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

STABILITY AND RESOLVENT ANALYSIS OF PLANE COUETTE FLOW IN
THE DISTINGUISHED LIMIT Re → ∞, α → 0 WITH Re · α = O(1)Toni Dokoza
Chair of Fluid Dynamics, Department of Mechanical Engineering
Technical University of Darmstadt
64287 Darmstadt, Germany
dokoza@fdy.tu-darmstadt.deMartin Oberlack
Chair of Fluid Dynamics, Department of Mechanical Engineering
Technical University of Darmstadt
64287 Darmstadt, Germany
oberlack@fdy.tu-darmstadt.de

ABSTRACT

most amplified response modes and their associated forcing
modes, which are coupled by the singular values. This work
aims to investigate the coherent structures through both the resolvent analysis and the linear stability theory with a laminarbase velocity profile for the limit of high Reynolds numbers
Re → ∞ and simultaneously small streamwise wavenumbers
α → 0 with the distinguished limit Reα = Reα = O(1).

In this work the plane Couette flow is being investigatedfor high Reynolds numbers Re → ∞ and small streamwise
wavenumbers α → 0 in the distinguished limit Reα = Reα =O(1) using the temporal linear stability theory as well as theresolvent analysis approach. The influence of Reα on both the
eigenfunctions obtained from the linear stability theory and therepsonse modes obtained from the resolvent analysis is investigated, while both modes are being compared in order to analyse the occurence of streamwise-elongated coherent structures
within the plane Couette flow.

GOVERNING EQUATIONSThe Navier-Stokes equations for the velocity fluctuations u′iaround a laminar base veloctiy profile in streamwise direction
x1 , given by U1 (x2 ) = x2 are given in wall-normal velocityvorticity formulation of the plane Couette flow for the wallnormal velocity fluctuation u′2 and the wall-normal vorticity

INTRODUCTION
It is well known that for a high enough Reynolds number
Re the plane Couette flow becomes turbulent and thus both
small- and large-scale structures are observed. In order to investigate the large streamwise-elongated structures, which canbe observed in plane Couette flow, the linear analysis of theNavier–Stokes equations in shear flows (Schmid & Henningson, 2001) provide useful insights. These linear analyses reveal that, for the plane Couette flow, the structures that are
most excitable are streamwise-constant. The observation of
streamwise-constant structures being most amplified has been
made for the laminar plane Couette flow among many others
by Gustavsson (1981) for example, as well as for its turbulent counterpart for which the linear analyses are performed
about the turbulent mean flow in Hwang & Cossu (2010a). In
turbulent flows, coherent structures can be understood as being permanently forced via convective nonlinear interactionsof the fluctuations. Thus it seems appropriate to analyse theflow behaviour as the response to a non-linear intrinsic forcing through a linearised operator, which is the key idea of theresolvent analysis, where all the non-linear fluctuation terms
in the Navier-Stokes equations for the perturbations are sum∂ u′marised as an intrinsic forcing term fi′ = −u′j ∂ x i , throughjwhich the Navier-Stokes equation can be rewritten in an inputoutput system. A singular-value-decomposition (SVD) is being performed in order to rewrite the resolvent operator in the

∂ u′

∂ u′

fluctuation η2′ = ∂ x1 − ∂ x3 . Hereby spatial directions are rep3
1
resented by xi with i = 1, 2, 3. Solid wall boundary conditions
for the wall-normal velocity-vorticity formulation are Dirichlet boundary conditions for u′2 and η2′ as well as the Neu⊺′mann boundary condition for u′2 , i.e. u′2 , ∂∂ ux2 , η2′ (x1 , x2 =2

±1, x3 ) = 0. A Fourier decomposition in streamwise and spanwise direction as well as in time is applied using
u′i = ũ′i (x2 )ei(αx1 +β x3 −ωt) ,(1)

η2′ = η̃2′ (x2 )ei(αx1 +β x3 −ωt) ,(2)with α an β being the streamwise and spanwise wavenumber,while ω represents the frequency of the fluctuations, whichleads to the governing equations as
2−iω (α + β

+

1
Re

= −iα

1

2

d 2 ũ′2)ũ′2 −dx22!
2

+ iαx2 (α + β

2d 2 ũ′2)ũ′2 −
dx22

d 4 ũ′2d ũ′− 2(α 2 + β 2 ) 22 + (α 2 + β 2 )2 ũ′2
4
dx2
dx2
d f˜′
d f˜1′− β 2 f˜2′ − iβ 3 ,
dx2
dx2

!

