# Exact coherent states with hairpin-like vortex structure in channel flow

Ashwin Shekar1  Michael D. Graham12<EMAIL_ADDRESS>1Department of Chemical
and Biological Engineering, University of Wisconsin-Madison, Madison, WI 53706
USA 2Kavli Institute for Theoretical Physics, University of California, Santa
Barbara, CA 93106 USA

###### Abstract

Hairpin vortices are widely studied as an important structural aspect of wall
turbulence. The present work describes, for the first time, nonlinear
traveling wave solutions to the Navier–Stokes equations in the channel flow
geometry – exact coherent states (ECS) – that display hairpin-like vortex
structure. This solution family comes into existence at a saddle-node
bifurcation at Reynolds number $\Rey=666$. At the bifurcation, the solution
has a highly symmetric quasistreamwise vortex structure similar to that
reported for previously studied ECS. With increasing distance from the
bifurcation, however, both the upper and lower branch solutions develop a
vortical structure characteristic of hairpins: a spanwise-oriented “head” near
the channel centerplane where the mean shear vanishes connected to counter-
rotating quasistreamwise “legs” that extend toward the channel wall. At
$\Rey=1800$, the upper branch solution has mean and Reynolds shear-stress
profiles that closely resemble those of turbulent mean profiles in the same
domain.

##  1 Introduction

Figure 1: (a) Theodorsen’s conceptualized picture of a hairpin 28. (b)
Isosurfaces of $\lambda_{2}$-criterion showing hairpin-like structures during
transition in boundary layers10. (c) Isosurfaces of vortex strength $Q$
illustrating hairpin vortices in direct simulation results for a turbulent
boundary layer39.

Ever since the observations of 28 (Figure 1), the role of _hairpin_ or
_horseshoe_ vortices in turbulent shear flows has been an important topic in
the study of turbulent structure 24, 1, 27. In a wall-bounded flow, hairpin
vortices are concentrated regions of vorticity whose shape is described by
their name, with the “legs” of the hairpin comprising counter-rotating
quasistreamwise vortices tilted upward away from the wall and joined by a
“head” of primarily spanwise vorticity far from the wall where the mean
velocity gradient (and thus mean spanwise vorticity) are small. Hairpin
vortices have been observed in many contexts; for example, Figure 1 shows
results from a simulation of transition, illustrating hairpin-like vortices
with heads near the edge of the boundary layer. 6, in a study of transitional
boundary layer flow, observe formation of hairpins just above the region where
adjacent streamwise streaks “pinch” together, and the robustness of this
process led the authors to suggest that it is a quasi-equilibrium feature of
the flow. Figure 1, from the boundary layer direct numerical simulation of 39
at transitional Reynolds numbers, shows isosurfaces of vortex strength that
exhibit a “forest” of vortices with clear hairpin structure. 25 also studied
the presence of hairpins in boundary layers at transitional Reynolds numbers
using interesting vortex eduction techniques. In fully developed turbulence,
hairpin vortex structures and packets thereof are closely related to the
formation of very large-scale uniform momentum zones 1. Additionally, vortices
of hairpin or related shape play a central role in the attached eddy
hypothesis 30, 22, 38, an important model for the statistical structure of
wall turbulence. The present work describes a traveling wave solution to the
Navier–Stokes equations in minimal channel flow that displays hairpin-like
vortices, demonstrating that such structures exist as autonomous, self-
sustaining patterns.

