SOLITARY WAVES IN A LAYER OF VISCOUS LIQUID Vhh" +@BULLET ] h'+.h-E (original) (raw)
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Numerical computation of solitary waves in a two-layer fluid
We consider steady two-dimensional flow in a two-layer fluid under the effects of gravity and surface tension. The upper fluid is bounded above by a free-surface and the lower fluid is bounded below by a rigid bottom. We assume the fluids to be inviscid and the flow to be irrotational in each layer. Solitary wave solutions are found to the fullynonlinear problem using a boundary integral method based on Cauchy integral formula. The behaviour of the solitary waves on the interface and free-surface is determined by the density ratio of the two fluids, the fluid depth ratio, the Froude number and the Bond numbers. The dispersion relation obtained for the linearised equations demonstrates the presence of two modes; a ‘slow’ mode and a ‘fast’ mode. When a sufficiently strong surface tension is present only on the free-surface, there is a region, or ‘gap’, between the two modes where no linear periodic waves are found. In-phase and out-of-phase solitary waves are computed in this spectral gap. Damped oscillations appear in the solitary waves’ tails when the value of the free-surface Bond number is either sufficiently small or large. The out-of-phase waves broaden as the Froude number tends towards a critical value. When surface tension is present on both surfaces, out-of-phase solitary waves are computed. Damped oscillations occur in the waves’ tails when the interfacial Bond number is sufficiently small. Oppositely oriented solitary waves are shown to coexist for identical parameter values
Fully nonlinear solitary waves in a layered stratified fluid
Journal of Fluid Mechanics, 2004
Fully nonlinear solitary waves in a layered stratified fluid, each layer with a constant Brunt-Väisälä frequency, are investigated. The stream function satisfies the Helmholtz equation in each layer and is expressed in terms of singularity distributions. As the Green function, a combination of Bessel functions of order zero, of the second and first kind is advocated. Computations performed for two-and three-layer cases show that the wave speed increases with increasing stratification of the top layer. The thickness of the pycnocline increases with wave amplitude when the top layer is homogeneous but decreases when the top layer is stratified. The wave width depends little on the pycnocline thickness. The fluid velocity may exceed the wave speed in the upper part of the water column when the top layer is stratified, but is always smaller than the wave velocity if the top layer is homogeneous. A large vertical excursion of the individual isopycnals contributes to a small Richardson number Ri. The smallest value of Ri is observed in the main body of the fluid. Solitary waves of increasing strength are investigated until the wave-induced fluid velocity equals the wave speed, or the minimal Ri becomes smaller than one quarter. The results may support experimental studies of breaking internal solitary waves.
Journal of Applied Mechanics and Technical Physics, 1994
A fluid flowing down a vertical surface due to gravity is an example of an active dissipative medium. The energy supply comes from the gravitational force while the dissipation comes from the viscous friction force. Studies of the linear stability of failing films of fluid [1, 2] have shown that smooth plane-parallel flow is unstable, no matter how small the Reynolds number. Usually motion in a film is considered laminar-wave flow up to Re-300-400 [3]. Almost all theoretical research has been done in the long-wavelength approximation, which corresponds to experimental data. It has been shown [4] that the long-wavelength approximation can be used up to Re ~ 1000 for ordinary fluids. This approximation allows the complete system of the Navier-Stokes equations to be simplified to a boundary-layer system. An integral method [5] has been proposed which gives a semi-parabolic velocity profile. The assumption of self-similarity has been verified [6]. A system of two equations has been derived within the framework of the integral approach [7, 8]: one for the instantaneous thickness and one for the flow rate of the fluid for moderate Reynolds numbers. Stationary nonlinear running waves of the first kind, which are similar in form to sinusoidal waves, have been found from this approach [7, 8]. Highly nonlinear solutions of this system-which correspond to waves with a smoothly sloping tail, a steep front, and a capillary ripple ahead of the wave-can only be solved numerically [9, 10]. The development stages of both stimulated and naturally occurring waves, including two types of attractors, were examined [11] by extending an earlier theory [8, 12]. The theoretical results agreed quantitatively with experiment in the main part of the wave, but the capillary ripple, predicted by the integral approach, was much stronger and had a higher oscillation amplitude than in experiments. Nonlinear theory has been examined and the velocity profde has been determined for waves of the first kind in the longwavelength approximation for stationary running waves [13]. Stationary nonlinear solutions, based on boundary-layer equations and integral equations for describing film flow, give good agreement with experiment [14]; a detailed comparison is in progress. A transient solution of the Navier-Stokes equations has been done in the long-wavelength approximation for the initial stage of wave development, up to the occurrence of reverse flow in the thinnest part of the t'tim [15]. Transient finite-element solutions for the complete system of Navier-Stokes equations have been found without any approximations [16]. A complete numerical solution using Galerkin's method has been presented for a stationary running wave in viscous fluid layers [17]. Calculations were done for various values of the dimensionless surface stress, including zero stress. Here a new pseudospectral method is presented for calculating the transient development of a wave within the framework of the long-wavelength approximation. It can compute the complete development of the wave until it becomes stationary. This stationary wave is compared with the solution to the stationary equations. The solution is extended parametrically to large Reynolds numbers in order to determine the effect of Reynolds number on the wave characteristics. Solutions found for the longwavelength approximation are compared with solutions [17] of the complete system of equations.
