justified

# Mechanism of Roll-Streak Structure Formation and Maintenance in Turbulent
Shear Flow

Brian F. Farrell2  Petros J. Ioannou1,2
<EMAIL_ADDRESS>Marios-Andreas Nikolaidis1  1Department of Physics,
National and Kapodistrian University of Athens, Athens, Greece 2Department of
Earth and Planetary Sciences, Harvard University, Cambridge, U.S.A.

###### Abstract

In wall-bounded shear flow the primary coherent structure is the streamwise
roll and streak (R-S). Absent of an associated instability the R-S has been
ascribed to non-normality mediated interaction between the mean flow and
perturbations. This interaction may occur either directly due to excitation of
a transiently growing perturbation or indirectly due to destabilization of the
R-S by turbulent Reynolds stresses. A fundamental distinction between the
direct and the indirect mechanisms, which is central to understanding the
physics of turbulence, is that in the direct mechanism the R-S is itself the
growing structure while in the indirect mechanism the R-S emerges as a self-
organized structure. In the emergent R-S theory the fundamental mechanism is
organization by the streak of Reynolds stresses configured to support its
associated roll by the lift-up process. This requires that a streak organizes
turbulent perturbations such as to produce Reynolds stresses configured to
reinforce the streak. In this paper we provide detailed analysis explaining
physically why this positive feedback occurs and is a universal property of
turbulence in shear flow. DNS data from the same turbulent flow as that used
in the theoretical study (Poisseullle flow at $R=1650$) is also analyzed
verifying that this mechanism operates in DNS.

###### keywords:

##  1 Introduction

While advances in experiment, simulation and theory continue to be made, the
physical mechanisms underlying turbulence in shear flows remains incompletely
understood. The problem of shear flow turbulence can be divided into two
components: transition from the laminar to the turbulent state and maintenance
of the turbulent state. The transition problem is posed by the lack of an
inflection in the velocity profiles of boundary layer flows. Inflections are
associated by the Rayleigh theorem with existence of robust instabilities that
continue in viscous flows from the inflectional instability of the same
velocity profile in an inviscid flow. The problem of robust disturbance growth
in perturbation stable shear flows was solved when it was recognized that the
non-normality of the underlying linear dynamics allows perturbation growth in
the absence of exponential instability. The concept of transient growth in
shear flow has roots in the classical work of Kelvin and Orr (Kelvin, 1887;
Orr, 1907). Although three dimensional perturbations in the form of a roll-
streak structure were observed in boundary layers (Townsend, 1956; Klebanoff
et al., 1962; Kline et al., 1967; Blackwelder & Eckelmann, 1979; Robinson,
1991) and related to the nonmodal lift-up growth mechanism (Landahl, 1980),
comprehensive observational evidence for the mechanism of nonmodal growth in
boundary layer flows awaited the advent of DNS at Reynolds numbers ${\cal
O}(1000)$, for which turbulence is maintained, and particle image velocimetry
(PIV) of turbulent shear flows. The methods of non-normal operator analysis
and optimal perturbation theory were first applied in the context of
laboratory shear flows to two dimensional disturbances (Farrell, 1988). It was
believed at the time that secondary instability of finite amplitude two
dimensional equilibria were the mechanism of transition (Pierrehumbert, 1986;
Bayly et al., 1988) and it was shown that these unstable two dimensional
nonlinear finite amplitude equilibria could be readily excited by even very
small optimal initial perturbations (Butler & Farrell, 1994). However, it
became increasingly apparent from observation and simulation that the finite
amplitude structures associated with transition are three dimensional and
analysis of three dimensional optimal perturbation growth followed (Butler &
Farrell, 1992, 1993; Farrell & Ioannou, 1993a, b; Reddy & Henningson, 1993;
Trefethen et al., 1993; Schmid & Henningson, 2001). These analyses revealed
that the optimally growing three dimensional structure is associated with
cross-stream/spanwise rolls and associated streamwise streaks and is related
to the linear lift up mechanism. The remarkable convected coordinate solutions
for perturbation growth in unbounded shear flow (Kelvin, 1887) allow closed
form solution for the scale independent structures producing optimal growth in
three dimensional shear flow (Farrell & Ioannou, 1993a, b). These closed form
optimal solutions in unbounded shear flow confirm the result found numerically
in bounded shear flows that for sufficiently long optimizing times streamwise
rolls produce optimal energy growth while for short optimizing times the
optimal perturbations are oblique wave structures that synergistically exploit
both the two dimensional shear and the three dimensional lift up mechanisms
producing vortex cores oriented at an angle of approximately $60$ degrees from
the spanwise direction. And indeed, the roll-streak and oblique accompanying
structure complex that is predicted to produce optimal growth by analysis of
non-normal perturbation dynamics of shear flows has been convincingly seen in
both observations and simulations (Klebanoff et al., 1962; Schoppa & Hussain,
2000; Adrian, 2007; Wu & Moin, 2009; Jiménez, 2013) and shown to be
essentially related to the non-normality of shear flow dynamics (Kim & Lim,
2000; Schoppa & Hussain, 2002).

The most direct mechanism exploiting non-normality to form roll-streak (R-S)
structures is to introduce an optimal perturbation into the flow, perhaps by
using a trip or other device (Butler & Farrell, 1992; Reddy & Henningson,
1993; Trefethen et al., 1993). A related approach is to stochastically force
the flow which can be analyzed using stochastic turbulence modeling (STM)
(Farrell & Ioannou, 1993d, c, 1994, 1998a, 1998b; Bamieh & Dahleh, 2001;
Jovanović & Bamieh, 2005; Hoepffner & Brandt, 2008). Because of the non-normal
nature of perturbation growth in shear flow, stochastic turbulence models are
closely related to optimal perturbation dynamics. In conventional stochastic
turbulence models the R-S is envisioned to arise from chance occurrence of
optimal or near optimal perturbations in the stochastic forcing (Bamieh &
Dahleh, 2001; Hwang & Cossu, 2010a, b; McKeon & Sharma, 2010). These
mechanisms exploit the linear non-normal growth process directly. Although the
R-S structures could arise from an optimal or near optimal perturbation
occurring by chance in the free stream or instigated by a mechanism such as a
trip or a boundary injection, the ubiquity of the R-S structure in shear at
low levels of free stream turbulence (Adrian, 2007; Wu & Moin, 2009) suggests
a more systematic origin and a number of ideas have been advanced to explain
the continuous generation and maintenance of this structure in turbulent shear
flow (Schoppa & Hussain, 2002; Panton, 2001). However, the ubiquity of streak
formation suggests, as argued by Schoppa & Hussain (2000), that some form of
instability process underlies the formation of streaks and that this
instability involves an intrinsic association between the R-S structure and
associated oblique waves. Such a three dimensional instability must differ
qualitatively from the familiar laminar shear flow instability given that it
would necessarily violate the Squire theorem and that extensive search had
failed to reveal a candidate instability in the linearized Navier-Stokes
equations. Nevertheless, its existence was frequently inferred from experiment
and simulation e.g. Andersson et al. (1999) who concluded that the evidence
“..corresponds to some fundamental mode triggered in the flat-plate boundary
layer when subjected to high enough levels of free-stream turbulence..”.
Efforts to identify the mechanism underlying instability of the R-S structure
include exponential instability mechanisms (Brown & Thomas, 1977) and the
Craik-Leibovich instability (Phillips et al., 1996). Proposed algebraic growth
mechanisms involve a streamwise average torque produced by interaction of
discrete oblique waves (Benney, 1960; Jang et al., 1986).

In a study of a turbulent shear flow Hamilton et al. (1995) noted that the
Reynolds stresses arising from turbulent perturbations were systematically
correlated with the R-S so as to maintain the roll in the manner of a feedback
process referred to as the self-sustaining process (SSP). This correlation was
subsequently attributed to the structure of the Reynolds stresses produced by
inflectionsl instability of the streak (Waleffe, 1997), modal critical layer
fluxes (Hall & Smith, 1991; Hall & Sherwin, 2010) and transiently growing
structures (Schoppa & Hussain, 2002). However, suppression of unstable modes
has been demonstrated to have essentially no effect on maintenance of the R-S
providing a constructive proof that inflectional instability is not
responsible for the SSP (Farrell & Ioannou, 2012a; Lozano-Durán et al., 2021).
This left collocation of transiently growing structures as being responsible
for providing the roll maintaining torques as suggested by Schoppa & Hussain
(2002). The central physical problem posed by this result is explaining how
the perturbation Reynolds stresses are maintained by non-normal growth
processes to have the correct amplitude and structure to provide the roll
collocated torques required by the instability of the pre-transitional
boundary layer R-S as well as the time dependent R-S of the SSP in the
turbulent state.

