3.1 Suction–blowing (SB) versus suction-only (SO) separation bubble

In this section, we compare the key features of these two separation bubbles. This is followed by a detailed analysis of the unsteadiness in the TSB-SO bubble.

3.1.1 Vortex topology

Figure 5 shows instantaneous snapshots of the vortex structures in the two separation bubbles via the second invariant of the velocity-gradient tensor

(3.1)$$\begin{eqnarray}\unicode[STIX]{x1D618}=-\frac{1}{2}\frac{\unicode[STIX]{x2202}u_{j}}{\unicode[STIX]{x2202}x_{i}}\frac{\unicode[STIX]{x2202}u_{i}}{\unicode[STIX]{x2202}x_{j}}=\frac{1}{2}(\unicode[STIX]{x1D6FA}_{ij}\unicode[STIX]{x1D6FA}_{ij}-\unicode[STIX]{x1D61A}_{ij}\unicode[STIX]{x1D61A}_{ij}),\end{eqnarray}$$

where $\unicode[STIX]{x1D6FA}_{ij}$ and $\unicode[STIX]{x1D61A}_{ij}$ are the antisymmetric and symmetric parts of the velocity-gradient tensor, respectively. It can be seen that many elongated low-speed regions are present in the separated shear layer. The size of these structures in the spanwise direction is relatively small near the separation point and they merge into larger structures as the flow separated. Each of them is surrounded by a group of streamwise-aligned hairpin-like structures. In the TSB-SB case these structures are broken by the blowing, and near-wall streaks form during its recovery to TBL after reattachment. In the TSB-SO cases the structures in the separated shear layer sustain for a long time and break around $x=450\unicode[STIX]{x1D703}_{o}$ at the time instant shown. A large vorticity packet is observed downstream, including small-scale turbulent eddies nested inside the packet.

Figure 5. Vortex structures at one time instant for two separation bubbles: (a) TSB-SB and (b) TSB-SO. Vortex structures are visualized by the isosurfaces of the second invariant of the velocity-gradient tensor $\unicode[STIX]{x1D618}=0.0165U_{o}^{2}/\unicode[STIX]{x1D703}_{o}^{2}$ (see text), coloured by the distance from the wall. The dark-grey isosurfaces are $u^{\prime }=-0.1U_{o}$.

3.1.2 Characteristics of the mean flow

Figure 6 shows contours of mean streamwise velocity $U/U_{o}$ in the $x$–$y$ plane together with several selected mean streamlines for both the separation bubbles. The characteristic properties of the mean separation bubble are also compared in table 3. The mean separation bubble in the TSB-SO configuration is significantly longer than that in the TSB-SB case but slightly thinner. Furthermore, the reversed flow in the TSB-SO configuration has a lower magnitude, and the streamwise velocity gradient near the reattachment point is also lower due to the absence of the forced impingement of the flow. The peak reversed flow amplitude, scaled with the far-field velocity at each streamwise location, is approximately 8 % in TSB-SO and 11 % in TSB-SB.

Figure 6. Contours of mean streamwise velocity with selected streamlines (solid): (a) TSB-SB and (b) TSB-SO.

Table 3. Characteristics of the mean separation bubble. The mean separation point $x_{sep}$ is where $C_{f}=2\unicode[STIX]{x1D70F}_{w}/U_{o}^{2}=0$ and $\text{d}C_{f}/\text{d}x<0$; the mean reattachment point $x_{ratt}$ is where $C_{f}=0$ and $\text{d}C_{f}/\text{d}x>0$; and $h_{sep}$ is the maximum distance between the contour line of $U=0$ and the wall.

The time-mean profiles of skin-friction coefficient ($C_{f}$) and pressure coefficient on the wall ($C_{p}$) for both the bubbles are compared in figure 7. In both cases the wall pressure agrees well with that for an inviscid solution up to the separation point. The suction-and-blowing configuration produces a bell-shaped inviscid wall pressure profile over the surface that consists of an APG followed by an FPG (Na & Moin Reference Na and Moin1998a; Abe Reference Abe2017; Wu & Piomelli Reference Wu and Piomelli2018). Pressure profiles of realistic separated flow are, however, quite different. For an airfoil at large angle of attack, a steep APG appears near the leading edge of the airfoil and gradually decreases towards the trailing edge. However, there is usually no region of FPG (Rinoie & Takemura Reference Rinoie and Takemura2004; Kim et al. Reference Kim, Yang, Chang and Chung2009; Asada & Kawai Reference Asada and Kawai2018). In diffusers with fixed opening angle in the streamwise direction (Kaltenbach et al. Reference Kaltenbach, Fatica, Mittal, Lund and Moin1999), strong APG occurs at the beginning of the deflected wall and decreases rapidly to ZPG downstream. Unless the wall itself is not parallel to the free-stream flow (e.g. a shock-induced separated flow at a compression corner), there is no common mechanism to have a driving mean flow that impinges towards the surface after the flow separates.

The $C_{p}$ in the TSB-SO configuration consists only of the APG as expected, and there is no local pressure peak due to the stagnation of the mean flow such as the one observed at $x/\unicode[STIX]{x1D703}_{o}\sim 390$ in the TSB-SB case. In both cases, the Clauser pressure-gradient parameter

(3.2)$$\begin{eqnarray}\unicode[STIX]{x1D6FD}=\frac{\unicode[STIX]{x1D6FF}^{\ast }}{\unicode[STIX]{x1D70F}_{w}}\frac{\text{d}P_{e}}{\text{d}x},\end{eqnarray}$$

where $(\cdot )_{e}$ denotes the quantity measured at the edge of the boundary layer, is $\unicode[STIX]{x1D6FD}=0.5$ at $x=60\unicode[STIX]{x1D703}_{o}$, $\unicode[STIX]{x1D6FD}=3$ at $x=100\unicode[STIX]{x1D703}_{o}$, and then rapidly increases to $\unicode[STIX]{x1D6FD}=85$ before $u_{\unicode[STIX]{x1D70F}}$ becomes smaller than $0.01U_{e}$ upstream of the separation bubble.

Figure 7. Streamwise profiles of $C_{f}$ and $C_{p}$. Lines with markers, TSB-SB; lines only, TSB-SO; dash-dotted lines in right panel, inviscid wall $C_{p}$.

Figure 8. Probability of occurrence of reversed flow as a function of position: (a) TSB-SB; (b) TSB-SO. Light solid lines are selected mean streamlines.

