2.1 Optimal transient growth

For more complex mean velocity profiles, Orr-like solutions can be computed numerically. We use the algorithm in Schmid & Henningson (Reference Schmid and Henningson2001) to find solutions that optimally amplify the total kinetic energy, including a $y$-dependent eddy viscosity to account for the nonlinear turbulent dissipation, and a velocity profile and kinematic viscosity corresponding to a turbulent channel at $Re_{\unicode[STIX]{x1D70F}}=u_{\unicode[STIX]{x1D70F}}h/\unicode[STIX]{x1D708}=2000$, as in del Álamo & Jiménez (Reference del Álamo and Jiménez2006) and Pujals et al. (Reference Pujals, García-Villalba, Cossu and Depardon2009). We use the approximation to the mean eddy viscosity $\unicode[STIX]{x1D708}_{T}$ of Cess (Reference Cess1958):

(2.7)$$\begin{eqnarray}\unicode[STIX]{x1D708}_{T}=\frac{\unicode[STIX]{x1D708}}{2}\left\{1+\frac{K^{2}Re_{\unicode[STIX]{x1D70F}}}{9}\left[2\unicode[STIX]{x1D702}-\unicode[STIX]{x1D702}^{2}\right]\left[3-4\unicode[STIX]{x1D702}+\unicode[STIX]{x1D702}^{2}\right]^{2}\left[1-\exp \left(\frac{-\unicode[STIX]{x1D702}Re_{\unicode[STIX]{x1D70F}}}{A}\right)\right]^{2}\right\}^{1/2}+\frac{1}{2},\end{eqnarray}$$

where $\unicode[STIX]{x1D702}=y/h$, and the parameters $A=25.4$ and $K=0.426$ are tuned to match the profile of Hoyas & Jiménez (Reference Hoyas and Jiménez2006). The mean velocity profile is obtained by integrating $\left(1-\unicode[STIX]{x1D702}\right){u_{\unicode[STIX]{x1D70F}}}^{2}/\unicode[STIX]{x1D708}_{T}$. Using the same profiles, del Álamo & Jiménez (Reference del Álamo and Jiménez2006) found that the maximum energy amplification occurs for $\unicode[STIX]{x1D706}_{x}\gg \unicode[STIX]{x1D706}_{z}$, but Jiménez (Reference Jiménez2018) remarked that these wavenumbers are associated with the slow amplification of the streamwise-velocity streaks through (2.5), and with very long evolution times that may be limited by the nonlinear or viscous effects mentioned above. We are here interested in situations such as (2.6), where the energy amplification is associated with the transient bursting of $v$ over a few shear times. These were shown by Jiménez (Reference Jiménez2018) to be strongest at $\unicode[STIX]{x1D706}_{x}\approx \unicode[STIX]{x1D706}_{z}$.

Figure 1(a) shows as shaded contours several snapshots of the most amplified transient mode with $\unicode[STIX]{x1D706}_{x}=\unicode[STIX]{x1D706}_{z}\approx 0.7h$, plotted at intervals of $S(y_{g})\unicode[STIX]{x0394}t\approx 0.23$, where $S(y_{g})$ is the shear at the centre of gravity of the $v^{2}$ profile at maximum amplification. It was shown in Jiménez (Reference Jiménez2013) that the dimensions of the most amplified inviscid $v$-modes in the logarithmic layer are proportional to $\unicode[STIX]{x1D706}_{0}=2\unicode[STIX]{x03C0}/(k_{x}^{2}+k_{z}^{2})^{1/2}$, and that their centre of gravity is at $y_{g}\approx 0.22\unicode[STIX]{x1D706}_{0}$, or $y_{g}\approx 0.16\unicode[STIX]{x1D706}_{x}$ for equilateral wavevectors. The same is true in figure 1(a), even if this case differs from Jiménez (Reference Jiménez2013) in using an eddy viscosity. The temporal evolution of the perturbations in Jiménez (Reference Jiménez2013) is controlled by $\unicode[STIX]{x1D706}_{0}S(y_{g})t/\unicode[STIX]{x1D706}_{x}$. In the range of wavelengths $250\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}}\lesssim \unicode[STIX]{x1D706}_{x}\lesssim 4h$, corresponding to $40\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}}\lesssim y_{g}\lesssim 0.64h$, we can assume that $S(y_{g})\propto u_{\unicode[STIX]{x1D70F}}/y_{g}\propto u_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D706}_{0}$, and that the scaling of the time is $\unicode[STIX]{x1D706}_{0}S(y_{g})t/\unicode[STIX]{x1D706}_{x}\propto u_{\unicode[STIX]{x1D70F}}t/\unicode[STIX]{x1D706}_{x}$. The result is that both the dimensions and the temporal evolution of the amplified structures scale with $\unicode[STIX]{x1D706}_{x}$. The relation between the inclination angle $\unicode[STIX]{x1D6F9}$ and the amplitude of $v$ was shown in Jiménez (Reference Jiménez2013) to be fairly independent of the wavenumbers, and even of the type of flow. These results are extended here to the three velocity components, and to the eddy-viscosity equations, and we indeed find that the optimal modes in the neighbourhood of $\unicode[STIX]{x1D706}_{x}\approx \unicode[STIX]{x1D706}_{z}$ scale well with $\unicode[STIX]{x1D706}_{x}$ in the range of wavenumbers mentioned above. For example, the shape and inclination angles of the solution in figure 1(a) are remarkably similar to those in figure 19 of Jiménez (Reference Jiménez2018), where the wavelengths are much larger, $\unicode[STIX]{x1D706}_{x}=3.9h,\unicode[STIX]{x1D706}_{z}=5.55h$, and the solid contour lines superimposed on the second snapshot in figure 1(a), which show the moment of maximum amplitude of a mode with $\unicode[STIX]{x1D706}_{x}=\unicode[STIX]{x1D706}_{z}\approx 0.35h$, also agree very well with those at the longer wavelengths. The self-similar behaviour ceases to hold at heights corresponding to the buffer layer, where the shear is no longer inversely proportional to $y$.