The cross stream-spanwise roll structure provides a powerful mechanism for
forming streamwise streaks in shear flows when continuously forced by an
oblique wave structure. However, observation of the spatial correlation of
perturbation structures in turbulent flow does not support the existence of
extensive interfering oblique plane waves. Nevertheless, if we observe a
turbulent shear flow in the cross stream-spanwise plane at a fixed streamwise
location we see that at any instant there is a substantial torque from
Reynolds stress divergence forcing cross stream-spanwise rolls. The problem is
that this torque is not systematic and so it vanishes in temporal or
streamwise average. However, in the presence of a perturbation streak the
symmetry in the spanwise direction is broken and the torque from Reynolds
stress divergence can become organized to produce the positive feedback
between the streak and roll required to destabilize this structure by
continuously and coherently exploiting the powerful non-normal R-S
amplification mechanism. The existence of this mechanism for destabilizing the
R-S in turbulence makes it likely that some dynamical perturbation complex
exists to exploit it. That this is so was demonstrated by deriving from the
Navier-Stokes equations a system of statistical state dynamics (SSD) equations
closed at second order, eigenanalysis of which reveals the instability
responsible for destabilizing turbulent shear flow to streak formation
(Farrell & Ioannou, 2012a; Farrell et al., 2017b). This fundamental
instability underlying shear flow turbulence had evaded detection in part
because it has no counterpart in the linearized Navier-Stokes equations.

This emergent instability can be understood as a streak formation mechanism in
which background turbulence, rather than itself comprising the perturbation
the non-normal growth of which constitutes the streak, instead the
stochastically occurring perturbations are organized and collocated by the
streak to form optimally growing structures with oblique wave form that force
the cross stream-spanwise roll by inducing a Reynolds stress torque linearly
proportional to streak amplitude and collocated with the streak resulting in
an emergent exponential instability of the combined roll-streak-turbulence
complex.

It has been demonstrated constructively using second order SSD analysis of the
N-S equations that a dynamical perturbation complex exploiting the high non-
normality of the R-S plus turbulence complex exists to destabilize the pre-
transitional boundary layer forming a modal R-S and that this mechanism
persists subsequent to transition to maintain the temporally fluctuating R-S
complex in the fully turbulent state through a parametric instabiity (Farrell
& Ioannou, 2012a; Farrell et al., 2017b). However, the specific mean R-S
components and their supporting perturbation structures as well as the origin
and the mechanism by which these structures produce torques correctly
collocated to explain the ubiquitous occurrence of the mean R-S in wall-
bounded shear flow turbulence remains to be fully elucidated and its
predictions tested against DNS data. In this work we analyze the mechanism by
which turbulent perturbations are organized by streaks to force and maintain
the R-S and compare with DNS to verify that the Reynolds stress
destabilization mechanism occurs and is consistent with the observed formation
and maintenance of the R-S structure in turbulent Poiseuille flow.

##  2 Analysis of the forces responsible for the formation and maintenance of
streamwise-mean rolls

The flow velocity is decomposed into streamwise mean, denoted by
$\mathbf{U}=(U,V,W)$, with $U$ being the mean streamwise velocity, in the $x$
streamwise direction; $V$ the mean cross stream velocity in the $y$ cross-
stream direction; and $W$ the mean spanwise velocity in the spanwise $z$
direction. The velocity deviations from the mean, are $\mathbf{u}=(u,v,w)$,
and referred to as the perturbations. The non-dimensionalized equations
governing the mean and the perturbations flow and pressure fields $(P,p)$ in a
channel take the form

$\displaystyle\partial_{t}\mathbf{U}$
$\displaystyle+\mathbf{U}\bm{\cdot}\bm{\nabla}\mathbf{U}+\nabla
P-R^{-1}\Delta\mathbf{U}=-\overline{\mathbf{u}\bm{\cdot}\bm{\nabla}\mathbf{u}}~{},$
(1a) $\displaystyle\partial_{t}\mathbf{u}$
$\displaystyle+\mathbf{U}\bm{\cdot}\bm{\nabla}\mathbf{u}+\mathbf{u}\bm{\cdot}\bm{\nabla}\mathbf{U}+\bm{\nabla}p-R^{-1}\Delta\mathbf{u}=-(\mathbf{u}\bm{\cdot}\bm{\nabla}\mathbf{u}-\overline{\mathbf{u}\bm{\cdot}\bm{\nabla}\mathbf{u}})~{}.$
(1b)
$\displaystyle\bm{\nabla}\bm{\cdot}\mathbf{U}=0~{},~{}~{}~{}\bm{\nabla}\bm{\cdot}\mathbf{u}=0~{},$
(1c)

with no slip boundary conditions at the channel walls. The overline denotes
the streamwise average and $R$ is the Reynolds number of the flow. In the
equation for the mean (1a)
$-\overline{\mathbf{u}\bm{\cdot}\bm{\nabla}\mathbf{u}}$ represents the
Reynolds stress induced forcing of the mean by the perturbations. Assuming
unit density we do not insist on the nomenclature distinction between force
and acceleration so that divergence of the Reynolds stress induces a
streamwise mean force per unit mass in Eq. (1a), which is independent of $x$,
resulting in streamwise velocity component acceleration

$F_{x}=-\partial_{y}(\overline{uv})-\partial_{z}(\overline{uw})~{},$ (2)

and cross-stream and spanwise velocity component accelerations:

$F_{y}=-\partial_{z}(\overline{vw})-\partial_{y}(\overline{v^{2}})~{}~{},~{}F_{z}=-\partial_{y}(\overline{vw})-\partial_{z}(\overline{w^{2}})~{}.$
(3)

We want to examine the maintenance of the streamwise mean roll circulation
with velocity components $V$, $W$ and streamwise mean vorticity
$\Omega_{x}=\partial_{y}W-\partial_{z}V$. From (1a) we obtain that
$\Omega_{x}$ is governed by:

$\displaystyle\frac{D\Omega_{x}}{Dt}=G_{x}+\frac{1}{R}\Delta_{2}\Omega_{x}~{},$
(4)

with $D/Dt=\partial_{t}+V\partial_{y}+W\partial_{z}$ the substantial
derivative of the streamwise vorticity under advection by the streamwise mean
flow $(V,W)$. On the RHS of Eq. (4) appears the dissipation term, with the
notation $\Delta_{2}=\partial_{y}^{2}+\partial_{z}^{2}$ for the two-
dimensional Laplacian, and the Reynolds stress roll vorticity forcing term
arising from the turbulent perturbations

$G_{x}=\partial_{y}F_{z}-\partial_{z}F_{y}=\left(\partial_{z}^{2}-\partial_{y}^{2}\right)\overline{vw}+\partial_{yz}\left(\overline{v^{2}}-\overline{w^{2}}\right)~{}.$
(5)

In the absence of $G_{x}$ the mean streamwise vorticity decays as it then
obeys the advection-diffusion equation

$\displaystyle\frac{D\Omega_{x}}{Dt}=\frac{1}{R}\Delta_{2}\Omega_{x}~{},$ (6)

which implies

$\displaystyle\frac{d}{dt}\int\Omega_{x}^{2}dydz=-\frac{1}{R}\int|\bm{\nabla}\Omega_{x}|^{2}dydz~{}~{},~{}\frac{d}{dt}\int(V^{2}+W^{2})dydz=-\frac{1}{R}\int\Omega_{x}^{2}dydz~{},$
(7)

indicating decay in time of both the square vorticity,
$\int\Omega_{x}^{2}dydz$, and the energy of the roll circulation
$\int(V^{2}+W^{2})dydz$, as noted by Moffatt (1990). This diagnostic clearly
implies that the formation and maintenance of the streamwise mean roll
requires a systematic source of streamwise mean vorticity that can only be
provided by the rotational component of the Reynolds stress vorticity forcing
$G_{x}$. In wall-bounded flows with cross-stream mean flow shear such rolls
would lead to formation by the lift-up process of low and high speed spanwise
inhomogeneities in the mean streamwise flow, $U$, referred to as streaks. This
streak component is defined as $U_{s}=U-[U]_{z}$ where $[U]_{z}$ is the
spanwise average of $U$. These considerations require that a Reynolds stress
torque exists to maintain a streamwise roll circulation and that a properly
collocated streamwise roll necessarily forces a collocated streak so that the
R-S is a dynamical consequence of a streamwise mean Reynolds stress torque.
The dynamical hypothesis of the SSP (Hamilton et al., 1995; Waleffe, 1997) is
that the streak instigates the torque that maintains both the roll and the
streak itself through the lift-up process. If this is so then explaining how
the streak gives rise to its self-sustaining roll-inducing torque is the
central dynamical problem posed by the maintenance of wall-bounded turbulence.
Given that the torques are not dependent on modal instability (Farrell &
Ioannou, 2012a) theoretical explanation of this SSP requires identifying how
transiently growing structures produce the required torque.

