Abstract: For ideal hydrodynamic modeling of earthquake-generated tsunamis, the principal features of tsunamis occuring in nature are abstracted to provide a fundamental case of a one-dimensional solitary wave of height a, propagating in a layer of water of uniform rest depth h for modeling the tsunami progressing in the open ocean over long range, with height down to a/h ≃ 10^(−4) as commonly known. The Euler model is adopted for evaluating the irrotational flow in an incompressible and inviscid fluid to attain exact solutions so that the effects of nonlinearity and wave dispersion can both be fully accounted for with maximum relative error of O(10^(−6)) or less. Such high accuracy is needed to predict the wave-energy distribution as the wave magnifies to deliver any devastating attack on coastal destinations. The present UIFE method, successful in giving the maximum wave of height (a/h = 0.8331990) down to low ones (e.g. a/h = 0.01), becomes, however, impractical for similar evaluations of the dwarf waves (a/h < 0.01) due to the algebraic branch singularities rising too high to be accurately resolved. Here, these singularities are all removed by introducing regularized coordinates under conformal mapping to establish the regularized solitary-wave theory. This theory is ideal to differentiate between the nonlinear and dispersive effects in various premises for producing an optimal tsunami model, with new computations all regular uniformly down to such low tsunamis as that of height a/h = 10^(−4).

Publication: Journal of Engineering Mathematics Vol.: 70 No.: 1-3 ISSN: 0022-0833

ID: CaltechAUTHORS:20110623-094411666

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Abstract: This is a series of studies on Wu’s conjecture and on its resolution to be presented herein. Both are devoted to expound all the comprehensive properties of Cauchy’s function f(z) (z = x + iy) and its integral J[f(z)] ≡ (2πi)^(−1) ∮_Cf(t)(t−z)^(−1)dt taken along the unit circle as contour C, inside which (the open domain D^+) f(z) is regular but has singularities distributed in open domain D^− outside C. Resolution is given to the inverse problem that the singularities of f(z) can be determined in analytical form in terms of the values f(t) of f(z) numerically prescribed on C (|t| = 1), as so enunciated by Wu’s conjecture. The case of a single singularity is solved using complex algebra and analysis to acquire the solution structure for a standard reference. Multiple singularities are resolved by reducing them to a single one by elimination in principle, for which purpose a general asymptotic method is developed here for resolution to the conjecture by induction, and essential singularities are treated with employing the generalized Hilbert transforms. These new methods are applicable to relevant problems in mathematics, engineering and technology in analogy with resolving the inverse problem presented here.

Publication: Acta Mechanica Sinica Vol.: 27 No.: 3 ISSN: 0567-7718

ID: CaltechAUTHORS:20110718-115514762

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Abstract: This article studies on Cauchy’s function f (z) and its integral, (2πi)J[f(z)] ≡ ∮f(t)dt(t−z) taken along a closed simple contour C, in regard to their comprehensive properties over the entire z = x + iy plane consisted of the simply connected open domain D + bounded by C and the open domain D^− outside C. (1) With f (z) assumed to be C^n (n < ∞-times continuously differentiable) ∀ z ∈ D^+ and in a neighborhood of C, f (z) and its derivatives f^(n)(z) are proved uniformly continuous in the closed domain D^+ = [D^+ + C]. (2) Cauchy’s integral formulas and their derivatives ∀z ∈ D^+ (or ∀z ∈ D^−) are proved to converge uniformly in D^+ (or in [D^ + C]), respectively, thereby rendering the integral formulas valid over the entire z-plane. (3) The same claims (as for f (z) and J[f (z)]) are shown extended to hold for the complement function F(z), defined to be C^n ∀z ∈ D^- and about C. (4) The uniform convergence theorems for f (z) and F(z) shown for arbitrary contour C are adapted to find special domains in the upper or lower half z-planes and those inside and outside the unit circle |z| = 1 such that the four generalized Hilbert-type integral transforms are proved. (5) Further, the singularity distribution of f(z) in D^− is elucidated by considering the direct problem exemplified with several typical singularities prescribed in D^−. (6) A comparative study is made between generalized integral formulas and Plemelj’s formulas on their differing basic properties. (7) Physical significances of these formulas are illustrated with applications to nonlinear airfoil theory. (8) Finally, an unsolved inverse problem to determine all the singularities of Cauchy function f(z) in domain D −, based on the continuous numerical value of f(z)∀z ∈ D+=[D++C], is presented for resolution as a conjecture.

Publication: Acta Mechanica Sinica Vol.: 27 No.: 2 ISSN: 0567-7718

ID: CaltechAUTHORS:20110718-112448228

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Abstract: This expository review is devoted to fish swimming and bird/insect flight. (a) The simple waving motion of an elongated flexible ribbon plate of constant width propagating a wave distally down the plate to swim forward in a fluid, initially at rest, is first considered to provide a fundamental concept on energy conservation. It is generalized to include variations in body width and thickness, with appended dorsal, ventral and caudal fins shedding vortices to closely simulate fish swimming, for which a nonlinear theory is presented for large-amplitude propulsion. (b) For bird flight, the pioneering studies on oscillatory rigid wings are discussed with delineating a fully nonlinear unsteady theory for a two-dimensional flexible wing with arbitrary variations in shape and trajectory to provide a comparative study with experiments. (c) For insect flight, recent advances are reviewed by items on aerodynamic theory and modeling, computational methods, and experiments, for forward and hovering flights with producing leading-edge vortex to yield unsteady high lift. (d) Prospects are explored on extracting prevailing intrinsic flow energy by fish and bird to enhance thrust for propulsion. (e) The mechanical and biological principles are drawn together for unified studies on the energetics in deriving metabolic power for animal locomotion, leading to the surprising discovery that the hydrodynamic viscous drag on swimming fish is largely associated with laminar boundary layers, thus drawing valid and sound evidences for a resounding resolution to the long-standing fish-swim paradox proclaimed by Gray (1936, 1968).

Publication: Annual Review of Fluid Mechanics Vol.: 43ISSN: 0066-4189

ID: CaltechAUTHORS:20110524-112751883

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Abstract: This work applies the regularized solitary wave theory to develop accurate computational method for evaluating the dwarf solitary waves, with amplitude-to-water depth ratio α ≤ 10⁻², as a useful model of one-dimensional tsunamis propagating in the open ocean. The algebraic branch singularities of these solitary waves magnifying with diminishing wave amplitude, making their computations insurmountable by existing methods, are removed by the regularized coordinates given by this new theory. Numerical examples show that this new method can produce accurate results even for α ≅ 10⁻³ or less.

Publication: Journal of Hydrodynamics Vol.: 22 No.: S1 ISSN: 1001-6058

ID: CaltechAUTHORS:20200226-133731469

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Abstract: In this paper, we investigate the locomotion of fish and birds by applying a new unsteady, flexible wing theory that takes into account the strong nonlinear dynamics semi-analytically. We also make extensive comparative study between the new approach and the modified vortex blob method inspired from Chorin’s and Krasny’s work. We first implement the modified vortex blob method for two examples and then discuss the numerical implementation of the nonlinear analytical mathematical model of Wu. We will demonstrate that Wu’s method can capture the nonlinear effects very well by applying it to some specific cases and by comparing with the experiments available. In particular, we apply Wu’s method to analyze Wagner’s result for a wing abruptly undergoing an increase in incidence angle. Moreover, we study the vorticity generated by a wing in heaving, pitching and bending motion. In both cases, we show that the new method can accurately represent the vortex structure behind a flying wing and its influence on the bound vortex sheet on the wing.

Publication: Journal of Computational Physics Vol.: 225 No.: 2 ISSN: 0021-9991

ID: CaltechAUTHORS:20200310-145803950

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Abstract: This paper extends the previous studies by Wu [Wu TY (2001) Adv Appl Mech 38:291–353; Wu TY (2005) Advances in engineering mechanics—reflections and outlooks. World Scientific; Wu TY (2006) Struct Control Health Monit 13:553–560] to present a fully nonlinear theory for the evaluation of the unsteady flow generated by a two-dimensional flexible lifting surface moving in an arbitrary manner through an incompressible and inviscid fluid for modeling bird/insect flight and fish swimming. The original physical concept founded by Theodore von Kármán and William R. Sears [von Kármán T, Sears WR (1938) J Aero Sci 5:379–390] in describing the complete vortex system of a wing and its wake in non-uniform motion for their linear theory is adapted and extended to a fully nonlinear consideration. The new theory employs a joint Eulerian and Lagrangian description of the wing motion to establish a fully nonlinear theory for a flexible wing moving with arbitrary variations in wing shape and trajectory, and obtain a fully nonlinear integral equation for the wake vorticity in generalizing Herbert Wagner’s [Wagner H (1925) ZAMM 5:17–35] linear version for an efficient determination of exact solutions in general.

Publication: Journal of Engineering Mathematics Vol.: 58 No.: 1-4 ISSN: 0022-0833

ID: CaltechAUTHORS:20170408-164240573

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Abstract: In this paper, we introduce a three-dimensional numerical method for computing the wake behind a flat plate advancing perpendicular to the flow. Our numerical method is inspired by the panel method of J. Katz and A. Plotkin [J. Katz and A. Plotkin, Low-speed Aerodynamics, 2001] and the 2D vortex blob method of Krasny [R. Krasny, Lectures in Appl. Math., 28 (1991), pp. 385--402]. The accuracy of the method will be demonstrated by comparing the 3D computation at the center section of a very high aspect ratio plate with the corresponding two-dimensional computation. Furthermore, we compare the numerical results obtained by our 3D numerical method with the corresponding experimental results obtained recently by Ringuette [M. J. Ringuette, Ph.D. Thesis, 2004] in the towing tank. Our numerical results are shown to be in excellent agreement with the experimental results up to the so-called formation time.

Publication: Communications in Computational Physics Vol.: 1 No.: 2 ISSN: 1815-2406

ID: CaltechAUTHORS:HOUccp06

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Abstract: This paper extends a previous study by Wu (Adv. Appl. Mech. 2001; 38:291-353) to continue developing a fully non-linear theory for calculation of unsteady flow generated by a two-dimensional flexible lifting surface moving in arbitrary manner through an incompressible and inviscid fluid for modelling bird/insect flight and fish swimming. The original physical concept elucidated by von Kármán and Sears (J. Aeronau Sci. 1938; 5:379-390) in describing the complete vortex system of a wing and its wake in non-uniform motion for their linear theory is adapted and applied to a fully non-linear consideration. The new theory employs a joint Eulerian and Lagrangian description of the lifting-surface movement to facilitate the formulation. The present investigation presents further analysis for addressing arbitrary variations in wing shape and trajectory to achieve a non-linear integral equation akin to Wagner's (Z. Angew. Math. Mech. 1925; 5:17-35) linear version for accurate computation of the entire system of vorticity distribution.

Publication: Structural Control and Health Monitoring Vol.: 13 No.: 1 ISSN: 1545-2255

ID: CaltechAUTHORS:WUTschm06

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Abstract: A unified perturbation theory is developed here for calculating solitary waves of all heights by series expansion of base flow variables in powers of a small base parameter to eighteenth order for the one-parameter family of solutions in exact form, with all the coefficients determined in rational numbers. Comparative studies are pursued to investigate the effects due to changes of base parameters on (i) the accuracy of the theoretically predicted wave properties and (ii) the rate of convergence of perturbation expansion. Two important results are found by comparisons between the theoretical predictions based on a set of parameters separately adopted for expansion in turn. First, the accuracy and the convergence of the perturbation expansions, appraised versus the exact solution provided by an earlier paper [1] as the standard reference, are found to depend, quite sensitively, on changes in base parameter. The resulting variations in the solution are physically displayed in various wave properties with differences found dependent on which property (e.g. the wave amplitude, speed, its profile, excess mass, momentum, and energy), on what range in value of the base, and on the rank of the order n in the expansion being addressed. Secondly, regarding convergence, the present perturbation series is found definitely asymptotic in nature, with the relative error δ(n) (the relative mean-square difference between successive orders n of wave elevations) reaching a minimum, δ_m, at a specific order, n=n_m, both depending on the base adopted, e.g. n_(m, α)=11-12 based on parameter α (wave amplitude), n_(m, β)=15 on β (amplitude-speed square ratio), and n_(m, ∈)=17 on ∈ (wave number squared). The asymptotic range is brought to completion by the highest order of n=18 reached in this work.

