Long Waves Of Finite Amplitude Research Articles

Exponential traveling-wave on a viscous fluid flowing down an inclined plane

An exponential traveling-wave solution of the free surface equation for a viscous film flowing down an inclined plane is presented. The free surface equation, valid for long waves of finite amplitude, is obtained from the general equation for waves of arbitrary amplitude, which is developed by the method of multiple-scales and considering a series solution expanded at two orders higher than the series used by Lin in 1974. Thus, confining ourselves to the case of weakly nonlinear wave motion, we use the Ince transformation method to find an exponential travelling-wave solution that depend of the physical parameters of the viscous fluid , the angle θ of the inclined plane and the wave parameters, which are subjected to three constraints between them. This solution show a free surface with a dynamic gap of thickness between the top and the bottom of the inclined plane. Also, we find a particular numerical example, for laminar flow, that may represent the evolution of the free surface h(x,t) of a viscous film flowing down an inclined plane, when a small quantity, of the same fluid, is added constantly to the primary fluid in the top of the inclined plane. An interesting result of this case show that the final state, when t → ∞, is also stationary and has the same behavior as the unperturbed primary flow, however the final layer of fluid has a greater thickness.

Surface tension effects on the nonlinear behavior of long waves in a two layer flow

The propagation of long waves of finite amplitude at the interface of two viscous fluids in the presence of interfacial tension is examined. The effect of capillarity on the shape of the waves at the interface of two superposed fluids is investigated for a wide range of density differences, viscosity ratios and imposed pressure gradients. It is found that in planar geometry surface tension stabilizes the interfacial disturbances. Attention is given to the case in which the upper fluid is more dense and comprises a thin film above the lower fluid. With the heavier fluid on the top the flow pattern is always unstable when surface tension effects are neglected. In this case the interfacial waves do not grow forever and reach a finite amplitude only when the interfacial tension is greater than a critical value.

Diagnostics of the medium structure by long wave of finite amplitude

The averaged systems of hydrodynamic equations for a structured medium in the Lagrangian and the Eulerian coordinates are discussed. In the general case, the equations cannot be reduced to the average hydrodynamic terms. Under propagation of long waves in media with structure, the non-linear effects appear and they are analyzed in the framework of the asymptotic averaged model. The heterogeneity in a medium structure always increases the non-linear effects for the long-wave perturbations. A new method for diagnostics of the properties of medium components by long non-linear waves is suggested (inverse problem). The mass contents of components in the media can be determined by this diagnostic method.

Two-dimensional modulation and instability of a short wave riding on a finite-amplitude long wave

We study the effects of a long wave of finite amplitude on a short wave propagating in a different direction. To facilitate analytical treatment, the long wave is modelled by Gerstner's exact solution, and Lagrangian coordinates are used. Focus is first on the moderately long-scale modulation within a duration comparable to the period of the long wave. The angle of inclination is allowed to be arbitrary. Extension to the evolution over much longer time and distance of propagation is then made for a short wave colinear with the long wave but modulated slowly in both longitudinal and transverse directions. The instability of the short wave due to oblique side bands is examined, and is found to be significantly enhanced by the presence of the long wave.

Roll waves on a layer of a muddy fluid flowing down a gentle slope—A Bingham model

To study rapid and laminar flow of a mud layer down a gentle slope, a shallow layer of Bingham fluid is considered. Attention is focused on long waves of finite amplitude. Convective inertia is kept in the approximate equations which are simplified by Kármán’s momentum integral approach with an assumed velocity profile. A linearized instability analysis is first carried out for periodic disturbances. The nonlinear development of unstable disturbances is computed numerically to study the formation of periodic shocks, or roll waves. Nonlinear motion caused by external influx to an otherwise steadily flowing mud layer is also investigated.

Propagation of long waves of finite amplitude in a liquid with polydispersed gas bubbles

Propagation of long waves of finite amplitude in a liquid with polydispersed gas bubbles

Nonlinear inertial effects on the instability of a single long wave of finite amplitude at the interface of two viscous fluids

The evolution of a single long wave of finite amplitude at the interface of two immiscible fluids of different viscosities and densities, between two horizontal plates is solved, using a boundary layer flow approximation for the equation of motion in each fluid layer. It is found that when the nonlinear inertial effects are taken into account in a moderate manner, at least in the frame of the boundary layer approximation, the initial unperturbed flow with smooth interface is stable to a single wave perturbation at the interface, even in the presence of adverse density and viscosity stratifications. However, when the nonlinear effects are increased in a specific way, and the magnitudes of the parameters involved are kept within the order of magnitude established for the present theory, an unstable flow configuration can be obtained.

