4.1. Mean flow characteristics

Mean flow characteristics for various cases, as mentioned in table 1, are obtained based on the ensemble average of the PIV realizations acquired using the conventional PIV system. Mean velocity vectors overlaid with the contours of $u_{rms}/U_{r}$ are shown in figures 3 and 4 for various GISB and PISB cases, respectively. One may notice in figures 3 and 4 that the maximum value of $u_{rms}/U_{r}$ occurs in the downstream region of the maximum height of a separation bubble. Similar to $u_{rms}/U_{r}$ values, $v_{rms}/U_{r}$ values are also found to be higher in this region (not shown here for brevity). This can be attributed to the transition-to-turbulent activity in the separated shear layer. These figures also show that the size of a separation bubble (i.e. both length and height) reduces with increasing FST intensity and Reynolds numbers, as can be deduced from the $U=0$ line and the mean dividing streamline for both the GISB and PISB cases. These experimental observations for the present PISB cases are found to be similar to the recent numerical works of Balzer & Fasel (Reference Balzer and Fasel2016) and Hosseinverdi & Fasel (Reference Hosseinverdi and Fasel2019), among others. In addition, the present measurements reveal that the distance between the point of separation and the streamwise location of the maximum height of a separation bubble obtained based on the mean dividing streamline is large, as compared with the distance obtained based on the $U=0$ line, as can be seen in these figures. Interestingly, we also find that the distance of the location of the point of inflection in the velocity profile from the wall is equal to the numerical value of the displacement thickness, as can be seen in figures 3 and 4.

The mean velocity vectors in figures 3 and 4 also indicate that there exists a point of inflection in the mean velocity profile within the separated region, as expected. The ratio of the mean velocity at the point of inflection, $U_{in}$, and the edge velocity, $U_{e}$, as shown in figure 5, is found to be nearly constant for all the cases considered here. In fact, some published data (Häggmark et al. Reference Häggmark, Bakchinov and Alfredsson2000; Balzer & Fasel Reference Balzer and Fasel2016; Hosseinverdi & Fasel Reference Hosseinverdi and Fasel2019), also reveal that this is indeed correct. Here, $x_{s}$ and $l_{b}(=x_{r} - x_{s})$ denote the point of separation and the streamwise length of a separation bubble, respectively; $x_{r}$ denotes the point of reattachment. We may note that we have followed the procedure of Häggmark (Reference Häggmark2000) to find the numerical values of $x_{s}$ and $x_{r}$ at the wall. It should be further noted that the point of inflection is estimated based on the different curve fits to the experimental data, and the error bars, as shown in figure 5, indicate one standard deviation with respect to the mean values of $U_{in}/U_{e}$, obtained using different curve fits. The ratio, $U_{in}/U_{e} \approx 0.5$, appears to remain unaffected by the FST levels at least up to 3.3 % for both the GISB and PISB cases.

Figure 5. Ratio of the mean velocity at the point of inflection, $U_{in}$, and the shear layer edge velocity, $U_{e}$, for different cases. Ratios determined from the data of Häggmark et al. (Reference Häggmark, Bakchinov and Alfredsson2000), Hosseinverdi & Fasel (Reference Hosseinverdi and Fasel2019) and Balzer & Fasel (Reference Balzer and Fasel2016) are also shown in this figure.

For better comparison and to be more specific regarding the bubble dimensions, the $U=0$ line and the mean dividing streamline, respectively, are shown in figures 6(a,b) and 6(c,d) for various GISB and PISB cases, respectively; the numerical values of the point of separation, the maximum height and the reattachment point are detailed in the table 2. These figures clearly show that the bubble height is reduced, and the point of reattachment is shifted towards the leading edge with an increasing level of FST. Without enhancing the FST level, reduction in bubble size is also observed with increasing $Re$ (see the GISB-N1 and GISB-N2 cases in figure 6a,b). However, in contrast to the PISB cases (figure 6c,d), the point of separation and the bubble outline remain unchanged nearly up to the maximum height of the bubble for the GISB cases with increasing FST and $Re$, as seen in figure 6(a,b). This can be attributed to the fact that the free-stream flow with different levels of FST does not contain any streaky structures and separates at the leading edge for the GISB cases, whereas the attached boundary layer that can be laden with streaky structures based on the level of FST separates from the wall for the PISB cases, as shown and discussed in § 4.4. Therefore, initially, the bubble outline for the GISB cases remains the same, but it changes later on due to the rapid transition-to-turbulent process in the separated shear layer triggered by a high level of FST. On the other hand, the transition-to-turbulent characteristics in the separated shear layer for the PISB cases depend not only on the different levels of FST but also on the nature of the attached boundary layer getting separated due to an adverse pressure gradient. Another contrasting observation between the GISB and PISB cases is that the ratio of the bubble length to its maximum height is approximately constant for the GISB cases, whereas this ratio is found to increase for the PISB cases with an increasing FST level (see table 2).

Figure 6. Comparison of the $U = 0$ line and the mean dividing streamline for various cases considered and their self-similar characteristic. Descriptions of symbols used in this figure are detailed in table 1. (a,b) The $U = 0$ line and the mean dividing streamline, respectively, for various GISB cases. (c,d) The $U = 0$ line and the mean dividing streamline, respectively, for various PISB cases. (e, f) Self-similar characteristics of the $U = 0$ line and the mean dividing streamline, respectively, for various GISB cases. (g,h) Self-similar characteristics of the $U = 0$ line and the mean dividing streamline, respectively, for various PISB cases. (i) Self-similar characteristics of the $U = 0$ line for the data of Simoni et al. (Reference Simoni, Lengani, Ubaldi, Zunino and Dellacasagrande2017). ( j) Self-similar characteristics of the mean dividing streamline for the data of Balzer & Fasel (Reference Balzer and Fasel2016).

Table 2. Various parameters of a separation bubble for different cases considered. Here, $x_{s}$, $x_{m}$, $x_{r}$, $y_{max}$ and $l_{b}$ indicate point of separation, streamwise location of the maximum height, point of reattachment, the maximum height based on $U = 0$ line and the length of a separation bubble, respectively. Large variation of $l_{b}/y_{max}$ may be noted for the PISB cases, as compared with the GISB cases.

