2.1. Experimental facility and measurement technique

The measurements were performed in the BL wind tunnel at KTH. The test section has cross-sectional area $0.5\times0.75$ m $^2$ and length 4.2 m. The axial fan (DC 15 kW) can produce airflow in the empty test section with speed up to 48 m s $^{-1}$ , and the cooling system of the wind tunnel is capable of maintaining a constant temperature in the test section within $\pm 0.07\,^\circ$ C. The streamwise turbulence intensity in the test section is less than 0.04 % in the core, with similar values for the cross-flow turbulence intensities at nominal speed 25 m s $^{-1}$ (cf. Lindgren & Johansson Reference Lindgren and Johansson2002).

The experiments were carried out over a 3.5 m long flat plate with a superelliptical leading edge according to

(2.1) \begin{equation} \left (\frac {x}{a} \right )^{m} + \left (\frac {y}{b} \right )^{n} = 1, \end{equation}

where the aspect ratio AR ( $=a/b$ ) is 20 : 1, with $a = 200\ \rm mm$ , and the exponents $(m,n) = (2.3,1.9)$ . The plate is positioned vertically in the test section to achieve the best spanwise homogeneity in the flow. With the installation of a trailing flap along with the adjustable side wall of the test section, a zero pressure gradient boundary layer flow with a relatively short pressure gradient region close to the leading edge was obtained. The free-stream velocity in the current experiments was kept constant at $U_\infty =5.9\ \rm m\ s^{-1}$ , measured above the plate sufficiently far downstream of the leading edge. A sketch of the experimental set-up is shown in figure 1.

Figure 1.

Sketch of the experimental set-up. Quantities $\tilde {x}$ denote distances from the leading edge; otherwise, the origin of the applied coordinate system for the measurements is the trailing edge of the FSVGs (cf. figure 2).

Hot-wire anemometry was employed throughout the campaign to measure time-resolved streamwise velocity signals along the flat plate. A single-sensor hot-wire probe operating in constant temperature mode was manufactured in-house, employing Wollaston wire composed of platinum with 10 % rhodium. The wire, of diameter 2.5 $\unicode{x03BC}$ m and length 0.5 mm, was fully etched before being soldered to the hot-wire prongs. Data were collected using a 16-bit NI USB-6215 DAQ system at sampling frequency 5 kHz. The probe was calibrated in the free stream inside the tunnel against the dynamic pressure obtained from a Prandtl tube connected to a Furness FCO560 differential manometer.

The TS waves were introduced by periodic blowing and suction at the wall through a spanwise slot at $\tilde {x}=135$ mm from the leading edge. The sinusoidal signal is generated using a function generator and supplied to an audio amplifier driving a loudspeaker, which is connected to the slot through tightly sealed silicon hoses. The amplitude of the disturbances from the slot can be quantified by measuring the AC output voltage from the amplifier to the loudspeaker.

Due to the viscous nature of the boundary layer instability, we here introduce the standard non-dimensional frequency $F$ according to

(2.2) \begin{equation} F=2\pi f \nu \times 10^6 /U_\infty ^2, \end{equation}

where $f$ is the dimensional frequency in Hz, and $\nu$ is the kinematic viscosity. For the current tests, a constant value $F=175$ was employed throughout. For the chosen $U_\infty$ , this positions branch II of the neutral stability curve at the centre of the measurement domain, allowing good resolution in the measurements for the transition region, which is expected to occur after branch II. The study of only one TS wave frequency is warranted since many studies have shown that if a modulated base flow is efficient in delaying transition for one TS wave frequency, then it is also efficient for others (cf. Shahinfar et al. Reference Shahinfar, Sattarzadeh, Fransson and Talamelli2012). In addition, from the stability analysis study by Siconolfi et al. (Reference Siconolfi, Camarri and Fransson2015b ), it is evident that with a streaky base flow, the neutral stability curve of the reference Blasius boundary layer will shift downstream at the same time as it is quenched towards lower frequencies as the streak amplitude increases. All this supports our decision to present only one TS wave frequency, since the rest is all about streaky base flow optimization; with one frequency, we can conclude that the stabilization is successful and that we can obtain transition delay. Testing more than one frequency is not warranted due to the time consumption. Considering all the cross-sectional planes that need to be taken in order to integrate the base flow modulation and the TS wave amplitude accurately, this would not add any extra value to the study.

