# Minimal seeds for turbulent bands in channel flow

E. Parente1,2<EMAIL_ADDRESS>J.-Ch. Robinet 1  P. De Palma2  S.
Cherubini2  1 DynFluid - Arts et Métiers Paris, 151 Bd de l’Hôpital, 75013
Paris, France 2 Dipartimento di Meccanica, Matematica e Management (DMMM),
Politecnico di Bari, Via Re David 200, 70126 Bari, Italy

###### Abstract

In this work, nonlinear variational optimization is used for obtaining minimal
seeds for the formation of turbulent bands in channel flow. Using nonlinear
optimization together with energy bisection, we have found that the minimal
energy threshold for obtaining spatially-patterned turbulence scales with
$Re^{-8.5}$ for $Re>1000$. The minimal seed is constituted by a localized
spot-like structure surrounded by a low-amplitude large-scale quadrupolar
structure filling the whole domain. This minimal-energy perturbation of the
laminar flow has dominant wavenumbers equal to $0.15$ and $4$ in the
streamwise and spanwise directions, respectively, and is characterized by a
more marked spatial localization when the Reynolds number increases. At
$Re\lesssim 1200$, the minimal seed evolves in time creating an isolated
oblique band. Whereas, for $Re\gtrsim 1200$, an almost spanwise-symmetric
evolution is observed, giving rise to two distinct bands. A similar evolution
is found also at low $Re$ for non-minimal optimal perturbations. This
highlights two different mechanisms of formation of turbulent bands in channel
flow, depending on the Reynolds number and initial energy of the perturbation.
The selection of one of these two mechanisms appears to be affected by the
probability of decay of the newly-created stripe, which increases with time,
but decreases with the Reynolds number.

##  1 Introduction

Since the experimental work of Reynolds (1883), who first observed subcritical
transition to turbulence in a wall bounded shear flow, the dynamics of
transition has remained an open problem of fluid dynamics. One of the most
triggering features of transitional flows is that turbulence does not arise at
the same time in the whole domain, being preceded by the formation of
localised flow structures that grow in amplitude and spread in space. The
first of these localized flow features to be observed were turbulent spots,
which have been investigated by many researchers in the past. Emmons (1951)
was the first to experimentally show that turbulent spots may trigger
turbulence; later this was confirmed experimentally and numerically in plane
channel flow analysing the spots characteristics (Carlson et al., 1982;
Henningson & Kim, 1991; Klingmann, 1992; Aida et al., 2010, 2011; Lemoult et
al., 2013, 2014). In pipe flows, other localized flow structures dubbed puffs
are observed both numerically and experimentally (Eckhardt et al., 2007; Avila
et al., 2011). Puffs are localised, downstream-travelling flow structures
within a laminar field, sustained by the energy provided by the neighbouring
laminar motion at the upstream end of the puff. They can decay, split or
merge, filling the laminar flow with turbulent fluctuations.
More recently, Duguet et al. (2010) have shown that, locally perturbing the
plane Couette flow, the fully turbulent state is preceded by the formation of
turbulent bands. The same behaviour was reported by Tao & Xiong (2013) and
Xiong et al. (2015), where it was observed that, after injecting a localised
perturbation, turbulent bands grow obliquely in the domain until decay or
breakdown to turbulence. In the latter case, it was assessed that the
turbulent bands lifetime is longer than that of turbulent spots. Besides
having a crucial role in turbulent transition, oblique turbulent bands
characterise also the (transiently) turbulent state in shear flows at low
Reynolds numbers, as shown by Tsukahara et al. (2005). In fact, in the pipe,
plane Couette and plane Poiseuille flow, there is a range of moderate Reynolds
numbers for which experiments and numerical simulations have shown that
turbulence is not homogeneous in space, since spatially-localized regions
characterised by laminar and turbulent behaviour coexist when the
statistically-steady turbulent state is reached. Indeed, elongated oblique
turbulent bands plunged in a laminar flow appear in plane Couette flow at
$Re>290$ (Prigent et al., 2002; Barkley & Tuckerman, 2005; Duguet et al.,
2010; Tuckerman & Barkley, 2011) and plane Poiseuille flow (Tsukahara et al.,
2014; Tuckerman et al., 2014; Tao et al., 2018; Shimizu & Manneville, 2019;
Kashyap et al., 2020)). In the channel flow, turbulent-laminar oblique
patterns were observed experimentally for $Re=1000$ by Carlson et al. (1982).
Numerical simulations reproduced these patterns also at lower Reynolds numbers
(Xiong et al. (2015), Tao & Xiong (2017), Kashyap et al. (2020)), estimating a
threshold value for observing turbulent bands at $Re\approx 660$, according to
Tao et al. (2018). Moreover, Song & Xiao (2020) recently reported the onset of
turbulent bands even at Reynolds number as low as $Re=500$, generated by
forcing the flow with a local perturbation with a sufficiently strong spanwise
inflection. They showed that this forcing method permits to generate bands at
very low values of the Reynolds number, for which bands previously appeared to
be not sustained. This procedure was motivated by the work of Xiao & Song
(2020), where the authors performed a linear stability analysis of the mean
velocity profile extracted in a small domain at the head of the turbulent
band. By means of stability analysis, it was suggested that spanwise
inflectional instability may be the mechanism involved in the growth and self-
sustaining process of turbulent bands.
In addition to the Reynolds number, the domain size plays an important role in
the growth and self-interaction of turbulent bands. When the considered domain
is sufficiently large, turbulent bands can grow for longer times, avoiding the
probability of interaction with themselves or other bands. For this reason,
several works were performed to study the influence of the domain in the onset
of turbulent bands. In order to reduce the computational cost, some numerical
studies cleverly considered computational domains tilted in the direction of
the bands, as done for the plane Couette flow by Barkley & Tuckerman (2005)
and for the plane Poiseuille flow by Tuckerman et al. (2014). Very recently,
Parente et al. (2021) carried out an optimal growth analysis in a tilted
domain with given angle. They showed that, although linear optimization is
able to recover the main wavenumbers of the DNS, nonlinear effects are
necessary for providing large-scale flow and spatial localization able to
generate a turbulent band. Although very interesting for studying the dynamics
of a single band, using the tilted domain constrains the turbulent bands to
develop at a fixed angle and avoids (or reduce) the interactions with other
bands, resulting in a consequently less rich dynamics. A more detailed
analysis about turbulent-laminar patterns in shear flows is tackled in the
review of Tuckerman et al. (2020).
In the literature, two strategies are typically used to obtain laminar-
turbulent bands. The first one consists in starting from a spatially-
homogeneous statistically-steady turbulent state, and slowly decreasing the
Reynolds number until patterned laminar regions plunged in the initially
turbulent flow begin to be observed (Tsukahara et al. (2005), Tsukahara et al.
(2005), Kashyap et al. (2020)). This method usually leads to the formation of
statistically-steady laminar-turbulent patterns. Another possible method
consists in perturbing the laminar flow with localised solutions having enough
energy to trigger localized regions of turbulence eventually evolving into
oblique stripes (Duguet et al. (2010), Aida et al. (2010), Aida et al. (2011),
Tao & Xiong (2013), Xiong et al. (2015)). When focusing on the second method,
the amplitude, the shape and the localization of these initial perturbations
should be carefully chosen to ensure their growth towards oblique bands.
Depending on their shape, perturbations with higher amplitude may decay, while
weaker perturbations may lead the flow to transition.
From a phase-space point of view, the problem of finding perturbations
eventually leading to these laminar-turbulent patterns consists in placing the
starting point of the trajectory outside the boundary of the basin of
attraction of the laminar solution. Notably, the most relevant point of this
boundary is its energy minimum, since it represents the minimal (in energy
norm) perturbation of the laminar state that can lead the flow to transition.
This point in the phase space has been dubbed, by Rabin et al. (2012), minimal
seed for turbulent transition. This energy minimum has been assessed for
several shear flows, allowing at finding minimal energy thresholds for
transition. In particular, in plane Couette flows, a minimal energy threshold
varying with the Reynolds number as $Re^{-2.7}$ has been found, in
quantitative agreement with experimental estimates for pipe flows. Whereas,
for the asymptotic-suction boundary layer Cherubini et al. (2015) found a
scaling law of this energy threshold of $Re^{-2}$. Vavaliaris et al. (2020)
recently performed similar computations for a non-parallel boundary-layer
flow. The determination of these energy thresholds is of primary importance
for control purposes, since passive or active control methods such as boundary
manipulation (Rabin et al., 2014) or profile flattening (Marensi et al., 2019)
able to increase this minimal energy would render these flows less prone to
transition. However, to the authors knowledge, the minimal seed computation
has never been carried out for the channel flow. Moreover, all the minimal
seed computations reported in the literature have been carried out in small
domains, not allowing to observe laminar-turbulent patterned flow states. In
large domains where laminar-turbulent oblique bands exist, the minimal
transition thresholds can be potentially different from that obtained in small
domains. Moreover, the analysis of these minimal perturbations and of their
evolution towards oblique bands can potentially unveil the main mechanisms
leading to the formation and sustainment of these laminar-turbulent patterns.
This work aims at finding minimal seeds for the generation of turbulent bands
in channel flow. The methodology to find these perturbations is based on the
nonlinear variational optimization first used in the boundary layer flow by
Cherubini et al. (2010a, 2011) and in pipe flow by Pringle & Kerswell (2010);
Pringle et al. (2012), and then coupled with energy bisection by Rabin et al.
(2012) for the plane Couette flow and by other authors for other shear flows
(Duguet et al., 2013; Cherubini et al., 2015; Vavaliaris et al., 2020). In all
these studies, the kinetic energy was considered as objective function of the
optimization and the optimal perturbations associated to the minimal input
energy were spatially localised. Similar behaviour was found in Couette flow
by Monokrousos et al. (2011) maximizing the time integral of the entropy
production. A detailed review of these methods is reported in Kerswell et al.
(2014); Kerswell (2018). In the present work, we carry out for the first time
the computation of minimal seeds in channel flow, for very large domains
allowing the creation of laminar-turbulent patterned final states such as
oblique stripes. The perturbation kinetic energy is used as objective function
of the nonlinear optimization, which appears an appropriate choice since many
literature studies report peaks of kinetic energy during the development of
turbulent bands (Tao et al., 2018). The minimal seeds obtained using this
optimization-and-bisect method are found to change with the Reynolds number,
presenting a power-law scaling of the initial energy $E_{0_{min}}\propto
Re^{-\gamma}$, with $\gamma$ being approximately four times larger than the
values reported in other shear flows in smaller domains. Moreover, two
distinct mechanisms of creation of turbulent bands are reported and discussed
in detail.
The paper is organised in the following way: in Section 2 the used methodology
is presented. In Section 3 the main results of the optimization are reported
and the nonlinear evolution in time of the optimal perturbation is shown. In
section 4 conclusions are drawn.

