176 



Introducing an assumption that the entrainment 

 equation of three-dimensional flow is related ex- 

 clusively to the streamwise quantities, Cumpsty and 

 Head (1967) employed the Head's entrainment function 

 for three-dimensional boundary layer calculations. 

 This is of course open to criticism. 



In Figure 9, Head's entrainment function and 

 experimental values, obtained from Eq. (14) , invok- 

 ing measured velocity profiles, are compared. It 

 can be mentioned that Head's function gives rather 



good mean lines both in relation to H 



6-6 



- H and 



F-H5_gj. The values of Hg_jj are not fairly related 

 to H in the fore part, where laminar flow may still 

 exist, and neither F to H|5_|5, in the aft part. The 

 former does not seriously effect F. We should bear 

 in mind here that the determination of boundary 

 layer thicknesses is not clear in the three- 

 dimensional case and accurate estimations of their 

 derivatives are very difficult. 



Himeno and Tanaka (1973) used the moment of 

 momentum equation as the third equation instead. 

 In this case, assumptions for the Reynolds' stress 

 are also required and significant improvements are 

 not always found. 



Summarizing the above discussions it can be 

 safely concluded that the integral method, where 

 either Mager's model or the wall-wake law is used 

 for velocity profile, Ludwieg-Tillmann' s equation 

 for skin friction, and entrainment equation for 

 auxiliary equation, is expected to yield meaningful 

 results. Moreover, it can be also pointed out that 

 improvements can be attained when the second order 

 approximation for the static pressure is taken into 

 account near the stern. However, in the region 

 where reverse crossflows or large crossflow angles 

 exist, although the boundary layer assOmption is 

 not violated, the integral method is no more 

 available. 



35 +3^ + (H+2)u 35 - Ki(eii-e22) 



wl e 



(17) 



9921 ^ 3622 , 2621 ^"e , 6ll 9"e 612, 

 8S 3ri Ug 3n Ue 3n 611 



2K1621 = T^2 / P"e ' 



(18) 



where 6jj, 612/ 021' ^ri'i 622 are momentum thickness 

 parameters defined by 



"e ^11 = ^ qi(ui-qi)dc , 



Up 012 = / q2CUi-qi)dC , 



Up 921 = /'^qi(Vi-q2)dt; , 

 = 



U2 622 = /'^q2(Vi-q2)di; 

 ^ 



(19) 



The entrainment equation is employed as the third 

 equation; 



_|(5-6!) - ^6! = F - (6-6!) (-K1+ 4 jf-) . (20) 



For the function F, the relation of Head is used, 

 which has already been examined. 



If Mager's velocity profiles are employed here, 

 boundary layer thickness parameters are given using 

 Oil, H and 3 , 



BOUNDARY LAYER CALCULATIONS 



According to the preceding conclusions , boundary 

 layer calculations were carried out by the integral 

 method and compared with experimental results. 



Basic Equations and Auxiliary Equations 



The integrated boundary layer equations are given 

 in streamline coordinates by 



621 = eiiE(H)tanB , 822 = eiiC(H)tan^6 

 612 = 9iiJ(H)tan6 , 62 = 6iiD{H)tan6 , 

 6-6* = 6iiN(H) , 



where 



C{H) 



24 



D(H) 



(H-1) (H+2) (H+3) (H+4) 



16H 



"(H-1) (H+3) (H+5) ' 



(21) 



15 _ 



H,.c 



.S.S.9 



5 - 



FIGURE 9. Comparison of Head's entrainment 

 function with experiments. 



0.05 



"H 



,5-S.ij 



STREAMLINE 



H.-tf 



