Figure 3 presents two distributions of the two-dimensional circular cylinder on the time-mean pressure coefficients at θ=±90o and 180o versus Re in the critical transition range. The two cases named the Case 14 and Case 17-1 [14] were conducted within different time periods under different weather conditions. The real-time pressure data were obtained from the pressure taps at the level 1 shown in Fig. 1a.
An interesting feature learned described below. Despite that the distributions of the two cases consistently indicate that the critical transition falls in a range of Re= 3.5∙105 to 4∙105, the time-mean pressure coefficients obtained appear to scatter widely between the two cases. By examining the real-time pressure signals in this range, it was further suggested that the flow characteristics be categorized in five regimes, which are numbered and
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indicated in different colors in Fig. 3. Meanwhile, Table 1 provides additional remarks on the flow characteristics of each regime. Similar categorization can be found in the literature [1, 2], however the characteristic features of each of the regimes summarized in Table 1 were concluded from the present observations.
It is noted in the distribution in Fig. 3b that the regime (4) signifying the transition from one-bubble to two-bubble sub-regimes is missing, implying that the flow characteristics relevant to this regime were not confirmed by the experimental data sampled over 120 seconds at a fixed Reynolds number. In fact, during the experiment, repeated efforts were attempted to identify the regime (4), but no success. The situation learned was that the regime (4) was so unstable that would not be able to sustain over a sampling time period of 120 seconds. In addition, a minute change in the Reynolds number around this range would result in a switching between the regimes (3) to (5), as the regimes (3) and (5) behaved much more stable than regime (4) did.
FIGURE 3. Variations of pressure coefficient versus Re for two experiments made at different periods of time (a) case 14 and (b) case 17-1. [14]
Table 1. Categorization of five sub-regimes in the critical transition range. [14]
No. Color of area Regime Remarks
1 Light blue Pre-critical regime The percentage of time that the vortex shedding frequency is detectable is lower than 80%.
2 Pink Transition from pre-
critical to one-bubble age
The variations of the time-mean pressure coefficients at Θ=±900 are significant.
3 Khaki Steady one-bubble
regimes
The variations of the time-mean pressure coefficients at Θ =±900 are small but the difference between them is significant.
4 Light purple Transition from one- bubble to two-bubble
regimes
Transition between the steady one-bubble and two-bubble regimes. The variations of the time-mean pressure coefficients at Θ =±900 are
significant.
5 Light red Steady two-bubble regime
The variations and difference of the time-mean pressure coefficients at Θ =±90 are small compared to those in the one-bubble regime.
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FIGURE 4. Distributions of the real-time (Cp1+90- Cp1-90, Cpb) values with regard to the four regimes identified, namely, the regimes of subcritical, transition from subcritical to one-bubble, steady one-bubble regime and steady two-bubble.
The data were obtained from Case 17-1 [14]
The real-time flow characteristics in the critical transition range can be exemplified by Fig. 4. In the figure, the distributions of the real-time (Cp1+90- Cp1-90, Cpb) valuesof the four regimes, namely, subcritical, transition from subcritical to one-bubble, steady one-bubble regime and steady two-bubble regimes, are included for comparison.
The data (Cp1+90- Cp1-90, Cpb) are reduced from the real-time pressure measurements of Case 17-1. The quantity of Cp1+90- Cp1-90, i.e., the difference between the real-time pressure coefficients at θ= 90o and -90o, infers the deg ree of bias with regard to the flow distribution around the model. On the other hand, the quantity of Cpb is an indicator as far as the magnitude of drag is concerned. In the subcritical regime, the Cpb value can be lower than -1, whereas in the two-bubble regime the Cpb value can be higher than -0.4. In contrast to the three cases of subcritical, one-bubble and two-bubble regimes shown in the figure, of which scatterings of the data points are limited in small regions, the case of transition from subcritical to one-bubble regime shows very pronounced scatterings in Cp1+90- Cp1-90 , implying that the flow behaved highly unsteady, even non-stationary. This is the situation under which high pressure fluctuation intensity would be measured at θ= 90o and -90o on the circular cylinder.
The critical transition phenomenon was also realized for the finite circular cylinder model. For example, a situation corresponding to the transition from the one-bubble to two-bubble regimes is illustrated in Fig.5 for Re=3.99ꓫ105. It is seen in Fig. 5a that the intermittent switching between the one-bubble and two-bubble states is remarkable in the real-time sense, revealed by both of the real-time pressure signals obtained at the upper (z=3.5 D) and middle (z=2 D) levels. Moreover, the switching at the two levels is realized in a synchronous manner, inferring that either of the one-bubble or two-bubble states persisted along the spanwise direction.
The phenomenon of bi-states switching observed in Fig. 5a can be further examined in Fig. 5b, in which two distributions depict the variations of the real-time (Cp+90-Cp-90, Cpb) values at the upper and middle levels. Two distinguishable differences learned from the two distributions can be pointed out here. Compared to the distribution of the middle level, the distribution of the upper level reveals that in the one-bubble state, higher drag was resulted near the free end. On the other hand, regarding the two-bubble state at the two levels, the two bubbles formed near the free end would be more symmetric and smaller in size, in comparison with those at the middle level.
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FIGURE 5. (a) The real-time pressure signals obtained at θ= 90o, -90o and 180o at the upper (z=3.5 D) and middle levels (z=2 D) of the finite circular cylinder model, at Re=3.991ꓫ104. (b) The real-time (Cp+90-Cp-90, Cpb) distribution