Figure 1.

Evolution of the amplitude and inclination angles of an optimal transient growth mode in a channel flow ($\unicode[STIX]{x1D706}_{x}=\unicode[STIX]{x1D706}_{z}$). (a) Snapshots of the velocity components (top to bottom, $u$, $v$, $w$), spaced by $St\approx 0.52$, from left to right. The dashed lines show the inclination angle computed as in Jiménez (Reference Jiménez2015). The contour levels are $u=4,v=1,w=3$ in uniform arbitrary units. (b) Amplitude as a function of the inclination angle. The markers are: ●, $u$; ▪, $v$; ▴, $w$; and correspond to the snapshots in (a).

The dashed lines in figure 1(a) are the vertically averaged inclination angles at each time, computed using the formula for single modes in Jiménez (Reference Jiménez2015); for a more general one see (3.11) below. They represent well the instantaneous structure of $v$, but they are biased towards the lower half of the structures for $u$ and $w$ because the averaging in (3.11) is weighted with the energy, which tends to be concentrated near the wall for these two components. This is also where the shear is highest, leading to flatter inclinations. Figure 1(b) shows the amplitude–inclination relation for the three velocity components in figure 1(a). The amplitude and inclination angles of the wavetrain are defined as in (Jiménez Reference Jiménez2015). Both definitions are extended in (3.10)–(3.11), later in this paper, to more general wavepackets. The trajectory for $v$ is reminiscent of (2.6), but the wall-parallel components tilt forwards much faster, as shown by the larger separation of their first few symbols, which are uniformly spaced in time. Inspection of figure 1(a) shows that this is also due to the weighting of the angle close to the wall, where the higher shear results in faster characteristic times. The upper part of the $u$ and $w$ structures does not tilt much faster than that of $v$, while the root of the latter tilts as fast as those of the other two components. The last half of the angle–amplitude evolution is also different for the three velocity components. The spanwise velocity has an amplitude peak at approximately the same time as $v$, which is not present for $u$. At the end of the evolution, $v$ decays at a rate dictated by the shear, while $u$ and $w$ only decay under the effect of the eddy viscosity. For inviscid perturbations in a uniform shear, the three velocity components have well-differentiated behaviours. During the burst, $v$ grows and decays in $t=O(S^{-1})$, as in (2.6), but $u$ and $w$ never decay (Jiménez Reference Jiménez2013). In all cases, continuity requires that the two wall-parallel components have to be proportional to each other once $v$ has died. An important message of figure 1 is that optimum growth solutions should only be taken as indicative. The properties enumerated in the last few sentences are common to any initial condition, while optimum growth is meaningless in an inviscid flow. The behaviour of $v$, whose equation (2.4) is autonomous, is essentially independent of the initial conditions, which mostly define the origin of time, but $u$ and $w$ satisfy the forced equation (2.5), whose evolution depends on $v$ and on the initial conditions. The optimum transient growth criterion selects the initial condition with a highest local maximum, but the $v$ burst and the $v$-less final decay are robust properties of most initial conditions involving backwards-leaning perturbations. Note also that the energy growth in figure 1(b) is not particularly large, and that the same is true of all the cases mapped in figure 29 of Jiménez (Reference Jiménez2018).

4.1 Wall-parallel velocities

Figure 7.

Joint p.d.f.s of $A$ and $\unicode[STIX]{x1D6F9}$ involving $\overline{u}_{3}$ and $\overline{w}_{3}$ for S1000. All amplitudes are normalised with their modal values: (a) $\unicode[STIX]{x1D6F9}_{\overline{u}}$ and $A_{\overline{u}}/A_{\overline{u}M}$, contour levels are $90\,\%$, $50\,\%$ and $10\,\%$ of the probability mass; (b) $\unicode[STIX]{x1D6F9}_{w}$ and $A_{\overline{w}}/A_{\overline{w}M}$, contours as in (a). (c–f) Black contours are the joint p.d.f. of $A_{\overline{v}}/A_{\overline{v}M}$ and $\unicode[STIX]{x1D6F9}_{\overline{v}}$, contours containing $99.9\,\%$, $99\,\%$, $90\,\%$ and $50\,\%$ of the probability mass. Shaded contours are the mean of several quantities conditioned to the value of $A_{\overline{v}}$ and $\unicode[STIX]{x1D6F9}_{\overline{v}}$. Conditional average of (c) $A_{\overline{u}}/A_{\overline{u}M}$, (d) $A_{\overline{w}}/A_{\overline{w}M}$, (e) $\unicode[STIX]{x1D6F9}_{\overline{u}}$ and (f) $\unicode[STIX]{x1D6F9}_{\overline{w}}$.

