3.1 Quasi-steady response

Figure 5 depicts the effect of forcing frequency on the mean and root-mean-square (rms) fields of the streamwise velocity component. The momentum coefficient for all the forced cases is kept constant at $c_{\unicode[STIX]{x1D707}}=5.8\times 10^{-8}$ ( $V_{pp}=3.5$  kV). The results pertaining to the baseline case (no forcing) reveal the presence of a separation bubble within $0.6\lesssim x/c\lesssim 0.95$ , marked in figure 5 using a dividing streamline and the estimated locations of mean separation and reattachment. The contours of rms streamwise velocity fluctuations indicate the growth of disturbances in the separated shear layer and in the reverse flow region close to the wall. Distinct triple peaks emerge in the streamwise fluctuating velocity profiles in the aft portion of the bubble, agreeing with the results from previous studies (Boiko et al. Reference Boiko, Grek, Dovgal and Kozlov2002; Boutilier & Yarusevych Reference Boutilier and Yarusevych2012). The dashed line in figure 5 traces the locations of the maxima in streamwise velocity fluctuations, and corresponds to the location of the second peak from the wall in $u_{rms}$ contours, located in the core of the separated shear layer where the highest growth of the disturbances is expected (Dovgal et al. Reference Dovgal, Kozlov and Michalke1994).

Figure 5.

Time-averaged ( $\langle u\rangle$ ) and fluctuating ( $u_{rms}$ ) streamwise velocity fields for different forcing frequencies. Time-averaged dividing streamline (thick solid line), loci of maximum streamwise fluctuations (dashed line), mean separation point ( $\blacklozenge$ ) and mean reattachment point (●).

Figure 6.

Time-averaged LSB topology as a function of forcing frequency. (a) Locations of separation point ( $\diamondsuit$ ), maximum bubble height (▫) and reattachment point (○). (b) Bubble length (▿) and maximum bubble height (▵). Solid black symbols denote no forcing and solid grey symbols denote continuous forcing.

Introducing continuous forcing at the carrier frequency ( $f^{+}=100$ ) diminishes both the streamwise and vertical extent of the bubble. Since the carrier frequency is well removed from the characteristic flow frequencies, the induced changes are attributed to the mean injection of momentum in the vicinity of the separation location (figure 2), which acts to delay separation. The effect of the forcing becomes more pronounced when the modulated excitation is used to introduce disturbances at lower frequencies within $2\leqslant f^{+}\leqslant 10$ , which are of the same order as the characteristic time scales of disturbances expected to develop in the separated shear layer. Forcing at these frequencies increases the amplitude of velocity fluctuations at a given downstream position in the separated shear layer, with the same levels of velocity fluctuations attained earlier upstream in smaller bubbles. Since the forcing amplitude is kept constant, it can be conjectured that disturbances introduced at particular frequencies are more amplified in the separated shear layer, leading to a more significant effect on the mean extent of the bubble. The variation in the location and size of the bubble with forcing frequency is presented in figure 6. The results illustrate that forcing frequency has a significant effect on the streamwise and vertical extent of the bubble, with the minimum bubble size attained at $f^{+}\approx 6$ .

To investigate the effect of forcing amplitude on the time-averaged flow topology, controlled disturbances of varying amplitude were introduced at a fixed frequency of $f^{+}=6$ . The effect on the streamwise velocity field is illustrated in figure 7, showing that the size of the separation bubble is progressively diminished by increasing excitation amplitude. Similar to figure 5, the diminishment of the separation bubble correlates with the upstream advancement of shear layer transition. Figure 8 presents the variation of the locations of mean separation, maximum bubble height and reattachment, as well as the streamwise and vertical extent of the bubble, for all the forcing amplitudes investigated. It can be seen that changes in the mean baseline flow are attained only above some amplitude threshold ( $c_{\unicode[STIX]{x1D707}}\approx 2\times 10^{-9}$ ), below which no substantial effect is observed. It is speculated that the minimum amplitude threshold corresponds to the amplitude of natural perturbations attained in the attached boundary layer at the location of the actuator. Once the threshold value is exceeded, the increase in excitation amplitude leads to delayed mean separation and the upstream advancement of the mean reattachment location (figure 8 a), leading to the overall decrease in bubble size (figure 8 b). Yarusevych et al. (Reference Yarusevych, Sullivan and Kawall2007) observed a similar effect of excitation amplitude on the laminar separation bubble in their experiments involving external acoustic forcing. The present results also show that the decrease in streamwise extent of the bubble with increasing forcing amplitude is nearly exponential (figure 8 b). In their direct numerical simulations, Marxen & Henningson (Reference Marxen and Henningson2011) also report a nearly exponential variation in the mean separation and reattachment locations, and, hence, the bubble length, with increasing amplitude of disturbances introduced upstream of separation.

Figure 7.

Time-averaged ( $\langle u\rangle$ ) and fluctuating ( $u_{rms}$ ) streamwise velocity fields for different forcing amplitudes. Time-averaged dividing streamline (thick solid line), loci of maximum streamwise fluctuations (dashed line), mean separation point ( $\blacklozenge$ ) and mean reattachment point (●).

Figure 8.

Time-averaged LSB topology as function of forcing amplitude. (a) Locations of separation point ( $\diamondsuit$ ), maximum bubble height (▫) and reattachment point (○). (b) Bubble length (▿) and maximum bubble height (▵). Solid black symbols denote no forcing.

