NAVY DEPARTMENT THE DAVID W. TAYLOR MODEL BASIN i WASHINGTON 7, D.C. A BRIEF SURVEY OF PROGRESS ON THE MECHANICS OF CAVITATION An Addendum to DTMB Report 712, On the Mechanism and Prevention of Gavitation by Phillip Eisenberg ! | { ‘ Revised Edition sa June 1953 Report 842 NS 715-102 A BRIEF SURVEY OF PROGRESS ON THE MECHANICS OF CAVITATION An Addendum to DTMB Report 712, On the Mechanism and Prevention of Cavitation by Phillip Eisenberg First Printing, October 1952 Revised Edition June 1953 Report 842 NS715-102 PREFACE This report was prepared for presentation to the Chesapeake Section of the Society of Naval Architects and Marine Engineers. It was presented in preliminary form at the Meeting of October 18, 1952. iii TABLE OF CONTENTS JNISHSHTO RYN GE Aa eee reee aC rr aL ES I eerie aro Se RR AA Ros nari A CREE aE FINGISIR © DW STO Ness ee rede sce cate cabelas aateuat ae d/Mematne eseteu iia cunerasode daawetabnwensa rede tec ec VASO (CAVITATION eae Sak reine a AS NSE ST BIR EE RAE ie THE INCEPTION OF CAVITATION TimieOsIN@ Waray TRSTTEE ES) egsccecoccssococbbxecucbsse ecoceoc0s0c cboasob ouocaedodooudsecoucebbavceacncdadockcde subcasocdasens Further Remarks on the Role of Turbulence and Boundary Layer Effects in Cavitating FIOWS ..........-..:::::ccecetseeeese eee ee ese teseeaeeres Cavitationsiniseparated oh lOws ve. swccsc sce: esrcees ceranerac ar. codsdcpevacectecsaceutaneecenenna Effects of Scale on the Inception of Cavitation ..................cccceesesscecetseeeeceeeneeeene A Remark on the Question of Gas Content and Nuclei Content .....................0 DYNAMICS OF TRANSIENT CAVITIES The Motion of Small Transient Cavities ............cccccccccccceccececessstecesssssssnsrseeeeseaaeeseeesens Analytical) Description Of the MOUIOM Teer erevcnsesccecectenceccererencesecessetres MECHANISM OF STEADY-STATE CAVITIES AND THEIR ANALYTICAL DESCRIPTION TTTNROGCINCHOTAT TREAMETE IES). cone scooaadacbotedebyoso050sb9s86009s56s80s99s4a050c02n96n5040pC0R edo spo coOAdEdoDoSBOsOnEACHOCSSE Some Remarks on Steady-State Cavities in Real Liquids ..............0...cce eee The Analytical Description of Steady-State Cavities ..........-::ceesceeeeteeeeneenees A Remark on the Drag of Cavitating Bodies ...............ccccccccesessecneeeeeeesteneeeneees pattie SOME RECENT WORK ON NONSTATIONARY CAVITIBS...........0.00:ccccccesseseeseeeeeess GONGISUDING RE MARIS eects core cat oc eee ab oee pc cee base Adis. eA On eh dae Sus ulate eee RAESEGE RUBIN © S ereticceces arene cece cece suite arabe sebacrins Saautsa asasvealsou neo vaceevaswedeteasaonensaees oueeauesoceersens fin ‘ary sibadich 46 etost om vas ‘ ead Tas Brean oe Pa penne th eos a ya bingy 2 a Hoa a AE Ne Mc ME i Diente ite ae hae aks Taavainayoe oj ¥ At , dite paras ta" hn SES ; nih ; ty y = Bas * a poo J SF A ean 1 Be roa My i uy Son hs Ree “4 in ay 7 Sp nate ei da pial? ani 3 Xo si oa 2 ais yi O HAH in LE ¢ Wika t vi 4 ma | a} ¥ ABSTRACT This report is intended to bring up to date the discussions of TMB Report 712 which summarized briefly the status of cavitation problems up to 1950. An attempt has been made to present a more unified outline of the various cavitation processes and to indicate, approximately, the progress on the various topics in this outline. Thus, the earlier discussions of the role of turbulence and of bound- ary layer effects in inception are extended. Some recent work on scale effects in inception is reviewed and some further thoughts on the influence of nuclei content are presented. Recent results of theoretical analysis by various writers on tran- sient cavities including the effects of compressibility and viscosity are briefly mentioned. Further remarks are included on steady-state cavities in real liquids, and recent theoretical wor':s on this type of cavitation are cited. In particular, mention is made of a recent linearized theory by Tulin, which will be of consider- able interest for technical applications. Some recent observations on the drag of cavitating bodies are summarized. Finally, it is observed that perhaps the great- est progress in this field in recent years has been in the recognition and more clear definition of the problems requiring investigation. INTRODUCTION Although the significance of cavitation phenomena in technical applications has been clearly recognized for many years, investigations of the various aspects were, until recent years, rather limited. In general, earlier researches were restricted to rather isolated problems with little evidence of recognition of the relationships between cavitation and other hydrody- namic phenomena or even between the various types of cavitation. However, modern require- ments in the design of hydrodynamic systems for use at increasingly greater speeds, higher temperatures, and lower pressures have emphasized the need for prevention of cavitation and the associated damage to materials, objectionable noise and induced vibration, losses in efficiency and increase in drag with cavitation onset. Moreover, there has been increasing interest in the design of systems in which normal operation occurs under fully cavitating con- ditions. These problems require a clear understanding of the physical and thermodynamical processes involved and not merely the assembling of systematic, empirical data; the phenomena are such that without this understanding even seemingly straightforward tests may be invalid or subject to misinterpretation when attempting to apply these data. As a result of these re- quirements and realizations, there has been within the past ten years, a significant increase in research activities on the mechanics of cavitation and in the analytic descriptions of the various types of cavitation. In addition to these motivations for cavitation research, the use of caviteted systems for accelerating chemical processes has been of interest for some years, and more recently, the recognition in the medical field of phenomena which may be included in the field of cavi- tation has provided yet another incentive for research. For example, interest in the inception and subsequent behavior of small, individual cavitation bubbles arises in connection with the general medical problem of gas separation in living tissue (see, e.g., Reference 1*) of which a special problem is the phenomenon of deep-sea divers’ ‘‘bends’’ in which gas separates from the blood stream during too-rapid decompression. Again, there is interest in the medical field in the type of cavity formed during the air-water entry of a projectile since the wounds produced by missiles evidently behave as such cavities during the entry phases (see, e.g., Reference 2). Although many questions remain to be investigated before the mechanism of cavitation phenomena can be more completely understood, much progress has been made toward rational descriptions of the processes in cavitating systems. This work includes not only the hydro- dynamics of flows with both liquid and gas or vapor phases, but also the physical chemistry of such multiphase flows in relation to the formation, collapse, and maintenance of cavities. In a previous report,* the writer attempted to outline very briefly and approximately the status of knowledge of various cavitation problems up to the year 1950. In that report, an attempt was also made to indicate some of the relationships between cavitation and other hydrodynam- ic phenomena and between the various types of cavitation in a somewhat more systematic way than had been done previously. Because of the constantly growing need among naval archi- tects and engineers for more precise data on cavitation, it seems worthwhile to bring up to date the discussions of Reference 8 with the purpose not of providing design data for specific problems but, primarily, of providing background information necessary in the development of rational design criteria, procedures, and evaluations. Although the present work is intended primarily as a survey of recent progress, an attempt will also be made to indicate the various problems and factors that, in the writer’s opinion, require further study and to indicate how these problems arise by following an outline based on the sequence of events in the establishment of cavitating flows. This requires de- finiuve terms for the ‘‘types’’ of cavitation, discussion of inception and of the processes in these various types of cavitation as well as of the types to be expected in any given system. Of necessity, some of the material in Reference 3 will be repeated, but, in general, liberal reference will be made to that work in preference to repetition. In this short paper, it will not be possible, of course, to give many details of recent investigations. However, it is hoped that not too many available references have been overlooked. ‘ *References are listed on page 21. TYPES OF CAVITATION In Reference 8, brief discussions were given and several illustrations shown of the types of multiphase, single or multicomponent flows which can be considered cavitational flows from either a hydrodynamical or thermodynamical point of view or both. For the present work, it will only be necessary to define the terms preferred by the writer in referring to the various types of cavitation: 1. TRANSIENT CAVITIES. This term will be applied to the small, individual cavitation bubbles which grow, sometimes oscillate, and eventually collapse and disappear. This type has been characterized as ‘‘burbling’’ or ‘‘bubble’’ cavitation by naval architects working on problems of cavitation of propellers and other ships’ appendages. The following types may be considered in the general class of free streamline flows and characterized as Kirchhoff flows or Helmholtz motions: 2. STEADY-STATE CAVITIES. This term is applied to the large, stationary cavities observed behind blunt bodies and very often on hydrofoil profiles having relatively sharp leading edges. The analytic description is not a function of time.