To investigate the self-similar nature of the bubble outlines, the streamwise and wall-normal coordinates of the $U=0$ line and the mean dividing streamline are normalized by the respective bubble length and the maximum height, as shown in figures 6(e) and 6( f) for the GISB cases, and in figures 6(g) and 6(h) for the PISB cases, respectively. One may notice that the bubble outline in terms of the mean dividing streamline shows a better collapse of the data for both the GISB and PISB cases (see figures 6f and 6h in comparison with figures 6e and 6g, respectively). An exact collapse of the data for the mean dividing streamline is found for the GISB cases, whereas a small deviation may be seen for the PISB cases. In fact, a similar collapse of the data for the $U=0$ line and the mean dividing streamline for various levels of FST for PISBs are also found for the experimental and numerical data of Simoni et al. (Reference Simoni, Lengani, Ubaldi, Zunino and Dellacasagrande2017) and Balzer & Fasel (Reference Balzer and Fasel2016), as shown in figures 6(i) and 6( j), respectively.

To investigate the nature of disturbance growth in a separation bubble, variations of the normalized peak $u_{rms}$ and the peak $v_{rms}$ in the streamwise direction are shown in figure 7. One may clearly notice an exponential growth regime exists for all the cases considered here. At the initial stage after separation for the GISB cases, there is a jump in the growth of $u_{rms,max}/U_{r}$, as seen at approximately $x/h$ = 0.5–1 in figure 7(a,b). This is found to be due to the presence of small shear layer vortices near to the blunt leading edge of the flat plate (see figure 8a,b, for example).

Figure 7. Streamwise variation of $u_{rms,max}/U_{r}$ and $v_{rms,max}/U_{r}$ for GISB cases (a,b,d,e) and PISB cases (c, f). Description of symbols is given in table 1. Separation and reattachment points for each case are shown by filled grey and black coloured symbols, respectively.

Figure 8. Time sequence of the roll-up process and vortex shedding in terms of the spanwise vorticity contours for various cases. Results are shown for the (a) GISB-N2 case, (b) GISB-B2 case, (c) PISB-N2 case, (d) PISB-B2 case.

The exponential growth of $v_{rms,max}/U_{r}$ in figure 7(d–f) is attributed to the significant increase of the $v$ fluctuation due to the vortex shedding associated with the separated shear layer. That is, the rapid increase of $v_{rms, max}/U_{r}$ can be considered as the starting location of the shear layer roll-up/vortex shedding. Kirk & Yarusevych (Reference Kirk and Yarusevych2017) also observed that the shear layer roll-up location is near to a point where the pressure fluctuation increases rapidly. Moreover, figure 7(d,e) shows that the onset of the exponential growth is shifted towards the leading edge under the high level of FST for the GISB cases, whereas the onset of the exponential growth is delayed and shifted in the downstream direction for the PISB cases, as seen in figure 7( f). This is because of the fact that the presence of a high level of initial disturbance in the free-stream flow leads to the rapid transition and shear layer roll-up for the GISB cases. On the other hand, the streaky structures are generated at an enhanced level of FST in the attached boundary layer prior to its separation for the PISB cases (as shown and discussed in § 4.4). These streaky structures delay the flow separation (e.g. Dellacasagrande et al. Reference Dellacasagrande, Barsi, Lengani, Simoni and Verdoya2020; Karp & Hack Reference Karp and Hack2020; Xu & Wu Reference Xu and Wu2021).

4.2. Unsteady flow characteristics in wall-normal ($x\unicode{x2013}y$) plane

The growth of disturbance in a separated shear layer eventually leads to the roll-up of the shear layer into a vortex that detaches from the shear layer and initiates vortex shedding at some downstream distance from the point of separation (e.g. Pauley et al. Reference Pauley, Moin and Reynolds1990; Watmuff Reference Watmuff1997). To investigate the detailed shedding characteristics of these vortices, the TR-PIV measurements were carried out in the wall-normal plane for both the GISB and PISB cases. Time sequences of such vortex shedding with and without an enhanced level of FST are shown in figure 8 for both the GISB and PISB cases. The displacement thickness curve denoted by a dashed line is seen to pass through the concentrated vorticity region for all the cases shown here. This indicates that the locus of the cores of shed vortices can be described by the displacement thickness curve, on an average. Unlike the PISB cases, the vorticity contours for the GISB-N2 and GISB-B2 cases indicate the presence of small vortices near the leading edge of the flat plate and large-scale vortices near the maximum height of the bubble (see the rectangular zone). The presence of such small vortices near the leading edge of a square cylinder is also reported by Brun et al. (Reference Brun, Aubrun, Goossens and Ravier2008), who call these small-scale vortices the KH vortices. For the present GISB cases, the separated shear layers containing such small vortices near the leading edge are called shear layer vortices. Figure 8(a,b) show that these small vortices eventually rolls up into large vortices near the maximum height of the bubble, which are referred to here as the shedding vortices.

Figure 9 shows the spectral analyses of $u$ and $v$ velocities at three different points (P1, P2, P3), as shown by white coloured solid symbols in figure 8. Figures 9(a–d) and 9(e–h) show the frequency spectra for $u$ and $v$ velocities, respectively. It should be noted that the frequency, $f$, has been normalized as $f^{*}={fh}/{U_{r}}$, and as $F^{*} (=2{\rm \pi} f\nu /U_r^2\times 10^6)$, where $\nu$ is kinematic viscosity of air. Figure 9 shows that the spectral analysis of $v$ velocity signal can clearly identify the underlying dominant peaks, as compared with the spectral analysis of $u$ velocity signal.

Figure 9. Power spectral density (PSD) of the fluctuating $u$ and $v$ velocity components at three different locations (P1, P2, P3), as shown by solid white symbols in the first panels of figure 8(a–d). (a–d) Estimated PSD of $u$ velocity component using Welch's method. (e–h) Estimated PSD of $v$ velocity component using Welch's method.

