117 



turbulent boundary layers [Bradshaw (1969, 1973), 

 So and Mellor (1972, 1973, 1975) , Meroney and 

 Bradshaw (1975) ,- Ramaprian and Shivaprasad (1977) ; 

 Shivaprasad and Ramaprian (1977) ] have indicated 

 that even mild (k6-0.01) longitudinal surface 

 curvature exerts a dramatic influence on the turbu- 

 lence structure. In particular, it is noted that 

 quantities such as the mixing length £, the struc- 



ture parameter aj 



-uv/q and the shear-stress 



correlation coefficient uv/(i'u /v^) are influenced 

 markedly, and experiments indicate that convex 

 streamline curvature leads to a reduction in these, 

 whereas concave curvature has an opposite effect. 

 The turbulence measurements on the modified spheroid 

 and the low-drag body appear to confirm these ob- 

 servatons although the relative influence of longi- 

 tudinal streamline curvature and transverse surface 

 curvature could not be separated readily. 



Bradshaw (197 3) has argued that whenever a thin 

 turbulent shear layer experiences an extra rate of 

 strain, i.e., in addition to the usual 3U/3y, the 

 response of the turbulence parameters is an order 

 of magnitude greater than one would expect from 

 an observation of the appropriate extra terms in 

 the mean-flow equations of momentum and continuity. 

 For THIN shear layers and SMALL extra rates of 

 strain he proposed a simple linear correction for 

 the length scale of the turbulence, viz. 



= 1 + 



3U/3y 



(5) 



where S,,-, is the length scale with the usual rate 

 of strain, 3U/3y, i is the length scale with the 

 extra rate of strain, e, and a is a constant of 

 the order of 10. For the axisymmetric boundary 

 layer being considered here, there are two extra 

 rates of strain: 



K U 

 1 + Ky 



due to the longitudinal curvature, and 



1 3r 



dr 

 U o 



1 + Ky r 3x r dx 



(6) 



(7) 



due to the convergence or divergence of the stream- 

 lines (in planes parallel to the surface) associated 

 with the changes in the transverse curvature. The 

 former is a shearing strain while the latter is a 

 plain strain, and it is not certain whether the 

 two effects can be added simply in using Eq. (5) 

 as recommended by Bradshaw (1973) . If this is the 

 case, however, we would expect a greater reduction 

 in Z in the tail region of the modified spheroid, 

 where k is positive and dr^/dx is negative, than 

 on the low-drag body, where < becomes negative and 

 would therefore tend to offset the influence of 

 the negative dr^/dx. Although the available data 

 appear to bear this out to some extent, a direct 

 comparison between Eqs . (5), (6), and (7) and the 

 data was not attempted, especially in view of 

 Bradshaw' s [Bradshaw and Unsworth (1976)] assertion 

 that Eq. (5) should be used in conjunction with a 

 simple rate equation which accounts for the up- 

 stream extra rate-of-strain history. He proposes 



1 + 



eff 

 3U/3y 



(8) 



d . eff 



dx eff' ~ 106 



(9) 



where e is the actual rate of strain, eeff is its 

 effective value and 106 represents the "lag length" 

 over which the boundary layer responds to a change 

 in e. In order to determine the merit of this 

 proposal, it is of course necessary to incorporate 

 it in an actual calculation and make a comparison 

 between the predictons and measurement. Such an 

 attempt has been made here. 



The functions i^ and G used in the present study 

 are shown in Figure 3. For the wake calculation, 

 the linear variation of l^ in the wall region is 

 replaced by the constant value of 0.09, as shown 

 by the dotted line in the figure. The local dis- 

 tribution of the length scale, I, is thus given by 

 Eqs. (6) through (9) while the diffusion function, 

 G, and the structure parameter, aj , retain their 

 thin-boundary-layer values. 



3. SOLUTION OF THE DIFFERENTIAL EQUATIONS 



A numerical method available for the solution of 

 equations corresponding to (1) , (3) , and (4) for 

 a thin two-dimensional boundary layer was modified 

 to introduce the longitudinal- and transverse- 

 curvature terms. Instead of incorporating the y- 

 momentum, Eq. (2) , into the solution procedure, 

 however, changes were made such that a prescribed 

 variation, across the boundary layer, of the pres- 

 sure gradient 3p/3x could be used. This implies 

 that the pressure field is known a priori. The 

 solution of Eqs. (1) , (3) , and (4) together with 

 Eqs. (6) , (7) , (8) , and (9) can then be obtained 

 through step-by-step integration by marching down- 

 stream from some initial station where the velocity 

 and shear-stress profiles are prescribed. A 

 staggered mesh, explicit numerical scheme, similar 

 to that used by Nash (1969) , was used to integrate 

 the equations in the domain between the first mesh 

 point away from the surface (or the wake center- 

 line) to some distance, typically 1.25 6, outside 

 the boundary layer and the wake. The fifteen mesh 

 points across the boundary layer are distributed 



and 



FIGURE 3. Distributions of empirical functions, Kq 

 and G. 



