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

SPANWISE WALL OSCILLATION APPLIED TO EXACT COHERENTSTATES OF PLANE COUETTE FLOW

Yacine Bengana
Department of Aeronautics
Imperial College London
South Kensington, London SW7 2AZ, UK
y.bengana14@imperial.ac.uk

Qiang Yang
State Key Laboratory of Aerodynamics
China Aerodynamics Research and Development Centre
Mianyang 621000, PR China
qiang.yang@skla.cardc.cn

Yongyun Hwang
Department of Aeronautics
Imperial College London
South Kensington, London SW7 2AZ, UK
y.hwang@imperial.ac.uk

ABSTRACT

unsteady shear layer (i.e. Stokes layer) that interacts with
the near-wall structures and can significantly reduce turbulent
skin-friction (Jung et al., 1992; Ricco et al., 2021). We defineskin-friction drag reduction as the relative difference of skinfriction coefficient of the controlled and uncontrolled states.
At low Reynolds numbers (Reτ ≃ 200, where Reτ is the friction Reynolds number), the spanwise wall oscillation can reduce the skin-friction drag up to 40% with the optimal period
T + ≈ 100. The skin-friction decreases as the amplitude of oscillation A+
w is increased monotonically.
The main goal of this study is to characterize the statespace dynamics of near-wall turbulence under spanwise wall
oscillation control. It has been shown that state-space dynamics may be an important concept to understand and optimize
the control of turbulent flow (Kawahara, 2005; Ibrahim et al.,2019). The state space is characterised by various invariant solutions (e.g. equilibria, traveling waves, periodic orbits, etc.),
which form the skeleton of its dynamics. In particular, the sustaining actuation like spanwise wall oscillation would deformthe entire state space. As such, the examination of how thecontrol changes the invariant solutions would provide physicalinsight into the state-space dynamics to understand and improve the given control strategy in wall-bounded shear flows.

The effect of spanwise wall oscillation on exact coherent
states in plane Couette flow is investigated for a large rangeof amplitudes (Aw ) and periods of oscillations (T ). The control forms a time-dependent shear layer, found responsible fora significant drag reduction of the upper branches of equilibrium solutions. On the other hand, it has only a minor effecton the lower branches. Increasing the oscillation amplitude decreases the skin-friction drag gradually and increases the stabilisation effect. The analysis of phase portraits reveals that thewall oscillation reduces the size of the turbulent region in state
space and damps the repelling strength of the upper branch solutions. Moreover, this modification of state-space dynamics
by spanwise wall oscillation stabilises most of the equilibrium
solutions’ unstable modes and shortens the turbulence lifetime
significantly.

INTRODUCTION
The skin-friction drag reduction in wall-bounded shear
flows is one of the main goals for many engineering designs.
Most of the skin-friction drag in such flows is related to theturbulence generation mechanism near the wall especially at
low Reynolds numbers. This mechanism, known as the selfsustaining process (Hamilton et al., 1995), is a quasi-cyclic
process of alternating quasi-streamwise vortices and high/lowspeed streaks. The quasi-streamwise vortices give rise to thestreaks through the lift-up mechanism (e.g. Butler & Farrell,1993). The streaks subsequently experience an instability ora transient growth, which causes their breakdown. Finally, after this breakdown, the quasi-streamwise vortices are regenerated. During the formation of the streaks, the lift-up effectbrings a high streamwise momentum to the wall, responsible
for a large part of skin-friction production (Kravchenko et al.,1993). Therefore, for the purpose of drag reduction at least at
low Reynolds numbers, it is relevant to act on the flow structures in the vicinity of the wall.
There are various strategies for turbulent skin-friction
drag reduction in near-wall turbulence. In this paper, we
will focus on the spanwise wall oscillation, which forms an