The dominant spectral peaks at points P1 and P2 for both the GISB-N2 and GISB-B2 cases are found to be 0.59, whereas the dominant peaks at point P3 for these cases are found to be 0.38, as clearly seen in figure 9(e, f). The peak at 0.59 is attributed to the presence of the shear layer vortices, as discussed above, and the peak at $\approx$0.38 is due to the shedding vortices. It is interesting to note that the peaks associated with both the shear layer vortices and the shedding vortices are not changed even at an enhanced level of FST for the GISB cases. Dominant shedding frequencies and the Strouhal number ($St_{\theta }$) based on $\theta$ and $U_e$ at the point of separation for various cases are also tabulated in table 3. For the PISB cases, the values of $St_{\theta }$ are found to be comparable with the literature (Pauley Reference Pauley1994; Watmuff Reference Watmuff1999; Rodríguez, Gennaro & Souza Reference Rodríguez, Gennaro and Souza2021). The peak frequencies for the PISB cases are found to decrease at an enhanced level of FST, as clearly seen in figure 9(g,h). This is attributed to the interaction of the boundary layer streaks with the fore part of the separation bubble, as discussed in § 4.4. This interaction can reduce the distance between the wall and the bubble outline, leading to a reduction in frequency.

Table 3. Measured dominant frequencies for various cases.

Considering the presence/absence of such wall-vortex interaction, a criterion based on integral flow parameters has been proposed to determine whether the shedding frequency will remain the same with and without an enhanced level of FST. The location of a vortex core from the wall can be approximated as $\delta ^*$, as discussed earlier. Similarly, on an average, the maximum wall-normal extension of a vortex may be approximated as the boundary layer thickness, $\delta$, which is defined as the wall-normal height where the local velocity equals $U_e$. Hence, the distance between the core of a vortex and its top edge can be considered as $(\delta -\delta ^*)$. A symmetrical line with respect to the $\delta ^*$ line can be considered as the bottom edge of the vortex. Therefore, the distance between such a symmetry line and the wall is $\delta ^*-(\delta -\delta ^*)$, and a vortex well above the wall is expected to be confined within the lines between $\delta$ and $\delta ^*-(\delta -\delta ^*)$. Three white lines representing $\delta$, $\delta -\delta ^*$ and $\delta ^*-(\delta -\delta ^*)$, based on the above discussions, are shown in the last panels of figure 8(a–d), respectively. The core of a vortex, its top and bottom edges are found to be confined well within the lines of $\delta$ and $\delta ^*-(\delta -\delta ^*)$ for the GISB-N2, GISB-B2 and PISB-N2 cases. Hence, when the line $\delta ^*-(\delta -\delta ^*)$ touches the wall, there is a possibility of a wall-vortex interaction. That is, $\delta ^*-(\delta -\delta ^*)$ $\leq$ 0 or ${2\delta ^*}/{\delta } \leq 1$ leads to a condition for a wall interaction. It should be noted that the $\delta ^*-(\delta -\delta ^*)$ line is not shown for the PISB-B2 case because this value is less than zero. However, figure 10 shows that the value of ${2\delta ^*}/{\delta }$ near the maximum height of the bubble is greater than 1 for the GISB-N2, GISB-B2 and PISB-N2 cases, but it is less than 1 for the PISB-B2 case, indicating the presence of the wall-vortex interaction for this case. The numerical data of Balzer & Fasel (Reference Balzer and Fasel2016) show that the value of ${2\delta ^*}/{\delta }$ is greater than 1 (see figure 10). This indicates the absence of the wall-vortex interaction, which led to a constant shedding frequency even with an enhanced level of FST in their case. In fact, for further confirmation of the above observation, the TR-PIV measurements were carried out for a different adverse pressure gradient set-up with the same flow condition (not reported here for brevity). Using similar analyses of the TR-PIV data, the present observation was found to be consistent with the new pressure gradient set-up as well.

Figure 10. Variation of ${2\delta ^*}/{\delta }$ along the length of a separation bubble for various cases.

To ensure the fact that the dominant frequencies in table 3 are associated with the shedding vortices, the POD analysis of the fluctuating $v$ velocity component (v-POD) has been carried out for both the GISB and PISB cases. The v-POD analysis has been performed in the spatial region represented by a rectangle for the GISB cases, as the vortex shedding is found to be active in this zone (see figure 8a,b). However, for the PISB cases, the v-POD analysis has been carried out over the entire spatial regions shown in figure 8(c,d).

Figure 11 shows the POD analysis of the TR-PIV data for various cases. The relative energy obtained from the v-POD analysis is shown in figure 11(a–d) for different cases. In all these cases, the energy of the first two modes is found to be nearly the same. This indicates that the first two modes are coupled, similar to the observation of Lengani et al. (Reference Lengani, Simoni, Ubaldi and Zunino2014). It is important to note that the first two modes are also found to be coupled in the PISB-B2 case, even though the wall influence is present, as discussed above. The first and second v-POD modes, as shown in figure 11(e–h), for various cases, reveal the same mode shapes with just a spatial shift. A similar shift in time can also be noticed for the corresponding time coefficients, $a_{1}$ and $a_{2}$, as shown in figure 11(i–l). A space–time shift of the POD modes with a nearly equal energy level indicates the presence of a travelling disturbance/wave, and the lack of exact symmetry in the space–time data is attributed to the fact that the associated convecting disturbance is a modulated travelling wave (e.g. Hasan & Sanghi Reference Hasan and Sanghi2007; Mandal et al. Reference Mandal, Venkatakrishnan and Dey2010). The spectra of $a_{1}$ and $a_{2}$, obtained using the Welch method, reveal nearly the same peak frequency, as shown in figure 11(m–p), for all the four cases. These peak frequency values estimated from the POD analysis for each case are also found to be consistent with the spectral analysis of the actual $v$ velocity signal in the vortex shedding region (see figure 9). Therefore, the first two dominant POD modes represent the vortex shedding around the maximum height of the bubble, revealing the fact that the vortex shedding mode due to the spatially evolving disturbances dominates the separated flow dynamics. Since the vortex shedding frequency decreases for the PISB-B2 case, it is interesting to investigate how the convection/phase velocity and wavelength of the shed vortices contribute to this change.

