Numerical computation of solitary waves in a two-layer fluid (original) (raw)
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Structure of internal solitary waves in two-layer fluid at near-critical situation
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Journal of engineering mathematics, 2002
Large-amplitude waves at the interface between two laminar immisible inviscid streams of different densities and velocites, bounded together in a straight infinite channel are studied, when surface tension and gravity are both present. A long-wave approximation is used to develop a theory for fully nonlinear interfacial waves allowing amplitudes as large as the channel thickness. The result is a set of evolution equations for the interfacial shape and the velocity jump across it. Traveling waves of permanent form are studied and it is shown that solitary waves are possible for a range of physical parameters. All solitary waves can be expressed implicitly in terms of incomplete elliptic integrals of the third kind. When the upper layer has zero density, two explicit solitary-wave solutions have been found whose amplitudes are equal to h/4 or h/9, where 2h is the channel thickness. In the absence of gravity solitary waves are not possible but periodic ones are. Numerically constructed solitary waves are given for representative physical parameters.
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Running interfacial waves in a two-layer fluid system subject to longitudinal vibrations
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We study the waves at the interface between two thin horizontal layers of immiscible fluids subject to high-frequency horizontal vibrations. Previously, the variational principle for energy functional, which can be adopted for treatment of quasi-stationary states of free interface in fluid dynamical systems subject to vibrations, revealed existence of standing periodic waves and solitons in this system. However, this approach does not provide regular means for dealing with evolutionary problems: neither stability problems nor ones associated with propagating waves. In this work, we rigorously derive the evolution equations for long waves in the system, find all time-independent-profile solitary waves (standing solitons are a specific case of these propagating waves), which exist below the linear instability threshold, and reveal standing and slow solitons to be always unstable while fast solitons are stable. Depending on initial perturbations, unstable solitons either grow in an explosive manner, which means layer rupture in a finite time, or falls apart into stable solitons. The results are derived within the long-wave approximation as the linear stability analysis for the flat-interface state [Lyubimov D V and Cherepanov A A 1987 Fluid Dynamics 21 849-854] reveals the instabilities of thin layers to be long-wavelength.
Fully nonlinear interfacial waves in a bounded two-fluid system
Proquest Dissertations and Theses Thesis New Jersey Institute of Technology 2003 Publication Number Aai3177204 Isbn 9780542164156 Source Dissertation Abstracts International Volume 66 05 Section B Page 2612 158 P, 2003
We study the nonlinear flow which results when two immiscible inviscid incompressible fluids of different densities and separated by an interface which is free to move and which supports surface tension, are caused to flow in a straight infinite channel. Gravity is taken into consideration and the velocities of each phase can be different, thus giving rise to the Kelvin-Helmholtz instability. Our objective is to study the competing effects of the Kelvin-Helmholtz instability coupled with a stably or unstably stratified fluid system (Rayleigh-Taylor instability) when surface tension is present to regularize the dynamics. Our approach involves the derivation of twoand three-dimensional model evolution equations using long-wave asymptotics and the ensuing analysis and computation of these models. In addition, we derive the appropriate Birkhoff-Rott integro-differential equation for two-phase inviscid flows in channels of arbitrary aspect ratios. A long wave asymptotic analysis is undertaken to develop a theory for fully nonlinear interfacial waves allowing amplitudes as large as the channel thickness. The result is a set of evolution equations for the interfacial shape and the velocity jump across the interface. Linear stability analysis reveals that capillary forces stabilize shortwave disturbances in a dispersive manner and we study their effect on the fully nonlinear dynamics described by our models. In the case of two-dimensional interfacial deflections, traveling waves of permanent form are constructed and it is shown that solitary waves are possible for a range of physical parameters. All solitary waves are expressed implicitly in terms of incomplete elliptic integrals of the third kind. When the upper layer has zero density, two explicit solitary-wave solutions BIOGRAPHICAL SKETCH
A NOTE ON SOLITARY WAVES WITH VARIABLE SURFACE TENSION IN WATER OF INFINITE DEPTH
Two-dimensional gravity-capillary solitary waves propagating at the surface of a fluid of infinite depth are considered. The effects of gravity and of variable surface tension are included in the free-surface boundary condition. The numerical results extend the constant surface tension results of Vanden-Broeck and Dias to situations where the surface tension varies along the free surface. 2000 Mathematics subject classification: primary 74J35, 76B45, 76D45, 76B15.