We next compute the amplitude and inclination angles of the band-pass-filtered wall-parallel velocities. Figures 7(a) and 7(b) show the joint p.d.f.s of $(\unicode[STIX]{x1D6F9},\,A)$ for $\overline{u}$ and $\overline{w}$, respectively. Their structures are preferentially tilted towards the direction of the shear, in contrast to the wall-normal velocity structures, which are almost equally distributed between forward and backward inclinations. This is more pronounced for the spanwise velocity, which is rarely tilted backwards (10 % probability), and subtler for the streamwise component, which is tilted backwards 40 % of the time. As with the wall-normal velocity, there is a statistical correlation between the amplitude and inclination of the spanwise velocity, but the same is not true for the streamwise velocity, for which the two quantities are essentially unrelated. Scrambling $A_{\overline{u}}$ with respect to $\unicode[STIX]{x1D6F9}_{\overline{u}}$ leaves their joint p.d.f. unchanged (not shown). The behaviour of the inclination angles and amplitudes of $\overline{u}$ and $\overline{w}$ in the upper part of their joint p.d.f.s is qualitatively similar to the linearised trajectories in figure 1(b), although the inclination angles differ quantitatively. For example, the spanwise velocity shares with the transient-growth model an amplitude ‘peak’ towards positive inclinations, but the tilting angle of the linearised peak is always larger than that of the direct numerical simulation by 0.1–0.2 radians. Using unconditional two-point correlations of the wall-normal and spanwise velocity components, Jiménez (Reference Jiménez2018) argued that the cross-stream velocities contributing to the correlation are part of a quasi-streamwise ‘roller’. Some of the features of those autocorrelation functions are shared by the intense core of the joint p.d.f.s in figures 6 and 7. For example, the autocorrelation of the wall-normal velocity in the logarithmic layer is approximately vertical ($\unicode[STIX]{x1D6F9}_{\overline{v}}\approx 0$), and the spanwise velocity is tilted forward by $\unicode[STIX]{x1D6F9}_{\overline{w}}\approx \unicode[STIX]{x03C0}/6$, in agreement with the amplitude peaks of the joint p.d.f.s of $\overline{v}$ and $\overline{w}$. Because the correlation function is dominated by strong events, these similarities are not surprising, but they confirm that the band-pass filter retains some of the structure of the intense events of the velocity.

To explore the relation between the different variables during the bursting cycle, we compute averages conditioned to ($\unicode[STIX]{x1D6F9}_{\overline{v}},A_{\overline{v}}$). The conditional mean amplitudes of $\overline{u}$ and $\overline{w}$ are presented in figure 7(c,d) and their conditional inclinations are shown in figure 7(e,f). If we assume that the CMVs in figure 6(e) reflect the temporal evolution of individual strong bursts, we can describe the burst by the evolution of the conditional mean values of the different quantities as we move from left to right along the upper edge of the p.d.f. of ($\unicode[STIX]{x1D6F9}_{\overline{v}},A_{\overline{v}}$). The burst starts from intense $\overline{u}$, with weaker $\overline{v}$ and $\overline{w}$. At this stage $\overline{u}$ and $\overline{v}$ are tilted upstream, but $\overline{w}$ is too weak for its inclination to be defined. The ambient shear then tilts everything forward, and $\overline{v}$ and $\overline{w}$ are amplified while the amplitude of $\overline{u}$ decreases. Beyond the point where $\overline{v}$ is vertical, the amplitude of the wall-normal velocity decreases again, but $\overline{w}$ reaches its maximum value. The interpretation of these observations will be deferred to § 4.2, after we have examined the conditional temporal behaviour of individual bursts.

4.2 Conditional mean evolutions

While in the previous section we inferred a hypothetical burst evolution from the statistical relations among the different quantities along the upper edge of the joint p.d.f. of $(\unicode[STIX]{x1D6F9}_{\overline{v}},A_{\overline{v}})$, we now turn our attention to the direct study of their conditional temporal evolution by examining structures that pass through the upper ‘tip’ of the joint p.d.f.,

(4.3a,b)$$\begin{eqnarray}A_{\overline{v}}>2A_{\overline{v}M},\quad \left|\unicode[STIX]{x1D6F9}_{\overline{v}}\right|<0.15,\end{eqnarray}$$

at some stage of their lives. We know from figure 6(e) that the standard deviation of the CMVs passing through these very intense events is low, and that the statistics along the upper edge of the p.d.f. are approximately the same as those of linearised Orr bursts. This suggests that they represent the evolution of individual bursts, at least as long as the amplitude threshold is chosen high enough to separate bursts from each other. The restriction in (4.3) to a small inclination angle is not strictly necessary, since it follows from figure 6 that intense wall-normal velocity structures are almost always vertical, but including it allows us to relax somewhat the requirement for large amplitudes, and to identify bursts that would be too weak otherwise. The analysis is most easily performed in a convective frame of reference, $(\unicode[STIX]{x1D709}=x-C_{\!\unicode[STIX]{x1D6EC}}t,\,z,t)$, where $C_{\!\unicode[STIX]{x1D6EC}}$ is the convection velocity used in (4.1). We identify the position and time, $(\unicode[STIX]{x1D709}_{c},z_{c},t_{c})$, of the maximum amplitude for each region as representative of the ‘peaking’ point, $(\unicode[STIX]{x1D709}_{c},z_{c},t_{c})=\text{argmax }A_{\overline{v}}(\unicode[STIX]{x1D709},z,t)$, of the burst, and use it as the centre for our conditionally averaged evolution $\langle \cdot \rangle _{B}$, defined as