Figure 11. The POD analysis of the fluctuating $v$ velocity for the data shown in figure 8(a–d). The POD analysis for the GISB cases was carried out in the selected rectangular zones, as shown in the first panels of figure 8(a) and figure 8(b). Figures in (a,e,i,m), (b, f, j,n), (c,g,k,o) and (d,h,l,p) correspond to GISB-N2, GISB-B2, PISB-N2 and PISB-B2 cases, respectively. (a–d) Relative energy of the POD modes for various cases. (e–h) The first and second POD modes. Symbols: —-, $U = 0$ line; - - -, mean dividing streamline. (i–l) Time coefficients of the first and second POD modes. (m–p) The PSD of the time coefficients of the first and second POD modes for different cases.

To find the convection velocity of the vortex shedding, the $v$ fluctuating velocity signal is extracted at $y=\delta ^*$ for three different streamwise locations (e.g. at $\Delta x$, 2$\Delta x$, 3$\Delta x$, with an appropriate value of $\Delta x$) in the vortex shedding region. Following (e.g. Boutilier & Yarusevych Reference Boutilier and Yarusevych2012), the cross-correlation analysis of the extracted time-series signals is then carried out to estimate the time taken by the extracted signals to travel from one location to another location. A similar analysis has also been carried out for the filtered velocity field to find out the phase velocity associated with the peak frequencies in figure 11(m–p). It should be noted that the filtered velocity field was obtained based on a 1 Hz band pass filter around the peak frequency. However, the convection velocity and the phase velocity, thus obtained, are found to be nearly constant even at an enhanced level of FST, as shown in figures 12(a) and 12(b), for the GISB and PISB cases, respectively. It is interesting to note that the convection velocity and the phase velocity remain the same either for the GISB or the PISB cases with and without an enhanced level of FST, even though the wall influence is present for the PISB-B2 case.

Figure 12. Streamwise variation of the convection and the phase velocity. Results are shown for the (a) GISB cases and (b) PISB cases.

Similarly, using the v-POD analysis in the shedding region, we estimated the wavelength of the shed vortices. As the positive and negative regions of the spatial v-POD modes (for example, see figure 11e–h) are associated with a shedding structure, the distance between peak-to-peak/crest-to-crest of the spatial data extracted along the streamwise direction (at approximately $y=\delta ^*$) provides the wavelength of the shed vortices (e.g. Lengani et al. Reference Lengani, Simoni, Ubaldi and Zunino2014). The normalized values of the wavelength and the wavenumber are given in table 4. Even at an enhanced level of FST, the wavelength of the shed vortices is found to be nearly the same for the GISB cases, whereas it is found to change for the PISB cases.

Table 4. Normalized wavelength and wavenumber estimated from the spatial POD modes for different cases.

4.3. Linear stability analysis

The analyses in the previous section show that the vortex shedding at an enhanced level of FST is not bypassed for both the GISB and PISB cases. It is then legitimate to ask whether a LSA, even for the case of an enhanced level of FST, can describe the disturbance characteristics. Various curve fits are often utilized to fit the measured mean velocity data in a separation bubble for the local stability analysis based on the parallel flow assumption (e.g. Boutilier & Yarusevych Reference Boutilier and Yarusevych2012; Balzer & Fasel Reference Balzer and Fasel2016). Even though the separated flow is not strictly parallel, the results of the LSA based on the parallel flow assumption closely represent the experimental observations (e.g. Diwan & Ramesh Reference Diwan and Ramesh2009; Boutilier & Yarusevych Reference Boutilier and Yarusevych2012; Diwan & Ramesh Reference Diwan and Ramesh2012). However, following the works of Boutilier & Yarusevych (Reference Boutilier and Yarusevych2010, Reference Boutilier and Yarusevych2012), we use the fit proposed by Dovgal et al. (Reference Dovgal, Kozlov and Michalke1994), that is,

(4.1)\begin{align} U(y_{1}) = [\mathrm{tanh}(a(y_{1} - d)) + \mathrm{tanh}(ad)]/[1 + \mathrm{tanh}(ad)] + b\sqrt{3}\mathrm{exp}\left[- 1.5\frac{y_{1}^{2}}{d^{2}} + 0.5\right]. \end{align}

Here, the constant $b$ is used to control the backflow in the velocity profiles, the constant, $d$, is the normalized distance of the inflection point from the wall, and the variable, $y_{1}$, is the normalized distance from the wall. The momentum thickness, $\theta$, and the shear layer edge velocity, $U_{e}$, are used as the length and the velocity scales for normalization, respectively. The constant, $a$, has to be fixed based on the fact that the equation, $\int ^{\infty }_{0}U(y_{1})[1-U(y_{1})]\,{\rm d}y_{1} = 1$, is satisfied.

Figure 13 shows the experimentally measured and the corresponding fitted velocity profiles for different cases. These velocity profiles are chosen at the streamwise locations where $v_{rms, max}$ begin to grow exponentially (see figure 7e, f). The curve-fitted velocity profiles closely follow the experimental data. Wall-normal locations of the mean dividing streamline ($y_d$) and the point of inflection ($y_{in}$) are also shown in figure 13. One may notice that the point of inflection lies above the mean dividing streamline for all the velocity profiles. This observation appears to be consistent for some other streamwise locations as well, as the ratio of $y_{in}$ to $y_{d}$ is found to be greater than 1 at those locations, as shown in figure 14. Based on the stability analyses of some analytical base flow profiles $U(y)$ in a separation bubble, Avanci, Rodríguez & Alves (Reference Avanci, Rodríguez and Alves2019) found $y_{in}\leq 0.9y_{d}$ for the absolute instability of the base flow profiles. Although the present numerical values of $y_{in}$ for the PISB cases are clearly greater than 0.9$y_{d}$, the values of $y_{in}$ for the GISB cases at some streamwise locations are found to be just marginally higher than the proposed critical value. Therefore, it is difficult to rule out the presence of absolute instability at least for the GISB cases, as the non-parallel effect is also significant for the GISB cases (see figure 3). The different dynamics and the characteristics between the GISB and PISB cases may also be related to the fact that a GISB is more prone to be absolutely unstable as compared with a PISB. However, further studies are essential for a strong conclusive outcome in this regard.

