Fluid Of Finite Depth Research Articles

Sub-inertial dynamics of density-driven flows in a continuously stratified fluid on a sloping bottom. I. Model derivation and stability characteristics

A theory is developed describing the sub–inertial baroclinic dynamics of bottom–intensified density–driven flows within a continuously stratified fluid of finite depth with variable bottom topography. The evolution of the density–driven current is modelled as a geostrophically balanced homogeneous flow, which allows for finite–amplitude dynamic thickness variations and for which the pressure fields in each layer are strongly coupled together. The evolution of the overlying fluid is governed by baroclinic quasi–geostrophic dynamics describing a balance between the production of relative vorticity, the vortex–tube stretching/compression associated with a deforming gravity current height, and the rectifying influence of a background topographic vorticity gradient. The model is derived as a systematic asymptotic reduction of the two-fluid system in which the upper fluid is described by the Boussinesq adiabatic equations for a continuously stratified fluid and a lower homogeneous layer described by shallow–water theory appropriate for an f -plane with variable bottom topography. The model is shown to possess a non–canonical Hamiltonian formulation. This structure is exploited to give a variational principle for arbitrary steady solutions and stability conditions in the sense of Liapunov. The general linear stability problem associated with parallel shear flow solutions is examined in some detail. Necessary conditions for instability are derived. The instability is convective in the sense that it proceeds by extracting the available gravitational potential energy associated with the lower–layer water mass sliding down the sloping bottom. For the normal–mode instability problem, a semicircle theorem is derived. The linear stability characteristics are illustrated by solving the normal–mode equations for a simple linearly varying lower–layer height profile. In the overlying fluid, the unstable normal modes correspond to amplifying bottom–intensified topographic Rossby waves.

Non–existence of truly solitary waves in water with small surface tension

This paper considers permanent capillary–gravity waves on the free surface of a two–dimensional incompressible inviscid fluid of finite depth. It is shown that there are no solitary-wave solutions of the exact governing equations of the flow that decay to zero exponentially at infinity if the surface tension coefficient is less than its critical value and lies in some intervals. The proof is based upon an estimate of a constant that is related to the approximation of the solution, if it exists, near its singularity. The approximation satisfies a fourth–order nonlinear ordinary differential equation when the solution is extended to the complex plane. Then the non–existence of truly solitary waves is obtained by using a contradiction on this constant.

Statistical theory of gravity and capillary waves on the surface of a finite-depth fluid

Statistical theory of gravity and capillary waves on the surface of a finite-depth fluid

Uniqueness and solvability in the linearized two-dimensional problem of a body in a finite depth subcritical stream

Uniqueness and solvability theorems are proved for the two-dimensional Neumann–Kelvin problem in the case when a body is totally submerged in a subcritical stream of finite depth fluid. A version of source method is developed to find a solution. The Green's identity coupling the solution with a solution of the problem with opposite stream direction is used to prove that the solution is unique.

Linear and Nonlinear Vortex Motion in an Asymmetric Balance Shallow Water Model

Abstract This work extends asymmetric balance (AB) theory to the beta plane (β-AB). The physical problem examined is the motion of a coherent vortex on a beta plane in a finite depth fluid in the absence of an environmental steering flow. A useful attribute of the β-AB formulation is that it allows one to separate the linear and nonlinear balance contributions to the vortex motion when the standard Rossby number is not small compared to unity. It is therefore well suited for testing the hurricane-motion paradigm proposed by Willoughby for equivalent barotropic dynamics. The β-AB model is formulated first for linear shallow water dynamics on a circular vortex forced by the meridional gradient of planetary vorticity (the beta effect) in an earth-based coordinate system. The linear dynamics precludes wave–wave and wave–mean flow interactions. From incipient vortices to hurricanes, the β-AB model correctly develops the wavenumber-one asymmetries (the “beta” gyres) necessary for vortex self-advection. Cyclonic...

