4.1. Effect of steps on mean flow

The time-averaged results are presented in this section. Figure 4 shows spanwise-averaged velocities upstream and downstream of the $h/\delta =0.49$ BFS for each of the three velocity components, as well as the crossflow $W_{cf}$ component. The $x$–$y$ plane results presented throughout the paper are reconstructed from the spanwise planes that were acquired at multiple streamwise locations. The results presented in figure 4 were acquired with a streamwise resolution of 6.35 mm. The crossflow velocity component is the velocity in the direction normal to the external inviscid streamline. The angle between the $W_{\parallel }$ and $W_{cf}$ components ranges from approximately 35$^{\circ }$ to 36$^{\circ }$ over the measurement region. The white lines in figure 4(a) indicate regions of reversed flow, where the velocity normal to the step ($U_{\perp }$) is less than or equal to 0. Also note that measurements were started approximately 2 mm downstream of the step, so the white region downstream of the step at $x-x_s=0$ indicates a lack of measurement data in that region. Similar to previous measurements (Eppink et al. Reference Eppink, Wlezien, King and Choudhari2019), the recirculation velocity field is fairly weak (the largest contour is $-$0.04) and extends approximately 30 mm downstream of the step (which corresponds to approximately 12.5$\delta$). Figure 4(d) indicates the presence of a crossflow-reversal region downstream of the step. The typical crossflow direction (in the absence of the step) is negative, but, due to the adverse pressure gradient produced by the step, a region of positive crossflow occurs near the wall. The region of positive crossflow extends farther than the recirculating region, to approximately 50 mm downstream of the step. Similar behaviour has also been observed downstream of a FFS for the same experimental set-up (Eppink Reference Eppink2020b), though the crossflow-reversal region was typically quite small in those cases. The direction and strength of the crossflow velocity component can be important in determining the behaviour of the crossflow instabilities.

Figure 4.

Spanwise-averaged velocity profiles upstream and downstream of the $h/\delta = 0.49$ step, for the (a) $U_{\perp }$, (b) $V$, (c) $W_{\parallel }$ and (d) $W_{cf}$ velocity components. White lines in (a) indicate regions of reversed flow ($U_{\perp }\leq 0$).

In general, over the range of step heights studied, there were not many significant or interesting differences observed in the mean flow behaviour. While it is true that an increase in the step height led to a slight increase in the separation bubble length, the coarse spacing of measurement planes does not allow an accurate measurement of that value, particularly over such a small range of heights, in which the differences are likely very small. However, we can state that there was no reversed flow present by $x-x_s=25$ mm for both of the $h/\delta =0.49$ cases, whereas for the two larger step heights, there was still some small amount of reversed flow at this location. For those two cases, the flow was fully reattached by the next measurement station, $x-x_s=38$ mm. The strength of the recirculation (i.e.  maximum negative velocity) did increase slightly with step height, from approximately 4 % to 4.7 % $U_{\perp _e}$, across the range of step heights studied. There were also some noticeable differences in the crossflow velocity component, as illustrated in figure 5 for several selected locations downstream of the step. Interestingly, all cases are very similar near the step, except for the $h/\delta =0.57$ case, in which the peak negative and positive crossflow values are lower than all other cases, particularly the negative peak. However, as one might intuitively expect, the strength of the positive crossflow region near the wall increases for the larger step heights as the flow progresses downstream of the step.

Figure 5.

Crossflow velocity profiles.

4.2. Effect of step height on SCF

Figure 6 shows the $U$- and $V$-perturbation profiles upstream and downstream of the $h/\delta =0.49$ BFS. The $U$-perturbation profiles are calculated from the r.m.s. of the steady disturbance velocity ($U'$) across the span (integrated across a wavelength range of 5 to 20 mm). The steady disturbance velocity ($U'$) is the time-averaged velocity at each location minus the spanwise-averaged velocity at the same wall-normal location. The $V$-perturbation profiles are computed in the same way using the $V$-velocity component. These profiles illustrate the behaviour of the SCF instability and are also typically referred to as the SCF profiles. The general behaviours of the $U$ and $V$-perturbation profiles are similar, but given the small amplitude of the $V$-component, it is much easier to accurately measure the $U$-component. Note that the bright spot near $x-x_s=10$ mm in figure 6(b) is due to noise in the data and is not a true measurement of the SCF profile. Focusing on figure 6(a), there is a clear peak upstream of the step, due to the incoming SCF instability. This peak decays somewhat in amplitude as it goes past the step, while a new peak begins to emerge near the wall starting approximately 20 mm downstream of the step. This lower peak increases in amplitude and appears to take over farther downstream, as the two peaks merge into one. The appearance of the lower peak occurs in the region of strong positive crossflow, indicating that it may be related to the occurrence of the crossflow reversal. Once again, this behaviour is similar to that seen downstream of the FFS (Eppink Reference Eppink2020b), in which double-peaked SCF modes were observed, presumably due to the highly inflectional profiles that occurred in the recirculation and crossflow-reversal flow regions.

Figure 6.

Stationary perturbation profiles, bandpass filtered from 5 to 20 mm, upstream and downstream of the $h/\delta = 0.49$ step, for the (a) $U$ and (b) $V$ velocity components.

The $U$-perturbation profiles at several selected locations are compared in figure 7 for each of the step heights studied. At the location nearest the step (figure 7a), the shapes are all very similar, but the amplitude of the SCF mode for the largest step height is noticeably smaller than that of the smaller step heights. This is interesting, but perhaps not surprising, given that this peak is likely associated with the negative peak in the crossflow velocity profile, which was noticeably lower for the larger step height (see figure 5). The weaker negative crossflow component results in a smaller wall-normal velocity gradient of this velocity component, which is directly related to the growth of the SCF instability. By the next measurement station, $x-x_s=12$ mm (figure 7b), the profiles and amplitudes have collapsed to similar values, which is also in line with the crossflow profile measurements shown in figure 5. At this point, a lower peak, near $y=1$ mm, is starting to become visible. By the next station, $x-x_s=25$ mm, the double-peaked profiles have developed, with the upper peak amplitudes very similar among the step heights, but with increasing amplitude of the lower peak values as the step height increases. At the final station shown, $x-x_s=38$ mm (figure 7d), only a single peak is visible in the profile, and the amplitude again increases with increasing step height. These observations are also consistent with the observed increase in positive crossflow strength near the wall with increasing step height (figure 5).

Figure 7.

Comparison of $U$-perturbation profiles at (a) $x-x_s=2$ mm, (b) $x-x_s=12$ mm, (c) $x-x_s=25$ mm and (d) $x-x_s=38$ mm.

