Fluid Of Finite Depth Research Articles

Steady three-dimensional water-wave patterns on a finite-depth fluid

The formation of doubly-periodic patterns on the surface of a fluid layer with a uniform velocity field and constant depth is considered. The fluid is assumed to be inviscid and the flow irrotational. The problem of steady patterns is shown to have a novel variational formulation, which includes a new characterization of steady uniform mean flow, and steady uniform flow coupled with steady doubly periodic patterns. A central observation is that mean flow can be characterized geometrically by associating it with symmetries. The theory gives precise information about the role of the ten natural parameters in the problem which govern the wave–mean flow interaction for steady patterns in finite depth. The formulation is applied to the problem of interaction of capillary–gravity short-crested waves with oblique travelling waves, leading to several new observations for this class of waves. Moreover, by including oblique travelling waves and short-crested waves in the same analysis, new bifurcations of short-crested waves are found, which give rise to mixed waves which may have complicated spatial structure.

Scattering of Surface Waves by the Edge of a Floating Elastic Plate

The diffraction of plane surface waves by a semi‐infinite floating plate in a fluid of finite depth is studied. An explicit analytical solution of the problem is obtained using the Wiener–Hopf technique. Simple exact formulas for reflection and transmission coefficients and their asymptotic expressions are derived. Results of numerical calculations using the obtained formulas are presented.

Method of Integral Equations in 2D and 3D Problems of Plate Impact on a Fluid of Finite Depth

This paper describes a method of integral equations for solution of the 2D and 3D problems of plate impact on an incompressible fluid of finite depth. The solutions of the equations obtained are investigated analytically and numerically. The behavior of the impact impulse is studied for various fluid depths and aspect ratios of the plate.

Generation of Magneto-Hydrodynamic Waves by Moving Pressure Distribution.

Two-dimensional magneto-hydrodynamic surface waves generated by a moving oscillatory pressure distribution in an inviscid, incompressible and electrically conducting fluid of finite uniform depth are investigated. In the ultimate steady state, there exist six progressive waves, four original gravity waves and two other magnetic waves, all of which are mainly due to magnetic force. The effect of this force on the waves produced is studied in detail. It is found that, due to this force, disturbance in the wake is increased.

Internal Waves Behind a Local Disturbance upon its Weakly Non-Stationary Motion in Stratified Fluid of Finite Depth

Internal Waves Behind a Local Disturbance upon its Weakly Non-Stationary Motion in Stratified Fluid of Finite Depth

Smoothly attaching bow flows with constant vorticity

AbstractThe free surface flow of a finite depth fluid past a semi-infinite body is considered. The fluid is assumed to have constant vorticity throughout and the free surface is assumed to attach smoothly to the front face of the body. Numerical solutions are found using a boundary integral method in the physical plane and it is shown that solutions exist for all supercritical Froude numbers. The related problem of the cusp-like flow due to a submerged sink in a comer is also considered. Vorticity is included in the flow and it is shown that the behaviour of the solutions is qualitatively the same as that found in the problem described above.

The theory of three-dimensional hetons and vortex-dominated spreading in localized turbulent convection in a fast rotating stratified fluid

The problem of lateral heat/buoyancy transport in localized turbulent convection dominated by rotation in continuously stratified fluids of finite depth is considered. We investigate the specific mechanism of the vortex-dominated lateral spreading of anomalous buoyancy created in localized convective regions owing to outward propagation of intense heton-like vortices (pairs of vortices of equal potential vorticity (PV) strength with opposite signs located at different depths), each carrying a portion of buoyancy anomaly. Assuming that the quasi-geostrophic form of the PV evolution equation can be used to analyse the spreading phenomenon at fast rotation, we develop an analytical theory for the dynamics of a population of three-dimensional hetons. We analyse in detail the structure and dynamics of a single three-dimensional heton, and the mutual interaction between two hetons and show that the vortices can be in confinement, splitting or reconnection regimes of motion depending on the initial distance between them and the ratio of the mixing-layer depth to the depth of fluid (local to bulk Rossby radii). Numerical experiments are made for ring-like populations of randomly distributed three-dimensional hetons. We found two basic types of evolution of the populations which are homogenizing confinement (all vortices are predominantly inside the localized region having highly correlated wavelike dynamics) and vortex-dominated spreading (vortices propagate out of the region of generation as individual hetons or heton clusters). For the vortex-dominated spreading, the mean radius of heton populations and its variance grow linearly with time. The law of spreading is quantified in terms of both internal (specific for vortex dynamics) and external (specific for convection) parameters. The spreading rate is proportional to the mean speed of propagation of individual hetons or heton clusters and therefore depends essentially on the strength of hetons and the ratio of local to bulk Rossby radii. A theoretical explanation for the spreading law is given in terms of the asymptotic dynamics of a single heton and within the frames of the kinetic equation derived for the distribution function of hetons in collisionless approximation. This spreading law gives an upper ‘advective’ bound for the superdiffusion of heat/buoyancy. A linear law of spreading implies that diffusion parameterizations of lateral buoyancy flux in non-eddy-resolving models are questionable, at least when the spreading is dominated by heton dynamics. We suggest a scaling for the ‘advective’ parameterization of the buoyancy flux, and quantify the exchange coefficient in terms of the mean propagation speed of hetons. Finally, we discuss the perspectives of the heton theories in other problems of geophysical fluid dynamics.

