Investigation of interaction between reflected shocks and growing perturbation on an interface

This paper reports the result of investigation into Richtmyer–Meshkov instability (RMI) resulting from multiple interactions of shock waves with the interface between two media of different densities. The instability growth rates were measured after the interactions of the mixing zone with the refracted shock and the first and the second shocks reflected from the endwall. It was shown that for the contribution of separate shock–interface interactions to the instability growth rate, the condition of additivity is not realized. The values of the factor , accounting for the decrease in the RMI growth rate due to the thickening of the mixing zone, have been determined for a continuous interface and for a turbulent mixing zone. Received 27 January 1998 / Accepted 10 June 1998     

柱形汇聚几何中内爆驱动金属界面不稳定性

采用自研的高保真度爆轰与冲击动力学程序,对柱形汇聚几何中内爆驱动金属材料界面不稳定性的动力学行为,进行了数值模拟研究。结果表明,首次冲击后至约12 μs,界面发展以RM（Richtmyer-Meshkov）不稳定性为主;12 μs后至冲击波聚心反弹加载前,界面聚心运动处于加速减速状态,界面发展由RT (Rayleigh-Taylor)不稳定性主导;冲击波聚心反弹加载后,界面发展又由RM不稳定性主导。另外,还研究了初始条件（初始振幅、初始波长、钢壳初始厚度和几何构型）对柱形内爆驱动金属材料界面不稳定性的影响。结果显示:初始振幅较大时振幅增长也较大;初始波长较小（模数较大）时振幅增长较小,而且存在一个截止波长;钢壳厚度会抑制扰动增长,也存在一个截止厚度;几何汇聚效应会使扰动增长速度更快。     

Modeling jet flows caused by the incidence of a shock wave on a profiled free surface

The problem of the incidence of a shock wave with a front-pressure amplitude of about 30 GPa at the profiled free surface of an aluminum sample is studied. It is shown that in the case of large perturbations (amplitude 1 mm and wavelength 10 mm), jet flows occur on the free surface. The data obtained are described using a kinetic fracture model that takes into account the damage initiation and growth in the material due to tensile stress and shear strain. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 1, pp. 16–23, January–February, 2007.     

Evolution of the continuous interface between gases of different density during the passage of a shock wave

The evolution of the interface between gases of different density following the passage of a shock wave has been experimentally investigated. It is shown that replacing the discontinuous change of density on the wavy contact discontinuity by a continuous change in a layer of finite thickness leads to a reduction in the amplitude growth rate in the initial stage of development of Richtmyer—Meshkov instability. The experimentally determined values of the amplitude growth rate reduction factor are satisfactorily described by a model to be found in the literature. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 18–26, March–April, 1994.     

Richtmyer–Meshkov instability on a low Atwood number interface after reshock

The Richtmyer–Meshkov instability after reshock is investigated in shock tube experiments at the Wisconsin Shock Tube Laboratory using planar laser imaging and a new high-speed interface-tracking technique. The interface is a mixture of helium and argon (50% each by volume) stratified over pure argon. This interface has an Atwood number of 0.29 and a near single-mode, two-dimensional, standing wave perturbation with an average amplitude of 0.35?cm and a wavelength of 19.4?cm. The incident shock wave of Mach number 1.92 accelerates the interface before reflecting from the shock tube end wall with M =?1.70 and accelerating the interface in the opposite direction. The amplitude growth after reshock is reported for variations in this initial amplitude, and several amplitude growth rate models are compared to the experimental growth rate after reshock. A new growth model is introduced, based on a model of circulation deposition calculated from one-dimensional gas dynamics parameters. This model compares well with the amplitude growth rate after reshock and the circulation over one-half wavelength of the interface after the first shock wave and after reshock.     

Development of the Richtmyer-Meshkov instability on an interface with continuous density variation

The development of the Richtmyer-Meshkov instability driven by a shock wave reflected from a rigid wall is investigated. It is shown that the perturbation amplitude growth rate depends on the nature of the shock wave refraction on the interface between gases of different density. In the case of regular-soft refraction the reduction in the growth rate of the Richtmyer-Meshkov instability associated with the continuous change in density on the interface is measured.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 52–58, September–October, 1996.     

