Each of the major concentrations of vorticity exhibited in Figures 2a and b will make different contributions to the time integrals of the horizontal and vertical force coefficients, int(Cxdt) and int(Cydt), depending upon the variation of their circulation and position with time during the wave cycle. Contributions of the principal vorticity concentrations A, B, C, D, A', and B' to the x and y components of the vorticity moments, i.e., (Mw)x and (Mw)y, are given in Figure 6.
The top plot of Figure 6 gives the variation of (Mw)x as a function of frame number N. Generally speaking, the major concentrations A and A' produce the largest moments. Moreover, the change of the vorticity moment with time, represented by the slope (D(Mw)x /DN) has markedly larger values for concentrations A and A' than those due to the remaining vorticity concentrations B, B', and C. Comparison with the image sequence of Figures 2a and b shows the physical basis for large changes of the moments due to A and A'. First, consider the rapid movement of vorticity concentration A back towards the cylinder, represented by frames N = 9 and 11 in Figure 2; in fact this downward movement continues at least through frame N = 12, as shown in the top left image of Figure 4. As a consequence, the values of moment arm y of the vorticity concentration A decrease, producing decreases in the value of (Mw)x. This observation is in accord with the pronounced negative value of the slope (D(Mw)x /DN) between frames N = 9 and 12 in the upper plot of Figure 6. Moreover, the actual value of the force coefficient Cx shown in Figure 4 attains its largest negative values over N = 11 to 13.
On the other hand, movement of vorticity concentration A' in the upward direction and its continuing increase in scale, evident in frames N = 17 and 19 in Figure 2b, as well as in images N = 20 to 23 (not shown herein) promote an increase in the value of the moment arm y and thereby moment (Mw)x of concentration A' with increasing N; this corresponds to a large positive slope D(Mw)x/DN over N = 17 to 23 in Figure 6. In turn, this slope corresponds to the positive peak of Cx over N = 18 to 23 in the top trace of Figure 4. To be sure, the concentration of vorticity B,D makes significant, but generally smaller, contributions, as indicated in the top plot of Figure 6.
A similar assessment can be undertaken for the individual contributions to the moment (Mw)y as shown in the bottom plot of Figure 6. The largest contributions to (Mw)y generally are from: concentrations A and A' over the same portions of the oscillation cycle as for the (Mw)x contributions of the top plot of Figure 6; and concentration B,D during the initial part of its trajectory about the cylinder. The major vorticity concentrations A and A' produce large positive values of moment (Mw)y at small and large values of N; likewise, concentration B,D yields large moments at intermediate values of N. Again, Figure 2 shows the underlying physics. For vorticity concentration A, frames 7, 9 and 11 suggest decreases in the positive value of (Mw)y , due both to a decrease in the peak value of vorticity and a decrease in the x moment arm. This negative slope D(Mw)y/DN produces, in accord with equation (1), a negative peak in the vertical force coefficient Cy indicated in the bottom plot of Figure 4. Similarly, as indicated in the bottom plot of Figure 6, at large values of N = 23 and 24 (not shown), vorticity concentration A' generates both large positive values of moment (Mw)y and slope of moment D(Mw)y/DN, relative to those of the other major vortices B and B'. Correspondingly, the plot of Figure 4 shows the onset of a positive peak of Cy. These observations are due to the movement of the concentration A' away from the surface of the cylinder; although frames N = 23 and 24 are not shown in the image sequence of Figure 2, the initial launching of concentration A' is evident in frame N = 19.
Finally, the bottom plot of Figure 6 also shows that vorticity concentration B,D makes substantial contributions to the moment (Mw)y over the range of intermediate values of N, say N = 14 to 19. Attainment of the maximum positive value of moment (Mw)y at N = 16 is due to movement of the large-scale concentration B,D to the left of the cylinder, as shown in Figure 2b. At this instant, the slope of the moment D(Mw)y/DN is approximately zero, and the value of Cy in Figure 4 is small, i.e., slightly positive.
On the basis of the foregoing observations of the dominant contributions of negative vortices A, A' and the positive vortex B,D to both (Mw)x and (Mw)y, we conclude that such contributions occur in the relatively early stages of development of these vorticity concentrations, corresponding to small displacements away from the surface of the cylinder. This effect, involving a relatively short moment arm and large circulation of the vorticity concentration, appears to be more prevalent than the large moment arm and reduced value of circulation of the vorticity concentrations associated with large orbital trajectories.
