Linear stern waves in finite depth channels

This paper formulates an analytically tractable problem for the wake generated by a long flat bottom ship by considering the steady free surface flow of an inviscid, incompressible fluid emerging from beneath a semi-infinite rigid plate. The flow is considered to be irrotational and two-dimensional so that classical potential flow methods can be exploited. In addition, it is assumed that the draft of the plate is small compared to the depth of the channel. The linearised problem is solved exactly using a Fourier transform and the Wiener-Hopf technique, and it is shown that there is a family of subcritical solutions characterised by a train of sinusoidal waves on the downstream free surface. The amplitude of these waves decreases as the Froude number increases. Supercritical solutions are also obtained, but, in general, these have infinite vertical velocities at the trailing edge of the plate. Consideration of further terms in the expansions suggests a way of canceling the singularity for certain values of the Froude number.

Wake angle for surface gravity waves on a finite depth fluid

Linear water wave theory suggests that wave patterns caused by a steadily moving disturbance are contained within a wedge whose half-angle depends on the depth-based Froude number $F_H$. For the problem of flow past an axisymmetric pressure distribution in a finite-depth channel, we report on the apparent angle of the wake, which is the angle of maximum peaks. For moderately deep channels, the dependence of the apparent wake angle on the Froude number is very different to the wedge angle, and varies smoothly as $F_H$ passes through the critical value $F_H=1$. For shallow water, the two angles tend to follow each other more closely, which leads to very large apparent wake angles for certain regimes.

Two-dimensional bow and stern flows in a finite-depth fluid with constant vorticity

This thesis is concerned with two-dimensional free surface flows past semi-infinite surface-piercing bodies in a fluid of finite-depth. Throughout the study, it is assumed that the fluid in question is incompressible, and that the effects of viscosity and surface tension are negligible. The problems considered are physically important, since they can be used to model the flow of water near the bow or stern of a wide, blunt ship. Alternatively, the solutions can be interpreted as describing the flow into, or out of, a horizontal slot. In the past, all research conducted on this topic has been dedicated to the situation where the flow is irrotational. The results from such studies are extended here, by allowing the fluid to have constant vorticity throughout the flow domain. In addition, new results for irrotational flow are also presented. When studying the flow of a fluid past a surface-piercing body, it is important to stipulate in advance the nature of the free surface as it intersects the body. Three different possibilities are considered in this thesis. In the first of these possibilities, it is assumed that the free surface rises up and meets the body at a stagnation point. For this configuration, the nonlinear problem is solved numerically with the use of a boundary integral method in the physical plane. Here the semi-infinite body is assumed to be rectangular in shape, with a rounded corner. Supercritical solutions which satisfy the radiation condition are found for various values of the Froude number and the dimensionless vorticity. Subcritical solutions are also found; however these solutions violate the radiation condition and are characterised by a train of waves upstream. In the limit that the height of the body above the horizontal bottom vanishes, the flow approaches that due to a submerged line sink in a $90^\circ$ corner. This limiting problem is also examined as a special case. The second configuration considered in this thesis involves the free surface attaching smoothly to the front face of the rectangular shaped body. For this configuration, nonlinear solutions are computed using a similar numerical scheme to that used in the stagnant attachment case. It is found that these solution exist for all supercritical Froude numbers. The related problem of the cusp-like flow due to a submerged sink in a corner is also considered. Finally, the flow of a fluid emerging from beneath a semi-infinite flat plate is examined. Here the free surface is assumed to detach from the trailing edge of the plate horizontally. A linear problem is formulated under the assumption that the elevation of the plate is close to the undisturbed free surface level. This problem is solved exactly using the Wiener-Hopf technique, and subcritical solutions are found which are characterised by a train of sinusoidal waves in the far field. The nonlinear problem is also considered. Exact relations between certain parameters for supercritical flow are derived using conservation of mass and momentum arguments, and these are confirmed numerically. Nonlinear subcritical solutions are computed, and the results are compared to those predicted by the linear theory.