In order to understand the mechanism of roll formation by perturbation
Reynolds stress we must first isolate the component of $\bm{F}=(F_{y},F_{z})$
responsible for forcing the roll circulation. It is clear from (5) that only
the rotational part of $\bm{F}$ is involved in roll formation, the substantial
part of $\bm{F}$ that is divergent being cancelled by the immediate
development of the pressure field required by continuity. The force $\bm{f}$
that remains from $\bm{F}$ to induce the roll circulation is obtained by a
Helmholtz-Hodge decomposition of the total Reynolds stress force $\bm{F}$ into
a divergent component, -$\bm{\nabla}\varphi$, which acts as a pressure force,
and a non-divergent component $\bm{f}=(f_{y},f_{z})$, so that:

$\bm{F}=\bm{f}-\bm{\nabla}\varphi~{},$ (8)

and $f_{y}=0$ at the boundaries of the channel. This Helmholtz-Hodge
decomposition in the channel domain is unique (Chorin & Marsden, 1997). The
non-divergent force field $\bm{f}$ has components

$\bm{f}=-\Delta_{2}^{-1}(\bm{\nabla}\times G_{x}\widehat{\mathbf{x}})]~{},§$
(9)

where $\widehat{\mathbf{x}}$ is the unit vector in the streamwise direction,
and $\Delta_{2}^{-1}$ the inverse of the two-dimensional Laplacian
$\Delta_{2}=\partial_{yy}+\partial_{zz}$, rendered unique by the boundary
conditions. The force-field $\bm{f}=(f_{y},f_{z})$ determines the roll
circulation, as it is the force-field that accelerates the $(V,W)$ velocity
field by:

$\partial_{t}V=f_{y}=-\Delta^{-1}_{2}\partial_{z}G_{x}~{}~{},~{}~{}\partial_{t}W=f_{z}=\Delta^{-1}_{2}\partial_{y}G_{x}~{}~{}.$
(10)

##  3 Properties of turbulent perturbation Reynolds stress in mean flows
without streaks

Consider a perturbation field $\mathbf{u}$ in a channel. The perturbation
dynamics linearized about the streamwise mean flow $U(y)\widehat{\mathbf{x}}$,
with $\widehat{\mathbf{x}}$ the unit vector in the streamwise direction, is
governed by the equations:

$\partial_{t}\mathbf{u}+U\partial_{x}\mathbf{u}+(\mathbf{u}\bm{\cdot}\nabla)U~{}\widehat{\mathbf{x}}+\nabla
p-R^{-1}\Delta\mathbf{u}=0~{},~{}~{}\nabla\bm{\cdot}\mathbf{u}=0~{},$ (11)

with no slip boundary conditions at the channel walls and periodic boundary
conditions in $x$ and $z$.

Because of the homogeneity in both the streamwise and spanwise direction we
can identify a component of the flow field satisfying (11) with the form

$(u_{s},v_{s})=\sin(k_{z}z)e^{ik_{x}x}(\widehat{u},\widehat{v}),~{}w_{s}=\cos(k_{z}z)e^{ik_{x}x}\widehat{w},~{}p_{s}=\sin(k_{z}z)e^{ik_{x}x}\widehat{p}~{},$
(12)

in which the Fourier components $\widehat{u}(y,t)$, $\widehat{v}(y,t)$,
$\widehat{w}(y,t)$ satisfy Eq. (11). Perturbations of the form (12) comprises
a superposition of two oblique plane waves in the $(x,z)$ plane with
wavevectors $(k_{x},±k_{z})$. Perturbations of the form (12) are referred to
as sinuous about the $z=0$ axis because the $v_{s}$ and $u_{s}$ components are
antisymmetric while the $w_{s}$ component is symmetric about $z=0$. In order
to complete the set of perturbations we choose as companion to the sinuous
perturbation field (12), the varicose perturbation field

$(u_{v},v_{v})=\cos(k_{z}z)e^{ik_{x}x}(\widehat{u},\widehat{v}),~{}w_{v}=-\sin(k_{z}z)e^{ik_{x}x}\widehat{w},~{}p_{v}=\cos(k_{z}z)e^{ik_{x}x}\widehat{p}~{}.$
(13)

In (13) the functions $(\widehat{u},\widehat{v},\widehat{w},\widehat{p})$ are
the same as those in Eq. (12), as any spanwise shift of the sinuous
perturbation field satisfies the perturbation equations in a spanwise uniform
mean flow, $U(y)$. Perturbations of the form (13), referred to as varicose
about the $z=0$ axis, have $v_{v}$ and $u_{v}$ components symmetric and
$w_{v}$ component antisymmetric about $z=0$.

Figure 1: The time mean streamwise flow $U_{m}(y)$ in a turbulent Poiseuille
half-channel at $R=1650$. Figure 2: Contours of the cross-stream velocity at
initial time of the first sinuous (left panel) and first varicose (right
panel) $T=10$ optimal with $k_{x}=3\alpha$. Both of these optimals achieve
energy growth $E_{opt}(T)=23.2$. Top panels: contours of cross-stream velocity
at $(x,y)$ cross section reveal the expected initial perturbation structure
leaning against the shear indicative of growth by the Orr mechanism. The mean
flow in Fig. 1 is shown by the white line. Bottom panels: contours of the
cross-stream velocity in the $(x,z)$ plane at the specified $y=0.225$. The
optimals have identical checker-board sinusoidal structure, with the varicose
shifted by a phase of $\pi/2$ in the spanwise direction.  Figure 3: The energy
growth as a function of time of the first five pairs of $T=10$ optimals on the
mean flow shown in Fig. 1 for perturbations with $k_{x}=3\alpha$ and at
$R=1650$. The mean flow has no streak and is exponentially stable.

Up to this point the symmetry axis $z=0$ is arbitrary, but with the emergence
of a streak it can be distinguished to be the spanwise location of the streak
center. When there is a streak $U_{s}(y,z)$ in the mean flow, which for
theoretical convenience will be made symmetric, the sinuous and varicose field
components will no longer be spanwise translations of each other and an
asymmetry between these two fields develops. Quantities derived exclusively
from sinuous perturbations will from now on be indicated with the subscript
$s$, while those of varicose form will be indicated with the subscript $v$. We
note the following general properties of the Reynolds stresses of sinuous and
varicose perturbation fields:

  1. (a)

When the mean flow has no streak and is spanwise homogeneous the sinuous and
varicose perturbation fields specified by (12) and (13) produce equal and
opposite force-fields $\bm{f}_{s}=-\bm{f}_{v}$. In this case it is immediate
that the Reynolds stresses of the sinuous

$\displaystyle\overline{v_{s}w_{s}}=\frac{1}{4}\sin(2k_{z}z)~{}{\rm
Re}(\widehat{v}\widehat{w}^{*})~{},~{}~{}\overline{v_{s}^{2}-w_{s}^{2}}=\frac{1}{2}\sin^{2}(k_{z}z)|\widehat{v}|^{2}-\frac{1}{2}\cos^{2}(k_{z}z)|\widehat{w}|^{2}~{}~{},$
(14)

and the varicose

$\displaystyle\overline{v_{v}w_{v}}=-\frac{1}{4}\sin(2k_{z}z)~{}{\rm
Re}(\widehat{v}\widehat{w}^{*})~{},~{}~{}\overline{v_{v}^{2}-w_{v}^{2}}=\frac{1}{2}\cos^{2}(k_{z}z)|\widehat{v}|^{2}-\frac{1}{2}\sin^{2}(k_{z}z)|\widehat{w}|^{2}~{}~{},$
(15)

induce equal and opposite torques
$G_{x}=\left(\partial_{z}^{2}-\partial_{y}^{2}\right)\overline{vw}+\partial_{yz}\left(\overline{v^{2}}-\overline{w^{2}}\right)$
and therefore equal and opposite force-fields $\bm{f}_{s}=-\bm{f}_{v}$.

Figure 4: Top panel: vectors of the total Reynolds stress forcing,
$\bm{F}=(F_{z},F_{y})$, given by (3), for the first sinuous optimal in the
mean flow shown in Fig. 1. Center panel: contours of the equivalent pressure
$\varphi$ of the irrotational part of the Reynolds-stress forcing shown in the
top panel. Vectors indicate the force field associated with this pressure
field. Bottom panel: vectors of the rotational component of the forcing,
$\bm{f}=(f_{z},f_{y})$, that is responsible for inducing the streamwise-mean
roll circulation. Shown is the forcing by the $T=10$ sinuous optimal with
$k_{x}=3\alpha$ at the initial time. Similar forcing structure characterizes
the optimal perturbation at later times.  Figure 5: Top panel: vectors of the
dominant Reynolds stress forcing (3), $\bm{F}_{2}=(F_{2z},F_{2y})$, with
$F_{2z}=-\partial_{z}w^{2}$ and $F_{2y}=-\partial_{y}v^{2}$ . Center panel:
contours of the equivalent pressure $\varphi_{2}$ opposing the irrotational
part of the Reynolds-stress forcing of the top panel with arrows showing the
force field associated with this pressure field. Bottom panel: vectors with
components of the rotational part of the forcing $\bm{f}_{2}=(f_{2z},f_{2y})$
that is primarily responsible for inducing the streamwise-mean roll
circulation. For the $T=10$ sinuous optimal with $k_{x}=3\alpha$ at the
initial time.

  2. (b)

In mean flows with a symmetric streak $U_{s}(y,z)$ exclusively sinuous or
exclusively varicose perturbations induce rotational force-fields $\bm{f}$,
with $y$ components that are symmetric about the symmetry axis of the streak,
$z=0$, and $z$ components that are antisymmetric, i.e.
$f_{y}(y,z)=f_{y}(y,-z)$ and $f_{z}(y,z)=-f_{z}(y,-z)$. This is a general
result arising from the symmetry properties of the velocity fields: under
reflection $z\to-z$ exclusively sinuous or varicose fields transform
$\overline{vw}$ antisymmetrically and $\overline{v^{2}-w^{2}}$ symmetrically,
implying from Eq. (5) that the torque $G_{x}$ transforms antisymmetrically,
$G_{x}\to-G_{x}$, and from Eq. (10) that the force field components transform
as: $f_{y}\to f_{y}$ , $f_{z}\to-f_{z}$.