Point sink flow in a linearly stratified fluid of finite depth

The evolution of selective withdrawal through a point sink of horizontally unbounded, linearly stratified fluid of finite depth is studied as an initial-value problem. Following the initiation of discharge from the sink, internal gravity wave modes propagate radially upstream to change the flow pattern. These modes are called cylindrical modes. We first consider the case of F→0 (where F is the Froude number) to get linearized governing equations, and seek a linear asymptotic solution for large times t* after starting the discharge, of the cylindrical modes in a stratified fluid where viscous and diffusive effects are negligible. The obtained solution shows that the strength of the modal front grows like t*1/3, unlike the case of two-dimensional modes whose strength at the front is kept constant. Numerical calculations are also performed to study the case of F>0. The results then indicate that all the cylindrical modes can propagate upstream for any F except infinity. The steady-state withdrawal-layer thickness and time to steady state are also investigated over the full parameter range considering the viscous and diffusive effects. The obtained results are then compared with both the analytical and experimental results of prior works.

Trapped modes supported by submerged obstacles

In this paper, trapped modes are shown to exist for a pair of bodies that are submerged in deep water. An inverse procedure is used whereby a flow that represents a local oscillation is constructed and closed streamlines that surround the singularities of the flow are interpreted as body boundaries. The velocity potential for the flow is constructed from a symmetric combination of horizontal and vertical dipoles. This combination of singular solutions of Laplace's equation satisfies the free-surface boundary condition, and the spacing and relative amplitudes of the singularities are related in such a way that there is no radiation of waves to infinity. The existence of closed streamlines that surround the singularities is demonstrated by investigating how the structure of the streamlines varies as the parameters of the flow change.

Free surface flow into a horizontal slot with zero gravity

The two-dimensional free surface flow of a finite-depth fluid into a horizontal slot is considered. For this study, the effects of viscosity and gravity are ignored. A generalized Schwarz–Christoffel mapping is used to formulate the problem in terms of a linear integral equation, which is solved exactly with the use of a Fourier transform. The resulting free surface profile is given explicitly in closed form.

Wave number of maximal growth in viscous magnetic fluids of arbitrary depth

An analytical method within the frame of linear stability theory is presented for the normal field instability in magnetic fluids. It allows us to calculate the maximal growth rate and the corresponding wave number for any combination of thickness and viscosity of the fluid. Applying this method to magnetic fluids of finite depth, these results are quantitatively compared to the wave number of the transient pattern observed experimentally after a jumplike increase of the field. The wave number grows linearly with increasing induction where the theoretical and the experimental data agree well. Thereby, a long-standing controversy about the behavior of the wave number above the critical magnetic field is tackled.

Numerical and asymptotic study of the two-dimensional problem of the hydroelastic behavior of a floating plate in waves

The problem of the behavior of a floating elastic plate in waves is solved numerically. The normal mode method is used. For a fluid of finite depth, the hydrodynamic coefficients are obtained in explicit form. Numerical results are compared with experimental data for the stress distribution in the plate and also with numerical results of other authors. The results are in good agreement for not very short waves. For incident waves whose wavelength is comparable with the length of the plate, a long-wave approximation of the solution is proposed. Within the framework of this approximation, the solution is given in analytical form.

Solitary waves in media with dispersion and dissipation (a review)

The latest results relating to the theory of nonlinear waves in dispersive and dissipative media are reviewed. Attention is concentrated on small-amplitude solitary waves and, in particular, on the classification of types of solitary waves, their conditions of existence, the evolution of local perturbations associated with the presence of solitary waves of various types, and problems of the existence of nonlinear waves localized with respect to a particular direction as the space dimension increases (spontaneous dimension breaking). As examples of dispersive and dissipative media admitting plane solitary waves of various types, we consider a cold collisionless plasma, an ideal incompressible fluid of finite depth beneath an elastic plate and with surface tension, and a fluid in a rapidly oscillating rectangular vessel (Faraday resonance). Examples of spontaneous dimension breaking are considered for the generalized Kadomtsev-Petviashvili equation.