* Such cavitation is re- ferred to among naval architects as ‘‘laminar’’ and, more recently, as ‘‘sheet?’ cavitation. It should be noted that stationary vortex cavitation may be included here, being a time- independent flow with circulation. 3. NONSTATIONARY CAVITIES. This term is applied to cavities resembling the steady- state cavity but varying with time as in the air-water entry of an air-dropped missile or of an initially submerged but accelerating body. The term ‘‘unsteady’’ is also often used for this type of flow. These terms provide a nonambiguous set both for physical descriptions and for the associated mathematical formulations. Although all three are free-boundary flows, in the first, the pressure at the boundary varies with time; in the second, the boundaries are free streamlines (constant velocity magnitude and, therefore, pressure); and in the third, the boundaries are such that the material lines are not necessarily free streamlines. THE INCEPTION OF CAVITATION INTRODUCTORY REMARKS Although it is usual to assume, in most engineering applications, that cavitation will begin when the local pressure reaches the vapor pressure corresponding to the temperature of the liquid, the actual conditions are much more complex and very often the errors resulting *It must be recognized here that when applying the term ‘‘steady-state’’ to cavities in real liquids reference is 3 made to the average description taken over a relatively long time interval. When examined in detail,~ such cavi- ties are very often observed to oscillate both in the streamwise direction and locally in the cavity surface. from this assumption cannot be tolerated. Depending upon the condition of the liquid (air content and air or vapor nuclei which may exist in a stable condition or be stabilized on solid particles, etc.), cavitation may begin above or below the vapor pressure. The role of air and vapor nuclei in the inception of cavitation in both pure and contaminated liquids was discussed briefly in Reference 3. Some effects associated with turbulence in a free stream and in a boundary layer, as well as conditions in a boundary layer that are of some importance in cavitation problems were also discussed. In the present report, these remarks will be elab- orated and extended somewhat, and some recently reported work relating to scaling of incep- tion will be reviewed. The problem of the limiting tensions which can be developed in liquids will not be discussed in this report~some of the recent ideas and methods having been reviewed briefly in Reference 3. Although these ideas, based on so-called ‘‘hole’’ and ‘‘nucleation’’ theories, lead to predictions that are of the order of magnitude of available experimental results, the theoretical models and approximations are not complete and, at best, can only result in prob- able predictions (in the statistical sense). In view of the experimental complications, e.g., the ever-present uncertainties associated with the behavior of the liquid at a wall, and of the properties of the wall itself, and the lack of sufficiently detailed theory needed for design of critical experiments, a completely satisfactory prediction procedure cannot as yet be estab- lished. Nevertheless, these ideas and methods are of immediate importance in helping to gain insight into the processes involved in cavitation even in ‘‘technical’’ liquids, and of continuing interest from the standpoint of eventual contribution to a unified theory of the mechanism of inception in any liquid, pure or contaminated. FURTHER REMARKS ON THE ROLE OF TURBULENCE AND BOUNDARY LAYER EFFECTS IN CAVITATING FLOWS In Reference 3, a rough calculation was carried out to illustrate the possibility of cavitation associated with the pressure fluctuations in a turbulent flow. It was shown, that in a boundary layer, the velocity fluctuations are such that the instantaneous pressures may vary sufficiently from the average value to result in a ‘‘microscale’’ cavitation which would occur in a more-or-less random fashion well before it could be observed visually. Thus, it is of some interest to be able to estimate the magnitude of the pressure fluctuations in turbulent flows of both the boundary layer type and wake and jet type. In these cases, however, the complete relations of turbulence phenomena to cavitation inception cannot be established until more is known of the turbulence mechanism itself. Although a great deal of work has been done in describing the velocity fluctuations, little effort has been devoted to investiga- tions of the pressure fluctuations. The extreme difficulty of the latter problem has restricted the work so far to isotropic turbulence and only statistical averages can so far be obtained. For the cavitation problem, it will probably be important to know in somewhat more detail the values of the pressure fluctuations, the scale(length), and ultimately a time scale. The estimates made in Reference 3 were based on results of Taylor* and Green> who found from computations based on models of isotropic turbulence that the pressure fluctuation p’ may be written Vr2 3 72 a /K u’2 Pp 9 tP where p is the mass density of the fluid and w”1s the velocity fluctuation. The value of K given by Taylor and by Green for a number of theoretical models is of the order of one. In 1950, Batchelor® considered this problem in the light of modern theories of turbulence and was able to derive the following relation for the fluctuating pressures in homogeneous, isotropic turbulence: yp? = 0.34 p2 (u’?) 2 In terms of Taylor’s formula, this corresponds to a value of K of 0.39. Since this magnitude is essentially of the order assumed in estimating the effects on cavitation, the previous com- putation need not be modified. Batchelor found further that the pressure scale is of the order of half the velocity scale. In addition to such computations, actual measurements of pressure fluctuations will be required.* Such measurements should include the temporal as well as spatial correlations with the ultimate aim of estimating the time available for formation of cavities. The average conditions in both laminar and turbulent boundary layers are also of some interest with respect to analysis of behavior of nuclei within the boundary layer. It was shown in Reference 3 that, aside from low pressures associated with turbulence which occur at a small distance from the boundary, the minimum pressures may be expected to occur at the boundary rather than elsewhere in the boundary layer. Furthermore, nuclei which are exposed to low pressures in the slowly moving fluid of the boundary layer have a longer time in which to grow beyond a critical size and start the cavitation process. Further work on boundary layer and wake flows will, of course, be necessary before the importance of the processes in such flows in relation to cavitation can be fully evaluated. Recent studies by Townsend’ have shown that the outer regions of the turbulent boundary layer resemble wake flows with a rather well-defined boundary between the turbulent and non- turbulent regions but that at any point in this outer region, the turbulence is intermittent. The picture presented is that of a wake with jets or fingers of turbulent fluid extending into the nonturbulent fluid. This phenomenon wil] be mentioned again in connection with the discussion of steady-state cavities in real liquids. *Such measurements are being taken as part of a program of studies of jet cavitation at the Iowa Institute of Hydraulic Research. Preliminary results were reported by Dr. H. Rouse at the Eighth International Congress on Theoretical and Applied Mechanics, Istanbul, Turkey, August, 1952 However, published results, which will be of considerable interest in relation to these questions, are not, as yet, generally available. CAVITATION IN SEPARATED FLOWS The foregoing remarks were concerned with the conditions near a hydraulically smooth surface having unseparated boundary layers. It was pointed out in Reference 3 that care must be exercised in the design of hydrodynamic systems to insure that separated regions do not occur. In such regions, the very low pressures which can be developed in the essentially vortex flows can lead to cavitation in spite of relatively high ambient pressures. This was illustrated for wake flows from experiments at the Taylor Model Basin and for locally separ- ated boundary layer flows on bodies of revolution from experiments at the Iowa Institute of Hydraulic Research. An example of cavitation in a separated boundary layer in the vicinity of the stagnation point is shown in Figure 1. In this case, the (probably laminar) boundary Figure 1 - Cavitation in the Separated Boundary Layer on the Upstream Side of a Toroidal Ring A half-toroid is fastened to a lucite plate