Tuck and Taylor 



The program has been tested by comparison with Flagg and 

 Newman's [ 1971] computations for a rectangular section, and gives 

 good agreement over the range of dimension of interest. For instance, 

 with a rectangle of total width 0,25 and a (submerged) draft 0. 1 in 

 water of depth 0.125, Flagg and Newman's [ 1971] computations give 

 C = 0. 598, while our program with 24 segments on the bottom of the 

 rectangle and 12 segnnents on each side gives C = 0.603. Although 

 this accuracy (1%) is already very good in the present application, 

 it can easily be and has been improved by use of a larger number of 

 segments, especially in the neighborhood of the corners. 



Another check is by means of asymptotic estimates for small 

 clearances (Taylor [1971]). A formula which is valid for arbitrary 

 sections, providing they have substantially vertical sides 2b units 

 apart and a substantially flat bottom c (« h,b) units from the water 

 bottom, is 



C=^. (6.9) 



For strictly rectangular sections, this formula may be improved by 

 estimation of the next term in an asymptotic expansion for small 

 c/h, giving 



C =:yi +lJ}iogA +lh _ ^ 



C TT ^ 4c TT ^ ' ^ ' 



For the rectangular section used as an example above, (6.9) gives 

 C = 0.625 while (6.10) gives C = 0.597. 



Indeed, it must be noted that rectangles are not a fair test for 

 the computer program, since the generating source strength becomes 

 infinite at the sharp corner. We should therefore expect far better 

 accuracy for smooth ship-like sections. For instance, the program 

 gives results with accuracies of better than 1% when applied to the 

 oval-shaped sections generated by a single isolated dipole in a channel 

 (Lamb [ 1932]). 



Figure 3 shows computations of C(x)/i for a Series 60, 

 block 0.80, tanker hull (Todd [ 1963]), with beam/draft ratio of 2.5 

 and length/beam ratio of 8.0, the ship length being Zi . The results 

 are for two depths of water only, with draft/depth ratios of 0.8 and 

 0.9. In neither of these cases is there a great deal of water beneath 

 the keel, but this is the interesting range, since it is necessary that 

 the clearance be relatively small to achieve significant flow blockage. 

 Thus at a draft /depth ratio of 0.4, the typical values of C/i are 

 already below 0. 125 over the whole length of the ship, which leads 

 to a maximum force (see the following section) less than a quarter 

 of that for the full -blocked situation. 



652 



