116 







_ y=s 



FIGURE 1 . Coordinate system and notation . 



U 

 hi 



3U 

 8x 



+ V- 



.3U 



3y 



uv 



phi 3x 



rhi 



3_ 

 3y 



hjrT 



-) 



hj 8x 3y hi p 8y 

 and the continuity equation is 



^(Ur) + ^ (rhjV) = 

 3x 3y 



(1) 



(2) 



(3) 



U and V are the components of mean velocity in the 

 X and y directions, respectively; hj = 1 + Ky, k 

 being the longitudinal surface curvature; x = -puv 

 + V 3U/3y, where p is density, v is viS'cosity and 

 -puv is the Reynolds stress; r = r^ + y cos 8 is the 

 radial distance measured from the body axis, 9 

 being the angle between the tangent to the surface 

 and the axis of the body; and p is the static pres- 

 sure. These equations resulted from order of mag- 

 nitude considerations and an examination of the 

 data from the modified spheroid experiments of 

 Patel, Nakayama, and Damian (1974) . Specifically, 

 from Eq. (2) we note that the static pressure varies 

 across the boundary layer and that the gradient of 

 the pressure in the direction normal to the surface 

 is associated primarily with the curvature of the 

 mean streamlines. 



Equations (1) , (2) , and (3) also apply to the 

 wake, with < = and 9 = (i.e. , r = y) . In place 

 of the no-slip boundary conditions on the body svx- 

 face, however, the conditions on the wake center- 

 line are 3U/3y = and t = 0. 



If the Reynolds stress is determined by a one- 

 equation model using the turbulent kinetic-energy 

 equation, as proposed by Bradshaw, Ferriss, and 

 Atwell (1967) , then the appropriate closure equa- 

 tion for the flow outside the viscous sublayer and 

 the blending zone is 



1 

 2a 1 



+ 1 



3x 



rG 



+ V 



3u 



3y 



kU 



3/2 



ai 



i 1/2 



= 



(4) 



with the usual mixing length. G and I are assumed 

 to be universal functions of y/6 , where 6 is the 

 boundary layer thickness. The particular forms of 

 these functions proposed by Bradshaw et al. (1967) 

 for a thin boundary layer have gained wide accep- 

 tance and have proved adequate for the prediction 

 of a variety of boundary layers developing under 

 the influence of different pressure gradients and 

 upstream history. In the adoption of this closure 

 model for the treatment of thick boundary layers 

 and wakes, however, it is necessary to consider the 

 influence of transverse and longitudinal surface 

 curvatures on the turbulence. 



Figure 2 shows the conventional transverse and 

 longitudinal curvature parameters for the modified 

 spheroid and low-drag body [Patel and Lee (1977)]. 

 The ratio of the boundary-layer thickness to the 

 transverse radius of curvature, 6/rQ, is seen to be 

 more than twice as large in the latter case as in 

 the former. In both cases, however, 6/ro is less 

 than 0.4 up to X/L = 0.75, so that the boundary 

 layers may be regarded as thin up to that station. 

 Over the rear one-quarter of the body length, the 

 influence of transverse curvature would prevail 

 not only through the geometrical terms in the mo- 

 mentum and continuity equations but also through 

 any direct effect on the turbulence. The precise 

 nature of the latter is not known at the present 

 time since the turbulence is also affected by the 

 longitudinal curvature of the streamlines associated 

 with the curvature of the surface as well as the 

 curvature induced by the rapid thickening of the 

 boundary layer over the tail. 



The longitudinal surface curvature parameter k6 

 is seen to be quite different for the two bodies. 

 In the case of the modified spheroid, the curvature 

 is convex up to X/L = 0.933 and zero thereafter, 

 while that of the low-drag body is initially convex 

 and becomes concave for X/L > 0.772. Several 

 recent studies with nominally two-dimensional thin 



O Modified Spheroid 

 A Low-Drag Body 



kS 



' ' *— ' 



0.4 0.5 0.6 0.7 0.8 0.9 1.0 



where ai is a constant (=0.15) , G is a diffusion 

 function and I is a length-scale function identified 



FIGURE 2 . Ratios of boundary-layer thickness to the 

 longitudinal and transverse radii of surface curvature. 



