12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

STRUCTURED INPUT-OUTPUT ANALYSIS OF OBLIQUE TURBULENTBANDS IN TRANSITIONAL PLANE COUETTE-POISEUILLE FLOW

Yu Shuai
Dept. of Mechanics & Eng. Sci.
Peking University
Beijing, 100871, China
1800011056@pku.edu.cn

Chang Liu
Dept. of Physics
University of California, Berkeley
Berkeley, CA, 94720, USAchang liu@berkeley.edu

ABSTRACT

Dennice F. Gayme
Dept. of Mechanical Eng.
Johns Hopkins University
Baltimore, MD, 21218, USA
dennice@jhu.edu

coherent structures in channel flows also highlight the large
domain sizes required to characterize the laminar-turbulent interface of these flow patterns (Schneider et al., 2010). Theselarge channel extents increase the computational costs of DNS
and complexity of experiments needed to further study thesestructures.
Input-output analysis of the linearized Navier-Stokes(LNS) equations (Jovanović & Bamieh, 2005) provides a computationally tractable tool that has shown promise in analyzing transitional wall-bounded shear flows. In particular, therecently introduced structured input-output analysis has beenshown to predict large amplification of similar oblique structures in transitional plane Couette and Poiseuille flow (Liu &Gayme, 2021). Both the large energy growth and featuresof the structures identified show good agreement with resultsfrom experiments (Prigent et al., 2003) and DNS (Kanazawa,
2018), as well as predictions of nonlinear optimal perturbation
analysis (Rabin et al., 2012; Farano et al., 2015).
In this work we apply structured input-output analysis
to transitional plane Couette-Poiseuille flow (CPF). We first
focus on the well studied intermediate case in which theshear and pressure gradient are equally weighted (Klotz et al.,2017a; Klotz & Wesfreid, 2017b; Klotz et al., 2021; Liu et al.,2021). Our results indicate that the oblique laminar-turbulent
patterns with nonzero streamwise and spanwise wavenumbers
show the highest gain under structured amplification. The
wavelengths of the oblique turbulent bands identified are consistent with recent experimental observations (Klotz et al.,2021). In addition, the dependence of the predicted inclination
angles on Reynolds number corresponds to previous findings
in experiments of plane Couette flow (Prigent et al., 2003). We
then analyze the how the Reynolds number scaling ∼ O(Reα )changes as a function of the shape of laminar base flow. The
scaling of the highest structured gain for the intermediate flow
regime is shown to be α = 1.3, which is halfway between therespective values of α = 1.1 for plane Couette flow and 1.5 forplane Poiseuille flow found in Liu & Gayme (2021). However,
the overall trends are not monotonic and the shape of the curve
appears to suggest changes in the most amplified flow structures as the the flow moves from background shear to pressure
gradient dominant.
Finally, we show that the structured-input output method
can be adapted to estimate the downstream advection speedof the oblique turbulent bands in plane Couette and Poiseuilleflow. The results compare favorably to measurements basedon oblique turbulent bands captured in DNS (Xiao & Song,
2020; Tuckerman et al., 2014; Lu et al., 2019). The combined
analysis demonstrates the wide range of quantities that can becharacterized by structured input-output analysis.

In this work, we apply structured input-output analysis
to study optimal perturbations and dominant flow patterns intransitional plane Couette-Poiseuille flow. The results demonstrate that this approach predicts the high structured gain of
perturbations with wavelengths corresponding to the obliqueturbulent bands observed in experiments. The inclination angles of these structures and their Reynolds number dependence
are also consistent with previously observed trends. Reynolds
number scalings of the maximally amplified structures for anintermediate laminar profile that is equally balanced between
plane Couette and Poiseuille flow show an exponent that is
at the midpoint of previously computed values for these twoflows. However, the dependence of these scaling exponents on
the shape of laminar flow as the relative contribution moves
from predominately plane Couette to Poiseuille flow is not
monotonic and our analysis indicates the emergence of different optimal perturbation structures through the parameter
regime. Finally we adapt our approach to estimate the advection speeds of oblique turbulent bands in plane Couette flow
and Poiseuille flow by computing their phase speed. The results show good agreement with prior predictions of the convection speeds of these structures from direct numerical simulations, which suggests that this framework has further potential in examining the dynamics of these structures.

