J. Fluid Mech.
(2023),
v
ol
.
959, A26, doi:10.1017/jfm.2023.149
Frequency-tuned surfaces for passive control of
wall-bounded turbulent flow – a resolvent
analysis study
Azadeh Jafari
1
,
†
, Beverley J. McKeon
2
and Maziar Arjomandi
1
1
School of Mechanical Engineering, University of Adelaide, Adelaide, SA 5005, Australia
2
Graduate Aerospace Laboratories, California Institute of Technology, 1200 East California Boulevard,
Pasadena, CA 91125, USA
(Received 16 September 2022; revised 17 January 2023; accepted 16 February 2023)
The potential of frequency-tuned surfaces as a passive control strategy for reducing drag
in wall-bounded turbulent flows is investigated using resolvent analysis. These surfaces
are considered to have geometries with impedances that permit transpiration and/or slip at
the wall in response to wall pressure and/or shear and are tuned to target the dynamically
important structures of wall turbulence. It is shown that wall impedance can suppress the
modes resembling the near-wall cycle and the very-large-scale motions and the Reynolds
stress contribution of these modes. Suppression of the near-wall cycle requires a more
reactive impedance. In addition to these dynamically important modes, the effect of wall
impedance across the spectral space is analysed by considering varying mode speeds and
wavelengths. It is shown that the materials designed for suppression of the near-wall modes
lead to gain reduction over a wide range across the spectral space. Furthermore, a wall with
only shear-driven impedance is found to suppress turbulent structures over a wider range
in spectral space
, leading to an overall turbulent drag reduction. Most importantly, the
present analysis shows that the drag-reducing impedance is non-unique and the control
performance is not sensitive to variations of the surface impedance within a favourable
range. This implies that specific frequency bandwidths can be targeted with periodic
material design.
Key words:
drag reduction
†
Email address for correspondence:
azadeh.jafari@adelaide.edu.au
© The Author(s), 2023. Published by Cambridge University Press. This is an Open Access article,
distributed under the terms of the Creative Commons Attribution licence (
http://creativecommons.org/
licenses/by/4.0/
), which permits unrestricted re-use, distribution and reproduction, provided the
original article is properly cited.
959
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A. Jafari, B.J. McKeon and M. Arjomandi
1. Introduction
Turbulent skin-friction drag is a major constituent of the total drag in many engineering
applications including air, sea, ground and fluid transportation. As an example,
approximately half of the total drag on an aircraft is due to skin-friction drag (Gad-el-Hak
1994
). Hence, due to the significant
eco
nomic and environmental benefits, reducing skin
friction has motivated considerable effort for the control of wall-bounded turbulence,
including active and passive control techniques. A particularly attractive passive control
concept, requiring no energy input and no complex control algorithms, is taking benefit
of the wall material properties through the interaction between the wall and the turbulent
flow, examples of which include perforated (Silvestri
et al.
2017
;Bhat
et al.
2021
; Jafari,
Cazzolato & Arjomandi
2022
), permeable (Breugem, Boersma & Uittenbogaard
2006
;
Kuwata & Suga
2017
;Suga
et al.
2018
;Chavarin
et al.
2020
,
2021
) and compliant
walls (Lee, Fisher & Schwarz
1993
; Xu, Rempfer & Lumley
2003
; Kim & Choi
2014
;
Xia, Huang & Xu
2017
). Based on the surface properties, these walls may suppress
and/or energise specific frequency bandwidths within the turbulent flow. While some of
the previous studies on permeable and compliant walls showed promising results, drag
increasing cases, arising from energised large-scale spanwise structures (Jiménez
et al.
2001
;Breugem
et al.
2006
; Kim & Choi
2014
; Luhar, Sharma & McKeon
2015
;Kuwata
&Suga
2017
), were also found. The motivation of this study is to explore walls that
could be tuned to passively suppress the dynamically important energetic structures of wall
turbulence without significantly amplifying other scales, such that these
fre
quency
-
tuned
walls could create an overall reduction in drag.