To investigate the development of disturbances in the separated shear layer, spectra of the wall-normal velocity fluctuations are presented in figure 9. The analysed data pertain to the locations of maximum velocity fluctuations (marked by dashed lines in figure 5) attained in the core of the separated shear layer due to the amplification of disturbances. All the forced cases correspond to a constant forcing amplitude of $c_{\unicode[STIX]{x1D707}}=5.8\times 10^{-8}$ . For the baseline case (no forcing), the results reveal the amplification of disturbances in a band of frequencies $4\lesssim F^{+}\lesssim 8$ , with the most amplified disturbances associated with $F^{+}\approx 6$ . The amplified band of frequencies is identifiable in the baseline spectra at and beyond $x/c\approx 0.82$ . As the disturbances develop downstream, the amplified frequency band broadens gradually, until a rapid broadening of the energy content over a wide range of frequencies occurs at $x/c\approx 0.87$ . This corresponds to the location of the maximum bubble height (figure 6 b), and marks the final stages of laminar to turbulent transition, followed by the mean flow reattachment shortly downstream (figure 5).

Figure 9.

Power spectral density of the fluctuating wall-normal velocity component ( $v^{\prime }$ ) probed at the wall-normal location of maximum fluctuations.

For continuous excitation ( $f^{+}=100$ ), a similar trend is observed in the spectra. The unstable frequency band broadens, but the most amplified disturbance frequency remains at $F^{+}\approx 6$ . However, when forcing is applied at frequencies close to the unstable frequency range, significant changes in the development of disturbances are observed. Specifically, the primary growth of disturbances occurs within a narrow range of frequencies associated with the excitation frequency and its harmonics. The disturbances attain stronger amplitudes farther upstream as the excitation frequency approaches that of the most amplified disturbances in the baseline case, similar to the trend in the reduction of the extent of the bubble size observed in figure 5. For $f^{+}=2$ , the most significant amplification is attained at higher harmonics of the excitation frequency, i.e. $F^{+}=4$ , 6 and 8. The most significant effect on the spectral energy content is observed for $f^{+}=6$ , where the amplification of the disturbances locks primarily onto $F^{+}=6$ , as well as the first two harmonics. Similar effect is observed at $f^{+}=10$ ; however, a significant amplification is seen at the lower frequency band $4\lesssim F^{+}\lesssim 8$ , similar to the amplified frequency band in the baseline case. The results indicate that the artificial disturbances are most amplified when their frequency falls within the band of the most unstable disturbances in the shear layer. In this case, higher amplitude velocity fluctuations are attained earlier due to the amplification of disturbances centred at the excitation frequency, resulting in the most significant diminishment of the separation bubble. In contrast, when the forcing frequency differs substantially from the most unstable frequencies, the development of disturbances is not augmented as effectively through the stability mechanism, and the transition process gradually switches to that driven by a broader frequency band amplification of natural disturbances in the separated shear layer.

Figure 10.

Instantaneous spanwise vorticity fields for different forcing frequencies. Contours based on the $\unicode[STIX]{x1D706}_{2}$ criterion are superimposed. Each snapshot is separated by 0.025 convective time scale ( $tU_{0}/c$ ).

Figure 10 depicts the unsteady nature of the separation bubble development using sequences of instantaneous spanwise vorticity fields pertaining to the baseline case and several forcing frequencies applied at a constant amplitude of $c_{\unicode[STIX]{x1D707}}=5.8\times 10^{-8}$ . For both the baseline and forced cases, the results show a roll-up of the separated shear layer in the aft portion of the bubble and periodic shedding of vortices. Comparing these results with figures 5 and 6, it can be concluded that the shear layer rolls up just upstream of the streamwise location of the maximum mean bubble height, where the most amplified disturbances reach significant amplitudes. The periodic nature of these structures, in conjunction with spectral results (figure 9) showing distinct spectral peaks in wall-normal velocity fluctuations where vortices form, suggest that the roll-up vortices develop as a consequence of the disturbance amplification in the shear layer. It is evident that the dynamics of the bubble is largely governed by the development of the shear layer vortices. The formation of these structures precipitates the mean flow reattachment by entraining higher momentum fluid from the outer flow into the near-wall region. The classical mean bubble topology, particularly in the aft portion of the bubble, is not reflective of the highly unsteady nature of the flow dominated by the shear layer vortices, and the concept of flow reattachment is applicable mainly in the time-averaged sense.

To aid vortex identification, contours of $\unicode[STIX]{x1D706}_{2}$ criterion magnitude (Jeong & Hussain Reference Jeong and Hussain1995) are plotted in figure 10. The dashed lines are drawn through the approximate location of the vortex cores in figure 10, providing estimates of the streamwise wavelength and convective velocity of the structures, related to the streamwise distance and slopes of the lines, respectively. The results indicate that the convective velocity of the structures remains approximately constant ( ${\approx}0.6U_{0}$ ) between the baseline and forced cases. In contrast, the streamwise wavelength scales inversely with the frequency of the most amplified disturbances in each case, as expected for travelling waves of constant propagation speed. The time sequences in figure 10 show that the roll-up location changes with excitation frequency, following the movement of the mean separation bubble in figure 5. The shear layer roll-up becomes more pronounced when the excitation is applied close to the naturally amplified frequencies. For all the cases, the vortices begin to break up downstream of the roll-up location, as can be deduced from the appearance of opposite sign vorticity regions within distinct vortex cores. The onset of this process is reflected in the broadening of the frequency content in the velocity spectra in figure 9, which occurs near the mean reattachment point. However, the break-up is delayed when the excitation is applied at the frequency close to that of the most unstable band in the base flow, as can be seen from a comparison of baseline (no forcing) and $f^{+}=6$ data in figure 10. It can be inferred from the results that forcing the flow at these frequencies produces earlier formation of more coherent spanwise vortices in the shear layer, bringing about the reduction in the extent of the separation bubble.

