小扰动界面形态对RM不稳定性影响的数值分析

采用Navier-Stokes 方程对入射激波及其反射激波连续诱导小振幅扰动界面的Richtmyer-Meshkov 不稳定性增长过程进行了二维数值模拟,分析了单模和随机多模两种不同初始形态的界面上钉结构和泡结构在反射激波作用前后的发展特性. 研究结果发现:单模扰动的初始界面形态对反射激波前、后界面的扰动增长都有影响,反射激波前的界面形态信息可以通过钉和泡结构之间的反转传递到反射激波过后. 扰动界面上钉结构的发展速度控制了界面混合区总体的发展速度,反射激波前界面上发展成具有完整冠部形态的钉,在反射激波后会反转成复杂的泡结构,此泡结构对反射激波后钉的发展不利. 随机多模界面显示了与单模界面类似的发展规律,但随机多模界面上的复杂泡结构分布的不对称性使得其对钉结构增长的拖曳效应相对要弱,这导致了相似扰动波长下多模随机界面的扰动发展相对单模界面扰动发展要快.      

NUMERICAL STUDY ON THE EFFECTS OF SMALL-AMPLITUDE INITIAL PERTURBATIONS ON RM INSTABILITY

采用Navier-Stokes 方程对入射激波及其反射激波连续诱导小振幅扰动界面的Richtmyer-Meshkov 不稳定性增长过程进行了二维数值模拟,分析了单模和随机多模两种不同初始形态的界面上钉结构和泡结构在反射激波作用前后的发展特性. 研究结果发现：单模扰动的初始界面形态对反射激波前、后界面的扰动增长都有影响,反射激波前的界面形态信息可以通过钉和泡结构之间的反转传递到反射激波过后. 扰动界面上钉结构的发展速度控制了界面混合区总体的发展速度,反射激波前界面上发展成具有完整冠部形态的钉,在反射激波后会反转成复杂的泡结构,此泡结构对反射激波后钉的发展不利. 随机多模界面显示了与单模界面类似的发展规律,但随机多模界面上的复杂泡结构分布的不对称性使得其对钉结构增长的拖曳效应相对要弱,这导致了相似扰动波长下多模随机界面的扰动发展相对单模界面扰动发展要快.     

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     

单模大扰动的Richtmyer-Meshkov不稳定性

采用高精度的多介质Ghost-Fluid方法,对马赫数为1.15的激波分别作用于单模大扰动Air-CO2、Air-SF6、Air-N2和Air-He界面后的Richtmyer-Meshkov不稳定现象进行了数值研究,得到了不同时刻扰动界面的演化图像,给出了流场的密度等值线和密度纹影图,同实验结果吻合较好。给出了界面的扰动增长随时间变化的情况,并同理论模型进行了对比。对激波从轻气体进入重气体的情况,扰动增长可采用Sadot模型描述线性阶段和早期非线性阶段;对于弱激波同密度接近的气体界面的相互作用,线性阶段时间较长,可用线性模型描述。     

单模态Richtmyer-Meshkov不稳定性数值模拟

采用VOF(Volume of Fluid)方法和PPM(Piecewise Parabolic Method)方法,发展了可用于可压缩多介质粘性流体动力学问题的数值模拟方法MVPPM(Multi-Viscous-Fluid Piecewise Parabolic Method)。利用MVPPM对多个具有不同初始扰动振幅的二维和三维单模态RM(Richtmyer-Meshkov)不稳定性模型进行了数值计算,并与理论模型的计算结果进行了比较。结果表明,无论二维还是三维情况,当初始扰动振幅相对于波长较小的时候,计算的扰动振幅和增长率与理论模型的计算结果一致。当初始扰动波长不变而振幅逐渐增大时,界面振幅和增长率也逐渐增大。对于具有相同初始扰动的情况,三维计算结果在线性段与二维计算结果相同,但是在非线性段比二维结果大,说明非线性和三维效应在RM不稳定性发展过程中起着重要作用。     

Specification and evolution of local (periodic) perturbations in experimental studies of the rayleigh-taylor instability in strong media

Various techniques for specifying the initial local or periodic perturbations on the boundaries of strong solids are discussed. The evolution of a local perturbation is studied experimentally. It is shown that the “mass inhomogeneity” of the perturbation zone and the geometrical dimensions of an insert play an important role in the perturbation transformation. Institute of Experimental Physics, Sarov 607190. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 2, pp. 171–176, March–April, 2000.     

