Small-amplitude perturbations in the three-dimensional cylindrical Richtmyer–Meshkov instability

Lombardini, M.; Pullin, D. I.

doi:10.1063/1.3258668

Cited by 41 publications

(33 citation statements)

References 29 publications

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“…The self-similar structure has been preliminary confirmed by simulations of a single converging shock. In particular, n has been computed before and after reflection at the centre Lombardini & Pullin (2009). Initializing the shock using Chisnell 's self-similar solution not only avoids spurious waves that would appear if setting up the shock as a Riemann problem solution for the strictly axisymmetric shock-implosion process, but also leaves the shock thickness as the only intrinsic length scale.…”

Section: Computational Approachmentioning

confidence: 99%

“…The early-time growth of these instabilities has been investigated in cyli ndrical (Bell 1951 ;Mikaelian 2005;Yu & Livescu 2008;Lombardini & Pullin 2009) and spherical geometries (Bell 195 I ;Plesset I 95.+;Mikaelian 1990;Kumar, Hornung & Sturtevant 200 3;Mankbadi & Balachandar 201 2). The stability analysis by Krechetnikov (2009) actually unifies some of the work previously cited by uncovering the interrelation between the RT and RM instabilities and the general effect of interfacial curvature.…”

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confidence: 99%

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Turbulent mixing driven by spherical implosions. Part 1. Flow description and mixing-layer growth

2014

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We present large-eddy si mulations (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 s hock Mach number ~1.2, the density ratio (precisely IAol ~ 0.67) and the perturbation shape (dominant spherical wavenumber .f 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. , 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 Be ll (Los Alamos Scientific Laboratory Report, LA-132 L, 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 Be ll-Plessel 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 ce > 10) mode numbers considered.

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Section: Computational Approachmentioning

confidence: 99%

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confidence: 99%

Turbulent mixing driven by spherical implosions. Part 1. Flow description and mixing-layer growth

2014

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“…In particular, we plan to analyze the Richtmyer-Meshkov flow that would be generated when an imploding wave impacts an inhomogeneous material or more simply an interface between two different materials (e.g., solid-solid or solid-gas interfaces). Previous publications by the authors already analyzed the Richtmyer-Meshkov flow at an impulsively accelerated planar interface between two elastic incompressible solids [23], obtaining stable behavior of the interface in any conditions, and for gas-gas interfaces in converging geometry [24], which can be unstable.…”

Section: Discussionmentioning

confidence: 99%

Converging shocks in elastic-plastic solids

2011

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We present an approximate description of the behavior of an elastic-plastic material processed by a cylindrically or spherically symmetric converging shock, following Whitham's shock dynamics theory. Originally applied with success to various gas dynamics problems, this theory is presently derived for solid media, in both elastic and plastic regimes. The exact solutions of the shock dynamics equations obtained reproduce well the results obtained by high-resolution numerical simulations. The examined constitutive laws share a compressible neo-Hookean structure for the internal energy e = e s (I 1 ) + e h (ρ,ς), where e s accounts for shear through the first invariant of the Cauchy-Green tensor, and e h represents the hydrostatic contribution as a function of the density ρ and entropy ς . In the strong-shock limit, reached as the shock approaches the axis or origin r = 0, we show that compression effects are dominant over shear deformations. For an isothermal constitutive law, i.e., e h = e h (ρ), with a power-law dependence e h ∝ ρ α , shock dynamics predicts that for a converging shock located at r = R(t) at time t, the Mach number increases as M ∝ [log(1/R)] α , independently of the space index s, where s = 2 in cylindrical geometry and 3 in spherical geometry. An alternative isothermal constitutive law with p(ρ) of the arctanh type, which enforces a finite density in the strong-shock limit, leads to M ∝ R −(s−1) for strong shocks. A nonisothermal constitutive law, whose hydrostatic part e h is that of an ideal gas, is also tested, recovering the strong-shock limit M ∝ R −(s−1)/n(γ ) originally derived by Whitham for perfect gases, where γ is inherently related to the maximum compression ratio that the material can reach, (γ + 1)/(γ − 1). From these strong-shock limits, we also estimate analytically the density, radial velocity, pressure, and sound speed immediately behind the shock. While the hydrostatic part of the energy essentially commands the strong-shock behavior, the shear modulus and yield stress modify the compression ratio and velocity of the shock far from the axis or origin. A characterization of the elastic-plastic transition in converging shocks, which involves an elastic precursor and a plastic compression region, is finally exposed.

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“…4.5, we plot the scaled growth rate history in hydrodynamics for different perturbation wavenumbers m = 16, 64, 256. We nondimensionalize both the time scale and growth rate, following Lombardini and Pullin [14]. We choose R 0 /a 0 m as our reference time scale, where a 0 is the unshocked sound speed.…”

Section: Solution Details For Rmi In Hydrodynamicsmentioning

confidence: 99%

“…Ponchaut et al [13] adopted the method of series expansion and developed a numerical, characteristics based solution which was also valid for weak shocks. In the context of the RMI, Lombardini and Pullin [14] developed the theory for asymptotic growth rate in density interface for a three dimensional cylindrical geometry. In a converging geometry, not only do we have the RMI due to initial shock impact, but also the acceleration of the interface leads to an Rayleigh-Taylor instability (RTI) phase which changes the growth rate of the interface perturbations.…”

Section: Introductionmentioning

confidence: 99%

Linear Simulations of the Cylindrical Richtmyer-Meshkov Instability in Hydrodynamics and MHD

Gao

Samtaney

2015

29th International Symposium on Shock Waves 2

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Linear Simulations of the Cylindrical Richtmyer-MeshkovInstability in Hydrodynamics and MHD Song GaoThe Richtmyer-Meshkov instability occurs when density-stratified interfaces are impulsively accelerated, typically by a shock wave. We present a numerical method to simulate the Richtmyer-Meshkov instability in cylindrical geometry. The ideal MHD equations are linearized about a time-dependent base state to yield linear partial differential equations governing the perturbed quantities. Convergence tests demonstrate that second order accuracy is achieved for smooth flows, and the order of accuracy is between first and second order for flows with discontinuities.Numerical results are presented for cases of interfaces with positive Atwood number and purely azimuthal perturbations. In hydrodynamics, the Richtmyer-Meshkov instability growth of perturbations is followed by a Rayleigh-Taylor growth phase.In MHD, numerical results indicate that the perturbations can be suppressed for sufficiently large perturbation wavenumbers and magnetic fields. 5 ACKNOWLEDGEMENTSSincere thanks to my supervisor, Prof. Ravi Samtaney, for his profound knowledge, patience and enthusiasm.Thanks to the members of my thesis examination committee, Prof. Sigurdur Thoroddsen and Prof.Georgiy Stenchikov, for their valuable comments and precious time.Thanks to Dr. Manuel Lombardini, for the interesting discussion and invaluable help. Thanks to Mr. Wei Gao, for his great encouragement.Thanks to my parents, for raising me up, and my unmet wife.Last but not least, special thanks to Dr. Fabrizio Bisetti, for the sharp lesson he has taught me. 6

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Small-amplitude perturbations in the three-dimensional cylindrical Richtmyer–Meshkov instability

Cited by 41 publications

References 29 publications

Turbulent mixing driven by spherical implosions. Part 1. Flow description and mixing-layer growth

Turbulent mixing driven by spherical implosions. Part 1. Flow description and mixing-layer growth

Converging shocks in elastic-plastic solids

Linear Simulations of the Cylindrical Richtmyer-Meshkov Instability in Hydrodynamics and MHD

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