Gravity-capillary waves in the presence of constant vorticity

Periodic and solitary gravity-capillary waves propagating at a constant velocity at the surface of a fluid of finite depth are considered. The vorticity in the fluid is assumed to be constant. Analytical solutions are presented for waves of small amplitude. For waves of large amplitude, numerical solutions are computed by boundary integral equation methods. The results unify previous findings for irrotational gravity capillary waves and gravity waves with constant vorticity. In particular solitary waves with oscillatory tails and branches of solutions which exist only for waves of large amplitude are found.

Two-dimensional evolution equation of finite-amplitude internal gravity waves in a uniformly stratified fluid

We derive a fully nonlinear evolution equation that can describe the two-dimensional motion of finite-amplitude long internal waves in a uniformly stratified three-dimensional fluid of finite depth. The derived equation is the two-dimensional counterpart of the evolution equation obtained by Grimshaw and Yi [J. Fluid Mech. 229, 603 (1991)]. In the small-amplitude limit, our equation is reduced to the celebrated Kadomtsev-Petviashvili equation.

Withdrawal from a fluid of finite depth through a line sink, including surface-tension effects

The steady withdrawal of an inviscid fluid of finite depth into a line sink is considered for the case in which surface tension is acting on the free surface. The problem is solved numerically by use of a boundary-integral-equation method. It is shown that the flow depends on the Froude number, FB=m(gH3B)−1/2, and the nondimensional sink depth λ=HS/HB, where m is the sink strength, g the acceleration of gravity, HB is the total depth upstream, HS is the height of the sink, and on the surface tension, T. Solutions are obtained in which the free surface has a stagnation point above the sink, and it is found that these exist for almost all Froude numbers less than unity. A train of steady waves is found on the free surface for very small values of the surface tension, while for larger values of surface tension the waves disappear, leaving a waveless free surface. It the sink is a long way off the bottom, the solutions break down at a Froude number which appears to be bounded by a region containing solutions with a cusp in the surface. For certain values of the parameters, two solutions can be obtained.

An analysis of two and three dimensional unsteady withdrawal flows, using shallow water theory

AbstractThis paper examines the predictions of shallow water theory for steady and unsteady withdrawal flows through an extended sink from fluid of finite depth. Two-dimensional plane flows and three-dimensional axi-symmetric flow through a circular drain are examined. Shallow water theory indicates the presence of limiting configurations, where the surface of the fluid collapses directly into the sink. In addition, this theory suggests that some previously computed steady solutions may be unstable.

Exact Solution of the Wave-Resistance Problem for a Submerged Cylinder. II. The Non-linear Problem

. We study the problem of wave resistance for a “slender” cylinder submerged in a heavy fluid of finite depth with the cylinder moving at uniform supercritical speed in the direction orthogonal to its generators. We look for a divergence‐free, irrotational flow; the boundary of the region occupied by the fluid (consisting of the free surface, the bottom and the obstacle profile) is assumed to belong to streamlines and the Bernoulli condition is taken on the free surface. The problem is transformed, via the hodograph map, into a problem set in a strip with a cut. By using a “hard” version of the inverse function theorem and by taking account of the results obtained in Part I (which we recall here), we prove the existence of a complex velocity function satisfying all the requirements of the problem. In particular, this function is continuous up to the surface of the obstacle, and the only possible singularities appear at the end‐points where the boundary is not smooth. Moreover, two stagnation points appear near to the extremities of the submerged body.

Evolution of long waves and chaos in magnetohydrodynamics

The evolution equation for a wavepacket travelling on the surface of a conducting fluid of finite depth in 2+1 dimensions in the long-wave approximation is studied in the context of magnetohydrodynamics. The amplitude equation thus obtained is the well-known Kadomtsev–Petviashvili equation with modified coefficients in the presence of a tangential magnetic field. By taking the double limit of this equation, Schrödinger–Poisson type equations are obtained. A nonlinear evolution equation is sought, in order to study Rayleigh–Taylor instability. It is shown that the magnetic field and surface tension have a stabilizing influence on the formation of bubbles arising owing to this instability. By incorporating forcing and damping terms in the Kadomtsev–Petviashvili equation, a condition for the existence of transverse homoclinic orbits giving rise to chaotic motions is obtained by using the Melnikov function method. It is shown that the chaotic motions can be suppressed by applying a suitably strong magnetic field. A study of subharmonic bifurcation leading to surface waves is also undertaken. The tangential magnetic field has a stabilizing influence on such motions.

Exact Evolution Equations for Surface Waves

This paper considers surface gravity-capillary waves in an ideal fluid of finite depth and generalizes exact evolution equations for free gravity waves obtained by Dyachenko et al. to those for forced gravity-capillary waves. The model derived here describes the time evolution of the free surface and the velocity potential evaluated at the free surface under external pressure forcing. Two integro-differential equations are written explicitly in terms of these two dependent variables, and no extra step is required to close the system. These equations are solved numerically for the particular case of stationary periodic waves, and the results compared with analogous ones available in the literature.