4.2.1 Spectral proper orthogonal decomposition

The Reynolds stresses suggest that the wake has spanwise movements which are likely a part of the quasi-periodic shedding cycle. Understanding the quasi-periodic vortex shedding cycle is critical to understanding the dynamics creating a common-up vortex pair in the separation bubble and a common-down pair downstream as well as the Reynolds stress distribution and wake recovery. Phase-averaging methods are difficult to use here due to the high broadband turbulence. For example, figure 13 shows a time series of spanwise velocity at $(X/H,Y/H,Z/H)=(4.0,0.5,0)$, for which the power spectrum shows a strong peak at the shedding frequency. There are clear large-scale fluctuations, but identifying a specific phase is difficult.

Figure 13.

Spanwise velocity time series at $(X/H,Y/H,Z/H)=(4.0,0.5,0)$. Time is non-dimensionalized by the shedding frequency.

We use SPOD to calculate modes at the shedding frequency that describe the vortex shedding dynamics. SPOD is a model order reduction technique that identifies modes evolving coherently in space and time at distinct frequencies (Lumley Reference Lumley1970; Towne, Schmidt & Colonius Reference Towne, Schmidt and Colonius2018). The advantage of SPOD over proper orthogonal decomposition is that SPOD isolates dynamics at a single frequency and therefore ignores broadband turbulence at other frequencies. We use SPOD to reconstruct the vortex shedding cycle from modes at the shedding frequency.

SPOD finds modes $\unicode[STIX]{x1D713}_{i}(x,f_{k})$ that are a function of space and operate at frequency $f_{k}$. The modes are calculated using the method described in Towne et al. (Reference Towne, Schmidt and Colonius2018) and summarized here. A time series of instantaneous flow snapshots are divided into $N_{w}$ fixed-length windows with 50 % overlap, each of length $T_{w}$. The data from each individual snapshot are arranged into a vector containing each velocity component at every point in the region of interest. The time Fourier transform is calculated on each window using a Hanning window with weights $w_{i}$, and the coefficients $\hat{q}_{k}^{(j)}$ at the chosen frequency $f_{k}$ are arranged into a matrix

(4.1)$$\begin{eqnarray}\displaystyle \hat{\unicode[STIX]{x1D64C}}_{f_{k}}=\sqrt{\frac{\unicode[STIX]{x0394}t}{sN_{w}}}[\hat{q}_{k}^{(1)},\hat{q}_{k}^{(2)},\ldots ,\hat{q}_{k}^{(N_{w})}], & & \displaystyle\end{eqnarray}$$

where $s=\sum _{i=1}^{T_{w}}w_{i}^{2}$. The cross-spectral density tensor at frequency $f_{k}$ is

(4.2)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D64E}_{f_{k}}=\hat{\unicode[STIX]{x1D64C}}_{f_{k}}\hat{\unicode[STIX]{x1D64C}}_{f_{k}}^{\ast }, & & \displaystyle\end{eqnarray}$$

where $\hat{\unicode[STIX]{x1D64C}}_{f_{k}}^{\ast }$ is the complex conjugate transpose of $\hat{\unicode[STIX]{x1D64C}}_{f_{k}}$. The SPOD modes $\unicode[STIX]{x1D713}_{i}(x,f_{k})$ are the eigenvectors of $\unicode[STIX]{x1D64E}_{f_{k}}$. A useful property is that the eigenvalues of $\unicode[STIX]{x1D64E}_{f_{k}}$ tell how much energy is contained in each mode.

Each mode contains all three velocity components and has real and imaginary parts that are phase-shifted a quarter cycle. Figure 14 shows the first SPOD mode at the shedding frequency for the plane located at $X/H=4$ in the wake of the $10^{\circ }$ bump. Because SPOD modes have arbitrary phase, the phase of the modes in figure 14 appear different from the SPOD modes in Ching et al. (Reference Ching, Elkins and Eaton2018b). SPOD modes operate at fixed frequency, so the modes show correlated fluctuations at the shedding frequency. The mode in figure 14 shows large-scale oscillations of each velocity component. The streamwise and wall-normal velocity components of the mode indicate spanwise oscillations, while the spanwise velocity appears to periodically sweep fluid across the centreplane near the wall.