The results discussed up to this point indicate that periodic disturbances introduced at a frequency close to the band of unstable frequencies in the baseline flow produce the most substantial influence on flow development. The most significant effect on separation bubble is observed when forcing is applied close to the frequency of the most amplified disturbances in the baseline flow. To examine this in more detail, a linear stability analysis was performed on the experimental data. The development of shear layer disturbances is modelled via the Orr–Sommerfeld equation (e.g. Schmid & Henningson Reference Schmid and Henningson2001) in the spatial stability formulation, i.e. modelling the convective amplification of disturbances. In the analysis, measured mean streamwise velocity profiles were used as the base flow, and the Orr–Sommerfeld equation was solved numerically using spectral colocation techniques based on Chebychev polynomials. The nonlinearity of the eigenvalue problem was treated using the companion matrix approach by Bridges & Morris (Reference Bridges and Morris1984).

Figure 11.

(a) Experimental spectra of $v^{\prime }$ and linear stability theory (LST) $N$ factors at $x/c=0.85$ . (b) Experimental and LST $N$ factors for $F^{+}=6$ . Dashed lines indicate uncertainty intervals for LST predictions. Experimental data pertain to the baseline case.

For validation purposes, figures 11(a,b) present a comparison between linear stability theory (LST) predictions and the baseline experimental data for the frequencies of the amplified disturbances and the amplification factors of the most amplified disturbances ( $N(x)=\ln (A(x)/A(x_{ref})$ , where $A$ is the disturbance amplitude and $x_{ref}$ is the reference location). As discussed by Boutilier & Yarusevych (Reference Boutilier and Yarusevych2013), LST predictions are known to be sensitive to the effects of measurement noise in the mean velocity fields. Here, the uncertainty in LST predictions is estimated by propagating the estimated experimental uncertainty through stability calculations. The obtained uncertainty bounds are marked by the shaded regions in figure 11, and the degree of variation in stability predictions due to experimental error is in agreement with that reported in Boutilier & Yarusevych (Reference Boutilier and Yarusevych2013). In figure 11(a), a spectrum of wall-normal velocity fluctuations shows the amplification of disturbances within $5\leqslant F^{+}\leqslant 10$ , with the highest energy content associated with disturbances at $F^{+}\approx 6$ . The overlaid plot of $N$ factors shows that this frequency range falls within the unstable frequency band from LST and the frequency of the most amplified disturbances matches the LST predictions to within experimental uncertainty. It should be noted that the exact comparison of experimental amplification factors and those from LST results cannot be performed since the disturbances in the experiment can only be detected well downstream of the neutral stability point, where they reach measurable amplitudes. Thus, following the conventional approach used in stability investigations (Schmid & Henningson Reference Schmid and Henningson2001), the amplification factors are matched at a reference location ( $x_{ref}/c\approx 0.805$ ), where the experimentally measured disturbance amplitudes reach $0.005U_{0}$ . The experimental data were bandpass filtered at $F^{+}=6\pm 0.2$ to compute amplification factors corresponding to the most amplified disturbances. As shown in figure 11(b), the experimental results capture well the initial nearly exponential growth of the disturbances and are in good agreement with the LST predictions in this region. As the growth of disturbances begins to saturate, the agreement with LST deteriorates since nonlinear effects become significant. Similar results were also obtained for the forced cases, confirming that LST analysis can be used to predict stability characteristics of the separation bubbles observed in the present study.

Figure 12.

LST analysis on the LSB for different forcing frequencies ( $f^{+}$ ). (a) Growth rate contours. Location of maximum growth rate ( $-\unicode[STIX]{x1D6FC}_{i}^{min}$ ) (solid white line), location of separation (dashed black line) and mid-location between separation and maximum bubble height (dashed white line). (b) Growth rate as function of forcing frequency at the mid-location between separation and maximum bubble height. (c) Frequency of most unstable mode (maximum growth rate) at mid-location between separation and maximum bubble height.

The effect of forcing on stability characteristics of the separation bubble is first examined in figure 12 for the constant amplitude forcing applied at different frequencies, with the excitation parameters matching those presented in figure 5. Figure 12(a) presents the contours of the local growth rate of disturbances ( $-\unicode[STIX]{x1D6FC}_{i}$ ), with negative values associated with convectively amplified disturbances, in line with the spatial formulation of LST used in this study. For the baseline case, the stability results show that the frequency of the most amplified disturbances (traced by a solid white line) increases gradually downstream of separation (dashed black line), levelling off at $F^{+}=6$ by the mid-point between separation and maximum bubble height position (white dashed line), where the amplification of disturbances at this frequency is first detected in the corresponding experimental measurements (figure 9). When forcing is applied, the growth rates associated with the most unstable disturbances in the separation bubble diminish notably but the corresponding frequencies are not affected to the same degree. This is illustrated in figures 12(b) and 12(c), which compare the growth rates and the frequencies of the most unstable disturbances, respectively, at the locations marked by the white dashed lines in figure 12(a). Figure 12(b) shows that the introduction of artificial disturbances leads to a significant decrease in the growth rate, proportional to the relative decrease in the size of the separation bubble produced by the forcing (figure 6). Specifically, at this streamwise location, the maximum normalised growth rate for the baseline case is $-\unicode[STIX]{x1D6FC}_{i}c\approx 30$ ; whereas, that for the forced case at $f^{+}=6$ decreases to about 20. The frequency of the most unstable disturbances also decreases (figure 12) since the bubble size is affected by the forcing. It reaches a minimum at approximately $f^{+}=6$ , where the most amplified frequency is reduced from $F^{+}|_{\unicode[STIX]{x1D6FC}_{i}^{min}}=6.65$ to $F^{+}|_{\unicode[STIX]{x1D6FC}_{i}^{min}}=5.75$ . Thus, the fact that the most significant effect on separation bubble was observed at the frequency matching that of the most amplified disturbances in the baseline flow is attributed to the relatively small change in the most unstable frequency produced by forcing at this amplitude.

Figure 13.

