Taylor Instability of the Interface between Superposed Fluids -Solution by Successive Approximations

Ingraham, R. L.

doi:10.1088/0370-1301/67/10/302

Cited by 33 publications

(7 citation statements)

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“…1,3-8 Perturbation methods 9 have been used by several authors to describe the weakly nonlinear regime of interfacial instabilities. [10][11][12][13][14] In particular, the first step has been carried out by Ingraham for the RayleighTaylor instability. In such methods, the small parameter is the initial wave steepness, a 0 k, where a 0 and k are the initial amplitude and the wave number of the interface, respectively.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear regime of a multimode Richtmyer–Meshkov instability: A simplified perturbation theory

2002

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In this paper we present a drastic simplification of the perturbation method for the Richtmyer–Meshkov instability developed by Zhang and Sohn [Phys. Fluids 9, 1106 (1997)]. This theory is devoted to the calculus of the growth rate of the perturbation of the interface in the weakly nonlinear stage. In the standard approach, expansions appear to be power series in time. We build accurate approximations by retaining only the terms with the highest power in time. This simplifies and accelerates the solution. High-order expressions are then easily reachable. Furthermore, computations for multimode interfaces become tractable. The accuracy of this approach is checked against two-dimensional numerical simulations. The selection mode process is studied and the phase between modes is shown to be as important as the wave number or the amplitude. Inferences for the intermediate nonlinear regime are also proposed. In particular, a class of homothetic configurations is inferred; its validity is verified with numerical simulations even as vortex structures appear at the interface.

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Section: Introductionmentioning

confidence: 99%

Nonlinear regime of a multimode Richtmyer–Meshkov instability: A simplified perturbation theory

2002

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“…These expressions agree with previous reported results for classical RTI. 61,62 A difference to highlight between RTI and MRTI is the following. In the case of RTI, the Fourier component ξ 2 and the O( 2 ) correction to ξ 1 are always negative [see Eqs.…”

Section: Double-harmonic Weakly-nonlinear Mrtimentioning

confidence: 99%

On a variational formulation of the weakly nonlinear magnetic Rayleigh–Taylor instability

Ruiz

2020

Physics of Plasmas

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The magnetic-Rayleigh-Taylor (MRT) instability is a ubiquitous phenomenon that occurs in magneticallydriven Z-pinch implosions. It is important to understand this instability since it can decrease the performance of such implosions. In this work, I present a theoretical model for the weakly nonlinear MRT instability. I obtain such model by asymptotically expanding an action principle, whose Lagrangian leads to the fully nonlinear MRT equations. After introducing a suitable choice of coordinates, I show that the theory can be cast as a Hamiltonian system, whose Hamiltonian is calculated up to sixth order in a perturbation parameter. The resulting theory captures the harmonic generation of MRT modes. In particular, it is shown that the saturation amplitude of the linear MRT instability grows as the stabilization effect of the magnetic-field tension increases. Overall, the theory provides an intuitive interpretation of the weakly nonlinear MRT instability and provides a systematic approach for studying this instability in more complex settings.

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“…We have shown that the governing equations, (8)-(11), (13)-(15), (18), (21), (23) and (26) in three dimensions depend neither on k x , nor on k y explicitly. One can see from these equations and (6), (7), (12), (16), (17), (21)…”

Section: Quantitative Theory Of Richtmyer-meshkov Instability 15mentioning

confidence: 99%

“…The second and third order perturbation solutions for the Rayleigh-Taylor instability (interfacial instability driven by a gravitational force) in two dimensions have been obtained by Ingraham [12] and Chang [5], respectively. Jacobs and Catton derived third order solutions for the Rayleigh-Taylor instability in three dimensions [13].…”

Section: Nonlinear Perturbation Solutions For Incompressible Fluids Imentioning

confidence: 99%

Quantitative theory of Richtmyer-Meshkov instability in three dimensions

Zhang¹,

Sohn

1999

Z. angew. Math. Phys.

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A material interface between two fluids of different density accelerated by a shock wave is unstable. This instability is known as the Richtmyer-Meshkov instability. Previous theoretical and numerical studies of this instability primarily focused on fluids in two dimensions. In this paper, we present studies of the Richtmyer-Meshkov instability in three dimensions. There are three main new results presented here: (1) The analysis of the linear theory of the RichtmyerMeshkov instability for both the reflected shock and reflected rarefaction wave cases.(2) Derivations of nonlinear perturbative solutions for an unstable interface between incompressible fluids (evaluated explicitly for the impulsive model through the third order). (3) A quantitative nonlinear theory of the compressible Richtmyer-Meshkov instability from early to later times. Our nonlinear theory contains no free parameter and provides analytical predictions for the overall growth rate, as well as the growth rates of bubble and spike, of Richtmyer-Meshkov unstable interfaces. Comparison to numerical simulations, which show excellent agreement to our theory, will be presented separately.Mathematics Subject Classification (1991). 76N15, 76L05, 76Bxx, 76E30.

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Taylor Instability of the Interface between Superposed Fluids -Solution by Successive Approximations

Cited by 33 publications

References 1 publication

Nonlinear regime of a multimode Richtmyer–Meshkov instability: A simplified perturbation theory

Nonlinear regime of a multimode Richtmyer–Meshkov instability: A simplified perturbation theory

On a variational formulation of the weakly nonlinear magnetic Rayleigh–Taylor instability

Quantitative theory of Richtmyer-Meshkov instability in three dimensions

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