Publication: Acta Mechanica Sinica Vol.: 21 No.: 6 ISSN: 0567-7718

ID: CaltechAUTHORS:20190816-151854232

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Abstract: A new theory is developed here for evaluating solitary waves on water, with results of high accuracy uniformly valid for waves of all heights, from the highest wave with a corner crest of 120∘ down to very low ones of diminishing height. Solutions are sought for the Euler model by employing a unified expansion of the logarithmic hodograph in terms of a set of intrinsic component functions analytically determined to represent all the intrinsic properties of the wave entity from the wave crest to its outskirts. The unknown coefficients in the expansion are determined by minimization of the mean-square error of the solution, with the minimization optimized so as to take as few terms as needed to attain results as high in accuracy as attainable. In this regard, Stokes’s formula, F^2μπ= tan μπ, relating the wave speed (the Froude number F) and the logarithmic decrement μ of its wave field in the outskirt, is generalized to establish a new criterion requiring (for minimizing solution error) the functional expansion to contain a finite power series in M terms of Stokes’s basic term (singular in μ), such that 2Mμ is just somewhat beyond unity, i.e. 2Mμ≃1. This fundamental criterion is fully validated by solutions for waves of various amplitude-to-water depth ratio α=a/h, especially about α≃0.01, at which M=10 by the criterion. In this pursuit, the class of dwarf solitary waves, defined for waves with α≤0.01, is discovered as a group of problems more challenging than even the highest wave. For the highest wave, a new solution is determined here to give the maximum height α_(hst( =0.8331990, and speed F_(hst) =1.290890, accurate to the last significant figure, which seems to be a new record.

Publication: Acta Mechanica Sinica Vol.: 21 No.: 1 ISSN: 0567-7718

ID: CaltechAUTHORS:20191009-152532150

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Abstract: This work is an exposition of the course with reflections for resolution to three recent studies of fluid mechanical problems. One is to develop a unified theory for solitary waves of all heights, from the highest wave with a corner crest of 1200 vertex angle down to very low ones of diminishing magnitude, with high accuracy based on the Euler model. This has been benefited from reflections on the pioneering works of Sir George G. Stokes (1880)[1] on the foundation of solitary wave theory. Another investigation is to pursue an extension of the linear unsteady wing theory of Theodore von Kármán and William Sears (1938)(2) to a nonlinear theory for lifting-surface with arbitrary time-varying shape, moving along arbitrary trajectory for modeling bird/insect flight and fish swimming. The original physical concept crystallized by von Kármán and Sears in elucidating the complete vortex system of a wing in non-uniform motion for their linear theory appears so clear that it is readily adapted here to a fully nonlinear consideration. Still another revisit is to examine the self-propulsion of ciliates, an interesting field opened by Sir G. I. Taylor (1951)[3]. Reflecting on the needs still remaining, this study has led to explore a conjecture whether the inviscid irrotational flow can be ubiquitous in the microscopic world of living micro-organisms like ciliates self-propelling at vanishing Reynolds numbers, yet still exhibiting phenomena all similar with those commonly observed in the macro-world. Here the intent is to delineate the evolving lines of thinking and deliberations rather than elaborating on substantial details.

ID: CaltechAUTHORS:20110825-142002173

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Abstract: This chapter discusses the theoretical modeling of aquatic and aerial animal locomotion, several objectives that are focused on exploring how, why, and under what premises such high efficiency and low-energy cost can be achieved in these diverse modes of locomotion as a result of a long history of convergent evolution. The chapter takes an integral viewpoint from the foundation built by the pioneering leaders in the field, such as Herbert Wagner, Theodore von Karman, William R. Sears, and Sir James Lighthill, followed by other researchers through developing various theoretical and experimental methods used in studies on the subject. In subdividing the various classes of hydrodynamic theories and the underlying physical conceptions, it is seen that the generalized slender-body theory is readily capable of expounding the complex interaction between the swimming body and the vortex sheets shed from the appended fins, caudal fins, or lunate tails. Some important nonlinear effects are considered, and separate resort is made for mechanophysiological studies on energetics and hydromechanics of fish propulsion involving the biochemical and mechanical conversions of energy.

Publication: Advances in Applied Mechanics Vol.: 38ISSN: 0065-2156

ID: CaltechAUTHORS:20200226-133731574

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Abstract: This chapter presents a unified theory for modeling water waves. The different types of waves in water, varying from ripples on a placid pond, breaking of shoaling waves on a beach, billows on a stormy sea and in the ocean interior, to geophysical waves and devastating tsunamis, are truly extensive. Euler's equations are adopted to describe three-dimensional, incompressible, inviscid long waves on a layer of water of variable depth, which may vary with the horizontal position vector. Four sets of theoretical models for describing fully nonlinear fully dispersive (FNFD) unsteady gravity-capillary waves on water of variable depth in terms of four sets of basic variables are obtained. It is found that for determining solutions to the model equations for initial-boundary value problems with external forcing by surface pressure and seabed motion, effective numerical schemes are useful. It is observed that for two-dimensional irrotational water waves in particular, an alternative closure relation is derived by applying Cauchy's contour integral formula. The modeling of FNFD waves in water of uniform depth is also elaborated.

Publication: Advances in Applied Mechanics Vol.: 37ISSN: 0065-2156

ID: CaltechAUTHORS:20200226-133731702

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Abstract: This chapter describes the coastal hydrodynamics of ocean waves on beach. A comprehensive study on modeling three-dimensional ocean waves coming from an open ocean of uniform depth and obliquely incident on beach with arbitrary offshore slope distribution, while evolving under balanced effects of nonlinearity and dispersion is presented. A family of beach configurations that is uniform in the long-shore direction as a first approximation for beaches with negligible long-shore curvature is considered. The beach slope variation is assumed to have such distributions that the ocean waves will evolve on beach without breaking. The overall approach adopted begins with development of a three-dimensional linear shallow-water wave theory, followed by taking, step by step, the nonlinear and dispersive effects into account. The linear theory is shown to provide a fundamental solution involving a central function, called the beach-wave function that delineates the evolution of the incoming train of simple waves during interaction with any beach belonging to this broad family of beach configurations. This linear theory can easily afford to cover such factors as oblique wave incidence, arbitrary distribution of offshore beach slope, and wavelength variations with respect to beach breadth.

Publication: Advances in Applied Mechanics Vol.: 37ISSN: 0065-2156

ID: CaltechAUTHORS:20200226-133732456

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Abstract: An expository review is given on various theories of modeling weakly to strongly nonlinear, dispersive, time-evolving, three-dimensional gravity-capillary waves on a layer of water. It is based on a new model that allows the nonlinear and dispersive effects to operate to the same full extent as in the Euler equations. Its relationships with some existing models are discussed. Various interesting phenomena will be illustrated with applications of these models and with an exposition on the salient features of nonlinear waves in wave-wave interactions and the related processes of transport of mass and energy.

Publication: Journal of Engineering Mechanics Vol.: 125 No.: 7 ISSN: 0733-9399

ID: CaltechAUTHORS:WUTjem99

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Abstract: This study considers the 3D runup of long waves on a uniform beach of constant or variable downward slope that is connected to an open ocean of uniform depth. An inviscid linear long-wave theory is 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. For waves at nearly grazing incidence, runup 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. 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 [degrees] 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 3D runup and reflection of long waves.

Publication: Journal of Engineering Mechanics Vol.: 125 No.: 7 ISSN: 0733-9399

ID: CaltechAUTHORS:ZHAjem99

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Abstract: Propagation of solitary waves in curved shallow water channels of constant depth and width is investigated by carrying out numerical simulations based on the generalized weakly nonlinear and weakly dispersive Boussinesq model. The objective is to investigate the effects of channel width and bending sharpness on the transmission and reflection of long waves propagating through significantly curved channels. Our numerical results show that, when travelling through narrow channel bends including both smooth and sharp-cornered 90°-bends, a solitary wave is transmitted almost completely with little reflection and scattering. For wide channel bends, we find that, if the bend is rounded and smooth, a solitary wave is still fully transmitted with little backward reflection, but the transmitted wave will no longer preserve the shape of the original solitary wave but will disintegrate into several smaller waves. For solitary waves travelling through wide sharp-cornered 90°-bends, wave reflection is seen to be very significant, and the wider the channel bend, the stronger the reflected wave amplitude. Our numerical results for waves in sharp-cornered 90°-bends revealed a similarity relationship which indicates that the ratios of the transmitted and reflected wave amplitude, excess mass and energy to the original wave amplitude, mass and energy all depend on one single dimensionless parameter, namely the ratio of the channel width b to the effective wavelength [lambda][sub]e. Quantitative results for predicting wave transmission and reflection based on b/[lambda][sub]e are presented.

Publication: Journal of Fluid Mechanics Vol.: 362ISSN: 0022-1120

ID: CaltechAUTHORS:SHIjfm98

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Abstract: An expository review is given on a theory of modeling fully nonlinear, fuly dispersive, time-evolving, three-dimensional gravity-capillary waves on water of uniform depth. Its relationship with some existing models will be discussed.

ID: CaltechAUTHORS:20121022-165111549

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Abstract: Joint theoretical and experimental studies are carried out to investigate the effects of channel cross-sectional geometry on long wave generation and propagation in uniform shallow water channels. The existing channel Boussinesq and channel KdV equations are extended in the present study to include the effects of channel sidewall slope at the waterline in the first-order section-mean equations. Our theoretical results show that both the channel cross-sectional geometry below the unperturbed water surface (characterized by a shape factor kappa) and the channel sidewall slope at the waterline (represented by a slope factor gamma) affect the wavelength (lambda) and time period (Ts) of waves generated under resonant external forcing. A quantitative relationship between lambda, Ts, kappa, and gamma is given by our theory which predicts that, under the condition of equal mean water depth and equal mean wave amplitude, lambda and Ts increase with increasing kappa and gamma. To verify the theoretical results, experiments are conducted in two channels of different geometries, namely a rectangular channel with kappa[equivalent]1, gamma=0 and a trapezoidal channel with kappa=1.27, gamma=0.16, to measure the wavelength of free traveling solitary waves and the time period of wave generation by a towed vertical hydrofoil moving with critical speed. The experimental results are found to be in broad agreement with the theoretical predictions.

Publication: Physics of Fluids Vol.: 9 No.: 11 ISSN: 1070-6631

ID: CaltechAUTHORS:TENpof97

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Abstract: In the present study, we examine the consistency and validity of the Boussinesq and KdV equations in describing nonlinear water waves generated by vertical slender bodies moving with near critical speed in a rectangular channel. Our study is focused on investigating the effect of disturbance length L on wave generation, and whether the two long wave models, which in theory require L to be much greater than water depth H, can actually be applied to cases where L/H = O(1). Our numerical results based on the KdV and Boussinesq wave models show that, if L is sufficiently long, the dominant factor affecting wave amplitude and period will be the ratio of the maximum disturbance width (i.e., beam of a vertical strut) over the channel width, while L has little effect. This confirms Ertekin''s (1984) and Mei''s (1986) earlier results on the "blockage coefficient". When L is of the same order of H, we found that, as L decreases, it weakens the forcing strength significantly. Results from our towing tank experiments with Froude number ranging from 0.8 to 1.07 revealed that the long wave models give good predictions for resonantly forced long waves even when L is slightly shorter than water depth.

Vol.: 3
ID: CaltechAUTHORS:20200306-162005103

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Abstract: For analyzing forced axisymmetric flow of a non-uniformly rotating, inviscid and incompressible fluid within a long tube of slowly varying radius, a theoretical model called the forced Korteweg-de Vries (fKdV) equation with variable coefficients is derived to calculate the amplitude function of the Stokes stream function. When the fluid system is placed under forcing by axisymmetric disturbance steadily moving with a transcritical velocity, new numerical results of flow streamlines are presented to show that well-defined axisymmetrical recirculating eddies can be periodically produced and sequentially emitted to radiate upstream of the disturbance, becoming permanent in form as a procession of vortex solitons. The Rankine vortex and the Burgers vortex are adopted as two primary flows to exemplify this phenomenon and it is shown that flow with a highly centralized axial vorticity is more effective in producing upstream-radiating vortex solitons.

Publication: Wave Motion Vol.: 24 No.: 3 ISSN: 0165-2125

ID: CaltechAUTHORS:20200226-133732516

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Abstract: The bidirectional long-wave model introduced by Wu (1994) and Yih & Wu (1995) is applied to evaluate interactions between multiple solitary waves progressing in both directions in a uniform channel of rectangular cross-section and undergoing collisions of two classes, one being head-on and the other overtaking collisions between these solitons. For a binary head-on collision, the two interacting solitary waves are shown to merge during a phase-locking period from which they reemerge separated, each asymptotically recovering its own initial identity while both being retarded in phase from their original pathlines. For a binary overtaking collision between a soliton of height α1 overtaking a weaker one of height α₁, the two solition peaks are shown to either pass through each other or remain separated throughout the encounter according as α₁/α₂ or <3, respectively. With no phase locking during the overtaking, the two solitary waves re-emerge afterwards with their initial forms recovered and with the stronger wave being advanced whereas the weaker one retarded in phase from their original pathlines. By extension, the theory is generalized to apply to uniform channels of arbitrary cross-sectional shape.

Publication: Acta Mechanica Sinica Vol.: 11 No.: 4 ISSN: 0567-7718

ID: CaltechAUTHORS:20200226-133730786

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Abstract: A corrected version of the Boussinesq equation for long water waves is derived and its general solution for interaction of any number of solitary waves, including head-on collisions, is given. For two solitary waves in head-on collision (which includes the case of normal reflection) the results agree with the experiments known.