Oblique interaction of internal solitary waves in a two-layer fluid of infinite depth
Fluid Dynamics Research, 2001
Oblique interaction of internal solitary waves in a two-layer uid system with inÿnite depth is studied. Two-dimensional Benjamin-Ono (BO) equation is solved numerically to investigate the strong interactions of the non-linear long waves whose propagation directions are very close to each other. Computations of time development are performed for two initial settings: the ÿrst one is superposition of two BO solitons with the same amplitude and with di erent propagation directions, and the second one is an oblique re ection of a BO soliton at a vertical wall. It is observed that the Mach re ection does occur for small incident angles and for some incident angles very large stem waves are generated.
SOLITARY WAVES IN A LAYER OF VISCOUS LIQUID Vhh" +@BULLET ] h'+.h-E
Solitary waves in a thin layer of viscous liquid which is running down a vertical surface under the action of gravity are investigated. The existence of such waves was demonstrated in the experiments of [i, 2]. The difficulties that must be faced in a theoretical computation were also noted in these studies. Below a solution of the problem of stationary waves is obtained by the method of expansion in the small parameter in two regions with subsequent matching and also by a numerical integration method. It is shown that in each case a solution of solitary wave type exists along with the single-parameter family of periodic solutions (parameter the wave number ~). On decreasing the wave number, the periodic waves go over into a succession of solitary waves. As the basis of the investigation we take the equation for the thickness of the layer h(~:), which is obtained by integrating the basic equations of motion of a viscous liquid transverse to the layer. In the integration it is assumed that the boundary-layer approximation can be used and a parabolic profile of the longitudinal velocity is taken. In the coordinate system attached to the wave this equation has the following form: Here c is the wave velocity; Uo and ao are the characteristic values of the velocity and the thickness of the layer. In the case of a solitary wave Uo and ao denote the mean values of the velocity and thickness of the unperturbed layer. The nonlinear periodic solutions of Eq. (i) were investigated in [3]. The method of Fourier series expansion was used and explicit expressions were obtained for the waveform, the phase velocity, and the layer thickness. For a fixed number of terms considered in this solution, the accuracy decreases with the decreasing wave number ~ due to the fact that for small values of ~ the wave profile is very different from a harmonic wave. As an example, some results of direct numerical integration of Eq. (I) in the case of periodic waves in a layer of water are shown in Fig. i for Re = 3aoUov-I = 24.41 and ~ = 0.107 (curve I) and = 0.051 (curve 3). The integration was done over a wavelength ~io ~ ~ ~ ~Io + 2~-~ The initial point ~1owas chosen at the crest of the wave h'(~1o) = 0; for given values of Re and ~, the values of c, ao, and the initial data h(~1o), h"(~o) were chosen in such a way that for ~ = ~o + 2~-~ the periodicity condition is satisfied. For the values of a lying close to the neutral stability curve in the Re, ~ plane the wave profiles are almost sinusoidal. The effect of the nonlinear terms in Eq. (i) increases with the decrease of a, and the profiles become noticeably deformed, acquiring the form of solitary waves. The computations were carried out with a small step along parameter a. For obtaining the wave for ~: = ~ + As the characteristics of the wave solution corresponding to ~ were Moscow.
Solitary interfacial hydroelastic waves
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2017
Solitary waves travelling along an elastic plate present between two fluids with different densities are computed in this paper. Different two-dimensional configurations are considered: the upper fluid can be of infinite extent, bounded by a rigid wall or under a second elastic plate. The dispersion relation is obtained for each case and numerical codes based on integro-differential formulations for the full nonlinear problem are derived. This article is part of the theme issue ‘Nonlinear water waves’.
Dynamics of gravity–capillary solitary waves in deep water
Journal of Fluid Mechanics, 2012
The dynamics of solitary gravity–capillary water waves propagating on the surface of a three-dimensional fluid domain is studied numerically. In order to accurately compute complex time-dependent solutions, we simplify the full potential flow problem by using surface variables and taking a particular cubic truncation possessing a Hamiltonian with desirable properties. This approximation agrees remarkably well with the full equations for the bifurcation curves, wave profiles and the dynamics of solitary waves for a two-dimensional fluid domain, and with higher-order truncations in three dimensions. Fully localized solitary waves are then computed in the three-dimensional problem and the stability and interaction of both line and localized solitary waves are investigated via numerical time integration of the equations. There are many solitary wave branches, indexed by their finite energy as their amplitude tends to zero. The dynamics of the solitary waves is complex, involving nonline...