The probability distribution of reverse flow occurring in the $x{-}y$ plane is shown in figure 8 for both bubbles. The point of the separation region in both cases shows a very steep gradient in probability, indicating a stationary separation point. However, the reattachment behaviour of the two bubble is quite different. In particular, unlike the TSB-SB bubble, for the TSB-SO bubble there is a large region beyond the reattachment point where reverse flow can occur up to 10 % of the time. This is because the reattachment of the mean flow in the case of the TSB-SO bubble is due to the transport of momentum by turbulence and not forced due to the imposed blowing at the top. In order to confirm this statement, we examine the balance of momentum flux (Abe Reference Abe2019) in the aft portion (i.e. the region between the bubble crest and the reattachment point) of both separation bubbles (figure 9). In the TSB-SB case, the convection term dominates the positive gain in the momentum balance near the $U=0$ ‘interface’, with a small turbulent transport term. In the TSB-SO case, on the contrary, the turbulent transport becomes the leading term in TSB-SO at such interface, and the mean flow only contributes near the edge of the boundary layer. On the wall, the region between $x/\unicode[STIX]{x1D703}_{o}=600$ and $x/\unicode[STIX]{x1D703}_{o}=706$ experiences backflow during 50 % to 30 % of the time for the TSB-SO bubble. The extent of this region in the TSB-SB case, in comparison, is only approximately $6\unicode[STIX]{x1D703}_{o}$ ($x/\unicode[STIX]{x1D703}_{o}\in [379,385]$). The periodic formation of a discrete vorticity packet causes remarkable intermittency in flow reattachment: for 30 % (10 %) of time the bubble is 20 % (40 %) longer than the mean. As will be described later in the paper, this feature of the reattachment is associated with a low-frequency ‘breathing’ or ‘flapping’ of the TSB-SO bubble.

Figure 9. Budget of the mean $x$-momentum balance. Dash-dotted line, advective term $-U_{j}\unicode[STIX]{x2202}U/\unicode[STIX]{x2202}x_{j}$; solid line, pressure term $-\unicode[STIX]{x2202}P/\unicode[STIX]{x2202}x$; dashed line, viscous term $\unicode[STIX]{x1D708}\unicode[STIX]{x2202}^{2}\overline{U}/\unicode[STIX]{x2202}x_{j}^{2}$; solid line with marker, Reynolds stress term $-\unicode[STIX]{x2202}\overline{u^{\prime }u_{j}^{\prime }}/\unicode[STIX]{x2202}x_{j}$; horizontal dotted line, local $y|_{U=0}$. (a) TSB-SB at $x/\unicode[STIX]{x1D703}_{o}=342$. (b) TSB-SO at $x/\unicode[STIX]{x1D703}_{o}=467$. The examining $x$-location is chosen as the middle point between the bubble crest and the mean reattachment point. The residual in both cases is smaller than $2\times 10^{-4}\,U_{o}^{2}/\unicode[STIX]{x1D703}_{o}$.

Eliminating the forced impingement due to the imposed blowing also significantly impacts the development of turbulence in the two flows as shown in figure 10. While the two cases share a similar turbulence profile near the region of initial separation, the TSB-SO case also exhibits another large area of high turbulence down near and above the mean reattachment point. As will be shown later in the paper, this large region of turbulent fluctuations is associated with unsteady motions that occur at a large time and spatial scale. In general, the observed Reynolds stress distributions obtained for the TSB-SO case are in much better qualitative agreement with the ones in flow separation over an airfoil (Jones et al. Reference Jones, Sandberg and Sandham2008; Balakumar Reference Balakumar2017) than those for the TSB-SB case. More details about the statistics of the TSB-SB bubble may be found in Wu, Meneveau & Mittal (Reference Wu, Meneveau and Mittal2019).

Figure 10. Time- and spanwise-averaged Reynolds normal stresses: (a–c) TSB-SB; (d–f) TSB-SO; (a,d) $\overline{u^{\prime }u^{\prime }}/U_{o}^{2}$; (b,e) $\overline{v^{\prime }v^{\prime }}/U_{o}^{2}$; and (c,f) $\overline{w^{\prime }w^{\prime }}/U_{o}^{2}$.

3.1.3 Low-frequency unsteadiness in the separation bubbles

In this section, we examine the two separation bubbles for the presence of a low-frequency ‘breathing’ or ‘flapping’ mode. In figure 11 we show contours of instantaneous streamwise velocity and the velocity vectors in the $x{-}y$ plane at centre span at two different time instants. The TSB-SO bubble shows large variation of shape in time: the bubble is long, enclosing a continuous backflow region (figure 11c) at some times, and has several distinct backflow regions at other times (figure 11d). The TSB-SB bubble, on the other hand, does not show any such large-scale changes in the size of the bubble with time (figure 11a,b). This behaviour also explains the differences in the probability distributions observed in figure 8 in the rear part of the TSB-SO bubble.

Figure 11. Instantaneous streamwise velocity and velocity vector in the $x$–$y$ plane at $z=L_{z}/2$: (a) TSB-SB, $tU_{o}/\unicode[STIX]{x1D703}_{o}\approx 9260$; (b) TSB-SB, $tU_{o}/\unicode[STIX]{x1D703}_{o}\approx 11\,170$; (c) TSB-SO, $tU_{o}/\unicode[STIX]{x1D703}_{o}\approx 15\,000$; and (d) TSB-SO, $tU_{o}/\unicode[STIX]{x1D703}_{o}\approx 16\,950$. For colour maps, refer to figure 6.

Figure 12. Time history of the total reversed flow area in the $x$–$y$ plane.

The change in the size of the separation bubble versus time serves as a simple parameter that measures large-scale unsteadiness of a separation bubble. The history of the total reversed flow area in the $x$–$y$ plane, i.e. $A_{r,xy}=\int _{\unicode[STIX]{x1D6FA}_{r}}\text{d}x\,\text{d}y$, where $\unicode[STIX]{x1D6FA}_{r}$ is the region where $\langle u\rangle _{z}<0$, is plotted in figure 12. A large-amplitude, low-frequency variation of the total backflow area is clearly observed for the TSB-SO bubble, whereas a similar behaviour is missing the TSB-SB bubble. Thus, taken together, figures 11 and 12 clearly suggest that, while the TSB-SO bubble exhibits a large-scale low-frequency unsteadiness of the type that has been characterized as the ‘breathing’ or ‘flapping’ mode in previous studies, the TSB-SB bubble does not show any such distinct behaviour.