INTRODUCTION
A coexistence of turbulent and laminar behavior has been
observed in wall-bounded shear flows as they transition froma laminar to a turbulent regime (Tuckerman et al., 2020). Thistransitional state has been observed to take the form of bandsof laminar and turbulent regions that form an oblique angle
with the streamwise direction (Prigent et al., 2003; Kanazawa,
2018). Numerical simulations (Tuckerman & Barkley, 2011;
Tuckerman et al., 2014; Fukudome & Iida, 2012) and experiments (Prigent et al., 2003) have indicated that both the wavelengths and inclination angles of these structures are Reynolds
number dependent. The sensitivity of the flow to perturbationsof this type has been exploited through the use of these flow
patterns as initial conditions to trigger turbulence in direct numerical simulations (DNS) (Tao et al., 2018).
While the prevalence and large amplification of obliqueturbulent bands in transitional wall-bounded shear flows has
been widely observed, a comprehensive understanding of theirunderlying dynamics and role in transition has yet to be realized. Study of these structures is complicated by the verylarge channel size (∼ O(100) channel half-heights) required
to observe them, see e.g., (Prigent et al., 2002; Tuckerman& Barkley, 2011; Kim et al., 2020). Computations of exact

1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

Figure 1. Illustration of the structured feedback interconnection between the modified frequency response operator H∇ :=b,∇b,∇b )H (blocks inside of red solid line ) and the structured uncertainty bdiag(∇u Ξ := diag(−bu Tξ , −bu Tξ , −bu Tξ ) (block inside of purple
bbbbb ub)T , (∇b vb)T , (∇b w)
b T ]T is the 9-by-1 vectorized
solid line ). The modeled forcing term bf ξ := [ fbx,ξ , fby,ξ , fbz,ξ ]T := buΞ∇u , where ∇u := [(∇
velocity gradient.
 
 
ub
 vb = Cb vb ,
by
ω
b
w

PROBLEM SETUP
We consider incompressible flow between two infinite
parallel plates with streamwise, wall-normal, spanwise andtime coordinates (x, y, z,t), respectively. The total velocity isdecomposed into a laminar base flow and fluctuations aboutthat base flow, i.e., utot = [U(y), 0, 0]T + u . The pressure issimilarly decomposed as ptot = P + p. We employ the laminarbase flow from Klotz & Wesfreid (2017b), which is given by

U(y) =

3(1 + η) 21−η(y − 1) +(y − 1) + 1.
42

where (Jovanović & Bamieh, 2005)
"
#
′′
b2
b4
0
−1 −ikxU ∇ + ikxU + ∇ /Rec
A :=M
b 2 /Re ,−ikzU ′
−ikxU + ∇




2
2
b2
b :=M −1 −ikx ∂y −(kx + kz ) −ikz ∂y , M := ∇ 0
B
ikz
0
−ikx
0 I


ik ∂ −ikz
1  2 x y2
Cb := 2k
+ kz 0  .
kx + kz2 x
ikz ∂y ikx(1)

The parameter η ∈ [−1, 1] determines the relative contributionof shear versus pressure gradient with η = −1 corresponding
to plane Couette flow with laminar profile U(y) = y, and η = 1
to plane Poiseuille flow with U(y) = 1.5y2 − 0.5. Here, thestreamwise velocity is normalized by the upper-plate speed
U ∗ , and length is normalized by the channel half-height h;leading to a wall-normal domain of y ∈ [−1, 1] and Reynolds∗
number Re := Uν h , where ν denotes the kinematic viscosity.
The dynamics of velocity fluctuations u := [u, v, w]T aregoverned by the Navier-Stokes equations (NSE). Following
Liu & Gayme (2021) we linearize the NSE about the laminar based flow and treat the nonlinear terms as forcing f :=−uu · ∇ u = [ fx , fy , fz ]T , which we model as T  