LST analysis on the LSB for different forcing amplitudes ( $c_{\unicode[STIX]{x1D707}}$ ). (a) Growth rate contours. Location of maximum growth rate ( $-\unicode[STIX]{x1D6FC}_{i}^{min}$ ) (solid white line), location of separation (dashed black line) and mid-location between separation and maximum bubble height (dashed white line). (b) Growth rate as function of forcing amplitude at the mid-location between separation and maximum bubble height (for consistency with the logarithmic scale no forcing has been placed at $c_{\unicode[STIX]{x1D707}}=10^{-11}$ ). (c) Frequency of most unstable mode (maximum growth rate) at mid-location between separation and maximum bubble height.

Figure 13 illustrates the effect of forcing amplitude on stability characteristics of the laminar separation bubble for the fixed excitation frequency of $f^{+}=6$ . It can be seen that no substantial changes in the growth rate or the associated frequency of the most amplified disturbances are attained at low excitation amplitudes (figure 13 b,c), with no significant changes to the mean bubble characteristics produced at these excitation conditions (figure 8). However, as the forcing amplitude exceeds $c_{\unicode[STIX]{x1D707}}=2\times 10^{-9}$ , it leads to changes in the mean bubble parameters (figure 8) and, hence, the stability characteristics (figure 13). Specifically, as the forcing amplitude increases above this threshold, both the maximum growth rate and the associated frequency decrease (figures 13 b–c), following the associated decrease in the bubble length and height (figure 8 b). At the highest excitation amplitudes examined, the frequency of the most amplified disturbances is reduced by approximately 20 % compared to the baseline case (no forcing).

A comparison of the flow measurements and stability results suggests that the most notable changes in stability characteristics correspond to the cases when artificial disturbances produce the most significant changes to the mean size of the separation bubble. This is illustrated in figures 14(a) and 14(b), which show the variation of the maximum growth rates and the associated frequencies with bubble length and height, respectively. The results demonstrate that both the frequency and the growth rate of the most amplified disturbances decrease as the size of the bubble decreases. A similar effect of artificial forcing has also been observed in the numerical simulations of Marxen & Rist (Reference Marxen and Rist2010), Marxen & Henningson (Reference Marxen and Henningson2011) and Marxen et al. (Reference Marxen, Kotapati, Mittal and Zaki2015) on a flat plate geometry subjected to an adverse pressure gradient, and was attributed to the mean flow deformation. It can be seen from figure 14(a,b) that the reduction in the size of the separation bubble (either length or height) compared to the baseline case produces a decrease in the relative maximum growth rate and the most amplified frequency. However, the growth rate decreases more rapidly in comparison to the frequency, with the relative decreases in the growth rate being, on the average, double that in the frequency for a given reduction in the bubble size.

Figure 14.

Frequency (▫) and growth rate (○) of most amplified disturbances as a function of (a) bubble length, (b) bubble height and (c) angle of separation ( $\unicode[STIX]{x1D6FE}$ ) for all tested cases. All values are normalised with the respective quantities pertaining to the no forcing case ( $\times$ ). Dashed lines correspond to a $45^{\circ }$ slope.

The observed weaker effect of mean flow deformation on the frequency of the most amplified disturbances in the separated shear layer is consistent with the recent results of Marxen et al. (Reference Marxen, Kotapati, Mittal and Zaki2015), who numerically modelled separation bubble control with a synthetic jet. Similar to the earlier conclusion made in the present study, this reduced sensitivity of the unstable frequency to the changes in the mean bubble size due to excitation explains why the optimum excitation effect has been observed in other flow control studies when periodic flow excitation was applied at the frequency of the most amplified disturbances in the base flow. For example, Yarusevych, Kawall & Sullivan (Reference Yarusevych, Kawall and Sullivan2006a ) and Yarusevych, Sullivan & Kawall (Reference Yarusevych, Sullivan and Kawall2006b ) observed that the optimal frequency of external acoustic excitation and that of the most amplified disturbances in the base flow agree to within approximately 10 %.

It can be inferred from the analysis of the present results that the diminishment of the bubble is produced due to the amplification of artificial disturbances in the separated shear layer, which is governed by the underlying stability of the mean flow. Marxen et al. (Reference Marxen, Kotapati, Mittal and Zaki2015) also attributed the most effective control strategy in their numerical simulation to a convective instability in the separated shear layer, although other instability mechanisms were also observed. The quantitative flow visualisations in figure 10 suggest that the mean flow reattachment is attained due to the enhanced entrainment of momentum by the shear layer vortices, shed in the aft portion of the bubble at the frequency of the most amplified disturbances. Thus, the upstream advancement of the mean reattachment by means of artificial two-dimensional disturbances requires that such disturbances attain critical amplitude and produce shear layer roll-up earlier upstream, which is observed in the present measurements (figures 5 and 10). Given the decrease in maximum growth rates with decreasing the size of the bubble, revealed by the stability analysis (figure 14 a,b), the implication here is that the disturbances attain higher amplitudes at the same locations within the bubble (figure 7) due to the higher initial amplitudes produced by the actuator. The increased stability of smaller bubbles is likely responsible for the gradual reduction in the mean deformation effect with the decrease of the separation bubble size seen in figure 14(a,b). Specifically, on the average, as the bubble size diminishes, the relative change in stability characteristics gradually decreases.

