Desai and Lieber 



then hold for all Reynolds numbers in the range < Re < 20. Table 2 contains 

 all of the significant information, a part of which is also presented in Figs. lA 

 through 2E. 



Bearing in mind that there is some arbitrariness involved in the choice of 

 RT = hf and its lower bound for a given Reynolds number, the corresponding 

 values from Table 2 for different Re show that even with increasing Re, the 

 physically infinite distance h^ decreases, while its lower bound at first in- 

 creases but ultimately decreases for the most part. This means that viscous 

 effects beco7ne increasingly localized near the cylinder as the Reynolds number 

 is increased. And this, of course, is confirmed by experimental observations. 

 The existence of asymptotic behavior of the first and second iteration solutions 

 for all Reynolds numbers considered is conclusive support of the validity of the 

 idea of a physically infinite distance as applied to this problem. Figures lA 

 through IE for drag, and Figs. 2C, 2D, and 2E for pressure, show that though 

 the solutions for various Reynolds numbers are obtained by applying the condi- 

 tions of Eqs. (2.62) and (2.63) at different distances h|, in general, the corre- 

 sponding points for drags and pressures when plotted against Re give smooth 

 curves which behave asymptotically with increasing Re in the range < Re < 20. 

 This further supports the validity of the idea of a physically infinite distance, 

 for otherwise the curves would not be smooth. However, the values of drag 

 plotted in these figures for Re > 4 could be lower than those shown if the physi- 

 cally infinite distance could be increased. This is indicated because, for Re > 4, 

 the values of drag plotted lie close to the bend in the drag v/s r plots for a 

 given Re and are not the asymptotically limiting values. This accounts for the 

 discrepancy for Re > 4. It also indicates the need to take more harmonics into 

 consideration. 



Figure 3D shows that the second iteration drag for Re > 4 becomes positive. 

 Referring to Table 2, we see that the viscous component CDV2 of this drag re- 

 mains negative whereas the pressure component CDP2 becomes positive. Be- 

 cause of possible numerical errors arising from the application of boundary 

 ditions at short distances, in order to avert numerical breakdown, it is not con 

 conclusive that CDP2 and CD2 are indeed positive. K they remain positive after 

 the possibility of computational inaccuracies is ruled out, in our opinion, this 

 would support the conjecture that, even at very low Reynolds number, higher 

 harmonics must be used in order to effectively apply higher order iterations to 

 improve accuracy in numerical representations of flow fields. 



Figures 21A through 34F give streamline plots for all the Reynolds num- 

 bers considered, except for Re = 2.1, 2.3, 2.4, and 2.5. There are two harmonics 

 and two iterations. Both the harmonics and their sum are plotted for each itera- 

 tion. Therefore, there are basically six plots for a given Reynolds number. How- 

 ever, when a vortex appears in the flow field, an enlarged plot of the vortex is 

 added to the set. The only exception to this is the case of Re = 20, where the 

 vortex in the flow field is not plotted. The reason is that the computer program 

 had to be modified at this stage, to plot the vortex. Figures 22C and 22D for 

 Re = 0.125 are the same, except that the latter shows a discontinuous behavior 

 in the streamline pattern. The reason for this is that when the two harmonics 

 are added together, due to the discontinuous behavior of the second harmonic at 



578 