Publication: Acta Mechanica Sinica Vol.: 11 No.: 3 ISSN: 0567-7718

ID: CaltechAUTHORS:20200226-133730878

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Abstract: This paper applies two theoretical wave models, namely the generalized channel Boussinesq (gcB) and the channel Korteweg–de Vries (cKdV) models (Teng & Wu 1992) to investigate the evolution, transmission and reflection of long water waves propagating in a convergent–divergent channel of arbitrary cross-section. A new simplified version of the gcB model is introduced based on neglecting the higher-order derivatives of channel variations. This simplification preserves the mass conservation property of the original gcB model, yet greatly facilitates applications and clarifies the effect of channel cross-section. A critical comparative study between the gcB and cKdV models is then pursued for predicting the evolution of long waves in variable channels. Regarding the integral properties, the gcB model is shown to conserve mass exactly whereas the cKdV model, being limited to unidirectional waves only, violates the mass conservation law by a significant margin and bears no waves which are reflected due to changes in channel cross-sectional area. Although theoretically both models imply adiabatic invariance for the wave energy, the gcB model exhibits numerically a greater accuracy than the cKdV model in conserving wave energy. In general, the gcB model is found to have excellent conservation properties and can be applied to predict both transmitted and reflected waves simultaneously. It also broadly agrees well with the experiments. A result of basic interest is that in spite of the weakness in conserving total mass and energy, the cKdV model is found to predict the transmitted waves in good agreement with the gcB model and with the experimental data available

Publication: Journal of Fluid Mechanics Vol.: 266ISSN: 0022-1120

ID: CaltechAUTHORS:TENjfm94.973

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Abstract: This paper investigates the basic stability properties of the standing soliton occurring in Faraday resonance. We start by constructing a complete picture of the linearized stability of the soliton solutions of the forced, damped, nonlinear Schrödinger (NLS) equation. The linear-stability analysis shows that a small region of the parameter space is stable. The stable region is bordered by regions in which the soliton is unstable to the continuous spectrum of the linear operator, unstable to discrete spectrum of the linear operator or unstable to both. We perform numerical simulations of the forced, damped, NLS equation to determine the evolution of the solution after the onset of instability. The simulations fully confirm the predictions of the linear theory in each of the various régimes. Lastly, broad experimental observations are carried out to investigate the evolution of the solitons in the laboratory. The observations qualitatively confirm all the predicted behavior for the evolution of the soliton in the stable and unstable parameter régimes when the amplitude of the forcing and the frequency detuning are small.

ID: CaltechAUTHORS:20200114-151136505

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Abstract: This talk will examine the stability properties of representative cases of nonlinear dispersive waves generated and sustained at resonance of physical systems capable of supporting solitary waves. The criteria are sought for realizing the remarkable phenomenon of periodic production of upstream-radiating solitary waves by critical disturbances moving steadily in a layer of shallow water as modeled by the forced KdV equation. Of primary interest are the distinctive features of instabilities of a few typical steady basic flows, the salient new characteristics of the associated eigenvalue problems, the relevant nonlinear effects, and the resulting bifurcation diagrams.

ID: CaltechAUTHORS:20201023-102058196

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Abstract: The generalized channel Boussinesq (gcB) two-equation model and the forced channel Korteweg-de Vries (cKdV) one-equation model previously derived by the authors are further analysed and discussed in the present study. The gcB model describes the propagation and generation of weakly nonlinear, weakly dispersiveand weakly forced long water waves in channelsof arbitrary shape that may vary both in space and time, and the cKdV model is applicable to unidirectional motions of such waves, which may be sustained under forcing at resonance of the system. These two models are long wave approximations of a hierarchy set of section-mean conservation equations of mass, momentum and energy, which are exact for inviscid fluids. Results of these models are demonstrated with four specific channel shapes, namely variable rectangular, triangular, parabolic and semicircular sections, in which case solutions are obtained in closed form. In particular, for uniform channels of equal mean water depth, different cross-sectional shapes have a leading-order effect only on the variations of a K-factor of the coefficient of the term bearing the dispersive effects in the model equations. For this case, the uniform-channel analogy theorem enunciated here shows that long waves of equal (mean) height in different uniform channels of equal mean depth but distinct K-shape factors will propagate with equal veolcity and with their effective wavelengths appearing K times of that in the rectangular channel, for which K=1. It also shows that the further channel shape departs from the rectangular, the greater the value of K. Based on this observation, the solitary and cnoidal waves in a K-shaped channel are compared with experiments on wave profiles and wave velocities. Finally, some three-dimensional features of these solitary waves are presented for a triangular channel.

Publication: Journal of Fluid Mechanics Vol.: 242ISSN: 0022-1120

ID: CaltechAUTHORS:TENjfm92

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Abstract: This is a study of the scattering and diffraction of a solitary wave by a surface-piercing vertical cylinder held fixed in shallow water. Particular interest is focused on the roles played by the nonlinear effects and the dispersive effects in this fully three-dimensional problem of strong interaction between a solitary wave and a solid structure. The theoretical model adopted here for predicting the scattering and propagation of three-dimensional long waves in shallow water is the generalized Boussinesq (gB) two-equation model, developed by Wu. Using this model, the predicted flow field, the free-surface elevations, the wave-induced forces acting on the cyiindcr during the wave impact, and the subsequent evolution of the scattered wave field are numerically evaluated. The numerical results show that the front of the scattered wave field propagates very nearly in a circular belt, which is concentric to the cylinder as an overall topographical structure. This remarkable asymptotic geometrical feature of the resulting scattered wave cannot be obtained without the basic equations being able to correctly model the three-dimensional effects, and without bias toward the direction of wave propagation. The role of the nonlinear, dispersive, and linear wave effects during the wave-structure interaction are discussed in detail.

Publication: Journal of Waterway, Ports, Coastal, and Ocean Engineering Vol.: 118 No.: 5 ISSN: 0733-950X

ID: CaltechAUTHORS:WANjwpcoe92

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Abstract: This paper explores the basic mechanism underlying the remarkable phenomenon that a forcing excitation stationary in character and sustained at near resonance in a shallow channel of uniform water depth generates a non-stationary response in the form of a sequential upstream emission of solitary waves. Adopting the forced Korteweg-de Vries (fKdV) model and using two of its steady forced solitary wave solutions as primary flows, the stability of these two transcritical steady motions is investigated, and their bifurcation diagrams relating these solutions to other stationary solutions determined, with the forcing held fixed. The corresponding forcing functions are characterized by a velocity parameter for one, and an amplitude parameter for the other of the steadily moving excitations.

Publication: Philosophical Transactions: Physical Sciences and Engineering Vol.: 337 No.: 1648 ISSN: 0962-8428

ID: CaltechAUTHORS:20141217-132715581

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Abstract: A layer of water in a cylindrical tank is known to be capable of sustaining standing solitary waves within a certain parametric domain when the tank is excited under vertical oscillation. A new mode of forced waves is discovered to exist in a different parametric domain for rectangular tanks with the wave sloshing across the short side of the tank and with its profile modulated by one or more hyperbolic-tangent, or kink-wave-like envelopes. A theoretical explanation for the kink wave properties is provided. Experiments were performed to confirm their existence.

Publication: Proceedings of the Royal Society of London. Series A, Mathematical, Physical and Engineering Sciences Vol.: 434 No.: 1891 ISSN: 0962-8444

ID: CaltechAUTHORS:20141215-104632186

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Abstract: In this joint theoretical, numerical and experimental study, we investigate the phenomenon of forced generation of nonlinear waves by disturbances moving steadily with a transcritical velocity through a layer of shallow water. The plane motion considered here is modelled by the generalized Boussinesq equations and the forced Korteweg-de Vries (fKdV) equation, both of which admit two types of forcing agencies in the form of an external surface pressure and a bottom topography. Numerical results are obtained using both theoretical models for the two types of forcings. These results illustrate that within a transcritical speed range, a succession of solitary waves are generated, periodically and indefinitely, to form a procession advancing upstream of the disturbance, while a train of weakly nonlinear and weakly dispersive waves develops downstream of an ever elongating stretch of a uniformly depressed water surface immediately behind the disturbance. This is a beautiful example showing that the response of a dynamic system to steady forcing need not asymptotically tend to a steady state, but can be conspicuously periodic, after an impulsive start, when the system is being forced at resonance. A series of laboratory experiments was conducted with a cambered bottom topography impulsively started from rest to a constant transcritical velocity U, the corresponding depth Froude number F = U/(gh[sub]0)^1/2 (g being the gravitational constant and h[sub]0 the original uniform water depth) being nearly the critical value of unity. For the two types of forcing, the generalized Boussinesq model indicates that the surface pressure can be more effective in generating the precursor solitary waves than the submerged topography of the same normalized spatial distribution. However, according to the fKdV model, these two types of forcing are entirely equivalent. Besides these and some other rather refined differences, a broad agreement is found between theory and experiment, both in respect of the amplitudes and phases of the waves generated, when the speed is nearly critical (0.9 < F < 1.1) and when the forcing is sufficiently weak (the topography-height to water-depth ratio less than 0.15) to avoid breaking. Experimentally, wave breaking was observed to occur in the precursor solitary waves at low supercritical speeds (about 1.1 < F < 1.2) and in the first few trailing waves at high subcritical speeds (about 0.8 < F < 0.9), when sufficiently forced. For still lower subcritical speeds, the trailing waves behaved more like sinusoidal waves as found in the classical case and the forward-running solitary waves, while still experimentally discernible and numerically predicted for 0.6 > F > 0.2, finally disappear at F ~= 0.2. In the other direction, as the Froude number is increased beyond F ~= 1.2, the precursor soliton phenomenon was found also to evanesce as no finite-amplitude solitary waves can outrun, nor can any two-dimensional waves continue to follow, the rapidly moving disturbance. In this supercritical range and for asymptotically large times, all the effects remain only local to the disturbance. Thus, the criterion of the fascinating phenomenon of the generation of precursor solitons is ascertained.

Publication: Journal of Fluid Mechanics Vol.: 199ISSN: 0022-1120

ID: CaltechAUTHORS:EEjfm89

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Abstract: Here we study the phenomenon of internal solitons generated periodically by a two-dimensional disturbance moving through a stratified fluid with a constant transcritical velocity. Experimentally, with a topography moving along the floor of a fluid layer as forcing disturbance, we found that every so often, a new internal solitary wave was generated to propagate ahead of the disturbance, forming in time a procession of upstream-moving solitons. The amplitude and period of generation of the solitons depend on the Froude number, the density stratification and forcing distribution. Theoretical predictions are made numerically based on the inhomogeneous Boussinesq model (IB) and the forced KdV model (IKdV). Broad agreement was found between theory and experiment.

ID: CaltechAUTHORS:20200226-133731284

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Abstract: This study investigates the recently identified phenomenon whereby a forcing disturbance moving steadily with a transcritical velocity in shallow water can generate, periodically, a succession of solitary waves, advancing upstream of the disturbance in procession, while a train of weakly nonlinear and weakly dispersive waves develops downstream of a region of depressed water surface trailing just behind the disturbance. This phenomenon was numerically discovered by Wu & Wu (1982) based on the generalized Boussinesq model for describing two-dimensional long waves generated by moving surface pressure or topography. In a joint theoretical and experimental study, Lee (1985) found a broad agreement between the experiment and two theoretical models, the generalized Boussinesq and the forced Korteweg de Vries (fKdV) equations, both containing forcing functions. The fKdV model is applied in the present study to explore the basic mechanism underlying the phenomenon. To facilitate the analysis of the stability of solutions of the initial-boundary-value problem of the fKdV equation, a family of forced steady solitary waves is found. Any such solution, if once established, will remain permanent in form in accordance with the uniqueness theorem shown here. One of the simplest of the stationary solutions, which is a one-parameter family and can be scaled into a universal similarity form, is chosen for stability calculations. As a test of the computer code, the initially established stationary solution is found to be numerically permanent in form with fractional uncertainties of less than 2% after the wave has traversed, under forcing, the distance of 600 water depths. The other numerical results show that when the wave is initially so disturbed as to have to rise from the rest state, which is taken as the initial value, the same phenomenon of the generation of upstream-advancing solitons is found to appear, with a definite time period of generation. The result for this similarity family shows that the period of generation, T[sub]S, and the scaled amplitude [alpha] of the solitons so generated are related by the formula T[sub]S = const [alpha]^-3/2. This relation is further found to be in good agreement with the first-principle prediction derived here based on mass, momentum and energy considerations of the fKdV equation.