##  2 Problem formulation

The considered flow is a plane channel flow at Reynolds number
$Re=U_{c}H/\nu$, with $U_{c}$ being the centreline velocity of the laminar
Poiseuille flow, $H$ being the half width of the channel and $\nu$ the
kinematic viscosity. The Reynolds number is varied by changing the pressure
gradient, while the volume flux remains constant, with bulk velocity
$U_{b}=3/2$. The dynamics of this flow is studied by decomposing the
instantaneous velocity field, $\textbf{u}=[u,v,w]^{T}$, into the laminar base
flow $\textbf{U}=[U(y),0,0]$ and a perturbation
$\textbf{u}^{\prime}=[u^{\prime},v^{\prime},w^{\prime}]^{T}$. The dynamics of
the perturbations of the laminar base flow is computed by solving the
perturbative nonlinear incompressible Navier-Stokes equations:

$\begin{cases}\displaystyle\frac{\partial u_{i}^{\prime}}{\partial
x_{i}}=0,\\\ \displaystyle\frac{\partial u_{i}^{\prime}}{\partial
t}+u_{j}^{\prime}\frac{\partial u_{i}^{\prime}}{\partial
x_{j}}+u_{j}^{\prime}\frac{\partial U_{i}}{\partial x_{j}}+U_{j}\frac{\partial
u_{i}^{\prime}}{\partial x_{j}}=-\frac{\partial p^{\prime}}{\partial
x_{i}}+\frac{1}{Re}\frac{\partial^{2}u_{i}^{\prime}}{\partial
x_{j}},\end{cases}$ (1)

with $\mathbf{U}=[U(y),0,0]$, $U(y)=1-y^{2}$ being the laminar streamwise
velocity profile, $p^{\prime}$ the pressure perturbation, while $x_{i}$ is the
Cartesian reference frame having components $x,y,z$, for the streamwise, wall-
normal and spanwise directions, respectively. No-slip boundary conditions are
imposed at the walls for the three velocity components, while periodicity is
fixed in the streamwise and spanwise directions.
In order to find the minimal seed for the considered flow, we first search for
the optimal perturbation $\textbf{u}^{\prime}$ at $t=0$, providing the maximum
value of the objective function at target time $T$. Following previous works
(Cherubini et al., 2010a; Pringle & Kerswell, 2010), we choose as objective
function the energy gain $G(T)=E(T)/E(0)$, where:

