(PHD, 2000)

Abstract: This study investigates the phenomena of evolution of two-dimensional, fully nonlinear, fully dispersive, incompressible and irrotational waves in water of uniform depth in single and in double layers. The study is based on an exact fully nonlinear and fully dispersive (FNFD) wave model developed by Wu (1997, 1999a). This FNFD wave model is first based on two exact equations involving three variables all pertaining to their values at the water surface. Closure of the system of model equations is accomplished either in differential form, by attaining a series expansion of the velocity potential, or in integral form by adopting a boundary integral equation for the velocity field. A reductive perturbation method for deriving asymptotic theory for higher-order solitary waves is developed using the differential closure equation of the FNFD wave theory. Using this method, we have found the leading 15th-order solitary wave solutions. The solution is found to be an asymptotic solution which starts to diverge from the 12th-order so that the 11th-order solution appears to provide the best approximation to the fully nonlinear solitary waves, with a great accuracy for waves of small to moderately large amplitudes. Two numerical methods for calculating unsteady fully nonlinear waves, namely, the FNFD method and the Point-vortex method, are developed and applied to compute evolutions of fully nonlinear solitary waves. The FNFD method, which is based on the integral closure equation of Wu’s theory, can provide good performance on computation of solitary waves of very large amplitude. The Point-vortex method using the Lagrange markers is very efficient for computation of waves of small to moderate amplitudes, but has intrinsic difficulties in computing waves of large amplitudes. These two numerical methods are applied to carry out a comparative study of interactions between solitary waves. Capillary-gravity solitary waves are investigated both theoretically and numerically. The theoretical study based on the reductive perturbation method provides asymptotic theories for higher-order capillary-gravity solitary waves. A stable numerical method (FNFD) for computing exact solutions for unsteady capillary-gravity solitary waves is developed based on the FNFD wave theory. The results of the higher-order asymptotic theories compare extremely well with those given by the FNFD method for waves of small to moderate amplitudes. A numerical method for computing unsteady fully nonlinear interfacial waves in two-layer fluid systems is developed based on the FNFD model. The subcritical and supercritical cases can be clearly distinguished by this method, especially for waves of amplitudes approaching the maximum attainable for the fully nonlinear theory.

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(PHD, 1998)

Abstract:

NOTE: Text or symbols not renderable in plain ASCII are indicated by […]. Abstract is included in .pdf document.

In this study, the two-dimensional steady bow flow in water of arbitrary finite depth has been investigated. The two-dimensional bow is assumed to consist of an inclined flat plate connected downstream to a horizontal semi-infinite draft plate. The bottom of the channel is assumed to be a horizontal plate; the fluid is assumed to be invicid , incompressible; and the flow irrotational. For the angle of incidence [alpha] (held by the bow plate) lying between 0° and 60°, the local flow analysis near the stagnation point shows that the angle lying between the free surface and the inclined plate, [beta], must always be equal to 120°, otherwise no solution can exist. Moreover, we further find that the local flow solution does not exist if [alpha] > 60°, and that on the inclined plate there exists a negative pressure region adjacent to the stagnation point for [alpha] < 30°. Singularities at the stagnation point and the upstream infinity are found to have multiple branch-point singularities of irrational orders.

A fully nonlinear theoretical model has been developed in this study for evaluating the incompressible irrotational flow satisfying the free-surface conditions and two constraint equations. To solve the bow flow problem, successive conformal mappings are first used to transform the flow domain into the interior of a unit semi-circle in which the unknowns can be represented as the coefficients of an infinite series. A total error function equivalent to satisfying the Bernoulli equation is defined and solved by minimizing the error function and applying the method of Lagrange’s multiplier. Smooth solutions with monotonic free surface profiles have been found and presented here for the range of 35° < [alpha] < 60°, a draft Froude number […] less then 0.5, and a water-depth Froude number […] less than 0.4.

The dependence of the solution on these key parameters is examined. As [alpha] decreases for fixed […] and […], the free surface falls off more steeply from the stagnation point. Similarly, as […] increases, the free surface falls off quickly from the stagnation point, but for decreasing […] it descends rather slowly towards the upstream level. As […] decreases further, difficulties cannot be surmounted in finding an exact asymptotic water level at upstream infinity, which may imply difficulties in finding solutions for water of infinite depth. Our results may be useful in designing the optimum bow shape.