−uuξ
fx,ξ
∇u


T
−uuξ
f ξ := (2)  ∇ v  =  fy,ξ  .
∇w
fz,ξ
−uuTξ

We employ no-slip boundary conditions vb(y = ±1) =by (y = ±1) = 0.
∂ vb/∂ y (y = ±1) = ω
We are interested in defining the block-diagonal gain
operator u Ξ := diag(−uuTξ , −uuTξ , −uuTξ ) that maximizes the response of the structured input-output system in Fig. 1.
To build this feedback interconnection we define H∇ :=b,∇b,∇b )H , where H (y; kx , kz , ω) := Cb(iωI − Ac)−1 Bbdiag(∇
is the spatio-temporal frequency response operator mapping bf ξ to bu for each triplet (kx , kz , ω) and I is the identity operator. The modified operator H∇ maps bf ξ :=[ fbx,ξ , fby,ξ , fbz,ξ ]T to the 9-by-1 vectorized velocity gradientbbb ub)T , (∇b vb)T , (∇b w)∇
u := [(∇b T ]T . Defining this modified operator allows us to evaluate the closed-loop system defined by the
feedback interconnection between H∇ and the block-diagonal
structured gain bu Ξ := diag(−bu Tξ , −bu Tξ , −bu Tξ ) shown in Fig. 1.
We then define the structured input-output response (Liu
& Gayme, 2021) as

This model preserves the structure of the nonlinear interactionsof the NSE since all elements outside of the block-diagonal
elements diag(−uuTξ , −uuTξ , −uuTξ ) are zero.
We employ the standard transformation to write
the governing equations in terms of the fluctuatingwall-normal velocity and vorticity [v, ωy ]T .
We thenc kx , kz , ω) :=perform
thetripleFouriertransform(·)(y;
R∞ R∞ R∞−i(kx x+kz z+ωt) dxdzdt. Here k and kxz−∞ −∞ −∞(·)(x, y, z,t)e
are the respective streamwise
andspanwisewavenumbers,
ω√is the frequency and i = −1. The transformed linearized
NSE driven by the modeled forcing f̂f ξ for each (kx , kz , ω)triplet is then given by∥H∇ ∥µ (kx , kz ) := sup µUb [H∇ (kx , kz , ω)] ,
ω∈R

(5)

Ξ

where

µUb [H∇ (·)] :=
Ξ



(3b)


1

 min{σ̄ [bu Ξ ]:bu Ξ ∈Ub Ξ ,det[I−H∇ (·)bu Ξ ]=0} ,b Ξ , det [I − H∇ (·)b0, if ∀ buΞ ∈ Uu Ξ ] ̸= 0




 
fbx,ξ
vb
vb
b 
c
b
iω
=A
+ B  fy,ξ  ,
by
by
ω
ω
fbz,ξ


(3a)

is the structured singular value of H∇ (kx , kz , ω) for eachb Ξ :=
triplet (kx , kz , ω) (Packard & Doyle, 1993). Here U

212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

(a)

(b)

(c)