Figure 2.

(a) The FSVGs consist of a triangular delta wing and a separate leg. (b) The assembled devices, mounted in a spanwise array. The streamwise cuts allow us to vary the FSVG spacing (however, not used here). The rail was then flush-mounted in a slot in the plate. ( $c$ ) A side-view sketch of the set-up and the employed coordinate system.

2.2. The FSVGs

The FSVGs are utilized to create SVGs in the boundary layer. The advantage of this type of vortex generator over the wall-mounted devices is their significantly reduced influence on the boundary layer. The vortex pairs generated by the FSVGs can have a higher intensity as they develop for a longer time in the free stream. In the current experiments, the FSVGs are made of two parts, triangular delta wings and separate legs, which are shown in figure 2(a). The delta wings are isosceles triangles with base length $B=6\ \rm mm$ and sweep angle $\beta =40^\circ$ . The legs have a trapezoidal shape with a cut-out to minimize the wake effect. Both parts are cut from stainless steel sheets using electrical discharge manufacturing, and glued together using a specifically designed device to ensure that the angle is close to the intended value (Lozano Reference Lozano2017). The thicknesses of the legs and the delta wings are 0.15 mm and 0.3 mm, respectively. The FSVGs are then installed into small sliding blocks that are placed on a rail to adjust the spacing ( $\Lambda _z$ ) between them. A plate to cover this rail is installed, and the assembly is inserted into the slot on the flat plate. The mounted FSVG devices including the cover plate are shown in a photo in figure 2(b). Five FSVGs were mounted on the flat plate, and all measurements were performed behind the central element of the array in order to achieve quasi-two-dimensional conditions. Once installed, a single plane measurement covering the wake of all five elements was performed in order to identify flawed FSVGs that might not have been installed or manufactured at the correct angle. These devices were then re-mounted or replaced iteratively, ensuring that all FSVGs behave similarly. More information on the manufacturing and mounting process is given in Cui (Reference Cui2023). A side-view sketch of the set-up is shown in figure 2(c), where the angle of attack ( $\alpha$ ) and the height of the vortex generators ( $h_v$ ) and the boundary layer thickness ( $\delta _{99}$ ) are introduced, as well as the origin of the coordinate system for the measurements.

2.3. Investigated FSVG parameters

The abundant number of FSVG parameters makes a complete parameter variation study impossible in an isolated work, meaning that some parameters were locked a priori to focus on what we believe are the most relevant for our purpose. Apart from the FSVG design parameters $\beta, B, h_v, \Lambda _z, \alpha$ introduced in figure 2, there are two other parameters influencing the strength of the vortices and the interaction with the boundary layer. These parameters are $U_\infty$ and $\delta _{99}$ , which are directly related to each other since the location of the FSVGs in the streamwise direction ( $\tilde {x}_{\textit {FSVG}}$ ) is locked (see figure 1). In the present study, $\beta$ , $B$ , $\Lambda _z$ , $U_\infty$ and hence $\delta _{99}$ are kept constant. Our focus has been on the influence of $\alpha$ and $h_v$ on the boundary layer interaction leading to the desired boundary layer modulation. The separate leg was kept the same except for its height above the wall, i.e. varied through $h_v$ .

Based on the range of vortex heights $h_v$ investigated by Siconolfi et al. (Reference Siconolfi, Camarri and Fransson2015b ), which varied from $1\delta _{99}$ to $2.5\delta _{99}$ , three $h_v$ values have been selected. On the other hand, the range of $\alpha$ has been chosen such that streaks with too-high amplitudes are avoided, as angle of attack 3 $^\circ$ with free-stream velocity 4 m s $^{-1}$ has been observed to generate streaks of 70 % amplitude in the work of Lozano (Reference Lozano2017). The spacing between the FSVGs, $\Lambda _z$ , has been set to 14 mm, based on previous research by Shahinfar et al. (Reference Shahinfar, Sattarzadeh, Fransson and Talamelli2012), Siconolfi et al. (Reference Siconolfi, Camarri and Fransson2015a ) and Lozano (Reference Lozano2017). The different FSVG cases investigated here are summarized in table 1.