Evolution of isolated vortices in a rotating fluid of finite depth

Laboratory experiments have shown that monopolar isolated vortices in a rotating flow undergo instabilities that result in the formation of multipolar vortex states such as dipoles and tripoles. In some cases the instability is entirely two-dimensional, with the vortices taking the form of vortex columns aligned along the direction of the ambient rotation at all times. In other cases, the vortex first passes through a highly turbulent three-dimensional state before eventually reorganizing into vortex columns. Through a series of three-dimensional numerical simulations, the roles that centrifugal instability, barotropic instability, and the bottom Ekman boundary layer play in these instabilities are investigated. Evidence is presented that the centrifugal instability can trigger the barotropic instabilities by the enhancement of vorticity gradients. It is shown that the bottom Ekman layer is not essential to these instabilities but can strongly modify their evolution.

Waves generated by a moving source in a two-layer ocean of finite depth

The velocity potentials of a point source moving at a constant velocity in the upper layer of a two-layer fluid are obtained in a form amenable to numerical integration. Each fluid layer is of finite depth, and the density difference between the two layers is not necessarily small. The far-field asymptotic behavior of the surface waves and internal waves are also derived using the method of stationary phase. They show that the wave system at the free surface or at the interface each contains contributions from two different modes: a surface-wave mode and an internal-wave mode. When the density difference between the two layers is small or the depth of the upper layer is large, the surface-wave mode mainly affects the surface waves while the internal-wave mode mainly affects the internal waves. However, for large density difference, both modes contribute to the surface wave or internal wave system. For each mode, both divergent and transverse waves are present if the total depth Froude number is less than a certain critical Froude number which is mode-dependent. For depth Froude number greater than the critical Froude number, only divergent waves exist for that mode. This classification is similar to that of a uniform fluid of finite depth, where the critical Froude number is simply unity. The surface waves and internal waves are also calculated using the full expressions of the source potentials. They further confirm and illustrate the features observed in the asymptotic analysis.

Cusp flows due to an extended sink in two dimensions

The steady two-dimensioanl potential flow of a finite-depth fluid into an extended or distributed sink, in which the free surface dips to form a cusp above the centre of the sink, is examined. The extended sink is a region where the vertical outflow velocity V is constant and uniform. Numerical solutions for the free-surface profiles are obtained by use of a boundary-integral technique. Solutions are only found for the supercritical case where the Froude numbers are greater than one. In the limiting case where the extended sink width tends to zero, the problem reduces to that of a line sink beneath the free surface, and comparisons are made to existing results for this type of flow.

Two-dimensional problem of a uniform flow of a two-layer fluid of finite depth past a circular cylinder

The linear steady problem of an irrotational uniform flow past a horizontal circular cylinder located in the upper or in the lower layer of a two-layer fluid is solved by the multipole-expansion method. The flow is perpendicular to the axis of the cylinder. The fluid is assumed to be inviscid and incompressible, and the flow in each layer is assumed to be potential. The upper layer can be bounded by a free surface or a solid lid, and the lower layer by a rigid horizontal bottom.

Linear problem of a hydrofoil moving under the free surface of a finite-depth fluid

A method of solving the plane linear problem of a steady-state irrotational flow about a body under the free surface of a heavy fluid of finite depth is developed. The boundary-value problem is formulated for a complex perturbed velocity and is reduced to a singular integral equation relative to the intensity of a vortex layer that models the hydrofoil. The kernel of the equation is the exact solution of the corresponding boundary-value problem for a vortex of unit intensity. The equation is solved by the discrete-vortex method. The effect of the parameters of the problem on the hydrodynamic characteristics of the elliptical hydrofoil and the shape of the free surface are estimated numerically.

Fully nonlinear solitary waves in continuously stratified incompressible Boussinesq fluids

