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The Dynamics of (Zavitation RubblesThree regimes of liquid flow over a body are defined,

namely: (a) oncavitating flow; (6) cavitatin g flow wi th arelat ively sm all nu mb er of cavi tat ion bubbles i n t he f ieldof flow; a n d ( c ) cav itat ing flow with a single large cavityabout the body. Th e assumpt ion i s mad e that , for th esecond r egim e of flow, t h e pressure coefficient i n t h e flowfield is no different f rom t h a t in th e nonc avitating flow.On th i s bas is , t he equat ion of mot ion for the growth andcollapse of a cavitati on bubble con taining vapor i s derivedan d applied t o experimental observat ions on suc h bubbles.The l imitat ions of this equat ion of mot ion ar e poin tedout , a nd inc lude th e effect of th e f inite rate of evaporat iona nd condensation, an d compressibi l i ty of vapor a ndliquid. A brief discuss ion of th e role of "nuclei" in t h eliquid in t h e rate of format ion of cavitat ion bubbles isalso given.

INTRODUCTIONA ISTINCTIVE feature of the hydrodynamics of liquids isthe possibility of the coexistence of a vapor or gas phasewith the liquid phase. Such two-phase flow is usuallycalled cavitating flow, although it could as well be characterizedas liquid flow with boiling. Cavitating flow has great theoreticalinterest in addit ion to the hydrodynamics involved because of therelation of this flow condition to the physical-chemical proper-ties of the liquid. Th e practical significance of cavitat ion is ofcourse clear. The drag of submerged bodies moving through aliquid rises when cavitat ion appears; similarly, the efficiency ofpumps, turbines, and propellers drops with the development ofcavitation; and the damage which may be produced by cavita-tion in these devices is well known.

PASADENA, CALIF.effects, follows the same laws as are familiar in air flow. If nowK is made smaller, a state of flow is atta ined in which a relativelysmall number of bubbles appear near the boundary of the body.This state of flow will be taken a s the second regime of flow. If K ,is further reduced, the number of bubbles increases, unt il eventu-ally they merge into one large cavity which completely encloses aportion of the body. The state of flow with a single cavity abou tthe body is the third flow regime, and may be called cavity flow.A further reduction of K brings about only an increase in the sizeof the cavity. These three flow conditions are illustrated inFig. 1.

The particular flow problem discumed in this paper is the flowof a liquid (water) over a submerged body which will be con- FIG. 1 VIEWS HOWINGHE THREEREGIMESF FLOW(In the top view, the cavitation parameter K = 0.40; in the center K =sidered to be a t rest. If pa denotes the static pressure, and Vo 0.28; and in the bottom K = 0.18.)the uniform flow velocity of t he liquid a t a great distance fromthe body, then the general character of the flow in SO far as cavi- In the cavity-flow regime, the boundary of the cavity may betation is concerned is correlated with the cavitation parameter taken with reasonably good approximation t o be a surface of con-

stan t pressure and of constant flow speed. Th e pressure andPo - " .,. . . . . . . . . . . . .[I] velocity in the flow field are fundamentally different from those= - -

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JOURNAL OF APPLIED MECHANICS SEPTEMBER., 1949

is independent of po and Vo. The present assumption consistsin the calculation of the static pressure p in the second flowregime with the appropriate values of po and Va from the pres-sure coefficient C, determined for noncavitating flow. Thisassumption that the pressure coefficient is essentially the samejust before the first few cavitation bubbles appear as it is afterof course is subject to experimental verification, and the neces-sary experiments are planned for the high-speed water tunnel inthe Hydrodynamics Laboratory of the California Institute ofTechnology. For the present, this assumption is considered areasonable one. I t may be remarked also that as the numberof bubbles increases with decreasingK, the pressure field shouldgo over into that characteristic of the cavity-flow field; but, inthe transition, the pressure distribution over the body shouldshow small-scale spatial variations between the limits of the pres-sure field of noncavitating flow and that of the fully developedcavity flow.