!

(3)

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022−iω η̃2′ + x2 iα η̃2′ + iβ ũ′2
+

1
Re

with

∂ 2η ′− 22 + (α 2 + β 2 )η̃2′∂ x2!
= iβ f˜1′ − iα f˜3′

(4)
L11 = −x2 iReαand

L21 = −iα ũ′1 +d ũ′2+ iβ ũ′3 = 0,
dx2

(5)

!

d 2 ũ′2d 2 ũ′2
2 ′
+
iRe
x
β
ũ
−
β 2 ũ′2 −
α
2
2
dx22
dx22
d 4 ũ′d 2 ũ′+ 42 − 2β 2 22 + β 4 ũ′2
dx2
dx2


d f˜′
d f˜1′Reα=−iα− β 2 f˜2′ − iβ 3
α
dx2
dx2

ω
−i Reα
α

M=

d2
−β2
dx22



0

0
Reα
α

!

(13)

In a final step we may rewrite (8) by introducing the resolvent
matrix H as′′q̃′ = Hf̃B = (−L − iω1 M)−1 f̃B .
(6)

(14)

Singular Value Decomposition
The goal of the resolvent analysis is to identify the dominant′directions along which f̃B can be most amplified through theresolvent operator H to form the corresponding responses inq̃′ . By using a singular value decomposition on the resolvent
operator H one can obtain these modes. The response modes
are considered to represent the occurring coherent structures,
observed in plane Couette flow, where only the knowledge ofthe mean velocity profile has to be taken from external sources.
The singular value decomposition rewrites the matrix H into

(7)


 σ

1
| |
|
− Φ1 −

..
H = USVT = Ψ1 Ψ2 ... Ψn    ...  .
.
| ||− Φn −σn(8)In this formulation S represents a diagonal matrix with thesingular values σn in decreasing order in its diagonal. The
matrix VT contains the forcing modes ⃗Φ j , while U contains
the respective response modes ⃗Ψ j . Both sets of singular vectors are guaranteed to be orthonormal bases and are ranked
according to their singular values. In other words, the resolvent analysis interprets left and right singular vectors q̃′ and′′f̃B of q̃′ = Hf̃B respectively as response and forcing modes,
with the magnitude-ranked singular values σi being the amplification (gain) for the corresponding forcing-response pair.
p
The resolvent operator is of lower rank if ∑ j=1 σ 2j ≈ ∑∞j=1 σ 2j ,where σ p ≫ σ p+1 and the number of these singular values p
is small. If the resolvent operator is rank-1, hence σ1 ≫ σ2 ,the response can be well predicted from the leading singular
vectors alone.

!
,

(9)

with L, representing the Orr-Sommerfeld and Squire operator,
is given as


L11 0
L=
,
L21 L22

(12)

2

 ′
 ′
f˜1
ũ
′
2 and f̃ =  f˜′ .
with q̃′ =
′
2
η̃2
f˜3′
Hereby the mass-matrix M reads


.



 ′
 ′
iα dxd 2 −iβũ1ũ′  = C ũ2′ = 1 
0 .
 β2
2
η̃2
β2
′
d
ũ3
iβ dx iαFor the resolvent analysis it is necessary to rewrite the partial
differential equations (3) and (4) in an input-output system as

Reα
α

!