Comments:

  1. 1.

The Reynolds stresses of sinuous and varicose perturbations (14) and (15) are
anisotropic and produce streamwise-mean torques when $k_{z}\neq 0$, i.e. when
two oblique perturbations with $k_{z}\neq 0$ and $k_{x}\neq 0$ interact.

  2. 2.

Property (a) implies that if a sinuous perturbation in a mean flow $U_{m}(y)$
induces a roll circulation, its companion varicose perturbation induces
exactly the opposite roll circulation, and consequently a statistically
unbiased and uncorrelated field of perturbations in a spanwise uniform mean
flow $U_{m}(y)$ can not induce a streamwise mean roll circulation. However, if
spanwise homogeneity is broken by the purposeful introduction of a sinuous or
varicose perturbation that breaks the spanwise homogeneity of the perturbation
field a roll circulation may be forced. The mechanism of oblique wave
interference forcing of roll circulation was proposed by Benney (1960) to be
responsible for the emergence of R-S by interfering T-S waves in the
experiments of Klebanoff et al. (1962).

  3. 3.

As discussed, a perturbation field that is spanwise homogeneous can not induce
coherent streanwise roll circulations. However, if the streamwise mean flow is
infinitesimally perturbed with a streak perturbation $\delta U_{s}$, a
homogeneous perturbation field will be distorted so as to break the symmetry
of sinuous and varicose components and roll circulations will be induced.
Remarkably, at all scales the perturbation field distorted by a streak results
in a rotational forcing configured to amplify the distorting streak
perturbation. Moreover, the roll forcing is not only correlated to amplify the
perturbing streak but also, for perturbationally small streak amplitude, to
result in roll forcing proportional to the instigating streak perturbation.
i.e. $\delta\Omega_{x}\propto\delta U_{s}$ leading to exponential instability
and the emergence of coherent finite amplitude streamwise R-S. This nonlinear
instability is a collective instability that has analytic expression only in
the equations for the statistics of the Navier-Stokes equations. This
instability was analyzed in the framework of a second-order closure of the
statistical-state dynamics in Farrell & Ioannou (2012b) and in the full
statistical dynamics of the Navier-Stokes equations in Farrell et al. (2017b).
This is the instability underlying the formation and maintenance of the R-S in
turbulent shear flow that was sought in the linear N-S equations without
success because it does not exist in the state component expression of the
dynamics.

Figure 6: Left panels: Contours in the $(x,y)$ plane of the cross-stream
velocity of the first sinuous $T=10$ optimal with $k_{x}=3\alpha$ at times
$t=5,10,15$ in the mean flow with no streak. The profile of the mean flow at
the cross-section is indicated with the white line. Right panels: Contours of
the low-speed streak and vectors at the corresponding times of the Reynolds
stress force field that induces the streamwise mean roll circulation in the
$(y,z)$ plane. This plot shows that the sinuous optimal at all times induces a
roll-circulation that tends through lift-up to form low and high speed
streaks. The corresponding varicose perturbation induces the exact opposite
circulation. The initial energy of the optimal is $E(0)=0.01$. Figure 7:
Vectors of the rotational component of the ensemble mean Reynolds stress
forcing $\langle\bm{f}\rangle=(\langle f_{z}\rangle,\langle f_{y}\rangle)$ of
the streamwise-mean roll circulation for the sinuous components of a
stochastically excited perturbation field in the turbulent mean flow shown in
Fig. 1 without a streak. The varicose part of the Reynolds stress induces in
this case the exact opposite circulation so that for the complete perturbation
field the net induced circulation is zero. The plot is for the $k_{x}=3\alpha$
perturbation component.  Figure 8: Time evolution of the acceleration of the
streamwise-mean streak $-\int_{0}^{1}dy~{}f_{y}U^{\prime}(y,0)$ at $z=0$ by
first four $T=10$, $k_{x}=3\alpha$ optimal perturbations. The Reynolds
stresses of the sinuous optimals (solid) produce a positive $f_{y}>0$, at the
symmetry axis near the wall, and consequently produce roll circulations that
tend to form a low-speed streak, while the varicose (dashed) induce equal and
opposite roll circulations that tend to form a high speed streak. The induced
streamwise acceleration by the optimals is negative at all times for the
sinuous perturbations and positive for the varicose. The initial energy of the
optimals is $E(0)=0.01$.  Figure 9: The cross-stream distribution of the time
averaged streak acceleration $\langle-f_{y}U^{\prime}(y,0)\rangle_{t}$ at the
streak symmetry axis induced by the first four $T=10$ optimals with streamwise
wavenumber $k_{x}=3\alpha$. Shown separately are the contributions from
perturbations, with sinuous (solid) and varicose (dashed) structure. The
initial energy of the optimals is $E(0)=0.01$. Figure 10: Left panel: Time
evolution of the acceleration of the streamwise-mean streak
$-\int_{0}^{1}dy~{}f_{y}U^{\prime}(y,0)$ at the symmetry axis $z=0$ induced by
the first sinuous (red) and its corresponding varicose (blue) optimal
perturbation as shown in Fig. 2. Most of the acceleration is due to the
contribution from the $\overline{v^{2}-w^{2}}$ Reynolds stress component
(dashed line), the smaller contribution of the $\overline{vw}$ component of
the Reynolds stress is indicated with the dotted line. The net acceleration
from this pair of optimals (black line) is zero. Right panel: The cross-stream
distribution of the time averaged acceleration $\langle-
f_{y}U^{\prime}(y,0)\rangle_{t}$. The contribution of $\overline{v^{2}-w^{2}}$
(dashed line) dominates that of the $\overline{vw}$ (dotted line). The initial
energy of the optimals is $E(0)=0.01$ and their streamwise wavenumber is
$k_{x}=3\alpha$. Figure 11: Top panels. Left: vectors of the rotational
component of the ensemble mean Reynolds stress forcing
$\langle\bm{f}\rangle=(\langle f_{z}\rangle,\langle f_{y}\rangle)$ of the
streamwise-mean roll circulation for the sinuous components of a
stochastically excited perturbation field with zero mean flow. Right: the
cross-stream component $\langle f_{y}(y,0)\rangle$ as function of $y$ at the
symmetry axis $z=0$ (black), the dashed-red line is the contribution to
$\langle f_{y}(y,0)\rangle$ from the $\overline{v^{2}-w^{2}}$ Reynolds stress,
and the dashed blue line is the contribution from the $\overline{vw}$ Reynolds
stress. Bottom panels: the corresponding $\langle\bm{f}\rangle$ for the
varicose perturbations, which induces the exact opposite circulation so that
for the complete perturbation field the net induced circulation is zero. The
plot is for the $k_{x}=3\alpha$ perturbation component, which has been
stochastically excited with spatially homogeneous forcing in the spanwise. In
the cross-stream direction the forcing was weighted by
$w=\tanh(y/(5\delta))+\tanh((L_{y}-y)/(5\delta))-1$, where $\delta$ is the
cross-stream grid spacing, in order to reduce the forcing adjacent to the
boundaries. The forcing amplitudes have been normalized by the energy input by
the stochastic excitation.

##  4 Roll circulation induced by the Reynolds stresses of sinuous and
varicose perturbations in a spanwise uniform flow

In the previous section it was shown that the Reynolds stresses of sinuous
perturbations in a spanwise uniform mean flow induce roll circulations that
are equal and opposite to those induced by their companion varicose
perturbations. We demonstrate in this section that sinuous perturbations with
symmetry axis $z=0$ induce roll circulations forming low-speed streaks at the
symmetry axis, while varicose perturbations induce high-speed streaks.

We choose as mean flow the spanwise-averaged turbulent flow $U_{m}(y)$
obtained from a turbulent channel Poiseuille DNS at $R=1650$ (cf. Fig. 1). Our
results have been confirmed to be insensitive to this specific choice. This
mean flow was chosen in order to facilitate comparison to DNS data. The
$(x,y,z)$ flow domain is
${\cal{D}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}[0,4\pi]\times[0,1]\times[-\pi/2,\pi/2]$,
and the perturbations $\mathbf{u}$ evolve according to (11), and satisfy
periodic boundary conditions in $x$ and $z$ and no slip boundary conditions at
$y=0,1$. The gravest streamwise wavenumber $k_{x}$ is $\alpha=2\pi/L_{x}$. Our
analysis will concentrate on the $k_{x}=3\alpha$ component of the flow because
in DNS this component’s Reynolds stresses contributed the most in inducing the
streamwise-mean roll circulation. The same qualitative behavior was obtained
for the other streamwise components of the flow, indicating that the results
discussed exhibit insensitivity to scale.