In the present paper an equation of motion wil l be developedfor a cavitation bubble in a flow regime of the second type. Thisequation of motion will be applied to an analysis of experimentalobservations made in the high-speed water tunnel. Since a diticuasion of these experiments has been given recently by Knappand Hollander (3), only general feat- will be mentioned here.

The cavitation experiments were made with a 1.5-caliber ogivefor which the noncavitating pressure distribution had beenmeasured, Fig. 2. Runs were made with tunnel velocities VC

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280 JOURNAL OF APPLIED MECHANICS SEPTEMBER, 1949

leads to a thickness of t he order of 6 X in. On the basis ofthis estimate, the effect of the boundary layer will be neglected. ,It should be noted that the present measurements extend tominimum bubble sizes larger than this boundary-layer thickness

where L is the latent heat of evaporation. Thus for a bubblewhich grows to a maximum radius R, = 0.10 in. in 20 frames(T = 10-3 sec), the mass of vapor is 1.17 X lod6grams, and Q =

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PLESSET-THE DYNAMICS OF CAVITATION BUBBLES 281temperature change, estimated in this same way, is

AT (collapse) = 1deg C = 1.8 deg FI t is apparent tha t these temperature changes are insignifi-cant so th at one may take the bubble boundary to have a con-stant temperature, essentially the same as the water tempera-ture, and a constant value of p,.This conclusion cannot be accepted unconditionally, how-ever, since evaporation, or condensation, is a process whichtakes place a t a finite rate and, if this rate is not sufficientlyhigh to keep up with the rate of volume change of t he bubble,the vapor in th e bubble will behave more like a permanent than acondensable gas. This effect definitely limits the range of valid-

ity of the particular assumption, p , = const, toward the endof the collapse phase where the radial velocity R increasesrapidly. This trend toward rapid increase in the calculatedradial velocity is illustrated in Fig. 8. The ra te of evaporation.

--here V = ~ B T I ~ ~ Ms the desired velocity to be associatedwith the rate of the evaporation or condensation process. Forthe present problem, a t 22.2 C = 72 F, V is approximately 150mps = 500 fps. Hence unless R is appreciably less than thisvalue, one may not assume the constant value for p,. Duringthe collapse, when R approaches or exceeds this value, the col-lapse velocity would tend to be decreased because the vaporwill begin to show a rising pressure as it behaves like a permanentgas.A further effect of interest is the shock loss which will appearin the vapor when R reaches the gas acoustic velocity. Th e ef-fects of compressibility both in the vapor and in the liquid willnot be considered here, although the problems posed by them are

of great interest. A solution of these problems will be decisivefor th e quantitative determination of the high pressures arisingtoward the end of the bubble collapse, the regrowth or subse-quent oscillations of the bubble. and the sound energy radi-ated.A ir Content in Bubble and Role of N uclei i n Formation ofBubbles. The assumption has been made that any air con-tained in the bubble does not affect th e dynamics of t he bubblegrowth and collapse over the range of bubble sizes which havebeen measured and analyzed here. This assumption might beconsidered questionable since the water-tunnel flow experimentsare made with water containing an appreciable concentration ofdissolved air. Furthermore in the region of flow in which thebubble behavior is studied, the liquid pressure is considerablybelow the liquid static pressure po a t a distance from the model.Hence one should expect th at the water is supersaturated withdissolved air and that diffusion of air in to the bubble wouldtake place.

An analytic solution for such a diiusion problem has beencarried out by P. S. Epstein and the author, t he details of whichwill be presented elsewhere. For the present discussion it isnecessary only to say that the diffusion process is so slow that itdoes not contribute appreciably to any alteration in the aircontent of the bubble.As will be pointed out later, the initial air content of a bubbleis very small so that over th e range of bubble sizes which are ob-served and to which the present calculations have been applied,the effect of the air may be neglected. I t must be emphasized,however, th at the small mass of air in the bubble plays a mostimportant role in the initial stages of bubble growth, and alsomay enter in the h a 1 stages of the bubble collapse. Th e initialstages of bubble growth in which th e air content would be of

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282 JOURNAL O F APPLIED MECHANICS . SEPTEMBER, 1949or may be superheated considerably, without the formation ofcavities and bubbles.