We consider a general framework in which walls are designed with geometries that
permit transpiration and/or slip in response to wall pressure and/or shear, and thus
passively interact with the turbulent flow. We seek to determine the potential of such
passive walls for reducing turbulent drag
, thus shedding light on future surface designs.
In the present study, surface impedance is employed to describe the interaction of the
frequency-tuned walls with the turbulent flow in terms of a modified wall-boundary
condition, i.e. an impedance wall formulation. The surface impedance is commonly used
as an effective boundary condition for acoustic analysis of the interactions of a surface with
an acoustic field. In classical definitions, surface impedance defines a linear relationship
between pressure and the flow velocity normal to the surface, i.e. a pressure-driven
impedance. The classical pressure-driven impedance has been used in the literature, in
the form of coupled wall-normal and pressure boundary conditions, for modelling of
the boundary layer stability and transition over perforated surfaces (Burden
1969
;Porter
1998
; Luhar
et al.
2015
). While the classical impedance formulation in previous studies
correlates the wall-normal velocity and pressure at the wall, it does not account for the
presence of viscous flow over the surface. It has been shown in the impedance eduction
measurements in the presence of grazing flow (Renou & Aurégan
2011
; Dai & Auregan
2016
; Boden
et al.
2017
) that surface impedance is also correlated with the wall shear.
Hence, the effect of surface impedance on the turbulent flow can only be fully described if
its correlation with wall shear, i.e. wall-shear-driven impedance, is also considered. This
is also of significance for design of control strategies as the studies in the literature suggest
a potential for passive control driven by wall shear. For example, Fukagata
et al.
(
2008
)
showed that a compliant wall which could be deformed by both streamwise wall shear
stress and pressure could create up to 8 % drag reduction in a turbulent channel flow.
Investigating the application of a wall-shear-driven compliant surface with in-plane wall
deformations, Józsa
et al.
(
2019
) also found that passive streamwise in-plane motions of
thewallcouldcreateupto3.8%dragreduction(
Re
τ
=
180), while passive spanwise
wall
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Frequency-tuned surfaces for control of wall turbulence
fluc
tu
a
tions increased skin friction by more than 50 %. To our knowledge, despite previous
studies on pressure-driven passive control, specifically passive walls such as permeable
and compliant walls, a generalised theoretical model for pressure/shear-driven impedance
walls is lacking.
This study develops a theoretical framework to investigate the interaction between
the frequency-tuned walls and turbulent flows incorporating both pressure- and
wall-shear-driven impedance. This framework is particularly beneficial for design of
passive frequency-tuned walls and provides an understanding of the combined effects
of passive pressure-driven and wall-shear-driven control approaches. Furthermore, this
bulk approach based on surface impedance permits modelling
of a surface with a general
geometry without the need to resolve geometric details. The latter not only requires large
and strenuous computations, but is also limited to the specific geometries considered.
Therefore, using the surface impedance formulation is advantageous for conducting a
generalised
and thorough analysis of the application of the described walls as a passive
control strategy. As the classical impedance formulation only considers pressure-driven
control, an improved generalised impedance formulation developed by Gabard (
2020
)
is adopted in the present study. The generalised impedance defines a linear correlation
between surface traction and flow velocity including the effects of mean shear at the
surface (Gabard
2020
), therefore incorporating both pressure- and wall-shear-driven
control schemes into the impedance tensor and considering a surface that allows either
or both transpiration and slip at the wall. This impedance formulation is introduced to
the resolvent analysis formulation of McKeon & Sharma (
2010
) to investigate the effect
of frequency-tuned surfaces on wall turbulence in the present study. By considering
both wall-shear- and pressure-driven impedance components, the developed framework
provides an improvement to the previous reduced-order models and can benefit design of
passive flow control strategies.
The remainder of this paper is organised as follows. The resolvent analysis and
impedance formulations are presented in §
2
. Section
3
describes the effect of wall
impedance on modes throughout the spectral space
, including those resembling the
near-wall cycle (as categorised by Smits, McKeon & Marusic
2011
) and very-large-scale
motions (VLSMs), with a streamwise length scale of
λ
x
>
5–10
δ
that appear in the
logarithmic region of the turbulent boundary layer at high Reynolds numbers and have a
modulating effect on smaller-scale turbulent activity (Mathis, Hutchins & Marusic
2009
;
Smits
et al.