Figure 13. Dovgal's curve fit to the measured velocity profiles at different locations. Symbols: $\circ$, measured data; —, Dovgal's curve fit; *, location of the inflection point ($y_{in}$); $\diamondsuit$, location of the mean dividing streamline ($y_{d}$). (a) Velocity profile for the GISB-N2 case at $x/h = 1.26$ with curve-fit constants, $a = 0.262$, $b = - 0.075$, $d = 22$. (b) Velocity profile for the GISB-B2 case at $x/h = 0.94$ with curve-fit constants, $a = 0.3002$, $b = - 0.09$, $d = 13.7$. (c) Velocity profile for the PISB-N2 case at $x/h = 50.03$ with curve-fit constants, $a = 0.3718$, $b = - 0.08$, $d = 8.353$. (d) Velocity profile for the PISB-B2 case at $x/h = 46.93$ with curve-fit constants, $a = 0.427$, $b = - 0.038$, $d = 4.096$.

Figure 14. A ratio of $y_{in}/y_{d}$ along the streamwsie direction. Results are shown for the (a) GISB cases and (b) PISB cases.

To determine the spatial linear stability characteristics of the velocity profiles, as shown in figure 13, the OSE and Rayleigh equations, as detailed in § 3.2, are solved considering parallel flow approximation. Figure 15 shows the growth rates ($-\alpha _{i}h$) variation with $f^{*}(=fh/U_{r})$ for the above velocity profiles. In this figure the normalized circular frequency, $\omega ^{*} (= \tfrac {1}{4}\delta _w({2{\rm \pi} f}/{U_m}))$, and the spectra obtained from the v-POD analysis are also superposed for comparison purposes. Here, $\delta_{w}$ and $U_{m}$ denote the vorticity thickness and the average of the minimum and maximum velocities in the separated shear layer, respectively (see Diwan and Ramesh Reference Diwan and Ramesh2009, for details). The peak frequencies obtained from the viscous and the inviscid stability analyses for the respective cases are found to be the same, with nearly comparable maximum growth rates for the viscous (OSE) and inviscid (Rayleigh) calculations. Also, the growth rates are nearly the same for both the GISB cases with and without an enhanced level of FST, whereas a reduction in growth rates can be seen for the PISB case with an enhanced level of FST, as compared with the case without an enhanced level of FST. This is attributed to the presence of streamwise velocity streaks, as discussed below in § 4.4; these streaks can modify the distance between the inflection point and the wall, eventually leading to a reduction of the growth rate with an increase of FST level. Most importantly, we find that the experimental peak frequencies closely follow the most unstable frequencies obtained from the LSA, even for an enhanced level of FST. Moreover, figure 15 reveals that, with and without an enhanced level of FST at the same free-stream velocity, the most unstable frequency/shedding frequency is nearly constant for the GISB cases, whereas it changes for the PISB cases.

Figure 15. Calculated spatial growth rates obtained from the LSA for the corresponding velocity profiles shown in figure 13. The PSD of POD time coefficients for the first and second POD modes, as shown in figure 11(m–p), are reproduced here for comparison purpose. Results are shown for the (a) GISB-N2 case, (b) GISB-B2 case, (c) PISB-N2 case, (d) PISB-B2 case.

The existence of periodic vortex shedding from the separated shear layer is often attributed to the KH instability mechanism (e.g. Watmuff Reference Watmuff1999; Spalart & Strelets Reference Spalart and Strelets2000; Simoni et al. Reference Simoni, Ubaldi, Zunino and Bertini2012a). A criterion based on the normalized circular frequency, $\omega ^{*}$, is often used to determine the presence of such a mechanism (e.g. Watmuff Reference Watmuff1999; Simoni et al. Reference Simoni, Ubaldi, Zunino and Bertini2012a,Reference Simoni, Ubaldi, Zunino, Lengani and Bertinib). The criterion is actually adopted from the inviscid stability analysis of a mixing layer velocity profile by Monkewitz & Huerre (Reference Monkewitz and Huerre1982) who found $\omega ^{*}=[0.21 \quad 0.22]$ for a wide range of velocity ratios. However, we find the value of $\omega ^{*}$ $\approx$ 0.21 for the GISB-N2, GISB-B2 and PISB-N2 cases, whereas it is approximately $\approx$ 0.18 for the PISB-B2 case. Based on the $\omega ^{*}$ criterion, the inviscid instability mechanism is bypassed for the PSIB-B2 case, even though the present linear stability analyses can closely describe the shedding frequency for this case as well. Therefore, it can be inferred that the $\omega ^{*}$ criterion is not valid for the PISB-B2 case. Furthermore, the wavenumber spectra for the unstable modes are also found to compare well with the experimentally estimated wavenumbers even at an enhanced level of FST, as shown in figure 16; here, the wavenumbers were estimated using the frequency and the phase velocity, as discussed in § 4.2. Similarly, the eigenmodes closely match with their experimental counterparts within the unstable band of the linear stability frequency, as shown in figures 17 and 18, for the GISB and the PISB cases, respectively. In fact, a better match can be seen at approximately the shedding frequency for all the cases. It should be emphasised that even though the shedding frequency changes at an enhanced level of FST for the PISB-B2 case, the LST mode shape still matches with its experimental counterpart. For all the cases, these figures also show that the eigenmodes obtained from the OSE and the Rayleigh equations are found to be the same except very close to the wall. The occurrences of the peak of the mode shapes at $y \approx \delta ^{*}$ also lead to the fact that the disturbances get amplified due to the inflectional instability mechanism, as the point of inflection in the velocity profile is found to be at $y_{in}= \delta ^{*}$ (Marxen, Lang & Rist Reference Marxen, Lang and Rist2012). Being guided by the work of Villermaux (Reference Villermaux1998), and using the inviscid LSA of a piecewise linear velocity profile in the presence of a wall, Diwan & Ramesh (Reference Diwan and Ramesh2009) arrived at