Figure 2. Contour plots of (a) log10 [∥H∇ ∥µ (kx , kz )], (b) log10 [∥H ∥∞ (kx , kz )] and (c) log10 [∥H∇ ∥∞ (kx , kz )] for plane CouettePoiseuille flow U(y) = 34 (y2 − 1) + 12 (y + 1) at Re = 610 and η = 0. Here, the black triangle (∆) marks λx ≈ 93 and λz ≈ 48 which
are the wavelengths of the observed oblique turbulent band at Re = 610 (Klotz et al., 2021). The black solid line ( ) λz = λx tan(27◦ )corresponds to a 27◦ oblique angle of the oblique turbulent band.
{diag(−buTξ , −buTξ , −buTξ ) : −buTξ ∈ CNy ×3Ny } is the set of wallnormally discretized structured uncertainties with the same
block-diagonal structure as bu Ξ , and Ny denotes the number of
Chebyshev collocation points excluding boundary points. H∇
denotes the discretized spatio-temporal frequency response
operator, σ̄ [·] represents the largest singular value of the argument, and det[·] is the determinant of the argument.

bands. These results also indicate that the change in the response is due to the structured feedback interconnection rather
than the modified frequency response operator. These differences between the structured and the traditional unstructured
input-output response is similar to that seen in previous studiesof plane Couette and Poiseuille flow, where the change in the
structures obtained under the structured feedback interconnection was related to a weakening of the lift-up mechanism that
dominates the unstructured response (Liu & Gayme, 2021).
We next study the inclination angles and length scalesof structures that maximize the structured gain for this intermediate case (η = 0) and examine and their dependence
on Reynolds number. We define this dominant angle as θ =tan−1 (kxM /kzM ), where kxM and kzM are defined as the wavenumbers associated with the largest structured gain

RESULTSIn this section, we compute the response in (5) using MATLAB with Chebyshev differentiation matrices from
Weideman & Reddy (2000). We employ Ny = 30 collocation points in the wall normal direction and 50 × 90 logarithmically spaced points in the spatial wavenumber domainkx ∈ [10−4 , 100.48 ] and kz ∈ [10−2 , 101.2 ]. This configuration
matches that in Liu & Gayme (2021), who found this numberof points sufficient for convergence of the results.
We first consider the intermediate case, the laminar profile with η = 0 in (1), which corresponds to equal weighting of the contributions from plane Couette and Poiseuilleflow. This profile is linearly stable for all Reynolds numbers (Balakumar, 1997). Fig. 2 (a) shows the corresponding ∥H∇ ∥µ (kx , kz ) at Re = 610, which was selected to matchthat in Klotz et al. (2021). Here, the peak region covers the
wavelength range λx ≈ 93 ± 5 and λz ≈ 48 ± 5. This region isconsistent with the oblique turbulent bands observed in Klotz
et al. (2021, Fig. 20), whose results are indicated as a black
triangles (∆) in all panels of Fig. 2. The shape of the peakregion in Fig. 2(a) also reflects the same inclination angle
θ := tan−1 (λz /λx ) = tan−1 (48/93) ≈ 27◦ (the black solid line
in Fig. 2) as the experimentally observed band. The consistency of our results with the literature shows that structured
input-output analysis captures both the length scales and inclination angles of dominant flow structures in this intermediatecase of plane CPF.
We next compare these results to similar quantities
that would be obtained through an unstructured analysis by computing ∥H ∥∞ (kx , kz ) := sup σ̄ [H(kx , kz , ω)] and(kxM , kzM ) := arg max ∥H∇ ∥µ (kx , kz ).

(6)

kx ,kzFig. 3(a) shows how these wavelengths vary as a function of
Reynolds number. Here it is clear that while both wavenumbers decrease as Re increases, kxM decreases faster. We quantify
this difference by computing scalings of these quantities with
Reynolds number using a fit of the form of Ci Reαi (i = 1, 2, 3).
The results show that kxM ∼ Re−0.41 and kzM ∼ Re−0.34 . Wenote that this scaling of kxM is consistent with previous analysis
in the optimal wavenumber range kx ≤ O(1) for oblique flow
structures that produce streamwise vortices in wall-boundedshear flows (Chapman, 2002). These trends are consistent witha reduction in the inclination angle as a function of Reynolds
number over this range.
Fig. 3(b) and (c) show ∥H∇ ∥µ for two additional
Reynolds numbers, respectively Re = 2000 and 20000. In
both panels the most energetic wavenumber pairs, respectively(kxM , kzM ) = (0.25, 0.67) and (0.08, 0.29) are indicated with ablack asterisk. These figures indicate that θ decreases fromapproximately 22◦ to 15◦ over this Reynolds number range,
and that both angles are lower than the value at Re = 610. In
addition, as Reynolds number increases, the peak response region extends visibly and shifts slightly more towards larger
streamwise wavelengths than toward the large spanwise ones.
This negative correlation is consistent with trends observed inprevious studies of laminar-turbulent patterns in plane Couette
flow (Prigent et al., 2003).