PROBLEM FORMULATION AND METHOD
We consider the canonical plane Couette flow of incompressible viscous fluid confined between two parallel impermeable plates. We denote x, y and z by the streamwise, wallnormal and spanwise directions, respectively. The upper andlower walls are located at y = ±h, and the flow is driven by
imposing the following boundary conditions: u|y=±h = ±Uw ,where u is the streamwise velocity and ±Uw the moving velocity of the upper and lower walls. We define the Reynolds
number as Re = Uw h/ν, where ν is the kinematic viscosity.
The numerical simulations are carried out using theNavier-Stokes solver Diablo (Bewley, 2014). In this code,
we use a second-order central differences scheme and Fourier
series with 2/3 dealiasing rule to discretize the wall-normal
and periodic directions respectively. A combination of the1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Crank-Nicolson and third-order Runge-Kutta method serves
for time integration. We considered two flow domainsΩN ≡ [Lx /h Ly /h Lz /h] = [1.75π 2 1.2π] and ΩHKW =[3.4662π 2 1.7391π]. The periodic boundary condition is
imposed in the wall-parallel directions. The spanwise walloscillation control is implemented, so that the two walls oscillate in phase in the spanwise direction according to
Ww (t) = Aw sin(ωt) = Aw sin(2πt/T ),(1)

where Ww is the spanwise velocity at the wall, Aw the amplitude, T the wall oscillation period, and ω the angular frequency. The parameters are scaled with inner viscous unitsof the uncontrolled turbulent state at Rep= 400: i.e. A+w =Aw /uτ and T + = Tu2τ /ν, where uτ = τw /ρ is the skinfriction velocity with τw being the wall-shear stress of theuncontrolled flow. The skin-friction coefficient is defined as
C f = 2(Reτ /Re)2 , where Reτ = uτ h/ν is the friction Reynolds
number.
In the present study, we consider four sets of equilibrium
solutions, i.e. EQ0 obtained in ΩN by Nagata (1990), EQ1,
EQ4 and EQ7 obtained in ΩHKW by Gibson et al. (2009). Theedge state is a periodic orbit (PO) identical to the gentle periodic solution found by Kawahara & Kida (2001). Depending on the skin-friction value, the solutions are classified into
lower (low friction) and upper (high friction) branches (seebelow for further details). The solutions are computed using the Newton-Krylov-Hookstep method, which minimizes
the relative error between the initial state and the final statetranslated in time and streamwise direction at t = T0 . When
there is no control, the equilibrium states considered are stationary. Therefore, the period T0 can arbitrarily be fixed, andT0 = 8h/Uw is used in the present study. However, under
spanwise wall oscillation control, the period needs to satisfy
T0 = nT , where a positive integer number n is fixed as n = 1
to minimize the computational cost. Once an equilibrium solution is obtained, its bifurcation is also studied. To construct
the bifurcation diagrams, the continuation with the Reynolds
number Re or the control parameters Aw and T of the solutions
is carried out using pseudo-arc-length continuation which improves the computation of the equilibrium solutions near turning points. Finally, the stability of each equilibrium solution is
examined using the standard Arnoldi iteration.

Figure 1. Bifurcation diagram of the equilibrium solutions
and periodic orbit: EQ0, black thick solid line; EQ1, red solidline; EQ4, pink dash-dotted line; EQ7, blue dashed line; PO,green dotted line. The vertical dotted line indicates Re = 400.