The observed effect of mean bubble deformation on stability characteristics is speculated to be dependent primarily on the distance between the separated shear layer and the wall. Dovgal et al. (Reference Dovgal, Kozlov and Michalke1994) and Diwan & Ramesh (Reference Diwan and Ramesh2009), using synthetic mean velocity profiles and experimental data, respectively, concluded that this is one of the main parameters affecting separated shear layer stability characteristics. As an approximation, the distance between the separated shear layer and the wall in a separation bubble can be represented by the angle between the dividing streamline and the wall ( $\unicode[STIX]{x1D6FE}$ ) in the wall-attached coordinates, with $\tan (\unicode[STIX]{x1D6FE})=h/l$ . The variation of maximum growth rates of disturbances and the associated frequencies with $\tan (\unicode[STIX]{x1D6FE})$ is presented in figure 14(c). The results demonstrate that the decrease in the separation angle leads to the decrease in the associated stability characteristics. The distance between the wall and the core of the separated shear layer can be approximated more precisely using a displacement thickness (Boutilier & Yarusevych Reference Boutilier and Yarusevych2012). At the same time, integral boundary layer characteristics at separation are also often used as characteristics length scales (Boiko et al. Reference Boiko, Grek, Dovgal and Kozlov2002). Since the momentum thickness does not vary substantially in the fore portion of the separation bubble, the shape factor ( $H$ ) can be used as a suitable dimensionless parameter, which also allows a comparative analysis with other studies. The effect of excitation on these integral parameters is illustrated in figure 15, which shows the expected decrease in both the momentum thickness and the shape factor proportional to the effect of excitation on the separation bubble size.

Figure 15.

Displacement thickness ( $\unicode[STIX]{x1D6FF}^{\ast }$ ) and shape factor $(H)$ as function of (a) forcing frequency and (b) forcing amplitude. Shaded grey areas denote uncertainty intervals.

Figure 16.

Comparison of the no forcing case (solid lines) at four streamwise locations to the Blasius (dashed line) profile at $x/c=0.59$ and hyperbolic tangent profile (dotted dashed line) at $x/c=0.735$ . (a) Streamwise velocity profiles and (b) LST growth rates.

A prominent effect of shape factor on stability characteristics is depicted in figure 16 for the no forcing case examined in the present investigation. The results demonstrate that both the maximum growth rate and the corresponding frequency decrease and the band of unstable frequencies narrows with decreasing shape factor. In figure 16(b), dispersion relations representative of the free shear layer (hyperbolic tangent profile) and the attached Blasius boundary layer are also plotted for comparison with the corresponding profiles shown in figure 16(a). It can be seen that the stability characteristics of the laminar separation bubble lie between the limiting cases of the attached, zero pressure gradient boundary layer and the free shear layer.

Figure 17.

(a) Maximum growth rate as function of the LSB shape factor for all tested cases and (b) most unstable frequency as function of the LSB shape factor for all tested cases. Scatter points are plotted for locations between separation and mid-point between separation and maximum height. For comparison, cases pertaining to the studies of Boutilier & Yarusevych (Reference Boutilier and Yarusevych2012) (○) and LeBlanc, Blackwelder & Liebeck (Reference LeBlanc, Blackwelder and Liebeck1989) (▫) are shown.

Figure 17 compares the variation of non-dimensional maximum growth rate and the frequency of the most amplified disturbances with the shape factor. The figure also includes the data from other studies involving laminar separation bubbles on different aerofoils, where relevant parameters could be extracted. The results show that using the displacement thickness as the reference length scale effectively collapses each data set on a linear trend. The observed data scatter lies within experimental uncertainty in the relevant parameters (e.g. figures 12(c), 13(c) and 15). This is particularly significant considering that the results from Boutilier & Yarusevych (Reference Boutilier and Yarusevych2012) pertain to a constant Reynolds number but different angles of attack, while both of these parameters were varied over a relatively wide range in LeBlanc et al. (Reference LeBlanc, Blackwelder and Liebeck1989). Although there are deviations between the individual linear fits, there is a clear increase in the growth rate with increasing the shape factor for different geometries and flow characteristics, and the proposed functional dependence of stability characteristics shows good robustness for a given aerofoil profile. This implies that, for a given geometry, the shape factor effectively governs the non-dimensional stability characteristics of the separated shear layer.

3.2 Transient response

The discussed relation between the development of artificial disturbances, stability characteristics of the bubble, and the attendant mean flow deformation obtained for the steady state cases highlights the importance of considering the transient flow development that ensues when controlled disturbances are either introduced or removed from the flow. The transient behaviour of the separation bubble is also of interest from the perspective of flow control development and implementation. Since disturbances affect the bubble dynamics and mean parameters, stability characteristics are expected to change when artificial disturbances are introduced or removed from the flow, thereby affecting the transient flow development until a quasi-steady state is reached. In addition, the transient development can be anticipated to depend on the initial mean flow state and, consequently, is likely to differ depending on whether artificial disturbances are introduced to the flow or removed from it. The transient dynamics of the separation bubble associated with the introduction of artificial disturbances and the termination of forcing is examined in this section using the case of excitation at $f^{+}=6$ and $c_{\unicode[STIX]{x1D707}}=5.8\times 10^{-8}$ ( $V_{pp}=3.5$  kV).

Figure 18.

Time-averaged streamwise velocity field ( $\langle u\rangle$ ), fluctuating streamwise ( $u_{rms}$ ) and wall-normal ( $v_{rms}$ ) velocity fields prior to and after the period of transient response to the introduction of forcing ( $f^{+}=6$ , $c_{\unicode[STIX]{x1D707}}=5.79\times 10^{-8}$ ). Time-averaged dividing streamline (thick solid line), mean separation point ( $\blacklozenge$ ) and mean reattachment point (●).

Figure 18 compares time-averaged flow field characteristics for the baseline case and the excited flow, which represent approximate quasi-steady state limits for the investigated transient separation bubble dynamics. As discussed earlier, the introduction of artificial disturbances results in higher velocity fluctuations being attained earlier upstream, leading to earlier and more coherent shear layer roll-up. As a consequence, both the length and height of the separation bubble are reduced. However, it can be seen from the root-mean-square fields that stronger velocity fluctuations, particularly in wall-normal velocity, are produced in the aft portion of the separation bubble in the baseline case. Notwithstanding the increased amplitude of incoming disturbances during the excitation, it is striking to note that the location of the maximum height of the bubble approximately corresponds to the same level of maximum velocity fluctuations in both the forced and unforced bubble. At the same time, for the excited flow, the magnitude of maximum vertical velocity fluctuations induced by shear layer vortices in the aft portion of the bubble is reduced due to the proximity of the wall.