through which this view was taken. The flow is from left to right. The two horseshoe-like vortices are swept down- stream, one leg of each being clearly seen outside the ring and the other leg passing through the hole but ob- scured by the cavitation on the ring itself. ' This photograph was obtained by Mrs. E. A. Sykes of the David Taylor Model Basin. NP21-49486_ layer separated in the region of the adverse pressure gradients approaching the toroidal ring. The cavitating region was swept downstream in the form of two horseshoe vortices with legs outside and inside the toroid. Conditions similar to these are evidently associated with cavitation about surface roughnesses. It is clear that the flow about roughness elements with sharply varying contours will separate and that cavitation may occur in the core of the separated regions. The strength of the vortices will evidently depend upon the velocity near the top of the roughness element so that the cavitation characteristics will depend upon the boundary layer configuration in the vicinity of the elements considered. Although much work has been done on the noncavitating flows about rough surfaces, results are not available in sufficient quantity to enable design criteria to be-established for roughness limits acceptable from the standpoint of cavitation prevention. Experiments have been made by Shalnev® for roughness elements in a restricted flow but lack of complete correlation with boundary layer characteristics make these results of limited usefulness. Nevertheless, they are the only published results known to the writer, and some empirical formulas are given which can be used when the roughness height is large compared with the boundary layer thickness. Other separation phenomena may be observed when fluid issues from a nozzle with an abrupt expansion. This was discussed in Reference 3 in connection with the discrepancies between:the results of Crump in a nozzle with a gradually expanding diffusor and of Numachi in a nozzle with an abrupt expansion. The trends observed by Numachi were confirmed by Crump in a subsequent TMB report? although the numerical values differ somewhat. The ob- servations of increasing tensions with increasing time of exposure to low pressures in Crump’s first nozzle have not as yet been explained. EFFECTS OF SCALE ON THE INCEPTION OF CAVITATION Only a few remarks were made in Reference 3 on the question of scaling cavitation phenomena. However, this problem is of much importance in view of the need for reliance on results from model experiments. It is desired to mention here only the problem of scaling of the inception of cavitation and, in particular, to review some very interesting results obtained at the California Institute of Technology during the last two years on the effects of geometri- cal scale on inception of cavitation. Before discussing the latter results, however, it seems worthwhile to review the physical picture of the role of nuclei in cavitation inception and to point out the expected consequences of thts concept. It is now generally accepted that cavitation in a fluid under reduced pressure or boiling in a heated liquid begins with the growth of microscopic nuclei containing gas phase (air or vapor, etc.) It is well known that the absence of such nuclei requires very large forces for rupture since the surface tension forces become very large. Thus in well degassed liquids one expects rupture forces of the order of those predicted by kinetic theoretical formulations. Experimental evidence has also been obtained that water saturated with air, but denucleated by application of very high pressures, exhibits high tensile strength (of the order of several hundred atmospheres). !4 Thus, the presence of nuclei is evidently necessary for the incep- tion of cavitation at pressures of the order of vapor pressure. . In many engineering applica- tions, it is usually sufficient to assume that cavitation will occur at the vapor pressure corre- sponding to the temperature of the liquid. This assumes that there are sufficient nuclei of large enough initial size to grow to observable size during the time of application of reduced pressure. In supersaturated liquids, it is easy to account for the presence and stability of such nuclei, but in saturated and undersaturated liquids it is necessary to account for such nuclei on the basis that they are stabilized on particles suspended in the liquid (see, e.g., Reference 3 and the discussion of Reference 10). As a consequence, depending upon the size and number of these nuclei, cavitation may be expected to begin above as well as below the vapor pressure, as was shown in Crump’s experiments with sea water and fresh water. A further consequence of this concept is that inception will depend upon the time of exposure of these nuclei to low pressures. Thus, for nuclei of a given size, the longer the time of ex- posure the higher inception pressures that would be expected. In addition to the effect of actual time of exposure to low pressure, another factor which must be considered when examining the effects of scale on inception is the dynamical behav- ior of the nuclei as a function of the rate of application of pressure, i.e., the pressure gradir ents. Thus, in scaling experiments in which the pressures are determined by the geometrical bounaary conditions, not only will the actual time of exposure to low pressures vary but also the pressure gradients.* Since, in general, with geometrically similar boundaries and the same velocity, the pressure gradients will increase with decreasing model size, the dynamical response of the nuclei will be initially retarded and the initial appearance of an observable inception will be delayed in the experiment of smaller size if the properties of the liquid are independent of scale, i.e., if nuclei size remains unchanged. From a most elementary view- point* it would appear, therefore, that in order to carry out experiments at reduced scale, the nuclei size must vary inversely with the model scale, i.e., in order for the nuclei to grow to an ‘‘observable’’ size (visual, aural, etc.), larger nuclei must be present in the small scale experiment than in the prototype experiment. This discussion has assumed that surface ten- sion effects are always important in the inception of cavitation. If the nuclei are initially so large that surface tension is not important in determining the rate of growth, then the effects of pressure gradients will not be controlling and only the time of exposure need be considered. However, if one further considers that only certain nuclei will grow to observable size, then the number of exposed nuclei must also be considered. Thus, on a reduced scale, the total number of nuclei in the liquid must be greater for the smaller scale than the larger since the region of low pressure is also reduced. For such considerations, the total number of nuclei must vary inversely as the cube of the linear ratio. ** . The experiments on the effects of geometrical scale on cavitation inception carried out at the California Institute of Technology are of particular interest in relation to the foregoing remarks. It will be observed that in the type of systems investigated in these experiments it was not possible to scale nuclei size and content to correspond to model scaling. Kermeen!! has conducted a large number of experiments on bodies of revolution with hemispherical noses and cylindrical afterbodies having diameters ranging from 1/4 to 2 inches. He has observed a definite dependence of the cavitation number for inception on the model size and upon the absolute flow velocity. Furthermore, as will be seen from his results, Figure 2, the data cannot be correlated on the basis of Reynolds number alone. Further studies have been conducted on hydrofoils by Parkin who also attempted an analysis of the observations based on the growth of nuclei assumed to be stabilized on solid particles in the fluid. 12 The results shown in Figure 2, which is only a sample of the data obtained by Kermeen, represent average values of a large number of data. The methods used in obtaining these data *Effects of viscosity, diffusion, etc., have been neglected in this discussion. **Investigations of the various factors discussed in the foregoing are now getting underway at the Taylor Model Basin. i Figure 2 - Cavitation Number K for Incipient Cavitation as a Function of Reynolds Number for Bodies with Hemis- pherical Noses These results were reported by R.W. Kermeen, California Institute of Technology, Reference 11. Cavitation Number K at Maximum Sound Arrows Indicate Model Diameters 2 OB LO 2.0 40 60 10 20 Rexlo7> are of some interest in connection with the remarks made in the subsequent section of this report. The curves shown were obtained by first allowing cavitation to become fully develop- ed at cavitation numbers well below the inception value and then raising the ambient pressure 665 ‘until cavitation just disappeared. This point was defined as the ‘“‘inception point.’’ Attempts to observe inception by the direct procedure of approaching the critical point starting with a noncavitating system evidently resulted in so large a scatter that it was difficult to obtain accurate correlation of the results. Kermeen remarks that a so-called ‘‘hysteresis’’ could often be observed, i.e., when approaching the inception point from a noncavitating condition, much lower critical cavitation numbers were observed than those at which cavitation disap- peared. Nevertheless, the results evidently indicated the trends clearly shown in Figure 2. These results are of further importance in that they tend to substantiate the physical picture of the role of nuclei in cavitation inception since the observed trends coincide with the ex- pected consequences of this concept. Although the results obtained by Kermeen and Parkin are consistent with the present ideas of the role of nuclei, there still remains the anomaly of the results obtained by Crump in which simultaneously increasing time of exposure to low pressures and decreasing pres- sure resulted in decreasing critical cavitation numbers.* On the other hand, it will be ob- served that in Crump’s experiments, the inception point was obtained by approaching. from a noncavitating condition, and that the same effect characterized as ‘‘hysteresis’’ was obtained when approaching from the fully cavitating condition. (The latter method was not considered as a criterion for inception in his experiments and no quantitative results were recorded.) *Whether Crump’s results can be reconciled in terms of the effects of the pressure gradients remains to be in- vestigated. » 10 Thus, some care must be exercised in relating the results of these different types of experi- ments to practical systems in which the actual onset is of interest. It would seem to be of rather significant interest to know the maximum possible critical cavitation number for any given system and, further, to know a ‘‘critical’’ cavitation number at which adverse effects (noise, loss in performance, etc.) first begin. (These numbers are not, of course, necessarily the same. As a matter of fact, this is only the question of the definition of cavitation onset since, presumably, nuclei will be affected by any pressure field whether or not they are seen or, as a result of the motion, radiate noise.) A REMARK ON THE QUESTION OF GAS CONTENT AND NUCLEI CONTENT It seems clear from the foregoing remarks and the discussions in Reference 3 that the inception of cavitation in technical liquids is intimately associated with the presence of nuclei whether as free gas or vapor bubbles or bubbles trapped on solid particles. That the role of gases in liquids is well recognized is reflected in the now almost standard procedure of measuring air-content in experiments in which cavitation phenomena are involved. However, the measurements that have been reported so far have all been of the total air and gas content, i.e., both dissolved, and entrained as individual bubbles. On the basis of the previous dis- cussion, it seems unlikely that completely dissolved gases can play an important role in the inception of cavitation* since, in this situation, very high tensions would be expected. Thus, it is the opinion of the writer that further progress on inception may depend, to a considerable extent, on the development of theoretical and experimental methods for the characterization and measurement of the nuclei content of liquids.** These studies should eventually result in descriptions of the size distribution and quantity of undissolved gas nuclei—in other words, a ‘‘spectrum’’ of nuclei. With such information available, it would then be possible to undertake correlation studies of inception of cavitation as a function of the nuclei spectrum. Judging from the ex- periments that have been made and remembering the many factors which must be considered (turbulence, physical-chemical properties of the liquid, other foreign materials, etc.), it seems not unlikely that under a given set of conditions, it will not be possible to prescribe a unique critical cavitation number but only a ‘‘most probable’? critical cavitation number. *Except insofar as the equilibrium conditions between dissolved gas and undissolved nuclei are concerned. **Such studies are now underway at the David Taylor Model Basin. - 11 DYNAMICS OF TRANSIENT CAVITIES THE MOTION OF SMALL TRANSIENT CAVITIES Since, in Reference 3, a brief outline of the experimental observations on transient cavities up to 1950 was given, only a few remarks will be made here in connection with an apparent difference in experience among different investigators on the rebound of such cavi- ties. For this discussion, we exclude observations in which magnetostriction oscillators or ultrasonic fields were used to cavitate the liquid. Also excluded are experiments in which small air bubbles were introduced to assist the formation of the cavity. In 1928, Mueller!? published prints of motion picture frames showing cavitation on a hydrofoil. These photographs showed clearly the growth and collapse of individual cavities, but no oscillations were observed. Harvey et al,**4 using rods withdrawn rapidly from a liquid, obtained several oscillations of the resultant cavity and attributed them to energy storage in air entrained in the bubble. Knapp and Hollander, !5 in a now classical series of photographs, showed several cycles in the oscillations of cavities in a flowing liquid. They argued that in their experiments the initial air content of the cavities was extremely small and attributed the rebound of the cavity primarily to the storage of energy in the liquid in elastic compres- Sion, with this stored energy subsequently producing the outward radial velocity. More recently, experiments by Harrison!® have shown that cavities formed in water of low air content do no¢ rebound, and the conclusion was reached that only cavities containing a large amount of gas will oscillate. In a private communication, Dr. M.S. Plesset informed the writer that the same results were recently obtained at the California Institute of Technol- ogy. It may, therefore, be concluded that in liquid of low air.content, the effects of compress- ibility and viscosity in both the liquid and gas phases in dissipating energy are such as to allow no rebound if there is only a small amount of permanent gas in the bubble. However, no consideration appears to have been given as to what this lower limit of initial air content in the nucleus must be to prevent rebounds. ANALYTICAL DESC RIPTION OF THE MOTION Work on the motions of a spherical cavity in an incompressible fluid in which the role of the gas within the bubble is neglected was reviewed in Reference 3. These theories give adequate descriptions of the motion for the largest part of the cycle of such bubbles but are inadequate toward the end of the collapse stage where compressibility and viscosity effects in both the liquid and the gas phase become of importance. This was shown by the computa- tions of Plesset as compared with the experimental results of Knapp and Hollander. Dis- crepancies noted between theory and observation for the collapse part of the cycle were attri- buted to wall effects--the bubbles having been formed near a model and these effects neglect- ed. This was clearly shown by Rattray1” in a dissertation in which he computed the motion of such cavities when near a plane wall. The analysis is too lengthy to give details here, 12 being carried out to several terms in an expansion of the velocity potential in Legendre poly- nomials. However, Rattray showed that for the above bubbles, the time of collapse could bé increased by as much as 20 percent for the assumed distance of approximately one-half to one diameter from the wall. It is clear that analyses based on empty cavities in an incompressible liquid cannot give rise to oscillations. Furthermore, with constant external pressure, the velocity of the bubble wall increases without limit as the bubble collapses. To obviate this result, Lord Rayleigh, who gave the first complete solution of the collapse of an empty, spherical cavity in an incompressible liquid!8 extended his computations to include the case of a cavity filled with a gas which is expanded and compressed isothermally and showed that the boundary oscillates between two positions, of which one is the initial position. Although the motion of the oscillating cavity in a real liquid is evidently complicated by the diffusion and vapor- ization processes and the problems of energy dissipation, such solutions, which are clearly oversimplifications, nevertheless, give a clear, quantitative picture of the hydrodynamics of the motion as long as the cavity radius is iarge compared with the minimum radius. In Reference 3, the writer pointed out that further extensions to include the problems of energy dissipation associated with the compressibility of the vapor and gas mixture and of the liquid would be of great practical as well as theoretical interest. For a bubble filled with a permanent gas being compressed and expanded adiabatically rather than isothermally, Ray- leigh’s case corresponds to the case or pulsation of a gas globe following an underwater ex- plosion (see, e.g., Reference 19). Although, in the latter problems, much progress had been made (up to the time Reference 3 was written) in describing an oscillatory motion with energy dissipation, the problem of the vapor condensation and formation prevents a complete analogy to gas-globe theory and a completely satisfactory description of the motion of cavities based on this theory had not been formulated. However, if one neglects the condensation problems, which are important only insofar as determining the conditions under which the vapor begins to act as a permanent gas, many of the results and methods first developed in the field of ex- plosion hydrodynamics may be applied to the present problem. This has been done within the last two years, first by Trilling?