ω∈R∥H∇ ∥∞ (kx , kz ) := sup σ̄ [H∇ (kx , kz , ω)], respectively shownω∈R

in Figs. 2(b) and (c). The figures show that both of thesequantities place more emphasis on the streamwise elongatedstructures associated with kx ≈ 0 than the oblique turbulent

312th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

(a)

(b)

(c)

(a) The most energetic wavenumber pairs from Re = 300 to 20000 with scalings kxM ∼ Re−0.41 and kzM ∼ Re−0.34 andFigure 3.
contour plots showing the relative location of regions associated with peak structured frequency response at (b) Re = 2000 and (c)20000 with η = 0. The blue asterisks (✳) in panels (b) and (c) symbol the wavenumber pairs (kxM , kzM ) = (0.25, 0.67) at Re = 2000 and(kxM , kzM ) = (0.08, 0.29) at Re = 20000 corresponding to the largest structured frequency response ∥H∇ ∥Mµ at Re = 2000. The blacksolid line ( ) represents λz = λx tan(22◦ ) in panel (b) and λz = λx tan(15◦ ) in (c) respectively.

The maximal response for a given Reynolds number
for the three quantities in Fig. 2 can be respectivelyMquantified as ∥H∇ ∥Mµ := max ∥H∇ ∥µ (kx , kz ), ∥H∇ ∥∞ :=We next explore how the scaling and structural features
vary as a function of the relative contributions of shear andpressure gradient forcing to the laminar profile shape. Fig.
4(b) shows the scaling exponents of the maximal structuredαfrequency response ∥H∇ ∥Mµ ∼ Re with laminar profiles defined by equation (1) with η = [−1, 1] computed at steps of
0.1 over the Reynolds number range Re ∈ [300, 4000]. Fig.
4(b) indicates that α grows from 1.1 to 1.35 as η increases
roughly from −1 to 0.3, and then declines at a low rate in
the range 0.3 ≤ η ≤ 0.7 before finally climbing to α = 1.5
at η = 1. While the end points are consistent with previous
scalings associated with plane Couette and Poiseuille flow,
the non-monotonic growth of the scaling exponent was unexpected. In order to analyze this behavior we explore thestructural features associated with the different regions of thisscaling plot in Fig. 5, which shows ∥H∇ ∥µ (kx , kz ) for laminar profiles corresponding to η = −1, −0.5, 0, 0.3, 0.5, 0.7, 0.9
and 1 for Re = 500.
Figs. 5 (a) – (c) indicate that there is a single peak frequency response region (kx , kz ) ≈ (10−1 , 1) associated withoblique flow structures when −1 ≤ η ≤ 0. In this region thekx ,kzmax ∥H∇ ∥∞ (kx , kz ), and ∥H ∥M∞ := max ∥H ∥∞ (kx , kz ). Thesekx ,kzkx ,kzquantities for the range Re ∈ [300, 4000] are depicted in Fig.
4(a), where we again obtain their respective Reynolds number scalings through a fit to Ci Reαi (i = 1, 2, 3). The resultsM
2
reveal that both ∥H∇ ∥M∞ and ∥H ∥∞ scale as Re , which isthe same scaling seen in both plane Couette and Poiseuilleflow, see e.g. Trefethen et al. (1993); Jovanović (2004). How1.3
ever, ∥H∇ ∥Mµ ∼ Re , and this lower scaling exponent versus the unstructured response is similar to previous results that
1.1 for plane Couette flow U(y) = y andshowed ∥H∇ ∥Mµ ∼ Re1.5 for plane Poiseuille flow U(y) = 1 − y2 (Liu∥H∇ ∥Mµ ∼ Re& Gayme, 2021). This lower amplification of the structured response is consistent with previous observations that the structured feedback weakens the amplification of the lift-up mechanism in a manner similar to that of nonlinear effects in plane
Couette and plane Poiseuille flows (Liu & Gayme, 2021).

