143 



decrease toward the edge of the boundary layer. The 

 values of aj also decrease in the inner region of 

 wake at X/L = 1.057 and 1.182 of Afterbody 1. It 

 should be pointed out that the measured values of 

 q" contain the free-stream turbulence fluctuation, 

 no attempt having been made to remove the free- 

 stream turbulence fluctuation from the measured 

 values of q^. The measured reduction of aj near 

 the outer edge of the boundary layer is in part 

 caused by the larger contribution of the free-stream 

 turbulence to q^ than to -u' V . Nevertheless, the 

 measured values of the turbulence structure param- 

 eter aj are quite constant across the inner portion 

 of the boundary layer where the effect of free- 

 stream turbulence is small. 



Eddy Viscosity and Mixing Length 



The measured distributions of shear stress -u' v' 

 and mean velocity gradient, Su^/ar, were used to 

 calculate the variations of eddy viscosity and 

 mixing length across the thick stern boundary layers 

 according to the following definitions 



and 



-u'v' 



3u 

 X 



37" 



^^laF^I 



3u 

 X 



3r 



(14) 



(15) 



eddy viscosity agree resonably well with the eddy- 

 viscosity model of Cebeci and Smith (1974, Eq. 3) 

 when the boundary layers are thin. However, as the 

 stern boundary layer thickens, the measured values 

 of e/Ug 6p* in the thick stern boundary layers are 

 only about 1/6 of the values for thin boundary 

 layers given by the Cebeci and Smith model (1974) . 

 The measured distributions of mixing length shown 

 in Figures 19 and 20 also agree quite well with the 

 thin boundary layer results of Bradshaw, Ferriss, 

 and Atwell (1967). Again as the boundary thickens, 

 the measured values of l/Sy. reduce drastically. 

 The values of S./6j. in the thick stern boundary 

 layers are only about 1/3 of those of the thin 

 boundary layers. Similar reductions of eddy vis- 

 cosity and mixing length in thick stern boundary 

 layers were also measured by Patel et al. (1974, 

 1977) . 



As the axisymmetric boundary layer thickens in 

 the stern region, the boundary layer thickness 6 

 and the displacement thickness &p* increase dras- 

 tically. However, the values of eddy viscosity and 

 mixing length do not have enough time to respond to 

 this change. Therefore, neither the eddy viscosity 

 model of Cebeci and Smith (1974) , nor the mixing 

 length results of Bradshaw, Ferriss, and Atwell 

 (1967) can be applied to the thick stern boundary 

 layer. 



TURBULENCE MODELS 



The experimentally-determined distributions of 

 eddy viscosity, e/Ugfip*, are shown in Figure 17 for 

 Afterbody 1 and in Figure 18 for Afterbody 2 , where 

 Ug is the potential-flow velocity at the edge of 

 the boundary layer and 6p* is the displacement 

 thickness (based on the planar definition, Eq. 10) . 

 Figures 19 and 20 show the experimentally-determined 

 distributions of mixing length i/S^, for the after- 

 bodies, where 6j- is the boundary-layer thickness 

 measured normal to the body axis. As shown in 

 Figures 19 and 20, the measured distributions of 



In most works, the basic assumption made in the 

 differential methods for calculating turbulent 

 boundary layers is that the mixing length or eddy 

 viscosity is uniquely related to the mean velocity 

 gradient and the boundary- layer thickness parameter 

 at a given location. So long as the boundary layer 

 is thin and the change in boundary- layer properties 

 due to the pressure gradient is gradual, this simple 

 assumption is know to be satisfactory [see e.g., 

 Cebeci and Smith (1974)]. When the past history of 

 boundary layer characteristics is important, Brad- 



1.4 



FIGURE 17. Measured distributions 

 of eddy viscosity for afterbody 1. 