On short-crested waves: experimental and analytical investigations
European Journal of Mechanics B-fluids, 1999
Analytical and experimental investigations were conducted on short-crested wave fields generated by a sea-wall reflection of an incident plane wave. A perturbation method was used to compute analytically the solution of the basic equations up to the sixth order for capillary-gravity waves in finite depth, and up to the ninth order for gravity waves in deep water. For the experiments, we developed a new video-optical tool to measure the full three dimensional wave field η(x, y, t). A good agreement was found between theory and experiments. The spatio-temporal bi-orthogonal decomposition technique was used to exhibit the periodic and progressive properties of the short-crested wave field. © 1999 Éditions scientifiques et médicales Elsevier SAS O. Kimmoun et al.
Nonlinear, periodic, internal waves
Fluid Dynamics Research, 1990
It is shown how to calculate large-amplitude, periodic; internal waves in a channel filled with continuously stratified fluid by using a Stokes-type amplitude expansion along the channel and a modal expansion across the channel. We obtain explicit solutions for the case where the density increases exponentially with depth. It is found that periodic waves in exponentially stratified fluid are waves of depression: as the ,save increases in amplitude the wave speed decreases and the mean density of the fluid at a given height decreases. The waves are limited in height by the formation of an eddy of fluid on the upper boundary above the trough of the wave. This is consistent with the description of waves of depression given by Amick and Toland (1984). There is no backflow before the eddy forms. By contrast, Amick and Toland (1984) have shown that solitary waves in the same system are waves of elevation, limited in height by a cusp in the streamlines at some interior point. A relationship is found between the mass flux and the mass displacement. The expressions necessary to calculate the potential and kinetic energies are given. A simple analytic solution for the internal wave is presented for a fluid with a weak exponential stratification, that is when the Boussinesq approximation is appropriate. For an Nth mode wave the limiting amplitude of the perturbation streamfunction is shown to he 1/N,rr This corresponds to a maximum streamline displacement of 2z/N'rr, where ~ (= 0.7391) is the solution to z-cos z.
On a new type of solitary surface waves in finite water depth
Many models of shallow water waves, such as the famous Camassa-Holm equation, admit peaked solitary waves. However, it is an open question whether or not the widely accepted peaked solitary waves can be derived from the fully nonlinear wave equations. In this paper, a unified wave model (UWM) based on the symmetry and the fully nonlinear wave equations is put forward for progressive waves with permanent form in finite water depth. Different from traditional wave models, the flows described by the UWM are not necessarily irrotational at crest, so that it is more general. The unified wave model admits not only the traditional progressive waves with smooth crest, but also a new kind of solitary waves with peaked crest that include the famous peaked solitary waves given by the Camassa-Holm equation. Besides, it is proved that Kelvin's theorem still holds everywhere for the newly found peaked solitary waves. Thus, the UWM unifies, for the first time, both of the traditional smooth waves and the peaked solitary waves. In other words, the peaked solitary waves are consistent with the traditional smooth ones. So, in the frame of inviscid fluid, the peaked solitary waves are as acceptable and reasonable as the traditional smooth ones. It is found that the peaked solitary waves have some unusual and unique characteristics. First of all, they have a peaked crest with a discontinuous vertical velocity at crest. Especially, unlike the traditional smooth waves that are dispersive with wave height, the phase speed of the peaked solitary waves has nothing to do with wave height, but depends (for a fixed wave height) on its decay length, i.e. the actual wavelength: in fact, the peaked solitary waves are dispersive with the actual wavelength when wave height is fixed. In addition, unlike traditional smooth waves whose kinetic energy decays exponentially from free surface to bottom, the kinetic energy of the peaked solitary waves either increases or almost keeps the same. All of these unusual properties show the novelty of the peaked solitary waves, although it is still an open question whether or not they are reasonable in physics if the viscosity of fluid and surface tension are considered.
Ocean Engineering, 2007
Experimental investigations on internal solitary wave (ISW) propagation and their reflection from a smooth uniform slope were conducted in a two-layered fluid system with a free surface. A 12-meter-long wave flume was in use which incorporated with: (1) a movable vertical gate for generating ISW; (2) six ultrasonic probes for measuring the fluctuation of an ISW; and (3) a steep uniform slope (from one of y ¼ 301, 501, 601, 901, 1201 and 1301) much greater than those ever published in the literature. This paper presents the wave profile properties of the ISW recorded in the flume and their nonlinear features in comparison with the existing Korteweg de Vries (KdV) and modified Korteweg-de Vries (MKdV) theories. Experimental results show that the KdV theory is suitable for most small-amplituded ISWs and MKdV theory is appropriate for the reflected ISWs from various uniform slopes. In addition, both the amplitude-based reflection coefficient and reflected energy approach a constant value asymptotically when plotted against the slope and the characteristic length ratio, respectively. The reflected wave amplitudes calculated from experimental data agree well with those reported elsewhere. The optimum reflection coefficient is found within the limit of 0.85 for wave amplitude, among the test runs from steep normal slope of 301 to inverse angle of 1301, and around 0.75 for the reflected wave energy, produced by an ISW on a vertical wall. r
Steady solution of the velocity field of steep solitary waves
Applied Ocean Research, 2018
The steady solutions of solitary waves are studied through the use of the Irrotational Green-Naghdi (IGN) equations for an incompressible and inviscid fluid. The steady solutions are obtained by Newton-Raphson method. We consider the solitary gravity waves of H/h = 0.30, 0.50, 0.70, 0.79, 0.8296, 0.833199 (where H is the wave amplitude and h is the water depth). Some experiments are conducted to test the results of the IGN equations. In particular, we focus on the wave profile and velocity field. We find that for the cases of the solitary waves of H/h ≤ 0.79, the IGN-5 results agree very well with the Euler solutions. For the solitary waves that close to the maximum amplitude, the converged IGN results are presented. It is also shown that high level IGN results on wave speed agree well with the results obtained by others.