Figure 19.

Evolution of spanwise vorticity ( $\unicode[STIX]{x1D714}_{z}$ ) in the transient period after initiation of forcing (on). Inset indicates a more detailed evolution in the period $2.25<t-t_{on}<4.65$ .

Figures 19 and 20 present sequences of vorticity snapshots spanning the transients encountered after the introduction of excitation and after its removal, respectively. The results in figure 19 confirm that introducing the artificial disturbances moves the roll-up location upstream and, initially, produces much stronger rollers, as can be seen at $(t-t_{on})/T=4.5$ and in the inset detailed sequence. The roll-up location then moves downstream, and the quasi-steady shedding is established at approximately $(t-t_{on})/T\approx 20$ . When the excitation is removed (figure 20), the shear layer roll-up moves downstream, moving outside of the field of view at $(t-t_{off})/T\approx 9$ . The bubble effectively bursts, which is followed by a gradual upstream advancement of the roll-up location that converges on a quasi-steady state similar to that observed in the baseline case without excitation at $(t-t_{on})/T\approx 30$ .

Figure 20.

Evolution of spanwise vorticity ( $\unicode[STIX]{x1D714}_{z}$ ) in the transient period after cessation of forcing (off). Inset indicates a more detailed evolution in the period $2.25<t-t_{on}<4.65$ .

Figure 21.

Transient response of the LSB topology to initiation of forcing. All quantities are based on a sliding average spanning 2.5 global convective time units. (a) Successive loci of $\unicode[STIX]{x1D6FF}^{\ast }$ (grey solid line) and (b) successive dividing streamlines (black solid line) are spaced with 0.875 convective time units. Start (thick dashed lines) and end (thick dash-dotted lines) of the transient response. Probe locations P1 (○) and P2 (▫) located at streamwise location of maximum bubble height. Trajectory of maximum $\unicode[STIX]{x1D6FF}^{\ast }$ during transient period (thick solid black line). (c) Successive streamwise velocity profiles (identical notation as in (a)). Black arrows denote increasing time.

The transient response of the LSB is investigated further by obtaining statistical estimates of flow characteristics via a sliding average within a time window of 2.5 global convective time scales in width ( $2.5c/U_{0}$ ). This approach allows elucidating global temporal changes while reducing the random noise in the data. The width of the sliding window is selected as a compromise between the temporal smoothing and noise reduction; nevertheless, the overall trends were verified to be largely insensitive to the width of the window within a band from 1 to 4 convective time scales. Figure 21 presents the variation of the displacement thickness and dividing streamline when the excitation is introduced. Changes in the mean streamwise velocity profiles are also shown in the figure. Here, the presented properties evolve from those in the unforced bubble (marked by dashed lines) to the quasi-steady bubble attained by continuous excitation (dash-dotted line) in time increments of $0.875c/U_{0}$ . The local displacement thickness is representative of the averaged height of the convectively amplified wave packet in the separated shear layer, with the location of the maximum displacement thickness (solid line in figure 21 a) approximately marking the onset of shear layer roll-up (cf. figure 19). Examination of the results in figure 21 suggests that the bubble does not respond monotonically to the onset of excitation. Shortly after the onset of excitation ( $t-t_{on}\lesssim 0.9c/U_{0}$ ), relatively small changes are observed within the bubble. This is followed by the rapid advancement of the roll-up location upstream (figure 21 a) accompanied by the upstream advancement of the mean reattachment location and the overall reduction in the size of the separation bubble (figure 21 b). After the bubble attains a minimum size, it expands back to the quasi-steady state characteristics representative of continuous excitation, in agreement with the movement of the roll-up location in figure 19.

Figure 22 presents changes in the same variables as those shown in figure 21 for the transient case ensuing when the excitation is stopped. The initial bubble characteristics prior to the onset of the transient are marked by dashed lines, while those established after the transient period are identified with dash-dotted lines. As in the case of the transient observed after the introduction of artificial disturbances, the bubble undergoes non-monotonic topological changes between the two quasi-steady states. Initially for ( $t-t_{off}\lesssim 1.5c/U_{0}$ ), relatively small changes are observed in the parameters. This is followed by a rapid downstream movement of the roll-up location (figures 20 and 22 a), with the mean bubble growing both in length and in height (figure 22 b). During the second half of the transient process, the mean bubble reaches its maximum extent and then contracts to the quasi-steady state sustained in the absence of excitation.

Figure 22.

Transient response of the LSB topology to cessation of forcing. Identical notation to figure 21.

A more close examination of temporal changes during these two transient processes is performed by considering time histories of window-averaged displacement thickness, and local velocity fluctuations. The results are presented in figures 23 and 24 using two probe locations identified in figures 21 and 22, which correspond to the streamwise location of the maximum mean bubble height for the initial and final configurations in each of the two transient cases respectively. The temporal changes are characterised on two time scales: one relative to the global convective time scale ( $c/U_{0}$ ), and another relative to the average shedding period of the shear layer vortices ( $T$ ), which effectively coincides with the forcing period. Figures 23(a) and 23(b) compare temporal variations in the window-averaged displacement thickness and its temporal rate of change, respectively. For the case when the excitation is turned on, the transient process lasts approximately 4 global convective time units ( $4c/U_{0}$ ) or approximately 23 shedding periods. When the excitation is turned off, the duration of the transient process increases by about 45 %, lasting approximately $5.5c/U_{0}$ or 33 shedding periods. A similar disparity in the duration of the transients associated with the control of highly separated shear flows has been reported in some previous studies summarised in table 2. For a variety of flow configurations, the ratio of the transient duration when the excitation is turned off to that when the excitation is introduced ( $\unicode[STIX]{x1D70F}_{off}/\unicode[STIX]{x1D70F}_{on}$ ) ranges from 1.25 to 2, with $\unicode[STIX]{x1D70F}_{off}/\unicode[STIX]{x1D70F}_{on}\approx 1.44$ in the present study. Although it was speculated that the transient flow development is related to mean flow stability to the introduced perturbations (Darabi & Wygnanski Reference Darabi and Wygnanski2004), this has not been investigated in detail to this point.