$E(t)=\frac{1}{2V}\int_{V}(u^{\prime}(t)^{2}+v^{\prime}(t)^{2}+w^{\prime}(t)^{2})dV$
(2)

is the kinetic energy at time $t$ and $V$ is the volume of the computational
domain. In order to find the initial perturbation $\textbf{u}^{\prime}(0)$
having given initial energy $E(0)=E_{0}$, providing the largest possible
energy $E(T)$ at the target time, an optimization loop is set using the
Lagrange multiplier technique (Zuccher et al. (2004), Pringle & Kerswell
(2010), Cherubini et al. (2011)). A Lagrangian functional is defined by
augmenting the objective function with the following constraints: i) the
optimal perturbation $\mathbf{u}^{\prime}(t)$ must be solution of the Navier-
Stokes equations at all times $t\in]0,T[$; ii) it must be divergence free at
all times $t\in[0,T]$; and iii) it must have energy norm equal to a given
value $E_{0}$ at $t=0$. With these constraints, the Lagrangian functional
reads:

$\begin{split}\mathcal{L}(u_{k}^{\prime},p^{\prime},u_{k}^{\dagger},p^{\dagger},u_{k}^{\prime}(0),u_{k}^{\prime}(T),\lambda)=\frac{E(T)}{E(0)}\\\
-\int_{0}^{T}\int_{V}u_{i}^{\dagger}\left(\frac{\partial
u_{i}^{\prime}}{\partial
t}+\frac{\partial(u_{i}^{\prime}u_{j}^{\prime})}{\partial
x_{j}}+\frac{\partial(U_{i}u_{j}^{\prime})}{\partial
x_{j}}+\frac{\partial(u_{i}^{\prime}U_{j})}{\partial x_{j}}+\frac{\partial
p}{\partial x_{i}}-\frac{1}{Re}\frac{\partial^{2}u_{i}^{\prime}}{\partial
x_{j}}\right)dVdt\\\ -\int_{0}^{T}\int_{V}p^{\dagger}\frac{\partial
u_{i}^{\prime}}{\partial
x_{i}}dVdt-E^{\dagger}\left(\frac{E_{0}}{E(0)}-1\right).\end{split}$ (3)

with $\mathbf{u}^{\dagger}$, $p^{\dagger}$ and $E^{\dagger}$ being the
Lagrangian multipliers (or adjoint variables).
To maximise the augmented functional $\mathcal{L}$ we evaluate its variation
with respect to the direct and adjoint variables and nullify it. The variation
of the Lagrangian functional with respect to the direct variables
$\textbf{u}^{\prime}$, $p^{\prime}$, provides the following adjoint equations:

$\begin{cases}\displaystyle\frac{\partial u_{i}^{\dagger}}{\partial
x_{i}}=0,\\\ \displaystyle\frac{\partial u_{i}^{\dagger}}{\partial
t}-u_{j}^{\dagger}\frac{\partial u_{i}^{\prime}}{\partial
x_{j}}+u_{j}^{\prime}\frac{\partial u_{i}^{\dagger}}{\partial
x_{j}}-u_{j}^{\dagger}\frac{\partial U_{i}}{\partial
x_{j}}+U_{j}\frac{\partial u_{i}^{\dagger}}{\partial x_{j}}=-\frac{\partial
p^{\dagger}}{\partial
x_{i}}+\frac{1}{Re}\frac{\partial^{2}u_{i}^{\dagger}}{\partial
x_{j}}.\end{cases}$ (4)

The optimization problem is then solved using a direct-adjoint looping
algorithm (as done in previous works by Pringle & Kerswell (2010), Cherubini
et al. (2010a), among others), which consists in integrating iteratively in
time the direct and adjoint equations between $0$ and $T$ to evaluate the
gradient of $\mathcal{L}$ with respect to $\mathbf{u}^{\prime}(0)$, which is
then used to update the initial perturbation using a gradient rotation
algorithm (Foures et al., 2013; Farano et al., 2016, 2017). Convergence is
attained when the variation of the gain between two successively iterations is
smaller then a chosen threshold, $\epsilon=5\times 10^{-8}$.
For computing the minimal seed, the variational optimization is coupled with
an energy-bisection procedure. Two different optimizations are initialised
with a random divergence-free perturbation for low $T$, with a value of
$E_{0}$ sufficiently high (low) to induce transition (relaminarization) at
longer times. The energy is bisected and the optimization repeated, using the
results of the previous optimisation as initial guess. Energy-bisection is at
first carried out for a low ($\mathcal{O}(10)$) target time in order to have a
very rough (but computationally cheap) upper estimate of the initial minimal
energy threshold $E_{0_{min}}$. Then, the target time is increased at
$\mathcal{O}(100)$ and the energy bisection is repeated, allowing to converge
towards $E_{0_{min}}$.
The optimization algorithm is implemented within the open source code
$Channelflow$ (channelflow.ch) (Gibson et al. (2021)). Both direct and adjoint
equations are solved using a Fourier-Chebychev discretization in space and a
third-order semi-implicit backward difference scheme for the time integration.
An influence matrix method with Chebyshev tau correction was used to enforce
no-slip boundary condition at the walls.
The domain size in the streamwise, normal and spanwise directions is
$L_{x}\times L_{y}\times L_{z}=250\times 2\times 125$, while the number of
grid points in the same directions are $N_{x}\times N_{y}\times
N_{z}=1024\times 65\times 1024$. This results in a numerical resolution
comparable with those used by Shimizu & Manneville (2019) and Kashyap et al.
(2020) for the same range of Reynolds numbers. Finally, it should be noticed
that the fact that optimizations are performed in a large domain involves
challenging computations in terms of memory, especially for the highest
considered target time, $T=150$. In fact, the direct-adjoint algorithm
requires storage of the flow field snapshots $\textbf{u}^{\prime}$ at each
time step, in order to evaluate the term coupling direct and adjoint variables
in the adjoint equations. For alleviating the storage requirement, a
checkpointing technique is used, similar to that used in Griewank & Walther
(2000) and Hinze et al. (2006).

##  3 Results

Nonlinear optimizations are performed for four values of
$Re=1000,1150,1250,1568$, chosen in the range of Reynolds numbers for which
the plane Poiseuille flow is linearly stable, while the turbulent state
appears in the form of turbulent-laminar patterns (Tao et al. (2018), Kashyap
et al. (2020)). As reported by Xiong et al. (2015), the onset of bands does
not lead to sustained turbulence for $Re<1000$. Whereas, for $Re\geq 1000$,
the bands can split, providing a mechanism for turbulence spreading leading to
the coexistence of laminar regions with inclined turbulent bands, which
persist up to $Re>3900$, for which only featureless turbulence is present.
Figure 1 (a) provides the influence of the target time on the optimal energy
gain for $Re=1150$ and given input energy $E_{0}=1.1\times 10^{-7}$. The
energy gain grows in time with an almost exponential trend for $T<40$ and for
$T>50$, reaching an amplification of three orders of magnitude for $T>80$. No
saturation of the final energy is observed for $T\leq 100$, since very long
(i.e., $\mathcal{O}(10^{3})$) times would be needed to fill the entire domain
with turbulent bands, as it will be shown in subsection 3.3.