Recently, Harvey (7) and subsequently Pease and Blinks (8)have shown experimentally that water saturated with air alsohas high tensile strength, provided it is denucleated. Harvey'smethod of denucleation of water saturated with air consists inputting the solution under high pressures (of the order of 10,000psi) for several minutes. The air nuclei are squeezed into solu-tion so that when the solution is brought back to atmosphericpressure it does not cavitate under the tensions which freelyproduced cavitation before the pressurization. These same pres-sure-treated air-water solutions also can be superheated by asmuch as 60 to 80 deg C without boiling.Presumably in ordinary untreated water the nuclei whichcontain gas and vapor are stabilized on small solid particles.The presence of a solid, or third phase, is indicated since the sur-face energy of a bubble bounded by a solid surface and a liquidsurface may be very low. Methods whereby the probable rate offormation of nuclei as determined by the surface energy may becalculated have been discussed by Becker and Doring (9) andKaischew and Stranski (10). The aim of the theory is to calcu-late the tensile strength of liquids, bu t it should be applicablealso to the statistics of the number of nuclei which should gromto macroscopic bubbles for given conditions of liquid temperatureand pressure.

CONCLUSIONThe main purpose of the present discussion, aside from touch-

ing upon problems which still await quantitative solution, hasbeen to point out the following: Liquid flow can be divided intothe three regimes mentioned; and, since the noncavitating regimeand the single-cavity regime may be considered to be on a quanti-tative basis, the main concern here has been a clarification of thesecond, or bubble, regime of flow. Also, it has been remarkedth at an interesting experiment would be the measurement of thepressure distribution over a body in this second regime of flow.It has been shown th at the macroscopic behavior of cavitationbubbles may be explained reasonably well by a fairly simpleequation.

Finally, it may be pointed out that the macroscopic behavior

of the bubbles formed in a boiling liquid may be considered aa en-tirely analogous to the cavitation bubbles more specifically con-sidered here. The growth of bubbles in a liquid has great interesta t present in the problem of increasing the heat transfer from aheated solid to a liquid.

The study was carried on in th e Hydrodynamics Laboratory ofthe California Ins tit ute of Technology. It forms a par t of theactivities of Contract NOrd-9612 which is jointly supported bythe Research and Development Division of the Bureau of Ord-nance and th e Fluid Mechanics Branch of t he Office of NavalResearch.Th e author wishes also to acknowledge th e assistance of Mr.I?. H. Brady and Mr. J. M. Green in th e numerical computations.

BIBLIOGRAPHY1 "Hydrodynamics," by H. Lamb, Dover Publications, NewYork,N. Y., sixth edition, 1945, chapt. 4, pp. 73-78.2 "Theoretical Hydrodynamics," by L. M. Milne-Thompson,Macmillan & Company, London, England, 1938, chapt. 13.3 "Laboratory Investigations of the Mechanism of Cavitation,"by R. T. Knapp and A. Hollander, Trans. ASAWE,vol. 70, 1948,p. 419.4 "Pressure Developed in a Liquid During the Collapse of aSpherical Cavity," by Lord Rayleigh, Philosophical Magazine, vol.34, 1917, pp. 94-98 (Collected Papers, vol. 6, pp. 504-507).5 "Hydrodynamics," by H. Lamb, Dover Publications. NewYork, N. Y., sixth edition, 1945, chapt. 5, art. 91s.6 "Properties of Ordinary Water-Substance," by N. E. Dorsey.Reinhold Publishing Corporation, New York, N. Y., 1940, table 250,pp. 575-576.7 "On Cavity Formation in Water," by E. Newton Harvey.W. D. McElroy, and A. H. Whiteley, Journal of Applied Physics,vol. 18, 1947, p. 162.8 "Cavitation From Solid Surfaces in the Absence of GasNuclei," by D. C. Pease and L. R. Blinks, Journal of Physical andColloid Chemistry, vol. 51, 1947,p. 556.9 "I