All velocity fluctuations ũ′i are generated from ũ′2 and η̃2′ using
the continuity equation (5), since u′2 and η2′ are considered asknown quantities. The veloctiy fluctuations u′i are then given
by

RESOLVENT ANALYSIS

′

(11)

−iReα dxd 2 − Reαα β 2 −iβ Reαα dxd 2
Reα
0− Reαα iα
α iβ

B=

−i

−iωMq̃′ = Lq̃′ + Bf̃ ,

d2
−β2
dx22

and the matrix B is given as

andd 2 η̃2′ωReαReα η̃2′ + x2 iReα η̃2′ +iβ ũ′2 −
α
α
dx22
Reα
(iβ f˜1′ − iα f˜3′ ).
+β 2 η̃2′ =
α

Reα
iβ
α

L22 = −x2 iReα +



∂ u′
dewith the Reynoldsnumber Re, where f˜i = −u′j ∂ x i
j k
notes the Fourier-transformed nonlinear forcing term
corre⊺sponding to the wavenumber vector k = α, β , ω . We arepresently particularly interested in large-scale coherent structures which are observed to occur for high Reynolds number flows and which admit very weak streamwise variation,
since these are structures found to be most amplified (Hwang& Cossu (2010b)). Hence, we intend to analyse the distinguished asymptotic limit of α → 0 and Re → ∞, which are thekey parameters to observe coherent structures in the form ofcounter rotating vortices occupying the whole channel width.
Thus we introduce Reα = Re · α, which is to be considered oforder O(1), as done in Yalcin et al. (2021) for the temporal linear stability of the asymptotic suction boundary layer. We may
neglect all terms of order O(α n ) with n > 1 in the equation (3)and the equation (4), leading to

!
d4
d2
d2
2
− β + 4 − 2β 2 2 + β 4
2
dx2
dx2
dx2

(10)

2

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

Figure 1. Share of first 10 singular values of the resolvent
operator over all singular values for ω = 0, β = 2, Reα = 1,
α = 0.0001.

Figure 2. Forcing modes and Response modes for each velocity fluctuation for ω = 0, β = 2, Reα = 1, α = 0.0001 areshown over the wall-normal coordinate x2 .

Numerical methods and discretisation
The input-output system (14) is being analysed numericallyusing Matlab. The matrix system (14) is discretised in wallnormal direction x2 using a Chebyshev-discretisation with
N = 251 collocation points, whereas the differentiation matrices are being built using the chebdif function by Weideman &Reddy (2000). The Dirichlet boundary and Neumann boundary conditions for the wall-normal velocity u′2 and the Dirichlet boundary condition for the wall-normal vorticity η2′ are implemented using the cheb4c function by Weideman & Reddy(2000). The resolvent operator is scaled to the kinetic energy
for wall-normal fluctuations as described in Schmid & Henningson (2001) as
Z Z

Z 1

α

−1

Ev =

1
β 2k2

Figure 3. First response modes ũ′1 , ũ′2 , ũ′3 for ω = 0, β =2, Reα = 1, α = 0.0001 are shown over the wall-normal direction x2 . Solid lines represent the real part of the solution, withdotted lines representing the imaginary part of the responsemodes.

!
∂ ′ 2
2 ′ 2
′ 2
u + k |u2 | + |η2 | dydαdβ ,∂ x2 2(15)

with k = α 2 + β 2 .

Results
Figure 1 shows the share of the first 10 singular values over all
singular values for ω = 0, β = 2, Reα = 1 and α = 0.0001.
The share of each singular value decreases over-exponentiallywith increasing modes. One can see that σ1 makes up to 82% of all singular values, thus a low-rank behaviour can be assumed, allowing the coherent structures to be well estimatedby only using the first response modes in the subsequent investigations. In Figure 2 the forcing modes are shown over thewall-normal direction x2 on the left and the corresponding response modes on the right side for ω = 0, β = 2, Reα = 1,
α = 0.0001 . It can be seen that the most energy can befound in the streamwise fluctuation u′1 for the considered distinguished limit. The magnitude of the wall-normal and spanwise fluctuations are of order O(α) compared to the streamwise fluctuation, thus the streamwise streaks can be observed.
Furthermore, we observe, that the energy within the forcing
is mainly given in the wall-normal and spanwise components,
while the energy of the response is stored in the streamwise
component. Spanwise and wall-normal forcing modes giverise to streamwise response modes, indicating the lift-up effect being dominant.
The first response modes ũ′1 , ũ′2 , ũ′3 for ω = 0, β = 2, Reα =1, α = 0.0001 are shown over the wall-normal direction x2 in
Figure 3. It once agains shows the most energy being stored in
the streamwise direction.
Figure 4 shows the response of the system as streamwise fluctuations over the the spanwise and wall-normal direction fora variation of α on the left side with a fixed Re and for various Re on the right side for a fixed α, where Reα on each

