Ship Maneuvering in Deep and Confined Waters 



friction line and a form factor, characteristic for the super- 

 velocities along the hull. This resistance now may be written 

 [X^]^-(,-Qp= 2 XyyU^, where u is the forward speed of the ship. In 

 confined waters there are additional supervelocities , the effect of 

 which is equivalent to a back- flow along the hull and waterway 

 bottonm, where another boundary layer is generated. The two bound- 

 ary layers will reduce the effective under-keel clearance, which 

 tend to increase the trim by stern. Separation and unsymmetrical 

 eddy -making within the boundary layers may initiate yawing ten- 

 dencies in straight running, or change the behaviour of the ship in 

 manoeuvres , 



Graff has suggested to consider part of the mean back flow, 

 AU5, to be due to the lateral restriction, and the other part, '^U^j 

 to be due to the finite depth, [83] , In normal applications AU is 

 small compared to u, so that 



X^ = iX,V'(l + i4Ub)(l ^^^) = ix:;,u2(l + K,)(l + KJ (9.1) 



The effects of a plane bottom at distance h below the ship 

 waterline and a pair of parallel vertical walls, each one at distance 

 W from the ship centreline, are those produced b^ an Infinite array 

 of Image bodies with spaclngs equal to 2h and 2W respectively. 

 At the double-body ship centreline the lateral perturbation velocities 

 cancel whereas the axial components add together. (This simple 

 concept Is not valid for W or h small compared to B or T, In 

 which case additional doublet distributions are required to prevent 

 a deformation of the body contour.) Graff choose to calculate an 

 approximate value of Kjj for an elliptic cylinder, extending from 

 the surface down to the bottom and having a beam given by the three- 

 dimensional form displacement. (Thus Ku Is dependent on canal 

 depth, although the fined calculation Is purely two-dimensional.) 

 For the calculation of Ku he used an equivalent spheroid and results 

 for supervelocities earlier published by Kirch, [ 84] . His final 

 results are given In graphs and compared with model measurements, 

 which confirm that this method offers acceptable values of resistance 

 allowances for moderate confinements. It Is thereby also possible 

 to define a suitable form of resistance derivatives to be evaluated 

 from model experiments from case to case. 



In particular, a limited re-analysls of some of the data 

 given by Graff Indicates that the resistance Increase In shallow 

 water will be proportional to the Increase of an under-keel clearance 

 parameter t, = T/(h-T). Further analysis of the results for slnkage 

 In shallow water according to Tuck's theory are likewise In favour 

 of the use of this parameter. (See below.) 



In waterways severely restricted In width as well as In depth 

 the Increase of resistance Is a complex function of blockage conditions, 



869 