Important insights into turbulent structure, especially near solid surfaces or
at low Reynolds number, have been gained in recent years by looking at
turbulence through the lens of nonlinear dynamical systems theory, which
relates observed dynamics to underlying state space structure. In all of the
canonical wall-bounded shear flow geometries (pipe flow, channel flow, plane
shear (Couette) flow), families of nonlinear traveling wave solutions to the
Navier-Stokes equations (NSE) that govern fluid motion have been discovered
32, 33, 34, 36, 15, 9, 8, 7, 37. They have also been found in the asymptotic
suction boundary layer flow 18. These solutions are often denoted “exact
coherent states” (ECS). Related, but more complex states have been found as
well, that are not pure traveling waves but rather “relative periodic orbits”
that are time-periodic modulo a phase shift in one of the translation-
invariant spatial directions 5. (A traveling wave is a relative equilibrium –
a state that is time-invariant modulo phase shift in the downstream
direction.) In minimal domains at Reynolds numbers near transition the
turbulent dynamics have been found to be organized, at least in part, around
these traveling wave and relative periodic orbit solutions (see, e.g. 13, 17,
21). The present work exclusively considers traveling wave exact coherent
states.

The basic scenario for appearance of these states is that as $\Rey$ increases,
they emerge in pairs with finite amplitude at so-called saddle-node
bifurcations. The solution with the higher drag is called an “upper branch”
(UB) solution while the other is a “lower branch” (LB). The lowest Reynolds
number at which these solutions appear is slightly below the value where
transition is observed in experiments. For channel flow, our focus here,
transition occurs at a Reynolds number $\Rey$ based on the laminar centerline
velocity of about $1000$, while various families of ECS come into existence
once $\Rey\gtrsim 660$ 21, 35. Other coherent states can arise through
secondary bifurcations off of the states that appear in this way.

The spatial structure of the heretofore-discovered ECS is pairs of counter-
rotating streamwise vortices, with the corresponding low-and high-speed
streaks associated with the advection of the mean shear by the vortices.
Letting $x$, $y$ and $z$ be the mean flow, wall-normal and spanwise
directions, respectively, and $L_{x}$ and $L_{z}$ the spatial periods of the
flow in $x$ and $z$, respectively, traveling wave velocity fields
$\boldsymbol{u}(x,y,z,t)=\boldsymbol{u}(x-ct,y,z)$ with wavespeed $c$ have
generally been sought in one of two invariant subspaces with respect to
symmetry in the $x-z$ plane. If $u,v$ and $w$ are the velocity components in
$x,y$, and $z$, solutions have been sought that satisfy either a simple
reflection symmetry across $z=L_{z}/2$,

$\begin{bmatrix}u&v&w\end{bmatrix}(x,y,z-L_{z}/2,t)=\begin{bmatrix}u&v&-w\end{bmatrix}(x,y,-(z-L_{z}/2),t)$
(1)

or a “shift and reflect” symmetry, where

$\begin{bmatrix}u&v&w\end{bmatrix}(x,y,z-L_{z}/2,t)=\begin{bmatrix}u&v&-w\end{bmatrix}(x+L_{x}/2,y,-(z-L_{z}/2),t).$
(2)

The latter case leads to sinuous vortex structures, while the former leads to
varicose or potentially sinucose ones 31. (We address this distinction below.)
The former symmetry is also that of an ideal hairpin vortex. In the case of
channel flow, reflection symmetry across the centerplane of the channel $y=0$
is also often imposed, in which case

$\begin{bmatrix}u&v&w\end{bmatrix}(x,y,z)=\begin{bmatrix}u&-v&w\end{bmatrix}(x,-y,z).$
(3)

Finally, a symmetry that has not received much prior attention, but which
turns out to be central to the present work, is the “$xz$-shift” symmetry
condition

$\begin{bmatrix}u&v&w\end{bmatrix}(x,y,z)=\begin{bmatrix}u&v&w\end{bmatrix}(x+L_{x}/2,y,z+L_{z}/2).$
(4)