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(PHD, 1996)

Abstract:

Part I This study considers the three-dimensional run-up of long waves on a horizontally uniform beach of vertically constant or variable slope which is connected to an open ocean of uniform depth. An inviscid linear long-wave theory is first applied to obtain the fundamental solution for a uniform train of sinusoidal waves obliquely incident upon a uniform beach of variable downward slope without wave breaking. The linearly superposable solutions provide a basis for subsequent comparative studies when the nonlinear and dispersive effects are taken into account, both separately and jointly, thus providing a comprehensive prospect of the extents of influences due to these physical effects. These comparative results seem to be new. By linear theory for waves at nearly grazing incidence, run-up is significant only for the waves in a set of eigenmodes being trapped within the beach at resonance with the exterior ocean waves. Fourier synthesis is employed to analyze a solitary wave and a train of cnoidal waves obliquely incident upon a sloping beach, with the nonlinear and dispersive effects neglected at this stage. Comparison is made between the present theory and the ray theory to ascertain a criterion of validity for the classical ray theory. The wave-induced longshore current is evaluated by finding the Stokes drift of the fluid particles carried by the momentum of the waves obliquely incident upon a sloping beach. Currents of significant velocities are produced by waves at incidence angles about 45° and by grazing waves trapped on the beach. Also explored are the effects of the variable downward slope and curvature of a uniform beach on three-dimensional run-up and reflection of long waves. When the nonlinear effects are taken into account, the exact governing equations for determining a moving inviscid waterline are introduced here based on the local Lagrangian coordinates. A special numerical scheme has been developed for efficient evaluation of these governing equations. The scheme is shown to have a very high accuracy by comparison with some exact solutions of the shallow water equations. The maximum run-up of a solitary wave predicted by the shallow water equations depends on the initial location of the solitary wave and is not unique in value because the wave becomes increasingly more steepened given longer time to travel in the absence of the dispersive effects; it is in general larger than that predicted by the linear long-wave theory. The farther the initial solitary wave of the KdV form is imposed from the beach, the larger the maximum run-up it will reach. The dispersive effects are also very important in two-dimensional run-ups in its role of keeping the nonlinear effects balanced at equilibrium, so that the run-ups predicted by the generalized Boussinesq model (Wu 1979) always yield unique values for run-up of a given initial solitary wave, regardless of its initial position. The result for the gB model is slightly larger than the wave run-up predicted by linear long-wave theory. The dispersive effects tend to reduce the wave run-up either for linear system or for nonlinear system. A three-dimensional process of wave run-up upon a vertical wall has also been studied. Part II This part is a study of nonlinear waves in a fluid-filled elastic tube, whose wall material satisfies the stress-strain law given by the kinetic theory of rubber. The results of this study have extended the scope of this subject, which has been limited to dealing with unidirectional solitary waves only (Olsen and Shapiro 1967), by establishing an exact theory for bidirectional solitons of arbitrary shape. This class of solitons has several remarkable characteristics. These solitons may have arbitrary shape and arbitrary polarity (upward or downward), and all propagate with the same phase velocity. The last feature of wave velocity renders the interactions impossible between unidirectional waves. However, the present new theory shows that bidirectional waves can have head-on collision through which our exact solution leaves each wave a specific phase shift as a permanent mark of the waves having made the nonlinear encounter. The system is at least tri-Hamiltonian and integrable. An iteration scheme has been developed to integrate the system. The system is distinguished by the fact that any local initial disturbance released from a state of rest will become two solitons traveling to the opposite direction, and shocks do not form if initial value is continuous.

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(PHD, 1996)

Abstract:

NOTE: Text or symbols not renderable in plain ASCII are indicated by […]. Abstract is included in .pdf document.

A weakly nonlinear and weakly dispersive oceanic internal long wave (ILW) model, in analogy with the generalized Boussinesq’s (gB) model, is developed to investigate generation and propagation of internal waves (IWs) in a system of two-layer fluids. The ILW model can be further derived to give a bidirectional ILW model for facilitating calculations of head-on collisions of nonlinear internal solitary waves (ISWs). The important nonlinear features, such as phase shift of ISWs resulting from nonlinear collision encounters, are presented. The nonlinear processes of reflection and transmission of waves in channels with a slowly varying bottom are studied.

The terminal effects of IWs running up submerged sloping seabed are studied by the ILW model in considerable detail. Explicit solution of the nonlinear equations are obtained for several classes of wave forms, which are taken as the inner solutions and matched, when necessary for achieving uniformly valid results, with the outer solution based on linear theory for the outer region with waves in deep water. Based on the nonlinear analytic solution, two kinds of initial run-up problems can be solved analytically, and the breaking criteria and run-up law for IWs are obtained. The run-up of ISWs along the uniform beach is simulated by numerical computations using a moving boundary technique. The numerical results based on the ILW model are found in good agreement with the run-up law of ISWs when the amplitudes of the ISWs are small.

The ILW model differs from the corresponding KdV model in admitting bidirectional waves simultaneously and conserving mass. This model is applied to analyze the so-called critical depth problem of ISWs propagating across a critical station at which the depths of the two fluid layers are about equal so as to give rise to a critical point of the KdV equation. As the critical point is passed, the KdV model may predict a new upward facing ISW relative to a local mean interface is about to emerge from the effects of disintegrating original downward ISW. This phenomenon has never been observed in our laboratory. Numerical results are presented based on the present ILW model for ISWs climbing up a curved shelf and a sloping plane seabed. It is shown that in the transcritical region, the behaviour of the ISWs predicted by the ILW model depends on the relative importance of two dimensionless parameters, […], the order of ISW wave slope, and s, the beach slope. For s >> […], the wave profile of ISWs exhibits a smooth transition across the transcritical region; for s << […], ISWs emerge with an oscillatory tail after passing across the critical point. Numerical simulations based on the ILW model are found in good agreement with laboratary observations.