Note that the low-frequency motion we observed here does not have a very significant separation in time scale with respect to the high-frequency ones. As we will describe in detail in the next section, the frequencies of the two motions differ only by a factor of 2–2.5. As summarized in the Introduction, researchers have reported low-frequency motions that occur at a time scale ranging from 6 to 35 times longer than the vortex shedding mode. One reason for this discrepancy may be that we are observing a different type of low-frequency motion in flow separation. It also is possibly related to the relatively low Reynolds number of the current flow since it is expected that, while the vortex shedding frequency scales with the inverse of the boundary layer thickness (which decreases with increasing Reynolds numbers), the low-frequency mode scales with the size of the separation bubble, which would be less dependent on the Reynolds number. Indeed, in the experiments of TSBs conducted by Weiss et al. (Reference Weiss, Mohammed-Taifour and Schwaab2015) and Mohammed-Taifour & Weiss (Reference Mohammed-Taifour and Weiss2016), the boundary Reynolds number was $Re_{\unicode[STIX]{x1D703}}=5000$ and they observed a low-frequency peak in the spectrum that was 35 times lower than the vortex shedding peak. Note that, in their experiments, which employed a ceiling with an expansion and mass removal followed by a contraction, there is an FPG following the APG but the net FPG is smaller than the net APG. Thus their configuration falls in between what we define as SB and SO configurations.

3.2 Characteristics of unsteadiness in the TSBs

We begin by examining more closely the reversed flow in the vicinity of the wall for the TSB-SO bubble. The spatio-temporal map of the spanwise-averaged streamwise velocity at the first grid point away from the wall is plotted in figure 13. For this separation bubble, besides the fluctuation of the incoming TBL at a very short time scale, a high-frequency unsteadiness is featured as parallel stripes separated in time by approximately $TU_{o}/\unicode[STIX]{x1D703}_{o}\approx O(400)$ in the map. Also, strong forward flow is observed to penetrate into the reversed flow region up to $x/\unicode[STIX]{x1D703}_{o}=400$, and this occurs on a time scale that is larger than that at the upstream region of the separation bubble. The low-frequency motion does not appear in the TSB-SB case, in which the reversed flow stripes exhibit the same temporal interval near the separation and reattachment regions.

Figure 13. Spatio-temporal map of the spanwise-averaged streamwise velocity at the first grid point away from the bottom wall for: (a) TSB-SB bubble; (b) TSB-SO bubble. Some region in the left panel is left blank intentionally so that the two panels have the same axis range, because the TSB-SB case was integrated to $t=14\,300\,\unicode[STIX]{x1D703}_{o}/U_{o}$ with a smaller domain in $x$.

Figure 14. Pre-multiplied energy spectrum of the streamwise velocity shown in figure 13: dash-dotted, TSB-SB; solid, TSB-SO. Each plot is shifted upwards by $7\times 10^{-7}$ units for clarity. The thin red vertical solid line marks the high frequency $f_{h}=0.0025U_{o}/\unicode[STIX]{x1D703}_{o}$; the dashed black vertical line represents the low frequency $f_{l}=0.001U_{o}/\unicode[STIX]{x1D703}_{o}$; and the dash-dotted blue vertical line is $f_{m}=0.002U_{o}/\unicode[STIX]{x1D703}_{o}$.

Note that the time scale shown here has no direct link to the ones described in figure 12: figure 13 represents the local (i.e. $x$) intermittency of the flow on the wall, describing the streamwise motion of a band-like, quasi-two-dimensional separated region. Figure 12, on the other hand, is an integral quantity over the entire $x$–$y$ domain describing the total reversed flow area. When a discrete vorticity packet sheds off from the separating shear layer, for instance, it will show as unsteadiness in figure 13 but not in figure 12 until the discrete vorticity packet decays and the reversed flow diminishes.

Figure 14 plots the pre-multiplied power density spectra of the velocity shown in figure 13. The data series is windowed and detrended and the analysis uses a total of 11 equal-length segments in time with 50 % overlap (Na & Moin Reference Na and Moin1998b; Abe Reference Abe2017). Each segment contains 896 samples (stored every 100 time steps). The resulting resolved frequency range is $0\leqslant f\unicode[STIX]{x1D703}_{o}/U_{o}\leqslant 0.129$. On the upstream end of the separation bubble (i.e. $x/\unicode[STIX]{x1D703}_{o}\approx 200$), a clear peak appears at a frequency of approximately $f_{h}\unicode[STIX]{x1D703}_{o}/U_{o}\approx 0.0025$. At the other end of the separation bubble ($x/\unicode[STIX]{x1D703}_{o}>400$) the spectrum is dominated by a low frequency corresponding to approximately $f_{l}\unicode[STIX]{x1D703}_{o}/U_{o}\approx 0.001$. The spectrum also indicates a peak corresponding to a frequency of $f_{m}\unicode[STIX]{x1D703}_{o}/U_{o}\approx 0.002$ in the region $(350{-}400)\unicode[STIX]{x1D703}_{o}$. We refer to this as the ‘mid-frequency’ but, as we will discuss later in the paper, this mode and the one at $f_{l}\unicode[STIX]{x1D703}_{o}/U_{o}\approx 0.0025$ seem to be driven by the same vortex rollup mechanism. Overall, the low and high frequencies show a clear separation in both scale ($f_{h}\unicode[STIX]{x1D703}_{o}/U_{o}\approx 0.0025$ versus 0.001) and location (high frequency for $x/\unicode[STIX]{x1D703}_{o}<250$ and low frequency for $x/\unicode[STIX]{x1D703}_{o}>250$), with the $x/\unicode[STIX]{x1D703}_{o}$ from 350 to 400 appearing as a region of transition between these two distinct regions of the flow. The low frequency coincides with the subharmonic of the medium-frequency motion, which indicates that the former could be a result of vortex pairing.