(4.4)$$\begin{eqnarray}\langle a\rangle _{B}(\unicode[STIX]{x1D6EF},Z,T)=\frac{1}{N}\mathop{\sum }_{c=1,\ldots ,N}a(\unicode[STIX]{x1D709}_{c}+\unicode[STIX]{x1D6EF},z_{c}+Z,t_{c}+T),\end{eqnarray}$$

where $N$ is the number of identified bursts for each filter (see table 3) and $a$ is either $A_{\overline{u},\overline{v},\overline{w}}$ or $\unicode[STIX]{x1D6F9}_{\overline{u},\overline{v},\overline{w}}$. To avoid ‘false positives’, we discard small bursts for which ${\mathcal{S}}<2.25\unicode[STIX]{x1D6EC}^{2}$, where ${\mathcal{S}}$ is the temporally averaged area enclosed in the $(x,z)$ plane by the threshold in (4.3). Independently of the chosen threshold, approximately 5 % of the identified bursts are temporally clustered in quick succession, reminiscent of the ‘packets’ of vortical structures reported by Adrian, Meinhart & Tomkins (Reference Adrian, Meinhart and Tomkins2000), but most cases cannot be easily grouped into such packets.

Table 3.

Number of bursts found in different band-pass-filtered time series. See table 1 for details of the simulations and § 4.2 for the details of the time–area normalisation.

Figure 8.

(a) Snapshot of the conditional bursts of $\overline{\boldsymbol{q}}_{2}$ at $S_{\!\unicode[STIX]{x1D6EC}}T=0$, the peaking time of $\overline{v}_{2}$. The contours, normalised with the maximum of each component, are: shaded, $A_{\overline{v}}=(0.6,0.8)$; solid, $A_{\overline{u}}=(0.8,0.9)$; dashed, $A_{\overline{w}}=(0.8,0.9)$. (b–e) Different aspects of the evolution of $A$ and $\unicode[STIX]{x1D6F9}$, conditioned to the presence of a burst, F2000. (b) Temporal evolution of the maximum amplitude, as a function of time: ——, conditional $\overline{u}_{2}$; - - - -, conditional $\overline{u}_{3}$. Symbols: ●, $\overline{u}$; ▪, $\overline{v}$; ▴, $\overline{w}$. (c) As (b), for the deviations of the inclination angle with respect to their mean, at the position of the maxima of each component at every instant. (d) Transient amplification of a linearised Orr burst. Conditions are as in figure 1(b), but the energies are computed over the range of $y$ corresponding to the $\unicode[STIX]{x1D6EC}_{2}$ filter. (e) The shaded contours are the joint p.d.f. of $(\unicode[STIX]{x1D6F9}_{\overline{v}},A_{\overline{v}})$ for $\unicode[STIX]{x1D6EC}_{2}$: —●—, evolution in (b,c), conditioned to (4.3); - –●- –, conditioned to the higher threshold (4.8); —▪— evolution in (f,g), conditioned to (4.9). The five markers are $S_{\!\unicode[STIX]{x1D6EC}}T=[-1.5,-0.3,0,0.3,1.5]$. (f,g) As in (b,c), but conditioned to a burst precursor.

Bursts defined in this way are spatio-temporal objects in $(\unicode[STIX]{x1D709},z,t)$. To normalise their density per unit duration–area, we assume that two bursts cannot occupy the same volume in space–time, and that they scale spatially with $\unicode[STIX]{x1D6EC}$ and temporally with the inverse of the average shear across their integration band (3.10):

(4.5)$$\begin{eqnarray}S_{\!\unicode[STIX]{x1D6EC}}=(U(y_{1}(\unicode[STIX]{x1D6EC}))-U(y_{0}(\unicode[STIX]{x1D6EC})))/(y_{1}(\unicode[STIX]{x1D6EC})-y_{0}(\unicode[STIX]{x1D6EC})).\end{eqnarray}$$

Their expected duration–size would then be proportional to $\unicode[STIX]{x1D6EC}^{2}/S_{\!\unicode[STIX]{x1D6EC}}$, and the number of detected bursts should collapse when their total duration–area is normalised in these units. Table 3 shows that this is approximately true, with a density close to $10\,\%$ except for the largest filter, which is too large for the scaling to hold. This density is of the same order as the volume fraction of the ‘Q’ structures studied in Lozano-Durán et al. (Reference Lozano-Durán, Flores and Jiménez2012).

Figure 8(a) shows a snapshot of the conditional burst of $\overline{\boldsymbol{q}}_{2}$ in F2000 at peaking time. Other filters collapse well with $\unicode[STIX]{x1D6EC}$, with average dimensions $(\ell _{\unicode[STIX]{x1D6EF}}/\unicode[STIX]{x1D6EC},\ell _{Z}/\unicode[STIX]{x1D6EC})\approx (2,1)$. The filtered spanwise velocity can be seen to be offset to one side of the streamwise and wall-normal components, while the latter fall on top of each other. If the average had been computed strictly as in (4.4), the conditional mean evolution would have two symmetric lobes of $A_{\overline{w}}$ centred on the burst, but this symmetry is statistical, and does not imply the symmetry of individual events. The equations of motion and boundary conditions of the channel are invariant to reflections with respect to $(x,y)$ planes, so that any solution $(u(x,y,z),v(x,y,z),w(x,y,z))$ implies that $(u(x,y,-z),v(x,y,-z),-w(x,y,-z))$ is also a solution. Individual Orr events are equally likely to be chiral-left or chiral-right, and the reflection symmetry can be applied to each snapshot to obtain an average that is more representative of individual events (Stretch Reference Stretch1990; Lozano-Durán et al. Reference Lozano-Durán, Flores and Jiménez2012). There is no unique way of deciding how to do this consistently for complete histories, and the criterion that we found to produce the least symmetric conditional mean was to always locate to the left of the burst the strongest maximum of $A_{\overline{w}}$ as it passes through the spatio-temporal neighbourhood of its peak:

(4.6a-c)$$\begin{eqnarray}|\unicode[STIX]{x1D6EF}|\leqslant 0.2\unicode[STIX]{x1D6EC},\quad |Z|\leqslant \unicode[STIX]{x1D6EC},\quad S_{\!\unicode[STIX]{x1D6EC}}|T|\leqslant 0.2.\end{eqnarray}$$

The conditional histories computed in this way only contain one region of strong $w$, as in figure 8(a), suggesting that most of the underlying events have only one strong spanwise velocity region, corresponding to a single streamwise roller. Double rollers are found more seldom. Visual inspection of a representative sample of cases shows that approximately 5 % of the bursts are almost symmetric, with two $w$ lobes.

Figure 8(b) shows the conditional evolution of the maximum amplitude of the three filtered velocity components as a function of time. Far from $S_{\!\unicode[STIX]{x1D6EC}}T=0$, the three components tend to their unconditional mean, which is used to normalise the plot. The wall-normal velocity, which is used to condition the temporal evolution, is amplified the most, but the streamwise and spanwise velocities are also amplified, peaking at $S_{\!\unicode[STIX]{x1D6EC}}T\approx \pm 0.3$, respectively before and after the peak of $\langle A_{\overline{v}}\rangle _{B}$, as also suggested by figure 7(d,e). The evolution of the deviation of the inclination angles from their unconditional means,

(4.7)$$\begin{eqnarray}\langle \unicode[STIX]{x1D713}_{\overline{a}}\rangle _{B}=\langle \unicode[STIX]{x1D6F9}_{\overline{a}}\rangle _{B}-\langle \unicode[STIX]{x1D6F9}_{\overline{a}}\rangle ,\end{eqnarray}$$

is presented in figure 8(c), which plots the inclination at the location of the maximum amplitude of each component. The evolution of the inclination angles of the streamwise and wall-normal velocity is from negative to positive, and tends to the unconditional mean beyond $|S_{\!\unicode[STIX]{x1D6EC}}T|\approx 1$. The changes in $\unicode[STIX]{x1D6F9}_{\overline{w}}$ are less pronounced, with values closer to its unconditional mean across the whole evolution.

A reference to figure 1 shows that this behaviour is qualitatively similar to linearised equilateral Orr bursts, but there are some interesting differences. The inclination angles of $\overline{u}$ and $\overline{v}$ are proportional to each other in the upper part of the p.d.f. of $(\unicode[STIX]{x1D6F9}_{\overline{v}},A_{\overline{v}})$, where the presumed quasi-linear cycle takes place, but the inclination angle of $\overline{w}$ is not. While $\unicode[STIX]{x1D6F9}_{\overline{u}}$ evolves from negative to positive in both figures 1 and 7(e), the inclination $\unicode[STIX]{x1D6F9}_{\overline{w}}$ is never negative in figures 7(f) and 8(c). Moreover the conditional amplification of $\overline{u}$, as given by the ratio between the highest and lowest conditional averages in figure 7(c), is almost three times higher than that for $\overline{w}$ in figure 7(d), while the opposite seems to be true in figure 1(b), and the maximum amplitudes of the two variables in figure 8(b) are approximately equal. The reason turns out to be the different spatial location of the various velocity components. We saw in figure 8(a) that, while the structures of $\overline{u}$ and $\overline{v}$ are spatially collocated in $(x,z)$, the $\overline{w}$-eddy is offset to one side, resulting in a weaker footprint in the conditional joint p.d.f.s. Even more important is the vertical offset. The amplitudes in figure 1(b) are integrated over the whole channel, but those in figure 7 are band-pass filtered. This includes the wall-parallel filtering by wavelength, which has no effect on the monochromatic wavetrain in figure 1, but also the restriction in (3.10)–(3.11) to a band of wall distances, which changes the balance among the different components as they drift vertically in and out of the filtering band. This effect is visually clear from figure 1(a), and figure 8(d) shows that it can be reproduced by restricting the amplitudes in figure 1(b) to the same band of wall distances as in figure 7. It was already mentioned when discussing figure 1 that the definition of the angles is biased by the different amplitude distribution of the three velocity components, and it is now clear that so is the definition of the band-pass amplitudes. Thus, while figures 7(c,d) and 8(b–d) might give the impression that the earlier part of the burst is dominated by the association of $u$ and $v$ (such as in a spanwise-oriented roller), while the later part is dominated by a streamwise roller (of $v$ and $w$), this interpretation could also be an artefact of how the flow is filtered. It should be remembered that linear models cannot change the wall-parallel wavevector of a single mode, nor its wall-parallel orientation, and changes in them have to be attributed to nonlinearity, or to the different amplification of different wavenumbers within a wavepacket, which probably would not apply to the relatively narrow wavebands here. This suggests that the interpretation of the phenomenon as a ‘rotating’ roller outlined above should be disregarded. Nevertheless, figures 8(d) and 1(b) are similar, even though the relative amplitudes of the velocity components are different in both figures due to the energy contained outside the $y$-band limits. Moreover, Jiménez (Reference Jiménez2013) showed that the cross-correlation of $v$ and $w$, integrated across the channel half-height, has a delayed peak when it is conditioned to a $v$-burst, albeit with slightly different delay between components. These evidences corroborate the causal relation between the burst and the increase in the magnitude of $w$. Other interpretations are also possible. For example, $A_{\overline{w}}$ could also be indicative of streak breakdown, which has been observed in buffer-layer streaks (Kline et al. Reference Kline, Reynolds, Schraub and Runstadler1967; Jiménez & Moin Reference Jiménez and Moin1991; Hamilton, Kim & Waleffe Reference Hamilton, Kim and Waleffe1995) or in transient growth modes in the transition of boundary layers in the presence of a streaky background (Brandt & Henningson Reference Brandt and Henningson2002).