Publication: Journal of Fluid Mechanics Vol.: 184ISSN: 0022-1120

ID: CaltechAUTHORS:WUTjfm87

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Abstract: The transport of particles by cilia lining vapor-filled tubes such as the trachea, bronchi, and upper bronchioles requires that the particles be carried by mucus [Sade et al, 1970], and the force that causes mucus flow is generated by the beat of the underlying cilia. The means by which the ciliary beat force is transmitted to the mucus is not clear. Lucas and Douglas [1934] proposed that cilia penetrate the mucus enough to push it forward in a conveyor-belt fashion. This view is consistent with recent EM studies [Yoneda, 1976; Reissig, Bang, and Bang, 1978] that suggest the serous layer is thinner than the length of a cilium (5-6 µm). In contrast, Ross and Corrsin [1974] developed a theoretical model for mucociliary transport based on the assumption that mucus persists as a "blanket" carried by the serous fluid, which is in turn propelled by the cilia. If one accepts current simplified models which take into account the viscoelastic properties of mucus, the Lucas and Douglas model is the more reasonable concept. Accordingly, the two most recent fluid mechanical models for mucociliary transport incorporate ciliary tip penetration as a central requirement. One [Blake and Winet, 1980] favors the average depth of penetration as the critical force-generating factor, whereas the other [Yates et al, 1980] favors the average number of cilia penetrating per wavelength. There appear to be no articles describing tests of these theoretical models with measurements of mucus and serous fluid below the air-mucus interface. The primary reasons for this deficit are the following: a) Epithelium viewed from the side must be folded over and placed in narrow chambers where mucus blankets tend to adhere to the glass walls, and b) epithelium viewed from above must be observed through mucus which refracts and scatters light unevenly such that one cannot resolved tracer particles in the mucus reflected or transmitted light. We chose to avoid these optical barriers by utilizing fluorescent tracer particles to investigate mucociliaryt flow profiles.

Publication: Cell Motility Vol.: 2 No.: S1 ISSN: 0271-6585

ID: CaltechAUTHORS:20200226-133731188

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Abstract: In 1965, a book was published with the title Research Frontiers in Fluid Dynamics, edited by Raymond Seeger and G. Temple. It was intended to give a panoramic view of some exciting vistas in fluid dynamics. It covered the following areas: (1) High-speed aerodynamics; (2) magnetophydrodynamics (MHD); (3) physics of fluids (low or high density, low or high temperature, etc); (4) constitutive properties of fluids (viscosity, viscoelasticity, etc.); (5) oceanography and meterology; (6) astrophysical and planetary fluid mechanics; and (7) mathematical aspects and numerical aspects. A few of these areas, such as MHD, have blossomed and faded away within a short decade. Some others, such as high-speed aerodynamics, have reached maturity and hope to keep their momentum. In the intervening years, we have witnessed that a number of fields in fluid mechanics have revived from their old times into a new life; still, some have emerged with brand new growth. For instance, the subject of long waves has had a colorful development, with the result of improving our understanding of at least seven different physical phenomena, though originally the solitary water wave was its home base. Low Reynolds number flows have again received new stimuli from many needed applications such as aerosol physics, two-phase flows, rheology, geophysics of the earth interior, as well as micro and molecular biology. Oil exploration has motivated various aspects of marine-related research and development, giving ever-increasing activities in ocean engineering. The energy program, a new glamorous field by its own importance, has brought forth investigations of fluid mechanical problems pertaining to nuclear, geothermal, solar, wind, ocean wave, and other forms of energy sources. Riding on the waves of these broad movements that have carried us thus far, we now hope to forecast the future of fluid mechanics research in 1986. We may like to put the focus at a slightly different depth and ask: What will be the most significant areas of fluid mechanics that by 1986 will enjoy the best prospects of vigorous development, most rewarding not only to the fluid dynamicist but also to mankind, and by then, still offer the expectation of longevity into the 1990's? The task is almost as hard as to make a prophecy on what the political world will be in 1986.

Publication: Journal of Engineering Mechanics Vol.: 107 No.: EM3 ISSN: 0733-9399

ID: CaltechAUTHORS:WUTjem81a

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Abstract: Water waves occurring in the ocean have a wide spectrum of wavelength and period, ranging from capillary waves of 1 cm or shorter wavelength to long waves with wavelength being large compared to ocean depth, anywhere from tens to thousands of kilometers. Of the various long-wavegenic sources, distant body forces can act as the continuous ponderomotive force for the tides. Hurricanes and storms in the sea can develop a sea state, with the waves being worked on by winds and eventually cascading down to swells after a long distance of travel away from their birthplace. Large tsunamis can be ascribed to a rapidly occurring tectonic displacement of the ocean floor (usually near the coast of the Pacific Ocean) over a large horizontal dimension (of hundreds to over a thousand square kilometers) during strong earthquakes, causing vertical displacements to ocean floor of tens of meters. Other generation mechanisms include underwater subsidence or land avalanche in the ocean and submarine volcanic eruption. Gigantic rockfalls and long-period seismic waves can also produce gravity waves in lakes, reservoirs, and rivers. Generation, propagation, and evolution of such long waves in the ocean and their effects in coastal waters and harbors is a subject of increasing importance in civil, coastal, and environmental engineering and science. Of the various long wave phenomena, tsunami appears to stand out in possessing a broad variation of wave characteristics and scaling parameters on the one hand, and, on the other, in having the capacity of inflicting a disastrous effect on the target area. In taking tsunamis as a representative case for the study of long waves in the ocean, it can be said that large tsunamis are generated with a great source of potential energy (as high as 10^15-10^16J ), though the detailed source motion of a specific tsunami is generally difficult to determine. The large size of source region implies that the "new born" waves would be initially long and the energy contained in the large wave-number part (k, nondimensionalized with respect to the local ocean depth, h) would be unimportant. Soon after leaving the source region, the low wave-number components of the source spectrum are further dispersed effectively by the factor sech kh into the even lower wave-number parts. Tsunamis thus evolve into a train of long waves, with wavelength continually increasing from about 50 km to as high as 250 km, but with a quite small amplitude, typically of 1/2 m or smaller, as they travel across the Pacific Ocean at a speed of 650 km/h-760 km/h. There is experimental evidence indicating that tsunamis continually, though slowly, evolve due to dispersion while propagating in the open ocean; this property has been observed by Van Dorn (16) from the data taken at Wake Island of the March 9, 1957 Aleutian tsunami. One of our primary interests is, of course, the evolution of tsumanis in coastal waters and their terminal effects. Large tsunamis can have their wave height amplified many fold in climbing up the continental slope and propagating into shallower water, producing devastating waves (up to 20 m or higher on record) upon arriving at a beach. The terminal amplification can be crucially affected by three-dimensional configurations of the coastal environment enroute to beach. These factors dictate the transmission, reflection, rate of growth, and trapping of tsunamis in their terminal stage. After the first hit on target, a tsunami is partly reflected to travel once over across the Pacific Ocean, with some degree of attenuation -- a process which is still unclear, but is generally known to be small. Based on observations, Munk (13) suggests the figure of the "decay time" (intensity reducing to 1/e) being about 112 day, and the "reverberation time" (intensity falling off to 10^-6) about a week, while the reflection frequency (across the Pacific) is around 1.7/day. To fix idea, the pertinent physical characteristics and their scaling parameters of a tsunami through its life span of evolution can be described qualitatively in Table I. From the aforementioned estimate we note that the dispersion parameter, h/[lambda], and the amplitude parameter, a/h, are both small in general. However, their competitive roles as rated by the Ursell number Ur, can increase from some small values in the deep ocean, typically of order 10^-2 for large tsunamis, by a factor of 10^3 upon arriving in near-shore waters. This indicates that the effects of nonlinearity (amplitude dispersion) are practically nonexistent in the deep ocean, but gradually become more important and can no longer be neglected when the Ursell number increases to order unity or greater during the terminal stage in which the coastal effects manifest. The small values of the dimensionless wave number, kh = 2[pi]h/[lamda] being in the range of 0.6-0.03 during travel in open ocean, suggests that a slight dispersive effect is still present and this can lead to an accumulated effect in predicting the phase position over very large distances of travel. The overall evolution of tsunamis, as only crudely characterized in Table 1, depends in fact on many factors such as the features of source motion, nonlinear and dispersive effects on propagation in one and two dimensions, the three-dimensional configuration of the coastal region, the direction of incidence, converging or diverging passage of the waves, local reflection and adsorption, density stratification in water, etc. While these aspects of physical behavior are akin to tsunamis, they are also relevant to the consideration of other long wave phenomena. With an intent to provide a sound basis for general applications to long wave phenomena in nature, this paper presents (in the section on three-dimensional long-wave models) a basic long-wave equation which is of the Boussinesq class with special reference to tsunami propagation in two horizontal dimensions through water having spatial and temporal variations in depth. Under certain particular conditions (such as the propagation in one space dimension, or primarily one space dimensional of long waves in water of constant depth) this equation reduces to the Korteweg-de Vries equation or the nonlinear Schrodinger equation. In these special cases we have seen the impressive developments in recent studies of the "soliton-bearing" nonlinear partial differential equations by means of such methods as the variational modulation, the inverse scattering analysis, and modern differential geometry (12,14,17). While extensions of these methods to more general cases will require further major developments, the present analysis and survey will concentrate on the three-dimensional (with propagation in two horizontal dimensions) effects under various conditions by examining the validity of different wave models (based on neglecting the effects of nonlinearity, dispersion, or reflection) in different circumstances. From the example of self focusing of weakly-nonlinear waves (given in the section on converging cylindrical long waves), the effects of nonlinearity, dispersion, and reflection will be seen all to play such a major role that the present basic equation cannot be further modified without suffering from a significant loss of accuracy.

Publication: Journal of Engineering Mechanics Vol.: 107 No.: EM3 ISSN: 0733-9399

ID: CaltechAUTHORS:WUTjem81b

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Abstract: A series of experiments have been carried out to investigate the muco-ciliary transport in the trachea of rat, rabbit and chicken with cinemicrographic analyses of the movement, beat pattern and metachrony of the tracheal cilia. As a principal test case of the cilia-tip-penetration-into-mucus hypothesis, detailed studies have been made on the transport of a biochemically viable mucous plug artificially introduced into the chicken trachea both with and without a pressure differential across the mucous plug. Guided by the result of these pilot experiments, a theoretical two-layer model of the muco-ciliary transport is introduced, in which the cilia are assumed to penetrate shallowly into the overlaying mucous layer during the effective stroke and to withdraw from the mucous layer during the recovery stroke. Both the mucus and the serous fluid are assumed to be Newtonian and with widely differ ing viscosities. Based on this model the mucus transport rate is found to depend linearly on the ciliary beat frequency and also on the time of ciliary tip penetration in the mucus. Results are also given on the propulsive force contributed by each individual cilium and an estimate of shear and shear rates within the mucous layer.

Publication: Biorheology Vol.: 17 No.: 1-2 ISSN: 1878-5034

ID: CaltechAUTHORS:20200226-133731089

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Abstract: In order to elucidate the general Stokes flow characteristics present for slender bodies of finite centre-line curvature the singularity method for Stokes flow has been employed to construct solutions to the flow past a slender torus. The symmetry of the geometry and absence of ends has made a highly accurate analysis possible. The no-slip boundary condition on the body surface is satisfied up to an error term of O(E^2 ln E), where E is the slenderness parameter (ratio of cross-sectional radius to centre-line radius). This degree of accuracy makes it possible to determine the force per unit length experienced by the torus up to a term of O(E^2). A comparison is made between the force coefficients of the slender torus to those of a straight slender body to illustrate the large differences that may occur as a result of the finite centre-line curvature.

Publication: Journal of Fluid Mechanics Vol.: 95 No.: 2 ISSN: 0022-1120

ID: CaltechAUTHORS:JOHjfm79

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Abstract: A fluid-mechanical model is developed for representing the mechanism of propulsion of a finite ciliated micro-organism having a prolate-spheroidal shape. The basic concept is the representation of the micro-organism by a prolate-spheroidal control surface upon which certain boundary conditions on the tangential and normal fluid velocities are prescribed. 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 good agreement with the theoretical prediction of the streamlines is found.

Publication: Journal of Fluid Mechanics Vol.: 80 No.: 2 ISSN: 0022-1120

ID: CaltechAUTHORS:KELjfm77

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Abstract: The problem of a uniform transverse flow past a prolate spheroid of arbitrary aspect ratio at low Reynolds numbers has been analysed by the method of matched asymptotic expansions. The solution is found to depend on two Reynolds numbers, one based on the semi-minor axis b, R[sub]b = Ub/v, and the other on the semi-major axis a, R[sub]a = Ua/v (U being the free-stream velocity at infinity, which is perpendicular to the major axis of the spheroid, and v the kinematic viscosity of the fluid). A drag formula is obtained for small values of R[sub]b and arbitrary values of R[sub]a. When R[sub]a is also small, the present drag formula reduces to the Oberbeck (1876) result for Stokes flow past a spheroid, and it gives the Oseen (1910) drag for an infinitely long cylinder when R[sub]a tends to infinity. This result thus provides a clear physical picture and explanation of the 'Stokes paradox' known in viscous flow theory.

Publication: Journal of Fluid Mechanics Vol.: 75 No.: 4 ISSN: 0022-1120

ID: CaltechAUTHORS:CHWjfm76

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Abstract: The present study furthcr explores the fundamental singular solutions for Stokes flow that can be useful for constructing solutions over a wide range of free-stream profiles and body shapes. The primary singularity is the Stokeslet, which is associated with a singular point force embedded in a Stokes flow. From its derivatives other fundamental singularities can be obtained, including rotlets, stresslets, potential doublets and higher-order poles derived from them. For treating interior Stokes-flow problems new fundamental solutions are introduced; they include the Stokeson and its derivatives, called the roton and stresson. These fundamental singularities are employed here to construct exact solutions to a number of exterior and interior Stokes-flow problems for several specific body shapes translating and rotating in a viscous fluid which may itself be providing a primary flow. The different primary flows considered here include the uniform stream, shear flows, parabolic profiles and extensional flows (hyperbolic profiles), while the body shapcs cover prolate spheroids, spheres and circular cylinders. The salient features of these exact solutions (all obtained in closed form) regarding the types of singularities required for the construction of a solution in each specific case, their distribution densities and the range of validity of the solution, which may depend on the characteristic Reynolds numbers and governing geometrical parameters, are discussed.