Figure 1: (a) Optimal energy gain versus target time $T$ for $Re=1150$ and
$E_{0}=1.1\times 10^{-7}$. (b) Minimal energy threshold ${E_{0}}_{min}$ for
transition to turbulence (red dots) and upper estimate obtained for $T=10$
(black dots). The black and red lines represent the power-law for low
($\mathcal{O}(10)$) and high ($\mathcal{O}(100)$) target times, respectively,
for $Re>1000$.

As mentioned before, the procedure of bisection of the initial energy is at
first carried out for $T=10$, in order to have a computationally cheap upper
bound for the computation of ${E_{0}}_{min}$, which will be then carried out
for $T=100$ or $T=150$, depending on the Reynolds number. For $Re<1568$, the
estimate of ${E_{0}}_{min}$ was obtained for $T=100$, since we have verified
that increasing the target time from $T=100$ to $T=150$ leads to slight
changes of the minimal energy, namely, from $5.5\times 10^{-7}$ to $5.6\times
10^{-7}$ for $Re=1000$. Whereas, at $Re=1568$ it has been necessary to
increase the target time to $T=150$ to achieve a sufficiently good
approximation of the threshold energy for generating bands. Figure 1 (b)
provides the minimal input energy able to induce transition towards the
turbulent bands, ${E_{0}}_{min}$ (red dots), together with its upper estimate
obtained for $T=10$ (black dots), for the four considered values of the
Reynolds number. In the range of $Re$ analysed, we tried to fit the minimal
input energy with a power law of the type $E_{0_{min}}\propto Re^{-\gamma}$,
but we obtained a satisfying fit only by restraining the power law to the
minimal seeds in the range $Re>1000$ (red dashed line, obtained for
$\gamma\approx 8.5$). A similar behaviour is observed for the upper estimate
of $E_{0_{min}}$ obtained for $T=10$ (black dashed line), although the
associated value of $\gamma$ is much lower. Incidentally, we observe that as
expected, larger threshold initial energies are obtained for the lower target
time $T=10$ (black line). The fact that the minimal-seed energy obtained for
$Re=1000$ appears not to be aligned with the fitting line recovered for larger
values of $Re$ might have been anticipated. As reported by Xiong et al.
(2015), $Re=1000$ is exactly the limit value of the Reynolds number for which
band splitting begins to be observed. Thus, at this threshold value of $Re$
the flow dynamics may present a transitional behaviour between two different
regimes, not fitting with that observed at larger values of $Re$. Finally, it
is also worth to notice that the exponent of the power law approximating the
minimal-energy threshold is much higher than those reported in previous works.
For the plane Couette flow, a minimal seed energy varying as $Re^{-2.7}$ was
reported by Duguet et al. (2013); for the asymptotic suction boundary layer, a
scaling of $Re^{-2}$ was found by Cherubini et al. (2015). This discrepancy
may be more likely linked to the much larger domain considered in the present
study, rather than to the different type of flow.

(a) $Re=1000$, $E_{0}=8.4\times 10^{-7}$ (b) $Re=1150$, $E_{0}=1.1\times
10^{-7}$ (c) $Re=1250$, $E_{0}=9.4\times 10^{-8}$ (d) $Re=1568$,
$E_{0}=3.7\times 10^{-8}$

Figure 2: Nonlinear optimal perturbation at time $t=0$ for different Reynolds
numbers with $U_{bulk}=3/2$ obtained at target time $T=10$ with the input
energy reported in figure 1 by the black dots. Isosurface of the streamwise
velocity (light grey for positive and black for negative values, $u=\pm
0.003$).

(a) $Re=1000$, $E_{0}=8.4\times 10^{-7}$ (b) $Re=1150$, $E_{0}=1.1\times
10^{-7}$ (c) $Re=1250$, $E_{0}=9.4\times 10^{-8}$ (d) $Re=1568$,
$E_{0}=3.7\times 10^{-8}$

Figure 3: Nonlinear optimal perturbations at time $t=T$ for different Reynolds
numbers with $U_{bulk}=3/2$ obtained at target time $T=10$ with the input
energy reported in figure 1 by the black dots. Isosurface of the streamwise
velocity (light grey for positive and black for negative values $u=\pm 0.02$).

###  3.1 Nonlinear optimal perturbations

The initial optimal solutions obtained for the target time $T=10$ at the
threshold energy able to trigger turbulence are shown in figure 2 for the four
$Re$ considered. In all cases, the optimal perturbation is spatially localised
in a small region of the domain, having shape similar to that of a spot, being
composed by alternating positive and negative finite-size streamwise streaks.
Despite spatial localization has been already observed in nonlinear
optimizations (for instance, see Cherubini et al. (2011), Monokrousos et al.
(2011), Pringle et al. (2012)), the structure of this optimal perturbation is
rather different to that of previously computed nonlinear optimal
perturbations. Whereas, it closely resembles the optimal wave packet recovered
by linear optimization and windowing, obtained for the boundary layer flow by
Cherubini et al. (2010b). This is probably due to the low target time and
initial energy used for this computations, which partially hinders the
development of nonlinear effects. At $t=0$, all the optimal perturbations
shown in figure 2 present vortical structures pointing in the opposite
direction to the shear (see the $x-y$ planes in the right bottom of each
frame). Whereas, at the target time, the perturbations reverse their
inclination, pointing in the same direction of the shear flow, as shown in
figure 3. This streamwise tilting is a common feature of optimal perturbations
in shear flows (Cherubini et al., 2010a; Pringle & Kerswell, 2010; Duguet et
al., 2013), and suggests that the Orr mechanism is a fundamental mechanism
involved in the early stage of transition to turbulence, with characteristic
time approximately equal to $10$ (Orr, 1907). One can also notice that, for
$T=10$, the shape of the optimal perturbations does not change much while
evolving from $t=0$ to the target time, as shown in figure 3, remaining
characterized by alternating positive and negative streaks localised in a
spot-like region.

(a) $t=0$ (b) $t=T$

Figure 4: Shaded isocontours of the crossflow energy $E_{cf}$ together with
the normalized $y$-integrated large-scale flow (vectors) characterizing the
nonlinear optimal for $Re=1150$, $E_{0}=1.1\times 10^{-7}$, $T=10$: (a)
initial optimal solution, (b) optimal solution at the target time.

