For aspect ratios for which the shedding asymptoted to a single frequency, c/t =3-5 exhibited the first mode of shedding with Stc = 0.6. This corresponded to a single leading-edge vortex convecting along the plate at any time. The feedback loop controlled the next shed vortex. The leading-edge shedding for larger aspect ratio ranges, c/t = 6-8, 9-10 and 13, represented the second, third and fourth shedding modes respectively. Hence there were (n =) two, three and four vortices above the plate for these flows (and the feedback loop controls the shedding of the second, third and fourth subsequently shed leading-edge vortex respectively.) The Strouhal number based on chord is approximately constant within each range and is given by 0.6n. These modes of shedding are shown by in the animations of vorticity plots in Movie 1. As observed by Ozono et al. (1992), the flow is more regular at the beginning of each step (i.e. c/t = 3, 6 and 9) and this deteriorates with increasing c/t until it jumps to the next step. This results in higher mean base suction and fluctuating lift forces at the begining of each step as shown in Table 1. This weakening of the vortex locking results from a competition between the natural shedding from the trailing edge and forcing from shed leading-edge vortices leading to a loss in the periodic nature of the base pressure at c/t = 5 and 8. This is consistent with the measurements of lift coefficient predicted by Ozono et al. (1992) and Ohya et al. (1992). Figure 5 show a trace of the base pressure at c/t = 7, 8 and 9 where the shedding changes mode.
Simulations were performed for a wide range of St for c/t = 6-16 and the time mean base pressure coefficient Cp was predicted. (This was calculated by determining the time mean of the pressure coefficient at the midpoint of the trailing edge.) These predictions together with the measurements of Mills et al. (1995) are presented in Figure 6; it can be seen that although the flow is locked on over a wide range of forcing frequencies, it is most receptive (as indicated by local maxima in base suction) at one, or at most two, frequencies for each c/t.
This receptivity is clarified by considering the base suction response for each particular aspect ratio as the forcing frequency is varied. Beginning with c/t = 6, a local maximum base suction is reached for St = 0.17. As c/t is increased to 7, the Strouhal number at which maximum base suction occurs drops to St = 0.16, then St = 0.14 at c/t=8 and St = 0.12 for c/t = 9. While the local base suction maximum is shifting to lower Strouhal numbers, another local peak emerges at St = 0.174 for c/t = 8. Then with increasing c/t, this peak shifts to St = 0.162 at c/t = 9, St =0.165 at c/t = 10, St = 0.155 at c/t = 11, St = 0.14 at c/t = 12 and St = 0.13 for c/t = 13. This pattern repeats with another local peak developing at St = 0.157 for c/t = 13 which moves to St = 0.155 at c/t = 14, St = 0.155 at c/t = 15 and St = 0.145 at c/t = 16. A similar trend is also evident in experimental mean base suction measurements by Mills et al. (1995) plotted alongside in Figure 6.
The predicted and observed mean pressure measurements show some differences presumably due to modelling assumptions. The experimental results were performed at approximately Re = 9,000 with an acoustic field used to lock the flow. The numerical simulations were carried out for Re = 400 and were restricted to two dimensions, which resulted in the vortices remaining more coherent along the plate surface (as opposed to the formation of hairpin structures in the observations) and exerting a larger influence at the trailing edge; consequently, the magnitude of the predicted base pressure is higher than the observation. Nevertheless, the predicted and observed flows show a distinct staging in the mean base suction with c/t and this will be the focus of further discussion.
To assist in comparing the flow states leading to different base pressure suctions, vorticity plots are presented for the cases where there are local maxima in base pressure. Figure 7 are plots taken at 90° into the sinusoidal forcing cycle when the perturbation is at its maximum in the upward direction. It can be seen that the peaks in base suction occur when the flow is locked into distinct shedding modes. For c/t = 6-9, c/t = 8-13 and c/t=13-16, the flow is locked into the n = 2, n = 3 and n = 4 modes respectively which correspond to the number of vortices on the side face. Also note that the leading edge and trailing edge are in the same phase of their shedding cycle suggesting that the phase of shedding influences the base pressure. At c/t = 8, 9 and 13, there are two local maxima in base suction which Figure 7 illustrates to correspond to two different shedding modes; for the higher mode shedding at the trailing edge is 180° out of phase relative to the lower mode. These plate aspect ratios in the transition region between shedding modes.