Publication: Journal of Fluid Mechanics Vol.: 67 No.: 4 ISSN: 0022-1120

ID: CaltechAUTHORS:CHWjfm75

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Abstract: The purpose of this paper is to present a new model for ciliary propulsion intended to rectify certain deficiencies in the existing theoretical models. The envelope model has been developed by several authors including Taylor (1951), Reynolds (1965), Tuck (1968), Blake (1971a,b,c) and Brennen (1974); it employs the concept of representing the ciliary propulsion by a waving material sheet enveloping the tips of the cilia. The principal limitations of this approach, as discussed in the review by Blake and Sleigh (1974), are due to the impermeability and no-slip conditions imposed on the flow at the envelope sheet (an assumption not fully supported by physical observations) and the mathematical necessity of a small amplitude analysis.

ID: CaltechAUTHORS:KELsfn75

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Abstract: A renewed interest in catamaran-type vehicles has emerged in the past decade in connection with the development of unconventional water crafts. This interest has been stimulated by the main advantages that this type of unconventional vehicles may be able to offer, namely, 1. a large, useful deck area (compared with that of conventional ships), 2. a favorable roll stability for various ocean tasks, 3. good seaworthiness characteristics in rough seas, especially when the waterplane area can be made small.

Vol.: 1
ID: CaltechAUTHORS:20150413-100505035

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Abstract: The Seventeenth General Meeting of the American Towing Tank Conference was held in Pasadena, California, from 18th to 20th June 1974, in Baxter Auditorium of the California Institute of Technology. The Conference consisted of Welcoming Messages, five Technical Sessions, a General Business Session and Closing Remarks. In addition, a reception-dinner program and a tour of several laboratories at the Institute were arranged for the participants and accompanying family members. The Conference was attended by about 90 delegates and observers, representing thirty-three different professional establishments and universities.

Vol.: 1
ID: CaltechAUTHORS:20150413-104047718

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Abstract: The present series of studies is concerned with low-Reynolds-number flow in general; the main objective is to develop an effective method of solution for arbitrary body shapes. In this first part, consideration is given to the viscous flow generated by pure rotation of an axisymmetric body having an arbitrary prolate form, the inertia forces being assumed to have a negligible effect on the flow. The method of solution explored here is based on a spatial distribution of singular torques, called rotlets, by which the rotational motion of a given body can be represented. Exact solutions are determined in closed form for a number of body shapes, including the dumbbell profile, elongated rods and some prolate forms. In the special case of prolate spheroids, the present exact solution agrees with that of Jeffery (1922), this being one of very few cases where previous exact solutions are available for comparison. The velocity field and the total torque are derived, and their salient features discussed for several representative and limiting cases. The moment coefficient C[sub]M = M/(8[pi][mu][omega sub 0]ab^2) (M being the torque of an axisymmetric body of length 2a and maximum radius b rotating at angular velocity [omega], about its axis in a fluid of viscosity [mu]) of various body shapes so far investigated is found to lie between 2/3 and 1, usually very near unity for not extremely slender bodies. For slender bodies, an asymptotic relationship is found between the nose curvature and the rotlet strength near the end of its axial distribution. It is also found that the theory, when applied to slender bodies, remains valid at higher Reynolds numbers than was originally intended, so long as they are small compared with the (large) aspect ratio of the body, before the inertia effects become significant.

Publication: Journal of Fluid Mechanics Vol.: 63 No.: 3 ISSN: 0022-1120

ID: CaltechAUTHORS:CHWjfm74

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Abstract: A new class of optimization problems arising in fluid mechanics can be characterized mathematically as equivalent to extremizing a functional in which the two unknown argument functions are related by a singular Cauchy integral equation. Analysis of the first variation of the functional yields a set of dual, nonlinear, integral equations, as opposed to the Euler differential equation in classical theory. A necessary condition for the extremum to be a minimum is derived from consideration of the second variation. Analytical solutions by singular integral equation methods and by the Rayleigh‐Ritz method are discussed for the linearized theory. The general features of these solutions are demonstrated by numerical examples.

Publication: ZAMM - Journal of Applied Mathematics Vol.: 53 No.: 11 ISSN: 0044-2267

ID: CaltechAUTHORS:20200226-133730986

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Abstract: The wall effects in cavity flows have been long recognized to be more important and more difficult to determine than those in single-phase, nonseparated flows. Earlier theoretical investigations of this problem have been limited largely to simple body forms in plane flows, based on some commonly used cavity-flow models, such as the Riabouchinsky, the reentrant jet, or the linearized flow model, to represent a finite cavity. Although not meant to be exhaustive, references may be made to Cisotti (1922), Birkhoff, Plesset and Simmons (1950, 1952), Gurevich (1953), Cohen et al. (1957, 1958), and Fabula (1964). The wall effects in axisymmetric flows with a finite cavity has been evaluated numerically by Brennen (1969) for a disk and a sphere. Some intricate features of the wall effects have been noted in experimental studies by Morgan (1966) and Dobay (1967). Also, an empirical method for correcting the wall effect has been proposed by Meijer (1967). The presence of lateral flow boundaries in a closed water tunnel introduces the following physical effects: (i) First, in dealing with the part of irrotational flow outside the viscous region, these flow boundaries will impose a condition on the flow direction at the rigid tunnel walls. This "streamline-blocking" effect will produce extraneous forces and modifications of cavity shape. (ii) The boundary layer built up at the tunnel walls may effectively reduce the tunnel cross-sectional area, and generate a longitudinal pressure gradient in the working section, giving rise to an additional drag force known as the "horizontal buoyancy." (iii) The lateral constraint of tunnel walls results in a higher velocity outside the boundary layer, and hence a greater skin friction at the wetted body surface. (iv) The lateral constraint also affects the spreading of the viscous wake behind the cavity, an effect known as the "wake-blocking." (v) It may modify the location of the "smooth detachment" of cavity boundary from a continuously curved body. In the present paper, the aforementioned effect (i) will be investigated for the pure-drag flows so that this primary effect can be clarified first. Two cavity flow models, namely, the Riabouchinsky and the open-wake (the latter has been attributed, independently, to Joukowsky, Roshko, and Eppler) models, are adopted for detailed examination. The asymptotic representations of these theoretical solutions, with the wall effect treated as a small correction to the unbounded-flow limit, have yielded two different wall-correction rules, both of which can be applied very effectively in practice. It is of interest to note that the most critical range for comparison of these results lies in the case when the cavitating body is slender, rather than blunt ones, and when the cavity is short, instead of very long ones in the nearly choked-flow state. Only in this critical range do these flow models deviate significantly from each other, thereby permitting a refined differentiation and a critical examination of the accuracy of these flow models in representing physical flows. A series of experiments carefully planned for this purpose has provided conclusive evidences, which seem to be beyond possible experimental uncertainties, that the Riabouchinsky model gives a very satisfactory agreement with the experimental results, and is superior to other models, even in the most critical range when the wall effects are especially significant and the differences between these theoretical flow models become noticeably large. These outstanding features are effectively demonstrated by the relatively simple case of a symmetric wedge held in a non-lifting flow within a closed tunnel, which we discuss in the sequel.

ID: CaltechAUTHORS:WUTiutam71

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Abstract: This paper attempts to determine the optimum profile of a two-dimensional plate that produces the maximum hydrodynamic lift while planing on a water surface, under the condition of no spray formation and no gravitational effect, the latter assumption serving as a good approximation for operations at large Froude numbers. The lift of the sprayless planing surface is maximized under the isoperimetric constraints of fixed chord length and fixed wetted arc-length of the plate. Consideration of the extremization yields, as the Euler equation, a pair of coupled nonlinear singular integral equations of the Cauchy type. These equations are subsequently linearized to facilitate further analysis. The analytical solution of the linearized problem has a branch-type singularity, in both pressure and flow angle, at the two ends of plate. In a special limit, this singularity changes its type, emerging into a logarithmic one, which is the weakest type possible. Guided by this analytic solution of the linearized problem, approximate solutions have been calculated for the nonlinear problem using the Rayleigh-Ritz method and the numerical results compared with the linearized theory.

Publication: Journal of Fluid Mechanics Vol.: 55 No.: 3 ISSN: 0022-1120

ID: CaltechAUTHORS:20120806-160202833

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Abstract: The phenomenon of wake formation behind a body moving through a fluid, and the associated resistance of fluids, must have been one of the oldest experiences of man. From an analytical point of view, it is also one of the most difficult problems in fluid mechanics. Rayleigh, in his 1876 paper, observed that "there is no part of hydrodynamics more perplexing to the student than that which treats of the resistance of fluids." This insight of Rayleigh is so penetrating that the march of time has virtually left no mark on its validity even today, and likely still for some time to come. The first major step concerning the resistance of fluids was made over a century ago when Kirchhoff (1869) introduced an idealized inviscid-flow model with free streamlines (or surfaces of discontinuity) and employed (for steady, plane flows) the ingenious conformal-mapping technique that had been invented a short time earlier by Helmholtz (1868) for treating two-dimensional jets formed by free streamlines. This pioneering work offered an alternative to the classical paradox of D’Alembert (or the absence of resistance) and laid the foundation of the free-streamline theory. We appreciate the profound insight of these celebrated works even more when we consider that their basic idea about wakes and jets, based on a construction with surfaces of discontinuity, was formed decades before laminar and turbulent flows were distinguished by Reynolds (1883), and long before the fundamental concepts of boundary-layer theory and flow separation were established by Prandtl (1904a). However, there have been some questions raised in the past, and still today, about the validity of the Kirchhoff flow for the approximate calculation of resistance. Historically there is little doubt that in constructing the flow model Kirchhoff was thinking of the wake in a single-phase fluid, and not at all of the vapor-gas cavity in a liquid; hence the arguments, both for and against the Kirchhoff flow, should be viewed in this light. On this basis, an important observation was made by Sir William Thomson, later Lord Kelvin (see Rayleigh 1876) "that motions involving a surface of separation are unstable" (we infer that instability here includes the viscous effect). Regarding this comment Rayleigh asked "whether the calculations of resistance are materially affected by this circumstance as the pressures experienced must be nearly independent of what happens at some distance in the rear of the obstacle, where the instability would first begin to manifest itself." This discussion undoubtedly widened the original scope, brought the wake analysis closer to reality, and hence should influence the course of further developments. An expanded discussion essentially along these lines was given by Levi-Civita (1907) and was included in the survey by Goldstein (1969). Another point of fundamental importance is whether the Kirchhoff flow is the only correct Euler (or outer) limit of the Navier-Stokes solution to steady flow at high Reynolds numbers. If so, then a second difficulty arises, a consequence of the following argument: We know that the width of the Kirchhoff wake grows parabolically with the downstream distance x, at a rate independent of the (kinematic) viscosity u. If Prandtl’s boundary-layer theory is then applied to smooth out the discontinuity (i.e. the vortex sheet) between the wake and the potential flow, one obtains a laminar shear layer whose thickness grows like (ux/U)^-1/2 in a free stream of velocity U. Hence, for sufficiently small u/U the shear layers do not meet, so that the wake bubble remains infinitely long at a finite Reynolds number, a result not supported by experience. (For more details see Lagerstrom 1964, before p. 106, 131; Kaplun 1967, Part II.) The weaknesses in the above argument appear to lie in the two primary suppositions that, first, the free shear layer enveloping the wake would remain stable indefinitely, and second (perhaps a less serious one), the boundary-layer approximation would be valid along the infinitely long wake boundary. Reattachment of two turbulent shear layers, for instance, is possible since their thickness grows linearly with x. By and large, various criticisms, of the Kirchhoff flow model have led to constructive refinements of the free-streamline theory rather than to a weakening of the foundation of the theory as a valuable idealization. The major development in this direction has been based on the observation that the wake bubble is finite in size at high Reynolds numbers. (The wake bubble, or the near-wake, means, in the ordinary physical sense, the region of closed streamlines behind the body as characterized by a constant or nearly constant pressure.) To facilitate the mathematical analysis of flows with a finite wake bubble, a number of potential-flow models have been introduced to give the near-wake a definite configuration as an approximation to the inviscid outer flow. These theoretical models will be discussed explicitly later. It suffices to note here that all these models, even though artificial to various degrees, are aimed at admitting the near-wake pressure coefficient as a single free parameter of the flow, thus providing a satisfactory solution to the state of motion in the near part of the wake attached to the body. On the whole, their utility is established by their capability of bringing the results of potential theory of inviscid flows into better agreement with experimental measurements in fluids of small viscosity. The cavity flow also has a long, active history. Already in 1754, Euler, in connection with his study of turbines, realized that vapor cavitation may likely occur in a water stream at high speeds. In investigating the cause of the racing of a ship propeller, Reynolds (1873) observed the phenomenon of cavitation at the propeller blades. After the turn of this century, numerous investigations of cavitation and cavity flows were stimulated by studies of ship propellers, turbomachinery, hydrofoils, and other engineering developments. Important concepts in this subject began to appear about fifty years ago. In an extensive study of the cavitation of water turbines, Thoma (1926) introduced the cavitation number (the underpressure coefficient of the vapor phase) as the principal similarity parameter, which has ever since played a central role in small-bubble cavitation as well as in well-developed cavity flows. Applications of free-streamline theory to finite-cavity flows have attracted much mathematical interest and also provided valuable information for engineering purposes. Although the wake interpretation of the flow models used to be standard, experimental verifications generally indicate that the theoretical predictions by these finite-wake models are satisfactory to the same degree for both wake and cavity flows. This fact, however, has not been widely recognized and some confusion still exists. As a possible explanation, it is quite plausible that even for the wake in a single-phase flow, the kinetic energy of the viscous flow within the wake bubble is small, thus keeping the pressure almost unchanged throughout. Although this review gives more emphasis to cavity flows, several basic aspects of cavity and wake flows can be effectively discussed together since they are found to have many important features in common, or in close analogy. This is in spite of relatively minor differences that arise from new physical effects, such as gravity, surface tension, thermodynamics of phase transition, density ratio and viscosity ratio of the two phases, etc., that are intrinsic only to cavity flows. Based on this approach, attempts will be made to give a brief survey of the physical background, a general discussion of the free-streamline theory, some comments on the problems and issues of current interest, and to point out some basic problems yet to be resolved. In view of the vast scope of this subject and the voluminous literature, efforts will not be aimed at completeness, but rather on selective interests. Extensive review of the literature up to the 1960s may be found in recent expositions by Birkhoff & Zarantonello (1957), Gilbarg (1960), Gurevich (1961), Wehausen (1965), Sedov (1966), Wu (1968), Robertson & Wislicenus (1969), and (1961).