Figure 4. Streamwise fluctuations are shown over the spanwise direction x3 and wall-normal direction x2 for (a) ω = 0,
β = 2, Reα = 1, α = 0.0001, (b) ω = 0, β = 2, Reα = 5,
α = 0.0005, (c) ω = 0, β = 2, Reα = 10, α = 0.001, (d) ω = 0,
β = 2, Reα = 1, Re = 10000, (e) ω = 0, β = 2, Reα = 5,
Re = 50000, (f) ω = 0, β = 2, Reα = 10, Re = 100000, as
pairs of vortices. Solid lines represent fluctuations with a positive sign, while dashed lines represent fluctuations with a negative sign.

row is equal on both the left and the right side. One can seethat an increase of Reα leads to a stronger inclination of thestreamwise structures within the spanwise and wall-normal direction, while the effect of an increasing Re and an increasing
α is the same, which can be seen through the fact that the coherent structures in (a) and (d), (b) and (e), (c) and (f) are thesame. Thus Reα effects the inclination of these strucutures
rather than α or Re alone. The Orr-mechansim yields an inclination of the structures in the direction of the propagatingwaves. Since for α ̸= 0, the waves are not propagating in thestreamwise direction alone anymore, an inclination within the
spanwise and wall-normal direction can be observed. The influence of β and Reα on σ1 and thus the energy of the systemfor ω = 0 is now being investigated. For this purpose Figure

3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
leading to the further expanded Orr-Sommerfeld and Squire
operators

β 2 ũ′2,α −−iω1 Reα−iReα x2
Figure 5. The first singular value σ1 is shown over β for ω =0 for a fixed α = 0.0001 and three different values for Re on
the left side and for a fixed Re = 10000 and three different
values for α on the right side.

d 2 ũ′2,αdx22+

d 2 ũ′2,αdx22

d 4 ũ′2,αdx24− 2β 2ũ′2,α = C1 eβ x2 + C2 eβ x2

Z x2
−1

e−2β x2 dx2

+ C4

with
Z x2

j

E j (Reα , β , ω1 , x2 ) =e(−1) β x2
−1


−2/3
· Ai (−i)1/3 Reα (Reα (ω1 − x2 ) + iβ 2 ) dx2 ,(21a)with j = 1, 2 andlEl (Reα , β , ω1 , x2 ) =e(−1) β x2
−1


−2/3
· Bi (−i)1/3 Reα (Reα (ω1 − x2 ) + iβ 2 ) dx2 ,
(16)

(21b)

with l = 3, 4. The eigenvalues ω1 are being solved from theeigenvalue problem using the Orr-Sommerfeld equation (18)based on a Matlab Chebyshev collocation code as done in
Schmid & Henningson (2001) with the boundary conditions


and

∂ u′

+β 2 η̃2′ = 0.

(20)!
e−β x2eβ x2
E1 (Reα , β , ω1 , x2 ) −E2 (Reα , β , ω1 , x2 )
2β
2β
!
eβ x2
e−β x2E3 (Reα , β , ω1 , x2 ) −E4 (Reα , β , ω1 , x2 ) ,
2β
2β

+ C3

+ iReα x2 β 2 ũ′2−iω1 Reα η̃2′ + x2 iReα η̃2′ +d 2 η̃2′dx22

The solution for ũ′2,α is obtained analytically from the OrrSommerfeld equation (18) in terms of Airy-functions Ai(z)and Bi(z) as!

d 2 ũ′2 d 4 ũ′2d 2 ũ′+− 2β 2 22 + β 4 ũ′2 = 0
2
4
dx2
dx2
dx2

+ β 4 ũ′2,α = 0 (18)

(19)

Z x2

−iReα x2

dx22

+β 2 η̃2′ = 0.