Figure 14.

(a) Real part of the $u$ component from the first SPOD mode at $f_{shed}$. Modes are calculated from planes at $X/H=4$ on the $10^{\circ }$ bump. The curve shows the bump profile projected on the plane. (b) Imaginary part of $u$ component. (c) Real part of $v$ component. (d) Imaginary part of $v$ component. (e) Real part of $w$ component. (f) Imaginary part of $w$ component.

4.2.2 Reconstruction method

It is instructive to reconstruct the oscillating velocity field to understand how the quasi-periodic fluctuation modes average to the complex mean flow observed in experiments and LES. In particular, the mean wake contains longitudinal vortex structures and a region of lower streamwise velocity which likely have spatial movements over the shedding cycle. It is difficult to learn how they evolve just by examining SPOD modes, so we form a reduced-order model of the quasi-periodic cycle using the first SPOD mode at the shedding frequency. This method is similar to the reconstruction with proper orthogonal decomposition modes done by Van Oudheusden et al. (Reference Van Oudheusden, Scarano, Van Hinsberg and Watt2005). We apply SPOD only at the shedding frequency, so $f_{k}$ corresponds to the shedding frequencies given in table 1. There are no significant peaks in power spectra at frequencies away from the shedding frequency peak, so it is sufficient that the reduced-order model contains only the shedding frequency.

The Fourier transform coefficients can be written as a linear combination of the SPOD modes. Using just one mode is equivalent to estimating the Fourier transform at $f_{shed}$ as

(4.3)$$\begin{eqnarray}\displaystyle \hat{q}(x,f_{shed})=\mathop{\sum }_{i=1}^{N_{w}}a_{i}(f_{shed})\unicode[STIX]{x1D713}_{i}(x,f_{shed})\approx a_{1}(f_{shed})\unicode[STIX]{x1D713}_{1}(x,f_{shed}), & & \displaystyle\end{eqnarray}$$

where $a_{i}(f_{shed})$ are coefficients of the modes. Reconstructing the cycle with just the first mode inherently assumes that the first SPOD mode represents shedding dynamics and the rest of the modes represent turbulence unrelated to shedding. Different SPOD modes can represent uncorrelated processes that occur at the same frequency, but the shedding cycle is highly correlated so it should be well represented by one mode.

Reconstructing the cycle using the SPOD mode at a single frequency implies that the power spectrum is a delta function at $f_{shed}$. In reality, because the cycle is quasi-periodic, the power spectrum has a range of frequencies that correspond to shedding. To ensure that the reconstructed cycle contains the energy of the frequency band corresponding to shedding, $a_{i}(f_{shed})$ is replaced with $\sqrt{P_{rec}/2}$, where $P_{rec}$ is the power contained in the reconstructed cycle integrated in space and averaged in time. The method of calculating $P_{rec}$ is given below. Without this substitution, the reconstructed cycle would contain only a fraction of the energy due to shedding and would be dependent on the frequency resolution. The SPOD mode $\unicode[STIX]{x1D713}_{1}(x,f_{shed})$ has a unitary 2-norm, so specifying $P_{rec}$ is sufficient to ensure the reconstructed cycle contains the appropriate energy.

Figure 15.

Power spectrum of velocity integrated over space and the three velocity components. The spectrum is normalized by total energy in the region of interest, $P_{tot}$. The data are from the $10^{\circ }$ bump at $X/H=4$ within the region of interest $-1.8<Z/H<1.8$ and $0<Y/H<1.8$. The dotted line shows the average power in the range $0.5f_{shed}<f<1.5f_{shed}$.

The power in the shedding cycle is approximated by computing the power spectrum within the domain and integrating over a narrow frequency range near the shedding frequency. For example, figure 15 shows the power spectrum in the plane located at $X/H=4$ of the $10^{\circ }$ bump within the region of interest $-1.8<Z/H<1.8$ and $0<Y/H<1.8$. There is a clear peak due to the quasi-periodic cycle, but the energy is spread over a range of frequencies. The shedding energy $P_{shed}$ is approximated as the integral of power over frequencies within the range $0.5f_{shed}<f<1.5f_{shed}$ and satisfying $P>\overline{P}_{r}$, where $\overline{P}_{r}$ is the mean of power over the frequencies $0.5f_{shed}<f<1.5f_{shed}$. These constraints ensure that only energy close to the shedding frequency and above broadband turbulence is considered. Table 2 shows the fraction of total energy associated with shedding for the planes in which the reconstructed cycle is shown.