(4.2)\begin{equation} \frac{f(y_{in}^{2}+\delta_{w}^{2})}{\nu} \sim \frac{U_{in}y_{in}}{\nu}\sqrt{\frac{y_{in}}{\delta_{w}}} \left(=\bar{R}\sqrt{\frac{y_{in}}{\delta_{w}}}\right) \end{equation}

for the most amplified linear stability frequency. The present data, with and without an enhanced level of FST, compare very well with their scaling relation, as shown in figure 19(a). However, as the present study reveals that $U_{in}/U_{e}$ = 0.5 (figure 5), and $y_{in} \approx \delta ^{*}$ (figures 3 and 4), the above scaling relation can be recast as

(4.3)\begin{equation} \frac{f({\delta^{*}}^{2}+\delta_{w}^{2})}{\nu} \sim \frac{U_{e}\delta^{*}}{\nu}\sqrt{\frac{\delta^{*}}{\delta_{w}}} \left(=\bar{R}_{m}\sqrt{\frac{\delta^{*}}{\delta_{w}}}\right). \end{equation}

This modified relation is also found to be universal, as it holds good not only for the present data but also for the available data in the literature, as shown in figure 19(b). In fact, with a proportionality constant, one can easily estimate the most unstable frequency for a given velocity profile from the modified scaling relation, as it is easier to find the values of $U_{e}$ and $\delta ^{*}$ as compared with the values of $U_{in}$ and $y_{in}$ from an experimental velocity profile.

Figure 16. Wavenumber spectrum obtained from the LSA and the estimated wavenumber obtained from the experimental data at $x/h$ = 1.23, 1.89, 50.03 and 46.93 for the (a) GISB-N2 case, (b) GISB-B2 case, (c) PISB-N2 case and (d) PISB-B2 case, respectively. Description of lines: solid line, viscous (OSE) solution; dashed line, inviscid (Rayleigh) solution. Symbols with error bars represent experimental data.

Figure 17. Comparison of the eigenmodes with their experimental counterparts for the GISB cases; here, $\mid u\mid (=\frac{\mid\hat{u}\mid}{\mid\hat{u}\mid_{max}})$ and $\mid v\mid(=\frac{\mid\hat{v}\mid}{\mid\hat{v}\mid_{max}}$) are the magnitudes of u and v eigenfunctions, respectively. Experimental eigenmodes are obtained from the r.m.s. values of the filtered velocities. Results are shown for the (a) GISB-N2 and (b) GISB-B2 cases. Description of lines: ${-}$, red, viscous (OSE) solution; ${--}$, red, inviscid (Rayleigh) solution. Symbols: ${\circ }$, red and ${\triangle }$, red, experimental data.

Figure 18. Comparison of the eigenmodes with their experimental counterparts for the PISB cases; here, $\mid u\mid (=\frac{\mid\hat{u}\mid}{\mid\hat{u}\mid_{max}})$ and $\mid v\mid(=\frac{\mid\hat{v}\mid}{\mid\hat{v}\mid_{max}})$ are the magnitudes of u and v eigenfunctions, respectively. Experimental eigenmodes are obtained from the r.m.s. values of the filtered velocities. Results are shown for the (a) PISB-N2 and (b) PISB-B2 cases. Description of lines: ${-}$, blue, viscous (OSE) solution; ${--}$, blue, inviscid (Rayleigh) solution. Symbols: ${\circ }$, blue and ${\triangle }$, blue, experimental data.

Figure 19. (a) Comparison of the most amplified frequency with the scaling relation of Diwan & Ramesh (Reference Diwan and Ramesh2009). (b) Comparison of the most amplified frequency with the modified scaling relation.

The growth rates, the wavenumber spectra and the eigenmodes (figures 15–19), as obtained from the solutions of the Rayleigh and the OSEs, are closely comparable. This is expected as the velocity profiles in separated flows are usually inflectional in nature (figure 5). The experimental data also compare well with these results. These observations imply that, for both the GISB and PISB cases, the inviscid inflectional instability mechanism dominates the disturbance growth in a separated shear layer even at an enhanced level of FST. Therefore, one needs to be careful enough to conclude that the linear stages of transition are bypassed just based on the fact that $\omega ^{*}$ is less than 0.21, as found for the PISB-B2 case (see figure 15), or the shedding frequency peak is not detectable in the spectra of the $u$ velocity signal, or the peak frequency reduces at an enhanced level of FST.

4.4. Unsteady flow characteristics in the wall-parallel ($x\unicode{x2013}z$) plane

To investigate the nature of the spanwise flow inside a separation bubble, the conventional PIV measurements have been carried out in the spanwise plane located at $y/h= 0.167$ ($y = 2$ mm) for both the GISB and PISB cases. One may note that this wall-normal height corresponds to $y/\delta ^*_{max}=0.2, 0.3, 0.15, 0.3$ for the GISB-N2, GISB-B2, PISB-N2 and PISB-B2 cases, respectively. Here, $\delta ^*_{max}$ represent the maximum displacement thickness value estimated from the wall-normal plane measurement. Figure 20(a–d) shows the typical four instantaneous PIV realizations in the spanwise plane for four different cases. Patterns of the instantaneous velocity contours and the fluctuating velocity vectors are not indicative of unsteady streaky structures for both the GISB cases. Instead, small $\varLambda$-like patterns in the instantaneous velocity contours are clearly visible for both the cases (i.e. for the cases with and without a turbulent generating grid in the test section). Our measurements at the lower Reynolds number indicate that these $\varLambda$-like patterns perhaps originate due to the distortion of the spanwise roller structures (not shown here for brevity). The streaky structures in a transitional boundary layer at an enhanced level of FST usually show elongated high- and low-velocity regions often accompanied by organized positive and negative $u$ fluctuations in the streamwise directions (e.g. Jacobs & Durbin Reference Jacobs and Durbin2001; Mandal et al. Reference Mandal, Venkatakrishnan and Dey2010; Balamurugan & Mandal Reference Balamurugan and Mandal2017). Since figure 20(c) does not show any such feature, one can be assured that the streaky structures are absent for the PISB-N2 case. On the other hand, one can clearly see the signature of streaky structures in figure 20(d) for the PISB-B2 case with an increasing level of FST, as the velocity contours reveal elongated high- and low-velocity regions dominated by organized positive and negative $u$ fluctuations (as the $w$ component is negligible), respectively. This indicates the clear existence of the K mode for the PISB-B2 case with an enhanced level of FST, similar to the numerically simulated ones reported by Balzer & Fasel (Reference Balzer and Fasel2016). These instantaneous observations are also statistically verified using the linear stochastic analyses detailed in Appendix A (see figure 25 in Appendix A).