Previous studies have shown that spanwise wall oscillation reaches the optimal of drag reduction at T + ≈ 100. Figure2(a) shows the bifurcation curves for EQ0 for different oscillation periods T + ∈ [0, 300] at a fixed wall oscillation amplitudeA+
w = 0.85. In this figure, we focus on the identification of thesaddle-node point. Therefore, for this figure, we use the viscous inner units of the uncontrolled saddle-node point for thescaling of the oscillation period. The control reduces the skinfriction drag for the upper branch significantly, while it has
a minor effect on the lower solution. The saddle-node point
is continuously pushed to higher Re as the oscillation period
T + is increased, from Rec = 127.3 at T + = 0 to Rec = 142.5
at T + = 211.9. We note that at T + = 264.9 the saddle-nodepoint decreases to Rec = 142.2 and increases only slightly for
T + > 106. Those observations indicate an optimal oscillation
period around T + = 100, consistent with the DNS data as wellas our previous bifurcation study of the ECS under spanwise
wall oscillation in Poiseuille flow (Yang et al., 2019). Figure2(b) shows the bifurcation curves of the equilibrium solution
EQ0 for different spanwise wall oscillation amplitudes at fixedwall oscillation period T + = 79.8. The solid line indicates theuncontrolled reference case. When increasing A+
w , the upper
branch shows a significant drag reduction. The strong control
effect on the upper branch involves a substantial change in thebifurcation diagram by introducing new bifurcations along theupper branch. On the contrary, the effect on the lower branch
is nearly negligible over almost the entire range of Re. The
effect of the control is, however, not monotonic. Indeed, near
the bifurcation point, the skin-friction drag of the lower branch
is slightly increased. Therefore, the two branches are pushed
towards each other until they merge and disappear. The consequence is that the control pushes the threshold of the saddlenode bifurcations to higher Reynolds numbers, which indicates a stabilisation of the equilibrium solutions. Figure 2(c)shows the bifurcation curves of the skin-friction coefficient to
the wall oscillation amplitude A+
w for various oscillation periods T + at a fixed Re = 400. The upper branch exhibits a significant reduction of skin friction when increasing A+
w , whichis consistent with figure 2(b). On the other hand, C f of thelower branch depends on each solution and the period of oscillation considered. Therefore the two branches converge andmeet at a given value of A+
w , forming a saddle-node bifurcation

RESULTS AND DISCUSSION
Bifurcation overview
The bifurcation diagram, represented by C f − Re, of theequilibrium solutions and edge state, is shown in figure 1.
Each solution arise from a saddle-node bifurcation. The solution EQ0, discovered by Nagata (1990), appears at the smallest
critical Reynolds number Rec ≈ 127, then followed by EQ7 at
Rec ≈ 167 from Gibson et al. (2009). The bifurcation curves
for EQ1 and EQ4 show several bifurcations. EQ1 exhibits aseries of sophisticated saddle-node bifurcations in which thefour branches twisted one another. The first saddle-node bifurcation occurs at Rec ≈ 386, and the second and third bifurcations occur on the upper branch at Rec = 408 and Rec = 435,
respectively. EQ4 arises from a saddle-node bifurcation at
Rec = 275, followed by a second from the lower and upper
branche at Rec = 494 and Rec = 438, respectively. The bifurcation curve of PO has a critical Reynolds number located at
Rec = 236. To analyse the action of control on the bifurcation
curves, we focus only on the equilibrium solution EQ0. The
other solutions show similar behaviours.

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

Figure 3. Spanwise Stokes layer for EQ0 and analytical solution at 8 equally separated phases. (a) Case A : Re = 400,+
T + = 159.7, A+
w = 2.7. (b) Case B : Re = 400, T = 159.7,+
Aw = 4.5.

Statistics
The spanwise wall oscillation form an unsteady near-wall
spanwise shear layer (i.e. Stokes layer) that interacts with thenear-wall structures. For the laminar case, the Stokes layer isrepresented analytically as (Schlichting, 1979)
W (y,t) = Aw e−η sin(2πt/T − η),(2)pwhere η = −y πRe/T drives the Stokes layer thickness
which depends on the oscillation period. We note that the amplitude act on the spanwise displacement of the wall. In thelaminar regime, the fluid enhanced by the spanwise wall oscillations and the streamwise flow are decoupled. Figure 3 shows
the Stokes layer at 8 equally separated phases for the cases A
and B indicated by the circle and triangle in figure 2(c), respectively. Here we fix the oscillation period T + = 159.7 and+consider two amplitudes A+
w = 2.7 (case A) and Aw = 4.5 (caseB). The analytical Stokes layer deviates from the equilibrium
state when the control is applied. We note that the deviation
decreases when close to the wall and becomes substantial awayfrom the wall. The Stokes layer of the equilibrium state is
fixed by the wall-normal gradient of the spanwise Reynolds
shear stress term ∂ v′ w′ /∂ y. The Reynolds shear stress has
been found to increase with T + and Re (Ricco & Quadrio,2008). Therefore, this justifies the improved agreement withthe analytical solution when the Stokes layer is thin at small
oscillation period T + (not shown here). We observe from figure 3 that the state with a higher drag (case A) differs significantly from the analytical solutions. Whereas, the state with alower drag (case B) shows a quite good agreement even if the

