3.1. Turbulence statistics

Figure 2 compares the contours of the streamwise mean velocity ($U$) among the three test cases. In the plots, the isopleths of $U=0$ and the stream function $\varPsi =0$, where ${\varPsi (x,y=Y)}\equiv \int _{Y}^0 U(x, y)\,{{\rm d} y}$, with $Y$ representing the designated vertical location, are overlaid to facilitate visualization of the mean flow topology. The former encompasses the region of flow reversal, while the latter marks the mean separating streamline (Castro & Haque Reference Castro and Haque1987; Mohammed-Taifour & Weiss Reference Mohammed-Taifour and Weiss2016; Fang & Tachie Reference Fang and Tachie2019b). The isopleths of $U=0$ and $\varPsi =0$ both start at the leading edge ($(x,y)=(0,0)$) and intersect with the wall at the same point. This intersection point is the mean reattachment point beyond which the mean flow reattaches onto the undersurface, thus its streamwise distance from the leading edge defines the mean reattachment length $L_r$. The values of $L_r$ in the smooth, sandpaper and cube roughness cases are $9.2h$, $7.3h$ and $7.7h$ (or equivalently, $4.6H$, $3.65H$ and $3.85H$), respectively. The present mean reattachment length ($L_r=4.6H$) in the smooth case is in good agreement with $5.0H$, $4.7H$ and $4.4H$, respectively, reported by Kiya & Sasaki (Reference Kiya and Sasaki1983), Djilali & Gartshore (Reference Djilali and Gartshore1991) and Moore et al. (Reference Moore, Letchford and Amitay2019) for rectangular bluff bodies subjected to uniform incoming flows. This implies a similarity between the mean flow beneath a semi-submerged bluff body subjected to negligible free-surface waves, and that around bluff bodies in unbounded uniform flows.

Figure 2. Contours of the streamwise mean velocity ($U$) for the (a) smooth, (b) sandpaper, and (c) cube roughness cases. The dashed, dash-dotted and solid isopleths are at $U=0$, $U=U_\infty$ and $\varPsi =0$, respectively. Symbols $+$ and $\times$ mark the locations of the maximum and minimum $U$, respectively, with the corresponding values written alongside.

From figure 2, the mean reattachment lengths in the rough cases are apparently shorter than that in the smooth case. It is also noted in figure 2 that a distinct mean shear layer emanating from the leading edge is flanked by regions of elevated streamwise mean velocity ($U>U_\infty$) and mean flow reversal ($U<0$). The maximum values of $U$ ($1.31U_\infty \unicode{x2013}1.32U_\infty$) are not altered significantly by the wall roughness, but the magnitude of the maximum mean flow reversal in the rough cases ($0.32U_\infty$) is 18 % smaller than that in the smooth case ($0.39U_\infty$).

Figure 3 compares the streamwise variation of the maximum Reynolds stresses in the vertical direction. In the region upstream of the bluff body, all Reynolds stresses retain their levels of the incoming free-stream turbulence state. This is a direct indication that with the present Froude number (0.23), free-surface waves neither alter the incoming turbulence level nor influence the flow separation beneath the semi-submerged bluff body. In the region of $x/h<4$, the peak values of $\overline {u'u'}$ in the sandpaper case are generally higher than those in the other two cases, but the peak values of $\overline {v'v'}$ and $\overline {u'v'}$ in all cases are comparable. Turbulence transition near the leading edge is promoted by sandpaper, but not by the cube roughness. This underscores the difference between irregular and regular roughness types in the sandpaper and cube roughness cases. Compared with the rough cases, the smooth case possesses higher peak values of all three Reynolds stresses in the region $x\in [6,12]$. It is also noted in figure 3 that the disparity observed in the Reynolds stresses for the sandpaper and cube roughness cases is most pronounced in the $\overline {u'u'}$ component for $x<10$.

Figure 3. Streamwise variation of the maximum (a) $\overline {u'u'}$, (b) $\overline {v'v'}$ and (c) $\overline {u'v'}$ in the vertical direction. Every tenth measurement point is plotted for clarity.

Following Moore et al. (Reference Moore, Letchford and Amitay2019), the averaged trajectory of the shear layer (denoted by $Y_S$) is defined as the vertical location of the maxima of the turbulence kinetic energy $E\equiv \frac {1}{2}(\overline {u'u'}+\overline {v'v'})$ (since the spanwise velocity was not measured). As presented in figure 4(a), the elevation of the mean shear layer away from the wall increases gradually in the first half of the mean separation bubble, and plateaus downstream of the centre of the mean separation bubble. This plateau is a manifestation of the trajectory of the separated shear layer diverging from the separating streamline in the second half of the mean separation bubble. This pattern is in line with the observations by Kiya & Sasaki (Reference Kiya and Sasaki1983) and Moore et al. (Reference Moore, Letchford and Amitay2019). The asymptotic value of $Y_S$ in the smooth case is in the range $(1.2\pm 0.1)h$, which is in good agreement with values reported by Kiya & Sasaki (Reference Kiya and Sasaki1983) and Moore et al. (Reference Moore, Letchford and Amitay2019). It is also observed that the variations of $Y_S$ in both the sandpaper and cube roughness cases are almost indistinguishable, which is in line with the similar mean reattachment length observed for the rough cases in figure 2. For $x/h>4$, the asymptotic values of $Y_S$ for the smooth and rough cases are different by 0.2$h$. This difference ($0.2h=6$ mm) is apparently much larger than the roughness heights. The fact that wall roughness shifts the centre of the shear layer towards the wall by more than the roughness height is direct evidence that wall roughness significantly affects the turbulence structures within the separation bubble.

Figure 4. (a) Streamwise variation of the vertical centre of the mean shear layer ($Y_S$) (i.e. the peak elevation of $\frac {1}{2}(\overline {u'u'}+\overline {v'v'})$). Vertical profiles of (b) $U$, (c) $\overline {u'u'}$, (d) $\overline {v'v'}$, and (e) $\overline {u'v'}$, are plotted against $(y-Y_S)/h$. In (a), the vertical lines mark the mean reattachment lengths. In (b–e), the profiles of the first three streamwise locations use the vertical scale on the right, and the marked two horizontal dashed lines are symmetrical about $y=Y_S$. Not all measurement points are plotted, for clarity.

Figures 4(b–e) show the vertical profiles of mean velocity and Reynolds stresses at different streamwise locations. Note that the vertical coordinates are shifted by $Y_S$ to facilitate comparison of flow statistics surrounding the separated shear layers in different cases. From figure 4(b), the profiles of streamwise mean velocity for all the test cases collapse well. It is therefore concluded that the streamwise mean velocity in the separated shear layer is not sensitive to the wall roughness. In contrast to the mean flow, the Reynolds stresses are susceptible to the wall roughness, as is shown evidently in figures 3 and 4(c–e). For instance, the peak magnitudes of the Reynolds stresses in the rough cases at $x/h=11.0$ are almost 25 % smaller than those in the smooth case.