Finally, conclusions are drawn from the results obtained in the present study based on the ILW model.
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(PHD, 1992)

Abstract:

A rectangular tank of high-aspect ratio contains a liquid of moderate depth. The tank is subjected to vertical, sinusoidal oscillations. When the frequency of forcing is nearly twice the first natural frequency of the short side of the tank, waves are observed on the free surface of the liquid that slosh across the tank at a frequency equal to one half of the forcing frequency. These sloshing waves are modulated by a slowly varying envelope along the length of the tank. The envelope of the sloshing wave possesses two solitary-wave solutions, the standing soliton corresponding to a hyperbolic-secant solution and the standing kink wave corresponding to a hyperbolic-tangent solution. The depth and width of the tank determine which soliton is present. In the present work, we derive an analytical model for the envelope solitons by direct perturbation of the governing equations. This derivation is an extension of a previous perturbation approach to include forcing and dissipation. The envelope equation is the parametrically forced, damped, nonlinear Schrodinger equation. Solutions of the envelope equations are found that represent the solitary waves, and regions of formal existence are discussed. Next, we investigate the stability of these solitary-wave solutions. A linear-stability analysis is constructed for both the kink soliton and the standing soliton. In both cases, the linear-stability analysis leads to a fourth-order, nonself-adjoint, singular eigenvalue problem. For the hyperbolic-secant envelope, we find eigenvalues that correspond to the continuous and discrete spectrum of the linear operator. The dependence of the continuous-spectrum eigenvalues on the system parameters is found explicitly. By using local perrturbations about known solutions and numerically continuing the branches, we find the bound-mode eigenvalues. For the kink soliton, continuous-spectrum branches are also found, and their dependence on the system parameters is determined. Bound-mode branches are found as well. In the case of the kink soliton, we extend the linear analysis by providing a nonlinear proof of stability when dissipation is neglected. We compute numerical solutions of the nonlinear Schrodinger equation directly and compare the results to the previous local analysis to verify the predicted behavior. Lastly, laboratory experiments were performed, examining the stability of the solitary waves, and comparisons are made with the foregoing work. In general, the agreement between the local analysis, the numerical simulations and the experiments is good. However, experiments and direct simulations show the existence of periodic solutions of the envelope equation when bound-mode instabilities are present.

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(PHD, 1990)

Abstract:

Part I

The forced Korteweg-de Vries model has been found satisfactory in predicting the periodic generation of upstream-advancing solitary waves by a bottom topography moving in a layer of shallow water with a steady transcritical velocity. It is also known that with certain characteristic forcing distributions, there exist waves, according to the fKdV model, which can remain steady in accompanying the characteristic forcing, provided such a wave exists initially, whereas for a different initial condition the phenomenon of periodic generation can still manifest itself. The stability of two such transcritically forced steady solitary waves is investigated, with their bifurcation diagrams determined with respect to the velocity and the amplitude of the forcing as parameters. The linear stability analysis is first carried out; it involves solving a singular, non-self-adjoint eigenvalue problem, which is examined by applying techniques of matched asymptotic expansions with suitable multiscales for singular perturbations, about the isolated bifurcation points of the parameters. The eigenvalues and eigenfunctions for the full range of the parameters are then obtained by numerically summing a power series expansion for the solution. The numerical results, which accurately match with the local analysis, show that the eigenvalues have only four branches σ = ±σ_{r} ±iσ_{i}. The real part σ_{r} is nonvanishing for the velocity less than a certain supercritical value and for the amplitude greater than a certain marginal bound except at a single point in the parametric plane at which the external forcings vanish, reducing the forced waves to the classical free solitary wave. Within this parametric range, the real part of the four eigenvalues is algebraically two to five orders smaller than the imaginary part σ_{i}, wherever σ_{i} exists; such a small σ_{r} indicates physically a weak exponential growth rate of perturbed solutions and mathematically the need of a very accurate numerical method for its determination. Beyond this parametric range, linear stability theory appears to fail because no eigenvalues can there be found to exist. In this latter case a non linear analysis based on the functional Hamiltonian formulation is found to prevail, and our analysis predicts stability. Finally, extensive numerical simulations using various finite difference schemes are pursued, with results providing full confirmation to the predictions made in various regimes by the analysis.

We consider the Korteweg-de Vries equation in the semi-infinite real line with a boundary condition at the origin. The numerical investigations of Chu et al.[2], are revisited and different new forms for the boundary forcing are assumed. In order to provide some qualitative description for the numerical simulations we develop a simple model based on the IST formalism. It is found that the model is also able to provide some quantitative predictions in agreement with the numerical results.

