Desai and Lieber 



boundary or to the 'wake.' Hence, to replace them by irrotational velocities 

 should be a closer approximation to the truth than to replace them by undis- 

 turbed velocities . . . And we may conclude that (16) will be a satisfactory ap- 

 proximation to the exact form (11) of the governing equation throughout the 

 whole of the speed- scale range, provided that a is the velocity potential func- 

 tion appropriate to a cylinder of the form which we are considering. For the 

 experimental evidence (5) indicates that the actual flow pattern at high speeds 

 does in fact approximate to irrotational flow pattern, except at points very 

 close to the boundary of the cylinder and in the 'wakes'." These quotations 

 show that although they have indicated that their equation (16) may be applicable 

 to high value of Reynolds number, they have not attached any significance to the 

 use of a potential solution as far as points very close to the cylinder and in the 

 wake are concerned. In fact, the idea of deviation from a potential flow is ab- 

 sent. Their work differs in other respects also. The method adopted by them 

 to solve their governing equation is quite different from ours. And consequently 

 there are no equations in their work like the subsidiary equations which we have 

 derived and used. The recognition that the results up to the first iteration must 

 give a value of drag higher than that observed in actual experiments by a con- 

 siderable margin, and that the second iteration is essential to account for this 

 difference, is absent in their work. In fact, their drag formula for a circular 

 cylinder gives a value which is less by 20% than Lamb's [18j and by 7% than 

 Bairstow's when R = 2. In principle, this should be the same as our first itera- 

 tion drag. But the value we have at Re = 1, i.e., at R = 2 since R = 2Re, is a 

 little higher than Bairstow's, as can be seen from Table 2 and Fig. IB. Since 

 we have shown that the first iteration values are lowered by taking into account 

 the second iteration, the discrepancy between their and our first iteration drags 

 when R = 2 shows that their value cannot be accurate. In their approximate solu- 

 tions to the governing equations they have applied the boundary condition at the 

 wall in such a way that it is satisfied at only a discrete number of points. This 

 is not the case with our method. 



A central feature of all the works which use the potential flow solution is 

 their use of the Boussinesq coordinate transformation from the Cartesian space 

 coordinates to the velocity potential a -stream function /3 coordinates. Burgers 

 neglects d^w/'d/S'^, where w = V^^, in the differential equation for co and hence 

 works ultimately with an equation different from the one which corresponds to 

 our base flow and the first iteration equation. Lewis, using Meksyn's analytical 

 methods for obtaining Green's function for the stream function removes the limi- 

 tations involved in the works of Southwell and Squire, and Burgers, but gives no 

 specific information about drag, pressure, etc. Pillow's work treats flow past 

 a parabola and uses similar techniques; consequently, a direct comparison be- 

 tween his and our work is not possible. However, to show the difference in 

 basic ideas involved in his and our work we quote the following from his work. 



A construction is given for the general solution of the Burgers 

 vorticity equation. Such a solution which satisfies the boundary 

 conditions at infinity provides a general outer solution for real 

 viscous flow past bodies, into which any inner solution must 

 ultimately match. 



582 