3.2. Unsteadiness of the separated shear layer

Figure 5 shows a typical example of the instantaneous streamwise velocity ($u$) field in the smooth case. Examples for the rough cases are qualitatively similar to figure 5, and therefore are not shown here for conciseness. The figure shows a distinct instantaneous shear layer emanating from the leading edge. This shear layer meanders as it evolves in the downstream direction, reflecting its unsteadiness. In this subsection, the unsteady characteristics of the separated shear layer and its sensitivity to different wall roughness conditions are elucidated.

Figure 5. Contours of a typical instantaneous streamwise velocity ($u$) for the smooth case. The dashed and solid isopleths are at $u=0$ and $u=U_\infty$, respectively. The inset magnifies the region near the leading edge.

3.2.1. Shear layer interface

In the context of the turbulent boundary layer (TBL), the mean shear is distributed across the entire boundary layer, but the instantaneous shear layers are concentrated at discrete wall-normal locations, creating the so-called uniform momentum zones in between (De Silva, Hutchins & Marusic Reference De Silva, Hutchins and Marusic2016; De Silva et al. Reference De Silva, Philip, Hutchins and Marusic2017). These instantaneous shear layers possess a constant streamwise velocity corresponding to local minima in the probability density function (p.d.f.) of the streamwise velocity (Adrian et al. Reference Adrian, Meinhart and Tomkins2000). The spatio-temporal and underlying structural mechanism of these instantaneous shear layers in the TBL have been studied extensively (Saxton-Fox & McKeon Reference Saxton-Fox and McKeon2017; Laskari et al. Reference Laskari, de Kat, Hearst and Ganapathisubramani2018). As exemplified in figure 5, the instantaneous separated shear layer is apparent from visual inspection, and deserves a closer quantitative examination. To this end, the p.d.f. of the instantaneous streamwise velocity field for the same snapshot in figure 5(a) is examined in figure 6.

Figure 6. Probability density functions (p.d.f.s) of $u$ in figure 5 and mean velocity $U$.

From figure 6, the p.d.f. of $u$ exhibits a steep increase as it increases to values larger than $U_\infty$, reflecting the near-uniform velocity in the region far away from the separated shear layer. On the other hand, the p.d.f. of $u$ possesses a plateau at a low value for $u< U_\infty$. This suggests that primarily, the instantaneous shear layer is confined to the regions $u< U_\infty$. This shear layer is clearly different from the instantaneous internal shear layer embedded in the TBL. Specifically, the internal shear layer in the TBL features a narrow thickness on the order of a wall unit (De Silva et al. Reference De Silva, Philip, Hutchins and Marusic2017), and is associated with a single velocity value corresponding to a local minimum in the p.d.f. of $u$. The geometry-induced shear layer in the present study features a thickness of the order of bluff body height, and therefore corresponds to a low-level plateau in the p.d.f. of $u$. It is deduced from figure 6 that the instantaneous shear layer is bounded by the isopleth of $u=U_\infty$. This deduction is consistent with the qualitative inspection in figure 5. Moreover, comparing the p.d.f.s of $u$ and $U$ in figure 6, the steep change in the p.d.f. at $U_\infty$ persists in both the mean and instantaneous flow. This is indicative of the persistence of the shear layer interface defined by the isopleth of $u=U_\infty$. This deduction is reminiscent of the conclusion by Fang & Tachie (Reference Fang and Tachie2019a) that the instantaneous shear layer induced by surface-mounted bluff bodies is bounded by the isopleth of the instantaneous streamwise velocity of the incoming flow at the bluff body height.

It should be noted that the above analysis is not applicable in the region very close to the leading edge (say $x/h<0.1$). From the inset in figure 5, the isopleth of $u=U_\infty$ folds downwards, and does not well bound the shear layer close to the leading edge. In the following, investigation of the unsteadiness of the shear layer is limited to the region where $x/h>0.1$. In figure 5, the isopleth of $u=U_\infty$ occasionally folds back onto itself to create enclosed ‘islands’ of low or high velocity, consistent with the topological characteristics of internal shear layer in the TBL (De Silva et al. Reference De Silva, Philip, Hutchins and Marusic2017). As time evolves, these islands become connected to the long isopleth of $u=U_\infty$, while new islands appear. Therefore, most of these islands are likely representative of a two-dimensional slice of a three-dimensional isosurface of $u=U_\infty$, and should be counted as the shear layer region. Eventually, the interface bounding the instantaneous separated shear layer is defined as the outermost envelope of the isopleth of $u=U_\infty$. For convenience, this interface is hereinafter denoted by $U_\infty ^i$, and the vertical location is described using $y_e(x,t)$ which is a function of streamwise location $x$ and time $t$. The space–time variation of $y_e$ reflects the unsteady nature of the shear layer.

Following the techniques established for the analyses of interfacial layers in the TBL (De Silva et al. Reference De Silva, Philip, Hutchins and Marusic2017) and jets (Westerweel et al. Reference Westerweel, Fukushima, Pedersen and Hunt2005), the p.d.f. of $y_e$ and the conditional-averaged streamwise velocity (denoted by $\langle u\rangle$) in the vertical neighbourhood of instantaneous $y_e$ at selected streamwise locations are examined. As seen in figure 7(a), the p.d.f. of $y_e$ is skewed towards the locations of $y_e-Y_S<0$, suggesting the frequent occurrence of $U_\infty ^i$ below the centre of the mean shear layer. The p.d.f. of $y_e$ spans wider vertical ranges at locations further downstream, and remains fairly similar between different test cases. This suggests that the vertical location of the shear layer varies over a wider range in the downstream locations. This variation of vertical location of the shear layer is not affected strongly by the wall roughness. From figure 7(b), the vertical profile of $\langle u\rangle$ is representative of an approximated instantaneous shear layer, and remains similar for different test cases.

Figure 7. Vertical profiles of (a) the p.d.f. of $y_e$, and (b) the conditionally averaged streamwise velocity $\langle u\rangle$ across $y_e$, at selected streamwise locations. In (a), the profiles of the first three $x/h$ use the scale on the left, while the other values use the scale on the right. Every third point is plotted, for clarity.

