Studies on the Motion of Viscous Flows— III 



PART 3 ' -'' 



RESULTS, DISCUSSION, AND CONCLUSIONS 



This part of the paper presents the results of computations of viscous flow 

 fields around a circular cylinder based on the analytical representations ob- 

 tained from the algorithm and its attendant linear substructure as they were ap- 

 plied in previous sections to complete the Navier-Stokes equations and boundary 

 conditions. These results are discussed with reference to the existing body of 

 literature which covers a range < Re < 20 of Reynolds number based on the 

 radius of the cylinder. Following the discussion, we present conclusions in the 

 light of the results and discussion. Numerical information is given in Table 2 

 and in a series of plots which are placed at the end of this paper. 



RESULTS 



The first and second iteration solutions are computed for 18 discrete val- 

 ues of the Reynolds number. The results of the computation are presented con- 

 cisely in Table 2. They are given in detail by plots contained in Figs. lA through 

 34F. These plots are divided into two sections. Figures lA through 20F give 

 information about the drag, pressure, separation, and behavior of solutions with 

 increasing Reynolds number and the radius at which the boundary conditions of 

 Eqs. (2.62), (2.63), (2.69), and (2.70) are applied. Figures 21A through 34F give 

 streamline plots showing the development of viscous flow fields with bound 

 vortices as the Reynolds number is increased from 0.05 to 20. 



In Fig. lA the total drag coefficient CD is plotted against the Reynolds num- 

 ber on a linear scale. Tritten's (1959) experimental results are included for 

 comparison. In Fig. IB logarithms of total drag coefficient CD are plotted 

 against the logarithm of the Reynolds number (-1.5 < logjo Re - l*^)? ^^^ the 

 results of Bairstow, Cave, and Lang (1923), and of Tritton (1959) are included 

 for comparison. Figure IC is exactly the same as IB, except that here the 

 least values of the first iteration drag coefficient CDi are plotted instead of the 

 total coefficient CD. Figure ID shows on a linear scale the plot of the second 

 iteration drag coefficient CD2 against the Reynolds number Re. Figure IE gives 

 an enlarged portion of the plot of CD against Re together with the results of 

 Lamb (1911), Kaplun (1957), and Tritton (1959) for comparison. Figure 2A gives 

 plots of the ratios -q, tj^, and 172 of pressure drag to viscous drag against Re. 

 Figure 2B gives plots of the angle of separation a^ obtained by the first itera- 

 tion and of the angle of separation a, obtained by the first and second iterations 

 against Re. Figures 2C and 2D give, respectively, the plots of the stagnation 

 pressures in front and at the rear of the cylinder against Re. In these figures 

 we have plotted PRESS-PREC2 instead of the stagnation pressure PRESS which is 

 the result of the first and second iterations together, because, due to its small- 

 ness, PREC2 could not be calculated accurately, and erroneous values were ob- 

 tained for it due to lack of precision in computation at that stage. Figure 2E 

 gives the plots of PRECl, PRETln,axj ^^^ PREPl^^in v/s Re. These quantities are 

 the constant, the first harmonic, and the second harmonic amplitudes of the 

 first iteration pressure field around the circular cylinder. 



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