Spatial structure of two-dimensional wave disturbances in a boundary layer

In [1] on the basis of a numerical integration of the Navier-Stokes equations the authors investigated the nonlinear evolution of two-dimensional disturbances of the traveling wave type in the boundary layer on a flat plate. The process of interaction of two waves with different wave numbers and initial amplitudes was examined. In this article the study of these interactions is continued. Special attention is paid to the spatial structure of the disturbances with respect to the cross-flow coordinate (with respect to the longitudinal coordinate the disturbances are assumed to be periodic) at various moments of time. It is shown that if the initial amplitude of one of the waves is sufficiently large, i.e., exceeds a certain threshold value, an undamped quasisteady regime is established during the interaction process. At lower amplitudes the process degenerates and the waves develop independently. In these two cases the evolution of the spatial distribution of the perturbation amplitudes is qualitatively different. In the first case the shape of the amplitude distribution varies only slightly with time, while in the second it depends importantly on the parameters of the wave numbers and the Reynolds number. When the parameters are such that one of the finite-amplitude waves is damped, its amplitude distribution rapidly evolves into the form characteristic of disturbances of the continuous spectrum of the linear stability problem.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 19–24, September–October, 1990.     

涂布流动的非线性阈值问题

本文研究了沿斜面流动薄层液体的非线性稳定性,即涂布流动的非线性稳定性问题。我们将周恒对平面Poiseuille流提出的弱非线性理论应用于涂布流动。文中对自由表面的世界条件提出了一个合理的简化方法,对亚临界时不同Reynolds数及扰动频率,求出了有限扰动的阈值。     

蜿蜒边界下平面流的线性流动稳定性

为便于数值分析,蜿蜒河流水动力和演变模型中一般隐性假设二次时均流-二次涡的关系与明渠流时均流-明渠湍流的关系相同,但由于高雷诺数下的DNS算力限制和实验尺度限制,这种隐含假设是否成立目前尚无相关湍流研究来支撑.文章试图通过分析明渠湍流和二次湍流发展初期的研究,侧面揭示其湍流结构的异同.通过对曲线正交坐标系下的平面二维NS方程使用双参数摄动的方法,建立了一种求解蜿蜒边界弱非线性层流的摄动解法,并推导得出一个适用于蜿蜒边界的EOS方程以及其特征值问题的解法.蜿蜒边界下弱非线性层流解为一系列蜿蜒谐波分量的叠加,其中线性部分使得两壁产生流速差,非线性部分随着雷诺数增大呈指数增长.水流的扰动增长率特征谱的第一模态与直道流相似,由3条曲线、4个波段合成,但其长波段和短波段的扰动流场与直道流不同,所有短波段的扰动流速近似于KH涡.蜿蜒边界对内部水流扰动有一定的选择性.偏角幅值越大扰动增长越快;蜿蜒波数的影响则为先增后减,有一个使扰动增长最快的蜿蜒波数.扰动流场由一个典型的TS波和一对波包形式的二次涡叠加而成,波包只有纵向流速分量,包络线由蜿蜒波数控制,波包内是与直道扰动波参数相同的TS波.     

Numerical simulation of the nonuniform breakup of capillary jets

The problem of the evolution of the surface of a jet up to the stage at which it breaks up into droplets is solved numerically for two initial wave disturbances. The wave number of one of these coincides with the wave number of the disturbance that grows most strongly according to the linear theory, while the wave number of the other is varied. The effect of the wave numbers and the amplitude ratio of the initial disturbances on the breakup time and the appearance of nonuniformity is investigated.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 12–17, March–April, 1993.     

Resonant interaction of wave packets in a boundary layer

The space-time evolution of resonance-coupled triads of wave packets in a Blasius boundary layer is studied within the framework of weakly nonlinear stability theory. The amplitude behavior of the packet envelopes is determined in relation to their initial shape, the carrier frequency and the region of propagation. As in the case of triads with a discrete spectrum, interaction leads to parametric pumping of the low-frequency fluctuations and explosive nonlinear growth of the packet maxima. The space-time evolution characteristics are expressed in the deformation of the shape and the spectra of the disturbance. Parts of the envelopes are amplified, depending on the local values of the parameters. This leads to sharp discrimination of the peaks and the equalization of their propagation velocities. These effects make it possible to explain the broadening of the spectrum, the stable distribution of the visualization pattern, and the appearance of irregularities in the oscillograms observed in the S transition. In order to analyze the nonlinear evolution of a disturbance initiated by an instantaneous point source, the interaction of a two-dimensional wave train with variable carrier frequency and pairs of three-dimensional low-frequency packets is examined. (The train frequency corresponds to the local maximum of the linear growth rate with respect to R.) The possibility of the progressive parametric excitation of fluctuations over the entire band of frequency parameters is established. This may explain the acceleration of the transition process in the presence of an impulsive disturbance of the boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 67–71, November–December, 1988.The authors are grateful to I. I. Maslennikov for useful discussions.     