° and later by Gilmore. 2+ Trilling derived the velocity and pressure fields about a bubble in a slightly compressible liquid using the acoustic approximation (i.e., only velocities that are first order small com- pared with the velocity of sound are considered) and obtained results which coincides with those of C. Herring?” 23 who carried out the same computation some years earlier, but only for the conditions at the bubble surface. Since these results are included in the later exten- sion of Gilmore,?! only one result of interest will be mentioned in connection with Trilling’s work. Computations of the collapse of a bubble supposed to be filled with a perfect gas showed that a series of shock waves are propagated into the gas. However, it was shown that the average variation at the bubble wall is very nearly the same as if the gas were compressed uniformly and isentropically. 13 Gilmore?! extended these computations to include higher order compressibility terms, as well as the effects of viscosity and surface tension. Instead of using the acoustic approxi- mation (i.e., all disturbances propagated with the velocity of sound c), Gilmore assumes the Kirkwood-Bethe hypothesis, 2* which assumes that disturbances are propagated with the velocity c + u, u being the local fluid velocity. Gilmore then derives the equation of motion of the bubble wall in the form:* Ru oY (y- ¥\,. 3 y2 (1-2) 7 fi,¥)\, RU GH _ wu de Gis 30 G) Gd G where RP is the radius of the bubble, C is the sonic velocity in the liquid at the bubble wall, PB. : H is the enthalpy difference (-| ée) between the liquid at pressure P at the bubble Peal tom wall and at pressure p_ at an infinite distance from the bubble, and U is the velocity of the bubble wall. The values of H and C are derived from the experimentally developed formula for isentropic compression of liquids. fds (8 =/[10 \F D_, * B Poo where B and n are constants for each liquid (for water, B = 3000 atm and n = 7), Thus, he finds that f +B \z5 C = C_|——_} 2” a ). se 183 and el Pg (oy 7a (n-1)p. \\p,, + B The effects of surface tension and viscosity are included in the boundary conditions by writing the pressure at the bubble wall as where P, is the pressure of the internal gas, o is the surface tension, and p is the dynamic viscosity. *Only a very few results of Gilmore’s paper are abstracted here. For full details see Reference Qe 14 Using the equation of continuity for a compressible fluid and assuming that the viscosity and compressibility of the liquid are small, he writes, finally, yon Cyn tl ie gon Ts If the internal pressure is constant and the surface tension and viscosity negligible,* the equation of motion can be solved analytically if only terms in dU/dRk and dH/dR are re- tained, thus, Gilmore finds BAC U\* Soe =) -(1-) (1+ *=—) Rk 3C 2p - P;) Neglect of the term U/3C yields the Rayleigh results. As R > 0, the Rayleigh theory (incompressible) gives U~ R~°/2, whereas Gilmore’s results gives U~R™ 1/2, 4.0 a Figure 3 - The Theoretical Wall-Velocity of = a Bubble in Water Collapsing under a at Constant Pressure Difference of EG 0.517 Atmospheres 5 0. 2 OY. These curves are reproduced from Gilmore, Refer- 8 5 a ence 21. ‘Present Theory”’ refers to Gilmore’s results, resen eory 0.2 F ———Herring Solution (First - Order See text. —-—Incompressible Compressible) © Schneider's Numerical Calculation 0,001 0.01 0.10 Radius Ratio, R/R, Figure 3 shows a comparison of Gilmore’s theory with the results of Rayleigh, Herring, and a numerical integration of the complete equation of motion carried out by Schneider. 25 It will be seen that the solutions approach each other for small ratios of the wall velocity to sound velocity but diverge rapidly as the sonic velocity is approached and finally exceeded. Gilmore also gives the equations of pressure and velocity throughout the fluid but did not compute numerical values. More recently, computations have been carried out by Poritsky2© and Shu27 to deter- mine specifically the effects of viscosity and surface tension, but assuming the fluid to be incompressible. In these papers, large effects of viscosity in retarding both the growth and * Although Gilmore only carried through the computation with surface tension and viscosity neglected, he ex- amined the effects of these variables and found bounds for the ratio R/R within which the effects could be neglected without affecting the motion except in the very last period of collapse (which will usually be of the order of a few microseconds). 15 collapse were found, while for surface tension, some cases are worked out to illustrate the effects in retarding growth and accelerating the collapse. MECHANISM OF STEADY-STATE CAVITIES AND THEIR ANALYTICAL DESCRIPTION INTRODUCTORY REMARKS A complete review and analysis of the experimental and analytical work on steady- state cavities would require a much more detailed and voluminous discussion than is intended here. Some results of more immediate interest in technical applications were presented in Reference 3. In this report, the discussion will again be confined to a few more-immediately applicable results and only mention made of the direction of other researches. The recognition of the applicability of free streamline theory (classical wake theory) to the steady-state cavity problem has greatly stimulated the mathematical work on such flows and has resulted in a rapidly growing literature in this field. The two-dimensional problem is now well understood and the theory:is available for the solution of flows about a large class of solid boundaries. However, very few numerical results are, as yet, available and will be required before application in engineering problems can be made. These results would be of much interest in connection with so-called ‘‘supercavitating”’ propellers, for example, as well as other problems requiring knowledge of the forces on fully cavitating hydrofoils. The suc- cessful treatment of the two-dimensional cases has been possible through the very powerful conformal mapping techniques. However, the extreme difficulty of the general problem (arising from the nonlinearity of the boundary conditions) has precluded general treatments, ‘and in the three-dimensional case, progress has been made only in the treatment of flows with axial sym- metry. In the following paragraphs, some additional thoughts on cavities in real liquids will be outlined and some recent results on the analytical treatment will be presented. SOME REMARKS ON STEADY-STATE CAVITIES IN REAL LIQUIDS The degree to which the theoretical models of steady-state cavities represent such flows in real liquids was discussed to some extent in Reference 3. The added complications of the properties of the liquid and of the surface conditions of the solid boundaries require further clarification and much work remains to be done in this direction. Furthermore, the question of the maintenance of such cavities in real liquids requires further investigation— for example, the vaporization process at the cavity wall, the processes in removal of all liquid phase from cavities in which cavitation first occurs only in the small scale eddies on the boundary of a viscous wake, and the processes of entrainment and condensation at the tail of cavities which have been observed to oscillate rather violently. The latter questions were also discussed to some extent in Reference 8 and, in this report, only a few additional 16 remarks on these problems will be made. So far, the transient cavities and the steady-state cavity flows have been discussed as entirely unrelated problems from a hydrodynamical point of view. One of the interesting and practically important questions which has not been considered to any extent is that of the conditions (both geometrical and physical) under which cavitation will occur in the form of a ~ large steady-state cavity or a mass of small, oscillating bubbles. In Reference 3, the writer expressed the view that the appearance of the cavitation is associated in part with the pres- sure gradients, but no satisfactory criteria are available as to the initial appearance or transi- tion from transient cavities to a large, steady-state cavity. A cavitated region made up entire- ly of transient bubbles may exhibit the properties of a steady cavity in that the average envel- ope of such a region does not vary with time. Another case of such ‘‘steady-state’’ cavities in which the average envelope remained unchanged, but in which rapid surface oscillations were observed without clear evidence of individual bubbles was discussed in Reference 3 in connection with the study of the development of cavitation in wakes. This case of cavitation in wakes raised the question of the processes involved in re- moving the liquid phase from the cavity and the spread of cavitation from the wake boundaries into the interior. It was pointed out in Reference 28 (and also discussed in Reference 8) that in the studies reported therein the cavities contained large quantities of liquid phase and that the liquid was not removed even at the lowest cavitation number reached (0.116). An example of such a flow is shown in Figure 4, which was obtained during the experiments reported in Reference 28. Subsequent experiments have disclosed certain cases in which the liquid phase was removed from the cavity. Under conditions not yet defined or even understood the liquid could be seen to move rearward out of the cavity. This process occurred at a speed slow enough to be followed by eye, leaving a fairly transparent cavity wall. This phenomenon has also been observed in the water tunnel of the Ordnance Research Laboratory at the Pennsyl- vania State College. Just what conditions determine the point at which equilibrium is upset between vaporization and possibly shearing motion at the boundary and the replenishment with re-entrant fluid from the tail of the cavity are not clear and require further investigation. Figure 4 - ‘‘Steady-State’’ Cavity behind a Hemisphere The photograph was taken with an exposure time of 1/10,000 second. 