Some Hydrodynamic Aspects of Ship Maneuverability 



' Vi-)jm(x) f^- vi-)h(x,t)| 



rit Bx / V3t 3 



where h(x,t) denotes the transverse displacement of the ship's centerline from 

 the original straight -ahead position. However, this approach does not produce 

 different results from the usual pseudo-steady state added mass, damping, and 

 restoring terms; its principal value is in showing that the pseudo-steady state 

 approach can be rationalized if the hull is sufficiently slender and if there is no 

 separation or shedding of vorticity along the length of the hull. 



Free-Surface Effects 



Ship hydrodynamicists need no reminder that the motion of a ship on the 

 surface of the water generates waves. Indeed the classical Kelvin wave system 

 generated by the steady motion of a ship is one of the most beautiful and intrigu- 

 ing phenomena of our field. Nor do we need to be reminded that the hydrody- 

 namic forces acting on the ship are influenced by the wave motion since wave 

 resistance is an obvious example of this fact. 



Michell's integral for the wave resistance of an idealized ship hull is so old 

 and well studied that it is surprising to find, with only two exceptions, no paral- 

 lel work in the case of the side force and moment on a yawed ship, even in the 

 simplest case of a steady drift angle. The force and moment on a yawed thin 

 ship or surface-piercing flat plate can be formulated in terms of a lifting- 

 surface integral equation, but the resulting kernel is complicated and no calcu- 

 lations have been attempted. However the following important qualitative con- 

 clusions can be established by this means: 



1. In the limit of low Froude numbers the kernel of the integral equation 

 tends to that associated with the ship plus its simple mirror image above the 

 free surface, corresponding to the lifting-surface problem of a flat wing of span 

 equal to twice the draft of the ship and chord equal to the length of the ship. 



2. In the limit of high Froude numbers the kernel of the integral equation 

 tends to that associated with the reversed image (negative angle of attack) of the 

 hull above the free surface, corresponding to the problem of a wing with a dis- 

 continuous asymmetrical twist. 



3. For sufficiently small aspect-ratio (i.e., small draft-length ratio) the 

 solution for finite Froude numbers tends to that associated with the low aspect- 

 ratio wing and its simple mirror image in an infinite fluid. Thus wave effects 

 should be small if the draft-length ratio is sufficiently small. 



This problem has been treated in some detail by Hu (1961). He expanded 

 the kernel of the integral equation in even powers of the Froude number thus 

 obtaining a sequence of iterative equations with the relatively simple kernel 

 corresponding to the zero Froude number limit. Subsequently he assumed that 

 the aspect ratio is small so that the integral equations can be inverted in closed 

 form. Computations on this basis show the side force and moment due to a 

 steady drift angle to be increasing functions of the Froude number. 



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