In Duguet & Schlatter (2013), it is discussed that oblique bands arise as a
result of advection of newly nucleated streaks in the direction of a large-
scale flow, which is oblique with respect to the $x$ direction. The local
orientation of the large-scale flow is thus responsible for the obliqueness of
the laminar-turbulent interface of growing incipient spots as well as for
maintaining turbulent stripes (Duguet & Schlatter, 2013). To ascertain whether
the computed nonlinear optimal perturbations contain the seed for the
development of turbulent bands, we compute the large-scale flow related to the
optimal disturbances by averaging the instantaneous velocity field in the wall
normal direction as $\overline{u_{i}}=\int_{-1}^{1}u_{i}dy$. Notice that
$\overline{u}$ is zero where the flow is laminar, close to zero where the flow
is turbulent, but it is non zero on laminar-turbulent interfaces, due to a
mismatch of the streamwise flow rates across them, linked to the presence of
overhang regions. In figure 4, we have reported the isocontours of the cross-
flow energy $E_{cf}=(1/2)\int(v^{2}+w^{2})dy$, surrounded by the large-scale
field $(\overline{u},\overline{w})$, for the initial optimal perturbation at
$Re=1150$, $E_{0}=1.1\times 10^{-7}$. The optimal flow field is characterized
by two different scales: a small-scale flow embedded within the spot-like
structure and a large-scale flow in the form of large vortices filling the
whole domain. The latter is characterised by a streamwise flow entering the
spot and a spanwise flow exiting it, constituting a quadrupolar structure.
Quadrupolar large-scale flow around spots or laminar-turbulent bands has been
observed in plane Couette flow (Schumacher & Eckhardt (2001); Lagha &
Manneville (2007)) and plane Poiseuille flow (Lemoult et al., 2014), due to
the shearing of the streamwise velocity and the breaking of the spanwise
homogeneity (Wang et al., 2020). Notice that this large-scale flow, which is
not associated with the presence of overhang regions, remains almost unchanged
at $t=0,T$ (compare figures 4 (a) and (b)).

(a) $t=0$ (b) $t=T$

Figure 5: Optimal perturbation for $Re=1150$, $U_{bulk}=3/2$, $T=100$ and
$E_{0}=1.1\times 10^{-7}$ at (a) $t=0$ and (b) $t=T$: isosurface of the
streamwise velocity (light grey for positive and black for negative values,
$u=\pm 0.003$ (a) and $u=\pm 0.12$ (b)).

Increasing the target time to $T=100$ for $Re=1150$, while keeping
$E_{0}=1.1\times 10^{-7}$, we obtain the optimal solution shown in figure 5
(a), which presents a localised structure similar to that computed for low
target times, despite being not perfectly symmetric and having thicker streaks
compared to those at lower target time. The optimal solution at the target
time, shown in figure 5(b), is very similar to that of a turbulent spot, now
clearly presenting an overhang region. Moreover, it is characterized by a
quadrupolar large-scale flow surrounding the spot-like small-scale
disturbances (not shown). Starting from this optimal (but yet not minimal)
solution, the optimize-and-bisect procedure is carried out at $T=100$ for
$Re=1150$. The same procedure is carried out for all the considered values of
$Re$, for obtaining the minimal seeds whose energy is reported in figure 1
(b).

(a) $t=0$ (b) $t=25$ (c) $t=50$ (d) $t=T$

Figure 6: Minimal seed for $Re=1150$, $T=100$ and $E_{0}=4.7\times 10^{-8}$ at
(a) $t=0$, (b) $t=25$, (c) $t=50$, (d) $t=T$: isosurface of the streamwise
velocity (light grey for positive and black for negative values, $u=\pm 0.003$
(a), $u=\pm 0.02$ (b-c) and $u=\pm 0.08$ (d).

###  3.2 Minimal seed at different Reynolds

(a) $Re=1000$, $E_{0}=5.5\times 10^{-7}$ (b) $Re=1150$, $E_{0}=4.7\times
10^{-8}$ (c) $Re=1250$, $E_{0}=2.9\times 10^{-8}$ (d) $Re=1568$,
$E_{0}=3.6\times 10^{-9}$

Figure 7: Minimal seed at different Reynolds numbers: isosurface of negative
streamwise velocity ($u=-0.0025$, light grey) and Q-criterion ($Q=0.003$)
coloured by the streamwise vorticity (positive red, negative blue).

(a) $Re=1000$, $E_{0}=5.5\times 10^{-7}$ (b) $Re=1150$, $E_{0}=4.7\times
10^{-8}$ (c) $Re=1250$, $E_{0}=2.9\times 10^{-8}$ (d) $Re=1568$,
$E_{0}=3.6\times 10^{-9}$

Figure 8: Minimal seed at target time $t=T$ for different Reynolds numbers:
isosurface of negative streamwise velocity ($u=-0.03$, light grey) and
Q-criterion ($Q=0.005$) coloured by the streamwise vorticity (positive red,
negative blue).

The minimal seed for turbulent bands obtained at $Re=1150$ for
$E_{0_{min}}=4.7\times 10^{-8}$ is reported in figure 6 (a). Noticeably, it
does not present significant differences compared to that at the same target
time with a higher input energy, provided in figure 5 (a). A three-dimensional
visualization, provided in figure 7 (b) shows the streamwise and spanwise
alternance of finite-size streaks together with spanwise-inclined vortical
structures closer to the wall. As shown in figure 6 (b-c), these finite-size
streaks increase their streamwise length, and the initial wave packet turns at
$t=25$ into four pairs of elongated streamwise streaks. At $t=50$, some
wiggling of the streaks can be recovered in the center of the wave packet,
which turns at target time into an arrow-shaped puff-like packet (figure 6
(d)), as also shown in the three-dimensional view of figure 8 (b). Comparing
figure 5 (b) with figure 6(d), one can also notice that, at target time, the
minimal perturbation differs from that computed for $E_{0}>{E_{0}}_{min}$,
having completely lost symmetry along the $z$ axis. A clear overhang region is
present as well, and the structure in the $x-y$ plane recalls that
experimentally observed in turbulent spots by Lemoult et al. (2013). Moreover,
the large-scale flow, shown in figure 9, maintains the previously observed
quadrupolar structure, presenting large-scale vortices almost symmetric in the
spanwise direction. The same large-scale quadrupolar structure surrounding the
minimal-energy wakepacket is observed also for the other considered Reynolds
numbers (not shown). Whereas, the small-scale minimal perturbations are found
to considerably change with the Reynolds numbers, as shown in figure 7. In
particular, a further localization of the initial wavepacket is observed for
increasing $Re$, leading to a minimal structure at $Re=1568$ which closely
resembles that found for the plane Couette and the asymptotic suction
boundary-layer flow in small domains (Duguet et al., 2013; Rabin et al., 2012;
Cherubini et al., 2015). Apart from the spatial localization, the main
structures observed within the minimal-seed wavepackets are essentially the
same, namely, finite-size streaks flanking upstream-tilted elongated vortices.
As a result of the discrepancies recovered at initial time, the minimal seeds
obtained at different $Re$ evolve differently in time, presenting an
increasing localization for larger values of $Re$, as provided in figure 8.
However, all of the wavepackets present an arrow-shaped structure, with the
downstream region essentially characterized by low-speed streaks, and the core
region filled with small-scale vortices together with some coherent streamwise
streaks. Moreover, the large-scale quadrupolar structure observed at $t=0$ is
maintained at target time for all the considered values of $Re$, as shown in
figure 9 (b) for $Re=1150$.

