Charaoteristias of Ship Boundary Layers 



Since in a boundary layer w is small in comparison with u 

 and V, and derivatives with respect to a and P are small in conn- 

 parison with derivatives with respect to y, we are justified in omitting 

 the derivatives with respect to a and § in (19), (20) and (21), In a 

 thick boundary layer it may be necessary to retain the terms K_y 

 and K Y in the denominators of (19), (20), and (21), but we shall 

 neglect these terms in the present treatnnent. Thus the expressions 

 for the vorticity components in a boundary layer become 



e=-|^-V (25) 



ri = |H + K^a (26) 



; = K,v - KgU . (27) 



Near the wall the y derivatives are doininant so that the expression 

 for the vorticity remains that given by (22). Farther into the boundary 

 layer, however, the terms K v, K u, K u, and K.v may become 

 appreciable when the curvatures are large, as at tne bilges of a ship 

 form. 



When t and r\ are known, the corresponding values of u 

 and V, obtained by integrating the differential equations (25) and (26), 

 are given by 



u = e ^ \ Tie 3 dy (28) 



V = - e"*^^^!" ee'^^'^'dy (29) 



somewhat more simply than by the Biot-Savart law. 



We can now obtain the components of curl co in the boundary 

 layer by replacing u, v, w by t, r\, t, in the right members of (25), 

 (26), and (27). Thus we obtain 



curl CO . e, = - -^ - K^r| (30) 



curl CO • e2= 1^ +K3e (31) 



curl CO . 63= K,Ti - K^e (32) 



463 



