Gadd 



Fig. 14 - Mathematical models of viscous effects 



waterlines and a linearly growing boundary layer with a constant-thickness 

 wake. If however X is close to L, as in Fig. 14b, the boundary-layer thickness 

 is assumed to be negligible until close to the stern, where a wedge-shaped re- 

 gion of separation occurs followed by a constant -thickness wake. It may be 

 shown that the theoretical wave resistance corresponding to this source distri- 

 bution is given by 



'W - 1 



R« 



= ^ r [l- exp(- KqD sec^e*)] cos^ G((9) dfi" , 



where 



G(0) = a S^f— b - ea [1 - cos (KgX sec 0)] 



+ eb sin (KqX sec 0) + e^ {l - cos [KqCL-X) sec 6]} , 



2 cos 9 

 = 1 + cos (KgL sec 9) ^r-r sin (KgL sec 9) 



and 



sin (KqL sec 0) 



2 cos 



[1 - cos (KqL sec e*)] 



Thus if X= 0, as in Fig. 14a, the e terms in G(^) vanish, and the wave resist- 

 ance is simply a linear combination of the inviscid resistance with - = and that 

 due to a uniform source distribution, as in Fig. 6. There is no interaction be- 

 tween the two components. To obtain the right kind of displacement thickness at 

 the stern we need e in the region of 0.003/A, and since this enters only as the 

 square it is likely to make only a negligible contribution to the wave resistance. 

 On the other hand if X is close to L, say 0.9L, e must be much bigger, say 

 0.03/A, if an appreciable growth of displacement thickness within the separated 

 region is to take place. Then, since e now enters in its first power, we see that 

 there is likely to be a much greater effect on wave resistance. Moreover this 

 latter effect is approximately independent of the small distance t (= L- X) from 

 the stern at which separation takes place, provided 1 1 is kept constant. Since 

 et defines the displacement thickness at the stern, or perhaps rather that part 

 of it which is due to the rapid growth of displacement thickness near the stern, 

 it could be determined experimentally fairly easily. Thus a simple method of 

 correcting theoretical wavemaking calculations for viscous effects may be to 

 locate concentrated sources of the proper strength at the stern. Alternatively 



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