Weak turbulence of capillary waves

In recent years the theory of weak turbulence, i.e. the stochastic theory of nonlinear waves [I, 9], has been intensively developed. In the theory of weak turbulence nonlinearity of waves is assumed to be small; this enables us, using the hypothesis of the random nature of the phases of individual waves, to obtain the kinetic equation for the mean squares of the wave aplitudes.In many cases of weak turbulence a situation arises where damping is considerable in the region of large wave numbers and is separated from the region where the basic energy of the waves is concentrated (as a result either of pumping or of the initial conditions) with a wide region of transparency. In [3,4] the hypothesis was stated that weak turbulence in these eases is completely analogous to hydrodynamic turbulence for large Reynolds numbers in the sense that in the region of transparency a univelsal spectrum is established which is determined onIy by the flow of energy into the region of large wave numbers. The spectrum of hydrodynamic turbulence Sk - k 5/s was obtained by A. N. Kolmogorov and A. M. Obukhov [5,6] from dimensional considerations. In the case of weak turbulence the spectrum - obtained as an exact solution of the stationary kinetic equation.Below the ease of weak turbulence of capillary waves on the surface of a liquid is considered.A kinetic equation is obtained for capillary waves. It is significant that in this case the basic contribution to interaction is provided by the process of the decomposition of a wave into two and by the process of two waves merging into one.It is shown that the collision term of the kinetic equation vanishes with the solution ek - k 7/4. Arguments are advanced in favor of the fact that this solution can be interpreted as a universal spectrum in the region of transparency.     

Nonlinear Wave Motions in Stratified Boundary Layers

We consider nonlinear wave motions in strongly buoyant mixed forced–free convection boundary layer flows. In the natural limit of large Reynolds number the nonlinear evolution of a single monochromatic wave mode is shown to be governed by a novel wave/mean-flow interaction in which the wave amplitude and the wave induced mean-flow are of comparable size. A nonlinear integral equation describing the bifurcation to finite-amplitude travelling wave solutions is derived. Solutions of this equation are presented together with a discussion of their physical significance. Received 10 December 1996 and accepted 14 April 1997     

Nonlinear analysis and solitary waves for two superposed streaming electrified fluids of uniform depths with rigid boundaries

The nonlinear modulation of the interfacial waves of two superposed dielectric fluids with uniform depths and rigid horizontal boundaries, under the influence of constant normal electric fields and uniform horizontal velocities, is investigated using the multiple-time scales method. It is found that the behavior of small perturbations superimposed on traveling wave trains can be described by a nonlinear Schrödinger equation in a frame of reference moving with the group velocity. Wave-like solutions to this equation are examined, and different types of localized excitations (envelope solitary waves) are shown to exist. It is shown that when these perturbations are neutrally stable and sufficiently long, solutions to the nonlinear Schrödinger equation may be approximated by the well-known Korteweg-de Vries equation. The speed of the solitary on the interface is seen to be reduced by the electric field. It is found that there are two critical values of the applied voltage that lead to (i) breaking up of the solitary waves, and (ii) bifurcation of solutions of the governing equations. On the other hand, the complex amplitude of standing wave trains near the marginal state is governed by a similar type of nonlinear Schrödinger equation in which the roles of time and space are interchanged. This equation, under a suitable transformation, is obtained as the Korteweg-de Vries equation with a variable coefficient. It is shown that this type of equations admit a solitary wave type of solutions with variable speed. Using the tangent hyperbolic method, it is observed that the wave speed increases as well as decreases, with the increase of electric field values, according to the chosen wavenumbers range. Finally, the nonlinear stability analysis is discussed in view of the coefficients of nonlinear Schrödinger equation to show the effects of various physical parameters, and also to recover the some limiting cases studied earlier in the literature.     