All of the SPOD modes at $f_{shed}$ together represent the energy at $f_{shed}$; thus, $P_{shed}$ must be multiplied by the fraction of energy in the first mode to approximate the power contained in the reconstructed cycle, $P_{rec}$. The eigenvalues of $S_{f_{shed}}$ have magnitude proportional to the energy in each mode, so the fraction of energy in the first mode is $\unicode[STIX]{x1D706}_{1}/\sum _{i}^{N_{w}}\unicode[STIX]{x1D706}_{i}$, where $\unicode[STIX]{x1D706}_{i}$ is the eigenvalue corresponding to mode $i$. Table 3 shows the fraction of energy in the first mode for each plane in which the reconstructed cycle is shown. The total energy of the reconstructed cycle is

(4.4)$$\begin{eqnarray}\displaystyle P_{rec}=\frac{\unicode[STIX]{x1D706}_{1}}{\displaystyle \mathop{\sum }_{i=1}^{N_{w}}\unicode[STIX]{x1D706}_{i}}P_{shed}. & & \displaystyle\end{eqnarray}$$

Since the reconstructed velocity is a real signal, its Fourier transform must contain delta functions at $f_{shed}$ and $f_{-shed}$ that are complex conjugates of each other. The Fourier transform of the reconstructed cycle with the appropriate power is

(4.5)$$\begin{eqnarray}\displaystyle \hat{q}(x,f_{shed})=\sqrt{\frac{P_{rec}}{2}}(\unicode[STIX]{x1D713}_{1}(x,f_{shed})\unicode[STIX]{x1D6FF}_{f_{shed}}+\unicode[STIX]{x1D713}_{1}(x,f_{shed})^{\ast }\unicode[STIX]{x1D6FF}_{f_{-shed}}). & & \displaystyle\end{eqnarray}$$

Taking the inverse transform and adding the mean velocity yields the reconstructed cycle

(4.6)$$\begin{eqnarray}\displaystyle u_{r}(x,t)=\overline{u}(x)+\sqrt{2P_{rec}}(\unicode[STIX]{x1D713}_{1}^{(\text{R})}\text{cos}(2\unicode[STIX]{x03C0}f_{shed}t)-\unicode[STIX]{x1D713}_{1}^{(\text{I})}\text{sin}(2\unicode[STIX]{x03C0}f_{shed}t)), & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D713}_{1}^{(\text{R})}$ and $\unicode[STIX]{x1D713}_{1}^{(\text{I})}$ are the real and imaginary parts of $\unicode[STIX]{x1D713}_{1}$ and $u_{r}(x,t)$ is the reconstructed cycle. In (4.6), $\overline{u}(x)$ and $u_{r}(x,t)$ contain all three velocity components at each point in the region of interest.

Table 2.

Fraction of energy associated with shedding, $P_{shed}/P_{total}$.

Table 3.

Fraction of energy contained in the first SPOD mode, $\unicode[STIX]{x1D706}_{1}/\sum _{i}\unicode[STIX]{x1D706}_{i}$.

This method assumes that only dynamics at the shedding frequency is important to the shedding cycle. In reality, is is possible that harmonics of the shedding frequency are also contained in the shedding cycle. However, the power spectrum of velocity integrated in space shown in figure 15 shows no peak at harmonics of the shedding cycle, which justifies reconstructing the cycle using a single mode at the shedding frequency.