To further demonstrate the dynamic significance of $U_\infty ^i$ to the separated shear layer, the possibility of reconstructing the streamwise Reynolds normal stress by a convolution of the conditionally averaged shear layer $\langle u\rangle (y-y_e)$ with the p.d.f. of $y_e$ is explored. This is expressed as

(3.1)\begin{equation} \langle u'u'\rangle(y)= \int_{-\infty}^0 \langle u\rangle^2 (y-y_e) \,\text{p.d.f.}(y_e) \,{\rm d} y_e - \left[ \int_{-\infty}^0 \langle u\rangle (y-y_e) \,\text{p.d.f.}(y_e) \,{\rm d} y_e\right]^2. \end{equation}

Figure 8 compares the reconstructed $\langle u'u'\rangle$ with the measured $\overline {u'u'}$ for different test cases. For the streamwise locations of $x/h = 0.5$ and $1.0$, the reconstructed $\langle u'u'\rangle$ is in good agreement with $\overline {u'u'}$ except for the near-wall region. Evidently, the reconstruction of (3.1) only sensitizes vortices near the shear layer interface. The similarity of $\langle u'u'\rangle$ and $\overline {u'u'}$ for $x/h\leqslant 1.0$ is because the separated shear layer is not saturated with a wide range of length scales of vortices in the region of early turbulence transition. For $x/h\geqslant 5$, on the other hand, $\langle u'u'\rangle$ agrees well with $\overline {u'u'}$ below the peak location of $\overline {u'u'}$, but exhibits noticeable differences from $\overline {u'u'}$ between the peak location of $\overline {u'u'}$ and the wall. These differences emphasize the confinement effect of the wall on the development of the shear layer. For $x/h\geqslant 8.0$, the peak values of $\langle u'u'\rangle$ in the rough cases are smaller than those in the smooth case, which is consistent with the observation from $\overline {u'u'}$ in figures 3(a) and 4(c).

Figure 8. Vertical profiles of $\langle u'u'\rangle$ constructed from (3.1) and measured $\overline {u'u'}$ for the (a) smooth, (b) sandpaper, and (c) cube roughness cases. Not all grid points are plotted, for clarity.

The temporal characteristics of the instantaneous shear layer interface $U_\infty ^i$ is investigated further using the frequency spectra of $y_e$ at different streamwise locations. Note that the spectral analysis in the present study is implemented using Welch's overlapped window techniques (Welch Reference Welch1967). The size and overlap of windows of Fourier transformation are 4.96 s (4000 snapshots) and 0.12 s (100 snapshots), respectively. As such, the resolved frequencies in the spectral analysis are multiples of $0.0126U_\infty /h$ (0.2018 Hz). For different test cases, as shown in figure 9, the oscillation of $U_\infty ^i$ downstream of the mean reattachment point is amplified at two distinct frequencies regardless of wall roughness conditions. These distinct frequencies observed in figure 9 are summarized in table 1 along with other frequencies identified in the subsequent figures to facilitate comparison. It is interesting to note in figure 9 that for the smooth case, the significance of $St_h=0.050$ is also visible in the first half of the mean separation bubble (see the local peak of the isopleth around $x/h\approx 1$). As for the rough cases, the significant frequencies identified downstream of the mean separation bubble do not manifest in the first half of the separation bubble. The underlying structural mechanism of these observations will be discussed in § 3.4.

Figure 9. Premultiplied frequency spectra of $y_e$ for the (a) smooth, (b) sandpaper, and (c) cube roughness cases. The horizontal dash-dot-dotted lines mark the characteristic frequencies, while the vertical dash-dotted and dashed lines mark, respectively, 50 % and 100 % of the mean reattachment length.

Table 1. Summary of significant Strouhal numbers ($St_h$) of different parameters/locations in different cases. Symbol – indicates non-existence, and superscript $^*$ marks the dominant Strouhal number.

3.2.2. Frequency migration

Figure 10 presents the streamwise variation of frequency spectra of the streamwise fluctuating velocity ($\phi _{uu}$) along the mean shear layer for different cases (see figure 4a). At the leading edge ($x/h=0$), the spectra for all test cases are of low magnitudes and do not possess any dominant frequency. This is in contrast to the observation by Moore et al. (Reference Moore, Letchford and Amitay2019) that for rectangular cylinders of aspect ratios less than 5, the dominant frequency associated with the VK vortex shedding in the wake region also manifests at the leading edge. In the present study, the streamwise length of the bluff body is 33 times its effective height (submergence depth), so instantaneous flow reattachment over the undersurface of the bluff body is guaranteed to exclude any influence of wake flow dynamics on the separated shear layer at the leading edge. Overall, low-intensity white noise at the leading edge is imposed by the incoming flow condition, and the frequency spectra presented in figure 10 showcase the development of a separated shear layer subjected to different wall roughness conditions.

Figure 10. Premultiplied frequency spectrum of the streamwise velocity ($\phi _{uu}$) at the streamwise locations $x/h=0.0$, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 and 1.0 along the peak locations of $E$ (see figure 4a) for the(a) smooth, (b) sandpaper, and (c) cube roughness cases. To facilitate visualization, the spectra are vertically offset by the values indicated on the right-hand side. Vertical dash-dotted lines mark characteristic $St_{h}$ numbers.

As seen in figure 10(a), slightly downstream of the leading edge at $x/h=0.1$, a broad peak centred around Strouhal number $St_h$ ($\equiv f h/U_\infty$) 3.60 appears. Further downstream, at $x/h=0.2$, a sharp dominant peak at $St_h=3.60$ becomes distinct, and is attributed to the KH instability. This is in good agreement with the observation by Kiya & Sasaki (Reference Kiya and Sasaki1983) that a dominant frequency $St_h\approx 3$ is induced by roll-up of the shear layer at $x/h=0.2$ for a plate at a similar Reynolds number. In contrast to the present results, Moore et al. (Reference Moore, Letchford and Amitay2019) observed that the KH instability induced by a bluff body of aspect ratio 5 at a similar Reynolds number manifests as a broader peak of spectra around a similar frequency ($St_h\approx 4$), but it is closer to the leading edge (at $x/h=0.046$). By comparing figures 10(b,c) with figure 10(a), in the two rough cases, sharp peaks of spectra do not occur near the leading edge, suggesting the bypass of KH instability.

It is also interesting to note in figure 10(a) that at $x/h=0.2$, there exists a subdominant peak around the first sub-harmonic ($St_h=1.80$) of the KH frequency. As $x/h$ increases from 0.2 to 0.4, the magnitude (measured by $f\phi _{uu}$) of the sub-harmonic at $St_h=1.80$ gradually dominates over the fundamental KH frequency, while the second sub-harmonic ($St_h=0.90$) appears and strengthens. The spectral peak associated with the KH instability ceases to exist at $x/h=0.5$. At locations further downstream, the dominance gradually switches from the first sub-harmonic ($St_h=1.80$) to the second sub-harmonic (${St_h=0.90}$). At $x/h=1.0$, the third sub-harmonic ($St_h=0.45$) dominates while exhibiting a relatively broad peak. Based on the above observations, it is concluded that downstream of the leading edge in the separated shear layer, the dominant frequency decreases discretely through higher ranks of sub-harmonics, while the associated peak becomes broader. This pattern of frequency migration is due to the mechanism of vortex pairing occurring repeatedly. Specifically, small-scale spanwise vortices initiate because of the roll-up of the shear layer due to the KH instability around $x/h=0.2$. Two adjacent vortices roll around each other as they are convected downstream, so as to form a single vortex of twice the size and half the frequency. This process repeats in the downstream direction leading to higher ranks of sub-harmonics of KH frequency. This repeating vortex pairing mechanism is well known to dictate the growth of the free shear layer (Winant & Browand Reference Winant and Browand1974; Ho & Huerre Reference Ho and Huerre1984), and is evidently also at play for a separated shear layer.