It is not surprising then that Figure 8, a plot of the Strouhal number based on chord corresponding to a local maximum in base suction as a function of c/t, shows a stepwise increase. There is good agreement between the predicted Strouhal numbers and the results of Mills et al. (1995) for the lower aspect ratios but the predicted values of the Strouhal numbers differ slightly more for the higher shedding modes (i.e., n = 3 and n = 4). In addition, in the experiments, the Strouhal number (based on thickness) for which the maximal base suction occurs tends to increase with mode number, while for the simulations show the opposite behaviour as shown in Table 2 below. A possible explanation for this is that the convection speed of the vortices may not follow the same trend in the two cases, as a result of different Reynolds number and three-dimensional effects. In the experiments of Nakamura et al. (1991) the trend of increasing Strouhal number for the higher modes was also observed while the two-dimensional unforced simulations of Ozono et al. (1992) and Ohya et al. (1992) showed the opposite behaviour in agreement with the current simulations. To help resolve this issue a program of three-dimensional simulations is currently being undertaken.
| c/t | St (numerical) | St (experimental) |
|---|---|---|
| 6 |
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| 10 |
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Ozono et al. (1992) observed in the case of flow around rectangular plates in the absence of external forcing that, at the start of each step, the flow is more strongly locked. A similar result is observed in the current predictions. With increasing c/t, it is found that there is less regularity in the vortex shedding until it eventually locks on to the frequency of the next step in Strouhal number based on chord. It can also be observed that, in those cases with forcing, at the beginning of each step, (i.e., c/t = 6, 10 and 14), the magnitude of base pressure is larger and decreases with increasing c/t until the flow state jumps to the next mode. This is evident in both the numerical results and the experimental results of Mills et al. (1995) both shown by Figure 6. If one considers the plots of vorticity in Figure 7, the vortex forming at the trailing edge of the plate appears strong at the beginning of each stage resulting in the larger pressure drop. As seen in Figure 7, the phase of shedding from the trailing edge matches at the peak base suction for varying c/t, this suggests that the trailing edge shedding is more receptive to the flow perturbations at the beginning of each step where the frequency is higher and with increasing c/t, it is less receptive to the lower frequency until jumping to the next mode.
Movie 3 contains a selection of sequential images showing the effects of the perturbation to the flow. Generally, the forcing causes the leading edge vortices to roll up tighter than the unperturbed flow. This has been observed by Sheridan et al. (1997). Movie 3a shows the flow at c/t = 10 with an external forcing at St = 0.12. At this relatively low forcing frequency, the trailing edge shedding is present but the vortices are relatively weak and the formation length is large. Also the leading and trailing edge vortices do not merge in the wake. This results in a weaker base suction. Movie3b represents the flow for a perturbation frequency of St = 0.165. This simulation has a strong trailing-edge shedding with a short formation length and the leading and trailing edge vortices merge in the wake. This results in a strong base suction. Movie 3c is for a perturbation frequency of St = 0.174 and c/t = 8. This result is similar to the previous except that it is shedding 180° out of phase relative to the previous simulation. This is common for the cases which result in local peaks in base suction at higher forcing frequencies (i.e., c/t = 8, 9 and 13). Movie 3d is for a perturbation frequency of St = 0.20 and c/t = 10. At the higher frequencies, trailing-edge shedding is suppressed and this results in a lower base suction. The shedding from the trailing edge is a result of the roll up of the boundary layer between vortices from the leading edge. At high frequencies, the leading-edge vortices are closer together and this restricts the amount circulation in the boundary layer resulting in a suppressed shedding from the trailing edge. The trailing-edge shedding is a strong absolute instability, presumably naturally unstable only over a narrow frequency range. At higher forcing frequencies the trailing edge shedding is not receptive to the external forcing. In addition, interference from leading-edge vortices as they pass the trailing-edge seems to cause trailing-edge shedding to be suppressed altogether.
The staging of the peaks in base pressure is similar
to the natural shedding case shown earlier. The numerical model does not
show the experimentally observed trend of increasing St at the start
of each stage. However, both results presented here and those of Mills
et al. (1995) show that the peak base suction is stronger at
the shorter plate lengths for each step. Since large base suctions are
associated with strong and tight vortex shedding from the trailing edge,
this result probably indicates that the natural frequency for trailing-edge
shedding is closer to the applied forcing frequency at the start of each
step and as the natural and applied frequencies become different the response
is less strong.
1. Introduction
3. Results and Discussion
Next Section : 4. Conclusions
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