Traveling wave exact coherent states have been computed in the channel flow
geometry by a number of authors 33, 34, 19, 21, 40, 23, 35, 20. None of these
have reported a hairpin vortex structure. In the plane Couette flow case,
Itano and Generalis 16 reported an ECS that they described as a hairpin vortex
state. (14 also found the same solution family.) Indeed, vortex lines for this
state do display a hairpin shape, but in their case the loop or “head” of the
hairpin is near the wall, where the shear rate is highest. Thus the local
vorticity is dominated by the mean shear, which is of course locally oriented
in the $z$-direction. In contrast, hairpin vortices in turbulent flows (e.g.
as in Figure 1) have a head that is a localized region of $z$-oriented
vorticity in a background of zero or very weak shear as arises near the
centerline in channel flow or the outer edge of a boundary layer. Furthermore,
inferring existence of hairpin vortices from vortex lines can be problematic:
_any_ vortex line that crosses a plane of reflection symmetry must do so
normally: for example, even a pair of quasistreamwise vortices that satisfies
(1) will display hairpin-shaped vortex lines. More generally, 24 points out
that “hairpin-shaped vorticity lines are common in any turbulent shear flow,
whether or not hairpin-shaped vortices are present”. Figure 4 of 24
illustrates this point. In our analysis below, we will infer hairpin-like
vortex structure from vortex strength $Q$ and direct examination of velocity
and vorticity fields.

As noted above, ECS play a role in organizing the state space dynamics of
turbulent flows. In shear flows whose laminar state is linearly stable (pipe
flow, plane Couette flow, channel flow at low Reynolds number), an important
role in organizing state space is also played by the boundary between initial
conditions that laminarize and those that become turbulent. At long times,
trajectories on this boundary approach _edge states_ 26. In boundary layer
flow, 4 showed the presence of two different edge states, one dominated by
streamwise vortex structures and one by transient localized hairpins. The
study of 6 described above also focuses on edge states and indicates that
hairpin-vortex-like structures can be found on the laminar-turbulent boundary.

The present work, which focuses on the channel flow geometry, reports for the
first time a family of traveling wave exact coherent states that displays a
hairpin-like vortex structure. Below we describe the bifurcation diagram for
this family, its connection to some previously known states, its structure,
and initial evidence indicating its relationship to fully turbulent mean
profiles.

##  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
centerline velocity $U_{c}$ as the characteristic length and velocity scales,
respectively, the nondimensionalized Navier-Stokes equations are then given as

$\frac{\partial\boldsymbol{u}}{\partial
t}+\boldsymbol{u}\cdot\nabla\boldsymbol{u}=-\nabla
p+\frac{1}{\Rey}\nabla^{2}\boldsymbol{u},\quad\nabla\cdot\boldsymbol{u}=0.$
(5)

Here, we define the laminar equivalent Reynolds number for the given
volumetric flux as $Re_{c}=U_{c}h/\nu$, where $\nu$ 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 $\Rey_{b}$ based on bulk
velocity is given by $\Rey_{b}=2\Rey/3$. Characteristic inner scales are the
friction velocity $u_{\tau}=(\bar{\tau}_{w}/\rho)^{1/2}$ and the near-wall
length scale or wall unit $\delta_{\nu}=\nu/u_{\tau}$, where $\rho$ is the
fluid density and $\bar{\tau}_{w}$ is the time- and area-averaged wall shear
stress. As usual, quantities nondimensionalized by these inner scales are
denoted with a superscript “+”. The friction Reynolds number is then defined
as $Re_{\tau}=u_{\tau}h/\nu=h/\delta_{\nu}$.

Computations are performed using the open source code ChannelFlow, which
performs direct simulations as well as computing exact coherent states by
Newton-Raphson iteration 11. In this study, we focus on the domain
$L_{x}\times L_{y}\times L_{z}=\pi\times 2\times\pi/2$, the same box size as
used in our prior ECS study 21. 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), as well as across the
centerplane of the channel $y=0$, as given by (3). The solutions of central
interest here additionally satisfy the $xz$-shift symmetry given by (4). There
are of course other channel flow ECS that do not obey these symmetries (e.g.
33, 34, 19, 12, 20). All turbulent trajectories reported here are computed
without imposing any symmetries on the flow field.

##  3 Results and Discussion

Figure 2: Bifurcation diagram in terms of (a) dissipation rate $D$ _vs._
$\Rey$ and (b) wave speed $c$ _vs._ $\Rey$. Inset in (a) is a zoom-in of the
box close to point d.