From figures 10(b) and 10(c), a dominant frequency along with its sub-harmonic do not emerge until $x/h=0.6$ and $0.4$ for the sandpaper and cube roughness cases, respectively. Downstream of these locations, the dominance gradually shifts from the dominant frequency to its sub-harmonic, similar to the smooth case. This is a clear indication that the presence of wall roughness interrupts the vortex pairing mechanism near the leading edge, where the separated shear layer is close to the roughness elements. It is also emphasized that the migration from high to low frequency in the rough cases is slower than that in the smooth case. For instance, between $x/h=0.4$ and 1.0, the dominant frequency decreases twofold from $St_h\approx 0.9$ to ${\approx }0.5$ in the rough cases, but it decreases fourfold from $St_h=1.80$ to $0.45$ in the smooth case.

Figure 11 characterizes the variation of $f\phi _{uu}$ along the mean shear layer over the entire measurement domain (see figure 4a). In general, high levels of $f\phi _{uu}$ appear in the form of horizontally aligned bands. This is attributed to vortices being convected in the downstream direction while maintaining their characteristic frequencies. As seen in figure 11, these horizontally aligned bands of $f\phi _{uu}$ appear at multiple discrete frequencies at any streamwise location, and the associated frequencies clearly migrate to lower frequencies in the first half of the mean separation bubble. This suggests the coexistence of vortex shedding at discrete frequencies through the vortex pairing mechanism exemplified in figure 10. It is important to note here that Moore et al. (Reference Moore, Letchford and Amitay2019) reported a continuous, as opposed to discrete, migration of dominant frequency from the KH frequency near the leading edge to the vortex shedding frequency in the wake region behind a bluff body of aspect ratio 5. This difference with the present result reflects the interruption of development of KH instability by the VK vortex shedding in the wake region.

Figure 11. Variation of $f\phi _{uu}$ along the centre of the mean shear layer for the (a) smooth, (b) sandpaper, and (c) cube roughness cases. The vertical dash-dotted and dashed lines mark, respectively, 50 % and 100 % of the mean reattachment length. The horizontal dash-dot-dotted lines mark characteristic frequencies.

It is interesting to see in figure 11 that the migration to lower frequency ceases around the mid-point of the mean reattachment length. The dominance of multiple frequencies is maintained in the second half of the separation bubble. The underlying mechanism is explained as follows. In the first half of the mean separation bubble, the separated shear layer is relatively thin and gradually deflects away from the wall. As such, the larger vortices associated with lower frequencies occur through the vortex pairing mechanism. In the second half of the mean separation bubble, the distance of the separated shear layer from the wall remains constant (see figure 4a), so further vortex pairing is suppressed by the wall confinement. Consequently, vortices are shed downstream of the mid-point of the mean separation bubble while maintaining their characteristic frequencies.

It is shown in figure 11 that successive frequencies do not have to be the sub-harmonic of the predecessor fundamental frequencies. This suggests the occurrence of pairing between two vortices at different frequencies. Specifically, the pairing of two vortices at frequencies $St_1$ and $St_2$ (whose time scales are $1/St_1$ and $1/St_2$, respectively) generates a vortex at frequency $1/(1/St_1+1/St_2)$ (whose time scale is $1/St_1+1/St_2$). For instance, as seen in figure 11(a) for the smooth case, the significance of frequencies $St_1=0.115$, $St_2=0.089$ and $St_3=0.050$ is evident. Although the harmonic of $St_3$ at $0.100$ is not discernible, $St_3$ is indeed $1/(1/St_1+1/St_2)$, suggesting that the vortices at $St=0.050$ are generated by the pairing of vortices at $St=0.115$ and vortices at $St=0.089$. A close examination of the identified significant frequencies in figures 11(b) and 11(c) indicates that the aforementioned vortex pairing mechanism for the smooth case is not valid for the rough cases. This is not surprising. Since the radius of vortices in the second half of the mean separation bubble is close to the height of the mean separation bubble, these vortices interact directly with the wall and therefore are more susceptible to wall roughness. In accordance with figures 10(b,c) and 11(b,c), the occurrence of vortex pairing in the rough wall cases is limited to a subrange in the first half of the mean separation bubble. Indeed, for the vortex pairing to occur, the separated shear layer needs to be sufficiently far away from the wall, and the vortices need to be sufficiently small to avoid direct interaction with the wall.

In figure 11(a), the dominance of $St_h=0.050$ in the smooth case is apparent. This frequency is similar to the vortex shedding frequencies ($St_h=0.05\sim 0.06$) observed by Kiya & Sasaki (Reference Kiya and Sasaki1983) and Moore et al. (Reference Moore, Letchford and Amitay2019). From figure 11(b), the dominant frequency in the sandpaper case is $St_h=0.63$, and its dominance does not persist beyond the mean reattachment point. As for the cube roughness case, the dominance of a particular frequency is less pronounced than the other two cases, and is at best identified to be between $St_h=0.094$ and $St_h=0.171$. This suggests a strong disruption of regular vortex shedding by the cube roughness. It is also noted in figure 11 that there exists a less prominent but clearly identifiable frequency ($St_h=0.025$ and $0.037$ in the sandpaper and cube roughness cases, respectively), lower than the dominant frequency in the smooth case. The interruption of vortex shedding/pairing by the wall roughness in the second half of the separation is responsible for the weaker Reynolds stresses than in the smooth case in the region of $x/h>5$ (see figure 3).

3.3. Unsteadiness of flow separation

Attention is now turned to the unsteady characteristics of flow separation. Figure 12 compares the premultiplied frequency spectra $f\phi _{uu}$ at $y/h=-0.09$ for different test cases. This is inspired by Kiya & Sasaki (Reference Kiya and Sasaki1985) who measured the unsteady velocity at $0.05h$ away from the wall to investigate the reverse-flow unsteadiness. As seen in figure 12(a) for the smooth case, the dominance of $St_h=0.089$ initiates around the mean reattachment point and persists in the downstream direction. This dominant frequency is close to $St_h=0.081$ observed by Kiya & Sasaki (Reference Kiya and Sasaki1985). In accordance with table 1, the dominant frequency $St_h=0.050$ in the mean shear layer becomes subdominant in the near-wall region around the mean reattachment point. Kiya & Sasaki (Reference Kiya and Sasaki1985) also noted the disparity between the dominant frequencies in the shear layer and flow reversal around the mean reattachment point, and interpreted this disparity as ‘dispersion of the large-scale vortices in strength and position’. In accordance with figures 12(a), 11(a) and 9(a), the vortex shedding motion at $St_h=0.089$ is closer to the wall than that at $St_h=0.050$. As such, the former is more influential in the near-wall region around the mean reattachment point while the later is more influential in the shear layer interface. By comparing figures 11(b) and 12(b) for the sandpaper case, the subdominant frequency $St_h=0.089$ in the mean shear layer becomes dominant in the near-wall region, whereas the dominance of $St_h=0.063$ observed in the mean shear layer is discernible in the near-wall region. As for the cube roughness case, based on figures 11(c) and 12(c), the dominance of $St_h=0.094$ persists in the mean shear layer and in the near-wall region.

