10 



HVDRODVN \MICS IN SHIP DESIGN 



Sec. -t 1.3 



Multiplying this by tlu- luilcr numher E, = 

 —0.3455 gives -0.223 psi ns the - ^p to be 

 expect<Kl at the corresponding orifice position on 

 tl>e model rudder. The submergence of the orifice, 

 witli tlie nuKJel at rest, is S/2o = 0.32 ft, and the 

 hydrostatic pressure there is 3.404/25 = 0.139 

 psi. Hence, from the equaUty p — p, = —^p, 

 (14.69 - y) - (14.G9 + 0.139) = -0.223, as 

 befor«, from which y = (0.223 - 0.139) = 0.084 

 psi below atmospheric pressure at the orifice. 

 This is tlie pressure that should be registered by 

 the pressure indicator on the model, on the basis 

 that the flow is djniamically similar. 



Another way of expressing the state of pressure 

 at the orifice P on the ship (or on the model) is 

 to say that it is equivalent to — 0.3455g. Assuming 

 that the test medium is the same, and that the 

 nature of the dynamic flow does not change (it 

 would change if cavitation set in on the ship but 

 not on the model), this pressure coefficient is 

 independent of the stream velocity. If a different 

 test medium is used, of a greater mass density in 

 order to obtain greater pressure differences, the 

 value of the pressure coefficient E„ = — 0.3455g 

 would still be the same, for dj'^namically similar 

 flow. In the same way, tests to determine the 

 pressure coefiicients over a series of points along 

 the section contour of a model rudder, mounted 

 by itself, could be made in air or they could be 

 run in mercury, whichever was most convenient. 



It is customarj', when tests are run in a different 

 medium, to set the speed at such a value that the 

 dynamic pressure of the test is the same numeri- 

 cally as the dynamic pressure in the medium in 

 which the full-scale body is to run. As the mass 

 density of mercury is about 13.60 times that of 

 fresh water, the velocity-squared term C/J would 

 have to be decreased by this ratio to keep the 

 dynamic pressure the same. This w ould involve 

 a red uction in velocity to a value of V2, 399/ 13. 60 

 = V 170.4 = 13.28 ft per sec, corresponding to 

 a ship speed of 7.86 kt for mercury instead of 

 29 kt for water. For a scale ratio of 25, and a 

 corresponding speed ratio of 5, the test speed for 

 the model would be 7.80/5 = 1.57 kt. 



If, as maj' be expected, the pressure coefficient 

 changes with rudder angle {(delta) and with the 

 thickness ratio t.x/c of the section, it is convenient 

 for teating and for design purposes to plot the 

 prciwure CKcflicicnt for any desired point or points 

 on a basis of rudder angle and of thickness ratio. 

 A Himilnr procedure i.s followed for other variables. 



If the liability of cavitation nrima in a study 



of the rudder section at the point 1', the captation 

 number ff(sigma) for the flow at that level is 

 readily calculated from the expression 



= P--« _E: 



(2.xix) 



Here the ambient pressure p. is 3.464 psi gage 

 or 14.69 -f- 3.464 = 18.15 psi absolute. The 

 vapor pressure e of fresh water at 59 deg F, with 

 a small amount of dissolved air, may be taken 

 from Table 47.a as about 0.25 psi absolute. The 

 dynamic pressure q is the same as before, 16.15 

 psi. The cavitation index is now 



_E= 



i: 7' 



18.15 - 0.25 

 16.15 



= 1.108 



This index is a measure of the pressure avail- 

 able to create a gradient which will (or will not) 

 cause the water to follow the rudder section. 

 Since it exceeds numerically the negative-pressure 

 coefficient —Ap/q = 0.3455 at the orifice P, 

 there is more than enough pressure available to 

 create a gradient which \vill turn the water in 

 toward the rudder. No cavitation is taking place 

 therefore at the point P, nor is any to be expected 

 at somewhat higher speeds. 



A look at pressure diagram 2 of Fig. 41.B, 

 plotted to scale in psi absolute, explains why this 

 is so. The numerator p — p^ — A73 in the pressure- 

 coefficient expression is represented by the 

 negative distance from P to H, or 12.57 — 18.15 = 

 — 5.58 psi. The terra p„ — e in the numerator of 

 the expression for the cavitation number is 

 represented by the distance from H to E. Cavi- 

 tation is not to be expected until the absolute 

 pressure at P drops to E, at which time both 

 numerators would be equal. Therefore, as long as 

 p — p„ is smaller numerically for a given speed 

 than p„ — c, cavitation does not occur at that 

 speed. This is also evident directly from the fact 

 that the absolute pressure p at the orifice P, 

 14.69 — 2.12 = 12.57 psi absolute, is far above 

 the vapor pressure e of the water, taken as 0.25 

 psi absolute. 



Unless one is working constantly in this field, 

 the calculation and use of pressure cix-fficicnts and 

 cavitation numbers can become confusing and 

 exasperating. It is recommended that, in these 

 cases, the calculations be supplemented by graphic 

 diagrams drawn ap|)roximatoly to scale, cor- 

 responding to those at 2 nud 3 in I'^ig. 11, B. 



