3.3.1 Skin-friction evolution

Figure 4 illustrates the temporal behaviours of the skin-friction coefficient $C_{f}=\bar{\unicode[STIX]{x1D70F}}_{w}/(1/2\unicode[STIX]{x1D70C}U_{c}^{2})$ normalized by initial values of each case using various $L_{s}$ values. Despite their distinct characteristics, the early-time behaviour of the skin friction of the P3 and P4 solutions is similar: (i) an initial stable period, (ii) a sharp increase, or a strong turbulent burst, following the stable period and (iii) transition to a fully turbulent flow (for P4, a flow is relaminarized beyond $L_{s}>0.0105$). This temporal behaviour indeed resembles a typical scenario observed in transition to turbulence (Schmid & Henningson Reference Schmid and Henningson2012). The strong turbulent burst, as it shall be called presently (Park et al. Reference Park, Shekar and Graham2018), is the process of escape out of the exact coherent solution along its most unstable manifold, comprising of the so-called linearly unstable stage followed by the nonlinear evolution stage (Itano & Toh Reference Itano and Toh2001). The end of this strong turbulent burst is then defined as the time when the skin friction reaches its maximum. It is evident that distinct transition scenarios are observed between the P3 and P4 solutions by the evolution of the skin friction due to inclusion of the slip surfaces.

Figure 4.

Early-time skin-friction coefficient profiles when using the (a) P3 and (b) P4 lower-branch travelling-wave solutions as initial conditions. Profiles are normalized by the initial skin-friction coefficient value for each case. The P3 and P4 lower-branch solutions are on the laminar–turbulent separatrix.

Notably, as shown in figure 4, the duration of the initial stable period decreases with slip length for the P3 solution, while the duration of this period increases with slip length for the P4 solution. It suggests that the P3 and P4 solutions experience early transition and delayed transition due to slip surfaces, respectively. For the P3 solution, the slip length appears to have a negligible effect on the reduction in maximum skin friction during the strong turbulent burst as the slip length is further increased. However, the behaviour of the strong turbulent burst for the P4 solution is quite different. The maximum skin friction reached during the strong turbulent burst following the initial stable period decreases with increasing slip length. This trend continues until a critical slip length ($L_{s}=0.0105$) after which there is no turbulent burst and the flow starts to laminarize after an initial stable period. If slip length is further increased, the initial stable period starts to reduce in duration and eventually the flow immediately laminarizes with negligible initial stable period for very large slip lengths. These observations for the P4 solution suggest that a slip surface appears to stabilize a flow and promote a return to the stable laminar state beyond the critical value $(L_{s}=0.0105)$. Thus, it is evident that the slip surfaces provide distinct effects on the laminar–turbulent separatrix of the P3 and P4 solutions.

To further characterize the temporal dynamics observed in the behaviours of the skin friction, figure 5(a,b) shows the maximum skin friction achieved during the turbulent bursting period and the duration of the initial stable period, respectively. Here, we define the bursting skin-friction coefficient as $C_{f,b}=C_{f,max}-C_{f,0}$, where $C_{f,max}$ is the maximum skin friction at the end of the bursting period and $C_{f,0}$ is skin friction at the initial time. The stable period, $T$, is defined as the duration for which skin friction continues to stay between $\pm 10\,\%$ of the initial skin-friction value depending on the direction of the trajectories: 10 % for bursting direction and $-10\,\%$ for laminarization direction. Both quantities are normalized by the associated values of the no-slip case. For the P3 solution, the magnitude of the turbulent burst remains relatively constant with slip length – the slip surface has a negligible effect on skin-friction reduction during the strong burst phase of the transition. The stable period slightly decreases with slip length and for a very substantial slip length of $L_{s}=0.02$, the stable period is reduced by approximately 10 % of the no-slip case. This observation indicates that slip surfaces cause the turbulent burst to occur sooner with almost same magnitude, promoting an early transition to turbulence. For the P4 solution, on the other hand, the magnitude of the turbulent burst decreases almost asymptotically with slip length up to a critical value of $L_{s}=0.0105$. After the critical slip length, there are no turbulent burst events because the flow becomes laminarized. Interestingly, the stable period of the P4 solution shows a non-trivial behaviour. It increases until reaching a critical value at $L_{s}=0.0105$ and starts to decrease because laminarization occurs sooner with slip length. Before the critical slip length, the slip length appears to play a role in delaying the transition to turbulence, as opposed to the P3 solution. It is worth noting that the dependence of the Reynolds number was tested by using different values of the Reynolds number in the transitional regime, where almost identical trends were produced for the P3 and P4 solutions (not shown).