Figure 8 illustrates the growth of the peak amplitude of the SCF for each case, including the $h/\delta =0.49,\ 40\,\mathrm {\mu }$m DRE case. The results are normalized differently in the two figures. In figure 8(a) the amplitudes are normalized by the SCF amplitude at the most upstream station (i.e.  upstream of the step) for each of the two DRE configurations. Since the upstream measurements were only performed for the two $h/\delta =0.49$ step height cases, the results from the other two step heights are normalized by the value that was obtained from the $h/\delta =0.49,\ 20\,\mathrm {\mu }$m DRE case, as the initial amplitudes should be the same. In figure 8(b) the amplitudes are normalized by the edge velocity at each station. One thing to note from this figure is that doubling the DRE height results in twice the amplitude of the SCF upstream of the step. The former normalization (figure 8a) allows for a direct comparison of the SCF growth, even with different initial SCF amplitudes. All four curves initially collapse pretty well and the amplitudes decrease slightly until $x-x_s=23$ mm, which corresponds to the end of the reversed flow region. For the $h/\delta =0.49,\ 20\,\mathrm {\mu }$m DRE case, the amplitude grows continuously downstream of $x-x_s=23$ mm. The growth rate is initially large just downstream of reattachment, but eventually levels off somewhat. For the $h/\delta =0.49,\ 40\,\mathrm {\mu }$m DRE case, the initial growth rate downstream of $x-x_s=23$ mm is similar to its counterpart with $20\,\mathrm {\mu }$m DREs, but slightly larger. However, there is a clear nonlinear effect when comparing these two cases, as the amplitude for the 40 $\mathrm {\mu }$m DRE case saturates near $x-x_s=60$, at an amplitude of approximately 6.5 % $U_e$, and decreases downstream. Some nonlinear behaviour is to be expected, since the initial SCF amplitude has doubled, and given that SCF vortices typically begin to display nonlinear behaviour at relatively low amplitudes. However, the relatively low saturation amplitude of approximately 6.5 % $U_e$ indicates that even with the larger SCF amplitude, SCF vortices are still not the dominant transition mechanism. The other two step heights show similar patterns to this case, with a peak in the SCF amplitude occurring near 60 mm downstream of the step, and with peak amplitudes increasing with increasing step height. The initial growth rate downstream of reattachment increases slightly as the step height is increased. This behaviour was also observed in figure 7. It is interesting that, for the largest step case ($h/\delta =0.57$), the amplitude initially saturates near $x-x_s=60$ mm, but then grows in amplitude at the farthest downstream station. The reason for this behaviour is unknown, and is only seen for this one case. The mean flow results for this case are not significantly different from that shown in figure 4 and does not provide any clue as to the reason for this behaviour.

Figure 8.

Peak SCF amplitude vs $x-x_s$; $U'_{rms}$ normalized by the (a) SCF amplitude upstream of the step, and (b) edge velocity.

4.3. Effect of step height on unsteady disturbance growth

While the effect of the BFS on the SCF instability is quite interesting and could warrant further investigation, it was shown in previous results (Eppink et al. Reference Eppink, Wlezien, King and Choudhari2018, Reference Eppink, Wlezien, King and Choudhari2019) that the unsteady instabilities that occur due to the presence of the step are the primary drivers of transition in these cases. The SCF instability was found to play a secondary role. To briefly summarize the previous findings, three different families of instabilities were identified downstream of the BFS, which were not present in the no-step case. These were identified as a travelling crossflow-like instability in the frequency range 80 to 200 Hz, a streamwise-travelling TS-like instability in the approximately 200 to 800 Hz range, and a streamwise-travelling shear-layer instability in the 800 to 1500 Hz range. When convenient throughout the paper, these will be referred to as the L, M and H-bands, respectively, for low, mid and high frequency. The SCF played a role in the transition process by causing a spanwise modulation of the instabilities. In addition, large fluctuations in velocity, referred to as ‘spikes’, were observed intermittently in the measurements downstream of reattachment and leading to transition. In the following sections we will attempt to further elucidate the features of the different unsteady instability modes, the effects of step height on those modes, discover the mechanisms through which they interact with the SCF instabilities, and attempt to further elucidate the velocity spike mechanism and the cause(s) of this mechanism.

In figure 9 spanwise-averaged velocity spectra of the three different velocity components are presented for each step height, at $x-x_s=38$ mm. These spectra were taken at the wall-normal location of the maximum fluctuation amplitude, though this $y$-location can vary somewhat across the frequency range. The peak near $f=100$ Hz is very clear in the chordwise velocity ($u'_{\perp }$) for every step height (figure 9a). However, this peak is not clearly visible in the $v'$ or $w'_{\parallel }$ velocity component spectra (figures (b,c)), except for the largest step height. Thus, the fluctuations for this low-frequency mode occur primarily in the chordwise velocity ($u'_{\perp }$). The higher-frequency fluctuations, above $f=200$ Hz, seem to occur primarily in the spanwise and wall-normal velocities, although some energy content is visible in the $u'_{\perp }$ spectra as well. It is clear from these figures that the fluctuation amplitude, in every velocity component, increases significantly as the step height is increased. It also appears that the lower-frequency mode, $f=80$ to 200 Hz, increases more, relative to the higher-frequency instabilities. Another interesting observation is the impact of the SCF amplitude on the fluctuation amplitude. With the doubling of the SCF amplitude for the lowest step height case, the spanwise-averaged fluctuation amplitude increased significantly. In fact, the spectra for the $h/\delta =0.49,\ 40\,\mathrm {\mu }$m DRE case overlap fairly well with the spectra for the $h/\delta =0.51,\ 20\,\mathrm {\mu }$m DRE case. This is convenient, as it provides a unique opportunity to study the impact of the SCF vortices nearly independently of the unsteady instability amplitude, as will be explored in § 4.6.

Figure 9.

Comparison of spanwise-averaged spectra at peak wall-normal location of fluctuations, $x-x_s=38$ mm, for the (a) $u'_{\perp }$, (b) $v'$ and (c) $w'_{\parallel }$ velocity components.