Figure 2. Bifurcation curves for EQ0. (a) C f − Re with different oscillation periods at A+
w = 0.85. Here wall units of
the uncontrolled case at the saddle-node point are used for thescaling of the wall oscillation period. (b) C f − Re with different oscillation amplitudes at T + = 79.8. The arrow indicates+the direction of increasing A+
w . (c) C f − Aw with different oscillation periods at Re = 400. The red circle and triangle indicate two cases (A and B) of parameters for which the statistics
are analysed.

+
in terms of A+
w . The critical Aw of the saddle-node bifurcationis reached at lower values when the wall-oscillation period T +is increased.

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

Figure 5. Phase portraits for controlled and uncontrolled
states at Re = 400. The control parameters are set to A+
w = 17.7
and T + = 14.3. (a) IU − Du plane where Dr is the rate of total dissipation of the uncontrolled laminar state. (b) Euu − Evv
plane. The black dashed and solid cyan lines represent the controlled and uncontrolled turbulent states. The open and closed
circles indicate the mean of those turbulent trajectories. The
open and closed symbols show the uncontrolled and controlledequilibrium states, respectively. Each controlled equilibrium
state is shown as the average over the wall-oscillation period.

Figure 4. Root mean square (r.m.s) velocity profiles for case++
A (Re = 400, T + = 159.7 and A+
w = 2.7). (a) urms , (b) vrms ,+′′and (c) −u v . The upper (lower) branch of the uncontrolled
state is represented by dashed-dotted (dashed) lines. The controlled state is shown with dotted lines at 8 equally separated
phases, and its mean profile is indicated by the solid line. The
components are scaled with viscous inner units of the uncontrolled DNS.

compare to the upper-branch solution, which indicates a reduction of the drag according to the FIK identity (Fukagata et al.,2002). Indeed the upper branch solution shows 35.8% of dragreduction. Those statistics features and bifurcation behaviour
are similar to the other equilibrium solutions in the HKW box(not shown).

amplitude of oscillations is twice smaller.
Figure 4 shows the root mean square (r.m.s) velocity com+′ ′+ponents, u+
rms , vrms , and the streamwise shear stress u v . The
dotted lines indicate eight equally separated phases over one
wall oscillation period. The peak in u+
rms of the upper branch
is near the wall and that of the lower branch is located in
the channel center. The profiles of state A (with Re = 400,
T + = 159.7 and A+
w = 2.7) appear between the lower and upper branches. Compared to the upper branch, the state A shows
an increase of u+
rms in the channel center and the near-wallpeak observed in the upper branch is significantly weakened.
+
Moreover, the control weakens the Reynolds shear stress u′ v′State-space dynamics
We will visualise and analyse, in this section, the state
space using phase portraits. Following previous studies (e.g.
Kawahara & Kida, 2001; Ibrahim et al., 2019), we introduce
the global mean kinetic energy (MKE):