Dynamic response of a beam on elastic foundation of finite depth under a moving force

In this paper, dynamic response of an infinitely long beam resting on a foundation of finite depth, under a moving force is studied. The effect of foundation inertia is included in the analysis by modelling the foundation as a series of closely spaced axially vibrating rods of finite depth, fixed at the bottom and connected to the beam at the top. Viscous damping in the beam and foundation is included in the analysis. Steady state response of the beam-foundation system is obtained. Detailed numerical results are presented to study the effect of various parameters such as foundation mass, velocity of the moving load, damping and axial force on the beam. It is shown that foundation inertia can considerably reduce the critical velocity and can also amplify the beam response.

A note on surface water waves for finite depth in the presence of an ice-cover

A class of I boundary value problems involving propagation of two-dimensional surface water waves, associated with water of uniform finite depth, against a plane vertical wave maker is investigated under the assumption that the surface is covered by a thin sheet of ice. It is assumed that the ice-cover behaves like a thin isotropic elastic plate. Then the problems under consideration lead to those of solving the two-dimensional Laplace equation in a semi-infinite strip, under Neumann boundary conditions on the vertical boundary as well as on one of the horizontal boundaries, representing the bottom of the fluid region, and a condition involving upto fifth order derivatives of the unknown function on the top horizontal ice-covered boundary, along with the two appropriate edge-conditions, at the ice-covered corner, ensuring the uniqueness of the solutions. The mixed boundary value problems are solved completely, by exploiting the regularity property of the Fourier cosine transform.

Exciting forces due to interaction of water waves with a submerged sphere in an ice-covered two-layer fluid of finite depth

Scattering of water waves by a sphere in a two-layer fluid, where the upper layer has an ice-cover modelled as an elastic plate of very small thickness, while the lower one has a rigid horizontal bottom surface, is investigated within the framework of linearized water wave theory. The effects of surface tension at the surface of separation is neglected. There exist two modes of time-harmonic waves - the one with lower wave number propagating along the ice-cover and the one with higher wave number along the interface. Method of multipole expansions is used to find the particular solution for the problem of wave scattering by a submerged sphere placed in either of the layers. The exciting forces for vertical and horizontal directions are derived and plotted against different values of the wave number for different submersion depths of the sphere and flexural rigidity of the ice-cover. When the flexural rigidity and the density of the ice-cover are taken to be zero, the numerical results for the exciting forces for the problem with free surface are recovered as particular cases. (C) 2011 Elsevier Ltd. All rights reserved.

Bernoulli free-boundary problems in strip-like domains and a property of permanent waves on water of finite depth

We study weak solutions for a class of free-boundary problems which includes as a special case the classical problem of travelling gravity waves on water of finite depth. We show that such problems are equivalent to problems in fixed domains and study the regularity of their solutions. We also prove that in very general situations the free boundary is necessarily the graph of a function.

Evolution equation for short surface waves on water of finite depth

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Wind-wave amplification mechanisms: possible models for steep wave events in finite depth

We extend the Miles mechanism of wind-wave generation to finite depth. A beta-Miles linear growth rate depending on the depth and wind velocity is derived and allows the study of linear growth rates of surface waves from weak to moderate winds in finite depth h. The evolution of beta is plotted, for several values of the dispersion parameter kh with k the wave number. For constant depths we find that no matter what the values of wind velocities are, at small enough wave age the beta-Miles linear growth rates are in the known deep-water limit. However winds of moderate intensities prevent the waves from growing beyond a critical wave age, which is also constrained by the water depth and is less than the wave age limit of deep water. Depending on wave age and wind velocity, the Jeffreys and Miles mechanisms are compared to determine which of them dominates. A wind-forced nonlinear Schrodinger equation is derived and the Akhmediev, Peregrine and Kuznetsov-Ma breather solutions for weak wind inputs in finite depth h are obtained.