Part II

There has been considerable interest recently on chaotic advection, for the first time explored in the context of Rayleigh-Bénard roll (2D) convection by the experimental work of J. Gollub and collaborators. When the Rayleigh number increases across a (supercritical) value, depending on the wavelength of the rolls, an oscillatory instability sets in. The flow near the onset of the instability can still be modelled by a stream function, which can be split into a time independent part plus a small time dependent perturbation. The motion of fluid particles can therefore be regarded as the flow for a near integrable, “one-and a half” degree of freedom Hamiltonian vector field, with the phase space corresponding to the physical domain. In absence of molecular diffusivity, the evolution of a certain region of phase space can thus be viewed as the motion of a dyed part of fluid, when the tracer is perfectly passive. The most important objects for a theory of transport are the invariant manifolds for the Poincaré map of the flow homoclinic to fixed points, which physically correspond to the stagnation points. As fluid particles cannot cross invariant lines, these curves constitute a sort of “template” for their motion. For the time independent flow, the invariant manifolds connect the stagnation points and define the roll boundaries. Thus, no transport from roll to roll can occur in this case. Switching the perturbation on, these connections are broken and the manifolds are free to wander along the array of rolls. We use segments of stable and unstable manifold to define the time dependent analogue of the roll boundaries. Transport of fluid across a boundary can then be attributed to the way a region bounded by segments of stable and unstable manifold, or “lobe,” is evolving under map iterations. This allows us to write explicit formulae for describing the fluid transport in terms of a few of these lobes, for a general cross section defining the Poincaré map. Using the symmetries of special cross sections, we are able to further reduce the number of necessary lobes to just one. Furthermore, these symmetries allow us to derive analytically a lower and upper bound for the first time tracer invades a roll, and a lower bound on the stretching of the interface between dyed and clear fluid. These results are independent of the fact that the perturbation is small. When this is the case however, the analytical tools of the Melnikov and subharmonic Melnikov functions are available, so that an approximation to the lobe areas and location and size of the island bands can be determined analytically. It turns out that in our case these approximations are quite good, even for relatively large perturbations. The results we have produced regarding the strong dependence of transport on the period of the oscillation suggest an effect for which no experimental verification is currently available. The presence of molecular diffusivity introduces a (long) time scale into the problem. We discuss the applicability of the theory in this situation, by introducing a simple rule for determining when the effects of diffusivity are negligible, and perform numerical simulations of the flow in this case to provide an example.

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(PHD, 1986)

Abstract:

A new phenomenon of the generation of internal solitons is investigated in this thesis by applying theoretical models and is observed in both numerical and experimental results. By imposing an external disturbance, such as a top surface pressure or a bottom bump, that moves with a constant velocity within a trans critical range after an impulsive start from rest, upon a two-layered or a continuously stratified fluid system, a series of solitons are generated, one after another periodically, each surging ahead of the disturbance in turn. Two theoretical models, belonging to the generalized Boussinesq class, are developed to investigate the generation of weakly nonlinear and weakly dispersive long waves and their evolution in an inviscid, immiscible, and incompressible stratified fluid system under the forcing of the external disturbances. The top surface may be either free or covered by a rigid horizontal plate. For the generalized Boussinesq class for two-layered fluid systems, we have derived the FOUR-equation model for the free top-surface case and the THREE-equation model for the rigid horizontal top-surface case; these are extensions of the one-layer homogeneous fluid system previously considered by Wu (1979). For primarily unidirectional motions a forced KdV equation is obtained which represents each normal mode of a two-layer system or a continuously stratified fluid system. Numerical schemes have been successfully developed to solve these equations. Experiments were performed to investigate this phenomenon,henomenon using fresh water to form the upper layer and brine the lower layer. The relationship between the main properties (the amplitude and the period of generation) of the generated solitons and the forcing function configurations is discussed along with comparisons of theoretical, numerical and experimental results. Qualitatively all the results are consistent in exhibiting the salient features of the resulting motion. Quantitatively the numerical results based on the continuously stratified fluid model seem to be more satisfactory than those given by the two-layered fluid model in comparison with the present experiments. The discrepancy between the theory and experiment is supposedly due to the viscous effects, which will be left for future work.

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(PHD, 1985)

Abstract:

Several theoretical models are developed to study generation of nonlinear dispersive long waves by moving disturbances. All these models belong to the same class as the original Boussinesq or KdV model. The newly developed models, now with external forcing functions added to the KdV equation and the pair of coupled Boussinesq equations, have been chosen for numerical investigations. A predictor-corrector method is adopted to develop the numerical schemes employed here. In order to make the region of computation reasonably small for the case with moving disturbances, a pseudo-moving frame and the sufficiently transparent open boundary conditions are devised. The numerically obtained surface elevations exhibit a series of positive waves running ahead of the disturbance over a wide range of transcritical speeds of the disturbance. The numerical results show that, for speeds close to the critical value, the generation of such waves appears to continue indefinitely. The numerically obtained wave resistance coefficient is compared to the results given by linear dispersive theory. Numerical solutions have been obtained using the KdV and Boussinesq models with surface pressure and bottom bump as forcing functions. Comparisons are made between these results for various cases. Experiments were conducted for a two-dimensional bottom bump moving steadily in shallow water of a towing tank. Experimental results so attained are compared with the numerical solutions, and the agreement between them is good in terms of both the magnitude and the phase of the waves for the range of parameters used in the current study.

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(PHD, 1984)

Abstract:

An investigation of a class of vertical axis wind turbines is carried out with the unsteady effects due to the rotating blade motion fully taken into account. The work is composed of two parts.