Figure 5.

(a) Maximum skin-friction coefficient $C_{f}$ in turbulent burst and (b) duration of initial stable period. The duration of stable period is defined by the time for which $C_{f}$ is between $\pm 10\,\%$ of $C_{f,0}$ and normalized by the duration of the initial stable period for the no-slip case. Lower and upper error bars correspond to values when using $\pm 5\,\%$ and $\pm 15\,\%$ of $C_{f,0}$, respectively.

3.3.2 Linear growth rate in transition

We estimate the growth rate of the linearly stable stage along the turbulent burst trajectories for the P3 and P4 solutions by introducing a time-dependent variable $s(t)=\Vert \boldsymbol{u}(t)\Vert -\Vert \boldsymbol{u}(t_{0})\Vert$, where $\Vert \boldsymbol{u}(t)\Vert$ and $\Vert \boldsymbol{u}(t_{0})\Vert$ are the $L^{2}$-norm of the velocity field at time $t$ and initial time $t_{0}$, respectively. Given the definition of the turbulent burst as an escape from the exact coherent solutions consisting of linearly unstable and nonlinear unstable stages, the linear growth rate can be estimated by fitting the linear portion of a bursting trajectory observed in $s(t)$ to an exponential function $Ae^{\unicode[STIX]{x1D70E}t}$, where $A$ is a constant related to the magnitude of the unstable eigenvector and $\unicode[STIX]{x1D70E}$ is the associated growth rate (Gibson, Halcrow & Cvitanović Reference Gibson, Halcrow and Cvitanović2008).

Figure 6.

Growth rates for the P3 (a,b) and P4 (c,d) solutions for various slip lengths. The growth rate can be approximated for short time during which the system behaves like exp($\unicode[STIX]{x1D70E}t$), where $\unicode[STIX]{x1D70E}$ is the linear growth rate.

Figure 6 shows the evolution of $s(t)$ and the growth rate of the P3 and P4 solutions for various slip lengths. The growth rate for the P3 solution tends to increase linearly with slip length, which is not all surprising given the trend seen in the skin friction in figure 4(a). Conversely, the growth rate for the P4 solution decreases almost asymptotically with slip length. This same trend was observed in the turbulent burst value in figure 5(a). Again, the escape from the P3 solution occurs earlier with increasing slip length, while the escape is delayed for the P4 solution. It should also be noted that the values of the growth rate for the P3 solution are lower than those for the P4 solution as the P3 is believed to be closer to laminar state (Park & Graham Reference Park and Graham2015). In addition, since the escape process from an exact coherent solution follows closely its unstable manifold associated with the most unstable eigenvalue, those growth rate values are comparable to the most unstable eigenvalues of these solutions (Park & Graham Reference Park and Graham2015; Park et al. Reference Park, Shekar and Graham2018).

Figure 7.

The phase-space projection of turbulent trajectories onto the energy input rate ($I$) and energy dissipation rate ($D$) for various $L_{s}$ at (a) $Re_{c}=1760$ and (b) $Re_{c}=1800$. The black dashed line shows an equilibrium line ($D=I$). P3 lower (♢) and upper (♦) and P4 lower (◃) and upper (◂) solutions are also shown (Park & Graham Reference Park and Graham2015). The contour shows the probability density function (PDF) of the $I-D$ state for the no-slip case at (a) $Re_{c}=1760$ and (b) $Re_{c}=1800$. Inset: the mean states of long-time flows at each respective $Re_{c}$ for $L_{s}=0.00$ (▫), $L_{s}=0.01$ (▵) and $L_{s}=0.02$ (○) overlaid on the no-slip PDF.