In this section we want to investigate the distinguished asymptotic limit with Reα = O(1) for the plane Couette flow using
the linear stability theory, where the Orr-Sommerfeld and the
Squire equation are obtained by setting the right-hand side of(6) and (7) to 0 due to considering only small fluctuations.
Furthermore we observe that in these equations terms of theorder ωα prevail, and hence, a low frequency assumption is implied with ω = ω1 α + O(α 2 ), leading to the expanded OrrSommerfeld and Squire equation given as
d 2 ũ′2dx22

d 2 ũ′2,α−iω1 Reα η̃2′ + x2 iReα η̃2′ + Reα iβ ũ′2,α −LINEAR STABILITY THEORYβ 2 ũ′2 −+ iReα x2 β 2 ũ′2,αand

5 shows the behaviour of σ1 over β for three different valuesof Reα for a fixed α = 0.0001 and three different values of Re
on the left side and for a fixed Re = 10000 and three different
values for α on the right side. It can be seen that σ1 first increases with β 1 for an increasing β before declining with β −2after reaching its peak. On the left hand side Reα is increased
by setting α = 0.0001 and increasing Re. It can be seen that
σ1 increases with increasing Reα for a fixed α. On the rightside the Reynolds number was set to Re = 10000, while α isbeing changed. Increasing Reα on this way, leads to a decreasing first singular value, agreeing with the findings of Hwang &Cossu (2010a) that streamwise-constant structures are being
most amplified.

−iω1 Reα!

⊺u′2 , ∂ x2 (x1 , x2 = ±1, x3 ) = 0, which may be written in ma2
trix form as

d 2 η̃2′Reαiβ ũ′2 −
α
dx22

  
  
C1
A11 A12 A13 A14
C1C2  A21 0 0 0  C2 





.
A(Reα , β , ω1 ) ·   = ·
C3
A31 A32 A33 A34  C3 
C4
A41 A42 0 0
C4(22)
Ai j represent the respective coefficients for the boundary con⊺
′ditions u′2 , ∂∂ ux2 (x1 , x2 = ±1, x3 ) = 0. For a non-trivial so2
lution the determinant of A has to vanish, yielding the dispersion relation(17)

The Orr-Sommerfeld equation (16) is only dependent on the
newly introduced variable Reα , whereas the Squire-equation(17) still contains α. Thus the wall-normal velocity fluctuation ũ′2 has to be of order O(α) compared to the wall-normal
vorticity fluctuation η̃2′ in order for all the terms to be of thesame order.
Therefore, we rewrite the wall-normal velocity fluctuation asũ2 = α ũ2,α , where α is used to rescale ũ2 . By this assumption α formally vanishes in the Squire equation (17) as well,det(A(Reα , β , ω1 )) = A14 A33 − A13 A34 = 0,

4

(23)

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

Figure 7. Analytically obtained eigenfunctions ũ′1 , ũ′2,α , ũ′3,αfor β = 2, Reα = 1 and for the most critical eigenvalue ω1 =−10.1588i are shown over the wall-normal direction x2 . Solidlines represent the real part of the solution, with dotted lines
representing the imaginary part of the eigenfunctions.

Figure 6. Most critical numerically obtained eigenvalue ω 1,i
is plotted over β for Reα = [1, 1.5] on the left side and plotted
over Reα for β = [1, 2] on the right side.

from which the eigenvalues are obtained numerically. It isnoticeable that using the linear stability theory the problem
is only dependent of Reα instead of an additional α, which
was the case for the resolvent analysis. In Figure 6 the mostcritical numerically obtained eigenvalue ω 1,i is shown over β
for Reα = [1, 1.5] on the left side and plotted over Reα for
β = [1, 2] on the right side. It can be seen that the most criticalnumerically obtained eigenvalue ω 1,i over β follows a similar
qualitative behaviour as the first singular value σ1 shown in
Figure 5. For larger values of Reα the most critical numericaleigenvalue ω 1,i converges towards a value close to 0.
In order to solve for the coefficients Ci in (20), first the Dirichlet and Neumann boundary condition for the lower wall at
x2 = −1 is being implemented, leading to∂ ũ′2,α∂ x2ũ′2,α (x2 = −1) = C1 e−β = 0,