Table 1.

FSVG configurations and geometrical parameters. The spanwise distance between the FSVGs was constant at $\Lambda _z=14\ \rm mm$ in all cases.

Figure 3.

Reference two-dimensional base flow case. (a) Experimental wall-normal mean velocity profiles for different streamwise and spanwise locations. The solid line corresponds to the Blasius solution. (b) Dimensional boundary layer displacement and momentum thicknesses along with the shape factor. See the text for the theoretical curve fitting and determination of the virtual origin from the leading edge. Data are presented in the streamwise extent of $Re_{\delta }$ from 294 to 568.

4.1. Mean flow modulation and quantification

In this subsection, the effect of different FSVG configurations on the mean flow is investigated. It is worth noting that due to the natural growth of the laminar boundary layer, even an FSVG with a delta wing of a geometrical angle of zero degrees will experience some effective angle of attack. The effective angle will always be positive, and as long as it is positioned outside the boundary layer, independent of the distance to the wall. It will, however, change with the streamwise location. Using the Blasius similarity solution for the wall-normal velocity component, the displacement velocity at the location of the FSVGs can be estimated to be 0.018 m s $^{-1}$ , giving induced angle of attack 0.17 $^\circ$ . It is worth mentioning that this angle is not far from the estimated manufacturing uncertainty $\pm 0.1^\circ$ as given in table 1.

Figure 5.

Streamwise mean velocity fields for C0, C1 and C3 (columns) at various downstream locations (rows). Dashed lines at $z/\Lambda =\pm 0.5$ indicate integration limits in (4.1).

The positive angle of attack of the delta wings results in a difference in pressure between the upper (suction) and lower (pressure) sides, which leads to the generation of leading-edge vortices, and hence to the existence of streamwise vorticity in the flow. Now, the presence of a counter-rotating vortex pair behind the delta wing results in wall-normal transport of fluid, with a common downwash on the centreline behind the delta wing that pushes high-momentum fluid towards the surface. At the same time, the vortex upwash, on the sides, tends to lift up low-momentum fluid towards the edge of the boundary layer. Hence streamwise streaks of alternating high and low speed are generated, with a single high-speed streak in line with the element, and two low-speed streaks on the sides.

Figure 5 shows the streamwise mean velocity in cross-sectional planes measured at various downstream locations behind the FSVG array for three out of the total five cases (cf. table 1). Case C0 (left-hand column of figure 5), featuring geometric angle $0^\circ$ , does not show a high-speed streak behind the element. Instead, a central low-speed streak is generated that decays with the downstream distance. This indicates that the production of streamwise vorticity due to the induced angle of attack is negligible, and instead the no-slip condition on the delta wing and subsequent wake behind it fully dominates. Another conclusion that can be drawn from C0 is that the angle of attack is in fact close to the intended value $0^\circ$ , making it likely that the other cases also feature the angle of attack that they are designed to have.

All cases with a positive angle of attack (C1–C4) show the generation of a central high-speed streak as per the aforementioned mechanism. The vortices slowly enter the boundary layer because of their slight downward trajectory as a result of the positive angle of attack, but also due to the boundary layer’s natural growth. The outcome is a high-speed streak in the centre, and two low-speed streaks on the sides. As expected, the formation of the high-speed streak inside the boundary layer begins earlier, with decreased height of the FSVGs above the wall since the vortices will interact with the boundary layer earlier. Furthermore, a larger angle of attack results in stronger vortices, and steepens somewhat the trajectory of the vortices towards the wall, resulting in a stronger and earlier impact on the boundary layer.