3.3.3 Phase-space dynamics with slip surfaces

Here, we investigate the phase-space dynamics for the P3 and P4 solutions over slip surfaces by projecting the dynamics onto the $I$–$D$ space, where $I$ and $D$ are the energy input rate and energy dissipation rate, respectively. For the current study (Poiseuille flow), the energy input rate is given as

(3.2)$$\begin{eqnarray}I=\frac{1}{2L_{z}}\int _{0}^{L_{z}}\int _{-1}^{1}(pu|_{x=0}-pu|_{x=L_{x}})\,\text{d}y\,\text{d}z,\end{eqnarray}$$

and the energy dissipation rate is given as

(3.3)$$\begin{eqnarray}D=\frac{1}{2L_{x}L_{z}}\int _{0}^{L_{z}}\int _{-1}^{1}\int _{0}^{L_{x}}(|\unicode[STIX]{x1D735}u|^{2}+|\unicode[STIX]{x1D735}v|^{2}+|\unicode[STIX]{x1D735}w|^{2})\,\text{d}x\,\text{d}y\,\text{d}z.\end{eqnarray}$$

The total energy of the flow is defined by

(3.4)$$\begin{eqnarray}E=\frac{1}{2L_{x}L_{z}}\int _{0}^{L_{z}}\int _{-1}^{1}\int _{0}^{L_{x}}(u^{2}+v^{2}+w^{2})\,\text{d}x\,\text{d}y\,\text{d}z\end{eqnarray}$$

and, thus, the rate of change in energy for the flow is equal to $\text{d}E/\text{d}t=I-D$. For exact coherent solutions, $D=I$.

Figure 7(a,b) shows the $I$–$D$ phase diagrams of transition and turbulent trajectories starting from P3 and P4 lower-branch solutions, respectively, along with their corresponding upper-branch solutions. A joint probability density function (PDF) of turbulent trajectories for no-slip case is also plotted. In figure 7(a), all trajectories from the P3 lower-branch solution with different slip lengths exhibit a qualitatively similar trend. Starting from the lower-branch solution, they approach the upper-branch solution, make a turn around the equilibrium line ($D=I$), and move toward a core region of PDF. However, as slip length is increased, the trajectory tends to approach the equilibrium line earlier and then move toward the core region. Therefore, larger slip lengths allow a flow to approach a turbulent state faster, promoting an earlier transition to turbulence, as seen in figure 4(a). Figure 7(b) shows the trajectories starting from the P4 lower-branch solution, exhibiting similar trends to those seen in the P3 solution up to the critical slip length $L_{s}=0.0105$ – the trajectory gets closer to the equilibrium line as slip length is increased. Around the critical slip length, however, trajectories are altered drastically, bypassing the P4 upper-branch solution and approaching the P3 upper-branch solution before laminarizing. Beyond $L_{s}=0.0105$, the trajectories immediately approach the laminar state with no bursting trajectories, as seen in figure 4(b).

It is evident that slip surfaces lead to modifications of the turbulent trajectories in the $I$–$D$ phase space. The core region of the turbulent trajectories with slip surfaces deviates from that of the no-slip case – it gets closer to the lower-branch solutions. These observations suggest that the distance between the turbulent state and the laminar–turbulent separatrix is reduced with the inclusion of slip surfaces. This is illustrated by the insets in figure 7(a,b) showing the mean state of the system for $L_{s}=0.00,0.01$ and $0.02$. As slip length is increased, the mean state of the system shifts closer to the lower-branch solutions. This reduced distance indicates a greater likelihood for the flow to approach the separatrix and, thus, a greater likelihood that the flow will laminarize. Similar behaviour has also been observed by applying opposition control schemes to wall-bounded turbulent flows (Ibrahim et al. Reference Ibrahim, Yang, Doohan and Hwang2018).