On nonlinear evolution equations of a three-dimensional capillary-gravity wave packet at the interface of two superposed fluids

Assuming amplitudes as slowly varying functions of space and time and using a perturbation method, two coupled nonlinear partial differential equations are derived that give the nonlinear evolution of the amplitude of a three-dimensional capillary-gravity wave packet at the interface of two superposed incompressible fluid layers of finite depths, including the effect of its interaction with a long gravity wave. Starting from these two coupled equations, a balanced set of modulation equations, both at nonresonance and at resonance, is derived. The balanced set of modulation equations, at nonresonance, reduces to a single nonlinear Schrödinger equation, if it is assumed that space variation of the amplitudes depends only on variation along an arbitrary fixed horizontal direction. Modulational instability conditions, both at resonance and at nonresonance, are also deduced. The advantage of the perturbation method adopted in the present problem, over the reductive perturbation method, is noticed.     

Richtmyer-Meshkov不稳定性流体混合区发展的实验研究

使用矩形激波管,在马赫数分别为$M=1.5$和1.7的条件下实验研究了气/液界面 上(即Atwood数$A$接近1时)由Richtmyer-Meshkov不稳定性引起的流体混合现象. 得到 了气/液界面上Richtmyer-Meshkov不稳定性后期流体混合区域宽度随时间的发展呈现出线 性关系的结果,即$h \propto t$. 比较了不同马赫数和初始扰动下的发展情况,发现当 马赫数增加时,同一时间混合区域 宽度随之增加,混合区域宽度增长变快;而相比于波长差别不大的弱多模态初始扰动(无人 为干扰界面), 当界面初始扰动获得随机外界干扰时,界面混合区域具有较大的宽度以及增 长速度. 并且增加激波马赫数和初始扰动多模态性,流体混合程度更为剧烈.     

Instability and long-time evolution of cnoidal wave solutions for a Benney–Luke equation

We investigate numerically the stability of periodic traveling wave solutions (cnoidal waves) for a generalized Benney–Luke equation. By using a high-accurate Fourier spectral method, we find different kinds of evolution depending on the period of the perturbation. A cnoidal wave solution with period T is orbitally stable with regard to perturbations having the same period T, within certain range of wave velocities. This is a fact proved recently by Angulo and Quintero [Existence and orbital stability of cnoidal waves for a 1D boussinesq equation, International Journal of Mathematics and Mathematical Sciences (2007), in press, doi:10.1155/2007/52020] and our numerical experiments are consistent with their theory. In the present work we show numerically that cnoidal waves with period T become unstable when perturbed by small amplitude disturbances whose period is an integer multiple of T. Particularly, if the period of the perturbation is 2T, the evolution of the deviation of the solution from the orbit of the cnoidal wave is found to be approximately a time-periodic function. In other cases, the numerical experiments indicate a non-periodic behavior.     

Dissipative interface waves and the transient response of a three-dimensional sliding interface with Coulomb friction

We investigate the linearized response of two elastic half-spaces sliding past one another with constant Coulomb friction to small three-dimensional perturbations. Starting with the assumption that friction always opposes slip velocity, we derive a set of linearized boundary conditions relating perturbations of shear traction to slip velocity. Friction introduces an effective viscosity transverse to the direction of the original sliding, but offers no additional resistance to slip aligned with the original sliding direction. The amplitude of transverse slip depends on a nondimensional parameter η=c_sτ₀/μv₀, where τ₀ is the initial shear stress, 2v₀ is the initial slip velocity, μ is the shear modulus, and c_s is the shear wave speed. As η→0, the transverse shear traction becomes negligible, and we find an azimuthally symmetric Rayleigh wave trapped along the interface. As η→∞, the inplane and antiplane wavesystems frictionally couple into an interface wave with a velocity that is directionally dependent, increasing from the Rayleigh speed in the direction of initial sliding up to the shear wave speed in the transverse direction. Except in these frictional limits and the specialization to two-dimensional inplane geometry, the interface waves are dissipative. In addition to forward and backward propagating interface waves, we find that for η>1, a third solution to the dispersion relation appears, corresponding to a damped standing wave mode. For large-amplitude perturbations, the interface becomes isotropically dissipative. The behavior resembles the frictionless response in the extremely strong perturbation limit, except that the waves are damped. We extend the linearized analysis by presenting analytical solutions for the transient response of the medium to both line and point sources on the interface. The resulting self-similar slip pulses consist of the interface waves and head waves, and help explain the transmission of forces across fracture surfaces. Furthermore, we suggest that the η→∞ limit describes the sliding interface behind the crack edge for shear fracture problems in which the absolute level of sliding friction is much larger than any interfacial stress changes.     