The unsteadiness exhibited in the spanwise-averaged field should not be interpreted as implying that the underlying mechanism is dominated by two-dimensional motions. Instead, it suggests that the spatial signature of the low-frequency motion is visible in the spanwise-averaged flow and this is indeed consistent with the notion of ‘breathing’ or expansion and contraction of the bubble. While figure 13 corresponds to the data extracted very near the wall, similar frequency peaks and scale separation are also observed within the separated shear layer. Figure 15(a) shows contours of the fluctuating pressure $p_{rms}^{\prime }$ and figure 15(b–d) shows the pre-multiplied energy spectrum of pressure fluctuations at several locations within the high-$p_{rms}^{\prime }$ regions. As for the velocity spectrum shown above, the pressure data series are windowed and detrended with 50 % overlap, and the spectra are averaged over all the grid points in the spanwise direction at each location. The peaks of the spectrum agree well with those for the reversed flow on the wall. One difference is that the low-frequency motion already appears at $x=200\unicode[STIX]{x1D703}_{o}$ in the separated shear layer, while it shows up much further downstream on the wall. This indicates that this low-frequency mode originates from the separated shear layer and impacts the near-wall flow after its effects spread downstream. Note that the ellipticity of the pressure does not seem to be the cause for this because such low-frequency motion is not observed in figures 13 and 14 near the separation point.

Figure 15. Pre-multiplied energy spectrum of the pressure fluctuation at several locations in the separated shear layer. (a) Contour map of $p_{rms}^{\prime }$; the cross markers indicate the locations where the spectra are obtained (that is, $x/\unicode[STIX]{x1D703}_{o}=207.3$ for (b), $310.3$ for (c) and $540.0$ for (d)). The square markers are the locations where the two-point correlation is evaluated; the solid line is the selected mean streamline passing the high-$p_{rms}^{\prime }$ regions. (b–d) Pressure spectra. The thin red solid vertical line marks the high-frequency $f_{h}=0.0025U_{o}/\unicode[STIX]{x1D703}_{o}$; the black dashed vertical line represents the low frequency $f_{l}=0.001U_{o}/\unicode[STIX]{x1D703}_{o}$; and the blue dash-dotted vertical line marks $f_{m}=0.002U_{o}/\unicode[STIX]{x1D703}_{o}$.

The Strouhal number defined as $St_{L_{sep}}=fL_{sep}/U_{o}$ is 1.125, 0.9 and 0.45 for $f_{h}$, $f_{m}$ and $f_{l}$, respectively, in the TSB-SO case. The high-frequency mode also appears in the TSB-SB case but the Strouhal number differs by a factor of 2.4 due to the different $L_{sep}$. In previous studies using the suction–blowing configuration (Na & Moin Reference Na and Moin1998a; Abe Reference Abe2017), a motion at $St\approx 0.35$ has been reported as the ‘shedding mode’. In our TSB-SB case, this value is close to 0.4, which agrees reasonably well with the literature. The fact that the same phenomenon (i.e. KH vortex shedding) in the two configurations gives different $St_{L_{sep}}$ indicates that $L_{sep}$ is not the best parameter to characterize the unsteadiness of a separation bubble. In the current study we associate the motion at frequency $f_{l}$ with the low-frequency motion even though its Strouhal number is higher than in some previous studies, because this motion agrees phenomenologically with the ‘breathing’ phenomenon associated with the low-frequency motion in earlier studies of incompressible separation bubbles (Pauley et al. Reference Pauley, Moin and Reynolds1990; Spalart & Strelets Reference Spalart and Strelets2000; Weiss et al. Reference Weiss, Mohammed-Taifour and Schwaab2015; Mohammed-Taifour & Weiss Reference Mohammed-Taifour and Weiss2016).

Previous studies have shown that the separated shear layer is similar to a canonical plane turbulent mixing layer (e.g. velocity and Reynolds stress scale well by the mixing layer scaling) and the high-frequency motion may be generated by KH instability (e.g. large-scale spanwise structures) (Na & Moin Reference Na and Moin1998a; Spalart & Strelets Reference Spalart and Strelets2000; Rist & Maucher Reference Rist and Maucher2002; Marxen & Henningson Reference Marxen and Henningson2010; Abe et al. Reference Abe, Mizobuchi, Matsuo and Spalart2012; Abe Reference Abe2017). To examine the degree of similarity to a mixing layer, we have examined the profile of the vorticity thickness $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}$ defined as

(3.3)$$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}=(U_{max}-U_{min})/(\unicode[STIX]{x2202}U/\unicode[STIX]{x2202}y)_{max},\end{eqnarray}$$

where $U_{max}$ and $U_{min}$ are the maximum and minimum time- and spanwise-averaged streamwise velocities in the two sides of the separated shear layer. We find that (see figure 16a) $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}$ exhibits a linear growth with $x$ after an initial transition at the beginning of the separation. The growth rate also agrees very well with the one found in previous studies on plane mixing layers (Liepmann & Laufer Reference Liepmann and Laufer1947; Brown & Roshko Reference Brown and Roshko1974; Pantano & Sarkar Reference Pantano and Sarkar2002; Abe Reference Abe2017).

Figure 16. (a) Profile of the normalized vorticity thickness: ——, TSB-SO; — ⋅ —,$\text{d}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}/\text{d}x=C_{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}}(U_{max}-U_{min})/(U_{max}+U_{min})$, $C_{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}}=0.16$. (b) Dominant frequency of mixing layer predicted by canonical mixing layer relationship: ——, $St_{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}}=0.25$; grey hatched region, $St_{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}}\in [0.2,0.3]$. The thin solid horizontal line marks the high frequency $f_{h}=0.0025U_{o}/\unicode[STIX]{x1D703}_{o}$; the dash-dotted horizontal line marks $f_{m}=0.002U_{o}/\unicode[STIX]{x1D703}_{o}$; and the dashed horizontal line represents the low frequency $f_{l}=0.001U_{o}/\unicode[STIX]{x1D703}_{o}$.

It has been widely reported in previous studies that the dominant frequency in a turbulent mixing layer corresponds to a Strouhal number $St_{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}}=f\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}/U_{sl}\approx 0.2{-}0.3$ (with $U_{sl}=(U_{max}+U_{min})/2$). Figure 16(b) plots the frequency predicted by this relationship in the current simulation where we have used the local $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}$. It can be seen that with $St_{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}}=0.25$ the scaling predicts a frequency $f\unicode[STIX]{x1D703}_{o}/U_{o}$ ranging from approximately 0.004 to 0.002 in the $200<x/\unicode[STIX]{x1D703}_{o}<250$ region, which is very much in line with the two observed high frequencies of $f_{m}\unicode[STIX]{x1D703}_{o}/U_{o}=0.002$ and $f_{h}\unicode[STIX]{x1D703}_{o}/U_{o}=0.0025$ in this region of the bubble. In particular, the profile shows one plateau upstream of $x/\unicode[STIX]{x1D703}_{o}=350$ and another downstream. This agrees very well with the locations where the spectral peaks appear in figure 14. This provides strong support for the notion that the mechanism of high- and medium-frequency unsteadiness is the KH instability of the mixing layer. Moreover, the plot shows that, despite a decrease in this predicted frequency with downstream distance, it never reaches the observed low frequency of $f_{l}\unicode[STIX]{x1D703}_{o}/U_{o}$ = 0.001. This suggests that some mechanism other than KH instability drives the low-frequency unsteadiness in the rear part of the separation bubble.