To quantify the modulation of the mean flow in a single non-dimensional quantity, the streak amplitude can be calculated. Various definitions can be found in the literature, but one that has proven to be meaningful in the context of attenuation of disturbances in the flow is the integrated streak amplitude, introduced by Shahinfar et al. (Reference Shahinfar, Fransson, Sattarzadeh and Talamelli2013):

(4.1) \begin{equation} A_{\mathit{ST}}^{{int}}(x)=\int _{-1/2}^{1/2} \int _0^{\eta ^*} \frac {|U(x,y,z)-U^z(x,y)|}{U_\infty } \, \mathrm{d}\eta\, \mathrm{d} \zeta . \end{equation}

Here, the non-dimensional variables are similar to the integrated TS wave amplitude definition (3.1) and $U^z$ is the spanwise-averaged velocity profile. Note that $ A_{ST}^{{int}}$ is independent of the chosen $\eta ^*$ as long as it is sufficiently far above $h_v$ , where the integrand is zero. Other definitions that utilize the normalized difference between the maximum and minimum velocities in the span (e.g. Andersson et al. Reference Andersson, Brandt, Bottaro and Henningson2001) have the drawback that they do not include a spanwise length scale. Hence information about the mean flow gradient $\partial U/\partial z$ , which is the quantity responsible for the boundary layer stabilization (Cossu & Brandt Reference Cossu and Brandt2004), is not included. Results for the integrated streak amplitude (4.1) of the investigated FSVG cases are shown in figure 6. Consistent with previous investigations, the streak amplitude grows as the induced velocity of the vortices causes the generation of streamwise streaks in the flow. The cases C0 and C1 show initially high values right behind the FSVGs, but as can be seen in figure 5, this is due to the no-slip wake effect of the geometry, which will slowly recover in the downstream direction (C0). In C1, the wake recovery is accelerated by the vortices, which not only neutralize the wake, but push high-momentum fluid towards the wall, creating a high-speed streak inside the boundary layer.

Figure 6.

Integrated streak amplitude values for the investigated cases: (a) full streak amplitude, performing the $\eta$ integral in (4.1) along the full wall-normal length ( $\eta ^*=9$ ); (b) inner boundary layer streak amplitude, with integration up to $\eta ^*=2.5=0.5\,\delta _{99}$ .

Compared to the streak amplitude evolution of classical wall-mounted MVGs in zero pressure gradient flows (e.g. Shahinfar et al. Reference Shahinfar, Fransson, Sattarzadeh and Talamelli2013; Sattarzadeh & Fransson Reference Sattarzadeh and Fransson2015), the streak amplitude due to the FSVGs grows more slowly and reaches a peak significantly later. In fact, the peak was not resolved in any of the investigated cases due to limits in traversing distance. Only in C1 can a peak be discerned in the most downstream region, while C2–C4 continue to grow. This observation is consistent with Siconolfi et al. (Reference Siconolfi, Camarri and Fransson2015b ), using free-stream vortices in DNS to generate a streaky base flow. A comparison is attempted in Appendix A, although several parameters differ. Nonetheless, qualitative agreement of the data can be observed. This discrepancy between MVGs and FSVGs can be explained by the fact that the vortices, being generated and located in the free stream, are sustained for significantly longer due to lower viscous dissipation in that region. Hence they can also modulate the boundary layer for a longer distance.

The streak amplitude is generally significantly higher than the optimal values for transition delay suggested in previous studies. Using blade-type MVGs, Shahinfar et al. (Reference Shahinfar, Fransson, Sattarzadeh and Talamelli2013) found that an integrated streak amplitude of $30$ % is optimal for transition delay. Even though stable streaks of an amplitude of 60 % can be obtained with MVGs (cf. figure 10 of Shahinfar et al. Reference Shahinfar, Fransson, Sattarzadeh and Talamelli2013), they are not robust against TS waves, and advance the transition location compared to the two-dimensional reference case already from approximately $A_{ST}^{{int}}=40$ % (cf. figure 20 of Shahinfar et al. Reference Shahinfar, Fransson, Sattarzadeh and Talamelli2013). The present results show that the streaks have amplitudes of up to 75 % in C4. But even C1 and C3 feature amplitudes of 50 % and 60 %, respectively, which according to MVG modulated boundary layer results would correspond to unsuccessful control cases. Nonetheless, all these cases are successful in stabilization and transition delay, as will be shown later in this paper. However, comparing the present data to the free-stream vortex results by Siconolfi et al. (Reference Siconolfi, Camarri and Fransson2015a ), who used a maximum-based streak amplitude definition, the amplitude values appear reasonable and close to the best case in their DNS (see Appendix A). This raises the question of whether the destabilization observed by Shahinfar et al. (Reference Shahinfar, Fransson, Sattarzadeh and Talamelli2013) is indeed due to the excess streak amplitude or rather the disturbance introduced by the MVG geometry. It was shown by Siconolfi et al. (Reference Siconolfi, Camarri and Fransson2015b ) that blade-type MVGs feature an absolute instability in their wake, which is likely more severe for larger and more intrusive geometries, which also generate stronger streaks. This suggests that the wake effect is the leading source of destabilization, rather than the intensity of the streaks.

