CL -2
0
2
0 2 4 6 8 10 12
t
I I I I I
2 4 6 8 10
FIGURE 19(a, b). For caption see facing page.
t
This rate is necessary for the drag and even the lift coefficients (due to pressure) to remain a t a fairly stationary level over a range of Reynolds numbers.
4.6. Pressure, lift and shedding
The bchaviour of the surface pressure distribution appears almost to be opposite to that of the surface vorticity distribution. Larger positive vorticity corresponds to
- 8
lower pressure. The position of minimal surface pressure, however, lags behind the corresponding position of (locally) maximal surface vorticity
.
Therefore, a bulge phenomenon or a closure point near the cylinder surface signifies a region of relatively high pressure while streamlines clustering round the surface signify a region of relatively low pressure. Contrasting figures 17 (a, b) and 16(a, b) shows the294 C.-C. Chang and R.-L. Chern
validity of the above remarks. Pressure distributions are actually the key elements to understanding the time variation of the lift coefficient and the vortex shedding phenomenon. Below we consider two cases in detail.
4.6.1. Re = 3000, a = 1
Recall that the potential flow is predominant for an initial moment of flow development. Right after the flow is started, the pressure on the back of the cylinder recovers quickly from substantially low values. The pressure is high near the front and back of the cylinder, and is low near its top and bottom. The front stagnation point S appears, and by t = 1 has an azimuthal angle clcarly less than 180". The last fact signifies that the front region of high pressure extends downstream along the upper cylinder surface (cf. figure 18a). In the meantime, a region of strong recirculation forms rapidly near the right upper side of the cylinder, indicating the formation of a local region of relatively low pressure nearby. These observations correspond to the fact that the lift in figure 14(b) is initially downward, increasing in magnitude with time, and begins to decrease when the recirculation becomes significant. The negative of the lift coefficient rises again when a sizable bulge phenomenon near the top of the cylinder makes its appearance, pushing against the region of strong recirculation. We also notice that the pressure near the right lower side of the cylinder has been relatively high, pushing the primary lower vortex E, downstream. Competition for shedding therefore exists between vortices in the lower and the upper wakes. The 'pushing effect' in the upper wake evidently outweighs that in the lower wake, leading eventually to the detachment of the first upper vortex. Note that in this case the forewake in the upper wake is not significant.
Figure 14(b) shows that a t t = 7.83, the negative of the lift coefficient reaches a substantial (local) maximum. On the other hand, figure 4(d, e ) indicates that E, is well detached from the cylinder by t = 8. Figures 17(a) and 18(b) indicate that a t t = 7.83 the front region of high pressure joins with that a t the back; a region near the right upper side of the cylinder is filled with fluid of high pressure. From t = 7.83, the negative of the lift coefficient decreases gradually and reaches a minimal value shortly after t = 12 (at about t = 12.2). Indeed, figure 17 ( a ) indicates that a t t = 12 fluid near the right lower side is of relatively high pressure while a region near the right upper side is filled with fluid of relatively low pressure. Recall that E, is about to be shed a t t = 12. The above observation has the following implication. The alternate formation of regions of relatively high pressure in the lower and the upper wakes is a conspicuous signature of the shedding of El, E, and perhaps vortices shedding later, marking extrema of opposite senses in a lift coefficient. It is therefore appealing to define the moment t = 7.83, that is, when the first substantial extremum of the lift coefficeint is attained, to be the shedding time
ti
of the vortex E, and t = 12.2 (estimated) to be the shedding time of the vortex E,. Evidently, the definition of shedding time is not sufficient to describe when a particular vortex begins to disengage itself rapidly from the cylinder. One possibility for this purpose is to define a disengaging timeth
to be the instant a t which the lift curve changes its sign of curvature and which is just before the negative of the lift coefficient reaches the maximum. This definition could be quite artificial, but is one obvious choice which has the meaning that the lift coefficient is on its way to reaching an extremum at the time defined. According to this definition, we find the disengaging timetb
= 5.90 for El and tb = 10.20 for E,.Vortex shedding from a circular cylinder 295
TABLE 2. Disengaging times of the first vortices shed
4.6.2. Re = 20000, a = 1
The initial development is quite close to the preceding case. A region of strong recirculation, however, appears a t a higher position relative to the horizontal axis through the cylinder centre. In other words, the P-phenomenon is significant. Figure 14 ( c ) shows that in contrast to the preceding case the lift coefficeint reaches a positive (upward) maximum at t = 4.17 when E, is about to be shed. Figure 17 ( b ) shows that at t = 5, a region near the right lower side of the cylinder contains fluid of relatively high pressure while a region of low pressure is still attached to the right upper side.
Figure 17 ( b ) shows further that a region of relatively high pressure exists next to the right and the upper sides of the cylinder. We see, therefore, another case of alternate formation of regions of relatively high pressure in the lower and the upper wakes, which correponds well to the alternate shedding of vortices. We may similarly find t i = 4.17,