In part one, a hydromechanical theory is developed which proceeds from the point of view of unsteady airfoil theory. A rotor comprised of a single blade is used and a two-dimensional analysis is applied to a cross section of the rotor in the limiting mode of operation wherein U « ΩR. Use of linearized theory and of the acceleration potential allows the problem to be expressed in terms of a Riemann-Hilbert boundary value problem. The method of characteristics is used to solve for the remaining unknown function. A uniformly valid first order solution is obtained in closed form after some approximation based on neglecting the variations in the curvature of the path. Explicit expressions of the instantaneous forces and moments acting on the blade are given and the total energy lost by the fluid and the total power input to the turbine are determined.

In part two, the lift acting on a wing crossing a vortex sheet is evaluated by application of a reciprocity theorem in reverse flow. This theorem follows from Green’s integral theorem and relates the circulation around a blade having impulsively crossed a vortex wake to the lift acting on a blade continuously crossing a vortex wake. A solution is obtained which indicates that the lift is composed of two parts having different rates of growth, each depending on the apparent flow velocity before and after the crossing.

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(PHD, 1982)

Abstract:

A new formulation of the pair of Boussinesq-class equations for modelling the propagation of three-dimensional nonlinear dispersive long water waves is presented. This set of model equations permits spatial and temporal variations of the bottom topography. Further, the two resultant equations may be combined into a single equation through the introduction of an irrotational layer-mean velocity. An exact permanent-form solution is derived for the combined equation, which is still of the Boussinesq-class and includes reflection. This solution for the surface height is found to describe a slightly wider wave than the permanent form solution to the uni-directional Korteweg-deVries Equation.

A numerical scheme using an implicit finite-difference method is developed to solve the combined equation for propagation over fixed sloping bottom topography. The scheme is tested for various grid sizes using the permanent-form solution, and an oscillatory tail is seen to develop as a result of insufficient mesh refinement.

Several cases of wave propagation over a straight sloping ramp onto a shelf are solved using the permanent-form solution as initial conditions and the results are found to be in good agreement with previous results obtained by using either the Boussinesq dual-equation set or the single Korteweg-deVries equation. The combined equation is used to solve the related problem in two horizontal dimensions of a wave propagating in a channel having a curved-ramp bottom topography. Depending on the specific topography, focussing or defocussing occurs and the crest is selectively amplified. Indications of cross-channel oscillation are presented. Linear, nondispersive theory is used to solve a case with identical topographical features and initial condition. The solutions using the simplified theory are found to be considerably different from the results for nonlinear, dispersive theory with respect to the overall three-dimensional wave shape as well as in the areas of crest amplification, soliton formation and cross-channel effects.

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(PHD, 1982)

Abstract:

A Petrov-Galerkin finite element formulation for first-order hyperbolic systems is developed generalizing the streamline approach which has been successfully applied previously to convection-diffusion and incompressible Navier-Stokes equations. The formulation is shown to possess desirable stability and accuracy properties.

The algorithm is applied to the Euler equations in conservation-law form and is shown to be effective in all cases studied, including ones with discontinuous solutions. Accurate and crisp representation of shock fronts in transonic problems is achieved.

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(PHD, 1977)

Abstract: NOTE: Text or symbols not renderable in plain ASCII are indicated by […]. Abstract is included in .pdf document. The singularity method for Stokes flow is used to examine the flow past slender bodies possessing finite centerline curvature, in a viscous, incompressible fluid without any appreciable inertia effects. The motion of a slender toroidal ring in Stokes flow is considered first. The symmetry of the geometry and absence of ends has made an accurate analysis possible; the result of this problem elucidates the general flow characteristics present for bodies moving in an arbitrary manner with a finite centerline curvature. Using the methods developed here it is possible to calculate the force/length to higher orders in the slenderness parameter, […], than has previously been possible. In particular, we find the Stokeslet strength with an error of […]. The solution of the torus problem serves as an effective guide in extending the theory to slender bodies of circular cross section with arbitrary centerline configurations and spheroidal ends. In all the cases considered, the no-slip boundary condition is satisfied by distributing appropriate Stokeslets, doublets, rotlets, sources, stresslets, and quadrupoles on the body centerline up to an error term of […], which is sufficient for practical application. From the general slender body analysis we find an integral equation which determines the Stokeslet strength up to the term of […]. The general theory is then applied to examine the propulsion of flagellated microorganisms, including an approximate solution for the interaction between cell body and flagella. A final brief note is made on the thrust enhancing capabilities of oscillating non-spherical cell bodies.

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(PHD, 1975)

Abstract:

Fluid mechanical investigations of ciliary propulsion are carried out from two points of view. In Part I, using a planar geometry, a model is developed for the fluid flow created by an array of metachronally coordinated cilia. The central concept of this model is to replace the discrete forces of the cilia ensemble by an equivalent continuum distribution of an unsteady body force within the cilia layer. This approach facilitates the calculation for the case of finite amplitude movement of cilia and takes into account the oscillatory component of the flow. Expressions for the flow velocity, pressure, and the energy expended by a cilium are obtained for small oscillatory Reynolds numbers. Calculations are carried out with the data obtained for the two ciliates Opalina ranarurn and Paramecium multimicronucleatum. The results are compared with those of previous theoretical models and some experimental observations.

