Some Hydrodynamic Aspects of Ship Maneuverability 



and thus the linearized side-force derivative, per unit drift angle, is given by 

 the integral 



r/2 



Y^ = ^ pgi. f sec^ |(6 sec^ 6 - 5) (P|- + Q|,) 



+ 2(sec2^ - l)(Ps'PxS" + Qs'QxS")! d^. 



where* 



Pg' + iQs' = JS'(x) exp(-ii^x sec 0) dx 

 P^g" + iQxs" - /''S"(x) exp(-ii^x sec 6*) dx . 



The integrals with respect to x are taken over the body length and 



S(x) = ^nr\x) 



is the sectional-area function of the submerged portion of the hull, with r(x) the 

 local radius. S'(x) and S"(x) denote the first and second derivatives respec- 

 tively. The practical value of these equations is limited by the fact that the 

 side-force integral diverges unless the ends of the body are cusped, i.e., both 

 the first and second derivatives of the sectional-area curve must vanish at the 

 ends. However it seems likely that this non -uniformity can be accounted for 

 (Tuck, 1964, Handelsman and Keller, 1966) and it would then be worthwhile to 

 generalize the side-force integral to more general and realistic hull forms with 

 full midbodies and a sharp trailing edge, to account analytically for the free- 

 surface effects which have been observed in experiments. 



Kochin's formulas have also been applied, but within the framework of thin- 

 ship theory, by Reinov and Shen Tszy-In (1964). They decompose the potential 

 into two terms, one due to thickness and one due to vorticity, and assume that 

 the vorticity can be approximated locally by the corresponding value for a wing 

 of low aspect-ratio in an infinite fluid. Integrals of the vortex distribution are 

 carried out only over the surface of the hull, with no account taken of the vortic- 

 ity in the wake. Unfortunately the results are presented graphically without suf- 

 ficient identification to permit comparison with other theoretical or experimen- 

 tal data. 



Viscous Effects 



In order to discuss the effects of viscosity we shall ignore the free surface, 

 or, more precisely, consider the simple image of the hull above the free surface 



*We note that the term with the factor 5 in this side -force integral may be iden- 

 tified with the wave resistance as determined from Michell's integral, and in- 

 deed there is a similarity in fornn between the complete side -force integral 

 and Michell's integral. 



221 