High-resolution measurements of two-dimensional instabilities and turbulence transition in plane mixing layers

 High-resolution two-dimensional (2D) measurements on a large plane mixing layer provide new quantitative information of its spatial and temporal evolution to turbulence. Periodic acoustic excitation with three frequencies was used to stabilize the fundamental instability of the mixing layer (roll-up) and its first and second subharmonics (vortex pairings). Phase-locked velocity measurements of the time evolution in 2D space (x, y, t) reveal accurate spatially resolved primary (2D) instabilities of the mixing layer and turbulence transition. The measurements unveil new quantitative details of the initial Kelvin–Helmholtz waves and their spatial and temporal evolution into vortex shedding and the effect of the second subharmonic on the first vortex pairing. The second-subharmonic effect hastens alternate first pairings of the rollers, with the result that pairing is completed at two downstream locations. The pairings that occur closer to the knife-edge are more organized (laminar) than those occurring farther downstream (transitional). This effect is corroborated using Taylor’s hypothesis to compute the vorticity distributions from the measured velocity field and a pseudo-spectral simulation of the temporal evolution of the mixing layer. Received: 26 March 1998/Accepted: 2 March 1999     

Measurements on supersonic free jets

Summary An experimental study of round supersonic air jets discharging into quiescent air is described. The initial stagnation enthalpy of the jets was equal to that of the atmosphere. Most of the experiments concerned a fully expanded jet with initial Mach number M _I =1.74. From the experimental results the turbulent Prandtl number and the turbulent coefficient of momentum transfer could be calculated in a large part of the mixing zone of the jets. A mixing parameter K has been introduced and calculated for the jets. The results of this investigation are compared with those obtained for low-speed jets.     

Growth rate predictions of single- and multi-mode Richtmyer�CMeshkov instability with reshock

The single- and multi-mode Richtmyer–Meshkov instabilities (RMI) with reshock are numerically analyzed in two- and three-dimensional domains. Four different types of air/SF ₆ interface shapes are investigated in a shock tube configuration, and the predicted post-reshock growth rates are compared with available empirical models of Mikaelian’s (Physica D 36(3):343–347, 1989) and Charakhch’an’s (J Appl Mech Tech Phys 41(1):23–31, 2000). The simulation of 3D multi-mode RMI shows good agreement with a past experimental study, but other interface types (2D single-mode, 2D multi-mode and 3D single-mode) result in different growth rates after reshock. Parametric studies are therefore performed to investigate the sensitivities of the post-reshock growth rates to model the empirical parameters. For single-mode RMI configurations, the interface shape is found to be only a weak function of the post-reshock growth rate, as also predicted by previous reshock models. The post-reshock growth rate shows a linear correlation to the velocity jump due to reshock; however, it is only about a half of the prediction of Charakhch’an’s model even though the growth before reshock compares well with pre-reshock models. The 3D single-mode post-reshock RMI growth rate is nearly 1.6 times larger than the 2D single-mode RMI. The parametric studies of multi-mode RMI show two distinctly different growth rates depending on the mixing conditions at reshock. If the interface remains sharp at the time of reshock, the post-reshock growth rate is as large as the single-mode cases. However, if the interface is mixed due to non-linear interactions of bubbles and spikes, the growth rates becomes slow and independent of the interface shapes. Overall, this study provides new insights into the flow features of reshocked RMI for different initial perturbation types.     

The effects of initial perturbation to mixing-layer noise

The far-field noise radiated from mixing layers is determined by the near-field flow dynamics which is sensitive to the initial perturbation of instability introduced physically or numerically. This study focuses on the effects of the phase delay in two initial perturbations, one at the fundamental wave number and the other at its subharmonic both calculated from linear instability analysis, on the sound generation in mixing layers. When different phase delays φ₁ changing from zero to 2π is applied on the fundamental mode, we observe different vortex merging processes (e.g. vortex pairing or tearing). The strong nonlinear interaction in the merging process generates most of the noise from mixing layers. There shows a pattern in a period of 2π for the response of far-field sound to the change of φ₁. Similar effects on the dynamics and acoustics can be achieved by adding different phase delays φ₂ to the subharmonic mode instead, however, the response repeats in a period of only π for φ₂. The effects of the combination of different phase delays to other parameters, including the amplitude and wave number for each perturbations, are also investigated. All the results indicate a critical role of nonlinearity in the sound generation mechanism of mixing layers.     