Another observation can be made when comparing the streak amplitude of case C0 to the MVG cases of e.g. Shahinfar et al. (Reference Shahinfar, Fransson, Sattarzadeh and Talamelli2013). Case C0 features a streak amplitude of more than 20 % due to the low-speed streak originating from the device wake. As will be shown in the next subsection, this case gives, however, a very modest damping of the TS waves, although a streak amplitude of 20 % is expected to be appropriate for TS wave attenuation (e.g. Shahinfar et al. Reference Shahinfar, Fransson, Sattarzadeh and Talamelli2013). This can be explained by the fact that the FSVG streaks are located mostly in the outer region of the boundary layer, while the largest amplitude of TS waves is known to be close to the wall (see figure 4 a). Wall-mounted devices, on the other hand, produce streaks (i.e. SVGs) near the surface. Due to the integral definition of the streak amplitude, it contains no information about the wall-normal location of the streaks. As a result, the values of the streak amplitude generated by MVGs and FSVGs are likely not directly comparable when it comes to boundary layer stabilization.

To overcome this apparent discrepancy, the inner boundary layer streak amplitude can be introduced (figure 6 b). Here, the wall-normal integral is performed only until $\eta ^*=2.5\delta =0.5\delta _{99}$ to more accurately represent the effect of the mean-flow modulation on the TS wave. For these integration limits, the values are closer to the proposed optimal value of 30 %, while the (almost) ineffective case C0 lacks sufficient inner streak amplitude. However, this quantity has not been calculated for MVG streaks since the streaks are naturally near the wall, so a direct comparison is not possible.

4.2. Stabilization in the linear regime

By ‘linear regime’ here we refer to the fact that disturbance amplitudes are kept sufficiently low, so the assumption of small amplitudes, as used in linear stability theory, is valid. One procedure to guarantee this is to tune the TS wave amplitude to be less than 0.5 % at branch II, i.e. $(A_{\mathit{TS}}/U_\infty )_{II} \leq 0.5\,\%$ (cf. Fransson et al. Reference Fransson, Brandt, Talamelli and Cossu2005), which was ensured here by iteratively tuning the voltage to the loudspeaker to an appropriate level.

Figure 7.

Normalized TS wave amplitude with the downstream distance for the different cases in the linear regime (low disturbance amplitude), $F=175$ .

The effect of the various FSVG configurations on the linear TS wave growth curves is shown in figure 7. All values are normalized with their respective first measurement point right behind the FSVGs, i.e. at $x=6\ \rm mm$ . The base flow shows the expected TS wave growth curve, featuring growth until $x=350\ \rm mm$ (branch II) and decay thereafter. All presented cases with modulated base flow show a reduction in the disturbance amplitude compared to the unmodulated case. The cases C1 and C3 feature the strongest attenuation of TS waves far downstream, while C3 performs better in the close region until approximately $x = 300\ \rm mm$ . The two cases with the highest streak amplitude, C3 and C4, both feature an inflection point at approximately 550 mm and 500 mm, respectively, which could indicate the onset of a base flow instability due to a too-high streak amplitude. The overall attenuation of C1 and C3 can be quantified by comparing the TS wave amplitude at the most downstream measured location ( $x = 596\ {\textrm {mm}}$ ) giving $([A_{\textit{TS}}^{{int}}]^{\textrm {3D}} / [A_{\textit {TS}}^{{int}}]^{\textrm {2D}})_{x=596\,\textrm {mm}} \approx 0.2$ , corresponding to a reduction in TS wave amplitude of 80 %.