Figure 17. (a) Contours of two-point correlation coefficient of streamwise velocity fluctuations with levels of correlation indicated by text. The reference locations are selected along a mean streamline passing the high-$p_{rms}^{\prime }$ regions and are indicated by square markers in figure 15(a). The contours for different reference locations are superposed. (b) The nominal radius of the roller vortex determined from the area enclosed by the $R_{11}=0.2$ contour line, i.e. $r/\unicode[STIX]{x1D703}_{o}=\sqrt{A/\unicode[STIX]{x03C0}}$.

Because the vortices generated by KH instability are spanwise rollers, we have examined the length scale of the flow in the separated shear layer, using the two-point autocorrelation coefficient in the $x$–$y$ plane,

(3.4)$$\begin{eqnarray}R_{11}(\boldsymbol{x}_{ref},\boldsymbol{x}_{ref}+\unicode[STIX]{x0394}\boldsymbol{x})=\frac{\langle u^{\prime }(\boldsymbol{x}_{ref})u^{\prime }(\boldsymbol{x}_{ref}+\unicode[STIX]{x0394}\boldsymbol{x})\rangle }{\langle u^{\prime }(\boldsymbol{x}_{ref})u^{\prime }(\boldsymbol{x}_{ref})\rangle },\end{eqnarray}$$

where $\unicode[STIX]{x0394}\boldsymbol{x}$ is the spatial separation between the reference location and the other locations in the computational domain. Several reference locations are chosen along a mean streamline passing through the high-$p_{rms}^{\prime }$ regions and the correlation is calculated between each of them and all the other locations in the flow fields. The contours of the correlation coefficients are shown in figure 17, where the $R_{11}$ contour lines for selected reference location are presented on the same plot. If we define non-negligible correlation as $R_{11}\geqslant 0.2$, it can be seen (figure 17b) that the spanwise roller starts to generate near $x/\unicode[STIX]{x1D703}_{o}=250$ and gradually grows in size. Between $x/\unicode[STIX]{x1D703}_{o}=$ 400 and 500, the vortex length scale increases abruptly, suggesting the formation of large-scale vortex structures, possibly associated with the merging of the spanwise rollers.

Contours of the second invariant of the velocity-gradient tensor at one selected time (corresponding to the time instant shown in figure 5b) are shown in figure 18 and in the turbulent shear layer, the spanwise rollers consist of small-scale eddies. A large-scale conglomeration of vortices is formed near $x/\unicode[STIX]{x1D703}_{o}=450$ at the time shown, and this seems to be in line with the notion that the large scales at this downstream location may be caused by a vortex merging process.

Figure 18. Contours of instantaneous secondary invariant of the velocity-gradient tensor in the $x$–$y$ plane $z=L_{z}/2$ at $tU_{o}/\unicode[STIX]{x1D703}_{o}=16\,950$. Only the positive $\unicode[STIX]{x1D618}$ are shown for clarity. The mean separating streamline is shown as the dashed line for reference.

It is, however, not clear whether this vortex merging is the mechanism underlying the low-frequency unsteadiness or whether the latter arises due to some other distinct mechanism. In particular, there are a number of possible candidate mechanisms for the low-frequency unsteadiness of the separation bubble, which include the following.

1. (a) Imbalance of entrainment from the recirculation region by the shear layer and the reinjection of fluid near the reattachment point (Eaton & Johnston Reference Eaton and Johnston1982); this makes the separating shear layer flap towards the wall and significantly slows down its convection downstream.

2. (b) Shear layer switches from a convective instability to a global instability, which leads to local amplification. In laminar flows, researchers have found that global instability occurs when the mixing layer is formed by counter-streams and when $R_{u}=(U_{max}-U_{min})/(U_{max}+U_{min})$ exceeds 1.315 (Huerre & Monkewitz Reference Huerre and Monkewitz1985; Strykowski & Niccum Reference Strykowski and Niccum1991), or the peak reversed flow amplitude exceeds 20 % of the free-stream velocity (Hammond & Redekopp Reference Hammond and Redekopp1998; Rist & Maucher Reference Rist and Maucher2002). In the current flow, the maximum value of $R_{u}$ is 1.25 and the peak reversed flow magnitude is $0.08U_{o}$. Thus, this is not likely to be the mechanism.

3. (c) Some mechanism unrelated to the KH vortices, which breaks down the sheet of the spanwise vortices.

In general, it is not straightforward to separate cause from effect in a fully saturated, highly turbulent flow with a large range of temporal and spatial scales. In the current study, we employ dynamic mode decomposition (DMD) (Schmid Reference Schmid2010; Chen, Tu & Rowley Reference Chen, Tu and Rowley2012) to examine the flow field and to extract features/modes of the flow that correspond to the dominant time scales in the flows. The topology of these extracted modes is then used as the basis to further investigate the underlying mechanism for the low-frequency ‘breathing’ mode.

Figure 19. (a) Eigenvalues of the DMD modes. (b) Spectrum of the DMD modes; the range of the $x$-axis is limited to the frequency range of concern. Modes whose $|\unicode[STIX]{x1D706}|<0.995$ are shown by blue markers in (a) and are not included in (b). The three thin vertical lines refer to figure 15. The red triangular markers show the selected DMD modes closest to the three target frequencies.