Experimental study of the Richtmyer-Meshkov instability induced by a Mach 3 shock wave

An experimental investigation of a shock-induced interfacial instability (Richtmyer-Meshkov instability) is undertaken in an effort to study temporal evolution of interfacial perturbations in the late stages of development. The experiments are performed in a vertical shock tube with a square cross-section. A membraneless interface is prepared by retracting a sinusoidally shaped metal plate initially separating carbon dioxide from air, with both gases initially at atmospheric pressure. With carbon dioxide above the plate, the Rayleigh-Taylor instability commences as the plate is retracted and the amplitude of the initial sinusoidal perturbation imposed on the interface begins to grow. The interface is accelerated by a strong shock wave (M = 3.08) while its shape is still sinusoidal and before the Kelvin-Helmholtz instability distorts it into the well known mushroom-like structures; its initial amplitude to wavelength ratio is large enough that the interface evolution enters its nonlinear stage very shortly after shock acceleration. The pre-shock evolution of the interface due to the Rayleigh-Taylor instability and the post-shock evolution of the interface due to the Richtmyer-Meshkov instability are visualized using planar Mie scattering. The pre-shock evolution of the interface is carried out in an independent set of experiments. The initial conditions for the Richtmyer-Meshkov experiment are determined from the pre-shock Rayleigh-Taylor growth. One image of the post-shock interface is obtained per experiment and image sequences, showing the post-shock evolution of the interface, are constructed from several experiments. The growth rate of the perturbation amplitude is measured and compared with two recent analytical models of the Richtmyer-Meshkov instability.PACS: 52.35.Py, 52.35.Tc     

Non-linear characteristics of Rayleigh–Taylor instable perturbations

The direct numerical simulation method is adopted to study the non-linear characteristics of Rayleigh-Taylor instable perturbations at the ablation front of a 200 μm planar CH ablation target. In the simulation, the classical electrical thermal conductivity is included, and NND difference scheme is used. The linear growth rates obtained from the simulation agree with the Takabe formula. The ampli- tude distribution of the density perturbation at the ablation front is obtained for the linear growth case. The non-linear characteristics of Rayleigh-Taylor instable perturbations are analyzed and the numerical results show that the amplitude distributions of the compulsive harmonics are very different from that of the fundamental perturbation. The characteristics of the amplitude distributions of the harmonics and their fast growth explain why spikes occur at the ablation front. The numerical results also show that non-linear effects have relations with the phase differences of double mode initial perturbations, and different phase differences lead to varied spikes.     

Spontaneous acoustic emission from strong shocks in solids

Summary  The role of free electrons in the stability of strong shock waves in metals under spontaneous acoustic emission is investigated. For that purpose, a three-term form of the equation of state is employed in order to describe the cold pressure, the thermal atomic pressure and the thermal pressure of free electrons. The equation of state enables the calculation of the sound velocity behind the shock, which in turn is utilized in the Dyakov–Kontorovich criteria for the shock stability. The integral over the Fermi–Dirac distribution function that describes the specific internal energy of free electrons is replaced by a model algebraic function that possesses correct asymptotic limits at low and high temperatures. It is shown that strong shock waves in all metals are prone to instability under spontaneous emission. However, the threshold for that instability is shifted to higher Mach numbers if free electrons are taken into account. It is further shown that the stabilizing effect of free electrons is vastly overestimated if the expressions for degenerate electron gas are employed for temperatures that are larger than the Fermi temperature. Received 22 November 1999; accepted for publication 12 July 2000     

Estimation of shock induced vorticity on irregular gaseous interfaces: a wavelet-based approach

We study the interaction of a shock with a density-stratified gaseous interface (Richtmyer–Meshkov instability) with localized jagged and irregular perturbations, with the aim of developing an analytical model of the vorticity deposition on the interface immediately after the passage of the shock. The jagged perturbations, meant to simulate machining errors on the surface of a laser fusion target, are characterized using Haar wavelets. Numerical solutions of the Euler equations show that the vortex sheet deposited on the jagged interface rolls into multiple mushroom-shaped dipolar structures which begin to merge before the interface evolves into a bubble-spike structure. The peaks in the distribution of x-integrated vorticity (vorticity integrated in the direction of the shock motion) decay in time as their bases widen, corresponding to the growth and merger of the mushrooms. However, these peaks were not seen to move significantly along the interface at early times i.e. t < 10 τ, where τ is the interface traversal time of the shock. We tested our analytical model against inviscid simulations for two test cases – a Mach 1.5 shock interacting with an interface with a density ratio of 3 and a Mach 10 shock interacting with a density ratio of 10. We find that this model captures the early time (t/τ ∼ 1) vorticity deposition (as characterized by the first and second moments of vorticity distributions) to within 5% of the numerical results. PACS 47.40.Nm; 47.20.Ma     

Polymeric liquids in extension: fluid mechanics or rheometry?

