2 Formulation

We consider incompressible Newtonian flow in the plane Poiseuille (channel) geometry, driven with a constant volumetric flux. The $x,y$ and $z$ coordinates are aligned with the streamwise, wall-normal and spanwise directions, respectively. Periodic boundary conditions are imposed in the $x$ and $z$ directions with fundamental periods $L_{x}$ and $L_{z}$ , and no-slip conditions are imposed at the walls $y=\pm h$ , where $h=L_{y}/2$ is the half-channel height. Using the half-height $h$ of the channel and the laminar centreline velocity $U_{c}$ as the characteristic length and velocity scales, respectively, the non-dimensionalized Navier–Stokes equations are then given as

(2.1) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\boldsymbol{u}}{\unicode[STIX]{x2202}t}+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}=-\unicode[STIX]{x1D735}p+\frac{1}{Re}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u},\quad \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0.\end{eqnarray}$$

Here, we define the laminar equivalent Reynolds number for the given volumetric flux as $Re_{c}=U_{c}h/\unicode[STIX]{x1D708}$ , where $\unicode[STIX]{x1D708}$ is the kinematic viscosity of the fluid. We fix the bulk velocity (volumetric flux) $U_{b}$ at the laminar value $2U_{c}/3$ , so the Reynolds number $Re_{b}$ based on bulk velocity is given by $Re_{b}=2Re/3$ . Characteristic inner scales are the friction velocity $u_{\unicode[STIX]{x1D70F}}=(\bar{\unicode[STIX]{x1D70F}}_{w}/\unicode[STIX]{x1D70C})^{1/2}$ and the near-wall length scale or wall unit $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}=\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}}$ , where $\unicode[STIX]{x1D70C}$ is the fluid density and $\bar{\unicode[STIX]{x1D70F}}_{w}$ is the time- and area-averaged wall shear stress. As usual, quantities non-dimensionalized by these inner scales are denoted with a superscript ‘ $+$ ’. The friction Reynolds number is then defined as $Re_{\unicode[STIX]{x1D70F}}=u_{\unicode[STIX]{x1D70F}}h/\unicode[STIX]{x1D708}=h/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$ .

Computations are performed using the open source code ChannelFlow, which performs direct simulations as well as computing exact coherent states by Newton–Raphson iteration (Gibson Reference Gibson2012). In this study, we focus on the domain $L_{x}\times L_{y}\times L_{z}=\unicode[STIX]{x03C0}\times 2\times \unicode[STIX]{x03C0}/2$ , the same box size as used in our prior ECS study (Park & Graham Reference Park and Graham2015). Fourier–Chebyshev–Fourier spectral spatial discretization is applied to all variables, using meshes with $N_{x}\times N_{y}\times N_{z}$ collocation points (in $x$ , $y$ , and $z$ ). A typical resolution used is $(N_{x},N_{y},N_{z})=(48,81,48)$ , which we have verified is sufficient for converged solutions in the Reynolds number range examined here.

The exact coherent states computed and studied here all satisfy reflection symmetry across $z=L_{z}/2$ , as given by (1.1), as well as across the centreplane of the channel $y=0$ , as given by (1.3). The solutions of central interest here additionally satisfy the $xz$ -shift symmetry given by (1.4). There are of course other channel flow ECS that do not obey these symmetries (e.g. Waleffe (Reference Waleffe2001, Reference Waleffe2003), Nagata & Deguchi (Reference Nagata and Deguchi2013), Gibson & Brand (Reference Gibson and Brand2014), Neelavara et al. (Reference Neelavara, Duguet and Lusseyran2017)). All turbulent trajectories reported here are computed without imposing any symmetries on the flow field.