Figure 8(b) includes results for two different filters, which scale well when normalised with the mean shear (4.5), and imply lifetimes of the order of $S_{\!\unicode[STIX]{x1D6EC}}T\approx 1$, measured at one-half the maximum amplification. Using slightly different definitions of mean shear and lifetime, Lozano-Durán & Jiménez (Reference Lozano-Durán and Jiménez2014) and Jiménez (Reference Jiménez2015) report the somewhat longer value $S_{\!\unicode[STIX]{x1D6EC}}T\approx 5$. In fact, the evolution of the inclination angle in figure 8(c) suggests that the conditional evolution only represents individual bursts when they are relatively near the conditioning time, $S_{\!\unicode[STIX]{x1D6EC}}|T|\lesssim 0.5$, beyond which both the angle and the amplitude tend to their unconditional values.

This is better seen in figure 8(f) which displays the $(\unicode[STIX]{x1D6F9}_{\overline{v}},A_{\overline{v}})$ trajectory of figure 8(b,c) on top of the joint $(\unicode[STIX]{x1D6F9}_{\overline{v}},A_{\overline{v}})$ p.d.f. The trajectory follows the direction of the CMVs near the conditioning point, but soon drops into the high-probability region at the core of the p.d.f. Note that the trajectory crosses the lowest possible point consistent with (4.3). Most eddies are in the core of the joint p.d.f., and any conditional mean is dominated by the points closest to the core. In a similar way, once the effect of the condition is lost, trajectories naturally fall towards the probability maximum. Thus, if condition (4.3) is substituted by the higher-amplitude one,

(4.8a,b)$$\begin{eqnarray}A_{\overline{v}}>3A_{\overline{v}M},\quad \left|\unicode[STIX]{x1D6F9}_{\overline{v}}\right|<0.15,\end{eqnarray}$$

the conditional trajectory changes to the qualitatively similar, but higher-intensity, trajectory plotted as dashed lines in figure 8(f).

Figure 9.

Wall-parallel plane of the streamwise velocity of F2000. The shading is $u^{+}$ at $y/h\approx 0.2$, ranging from $u^{+}\approx -5$ (blue) to $u^{+}\approx 5$ (red). The line contours are $A_{\overline{u}_{2}}$ for the same snapshot, representing 50 % and 75 % of the absolute maximum of $A_{\overline{u}_{2}}$ at the plane.

Perhaps the most interesting feature of the conditional evolution is the peak of $\overline{u}$ preceding the burst, also represented by the high values of the conditional amplitude of $\overline{u}$ over the left part of the joint p.d.f. of $(\unicode[STIX]{x1D6F9}_{\overline{v}},A_{\overline{v}})$ in figure 7(c,e). Note that a strong $A_{\overline{u}}$ should not be interpreted as a especially strong streak of the streamwise velocity, because the band-pass filter is designed to capture the energy of $v$, and the structures of $u$ are between six and ten times longer than those of $v$ (Jiménez Reference Jiménez2018). The band-pass-filtered amplitude of a uniform infinite streak is zero, and we should consider $A_{\overline{u}}$ as a measure of the inhomogeneity of the streak, e.g. due to break-up or to ‘meandering’. Streak inhomogeneities have often been identified as important features of the logarithmic layer cycle (Flores & Jiménez Reference Flores and Jiménez2010; de Giovanetti, Sung & Hwang Reference de Giovanetti, Sung and Hwang2017). The difference between $u$ and $A_{\overline{u}}$ is illustrated in figure 9, which shows both quantities in a wall-parallel plane of F2000. The streamwise velocity perturbations are shaded at $y/h\approx 0.2$, and the line contours are strong regions of $A_{\overline{u}_{2}}$. It is clear from the figure that both flow features are markedly different, and should not be confused.