Publication: Annual Review of Fluid Mechanics Vol.: 4ISSN: 0066-4189

ID: CaltechAUTHORS:WUTarfm72

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Abstract: This paper is intended to evaluate the wall effects in the pure-drag case of plane cavity flow past an arbitrary body held in a closed tunnel, and to establish an accurate correction rule. The three theoretical models in common use, namely, the open-wake, Riabouchinsky and re-entrant-jet models, are employed to provide solutions in the form of some functional equations. From these theoretical solutions several different rules for the correction of wall effects are derived for symmetric wedges. These simple correction rules are found to be accurate, as compared with their corresponding exact numerical solutions, for all wedge angles and for small to moderate 'tunnel-spacing ratio' (the ratio of body frontal width to tunnel spacing). According to these correction rules, conversion of a drag coefficient, measured experimentally in a closed tunnel, to the corresponding unbounded flow case requires only the data of the conventional cavitation number and the tunnel-spacing ratio if based on the open-wake model, though using the Riabouchinsky model it requires an additional measurement of the minimum pressure along the tunnel wall. The numerical results for symmetric wedges show that the wall effects invariably result in a lower drag coefficient than in an unbounded flow at the same cavitation number, and that this percentage drag reduction increases with decreasing wedge angle and/or with decreasing tunnel spacing relative to the body frontal width. This indicates that the wall effects are generally more significant for thinner bodies in cavity flows, and they become exceedingly small for sufficiently blunt bodies. Physical explanations for these remarkable features of cavity-flow wall effects are sought; they are supported by the present experimental investigation of the pressure distribution on the wetted body surface as the flow parameters are varied. It is also found that the theoretical drag coefficient based on the Riabouchinsky model is smaller than that predicted by the open-wake model, all the flow parameters being equal, except when the flow approaches the choked state (with the cavity becoming infinitely long in a closed tunnel), which is the limiting case common to all theoretical models. This difference between the two flow models becomes especially pronounced for smaller wedge angles, shorter cavities, and with tunnel walls farther apart. In order to gauge the degree of accuracy of these theoretical models in approximating the real flows, and to ascertain the validity of the correction rules, a series of definitive experiments was carefully designed to complement the theory, and then carried out in a high-speed water tunnel. The measurements on a series of fully cavitating wedges at zero incidence suggest that, of the theoretical models, that due to Riabouchinsky is superior throughout the range tested. The accuracy of the correction rule based on that model has also been firmly established. Although the experimental investigation has been limited to symmetric wedges only, this correction rule (equations (85), (86) of the text) is expected to possess a general validity, at least for symmetric bodies without too large curvatures, since the geometry of the body profile is only implicitly involved in the correction formula. This experimental study is perhaps one of a very few with the particular objective of scrutinizing various theoretical cavity-flow models.

Publication: Journal of Fluid Mechanics Vol.: 49 No.: 2 ISSN: 0022-1120

ID: CaltechAUTHORS:WUTjfm71

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Abstract: This note seeks to evaluate the self-propulsion of a micro-organism, in a viscous fluid, by sending a helical wave down its flagellated tail. An explanation is provided to resolve the paradoxical phenomenon that a micro-organism can roll about its longitudinal axis without passing bending waves along its tail (Rothschild 1961, 1962; Bishop 1958; Gray 1962). The effort made by tho organism in so doing is not torsion, but bending simultaneously in two mutually perpendicular planes. The mechanical model of the micro-organism adopted for the present study consists of a spherical head of radius ɑ and a long cylindrical tail of cross-sectional radius b, along which a helical wave progresses distally. Under the equilibrium condition at a constant forward speed, both the net force and net torque acting on the organism are required to vanish, yielding two equations for the velocity of propulsion, U, and the induced angular velocity, Ω, of the organism. 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. In fact, there exists 1m optimum head-tail ratio ɑ/bat which the propulsion velocity U reaches a maximum, holding the other physical parameters fixed. The power required for propulsion by means of helical waves is determined, based on which a hydromechanical efficiency η is defined. When the head-tail ratio ɑ/b assumes its optimum value and when b is very small compared with the wavelength λ, η ≃ Ω/ω approximately (Ω being the induced angular velocity of the head, ω the circular frequency of the helical wave). This η reaches a maximum at kh ≃ 0.9 (k being the wavenumber 2π/λ, and h the amplitude of the helical wave). In the neighbourhood 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 (c = ω/k being the wave phase velocity), and the efficiency in 0.14 < η < 0.24, as kb varies over 0.03 < kb < 0.2, a range of practical interest. Furthermore, a comparison between the advantageous features of planar and helical waves, relative to each other, is made in terms of their propulsive velocities and power consumptions.

Publication: Proceedings of the Royal Society of London. Series B, Biological Sciences Vol.: 178 No.: 1052 ISSN: 0962-8452

ID: CaltechAUTHORS:20150211-145731256

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Abstract: The optimum shape problems considered in this part are for those profiles of a two-dimensional flexible plate in time-harmonic motion that will minimize the energy loss under the condition of fixed thrust and possibly also under other isoperimetric constraints. First, the optimum movement of a rigid plate is completely determined; it is necessary first to reduce the original singular quadratic form representing the energy loss to a regular one of a lower order, which is then tractable by usual variational methods. A favourable range of the reduced frequency is found in which the thrust contribution coming from the leading-edge suction is as small as possible under the prescribed conditions, outside of which this contribution becomes so large as to be hard to realize in practice without stalling. This optimum solution is compared with the recent theory of Lighthill (1970); these independently arrived-at conclusions are found to be virtually in agreement. The present theory is further applied t0 predict the movement of a porpoise tail of large aspect-ratio and is found in satisfactory agreement with the experimental measurements. A qualitative discussion of the wing movement in flapping flight of birds is also given on the basis of optimum efficiency. The optimum shape of a flexible plate is analysed for the most general case of infinite degrees of freedom. It is shown that the solution can be determined to a certain extent, but the exact shape is not always uniquely determinate.

Publication: Journal of Fluid Mechanics Vol.: 46 No.: 3 ISSN: 0022-1120

ID: CaltechAUTHORS:WUTjfm71b

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Abstract: This paper seeks to evaluate the swimming flow around a typical slender fish whose transverse cross-section to the rear of its maximum span section is of a lenticular shape with pointed edges, such as those of spiny fins, so that these side edges are sharp trailing edges, from which an oscillating vortex sheet is shed to trail the body in swimming. The additional feature of shedding of vortex sheet makes this problem a moderate generalization of the paper on the swimming of slender fish treated by Lighthill (1960a). It is found here that the thrust depends not only on the virtual mass of the tail-end section, but also on an integral effect of variations of the virtual mass along the entire body segment containing the trailing side edges, and that this latter effect can greatly enhance the thrust-making. The optimum shape problem considered here is to determine the transverse oscillatory movements a slender fish can make which will produce a prescribed thrust, so as to overcome the frictional drag, at the expense of the minimum work done in maintaining the motion. The solution is for the fish to send a wave down its body at a phase velocity c somewhat greater than the desired swimming speed U, with an amplitude nearly uniform from the maximum span section to the tail. Both the ratio U/c and the optimum efficiency are found to depend upon two parameters: the reduced wave frequency and a 'proportional-loading parameter', the latter being proportional to the thrust coefficient and to the inverse square of the wave amplitude. The basic mechanism of swimming is examined in the light of the principle of action and reaction by studying the vortex wake generated by the optimum movement.

Publication: Journal of Fluid Mechanics Vol.: 46 No.: 3 ISSN: 0022-1120

ID: CaltechAUTHORS:WUTjfm71c

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Abstract: The most effective movements of swimming aquatic animals of almost all sizes appear to have the form of a transverse wave progressing along the body from head to tail. The main features of this undulatory mode of propulsion are discussed for the case of large Reynolds number, based on the principle of energy conservation. The general problem of a two-dimensional flexible plate, swimming at arbitrary, unsteady forward speeds, is solved by applying the linearized inviscid flow theory. The large-time asymptotic behaviour of an initial-value harmonic motion shows the decay of the transient terms. For a flexible plate starting with a constant acceleration from at rest, the small-time solution is evaluated and the initial optimum shape is determined for the maximum thrust under conditions of fixed power and negligible body recoil.

Publication: Journal of Fluid Mechanics Vol.: 46 No.: 2 ISSN: 0022-1120

ID: CaltechAUTHORS:WUTjfm71a

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Abstract: This report is intended as a companion to Report No. E-111A.5, "Wall Efects in Cavity Flows", by Wu, Whitney and Lin. Some simple rules for the correction of wall effect are derived from that theoretical study. Experiments designed to complement the theory and to inspect the validity of the correction rules were then carried out in the high-speed water tunnel of the Hydrodynamics Laboratory, California Institute of Technology. The measurements on a series of fully cavitating wedges at zero angle of attack suggested that of the theoretical models that due to Riabouchinsky is superior. They also confirmed the accuracy of the correction rule derived using that model and based on a measurement of the minimum pressure along the tunnel wall.

ID: CaltechAUTHORS:WHIcitr70

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Abstract: The wall effects in cavity flows past an arbitrary two-dimensional body is investigated for both pure-drag and lifting cases based on an inviscid nonlinear flow theory. The over-all features of various theoretical flow models for inviscid cavity flows under the wall effects are discussed from the general momentum consideration in comparison with typical viscous, incompressible wake flows in a channel. In the case of pure drag cavity flows, three theoretical models in common use, namely, the open-wake, Riabouchinsky and re-entrant jet models, are applied to evaluate the solution. Methods of numerical computation are discussed for bodies of arbitrary shape, and are carried out in detail for wedges of all angles. The final numerical results are compared between the different flow models, and the differences pointed out. Further analysis of the results has led to development of several useful formulas for correcting the wall effect. In the lifting flow case, the wall effect on the pressure and hydrodynamic forces acting on arbitrary body is formulated for the choked cavity flow in a closed water tunnel of arbitrary shape, and computed for the flat plate with a finite cavity in a straight tunnel.

ID: CaltechAUTHORS:E-111-A5

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Abstract: The solution of the two-dimensional gravity waves in a plane stratified ocean previously calculated by the authors is rectified. By formulating a corresponding initial value problem, the steady-state solution becomes completely determinate without using the radiation condition.

Publication: Physics of Fluids Vol.: 10 No.: 3 ISSN: 0031-9171

ID: CaltechAUTHORS:20120830-104637936

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Abstract: The problem of a two-dimensional cavity flow of an ideal fluid with small unsteady disturbances in a gravity free field is considered. By regarding the unsteady motion as a small perturbation of an established steady cavity flow, a fundamental formulation of the problem is presented. It is shown that the unsteady disturbance generates a surface wave propagating downstream along the free cavity boundary, much in the same way as the classical gravity waves in water, only with the centrifugal acceleration owing to the curvature of the streamlines in the basic flow playing the role of an equivalent gravity effect. As a particularly simple example, the surface waves in a hollow potential vortex flow is calculated by using the present theory.