Long waves of finite amplitude in polydispersed gas suspensions

Long waves of finite amplitude in polydispersed gas suspensions

Finite-Amplitude Long Waves on Coastal Currents

Abstract Two models of coastal currents are described that allow fully nonlinear wavelike solutions for the limit of long waves. The first model is an adaptation of a model used by Yi and Warn for finite-amplitude βplane Rossby waves in a channel. It utilizes a particular choice of continental shelf topography to obtain a nonlinear evolution equation for long waves of finite amplitude. The second model describes the waves that form at the vorticity interface between two regions of constant potential vorticity. Again a nonlinear evolution equation is obtained for long waves of finite amplitude. For both model equations, numerical results are presented and compared with the corresponding results for the BDA equation, which is the weakly nonlinear limit for both models.

Propagation of long waves of finite amplitude at the interface of two viscous fluids

The propagation of long waves of finite amplitude at the interface of two viscous fluids has been studied theoretically. For plane Couette-Poiseuille flow of two superposed layers of fluids of different viscosity, an equation is derived to determine the development in time of the shape of these finite amplitude waves. The influence of the viscosity ratio, the density difference of the fluids and an imposed pressure gradient have been investigated.

The Propagation of Long Large Amplitude Internal Waves

This study is concerned with long internal gravity waves in a stratified fluid contained between rigid horizontal boundaries. For a general stratification, long waves of finite amplitude will tend to distort, and no permanent wave shape will result. In certain important cases, however, steady waveforms are found to be possible. The properties of such waves are investigated, and their relationship to the solutions provided by the weakly nonlinear theory is studied.

The initial-value problem for long waves of finite amplitude

Derived herein is a set of partial differential equations governing the propagation of an arbitrary, long-wave disturbance of small, but finite amplitude. The equations reduce to that of Boussinesq (1872) when the assumption is made that the disturbance is propagating in one direction only. The equations are hyperbolic with characteristic curves of constant slope. The initial-value problem can be solved very readily by numerical integration along characteristics. A few examples are included.

CNOIDAL WAVES IN SHALLOW WATER

The propagation of long waves of finite amplitude in water with depth to wavelength ratios less than about one-tenth and greater than about one-fiftieth can be described by cnoidal wave theory. To date little use has been made of the theory because of the difficulties involved in practical application. This paper presents the theory necessary for predicting the transforming characteristics of long waves based on cnoidal theory. Basically the method involves calculating the power transmission for a wave train m shallow water from cnoidal theory and equating this to the deep water power transmission assuming no reflections or loss of energy as the waves move into shoaling water. The equations for wave power have been programmed for the range of cnoidal waves, and the results are plotted in non-dimensional form.

Nonlinear Oscillations of a Particle in a Long Wave in a Rotating, Viscous Fluid

Analysis is made of the effects of a long wave of finite amplitude on the oscillation of a particle in a rotating, viscous fluid. It is shown that the zeroth-order solution of the Lagrangian equations of motion consists of the pressure wave and inertial frequencies, whereas the first-order solution gives three frequencies, namely the sum and difference of the pressure-wave and inertial frequencies, and a frequency of twice the pressure-wave frequency. The effects of the frictional force are to damp out the inertial oscillation but leave the other physically significant oscillations intact. Furthermore, it eliminates the solution which contributes to the amplification of the amplitude of particle trajectories in an inviscid fluid. A comparison of the frequencies predicted by the theory with those obtained from the flights of constant-pressure balloons is made.

Nonlinear oscillations of a particle in a long wave in a rotating fluid

The effects of a long wave of finite amplitude on the oscillation of a particle in a rotating fluid are examined by solving the Lagrangian equations of motion. It is shown that the motion of the particle is nonoscillatory or oscillatory, depending respectively on whether the forcing frequency kUr is equal to the multiples of the natural frequency or not. In the former case, the zeroth-order solution consists of the natural and forcing frequencies, whereas the first-order solution gives three new frequencies. In the latter case, the zeroth-order solution consists of the natural frequency with linearly growing amplitude, whereas the first-order solution gives a high frequency oscillation of twice the natural frequency with linearly growing amplitude. A comparison of the frequencies predicted by the zeroth-order and first-order solutions and those calculated from the trajectories of 300-mb constant-pressure balloons is made.