Figure 20. Four typical instantaneous PIV realizations in the spanwise plane for four different cases (a–d). Fluctuating velocity vectors are overlaid with the contours of the instantaneous streamwise velocity in the spanwise plane. Black solid and dashed lines show approximate separation and reattachment locations for each case. The zoomed view of the $\varLambda$-like structure for the cases of GISB-N2 and GISB-B2 are shown by the subfigures in (a,b), respectively.

The above results for the PISB at an enhanced level of FST clearly show the high- and low-velocity streaks in a separated boundary layer. However, the role of these streaky structures in the secondary instability process of a roller vortex has not been addressed in the literature, to the best of our knowledge. For such an investigation, the TR-PIV measurements in the spanwise plane at different wall-normal locations are carried out for the PISB cases.

The time sequences of the fluctuating velocity fields at two different wall-normal locations, $y/h=0.8$ and $y/h=1.3$ that corresponds to $y/\delta ^*_{max}= 0.9, 1.3$, are shown in figures 21(a) and 21(b), respectively, for the PISB-N2 case. This figure shows some periodical patches of intense velocity fluctuations embedded in the spanwise roller, as identified from the $u$ velocity contours. These patches are often accompanied by some counter-rotating vortices as identified by the swirling strength contour, which is the imaginary part of the complex eigenvalue of the local velocity gradient tensor (Chong, Perry & Cantwell Reference Chong, Perry and Cantwell1990; Adrian, Christensen & Liu Reference Adrian, Christensen and Liu2000). However, this indicates the secondary instability of the separated shear layer. The spanwise wavelengths ($\lambda _z$) of these patches (see figure 21a) at $y/h=0.8$ and $y/h = 1.3$ are found to be $1.9(\pm 0.3)$, and $2(\pm 0.3)$, respectively. Considering the streamwise wavelength ($\lambda _x$) from table 4, the ratio, $\lambda _z/\lambda _x$, is found to be 0.64 and 0.68 at $y/h=0.8$ and $1.3$, respectively. These values are within the range (0.41–1) reported in the literature (e.g. Marxen et al. Reference Marxen, Lang and Rist2013; Michelis, Yarusevych & Kotsonis Reference Michelis, Yarusevych and Kotsonis2018b; Michelis, Kotsonis & Yarusevych Reference Michelis, Kotsonis and Yarusevych2018a; Hosseinverdi & Fasel Reference Hosseinverdi and Fasel2019) for the PISBs. Interestingly, the present data compare well with the translative instability value ($\lambda _z/\lambda _x=0.67$) of a mixing layer (Pierrehumbert & Widnall Reference Pierrehumbert and Widnall1982). This secondary instability is due to the stretching of the spanwise vortex in the streamwise direction. The TR-PIV measurements in the wall-normal plane clearly reveal that the shear layer roll-up is not bypassed even at an enhanced level of FST, and the shedding frequency can also be described by the linear instability analysis. Then an obvious question is: What is the role of a boundary layer streak in a separated shear layer? To elucidate this, time sequences of some TR-PIV realizations at two different wall-normal locations $y/h=0.4$ and $0.8$ that corresponds to $y/\delta ^*_{max}$= 0.8 and 1.6, are shown in figures 22(a) and 22(b), respectively. An oscillating low-speed streak, as indicated by a black arrow, can clearly be seen in figure 22(a). From such similar sinuous streak oscillations, Lengani et al. (Reference Lengani, Simoni, Ubaldi, Zunino and Bertini2017) concluded that the streak oscillation leads to the streak breakdown, initiating the transition process in a PISB subjected to an enhanced level of FST. On the other hand, the present oscillating streak is not seen to break down to progressively smaller scales. Interestingly, the measurements at a higher wall-normal location (at $y/h=0.8$) do not reveal any streaky structure (see figure 22b). However, one can clearly see some packets of intense fluctuations, as shown in figure 22(b). These small intense fluctuations are accompanied by some counter-rotating vortices, as identified by the swirling strength contours. The spatial distribution of these packets clearly reveals a $\varLambda$-like inclined structure in the spanwise plane (marked by a white coloured line), which seems to have originated from the spanwise roller. This suggests that the streaky structures are perhaps responsible for the distortion of the spanwise roller.

Figure 21. An instability in the spanwise plane for the PISB-N2 case. (a,b) Fluctuating velocity components of $u$ and $w$ are superimposed with the $u$ fluctuating velocity contour at $y/h=0.8$, $y/h=1.3$, respectively. Swirling strength contour lines (blue coloured) are also shown over the contour of the $u$ fluctuating velocity in figure (b).

Figure 22. An instability in the spanwise plane for the PISB-B2 case. (a,b) Fluctuating velocity vectors of $u$ and $w$ are superimposed with the contours of $u$ fluctuation at $y/h=0.4$ and $y/h=0.8$, respectively. Line contours of the swirling strength (in blue) are also shown over the contour of $u$ fluctuation in (b). The arrows in the first panels of (a,b) indicate an oscillating streak and the small vortices, respectively.

