Development of LSB in the transition of drag crisis The unsteady characteristics of LSB through the

transition regime are discussed in this section. Real-time pressure measurements made are found informative to unveil the unsteady characteristics of LSB. First of all, Figure 11 provides an example concerning the space-time variations of the pressure coefficients obtained on the upper and lower surfaces of the model at Re = 5 x 10⁴. Moreover, a plot depicting the instantaneous pressure

distribution around the model extracted from the real-time pressure signals at t= 30 s is presented in Figure 12. A summary of the appearances of Figures 11 and 12 unveils an interesting aspect of unsteady aerodynamic force, which is described as follows. Under this flow condition, LSB did not appear synchronously on the upper and lower surfaces. Thus, pronounced fluctuating forces would be generated. Meanwhile, it is learned Figure 11 that the separation and reattachment points of LSB were located at

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x/c = 0.2 and x/c = 0.5, respectively. This observation is consistent with the visualization result in the Figure 9.

An intermittency factor IF was defined to quantify the unsteady presence of LSB on the model surface. It is the ratio of the time of LSB existed on the model to the total time measured; namely, at an instant, 0 was assigned if there was no presence of LSB or 1 was assigned if LSB was present. It is noted that the LSB appears normally in a streamwise region from 0.2 to 0.5 chord length on either of the lower or upper surface. Accordingly, we checked the real-time pressure signals obtained in this chordwise region. The fraction of time corresponding to the presence of LSB was counted, as inferred by the real-time pressure coefficient lower than a given threshold value. Such a method was reported by Chopra & Mittal (2016) to reduce the intermittency factor of LSB on the surface of a circular cylinder.

An example of identifying the presence of LSB on the upper and lower surfaces at x/c = 0.4 at Re = 5 x 10⁴ are

illustrated in Figure 13 (a) and (b), respectively. Moreover, Figure 14 present the real-time CP distributions on the upper surface of the model at t= 10.8 s, 42.7 s and 45.1 s for comparison and discussion. It is clear that LSB was present at t = 10.8 s and t =45.1 s which have the values of CP lower than the threshold value, but no LSB present at t

= 42.7 s. The pressure distributions are noted similar to those reported by O’Meara and Muller (1987). It should be noted that the threshold value was chosen independently for every single case in this study.

The real-time pressure measurements were repeated three times for each of the cases studied in the transition range in order to validate the stationarity of the data measured. The results of four cases shown in the Table 1 indicate that while the intermittency factors of LSB obtained on the upper and lower surfaces might vary individually, but their averaged values appeared quite consistent.

(a)

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(b)

Figure 11 Space-time diagram of pressure coefficient CP on (a) upper surface (b) lower surface at Re=5x10⁴.

Figure 12 Pressure distributions on upper and lower surfaces measured at t = 30 s with Re=5 x 10⁴

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(a)

(b)

Figure 13 Time diagram of pressure coefficient CP at x/c = 0.4 on (a) upper surface (b) lower surface with Re = 5x10⁴

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Figure 1 Pressure distributions on the upper surface measured at t = 10.8 s, t = 42.7 s and t = 45.1 s at Re = 5x10⁴

Table 1 Intermittency factor of LSB on both surfaces over the drag crisis phenomenon

Variations of the intermittency factor IF, in terms of (1-IF), for the upper and lower surfaces and CD against Re are shown in Figure 15. It is seen that the trends of variations of both quantities against Re look similar.

Therefore, it is nature to link the drag crisis phenomenon with the intermittency of the LSB. The correlation was

pointed out in Choppa & Mittal (2017) and Deshpande, et al. (2017) for circular cylinder and sphere models.

In the literature, the effects of LSB on the aerodynamic performance were reported differently with regard to different airfoil configurations. Aholt &

Finaish (2011) noted that eliminating LSB could make the

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aerodynamic efficiency of airfoil improved with the reduction of drag up to 30%. Muller & Batill (1982) conducted experiments on NACA 663-018 airfoil and found that the drag coefficient of the airfoil increased due

to the appearance of LSB. On the contrary, Thake (2011) reported that due to the formation of LSB, a significant increase of lift and a reduction of drag were realized over a range of AOA for a NACA 643-618 airfoil.

Figure 15 CD and (1-IF) diagrams with respect to Re.

To illustrate the characteristics of Cl in the drag crisis regime of the pressure model, Figure 16 compares the distributions of 1-IF on the upper and lower surfaces with Cl, which was reduced from the pressure measurements on the model. As seen, the non-zero Cl values in the transition regime is correlated with the difference of the 1-IF values obtained from the upper and lower surfaces.

The LSBs on upper and lower surfaces reached to their full appearances at the same Re near the end of the transition range, although they behaved differently in the first half of the transition regime. This finding in the present study is noted in contrast to the result of Yousefi &

Razeghi (2018) with regard to the simulation of flow over a NACA airfoil at critical Reynolds numbers. In their study, the airfoil always experienced CL=0 at all Re studied. On the other hand, Muller & Batill (1982) noted an asymmetry

in the lift curve, particularly at small AOA, due to the asymmetry of the airfoil profile. It should be noted that the present teardrop airfoil has a very high ratio of the maximum thickness to the chord length which is 0.4, therefore it is very sensitive to the existence of LSB that can modify the pressure distribution on the surface effectively.

It is worthwhile to point out that the Cl data reduced from the pressure measurements are in good agreement with those obtained by the balance, CL. Figure 17 compares the lift coefficients reduced from the balance measurements and the pressure experiment at AOA=0 over the transition range of Re, which show a good agreement over the entire range of Re studied, particularly at higher Re post the transition.

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Figure 16 Cl and (1-IF) diagram with respect to Re at AOA=0^o.

Figure 17 CL and Cl diagrams with respect to Re at AOA = 0^o

3.3 Relation between drag crisis and drag pressure