Figure 12. Streamwise variation of $f\phi _{uu}$ at $y/h=-0.09$ for the (a) smooth, (b) sandpaper, and (c) cube roughness cases. The vertical dash-dotted and dashed lines mark, respectively, 50 % and 100 % of the mean reattachment length. The horizontal dash-dot-dotted lines mark characteristic frequencies.

It is also worthwhile comparing the results in figure 12 with Cimarelli et al. (Reference Cimarelli, Leonforte and De Angeli2018) on a DNS study of flow separation around a bluff body with aspect ratio 5 at a Reynolds number ($Re_h=1500$ based on half of body height) one-tenth of the present study. Cimarelli et al. (Reference Cimarelli, Leonforte and De Angeli2018) examined the variation of dominant time scales (reciprocal of frequencies) along a near-wall mean streamline traced back near the mean reattachment point. The near-wall mean streamline of the present study is at a constant vertical elevation except for the regions close to the separating and reattaching points. This allows a direct comparison with the statistics along near-wall mean streamlines in Cimarelli et al. (Reference Cimarelli, Leonforte and De Angeli2018). Specifically, Cimarelli et al. (Reference Cimarelli, Leonforte and De Angeli2018) showed that the time scale in the near-wall region of the second half of the mean separation bubble is similar to that in the shear layer, which is consistent with the present observations from figures 11(a) and 12(a). Cimarelli et al. (Reference Cimarelli, Leonforte and De Angeli2018) also observed that as the separating point is approached along the near-wall mean streamline, the time scale gradually increases in the first half of the mean separation bubble. This is in sharp contrast to the present results. As seen in figure 12, regardless of the wall roughness condition, the dominant frequencies exhibit an abrupt change near the middle point of the mean reattachment length. This disparity with Cimarelli et al. (Reference Cimarelli, Leonforte and De Angeli2018) is due to the differences in Reynolds number and aspect ratio of bluff bodies.

In Kiya & Sasaki (Reference Kiya and Sasaki1983) and Moore et al. (Reference Moore, Letchford and Amitay2019), the integral time scales from their pointwise measurements at various locations of the flow separation bubble were analysed. In the present study, whole-field time-resolved velocity data are exploited to investigate the time scales of the entire separation bubble. Here, the integral time scale ($\tau$) is calculated by integrating the temporal autocorrelation of $u'$ from the time displacement of zero to the point associated with the first zero correlation, and the results are presented in figure 13. Outside the mean separation bubble (see the lower left corners of the plots), the high values of integral time scales are associated with incoming turbulence, and are not of particular importance. It is, however, worth noting that along the mean separated shear layer, the values of the integral time scale generally increase in the downstream direction, and are of the order of $h/U_\infty$ near the mean reattachment point. These are similar to the observations made by Kiya & Sasaki (Reference Kiya and Sasaki1983) and Moore et al. (Reference Moore, Letchford and Amitay2019). In the near-wall region, on the other hand, the values of integral time scale tend to decrease in the downstream direction. In particular, there exists a distinct region of elevated integral time scale in the first half of the mean reverse flow area, where the streamwise Reynolds normal stress, $\overline {u'u'}$, is also significant (see figure 4b). These patterns of integral time scale in the near-wall region are persistent irrespective of the wall roughness condition. It is noted that the peak levels of the integral time scale in the first half of the mean reverse flow area for the sandpaper case are lower than those for the other two cases.

Figure 13. Contours of the integral time scales $\tau$ for the (a) smooth, (b) sandpaper, and (c) cube roughness cases.

The observations from figures 12 and 13 suggest the existence of low-frequency motion residing exclusively in the first half of the mean separation bubble. This is reminiscent of the low-frequency flapping motion commonly observed in turbulent separation bubbles (Eaton & Johnston Reference Eaton and Johnston1982; Kiya & Sasaki Reference Kiya and Sasaki1983; Cherry et al. Reference Cherry, Hillier and Latour1984; Tafti & Vanka Reference Tafti and Vanka1991; Cimarelli et al. Reference Cimarelli, Leonforte and De Angeli2018). Kiya & Sasaki (Reference Kiya and Sasaki1983, Reference Kiya and Sasaki1985) noted the existence of a peak frequency between $St_h=0.012$ and $St_h=0.02$ at $x/L_r=0.2$. Kiya & Sasaki (Reference Kiya and Sasaki1985) presented the conditional-averaged streamwise velocity based on the surface pressure at $x/L_r=0.2$, and observed that this low-frequency pressure is associated with vertical flapping (hence the term ‘flapping motion’) of the shear layer along with the thickening/thinning of flow reversal. Prompted by the results from figure 13 and results from Kiya & Sasaki (Reference Kiya and Sasaki1983, Reference Kiya and Sasaki1985), the frequency spectra of the reverse flow area in the first half of the mean separation bubble are further examined, and expressed as

(3.2)\begin{equation} A(t)=\int_{-\infty}^0 \int_0^{0.5L_r} \mathscr{H}[u(x,y,t)]\, {{\rm d} x} \,{{\rm d} y}. \end{equation}

Here, $\mathscr {H}[u(x,y,t)]$ is 0 and 1 for positive- and negative-valued $u(x,y,t)$, respectively. Note that the integral range of $x$ is between 0 and $0.5L_r$. It is noted that the above equation with an integral range of $x$ over the entire measurement domain has also been evaluated, and the results are not as meaningful as those evaluated within $0 \leqslant x \leqslant 0.5L_r$.