3 Results and discussion

Figure 2 is the bifurcation diagram for the ECS solutions that we have computed in this study, shown in terms of dissipation rate $D$ and streamwise wave speed $c$ versus $Re$ . Two solution families are shown. The one of primary interest is shown in purple; it arises in a saddle-node bifurcation at $Re=666$ (point a in figure 2). This solution family satisfies both the $z$ -reflection symmetry (1.1) and the $xz$ -shift symmetry (1.4) and will be denoted S1. ECS with this set of symmetries have been reported in channel flow by Wall & Nagata (Reference Wall and Nagata2016) (their MS-S, MS-A and TW2 families), but in general have received little attention in the literature. The vortex structure of this state at the bifurcation point is illustrated in figure 3(a), which shows a surface of vortex strength, $Q=(\Vert \unicode[STIX]{x1D734}\Vert ^{2}-\Vert \unicode[STIX]{x1D71E}\Vert ^{2})/2$ in the bottom half of the channel ( $y<0$ ), where $\unicode[STIX]{x1D734}$ and $\unicode[STIX]{x1D71E}$ are the local vorticity and strain rate, respectively and $\Vert \cdot \Vert$ denotes Frobenius norm. This structure consists of two pairs of staggered counter-rotating streamwise vortices that are mirror symmetric across $z=L_{z}/2$ . By (1.3), a $y-$ reflection-symmetric vortex structure arises in the top half of the channel ( $y>0$ ). Figure 3(b) shows the streamwise velocity at $y^{+}=20$ , clearly illustrating the high degree of symmetry of the flow field and the ‘sinucose’ nature of the flow pattern – there is a $z$ -reflection-symmetric pair of sinuous low-speed streaks (dark). Observe that the low-speed streaks (dark) ‘pinch’ together at $x\approx 0.75$ . As mentioned above, Duguet et al. (Reference Duguet, Schlatter, Henningson and Eckhardt2012) observe in their study of edge states in boundary layer transition the formation of transient hairpins above pinches. We return to this point below.

Figure 2.

Bifurcation diagram in terms of (a) dissipation rate $D$ versus  $Re$ and (b) wave speed $c$ versus  $Re$ . Inset in (a) is a zoom in of the box close to point d.

Figure 3.

(a) Vortical structure of the S1 solution at the saddle-node bifurcation at $Re=666$ (point a in figure 2). Isosurface of $Q=0.05$ is shown with colours indicating distance from wall – blue near wall, red near centre. A colour contour plot of streamwise velocity at $x=0.75$ is also shown. (b) Streamwise velocity $u$ at $y^{+}=20$ .

On the upper (high dissipation) branch of S1, which we denote S1UB, a pitchfork bifurcation occurs at $Re=855$ (point b in figure 2). The bifurcating solutions lack the $xz$ -shift symmetry, as shown by the streak structure for a solution on this branch at $Re=935$ on figure 4(a) (point c on the bifurcation diagram). This solution family is the P4 family reported by Park & Graham (Reference Park and Graham2015); indeed the way we found S1 was by continuation of P4. As $Re$ increases along the P4 solution branch, the two mirror-symmetric low-speed streaks merge into one; the structure evolves from sinucose to varicose. Figure 4(b) shows this varicose structure at point d on the bifurcation diagram. Upon continuation to higher $Re$ , P4 turns back on itself several times in saddle-node bifurcations resulting in a rather tangled bifurcation diagram.

Figure 4.

Streamwise velocity at $y^{+}=20$ on P4: (a) near the pitchfork at $Re=935$ (point c in figure 2), (b) at $Re=1800$ (point d).

Figure 5.

(a) Structure on S1UB at $Re=1800$ (point e). Isosurface of $Q=0.072$ with slice of $u$ at $x=2$ is shown. The maximum value of $Q$ on the $z$ midplane is $Q=0.096$ . (b) Streamwise velocity at $y^{+}=20$ . (c) Streamwise velocity $u$ at position shown in (a) at $x=2$ with arrows of in-plane velocity vectors.

Figure 6.

Contour of $z$ -vorticity on the $z$ -midplane with arrows indicating the velocity field in a frame moving at (a) $v_{x}=0.76$ for S1UB at $Re=1800$ and (b) $v_{x}=0.74$ for S1 at the saddle-node bifurcation point $Re=666$ . In these plots the region between $y=-0.6$ and $y=0$ (the centreplane) is shown.

Now we return our attention to S1. The upper branch (higher dissipation) solution displays an initial sharp increase in $D$ with distance from the bifurcation point and the structure that emerges on this branch is very interesting and important. Recall the vortical structure of S1 at its inception, as shown in figure 3(a). This is a counter-rotating streamwise vortex structure, as expected for near-wall turbulence. By contrast, figure 5(a) shows the vortical structure of S1UB at $Re=1800$ (point e on the bifurcation diagram). By this point, the structure has evolved to become a self-sustaining combination of streamwise vortex structure (blue and green) near the wall and hairpin-like heads (red) near the centreplane. It is a microcosm of near-wall turbulence, where streamwise vortices dominate the structure close to the wall, with hairpins becoming dominant further away (Wu & Moin Reference Wu and Moin2009). Here the heads seem to be pulling ahead of the legs a bit, perhaps because of forward motion induced by the presence of a mirror-image hairpin head on the other side of the centreplane $y=0$ . Consistent with the observations of Duguet et al. (Reference Duguet, Schlatter, Henningson and Eckhardt2012), the hairpins appear above pinches in the streak structure, the most apparent of which is at $x\approx 2.5$ . The pinch here is less pronounced than that observed in figure 3(b). By symmetry, there are also pinches at $x\approx 1$ between the low-speed streaks in the domain shown and those in the spanwise neighbouring domains. Correspondingly, there are hairpin-like structures above these pinches. Half of each of these can be seen in figure 5(a). Overall, this ECS displays a staggered array of these structures.