The growth of some of the individual frequency bands are plotted in figure 10. These amplitudes were determined by averaging the spectra across the span, integrating the averaged power spectra over the frequency band of interest, and choosing the peak amplitude from the resulting disturbance profile. This procedure was followed for each plane of data, and the results are plotted vs streamwise distance downstream of the step. Based on observations made from the current work, the frequency bands for the two higher-frequency disturbances were adjusted slightly from what was used in previous work. Thus, the middle-frequency band ranges from 200 to 750 Hz, and the high-frequency band ranges from 750 to 1250 Hz. For the lower-frequency band (figure 10a), the $u'_{\perp }$-velocity component is used, while the $w'_{\parallel }$-velocity component is used for the higher-frequency bands. This was done based on the observation that the lower-frequency disturbance primarily occurs in the chordwise velocity component, while the higher frequencies occur primarily in the spanwise velocity component. In general, this behaviour should be expected for the types of modes that have been identified. With $\partial v'/\partial y$ negligibly small, to satisfy continuity, the amplitudes of $u'$ and $w'$ are linked based on the relative values of the streamwise and spanwise wavenumbers ($\alpha u'=-\beta w'$). For a travelling crossflow instability, the spanwise wavenumber ($\beta$) will be substantially larger than the streamwise wavenumber ($\alpha$), which would require $u'>w'$. The opposite is true for the TS and KH (higher-frequency) instabilities, thus, $w'>u'$. The dependence of unsteady instability amplitude on the step height is clear from these figures. There is also a clear dependence on the SCF vortex amplitude, as the fluctuation amplitudes in the $h/\delta =0.49$, 40 $\mathrm {\mu }$m DRE case are substantially (1.5 to 2 times) larger than for the 20 $\mathrm {\mu }$m DRE case. As evident from the spectra in figure 9, the amplitudes for the $h/\delta =0.49$, 40 $\mathrm {\mu }$m DRE and $h/\delta =0.51$ cases are of very similar amplitude for most locations and frequency bands. The largest discrepancy occurs in the highest frequency band (figure 10c).

Figure 10.

Peak amplitude of the spanwise-averaged unsteady modes vs $x-x_s$ for frequency ranges of (a) 80 to 200 Hz, (b) 200 to 750 Hz and (c) 750 to 1250 Hz.

4.4. Unsteady disturbance development

In this section the spatial features and downstream development of the unsteady disturbances will be examined in detail. With the limited spatial information available in previous studies (Eppink et al. Reference Eppink, Wlezien, King and Choudhari2018, Reference Eppink, Wlezien, King and Choudhari2019), we were able to identify many of the relevant features of each of the unsteady instabilities. However, time-resolved PIV provides even more detailed information from which we can gain further insight, particularly into the spanwise and 3-D behaviour of the instabilities as they interact with the SCF vortices. This section will primarily focus on the two smaller step heights, $h/\delta =0.49$ and 0.51, with 20 $\mathrm {\mu }$m DREs, but similar qualitative features were seen for all cases.

The SPOD analysis, as described in § 3.2, was performed for all cases to obtain modes for each selected frequency. Some SPOD energy spectra for the $h/\delta =0.51$ step are shown in figure 11. In these figures, $\lambda$ corresponds to the eigenvalue of each SPOD mode. Figure 11(a) shows the first mode spectra for the first six streamwise locations downstream of the step. At the location closest to the step, the lower-frequency peak at approximately 100 Hz contains the most energy, and there is an additional secondary peak centred around 1 kHz. Moving downstream, the lower-frequency peak grows in amplitude, while the higher-frequency peak grows as well and broadens to lower frequencies. Figures 11(b) and 11(c) show the first five SPOD modes at two different streamwise locations, $x-x_s=12$ and 50 mm. At $x-x_s=12$ mm, most of the energy in the flow is captured by the first mode for each frequency. At $x-x_s=50$ mm, the second and third modes, and even the fourth mode for some frequencies, contain broad peaks and features and appear to contain a significant amount of energy. For the highest frequency band, $f\approx 750$ to 1250 Hz, the first SPOD mode contains almost all of the energy, whereas the higher modes are important for the two lower-frequency bands. Thus, for most of the discussion, we will focus primarily on the first SPOD mode, but, farther downstream, the discussion will include the second mode as well.

Figure 11.

Spectral POD energy spectra for the $h/\delta$ = 0.51 case: (a) first mode spectra at multiple streamwise locations, (b) first five modes at $x-x_s=12$ mm, (c) first five modes at $x-x_s=50$ mm. Light shaded areas in (b,c) represent 95 % confidence intervals, and are only included for the first three modes.

As mentioned earlier, the SPOD modes contain phase and amplitude information. Thus, they can be represented as time-resolved modes by cycling through the phase from 0 to 2${\rm \pi}$. Videos for each of the modes discussed in this section are included in the supplemental material. For purposes of representation in the paper, a snapshot in time (i.e.  the amplitude distribution for a particular phase) is shown for each of the modes under investigation. The first mode snapshots for the lowest frequency band, at $f=117$ Hz, are shown in figure 12 for the first six streamwise locations. These figures contain results for all three velocity components. Similar to what was seen in previous measurements, the first mode at this frequency corresponds to a spanwise-travelling wave, with a fairly large wavelength of approximately 40 mm. The large structures are evident in all three velocity components. By 38 mm downstream of the step, some spanwise modulation of the instability is beginning to become apparent, and this modulation becomes more obvious farther downstream. Beginning at the fourth station shown, $x-x_s=50$ mm (figure 12d), three spanwise locations are evident at which the main structure of the large-wavelength travelling mode becomes interrupted by a localized disturbance of opposite sign near the wall. This modulation is especially clear in the $v'$-velocity component (figure 12(IId)), at $z\approx -6$, 5 and 15 mm. For this particular snapshot, the behaviour is most evident at $z\approx -6$ and 15 mm, but all three become evident at different times throughout the cycle (this is clear from the video). Inspection of the mean flow reveals that these locations correspond to regions of large positive spanwise gradients of the $V$-velocity ($\partial V/\partial z$), as well as large negative spanwise gradients of the $U$-velocity ($\partial U/\partial z$), caused by the SCF vortices. This provides evidence that the modulation of this low-frequency disturbance is due to a complex interaction between the travelling crossflow vortices and the SCF vortices, which cause the large spanwise velocity gradients in the mean flow. For example, before the instability becomes modulated, when the $v'$-velocity component of this disturbance is negative, the $w'$-velocity component very near the wall is negative as well (see figure 12(II–IIIb)). This is more evident for the $w'$ component than the $w'_{\parallel }$ component (for an illustration of this, see figure 26a). The negative $w'$ component, when passing by the region of large positive $\partial V/\partial z$, will pull the larger $V$ velocity to the left, resulting in the positive $v'$ peak seen at $z\approx -6$ mm. This interaction seems to share some similarities with the interaction described by Bonfigli et al. (Reference Bonfigli, Kloker and Wagner2003), between a stationary and travelling crossflow mode of similar wavelength and amplitude. In the present case, the travelling crossflow vortex wavelength is larger than the stationary wavelength, which lends itself to the formation of asynchronous events (e.g.  occurring non-simultaneously across the SCF vortices). This will be discussed further in the following sections as it relates to the spike events. Another important difference between the present case and that described by Bonfigli et al. (Reference Bonfigli, Kloker and Wagner2003) is that the SCF amplitudes are significantly lower in the present case.

Figure 12.

Spectral POD mode snapshots for the first $f=117$ Hz mode, $h/\delta = 0.51$, at $x-x_s =$ (a) 12 mm, (b) 25 mm, (c) 38 mm, (d) 50 mm, (e) 63 mm, (f) 76 mm.