This study is concerned with a fully nonlinear theoretical treatment of internal solitary waves in continuously stratified, incompressible, inviscid, shear-free Boussinesq fluids. Results are presented for wave propagation in both deep and shallow fluids with four different ambient stability profiles. Only the dominant mode with the greatest wave speed is considered. The morphology of finite-amplitude internal solitary waves in Boussinesq fluids is shown to be very sensitive to the precise form of the stability profile. The calculations indicate that a wave of maximum amplitude, which is less than the total fluid depth, exists for all internal solitary waves in continuously stratified Boussinesq fluids of finite depth. There is apparently no upper limit on the amplitude of internal solitary waves in many physically realistic unbounded fluids. Large amplitude waves of this type are mutually similar in form and the morphology of these waves appears to be independent of the ambient stability profile in the waveguide layer. It is shown that the properties of highly nonlinear waves with recirculating flow depend on the density distribution and vorticity of the trapped fluid inside the closed circulation cell. Fluid velocity components associated with the wave motion are evaluated and used to calculate the surface perturbation pressure. The surface perturbation pressure signature for internal solitary waves is found to change with the onset of recirculation from a single-crested profile at small wave amplitudes to a bimodal profile at large wave amplitudes. Results for solitary waves in finite-depth fluids differ from those found for deep fluids in that the surface perturbation pressure at the center of the wave eventually changes sign as wave amplitude increases.

Withdrawal of a fluid of finite depth through a line sink with a cusp in the free surface

The steady withdrawal of a fluid of finite depth into a line sink is considered. The problem is solved numerically by a boundary integral equation method. It is shown that the flow depends on the Froude number F=m( gH 3) −1/2 and the nondimensional sink depth β=H S/ H, where m is the sink strength, g the acceleration of gravity, H is the total depth and H S is the depth of the sink. For given values of β and F there is a one-parameter family of solutions with a cusp on the free surface above the sink. It is found that in general there is a train of steady waves on the free surface. For particular values of the parameters the amplitude of the waves vanishes and the solutions reduce to those computed by Vanden-Broeck and Keller. These findings confirm and generalize the calculations of Vanden-Broeck where the free surface was covered by a lid everywhere but close to the sink.

Hydrodynamics of particles embedded in a flat surfactant layer overlying a subphase of finite depth

The motion of membrane-bound objects is important in many aspects of biology and physical chemistry. A hydrodynamic model for this Fconfiguration was proposed by Saffman & Delbrück (1975) and here it is extended to study the translation of a disk-shaped object in a viscous surface film overlying a fluid of finite depth H. A solution to the flow problem is obtained in the form of a system of dual integral equations that are solved numerically. Results for the friction coefficient of the object are given for a complete range of the two dimensionless parameters that describe the system: the ratio of the sublayer (η) to membrane (ηm) viscosities, Λ=ηR/ηmh (where R and h are the object radius and thickness of the surface film, respectively), and the sublayer thickness ratio, H/R. Scaling arguments are presented that predict the variation of the friction coefficient based upon a comparison of the different length scales that appear in the problem: the geometric length scales H and R, the naturally occurring length scale [lscr ]m=ηmh/η, and an intermediate length scale [lscr ]H= (ηmhH/η)1/2. Eight distinct asymptotic regimes are identified based upon the different possible orderings of these length scales for each of the two limits Λ[Lt ]1 and Λ[Gt ]1. Moreover, the domains of validity of available approximations are established. Finally, some representative surface velocity fields are given and the implication of these results for the characterization of hydrodynamic interactions among membrane-bound proteins adjacent to a finite-depth sublayer is discussed briefly.

Phase velocities of surface waves in a sea of finite depth

Dispersion relations and phase velocities of surface gravity waves, with their non-linearity being considered, have been derived numerically from an equation for a nonvortical incompressible fluid of finite depth. For all the depths being considered, the dispersion relations are readily realized for the wavenumbers smaller than the wavenumber of the basic harmonic. Then the acquired relations tend to increasingly deviate from the linear dispersion relations. This is the case for all the depths and wave steepnesses under discussion. When the depth of the sea diminishes, the deviations dramatically increase in both cases, when waves grow steeper and when the form of the wave spectrum changes from wide to narrow. This also holds for the phase velocity of waves. Numerical results are matched up with the experimentally derived data. For calculations, JONSWAP spectra of various width have been used.

Waves due to initial disturbances at the inertial surface in a stratified fluid of finite depth

This paper is concerned with a Cauchy‐Poisson problem in a weakly stratified ocean of uniform finite depth bounded above by an inertial surface (IS). The inertial surface is composed of a thin but uniform distribution of noninteracting materials. The techniques of Laplace transform in time and either Green′s integral theorem or Fourier transform have been utilized in the mathematical analysis to obtain the form of the inertial surface in terms of an integral. The asymptotic behaviour of the inertial surface is obtained for large time and distance and displayed graphically. The effect of stratification is discussed.