2011
). Section
4
compares control approaches based on shear-driven and
pressure-driven impedances, and the effect of Reynolds numbers on the results is discussed
in §
5
. Further discussions on design of frequency-tuned surfaces are presented in §
6
.
Finally, conclusions are drawn in §
7
.
2. Methodology
This section describes the modelling approach implemented for investigation of the
effect of frequency-tuned surfaces on a fully developed turbulent channel flow. The
frequency-tuned surface is introduced via an impedance formulation for the boundary
conditions for the Navier–Stokes equations (NSEs). The induced change in the turbulent
flow structure is analysed through the resulting change in the structure and amplification
of the resolvent modes determined from resolvent analysis which are compared
with the
modes of the uncontrolled flow (a smooth impermeable wall). An overview of the resolvent
formulation is provided in §
2.1
, and the impedance boundary condition accounting for the
frequency-tuned surfaces is described in §
2.2
. The numerical implementation is described
in §
2.3
, and finally verification of the conducted modelling is presented in §
2.4
.
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A. Jafari, B.J. McKeon and M. Arjomandi
2.1.
Resolvent analysis
Resolvent analysis interprets the Fourier transformation of NSEs as a forcing-response
system with feedback. In this formulation, the linear terms of the NSEs are driven by the
feedback forcing, i.e. the
non
lin
ear terms of the NSEs, to generate a velocity and pressure
response. A low-order representation of the flow field is provided based on a gain-based
decomposition of the forcing-response transfer function. This low-order formulation has
been shown to reproduce the key structural features of wall turbulence (McKeon
2017
).
Specific resolvent modes have been associated with dynamically important structures of
wall turbulence such as the near-wall cycle and VLSMs (Moarref
et al.
2013
;Sharma
& McKeon
2013
; McKeon
2017
). It has been shown that these modes can be used as
low-order models for assessment of control techniques (Luhar, Sharma & McKeon
2014
b
;
Luhar
et al.
2015
; Nakashima, Fukagata & Luhar
2017
; Toedtli, Luhar & McKeon
2019
;
Chavarin
et al.
2021
). The reader is referred to McKeon (
2017
) and Toedtli
et al.
(
2019
)
for an in-depth discussion of the resolvent analysis and its application for evaluation of
control techniques.
For a fully developed turbulent channel flow that is stationary in time
t
and homogenous
in streamwise
x
and spanwise
z
directions, the Fourier-transformed NSEs after Reynolds
decomposition can be expressed
as
u
k
p
k
=
−
i
ω
I
0
−
L
k
−
∇
k
∇
T
k
0
−
1
I
0
f
k
=
H
k
f
k
.
(2.1)
Here,
u
=
[
u
,v,
w
]
T
represents the streamwise
u
, wall-normal
v
and spanwise
w
velocity
fields, and
p
is the pressure field.
I
and
ω
are the identity matrix and angular frequency,
respectively. Each
wavenumber–frequency combination
k
=
(κ
x
,κ
z
,ω)
represents a flow
structure, or mode, with streamwise and spanwise wavelengths
λ
x
=
2
π
/κ
x
and
λ
z
=
2
π
/κ
z
. These modes propagate downstream at streamwise wave speed
c
=
ω/κ
x
(the
wave speed normalised with friction velocity is
c
+
).
Also,
∇
k
=
[i
κ
x
,∂/∂
y
,
i
κ
z
]
T
and
∇
T
k
represent the gradient and divergence operators (where
T
shows the transpose) and
L
k
is
the linear Navier–Stokes operator. As shown in (
2.1
), the resolvent operator,
H
k
, maps the
non
lin
ear forcing
f
k
=
(
−
u
·∇
u
)
k
to a velocity
u
k
and pressure response
p
k
, where the
Fourier coefficient
u
k
and
p
k
denote the wall-normal variation in magnitude and phase of
the velocity and pressure field for each mode
k
. The special case of
k
=
(
0
,
0
,
0
)
represents
the mean velocity profile
u
0
=
[
U
(
y
),
0
,
0]
T
. Note that all parameters are normalised with
the friction velocity and half-channel height.