(a) $t=0$ (b) $t=T$

Figure 9: Isocontours of the crossflow energy $E_{cf}$ with the normalized
$y$-integrated large scale flow (vectors) for $Re=1150$, $E_{0}=4.7\times
10^{-8}$, $T=100$: (a) initial optimal solution, (b) optimal solution at the
target time.

The premultiplied energy spectra of the minimal seed obtained at $Re=1150$ are
reported in figure 10 by the coloured contours. The streamwise perturbation
energy peaks for $\lambda_{z}\approx 2.45$ and $\lambda_{x}\approx 3.78$,
whereas the wall-normal and spanwise perturbation energy have largest
amplitude at $\lambda_{z}\approx 1$, $\lambda_{x}\approx 4$ with a secondary
peak for $\lambda_{z}\approx 2$ and $\lambda_{x}\approx 25$. The lowest of
these wavelengths are consistent with those reported by Lemoult et al. (2014)
during the development of turbulent spots at similar Reynolds numbers, and are
linked to the finite size of the streaks. Whereas, the largest wavelengths are
directly linked to the streamwise and spanwise size of the wavepacket. Very
similar spectra are found to characterize the nonlinear optimal perturbations
at higher initial energy (solid contours in figure 10). The premultiplied
energy spectra of the minimal seeds obtained for the other considered values
of $Re$ are reported in figure 11 by the shaded contours ($Re=1000$) and the
dashed lines (black for $Re=1150$, blue for $Re=1250$, green for $Re=1568$).
As a consequence of the increased spatial localization, the overall
distribution of the energy spectra is displaced towards higher values of
$\lambda_{x},\lambda_{z}$ when $Re$ increases. This effect is coupled with a
narrowing and displacement of the spectra towards higher values of $y$, which
can be a consequence of an increased localization also in the wall-normal
direction. The primary and secondary peak values, reported in table 1 are also
considerably influenced by the Reynolds number.

(a) $k_{x}E_{uu}(k_{x})$ (b) $k_{x}E_{vv}(k_{x})$ (c) $k_{x}E_{ww}(k_{x})$ (d)
$k_{z}E_{uu}(k_{z})$ (e) $k_{z}E_{vv}(k_{z})$ (f) $k_{z}E_{ww}(k_{z})$

Figure 10: Logarithm of the premultiplied spectral energy versus the wall-
normal distance $y^{+}$ for the initial optimal solution at $Re=1150$ and
$T=100$ for $E_{0}=1.1\times 10^{-7}$ (coloured contours) and $E_{0}=4.7\times
10^{-8}$ (black contours). The white dots indicate the energy peaks.

(a) $k_{x}E_{uu}(k_{x})$ (b) $k_{x}E_{vv}(k_{x})$ (c) $k_{x}E_{ww}(k_{x})$ (d)
$k_{z}E_{uu}(k_{z})$ (e) $k_{z}E_{vv}(k_{z})$ (f) $k_{z}E_{ww}(k_{z})$

Figure 11: Logarithm of the premultiplied spectral energy versus the wall-normal distance $y^{+}$ for the initial optimal solution at different Reynolds numbers at $T=100$: $Re=1000$ for $E_{0}=5.5\times 10^{-7}$ (coloured contours), $Re=1150$ for $E_{0}=4.7\times 10^{-8}$ (black contours), $Re=1250$ for $E_{0}=2.9\times 10^{-8}$ (blue contours) and $Re=1568$ for $E_{0}=5.8\times 10^{-9}$ (green contours). The symbols indicate the peaks of the energy, also reported in table 1. $Re$ |  $(\lambda_{x})_{u}$ |  $(\lambda_{z})_{u}$ |  $(\lambda_{x})_{v}$ |  $(\lambda_{z})_{v}$ |  $(\lambda_{x})_{w}$ |  $(\lambda_{z})_{w}$
---|---|---|---|---|---|---
$1000$ |  $4.897\ (2.390)$ |  $2.117\ (3.307)$ |  $4.995\ (38.45)$ |  $1.049\ (2.198)$ |  $4.995\ (36.34)$ |  $2.080\ (1.042)$
$1150$ |  $3.784\ (1.618)$ |  $2.449\ (1.093)$ |  $4.163\ (25.56)$ |  $1.105\ (2.049)$ |  $4.028\ (25.56)$ |  $1.125\ (2.218)$
$1250$ |  $1.287\ (2.236)$ |  $2.153\ (2.114)$ |  $24.98\ (4.991)$ |  $1.049\ (0.508)$ |  $4.625\ (24.16)$ |  $1.105\ (2.639)$
$1568$ |  $3.244\ (1.916)$ |  $2.838\ (1.217)$ |  $3.518\ (18.75)$ |  $0.790\ (1.992)$ |  $3.6421\ (20.12)$ |  $1.759\ (0.958)$
Table 1: Streamwise and spanwise wavelengths $\lambda_{x,z}$ associated with
the primary and secondary (in brackets) peaks of the premultiplied energy
spectra of $u,v,w$, shown in figure (11) for different Reynolds numbers.

###  3.3 Minimal seed evolution in time

In this section, we analyse the time evolution of the minimal seeds towards
the turbulent bands. In figure 12 (a), the time evolution of the kinetic
energy obtained from direct numerical simulations initialised with the minimal
seeds, is reported. In all cases, the kinetic energy strongly increases in
time until saturating towards a statistically constant value. For the lowest
considered Reynolds numbers, we observe a rapid initial increase of the
kinetic energy, followed by a slow phase of saturation of the energy. Whereas,
for larger $Re$, the initial growth is slower and leads to lower values of the
kinetic energy at small time. One can notice once again that at $Re=1000$ the
flow appears to behave rather differently from what observed at larger
Reynolds numbers. However, the minimal seeds at larger $Re$, despite having
lower initial energy, tend towards higher values of the kinetic energy at
large times, suggesting that for larger Reynolds numbers, turbulence
eventually occupies a larger portion of the domain.