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
wavespeed $c$ _vs._ $\Rey$. Two solution families are shown. The one of
primary interest is shown in purple; it arises in a saddle-node bifurcation at
$\Rey=666$ (point a in Figure 2). This solution family satisfies both the
$z$-reflection symmetry (1) and the $xz$-shift symmetry (4) and will be
denoted S1. ECS with this set of symmetries have been reported in channel flow
by 35 (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 3a, which shows a surface of vortex
strength,
$Q=\frac{1}{2}(\|\boldsymbol{\Omega}\|^{2}-\|\boldsymbol{\Gamma}\|^{2})$ in
the bottom half of the channel ($y<0$), where $\boldsymbol{\Omega}$ and
$\boldsymbol{\Gamma}$ are the local vorticity and strain rate, respectively
and $||\cdot||$ denotes Frobenius norm. This structure consists of two pairs
of staggered counterrotating streamwise vortices that are mirror symmetric
across $z=L_{z}/2$. By (3), a $y-$reflection-symmetric vortex structure arises
in the top half of the channel ($y>0$). Figure 3 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 (blue). Observe
that the low speed streaks (blue) “pinch” together at $x\approx 0.75$. As
mentioned above, 6 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 3: (a) Vortical structure of the S1 solution at the saddle node
bifurcation at $\Rey=666$ (point a in figure 2). Isosurface of $Q=0.05$ is
shown with colors indicating distance from wall – blue near wall, red near
center. A color 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 $\Rey=855$ (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 $\Rey=935$ on Figure 4a (point c on
the bifurcation diagram). This solution family is the P4 family reported by
21; indeed the way we found S1 was by continuation of P4. As $\Rey$ increases
along the P4 solution branch, the two mirror symmetric low-speed streaks merge
into one; the structure evolves from sinucose to varicose. Figure 4b shows
this varicose structure at point d on the bifurcation diagram. Upon
continuation to higher $\Rey$, 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
$\Rey=935$ (point c in Figure 2), (b) at $\Rey=1800$ (point d).

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. This is a counter-rotating streamwise vortex
structure, as expected for near-wall turbulence. By contrast, Figure 5 shows
the vortical structure of S1UB at $\Rey=1800$ (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 and green) near the wall
and _hairpin -like_ heads (red) near the centerplane. It is a microcosm of
near-wall turbulence, where streamwise vortices dominate the structure close
to the wall, with hairpins becoming dominant further away 39.  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 centerplane $y=0$. Consistent with the observations of 6, 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. By symmetry, there are also pinches at $x\approx 1$ between the low speed
streaks in the domain shown and those in the spanwise neighboring domains.
Correspondingly, there are hairpin-like structures above these pinches. Half
of each of these can be seen in Figure 5. Overall, this ECS displays a
staggered array of these structures.

From Figure 5, we can estimate the width of the hairpin-like structure as the
spacing between the streaks in the pinch region, which is about $1$ in outer
units. Similarly, from Figure 6 we can estimate the distance of the vortex
head from the wall in the same units as $0.9$. At $\Rey=1800$, the friction
Reynolds number $\Rey_{\tau}\approx 95$. 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 about $95$ and $86$ wall units,
respectively. For comparison, 1 describes a “mature” hairpin vortex packet as
having similar size, with both width and height of about 100 wall units.

Figure 5: (a) Structure on S1UB at $\Rey=1800$ (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.

Further indication of the hairpin-like structure displayed by this ECS is
illustrated in Figure 6, 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 centerplane 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 16; 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 $\Rey=1800$,
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, Fig. 6
shows the same quantities plotted for the structure at $\Rey=666$, i. e. right
at the saddle-node bifurcation point. Consistent with the $Q$ structure shown
in Fig. 3, no localization of $z$-vorticity is seen.

