Sec. 41.4 



GENERAL LIQUID-FLOW FORMULAS 



11 



It is also a help to remember that the -pressure 

 coefficient E„ is a function of body shape, because it 

 embodies the pressure p which is found or meas- 

 ured on a body under given flow conditions. The 

 cavitation number is a function of the flow conditions 

 and of the liquid in the stream; this can be re- 

 membered because it embodies the vapor pressure 

 e. The critical cavitation number (tcr occurs 

 when the cavitation number of the flow drops 

 to the same numerical value as the pressure 

 coefficient of the body at the point of lowest 

 absolute pressure on the body. 



It is customary in some quarters, when con- 

 venience dictates the change, to derive the Euler 

 number E„ and the cavitation number a by using 

 values of head instead of pressure. The expression 

 for the former then becomes 



-En = 



h - h^ 

 2g 



(41. i) 



where Ul/(2g) is the dynamic or velocity head 

 in the undisturbed liquid. With a vapor-pressure 

 head hv , the cavitation number is 



h„ — fey 



29 



(41. ii) 



Employing the same data as for the preceding 

 examples, and taking g as 32.174 ft per sec^, the 

 dynamic or velocity head is 



hu = 



2g 



2,399 



2(32.174) 



= 37.28 ft. 



The measured gage pressure at the orifice P 

 of 2.12 psi below atmospheric corresponds to a 

 negative head of -2.12/0.433 = -4.896 ft. The 

 pressure coefficient is then: 



Ap 



g 



-8.00 - 4.896 

 37.28 



= —0.346, as before. 



The head corresponding to the atmospheric 

 pressure of 14.69 psi is 14.69/0.433 = 33.93 ft. 

 The cavitation index a is based upon a vapor 

 head of 0.25/0.433 or 0.57 ft of water. The 

 index is then 



h^ 



2g 



(33.93 -f 8.0) - 0..57 

 37.28 



= 1.109 



This is equal to the value previously derived. 



41.4 The Froude Number and the Taylor 

 Quotient. If it is assumed that at an excessive 



trim by the bow, due possibly to damage, the 

 topmost rudder sections in Fig. 41. A lie at the 

 surface of the water instead of below it, then 

 wavemaking and gravity effects come into play 

 and the Froude number is appHcable. Here 



Assuming the length of these rudder sections 

 as 14.6 ft, and the reduced speed as 12 kt, or 

 20.27 ft per sec, the Froude number for the rudder 

 only is 



20.27 



F„ = 



V32. 174(14.6) 



0.935 



The corresponding Taylor quotient for the 

 rudder only, based upon speed in kt and length in 

 ft, is 



V 12 



r„ = 



VL \/14.6 



= 3.14 



The Froude number F„ and the Taylor quotient 

 Tj are related to each other by the constant ratio 



„ 1.6889 .„. 

 ^ » = — ^ (-f J 



and 



T = ^ (F) 

 1.6889^ "'^ 



where 1 kt is 1.6889 ft per sec. If g is 32.174 ft 

 per sec', as in the example given, then F„ = 

 0.2978r, and T, = 3.358F„ . For mental calcula- 

 tions or rough approximations, F„ may be taken 

 as 0.3T, and T, as (10/3)F„ . 



There are certain applications, such as in 

 planing craft, in which the conventional Froude 

 number is modified by using the beam B as the 

 length dimension rather than the length L, or in 

 which ¥^'^ is used for L as the length dimension. 

 The derivation of the numerical values for the 

 modified F^ and Fv numbers is obvious from the 

 corresponding expressions 



F, = 



V 



and Fy = 



V 



VgV 



There are set down in Table 41. b a series of 

 Froude numbers F„ , covering a range of speed 

 from 2 through 100 ft per sec, equivalent to 1.18 

 through 59.21 kt, and embracing a range of 

 model and ship lengths from 5 through 1000 ft. 

 A first approximation to the Froude number for 

 a given speed and length, based upon a standard 

 g-value of 32.174 ft per sec', is obtained by 

 inspection from this table. More accurate values 

 are calculated by using the formulas in preceding 

 paragraphs of this section. 



