174 



i" STREAMLINE / L^^»_i J L J\ 



STREAMLINE 

 2'/? N0.3 



**wiO 



FIGURE 6. Crossflow profiles in V^ 



the boundary layer on the AFT hull fci, ^ r 



surface (GBT-125) . "^ ' " 



NO. 5 



•0.1 0.1 



O.I 



-O.I 



able for the velocity profiles of ship-like bodies 

 as far as streamwise components. 



Figure 5 shows crossflow profiles measured on 

 aft parts of a model. As easily observed, there 

 are some profiles which have reverse type (S-shaped) 

 profiles. For most of remaining parts r the cross- 

 flow angles are very small and do not show reverse 

 type profiles. Because Eq. (4) has only one inflec- 

 tion point, such S-shaped profiles can not be repre- 

 sented by it. 



To represent even reverse crossflows, more gen- 

 eral polynomial expressions are proposed [e.g., 

 Eichelbrenner (1973), Okuno (1977)]. However, they 

 require additional equations or boundary conditions 

 and it is reported they do not always yield improve- 

 ments [Okuno (1977) ] . This is because the cross- 

 flow does not always have such universal profiles 

 near the stern. 



On the other hand, the three-dimensional veloc- 

 ity profiles based on Coles' wall-wake law can be 

 represented by 



g , g are variable parameters, for wake parts, u^ 

 is the friction velocity, and k, B are constants. 



f given by Eq. (10) is called the wake function. 

 Figure 7 shows the existence of such parts in case 

 of ship-like bodies also. Velocity profiles 

 deviate from linearity when approaching the outer 

 edge of the boundary layer. Velocity profiles, 

 represented by Eqs. (8) and (9) , are compared with 

 measured profiles. Here parameters g-^ and g2 are 

 determined by the condition that q, equals Ug and 

 q2 equals zero at the boundary and u^ is determined 

 by a least-squares fit to the measured profiles. 

 The values of Clauser, 5.6 and 4.9, were used for 

 1/k and B respectively. Good reproductions are 

 examined except crossflow representations. 



As to crossflow profiles, the situation is not 

 much improved from Mager's model; reverse crossflow 

 observed in experiments can also not be represented 

 by the wall-wake law. The finite-difference method 

 may be a possible step toward representation of any 

 type of velocity profiles. 



^/ "e = ^ ^C 



^TC 



)cose + 



^i^i'f 



(8) 



Local Skin Friction 



V"e 



where 



and 



^0< 



^TC 



^)sin6+g2fi(f 



'TC, 



f ( -^ ) = - log ( — ^ ) + B , 



V K ^10 V 



fj( I ) = I [l-cos( Tr| )] , 



u^ = ( T^ / p) 



1/. 



(9) 



(10) 



(11) 



In the case of turbulent flow, most of the friction 

 is due to the turbulence (Reynolds' stress). For 

 this reason it is necessary to introduce additional 

 equations to determine it in closed form. 



Ludwieg and Tillmann's semi-empirical equation 

 for the skin friction [Ludwieg and Tillmann (1949)] 

 is most commonly used; it is 



T /po2 = 0.123xl0"°-^''^" 

 wj "^ e 



u e,, 



e 1 1 



-0.268 



(12) 



Because Eq. (12) is obtained from two-dimensional 

 experiments, the validity should be examined when 

 applied to three-dimensional flow. 



When Coles ' wall-wake law is employed for the 

 velocity profile, the skin friction can be deter- 