From figure 5(b), we can estimate the width of the hairpin-like structure as the spacing between the streaks in the pinch region, which is approximately $1$ in outer units. Similarly, from figure 6(a) we can estimate the distance of the vortex head from the wall in the same units as $0.9$ . At $Re=1800$ , the friction Reynolds number $Re_{\unicode[STIX]{x1D70F}}\approx 95$ . As this quantity gives the scaling between outer and inner units, we see that the vortex structure here has spanwise and wall-normal dimensions of approximately $95$ and $86$ wall units, respectively. For comparison, Adrian (Reference Adrian2007) describes a ‘mature’ hairpin vortex packet as having similar size, with both width and height of approximately 100 wall units.

Further indication of the hairpin-like structure displayed by this ECS is illustrated in figure 6(a), which shows the velocity field and $z$ -vorticity on the symmetry plane $z=L_{z}/2$ in a reference frame moving with the transverse vortex core. Toward the upper right of the plot the localized vortical motion corresponding to the hairpin head is clearly seen. It should be emphasized that this motion occurs just below the centreplane of the channel, where the mean shear and thus the mean $z$ -vorticity vanish. This is in sharp contrast to the putative hairpin states described by Itano & Generalis (Reference Itano and Generalis2009); as noted above, those structures were only identified as hairpins based on the shape of the vortex lines, and additionally the hairpin shape of these lines was observed near the boundaries where there is a strong mean shear. In contrast, at $Re=1800$ , S1UB displays a clear localized region of cross-stream vorticity in a nearly shear-free mean flow that is connected to streamwise vortex motions closer to the wall. To our knowledge this is the first observation of a persistent hairpin-like vortex structure in an exact coherent state. Finally, figure 6(b) shows the same quantities plotted for the structure at $Re=666$ , i.e. right at the saddle-node bifurcation point. Consistent with the $Q$ structure shown in figure 3(a), no localization of $z$ -vorticity is seen.

Completing our presentation of the structure of S1UB at $Re=1800$ , figures 7(a) and 7(b) show the sense of rotation of the legs and head of the hairpin-like structure, respectively. Additionally, a strong Q2 ejection event ( $u^{\prime }<0,v^{\prime }>0$ ) can be seen behind and below the vortex head. For comparison, figure 7(c) shows again the Theodorsen picture; arrows have been added to all of these panels to indicate that S1UB does indeed display the same qualitative vortex structure as the original Theodorsen hairpin.

In addition to the vortex head and leg structure and the Q2 event behind the head just described, Adrian (Reference Adrian2007) asserts, based on conditional averaging of the structure associated with Q2 events, that there are two additional ingredients to the signature of a hairpin vortex, a stagnation point separating Q2 and Q4 ( $u^{\prime }>0,v^{\prime }<0$ ) regions, corresponding to an inclined shear layer behind the hairpin head, and a low-speed streak well behind the vortex. These features are absent from the S1 ECS. Regarding the inclined shear layer and Q2/Q4 stagnation point, this is probably precluded by the reflection symmetry of the channel geometry and the proximity of the vortex heads to the centreplane. Regarding the trailing low-speed streak, this is precluded by the symmetries (1.1) and (1.4), which yield a staggered arrangement of hairpin-like vortices. It should also be emphasized that the ECS structures described here are not conditionally averaged – it is not obvious that instantaneous hairpin-like structures in flow will display all the features of a conditionally averaged one. Nevertheless, it would be interesting to learn whether ECS exist that have aligned rather than staggered hairpin-like structures –these are likely to bear more similarity to the hairpins described by Adrian than is found here.

Nevertheless, as noted above, the scale of the S1 ECS vortex structure is similar to that reported by Adrian (Reference Adrian2007). Furthermore, it is consistent with observations of hairpins in the transition context as exemplified by the observations of Duguet et al. (Reference Duguet, Schlatter, Henningson and Eckhardt2012), where hairpins arise above the pinch region between two low-speed streaks. Perhaps the S1 ECS is more related and relevant to hairpins observed in transition than to those observed in fully turbulent boundary layers.

Figure 7.

(a) In-plane velocity fluctuations on the legs of the vortex structure at $x=1.82$ of S1UB, $Re=1800$ , (b) Out of plane velocity fluctuations on the head of the vortex structure at $x=2.43$ of S1UB, $Re=1800$ and (c) Theodorsen’s picture of a hairpin vortex.