We consider the Reynolds stresses induced by the optimal perturbations on this
mean flow. A $T$-time optimal perturbation is the unit energy initial
perturbation, $\mathbf{u}(\mathbf{x},0)$, that leads to the largest energy
growth at time $T$, with energy growth:

$E_{opt}(T)={\rm
max}_{\mathbf{u}(\mathbf{x},0)}~{}\left(\frac{\int_{\cal{D}}d^{3}\mathbf{x}~{}|\mathbf{u}(\mathbf{x},T)|^{2}}{\int_{\cal{D}}d^{3}\mathbf{x}~{}|\mathbf{u}(\mathbf{x},0)|^{2}}\right)~{},$
(16)

(cf. Farrell (1988); Butler & Farrell (1992); Farrell & Ioannou (1996a)).
These perturbations provide an optimal orthogonal decomposition of the
perturbation field according to their growth over time $T$ in this flow. The
optimal perturbations in this mean flow, which is homogeneous both in the
streamwise and spanwise direction, are oblique plane waves of the form
$\mathbf{u}(y,t)e^{i(k_{x}x+k_{z}z)}$, characterized by their streamwise
Fourier wavenumber $k_{x}$ and spanwise wavenumber $k_{z}$. These oblique
plane waves can be combined to form symmetric and antisymmetric pairs in the
cross-stream $v$ velocity about $z=0$, which is the symmetry axis of the
streak that will be introduced in the sequel. In this way the optimals of this
spanwise uniform flow can be partitioned into sinuous and varicose
perturbations with structure given by Eqs. (12) and (13). The structure of the
cross-stream velocity of the first pair of sinuous and varicose $T=10$
optimals with $k_{x}=3\alpha$ are shown in Fig. 2. Because of the spanwise
translation symmetry of the mean flow each $(k_{x},k_{z})$ pair of optimals
with the same cross-stream structure will have identical energy evolution. For
example, the energy growth of the first five $k_{x}=3\alpha$ pairs of sinuous
and varicose $T=10$ optimal perturbations are shown in Fig. 3.

Figure 12:  Contours of the velocity of the equilibrium low-speed streak
(upper panel) and high-speed streak (lower panel) in turbulent Poiseuille flow
at $R=1650$ in a channel with $L_{x}=4\pi$ and $L_{z}=\pi$. The spacing of the
contours is $0.04$. Figure 13: Time evolution of the energy of the first
(solid lines) and second (dashed lines) $T=10$ optimals in a flow with a low
speed streak (left panel) and a high-speed streak with the same structure
(right panel) for streak amplitude $\varepsilon=0.4$ and $\varepsilon=1$. The
energy growth of the optimals in the mean flow with $\varepsilon=\pm 1$ are
indicated with red, those with $\varepsilon=\pm 0.4$ with blue, and those with
no streak $\varepsilon=0$ with black. The first optimal has sinuous structure
while the second has varicose structure. This figure shows that the spanwise
shear increases the energy growth of both perturbations but that the low-speed
streak supports substantially greater optimal growth compared to that
supported by the high speed streak. Perturbations have $k_{x}=3\alpha$. Figure
14:  Contours of the cross-stream velocity of the first sinuous (left panel)
and first varicose (right panel) $T=10$ optimal for $k_{x}=3\alpha$ at the
initial time with mean flow $U(y,z)=U_{m}(y)+U_{s}(y,z)$ in which $U_{m}(y)$
is the spanwise averaged turbulent mean flow shown in Fig. 1 and $U_{s}$ is
the equilibrium low-speed streak shown in Fig. 12 (upper panel). The presence
of the streak centers and confines the optimals at the streak minimum and
breaks the symmetry between the sinuous and varicose pairs. The sinuous
achieves the highest energy growth of $50$ at $t=10.4$, while the varicose
reaches energy growth $29$ at $t=8.8$. Top panels: contours of cross-stream
velocity in an $(x,y)$ cross section show the typical initial configuration of
an optimal perturbation which is that it leans against the shear as expected
for growth by the Orr mechanism. The profile of the mean flow is shown by the
solid white line. Bottom panels: contours of the cross-stream velocity in the
$(x,z)$ plane at $y=0.275$. The optimals are obliques waves configured to
transfer mean flow energy from the spanwise shear to the perturbation. The
profile of the mean flow is the solid white line. The flow is at $R=1650$ in a
doubly periodic channel with $L_{x}=4\pi$, $L_{z}=2$, and $\alpha=2\pi/L_{x}$.
Figure 15: Left panels: Contours in the $(x,z)$ plane of the cross-stream
velocity of the first sinuous $T=10$ optimal with $k_{x}=3\alpha$ at times
$t=5,10,15$ in the mean flow with the equilibrium low-speed streak. The
profile of the streak at the cross-section is indicated with the white line.
Right panels: Contours of the low-speed streak and vectors at the
corresponding times of the Reynolds stress force field that induces the
streamwise mean roll circulation in the $(y,z)$ plane. The maximum magnitude
of the force occurs here at the symmetry axis of the streak. This plot shows
that the sinuous optimal at all times induces a roll-circulation that tends
through lift-up to amplify the low-speed streak. The initial energy of the
optimal is $E(0)=0.01$. Figure 16: As in Fig. 15 for the first varicose
optimal. This plot shows that the varicose optimal at all times induces a
roll-circulation that tends through lift-up to destroy the low-speed streak.
However the varicose mode induces a weaker circulation than the sinuous and
the net circulation from the first sinuous and varicose optimals results in
amplification of the streak as shown in Fig. 17. The maximum magnitude of the
force does not occur at the symmetry axis of the streak.The initial energy of
the optimal is $E(0)=0.01$. Figure 17: As in Fig. 15 but vectors of the total
force $\mathbf{f}=(f_{y},f_{z})$ obtained by adding the force by the first
sinuous optimal to the force by the first varicose optimal. The initial energy
of the optimals is $E(0)=0.01$. Figure 18: As Fig. 14 but for the first
sinuous and varicose $T=10$ optimals for the mean flow
$U(y,z)=[U]_{z}-U_{s}(y,z)$, with high-speed streak the mirror image of the
low-speed streak of Fig. 12. The first optimal is still sinuous achieving
energy growth $32$ at $t=8.4$ while the second optimal is varicose and it
evolves almost identically with the sinuous achieving maximum energy growth
$29$ at $t=8.6$. Note that sinuous and the varicose first optimals have almost
the same oblique wave structure and that both perturbations are located at the
wings of the streak.  Figure 19: Contours of the equilibrium low-speed streak
and vectors of the ensemble mean force field, $\langle\mathbf{f}\rangle$, when
the $k_{x}=3\alpha$ perturbation field is stochastically excited white in
energy in a mean flow with the low-speed with $\varepsilon=1$ (upper panel)
and with the high-speed streak with $\varepsilon=-1$ (lower panel). The
response is normalized by the energy introduced in the flow, ${\rm
trace}(\bm{\mathsf{Q}})$, where $\bm{\mathsf{Q}}$ is the forcing covariance in
energy coordinates.

To understand the streamwise-mean roll circulation induced by these optimal
perturbations we first plot the divergent total force-field
$\bm{F}=(F_{y},F_{z})$ produced by the Reynolds stresses of the sinuous
optimal perturbation at the initial time in Fig. 4 (top panel) and perform a
Helmholtz-Hodge decomposition of this force-field in order to determine the
characteristics of the residual field $\bm{f}$ that determines the forcing of
the roll circulation. The total force field $\bm{F}$ with components
$F_{y}=-\partial_{z}\overline{vw}-\partial_{y}\overline{v^{2}}$ and
$F_{z}=-\partial_{y}\overline{vw}-\partial_{z}\overline{w^{2}}$ is strongly
divergent at the symmetry axis $z=0$ and is almost completely aligned in the
spanwise direction, $\hat{\mathbf{z}}$, i.e.
$\bm{F}\approx-\partial_{z}\overline{w^{2}}~{}\hat{\mathbf{z}}$. For sinuous
perturbations $-\overline{w^{2}}$ has a minimum at $z=0$ and therefore the
induced $\bm{F}$ is divergent in the region about $z=0$, as is evident in Fig.
4 (top panel). The opposite situation occurs for varicose perturbations. The
divergent component of $\bm{F}$ is opposed by the pressure field, $\varphi$,
that develops. Contours of the pressure, $\varphi$, and the resulting force
field opposing the divergent component of the Reynolds stress force,
$-\bm{\nabla}\varphi$, are shown in Fig. 4 (center panel). The force from the
pressure field is isotropic, and given that $\bm{F}$ is divergent along $z=0$,
the pressure force will be directed towards the channel boundary near the wall
and away far from the wall. This implies that as $\bm{F}$ has a very small
component in the cross-stream direction the residual $\bm{f}$ will have a
strong positive cross-stream component $f_{y}>0$ at the symmetry axis. Further
from the wall the residual force direction reverses, and the overall force
field $\bm{f}$ has quadrupole structure at the symmetry axis, as seen in Fig.
4 (lower panel).