Study of the membrane effect on turbulent mixing measurements in shock tubes

The effect of the thin membrane on the time evolution of the shock wave induced turbulent mixing between the two gases initially separated by it is investigated using two different sets of experiments. In the first set, in which a single-mode large-amplitude initial perturbation was studied, two gas combinations (air/SF and air/air) and two membrane thicknesses were used. The main conclusion of these experiments was that the tested membrane has a negligible effect on the evolution of the mixing zone, which evolves as predicted theoretically. In the second set, in which similar gas combinations and membrane thicknesses were used, small amplitude random-mode initial perturbation, caused by the membrane rupture, rather than the large amplitude single-mode initial perturbation used in the first set, was studied. The conclusions of these experiments were: (1) The membrane has a significant effect on the mixing zone during the initial stages of its growth. This has also been observed in the air/air experiment where theoretically no growth should exist. (2) The membrane effect on the late time evolution, where the mixing zone width has reached a relatively large-amplitude, was relatively small and in good agreement with full numerical simulations. The main conclusion from the present experiments is that the effect of the membrane is important only during the initial stages of the evolution (before the re-shock), when the perturbations have very small amplitudes, and is negligible when the perturbations reach relatively large amplitudes. Received 29 August 1998 / Accepted 25 December 1998     

Characterizing JONSWAP rogue waves and their statistics via inverse spectral data

Rogue waves in random sea states modeled by the JONSWAP power spectrum are high amplitude waves arising over non-uniform backgrounds that cannot be viewed as small amplitude modulations of Stokes waves. In the context of Nonlinear Schrödinger (NLS) models for waves in deep water, this poses the challenge of identifying appropriate analytical solutions for JONSWAP rogue waves, investigating possible mechanisms for their formation, and examining the validity of the NLS models in these more realistic settings. In this work we investigate JONSWAP rogue waves using the inverse spectral theory of the periodic NLS equation for moderate values of the period. For typical JONSWAP initial data, numerical experiments show that the developing sea state is well approximated by the first few dominant modes of the nonlinear spectrum and can be described in terms of a 2- or 3-phase periodic NLS solution. As for the case of uniform backgrounds, proximity to instabilities of the underlying 2-phase solution appears to be the main predictor of rogue wave occurrence, suggesting that the modulational instability of 2-phase solutions of the NLS is a main mechanism for rogue wave formation and that heteroclinic orbits of unstable 2-phase solutions are plausible models of JONSWAP rogue waves. To support this claim, we correlate the maximum wave strength as well as the higher statistical moments with elements of the nonlinear spectrum. The result is a diagnostic tool widely applicable to both model or field data for predicting the likelihood of rogue waves. Finally, we examine the validity of NLS models for JONSWAP data, and show that NLS solutions with JONSWAP initial data are described by non-Gaussian statistics, in agreement with the TOPEX field studies of sea surface height variability.     

Numerical simulation of fully nonlinear irregular wave tank in three dimension

A fully nonlinear irregular wave tank has been developed using a three‐dimensional higher‐order boundary element method (HOBEM) in the time domain. The Laplace equation is solved at each time step by an integral equation method. Based on image theory, a new Green function is applied in the whole fluid domain so that only the incident surface and free surface are discretized for the integral equation. The fully nonlinear free surface boundary conditions are integrated with time to update the wave profile and boundary values on it by a semi‐mixed Eulerian–Lagrangian time marching scheme. The incident waves are generated by feeding analytic forms on the input boundary and a ramp function is introduced at the start of simulation to avoid the initial transient disturbance. The outgoing waves are sufficiently dissipated by using a spatially varying artificial damping on the free surface before they reach the downstream boundary. Numerous numerical simulations of linear and nonlinear waves are performed and the simulated results are compared with the theoretical input waves. Copyright © 2006 John Wiley & Sons, Ltd.     

Packets of wave perturbations in a Blasius flow

Asymptotic methods are used in order to study the development of quasiharmonic perturbations in the linear and slightly nonlinear problems of the stability of locally parallel Blasius flow. The connection between complex frequencies and wave numbers is analyzed in dependence on the initial spectrum and the local Reynolds numbers R, the law of propagation of the packets is determined, the equation of the amplitudes of their envelopes is constructed and solved, and an explanation is given of the conditions for the activation of regimes of the regular and irregular types. An interpretation is given of the observed properties of the initial stage in the evolution of the pulse in the boundary layer. It is concluded that the model has a limited ability to explain the mechanisms of stochastization.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 32–38, January–February, 1988.     

Evolution of a 2-D disturbance in a supersonic boundary layer and the generation of shocklets

Through direct numerical simulation, the evolution of a 2-D disturbance in a supersonic boundary layer has been investigated. At a chosen location, a small amplitude T-S wave was fed into the boundary layer to investigate its evolution. Characteristics of nonlinear evolution have been found. Two methods were applied for the detection of shocklets, and it was found that when the amplitude of the disturbance reached a certain value, shocklets would be generated, which should be taken into consideration when nonlinear theory of hydrodynamic stability for compressible flows is to be established.     