Figure 12: Time evolution of the kinetic energy for the minimal seeds obtained
for the different considered Reynolds numbers.

(a) $t=200$ (b) $t=500$ (c) $t=900$ (d) $t=1500$ (e) $t=2500$ (f) $t=4000$

Figure 13: Isocontours of the crossflow energy $E_{cf}$ together with the
normalized $y$-integrated large scale flow (vectors) for several instantaneous
fields ($Re=1150$, $E_{0}=4.7\times 10^{-8}$, $T=100$).

(a) $t=150$ (b) $t=500$ (c) $t=900$ (d) $t=1500$ (e) $t=2500$ (f) $t=3000$

Figure 14: Isocontour of the crossflow energy $E_{cf}$ together with the
normalized $y$-integrated large scale flow for several instantaneous fields
($Re=1000$, $E_{0}=5.5\times 10^{-7}$, $T=100$).

(a) $t=200$ (b) $t=500$ (c) $t=900$ (d) $t=1200$ (e) $t=1500$ (f) $t=3000$

Figure 15: Isocontour of the crossflow energy $E_{cf}$ together with the
normalized $y$-integrated large scale flow for several instantaneous fields
($Re=1250$, $E_{0}=2.9\times 10^{-8}$, $T=100$).

(a) $t=200$ (b) $t=300$ (c) $t=350$ (d) $t=500$ (e) $t=600$ (f) $t=900$

Figure 16: Isocontour of the crossflow energy $E_{cf}$ together with the
normalized $y$-integrated large scale flow for several instantaneous fields
($Re=1568$, $E_{0}=3.6\times 10^{-9}$, $T=150$).

This can be verified by analysing the time evolution of the crossflow energy
and $y$-averaged flow fields for the minimal seeds at different Reynolds
numbers. For $Re=1150$, figure 13 shows that, as previously noticed (see also
figure 6), the localised minimal solution breaks rapidly its symmetry along
the spanwise direction, presenting a clearly asymmetric (but still spatially
compact) structure at $t=500$, as shown in figure 13 (b). This asymmetric
wavepacket evolves via nucleation of new streaky structures (see Parente et
al. (2021) concerning the mechanism of creation of the streaks) in the
direction of the inclined laminar-turbulent interface, clearly forming a
singular turbulent band, as it can be observed for $t=900$. The newly-formed
turbulent band continues growing in an oblique direction with angle $\approx
28^{\circ}$ until reaching the periodic boundaries, where it interacts with
itself ($t=1500$). This triggers splitting of the previously isolated band
($t=2500$), which saturates reaching a laminar-turbulent pattern filling the
whole domain at $t=4000$. The same behaviour has been observed by Tao & Xiong
(2013) and Xiong et al. (2015) by injecting in a plane Poiseuille flow a
”seed” of the turbulent bands, similar in shape to the instantaneous field at
$t=500$ in figure 13. Analysing the large-scale flow, we can observe the
formation of a small recirculation zone upstream of the spot during its
evolution. Moreover, when the bands are formed, the large-scale flow is found
to turn clockwise around bands with positive angle and anti-clockwise around
bands with negative angle. In fact, all the bands are formed in correspondence
with the shear layer which divides the different vortices. A rather similar
behaviour is observed at $Re=1000$, as shown in figure 14. Despite at small
times ($t=150$) the minimal seed has evolved into an almost symmetric V-shaped
spot, one of its two legs weakens in time ($t=500$) and completely disappears
at $t=900$, evolving into a single band as observed for $Re=1150$. Whereas,
looking at the evolution in time for the cases at higher Reynolds number, the
flow presents the same behaviour observed experimentally and numerically when
turbulence is triggered by a spot (Carlson et al. (1982), Henningson & Kim
(1991), Aida et al. (2010), Aida et al. (2011)). In fact, the localised
perturbation initially evolves in the domain forming a turbulent spot (see
figure 5), turning into a V-shape at $t=500$, as shown in figure 15. At this
time, two distinct fronts of the spots can be observed, which evolve in two
symmetric bands with angle $\approx\pm 45^{\circ}$ growing obliquely in the
domain, as shown at $t=900$. At $t=1200$, they start to interact which each
other, forming a spatio-temporally complex final state composed by a
coexistence of turbulent and laminar patterns ($t=3000$). Qualitatively the
same behaviour is observed at $Re=1568$, as shown in figure 16, although the
spatial spreading of the bands appears to be more rapid than at lower $Re$,
despite the initial energy of the perturbation is lower. Also in these cases,
the bands are found to form right in the mixing layer between two large-scale
counter-rotating vortices. Notice also that the same quasi-symmetric behaviour
can be observed at lower $Re$, for a larger initial energy. In fact, the
nonlinear optimal perturbation computed for $Re=1150$ and $E_{0}=4.7\times
10^{-8}>{E_{0}}_{min}$ evolves in two distinct bands, showing a time evolution
corresponding to that of the minimal seed at largest $Re$ (not shown).
An explanation of this behaviour can be attempted by recalling that, in the
channel flow, turbulent stripes have a probability of decay that increases
with time, and that decreases with the Reynolds number (Paranjape, 2019).
Thus, as all minimal seeds present an almost spanwise-symmetric structure, two
proto-bands begin to be created at the edges of the large-scale vortices
characterising the minimal seed. However, the probability of decay of these
bands is higher for low Reynolds number, and increases in time, so when $Re$
is sufficiently low one of these bands rapidly dies out, leading to the
development of one isolated band. Increasing $Re$, the probability of decay of
an initial band is lower, while the probability of splitting increases. Thus,
both oblique bands originated at the sides of the minimal seed survive longer
in time, until they split and interact, rapidly leading to the establishment
of a spatio-temporally complex final state. Notice that injecting more initial
energy at low value of $Re$ has the same effect of increasing $Re$. In fact,
an optimal perturbation with $E_{0}>{E_{0}}_{min}$ is less spatially
localized, and is able to reach a much larger kinetic energy at $T=100$,
leading to more spatially-extended and energetic proto-bands, which allows
their sustainment for a longer time.

(a) $t=0$ (b) $t=100$ (c) $t=500$ (d) $t=900$

Figure 17: Spanwise distribution of the production ($P$, red line) and
dissipation ($-\epsilon$, black line) terms integrated in $x-y$ planes for
different instantaneous fields obtained evolving in time the minimal seed for
$Re=1150$, $E_{0}=4.7\times 10^{-8}$, $T=100$.