(a)

(b)

Figure 4. (a) Reynolds number dependence of the maximal structured and unstructured frequency response associated with scaling
1.3
M
2
M
2
∥H∇ ∥Mµ ∼ Re , ∥H∇ ∥∞ ∼ Re and ∥H ∥∞ ∼ Re at Re ∈ [300, 4000] and η = 0. (b) The dependence of scaling exponents α3(1+η)

αin ∥H∇ ∥M(y2 − 1) + 1−ηµ ∼ Re on the parameter η determining the shape of laminar base flow U(y) =42 (y − 1) + 1. The
asterisks in panel (b) represent α at η = −1, −0.9, ..., 0.9, 1. The red asterisks (✳) mark η = −1, −0.5, 0, the magenta ones (✳) mark
η = 0.3, 0.5, 0.7, 0.9, 1, and the black ones (✳) indicate the remaining values of η.

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
Table 1. Advection speeds c of laminar-turbulent patterns in plane Couette and Poiseuille flows by predictions of structured inputoutput analysis and results of direct numerical simulations. Here the paper sources where we obtain DNS results of advection speeds
are consistent with those of wavenumbers in the third column.

LaminarReynolds

Wavenumbers

Estimations of c

DNS Results of cProfileNumber(kx , kz ) andby Structured

and Their Sources

U(y)

Re

Their Sources

I/O Analysis(From Papers in Col. 3)

1 − y2750(0.15, 0.10) (Fig. 1(a), Xiao & Song, 2020)

0.92

0.85 (Fig. 1(b))

1
3 2
2 − 2y1100(0.06, 0.14) (Fig. 2(a), Tuckerman et al., 2014)≈00.07 (Fig. 4(b))

y317(0.04, 0.05) (Fig. 3(b), Lu et al., 2019)≈0≈ 0 (Fig. 4)

can be represented by integrating its Fourier components overreal temporal frequencies (Trefethen et al., 1993). The phasespeed c of a given oblique flow structure with (kx , kz ) ̸= 0 is
obtained as

scaling is increasing at a roughly constant rate and the angle associated with these structures appears to increase with
η. Panels (d) – (h) indicate that a new wavenumber region (kx , kz ) ≈ (1, 0) corresponding to large structured frequency response emerges in the slowly decreasing (roughly
flat region in Fig. 4(b). Given that the laminar flow is
no longer linearly stable for all Reynolds numbers for η >0.309 (Balakumar, 1997), these are likely associated with
the Tollmien-Schlichting (TS) waves that become unstable athigher Reynolds numbers. This conjecture is supported by
the fact that the TS wavenumbers ranges are consistent withthese regions. The definition of this secondary TS related peak
becomes sharper for η > 0.7 as does the peak range of theoblique structures. The angle associated with these structuresand shape of this region also changes for η > 0.7, which may
indicate a change in the characteristic structures in this parameter regime. A full characterization of how the structural features of the flow change with η is a direction of ongoing work.
Finally, we adapt the structured input-output approach tocompute the phase speed associated with the maximum response, which we interpret as the convective velocity of thesestructures. We limit the analysis to the case where the temporal frequency ω ∈ R, since any time-dependent flow structurec(kx , kz ) = −1
arg max µ [H (kx , kz , ω)] .
kx ω∈R Ub Ξ ∇(7)In order to validate the approach we focus our investigation on
the phase speeds of flow patterns in plane Couette and plane
Poiseuille flow, which have been previously studied (Tuckerman et al., 2014; Xiao & Song, 2020; Lu et al., 2019). Table.
1 provides a comparison between the advection speeds of thelaminar turbulent (oblique bands) predicted from our approachto results computed from the DNS of Lu et al. (2019); Tuckerman et al. (2014) and Xiao & Song (2020). The obliqueturbulent band wavenumbers (kx , kz ) reported were extracted
from snapshots of these DNS results, specifically from Lu
et al., 2019, Fig. 3 (b); Tuckerman et al., 2014, Fig. 4 (a);Xiao & Song, 2020, Fig. 1 (a), respectively. In each case, thetable indicates the laminar base flows and Reynolds numbers
from the references. We note that Tuckerman et al. (2014) andLu et al. (2019) use the same characteristic length and velocity