Experimental observations of watertable waves in an unconfined aquifer with a sloping boundary

Observations of horizontal and vertical variations in piezometric head in a homogeneous, laboratory aquifer are presented and discussed. The observed fluctuations are induced by a simple harmonic oscillation in the clear water reservoir acting across a sloping boundary. The data qualitatively supports existing theories in that higher harmonics are generated in the active forcing zone and that a significant increase in the inland, asymptotic watertable over height (relative to that found for the vertical boundary case) is observed. The observed overheight is shown to be accurately reproduced by existing small-amplitude perturbation theory. Detailed measurements in the vicinity of the sloping boundary reveal that the signal of generated higher harmonics is strongest near the sand surface and that vertical flows are significant in this region. The aquifer is of finite-depth and is influenced by capillary effects, the experimental data therefore exposes limitations of theories which are based on the assumption of a shallow aquifer free of capillary effects. The dispersive properties of the measured pressure wave in the aquifer are comparable to those found from field observations and likewise do not agree with those predicted by the capillary free, shallow aquifer theory. Although some improvement is obtained, discrepancies between the data and theory persist even when a finite-depth aquifer and capillary effects are considered in the theoretical model. Further sand column experiments eliminate a truncated capillary fringe as a possible contributor to these discrepancies. However, the neglect of horizontal flows in the fringe may have caused the discrepancies. (C) 2004 Elsevier Ltd. All rights reserved.

A new method for the estimation of oceanic mixed-layer depth using shipboard X-band radar images

Shipboard X-band radar images acquired on 24 June 2009 are used to study nonlinear internal wave characteristics in the northeastern South China Sea. The studied images show three nonlinear internal waves in a packet. A method based on the Radon Transform technique is introduced to calculate internal wave parameters such as the direction of propagation and internal wave velocity from backscatter images. Assuming that the ocean is a two-layer finite depth system, we can derive the mixed-layer depth by applying the internal wave velocity to the mixed-layer depth formula. Results show reasonably good agreement with in-situ thermistor chain and conductivity-temperature-depth data sets.

Finite time blow-up and breaking of solitary wind waves

The evolution of surface water waves in finite depth under wind forcing is reduced to an antidissipative Korteweg-de Vries-Burgers equation. We exhibit its solitary wave solution. Antidissipation accelerates and increases the amplitude of the solitary wave and leads to blow-up and breaking. Blow-up occurs in finite time for infinitely large asymptotic space so it is a nonlinear, dispersive, and antidissipative equivalent of the linear instability which occurs for infinite time. Due to antidissipation two given arbitrary and adjacent planes of constant phases of the solitary wave acquire different velocities and accelerations inducing breaking. Soliton breaking occurs in finite space in a time prior to the blow-up. We show that the theoretical growth in amplitude and the time of breaking are both testable in an existing experimental facility.

Optimized aperture shapes for depth estimation

The finite depth of field of a real camera can be used to estimate the depth structure of a scene. The distance of an object from the plane in focus determines the defocus blur size. The shape of the blur depends on the shape of the aperture. The blur shape can be designed by masking the main lens aperture. In fact, aperture shapes different from the standard circular aperture give improved accuracy of depth estimation from defocus blur. We introduce an intuitive criterion to design aperture patterns for depth from defocus. The criterion is independent of a specific depth estimation algorithm. We formulate our design criterion by imposing constraints directly in the data domain and optimize the amount of depth information carried by blurred images. Our criterion is a quadratic function of the aperture transmission values. As such, it can be numerically evaluated to estimate optimized aperture patterns quickly. The proposed mask optimization procedure is applicable to different depth estimation scenarios. We use it for depth estimation from two images with different focus settings, for depth estimation from two images with different aperture shapes as well as for depth estimation from a single coded aperture image. In this work we show masks obtained with this new evaluation criterion and test their depth discrimination capability using a state-of-the-art depth estimation algorithm.

The ranks of the homotopy groups of odd degree of a finite complex

Special Issue in honor of Prof. Hans-Bjørn Foxby