where 



and 



Numerical Solutions 



n = 



^\ /V0\" 



L/ \V 



( n H+ H+ 1 ) 



2/0 



(H-l)(H+2)u' 



S = (i -) d^= boundary -layer thickness, 



^ = distance measured normal to surface, 



u = resultant velocity in the boundary parallel to surface, 



= \ In L _ -!i\ j^ = boundary- layer momentum thickness, 



n is related to the local friction formula, i.e., 



%..'■ 



a(H) 



^\-" 



(4) 



The solution to these equations will lead to the determination of , the 

 boundary- layer momentum thickness, /3 the angle between the limiting stream- 

 lines at the wall and the external streamlines, and H the shape parameter. It 

 is clear that Q. , 3 and h are determined from two sets of variables. The first 

 set of quantities are dependent on the Reynolds number and are, for instance, n 

 and a(H). The second set of quantities are dependent on the "outer" flow quanti- 

 ties, which are determined by obtaining the potential flow on the ship surface. 

 These are dependent on the Froude number and the ship geometry, and are, for 

 instance, u , o , w as a function of s, m, x, z. 



By comparing the empirical relation of Ludwieg and Tillman [4] for the 

 local friction coefficient in a pressure gradient with Eq. (4) the following values 

 of a(H) and n are determined: 



a(H) = 0. 246 x 10 

 n = -0. 268 . 



(5) 



However, this relationship, when used to obtain c^ for flat plates, with H de- 

 termined from the comprehensive analysis of Landweber (3), gives poor agree- 

 ment with the Schoenherr friction curve at Reynolds number corresponding to 



1630 