Next, the two higher-frequency bands, which correspond to (initially) primarily streamwise-travelling 2-D instabilities, will be examined. The first SPOD modes for a frequency of $f=517$ Hz, corresponding to the M-band TS-instability, are shown in figure 13. The motion of this instability, evident from the videos, indicates that it is primarily travelling through the plane. At the location nearest the step, $x-x_s=12$ mm, the measurement is noisy due to the low-disturbance amplitude relative to the noise level, but even so, a nearly 2-D structure is evident. Moving downstream, it becomes evident that these structures are being impacted by the SCF vortices, as the spanwise modulation becomes stronger in all three velocity components. As the modulation develops downstream, the $v'$- and $w_{\parallel }'$-components begin to develop in a manner that introduces localized regions of streamwise vorticity in the flow, corresponding to the SCF vortex spacing. To further illustrate this, results are shown in figure 14 for the planes at $x-x_s=38$ and 50 mm. In figures 14(a) and 14(b), the time-averaged $U$-velocity field is shown at both locations, along with velocity vectors corresponding to the SCF vortices. To acquire the vectors in these plots, first the wave angle of the SCF vortex is determined by finding the angle at which the crossflow component goes to 0 at the inflection point. This crossflow component, $W_{s}$, is then used, along with the $V'$ values, to obtain the vectors. The bottom two plots in this figure show snapshots from the $f=517$ first SPOD mode, along with the velocity vectors obtained from the $v'$ and $w'_s$ components of the mode. For consistency, the velocity components used to compute the SPOD mode are also rotated using the angle found above to compute $W_s$. At the first streamwise location, $x-x_s=38$ mm, two SCF vortices are evident, centred near $z=-5$ and 5 mm, and rotating in the counter-clockwise direction. Similarly, for the SPOD snapshot shown below (figure 14c), the modulation of the mode has introduced counter-clockwise rotation in locations corresponding very closely to the SCF vortices. Farther downstream, at $x-x_s=50$ mm, the flow is recovering from the step, and due to the adverse-to-favourable pressure gradient changeover, there are now two sets of SCF vortices. One set, located near $y=1.5$ mm, is rotating in the clockwise direction, while the set nearest the wall, corresponding to the same $z$-locations as before ($z\approx -5$, 5 and 15 mm), are still rotating in the counter-clockwise direction. For the SPOD snapshot at this location (figure 14d), there is again a localized region of rotation that aligns with this set of counter-clockwise rotating SCF vortices. The development and alignment of this vorticity with the SCF vortices is believed to play a crucial role in the ultimate breakdown of the flow to turbulence, as will be discussed in the following section.

Figure 13.

Spectral POD mode snapshots for the first $f=517$ Hz mode, $h/\delta = 0.51$, at $x-x_s =$ (a) 12 mm, (b) 25 mm, (c) 38 mm, (d) 50 mm, (e) 63 mm, (f) 76 mm.

Figure 14.

(a,b) Time-averaged $U$-velocity contours, with SCF vortex vectors ($W_{s},V'$) at (a) $x-x_s=38$ mm and (b) $x-x_s=50$ mm. (c,d) Snapshots of the first SPOD mode $v'$-component for $f=517$ Hz, with vectors of SPOD mode $v'$ and $w'_s$ components at (c) $x-x_s=38$ mm and (d) $x-x=50$ mm.

Similar SPOD results for the H-band are shown in figure 15. This mode resembles the 2-D behaviour of the M-band at the first two stations, $x-x_s=12$ and 25 mm (figure 15a,b). This mode was initially identified as a KH instability. However, even near the step, the higher-frequency band disturbance exhibits more modulation than the M-band (this is also more evident in the videos). By $x-x_s=38$ mm, the differences between the two frequency bands are becoming more apparent. All three velocity components begin to exhibit some differences in the modulation patterns for the H-band instability. In particular, the modulation appears stronger in all three velocity components. The videos also reveal that the propagation of the higher-frequency instability now appears to be primarily in the spanwise direction, as opposed to out-of-plane. These results indicate that farther downstream of the step, the higher-frequency fluctuations correspond predominantly to a 3-D instability, rather than the 2-D KH instability that was initially identified near the step. Similar types of behaviour were observed in the hot-wire measurement campaign, but there was not enough information to delve further into this behaviour. Based on the current results, there are two different instabilities that exist in the H-band frequency range – a 2-D instability near the step, and a 3-D (oblique) instability farther downstream of the step ($x-x_s>50$ mm). The 3-D behaviour becomes even more apparent as the flow progresses downstream.

Figure 15.

Spectral POD mode snapshots for the first $f=963$ Hz mode, $h/\delta = 0.51$, at $x-x_s=$ (a) 12 mm, (b) 25 mm, (c) 38 mm, (d) 50 mm, (e) 63 mm, (f) 76 mm.

Recall from the SPOD spectra (figure 11c) that locations downstream of $x-x_s=50$ mm exhibit some energy in the higher SPOD modes for the L-and M-bands, while the H-band contains almost all of the energy in the first mode. Snapshots from the second SPOD modes for the L- and M-band frequencies are shown in figures 16 and 17, respectively, for the three locations downstream of $x-x_s=50$ mm. The lower-frequency disturbance exhibits an interesting modulation of the flow, with wavelength corresponding to the primary SCF vortex wavelength. It is not yet clear what mechanism this fluctuation corresponds to. The contours do not appear to be moving in the plane, as would be expected of a travelling crossflow vortex. Rather, the contours simply alternate between regions of positive and negative $u'$-velocity, as can be seen in the supplemental videos for figure 16. It is possible that this fluctuation corresponds to a low-frequency flagging motion of the SCF vortices (Serpieri & Kotsonis Reference Serpieri and Kotsonis2016). It is interesting to note that this type of fluctuation pattern is not limited to this L-band frequency range, but, in fact, similar patterns are seen at even lower frequencies. It is evident from the SPOD spectra that the energy level of the second mode at 117 Hz is similar to that for the first mode at lower frequencies. This behaviour warrants further investigation, but is left for future research.

Figure 16.

Spectral POD mode snapshots for the second $f=117$ Hz mode, $h/\delta = 0.51$, at $x-x_s =$ (a) 50 mm, (b) 63 mm, (c) 76 mm.

Figure 17.

Spectral POD mode snapshots for the second $f=517$ Hz mode, $h/\delta = 0.51$, at $x-x_s =$ (a) 50 mm, (b) 63 mm, (c) 76 mm.

There is an interesting observation to be made from the second SPOD modes for the middle-frequency band, as well. Comparing figure 17 to 15(d–f), the modulation patterns appear very similar. In fact, the time-resolved behaviour of these second SPOD modes (evident from the videos) confirm a similar in-plane (spanwise) motion to that seen for the first SPOD modes of the higher-frequency band at these locations. These similarities indicate that the frequency band of the spanwise-travelling higher-frequency instability begins to spread to lower frequencies and overlap with the M-band at these downstream locations.

