Sec. 41.9 



GENERAL LIQUID-FLOW FORMULAS 



Resultant Velocitij at Point P in j<{^a2^ Ship Speed U, 



Plane of Poqe ' ^ i 



Fig. 41.1 Construction of 3-Diml Ovoid by Inserting 

 Single 3-Diml Source in Uniform Stream 



that in Fig. 41.1 or Fig. 67. H, formed by placing 

 a single 3-diml source in a uniform stream, 

 approaches as an asymptote a limiting value of 

 2Rq at an infinite distance downstream. 



As an example of the use of the formulas, 

 suppose that it is desired to develop the coordi- 

 nates of an ovoid shape for the bulb bow of the 

 ABC ship of Part 4, resembling the ovoid for 

 which a fore-and-aft section is drawn in Fig. 67. H. 

 Assume that the stream velocity U„ is equal to 

 the designed ship speed of 20.5 kt or 34.62 ft per 

 sec, and that at 10 ft abaft the nose it is desired 

 that the ovoid shape have a transverse radius of 

 6.5 ft. 



Fig. 41.1 is a longitudinal section through the 

 axisymmetric ovoid of this example, indicating 

 the initial dimensions given and including the 

 ovoid shape derived by the methods described 

 here. 



The equation of the ovoid, from Eq. (41.xxxviii), 



R' = 



2m(I - cos B) 

 C/„ sin^ d 



R = 



sin 6 



4 



' 2m(l - cos e) 



23 



To satisfy the condition that the ovoid is to have 

 a transverse "base" radius Rb of 6.5 ft, at a 

 distance of 10 ft abaft its nose. 



and 



Yj Rb cos 63 — 10 



Rb sin Ob = 6.5 



where Rb and 63 are the coordinates of the rim 

 of the ovoid at its base. Substituting in the 

 foregoing for Rb in terms of 6b and m, gives 



1 



tan 6b 



2m(l - cos 61b) 



(7. 



= 10 



and 



4 



2m(l — cos Qb) 



= 6.5 



From the second equation, 



42.25C7, 



2(1 - cos Bb) 



Substituting this expression for m in the first 

 equation gives 



4.596 



6.5 



Vl - cos 6b tan 6b 



= 10 



Solving for 6b by trial and error gives 6b = 135 

 deg. Then 



42.25(7. 



2(1 - cos 6b) 

 (42.25) (34.62) 



= 428.4 ft' per sec. 



(2) (1.707) 

 The equation of the desired ovoid is 

 (2)(428.4)(1 - cos 6) _ 24.75(1 



R' = 



cos 6) 



34.62 sin' 6 



sm" 6 



The distance of the nose of the ovoid from the 

 coordinate origin is 



Knowing the value of the 3-diml source strength 

 m, the radial and tangential velocity components 

 for any point P in the field, beyond the ovoid 

 surface, at a radius R from the source and an 

 angle 6 from the axis, are found by substituting 

 the proper values of m, R, cos 6, and sin 6 in 

 Eqs. (41.xxxvi) and (41.xxxvii), with the fixed 

 value of [/co . Combining these radial and axial 

 components vectorially gives the velocity and 