ID: CaltechAUTHORS:HydroLabRpt97-7

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Abstract: When Prof. H. W. Lerbs and Prof. G. Weinblum asked me to prepare a general and broad survey talk on the subject "Propellers and Propulsion" for this International Symposium of Hamburgischen Schiffbau-Versuchsanstalt, I was pleased by having this opportunity to extend my personal congratulations and to participate in this happy event. In view of the fact that this subject has a vast scope containing many special problems which have been under rapid development, I am fully aware of the challenge to prepare a thorough survey, even with the previous excellent review of the state-of-the-art by Prof. Lerbs (1955a, see Reference). Undoubtedly, my effort would be limited by the physical access to the informaiton and literatures not generally available, so I would entitle my talk as "Some recenty developments in propeller theory".

ID: CaltechAUTHORS:HydroLabRpt-E-97-6

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Abstract: The fundamental solution of the gravity waves due to a two‐dimensional point singularity submerged in a steady free‐surface flow of a stratified fluid is investigated. A linearized theory is formulated by using Love's equations. The effect of density stratification p_0(y) and the gravity effect are characterized by two flow parameters σ = −(dp_0∕dy)∕p_0 and λ = gL∕U^2, where λ^(1/2) may be regarded as the internal Froude number if L assumes a characteristic value of σ^(−1). Two special cases of σ and λ are treated in this paper. In the first case of constant σ (and arbitrary λ) an exact mathematical analysis is carried out. It is shown that the flow is subcritical or supercritical according as λ > or < (1/2), in analogy to the corresponding states of channel flows. In addition to a potential surface wave, which exists only for λ > (1/2), there arises an internal wave which is attenuated at large distances for λ > (1/4) and decays exponentially for λ < (1/4). In the second example an asymptotic theory for large λ is developed while σ(y) may assume the profile roughly resembling the actual situation in an ocean where a pronounced maximum called a seasonal thermocline occurs. Internal waves are now propagated to the downstream infinity in a manner analogous to the channel propagation of sound in an inhomogeneous medium.

Publication: Physics of Fluids Vol.: 7 No.: 8 ISSN: 0031-9171

ID: CaltechAUTHORS:20120830-161623975

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Abstract: The fundamental solution of the gravity waves due to a two-dimensional point singularity submerged in a steady free surface flow of a stratified fluid is investigated. A linearized theory is formulated by using Love's equations. The effect of density stratification p[sub]o(y) and the gravity effect are characterized by two flow parameters [sigma] = -(dp[sub]o/dy)/p[sub]o and [lambda] = gL/U^2, where [lambda]^-1/2 may be regarded as the internal Froude number if L assumes a characteristic value of [sigma]^-1. Two special cases of [sigma] and [lambda] are treated in this paper. In the first case of constant [sigma] (and arbitrary [lambda]) an exact mathematical analysis is carried out. It is shown that the flow is subcritical or supercritical according as [lambda] > or < 1/2, in analogy to the corresponding states of channel flows. In addition to a potential surface wave, which exists only for [lambda]>1/2, there arises an internal wave which is attenuated at large distances for [lambda] > 1/4 and decays exponentially for [lambda] < 1/4. In the second example an asymptotic theory for large [lambda] is developed while [sigma](y) may assume the profile roughly resembling the actual situation in an ocean where a pronounced maximum called a seasonal thermocline occurs. Internal waves are now propagated to the downstream infinity in a manner analogous to the channel propagation of sound in an inhomogeneous medium.

ID: CaltechAUTHORS:HydroLabRpt97-5

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Abstract: In Part 1 of this paper a free-streamline wake model mas introduced to treat the fully and partially developed wake flow or cavity flow past an oblique flat plate. This theory is generalized here to investigate the cavity flow past an obstacle of arbitrary profile at an arbitrary cavitation number. Consideration is first given to the cavity flow past a polygonal obstacle whose wetted sides may be concave towards the flow and may also possess some gentle convex corners. The general case of curved walls is then obtained by a limiting process. The analysis in this general case leads to a set of two funnctional equations for which several methods of solutioii are developed and discussed. As a few typictbl examples the analysis is carried out in detail for the specific cases of wedges, two-step wedges, flapped hydrofoils, and inclined circular arc plates. For these cases the present theory is found to be in good agreement with the experimental results available.

Publication: Journal of Fluid Mechanics Vol.: 18 No.: 1 ISSN: 0022-1120

ID: CaltechAUTHORS:WUTjfm64

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Abstract: Recently an exact theory for the cavity flow past an obstacle of arbitrary profile at an arbitrary cavitation number has been developed by adopting a free-streamline wake model. The analysis in this general case leads to a set of two functional equations for which several numerical methods have been devised; some of these methods have already been successfully carried out for several typical cases on a high speed electronic computer. In this paper an approximate numerical scheme, somewhat like an engineering principle, is introduced which greatly shortens the computation of the dual functional equations while still retaining a high degree of accuracy of the numerical result. With such drastic simplification, it becomes feasible to carry out this approximate mrmerical scheme even with a hand computing machine.

ID: CaltechAUTHORS:HydroLabRpt111-1

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Abstract: In Part I of this paper a free-streamline wake model was introduced to treat the fully and partially developed wake flow or cavity flow past an oblique flat plate. This theory is generalized here to investigate the cavity flow past an obstacle of arbitrary profile at an arbitrary cavitation number. Consideration is first given to the cavity flow past a polygonal obstacle whose wetted sides may be concave towards the flow and may also possess some gentle convex corners. The general case of curved walls is then obtained by a limiting process. The analysis in this general case leads to a set of two functional equations for which several methods of solution are developed and discussed. As a few typical examples the analysis is carried out in detail for the specific cases of wedges, two-step wedges, flapped hydrofoils, and inclined circular arc plate. For these cases the present theory is found in good agreement with the experimental results available.

ID: CaltechAUTHORS:HydroLabRpt97-4

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Abstract: A perturbation theory is applied to investigate the small-time behavior of unsteady cavity flows in which the time-dependent part of the flow may be taken as a small-time expansion superimposed on an established steady cavity flow of an ideal fluid. One purpose of this paper is to study the effect of the initial cavity size on the resulting flow due to a given disturbance. Various existing steady cavity-flow models have been employed for this purpose to evaluate the initial reaction of a cavitated body in an unsteady motion. Furthermore, a physical model is proposed here to give a proper representation of the mechanism by which the cavity volume may be changed with time; the initial hydrodynamic force resulting from such change is calculated based on this model.

ID: CaltechAUTHORS:HydroLabRpt97-3

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Abstract: A perturbation theory is applied to investigate the small-time behavior of unsteady cavity flows in which the time-dependent part of the flow may be taken as a small-time expansion superimposed on an established steady cavity flow of an ideal fluid. One purpose of this paper is to study the effect of the initial cavity size on the resulting flow due to a given disturbance. Various existing steady cavity-flow models have been employed for this purpose to evaluate the initial reaction of a cavitated body in an unsteady motion. Furthermore, a physical model is proposed here to give a proper representation of the mechanism by which the cavity volume may be changed with time; the initial hydrodynamic force resulting from such change is calculated on the basis of this model.

Publication: Archive for Rational Mechanics and Analysis Vol.: 14 No.: 1 ISSN: 0003-9527

ID: CaltechAUTHORS:20200114-145734220

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Abstract: A wake model for the free-streamline theory is proposed to treat the two-dimensional flow past an obstacle with a wake or cavity formation. In this model the wake flow is approximately described in the large by an equivalent potential flow such that along the wake boundary the pressure first assumes a prescribed constant under-pressure in a region downstream of the separation points (called the near-wake) and then increases continuously from this under-pressure to the given free-stream value in an infinite wake strip of finite width (the far-wake). Application of this wake model provides a rather smooth continuous transition of the hydrodynamic forces from the fully developed wake flow to the fully wetted flow as the wake disappears. When applied to the wake flow past an inclined flat plate, this model yields the exact solution in a closed form for the whole range of the wake under-pressure coefficient.

Publication: Journal of Fluid Mechanics Vol.: 13 No.: 2 ISSN: 0022-1120

ID: CaltechAUTHORS:WUTjfm62

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Abstract: A wake model for the free-streamline theory is proposed to treat the two-dimensional flow past an obstacle with a wake or cavity formation. In this model the wake flow is approximately described in the large by an equivalent potential flow such that along the wake boundary the pressure first assumes a prescribed constant under-pressure in a region downstream of the separation points (called the near-wake) and then increases continuously from this under-pressure to the given free stream value in an infinite wake strip of finite width (the far-wake). The boundary of the wake trailing a lifting body is allowed to change its slope and curvature at finite distances from the body and is required to be parallel to the main stream only asymptotically at downstream infinity. The pressure variation along the far-wake takes place in such a way that the upper and lower boundaries of the far-wake form a branch slit of undetermined shape in the hodograph plane. One advantage of this wake model is that it provides a rather smooth continuous transition of the hydrodynamic forces from the fully developed wake flow to the fully wetted flow as the wake disappears. When applied to the wake flow past an inclined flat plate, this model yields the exact solution in a closed form for the whole range of the wake under-pressure coefficient. The separated flow over a slightly cambered plate can be calculated by a perturbation theory based on this exact solution.

ID: CaltechAUTHORS:EngDivRpt97-2

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Abstract: A slender-body theory for the flow past a slender, pointed hydrofoil held at a small angle of Attack to the flow, with a cavity on the upper surface, has been worked out. The approximate solution valid near the body is seen to be the sum of two components. The first consists of a distribution of two-dimensional sources located along the centroid line of the cavity to represent the variation of the cross-sectional area of the cavity. The second component represents the crossflow perpendicular to the centroid line. It is found that over the cavity boundary which envelops a constant pressure region, the magnitude of the cross-flow velocity is not constant, but varies to a moderate extent. With this variation neglected only in the neighbourhood of the hydrofoil, the cross-flow is solved by adopting the Riabouchinsky model for the two-dimensional flow. The lift is then calculated by integrating the pressure along the chord; the dependence of the lift on cavitation number and angle of attack is shown for a specific case of the triangular plan form.

Publication: Journal of Fluid Mechanics Vol.: 11 No.: 2 ISSN: 0022-1120

ID: CaltechAUTHORS:CUMjfm61

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Abstract: The purpose of' this paper is to study the basic principle of fish propulsion. As a simplified model, the two-dimensional potential flow over a waving plate of finite chord is treated. The solid plate, assumed to be flexible and thin, is capable of performing the motion which consists of a progressing wave of given wavelength and phase velocity along the chord, the envelope of the wave train being an arbitrary function of the distance from the leading edge. The problem is solved by applying the general theory for oscillating deformable airfoils. The thrust, power required, and the energy imparted to the wake are calculated, and the propulsive efficiency is also evaluated. As a numerical example, the waving motion with linearly varying amplitude is carried out in detail. Finally, the basic mechanism of swimming is elucidated by applying the principle of action and reaction.

Publication: Journal of Fluid Mechanics Vol.: 10 No.: 3 ISSN: 0022-1120

ID: CaltechAUTHORS:WUTjfm61

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Abstract: The purpose of this paper is to study the basic principle of fish propulsion. As a simplified model, the two-dimensional potential flow over a waving plate of finite chord is treated. The solid plate, assumed to be flexible and thin, is capable of performing the motion which consists of a progressing wave of given wave length and phase velocity along the chord, the envelope of the wave train being an arbitrary function of the distance from the leading edge. The problem is solved by applying the general theory for oscillating deformable airfoils. The thrust, power required, and the energy imparted to the wake are calculated, and the propulsive efficiency is also evaluated. As a numerical example, the waving motion with linearly varying amplitude is carried out in detail. Finally, the basic mechanism of swimming is elucidated by applying the principle of action and reaction.

ID: CaltechAUTHORS:97-1

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Abstract: A linearized theory is applied here to investigate the viscous effect on water waves generated and maintained by a system of external disturbances which is distributed over the free surface of an otherwise uniform flow. The flow is taken to be in the steady state configuration. The analysis is carried out to yield the asymptotic expressions for the surface wave when the Reynolds number of the flow is either large or small.

ID: CaltechAUTHORS:EngDiv85-8

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Abstract: The problem under consideration is that of two-dimensional gravity waves in water generated by a surface disturbance which oscillates with frequency Ω/2π and moves with constant rectilinear velocity U over the free water surface. The present treatment may be regarded as a generalization of a previous paper by De Prima and Wu (Ref. 1) who treated the surface waves due to a disturbance which has only the rectilinear motion. It was pointed out in Ref. 1 that the dispersive effect, not the viscous effect, plays the significant role in producing the final stationary wave configuration, and the detailed dispersion phenomenon clearly exhibits itself through the formulation of a corresponding initial value problem. Following this viewpoint, the present problem is again formulated first as an initial value problem in which the surface disturbance starts to act at a certain time instant and maintains the prescribed motion thereafter. If at any finite time instant the boundary condition is imposed that the resulting disturbance vanishes at infinite distance (because of the finite wave velocity), then the limiting solution, with the time oscillating term factored out, is mathematically determinate as the time tends to infinity and also automatically has the desired physical properties. From the associated physical constants of this problem, namely Ω, U, and the gravity constant g, a nondimensional parameter of importance is found to be a = 4ΩU/g. The asymptotic solution for large time shows that the space distribution of the wave trains are different for 0 < a < 1 and a> 1. For 0 < a < 1 and time large, the solution shows that there are three wave trains in the downstream and one wave in the upstream of the disturbance. For a > 1, two of these waves are suppressed, leaving two waves in the downstream. At a = 1, a kind of "resonance" phenomenon results in which the amplitude and the extent in space of one particular wave both increase with time at a rate proportional to t^(1/2). Two other special cases: (1) Ω → 0 and U > 0, (2) U = 0, Ω > 0 are also discussed; in these cases the solution reduces to known results.

