Studies on the Motion of Viscous Flows — III 



refer to the Reynolds number at which there is no vortex but above which sepa- 

 ration begins and a vortex forms. Hence Taneda's value of 2.5, which is just 

 above 2.3 of the present work, confirm the linear substructure theory. . .c-- 



Figures 2C and 2D give respectively the stagnation pressures in front and 

 in the rear of a circular cylinder for a first iteration. The qualitative behavior 

 of these curves is again the same as found by Grove et al. [47], Thom [66], and 

 Roman [80,81]. The pressures behave asymptotically and tend to a constant 

 value with increasing Re in the range investigated. This is one of the main ob- 

 servations of Grove et al. They [47] state: "First, that the rear pressure 

 quickly reaches a limit of approximately -0.45 as the Reynolds number is in- 

 creased; and second, that this limit is attained, for all practical purposes, at 

 a Reynolds number as low as 25." From Fig. 2D it can be seen that the pres- 

 sure becomes essentially constant from Re = 4, i.e.. Re = 8 onwards. Thus 

 there is complete qualitative agreement. Figures 3E, 3F, 4E, 4F, etc. give the 

 distribution of total pressure, ideal pressure, first iteration pressure, second 

 iteration pressure, and their harmonic components on the surface of the cylin- 

 der for different values of Re. The total pressure, as stated earlier, is given 

 as the sum of the ideal pressure, the first iteration pressure, and the second 

 iteration pressure. Figure 2E shows how the amplitudes of the harmonic com- 

 ponents of the first iteration pressure varies with Reynolds number. It can be 

 seen that the amplitude PRETin,ax is at first very large, then decreases rapidly 

 in a narrow range of the Reynolds number, and thereafter behaves asymptotically 

 with increasing Re, tending to a constant value. The other two components, viz., 

 the constant PRECi and the amplitude PREPln^in a^^so behave asymptotically with 

 increasing Re, but they do not attain very high values like PRETi^ax for small 

 Re. This then means that, for small Re, the harmonic PRETi in cos d dominates 

 so that the total first iteration pressure PRESi and hence the total pressure 

 PRESS behaves essentially as this harmonic. On the other hand, for high Re, the 

 harmonic PREPl in cos 2e and the constant PRECl make significant contributions 

 to the pressure PRESI, and hence to the total pressure PRESS. The qualitative 

 behavior in Figs. 3E, 3F, 4E, 4F, etc. is as observed in experiments. 



Figures 3D, 4D, etc. give an idea of the error involved in the assumption 

 that Bn(r,t) = for n > 3 for different values of Re. They show that ERi 

 takes extreme values at the two boundaries. Further, the absolute magnitudes 

 of these extreme values which are at first small increase with increasing Re. 

 However, bearing in mind that these curves are plotted against S and that the 

 actual distance is given by r = ee<=s-i, we see that in most of the flow field the 

 absolute magnitude of ERi remains very small compared to the absolute magni- 

 tudes of these extreme values. The assumption seems to be justified, though 

 these curves do give an indication that for larger values of Re one may have to 

 introduce corrective measures for this error. 



In the present work we have not gone into an investigation of the time- 

 dependent motion. Consequently, the literature available on this aspect of the 

 flow field is not discussed. The experimental works of Grove et al. [47], 

 Gerrard [85], Roshko [86,87,88], Relf and Simmons [89], Tyler [90], Hollingdale 



587 