4.5. Three-dimensional reconstruction of the unsteady instabilities

We are interested in understanding the full 3-D picture of these instabilities, but this is difficult given that the current measurements are all planar. However, as described in § 3.2, with some limited knowledge of the streamwise phase speed of the instabilities, it is possible to do a qualitative reconstruction based on the SPOD modes. As mentioned earlier, the measurements were acquired with a streamwise spacing of 6.35 mm for the smallest step height ($h/\delta =0.49,$ 20 $\mathrm {\mu }$m DREs), which is enough to resolve the streamwise wavelengths of the instabilities up to approximately 300 Hz. Based on the hot-wire measurements, the streamwise wavelength of the 260 Hz mode is approximately 21 mm, which results in at least three points per wavelength for this mode. Since the L-band instability primarily travels in the spanwise direction (normal to the $x$-direction), resulting in a near-infinite streamwise wavelength, the resolution is more than sufficient to reconstruct this mode.

Figure 18 depicts the reconstruction of the first 117 Hz SPOD modes to yield a snapshot in time. The full video is included in the supplemental material. Figures 18(a) and 18(b) include isosurfaces of positive (red) and negative (blue) $u'_{\perp }$ velocity, along with isosurfaces of positive $\partial V/\partial z$, shown in green. Initially, the large spanwise wavelength of this instability is visible near the step. By approximately 40 mm downstream of the step, the modulation due to the interaction with the SCF vortices is beginning to become apparent. As discussed earlier, this interaction results in regions of localized velocity fluctuation of opposite sign in the middle of the larger regions of negative or positive $u'$ or $v'$, when these regions pass by areas of large positive $\partial V/\partial z$ (or negative $\partial U/\partial z$). Since the wavefront of this instability is nearly oriented in the streamwise direction, these localized regions occur almost instantaneously at all streamwise locations, which results in an interesting instantaneous 3-D modulation of the mean flow.

Figure 18.

Reconstructed SPOD data snapshot consisting of positive (red) and negative (blue) isosurfaces of the $u_{\perp }$-velocity component, $u_{\perp }=\pm 0.05$ m s$^{-1}$, for the $h/\delta =0.49$, 20 $\mathrm {\mu }$m DRE case. Results shown for (a) 117 Hz first SPOD mode, (b) 117 Hz first SPOD mode with green isosurfaces of positive $\partial V'/\partial z$ velocity gradient.

Similar results are shown in figure 19(a) for the 260 Hz mode. Given the limited streamwise resolution of these measurements, it was necessary to choose a frequency towards the lower end of the M-band. Green isosurfaces of positive $V'$-velocity are also included in this figure to aid in the discussion of the interaction with the SCF vortices. Shortly downstream of the step ($x-x_s\approx 15$ mm), the blue and red isosurfaces appear to be fairly two dimensional, with the wavefront aligned with the step, but as they convect downstream, they become more and more inclined relative to the sweep angle. The regions of the red and blue isosurfaces that fall under the green band become increasingly steeper in inclination from $x-x_s=25$ to 90 mm. Thus, the modulation of this instability appears to be primarily a consequence of the well-known lift-up effect (Theodorsen Reference Theodorsen1955), where the positive $V'$-velocity component pushes the instability up into regions of higher momentum fluid, causing the disturbance in these localized regions to convect downstream faster. This 3-D visualization of the instability provides a simple explanation for how the interaction of the 2-D TS-instability can become modulated due to an interaction with the SCF vortices.

Figure 19.

Reconstructed SPOD data snapshot consisting of positive (red) and negative (blue) isosurfaces of the $u_{\perp }$-velocity component, $u_{\perp }=\pm 0.05$ m s$^{-1}$, along with green isosurfaces of positive stationary $V'$-disturbance velocity, $V'=0.01$ m s$^{-1}$, for the $h/\delta =0.49$, 20 $\mathrm {\mu }$m DRE case. Results shown for (a) 260 Hz first SPOD mode, and (b) 260 Hz second SPOD mode.

Unfortunately, this same procedure cannot be used to reconstruct the H-band instability due to the limited spatial resolution and the smaller wavelengths that occur at higher frequencies. However, recall that the second SPOD mode for the M-band (figure 17) exhibited similar behaviours to the first SPOD modes at the higher frequencies (figure 15), downstream of approximately $x-x_s=50$ mm. Thus, reconstructing this second mode may give some additional insight into the behaviour of the high-frequency instability. The result from this reconstruction is shown in figure 19(b). In this case, once the mode is strong enough (near $x-x_s=40$ mm), the instability appears as an oblique mode travelling downstream and outboard, at approximately 60$^{\circ }$ relative to the streamwise direction. This illustrates a markedly different behaviour from the first mode at this frequency and suggests that the second mode may be indicative of a travelling crossflow instability. Evidently, this instability does not appear to experience a modulation due to the lift-up effect induced by the SCF vortices, as was observed for the first SPOD mode. However, it does interact with the SCF vortices in the sense that there are regions of increased fluctuation amplitude (see, for instance, figure 17(b) and the accompanying video, where the fluctuation amplitude is larger in several regions: $z=-5$ to 0 mm, 5 to 10 mm and 15 to 20 mm). Even though this is not a reconstruction of the higher-frequency instability, it is likely a good representation, given the obvious similarities between the second SPOD mode for the M-band and the first SPOD mode for the H-band. In fact, for the phase-locked measurements that were performed at 1 kHz for the much larger step height ($h/\delta =0.73$), we see a very similar behaviour. For these data, phenomena were observed to occur at two different streamwise wavelengths in the phase-averaged results, so a wavenumber filter was applied to separate the very large streamwise wavelengths from the smaller streamwise wavelengths. The results after the filtering was applied are shown in figure 20(a,b), as well as the accompanying supplemental videos. When the smaller wavelengths are filtered out (figure 20a), we see a very similar oblique structure to what was observed in figure 19(b). The agreement is quite remarkable, especially considering the very different reconstruction techniques that were used to visualize the data, as well as the difference in frequency, step height and forcing conditions. The wave angle for this case was also estimated to be approximately 62$^{\circ }$, in very good agreement with the results from the 260 Hz second SPOD mode. However, when the larger streamwise wavelengths are filtered out, we see a very clear spanwise roller structure, indicative of a KH-type instability, which is indeed verified by looking at the $Q$-criterion, which is plotted in figure 20(c). This figure illustrates that vortices are shed from the step at the forcing frequency of 1 kHz, but they start to die out shortly downstream of $x-x_s=20$ mm, near where the oblique structures start to become apparent. Note that when the $Q$-criterion was plotted for the reconstructed 260 Hz first SPOD mode, there was no clear evidence of vortex shedding from the step, in agreement with the initial categorization of this mode as a TS-instability.

Figure 20.