To better understand the interaction of a streaky structure with the spanwise roller, another time sequence of the instantaneous velocity fields is shown in figure 23(a). A streaky structure around $z/h$ = 0 can clearly be seen in this figure until $x/h\approx 52$. The streaky structure is followed by a $\varLambda$-like spanwise structure, which is indicated by a black line in figure 23(a). To decipher the hidden information in figure 23(a), the fluctuating velocity components are filtered at the low-frequency band ($f^*<0.03$) and band pass filtered at approximately the peak frequency of $f^*=0.18$. It should be noted that the choice of $f^*<0.03$ for the low-frequency band corresponds to $2{\rm \pi} f \nu \times 10^6 /U^2_e \leq 35$, as this value corresponds to low-frequency fluctuations due to the streaky structures (see Westin et al. Reference Westin, Boiko, Klingmann, Kozlov and Alfredsson1994; Balamurugan & Mandal Reference Balamurugan and Mandal2017, for details). The filtered velocity fields corresponding to the low and peak frequency bands are shown in figures 23(b) and 23(c), respectively. Even though the instantaneous flow field shows the streak-shedding structure interaction, this filtering band isolates the streaky structure and the $\varLambda$-like structures, as shown in figures (b) and (c), respectively. For better illustration of the $\varLambda$-like structures, white and black coloured lines are drawn over the positive and negative $u$ fluctuating velocity, respectively, as shown in figure 23(c). The same lines are also drawn in figure 23(a) at the same location. These lines indicate the hidden $\varLambda$-like structure in the instantaneous flow field. This suggests that a streaky structure does not take part in the transition process through the streak breakdown process at least for the FST level of $3.3\,\%$. Instead, it takes part in the secondary instability process of a separated shear layer by distorting the spanwise roller, which is also present at an enhanced level of FST. Furthermore, this figure also indicates that the KH instability mechanism is still dominant at an enhanced level of FST.

Figure 23. Fluctuating velocity vectors ($u-w$ component) superimposed with a $u$ fluctuating velocity contour for the PISB-B2 case at $y/h=0.4$. (a) Instantaneous measurement. (b) Filtered data at $f^*<0.03$. (c) Filtered data at $f^*=0.18$.

For further clarification of the role of streaky structures in the formation of a $\varLambda$-like structure, we calculated the spectral disturbance kinetic energy (SDKE) for the PISB-B2 case following Hosseinverdi & Fasel (Reference Hosseinverdi and Fasel2019). The filtered fluctuating velocity components at the low-frequency band ($f^*<0.03$) and the band pass filtered components at approximately the peak frequency of $f^*=0.18$ are used to identify the contribution of the streaks and the KH instability/shedding structure for the transition process, respectively; the filtered fluctuating velocities in the streamwise, wall-normal and spanwise directions are denoted as $u_{filtered}, v_{filtered}, w_{filtered}$, respectively. However, the SDKE has been calculated for both the $x\unicode{x2013}y$ and $x\unicode{x2013}z$ planes, and the spanwise plane has been considered at $y/h=0.4$, as the clear interaction of the streaks with the spanwise rollers can be seen in this plane (figure 23). For the $x\unicode{x2013}y$ plane, the disturbance kinetic energy, $E_{xy}$, has been calculated as

(4.4)\begin{equation} E_{xy}=\frac{1}{2U_{r}^2\delta}\int_{0}^{\delta}[ u_{filtered}^2(y)+ v_{filtered}^2(y)]\,{{\rm d} y}, \end{equation}

where $U_r$ and $\delta$ are the reference velocity and boundary layer thickness, respectively. Similarly, for the $x\unicode{x2013}z$ plane at $y/h=0.4$, the disturbance kinetic energy, $E_{xz}$, has been calculated as

(4.5)\begin{equation} E_{xz}=\frac{1}{2U_{r}^2(z_{1}-{z_0})}\int_{z_0}^{z_1}[ u_{filtered}^2(z)+ w_{filtered}^2(z)]\,{\rm d}z, \end{equation}

where $z_{0}$ and $z_{1}$ are the end points in the $x\unicode{x2013}z$ plane.

Figure 24 shows the variations of $E_{xy}$ and $E_{xz}$ in the streamwise direction. To investigate whether the energy growth is exponential or transient, the variations have been plotted both in semi-log and linear scales, as shown in figures 24(a) and 24(b), respectively. Note that the unfilled symbols correspond to the low-frequency disturbance, which is associated with the streaky structures, and the filled symbols correspond to the peak frequency, which is associated with the KH instability/shedding structure. Figure 24(a) shows that the energy associated with the filtered data at approximately the shedding/peak frequency of $f^*=0.18$ grows exponentially in the streamwise direction. For transient energy growth, it is expected to grow linearly, as commonly seen for the streaky structures in attached boundary layers at an enhanced level of FST (e.g. Matsubara & Alfredsson Reference Matsubara and Alfredsson2001; Balamurugan & Mandal Reference Balamurugan and Mandal2017). But figure 24(b) does not show such linear growth. In fact, a small exponential growth can be observed in the $x\unicode{x2013}y$ plane data even for the low-frequency boundary layer streaks (see figure 24a). A similar variation is also observed in the numerical study of Hosseinverdi & Fasel (Reference Hosseinverdi and Fasel2019). However, figure 24 shows that the energy associated with the low-frequency streaky structures is higher than the vortex shedding mode. Nevertheless, the exponential growth associated with vortex shedding is not suppressed or bypassed due to the presence of streaky structures. This indicates that the streaky structures do not play a significant role in bypassing the vortex shedding, rather it distorts the vortex shedding structure of the spanwise roller.

Figure 24. Comparison of integrated spectral disturbance kinetic energy growth for the PISB-B2 case. (a) Log scale. (b) Linear scale.

The above transition scenario cannot be generalized as it may depend on the disturbance amplitude at the point of separation. However, the present study at least ensures that the vortex shedding/the KH mechanism will not be bypassed if $u_{rms,max}/U_e\leq 0.1$ at the point of separation as the value of $u_{rms,max}/U_{e}$ is found to be 0.1 at the point of separation for the PISB-B2 case.