The initial amplification due to the presence of the FSVGs can be analysed by comparing the second most upstream point in the $A_{\mathit{TS}}^{{int}}$ evolution in figure 7. It shows that the initial amplification is clearly correlated with the distance of the delta wings to the wall $h_v$ . The case with the devices closest to the wall features the strongest initial growth, while for the case far away, it is marginal. The angle of attack of the delta wing has a negligible influence. However, the initial amplification seems to be harmless at least for the configurations studied here. As will be shown in the next subsection, even the strongest observed initial amplification (C1) is not sufficient to trigger the transition to turbulence prematurely.

In a numerical study, Martín & Paredes (Reference Martín and Paredes2017) analysed the optimal wall-normal distance of introduced vortices to maximize streak generation and disturbance attenuation. Their parameters, inspired by Siconolfi et al. (Reference Siconolfi, Camarri and Fransson2015a ), included free-stream velocity 7.7 m s^-1, and vortex introduction point at 52 mm. They identified an optimal wall-normal distance $h_v=2.18\delta _{99}$ , or 3.4 mm from the wall. Matching these optimal parameters in the present experiment for direct comparison is challenging, if not impossible. Both Siconolfi et al. (Reference Siconolfi, Camarri and Fransson2015a ) and Martín & Paredes (Reference Martín and Paredes2017) fixed the vortex introduction at $x = 52$ mm, and in the latter work, the circulation was adjusted to maintain a 20 % maximum streak amplitude. At this location, the boundary layer thickness $\delta _{99}$ is only 1.6 mm. Experimental constraints, including precise delta wing mounting, require feasible parameter selection. Studies on passive flow control highlight the importance of both streak amplitude and its streamwise location relative to the neutral stability curve for transition delay. In Siconolfi et al. (Reference Siconolfi, Camarri and Fransson2015a ) (E2) and Martín & Paredes (Reference Martín and Paredes2017) ( $h_v = 2.18$ ), a 20 % streak amplitude was chosen based on the successful MVG case C01 by Shahinfar et al. (Reference Shahinfar, Fransson, Sattarzadeh and Talamelli2013), with maximum streak locations at $Re_\delta =726$ and 650, respectively. In the present experiment, $Re_\delta =567$ for C01, but with a higher streak amplitude ( $\sim25$ %). This lower $Re_\delta$ and increased amplitude with respect to the numerical works are favourable, provided that instability does not occur. It is noteworthy that the successful C01 by Shahinfar et al. (Reference Shahinfar, Fransson, Sattarzadeh and Talamelli2013) has a maximum streak amplitude at $Re_\delta \approx 600$ . In the experiment, the streak amplitude distribution can be adjusted via delta wing angle of attack, height, and FSVG streamwise position, but neither maximum amplitude nor location can be held constant. Thus the systematic numerical studies cannot be validated directly in experiments, though the reverse is possible.

4.3. Transition delay

It is worth pointing out that even though TS wave attenuation can be accomplished, it is not certain that it can lead to transition delay. In order to test if transition delay is achievable with the given set-up, the amplitude of the loudspeaker was increased in the reference case (i.e. without FSVGs) until transition occurred within the measurement region.

Figure 8.

(a) Normalized disturbance energy (4.2) versus the downstream distance for the different cases in the transition case (large disturbance amplitude). (b) Same data as in (a), but log plot and growth curves normalized by corresponding first upstream measured point at $x = 6\ \rm mm$ .

Here, we define the normalized wall-normal integrated energy $E^{\textit {int}}$ , which is a more appropriate measure as compared to the TS wave amplitude, when analysing high-amplitude disturbance growth due to the broad-band nature of transition and turbulence. This energy measure is defined as

(4.2) \begin{equation} E^{{int}}(x) = \int _{-1/2}^{1/2}\int _0^{\eta ^\ast } \left ( \frac {u_{{rms}}}{U_\infty } \right )^2\,{\mathrm{d}}\eta\, \mathrm{d} \zeta, \end{equation}

where subscript rms indicates the r.m.s. value of the velocity signal, and int refers to wall-normal integration up to $\eta ^\ast = 9$ (similar to the truncated value used in (3.1) and (4.1)). In figure 8(a), $E^{{int}}$ is plotted versus the downstream distance for all configurations except C0, which showed a relatively weak damping effect of TS waves in the linear regime. The data show a success of transition to turbulence delay using FSVGs. In the reference, uncontrolled case, one can see that the onset of transition to turbulence takes place at approximately $x=400\ \rm mm$ whereafter $E^{{int}}$ grows sharply. In all the configurations C1–C4, there is a delay in transition, with C1 performing the best. The disturbance energy for C1 at $x = 596\ \rm mm$ is actually lower than in the first measured upstream location at $x = 6\ \rm mm$ due to the stabilization. It is noteworthy that at $x = 596\ \rm mm$ , the energy $E^{{int}}$ is approximately 500 times larger in the reference case with respect to C1.