Table 2.

Comparison of previous studies involving transient flow development in control of separated flows.

Inspection of figure 23 suggests that significant changes in bubble characteristics begin at earlier times when the excitation is introduced, compared to the case when the excitation is turned off. In figure 23(b), the maximum rate of change in $\unicode[STIX]{x1D6FF}^{\ast }$ is obtained approximately 10 shedding cycles after the onset of excitation and a nearly constant maximum rate of change is sustained for over 5 shedding cycles. When the excitation is turned off, a comparable maximum rate of change in $\unicode[STIX]{x1D6FF}^{\ast }$ is attained after approximately 15 cycles, followed by a gradual decay in the slope. Consequently, a shorter transient duration is attained in the former case.

The time traces of the vertical velocity in figure 24 provide added insight into the temporal changes in the bubble. The traces show that the aft portion of the bubble does not encounter appreciable changes for a few shedding periods after the onset of the excitation, with the time delay, on the average, representing the convective time it takes the introduced disturbances at the actuator position to reach the probe locations. At the onset of the excitation, the initial pulse produces a strong perturbation seen in the traces from the probes at $(t-t_{on})/T\approx 2.5$ , with the amplitude of subsequent periodic fluctuations decaying gradually. This suggests a much stronger amplification of the leading disturbances at the onset of excitation, which is also seen in figure 19. When the excitation is turned off, the amplitude of fluctuations begins to decay at $(t-t_{off})/T\approx 4$ , implying that the higher-amplitude velocity fluctuations induced by the forcing are sustained for about 4 cycles after the forcing is turned off (2 cycles longer than the delay expected based on the propagation speed of disturbances).

Figure 23.

(a) Evolution of boundary layer displacement thickness in the transient period after initiation of forcing (on) and cessation of forcing (off) and (b) rate of change of $\unicode[STIX]{x1D6FF}^{\ast }$ in the respective transient periods. All values are based on a sliding average spanning 2.5 convective units. Vertical dashed line denotes start of the transient period. The probe locations are indicated in figures 21(b) and 22(b).

Figure 24.

Evolution of wall-normal velocity ( $v$ ) sampled at probe locations P1-P2 in the transient period after initiation of forcing (on) and probe locations P3-P4 in the transient period after cessation of forcing (off). The probe locations are indicated in figures 21(b) and 22(b).

Figure 25.

Evolution of growth rate pertaining to mode frequency of $f^{+}=6$ in the transient period after (a) initiation of forcing (on) and (b) cessation of forcing (off). Data are calculated at the mid-location between separation and maximum height of the baseline bubble ( $x/c\approx 0.73$ ).

Clearly, the observed transient bubble dynamic behaviour is related to the development of the shear layer disturbances. Additionally, it follows from the previous discussion that the amplification rate of the disturbances in the separated shear layer is expected to change during the transient. To shed light on the physical mechanism behind the transient flow development associated with the introduction and removal of harmonic excitation, it is instructive to consider the changes in the stability characteristics of the separated shear layer. Figure 25 shows the variation of the growth rate associated with the average shear layer shedding frequency, which also matches the excitation frequency. The results are obtained based on the linear stability analysis of the sliding averages of the velocity fields within the transient, and serve as a first-order approximation for the overall variation in the growth rate. The data confirm that substantial changes in the growth rate occur during the transient. Specifically, when the excitation is introduced (figure 25 a), the bubble becomes more stable and, as its length and height are gradually reduced, the growth rate diminishes. The opposite trend is seen for the case when the excitation is turned off (figure 25 b).

These changes in stability characteristics provide insight into the attendant transient flow development. In particular, at the moment of introduction of the excitation, the bubble has not yet reacted and complies to the stability characteristics of the unforced flow. The bubble is rather unstable and hence the injection of higher amplitude initial disturbances at the most unstable frequency leads to a much earlier shear layer roll-up and the formation of a strong train of initial shear layer vortices (figure 19). This leads to a significant increase in the entrainment of higher momentum fluid from the outer flow, causing the bubble to contract rapidly (figures 21 and 23). However, as the bubble contracts, the flow becomes more stable and the amplification of the following disturbances reduces. As such, the smallest bubble state cannot be sustained due to the lower amplification rates experienced by the subsequent disturbances and consequently lower momentum entrainment. As a result, the bubble expands in the second part of the transient (figure 21) to settle on a quasi-steady configuration. In contrast, for the transient associated with the removal of excitation (figure 25 b), the bubble is more stable at the onset of the transient compared to the bubble forming in the absence of forcing. When the excitation that sustains such a state of the bubble is removed, the higher-amplitude harmonic disturbances are replaced by a broader spectrum of lower-amplitude natural perturbations, leading to much weaker velocity fluctuations being attained in the aft portion of the bubble (figure 20). As a result of the weakened momentum entrainment, the same mean bubble topology can no longer be maintained and the bubble effectively bursts (figure 20), i.e. the size of the bubble increases rapidly (Marxen & Henningson Reference Marxen and Henningson2011). Although the shear layer in the burst bubble becomes more unstable, the lower initial amplitude of the disturbances leads to a delayed shear layer roll-up. The bursting also significantly weakens the upstream feedback loop, as hypothesized by Dovgal et al. (Reference Dovgal, Kozlov and Michalke1994) and Boiko et al. (Reference Boiko, Grek, Dovgal and Kozlov2002), where pressure fluctuations produced due to the shedding of the shear layer vortices in the aft portion of the bubble feed back upstream and influence the initial amplitude of the disturbances. The existence of such a feedback is supported by studies on aerofoil tonal noise (e.g. Desquesnes, Terracol & Sagaut Reference Desquesnes, Terracol and Sagaut2007), since periodic passage of shear layer vortices over the trailing edge leads to relatively strong harmonic acoustic emissions establishing a stronger upstream feedback loop and locking shear layer disturbances more closely to the shedding frequency (Pröbsting & Yarusevych Reference Pröbsting and Yarusevych2015). While the influence of such tonal emissions is insignificant compared to that of artificial disturbances for the excited flow, it can alter the characteristics of the naturally developing bubble (Pröbsting & Yarusevych Reference Pröbsting and Yarusevych2015). Once the shedding resumes, the feedback loop is re-established, and the bubble contracts to a quasi-stable state (figures 20 and 22). Contrasting the two transient cases in this context suggests that the different stability characteristics of the initial state of the bubble, coupled with the importance of the upstream feedback loop in the absence of excitation, are largely responsible for the significant differences in the duration of the transients observed.