Figure 8.

(a) Vortical structure of S1LB at $Re=3000$ (point i). Isosurface of $Q=0.014$ is shown. (b) Streamwise velocity $u$ at $y^{+}=20$ .

Figure 9.

Contour of $z$ -vorticity on the $z$ -midplane with arrows indicating the velocity field in a frame moving at $v_{x}=0.775$ for S1LB at $Re=3000$ . In this plot the region between $y=-0.6$ and $y=0$ (the centreplane) is shown.

Figure 10.

Mean profiles for ECS and turbulent flow (direct numerical simulation) at $Re=1800$ : (a) mean velocity in inner units, (b) Reynolds shear stress in inner units, (c) mean velocity in outer units and (d) Reynolds shear stress in outer units. For the ECS, the parenthesized letter in the legend indicates the corresponding point in figure 2.

In prior observations, hairpin vortices have been observed as transient, spatio-temporally evolving structures. In contrast, the hairpin-like ECS structure observed in figure 5 is, in a travelling reference frame, an equilibrium structure. We believe that the reason it was possible to find such a structure is the channel geometry and the corresponding centreplane symmetry; this keeps the hairpin from continually drifting away from the wall as occurs in a boundary layer – the channel geometry has allowed us to ‘trap’ the hairpin structure, confining it to a finite distance from the wall and allowing it to propagate as a traveling wave. An interesting topic of future work would be to seek ECS with streamwise aligned, rather than staggered, hairpin-like vortices.

The above discussion has focused on the evolution of S1 from the bifurcation point in the direction of increasing dissipation. For completeness, we show in figures 8 and 9 the structure that evolves along the lower branch solution S1LB. Characteristic of lower branch solutions, the overall vortex and streak structure becomes much weaker with distance from the saddle-node bifurcation point, and while a $z$ -oriented vortex structure also emerges on this branch, it is very weak, and not well defined until higher $Re$ than on the upper branch. This is why we have illustrated the structure at $Re=3000$ rather than the value of $1800$ that we used for the upper branch. Observe that that the $Q$ value chosen for figure 8(a) is 0.014, much smaller than the value 0.072 used to visualize the upper branch (figure 5 a) and the waviness of the streaks shown in figure 8(b) is very weak – they are very nearly streamwise invariant.

An important issue is the relationship between ECS and turbulent dynamics. Figure 10 shows profiles of area-averaged mean velocity and Reynolds shear stress at $Re=1800$ for the states indicated with red dots on figure 2; these include solutions from both the P4 and S1 families. Also shown are profiles from turbulent trajectories obtained via direct numerical simulation (DNS) for the same domain and Reynolds number. Both S1UB and P4UB1 have mean velocity profiles that are fairly close to that of the turbulence, though the latter has inflectional kinks that are absent in S1UB. In inner units, the Reynolds shear stress for S1UB is in quantitative agreement with the turbulent result near the wall and yields very nearly the same peak position $y^{+}\approx 25$ . For $y^{+}\gtrsim 20$ , S1UB has slightly higher Reynolds shear stress than the turbulence. In outer units, the quantitative resemblance is not as strong, but is still reasonably good. In any case, S1UB more closely resembles the turbulence than do any of the other ECS reported here.

Another illustration of the relationship between ECS and turbulence is given in figure 11, which shows projections of ECS and trajectories onto selected variables: area-averaged wall shear rate, dissipation rate $D$ and disturbance kinetic energy $KE=(\Vert \boldsymbol{u}-\boldsymbol{u}_{laminar}\Vert ^{2})/2$ . In this projection, all of the ECS plotted appear to be embedded in the same region of state space as the turbulent attractor. Also shown in yellow in figure 11 is a trajectory starting from S1LB plus a random perturbation. It starts to approach P4LB2 before bursting towards the turbulent core. This observation points at a possible heteroclinic connection between the S1LB and P4LB2 and at a role that hairpins may play in the transition to turbulence.

Finally, while hairpin structures are not prominent in DNS under the conditions considered here, they do arise occasionally, as seen in the simulation snapshot shown in figure 11(c).

Figure 11.

(a) State-space visualization of DNS trajectories at $Re=1800$ , projected onto three dimensions: disturbance kinetic energy(KE), energy dissipation rate(D) and wall shear rate. The labelled symbols are computed ECS and the black lines are the DNS trajectories at $Re=1800$ . (b) 2D projection of (a). Also shown is a trajectory starting from S1LB (yellow). (c) Isosurface of $Q=0.072$ at an instant during DNS at $Re=1800$ . Shown is the top half of the domain, rotated to be consistent with other figures. For clarity, two periods in $z$ are shown.