We use a transient 3D free surface finite element method to simulate flow of entangled polymer fluids in the dual cylinder wind-up extensional rheometer. The constitutive equations are K-BKZ integral representations of the Doi–Edwards models with and without the independent alignment approximation (IA). It is demonstrated that the actual kinematics in this rheometer is a mixture of planar and uniaxial extension. Moreover, the ratio of planar to uniaxial deformation is highly dependent upon whether IA is invoked. Without IA, the flow has a tendency toward planar extension, while it tends to be more uniaxial with IA invoked. As a second illustration of the techniques, we simulate the phenomenon of delayed rupture after rapid extension of entangled polymer systems. It is demonstrated that this phenomenon can be explained on the basis of the Doi–Edwards model in terms of a Considere-type instability after chain stretch relaxation.     

Head-on Collision of a Detonation with a Planar Shock Wave

The phenomenon that occurs when a Chapman–Jouguet (CJ) detonation collides with a shock wave is discussed. Assuming a one-dimensional steady wave configuration analogous to a planar shock–shock frontal interaction, analytical solutions of the Rankine–Hugoniot relationships for the transmitted detonation and the transmitted shock are obtained by matching the pressure and particle velocity at the contact surface. The analytical results indicate that there exist three possible regions of solutions, i.e. the transmitted detonation can have either strong, weak or CJ solution, depending on the incident detonation and shock strengths. On the other hand, if we impose the transmitted detonation to have a CJ solution followed by a rarefaction fan, the boundary conditions are also satisfied at the contact surface. The existence of these multiple solutions is verified by an experimental investigation. It is found that the experimental results agree well with those predicted by the second wave interaction model and that the transmitted detonation is a CJ detonation. Unsteady numerical simulations of the reactive Euler equations with both simple one-step Arrhenius kinetic and chain-branching kinetic models are also carried out to look at the transient phenomena and at the influence of a finite reaction thickness of a detonation wave on the problem of head-on collision with a shock. From all the computational results, a relaxation process consisting of a quasi-steady period and an overshoot for the transmitted detonation subsequent to the head-on collisions can be observed, followed by the asymptotic decay to a CJ detonation as predicted theoretically. For unstable pulsating detonations, it is found that, due to the increase in the thermodynamic state of the reactive mixture caused by the shock, the transmitted pulsating detonation can become more stable with smaller amplitude and period oscillation. These observations are in good agreement with experimental evidence obtained from smoked foils where there is a significant decrease in the detonation cell size after a region of relaxation when the detonation collides head-on with a shock wave.     

Application of maximum entropy method for droplet size distribution prediction using instability analysis of liquid sheet

This paper describes the implementation of the instability analysis of wave growth on liquid jet surface, and maximum entropy principle (MEP) for prediction of droplet diameter distribution in primary breakup region. The early stage of the primary breakup, which contains the growth of wave on liquid–gas interface, is deterministic; whereas the droplet formation stage at the end of primary breakup is random and stochastic. The stage of droplet formation after the liquid bulk breakup can be modeled by statistical means based on the maximum entropy principle. The MEP provides a formulation that predicts the atomization process while satisfying constraint equations based on conservations of mass, momentum and energy. The deterministic aspect considers the instability of wave motion on jet surface before the liquid bulk breakup using the linear instability analysis, which provides information of the maximum growth rate and corresponding wavelength of instabilities in breakup zone. The two sub-models are coupled together using momentum source term and mean diameter of droplets. This model is also capable of considering drag force on droplets through gas–liquid interaction. The predicted results compared favorably with the experimentally measured droplet size distributions for hollow-cone sprays.     

受径向振荡激励的黏弹性液滴稳定性分析

当液滴受到外部周期性的径向激励时, 在其表面会形成驻波模式的不稳定, 这就是在球面上的Faraday不稳定问题. 不稳定的表面波的振荡频率根据流体物性参数和所施加激励条件的不同呈现为谐波或是亚谐波模式的振荡. 本文基于线性小扰动理论, 研究了受径向振荡体积力的黏弹性液滴表面波的不稳定性. 振荡的体积力导致动量方程为含有时间周期系数的Mathieu方程, 系统因此变成参数不稳定问题, 采用Floquet理论进行求解. 本模型中将黏弹性的特征处理为与流变模型参数相关的等效黏度, 从而简化了问题的求解. 基于对中性稳定曲线及增长率的分析, 研究了黏弹性参数对液滴稳定性的影响. 结果表明零剪切黏度和应变驰豫时间的增加具有抑制液滴表面波增长的作用, 提高了使液滴表面发生谐波不稳定的激励幅值. 随着振荡幅值的增加, 增长率不稳定的区域减少, 且随着振荡频率的增加, 液滴表面波增长率减小. 通过对增长率的分析可以得出, 应力松弛时间的增加使得增长率增加, 从而促进了液滴表面波的增长.      