Figure 26.

(a) Snapshots of wall-normal velocity ( $v$ ) in the transient period after initiation of forcing. (b) Evolution of integrated wall-normal velocity ( $\int v\,\text{d}y$ ) in the transient period after forcing initiation (on). Dotted dashed lines correspond to location of labelled vortices.

Figure 27.

(a) Snapshots of wall-normal velocity ( $v$ ) in the transient period after cessation of forcing. (b) Evolution of integrated wall-normal velocity ( $\int v\,\text{d}y$ ) in the transient period after forcing cessation (on). Dotted dashed lines correspond to location of labelled vortices.

The spatio-temporal evolution of the vortices during the two transients is examined more quantitatively in figures 26 and 27, where the vortices are traced using the wall-normal velocity. More specifically, the snapshots in figures 26(a) and 27(a) pertain to instantaneous wall-normal velocity contours while the trace maps in figures 26(b) and 27(b) are constructed using the integral of wall-normal velocity in the wall-normal direction ( $\int v\,\text{d}y$ ). The integral quantity ( $\int v\,\text{d}y$ ) serves as a measure of the intensity of wall-normal velocity fluctuations induced by the passing structures at a given streamwise location. As expected, the periodically shed vortices produce a distinct periodic train of positive and negative velocity regions, shown in figure 26(a), where approximate locations of individual vortical structures of interest are marked by dashed-doted lines for illustration purposes. Respectively, the produced convective ridges between minima and maxima in integral wall-normal velocity, marked by the dashed-doted lines in figure 26(b), follow the streamwise evolution of the structures. The ridge that intersects $t=0$ corresponds to the vortex that originates from a disturbance produced at the actuator location ( $x/c=0.6$ ) at the onset of the transient. For the transient encountered after the excitation is introduced (figure 26), it can be seen that the vortex produced by the first pulse ( $V_{0}$ ) is notably stronger, compared to the structures formed prior to the onset of actuation. However, the growing perturbation introduced by the first pulse induces a stronger preceding vortex ( $V_{-1}$ ), compared to the amplified background disturbances. This stronger vortex reaches the mean reattachment location of the initial mean bubble ( $x/c\approx 0.95$ ) at $t-t_{on}\approx 3T$ , which corresponds to the time when measurable changes in bubble characteristics begin in figure 23. The vortices $V_{0}$ , $V_{1}$ , $V_{2}$ , $V_{3}$ and $V_{4}$ , which are shed between the onset of excitation and the time vortex $V_{-1}$ starts to affect the initial trajectory of the separated shear layer, are characterised by similar magnitude of vertical velocity fluctuations. As these structures travel downstream and enhance the entrainment of higher momentum fluid in the aft portion of the bubble within $3T\lesssim t-t_{on}\lesssim 9T$ , significant changes in the bubble characteristics are observed (figure 23). As a result of the ensuing changes to the bubble and the associated decrease in the growth rate (figure 25 a), the strength of the subsequent vortices gradually decreases (figure 26), with a quasi-steady state reached at $t-t_{on}\approx 20T$ corresponding to the time when the bubble characteristics stop changing in figure 23.

Figure 27 illustrates the spatio-temporal development of the shear layer vortices during the transient associated with the cessation of excitation. The vortex labelled $V_{0}$ is the structure that develops from the last pulse of the actuator, while $V_{-1}$ , $V_{-2}$ are preceding vortices. It can be seen that, in contrast to the transient associated with the introduction of the forced disturbance, where a significant upstream effect of the first controlled disturbance resulted in the formation of a stronger preceding vortex ( $V_{-1}$ in figure 26), a much weakened effect of the last strong pulse ( $V_{0}$ ) on the following unforced disturbance is observed in figure 27. The considerably weaker disturbance following the last controlled perturbation stems from the significantly increased stability of the excited flow (figure 25 b). The direct consequence of this effect is the delayed temporal response of the flow to the cessation of forcing, compared to the initiation of forcing. Specifically, in figure 27, the last strong vortex interacting with the reattachment region is $V_{0}$ at $t-t_{off}\approx 4T$ and the first disturbances of significantly lower amplitude reach this region one shedding period later at $t-t_{off}\approx 5T$ . In figure 26, the first high amplitude vortex that interacts with the reattachment region is $V_{-1}$ at $t-t_{off}\approx 3T$ . As such, the difference of two shedding periods in arrival time of the leading altered-amplitude disturbances between the two cases corresponds directly to the difference between the times significant changes are observed in figure 23. Following $V_{0}$ , the results in figure 27 show a dramatic decrease in the magnitude of vertical velocity fluctuations, with no sign of roll-up observed within the field of view. This confirms that the bursting of the bubble is produced because the much smaller background disturbances remaining after the cessation of excitation reach insufficient amplitudes to sustain the initial state of the bubble. As discussed previously, this is related to the lower growth rate experienced by the disturbances in this bubble configuration (figure 25 b).