The time histories of the instantaneous horizontal Fx and vertical Fy forces were acquired simultaneously with the instantaneous images of the flow patterns described in the previous section. The force coefficients Cx and Cy are shown in Figure 4 as a function of frame number N and the corresponding elapsed time t. The force coefficient is defined as Cx = Fx/[1/2rU^2*Dl], in which r is the fluid density, U is the characteristic velocity of the flow, l is the length of the cylinder, D is the cylinder diameter, and similarly for Cy. Patterns of instantaneous vorticity, taken from the sequence of images described in the previous section, are related to the occurrence of minima and maxima of the force coefficients Cx and Cy in Figure 4.
The maximum-negative value of Cx occurs in the vicinity of N = 12. At this instant, the large-scale negative vortex A is on the return portion of its trajectory towards the cylinder and is undergoing initial stages of severe distortion during its interaction with the cylinder. The maximum positive value of Cx, which occurs (initially) at approximately N = 18, coincides with the continued development of the large-scale negative vortex A' along the top surface of the cylinder, as it moves in the clockwise direction above the surface of the cylinder. The remaining concentrations of vorticity, which, of course, contribute to Cx, develop in the fashion described in the previous section.
Concerning the vertical force coefficient Cy, the maximum-negative value coincides with movement of the vorticity concentration A up and away from the surface of the cylinder. On the other hand, the maximum-positive value of Cy occurs when vorticity concentration A' attains its position at the top of the cylinder during its clockwise migration about the cylinder surface, while the remaining vorticity concentrations develop as previously described.
The relationship between the space-time evolution of the instantaneous velocity field about the cylinder and the effective force F acting on the cylinder can be described using a concept described by Wu (1981) and Lighthill (1986). In essence, the vector force acting on a body can be expressed as the time derivative of the spatial integral of the moment of vorticity according to:
