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 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[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: 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 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: 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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