175 



SIHEAJtlNE NO.i 





^ 



FIGURE 7. Velocity profiles represented by Cole's 

 wall-wake law. 



obtained from Ludwieg-Tillmann' s formula, Eq. (12), 

 and those from friction velocity, Eq. (11) . For 

 estimations of the latter two values, measured 

 velocity profiles are invoked. Calculated results 

 are also shown here for later discussions. 



The values of Ludwieg-Tillmann' s formula produce 

 fairly good agreements with those directly measured, 

 which implies that Ludwieg-Tillmann' s expression 

 is also good for three-dimensional flow. 



Entrainment Equation 



In streamline coordinates, the continuity equation 

 is given by 



^(qih2) + 3^(q2hl) + hih2^3 = 0. (13) 

 12 ^ 



Integrating with respect to x. , from zero to 5 , 

 gives 



3 



hjdxi '■ h20X2 



= F 



<^-^l'^hl 



3h, 



jh^ 3Xj 



1 3Ue 



U h, 3x, 



el 1 



} 



where F is the entrainment function given by 



W- 



36 36 



1 



(14) 



(15) 



Equation (14) is often used as the third (auxiliary) 

 equation when the boundary layer calculation is 

 carried out by the integral method. Here, F should 

 also be given in someway in closed form. 



In two-dimensional flow, Eq. (14) is reduced to 



mined from the friction velocity. But it should 

 also be examined experimentally. 



In Figure 8, three kinds of experimental values 

 of skin friction are compared along streamline 

 Nos. 9, 11, and 18; directly measured values, those 



Head (1960) gave a relationship between F and 

 (6-6*)/6ii (SHg_5*) which was examined by two- 

 dimensional experiments . 



(16) 



♦ DIRECTLY MEASURED 



^ ESTIMATED FROM COLES' WALL -WAKE LAW 



o DO. BY LUDWIEG-TILLMANN'S EORMULA 



S I 



9 



3.0 



2.0 



3.0 



2.0 



R„=106 

 e 



STREAMLINE N0.11 



■ CALCULATED 



STREAMLINE NO. 18 



F.P. 



1 FIGURE 8. Comparisons of local 



A. P. skin friction (GBT-125). 