3.3.4 Flow structures

We investigate the effects of slip surfaces on flow structures for which the swirling strength $\unicode[STIX]{x1D706}_{ci}$, the imaginary part of the complex conjugate eigenvalues of the velocity gradient tensor (Zhou et al. Reference Zhou, Adrian, Balachandar and Kendall1999), are calculated. Figures 8 and 9 show contours of the swirling strength for the lower half of the channel for the P3 and P4 solutions, respectively, at the end of the turbulent bursting trajectory. The contours represent isosurfaces of $50\,\%$ of the maximum swirling strength for each solution, which is also given in the figure. The wall-normal velocity is represented by the colour contours flooding the isosurfaces. These colour contours provide insights into the effect of slip surface on the vertical motion of these vortical structures.

Figure 8.

Vortical structures of the P3 lower-branch solution at $Re_{c}=1760$ for slip lengths (a) $L_{s}=0.00$, (b) $L_{s}=0.008$, (c) $L_{s}=0.01$ and (d) $L_{s}=0.02$. The maximum swirling strength for each case is presented in parentheses. The multicoloured tubes are isosurfaces of $1/2$ of maximum swirling strength. The contours flooding the isosurfaces represent the wall-normal velocity. Minimum and maximum values of wall-normal velocity represented by the contours correspond to $-0.1$ (blue) and 0.1 (red), respectively. The grey isosurface represents the critical layer, where local streamwise velocity is equal to the wave speed of the travelling-wave solution.

As seen in Park & Graham (Reference Park and Graham2015), the P3 lower-branch solution displays large vortex cores near the channel centre – this solution is called the core mode. As the bursting process proceeds, the large vortices appear to be broken into smaller ones, move toward the wall and spread across the span of the domain. Eventually, there is a large population of small vortex cores across the channel, as shown in figure 8. It appears that the general shape and position of the structures is mostly unchanged for all slip lengths studied, while the maximum swirling strength is slightly reduced with slip length. The majority of the wall-normal velocities associated with these vortex structures are close to zero at this instant, while a very strong negative wall-normal velocity is located around vortex structures during bursting trajectories.

Figure 9.

Vortical structures of the P4 lower-branch solution at $Re_{c}=1800$ for slip lengths (a) $L_{s}=0.00$, (b) $L_{s}=0.004$, (c) $L_{s}=0.008$ and (d) $L_{s}=0.0105$. The maximum swirling strength for each case is presented in parentheses. The multicoloured tubes are isosurfaces of $1/2$ of maximum swirling strength. The contours flooding the isosurfaces represent the wall-normal velocity. Minimum and maximum values of wall-normal velocity represented by the contours correspond to $-0.1$ and 0.1, respectively. The grey isosurface represents the critical layer, where local streamwise velocity is equal to the wave speed of the travelling-wave solution.

In figure 9, the vortical structures of the P4 solution are presented. As shown by Park & Graham (Reference Park and Graham2015), the P4 lower-branch solution displays vortices of $\unicode[STIX]{x1D6EC}$-like structures without a head connecting the legs. The vortex cores are located near the critical layer where the local streamwise velocity matches the wave speed – this solution is called critical-layer mode. Recently, this solution family was continued at higher Reynolds numbers, showing hairpin-like vortex structures (Shekar & Graham Reference Shekar and Graham2018). With slip surfaces, there still appear to be leg structures with slightly upward side branches, which is also observed in a trajectory along the most unstable manifold of the P4 lower-branch solution (Park et al. Reference Park, Shekar and Graham2018). Its structures are significantly longer than ones of the P3 solution and are accompanied by a small number of smaller vortex cores around these long vortex structures. Vortex cores of the P4 solution seem to not extend into the channel centre but remain mostly streamwise oriented. As slip length is increased, the maximum swirling strength is drastically reduced (i.e. $50\,\%$ at $L_{s}=0.0105$). In particular, from the colour contours of the wall-normal velocity associated with long vortex structures, the vortex structures are shifted upward from the wall as slip length is increased, leaving a quiescent region near the wall.

