scholarly journals Efficient perturbation methods for Richtmyer–Meshkov and Rayleigh–Taylor instabilities: Weakly nonlinear stage and beyond

2003 ◽  
Vol 21 (3) ◽  
pp. 321-325 ◽  
Author(s):  
M. VANDENBOOMGAERDE ◽  
C. CHERFILS ◽  
D. GALMICHE ◽  
S. GAUTHIER ◽  
P.A. RAVIART
Keyword(s):  
Growth Rate ◽  
Perturbation Method ◽  
Numerical Simulations ◽  
Perturbation Methods ◽  
High Order ◽  
Standard Approach ◽  
Weakly Nonlinear ◽  
Nonlinear Stage ◽  
Selection Mode ◽  
Rayleigh Taylor Instability

The simplified perturbation method of Vandenboomgaerdeet al.(2002) is applied to both the Richtmyer–Meshkov and the Rayleigh–Taylor instabilities. 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 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. For the Richtmyer–Meshkov instability, multimode configurations become tractable and the selection mode process can be studied. 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. This kind of method can also be used for the Rayleigh–Taylor instability. Some examples are presented.

scholarly journals Nonlinear stage in the development of hydrodynamic instability in laser targets

1990 ◽  
Vol 8 (1-2) ◽  
pp. 173-182 ◽  
Author(s):  
E. G. Gamaly ◽  
I. G. Lebo ◽  
V. B. Rozanov ◽  
A. P. Favorsky ◽  
A. O. Fedyanin ◽  
...  
Keyword(s):  
Growth Rate ◽  
Laser Irradiation ◽  
Hydrodynamic Instability ◽  
Short Wave ◽  
Numerical Code ◽  
Linear Stage ◽  
Long Wave ◽  
Nonlinear Stage ◽  
Rayleigh Taylor Instability ◽  
Image Position

The development of a hydrodynamic instability in laser targets is studied by means of a 2-D numerical code “ATLANT”. During the linear stage, the perturbations grow as:In the nonlinear stage the growth rate of the Rayleigh-Taylor instability is reduced, and new harmonics are generated. The effect of the nonuniformity of the laser irradiation has been investigated for long-wave and short-wave perturbations. The growth rate of short-wave perturbations may be effectively decreased by means of symmetrical pre-pulses.

scholarly journals Ablation effects in weakly nonlinear stage of the ablative Rayleigh–Taylor instability

1996 ◽  
Vol 14 (1) ◽  
pp. 45-54
Author(s):  
Susumu Hasegawa ◽  
Katsunobu Nishihara
Keyword(s):  
Perturbation Theory ◽  
Mode Coupling ◽  
Wave Amplitude ◽  
Taylor Instability ◽  
Weakly Nonlinear ◽  
Effective Gravity ◽  
Nonlinear Stage ◽  
Excited Wave ◽  
Wave Numbers ◽  
Rayleigh Taylor Instability

Weakly nonlinear stage of the ablative Rayleigh-Taylor instability has been studied by the perturbation theory. Mode coupling of linear growing waves with wave numbers kA and kB drives new excited waves with wave numbers k0 (= kA ± kB, 2kA, 2kB). We have investigated time evolution of the excited waves and found that the ablation effect plays an important role even in the nonlinear stage to reduce amplitude of the excited waves. Differences between an ablation surface and a classical contact surface have been discussed. Dependence of the excited wave amplitude on the wavenumber k0, the ablation velocity va, and the effective gravity g is also investigated.

Richtmyer–Meshkov instability on a quasi-single-mode interface

2019 ◽  
Vol 872 ◽  
pp. 729-751 ◽  
Author(s):  
Yu Liang ◽  
Zhigang Zhai ◽  
Juchun Ding ◽  
Xisheng Luo
Keyword(s):  
Nonlinear Model ◽  
Small Amplitude ◽  
Linear Growth ◽  
Single Mode ◽  
High Order ◽  
Initial Amplitude ◽  
Previous Theory ◽  
Weakly Nonlinear ◽  
Nonlinear Stage ◽  
Amplitude Growth

Experiments on Richtmyer–Meshkov instability of quasi-single-mode interfaces are performed. Four quasi-single-mode air/$\text{SF}_{6}$ interfaces with different deviations from the single-mode one are generated by the soap film technique to evaluate the effects of high-order modes on amplitude growth in the linear and weakly nonlinear stages. For each case, two different initial amplitudes are considered to highlight the high-amplitude effect. For the single-mode and saw-tooth interfaces with high initial amplitude, a cavity is observed at the spike head, providing experimental evidence for the previous numerical results for the first time. For the quasi-single-mode interfaces, the fundamental mode is the dominant one such that it determines the amplitude linear growth, and subsequently the impulsive theory gives a reasonable prediction of the experiments by introducing a reduction factor. The discrepancy in linear growth rates between the experiment and the prediction is amplified as the quasi-single-mode interface deviates more severely from the single-mode one. In the weakly nonlinear stage, the nonlinear model valid for a single-mode interface with small amplitude loses efficacy, which indicates that the effects of high-order modes on amplitude growth must be considered. For the saw-tooth interface with small amplitude, the amplitudes of the first three harmonics are extracted from the experiment and compared with the previous theory. The comparison proves that each initial mode develops independently in the linear and weakly nonlinear stages. A nonlinear model proposed by Zhang & Guo (J. Fluid Mech., vol. 786, 2016, pp. 47–61) is then modified by considering the effects of high-order modes. The modified model is proved to be valid in the weakly nonlinear stage even for the cases with high initial amplitude. More high-order modes are needed to match the experiment for the interfaces with a more severe deviation from the single-mode one.