(a) $t=0$ (b) $t=100$ (c) $t=500$ (d) $t=900$

Figure 18: Spanwise distribution of the production ($P$, red line) and
dissipation ($-\epsilon$, black line) terms integrated in $x-y$ planes for
different instantaneous fields obtained evolving in time the minimal seed for
$Re=1250$, $E_{0}=2.9\times 10^{-8}$, $T=100$.

To corroborate this conjecture, we make use the Reynolds-Orr equation to
evaluate the production and dissipation of kinetic energy as, respectively:

$P=-{u}_{i}^{\prime}{u}_{j}^{\prime}\frac{\partial{U}_{i}}{\partial
x_{j}},\quad\quad\epsilon=\frac{2}{Re}{s_{ij}^{\prime}s_{ij}^{\prime}}\quad\text{with}\quad
s_{ij}^{\prime}=\frac{1}{2}\left(\frac{\partial{u}_{i}^{\prime}}{\partial
x_{j}}+\frac{\partial{u}_{j}^{\prime}}{\partial x_{i}}\right),$ (5)

where the Einstein summation convention has been used. One can compare the
time evolution of the production and dissipation terms, integrated in $x-y$
planes, for the minimal seeds at $Re=1150$ and $Re=1250$, provided in figure
17 and 18. For both Reynolds numbers, at $t=0$ the production presents one
single peak almost in the center of the spot ($z\approx 70$), which is found
to exceed dissipation of almost an order of magnitude. Production of kinetic
energy leads to a slight increase of the spanwise size of the wavepacket
($t=100$), together with a further increase of the kinetic-energy production,
probably due to the nucleation of new streaks which produce kinetic energy
thanks to the lift-up effect. Due to the breakdown of the structures inside
the spot, dissipation increases as well, reaching almost the same value than
the production term. Notice also that a weak secondary peak begins to be
visible in the production term. At $t=500$, the spot has strongly increased
its size in the spanwise direction, presenting an almost symmetric shape with
two peaks at $z\approx 55-60$ and $z\approx 80-85$. However, at $t=900$, the
evolution of the production and dissipation terms begins to strongly differ
between the two considered Reynolds numbers. For $Re=1250$, the spanwise
distribution of the dissipation and production terms remains almost spanwise
symmetric, centered at $z\approx 70$, with two distinct laminar-turbulent
fronts at $z\approx 20$ and $z\approx 110$ (see figure 18). Whereas, for
$Re=1150$, the left-most part of the packet has almost faded away, while the
right-most one has expanded up to $z\approx 110$, as shown in figure 17. The
analysis of the production and dissipation terms clearly indicates that the
minimal seed for turbulent bands leads to the generation of two almost
symmetric regions of high production and dissipation, which can be seen as two
distinct proto-bands. However, when the Reynolds number is lower, the weaker
of these two proto-bands rapidly decays, leading to the development of an
isolated band. Whereas, at larger Reynolds number, both bands survive for a
sufficiently long amount of time to begin interacting between themselves.

##  4 Conclusion

In this work we have investigated the minimal-energy perturbations for the
generation of turbulent bands in plane Poiseuille flow. A nonlinear
optimization maximising the kinetic energy at a given target time, coupled
with initial energy bisection, has been used. The optimization was performed
in very large domains, for a range of Reynolds number for which turbulent
bands are sustained and lead to a spatio-temporally complex turbulent-laminar
final state, namely $Re=1000,1150,1250,1568$ (the lowest value representing
the threshold $Re$ for which bands splitting and turbulence spreading can be
observed). The influence of the Reynolds number on the minimal energy
threshold for generating turbulent bands ($E_{{0}_{min}}$), is analysed. In
accordance with previous works carried out on other shear flows in small
domains, the minimal seed has been found to scale with $Re$ following a power-
law $E_{{0}_{min}}\propto Re^{-\gamma}$, although a sufficiently good fit is
found only by restraining the analysis to $Re>1000$. However, the value of
$\gamma$ recovered in the present work ($\approx 8.5$) is approximately $4$
time larger than the values reported in previous works ($\gamma\approx 2.7$ in
Duguet et al. (2013), $\gamma\approx 2$ in Cherubini et al. (2015)), probably
due to the much larger size of the considered domain.
For all the Reynolds numbers analysed, the minimal-energy perturbation able to
generate turbulent bands is a spatially localised spot-like structure composed
by finite-size streaks and elongated vortices. A more marked localization of
the minimal seed is found when $Re$ increases. As previously reported for the
channel flow in the presence of spatially-localised spots, a large-scale flow
having a quadrupolar structure has been found to surround the small-scale
localised minimal perturbations. These minimal perturbations have dominant
wavelengths $\approx 4$ and $\approx 1$ in the streamwise and spanwise
directions, respectively. Nonlinear optimal perturbations with energy higher
than minimal, are characterized by similar shape and wavenumbers.
The evolution of the minimal seeds towards the turbulent bands has been
investigated. For $Re<1250$, the minimal seeds evolve in time creating an
isolated oblique band. Whereas, for $Re\geq 1250$, it gives rise to two
distinct bands which grow almost symmetrically in the spanwise direction. This
almost symmetrical evolution is observed also at lower $Re$ for non-minimal
optimal perturbations. An analysis of the production and dissipation of
kinetic energy integrated in the streamwise and wall-normal directions shows
that in all cases the initial spot-like perturbation evolves in an almost
symmetric fashion, giving rise to two proto-bands at the edges of the large-
scale flow characterizing the minimal seeds. However, since the probability of
decay of bands increases in time and is higher for low Reynolds number, when
$Re$ is sufficiently low one of these bands rapidly dies out, leading to the
development of one isolated band. Whereas, for larger values of $Re$, the
probability of decay of an initial band is lower, while the probability of
splitting increases. Thus, both oblique bands originated at the sides of the
minimal seed survive longer in time, until they split and interact, rapidly
leading to the establishment of a spatio-temporally complex final state.
Injecting more initial energy at low value of $Re$ has the same effect of
increasing $Re$, since a more spatially-extended disturbance with higher
kinetic energy is reached at a small time, leading to longer and more
energetic proto-bands, able to be sustained for a longer time.
This work elucidates two (apparently distinct) minimal-energy mechanisms for
the generation of turbulent bands in channel flow. It appears that both the
initial and the final states are very sensitive to the energy and Reynolds
numbers characterising the flow, highlighting the complexity of the laminar-
turbulent patterned state and its initial seed. The selection of one of these
two mechanisms appears to be affected by the probability of decay of the
newly-created stripe, which increases with time, but decreases with the
Reynolds number. Future work will aim at extending the present investigation
to other shear flows presenting spatially-patterned turbulence.

## Acknowledgements

This work was granted access to the HPC resources of IDRIS under the
allocation 2020-A0072A06362 and A0092A06362 made by GENCI.

## Declaration of Interests

The authors report no conflict of interest

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