Figure 14 compares the premultiplied frequency spectra of $A(t)$ for different test cases. The dominant frequency at $St_h=0.025$ is evident for the smooth case. This frequency is close to the frequency of flapping motion of separation bubbles reported commonly in the literature (Eaton & Johnston Reference Eaton and Johnston1982; Kiya & Sasaki Reference Kiya and Sasaki1983; Cherry et al. Reference Cherry, Hillier and Latour1984; Tafti & Vanka Reference Tafti and Vanka1991; Cimarelli et al. Reference Cimarelli, Leonforte and De Angeli2018). It is therefore concluded that the reverse flow area in the first half of the mean separation bubble exhibits a low-frequency sequence of enlargement and shrinkage. Cimarelli et al. (Reference Cimarelli, Leonforte and De Angeli2018) also argued that the low-frequency flapping motion is ‘strictly related to the behaviour of the secondary vortex’, which occurs in the first half of the primary mean separation bubble. This confinement of low-frequency flapping motion to the first half of the mean separation bubble marks a significant difference between the flapping motions of separation bubbles induced by bluff bodies in uniform flow and those induced by surface-mounted bluff bodies subjected to incoming TBL investigated by Pearson et al. (Reference Pearson, Goulart and Ganapathisubramani2013), Fang & Tachie (Reference Fang and Tachie2019b, Reference Fang and Tachie2020) and Fang et al. (Reference Fang, Tachie, Bergstrom, Yang and Wang2021). For instance, Fang et al. (Reference Fang, Tachie, Bergstrom, Yang and Wang2021) reported that the reverse flow area of the entire separation bubble over a forward-facing step (FFS) in a turbulent channel flow experiences quasi-periodic expansion/contraction at multiple frequencies. These frequencies coincide with the fundamental and harmonic frequencies of incoming hairpin structures, but are significantly different from the typical frequencies of flapping motion due to development of separation shear layers (Eaton & Johnston Reference Eaton and Johnston1982; Kiya & Sasaki Reference Kiya and Sasaki1983; Cherry et al. Reference Cherry, Hillier and Latour1984; Tafti & Vanka Reference Tafti and Vanka1991; Cimarelli et al. Reference Cimarelli, Leonforte and De Angeli2018). Fang & Tachie (Reference Fang and Tachie2019b) and Fang et al. (Reference Fang, Dow, Tachie, Malenchak and Wang2022) demonstrated that the dominant frequencies of reverse flow areas reflect the interaction of incoming large-scale motion (Adrian et al. Reference Adrian, Meinhart and Tomkins2000) and hairpin structures with the FFS. For example, Fang et al. (Reference Fang, Dow, Tachie, Malenchak and Wang2022) showed that as a hairpin structure is leaning over the step, it induces a pair of vertical counter-rotating vortices on the frontal surface, while an opposite-signed pair of counter-rotating vortices forms near the wall over the step. As a consequence, an enlarged reverse flow area upstream of the step is typically associated with a reduced reverse flow area over the step. In the present study, however, the incoming flow is of low turbulence intensity, and turbulence transition occurs only downstream of the leading edge. As such, the flapping motion of flow separation reflects the interaction of vortical structures originating in the shear layer with flow reversal, which will be clarified in the next subsection. It is worth noting in figure 14 that the dominant frequency of reverse flow area in the first half of the mean separation bubble for the sandpaper case is the same ($St_h=0.025$) as that in the smooth case, whereas that in the cube roughness case is at a relatively higher value, i.e. $St_h=0.050$.

Figure 14. Premultiplied frequency spectra of reverse flow area ($f\varPhi _A$) in the first half of the mean reattachment length.

3.4. Spectral proper orthogonal decomposition analysis

Thus far, it has been demonstrated that the incoming free-surface flow is effectively a free-slip boundary, so the present smooth case exhibits many similarities with the turbulent separations induced by a bluff body exposed to incoming uniform flows (Kiya & Sasaki Reference Kiya and Sasaki1983, Reference Kiya and Sasaki1985). To identify the dominant frequencies along with their associated vortical structures, Kiya & Sasaki (Reference Kiya and Sasaki1983, Reference Kiya and Sasaki1985) used statistical techniques such as conditional average, frequency filtering and auto/cross-correlation to analyse pointwise measurement data of velocity and wall pressure. In this subsection, an investigation of both the spatial and temporal characteristics of the pertinent vortical structures is conducted to exploit fully the time-resolved whole-field measurements using spectral proper orthogonal decomposition (SPOD) (Towne, Schmidt & Colonius Reference Towne, Schmidt and Colonius2018).

In the framework of SPOD, the fluctuating velocity field is decomposed into different modes at different frequencies as follows:

(3.3a)\begin{align} \boldsymbol{u}'(\boldsymbol{X},t) &= \sum_{j=1}^\infty \hat{\boldsymbol{u}}_j(\boldsymbol{X}) \exp({{\rm i}2{\rm \pi} f_j t}) + \text{c.c.} \end{align}

(3.3b)\begin{align} &= \sum_{j=1}^\infty \sum_{k=1}^M a^{(k)}_j \boldsymbol{\varPhi}^{(k)}_j (\boldsymbol{X}) \exp({{\rm i}2{\rm \pi} f_j t}) +\text{c.c.}, \end{align}

where ${\rm i}\equiv \sqrt {-1}$ and c.c. means the complex conjugate counterpart. Here, (3.3a) is the Fourier transform of the velocity field exploiting the homogeneity in time. In practice, the Fourier transform is performed employing Welch's overlapped window technique (Welch Reference Welch1967), allowing multiple realizations for ensemble average. In (3.3b), the field of the Fourier-transformed coefficient for the $j$th frequency $\hat {\boldsymbol {u}}_j(\boldsymbol {X})$ is further decomposed into POD modes using singular value decomposition (SVD), in analogy to the space-only POD (see Meyer, Pedersen & Özcan Reference Meyer, Pedersen and Özcan2007; Fang & Tachie Reference Fang and Tachie2019b). In (3.3b), $M=200$ is the number of POD modes, which is determined by the number of windows of the Fourier transformation. Additionally, $\boldsymbol {\varPhi }^{(k)}_j(\boldsymbol {X})$ represents the $k$th POD mode for the $j$th frequency, and its corresponding coefficient is denoted by $a^{(k)}_j$. The POD modes are orthonormal to each other, i.e. $\boldsymbol {\varPhi }^{(p)}_j(\boldsymbol {X})[\boldsymbol {\varPhi }^{(q)}_j(\boldsymbol {X})]^{\rm T}=\delta _{pq}$ (where superscript $^{\rm T}$ is the complex conjugate transpose, and the summation convention is not implied), so $\lambda _j^{(k)}\equiv |a^{(k)}_j|^2$ measures the contribution of $\boldsymbol {\varPhi }^{(k)}_j(\boldsymbol {X})$ to the integration of in-plane turbulence kinetic energy ($E$) over the analysed area. The rank of POD mode is thus arranged in descending order according to its energy $\lambda _j^{(k)}$. In essence, each SPOD mode represents a spatial structure varying over a designated temporal period (i.e. reciprocal of the corresponding frequency) described by

(3.4)\begin{equation} \boldsymbol{\varXi}^{(k)}_j(\boldsymbol{X},\alpha) \equiv \boldsymbol{\varPhi}^{(k)}_j(\boldsymbol{X}) \exp({{\rm i}2{\rm \pi} f_j t}) + \text{c.c.},\end{equation}

where $\alpha \equiv 2{\rm \pi} f_j t$ defines the phase angle.