To quantitatively investigate the effect of the slip surfaces on the vertical shifting of vortical structures, the wall-normal location for the maximum value of the area-averaged swirling strength is plotted in figures 10(a) and 10(c) for P3 and P4 solutions, respectively. Bulk swirling strength is also shown for the P3 and P4 solutions in figures 10(b) and 10(d), respectively. The wall-normal location of the maximum of the average swirling strength for the P3 solution is almost constant for all slip lengths studied, as shown in figure 10(a). Figure 10(b) also shows that the effect of slip length on the bulk swirling strength is minimal, suggesting the slip surfaces have a minimal effect on the vortex structures associated with the P3 solution. However, the vortex structure dynamics for the P4 solution is quite different than that of the P3 solution. There are much larger deviations in height of maximum swirling strength as slip length is increased. Beyond $L_{s}=0.004$, the vortex structures are significantly shifted upward away from the wall, moving toward the channel centre. The bulk swirling strength is also significantly decreased as slip length increases as seen in figure 10(d). The trend appears to be asymptotic with slip length. Slip surfaces seem to have a profound weakening effect on the P4 vortex structures. These observations for the P4 solution are in good agreement with the experimental study (Zhang et al. Reference Zhang, Tian, Yao, Hao and Jiang2015), where the mechanisms for drag reduction over superhydrophobic surfaces were investigated for a turbulent boundary layer flow. They performed experiments to show that the drag reduction is caused by a combination of slip at the surface and modifications to the turbulent structures – vortices are weakened and lifted away from the surfaces. The combination of these modifications to the vortical structures (i.e. weakened and shifted vortices) of the P4 solution help to explain why, beyond the critical slip length, there is no turbulent burst and flow laminarizes. It appears that slip surfaces modify the turbulent structures such that the self-sustaining cycle of near-wall turbulence is disrupted and turbulence can no longer be sustained.

Figure 10.

(a,c) The wall-normal location corresponding to the maximum of the area-averaged swirling strength for the P3 and P4 solutions, respectively. The dashed line corresponds to the channel centre for the P3 and P4 solutions. (b,d) The bulk swirling strength as a function of slip length for the P3 and P4 solutions, respectively. All values are calculated at the time at which the maximum $L^{2}$-norm of velocity is reached.

3.3.5 Spatio-temporal dynamics

We now attempt to illuminate the mechanisms behind early or delayed transition to turbulence observed in the P3 and P4 solutions. It has been shown that at the onset of turbulence, the spatio-temporal dynamics provides a coherent mechanistic basis for the dynamics of transition to turbulence (Avila et al. Reference Avila, Moxey, de Lozar, Avila, Barkley and Hof2011; Barkley et al. Reference Barkley, Song, Mukund, Lemoult, Avila and Hof2015; Shih et al. Reference Shih, Hsieh and Goldenfeld2016) – puffs and slugs or directed percolation have been explored in this regard. Similarly, the spatio-temporal dynamics of the exact coherent solutions also appears to provide a coherent mechanistic basis for effects of slip surfaces on the transition observed in the P3 and P4 solutions.

For spatio-temporal dynamics, we incorporate a Fourier decomposition of the velocity field. Similar to Wang et al. (Reference Wang, Gibson and Waleffe2007), the velocity field of the travelling-wave solutions can be Fourier decomposed in the $x$ direction as

(3.5)$$\begin{eqnarray}\boldsymbol{u}(x,y,z,t)=\boldsymbol{u}_{0}(y,z,t)+\mathop{\sum }_{n=1}^{\infty }(\text{e}^{\text{i}n\unicode[STIX]{x1D703}}\boldsymbol{u}_{n}(y,z)+\text{c.c.}),\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}$ is the fundamental wavenumber in the streamwise direction, $\unicode[STIX]{x1D703}=\unicode[STIX]{x1D6FC}(x-ct)$, $c$ is the constant wave speed and c.c. denotes complex conjugates. For this study, we define the 0-mode as $\boldsymbol{u}_{0}(y,z,t)=(u_{0},v_{0},w_{0})$. The streamwise fluctuation $u^{\prime }$ can then be calculated by $u_{0}(y,z,t)-\overline{u}(y)$, where $\overline{u}(y)$ is the time- and area-averaged mean velocity. Note that the wall-normal velocity fluctuation $v^{\prime }(y,z,t)=v_{0}(y,z,t)$ because $\overline{v}(y)=0$. These 0-mode velocity fluctuations $u^{\prime }$ and $v^{\prime }$ provide information about streaky flow and rolls, respectively (Wang et al. Reference Wang, Gibson and Waleffe2007). Although $u^{\prime }$ and $v^{\prime }$ represent streamwise-averaged fluctuations, these quantities still enable identifying the sweep and ejection events based on the quadrant analysis (Picella et al. Reference Picella, Robinet and Cherubini2019).

