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: 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: 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: 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: 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: 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 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: 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: 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 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: 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: 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 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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