139 



0.2 



04 



0.6 



0.8 



1.0 



FIGURE 11. Measured mean axial velocity distributions 

 across near wake of afterbody 2 . 



r +6 ^ 



n = 



1 - 



u (r) 



X 



U (r) 



u (r) 



X 



U (r) 



X 



rdr 



(13) 



The measured and computed values of 6* and &^ are 

 shown in Figure 12 for Afterbody 1 and in Figure 

 13 for Afterbody 2. The measured values of 6f and 

 6 J- for X/L > 0.90 are slightly larger than the 

 computed values for both bodies. 



The transverse curvatures of the boundary-layer 

 flow with respect to the body radius, (r^ + 6*)/rQ 

 and (rQ + (5j,)/rQ, are also shown in Figures 12 and 

 13. A drastic increase of the values of {r^ + 6*)/r_ 

 and (r^ + 6j.)/r^ occurs at X/L = 0.9, indicating the 

 important effect of transverse curvature on the 

 stern. The longitudinal curvature of the body is 

 denoted by K^ = (d^ro/dx^) [l + (dr^/dx^) ]-3/2 ^^^ 

 the longitudinal curvature of the displacement body 

 is denoted by K^ = (d^r^j/dx^) [1 + (dr^/dx) 2] "3/2. 

 A positive sign for K^ or K^ indicates concave sur- 

 face. The values of K^^r and K^jr are shown 

 in Figures 12 and 13. There is a significant dif- 

 ference between K^ and K, in the thick boundary 

 layer region. In each case, the curvature of the 

 displacement body is convex up to X/L = 0.92, then 

 changes to concave and remains concave throughout 

 the entire thick boundary-layer and near-wake region. 

 The curvature of the body surface is convex up to 

 X/L =0.96. As already shown in Figures 4 and 6, 

 the measured distributions of static pressure and 



hence the curvatures of the mean streamlines are 

 much more closely related to the displacement body 

 than to the actual body. The magnitudes of the 

 maximum concave and convex radii of curvature of 

 the displacement bodies are estimated to be 8 rj^^j^ 

 and 30 r^ax fo^ Afterbody 1 (Figure 12) and 7 rj^ax 

 and 8 r^iax fo^ Afterbody 2 (Figure 13) , respectively. 

 The magnitudes of the radii of curvature of the 

 mean streamlines outside of the displacement body 

 are expected to be larger than 10 r^ax- 



7. 



MEASURED TURBULENCE CHARACTERISTICS 



The cross-wire probe was used to measure the tur- 

 bulence characteristics in the thick boundary layer. 

 The measured Reynolds stresses and the measured 

 mean velocity profiles were used to obtain eddy 

 viscosity and mixing length. 



Measured Reynolds Stresses 



The turbulence characteristics in the thick boundary 

 layer can be represented by the distributions of 



u'2, ^ " 



Reynolds stresses, namely, -u' v 



and 



w'' 



where u' 



and w' are the turbulence fluc- 



tuations in the axial, radial, and azimuthal direc- 

 tions, respectively. Figures 14 and 15 show the 

 measured distribution of Reynolds stress -u' v'/Uq^ 

 and three components of turbulence intensity at 

 several axial locations along the two afterbodies. 

 In general, for a given location, the intensity of 

 the axial turbulence-velocity component has the 

 highest value and the intensity of the radial com- 

 ponent has the smallest value. The degree of 

 anisotropy decreases as the stern boundary layer be- 

 comes thicker. Furthermore, the increased boundary- 

 layer thickness is accompanied by a reduction of 

 turbulence intensities and a more uniform distribu- 

 tion of turbulence intensities in the inner region. 

 The variation along the body of the radial location 

 of the maximum values of the measured Reynolds stress 

 -u'v'/U_^ layer is small. The spatial resolution of 

 the cross-wire probe is not fine enough to measure 

 the Reynolds stress distributions in the inner re- 

 gion when the boundary layer is thin. As the stern 

 boundary layer increases in thickness, the location 

 of maximum Reynolds stress moves away from the wall 

 (Figures 14 and 15) . The values of Reynolds stress 

 -u' v' decrease quickly from the maximum value to 

 zero at the edge of the boundary layer. As shown 

 in Figures 14 and 15 , the shape of the Reynolds 

 stress distribution curves in the outer region is 

 quite similar for all the thick boundary layers. 

 It is interesting to note that the shapes of the 

 Reynolds stress distributions in the inner regions 

 are different from those measured in the wake at 

 X/L = 1.057 and 1.182 (Figures 14 and 15); this is 

 a typical characteristic of a developing wake 

 [Chevray (1968)]. The Reynolds stresses experience 

 a drastic reduction in magnitude near the edge of 

 the boundary layer. 



A turbulence structure parameter defined by aj 

 = -u'v'/q^, where q^ = uj'^ + vj'^ + w'2, is of 

 interest. The measured distributions of aj are 

 shown in Figure 16. Most thin boundary layer data 

 show that ai is almost constant (aj :: 0.15) between 

 0.05 and 0.86. The present thick stern axisymmetric 

 data shown in Figure 16 indicate that aj is almost 

 constant up to 0.6 6j_, and the magnitudes of aj 



