145 



010 



0.08 



0.06 



0.04 



0.02 



0.2 



0.4 



0.6 



0.8 



1.0 



1.2 



1.4 



FIGURE 20. Measured distribu- 

 tions of mixing length for 

 afterbody 2 . 



tion will undergo rapid changes in basic forms. The 

 one known for certain is the empirical function for 

 mixing length. Therefore, it may be difficult to 

 compute the rate of change of the turbulence energy 

 or the extra rate of strain in the region. 



Fortunately the present measured distributions 

 of Reynolds stresses shown in Figures 14 and 15 are 

 quite similar in the outer region and differences 

 appear in the inner region where the turbulence is 

 reduced in intensity and more homogeneous. In such 

 an axisymmetric flow configuration, the character- 

 istic length scale is more closely related to the 

 entire turbulence annulus between the body surface 

 and the edge of the boundary layer rather than the 

 radial distance between the two. Therefore, we 

 propose that the mixing length of an axisymmetric 

 turbu-length boundary layer is proportional to the 

 square root of this area when the thickness in- 

 creases drastically at the stern-. 



^(^o + 



6)2 - 



In order to examine this simple hypothesis , the 

 present measured values of S,//I1q 



+ 6r)2 



"^o^ 



together with the data of Patel et al. (1974, 1977) 

 are shown in Figure 21. The solid line is the best 

 fit of the present data. The present values of £ 

 are slightly greater than those for Patel 's modified 

 spheroid (1974) and are slightly lower than those 

 for Patel's low-drag body (1977). The data in 

 Figure 21 support this simple hypothesis although 

 the data are quite scattered due to large varia- 

 tions of stern configurations and Reynolds number, 

 and probable measuring errors. 



The existing thin turbulent boundary-layer dif- 

 ferential methods can be applied to the forward 

 portion of the axisymmetric body up to the station 

 where the boundary layer thickness increases to 

 .about 20 percent of the body radius. Further down- 



stream, the apparent mixing length of the thick 

 axisymmetric stern boundary layer ({,) can be roughly 

 approximated by the mixing length for a thin flat 

 boundary layer ( 8, ) by 



(r + 6,)- 



3.336 



(16) 



which is the solid line of Figure 21. At the aft 

 end of the stern r^ is zero and the value of l/Z^ 

 is 1/3.33. This simple approximation of the mixing 

 length for thick axisymmetric stern turbulent bound- 

 ary layers can be incorporated into most existing 

 differential methods. As noted earlier, the mea- 

 sured axial velocities inside the thick boundary 

 layer (especially in the inner region) are smaller 

 than the computed values (Figures 8 and 10) . The 

 present CS method overestimates the magnitude of 

 eddy viscosity (Eq. 3) for the thick stern boundary 

 layer. While the mixing length approximations ob- 

 tained in the present investigation can be incorpo- 

 rated into the CS method to predict more accurately 

 the thick stern boundary- layer velocities, further 

 refinement of the theoretical methods is desirable. 



9. CONCLUSIONS 



In this paper, we have described recent experimental 

 investigations of the thick turbulent boundary lay- 

 ers on two axisymmetric sterns without shoulder flow 

 separation. A comprehensive set of boundary layer 

 measurements, including mean and turbulence veloc- 

 ity profiles and static pressure distributions, are 

 presented. Two major conclusions can be drawn: 



The Lighthill/Preston displacement body concept 

 has been proven experimentally to be an efficient 

 and accurate tool for treating the viscid and in- 