4.2.3 Reconstructed cycle analysis

The cycle reconstruction was done in several $y{-}z$ and $x{-}z$ planes. For all $y{-}z$ planes, the SPOD modes and reconstructed cycle are calculated in the region of interest $-1.8<Z/H<1.8$ and $0<Y/H<1.8$ to avoid including the top and sidewall boundary layers. In the $x{-}z$ plane, the region of interest is $-1.8<Z/H<1.8$ and $-2<X/H<12$. Inclusion of the top and sidewall boundary layers would add more broadband energy to the power spectrum, which makes identifying $P_{shed}$ more ambiguous. For the $10^{\circ }$ bump, SPOD was done on instantaneous velocities recorded every 10th time step, corresponding to a sampling rate 41 times the shedding frequency. For the $50^{\circ }$ bump, SPOD was done on the velocities every 14th time step, which is 24 times the shedding frequency. Each SPOD window contains 128 instantaneous flow snapshots. The SPOD modes are not sensitive to the window length or the precise frequency within 10 % of the shedding frequency.

Figure 16.

(a–h) The $10^{\circ }$ bump reconstructed cycle using SPOD at $X/H=4$. Contours are streamwise velocity and vectors are in-plane velocity. Phases are at evenly spaced intervals of $1/8$ of the cycle. The curves show the bump profile projected on the plane.

Figure 17.

Schematic of wake vortex dynamics.

Figure 18.

(a–h) The $50^{\circ }$ bump reconstructed cycle using SPOD at $X/H=4$. Contours are streamwise velocity and vectors are in-plane velocity. Phases are at evenly spaced intervals of $1/8$ of the cycle. The curves show the bump profile projected on the plane.

Figure 16 shows eight phases from the reconstructed cycle of the $10^{\circ }$ bump at $X/H=4$, where subsequent images are spaced $1/8$ of the cycle apart. In figure 16(a–d), the streamwise velocity in the wake gradually decreases and the wake shifts from left to right. In figure 16(d–h), the streamwise velocity first increases, then decreases again and the wake shifts back to the left. The wake strength therefore ‘pulses’ twice each cycle and the wake oscillates in the spanwise direction. The in-plane velocities show a counterclockwise vortex developing in figure 16(c–e), then in figure 16(f–a) the counterclockwise vortex disappears and a clockwise vortex develops. The vortex roll-up coincides with the wake shifting to the side, and the movement is in the direction that would be induced by an image vortex (Marshall Reference Marshall2001). The image vortex of a clockwise vortex near the wall moves the vortex in the negative $Z$ direction, and vice versa. While image vortices are derived from potential flow, it has been shown that longitudinal vortices near a wall have close agreement with the potential flow solution (Eibeck & Eaton Reference Eibeck and Eaton1987). The viscous effects typically are contained in a thin layer of vorticity very close to the wall, that typically has little effect on the vortex trajectory. Figure 17 is a schematic of the wake vortex dynamics, showing alternating vortices moving across the centreline as they are advected downstream. The reconstructed shedding cycle here is consistent with the experimental results from Ching et al. (Reference Ching, Elkins and Eaton2018b).

Although the $10^{\circ }$ bump is only slightly asymmetric, there is asymmetry evident in the reconstructed cycle. As the streamwise velocity pulses twice over the cycle, it has troughs of low streamwise velocity when the wake is shifted furthest from the centreline in figure 16(d,h). The streamwise velocity is lower when the wake is on the trailing-edge side (figure 16d) than the leading-edge side (figure 16h). The asymmetry is also evident in the streamwise component of the SPOD mode (figure 14). The vortices do not show significant asymmetry at the $10^{\circ }$ angle.

Figure 18 shows the reconstructed cycle from the $50^{\circ }$ bump at the same location as for figure 16. Whereas in the $10^{\circ }$ bump the streamwise velocity in the wake ‘pulses’ twice each cycle, there is only one pulse in the $50^{\circ }$ bump. The wake deficit on the centreline also decays faster than in the $10^{\circ }$ bump. The $50^{\circ }$ bump wake oscillates in the spanwise direction, but the movements are considerably diminished in comparison to the $10^{\circ }$ bump. The $50^{\circ }$ bump does not have alternating opposite-sign vortices like the $10^{\circ }$ bump; the counterclockwise vortex in the $50^{\circ }$ bump has similar strength to the vortices in the $10^{\circ }$ bump, but the clockwise vortex never develops into a full vortex. Sensitivity to bump angle observed in the mean fields appears to be caused by a lack of a coherent clockwise vortex in the quasi-periodic cycle and weaker spanwise movements.

Figure 19.