In Part II a model is developed for representing the mechanism of propulsion of a finite ciliated micro-organism having a prolate spheroidal shape. The basic concept of the model is to replace the micro-organism by a prolate spheroidal control surface at which certain boundary conditions on the fluid velocity are prescribed. These boundary conditions, which admit specific tangential and normal components of the flow velocity relative to the control surface, are proposed as a reasonable representation of the overall features of the flow field generated by the motion of the cilia system. Expressions are obtained for the velocity of propulsion, the rate of energy dissipation in the fluid exterior to the cilia layer, and the stream function of the motion. The effect of the shape of the organism upon its locomotion is explored. Experimental streak photographs of the flow around both freely swimming and inert sedimenting Paramecia are presented and compared with the theoretical prediction of the streamlines.

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(Engineer, 1973)

Abstract:

The gross effect of boundary layer separation on the flow field of stratified flow over a barrier was studied by means of the integral method of Lees and Reeves. The complete integral formulation of both inner and outer flow field of stratified flow over a barrier was obtained. Furthermore, an iteration scheme of computation is proposed for the simple case of incompressible homogeneous flow over a barrier with viscous-inviscid interaction included. However, in viewing the increasing importance, a considerable amount of work remains to be done on this problem.

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(PHD, 1971)

Abstract:

A linear theory is developed for the steady free surface flow of a viscous fluid past a general system of submerged flow disturbances (a point mass source and three orthogonal point forcelets). The viscous character of the flow is approximated by using the Oseen linearization of the Navier-Stokes equations.

Solution of the fundamental problem (point flow disturbances) using double Fourier transforms furnishes formal representations of all the interesting flow quantities: the wave height, the three components of the perturbation velocity, and the dynamic pressure. Asymptotic expansions are presented for the ‘free’ or propagating parts of the flow quantities as they would appear far downstream.

Centerplane distributions of the flow disturbance singularities are used to model the flow about a symmetric thin ship. From the application of the momentum theorem, general formulae are derived for the total fluid drag on a ship in a viscous flow. These results are then specialized for use with the Oseen equations. The wave resistance formulae are of particular interest because they contain the strengths of the three forcelet distributions as well as the mass source distribution.

A numerical example of a wave resistance calculation is presented in which the four distribution functions are prescribed. The results are compared to known experimental curves. These indicate that significant features in the character of ship wave resistance can be qualitatively described by including the strengths of local viscous forces acting on the body
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(PHD, 1971)

Abstract:

A hydromechanical theory is developed for cycloidal propellers for two limiting modes of operation wherein U » ΩR and U « ΩR, with U the rectilinear propeller speed (speed of advance) and ΩR the rotational blade speed. A first order theory is developed from the basic principles of the kinematics and dynamics of fluid motion and proceeds from the point of view of unsteady hydrofoil theory.

Explicit expressions for the instantaneous forces and moments produced by blade motions are presented. On the basis of these results an optimization procedure is carried out which minimizes the energy loss under the constraint of specified mean thrust. Under optimal conditions the propeller is found to possess high Froude efficiencies in both the high and low speed modes of propulsion. This efficiency is defined as the ratio of the average useful work obtained during one cycle of propeller operation to the average power input required to sustain the motion of the propeller during the cycle.

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(PHD, 1971)

Abstract:

NOTE: Text or symbols not renderable in plain ASCII are indicated by […]. Abstract is included in .pdf document.

The helical motion of an infinitely long flagellum with a cross-sectional radius b, along which a helical wave of amplitude h, wave-length [lambda] and phase velocity c is propagated, has been analyzed by using Stokes’ equations in a helical coordinate system (r,[xi],x). In order to satisfy all the boundary conditions, namely the no-slip condition on the flagellum surface and zero perturbation velocity at infinity, the flagellum must propel itself with a propulsion velocity U in the opposite direction to the phase velocity c. For small values of kb (where k = 2[pi]/[lambda] is the wave number), by a single-harmonic approximation for the outer region (r > h), the ratio of the propulsion velocity U to the phase velocity c is found to be[…], where Kn is the modified Bessel function of the second kind.

A modified and improved version of the Gray and Hancock method has been developed and applied to evaluate helical movements of a freely swimming microorganism with a spherical head of radius a and a tail of finite length and cross-sectional radius b. The propulsion velocity U and the induced angular velocity [omega] of the organism are derived. In order that this type of motion can be realized, it is necessary for the head of the organism to exceed a certain critical size, and some amount of body rotation is inevitable. For fixed kb and kh, an optimum head-tail ratio a/b, at which the propulsion velocity U reaches a maximum, has been discovered. The power required for propulsion by means of helical waves is determined, based on which a hydromechanical efficiency [eta] is defined. This [eta] reaches a maximum at kh […] 0.9 for microorganisms with optimum head-tail ratios. In the neighborhood of kh = 0.9, the optimum head-tail ratio varies in the range 15 < a/b < 40, the propulsion velocity in 0.08 < U/c < 0.2, and the efficiency in 0.14 < [eta] < 0.24, as kb varies over 0.03 < kb < 0.2.