We perform SPOD for the entire measurement area for all three test cases. Figure 15 compares the premultiplied frequency spectra of the first two SPOD modes. In general, for low frequencies (say $St_h<0.2$), the energies held by the first modes are much higher than those by the second modes. This emphasizes the coherency of turbulent motions (or ‘low-rank behaviour’ as termed by Schmidt et al. Reference Schmidt, Towne, Rigas, Colonius and Brès2018) at these low frequencies. In view of this, attention is limited to the first modes in the subsequent analyses. As mentioned earlier, the significant frequencies from the preceding discussions are summarized in table 1. As seen in figure 15(a) and table 1 for the smooth case, three dominant frequencies of similar energy levels are noted (i.e. $St_h=0.050$, 0.089 and 0.115). These three frequencies have been identified using the velocity spectra in the mean shear layer, while the first two frequencies (i.e. $St_h=0.050$ and 0.089) have also been identified in the region close to the mean reattachment point and in the instantaneous shear layer interface. As for the two rough cases, the dominant frequencies of the first SPOD modes do not always manifest in the aforementioned localized indicators. This suggests that either strong vortical structures do not necessarily exert strong influence everywhere, or some weak vortical structures are in fact locally dominant. The latter is exactly the situation with the flapping motion in the first half of the mean separation bubble, since in none of the cases does SPOD identify the low frequency corresponding to the flapping motion. It is also important to note from table 1 and figure 12 that the dominant frequencies close to the mean reattachment point in the near-wall region (at $y/h=-0.09$) for different test cases are fairly close to each other ($St_h\approx 0.09$). Therefore, the vortex shedding motion at a frequency higher than $St_h\approx 0.09$ is detached from the wall, whereas that at a frequency lower than $St_h\approx 0.09$ is touching the wall.

Figure 15. Premultiplied frequency spectra of the first and second SPOD modes ($f\lambda ^{(1)}$ and $f\lambda ^{(2)}$, respectively) for the (a) smooth, (b) sandpaper, and (c) cube roughness cases. The vertical dash-dotted lines mark characteristic frequencies of the first mode. The vertical axes use arbitrary scales.

Figure 16 compares the first POD modes at the frequencies $St_h=0.115$, 0.115 and 0.127 for the smooth, sandpaper and cube roughness cases, respectively. These plots are representative of significant frequencies higher than $St_h\approx 0.09$ identified in figure 15 for different cases, and are used to examine the vortical structures that are detached from the wall. Figure 16 shows $\varXi$ (see (3.4)) at an arbitrary phase angle $\alpha$, while supplementary movie 1 available at https://doi.org/10.1017/jfm.2022.661 shows the complete period of these SPOD modes, with $\alpha$ varying from 0 to $2{\rm \pi}$. In general, these modes exhibit consecutive slanted patches of alternating positive and negative streamwise fluctuating velocity. The inclination angles of these patches are about $45^\circ$ regardless of the wall roughness conditions. This angle ($45^\circ$) is the same as the angle of vortical structures conditioned on the wall pressure at the mean reattachment point observed by Kiya & Sasaki (Reference Kiya and Sasaki1983, Reference Kiya and Sasaki1985). Within the streamwise range of the mean separation bubble, each slanted patch possesses two local peaks that are well separated by the isopleth of $U=0$, and the upper peak is stacked over the lower peak of the adjacent downstream patch. As seen in supplementary movie 1 for figure 16, a train of vortices moves in the downstream direction in the close vicinity of the isopleth of $U=U_\infty$, and is attributed to the vortex shedding motion that originated in the separated shear layer. (An animation of the first SPOD mode at $St_h=3.6$ for the smooth case is also supplied as supplementary movie 2; it signifies the vortical structures associated with the KH instability.)

Figure 16. The first POD mode at (a) $St_h=0.115$ for the smooth case, (b) $St_h=0.115$ for the sandpaper case, and (c) $St_h=0.127$ for the cube roughness case, at an arbitrary phase angle. The dashed and dash-dotted isopleths are at $U=0$ and $U=U_\infty$, respectively. Blue and red indicate negative and positive streamwise velocity, respectively. Dashed ellipses mark characteristic vortices. An animated version is also provided as supplementary movie 1.

In figure 16, two opposite-signed vortices (marked as A$_1$ and A$_2$) are centred on the isopleth of $U=0$, while the vortex B$_1$ is centred on the isopleth of $U=U_\infty$. (For clarity, vortices centred on the isopleths of $U=0$ and $U=U_\infty$ are named as A$_i$ and B$_i$, respectively, where subscript $i$ indicates the order of appearance in discussion.) Vortex B$_1$ is the first vortex shed in the separated shear layer at this frequency. The onset of each shed vortex goes through the following process. Vortices A$_1$ and B$_1$ are counter-rotating, consequently strong mode velocities are induced in between. This strong mode velocity also forms as an upstream part of vortex A$_2$. As the phase angle increases, vortices A$_2$ and B$_1$ are convected downstream while growing in size and maintaining their relative streamwise locations. Downstream of the mean reattachment point, however, only vortex B$_1$ (and its successors) remains. Meanwhile, alternating-signed vortices (see A$_3$ and A$_4$ in supplementary movie 1) appear along the frontal half of the isopleth of $U=0$. As vortex A$_1$ is convected to beyond the mid-point of the mean reattachment length, vortex B$_2$ appears. The above process repeats itself except that the newly formed vortices are of opposite signs with their predecessors at the same locations. This process clearly demonstrates that the vortical structures encompassing the flow reversal are linked dynamically to the vortex shedding in the separated shear layer. By multiplying the wavelength with the corresponding mode frequency, the pertinent convective velocity of the vortex shedding shown in figure 16 is estimated to be approximately $0.6U_\infty$, which is close to the convective velocity of vortex shedding ($0.5U_\infty$) reported by Kiya & Sasaki (Reference Kiya and Sasaki1983, Reference Kiya and Sasaki1985).

Figure 17 along with supplementary movie 3 examines the first POD modes at ${St_h=0.089}$ for the three cases. Similar to figure 16, figure 17 reveals that the similar evolution process of alternating-signed vortices moving along the isopleths of $U=0$ and $U=U_\infty$. This demonstrates that multiple vortex shedding motions at different frequencies are nested with the vortex cores aligned along the isopleth of $U=U_\infty$ in the separated shear layer. This conclusion is also in line with the deduction from § 3.2.1 that the instantaneous shear layer is bounded primarily by the isopleth of $u=U_\infty$. By comparing figures 16 and 17 along with supplementary movies 1 and 3, the variations of the vortical structures at lower frequencies are summarized as follows. The interaction of vortices A$_i$ and B$_i$ occurs in the more downstream locations, while the sizes of all visible vortices downstream of the mean reattachment point are larger. The elevated mode velocity of the vortices is closer to the wall. This explains why the dominant frequencies of $St_h\approx 0.09$ at $y/h=-0.09$ manifest near the mean reattachment point for all test cases (see figure 12). Additionally, strong mode velocity occurs between the isopleths of $U=0$ and $U=U_\infty$, while new vortices (see A$_3$ and A$_4$ in supplementary movies 1 and 3) appear near the leading edge. The vortices A$_i$ near flow reversal in the cube roughness case are closer to the wall than those in the other two cases. This is attributed to the distortion of the near-wall side of the vortex by the wall roughness. The wavelength of vortex shedding motion in the second half of the mean separation bubble is approximately $6h$ for all test cases, and the pertinent convective velocity is estimated to be $0.5U_\infty$. The vortices downstream of the mean reattachment point are deformed by the wall and exhibit longer characteristic wavelength than those in the second half of the mean separation bubble.

