Landwebev 



These results for only two points already indicate that neither of the 

 above formulas gives good agreement, although Young's seems to be 

 preferable. It is planned to continue the analysis of Pavamani's data 

 with the aims of representing his shear-stress data by an alternative 

 formula, and to coinpare his measurements with computed values of 

 the boundary- layer characteristics. 



V. SHIP BOUNDARY-LAYER PHENOMENA 



At a ship's bow certain streamlines of the outer flow pass 

 downwards along a side, turn around the bilge, and continue along 

 the underside of the hull. Because of the large curvature at the 

 bilges, the cross-flow angle in the boundary layer may become large 

 and the resulting secondary flow has been observed to roll- up into 

 a pair of so-called bilge vortices [ 22] , Clearly the small cross-flow 

 assumption is not suitable for treating this phenomenon. 



It has been observed that these bilge vortices can be eliminated 

 by attaching a large bulb to the bow [ 23] , A possible explanation of 

 this effect is that the curvature reversals as an outer streamline 

 passes from the bulb to the bow, and then around a bilge result in an 

 S-shaped velocity profile, i.e. , one in which the sign of the cross - 

 flow angle changes in passing from the outer limit of the boundary 

 layer to the wall. In any case, since bows are frequently designed 

 so that streamlines would undergo changes in the sign of the curva- 

 ture, S-shaped velocity profiles would occur, so that the assumption 

 of monotonically varying cross-flow angle, and in particular its fre- 

 quently assumed quadratic variation, would be improper. 



The surface wave profile along the hull affects the boundary 

 layer in two ways, as has been shown by Chow [ 6] , Climbing from 

 a wave trough to a crest is equivalent to passing through a region of 

 adverse pressure gradient. If the free-surface slope is large enough 

 and continues long enough, separation will ensue near the free sur- 

 face. Secondly, the curvature at a surface-wave crest along the hull 

 tends to generate a secondary flow. Chow [ 6] has attributed a second 

 zone of separation at some distance beneath the free surface to this 

 phenomenon. 



The conjectured mechanism of the effect of a surface wave 

 on a boundary layer was confirmed by Tzou [ 7] . He simulated the 

 free surface by a sinusoidal ceiling in a wind tunnel, and observed 

 and photographed the flow directions in the boundary layer of a 

 vertical ogival strut, as indicated by an array of fine threads sup- 

 ported at various distances from the wall. He also verified the effect 

 by solving the Navier- Stokes equations and the equation of continuity 

 numerically, by a combination of a finite-difference method together 

 with the Blasius solution for a flat plate, for a simplified model of 

 his experiment. These results indicate once again the unsuitability 

 of the small cross-flow assurnption for ship boundary layers. 



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