Sec. 47.10 



SHIP AND PROPELLER CAVITATION 



155 



used in place of the virtual P/D, and if the 

 expanded area A e replaces the developed area Ad . 

 W. H. Bowers, of the TMB staff, devised a 

 number of formulations whereby a quick approxi- 

 mation could be made by a ship designer of the 

 rate of propeller rotation beyond which the thrust 

 is affected by blade cavitation. One method of 

 rapidly solving the equation given hereunder was 

 a circular slide rule devised by Bowers and L. W. 

 Sprinkle in 1947. It is not possible to describe 

 the exact method here but the criterion employed 

 by Bowers took the form, expressed in the symbols 

 of this book: 



37,500(1 - Suf'^QiA + hn - hy) 



PD 



(47.ii) 



where n is in rpm and all other linear dimensions 

 are in ft. 



To illustrate the appUcation of this formula, 

 take the case of the propeller designed for the 

 ABC ship in Chap. 70 of Part 4, and shown in 

 Fig. 78.L. The necessary basic data for the trial 

 speed of 20.5 kt and for an assumed typical blade 

 section at 0.8fiMa» are as follows: 



Real-slip ratio, Sr (actually Skt), from 

 Fig. 78.Nb 0.238 



Atmospheric-pressure head Ha , assumed 33.0 ft 



Hydrostatic head at 12 o'clock blade 

 position, hn , equal to [26 — 10.5 — 

 (0.8)10], for 0.8i2M.x 7.5 ft 



Vapor-pressure head hr , assumed 0.8 ft 



Value of {Ha + hn — hv) = 



(33.0 + 7.5 - 0.8) 39.7 



Mean-width ratio, Cm/D 0.211 



Blade-thickness fraction to/D, from 



Fig. 78.L 0.049 ft 



Pitch P, assumed as 0.982Z) at O.SKm.. , 

 from Fig. 78.L 19.64 ft 



Diameter D 20.0 ft 



Substituting in Eq. (47.ii) 



37,500(1 - 0.238)'*''''" 



<-')(ii) 



(19.64)20 



= 14,399 



whereupon n in rpm is 119.7. This is well above 

 the expected rate of rotation of 97.2 rpm from 

 Sec. 70.26. 



47.10 Predicting Hub Cavitation and Hub 

 Vortexes or Swirl Cores. The phenomenon of 

 cavitation abaft a rotating screw-propeller hub, 

 and the production of hub vortexes or swirl cores, 

 are discussed and illustrated in Sec. 23.14 on pages 

 337-339 of Volume I. This is an important 

 feature of the behavior of high-speed vessels 

 because of the damaging effect of this cavitation 

 upon appendages which lie in its path. 



Within the period 1945-1955 it has been 

 possible to produce and to photograph cavitation 

 of this type in variable-pressure water tunnels, 

 circulating-water channels, and model basins. It 

 appears from observation of model propellers at 

 atmospheric pressure in the latter two types of 

 facility that the hub vortex is a rather unstable 

 affair, appearing and disappearing with apparently 

 no change in test conditions. There is reason to 

 believe that it is far more stable in the full scale. 



On the basis of the swirl-core theory of Vi§kovic 

 it should eventually be possible to predict the 

 presence of a swirl core and its approximate 

 diameter behind the taper end of the hub fairing 

 of a screw propeller. Assuming that it is possible to 

 impart to the water in contact with the outside 

 of the hub, in way of the blade, a tangential 

 velocity equal to that of the hub surface, the 

 resulting centrifugal force at a smaller radius R 

 may be balanced against the probable vapor and 

 gas pressure in the water. For propeller hubs not 

 deeply submerged in sea water this lower limit 

 may be taken as somewhere between 0.5 and 

 1.0 lb per sq in. The ambient pressure pa> is the 

 atmospheric pressure Pa existing at the time 

 plus the hydrostatic pressure Ph to the shaft axis, 

 measured to the actual water (wave) surface over 

 the wheel. 



The equation of motion of a water particle in 

 a vortex coil, whirling around a vortex core, is 



1 ap 

 pdR 



where U is the local velocity and R is the local 

 radius [Eisenberg, P., TMB Rep. 712, Jul 1950, 

 pp. 4-5; FHA, 1934, pp. 213-214]. From this 

 there may be derived the relationship 



p at center of vortex core = p„ — 





(47.iii) 



where F (capital gamma) is the circulation in the 

 vortex coil, surrounding the core, p„ is the ambient 

 pressure in the undisturbed liquid at the same 

 depth as the center of the core, and Rq is the 