412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

dEU Uw τuw 1
=
−
dt
h } V
| {z
|

Z

ν
∂U
−⟨u′ v′ ⟩x,z
dV −
∂y
V
Ω
{z
} |

Z 

Ww τww 1
−
+
h } V
| {z
|

Z

Z 

IU

IW

Ω

Puv

ν
∂W
−⟨v′ w′ ⟩x,z
dV −
∂
y
V
Ω
{z
} |

∂U
∂y
{z

2

DU

Ω

Pvw

∂W
∂y
{z

DW

dV
}
2
dV
}
(3)

with the MKE given by

EU =

1
V

Z

U · U dV,

(4)Ωwhere Ω (ΩN or ΩHKW ) is the given computational
domain, Uw and Ww are the streamwise and spanwise velocities at the wall, and τuw = ν⟨∂U/∂ y|±h ⟩x,z and τww =ν⟨∂W /∂ y|±h ⟩x,z the mean streamwise and spanwise shear
stress, respectively. The average over the streamwise and spanwise directions is indicated by ⟨·⟩x,z . The terms, IU , Puv andDU in the first line of equation (3), are the rate of energy input, the rate of energy transport by streamwise mean shear toturbulent kinetic energy (TKE), and the dissipation rate by thestreamwise mean velocity, respectively. Likewise, the threeother terms in the second line, i.e. IW , Pvw and DW , correspond to the contributions from the spanwise wall oscillation.
The volume-averaged equation of TKE writes as

dEu
1
=
dt
V
|

Z

1
+
V
|

Z

Figure 6. Spectra of complex-conjugate eigenvalues com+pared between uncontrolled and controlled (A+w = 17.7, T =14.3) states. (a) the lower branch and (b) the upper branch of
the equilibrium solution EQ4 at Re = 400.

∂U−⟨u v ⟩x,z
dV
∂y
Ω
{z
}
′ ′

Puv

∂W
ν
−⟨v′ w′ ⟩x,z
dV −
∂y
V
Ω
{z
} |
Pvw

(5)

Z

(∇u′ )2 dVΩ{z}spanwise energy input IW feeds all the solutions (closed symbols). Figure 5(a) shows that the control pushes all the solutions towards the laminar state in the direction of decreasing IU and Du . Indeed, at these specific parameters, all
the solutions indicate a reduction of drag (not shown). The
upper branches and the mean turbulent state show a significant control effect compared to the lower solutions. Indeed,
the lower branches EQ4L and EQ7L show 7.3% and 3.2% ofdrag reduction (DR), and the upper branches EQ4U and EQ7U
show 25.8% and 33.7% of DR respectively, and the mean turbulent state indicates 10.9% of DR. The phase portrait in
Euu − Evv plane shown in figure 5(b) indicates similar results.
The decrease in Evv is the consequence of a reduction in thewall-normal velocity fluctuation resulting from the control andpushing the solutions towards the laminar state. Generally, thecontrol significantly affects the upper branches and has only amodest effect on the lower branches. The phase portraits indicate that the spanwise wall oscillation reduces the spread of
the turbulent state in phase space. Furthermore, the state-space
distances between the upper and lower branch of each solution
are reduced, indicating stabilisation of turbulence.
More generally, Figure 5 shows a reduction in the size ofthe turbulent state region in state-space. This observation may
indicate that the control damps the repelling strength of eachequilibrium. Therefore, we further investigate this reduction in
the turbulent state region by checking the eigenspectra of theDuwith Eu = Euu + Evv + Eww is the total turbulent energy
fluctuation averaged over the domain, written as(Euu , Evv , Eww ) =

1
V

Z
V

(u′2 , v′2 , w′2 )/Uw2 dV,(6)

The first two terms, Puv and Pvw in equation (5), areidentical to those of (3), however in equation (5) they correspond to the rate of turbulence production by streamwise andspanwise mean shear. The last term Du is the turbulent kinetic
energy dissipation. For details about the derivation of (3) and(5) in the context of spanwise wall oscillation in channel flow,
the reader may refer to Ricco et al. (2012). Finally, the rate of
total energy input and dissipation are given by I ≡ IU + IW
and D ≡ DU + DW + Du , respectively.
We report in figure 5 the phase portraits in Du and theEuu − Evv plane, respectively. We selected the solutions from
ΩHKW and we fix Re = 400 and the control parameters to+A+
w = 17.7 and T = 14.3. We present the equilibrium solutions and periodic edge state by averaging over a single walloscillation period while we show the turbulent state as timedependent. Naturally, when the control is applied, non-zero