(24)

−β

(25)

(x2 = −1) = β C1 eβ

+ C2 e = 0,

Finally the boundary condition for the upperwall at x2 = 1 forη̃2′ is being implemented, resulting in
C6 =
− C4

1

Ai (Z(x2 )) ũ′2,α dx2 ,iũ′1 = − η̃2′ ,βiiũ′3 = α ũ′2,α + α η̃2′ .
β
β

(30a)
(30b)

Inserting (20) in ũ′2 = α ũ2,α one can see that α acts as a magnitude amplifier in the wall-normal fluctuation ũ′2 , while its
qualitative behaviour is only dependent of Reα . From (27) it
can be seen that the wall-normal vorticity fluctuation is only
dependent of Reα in both its qualitative and quantitative behaviour. From (30a) one can see that the streamwise velocity
fluctuation ũ′1 is only dependent of the wall-normal vorticity
fluctuation η2′ , thus also only dependent of Reα . Furthermore
one can see in (30b) that one could rewrite ũ′3 = α ũ′3,α , withIn order to solve for the wall-normal vorticity η̄2′ analytically,
the Squire equation (19) can be used, resulting inũ′3,α = βi ũ′2,α + βi η̃2′ , thus α would also only acts as a magnitude amplifier for ũ′3 , while its qualitative behaviour is only
dependent of Reα .
Figure 7 shows the analytically obtained eigenfunctions for there-scaled wall-normal velocity fluctuation ũ2,α as solved for in(20), where the wall-normal vorticity fluctuation η̃2′ together
with (30) was used in order to obtain the streamwise velocity
fluctuation ũ′1 and the re-scaled spanwise velocity fluctuationũ′3,α for β = 1, Reα = 1 and the most critical eigenvalue being
found numerically using (23) found at ω1 = −10.1588i. In
Figure 8 the streamwise velocity fluctuation is shown over the
spanwise and wall-normal direction for (a) β = 2, Reα = 1,
ω1 = −10.1588i, (b) β = 2, Reα = 5, ω1 = −2.0508i and (c)β = 2, Reα = 10, ω1 = −1.0522i as pairs of vortices. It can beseen that an increasing value of Reα causes a stronger inclination of these vortices in the wall-normal and spanwise plane,
while the influence of Reα on the inclination is stronger thenfor the resolvent modes shown in Figure 4.

η̃2′ = C5 Ai (Z(x2 )) + C6 Bi (Z(x2 ))
(27)

Z 1
iAπβ Reα+
Ai (Z(x2 )) ·Bi (Z(x2 )) ũ′2,α dx2(−iReα )1/3
−1
Z 1
−Bi (Z(x2 ))Ai (Z(x2 )) ũ′2,α dx2 ,
−1

−2/3

with Z(x2 ) = (−i)1/3 Reα (Reα (ω1 − x2 ) + iβ 2 ). First the
Dirichlet boundary condition for the lower wall at x2 = −1 forη̃2′ is being implemented, leading toBi (Z(x2 = −1))
C = −γC6 .
Ai (Z(x2 = −1)) 6−1with κ1 = −γBi (Z(x2 = 1)) + Ai (Z(x2 = 1)). The streamwise
and spanwise fluctuations ũ′1 , ũ′3 can be obtained using the coninuity equation (5), resulting in

e−β E4 (Reα , β , ω1 , 1) − eβ E3 (Reα , β , ω1 , 1)
C4 . (26)eβ E1 (Reα , β , ω1 , 1) − e−β E2 (Reα , β , ω1 , 1)

C5 = −Z 1

iπβ ReακAi (Z(x2 = 1))
Bi(Z(x2 ))ũ′2,α dx2(−iReα )1/3
−1#
Z

−κBi (Z(x2 = 1))

so that C1 = C2 = 0. The Dirchlet boundary condition for theupper wall at x2 = 1 then leads to

