the parametric study. Section 5 contains the results of the parametric study, separated in different subsection referring to each of the parameters evaluated (i.e., geometry parameters, Atwood ratio, initial shock Mach number, and yield stress). Each case is supported by tables, time evolution graphics, and contour plots that help illustrate the main conclusions derived from the analysis. Finally, in Section 6, the previous results, in which the yield stress has been controlled assuming the hypothesis of perfect plasticity, are compared to the interface behavior obtained when a plasticity model calibrated with experimental results is used.
Section snippets
Equations and numerical implementation
In an Eulerian frame of reference and making use of Cartesian coordinates , the equations that describe the motion of elastic–plastic solid media arewhere ρ, , , and e are the density, velocity, Cauchy stress, and specific internal energy, respectively. The deformation history in the solid is tracked by the so-called elastic inverse deformation
Computational domain. Boundary and initial conditions
A square computational domain with Cartesian grid cells of aspect ratio unity is employed for simulating a quadrant of a full cylindrical simulation. Reflective conditions are imposed in the left and bottom domain boundaries while zero-gradient boundary conditions are enforced in the upper and right boundaries in a way such that the axis is located at . The number of cells and levels of refinement employed in our simulations is discussed in the next section alongside the
Baseline problem description and convergence analysis
This section describes the time evolution of a reference problem with initial parameters n=4, , (i.e., ), Ms=1.65, and (i.e., ). The initial density and pressure of the gas (, ) correspond to the compression at room temperature of helium. Although these values may not be accessible in experiments, CH foams (such as the ones used in Lanier et al., 2003) are available in this density range and their compression
Interface parametric analysis
This section describes the behavior of the solid–gas interface under variation of initial and material parameters. The study of the interface evolution in the impulsive and linearized analytical model for elastic solids (López Ortega et al., 2010, Plohr and Plohr, 2005, Piriz et al., 2006) reveals that the growth rate of the interface is mainly affected by the initial geometry (wave number and amplitude), the density ratio of the materials, and the initial velocity of the interface. In the
Comparison with calibrated plasticity model
In this section, we discuss results obtained using a plasticity model for OFHC copper that has been calibrated to yield minimum error with respect to tensile one-dimensional experiments at different strain-rates and temperatures. Following Barton and Romenski (2012), a model based upon dislocation kinetics is assumed to bewhere τ0 is a reference relaxation time, N0 is the initial dislocation density, M a multiplication parameter, D is
Conclusion
We performed, making use of a newly developed multi-material Eulerian numerical method for solid and fluid media, a parametric study of the effect of multiple geometrical and material parameters on the Ritchmyer–Meshkov instability for solid–gas interfaces in cylindrical geometries. This configuration is relevant in the study and development of inertial confinement fusion (ICF).
The numerical methodology employed has certain limitations associated with rupture of symmetry, conservation of mass,
References (48)
- et al.
Eulerian adaptive finite-difference method for high-velocity impact and penetration problems
J. Comput. Phys.
(2013) - et al.
Local adaptive mesh refinement for shock hydrodynamics
J. Comput. Phys.
(1989) - et al.
Efficient implementation of weighted ENO schemes
J. Comput. Phys.
(1996) - et al.
Numerical simulation of deflagration-to-detonation transition: the role of shock–flame interactions in turbulent flames
Combust. Flame
(1999) - et al.
Ghost fluid method for strong shock impacting on material interface
J. Comput. Phys.
(2003) - et al.
Numerical simulation of elastic–plastic solid mechanics using an Eulerian stretch tensor approach and HLLD Riemann solver
J. Comput. Phys.
(2014) Turbulent mixing generated by Rayleigh–Taylor and Richtmyer–Meshkov instabilities
Physica D: Nonlinear Phenom.
(1989)The role of mixing in astrophysics
Astrophys. J.
(2000)- et al.
On computational modelling of strain-hardening material dynamics
Commun. Comput. Phys.
(2012) The Richtmyer–Meshkov instability
Ann. Rev. Fluid Mech.
(2002)
Construction and application of an AMR algorithm for distributed memory computers
Use of the Richtmyer–Meshkov instability to infer yield stress at high-energy densities
Phys. Rev. Lett.
Dynamics of impulsive metal heating by a current and electrical explosion of conductors
J. Appl. Mech. Tech. Phys.
Postponement of saturation of the Richtmyer–Meshkov instability in a convergent geometry
Phys. Rev. Lett.
Instabilities and clumping in SN 1987A. I—early evolution in two dimensions
Astrophys. J.
Linear interaction analysis for Richtmyer–Meshkov instability at low Atwood numbers
Phys. Fluids
Flow stress and constitutive model of OFHC Cu for large deformation, different temperatures and different strain rates
Baozha Yu Chongji/Explos. Shock Waves
Multimode seeded Richtmyer–Meshkov mixing in a convergent, compressible, miscible plasma system
Phys. Plasmas
Inertial Confinement Fusion: The Quest for Ignition and Energy Gain using Indirect Drive
Progress toward ignition and burn propagation in inertial confinement fusion
Phys. Today
Small-amplitude perturbations in the three-dimensional cylindrical Richtmyer–Meshkov instability
Phys. Fluids
Startup process in the Richtmyer–Meshkov instability
Phys. Fluids
Large eddy simulations of the Richtmyer–Meshkov instability in a converging geometry
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This work was supported by the Department of Energy National Nuclear Security Administration under Award no. DE-FC52-08NA28613.