Ship Maneuvering in Deep and Confined Waters 



is far from say a simple axisymmetric cone. The modern half-spade 

 rudder on a fixed horn (the Mariner -type) is a hybrid of the all- 

 movable and the flapped types , and other comnnon forms all have 

 their special characteristics. The procedure here adopted is not 

 a substitute for the detailed calculations necessary for a certain 

 project design, but it will furnish a good estimate of control forces 

 and make possible the extended use of model results referred to 

 above. 



The Rudder or "Control" Derivatives 



It will be assumed that for each rudder configuration may be 

 defined "equivalent" values of rudder area, rudder aspect ratio, 

 rudder angle and rudder advance velocity. 



A detailed study of the velocity field in the slipstream of a 

 propelled tanker model and of the pressure distribution over the 

 rectangiilar rudder fitted to this model was reported by Lotveit, [ 62] . 

 The distorsion of the spanwise loading due to slip- stream rotation 

 was clearly demonstrated, but the diagrams did not indicate any 

 definite influence of the rudder image in the hull and free surface; 

 the gap distance from top of rudder to stern profile was some 12 per 

 cent of rudder height. Straightforward calculations of rudder lift 

 from known relations of lift curve slope versus geometric aspect 

 ratio and an average advance velocity based on the simple momentum 

 theory proved to give good agreement with the rudder forces measured 

 by a force balance or integrated fronn the pressure field. 



Unfortunately in this case no simultaneous measurements 

 were made of the total hull-and-rudder forces, and there is still a 

 lack of such data for normal surface ship forms. However, already 

 from the old experiments by Baker and Bottomley [ 63] it was seen 

 that the total force due to rudder deflection was increased by some 

 40 per cent in presence of a deep cruiser stern close above the 

 rudder, and that a third of the total force then was carried by the 

 hull. 



Let b be the height of the rudder at the stock, or the higher 

 value forward of it, and let a be the depth to top of rudder at the 

 same station. With a projected area A^ of the rudder the aspect 

 ratio of rudder + plane image is equal to 7f = 2b /A^, The lift 

 curve slope a-x is taken from the theoretical curve derived from the 

 Weissinger theory [ 64] , or from empirical curves available. 



The geometrical aspect ratio usually is of the order of 1.5, 

 i.e. the rudder is not a low-aspect- ratio fin, but it seems still to be 

 possible to make use of the results for wing-body interferences 

 applicable to such fins. In particular, the ratio of the lift on a rigid 



combination of a wing and a cylindrical central body, Laa , to the 

 lift of the abridged wing alone, Lg , is si 



> imply given by 



857 