C3 =

(29)
"

(28)

5

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
and the spanwise velocity fluctuation fluctuation ũ′2 and ũ′3 ,while the wall-normal vorticity η̃2′ and thus the streamwise
fluctuation ũ′1 are only dependent on Reα for the distinguishedlimit with high Reynolds numbers Re → ∞ and small streamwise wavenumbers α → 0. Once again an increase of Reαlead to inclined structures within the wall-normal and spanwise plane. The eigenfunctions obtained from the linear stability theory and the response modes from the resolvent analysis
were compared, where it could be seen that in both cases themost energy can be found in the streamwise direction, whereas
the amplitudes of the velocity fluctuations in wall-normal andspanwise direction are in the order of O(α) smaller comparedto the streamwise velocity fluctuations.
A first extension to this work is planned by also considering aturbulent mean velocity profile obtained from DNS data. Additionally the response of the resolvent system with frequencies close to the eigenvalues can be investigated in order toanalyse possible pseudo-resonance phenomena. Furthermore
various modifications, such as the plane Couette flow withwall-transpiration can be analysed in the future, where the influence of the wall-transpiration velocity on the stability of thecoherent structures could be a focus of the analysis.

Figure 8. Streamwise fluctuations are shown over the spanwise direction x3 and wall-normal direction x2 for (a) β = 2,
Reα = 1, ω1 = −10.1588i, (b) β = 2, Reα = 5, ω1 = −2.0508i
and (c) β = 2, Reα = 10, ω1 = −1.0522i as pairs of vortices.
Solid lines represent fluctuations with a positive sign, while
dashed lines represnent fluctuations with a negative sign.

CONCLUSION AND OUTLOOKIn this work the resolvent analysis was used in order to investigate the linear amplification mechanisms leading to coherent structures in laminar plane Couette flow with emphasison the distinguished asymptotic case with high Reynolds numbers Re → ∞ and small streamwise wavenumbers α → 0, with
Reα = Re · α = O(1).
Through a SVD decomposition of the input-output system (14)the most amplified structures and the energy of the systemwere analysed with using the knowledge of the laminar basevelocity profile only. The influence of the streamwise andspanwise wavenumbers α and β , as well as the Reynolds number Re on the first singular value σ1 , which represents the energy of the system, were investigated. The influence of Reα
on the streamwise velocity fluctuation was investigated, where
it could be seen that an increase of Reα lead to stronger inclinations of the streamwise velocity fluctuation within thewall-normal and spanwise plane due to the Orr-mechanism.
Additionally the applicability of the distinguished limit with
Reα = Re · α = O(1) for the linear stability theory was examined, where the eigenvalues of the Orr-Sommerfeld equation
could be obtained only depending on Reα , while analytical expressions for the eigenfunctions could be obtained using both
the Orr-Sommerfeld and Squire equation. It could be seen that
α only acts as a magnitude amplifier for both the wall-normal

REFERENCES
Gustavsson, L. 1981 Energy growth of three-dimensional disturbances in plane poiseuille flow. Journal of Fluid Mechanics 224, 241–260.
Hwang, Yongyun & Cossu, Carlo 2010a Amplification of coherent streaks in the turbulent couette flow: an input–output
analysis at low reynolds number. Journal of Fluid Mechanics 643, 333–348.
Hwang, Yongyun & Cossu, Carlo 2010b Linear non-normalenergy amplification of harmonic and stochastic forcing in
the turbulent channel flow. Journal of Fluid Mechanics 664,
51–73.
Schmid, Peter & Henningson, Dan 2001 Stability and Transition in Shear Flows, , vol. 142. Springer.
Weideman, J. A. & Reddy, S. C. 2000 A matlab differentiation
matrix suite. ACM Trans. Math. Softw. 26 (4), 465–519.
Yalcin, Alparslan, Turkac, Y. & Oberlack, Martin 2021 On thetemporal linear stability of the asymptotic suction boundary
layer. Physics of Fluids 33, 054111.

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