Figure 11.

The streamwise-averaged streamwise velocity fluctuations $u^{\prime }$ at $z^{+}=L_{z}^{+}/4$ as a function of wall-normal distance for the P3 solution (a–c) and the P4 solution (d–f) for various slip lengths. The dashed white line corresponds to time of transition ($C_{f}=1.1C_{f,0}$). The symbols ○ and ▫ are indicated for the minimum and maximum $u^{\prime }$ at their wall-normal locations, respectively. For the P3 solution, ($u_{min}^{\prime },u_{max}^{\prime }$) $=$ ($-0.0981$, 0.0461) for (a), ($-0.1110$, 0.0569) for (b), ($-0.1223$, 0.0643) for (c). For the P4 solution, ($u_{min}^{\prime },u_{max}^{\prime }$) $=$ ($-0.0665$, 0.1259) for (d), ($-0.0248$, 0.0917) for (e), ($-0.0262$, 0.1008) for (f). Note that these values remain almost constant before the time of transition.

Figure 12.

The streamwise-averaged wall-normal velocity fluctuations $v^{\prime }$ at $z^{+}=L_{z}^{+}/4$ as a function of wall-normal distance for the P3 solution (a–c) and the P4 solution (d–f) for various slip lengths. The dashed white line corresponds to time of transition ($C_{f}=1.1C_{f,0}$). The symbols ○ and ▫ are indicated for the minimum and maximum $v^{\prime }$ at their wall-normal locations, respectively. For the P3 solution, (a) $v_{min}^{\prime }=-0.0107$, (b) $v_{min}^{\prime }=-0.0139$, (c) $v_{min}^{\prime }=-0.0158$. For the P4 solution, (d) $v_{max}^{\prime }=0.0181$, (e) $v_{max}^{\prime }=0.0131$, (f) $v_{max}^{\prime }=0.0147$. Note that these values remain almost constant before the time of transition.

Figures 11 and 12 show the spatio-temporal dynamics of streamwise and wall-normal velocity fluctuations at a spanwise location ($z^{+}=L_{z}^{+}/4$) for different slip lengths, respectively, along with the dashed line corresponding to time of transition. This spanwise location is chosen because it is the location where the core of vortex structures is located – the other location ($z^{+}=3L_{z}^{+}/4$) where the core of vortex structures is also located was studied, giving essentially identical results.

Both solutions exhibit similar behaviours on the streamwise fluctuations. For the P3 solution, as shown in figure 11(a–c), the near-wall low-speed streak (negative $u^{\prime }$) appears to be enhanced as slip length is increased ($u_{min}^{\prime }$ decreases from $-$0.0981 to $-$0.1223), while for the P4 solutions, the near-wall low-speed streak is slightly extended upward but remains relatively quiescent ($u_{min}^{\prime }\approx -0.025$ with slip surfaces) compared to the P3, as seen in figure 11(d–f). Near the channel centre, there are positive streamwise fluctuations for both solutions. In particular, a height of peak streamwise fluctuations for the P3 solution corresponds to the region where the core of the vortex structures is localized.