Phase-averaged results for the $h/\delta = 0.73$ step with 1 kHz forcing. Isosurfaces of positive (red) and negative (blue) $u'$-velocity (a,b). Results are filtered based on streamwise wavelength: (a) larger wavelength (>10 mm) and (b) smaller wavelength (5.8 to 10 mm). (c) Plot of $Q$-criterion results computed from the smaller wavelength filtered flow field, isosurfaces of $Q=1$.

4.6. Interaction of unsteady and steady instabilities

In this section we explore the interaction of the unsteady instabilities that eventually leads to breakdown to turbulence. In previous measurement campaigns (Eppink et al. Reference Eppink, Wlezien, King and Choudhari2018, Reference Eppink, Wlezien, King and Choudhari2019), large amplitude velocity fluctuations were observed, which were not high-frequency events, but eventually led to high-frequency fluctuations, indicative of breakdown. These so-called ‘spikes’ occurred only in certain spatial regions, and resulted in skewed signals because they occurred mainly in a certain direction (either negative or positive $u'$ depending on the spatial location). This mechanism is referred to as the spike mechanism because of how it appears in the raw velocity signals. One such highly skewed velocity signal is shown in figure 21 as an example. These large fluctuations in velocity grow in amplitude downstream, and reach very large values, sometimes 20 % to 30 % of $U_e$, before beginning to exhibit high-frequency fluctuations indicative of breakdown. Since this mechanism appears to be the primary path to breakdown in these BFS cases, it warrants further investigation. We will first examine the behaviour of these so-called spike events, and then explore the mechanisms that lead to the development of these large fluctuations.

Figure 21.

Example of highly skewed velocity signal taken at $x-x_s=50$ mm downstream of the $h/\delta =0.49$ BFS.

The spikes are identified using the method described in § 3.3, using the location of maximum negative skewness near $z=-5$ mm (identified as location 1 in figure 22). Once the spike times have been identified, an average is taken of the snapshots at these times, for each of the velocity components. The results from this analysis are shown in figures 22 and 23 for the smallest and largest step heights, respectively. The streamwise location of these figures are slightly different, since spikes began to occur closer to the step for the larger step height. The $v'$ and $w'$ figures also include vectors computed using the two in-plane velocity components. These are shown to illustrate the regions of counter-clockwise rotation that are accentuated by the spike events. There is an obvious pattern, as seen in previous work (Eppink et al. Reference Eppink, Wlezien, King and Choudhari2018), that the large velocity fluctuations occur with a spanwise spacing corresponding to the SCF mode wavelength (${\approx }11$ mm).

Figure 22.

Time average of the spike-event snapshots, $h/\delta = 0.49$, $x-x_s=44$ mm, for the (a) $u'$, (b) $v'$ and (c) $w'$ velocity components, along with vectors of $w'$ and $v'$.

Figure 23.

Time average of the ‘spike’-event snapshots, $h/\delta = 0.57$, $x-x_s=38$ mm, for the (a) $u'$, (b) $v'$ and (c) $w'$ velocity components, along with vectors of $w'$ and $v'$.

Figure 24 illustrates another interesting aspect of these spike events, which is their relationship to the SCF vortices. In the top figures (a,b), the SCF vortex vectors are shown on top of the modulated mean flow. For both cases shown, there are counter-clockwise rotating vortices noticeable near the wall at $z\approx -5$, 5 and 15 mm, similar to the results shown in figure 14 for the $h/\delta =0.57$ case. The regions of counter-clockwise rotation that are evident in the fluctuating components of the spike event (figures 22 and 23) align closely with the SCF vortices. This is also shown more clearly by examining the two bottom figures (figure 24c,d), in which colour contours show the skewness computed from the time series of the $v'$-velocity component at each location, while the line contours show the $V'$-velocity, i.e. the steady disturbance velocity computed by subtracting the spanwise-averaged $V$-profile from the time-averaged $V$-profile at each spanwise location. The skewness (colour) contours correspond very closely to the average spike $v'$ contours shown in figures 22(b) and 23(b). It is clear from these figures that the large velocity fluctuations corresponding to the spike events align very well with the steady $V'$ velocity disturbances. This relationship between the occurrence of the spike events and the SCF vortices will be explored in the following discussions.

Figure 24.

(a,b) Contours of time-averaged $U/U_e$ velocity, with SCF vectors ($W_s,V'$). (c,d) Skewness of $v'$-velocity (colours) along with $V'$-disturbance velocity (lines). Positive $V'$ contours are in red, while negative contours are in blue. Results shown for (a,c) $h/\delta =0.49,\ x-x_s=44$ mm and (b,d) $h/\delta =0.57,\ x-x_s=38$ mm.

For the larger step height, more of the large positive or negative velocity fluctuations were observed to occur asynchronously, rather than simultaneously across multiple wavelengths. This is likely one of the reasons for the differences that can be observed between figures 22 and 23, particularly the difference in the spanwise uniformity of the amplitude peaks. The spike times used for these averages were determined from the data at a single point (1). For further analysis of this mechanism, spike times were determined from each location of maximum negative skewness in $u'$. These locations are denoted 1, 2 and 3 in figure 22(a). The spike times were identified for each of the three locations as described previously. Then, we identified spikes that occurred simultaneously or nearly simultaneously (within 2 time steps, or 0.0004 s) for any combination of the three locations that included location 1. This analysis was performed for each case, and the number of spikes that fit into each category were counted and normalized by the total number of spikes identified for each case. This analysis was performed at the farthest upstream location at which the spikes became apparent, which was $x-x_s=38$ mm for all cases except $h/\delta =0.49$, 20 $\mathrm {\mu }$m DREs, which was at $x-x_s=44$ mm. The results are shown in figure 25. There are clear differences depending on, not only the step height, but also the initial SCF amplitude. For the smallest step height and smaller initial SCF amplitude, most of the spikes occur simultaneously at all three locations. However, when the initial SCF amplitude is doubled, the spikes become more asynchronous. The results also change dramatically with step height. For the two largest step heights, more spikes occur asynchronously, either only at location 1, or at 1 and 2, with very few occurring at all three locations simultaneously. Note that this analysis technique is not ideal, as it depends on the selection of a threshold, and, thus, the results may vary depending on the chosen threshold. Despite this shortcoming, it helps to clearly identify the differences in these events as the step height or SCF amplitude changes.

Figure 25.

Fraction of each type of spike occurring for each case. The numbers in the legend refer to the locations in which the spikes occur simultaneously, as identified in figure 23(a).