Crucial to the above argument is the dominance of the force field induced by
the $\overline{v^{2}-w^{2}}$ components of the Reynolds stress over the
contribution of the $\overline{vw}$ Reynolds stress. This is shown in Fig. 5
where the Helmholtz-Hodge decomposition of the force field induced by this
Reynolds stress with components
$\bm{F}_{2}=(F_{2y},F_{2z})=(-\partial_{y}\overline{v^{2}},-\partial_{z}\overline{w^{2}})$
can be compared to the decomposition of the full force field field $\bm{F}$ in
Fig. 4. The dominance of the $\overline{v^{2}-w^{2}}$ Reynolds stress is
maintained during the whole evolution of the optimal and is the reason, as we
will see, underlying the direction of the induced roll-circulation of the
sinuous and the varicose perturbations. It should be noted that the smaller
contribution from the $\overline{vw}$ Reynolds stress changes structure and
sign during the evolution. We find that dominance of the
$\overline{v^{2}-w^{2}}$ Reynolds stress also characterizes the DNS
statistics.

The force field structure of $\bm{f}$ persists throughout the evolution of the
optimal, as shown in Fig. 6, and each developing optimal forces a coherent
roll circulation that through lift-up forms a low-speed streak near the wall
at the symmetry axis. This particular circulation structure results from the
sinuous form of the perturbations and is not limited to the first sinuous
optimal. We demonstrate that by calculating the ensemble mean force
$\langle\bm{f}\rangle$ that arises when all the sinuous components of the flow
are equally excited stochastically. The ensemble mean statistics are
calculated by determining the ensemble mean spatial covariance
$\bm{\mathsf{C}}$ of the velocity components which satisfies at statistical
equilibrium the Lyapunov equation

$\bm{\mathsf{A}}\bm{\mathsf{C}}+\bm{\mathsf{C}}\bm{\mathsf{A}}^{\dagger}=-\bm{\mathsf{Q}}~{},$
(17)

where $\bm{\mathsf{A}}$ is the linear operator governing the perturbation
dynamics in Eq. (11), and $\bm{\mathsf{Q}}$ is the spatial covariance of the
delta-correlated stochastic forcing, chosen here to be the identity in energy
coordinates so that equal energy input is imparted to all degrees of freedom
(cf. Farrell & Ioannou (1993d, 1996a)). The stability of $\bm{\mathsf{A}}$
ensures that the statistical steady state exists. From the covariance the
ensemble mean Reynolds stresses can be obtained and from them the ensemble
mean roll forcing and the lift-up induced streamwise acceleration cf. Farrell
& Ioannou (2012a).

The ensemble mean force $\langle\bm{f}\rangle$ from the sinuous components of
the stochastically excited flow is shown in Fig. 7. This is the force field
averaged over time if all sinuous optimals are excited and their Reynolds
stress contribution superposed. Fig. 7 shows that the ensemble mean Reynolds-
stress force-field from the sinuous perturbations induces roll circulations
that tend to form a low-speed streak near the wall at the symmetry axis.
Varicose perturbations induce the exact opposite roll-circulations and is not
shown. The resulting roll circulation at the center $z=0$ is similar to the
roll circulation induced by the first sinuous optimal. However the
stochastically excited flow results in a roll circulation that is concentrated
at the symmetry axis and has smaller spanwise wavenumber. This is caused by
cancellations of the induced circulations that occur far from the symmetry
axis as the roll circulations from the various sinuous optimals add
constructively at the symmetry axis and destructively away from it. This
phenomenon of localization of the roll circulation is observed during
transition to turbulence. Generically, streaks emerge during transition as an
exponential instability of the interaction of the mean flow and the
perturbation field in the background of free-stream turbulence, an instability
that has analytical expression in the framework of second order closure of the
statistical dynamics of the channel flow (S3T) (Farrell & Ioannou, 2012b;
Farrell et al., 2017b). The streaks that initially emerge are almost
sinusoidal and inherit the spanwise wavenumber of the most unstable mode
arising in the S3T closure equations. But as transition proceeds and the
streaks grow the spanwise length-scale increases dramatically as a broad
spectrum perturbation field starts being sustained with its Reynolds stresses
contributing to localizing the roll circulation.

To quantify the development over time of the rotational force $\bm{f}$
generating streaks by lift-up, the time development of quantity
$-f_{y}U^{\prime}(y,0)$ at the streak symmetry axis (with
$U^{\prime}(y,0)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\partial_{y}U$ at
the symmetry axis) is shown in Fig. 8 and Fig. 9. This quantity measures the
lift-up induced acceleration of the streamwise mean velocity at the symmetry
axis. Fig. 8 shows the time evolution of the net induced streamwise-mean
streak $-\int_{0}^{1}dy~{}f_{y}U^{\prime}(y,0)$ at the symmetry axis by the
first four pairs of sinuous and varicose optimals. The figure shows that at
all times the sinuous optimals induce a low speed streak, while the
corresponding varicose optimals induce an equal high-speed streak. The time
averaged cross-stream distribution of the induced streak acceleration
$\langle-f_{y}U^{\prime}(y,0)\rangle_{t}$ at the symmetry axis by the evolving
first four pairs of sinuous and varicose optimals is shown in Fig. 9. That the
contribution of the Reynolds stress component $\overline{v^{2}-w^{2}}$
dominates that of $\overline{vw}$ during the development of the first pair of
sinuous and varicose optimals is shown in Fig. 10.

We have seen that key property of the sinuous and varicose perturbations to
determining the direction of their induced roll forcing is the dominance of
the $F_{z}=-\partial_{y}\overline{vw}-\partial_{z}\overline{w^{2}}$ Reynolds
stress force component over the
$F_{y}=-\partial_{z}\overline{vw}-\partial_{y}\overline{v^{2}}$ component.
Dominance of $F_{z}$ is due to dominance of the $\overline{w^{2}}$ over
$\overline{v^{2}}$, which in turn is due to the $\overline{v^{2}}$
fluctuations being preferentially suppressed compared to the
$\overline{w^{2}}$ by the solid boundary condition at the wall. This asymmetry
between $\overline{w^{2}}$ and $\overline{v^{2}}$ near a boundary is a
property connected to the presence of the wall and is not fundamentally
dependent on the presence of wall-normal shear or the presence of a particular
type of perturbation, such as an optimally growing perturbation, other than
its being sinuous or varicose. We demonstrate this by showing that
stochastically excited sinuous perturbations in a channel with zero mean flow
produce a roll-inducing Reynolds stress force field. This resulting roll-
inducing force field, shown in Fig. 11, has the characteristic quadrupole
structure and is concentrated near the walls as is the case for optimal
perturbations of sinuous and varicose form. When the mean flow has wall-normal
shear the roll-inducing force field is concentrated in the shear region, as
shown in Fig. 7, and in this case it is the energy bearing optimal
perturbations, and their $\overline{v^{2}}$ field decaying faster than their
$\overline{w^{2}}$ field near the wall, that dominate the perturbation field
and are responsible for the structure of the resulting Reynolds stress roll-
forcing.

Figure 20: The magnitude of the maximum ensemble mean force
$\mathbf{f}=(f_{y},f_{z})$ produced by the Reynolds stresses as a function of
the amplitude of the streak $\varepsilon$. This maximum occurs at the symmetry
axis of the streak. The roll-circulation induced is in a direction to
accelerate both the low-speed, $\varepsilon>0$, and the high-speed streaks,
$\varepsilon<0$. The dashed green line is the force induced when the high-
speed streak has the structure of the equilibrium high-speed streak shown in
Fig. 12 (lower panel). The $k_{x}=3\alpha$ perturbations are stochastically
excited by a stochastic excitation that injects ${\rm trace}(Q)$ units of
energy density per unit time.

##  5 Roll circulation induced by the Reynolds stresses of sinuous and
varicose perturbations in the presence of a streak

We have demonstrated that oblique waves, individual optimal perturbations,
sums of optimal perturbations and general stochastically forced perturbations
with sinuous form force roll circulations as do oblique waves, optimals, sums
of optimals and general perturbations with varicose form. However, the
tendency of sinuous perturbations to form rolls configured to force low speed
streaks is exactly canceled in a spanwise statistically homogeneous flow by
the tendency of varicose optimals to form rolls with opposite sign configured
to force high speed streaks. It remains to show how the imposition of a
perturbation streak breaks the cancellation of tendencies in forcing between
the sinuous and varicose components resulting in the instability that is
responsible for the formation and maintenance of the R-S in shear flow.

In this section we discuss the Reynolds stresses induced by perturbations in
mean flows with a streak of the form $U=U_{m}(y)+\varepsilon U_{s}(y,z)$, with
$\varepsilon$ a parameter modulating the amplitude and sign of the streak. The
streamwise mean flow $U_{m}(y)$ is the half-channel time mean flow in
Poiseuille turbulence at $R=1650$, shown in Fig. 1, and $U_{s}(y,z)$ is the
equilibrium low-speed streak shown in Fig. 12 (upper-panel) which is obtained
by time averaging the low-speed streaks of the turbulent flow. With this
choice of $U_{s}$, when $\varepsilon>0$ low-speed streaks are introduced in
the flow with the shape of the low-speed streak, and when $\varepsilon<0$
high-speed streaks are introduced that are mirror images of the corresponding
low-speed streaks. We choose this family of high-speed streaks, instead of
those that arise from the equilibrium high-speed streak (cf. lower panel of
Fig. 12) in order to reduce the number of parameters. Comparisons were made
with high-speed streaks in the shape of the equilibrated high-speed streak of
Fig. 12. The results presented here were found to be robust and not sensitive
to this detail.