(a) The $10^{\circ }$ bump reconstructed cycle using SPOD at $X/H=1.5$ showing phase when counterclockwise vortex is coherent. Contours are streamwise velocity and vectors are in-plane velocity. The curve shows the bump profile projected on the plane. (b) Phase when clockwise vortex is coherent.

Figure 19 shows two phases from the quasi-periodic cycle reconstruction at the plane $X/H=1.5$ from the $10^{\circ }$ bump, which is within the separation bubble. The phases shown are chosen such that figure 19(a) is a phase when the counterclockwise vortex is coherent and figure 19(b) is half the cycle later when the clockwise vortex is coherent. Note that the clockwise vortex is right of the centreline and the counterclockwise vortex is left of the centreline. Downstream at $X/H=4$, figure 16 shows the vortices on opposite sides; the clockwise vortex is to the left of the centreline and the counterclockwise vortex is to the right of the centreline. The vortices therefore cross the centreline as they are advected downstream. This explains the observation of a common-up vortex pair in the mean fields within the separation bubble but a common-down pair downstream. The opposite-sign vortex pairs in the mean flow never exist in instantaneous field, but are a result of averaging over the quasi-periodic cycle. The clockwise vortices develop on the positive $Z$ side of the bump and the counterclockwise vortices on the opposite side, so they average to a common-up pair in the separation bubble. As the vortices move downstream, they cross the centreline as though pushed by image vortices. Downstream of the separation bubble, they average to a common-down pair.

Figure 20.

(a–f) The $10^{\circ }$ bump reconstructed cycle using SPOD at $Y/H=0.5$ showing contours of streamwise velocity and in-plane velocity streamlines. Phases are at evenly spaced intervals of $1/6$ of the cycle.

The quasi-periodic cycle reconstruction was done in an $x{-}z$ plane at $Y/H=0.5$, with phases shown in figure 20 from the $10^{\circ }$ bump. Streamlines show wall-normal vortex structures in the separation bubble. However, downstream of the separation bubble, there is no evidence of wall-normal vortices in the reconstructed cycle or instantaneous fields. Due to the bump geometry and pressure gradients, boundary layer vorticity in the wall-normal direction is turned after detachment to create streamwise vorticity, which forms the large streamwise vortices (Ching et al. Reference Ching, Elkins and Eaton2018a).

The separation bubble and wake oscillate with spanwise motions as vortices are shed and carried downstream in figure 20. The streamwise velocity oscillations viewed in the $Y/H=0.5$ plane are similar to shedding from a cylinder (Kravchenko & Moin Reference Kravchenko and Moin2000; Parnaudeau et al. Reference Parnaudeau, Carlier, Heitz and Lamballais2008). However, while a cylinder sheds vortices parallel to the cylinder axis, the bump flow sheds longitudinal vortices.

In figure 11 it was noted that streamwise velocity fluctuations far downstream have two peaks on either side of the centreline. Fluctuations due to the quasi-periodic shedding cycle explain the double peaks. The variance of the velocity in the reconstructed cycle can be derived from (4.6) as

(4.7)$$\begin{eqnarray}\displaystyle \overline{u_{r}^{\prime 2}}=P_{rec}[(\unicode[STIX]{x1D713}_{1}^{(\text{R})})^{2}+(\unicode[STIX]{x1D713}_{1}^{(\text{I})})^{2}]. & & \displaystyle\end{eqnarray}$$

Figure 21 shows spanwise profiles of streamwise velocity fluctuations at $X/H=6$ and $Y/H=0.5$ from LES statistics and from the reconstructed cycle. The reconstructed cycle has lower $u_{rms}$ because it only accounts for fluctuations due to the shedding cycle whereas the LES statistics contain fluctuations at all frequencies. The difference between the two profiles is therefore due to small-scale turbulence. Both profiles have two peaks off the centreline, and the difference between $u_{rms}$ at the centreline and at the peaks is similar for the two profiles, suggesting that fluctuations from the quasi-periodic shedding cycle are responsible for the double peak. Matsumura and Antonia (Reference Matsumura and Antonia1993) similarly found that the double peak in streamwise fluctuations in the far wake of a cylinder was caused by the quasi-periodic shedding cycle. For a wall-mounted square cylinder, Bourgeois, Sattari & Martinuzzi (Reference Bourgeois, Sattari and Martinuzzi2011) observed that the coherent fluctuations due to the shedding cycle had a much larger contribution to the double peak than incoherent fluctuations due to small-scale turbulence.