3.3 DMD-based analysis of the three-dimensional flow field

The basic idea of DMD analysis is to extract a low-dimensional description of a linear transformation that maps any snapshot of data from a dynamical system into the subsequent snapshot. The eigenvalue decomposition of this linear transformation then provides frequency information as well as the corresponding spatial structures. Furthermore, a projection of the snapshots onto these structures gives the amplitude (contribution) information (Schmid Reference Schmid2010; Sayadi et al. Reference Sayadi, Schmid, Nichols and Moin2014). Eventually, the snapshots can be approximated using the DMD modes as

(3.5)$$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{r}(\mathbf{x},t)=\mathop{\sum }_{k=1}^{r}b_{k}\unicode[STIX]{x1D713}_{k}(\mathbf{x})\exp (\unicode[STIX]{x1D714}_{k}t),\quad k=1,2,\ldots ,r,\end{eqnarray}$$

in which $\unicode[STIX]{x1D6F7}_{r}$, $\unicode[STIX]{x1D713}_{k}$ and $b_{k}$ are the field reconstructed by the $r$ DMD modes, the $k$th spatial DMD mode (shape of the mode) and the magnitude of the $k$th DMD mode, respectively. The term $\exp (\unicode[STIX]{x1D714}t)=\unicode[STIX]{x1D706}^{t/\unicode[STIX]{x0394}t}$, where $\unicode[STIX]{x1D706}$ is the eigenvalue of the modes, describes the growth/decay and oscillation of the mode.

In this study, our objective of using DMD is to extract the modes corresponding to the key frequencies identified in the earlier part of the paper and to examine the topology of the modal reconstructions based on these modes as a means for identifying potential generation mechanisms. Discrete Fourier transform (DFT) based techniques such as notch filtering could be used for the purpose of modal reconstruction, but these DFT techniques require the data series to cover an integer number of the corresponding periods, and suffer from spectral leakage (Sayadi et al. Reference Sayadi, Schmid, Nichols and Moin2014). Proper orthogonal decomposition (POD) analysis is another decomposition approach that could be used, but POD modes may contain multiple frequencies and therefore are not suitable for our analysis.

Figure 20. Isosurfaces of the real part of the high-frequency DMD mode $f_{h}\unicode[STIX]{x1D703}_{o}/U_{o}=2.49\times 10^{-3}$, $St_{L_{sep}}=1.125$: (a) $u_{dmd}/U_{o}=\pm 0.02$; (b,c) $v_{dmd}/U_{o}=\pm 0.01$, top and side views, respectively. The solid line in (b) is for $U=0$.

Figure 21. Isosurfaces of the real part of the mid-frequency DMD mode $f_{m}\unicode[STIX]{x1D703}_{o}/U_{o}=1.97\times 10^{-3}$, $St_{L_{sep}}=0.90$. Colour/line style refers to figure 20.

The total sample for the DMD analysis extends over a period of $11\,230\,\unicode[STIX]{x1D703}_{o}/U_{o}$, which corresponds roughly to 11.2 periods of the observed low frequency. Snapshots are extracted at every 400 time steps; this corresponds to a sampling frequency of $0.0642U_{o}/\unicode[STIX]{x1D703}_{o}$ and results in the extraction of a total of 721 snapshots. Each sample consists of the three velocity components $u$, $v$ and $w$. In order to capture the key regions of the separated and reattaching flow, the spatial region over which the data are extracted covers $0\leqslant x/\unicode[STIX]{x1D703}_{o}\leqslant 700$ and $0\leqslant y/\unicode[STIX]{x1D703}_{o}\leqslant 100$.

In order to reduce the overall size of the dataset used in the DMD analysis to manageable levels, we employ a spatial subsampling where we first filter the velocity components by a box filter with width of $3\unicode[STIX]{x1D703}_{o}$, and then extract every eighth grid point in the $x$ and $z$ directions and every other point in $y$. This procedure leads to a total of 2.35 million samples per snapshot per observable, which is approximately 0.8 % of the total mesh nodes in the sample region. Finally, to avoid overfitting the complex dynamics of the fully turbulent field, instead of using the velocity data directly, we employ a 361-mode POD projection (one real mean mode and 180 complex-conjugate complex pairs), which serves as a spatio-temporal filter. This projection retains 98 % of the total kinetic energy, and 94 % of the turbulent kinetic energy of the flow.

The eigenvalues and (discrete) spectrum of the DMD modes are shown in figure 19. Since the flow is in a statistically saturated state, the eigenvalues lie mostly on or slightly inside the unit circle on the complex plane. Furthermore, due to the broadband nature of the flow, the spectrum exhibits a relatively continuous distribution of modes without any distinct and isolated peaks. This indicates that a significant number of modes are required to accurately reconstruct the dynamics of this flow. In the following, we focus on three modes whose frequencies are closest to those identified as the high ($f\unicode[STIX]{x1D703}_{o}/U_{o}=0.0025$), medium ($f\unicode[STIX]{x1D703}_{o}/U_{o}=0.002$) and low ($f\unicode[STIX]{x1D703}_{o}/U_{o}=0.001$) frequencies in the previous sections, i.e. $f_{h}\unicode[STIX]{x1D703}_{o}/U_{o}=2.49\times 10^{-3}$, $f_{m}\unicode[STIX]{x1D703}_{o}/U_{o}=1.97\times 10^{-3}$ and $f_{l}\unicode[STIX]{x1D703}_{o}/U_{o}=1.03\times 10^{-4}$.

Figures 20 and 21 show DMD modes corresponding to the high- and mid-frequency spectral peaks. The key characteristic of these two modes that is visible from these plots is the highly regular arrangement of alternating structures separated in the streamwise direction. These structures also exhibit a considerable degree of coherence in the spanwise direction (figures 20c and 21c). The high-frequency mode appears similar to the $\unicode[STIX]{x1D706}$-vortex structure formed by the oblique modes in a transitional boundary layer (Schmid & Henningson Reference Schmid and Henningson2012). Both these features correspond well to the notion that these two modes are associated with the KH instability, and the modal structure corresponds primarily with spanwise vortex rollers.

Figure 22. Isosurfaces of the real part of the low-frequency DMD mode $f_{l}\unicode[STIX]{x1D703}_{o}/U_{o}=1.03\times 10^{-3}$, $St_{L_{sep}}=0.45$. Colour/line style refers to figure 20.

The topology of the low-frequency mode (shown in figure 22) is quite different from that of the high-frequency modes discussed above and appears to be dominated by highly elongated streamwise structures. For example, the structure located near $z=0$ starts from $x/\unicode[STIX]{x1D703}_{o}=40$ and extends to $x/\unicode[STIX]{x1D703}_{o}=410$; another one generated around $[x,z]/\unicode[STIX]{x1D703}_{o}=[100,86]$ extends to $x/\unicode[STIX]{x1D703}_{o}=270$ (figure 22a). Note that the simulation is periodic in the spanwise direction so that the structure at $z=L_{z}$ connects with the one at $z=0$. Further downstream in the region where the free-stream APG vanishes, the streamwise structures connect and form larger structures. However, their two-dimensionality along the streamwise direction seems to be preserved. When compared with the high-frequency modes, the low-frequency mode has a much larger wavelength in the streamwise direction (figure 22a). The size of the structures grows abruptly near $x/\unicode[STIX]{x1D703}_{o}=400$ (figure 22b,c).