ID: CaltechAUTHORS:WuEngDivRpt85-3

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Abstract: The wave profile generated by an obstacle moving at constant veiocity U over a water surface of infinite extent appears to be stationary with respect to the moving body provided, of course, the motion has been maintained for a long time. When the gravitational and capillary effects are both taken into account, the surface waves so generated may possess a minimum phase velocity c[sub]m characterized by a certain wave length, say [lambda][sub]m (see Ref. 1, p. 459). If the velocity U of the solid body is greater than c[sub]m, then the physically correct solution of this two-dimensional problem requires that the gravity waves (of wave length greater than [lambda][sub]m) should exist only on the downstream side and the capillary waves (of wave length less than [lambda][sub]m) only on the upstream side. If one follows strictly the so-called steady-state formulation so that the time does not appear in the problem, one finds in general that it is not possible to characterize uniquely the mathematical solution with the desired physical properties by imposing only the boundedness conditions at infinity. [Footnote: In the case of a three-dimensional steady-state problem, even the condition that the disturbance should vanish at infinity is not sufficient to characterize the unique solution.] Some stronger radiation conditions are actually necessary. In the linearized treatment of this stationary problem, several methods have been employed, most of which are aimed at obtaining the correct solution by introducing some artificial device, either of a mathematical or physical nature. One of these methods widely used was due to Rayleigh, and was further discussed by Lamb. In the analysis of this problem Rayleigh introduced a "small dissipative force", proportional to the velocity relative to the moving stream. This "law" of friction does not originate from viscosity and is hence physically fictitious, for in the final result this dissipation factor is made to vanish eventually. In the present investigation, Rayleigh's friction coefficient is shown to correspond roughly to a time convergence factor for obtaining the steady-state solution from an initial value problem. (It is not a space-limit factor for fixing the boundary conditions at space infinity, as has usually been assumed in explanation of its effect). Thus, the introduction of Rayleigh's coefficient is only a mathematical device to render the steady-state solution mathematically determinate and physically acceptable. For a physical understanding, however, it is confusing and even misleading; for example, in an unsteady flow case it leads to an incomplete solution, as has been shown by Green. Another approach, purely of a physical nature, was used by Michell in his treatment of the velocity potential for thin ships. To make the problem determinate, he chose the solution which represents the gravity waves propagating only downstream and discarded the part corresponding to the waves traveling upstream. For two-dimensional problems with the capillary effect, this method would mean a superposition of simple waves so as to make the solution physically correct. Some other methods appear to be limited in the necessity of interpreting the principal value of a certain kind of improper integral. In short, as to their physical soundness and mathematical rigor, or even to their merits or demerits, the preference of one method over the others has remained nevertheless a matter of considerable dispute. Only until recently the steady-state problem has been treated by first formulating a corresponding initial value problem. A brief historical sketch of these methods is given in the next section. The purpose of this paper is to try to understand the physical mechanism underlying the steady configuration of the surface wave phenomena and to clarify to a certain extent the background of the artifices adopted for solution of steady-state problems. The point of view to be presented here is that this problem should be formulated first as an initial value problem (for example, the body starts to move with constant velocity at a certain time instant), and then the stationary state is sought by passing to the limit as the time tends to infinity. If at any finite time instant the boundary condition that the disturbance vanishes at infinity (because of the finite wave velocity) is imposed, then the limiting solution as the time tends to infinity is determinate and bears automatically the desired physical properties. Also, from the integral representation of the linearized solution, the asymptotic behavior of the wave form for large time is derived in detail, showing the distribution of the wave trains in space. This asymptotic solution exhibits an interesting picture which reveals how the dispersion* generates two monochromatic wave trains, with the capillary wave in front of, and the gravity wave behind, the surface pressure. *[Footnote: By dispersive medium is meant one in which the wave velocity of a propagating wave depends on the wave length, so that a number of wave trains of different wave lengths tends to form groups, propagating with group velocities which are in general different from the phase velocities of individual wave trains. In case of waves on the water surface, both the gravity and surface tension are responsible for dispersion.] The special cases U< c[sub]m and U = c[sub]m are also discussed. The viscous effect and the effect of superposition are commented upon later. Through this detailed investigation it is found that the dispersive effect, not the viscous effect plays the significant role in producing the final stationary wave configuration.

ID: CaltechAUTHORS:HydroLabRpt21-23

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Abstract: The lifting problem of fully cavitated hydrofoils has recently received some attention. The nonlinear problem of two-dimensional fully cavitated hydrofoils has been treated by the author, using a generalized free streamline theory. The hydrofoils investigated in Ref. 1 were those with sharp leading and trailing edges which are assumed to be the separation points of the cavity streamlines. Except for this limitation, the nonlinear theory is applicable to hydrofoils of arbitrary geometric profile, operating at any cavitation number, and for almost all angles of attack as long as the cavity wake is fully developed. By using an elegant linear theory, Tulin has treated the problem of a fully cavitated flat plate set at a small angle of attack and operated at arbitrary cavitation number. In the case of hydrofoils of arbitrary profile operating at zero cavitation number, some interesting simple relationships are given by Tulin for the connection between the lift, drag and moment of a supercavitating hydrofoil and the lift, moment and the third moment of an equivalent airfoil (unstalled). In the present investigation, Tulin's linear theory is first extended to calculate the hydrodynamic lift and drag on a fully cavitated hydrofoil of arbitrary camber at arbitrary cavitation number. A numerical example is given for a circular hydrofoil subtending an arc angle of 160, for which the corresponding nonlinear solution is available. A direct comparison between these two theories is made explicitly for the flat plate and the circular arc hydrofoil. Some important aspects of the results are discussed subsequently.

ID: CaltechAUTHORS:HydroLabRpt21-22

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Abstract: This paper investigates in a rather idealized way the different properties of fully cavitating and fully wetted hydrofoils in order to clarify the relative hydrodynamic merits of each insofar as this is possible in the present state of the art. The discussion is mainly based on a recent theory, together with some experimental data, on the hydrodynamics of two-dimensional fully cavitating hydrofoils. A number of quantitative comparisons between the fully cavitating and fully wetted two-dimensional foils have been made to bring out the different effects of such design parameters as attack angle, camber, submergence and speed on the hydrofoil in the two regimes. In addition, some of the effects which modify the two-dimensional comparison are surveyed and roughly estimated wherever possible. The consequences of air ventilation (which is closely related to fully cavitating flow) are discussed, especially as applied to the supporting struts, from the standpoint of whether or not it should be avoided. Finally, after a few remarks on some practical aspects of the problem, a rough comparison is made from the economy point of view to indicate by an example how a criterion might be chosen for one or the other type of operation. From this and the preceding calculations it is conjectured that there is strong reason to believe that the fully cavitating type of operation will be advantageous in some circumstances, but it is emphasized that more experience must be accumulated for operation in both regimes before any practical criteria can be specified.

ID: CaltechAUTHORS:WUThydrolabrpt47-4

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Abstract: The problem of cavity flows received attention early in the development of hydrodynamics because of its occurrence in high speed motion of solid bodies in water. Many previous works in this field were mainly concerned with the calculation of drag in a cavitating flow. The lifting problem with a cavity (or wake) arose later in the applications of water pumps, marine propellers, stalling airfoils, and hydrofoil crafts. Although several formulations of the problem of lift in cavity flows have been pointed out before, these theories have not yet been developed to yield general results in explicit form so that a unified discussion can be made. The problems of cavitating flow with finite cavity demand an extension of the classical Helmholtz free boundary theory for which the cavity is infinite in extent. For this purpose, several self-consistent models have been introduced, all aiming to account for the cavity base pressure which is in general always less than the free stream pressure. In the Helmholtz-Kirchhoff flow these two pressures are assumed equal. Of all these existing models, three significant ones may be mentioned here. The first representation of a finite cavity was proposed by Riabouchinsky in in which the finite cavity is obtained by introducing an "image" obstacle downstream of the real body. A different representation in which a reentrant jet is postulated was suggested by Prandtl, Wagner, and was later considered by Kreisel and was further extended by Gilbarg and Serrin. Another representation of a free streamline flow with the base pressure different from the free stream pressure, was proposed recently by Roshko. In this model the base pressure in the wake (or cavity) near the body can take any assigned value. From a certain point in the wake, which can be determined from the theory, the flow downstream is supposed to be dissipated in such a way that the pressure increases gradually from the assigned value to that of the free stream in a strip parallel to the free stream. Apparently this model was also considered independently by Eppler in some generality. Other alternatives to these models have also been proposed, but they do not differ so basically from the above three models that they need to be mentioned here specifically. The mathematical solutions to the problem of flow past a flat plate set normal to the stream have been carried out for these three models. All the theories are found to give essentially the same results over the practical range of the wake underpressure. That such agreement is to be expected can be indicated, without the detailed solutions for the various models, from consideration of their underlying physical significance, as will be discussed in the next section. In the present work the free streamline theory is extended and applied to the lifting problem for two-dimensional hydrofoils with a fully cavitating wake. The analysis is carried out by using the Roshko model to approximate the wake far downstream. The reason for using this model is mainly because of its mathematical simplicity as compared with the Riabouchinsky model, or the reentrant jet model. In fact, it can be verified that these different models all yield practically the same result, as in the pure drag case; the deviation from the results of one model to another is not appreciable up to second order small quantities. The mathematical considerations here, as in the classical theory, depend on the conformal mapping of the complex velocity plane into the plane of complex potential. By using a generalization of Levi-Civita's method for curved barriers in cavity flows, the flow problem for curved hydrofoils is finally reduced to a nonlinear boundary value problem for an analytic function defined in the upper half of a unit circle to which the Schwarz's principle of reflection can be applied. The problem is then solved by using the expansion of this analytic function inside the unit circle together with the boundary conditions in the physical plane. In order to avoid the difficulty in determining the separation point of the free streamline from a hydrofoil with blunt nose, the hydrofoils investigated here are those with sharp leading and trailing edges which are assumed to be the separation points. Except for this limitation, the present nonlinear theory is applicable to hydrofoils of any geometric profile, operating at any cavitation number, and for almost all angles of attack as long as the wake has a fully cavitating configuration. As two typical examples, the problem is solved in explicit form for the circular arc and the flat plate for which the various flow quantities are expressed by simple formulas. From the final result the various effects, such as that of cavitation number, camber of the profile and the attack angle, are discussed in detail. It is also shown that the present theory is in good agreement with the experiment.

ID: CaltechAUTHORS:HydroLabRpt21-17

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Abstract: The effect of the free surface on the pressure distribution on the upper side of a shallow-running hydrofoil is considered from a general point of view. Previous theoretical and experimental work is reviewed in order to compare the range of flow variables for which each treatment of the surface proximity problem is valid. A qualitative theoretical expression for the pressure is developed. This result shows the relative importance of the pertinent parameters and it is shown to agree qualitatively with previous experiments as well as with new pressure measurements made in the Free Surface Water Tunnel. The above considerations reinforce the view generally held in the past, that the methods of potential theory when properly applied to hydrofoils at shallow submergences may be expected to lead to valid and useful results.

ID: CaltechAUTHORS:PARhydrolabrpt47-2

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Abstract: The program of this report is as follows: After a brief survey of the available theoretical and experimental information on the characteristics of hydrofoils, the theory for a hydrofoil of finite span will be formulated. The liquid medium is assumed to be incompressible and nonviscous and of infinite depth. The basic concept of the analysis is patterned after the famous Prandtl wing theory of modern aerodynamics in that the hydrofoil of large aspect ratio may be replaced by a lifting line. The lift distribution along the lifting line is the same as the lift distribution, integrated with respect to the chord of the hydrofoil, along the span direction. The induced velocity field of the lifting line is then calculated by proper consideration of lift distribution along the lifting line, free water surface pressure condition and wave formation. The "local velocity" so determined for flow around each local section perpendicular to the span of the hydrofoil can be considered as that of a two-dimensional flow around a hydrofoil without free water surface. The only additional feature of the flow in this sectional plane is the modification of the geometric angle of attack, as defined by the undisturbed flow, to the so-called effective angle of attack on account of the local induced velocity. Thus the local sectional characteristics to be used can be taken as those of a hydrofoil section in two-dimensional flow without free water surface but may involve cavitation. More precisely, the hydrofoil section at any location of the span has the same hydrodynamic characteristics as if it were a section of an infinite span hydrofoil in a fluid region of infinite extent at a geometric angle of attack equal to ae, together with proper modification of the free stream velocity. Such characteristics may be obtained by theory or by experiment and should be taken at the same Reynolds number and cavitation number. With this separation of the three-dimensional effects and the two-dimensional effects, the effects of Froude number are singled out. Thus a systematic and efficient analysis of the hydrofoil properties can be made.

ID: CaltechAUTHORS:WUYhydrolabrpt26-8

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