17 It will be observed in Figure 4 that cavitation appears to be intermittent even though this case is one in which the cavitation number is well below the values associated with the shedding of well defined vortices (see Reference 3). This intermittency further suggests the close connection with the turbulent wake flow mentioned previously. Furthermore, the ‘“‘billowing’’ nature of the surface resembles the descriptions of the turbulent wake as given by Townsend. Nevertheless, the average envelopes of such regions evidently behave as steady-state cavities. Thus, the problem of cavitation in such turbulent regions and adequate descriptions of the processes remain to be investigated and reconciled with available theory. Otherwise stated, the problem is one of describing the transition from a cavitating wake flow to free streamline flow. * THE ANALYTICAL DESCRIPTION OF STEADY-STATE CAVITIES Some results of free streamline theory for two-dimensional flows were given in Refer- ence 38. Further work continues on these flows based on the Riabouchinsky and the re-entrant jet models but the results have not been applied in a sufficient number of technically impor- tant cases. Discussions of these flows will be found in References 3, 29, 30, and 31 which are cited here not only for their own content but also for the bibliographies and references given therein. Theoretical work on three-dimensional cavities has been concerned almost exclusively with existence and uniqueness of solutions for axially symmetric flows. Recent work in this direction is given in References 32 and 33 which also summarize previous work on this prob- lem. Various attempts to apply numerical procedures to the solution of specific problems have been made but have evidently been unsuccessful in producing accurate or even physi- cally realistic results and will not be discussed here (a brief account of some computations will be found in Reference 29). In addition to the above attempts to develop exact solutions of free streamline flows, a recent theory which avoids the necessity for artificial mathematical models deserves spe- cial mention. M.F. Tulin, in a recent report,?4 has developed a linearized theory for two- dimensional cavity flows about slender, symmetric bodies. An important feature of the re- sults is that the calculation of drag and cavity shapes of arbitrary slender bodies is reduced to quadratures with the resultant attractiveness for use in actual applications. In this report, only some comparisons between Tulin’s results and the exact results for the Riabouchinsky model will be presented to indicate the range of applicability of this theory. The method of linearization is similar to that of the linearized airfoil theory. The results chosen for illus- tration are from computations for wedge profiles. Tulin gives the following results for the *From another point of view, such cavitating turbulent wakes might be used in studying the pressure fluctua- tions associated with turbulence, as has been pointed out by others, as well. 18 relation between cavitation number and cavity length (any profile), (0) g -s) Yo Yi-t dt ee WU Gi VE =O 2 the origin of coordinates being chosen at the trailing edge (base) of the profile: Here, o is the cavitation number, / is the cavity length, c is the body length, and y, is the body ordinate. His result for the drag coefficient as a function of wedge half-angle y is 8 Ons = General results for cavity shape and drag are given in Reference 34. Figure 5 shows the comparison between the drag coefficients from Tulin’s theory and the exact theory for various wedge angles at zero cavitation number. Figure 6 shows the comparison of cavity lengths from Tulin’s theory and the exact Riabouchinsky model for various cavitation numbers for a 30-degree wedge. The success of this theory promises to be of much importance in engineer- ing applications since Tulin has been able to extend the method to the computation of cavi- tating flows about lifting surfaces in both the steady and unsteady cases. The computations for the latter problems are now being carried out. He has also developed a linearized theory for axisymmetric, three-dimensional flows, but the results are not yet in a form suitable for numerical computation. Other results of immediate interest in technical application include work on wall ef- fects in steady cavitational flows. In Reference 35, the authors consider the two-dimensional cavity flow about tandem laminae situated normal to the flow direction in a straight-sided chan- mel of finite width and in a free jet of finite width. Results are given for finite as well as zero cavitation numbers. These results are of much interest in the design of water tunnels for studies of finite cavities of very small cavitation number, it being shown that the ‘‘wall’’ effects are much less severe with free jets than with rigid boundaries. Moreover, for a given cavitation number, there is a limiting value of the ratio of lamina width to channel width for rigid channels which cannot be exceeded (a ‘‘choking’’ phenomenon). Another problem which arises in flows with circulation is that of the stability of a cavitating vortex. This question is of some interest in connection with the effects of cavi- tation in the tip vortices of hydrofoils and propellers. Although no changes in the flow con- ditions at the hydrofoil or propeller blade will occur as long as the cavitated region remains detached, the behavior of such cavitating vortices will be of interest in gaining an under- standing of the changes and effects that result when the cavitated region becomes attached to the blade edge. An analysis of standing waves of infinitesimal amplitude on a vortex core has been carried out by Ackeret>® and his results verified, at least qualitatively, in experi- ments by Lerbs.?” More recently, Binnie?® has considered the problem in a more general way and has derived the properties of traveling waves, as well, giving numerical results for sever- al cases. It will be recalled, as Binnie points out, that this is a classical problem of 19 0.12 r 30° ONO} ee [ee ee | ARISES ENL| o— L 0.08 ;— a | 0.06 L t Riabouchinsky Model 0.04 Linearized Theory 0.02 ) 004. 008 ole Ol6 020. 024 Cavitation Number o Figure 6 - Cavity Length as a Function of at Cavitation Number for a 30-Degree Wedge Oo 0.1 0.2 0.3 0.4 Wedge Half Angle y (After Tulin, Reference 34) Figure 5 - Cavity Drag Coefficients for Wedge Profiles at Zero Cavitation Number (After Tulin, Reference 34) W. Thomson (Lord Kelvin).39 Interest in these results extend beyond their intrinsic content Since such oscillations may lead to vibrations of the propeller blades. Actual investigations of such interactions remain to be carried out, however. A REMARK ON THE DRAG OF CAVITATING BODIES Although the drag of bodies with fixed points of separation of the cavity (flat plates, discs, cones) can evidently be approximated by an equation of the form (see Reference 3) C plo) = Cp(0) (1+0) where C'p(c) is the drag coefficient for cavitation number a, this does not appear to be the case in actual flows for bodies with longitudinal curvature. The linear increase of drag with cavitation number shown in experiments is in good agreement with theory, but the slope of the curve evidently depends on the form. For results available so fat, it appears that the drag can be closely approximated by the formula C plo) = Cp(0) (1 +e) 20 Cavity Drag Coefficients for Various Bodies Mo del C0) aay Reynolds Number Disc 1.0 2.024 Hemisphere 2:1 Semiellipsoid and 2 Caliber Ogive Circular Cylinder The accompanying table summarizes the results examined by the writer. The value of Cp(0) for the disc is the result obtained by Plesset and Shaffer*® by assuming that the pressure distribution in the meridian plane is the same as that of the two-dimensional compu- tation. The values of (’)(0) for the hemisphere, ellipsoid, and ogive are extrapolated from the experimental data of Reference 28, from which the results for a for these bodies and the disc were also obtained. The value of C'p(0) for the circular cylinder is from the computation of Brodetsky.4! The value « = 0.73 for the circular cylinder is given by Birkhoff?9 based on the experiments of Martyrer. The other values of a for the circular cylinder are based on