Figure 21.

Profile of streamwise r.m.s. velocity at $X/H=6$ and $Y/H=0.5$ from LES statistics and from the reconstructed cycle.

4.2.4 Conditional averaging

The quasi-periodic cycle reconstruction ignores turbulence at scales outside the shedding frequency, so it does not show small-scale turbulent structures that are carried by the quasi-periodic shedding cycle. The separation bubble is highly turbulent and contains a wide range of scales, whereas the free stream has much lower turbulence. Fluid in the wake that was shed from the separation bubble would therefore be expected to have higher turbulence than free-stream fluid swept into the wake by quasi-periodic motions. We use conditional averaging to examine the relationship between small-scale turbulence and the quasi-periodic cycle.

Figure 22.

Instantaneous isosurfaces of $Q$ criterion at $Q(H^{2}/U_{bulk}^{2})=20$ from the $50^{\circ }$ bump. Flow is left to right.

Figure 22 shows instantaneous isosurfaces of $Q$ criterion (Hunt, Wray & Moin Reference Hunt, Wray and Moin1988) for a single snapshot from the $50^{\circ }$ bump. Regions of high turbulence appear between low-turbulence regions in the wake with a length scale of approximately $2H$ to $4H$. A distance of $4H$ at the bulk velocity corresponds to non-dimensional frequency of $0.25$, which is 1.6 times the shedding frequency (table 1). This suggests that the highly turbulent regions are related to the shedding cycle. The velocity in the wake is slower than the bulk velocity, so structures shed at the shedding frequency should appear closer together than if they were advected at free-stream velocity.

Figure 23.

(a) Conditionally averaged flow snapshots at $X/H=4$ with 20 % lowest condition (low turbulence) from $50^{\circ }$ bump. Contours are streamwise velocity and vectors are in-plane velocity. The curve shows the bump profile projected on the plane. (b) Conditionally averaged flow snapshots with 20 % highest condition (high turbulence). (c) Phase from reconstructed cycle with lowest least-squares difference from (a). (d) Phase from reconstructed cycle with lowest least-squares difference from (b).

Conditional averaging is used in the $y{-}z$ plane located at $X/H=4$ to relate the turbulent regions to the reconstructed cycle. We use the squared streamwise vorticity, $\unicode[STIX]{x1D714}_{x}^{2}$, averaged over a spatial region of interest as a quantitative proxy for the instantaneous turbulence level in the wake (4.8):

(4.8)$$\begin{eqnarray}\displaystyle C_{1}=\frac{1}{A_{1}}\int _{A_{1}}\unicode[STIX]{x1D714}_{x}^{2}\,\text{d}A. & & \displaystyle\end{eqnarray}$$

The region of interest, $A_{1}$, is chosen as $-1.8\leqslant Z/H\leqslant 1.8$ and $0\leqslant Y/H\leqslant 1.8$. Conditional averaging is done over the snapshots with the 20 % lowest $C_{1}$ on the $50^{\circ }$ bump in figure 23(a) and 20 % highest $C_{1}$ in figure 23(b). When turbulence is low, the wake has higher streamwise velocity and no vortex structure. When the turbulence is high, the wake has lower streamwise velocity, is shifted to the right and has a counterclockwise vortex. Figure 23(c,d) shows the two phases from the reconstructed cycle with the lowest least-squares difference from figure 23(a,b). The results from conditional averaging are strikingly similar to phases from the reconstructed cycle.

One potential concern is that the condition $C_{1}$ is higher when the large vortex structure in figure 23(b,d) is present because the vortex structure has higher vorticity than the flow in figure 23(a,c). However, when the mean squared vorticity is calculated on the averaged fields in figure 23(a,b), the conditionally averaged low-turbulence field actually has higher $C_{1}$ than the high-turbulence field. While instantaneous snapshots have high vorticity magnitude wherever there is turbulence in the domain, the conditionally averaged fields have highest vorticity magnitude very close to the wall. The conditionally averaged low-turbulence field has slightly higher vorticity near the wall, leading to a higher $C_{1}$. However, the small-scale structures have much larger contribution to the condition. Therefore, the vortex structure has negligible contribution to $C_{1}$ in comparison to turbulence.