Most interestingly, the wall-normal fluctuations show the opposite behaviours. In figure 12(a–c), the wall-normal fluctuations of the P3 solution are all negative across the channel height and become more negative in the buffer layer with increasing slip length ($v_{min}^{\prime }$ decreases from $-0.0107$ to $-0.0158$). The combination of increased streamwise velocity near the channel centre and strong negative wall-normal velocity in the buffer layer seems to promote an instability to the vortex structures at the channel centre. Specifically, this instability causes the large vortex structures to propagate toward the wall much sooner where they break up as the slip length is increased. These effects seem to lead to an early transition for the P3 solution. On the other hand, for the P4 solution, figure 12(d–f) show all positive wall-normal fluctuations across the channel height and peak values are located around $y^{+}=20{-}40$, where the core of the vortex structures is localized. As slip length increases, the peak value decreases as opposed to the P3 solution (from $v_{max}^{\prime }=0.0181$ to 0.0147). The combination of the very quiescent flow and positive wall-normal velocity around the vortex structures appears to allow the the vortex structures to be more stable and persist for a longer time before transition to turbulence occurs. Those effects appear to delay the transition. It is interesting to note that based on the quadrant analysis, slip surfaces are likely to promote stronger wall-toward motions (Q4-like) at the channel centre and inward interactions (Q3) in the buffer layer for the P3 solution, while for the P4 solution, the surfaces tend to promote longer ejection events (Q2) near the critical layer and wall-away motion (Q1-like) at the channel centre. This quadrant analysis may help to elucidate distinct transition dynamics for the P3 and P4 solutions – the combination of the core mode and strong sweep events seems to promote early transition, while the combination of the critical-layer mode and extended ejection events seems to delay the transition.

Finally, we can make the link between transition dynamics and flow structures, as the latter can be thought of as different disturbances in a receptivity process of transition (Kachanov Reference Kachanov1994). Flow structures associated with the core mode (P3 solution) seem to resemble those found in free-stream turbulence (i.e. more densely located at the centre of the channel or the edge of the boundary layer) as seen in figure 8 – they can be considered external perturbations. On the other hand, the P4 solution (critical-layer mode) exhibits $\unicode[STIX]{x1D6EC}$-shaped vortical structures, confined near the wall, resembling three-dimensional flow structures developed from Tollmien–Schlichting waves – they can be considered internal perturbations. Depending on different disturbances, transition scenarios appear to be very different. Sayadi, Hamman & Moin (Reference Sayadi, Hamman and Moin2013) compared the dynamics of different transition scenarios, namely, H-type, K-type and bypass transition. In the H- and K-type transitions, characterized by $\unicode[STIX]{x1D6EC}$-shaped vortical structures, a large overshoot in the skin-friction coefficient was observed when transition starts to occur. However, for the bypass transition, this overshoot was absent, giving a smooth transition to turbulence. This distinct overshoot is present in the skin-friction coefficient profile for the P4 solution and absent in that for the P3 solution as shown in figure 4. It is worth noting that, similar to the P3 solution, Wu & Moin (Reference Wu and Moin2009) introduced free-stream turbulence to a boundary layer and showed that the bypass transition was triggered. They showed that the typical near-wall Tollmien–Schlichting waves were not the initiating mechanism for transition and in this case was attributed to a bypass route triggered from the free stream. The evolution of the P3 solution is similar in that the structures originate in the centre of the channel (i.e. free stream) and propagate downward toward the near-wall region before breaking down, subsequently triggering transition. In addition, the shape of the vortical structures for the P4 solution are similar to those found in H- and K-type transitions. A similar observation for the effects of slip surfaces on the H- and K-type transitions was also made by Picella et al. (Reference Picella, Robinet and Cherubini2019), where the overshoot in skin friction was present and transition was delayed by the slip surface with the modal perturbation. However, they also showed that the non-modal perturbation, similar to those perturbations that cause bypass transition, was unaffected by the slip surfaces, contrary to the early transition behaviour observed in the P3 solution of the current study. Differences in the transition behaviour of the non-modal perturbation of Picella et al. (Reference Picella, Robinet and Cherubini2019) and the P3 solution of the current study could stem from assumptions made in the linearization of the Navier–Stokes equations or, possibly, differences between the non-modal perturbation and the P3 solution. The differences in the skin-friction coefficient and vortical structures between the P3 and P4 solutions may provide clear and plausible mechanisms responsible for early transition for P3 and delayed transition for P4 due to slip surfaces, for which further investigation is yet needed.