New explicit and exact traveling wave solutions for two nonlinear evolution equations

In this paper, with the aid of computer symbolic computation system such as Maple, an algebraic method is firstly applied to two nonlinear evolution equations, namely, nonlinear Schrodinger equation and Pochhammer–Chree (PC) equation. As a consequence, some new types of exact traveling wave solutions are obtained, which include bell and kink profile solitary wave solutions, triangular periodic wave solutions, and singular solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics.     

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.     

The evolution and interaction of Marangoni-Bénard solitary waves

Marangoni-Bénard convection is the process by which oscillatory waves are generated on an interface due to a change in surface tension. This process, which can be mass or temperature driven is described by a perturbed Korteweg-de Vries (KdV) equation. The evolution and interaction of solitary waves generated by Marangoni-Bénard convection is examined. The solitary wave with steady-state amplitude, which occurs when the excitation and friction terms of the perturbed KdV equation are in balance is found to second-order in the perturbation parameter. This solitary wave has a fixed amplitude, which depends on the coefficients of the perturbation terms in the governing equation. The evolution of a solitary wave of arbitrary amplitude to the steady-state amplitude is also found, to first-order in the perturbation parameter. In addition, by using a perturbation method based on inverse scattering, it is shown that the interaction of two solitary waves is not elastic with the change in wave amplitude determined. Numerical solutions of the perturbed KdV equation are presented and compared to the asymptotic solutions.     

Nonlinear Energy Transfer over the Wave Spectrum in Water Covered by Solid Ice

On the basis of the hydrodynamic equations for nonlinear elastic-gravity waves beneath a solid ice cover and their Hamiltonian representation, a three-wave kinetic equation for the time evolution of the wave spectrum is formulated. The properties of the kernel of the kinetic integral describing the nonlinear interactions between wave triplets are investigated. An algorithm for numerically calculating the kinetic integral is developed. The rate of nonlinear energy transfer over the wave spectrum is estimated quantitatively and its most important characteristics are found.     

Nonlinear evolution analysis of T-S disturbance wave at finite amplitude in nonparallel boundary layers

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Long time interaction of envelope solitons and freak wave formations

This paper concerns long time interaction of envelope solitary gravity waves propagating at the surface of a two-dimensional deep fluid in potential flow. Fully nonlinear numerical simulations show how an initially long wave group slowly splits into a number of solitary wave groups. In the example presented, three large wave events are formed during the evolution. They occur during a time scale that is beyond the time range of validity of simplified equations like the nonlinear Schrödinger (NLS) equation or modifications of this equation. A Fourier analysis shows that these large wave events are caused by significant transfer to side-band modes of the carrier waves. Temporary downshiftings of the dominant wavenumber of the spectrum coincide with the formation large wave events. The wave slope at maximal amplifications is about three times higher than the initial wave slope. The results show how interacting solitary wave groups that emerge from a long wave packet can produce freak wave events.Our reference numerical simulation are performed with the fully nonlinear model of Clamond and Grue [D. Clamond, J. Grue, A fast method for fully nonlinear water wave computations, J. Fluid Mech. 447 (2001) 337–355]. The results of this model are compared with that of two weakly nonlinear models, the NLS equation and its higher-order extension derived by Trulsen et al. [K. Trulsen, I. Kliakhandler, K.B. Dysthe, M.G. Velarde, On weakly nonlinear modulation of waves on deep water, Phys. Fluids 12 (10) (2000) 2432–2437]. They are also compared with the results obtained with a high-order spectral method (HOSM) based on the formulation of West et al. [B.J. West, K.A. Brueckner, R.S. Janda, A method of studying nonlinear random field of surface gravity waves by direct numerical simulation, J. Geophys. Res. 92 (C11) (1987) 11 803–11 824]. An important issue concerning the representation and the treatment of the vertical velocity in the HOSM formulation is highlighted here for the study of long-time evolutions.     

Multiple scales analysis of wave�Cwave interactions in?a?cubically nonlinear monoatomic chain

The interaction of waves in nonlinear media is of practical interest in the design of acoustic devices such as waveguides and filters. This investigation of the monoatomic mass?Cspring chain with a cubic nonlinearity demonstrates that the interaction of two waves results in different amplitude and frequency dependent dispersion branches for each wave, as opposed to a single amplitude-dependent branch when only a single wave is present. A theoretical development utilizing multiple time scales results in a set of evolution equations which are validated by numerical simulation. For the specific case where the wavenumber and frequency ratios are both close to 1:3 as in the long wavelength limit, the evolution equations suggest that small amplitude and frequency modulations may be present. Predictable dispersion behavior for weakly nonlinear materials provides additional latitude in tunable metamaterial design. The general results developed herein may be extended to three or more wave?Cwave interaction problems.