118 



non-uniformly to provide a greater concentration 

 near the wall and the wake centerline. Instead of 

 carrying out the integration of the equations up 

 to the wall, i.e., through the viscous sublayer 

 and the blending zone, the numerical solution at 

 the first mesh point, located in the fully turbulent 

 part of the boundary layer, is matched to the wall 

 using the law of the wall. In the extension of the 

 method to the wake, the matching between the first 

 mesh point and the wake centerline is accomplished 

 by using the conditions 8U/3y = and T = on the 

 centerline. The main differences between the 

 boundary layer and wake calculation procedures are 

 therefore the treatment of the flow between the 

 first mesh point and the wall or the wake center- 

 line, and the change in l^ at the tail. Note that 

 the local value of I in the boundary layer as well 

 as the wake is different from 8,q due to the lag, 

 Eq. (8) . The length scale recovers the reference 

 distribution Iq asymptotically in the far wake. 

 Since the near wake data from the low-drag body 

 indicated that most of the adjustment from the 

 boundary layer to the far wake is accomplished over 

 roughly five initial wake thicknesses, the lag 

 length for the wake calculation was taken to be 

 5 S, rather than 10 6 used for the boundary-layer 

 calculation on the basis of Bradshaw's (1973) sug- 

 gestion. Since the extra rates of strain vanish 

 at the tail (k = 0, dr^/dx = 0), the length scale 

 approaches the Iq distribution at about five wake 

 radii downstream of the tail. 



Preliminary calculations performed with the dif- 

 ferential method described above quickly indicated 

 that the extra rates of strain in both experiments 

 were much larger than those examined by Bradshaw 

 (1973) in support of the linear length-scale 

 correction formula of Eq. (8) . In fact, the use 

 of the linear formula led to a rapid decrease in I 

 and indicated almost total destruction of the 

 Reynolds stress across the boundary layer in the 

 tail region and the near wake. In view of this, 

 recourse was made to a non-linear correction formula 

 in the form 



^= u 



eff 

 3U/3y 



(8a) 



which reduces to the linear one, Eq. (8) , for 

 small extra rates of strain. Equations (1) , (3) , 

 and (4), together with (6), (7), (8a), and (9), 

 were then solved with the following inputs: 



A: the measured wall pressure distribution C„ 



pw 



(i.e. , no normal pressure variation) and 



i(y/S) = lo(Y/S) 

 B: the measured Cp„ with S.(y/6) corrected for 



only the longitudinal curvature (e = ejj) 

 C: the measured Cp„ with Jl(y/6) corrected for 



only the streamline convergence (e = e^-) 

 D: as above, but with e = ej^ + e^ 

 E: using e = e^j^ + e^ in Eqs. (8a) and (9) , and 



a variable 3p/3x across the boundary layer 



evaluated by assxoming a linear variation in 



p from y = to y = 6 and using the measured 



values of Cp„, Cp^ and 6. 



Thus, case A corresponds to an axisymmetric bound- 

 ary layer with thin, two-dimensional boundary-layer 

 physics. The other cases enable the evaluation of 

 the relative influence of the extra rates of strain 

 as well as the static pressure variation through 



the boundary layer. The calculations were started 

 with the velocity and shear-stress profiles mea- 

 sured at X/L = 0.662 on the modified spheroid and 

 at X/L = 0.601 on the low-drag body. 



4. COMPARISONS WITH EXPERIMENT 



The major results of the calculations are summarized 

 in Figure 4(a-k) for the low-drag body and in 

 Figure 5(a-h) for the modified spheroid. However, 

 in the latter case the calculations are restricted 

 to the boundary layer since detailed measurements 

 were not made in the wake. Both figures contain 

 comparisons between the experimental and calculated 

 velocity, shear-stress, and mixing-length profiles 

 at a few representative axial stations as well as 

 the development of the integral parameters, 62, A2/ 

 H, H, and Cf, with axial distance. These parameters 

 are defined by 



5i = ; 



dy. 



^2 = ^0 ^y- 







H 



5l/62 (10) 



U 



/. (1 - 7^)rdy, A2 = / 



H = A1/A2 



6 U_ 



u. 



(1 - ^)rdy. 



and 



hpW^ 



(11) 



(12) 



Where Ug is the velocity component at the edge of 

 the boundary layer and wake (y = 6) , tangent to the 

 body surface for the boundary layer and parallel 



%o'%o 



0.0005 0.0010 0.0015 



FIGURE 4(a). Comparison of measurements with the solu- 

 tion of the differential equations, low-drag body. Ve- 

 locity and shear stress profiles at X/L = 0.920. 