Upon introduction of a spanwise symmetric streak the perturbations were found
to localize about the center of the streak. The sinuous and varicose
perturbations continue to induce roll circulations predominantly from the
Reynolds stress associated with $\overline{v^{2}-w^{2}}$, as previously
described, and as previously described these roll circulations induce
respectively low and high-speed streak perturbations. However, the
introduction of the streak creates a crucial difference which is an imbalance
between the sinuous and varicose perturbations. With no streak the net
circulation induced by corresponding sinuous and varicose perturbations
cancel. In the presence of a low-speed streak the sinuous perturbations are
favored over the varicose and the induced net roll circulation tends to
reinforce the low-speed streak, while in the presence of a high-speed streak
the varicose perturbations are favored resulting in a net roll circulation
that tends to reinforce the high-speed so that in both cases the Reynolds-
stress induced circulation reinforces the pre-existing streak. We will
demonstrate this dynamics for the first pair of optimal perturbations in the
flow and for the ensemble response when a flow with a streak is stochastically
excited by temporally and spatially uncorrelated forcing.

We choose $k_{x}=3\alpha$ perturbations. This choice of streamwise wavenumber
was made because it is the wavenumber that induces the strongest roll
circulation both in DNS and when the flow with the equilibrium low-speed
streak is stochastically excited. This choice of streamwise wavenumber for the
optimal does not change qualitatively the results that are presented. The
energy growth of the first pair of $T=10$ sinuous and varicose optimals is
shown in Fig. 13 for low-speed streaks (left panel) and their mirror high-
speed streaks (right panel) with with streak amplitude $\varepsilon=0,\pm
0.4,\pm 1$. This figure shows that the optimal perturbation growth increases
as the amplitude of the streak increases and that the increase is substantial
when the streak is low-speed and marginal when the streak is high-speed.
Energy transfer from the mean spanwise shear to the perturbations,
$-\int_{\cal D}dydz~{}\overline{uw}U_{z}$, is the energy source that accounts
for the increased perturbation growth in the presence of the streak, and
especially so when a low-speed streak is present because flows with low-speed
streaks have a relatively smaller wall-normal shear and the perturbations are
less readily sheared over by the wall-normal shear, which limits their
potential growth.

Differences in the growth of perturbations in flows of the form
$U_{m}(y)\pm\varepsilon U_{s}(y,z)$ is expected, because the flows are not
mirror images of each other. But such pronounced asymmetry in the growth of
optimal perturbations in low-speed and their mirror high-speed streaks is
surprising and has dynamical implications. It implies, as we will show, that
high-speed streaks are supported weakly by their Reynolds stresses, which
provides an explanation for the dominance of low-speed streaks in wall-bounded
turbulence. Fig. 13 also shows that in low-speed streak flows the sinuous
optimal perturbations achieve substantially greater energy growth than their
companion varicose perturbations. The structure of the top pair of optimals is
shown in Fig. 14. As $\varepsilon$ increases the perturbations become
increasingly localized at the center of the low-speed steak. Note that the
sinuous perturbations are concentrated at the wings of the streak, where there
is larger spanwise mean shear. This is consistent with the observation that
sinuous perturbations grow more than the corresponding varicose perturbation,
which are concentrated at the center of the streak where the shear is small.

Figure 21: Contours of the low-speed streak (solid: negative values, dashed:
positive values) and vectors of the force field, $\langle\mathbf{f}\rangle$,
induced by the time mean Reynolds stresses in the DNS low-speed streak. Upper
panel: the force field induced by the sinuous components of the flow. In
agreement with the discussion, they tend to support the low-speed streak.
Lower panel: the force field induced by the varicose components of the flow,
which tends to destroy the low-speed streak. The force field of the total
perturbation field is shown in Fig. 22 (upper-panel). Figure 22: Vectors of
the force field, $\langle\mathbf{f}\rangle$ induced by the time mean Reynolds
stresses in the DNS low-speed streak (upper panel) and the high-speed streak
(lower panel). The contours show the respective profile of the average low and
high speed streaks in the DNS Top panel: solid line contours for negative
values, dashed for positive values. Bottom panel: solid line contour for
positive values, dashed for positive values. The contour level is 0.03, cf.
Fig. 12. Figure 23: The cross-stream distribution of the net mean acceleration
$-f_{y}U^{\prime}(y,0)$ from the sinuous (red lines) and varicose (blue lines)
$k_{x}=3\alpha$ perturbations in the time-averaged low-speed streak (left
panel) and in the time averaged high-speed streak (right panel). The streaks
are shown in Fig. 12. The dashed lines shows the contribution to the induced
acceleration from the Reynolds stress $\overline{v^{2}-w^{2}}$ and the dotted
lines the contribution from the $\overline{vw}$ Reynolds stress. The net
Reynolds stress acceleration from the total perturbation field tends to
support both the low-speed streak and the high-speed streak. Data from a DNS
simulation.

The time development of the sinuous optimal structure and plots of the
resulting rotational Reynolds stress force field $\bm{f}$ are shown in Fig.
15. This figure shows that the sinuous optimal induces a coherent roll-
circulation that tends to reinforce the low-speed streak. The reverse roll
circulation is induced by the companion varicose optimal, shown in Fig. 16.
However, in the presence of the low-speed streak the streak opposing
circulation of the varicose perturbation is weaker than the streak amplifying
circulation of the sinuous optimal. Hence, in the presence of even the
slightest streak, the net circulation from the top pair of optimals tends to
reinforce the low-speed streak, as indicated in Fig. 17. The reverse situation
occurs in the high-speed streak. In that case although at $\varepsilon=-1$ the
sinuous optimal grows slightly more than the varicose, the Reynolds stresses
from the varicose dominate and the net circulation tends to reinforce the
high-speed streak. The optimals in this case have very similar structure (cf.
Fig. 18) and nearly identical evolution and the net circulation induced from
the superposition of the opposing and almost identical sinuous and varicose
components is weak compared with the circulation induced by the optimals in
the low-speed streak flow.

We confirm the robustness of our conclusions about the roll forcing pattern
that was obtained by analysis of the top pair of optimals by considering the
general case of roll forcing by the ensemble mean Reynolds stresses of the
perturbation field resulting when all degrees of freedom are excited
stochastically and equal in energy. The ensemble mean force field $\bm{f}$
induced when there is a low-speed streak present in the flow is shown in Fig.
19 (upper-panel), showing that in low-speed streak flows the sinuous
perturbation component dominates the statistics. In a high-speed streak flow,
Fig. 19 (lower-panel), the induced roll circulation reverses with the varicose
modes dominating the statistics. Note that the induced force is stronger when
there is a low-speed streak. The dependence of the amplitude of the induced
force-field on the amplitude of the streak is plotted in Fig. 20 from which it
is apparent that a low-speed streak is more vigorously supported by the
Reynolds - stresses and this remains true when the calculation is repeated
using the high-speed streak with the structure of the equilibrium high-speed
streak that that is obtained in DNS (cf. lower panel of Fig. 12). The
direction of the induced force, which always tends to increase the pre-
existing streak, and the linear dependence of the induced force on streak
amplitude, when the the amplitude of the streak is small, indicates that the
interaction between streaks and perturbations is consistent with the necessary
requirement for producing an exponential instability. This instability has
been analytically studied using the S3T form of statistical state dynamics
(Farrell & Ioannou, 2012a; Farrell et al., 2017b).

##  6 Comparison with DNS data

Abbreviation | $[L_{x},L_{y},L_{z}]$ | $[L^{+}_{x},L^{+}_{y},L^{+}_{z}]$ | $N_{x}\times N_{z}\times N_{y}$ | $R_{\tau}$ | $R$
---|---|---|---|---|---
NL100 | $[4\pi\;,\;2\;,\;\pi]$ | [1264 , 201 ,316] | $128\times 63\times 97$ | $100.59$ | 1650
Table 1:  The simulation of the plane parallel Poiseuille flow was performed
in a channel of streamwise, wall-normal and spanwise length
$[L_{x},L_{y},L_{z}]$, with periodic boundary conditions in $x$ and $z$ and no
slip boundary conditions at the channel walls $y=0$ and $y=2$. Lengths have
been made nondimensional by $h$ the channel half-width, velocities by $\langle
U\rangle_{c}$, the center velocity of the time-mean flow, and time by
$h/\langle U\rangle_{c}$. The Reynolds number is $R=\langle
U\rangle_{c}h/\nu$, with $\nu$ the kinematic viscosity.
$[L^{+}_{x},L^{+}_{y},L^{+}_{z}]$, indicate the domain size in wall-units.
$N_{x}$, $N_{z}$ are the number of Fourier components after dealiasing and
$N_{y}$ is the number of Chebyshev components. $R_{\tau}=u_{\tau}h/\nu$ is the
Reynolds number of the simulation based on the friction velocity
$u_{\tau}=\sqrt{\nu\left.\textrm{d}\langle U\rangle/\textrm{d}y\right|_{\rm
w}}$,where $\left.\textrm{d}\langle U\rangle/\textrm{d}y\right|_{\rm w}$ is
the shear at the wall.