The side view of this mode shows that there is also a significant spanwise-oriented signature in this mode. Furthermore, even structures that show a streamwise-oriented topology can ‘shed’ in a spanwise homogeneous manner, resulting in a noticeable spanwise-averaged signal, and that is likely what is happening in this flow: figure 23 shows the streamwise velocity component for the low-frequency DMD mode at four equally spaced phases in one cycle. The shedding of structures can be observed in the $x$–$y$ slice contour, while the streamwise coherence persists in the $x$–$z$ slices.

Figure 23. Contour of the streamwise velocity of the low-frequency DMD mode (refer to figure 22) at four equally spaced phases in one cycle (i.e. period of $T$).

3.4 Görtler instability as a mechanism for the low-frequency mode

The topology of the low-frequency mode suggests that it could be related to Görtler vortices, which appear as streamwise-elongated structures in the boundary layers over concave walls. These Görtler vortices are generated by a centrifugal instability that destabilizes the flow when the direction of the wall-normal velocity gradient is opposite to the centrifugal force associated with the curvature of the streamlines (Görtler Reference Görtler1954), and previous studies on separated flows have suggested the presence of Görtler-type vortices (Settles, Fitzpatrick & Bogdonoff Reference Settles, Fitzpatrick and Bogdonoff1979; Loginov, Adams & Zheltovodov Reference Loginov, Adams and Zheltovodov2006; Priebe et al. Reference Priebe, Tu, Rowley and Martín2016). Numerous previous studies on Görtler vortices have proposed a variety of threshold criteria for Görtler instability as well as the characteristic wavelength of the generated structures. In general, it is widely accepted that Görtler instability appears when the Görtler number, defined as

(3.6)$$\begin{eqnarray}G_{T}=\frac{U_{e}\unicode[STIX]{x1D703}}{\unicode[STIX]{x1D708}}\sqrt{\frac{\unicode[STIX]{x1D703}}{R}},\end{eqnarray}$$

where $U_{e}$, $\unicode[STIX]{x1D703}$ and $R$ are the free-stream velocity at the edge of the boundary layer, the local momentum thickness and the radius of curvature of the mean flow, respectively, exceeds 0.3 (Görtler Reference Görtler1954; Smith Reference Smith1955). The most amplified wavelength in the spanwise direction is found to be $\unicode[STIX]{x1D706}_{T}$, corresponding to $U_{e}\unicode[STIX]{x1D706}_{T}/\unicode[STIX]{x1D708}\sqrt{\unicode[STIX]{x1D706}_{T}/R}\approx 220{-}270$ (Smith Reference Smith1955; Floryan & Saric Reference Floryan and Saric1982; Luchini & Bottaro Reference Luchini and Bottaro1998) or $\unicode[STIX]{x1D706}_{T}\approx \unicode[STIX]{x1D6FF}-2\unicode[STIX]{x1D6FF}$ (Smits & Dussauge Reference Smits and Dussauge2006). However, the above scaling laws are for laminar flows and their applicability to turbulent flows is unclear. Tani (Reference Tani1962) proposed that the criterion for the onset of the Görtler instability is valid for turbulent flows provided the molecular viscosity is replaced by the eddy viscosity. It has also been suggested that $\unicode[STIX]{x1D6FF}/R\geqslant 0.01$ is the applicable criterion instigating Görtler instability for the case of TBLs (Hoffmann, Muck & Bradshaw Reference Hoffmann, Muck and Bradshaw1985; Floryan Reference Floryan1991).

Figure 24. (a) Selected streamlines (solid, dashed and dash-dotted, respectively) for Görtler instability analysis; (b) Görtler number; (c) $\unicode[STIX]{x1D6FF}/R$; and (d) most amplified spanwise wavelength predicted by $U_{e}\unicode[STIX]{x1D706}_{T}/\unicode[STIX]{x1D708}_{tot}\sqrt{\unicode[STIX]{x1D706}_{T}/R}=250$. The red solid line with crosses is $\unicode[STIX]{x1D6FF}/\unicode[STIX]{x1D703}_{o}$ for reference. Black markers are the length scale in the separated shear layer measured by the streamwise fluctuating velocity (figure 26).

To examine these proposed criteria for the present flow, we calculate the effective turbulent eddy viscosity as (Spalart & Strelets Reference Spalart and Strelets2000)

(3.7)$$\begin{eqnarray}\unicode[STIX]{x1D708}_{t,eff}=-\frac{\overline{u_{i}^{\prime }u_{j}^{\prime }}\unicode[STIX]{x1D61A}_{ij}}{2\unicode[STIX]{x1D61A}_{ij}\unicode[STIX]{x1D61A}_{ij}},\end{eqnarray}$$

and use $\unicode[STIX]{x1D708}_{tot}=\unicode[STIX]{x1D708}+\unicode[STIX]{x1D708}_{t,eff}$ to estimate the Görtler number. The analysis is applied in the region of mean streamline concavity shown in figure 24. It can be seen that, when the boundary layer separates from the wall, the Görtler number ranges from approximately 5 to 21 in the near-wall region and $\unicode[STIX]{x1D6FF}/R$ ranges from 0.08 to 0.5. These values are well above the threshold proposed in previous studies, indicating the viability of the occurrence of Görtler instability. The most amplified spanwise wavelength (see figure 24d) is predicted to range from approximately $25\unicode[STIX]{x1D703}_{o}$ to $50\unicode[STIX]{x1D703}_{o}$ at $x=180\unicode[STIX]{x1D703}_{o}$, which is in the range of $(1{-}2)\unicode[STIX]{x1D6FF}$. The low-frequency DMD mode exhibits approximately four pairs of counter-rotating streamwise vortices in the spanwise direction near $x=200\unicode[STIX]{x1D703}_{o}$ (figure 22a). This corresponds to a wavelength of $0.9\unicode[STIX]{x1D6FF}$, consistent with the predicted range of the most amplified wavelength of possible Görtler instability modes.