scholarly journals Nonlinear stage in the development of hydrodynamic instability in laser targets

1990 ◽  
Vol 8 (3) ◽  
pp. 399-407 ◽  
Author(s):  
E. G. Gamaly ◽  
A. P. Favorsky ◽  
A. O. Fedyanin ◽  
I. G. Lebo ◽  
E. E. Myshetskaya ◽  
...  
Keyword(s):  
Growth Rate ◽  
Laser Irradiation ◽  
Hydrodynamic Instability ◽  
Short Wave ◽  
Taylor Instability ◽  
Numerical Code ◽  
Linear Stage ◽  
Nonlinear Stage ◽  
Rayleigh Taylor Instability

The development of hydrodynamic instability in laser targets is studied by means of the 2D numerical code “ATLANT.” At the linear stage, perturbations grow as At the nonlinear stage, the growth rate of Rayleigh-Taylor instability is reduced and new harmonics are generated. The effect of the nonuniformity of laser irradiation has been investigated for long- and shortwave perturbations. The growth rate of short-wave perturbations may be effectively decreased by means of symmetrical prepulses.

Saturation mechanism and generated viscosity in gravito-turbulent accretion disks

2020 ◽  
Vol 640 ◽  
pp. A53
Author(s):  
L. Löhnert ◽  
S. Krätschmer ◽  
A. G. Peeters
Keyword(s):  
Growth Rate ◽  
Numerical Simulations ◽  
Accretion Disks ◽  
Gravitational Instability ◽  
Turbulent Viscosity ◽  
Radial Distance ◽  
Maximum Growth ◽  
Wave Number ◽  
Radiative Cooling ◽  
Mixing Length

Here, we address the turbulent dynamics of the gravitational instability in accretion disks, retaining both radiative cooling and irradiation. Due to radiative cooling, the disk is unstable for all values of the Toomre parameter, and an accurate estimate of the maximum growth rate is derived analytically. A detailed study of the turbulent spectra shows a rapid decay with an azimuthal wave number stronger than ky−3, whereas the spectrum is more broad in the radial direction and shows a scaling in the range kx−3 to kx−2. The radial component of the radial velocity profile consists of a superposition of shocks of different heights, and is similar to that found in Burgers’ turbulence. Assuming saturation occurs through nonlinear wave steepening leading to shock formation, we developed a mixing-length model in which the typical length scale is related to the average radial distance between shocks. Furthermore, since the numerical simulations show that linear drive is necessary in order to sustain turbulence, we used the growth rate of the most unstable mode to estimate the typical timescale. The mixing-length model that was obtained agrees well with numerical simulations. The model gives an analytic expression for the turbulent viscosity as a function of the Toomre parameter and cooling time. It predicts that relevant values of α = 10−3 can be obtained in disks that have a Toomre parameter as high as Q ≈ 10.

scholarly journals A new upper bound for the largest growth rate of linear Rayleigh–Taylor instability

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Changsheng Dou ◽  
Jialiang Wang ◽  
Weiwei Wang
Keyword(s):  
Surface Tension ◽  
Growth Rate ◽  
Upper Bound ◽  
Viscous Fluids ◽  
Instability Condition ◽  
Interface Surface ◽  
Incompressible Viscous Fluids ◽  
Rayleigh Taylor Instability ◽  
Surface Tensor ◽  
Rt Instability

AbstractWe investigate the effect of (interface) surface tensor on the linear Rayleigh–Taylor (RT) instability in stratified incompressible viscous fluids. The existence of linear RT instability solutions with largest growth rate Λ is proved under the instability condition (i.e., the surface tension coefficient ϑ is less than a threshold $\vartheta _{\mathrm{c}}$ ϑ c ) by the modified variational method of PDEs. Moreover, we find a new upper bound for Λ. In particular, we directly observe from the upper bound that Λ decreasingly converges to zero as ϑ goes from zero to the threshold $\vartheta _{\mathrm{c}}$ ϑ c .

The Best Perturbation Method for the Parameter Inversion of Two-Dimension Convection-Diffusion Equation

2013 ◽  
Vol 380-384 ◽  
pp. 1143-1146
Author(s):  
Xiang Guo Liu
Keyword(s):  
Inverse Problem ◽  
Diffusion Equation ◽  
Perturbation Method ◽  
Numerical Simulations ◽  
Numerical Solution ◽  
Two Dimensional ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Parameter Inversion ◽  
Parametric Inversion

The paper researches the parametric inversion of the two-dimensional convection-diffusion equation by means of best perturbation method, draw a Numerical Solution for such inverse problem. It is shown by numerical simulations that the method is feasible and effective.

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