Figure 24.

(a) Conditionally averaged flow snapshots at $X/H=4$ with 40 % lowest $C_{2}$ from the set of snapshots with 30 % highest $C_{1}$ from $10^{\circ }$ bump. Contours are streamwise velocity and vectors are in-plane velocity. The curve shows the bump profile projected on the plane. (b) Conditionally averaged flow snapshots with 40 % highest $C_{2}$ from the set of snapshots with 30 % highest $C_{1}$. (c) Phase from reconstructed cycle with lowest least-squares difference from (a). (d) Phase from reconstructed cycle with lowest least-squares difference from (b).

Conditional averaging on the $10^{\circ }$ bump is more complicated because the quasi-periodic cycle is nearly symmetric, so opposite-sign vortex structures have nearly the same turbulence and would be averaged together if the same method were used. To distinguish the clockwise and counterclockwise vortices, a second condition, $C_{2}$, defined in (4.9) is used to quantify whether the streamwise velocity in the wake is shifted to the right or left:

(4.9)$$\begin{eqnarray}\displaystyle C_{2}=\int _{A_{2}}u\,\text{d}A-\int _{A_{3}}u\,\text{d}A, & & \displaystyle\end{eqnarray}$$

where $A_{2}$ is region of interest $0\leqslant Z/H\leqslant 0.5$ and $0\leqslant Y/H\leqslant 0.5$ and $A_{3}$ is region of interest $-0.5\leqslant Z/H\leqslant 0$ and $0\leqslant Y/H\leqslant 0.5$. Region of interest $A_{2}$ is to the right of the centreline and $A_{3}$ is to the left, so the subtraction in (4.9) is a measure of asymmetry. The method of conditional averaging for the $10^{\circ }$ bump is as follows. The 30 % of snapshots with highest $C_{1}$ form the set of high-turbulence snapshots. From that set, the 40 % of snapshots with the most negative $C_{2}$ are averaged and shown in figure 24(a), and the 40 % of snapshots with highest $C_{2}$ are averaged and shown in figure 24(b).

Figure 24(c,d) shows the two phases from the reconstructed cycle of the $10^{\circ }$ bump with the lowest least-squares difference with the conditionally averaged images. Conditional averaging on high-turbulence snapshots of the $10^{\circ }$ bump leads to almost identical results to the reconstructed cycle. A counterclockwise vortex is observed when the wake is shifted to the right, and vice versa, meaning that the high turbulence level is observed at the specific phases where a strong longitudinal vortex is present. Figure 25 shows the result of conditional averaging the 30 % of snapshots with lowest $C_{1}$ (low turbulence). The wake streamwise velocity is significantly higher and no vortex structure is observed. There is no phase in the reconstructed cycle similar to the low-turbulence conditional average because the wake may be shifting right or left when the plane has low turbulence, and figure 25 is a result of averaging over both directions of movement.

Figure 25.

Conditionally averaged flow snapshots with 30 % lowest $C_{1}$ at $X/H=4$ from the $10^{\circ }$ bump. The curve shows the bump profile projected on the plane.

This method of conditional averaging using turbulence as the condition reveals that the vortices are associated with high turbulence. The similarity between velocity fields from conditional averaging and the reconstructed cycle implies a strong link between turbulence and the quasi-periodic cycle. As the bump sheds a vortex, high-turbulence fluid from the separation bubble is released into the wake and carried downstream with the vortex. Between vortex shedding events, free-stream fluid with low turbulence sweeps into the wake from the sides. The $50^{\circ }$ bump sheds a single streamwise vortex per cycle while the $10^{\circ }$ bump sheds two vortices, so the $50^{\circ }$ bump releases a single burst of high turbulence while the $10^{\circ }$ bump releases two bursts per cycle. This undoubtedly contributes to the significantly higher TKE in the wake of the $10^{\circ }$ bump, seen in figure 10.