In order to separate the data in figure 8(a) and allow for a detailed analysis of the tiny differences in the growth curves, the energy is replotted in figure 8(b) with a log scale and with data normalized with the first upstream measured point. Here, we recall from figure 6 that the streak amplitude overall increases in the following order: C2, C1, C3 and C4. Keeping this in mind when analysing figure 8(b), some conjectures can be made. Case C1 performs better than C2, and it can be argued that the streak amplitude in C2 is below the optimal one for transition delay using FSVGs. Case C3 also performs worse than C1, which might be a result of too-high amplitude streaks that become unstable. Case C4, with an even higher streak amplitude than C3, transitions earlier than C3. Interestingly, however, C4 has a stronger stabilizing effect than C2 on TS waves due to the higher streak amplitude, but C2, with more stable streaky base flow, is a better configuration for transition delay.

The observation that the streak amplitude is a delicate quantity is in agreement with results from Shahinfar et al. (Reference Shahinfar, Fransson, Sattarzadeh and Talamelli2013), where it was shown that an optimum exists. Below the optimum, the streaks are not sufficiently strong to attenuate the TS waves effectively and above it, the high-amplitude streaks can become unstable. However, their suggested streak amplitude optimum is 30 % (for wall-mounted MVGs), whereas the current data suggest a value of approximately 50 % for FSVGs. This was already discussed in § 4.2. In figure 8(b), the success of the control can be quantified by comparing C1 with the reference case at $x\approx 515\ \rm mm$ . With the FSVGs, the disturbance level stays essentially constant down to approximately $x = 250\ \rm mm$ , whereafter it starts decaying until approximately $x = 515\ \rm mm$ . At this downstream location, the disturbance level for C1 is approximately 17 times smaller than in the reference case. Comparing the onset of transition for the reference case at $x = 400\ \rm mm$ , one can estimate the transition delay to be at least 40 % by extrapolating the disturbance growth curve for C1 to the same level as the reference case at $x = 400\ \rm mm$ , taking the distance from the virtual origin at $x\approx -215\ \rm mm$ (cf. figure 8 b).

Figure 9.

Normalized disturbance energy with increasing voltage supplied to the speaker, leading to a gradual increase in the initial amplitude of the perturbation.

As the measurement region is not long enough to fully cover the transitional region for the controlled cases, another way to present the transition delay achieved with the FSVGs can be employed. Figure 9 also shows $E^{{int}}$ , but at a fixed streamwise location while the voltage provided to the loudspeaker is gradually increased. The measurement was performed at the most downstream location ( $x=596$ mm) in the boundary layer, and results are shown for the two most successful cases from figure 8, namely C1 and C3, and compared to the uncontrolled case. Note that the speaker voltage is a set-up-dependent variable, as different speakers and different tubing connections will affect the generated perturbations. Both controlled cases show a stabilization of the boundary layer compared to the reference case by exhibiting the abrupt increase in $E^{{int}}$ , associated with transition onset, at considerably higher voltages fed to the loudspeaker. It can be seen that in agreement with the previous results, C1 is able to stabilize the flow most successfully, making the controlled boundary layer over 110 % more persistent to the imposed disturbances (in terms of loudspeaker voltage, 1.5 V / 0.7 V at onset) compared to the reference case. One can observe that the onset of transition between the two controlled cases, C1 and C3, takes place at an almost equal voltage. What makes C1 more successful over C2 here is the lower growth rate of $E^{{int}}$ with increasing disturbance amplitudes. The slightly scattered data can be attributed to unconverged measurements in the highly intermittent transitional flow.