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 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.

Harmonic generation by nonlinear self-interaction of a single internal wave mode

2017 ◽  
Vol 828 ◽  
pp. 630-647 ◽  
Author(s):  
Scott Wunsch
Keyword(s):  
Steady State ◽  
Small Amplitude ◽  
Internal Tide ◽  
Single Mode ◽  
Steady State Solution ◽  
Wave Mode ◽  
Internal Tides ◽  
Weakly Nonlinear ◽  
Harmonic Response ◽  
Weakly Nonlinear Theory

Weakly nonlinear theory is used to explore the dynamics of a single-mode internal tide in variable stratification with rotation. Nonlinear self-interaction in variable stratification generates a perturbation which is forced with double the original frequency and wavenumber. The dynamics of the perturbation is analogous to a forced harmonic oscillator, with the steady-state solution corresponding to a bound harmonic matching the forcing frequency and wavenumber. When the forcing frequency is near a natural frequency of the system, even a small-amplitude (nearly linear) internal tide may induce a significant harmonic response. Idealized stratification profiles are utilized to explore the relevance of this effect for oceanic $M_{2}$ baroclinic internal tides, and the results indicate that a rapidly growing harmonic may occur in some environments near the Equator, but is unlikely at higher latitudes. The results are relevant to recent observations of $M_{4}$ (harmonic) internal tides in the South China Sea and elsewhere. More generally, nonlinear self-interaction may contribute to the transfer of energy to smaller scales and the dissipation of baroclinic internal tides, especially in equatorial waters.

scholarly journals An experimental study of the high Mach number and high initial-amplitude effects on the evolution of the single-mode Richtmyer–Meshkov instability

2003 ◽  
Vol 21 (3) ◽  
pp. 341-346 ◽  
Author(s):  
O. SADOT ◽  
A. RIKANATI ◽  
D. ORON ◽  
G. BEN-DOR ◽  
D. SHVARTS
Keyword(s):  
Shock Wave ◽  
Experimental Study ◽  
Mach Number ◽  
Small Amplitude ◽  
Single Mode ◽  
Incident Shock ◽  
Initial Amplitude ◽  
High Mach Number ◽  
High Mach ◽  
Late Stages

The present article describes an experimental study that is a part of an integrated theoretical (Rikanatiet al.2003) and experiential investigation of the Richtmyer–Meshkov (RM) hydrodynamic instability that develops on a perturbed contact surface by a shock wave. The Mach number and the high initial-amplitude effects on the evolution of the single-mode shock-wave-induced instability were studied. To distinguish between the above-mentioned effects, two sets of shock-tube experiments were conducted: high initial amplitudes with a low-Mach incident shock and small amplitude initial conditions with a moderate-Mach incident shock. In the high-amplitude experiments a reduction of the initial velocity with respect to the linear prediction was measured. The results were compared to those predicted by a vorticity deposition model and to previous experiments with moderate and high Mach numbers done by others and good agreement was found. The result suggested that the high initial-amplitude effect is the dominant one rather than the high Mach number effect as suggested by others. In the small amplitude–moderate Mach numbers experiments, a reduction from the impulsive theory was noted at late stages. It is concluded that while high Mach number effect can dramatically change the behavior of the flow at all stages, the high initial-amplitude effect is of minor importance at the late stages. That result is supported by a two-dimensional numerical simulation.

scholarly journals High-order differentiable autoencoder for nonlinear model reduction

2021 ◽  
Vol 40 (4) ◽  
pp. 1-15
Author(s):  
Siyuan Shen ◽  
Yin Yang ◽  
Tianjia Shao ◽  
He Wang ◽  
Chenfanfu Jiang ◽  
...  
Keyword(s):  
Model Reduction ◽  
Nonlinear Model ◽  
High Order ◽  
Nonlinear Model Reduction

Three-dimensional tidal sand waves

2009 ◽  
Vol 618 ◽  
pp. 1-11 ◽  
Author(s):  
PAOLO BLONDEAUX ◽  
GIOVANNA VITTORI
Keyword(s):  
Stability Analysis ◽  
Nonlinear Interaction ◽  
Small Amplitude ◽  
Three Dimensional ◽  
Time Development ◽  
Resonance Mechanism ◽  
Weakly Nonlinear ◽  
Sand Waves ◽  
Tidal Sand ◽  
Harmonic Components

The process which leads to the formation of three-dimensional sand waves is investigated by means of a stability analysis which considers the time development of a small-amplitude bottom perturbation of a shallow tidal sea. The weakly nonlinear interaction of a triad of resonant harmonic components of the bottom perturbation is considered. The results show that the investigated resonance mechanism can trigger the formation of a three-dimensional bottom pattern similar to that observed in the field.

Richtmyer–Meshkov instability on two-dimensional multi-mode interfaces

2021 ◽  
Vol 928 ◽  
Author(s):  
Yu Liang ◽  
Lili Liu ◽  
Zhigang Zhai ◽  
Juchun Ding ◽  
Ting Si ◽  
...  
Keyword(s):  
Nonlinear Model ◽  
Initial Conditions ◽  
Superposition Principle ◽  
Mode Competition ◽  
Two Dimensional ◽  
Competition Effect ◽  
Density Ratios ◽  
Multi Mode ◽  
Amplitude Growth ◽  
Perturbation Growth

Shock-tube experiments on eight kinds of two-dimensional multi-mode air–SF $_6$ interface with controllable initial conditions are performed to examine the dependence of perturbation growth on initial spectra. We deduce and demonstrate experimentally that the amplitude development of each mode is influenced by the mode-competition effect from quasi-linear stages. It is confirmed that the mode-competition effect is closely related to initial spectra, including the wavenumber, the phase and the initial amplitude of constituent modes. By considering both the mode-competition effect and the high-order harmonics effect, a nonlinear model is established based on initial spectra to predict the amplitude growth of each individual mode. The nonlinear model is validated by the present experiments and data in the literature by considering diverse initial spectra, shock intensities and density ratios. Moreover, the nonlinear model is successfully extended based on the superposition principle to predict the growths of the total perturbation width and the bubble/spike width from quasi-linear to nonlinear stages.

scholarly journals Generation of Wave Groups by Shear Layer Instability

Fluids ◽  
2019 ◽  
Vol 4 (1) ◽  
pp. 39 ◽  
Author(s):  
Roger Grimshaw
Keyword(s):  
Growth Rate ◽  
Group Velocity ◽  
Linear Growth ◽  
Linear Stability Theory ◽  
Linear Growth Rate ◽  
Wave Groups ◽  
Weakly Nonlinear ◽  
Complex Valued ◽  
Fundamental Requirement ◽  
Temporal Growth Rate

The linear stability theory of wind-wave generation is revisited with an emphasis on the generation of wave groups. The outcome is the fundamental requirement that the group move with a real-valued group velocity. This implies that both the wave frequency and the wavenumber should be complex-valued, and in turn this then leads to a growth rate in the reference frame moving with the group velocity which is in general different from the temporal growth rate. In the weakly nonlinear regime, the amplitude envelope of the wave group is governed by a forced nonlinear Schrödinger equation. The effect of the wind forcing term is to enhance modulation instability both in terms of the wave growth and in terms of the domain of instability in the modulation wavenumber space. Also, the soliton solution for the wave envelope grows in amplitude at twice the linear growth rate.

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