Figure 17. The first POD mode at (a) $St_h=0.089$ for the smooth case, (b) $St_h=0.089$ for the sandpaper case, and (c) $St_h=0.089$ for the cube roughness case, at an arbitrary phase angle. The dashed and dash-dotted isopleths are at $U=0$ and $U=U_\infty$, respectively. Blue and red indicate negative and positive streamwise velocity, respectively. Dashed ellipses mark characteristic vortices. See also supplementary movie 3.

Figure 18 compares the first SPOD modes at the only significant frequencies lower than $St=0.089$ in different test cases (see figure 15). As seen in figure 18(a) for the smooth case, the first vortex in the separated shear layer (near the isopleth of $U=U_\infty$) appears near the mean reattachment length. On the other hand, vortices along the isopleth of $U=0$ occur with larger sizes and further downstream locations than those in figures 16 and 17. The supplementary movie (4) for figure 18(a) shows that as the phase angle increases, vortex A$_2$ approaches the mean reattachment point, and its vortex core becomes discernible while a vortex appears around $x/h=7$ on the isopleth of $U=U_\infty$. Meanwhile, vortex A$_2$ grows larger, and a pair of counter-rotating vortices (A$_2$ and B$_1$) appears in the second half of the mean separation bubble. This process emphasizes that a vortex in the separated shear layer is always accompanied with an opposite-signed vortex in the area of mean flow reversal.

Figure 18. The first POD mode at (a) $St_h=0.050$ for the smooth case, (b) $St_h=0.063$ for the sandpaper case, and (c) $St_h=0.037$ for the cube roughness case, at an arbitrary phase angle. The dashed and dash-dotted isopleths are at $U=0$ and $U=U_\infty$, respectively. Blue and red indicate negative and positive streamwise velocity, respectively. Dashed ellipses mark characteristic vortices. See also supplementary movie 4.

By comparing figures 18(b,c) with figure 18(a), it is evident that wall roughness exerts significant influence on the vortical structures at these low frequencies. In both rough cases, the pattern of a pair of counter-rotating vortices, one on the isopleth of $U=U_\infty$, and the other near the wall, are still visible. In the sandpaper case, vortex A$_1$ near the leading edge is inclined at a noticeable angle (see the dash-dot-dotted line in figure 18b) with the isopleth of $U=0$, whereas that in the smooth case is aligned (measured by the interface where the streamwise mode velocity changes sign) tangential with the isopleth of $U=0$. The orientation of vortex A$_1$ in the smooth case induces a near-wall area of elevated streamwise velocity that extends parallel with the wall, so that this lower part of the vortex is directly exposed to wall resistance. The orientation of vortex A$_1$ in the sandpaper, on the other hand, helps to move the location of elevated streamwise velocity away from the rough wall, thereby minimizing the contact length. This reduces the distortion of the vortex by wall roughness, so that the structure presented in figure 18(b) for the sandpaper case is survived. As seen in figure 18(c) and supplementary movie 4 for the cube roughness case, vortex A$_i$ is barely convected downstream and is destroyed abruptly, while the patch of negative/positive streamwise mode velocity downstream of it grows larger.

The results shown in figures 16, 17 and 18 can also provide some clarification to the well-recognized feedback influence of vortex shedding in flow separations. As commented by Bradshaw & Wong (Reference Bradshaw and Wong1972): ‘A key feature of the flow is found to be the splitting of the shear layer at reattachment, where part of the flow is deflected upstream into the recirculating flow region to supply the entrainment; the part of the flow that continues downstream suffers a pronounced decrease in eddy length scale, evidently because the larger eddies are torn in two’. A similar idea was adopted by Cimarelli et al. (Reference Cimarelli, Leonforte and De Angeli2018), where they further concluded that the upstream convection of structures by the mean flow reversal contributes to a disturbance deflected upstream near the separating point. The results in figures 16, 17 and 18 suggest that the feedback influence of vortex shedding does exist, but it does not require any vortical structures to be convected upstream. In figure 16, the mode velocities flanked by vortices A$_1$ and B$_1$ are amplified, while vortex A$_1$ is convected downstream. In figures 17 and 18(a,b), vortex A$_1$ further induces another vortex in its upstream location in the mean flow reversal area, and generates elevated mode velocity in the channel flanked by the isopleths of $U=0$ and $U=U_\infty$. As such, the process of vortex shedding in the separated shear layer is accompanied with patch(es) of intensified fluctuating velocities flanked by the isopleths of $U=0$ and $U=U_\infty$ upstream of it. The patch(es) is(are) perhaps what Bradshaw & Wong (Reference Bradshaw and Wong1972) referred to as ‘part of the flow is deflected upstream’. The patches indeed ‘supply the entrainment’ due to the strong mode velocities deflected towards the flow separation at the interfaces.

The aforementioned feedback influence of vortex shedding is more significant for lower frequencies (especially $St_h<0.09$) and is sensitive to the wall roughness condition. From figure 9, it is seen that the vortex shedding motion at the lower frequency ($St_h=0.050$) in the smooth case exerts influence on the shear layer interface in more upstream locations than that at the higher frequency ($St_h=0.089$). In contrast, the vortex shedding motions at lower frequencies in the rough cases do not manifest in the first half of the mean separation bubble. This is because the vortices near the mean flow reversal area are destroyed by the wall roughness, so that the feedback influence of vortex shedding is interrupted.

To investigate the underlying mechanism of flapping motion, figure 19 and supplementary movie 5 present the first SPOD mode at the dominant frequency of reverse flow areas in the first half of the mean separation bubble for different cases (figure 14). The flapping motion represents a low-frequency large-scale structure encompassing the entire separation bubble. This structure is relatively weak compared to the vortex shedding motions in the separated shear layer, thus its frequency does not manifest as local peaks in the SPOD spectra (figure 15). However, in the first half of the mean separation bubble, where the turbulence intensity induced by the vortex shedding motions is relatively weak, this large-scale structure, although weak, is influential and manifests as a low-frequency oscillation of reverse flow areas (figure 14). From the supplementary movie (5) for figure 19, for all test cases, a vortex appears in the area of mean flow reversal, and is subsequently deflected away from the mean separation bubble downstream of the mean reattachment point. The flapping motion of flow separation thus reflects the alteration of reverse flow area by the accompanied fluctuating velocity of vortices. This analysis is consistent with the deduction by Kiya & Sasaki (Reference Kiya and Sasaki1983) that the low-frequency flapping motion leads to an abrupt shedding of an extremely large vortex from flow separation. A similar conclusion was made by Hillier & Cherry (Reference Hillier and Cherry1981), Tafti & Vanka (Reference Tafti and Vanka1991) and Yang & Voke (Reference Yang and Voke2001).

Figure 19. The first POD mode at (a) $St_h=0.025$ for the smooth case, (b) $St_h=0.025$ for the sandpaper case, and (c) $St_h=0.050$ for the cube roughness case, at an arbitrary phase angle. The dashed and dash-dotted isopleths are at $U=0$ and $U=U_\infty$, respectively. Blue and red indicate negative and positive streamwise velocity, respectively. See also supplementary movie 5.