Instability of a plane interface between two immiscible conducting viscous fluids in a normal electrostatic field

The dispersion relation for motions of a charged plane interface between two viscous incompressible immiscible conducting fluids is analyzed numerically for finite values of all the parameters involved. It is shown that in addition to the well-known aperiodic (Tonkes-Frenkel’ type) instability for certain values of the physical parameters an oscillatory instability with periodically growing amplitude may be realized in the system. Yaroslavl’. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 116–123, November–December, 1998.     

Nonlinear MHD Kelvin-Helmholtz instability in a pipe

Nonlinear MHD Kelvin-Helmholtz (K-H) instability in a pipe is treated with the derivative expansion method in the present paper. The linear stability problem was discussed in the past by Chandrasekhar (1961)^[1] and Xu et al. (1981).^[6]Nagano (1979)^[3] discussed the nonlinear MHD K-H instability with infinite depth. He used the singular perturbation method and extrapolated the obtained second order modifier of amplitude vs. frequency to seek the nonlinear effect on the instability growth rate γ. However, in our view, such an extrapolation is inappropriate. Because when the instability sets in, the growth rates of higher order terms on the right hand side of equations will exceed the corresponding secular producing terms, so the expansion will still become meaningless even if the secular producing terms are eliminated. Mathematically speaking, it's impossible to derive formula (39) when γ ₀ ² is negative in Nagano's paper.^[3]Moreover, even as early as γ ₀ ² → O⁺, the expansion becomes invalid because the 2nd order modifier γ₂ (in his formula (56)) tends to infinity. This weakness is removed in this paper, and the result is extended to the case of a pipe with finite depth. Theproject is supported by the National Natural Science Foundation of China.     

Effect of disjoining pressure on the instability of a charged thin liquid film on a spherical rigid core

It is shown that the critical Rayleigh number which characterizes the stability of a thin charged viscous fluid film on the surface of a rigid spherical core develops rapidly with decrease in the film thickness to 100 nm when the effect of the disjoining pressure becomes significant. The dependence of the instability growth rate on the thickness of the fluid layer is obtained by analyzing the dispersion relation numerically. Yaroslavl’. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 102–106, January–February, 1999.     

被均匀流缓慢调制的有限水深毛细重力波

非线性的存在会产生高次谐波,这些谐波又反作用于原来的低次谐波,使波幅发生缓慢变化,从而产生缓慢调制现象．这里从考虑均匀流作用下的毛细重力水波基本方程出发,在不可压缩、无旋、无黏条件假设下,使用多重尺度分析方法推导出了在均匀流影响下有限深水毛细重力波振幅所满足的非线性Schr?dinger方程(NLSE)．分析了NLSE解的调制不稳定性．给出了毛细重力波调制不稳定的条件和钟型孤立波产生的条件．分析了无量纲最大不稳定增长率随无量纲水深和表面张力的变化趋势．同时给出了无量纲不稳定增长率随无量纲微扰动波数变化的曲线,呈现出了先增大后减小的趋势．最后指出均匀顺流减小了无量纲不稳定增长率及最大增长率,逆流增大了它们．由表面张力作用产生的毛细波及重力与表面张力共同作用产生的毛细重力波,与流的相互作用可以改变海表粗糙度和海洋上层流场结构,进而影响海气界面动量、热量及水汽的交换．了解海表这些短波动力机制,对卫星遥感的精确测量、海气相互作用的研究及海气耦合模式的改进等有重要意义．      

Strength of water under pulsed loading

Experiments were performed to study the strength of water under conditions of pulsed extension, which is typical of the interaction between a triangular compression pulse and a free surface. The tests were performed in a wide (40–1000 MPa) range of rariation in the amplitude of the compression pulse at deformation rates of 10⁴−10⁵ sec⁻¹. It is found that as the compression-pulse amplitude increases from 150 to 1050 MPa, the strength of water decreases from 46 to 22 MPa. The deformation rate was found to have little effect on the strength. The possibility of using the model of homogenous nucleation (formation of cavitation nuclei) to interpret the data obtained is discussed. Institute of Chemical Physical, Russian Academy of Sciences, Chernogolovka 142432. Translated from Prikladmaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 4, pp. 198–205, July–August, 2000.     

Formation mechanism of cavitating spalls

The dynamics of formation of cavitation zones in a liquid upon reflection of a shock pulse from the free surface is studied numerically in a one-dimensional formulation using the Iordanskii-Kogarko-van Wijngaarden two-phase model. It is shown that the formation of a system of cavitation zones (clusters) with a dynamically increasing volume concentration of the gas phase near the free surface is due to oscillations of the structure of the rarefaction wave profile. The fast relaxation of tensile stresses in the cavitation zone ends in the formation of a quasistationary mass-velocity field, which provides for almost unbounded growth of cavitation bubbles in subsurface clusters and explains the occurrence of the spall layers observed in experiments. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 2, pp. 65–73, March–April, 2008.