It is noted that, downstream of $x=200\unicode[STIX]{x1D703}_{o}$, the mean streamlines become convex in shape but the sign of the velocity gradient remains unchanged except for the very near-wall region inside the separation bubble. Thus, while the conditions required for the Görtler instability no longer exist in this region, Görtler vortices that were generated upstream of the region continue to develop and grow on this region. As observed from the structures of the low-frequency DMD modes, the scale of the elongated streamwise structures grows in the streamwise direction and the structures break down around $x=400\unicode[STIX]{x1D703}_{o}$.

3.5 Breakdown of Görtler vortices and turbulence structures

In this section we examine the elongated streamwise structures in the TSB and their possible relationship to structures in the incoming TBL upstream of the separation bubble. The prominence of large-scale streamwise structures in ZPG TBLs at high Reynolds numbers is well known (Adrian, Meinhart & Tomkins Reference Adrian, Meinhart and Tomkins2000; Ganapathisubramani, Longmire & Marusic Reference Ganapathisubramani, Longmire and Marusic2003; Hutchins & Marusic Reference Hutchins and Marusic2007), and these structures are amplified in the presence of an APG (Skote, Hanningson & Henkes Reference Skote, Hanningson and Henkes1998; Lee & Sung Reference Lee and Sung2009; Harun et al. Reference Harun, Monty, Mathis and Marusic2013). In studies on flow separation that employ suction and blowing (Abe et al. Reference Abe, Mizobuchi, Matsuo and Spalart2012; Abe Reference Abe2017), large-scale structures identified by the streamwise velocity fluctuations were also observed but were attributed to the lifted and energized outer layer structures after flow separation. Our results of the DMD analysis, on the other hand, indicate that the streamwise-elongated structures are mostly amplified when the flow separates and streamline curvature becomes significant. This, however, does not necessarily imply that there is no link between the low-speed regions (‘streaks’) in the incoming boundary layer and the large-scale streamwise-oriented structures we have identified as Görtler vortices. The low-speed regions could serve as the perturbations for the initial development of the Görtler vortices. Similar to Priebe et al. (Reference Priebe, Tu, Rowley and Martín2016), we identified a time-delayed correlation between the appearance of a low-speed region in the incoming boundary layer and the occurrence of the Görtler vortices (not shown). This could be the reason why the occurrence of the elongated structures in the spanwise direction is highly unsteady, instead of being at fixed locations when the Görtler instability is triggered by prescribed upstream disturbances (see Floryan & Saric Reference Floryan and Saric1982; among others).

Figure 25. Top view of the instantaneous vortical structures shown by the isosurfaces of the secondary invariant of the velocity-gradient tensor $\unicode[STIX]{x1D618}$, coloured by the distance from the wall. The dark-grey isosurfaces are $u^{\prime }=-0.1U_{o}$. For an isometric view and colour map, refer to figure 5(b).

Figure 25 shows the top view of the turbulent structures exhibited in figure 5. The streamwise alignment of the structures ($x<450\unicode[STIX]{x1D703}_{o}$) is quite evident and the scale of the low-speed region increases as the flow travels downstream (e.g. the dark grey isosurface of $u^{\prime }=-0.1U_{o}$ at $x/\unicode[STIX]{x1D703}_{o}=300$ is much thicker than the ones at $x/\unicode[STIX]{x1D703}_{o}=100$ and 200, which appear to be thin rod-like regions). The growth of Görtler vortices in a laminar flow over a concave wall has been examined in previous studies (Bippes & Görtler Reference Bippes and Görtler1972; Swearingen & Blackwelder Reference Swearingen and Blackwelder1983; Smith & Walker Reference Smith and Walker1989; Li & Malik Reference Li and Malik1995) and one observation is that the continuous ejection of fluid from the low-speed region by the counter-rotating vortices creates a marked retardation of flow in the upwash region and an APG in the streamwise direction, which has the dual effect of both forcing the vortices away from the wall and also increasing the spanwise distance between the vortices (Bippes & Görtler Reference Bippes and Görtler1972; Smith & Walker Reference Smith and Walker1989). The behaviour of the streamwise structures shown in our simulation is in line with this observation.

Figure 26. Pre-multiplied spanwise energy spectrum of the streamwise fluctuating velocity, examined at several streamwise locations. Each map is normalized by its maximum, and contours are shown at level 0.8, 0.6, 0.4 and 0.2, from dark to light, respectively.

The scale of the streamwise structures in the spanwise direction is examined using the pre-multiplied energy spectrum of $u^{\prime }$, and figure 26 shows the spectrum at several streamwise locations. The growth of the spanwise length scale is clearly seen. The spanwise length scale of these structures is approximately $20\unicode[STIX]{x1D703}_{o}$ to $30\unicode[STIX]{x1D703}_{o}$ in the upstream side of the separation bubble, which agrees reasonably well with the most unstable wavelength predicted by the Görtler instability (refer to figure 24d). From $x=300\unicode[STIX]{x1D703}_{o}$ the wavelength changes to $60\unicode[STIX]{x1D703}$, consistent with the observation of the low-frequency DMD mode, which shows that the structures merge after the crest of the separation bubble.

In previous studies, it has been proposed that Görtler vortices may break down due to four processes: (a) Tollmien–Schlichting waves. (b) Nonlinear development of Görtler vortices and energy cascades down from the fundamental into the higher harmonics and the mean flow. (c) Secondary instabilities that generate a retarded region in the upstream flow region between Görtler vortices and the formation of transverse vortices (Bippes & Görtler Reference Bippes and Görtler1972; Hall Reference Hall1982; Smith & Walker Reference Smith and Walker1989). Linear stability theory has revealed two types of secondary instabilities that help break down the Görtler vortices in laminar flows: a sinuous (odd) mode that perturbs the Görtler vortices in a wavy manner, and a varicose (even) mode that breaks up Görtler vortices into a series of ‘knotty’ structures (see Swearingen & Blackwelder Reference Swearingen and Blackwelder1983; Li & Malik Reference Li and Malik1995; among others). None of these processes is clearly evident in the single DMD mode reconstruction, and the structures in the flow velocity seem to exhibit both streamwise waviness and breakup. Which mode (if any) dominates the breakup of vortices in this flow is unclear from the current simulations. (d) The vortex merging/pairing process. As discussed regarding the frequency change (figure 14) and the vortex development (figures 17 and 18), it is possible that the breakup of the Görtler vortices is associated with the merging/pairing of the spanwise roller vortices. The DMD mode at the low frequency does exhibit a spanwise staggered signature (figure 22b), which provides further support for this possibility. However, this correlation does not necessarily imply causation.