Konstantinov’s experiments,+? which show differences depending on Reynolds number (based on cylinder diameter). For comparison, the range of Reynolds numbers in Martyrer’s tests is also.shown. It should be noted that Konstantinov’s results are for constant Reynolds number, whereas in Martyrer’s tests the Reynolds number varied as the cavitation number was varied. There may be some question, however, as to the accuracy of Konstantinov’s results since the forces were found by integrating pressure distributions rather than by direct measurement. SOME RECENT WORK ON NONSTATIONARY CAVITIES In addition to the steady-state theory for lifting surfaces, it will eventually be of con- siderable interest to have results for nonstationary cavities for such cases (the interest for nonlifting bodies is already well established). For example, the cavity on a blade of a sur- face ship propeller of large diameter will grow and contract depending on the position of the blade during rotation. Results for two-dimensional unsteady motions without circulation have been obtained by Gilbarg*? for polygonal obstacles. This problem differs from the steady- state case in that the free boundary is a material line and not, in general, a streamline. Gil- barg replaced the latter requirement by the approximating condition that the free boundary is a streamline, and then used standard conformal mapping techniques to obtain solutions for cavities behind a flat plate normal to the flow direction (symmetric cavities). He showed that these solutions are exact for unsteady flows whose free boundaries are of constant shape. In particular, two classes were distinguished: one in which the cavities have a cusped end 21 or a stagnation point (the latter found also by von Kérm4n** and included in the more general class of cavities of constant shape found by Gilbarg); the second comprises cavities for which the free streamlines cross the axis of symmetry and are, thus, not physically realistic. In addition to the above detailed computations, he briefly indicated the method of extension of these results to arbitrary polygonal shapes and also to asymmetric polygonal shapes. CONCLUDING REMARKS An attempt has been made to present a more unified outline of the various cavitation processes and to indicate, approximately, the progress on the various topics in this outline. It is hoped, however, that some old problems which still require investigation and some new ones have been more clearly pointed out in the course of this survey. Although no direct mention of damage or cavitation prevention has been made, the remarks of Reference 8 re- garding the mechanisms need no material modification. However, direct work on damage of specific materials is still of much importance and programs are underway in various labora- tories. As a final remark, it might be observed that perhaps the greatest progress in this field in recent years has been in the recognition and more clear definition of the problems requiring investigation. REFERENCES 1. Harvey, E. Newton, Barnes, K.D., McElroy, W.D., Whiteley, A.H., Peese, D.C., and Cooper, K.W., ‘“‘Bubble Formation in Animals,’’ Jour. of Cellular and Comp. Physiology, Vol. 24, No. 1, August 1944, pp. 1-34. 2. Harvey, E. Newton, Whiteley, Arthur A., Grundfest, H., and McMillen, J.H., ‘‘Piezo- electric Crystal Measurements of Pressure Changes in the Abdomen of Deeply Anaesthetized Animals during Passage of a High-Velocity Missile,’’ The Military Surgeon, Vol. 98, No. 1, January 1946, pp. 509-528. 3. Eisenberg, Phillip, ‘‘On the Mechanism and Prevention of Cavitation,’’ TMB Report 712, July 1950. 4. Taylor, G.I., ‘‘The Mean Value of the Fluctuations in Pressure and Pressure Gradient in a Turbulent Fluid.’’ Proc. Camb. Phil, Soc., 1936, Vol. 32, pp. 380-384. 5. Green, A.E., ‘‘The Mean Value of the Fluctuations in Pressure and Pressure Gradient in a Turbulent Fluid,’’ Proc. Camb. Phil. Soc., 1988, Vol. 34, pp. 534-539. 6. Batchelor, G.K., ‘‘Pressure Fluctuations in Isotropic Turbulence,’’ Proc. Camb. Phil. Soc., Vol. 47, April 1951, pp. 359-374. 7. Townsend, A.A., ‘‘The Structure of the Turbulent Boundary Layer,’’ Proc. Camb. Phil. Soc., Vol. 47, Pt. 2, April 1951, pp. 375-395. 22 8. Shalnev, K.K., ‘‘Cavitation of Surface Roughnesses,’’ Jour. of Theoretical Physics, U.S.S.R., Vol. 21, No. 2, 1951, pp. 206-220 (in Russian). 9. Crump, S.F., ‘‘Critical Pressures for the Inception of Cavitation in a Large-Scale Numachi Nozzle as Influenced by the Air Content of the Water,’? TMB Report 770. July 1951. 10. Epstein, P.S., and Plesset, M.S., ‘‘On the Stability of Gas Bubbles in Liquid-Gas Solutions,’’ Jour. Chem. Phys., Vol. 18, No. 11, November 1950, pp. 1505-1509. 11. Kermeen, R.W., ‘‘Some Observations of Cavitation on Hemispherical Head Models,”’ Hydrodynamics Laboratory, California Inst. of Tech. Report No. E-35.1, June 1952. 12 Parkin, Blaine R., ‘“‘Scale Effects in Cavitating Flow,’’ Hydrodynamics Laboratory, California Inst. of Tech. Report No. 21-8, July 31, 1952. 13. Mueller, J., “Uber den gegenwartigen Stand der Kavitationsforschung,’’ Die Natur- wissenschaften, Vol. 22, June 1928, pp. 423-426. 14. Harvey, E. Newton, McElroy, Wm. D., and Whiteley, A.H., ‘‘On Cavity Formation in Water,’’ Jour. of Appl. Phys., Vol. 18, February 1947, pp. 162-172. 15. Knapp, R.T., and Hollander, A., “‘Laboratory Investigations of the Mechanism of Cavi- tation,’’ Trans., A.S.M.E., Vol. 70, No. 5, July 1948, pp. 419-435. 16. Harrison, Mark, ‘‘An Experimental Study of Single Bubble Cavitation Noise,’’ TMB Report 815, June 1952. 17. Rattray, Maurice, Jr., ‘‘Perturbation Effects in Bubble Dynamics,’’ California Inst. of Tech. Doctoral dissertation, issued as Hydrodynamics Laboratory Report under ONR contract N Gone-24420 (NR-062-059), undated but issued in 1951. 18. Lord Rayleigh, ‘‘On the Pressure Developed in a Liquid during the Collapse of a Spherical Cavity,’’ Phil. Mag., Vol. 34, 1917, pp. 94-98. 19. Kennard, E.H., ‘‘Report on Underwater Explosions,’’ TMB Report 480, October 1941. 20. Trilling, Leon, ‘‘The Collapse and Rebound of a Gas Bubble,’’ Jour. of Appl. Phys., Vol. 23, No. 1, January 1952, pp. 14-17. 21. Gilmore, Forrest R., ‘‘The Growth or Collapse of a Spherical Bubble in a Viscous Compressible Liquid,’’ California Inst. of Tech., Hydrodynamics Laboratory Report No. 26-4, April 1, 1952. 22. Herring, C., ‘‘Theory of the Pulsations of the Gas Globe Produced by an Underwater Explosion,’’ OSRD (U.S.A.) Report 236, 1941. 23. Cole, R.H, ‘‘Underwater Explosions,’’ Princeton Univ. Press, 1948, pp. 305-307. 24. Kirkwood, J.G., and Bethe, H.A., ‘‘The Pressure Wave Produced by an Underwater Explosion,’’? OSRD (U.S.A.) Report 588, 1952. (A brief account is given in Reference 23, pp. 28-45, 102-109, 114-126, 425-426.) 23 25. Schneider, A.J.R., ‘‘Some Compressible Effects in Cavitation Bubble Dynamics,”’ Doctoral dissertation, California Inst. of Tech., 1949. 26. Poritsky, H., ‘‘The Collapse or Growth of a Spherical Bubble or Cavity in a Viscous Fluid,’? Proc. First U.S. Nat. Cong. Appl. Mech., A.S.M.E., 1952, pp. 813-821. 27. Shu, S.S., ‘‘Note on the Collapse of a Spherical Cavity in a Viscous Incompressible Fluid,’ Proc. First U.S. Nat. Cong. Appl. Mech., A.S.M.E., 1952, pp. 822-825. 98. Eisenberg, Phillip, and Pond, Hartley L., ‘‘Water Tunnel Investigations of Steady- State Cavities,’? TMB Report 668, October 1948. 99. Birkhoff, Garrett, ‘‘Hydrodynamics,’’ Princeton University Press, 1950. 30. Gilbarg, David, and Serrin, James, ‘‘Free Boundaries and Jets in the Theory of Cavi- tation,’’ Jour. Math. and Phys., Vol. 29, No. 1, April 1950, pp. 1-12. 31. Serrin, James B., Jr., ‘‘Existence Theorems for Some Hydrodynamical Free Boundary Problems,’’ Jour. Rational Mech. and Analysis, Vol. 1, No. 1, January 1952. 32. Gilbarg, David, ‘‘Uniqueness of Axially Symmetric Flows with Free Boundaries,”’ Jour. Rational Mech. and Analysis, Vol. 1, No. 2, April 1952. 33. Garabedian, P.R., Lewy, H., and Schiffer, M., “Axially Symmetric Cavitational Flow,”’ Applied Math. and Statistics Lab., Tech. Report No. 10, Stanford Univ., April 25, 1952. 34. Tulin, M.P., ‘‘Steady Two-Dimensional Cavity Flows about Slender Bodies,’’? TMB Report 834, May 1953. 35. Birkhoff, G., Plesset, M., and Simmons, N., ‘‘Wall Effects in Cavity Flow - I and II,”’’ Quart. Jour. of Math., Pt. I: Vol. VIII, No. 2, July 1950, pp. 151-168; Pt. II: Vol. IX, No. 4, January 1952, pp. 413-421. 36. Ackeret, J., “Uber stationare Hohlwirbel,”’ Ingenieur-Archiv, Vol. 1, 1930, pp. 399-402. 37. Lerbs, H., “‘Photographische Aufnahmen der Kavitation im Randwirbel eines Trag- flugels,’’ Werft-Reederei-Hafen, Heft 1, 1 January 1932, pp. 1-2. 38. Binnie, A.M., ‘‘The Theory of Waves Traveling on the Core of a Swirling Liquid,’’ Proc. Roy. Soc. Lond., Ser. A, Vol. 205, No. 1083, 7 March 1951, pp. 530-540. 39. Thomson, W., ‘‘Vibrations of a Columnar Vortex,’’ Phil. Mag., Vol. 10, No. 5, 1880, pp. 155-168. : 40. Plesset, Milton S., and Shaffer, Philip A., ‘‘Drag in Cavitating Flow,’’ Rev. Mod. Phys., Vol. 20, No. 1, January 1948, pp. 228-231. 41. Brodetsky, S., ‘‘Discontinuous Fluid Motion Past Circular and Elliptic Cylinders,”’ Proc. Roy. Soc. Lond., Ser. A, Vol. 102, No. A718, February 1923, pp. 542-553. 24 42. Konstantinov, W.A.. ‘Influence of the Reynolds Number on the Separation (Cavitation) Flow,’ Izvestiia Akademii Nauk SSSR, Otdelenie Tekhnicheskikh Nauk, 1946, No. 10, pp. 1355-1373, TMB Translation 233, November 1950. 