The evolution laws of dilatational spherical and cylindrical weak nonlinear shock waves in elastic non-conductors

The theory of singular surfaces yields a set of coupled evolution equations for the shock amplitude and the amplitudes of the higher order discontinuities which accompany the shock. To solve these equations, we use perturbation methods with a perturbation parameter characterising the initial shock amplitude. It is shown that for decaying shock waves, if the accompanying second order discontinuity is of order one, the straightforward perturbation procedure yields uniformly valid solutions, but if the accompanying second order discontinuity is of order , the method of multiple scales is needed in order to render the perturbation solutions uniformly valid with respect to the distance of travel. We also construct shock wave solutions from modulated simple wave solutions which are obtained with the aid ofHunter & Keller's Weakly Nonlinear Geometrical Optics method. The two approaches give exactly the same results within their common range of validity. The explicit evolution laws thus obtained enable us to see clearly how weak nonlinear curved shock waves are attenuated because of the effects of geometry and material nonlinearity, and on what length scale these effects are most pronounced. Communicated by C. C. Wang     

Finite-amplitude self-oscillations in a rotating Hagen-poiseuille flow

The nonlinear stability of a viscous incompressible flow in a circular pipe rotating about its own axis is investigated. A solution of the initial—boundary value problem for the unsteady three-dimensional Navier—Stokes equations is found by means of the Bubnov—Galerkin method [1–5]. A series of methodological investigations were made. The nonlinear evolution of the periodic self-oscillating regimes is studied, and their characteristic stabilization times, amplitudes, and other integral and fluctuational characteristics are found. The secondary instability of these finite-amplitude wave motions is examined. It is established that the secondary instability is initially weak and linear in character; the corresponding growth times are approximately an order greater than for the primary perturbations. There is the possibility of a sharp, explosive restructuring of the motion when the secondary perturbations reach a certain critical amplitude. A survival curve [5] is constructed, which makes it possible to determine the preferred perturbation, distinguishable from the rest if the initial perturbation amplitudes are equal, and the critical amplitude values starting from which the other perturbations may prevail even over the preferred one. The range of these surviving perturbations is obtained. It is shown that as a result of the non-linear interaction of several perturbations at low levels of supercritlcality periodic motion in the form of a single traveling wave is generated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 22–28, July–August, 1985.     

Numerical simulations of the process of multiple shock–flame interactions

Based on a weighted essentially nonoscillatory scheme, the multiple interactions of a flame interface with an incident shock wave and its reshock waves are numerically simulated by solving the compressible reactive Navier–Stokes equations with a single-step Arrhenius chemical reaction. The two-dimensional sinusoidally perturbed premixed flames with different initial perturbed amplitudes are used to investigate the effect of the initial perturbation on the flame evolutions. The results show that the development of the flame interface is directly affected by the initial perturbed amplitudes before the passages of reshock waves, and the perturbation development is mainly controlled by the Richtmyer–Meshkov instability(RMI). After the successive impacts of multiple reshock waves, the chemical reaction accelerates the consumption of reactants and leads to a gradual disappearance of the initial perturbed information. The perturbation developments in frozen flows with the same initial interface as those in reactive flows are also demonstrated.Comparisons of results between the reactive and frozen flows show that a chemical reaction changes the perturbation pattern of the flame interface by decreasing the density gradient,thereby weakening the baroclinic torque in the flame mixing region, and therefore plays a dominant role after the passage of reshock waves.     

Interface instability of methane hydrate stability zone in permafrost deposits

The method of linear stability analysis is applied to determine the necessary conditions of existence of natural hydrate deposits under Arctic permafrost. It is found that the existence is limited to relatively small thicknesses of the hydrate stability zone and the underlying methane-saturated layer, low-to-moderate permeability of the deposits, and low-to-moderate hydrate saturation. At larger thicknesses, higher permeability, or higher saturation, the interface between the two layers is unstable to small-amplitude perturbations. The results do not support the hypothesis that the interface instability may lead to accelerating meltdown of natural methane hydrates in response to increase in global temperature.     

Elementary solutions for streaky structures in boundary layers with and without suction

The temporal evolutions of small, streamwise elongated disturbances in the asymptotic suction boundary layer (ASBL) and the Blasius boundary layer (BBL) are compared. In particular, initial perturbations localized (δ-functions) in the wall-normal direction are studied, corresponding to an axi-symmetric jet coming out of a plane parallel to the flat plate. Analytical solutions are presented for the wall-normal and streamwise velocities in the ASBL case whereas both analytical and numerical methods are used for the BBL case. The initial position of the perturbation and its spanwise wave number are varied in a parameter study. We present results of maximum amplitudes obtained, the time to reach them, their position and optimal spanwise scales. Free-stream disturbances are shown to migrate towards the wall and reach their (negative) optimum inside the boundary layer. The migration is faster for the ASBL case and a larger amplitude is reached than for the BBL. For perturbations originating inside the boundary layer the amplitudes are overall larger and show the phenomenon of overshoot, i.e. positive amplitudes moving out of the boundary layer. The overall largest amplitudes are obtained for the BBL case, as in other studies, but it is shown that for free-stream disturbances initiated somewhere downstream the leading edge streak growth may be amplified due to suction since in the BBL the disturbance mainly advects above the boundary layer.