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Conclusion
We investigated, in this study, the effect of spanwise wall
oscillation control on exact coherent states computed in two
different domains. We found that the upper branches of the solutions are stabilized, and their skin-friction drag is reduced.
On the other hand, the lower branches are merely affectedby the control. The stabilisation of the upper-branch solution translates by the movement of the saddle-node bifurcation
point of the solutions to higher Reynolds numbers. When wevary the oscillation amplitude, this behaviour is observed for
all the solutions. However, when varying T + , such a trend isreversed for EQ0 when T + ≥ 100, consistent with the observation in the previous DNS. We observed in the phase portraits
that the spanwise wall oscillation reduces the size of the state
space region as well as the repelling strength of the upperbranch equilibrium states. The change in the state-space dynamics relates to the significant turbulence lifetime decrease.
Finally, the suppression of the lift-up effect may be the principal stabilisation mechanism of exact coherent structures.

Figure 7. Probability distribution of turbulence to sustain
over time at Re = 400 and A+
w = 1.2 for different values ofwall oscillation period T + . The square symbols indicate theuncontrolled reference case at A+
w = 0. The black arrow indicates the increasing direction of T + .

Acknowledgments
Y. B. and Y. H. acknowledge the support of the Engineering and Physical Sciences Research Council (EPSRC;
EP/T009365/1) in the UK. Q. Y. acknowledges the Natural Science Foundation of China (11802322, 92052301) andthanks Prof. Y. M. Chung for useful discussions.

exact coherent states. Figure 6 shows the complex-conjugate
eigenvalues of the lower and upper branches of EQ4. The realand imaginary parts indicate the growth rate and frequencyof the eigenmodes, respectively. The lower branch solutions
show a small drag reduction (or even drag increase) and pairwith a negligible influence on the growth rates of the leading
eigenmodes, indicating a weak effect of spanwise wall oscillation control. However, we observe for the upper branch solutions of both cases, that the growth rates of the eigenmodes follow that of the drag when control is active. Indeed, the growthrate decreases where the control provokes a reduction of drag,indicating a stabilisation.

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Turbulence in Couette flow with the given flow domains
is known as a ‘chaotic saddle’, and it is characterised by finite lifetime (Eckhardt et al., 2007a). The lifetime of the uncontrolled turbulent state also increases very rapidly with the
Reynolds number (Eckhardt et al., 2007b). The state-spacecharacteristics discussed in figures 5 and 6 suggest a strong
similarity between the effect of the control and that of lowering
the Reynolds number, indicating a reduction of turbulence lifetime when control is active. Therefore, the overall influence ofthe state space by the spanwise wall oscillation is finally examined by computing the lifetime of turbulence subject to thegiven control. To evaluate the lifetime, we carried out groupsof 100 controlled and uncontrolled DNS at Re = 400. The
initial conditions of the simulations were first obtained from100 states computed at a slightly higher Reynolds number,
Re = 415. The lifetime is defined as a probability distribution+
for the amplitude A+
w = 1.2 and oscillation periods T = 57.2,
71.5, 85.8, and 114.4. To monitor the lifetime of each simulation, we fix a threshold of the turbulent energy fluctuation
Evv + Euu < 10−5 , below which turbulence will decay to laminar flow. Figure 7 shows the probability for turbulence to sus+
tain up to a given time t as a function of A+
w and T . We
first observe the classic exponential distribution of the lifetime,
confirming that this is a memoryless process (Eckhardt et al.,2007a). It is found that the spanwise wall oscillation significantly reduces the lifetime of turbulence at A+
w = 1.2, as thewall-oscillation period T + is increased. The control clearlyenhances flow relaminarisation.

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Ricco, Pierre, Skote, Martin & Leschziner, Michael A 2021 A
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Flow 29 (4), 891–902.

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