The modified version of the Gray and Hancock method has also been utilized to describe the locomotion of spirochetes. It is found that although a spirochete has no head to resist the induced viscous torque, it can still propel by means of helical waves provided that the spirochete spins with an induced angular velocity [omega]. Thus the ‘Spirochete paradox’ is resolved. In order to achieve a maximum propulsion velocity, it is discovered that a spirochete should keep its amplitude-wavelength ratio h/[lambda] around 1:6 (or kh […] 1). At kh = 1, the propulsion velocity varies in the range 0 < U/c < 0.2, and the induced angular velocity in 0.4 < [omega]/[low-case omega] < 1 (where [low-case omega] = kc is the circular frequency of the helical wave), as the radius-amplitude ratio varies over 0 < b/h < 1.

A series of experiments have been carried out to determine by simulation the relative importance of the so-called ‘neighboring’ effect and ‘end’ effect, and results for the case of uniform helical waves are presented
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(PHD, 1969)

Abstract:

The problem considered is that of minimizing the drag of a symmetric plate in infinite cavity flow under the constraints of fixed arclength and fixed chord. The flow is assumed to be steady, irrotational, and incompressible. The effects of gravity and viscosity are ignored.

Using complex variables, expressions for the drag, arclength, and chord, are derived in terms of two hodograph variables, Γ (the logarithm of the speed) and β (the flow angle), and two real parameters, a magnification factor and a parameter which determines how much of the plate is a free-streamline.

Two methods are employed for optimization:

__The parameter method.__Γ and β are expanded in finite orthogonal series of N terms. Optimization is performed with respect to the N coefficients in these series and the magnification and free-streamline parameters. This method is carried out for the case N = 1 and minimum drag profiles and drag coefficients are found for all values of the ratio of arclength to chord.__The variational method.__A variational calculus method for minimizing integral functionals of a function and its finite Hilbert transform is introduced, This method is applied to functionals of quadratic form and a necessary condition for the existence of a minimum solution is derived. The variational method is applied to the minimum drag problem and a nonlinear integral equation is derived but not solved.

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(PHD, 1968)

Abstract:

Part I

The slow, viscous flow past a thin screen is analyzed based on Stokes equations. The problem is reduced to an associated electric potential problem as introduced by Roscoe. Alternatively, the problem is formulated in terms of a Stokeslet distribution, which turns out to be equivalent to the first approach.

Special interest is directed towards the solution of the Stokes flow past a circular annulus. A “Stokeslet” formulation is used in this analysis. The problem is finally reduced to solving a Fredholm integral equation of the second kind. Numerical data for the drag coefficient and the mean velocity through the hole of the annulus are obtained.

Stokes flow past a circular screen with numerous holes is also attempted by assuming a set of approximate boundary conditions. An “electric potential” formulation is used, and the problem is also reduced to solving a Fredholm integral equation of the second kind. Drag coefficient and mean velocity through the screen are computed.

Part II

The purpose of this investigation is to formulate correctly a set of boundary conditions to be prescribed at the interface between a viscous flow region and a porous medium so that the problem of a viscous flow past a porous body can be solved.

General macroscopic equations of motion for flow through porous media are first derived by averaging Stokes equations over a volume element of the medium. These equations, including viscous stresses for the description, are more general than Darcy’s law. They reduce to Darcy’s law when the Darcy number becomes extremely small.

The interface boundary conditions of the first kind are then formulated with respect to the general macroscopic equations applied within the porous region. An application of such equations and boundary conditions to a Poiseuille shear flow problem demonstrates that there usually exists a thin interface layer immediately inside the porous medium in which the tangential velocity varies exponentially and Darcy’s law does not apply.

With Darcy’s law assumed within the porous region, interface boundary conditions of the second kind are established which relate the flow variables across the interface layer. The primary feature is a jump condition on the tangential velocity, which is found to be directly proportional to the normal gradient of the tangential velocity immediately outside the porous medium. This is in agreement with the experimental results of Beavers, et al.

The derived boundary conditions are applied in the solutions of two other problems: (1) Viscous flow between a rotating solid cylinder and a stationary porous cylinder, and (2) Stokes flow past a porous sphere.

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(PHD, 1968)

Abstract:

The subject under investigation concerns the steady surface wave patterns created by small concentrated disturbances acting on a non-uniform flow of a heavy fluid. The initial value problem of a point disturbance in a primary flow having an arbitrary velocity distribution (U(y), 0, 0) in a direction parallel to the undisturbed free surface is formulated. A geometric optics method and the classical integral transformation method are employed as two different methods of solution for this problem. Whenever necessary, the special case of linear shear (i.e. U(y) = 1+ϵy)) is chosen for the purpose of facilitating the final integration of the solution.

The asymptotic form of the solution obtained by the method of integral transforms agrees with the leading terms of the solution obtained by geometric optics when the latter is expanded in powers of small ϵ r.