43. Gilbarg, David, ‘‘Unsteady Flows with Free Boundaries,”’ Zeit. fiir angewandte Math. und Phys., Vol. [II], 1952, pp. 34-42. 44. von Kérm4n, Theodor, ‘‘Accelerated Flow of an Incompressible Fluid with Wake Formation,’’ Annali di Matematica, Ser. 4, Tom. 29, Bologna, 1949. 25 INITIAL DISTRIBUTION Copies 19 Chief, Bureau of Ships, Technical Library (Code 327), for distribution: Technical Library Civilian Consultant to Chief (Code 106) Research and Development (Code 300) Applied Science (Code 370) Noise, Shock, and Vibration Branch (Code 371) Ship Design (Code 410) Preliminary Design and Ship Protection (Code 420) Preliminary Design (Code 421) Model Basin Liaison (Code 422) Submarines (Code 515) Minesweeping (Code 520) Torpedo Countermeasures (Code 5201) Propellers and Shafting (Code 554) Bee EPP HE dOP HEHE DOH oO 3 Chief, Bureau of Ordnance, Underwater Ordnance 2 Code Re6 1 Code Re3 3 Chief, Bureau of Aeronautics, for distribution: 2 Aero and Hydrodynamics Branch (DE-8) 1 Applied Mathematics Branch (RS-7) 7 Chief of Naval Research, for distribution: 3 Fluid Mechanics Branch (Code N 426) 1 Mathematics Sciences Division (Code 4380) 1 Mathematics Branch (Code 432) 1 Naval Sciences Division (Code 460) 1 Undersea Warfare Division (Code 466) 1 Director, Office of Naval Research, Branch Office, 346 Broadway, New York 13, N.Y. 1 Officer in Charge, Office of Naval Research, Branch Office, London, England, Navy 100, Fleet Post Office, New York, N.Y. 1 Director, Office of Naval Research, Branch Office, 1030 E. 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UI S}DeIJo IOAB] Ayepunog jo pus’ eduezNqiny JO e][01 94} JO SUOISSNOSIp JoljIvo ey} ‘snug, ‘eul;{no sty) ut sotdo, snoteA ey) UO Sseidoaid ey} ‘kyoyeutxoidde ‘eyeoIpul 07 pue sesseooid uONezIAO SNOTBA oY} JO eUI]]NO peljiun e1oW v yueseid O7 ope ueeq sey ydwey}e Uy GaI4ISSVTONN (‘pe *aoy) *syoa “sdy “out “d 6g ‘IIT "SG6T ouNp ‘uo{dUIYSEM *saoquesty diqjiyd Aq ‘uorqey1ABD jo uoUeAeIg pues Wstue “YOON 04) UO ‘BTL Wodey GNLG oO} wnpueppy uy - NOILVLIAVO 40 SOINVHOGUW GHL NO SSHUDOUd AO AMAUNS AUTUA V ‘PS “ydoy ulsed [OpoW Joj[ABL *M PIABG GOT-STLSN diyityd ‘siequest gy teXkey] Arepunog eouepnqiny, MOTH uOoI}e}IABD GOT-STLSN diyityd ‘diequestq jokey, Arepunog eoueTnginy, Mota UOTBIABD — i= rial od He “Il al v $s 6 “It Ayjoiq ore Ayisoosta pue AqipIqissesduoo jo syoejjo oy] Sutpnqout SONIAVO JUSISUBI] UO SJe}IIM SNolIVA Aq SIsAjeue [edTJEI0eT) JO S][Nsel yuo00% “pojueseid oie 4U9]U0 Ieyonu jo soUENTJUL 9y} UO sqysNoy} Jey}INj eulOS pu’ peMetAed SI UOTdeoUT UI S}OeJJe oTvos UO YIOM JUEDe1 EUIOS ‘pepue}Xxe oie UONdooUT UI SyOejJo JeART Ayepunog Jo puv eouenqing jo E01 ey} JO SUOISSNOSIp Jel]Ive ey} ‘snyg, “eul;jno sty} ut sotdoy snoleA ey) uo sseisoid 84} ‘kjeyeurtxoidde ‘ayvo1pul 0} pue Sesseooid uoIzezIABO SNOTIBA 984} jo oUulyjNO peljiun o1oul v queseid oj epeul ueeq sey ydweye uy GaIHISSV TONN (spo ‘Aey) ‘sjor ‘*sdy “pout -d 6g ‘TIT "EG6T oun ‘uojsUIYSEM ‘Sroquestq dipjiyd Aq ‘uoryez1ABD JO uotjueAeig pus wsIuE -yooW 4} UO ‘BIL wWodey ANLG o} Wnpueppy uy -NOILVLIAVO 40 SOINVHOUN AHL NO SSHYDOUd AO AWAUNS ATU V ‘SPR “jdey ‘ulseg [opoW sojAVT, *M plAed Ayerq exe AyIsoosta pue AqITIqIssesduod Jo s}9ejJe EY} SuIpnyout SONIABO JUEISUBI] UO S1E}1IM SNoLeA Aq sisA[eue [BoTJeI100q} JO S}[Nsol JueDeY “poyueseid ore 4Ue}UOD TeTONU Jo eoUENTJUL OYy UO sjyFnoy} eyjINJ owls pus peMeTAed ST UOdedU! Ul sjD9jJo eTBOS UO YIOM JUEDeI EUIOG “pepue;xe oie UOTdeouT UI S}0e}jJ0 eA] Ayepunog jo pue eoue]Nqain} Jo efO1 oy) JO SUOISSNOSIp Jel]Ive ey) ‘snyy, ‘eUury}NO siq} ul sotdo} snotweA ey} uo ssoidoid oy} ‘Kyeyeurxoidde ‘eyeoIpul 07 pue sesseooid uoryezIAvo snolieA ey} Jo our;No petjiun os10w v 4yueseid Oj opel useq sBYy ydweyqe UY GaldISSVTONN (‘pe ‘aoy) *sjoa ‘*sdy “pour -d 6g ‘TIT "eG6L ounp ‘uojsuTYSeM “Srequestgq diq[iyd Aq ‘uoryey1ABO Jo uoUeAeIg pue WsTue -YOoW 04} UO ‘BTL OdeYy ANALG %} wnpueppy uy - NOILVLIAVO 40 SOINVHOGN FHL NO SSAaNOUd AO AGAUNS ATU V "Gh8 “doy “UIs [OpOW JOTARL *M PIA ‘UOMVSYSOAUL Suinbea swe;qoid eyy JO UOyIUTJEp eEe[9 e1oW pues UOT}IUdODe;I ay) ul useq sBy sive yUED0I UT pel Sty) UT ssoidoid ysqyveId ey) sdeyiod yey} peAresqo SI 41 ‘ATUL “poezZVWUINS o1v SOTPOg FurzeIIAO Jo Sap sy) UO SUOT}BAIOSGO JUEDeI1 SWIOS ‘sUOT}BO -1jdde jeo1uyos} 10} ySe1e]UT eTqeJeptsuoo jo eq {{IM yoTyM ‘uIpNY, Aq A1cey} pozliveuly yuCdeI @ JO epeul ST UOUOU ‘Je{NOIVd U] “ped ere WOTZeIIABO Jo edAq SITY} UO SyIOM [VoI}0100q} queoei pue ‘spinbi] [vol Ul SeIAvO 9749S-Apeeys UO pepNfOUl O18 SyIeWeI 18yIIN| “pouotyuoU UOedSeAUI Sulinbei sweyqoid oy) JO UONIUIJep Ivejo eioW pue UOTyIUsODeI aq} url useq sey sieved qUedaI UT pjeYy Sty) Ul sseisoid yseyveId oy, sdeysed yey) peAdesqo st 41 ‘AT[eUI “peziewwNs exe setpog SuljyeqIAvo Jo Feip ayy uo suUOTAIesqoO yUued01 sUlOg “sUOeO -1jdde jwo1uyoe) 10} ySeie]UI a[qeiapIsuod jo eq [TIM YyoTyM ‘ulpng, Aq Aroey) poztaveul] yueoel B% JO epeul ST UOMUEUW ‘Ie[NOWIed U] *pozIO are UOTyeIIABO Jo edAy SIYy UO SYyIOM [BOTJeI00q3 queoei pus ‘spinbi] [vei Ul SeNIAvo 074)S-Apea}s UO papnpoUl o1v SYyIVWeI JoyZINGy “pouotyueu ‘UOIVINSEAUL Sutinbes swe,qosd OY} JO UOTIUTJEp IvEe[o eIOUI puw UOTyTUSODeI aq} Ul ueeq Sey sIvoA qUeDeI UT pjoly SIyy ul Sseisoid 4se}¢013 oy, sdeysod yey} peAdesqo St 41 ‘AT[VULY “poezWVWUNS ev SeIpoOg FurtyzezIAVo JO Fvap ayy Ve SUOMBAIJESGO yUEDeI SUIOG “suUOT]BO -1[dde Jwotuyoe) JOj ySe1eyUI O[Q¥JEpIsuod jo oq {IM Yorym ‘uipny, Aq Aroey) peziseeut] yueoeI % JO Opell SI UOTUOU! ‘ye;NoKYIed U] “pez E18 UOTWeIIABO Jo adAy STYy} UO SYIOM [BOTJEIDeYy queoe1 pus ‘spinbi] [eel Ul SetIAeo e7e%}s-Apee}S UO pepNpouUl ole SHIeUIE] JeyIINY “peuorjueu ‘UOIJSHSOAUI Sutiinboi sue,~qoid 9} JO UONIUIjep 1veTO Slow puB UO UdODeI 9yi Ul useq sey sive yUeDdI UT Plot] SIyy ul sseisoad 4seqy8018 984) sdeysed 4By peAdosqo St 41 ‘Aj[@Uly “poziyewuMs ove Sotpog SutyeqiAvo jo Feip ayy UO SUOTVAJOSGO JUEDeI eUIOG ‘“sUOTO -1jdde yeotuqoe} Joj ySeie}UT o[qvIepisuod Jo oq [TIM YoryAy ‘uipny, Aq Aloo peziaveuly] queoes @ JO ape SI UoNUew ‘IejNoNAed Ul “polo ere UOTYe}IABO Jo odAy STy} UO SYIOM [BOOI00q} qusoe1 pue ‘spinbi] [gel UI SeNIAZO o7e4s-Apeojs UO poepN{OUI 918 SHIVUOI JoyJINY| “pouoyueu BOI-S1L SN diy ‘siequestq jokey Arepunog eouetnqiny, Mold UOByIABO GOT-STLSN diytyd ‘diequestq jeXe, Arepunog eoue[nqiny, MOTH UOKNeIIABD Ayjotaq ore Aqtsoosta pue AqI{Iqisseiduoo Jo sqdejje oy} duIpnpoul SONU JUETSUBI) UO SIO}TIM SnoUeA Aq SIsA[eUe [VdTJeI00TY JO S][Nsel qus.ey “poyueseid ose qu0\UOD Te[ONU JO eOUENTJUT oy} UO Sqqsnoy}. 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UI sjDeJJo eTvOS UO YOM 4UEDe1 BIOS “pepuezxe o1B UOTdeoUT UI S}0e}jJO Joke] Ayepunog jo pue eoue]nqiny Jo efO1 ey) JO SUOISSNOSIp Jol]Ivo ey ‘snyJ, ‘eul;yNo sty) ur sotdo} snoteA 04) uo sseidoid oy} ‘yjoqyeutxoidde ‘ezzoIpul 0} pue sesseooid UONBYIAVD SNOTIBA OYy JO oUIT]NO peljIun e1ow B yuUeseld 0} OpeUl ueeq Sey qduieyye uy GaIAISSVTONN (‘pe ‘aoy) ‘sjor ‘*s3y “yout -d 6g ‘TIT "EGE oune ‘uo\{sUTYSeM *Saoques1q diypiyd Aq ‘uoryeqIABH jo uotueAeig pues wstus -YOOW ou} UO ‘BTL Wodey ANLd } wnpueppy uy - NOILVLIAVO dO SOINVHOUN BHL NO SSHUDOUd AO AMAUNS AGIUE V ‘GbR “ydey cuIseg [Opo| s0[AVL *M PIAeG ‘UONBISEAUT SuUINbes swerqosd oy Jo uOyTUIEp Aveo e10W pue UOTTUTODeI et) UT useq sBy sive queDdeI UT pjoy Sty} Ul ssoadod 4sqyvoId oy) sdeysed yum PeAIesqo SI 41 ‘A][RUIY “poztwUUNs ore selpog FurIyVIIAvO Jo Feap oy} Uo sUOMBAIESgo yUEDeI COS *suoneo -yjdde [woluyoo, 10} ySosezUT eTqvIepIsuoD Jo oq [JIM Yorym ‘urn, Aq Arey} peztaveury yUodeI B JO Opell ST UOTUOW ‘IB[NoWIVd UT *pejzIO e1e UOIYezTABO JO edAy SIq} UO SyIOM [BoeI00q} queded pus “spinbl] [eed Ur seIA¥ 67%4S-ApBo}s UO pepNpoUr crv SyIeUOI JEyIINY “peuonuoUt ‘uoTyesSeAUl JuInbes suetqosd ey) Jo UOIIUTJep Jve]o e1OW pue UuoTIUsOD0I 8y} UI Ueeq Sey siveA quedeI UT pel] Sty) UI SseaSoid 4sqyve—IF ey} sdeyied yey) peasesqo st 41 ‘AT[BUIY “poztwwUNs ore setpoq FuryeqtAvo Jo FeIp ey} UO SUOMAIESGO 4UeD0I eWIOg “sUOTRD -t[ddw yeotuyoe, Jj yseseqUI aTqesopIsuoo Jo eq THEM yoryay Surpny, Aq Arcey peztaeoury queses 8 JO epeul ST UoTUEW “IeynoNIed UT *payto exe UOTyeITAO Jo edéy sty) UO SYIOM [BO e100} queoei pus ‘spinbi] [wel UL setIAvo ayu9s-Apeaqs uO pepnfout ale SYIBUSI JOyIIN] + “peuonuew UONBANSEAUL Dulsinbes suleyqoid oy} Jo uONLUTEp 1ve]9d e1oUl pue uoTyIUdODe1 ey UI ueeq sey Sivek quedo, UI pel] Sty) ul ssoadoad 4sejeeaid 04) sdeysed 424) peasesgo st 41 ‘AT[BULY “pezlawuuns oe setpoq SuryeqtAvo jo deip oy} ve suoMeAIosqo 4uddeI eUOG *suolyeo -t[ddw ]zo1uyoe} Joy yse1e}Ut a{qesopIsuoo jo eq TIM yorqm ‘urpny, Aq Arce) pezteeuty quesel @ JO epBul ST UONUeU! “IejNoNsed UT “peyto ese UOTWeIIAGO Jo adAy Sty] UO SyIOM [89 MeI00q3 queoei pus ‘spinbi] [vor ul settAvo 03%4s-Apeojs uO pepnfour ev syzvUOI JoyyIN “peuoyuew ‘WONeSySeAUT Sutinbes sueyqoad oy} Jo UOMTUTJEp avoqo oJOUL pue uonyrusooe1 84} Ul Useq sey sIBed JUedAaI UT pjolyy sty) UL SseaSoid yseivols 04} sdeyied yey) peAsesgo st 41 ‘AT[VUL “poztwwUNs ere selpoq FuryeyIAvo Jo Feap oyy uo SUOTJVAJOSGO JUeDeI BUIOG “sUOTwO -tjdde [woruyoe) toy yse1eqUt e]qeJopIstioo Jo oq {TIM yorya: ‘urpng, Aq Arcey) pez iaveuly queses B JO epeul st UONUeU “IepNoIed Uy “pozlo e1v UOTyRIIABO Jo adAy sty] UO SyIOM [80010043 quedel pue ‘spnbyy] [eel ul setjIAvo e7uqs-Apeojs uo pepnfout ore SYIVWOI JOYANY ‘poeuonuow Cr eet Peet 4 nit