Observational support for the theoretical arguments that we have presented was
obtained in the DNS data of Poiseuille flow at $R=1650$. Details of the
simulation are given in Table 1. From the DNS we have obtained the average
low-speed streak and average high-speed streak, shown in Fig. 12, along with
their associated perturbation field statistics. The streaks, which were nearly
symmetric in the spanwise direction, were symmetrized and the average Reynolds
stresses of the sinuous and varicose components of the perturbation field were
obtained. We present here the roll-circulation induced by the $k_{x}=3\alpha$
component of the perturbation field that produces the largest streamwise-mean
torque in DNS.

The roll-inducing force field $\bm{f}$ produced by the sinuous component of
the perturbation field of the low-speed streak is shown in Fig. 21 (upper
panel) and that produced by the varicose component in Fig. 21 (bottom panel).
The sinuous component of the DNS tends to amplify the low-speed streak, while
the varicose tends to induce a weaker reverse circulation, in agreement with
the theoretical discussion in the earlier sections. In Fig. 22 (top panel) the
roll-inducing component of the force field, $\bm{f}$, produced by the entire
perturbation field is shown, which, as discussed above, supports the low-speed
streak. It was confirmed that the rotational force $\bm{f}$ of the total
perturbation field is very close to the sum of the rotational forcing by the
sinuous and varicose components. This indicates that there is no substantial
correlation between the sinuous and varicose components of the flow giving
rise to cross-term contributions to the Reynolds stresses, so that the total
rotational force $\bm{f}$ and that obtained from the analysis performed in
this paper, in which the sinuous and varicose components independently add, is
justified. In Fig. 22 (bottom panel) is shown the roll-inducing force field
$\bm{f}$ of the total perturbation field in the high-speed streak, which as
expected supports the high-speed streak, while the support is weaker than that
of the $\bm{f}$ in the low-speed streak, as in our previous discussion.

In Fig. 23 we plot the contribution of the $\overline{v^{2}-w^{2}}$ and
$\overline{vw}$ Reynolds stresses to the cross-stream acceleration of the
sinuous and varicose components of the perturbation field in the mean low-
speed streak (left panel) and the mean high-speed streak (right panel). As
expected dominant contribution to $\bm{f}$ is confirmed to be due to
$\overline{v^{2}-w^{2}}$.

##  7 Discussion

The study of turbulence dynamics has roots in interpretation of spatial and
temporal spectra (Taylor, 1935, 1938; Kolmogorov, 1941). At the time, data
were available from two point space or time correlations which allowed spectra
to be determined. Unfortunately this required the phase of Fourier components
to be left indeterminant and so implicitly the maximum entropy assumption of
random phase was made. The result was that turbulence theory neglected the
role of coherent structures the study of which requires determining the phase
of Fourier components. The origin of structures elicited by the observational
techniques available was commonly ascribed to modal instability based on
Rayleigh’s theory of modes (Rayleigh, 1880, 1896). However, smoke and hydrogen
bubbles tended to reveal three dimensional structures that did not correspond
with modes of maximal growth rate and in fact the canonical laboratory shear
flows did not support growing modes. The first 3D turbulence to be
comprehensively observed was that of the baroclinic turbulence of the
midlatitude atmosphere. This was accomplished soon after the invention of the
vacuum tube which allowed telemetry from weather ballooons to be made over the
entire troposphere. Twice daily radiosonde observations covering the entire
Northern Hemisphere soon followed. The primary coherent structures revealed by
this comprehensive data set was the midlatitude cyclone and the jet stream.
The midlatitude cyclone was quickly identified with an unstable mode of the
equations of motion linearized about the mean jet flow (Charney, 1947; Eady,
1949) while the coherent jet stream structure was determined to result from
the cooperative interaction between the midlatitude cyclones and the jet, the
requisite upgradient momentum flux required for the cyclone perturbations to
maintain the jet being ascribed to “negative viscosity” (Jeffreys, 1933;
Starr, 1953, 1968). It is remarkable that the two primary mechanisms of
coherent structure formation in turbulence, selective perturbation growth in
the linearized equations of motion and cooperative nonlinear interaction
between essentially stochastic turbulent perturbations and the coherent
structure had already been recognized within a decade after comprehensive
observations of a turbulent flow became available. While the fundamental
energy source for maintaining the coherent structure by these two dynamical
mechanisms had been identified, the predictions of the analysis of these
mechanisms was soon shown to be at variance with observation: the growth of
cyclones did not correspond with the growth of modes and the satisfaction of
necessary conditions for instability did not allow prediction of cyclone
formation. In the case of the second mechanism, while negative viscosity
provided a suggestive analogy for the upgradient eddy momentum fluxes giving
rise to the polar jet; as a parameterization in the equations of motion it did
not result in an analytic theory that predicted jet stream formation in
agreement with observations. Lack of agreement of observed cyclogenesis with
modal theory was subsequently explained by identification of cyclone formation
with non-normal transient growth of stable optimally growing perturbations
arising in the turbulence (Farrell, 1982, 1984, 1985, 1989a, 1989b) in which
case the dynamics is properly analyzed using generalized stability theory
(GST) (Farrell & Ioannou, 1996a, b). Understanding the mechanism of jet stream
formation has roots in the identification of equilibrium structures in
stochastic turbulence models (DelSole & Farrell, 1996) and obtained full
analytic development with the advent of statistical state dynamics methods
(Farrell & Ioannou, 2003, 2007; Srinivasan & Young, 2012; Parker & Krommes,
2013, 2014; Constantinou et al., 2016; Farrell et al., 2017a; Farrell &
Ioannou, 2019). In wall-bounded shear flow turbulence theory the non-normal
transient growth mechanism for analyzing coherent structure formation was
introduced in the case of 2D structures in Farrell (1988) and in the case of
3D structures in Butler & Farrell (1992, 1993); Farrell & Ioannou (1993a, b);
Reddy & Henningson (1993); Trefethen et al. (1993); Schmid & Henningson
(2001). Formation of the R-S by a cooperative instability mechanism was
advanced by Hamilton et al. (1995); Waleffe (1997) and the operative mechanism
was identified in Farrell & Ioannou (2012a); Farrell et al. (2017b). More
recently methods based on GST have been advanced ascribing the R-S to the
growth of individual optimals (Jiménez, 2013, 2018) and alternatively to the
excitation of the zero frequency resolvent (McKeon & Sharma, 2010; McKeon,
2017). However, in the case of 3D wall-bounded shear flow it has not been
determined whether and in which cases the non-normal transient growth
mechanism explains an observed R-S and in which cases the observation is
explained by the cooperative instability mechanism, that is whether SSD
analysis rather than non-normal transient growth analysis using GST is
appropriate.

##  8 Conclusions

Our analysis reveals the remarkable universal tendency of sinuous
perturbations in channel flows to induce through their Reynolds stresses roll
circulations and also for varicose perturbations to induce roll circulations
of opposite sign. This happens even in a stochastically forced channel with no
mean flow. We have ascribed this result to the presence of solid wall
boundaries which constrain the flow so that the wall normal velocity
fluctuations decay to zero faster than the spanwise velocity fluctuations.
However, when the mean flow has no spanwise variation the sinuous and varicose
perturbations induce roll circulations that cancel each other. Furthermore, we
have shown that even a streak at perturbative amplitude can organize turbulent
perturbation Reynolds stresses so as to amplify the perturbing streak which
destabilizes the R-S in turbulent shear flows. The reason for this universal
streak amplification property is revealed by the partitioning of the turbulent
perturbation field into sinuous and varicose components. With this partition
the breaking of the tendency for cancellation of the opposing sinuous and
varicose structure Reynolds stresses in a spanwise homogeneous perturbation
field is found to be broken by the imposition of the streak resulting in the
sinuous component dominating in the case of a low speed streak and the
varicose component dominating in the case of a high speed streak. This result
when coupled with the fact that the sinuous perturbations are favored in
growth by the low speed streak while the varicose are favored by the high
speed streak provides an analytic explanation for the universal streak
amplification property in shear flow. We find also that the Reynolds stress
forcing of the roll circulations of low-speed streaks is stronger than that of
high-speed streaks, indicating that a contributing factor for the observed
relative weakness of high-speed streaks in wall-turbulence is the weaker
forcing by the Reynolds stresses that the high speed streak organizes.

As the origin and maintenance of the R-S is central to the theory of
turbulence in shear flow, in this work we have performed an in depth analysis
of the physical mechanism by which the R-S arises in Poisseullle flow at
$R=1650$. We find that the R-S arises by organization of Reynolds stresses
producing roll inducing torques resulting in an SSP in agreement with
predictions of the cooperative SSD mechanism. In addition we have shown that
turbulent Reynolds stresses as predicted and required by this SSD cooperative
perturbation-roll-streak mechanism are observed to be in agreement with DNS
data of turbulence of the same flow which supports the conclusion that this
mechanism we have identified continues at finite amplitude to support the
maintenance of the turbulent state.

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