To further study these differences, we turn to the conditional POD analysis. For brevity, the following analysis focuses solely on the larger step height ($h/\delta =0.57$) at $x-x_s=38$ mm. We chose to focus on this specific location for this analysis because it is the most upstream location at which the spikes were first observed. The conditional POD analysis was performed for three out of the four spike behaviours identified in figure 25, corresponding to the (1), (1,2) and (1,2,3) cases. For this analysis, signal lengths of 31 time steps were used, with the identified spike event occurring right in the middle, at the 16th time step. Two different snapshots from the first modes identified from this analysis are shown, corresponding to time step 1 (figure 26) and time step 16 (figure 27). The entire step time series for each case is included as a supplemental video. In addition, a video of the first mode for the (1,3) case is included as well, though it is not included in the discussion for brevity, as it is similar to the (1) and (1,2) cases. The relative values of the eigenvalues for the POD modes can give an idea of how much of the energy is contained in the first mode. For these cases, the ratio of the first eigenvalue to the second is 3.3, 3.1, 2.1 and 2.8 for the (1), (1,2), (1,2,3) and (1,3) cases, respectively. Thus, the first mode contains two to three times more energy than any of the higher modes. In terms of total kinetic energy, the first modes in these cases make up 17 %, 17.2 %, 19.7 % and 37 %, respectively, of the total kinetic energy. These values indicate that the first modes represent a significant portion of the total fluctuations occurring during these events, and, thus, provide a good representation of the underlying dynamics of the event.

Figure 26.

Beginning snapshot of the first conditional POD mode, $h/\delta = 0.57$, $x-x_s=38$ mm. Simultaneous spike locations from figure 22: (a) (1), (b) (1,2) and (c) (1,2,3).

Figure 27.

Middle (spike) snapshot of the first CSTPOD mode, $h/\delta = 0.57$, $x-x_s=38$ mm. Simultaneous spike locations from figure 22: (a) (1), (b) (1,2) and (c) (1,2,3).

In figures 26(a) and 26(b) the structures that are apparent 15 time steps ahead of the actual spike event resemble those of the low-frequency, spanwise-travelling wave, as illustrated in figure 12. Note that the $w'$-velocity components look slightly different, and the reason is that we are now plotting the $w'$ component instead of the $w'_{\parallel }$ component. The reason for this is to better illustrate the streamwise vorticity that results from these disturbances. The main difference between figures 26(a) and 26(b) is that the phase of the low-frequency (long spanwise wavelength) instability is different, which helps to explain why the spikes occur in only specific locations. Moving to the actual time of the spike for those two cases, shown in figures 27(a) and 27(b), the low-frequency instability structure is still noticeable, particularly in the $u'$-velocity component. Again, this instability is at different phases when compared between the two cases. In these figures, particularly in the $v'$- and $w'$-velocity components, some aspects of the higher-frequency 2-D instabilities are noticeable, particularly the H-band. Note that for the cases where the spikes were identified only at location 1 (figure 27a), there is still a localized increase in the $u'$- and $v'$-velocity components near location 2, but the peak amplitudes are lower than at location 1, and apparently not large enough to reach the ‘spike’ threshold. In contrast to the first two cases, the third case (1,2,3) appears to be more reliant on the higher-frequency 2-D instabilities than the lower-frequency travelling mode. This is evident from the very beginning of the first CSTPOD mode (figure 26c), where the larger spanwise wavelength structures indicative of the lower-frequency instability are no longer evident. Rather, the modulated 2-D structures indicative of the higher-frequency modes are more apparent (for instance, compare with figure 13(IId)). These results reveal that the different types of instabilities all play a role in the occurrence of these large velocity fluctuations, and that asynchronous spikes occur when the lower-frequency spanwise-travelling mode is prevalent. The spikes are found to occur when the instabilities reach the phases in their cycles in which they add constructively to each other, which in turn creates the large velocity fluctuations at certain locations. This is evident from the videos of the CSTPOD first modes, which show occurrence of the different types of instabilities and their overlap. This will also be shown more clearly in the following discussion, but the question still remains, why do the spikes only occur in a certain direction (i.e.  positive or negative) depending on the location and the velocity component. Presumably, this type of constructive interaction should result in both positive and negative spikes (i.e.  an unskewed signal), at the locations of largest fluctuation amplitude, but this is clearly not the case. This question will also be explored in the following discussion.

First, to illustrate the prevalence of the different types of instabilities, we performed the same CSTPOD analysis as in the previous section, but on the bandpass-filtered datasets for each of the three frequency bands. Videos of the first mode for each of these are included in the supplemental material. To simply show the prevalence of each instability type, the velocity at the location of peak fluctuation amplitude was extracted for each time step in the CSTPOD mode. These values were multiplied by the average eigenvalue of the first mode for each set of spikes, in order to give an actual estimate of relative velocity amplitudes. These results are shown for two cases: the (1) case with asynchronous spikes only at location 1, and the (1,2,3) case where spikes occurred simultaneously at all three locations. In the first case (figure 28a), the amplitudes of all three frequency bands reach fairly similar magnitudes by the time the spike occurs ($t-t_s=0$), though the H-band is slightly lower in amplitude compared with the other two. In contrast to this case, in figure 28(b), it is clear that the L-band plays almost no role, as suspected based on the previous discussion, and the higher-frequency modes are almost equivalent in amplitude. It is interesting to note from these figures, as well as the accompanying videos, that the higher-frequency instabilities can be seen to gradually increase in amplitude leading up to the actual ‘spike’ event. Thus, it is possible that these events occur mainly in the middle of the passage of a wave-packet, where the instabilities happen to align and reach large enough amplitudes to cause this event to occur. This, in addition to the broadband nature of the disturbances, would help to explain the seemingly sporadic nature of these events. These results also help to explain the apparent change in behaviour as the step height is increased, as illustrated in figure 25. The asynchronous spikes would be expected to become more prevalent as the lower-frequency band disturbance becomes more prevalent, which indeed, is the case when the step height is increased.

Figure 28.

Plot of $u'$-velocity traces from first CSTPOD mode at location of peak fluctuation amplitude, $h/\delta = 0.57$, $x-x_s=38$ mm. Simultaneous spike locations from figure 22: (a) (1) and (b) (1,2,3).

This leads us to the question of why do the ‘spikes’ have a preferred positive or negative direction, depending on $y$ and $z$ location? It is interesting to note that, upstream of this $x-x_s=38$ mm location for the $h/\delta =0.57$ step height, at the next nearest measurement location, $x-x_s=25$ mm, no ‘spikes’ were evident. Both positive and negative peaks in velocity were observed nearly equally, and even similar phenomena of the constructive interaction of the different instability types were observed for both positive and negative peaks at the same location(s). The beginning and middle snapshots from the first CSTPOD mode for these two cases are shown in figure 29. The positive and negative peak cases are nearly identical, just 180$^{\circ }$ out of phase. The first snapshot indicates the prevalence of the lower-frequency spanwise-travelling mode, due to the prominence of the large spanwise wavelength structures. Similar to the results shown for the plane at $x-x_s=38$ mm (figure 26a,b), the low-frequency instability structures in figures 29(a) and 29(b) are similar, but the phase is different. Thus, the main determining factor on whether the velocity peak at the prescribed location will be negative or positive is the initial phase of the low-frequency instability. The structures that occur during the peak (figure 29c,d) are similar to those seen farther downstream (figure 27c), in that the 2-D instabilities have become apparent, but the biggest difference is that the structures farther upstream are not nearly as modulated in span. Videos of the first CSTPOD mode for these two cases are also included in the supplemental material. In these videos, one can clearly observe the initial presence of the low-frequency spanwise-travelling mode, followed by the superposition of the higher-frequency 2-D modes, until they constructively add together, leading to the peak in either positive or negative velocity. However, the spike mechanism, observed farther downstream, is not observed here.