The previous discussion shows that a burst of $\overline{v}$ is preceded by backwards-leaning perturbations of $\overline{u}$. The converse, that backwards-leaning perturbations of $\overline{u}$ act as precursors of the bursts of $\overline{v}$, is tested in figure 8(f,g) by the conditional evolution of bursts conditioned to

(4.9a,b)$$\begin{eqnarray}A_{\overline{u}}>1.8A_{\overline{u}M},\quad \unicode[STIX]{x1D6F9}_{\overline{u}}<-0.4,\end{eqnarray}$$

which is on the left edge of the joint $(\unicode[STIX]{x1D6F9}_{\overline{v}},A_{\overline{v}})$ p.d.f., but relatively far from its top. As shown in the figure, the conditional burst develops as predicted, with qualitatively similar temporal relations and delays among the different components as in figure 8(b,c). It is particularly striking that, even if figure 8(e,f) is conditioned on $\overline{u}$ rather than on $\overline{v}$, the amplification of the latter is even stronger than that of the former, suggesting that at least the left part of the joint p.d.f. represents well-organised bursts. A similar conclusion was reached by Jiménez (Reference Jiménez2015), who showed that bursts could be linearly ‘predicted’ from conditions up to about half their lifetime before the peak amplification of $v$. The mean trajectory in $(\unicode[STIX]{x1D6F9}_{\overline{v}},A_{\overline{v}})$ space of the evolutions initialised within (4.9) is plotted with squares in figure 8(e). Attempts to use the later peak of $A_{\overline{w}}$ to ‘postdict’ the burst of $\overline{v}$ were not successful. This may be interpreted as that strong, forward-leaning perturbations of $\overline{w}$ are not generated solely by Orr bursts. If this kind of $w$-perturbations can be generated by some other dynamics, denoted in this paragraph as ‘B’, it can only be assured that Orr bursts lead to forward-leaning $w$ perturbations, as well as do ‘B’ events; but identifying forward-leaning $w$ perturbations is not enough to determine if they come from Orr bursts, or from ‘B’ dynamics.

5.1 Space–time organisation of sweeps and ejections

It remains unclear from the previous discussion whether the secondary $v$ structure that appears to one side of the conditioning burst in figures 11 and 13(a,b) is an independent Orr burst, a true companion of the conditioning burst or another unrelated structure. However, it seems unlikely that $v$ ‘monopoles’ of any given sign, strong enough to be reflected in the conditional average, appear isolated in the flow, because continuity requires that the mean wall-normal velocity over patches of dimensions comparable to the distance from the wall should vanish. Our baseline hypothesis, motivated in part by the similarity of the conditional averages in figure 11 to the conditional Qs in Lozano-Durán et al. (Reference Lozano-Durán, Flores and Jiménez2012) and Lozano-Durán & Jiménez (Reference Lozano-Durán and Jiménez2014), is that bursts appear in side-by-side pairs of opposite sign, and that the secondary structure in figures 11 and 13 is just a poorly centred reflection of the opposite-signed partner of the burst being used to condition the evolution.

Figure 16.

Probability mass functions (p.m.f.s) of the number of partners of each structure. (a) Sketch of the definitions of pair, partner and number of partners (node degree). (b) The p.m.f. of the number of sweep partners for ejections (open symbols) and ejection partners for sweeps (filled symbols). (c) The p.m.f. of the number of ejection–ejection partners (open symbols) and sweep–sweep partners (filled symbols). (b,c) Symbols: ●, $\unicode[STIX]{x1D6EC}_{1}$; ▪, $\unicode[STIX]{x1D6EC}_{2}$; ▴, $\unicode[STIX]{x1D6EC}_{3}$; ▾, $\unicode[STIX]{x1D6EC}_{4}$. The solid line is the corresponding distribution for a Poisson process.

To verify this, we compute the distance between the centres of each detected sweep and its closest ejection, and vice versa, defined in the mean-convective frame of reference:

(5.5)$$\begin{eqnarray}L_{2}=\left[(\unicode[STIX]{x0394}tu_{\unicode[STIX]{x1D70F}})^{2}+(\unicode[STIX]{x0394}z)^{2}+(\unicode[STIX]{x0394}\unicode[STIX]{x1D709})^{2}\right]^{1/2}.\end{eqnarray}$$

Lozano-Durán et al. (Reference Lozano-Durán, Flores and Jiménez2012) and Dong et al. (Reference Dong, Lozano-Durán, Sekimoto and Jiménez2017) give similar statistics for unconditional Qs, although without the inclusion of the time difference. In their case, they require restrictions on the relative size of the eddies to avoid pairs with very different components, but this is made unnecessary in our case by only considering structures obtained with a particular filter. The definition of a pair is not straightforward, and figure 16(a) shows a two-dimensional sketch of possible complications. Each sweep, such as $s_{1}$, chooses as partner its closest ejection, such as $e_{1}$, and may in turn be chosen as partner by other ejections. Partnership is not symmetric. A given sweep may be chosen as partner by several ejections, such as $e_{2}$–$e_{4}$ in the case of $s_{3}$, and each of those may or may not be the ejection chosen by the sweep as its closest partner. Couples which choose each other are defined as pairs, such as ($e_{1}$, $s_{1}$) and ($s_{3}$, $e_{4}$) in figure 16(a), and this relation is mutual and unique. Each structure can be part of a pair, or of none at all, but being paired does not preclude having been chosen as a partner by other structures. Examples of unpaired structures are $e_{2}$, $e_{3}$ and $s_{2}$, which have not been chosen by their preferred partner, and a paired structure with a large number of partners (or node degree, in graph notation) is $s_{3}$. Approximately 60 % of the sweeps are in pairs, and 14 % of them are in exclusive pairs (see table 4).

There are no obvious statistical differences between the components of exclusive and non-exclusive pairs, but pairs are not random occurrences. The probability distribution of the degree of the sweeps and ejections is shown in figure 16(b), where it is compared with the expected density function for Poisson-distributed structures, represented by the solid line. It is clear that sweeps and ejections are clustered in the sense of having a higher number of partners of the opposite type than the Poisson model, obtained by randomising the position of the structures. On the other hand, figure 16(c) shows the distribution of the degree of structures of the same type, which is considerably steeper than that for the Poisson process. These two behaviours can be informally summarised as that bursts of opposite type ‘attract’ each other, while those of the same sign ‘repel’, and can probably be explained by the aforementioned effect of continuity, which requires that the net wall-normal mass flux over large areas should tend to vanish.