Completing our presentation of the structure of S1UB at $\Rey=1800$, Figures 7
and 7 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 shows again the Theodorsen picture; arrows have been
added to all of these figures 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, 1 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
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 centerplane. Regarding the trailing low
speed streak, this is precluded by the symmetries (1) and (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 1. Furthermore, it is consistent with observations
of hairpins in the transition context as exemplified by the observations of 6,
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 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
$\Rey=1800$ and (b) $v_{x}=0.74$ for S1 at the saddle-node bifurcation point
$\Rey=666$. In these plots the region between $y=-0.6$ and $y=0$ (the
centerplane) is shown.

Figure 7: (a) In-plane velocity fluctuations on the legs of the vortex
structure at $x=1.82$ of S1UB, $\Rey=1800$, (b) Out of plane velocity
fluctuations on the head of the vortex structure at $x=2.43$ of S1UB,
$\Rey=1800$ and (c) Theodorsen’s picture of a hairpin vortex.

Figure 8: (a) Vortical structure of S1LB at $\Rey=3000$ (point i). Isosurface
of $Q=0.014$ is shown. (b) Streamwise velocity $v_{x}$ 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 $\Rey=3000$. In
this plot the region between $y=-0.6$ and $y=0$ (the centerplane) is shown.

Figure 10: Mean profiles for ECS and turbulent flow (DNS) at $\Rey=1800$: (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 on Figure 2.

In prior observations, hairpin vortices have been observed as transient,
spatiotemporally evolving structures. In contrast, the hairpin-like ECS
structure observed in Figure 5 is, in a traveling 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 centerplane
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 $\Rey$ than on the upper
branch. This is why we have illustrated the structure at $\Rey=3000$ rather
than the value of $1800$ that we used for the upper branch. Observe that that
the $Q$ value chosen for Fig. 8 is 0.014, much smaller than the value 0.072
used to visualize the upper branch (Figure 5) and the waviness of the streaks
shown in Fig. 8 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 $\Rey=1800$ 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
$KE=\frac{1}{2}||\boldsymbol{u}-\boldsymbol{u}_{\textrm{laminar}}||^{2}$. 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 green 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.

Figure 11: (a) State space visualization of DNS trajectories at $\Rey=1800$,
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 $\Rey=1800$. (b) 2D projection
of (a). Also shown is a trajectory starting from S1LB (green). (c) Isosurface
of $Q=0.072$ at an instant during DNS at $\Rey=1800$. Shown is the top half of
the domain, rotated to be consistent with other figures. For clarity, two
periods in $z$ are shown.

##  4 Conclusions

Prior studies have created a catalogue of nonturbulent recurrent solutions to
the Navier–Stokes equations – exact coherent states – that capture key
features of near-wall turbulence, including streamwise vortices 33, bursts 29
and localized spots 2 or puffs 3. The present work adds an important entry to
this catalogue. Hairpin vortices have long been observed within turbulent
flows in various contexts, from boundary layer transition to fully-developed
flows.  Both the upper and lower branches of the S1 solution family reported
here for the minimal channel flow geometry display a clear hairpin-like
structure, with a spanwise-oriented head connected to quasistreamwise-oriented
legs. Additionally, in the Reynolds number regime studied, the upper branch
state has mean velocity and Reynolds shear-stress profiles that are
quantitatively similar to the turbulent profiles in a channel of the same
(minimal) dimensions.

The present results are of course obtained for a system that is highly
constrained by symmetry and periodicity conditions. Nevertheless, the
observations described here further strengthen the growing body of work
connecting turbulent dynamics to simpler underlying state-space structures. By
understanding the latter, we are increasingly gaining insight into the former.

## Acknowledgments

This work was supported by the National Science Foundation through grant
CBET-1510291 (Fluid Dynamics Program) and PHY-1125915 (Kavli Institute for
Theoretical Physics), and the Air Force Office of Scientific Research through
grant FA9550-15-1-0062 (Flow Interactions and Control Program). The authors
are grateful to John Gibson for development and distribution of ChannelFlow
and to Bruno Eckhardt for helpful discussions.

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