Figure 29.

Beginning (a,b) and middle (c,d) snapshots of the first conditional POD mode, $h/\delta$ = 0.57, $x-x_s=25$ mm: (a,c) positive and (b,d) negative peaks in $u'$ near $z=2$, $y=1.5$ mm.

One way that the relative importance of the SCF amplitude vs the unsteady instability amplitude can be studied is by looking at the two cases mentioned earlier: the $h/\delta =0.49$ step with 40 $\mathrm {\mu }$m DREs, and the $h/\delta =0.51$ step with 20 $\mathrm {\mu }$m DREs. Since these two cases have very similar levels of unsteady instabilities (figure 9), and different SCF amplitudes (figure 8b), a comparison of these cases will help to determine whether the SCF amplitude plays a significant role in the occurrence of this ‘spike’ mechanism. It is also helpful that there is only a small difference in step height between these two cases, as the differences in mean flow (apart from those caused by the SCF) are quite small. The most noticeable difference can be seen in figure 5, in which the region of crossflow reversal (positive crossflow near the wall) is larger and stronger for the larger step height. It turns out that, in contrast to the $h/\delta =0.49$ case with 20 $\mathrm {\mu }$m DREs, both of these cases exhibit large amplitude one-sided ‘spikes’ in velocity beginning with the measurement plane at $x-x_s=38$ mm. The number of spikes that occurred at this location for each of these two cases was determined using the procedure described earlier. For the lower step height with larger SCF amplitude, 262 spikes were identified out of the 10 000 time steps, compared with 180 for the $h/\delta =0.51$ step. Thus, there were approximately 50 % more spikes identified for the case with larger SCF amplitude. Additionally, the average spike amplitude was 1.25 m s$^{-1}$ for the larger amplitude SCF case, and 1 m s$^{-1}$ for the $h/\delta =0.51$ case. This result shows that the SCF vortice are indeed an important factor in the appearance of this ‘spike’ mechanism.

Summarizing the most important observations thus far: the unsteady instabilities that are introduced by the step become modulated as they progress downstream of the step, due to the interaction with the SCF vortices. Each of the three families of instabilities are modulated in different ways, but they all exhibit some amount of modulation that corresponds to the spanwise wavelength of the SCF mode. Because of this modulation, there are certain points during the phase of each of these instabilities when they can align in a constructive way with each other to enhance the amplitude of the fluctuation, and, when the modulation is large enough, to create streamwise vorticity. By itself, this constructive superposition of modes should result in positive and negative peaks occurring equally. However, because of the existence of the crossflow velocity component within the boundary layer, there is a particular direction of streamwise vorticity that is favoured. In addition, the modulation occurs in such a manner that the regions of streamwise vorticity align with the already existent SCF vortices. This is made most evident by observing the alignment of the $v'$-velocity component (or the skewness of $v'$) during a spike event with the $V'$-steady disturbance velocity. Thus, it is believed that the superposition of unsteady instability modes, in a certain arrangement, leads to a situation where there is a brief, but very strong, interaction with the SCF modes, leading to the large velocity spikes that are observed. It is not clear from the present data (since we are limited to 2-D slices of the flow field) whether the extra vorticity that is created results only in streamwise-local events, or if streamwise vortices are shed as a result of the interaction. In either case, the velocity fluctuations reach such large amplitudes farther downstream that they eventually lead to high-frequency secondary instabilities, as was observed in the hot-wire measurements, and ultimately breakdown of the flow, and it is clear that this is the primary mechanism leading to transition. An interesting feature of these events is that, since they are basically enhancing the strength of the existing SCF vortices, they lead to larger modulation of the flow, increasing the local spanwise and wall-normal velocity gradients for a short period of time, which leads to the appearance of the high-frequency secondary instabilities.

Some similarities to these observations can be found in the previous computational studies that have considered SCF interactions with unsteady instabilities. As mentioned previously, Bonfigli et al. (Reference Bonfigli, Kloker and Wagner2003) described the formation of IVS that resulted from the mutual induction of the SCF and travelling crossflow vortices of similar wavelength. They associated these IVS with the generation of spikes in the velocity signal. A similar mechanism may be occurring in the present work, though there are some significant differences: the travelling crossflow instability is clearly not the only unsteady instability that is present and interacting with the SCF vortices, and the travelling and SCF instabilities possess significantly different wavelengths. These different wavelengths lead to the asynchronous appearance of the velocity spikes across the SCF wavelengths, particularly when the travelling crossflow instability is more dominant than the other unsteady instabilities. This behaviour is similar to the asynchronous secondary instability behaviour caused by the de-tuned modes in a TS-SCF interaction (Wassermann & Kloker Reference Wassermann and Kloker2005).

As an attempt to quantify the location and level to which breakdown was occurring for each case, the intermittency was computed as described in § 4.3. The results are shown in figure 30 for each case. As expected, the intermittency begins to increase earlier as the step height is increased. For the smallest step height ($h/\delta =0.49$), the intermittency is nearly zero even at the farthest downstream measurement location. The next largest step height ($h/\delta =0.51$) begins to increase above zero intermittency starting near 60 mm downstream of the step. For the $h/\delta =0.49$, 40 $\mathrm {\mu }$m case, the intermittency grows even sooner, slightly reaching above zero starting at $x-x_s=50$ mm. While the intermittency remains lower for this case compared with the largest step height, the two reach comparable values near 50 % to 55 % by the farthest downstream measurement location.

Figure 30.

Intermittency vs $x-x_s$ for each case.

Once again, a comparison of the two cases with similar unsteady instability amplitude ($h/\delta =0.49$, 40 $\mathrm {\mu }$m DREs, and $h/\delta =0.51$) is useful in determining the relative importance of the unsteady instabilities vs the SCF. It is clear, yet again, that the SCF vortices do play a role in speeding up the transition process, due to the increased intermittency of the former case compared with the latter. This role could be through the enhancement of the spike mechanism, as discussed previously. An additional consideration is that the enhanced modulation of the mean flow could result in the earlier appearance of the secondary instabilities, since the spike mechanism would not need to reach as large of an amplitude to achieve the same localized velocity gradients.