123 



ment in the prediction of the velocity profile in 

 the case of the modified spheroid, but its influence 

 is small, and confined to the outer part of the 

 boundary layer, in the case of the low-drag body. 



Examination of the velocity and shear-stress 

 profiles at several axial stations shown in Figures 

 4a-f and 5a-c suggests that the incorporation of 

 the non-linear length-scale correction of Eq. (8a) , 

 the associated rate Eq. (9) and the static-pressure 

 variation in the equations of the thick boundary 

 layer, which already include the direct longitudinal 

 and transverse curvature terms, leads to satis- 

 factory overall agreement with the data for both 

 bodies. It is particularly noteworthy that the 

 velocity and shear stress distributions in the 

 far wake (X/L = 2.472) of the low-drag body are 

 predicted with good accuracy. The level of 

 agreement can obviously be improved further by 

 appropriate modifications in the empirical functions 

 in the turbulent kinetic-energy equation and changes 

 in the lag-length used in the length-scale equation. 

 The predictions of the shear stress profiles are 

 consistent with those of the mixing-length distri- 

 butions shown in Figures 4g and 5e insofar as lower 

 shear stresses correspond to an over correction in 

 the mixing length. These comparisons provide 

 further insight into the manner in which the length 

 scale must be modified to improve the correlation 

 between the calculation method and experiment. It 

 is apparent that the consistent discrepancy between 

 the calculated and measured velocity and shear- 

 stress profiles near the outer edge of the boundary 

 layer and wake stems from a poor representation of 

 the length scale distribution. 



It is interesting to note that, for both bodies 

 the calculation precedure predicts normal components 

 of mean velocity which are of the same order of 

 magnitude as those measured. The relatively close 

 agreement between the predictions and experiment 

 for both components of velocity is perhaps a good 

 indication of the axial symmetry achieved in the 

 experiments. The large values of the normal veloc- 

 ity and the influence of static pressure variation 

 noted above would appear to indicate that incorpora- 

 tion of the y-momentum equation in the calculation 

 procedure would be worthwhile. Note that this has 

 been avoided in the present calculations by using 

 the measured pressure distributions at the surface 

 and the outer edge of the boundary layer. 



Finally, the comparisons made in Figures 4 (i-k) 

 and 5 (e-h) with respect to the integral parameters 

 show several interesting and consistent features. 

 It is observed that the prediction of the physical 

 thickness of the boundary layer and the wake is 

 insensitive to the changes in I as well as the in- 

 clusion of static pressure variation. The under 

 estimation of the thickness is associated with the 

 discrepancy, noted earlier, in the velocity profile 

 near the outer edge of the boundary layer and wake . 

 The planar momentum thickness 62 and the momentum- 

 deficit area A2 are also insensitive to changes in 

 I. The variation of static pressure across the 

 boundary layer appears to make a small but notice- 

 able contribution to the development of A 2 in both 

 cases. However, it is not large enough to account 

 for the differences between the calculations and 

 experiment. The predictions of the shape parameters, 

 H and H, presented in Figures 4j and 5g, appear to 

 be satisfactory, especially in view of the rather 

 large scale of the plots. Nevertheless, there is 

 a systematic difference between the data and the 



calculation in the tail region and wake of the low- 

 drag body. As indicated earlier, this can be im- 

 proved by modifications in the empirical functions 

 and the lag length. The predictions of the wall 

 shear stress, shown in Figures 4k and 5h, indicate 

 that the present method gives acceptable results 

 for both bodies. 



5. COMPARISONS WITH THE INTEGRAL APPROACH 



An integral method for the calculation of a thick 

 axisymmetric boundary layer was described by Patel 

 (1974) and its extension to the wake was proposed 

 by Nakayama, Patel, and Landweber (1976b). A few 

 possible improvements in this method were examined 

 recently relative to the description of the velocity 

 profiles in the near wake and these are discussed 

 by Patel and Lee (1977) . The most recent version 

 of this method has been used here to calculate the 

 development of the boundary layer and the wake of 

 the low-drag body in order to assess its performance 

 relative to the experimental data (which were not 

 available at the time the method and its extension 

 were proposed) and the more elaborate differential 

 method . 



The results of the calculations are shown in 

 Figure 6. It is seen that the performance of the 

 integral method is comparable with that of the 

 differential method (compare Figures 4h-k with 

 6a-d) with respect to the prediction of the bound- 

 ary layer up to the tail. The prediction of the 

 near wake is, however, distinctly inferior to that 

 of the differential method, particular with respect 

 to the physical thickness S and momentum deficit 

 area A2. The main conclusion to emerge from these 

 calculations is that the integral method is capable 

 of giving a good overall description of the flow 

 features with considerably less computing effort. 

 The differential approach is to be preferred, how- 

 ever, since it affords the opportunity for further 

 refinement and gives greater details which may be 

 necessary for many applications. A more thorough 

 discussion of the integral method and its short- 

 comings is given in Patel and Lee (1977) . 



6. CONCLUSIONS 



From the present solutions of the differential 

 equations, using the (one-equation) turbulent 

 kinetic-energy model of Bradshaw, Ferriss, and 

 Atwell (1957) , it is clear that methods developed 

 for thin shear layers cannot be relied upon to pre- 

 dict the behavior of the thick boundary layer and 

 wake of a body of revolution. Although these cal- 

 culations have demonstrated that the boundary-layer 

 calculation can be readily extended to the wake 

 and that a fairly satisfactory prediction procedure 

 can be developed by incorporating ad hoc corrections 

 to the model for the extra rates of strain, along 

 the lines recommended by Bradshaw (1973) , it is 

 indeed surprising that such modifications, proposed 

 originally for small extra rates of strain and thin 

 shear layers, work so well for the two bodies which 

 are substantially different in shape. In keeping 

 with recent trends in the formulation of turbulence 

 models , one inquires whether thick axisymmetric 

 boundary layers and near wakes ought to be treated 

 by the so-called two-equation models . From the 

 rapid changes in the mixing-length indicated by 