Draining of a fluid of finite depth into a vertical slot

The draining of a fluid of finite depth into a vertical slot is considered. It is shown that the flow depends on the flow rate and depth, and on the size of the gap. At a particular flow rate there is a gap size at which the flow separates in the middle into two “waterfalls,” but for smaller gap sizes there are shown to be two possible flow configurations, one in which the two converging streams form a single downward flow into the slot, and another in which the two streams join and then divide into two separate falling streams further down into the slot.

Mass transport in viscous flow under a progressive water wave

We consider two-dimensional free surface flow caused by a pressure wavemaker in a viscous incompressible fluid of finite depth and infinite horizontal extent. The governing equations are expressed in dimensionless form, and attention is restricted to the case δ[Lt ]ε[Lt ]1, where δ is the characteristic dimensionless thickness of a Stokes boundary layer and ε is the Strouhal number. Our aim is to provide a global picture of the flow, with emphasis on the steady streaming velocity.The asymptotic flow structure near the wavenumber is found to consist of five distinct vertical regions: bottom and surface Stokes layers of dimensionless thickness O(δ), bottom and surface Stuart layers of dimensionless thickness O(δ/ε) lying outside the Stokes layers, and an irrotational outer region of dimensionless thickness O(1). Equations describing the flow in all regions are derived, and the lowest-order steady streaming velocity in the near-field outer region is computed analytically.It is shown that the flow far from the wavemaker is affected by thickening of the Stuart layers on the horizontal length scale O[(ε/δ)2], by viscous wave decay on the scale O(1/δ), and by nonlinear interactions on the scale O(1/ε2). The analysis of the flow in this region is simplified by imposing the restriction δ=O(ε2), so that all three processes take place on the same scale. The far-field flow structure is found to consist of a viscous outer core bounded by Stokes layers at the bottom boundary and water surface. An evolution equation governing the wave amplitude is derived and solved analytically. This solution and near-field matching conditions are employed to calculate the steady flow in the core numerically, and the results are compared with other theories and with observations.

Ship waves on a viscous fluid of finite depth

In this paper we are concerned with the wave generation by a singular forcelet in a viscous fluid of finite depth, where the singularity is located far from the bottom and not very near the free surface. In the first part of this work, the image system of an Oseenlet bounded by a no-slip wall, is considered. It is found that the resultant velocity field can be described by a planar distribution of vertical Oseen doublets and a negative Oseenlet located at the mirror point of the singularity with respect to the plane wall. In the second part of the work we deal with the generation of waves by these solutions. By imposing the linearized free-surface conditions on the solutions obtained from the first part, the wave generated is shown to exhibit the Kelvin ship wave pattern that agrees with observation. The effects of water depth and of submergence on the wave amplitude are also investigated.

Calculation of the far field finite depth green's function

This note presents a numerical method for calculating the far field part of the Green's function for a source in steady translation below the surface of fluid of finite depth. The principal steps of the numerical scheme are outlined, with emphasis on those aspects which are needed for a computer implementation of the scheme. Numerical results are given to show the behavior of the function in various parts of the computation domain, the computer requirements of the method, and its expected accuracy.

An analytical model of the Comet's collision with Jupiter

An important feature observed in the wake of the Jupiter-comet clash was the appearance of the ring structure axisymmetrically positioned around the center of the impact. The persistent expansion of the dark rings and its speed indicated an outward propagating gravity wave (Benka, 1995). We employ an analytical model of constant density, uniform finite depth and inviscid fluid layer to investigate the wave motion produced by the impact of Comet Shoemaker-Levy 9 on the Jovian atmosphere. The relevant thermal effects are neglected and an explosion resulting from the collision is then described by an initial impulsive pressure at the surface of the Jovian atmosphere. Under the assumption that all the kinetic energy of a comet fragment is completely converted into the energy of wave motions in the Jovian atmosphere, an analytical formula describing the relationship between the resulting wave motion in the atmosphere and the parameters of a comet fragment (the radius, density and speed) is derived. Results from the present simple analytical model give a qualitative agreement with observations regarding the distance and speed of the waves.