The overall effect of the shear is to confine the wave field on the downstream side of the disturbance to a region which is smaller than the wave region in the case of uniform flows. If U(y) vanishes, and changes sign at a critical plane y = y_{cr} (e.g. ϵy_{cr} = -1 for the case of linear shear), then the boundary of this asymmetric wave field approaches this critical vertical plane. On this boundary the wave crests are all perpendicular to the x-axis, indicating that waves are reflected at this boundary.

Inside the wave field, as in the case of a point disturbance in a uniform primary flow, there exist two wave systems. The loci of constant phases (such as the crests or troughs) of these wave systems are not symmetric with respect to the x-axis. The geometric optics method and the integral transform method yield the same result of these loci for the special case of U(y) = U_{o}(1 + ϵy) and for large Kr (ϵr ˂˂ 1 ˂˂ Kr).

An expression for the variation of the amplitude of the waves in the wave field is obtained by the integral transform method. This is in the form of an expansion in small ϵr. The zeroth order is identical to the expression for the uniform stream case and is thus not applicable near the boundary of the wave region because it becomes infinite in that neighborhood. Throughout this investigation the viscous terms in the equations of motion are neglected, a reasonable assumption which can be justified when the wavelengths of the resulting waves are sufficiently large.

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(PHD, 1967)

Abstract: NOTE: Text or symbols not renderable in plain ASCII are indicated by […]. Abstract is included in .pdf document. The method of matched singular perturbation expansions is used to solve the problem of a steady two-dimensional flow of a perfect fluid with a free surface under the influence of gravity. A flat plate of length […] is inclined at an angle [alpha] to the horizontal and its trailing edge is immersed to a depth h below the surface of an otherwise uniform stream of infinite depth, the velocity at upstream infinity being U. A parameter […] (Froude number […]) is assumed small so that the flow separates smoothly at the leading and trailing edges, giving rise to an upward jet and gravity waves in the downstream. An inner solution for the velocity field is obtained which is valid near the plate and an outer solution which holds far away. These are determined through the orders 1,[beta log beta], [beta], [beta^2 log^2 beta], [beta^2 log beta] up to order [beta^2], and are matched with one another to these orders. In contrast with linearized planing theory, the depth of submergence can be prescribed as a parameter. The lift coefficient is calculated for several values of [alpha], […] and [beta]. The results reduce to known ones in certain limits.

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(PHD, 1963)

Abstract:

I. Some initial value problems are studied regarding the radiation and scattering of gravity waves by finite bodies in an infinitely deep ocean. Emphasis is placed on the case where a finite number of thin plates lie on a vertical line, for which the general solution is obtained by transforming the boundary value problem to one of the Riemann-Hilbert type. Explicit investigations are made for the large time behavior of the free surface elevation for the case of a rolling plate, and for the Cauchy-Poisson problems in the presence of a stationary plate. By taking the limit as t → ∞, the steady state solution is derived for a harmonic point pressure acting on the free surface near a vertical barrier. Finally a formal asymptotic representation of the free surface elevation is given for large time when the geometry of the submerged bodies is arbitrary.

- The subject gravity waves in the two dimensional flow of a vertically stratified fluid is investigated with regard to the dynamic effects of a submerged singularity. Love’s linearized equations are adopted as the basis for the theory. Two specific cases are treated according as the parameter N^2 being a constant or a function of depth, where
N^2 = g/P_o dP_o/dy

characterizes the density variation in the fluid. The first example of constant N^2 is physically a hypothetical case but can be given an exact mathematical analysis; it is found that in a deep ocean with such a density variation the interval waves are local in nature, i.e., their amplitudes diminish to zero as the distance from the singularity becomes very large. In the second example an asymptotic theory for small Froude number, U^2/gL « 1, is developed when N^2(y) assumes the profile roughly resembling the actual situation in an ocean where a pronounced maximum called a seasonal thermocline occurs. Internal waves are now progagated to the downstream infinity in a manner analogous to the channel propagation of sound in an inhomogenous medium.

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(PHD, 1962)

Abstract:

This investigation deals with a perturbation theory for unsteady cavity flows in which the time-dependent part of the flow may be considered as a small perturbation superimposed on an established steady cavity flow of an ideal fluid, the gravity effect being neglected in this study. In order to make a comparison between the various existing steady-cavity-flow models when applied to unsteady motions, some of these models have been employed to evaluate the small time behavior of, and the initial reaction to an unsteady disturbance. Furthermore, the mechanism by which the cavity volume may be changed with time is studied and the initial hydrodynamic force resulting from such change is calculated. The second kind of unsteady cavity flow problems treated here is characterized by the fact that the disturbances are limited to be small for all time instants. Based on a systematic linearization with respect to the steady basic flow, a general perturbation theory for unsteady cavity flows is formulated. From this perturbation theory the generation of surface waves along the cavity boundary is revealed, much in the same way as the classical gravity waves in water, except with the centrifugal acceleration due to the curvature of the free-streamlines in the basic flow playing the role of an equivalent gravity effect.

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