It is interesting that the probability of having $N>3$ partners of the opposite type in figure 16(b) follows quite closely a geometric distribution $P(N)\sim q^{N}$ with $q=0.4$, represented by the dashed line. This is intriguing, because it implies that the probability of being chosen as a partner is independent of how many partners a structure already has, probably implying that some structures have to choose a distant partner because they cannot find a closer one. This, in turn, suggests that bursts that are not part of a local pair are relatively isolated from prospective partners of the opposite sign. The same is not true for the same-sign partners in figure 16(c) which do not have an exponential tail or, if they do, it is one with a much smaller parameter than in the previous case $(q\approx 0.04)$. The default model proposed by Lozano-Durán et al. (Reference Lozano-Durán, Flores and Jiménez2012) from the analysis of the $uv$ Qs is that Qs tend to cluster into streamwise trains of tight pairs. The previous discussion suggests that the organisation into trains is better developed than the one in pairs, or at least that many pairs are asymmetric, with one of the partners too weak to be identified as such by the analysis.

Figure 17.

The p.d.f.s of the distance between components of (a–d) sweep–ejection pairs and (e–h) sweep–sweep pairs. (a,e) The p.d.f.s of the space–time distance $L_{2}$, defined in (5.5). The p.d.f.s of (b,f) $|\unicode[STIX]{x0394}tu_{\unicode[STIX]{x1D70F}}/y_{c}|$, (c,g) $|\unicode[STIX]{x0394}z/y_{c}|$ and (d,h) $|\unicode[STIX]{x0394}\unicode[STIX]{x1D709}/y_{c}|$. In all cases, lines are: ——, $\unicode[STIX]{x1D6EC}_{1}$; - - - -, $\unicode[STIX]{x1D6EC}_{2}$; — ⋅ —, $\unicode[STIX]{x1D6EC}_{3}$; $\cdots \cdots$, $\unicode[STIX]{x1D6EC}_{4}$; —●—, p.d.f. of the intra-pair distance for an equivalent Poisson process.

Figure 17 shows the probability distribution of the distance between the components of established pairs, compared with its Poisson counterpart. Figure 17(a–d) refers to sweep–ejection pairs and figure 17(e–h) to pairs of the same type. More than 90 % of the normalised distance between sweeps and ejections fall within $L_{2}/y_{c}\in (0.5,2)$, with its mode at $L_{2}/y_{c}\approx 1.1$. Its p.d.f. is significantly different from the Poisson distribution, shown in the figure as a line with filled circles. Figure 17(b–d) shows the contributions to $L_{2}$ of its different components. The main one is from the spanwise distance, which peaks at approximately $\unicode[STIX]{x0394}z/y_{c}\approx 1$, similar to the spanwise offset between the main and secondary eddies in figures 11 and 13. The streamwise component also plateaus for $\unicode[STIX]{x0394}\unicode[STIX]{x1D709}/y_{c}\lesssim 1$, but the temporal offset peaks for contemporaneous events $(\unicode[STIX]{x0394}t=0)$ and decays quickly. As with the total $L_{2}$ distance, the Poisson model fails for all components. Events tend to be spatially farther from each other than those for the Poisson model, but closer in the time of their peak development. The former is most probably due to the finite size of the objects being considered. Figure 11 shows that the $(x,z)$ diameter of the sweeps and ejections is approximately $y_{c}$, and it is not surprising that there are few pairs in figure 17 spatially closer than that distance. On the other hand, this restriction does not apply to time, and figure 17(b) shows that pairs preferentially evolve in synchrony, as in figure 13.

Figure 17(e–h) tells a slightly different story about homogeneous pairs. In the first place, although the average distances are approximately the same, the homogeneous pairs do not show the same probability deficit at short distances as in the heterogeneous case. Sweeps are closer to each other than their size would appear to allow. The reason can be understood from the individual p.d.f.s in figure 17(f–h). While the exclusion in dissimilar pairs takes place in space, similar pairs are mainly spaced in time. The p.d.f. for $\unicode[STIX]{x0394}z$ peaks at the origin, in agreement with the model of a train of aligned pairs, and $\unicode[STIX]{x0394}\unicode[STIX]{x1D709}$ is also only weakly constrained at short distances, but these events do not happen simultaneously, and figure 17(f) reveals a clear deficit for $\unicode[STIX]{x0394}tu_{\unicode[STIX]{x1D70F}}/y_{c}\lesssim 0.5$, or $S\unicode[STIX]{x0394}t\lesssim 2$. This is of the order of the lifetime of the burst, suggesting that, although consecutive pairs may align along streaks, they are created asynchronously, so that neighbouring sweeps are typically at different stages of their life. Note that this would be consistent with the observation in figure 14 that sweeps tend to form at the ‘active’ front of the streak, and suggests a model in which the trains of bursts propagate incrementally while they extend the streak length. Although it has to be emphasised that, as we have mentioned several times in this discussion, these bursts are only very seldom symmetric hairpins, one cannot avoid noting the similarities between this propagation model and the incremental formation of hairpin packets analysed by Zhou et al. (Reference Zhou, Adrian, Balachandar and Kendall1999) in the absence of turbulent background activity.