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 5. Plots of log10 [∥H∇ ∥µ (kx , kz )] at Re = 500 with (a) η = −1, (b) η = −0.5, (c) η = 0, (d) η = 0.3, (e) η = 0.5, (f) η = 0.7,(j) η = 0.9, and (h) η = 1. The shape of laminar flow η associated with panels (a)–(c) correspond to the red asterisks (✳) while panels(d)–(h) match with the magenta asterisks (✳) in Fig. 4 (b).

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Osaka, Japan, July 19–22, 2022
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scales as our analysis so their laminar profiles directly correspond to the cases where η = 1 and η = −1 respectively, so we
can directly compare to their results by matching the Reynolds
numbers. On the other hand, the flow configuration in Lu et al.
(2019) differs from ours, which leads to a different laminar
profile. In order to compare with those results, we renormalize∗ and redevelocity using the centerline speed of base flow Ucenfine the laminar plane Poiseuille flow as U(y) = 1 − y2 and theU∗ hReynolds number as Re := cenν .
The results in Table 1 indicate that the estimated values
have a resonable margin of error when compared to the DNS
results. This success in predicting the traveling speeds of thegiven structures indicates that our framework may be usefulin estimating the convection rate of different flow structures.
Although further analysis is needed to fully validate this approach for a wider range of structures, these preliminary results suggest the promise of this framework in probing the dynamics of the oblique turbulent bands.

CONCLUSIONSIn this paper, we apply structured input-output analysis
to study the oblique turbulent bands that have been observedin transitional plane Couette-Poiseuille flow (CPF). The results reveal that the wavenumbers of these structure are associated with large gain under structured feedback forcing. As
Reynolds number increases, these most energetic streamwise
and spanwise wavenumbers decrease with different scalings
kxM ∼ Re−0.41 and kzM ∼ Re−0.34 , thereby resulting in the decline of oblique angles and extension of the wavenumber region associated with the large response. The Reynolds number scaling of the maximal structured amplification ∥H∇ ∥Mµassociated with dominant oblique patterns changes from 1.1 to
1.5 as the laminar profile changes from plane Couette flow toplane Poiseuille flow, with an intermediate CPF flow value of
1.3. While the end points are consistent with previous results
and the intermediate flow falls at the midpoint the increase is
not monotonic and the analysis suggests that different structural features emerge as the pressure gradient contribution to
the laminar profile is increased. Further characterization ofthis behavior is a direction for future work.
Finally we adapt the framework to predict the advection speeds of oblique turbulent bands for plane Couette andPoiseuille flow configurations. The results are consistent with
DNS and indicate the promise of this approach in further analyzing the dynamics of these transitional structures.

ACKNOWLEDGEMENTS
The authors gratefully acknowledge partial support fromthe US National Science Foundation (CBET 1652244).

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