The resolvent operator
H
k
depends on the linear operator
L
k
,
where
L
k
=
⎡
⎣
−
i
κ
x
U
+
Re
−
1
τ
∇
2
k
−
∂
U
/∂
y
0
0
−
i
κ
x
U
+
Re
−
1
τ
∇
2
k
0
00
−
i
κ
x
U
+
Re
−
1
τ
∇
2
k
⎤
⎦
.
(2.2)
Here,
Re
τ
=
u
τ
H
/ν
is the friction Reynolds number based on the
half
-
chan
nel height
H
and
∇
2
k
=
[
−
κ
2
x
+
∂
2
/∂
y
2
−
κ
2
z
] is the Fourier-transformed Laplacian.
A discretised singular value decomposition (SVD) of
the resolvent operator yields a
set of orthonormal forcing
f
k
,
m
and response modes [
u
k
,
m
,
p
k
,
m
]
T
ordered based on the
input–output gain
σ
k
,
m
. To ensure orthonormality of the resulting forcing and response
modes under an
L
2
energy norm, the resolvent operator of (
2.1
) is scaled such
that
[
W
u
0
]
u
k
p
k
=
[
W
u
0
]
H
k
W
−
1
f
W
f
f
k
,
(2.3)
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Frequency-tuned surfaces for control of wall turbulence
or
W
u
u
k
=
H
k
S
W
f
f
k
.
(2.4)
Here,
H
k
S
is the scaled resolvent operator
;
W
u
and
W
f
are diagonal matrices containing
numerical quadrature weights, which ensure that the SVD of the resolvent operator,
H
k
S
=
m
ψ
k
,
m
σ
k
,
m
φ
∗
k
,
m
(2.5)
yields forcing modes
f
k
,
m
=
W
−
1
f
φ
k
,
m
and velocity response modes
u
k
,
m
=
W
−
1
u
ψ
k
,
m
with unit energy over the channel cross-section. Hence, (
2.1
)–(
2.5
) show that forcing
in the direction
of the
m
th singular forcing mode with unit amplitude
f
k
,
m
creates a
response in the direction of the
m
th singular response mode amplified by the singular
value, i.e.
σ
k
,
m
[
u
k
,
m
,
p
k
,
m
].
As shown by McKeon & Sharma (
2010
), the forcing-response transfer function tends to
be low rank at the
wavenumber–frequency combinations associated with the energetic
structures in wall turbulence. Since a rank-1 approximation of the resolvent operator
H
k
S
≈
ψ
k
,
1
σ
k
,
1
φ
∗
k
,
1
is shown to represent the characteristics of the most energetic modes
of wall-bounded turbulence (Moarref
et al.
2013
), the rank-1 approximation is retained
for the remainder of this study (refer to the Appendix for justification of this assumption
and analysis of higher ranks) and for convenience the subscript 1 is dropped. The rank-1
velocity and pressure fields will be referred to as the ‘resolvent modes’ and the rank-1
singular value
σ
k
,
1
referred to as ‘amplification’ or ‘gain’ for the remainder of this article.
The effect of wall impedance on the turbulent flow will be described in terms of velocity
and pressure response for singular modes of dynamic significance in
wall
tur
bu
lence. In
the present approach, only the shape and amplification of resolvent modes determined
from the SVD are analysed, which is equivalent to considering unit amplitude forcing for
all
k
.
To evaluate the potential of frequency-tuned surfaces for the control of
wall
tur
bu
lence,
their impact on the Reynolds stress generation is also investigated. As discussed by
Luhar
et al.
(
2015
), a suppression in the generation of Reynolds stress can be achieved
through: (
a
) a reduction in the magnitude of or a change in the form of the
non
lin
ear
forcing that leads to a reduction of the magnitude of velocity response, (
b
) a reduction
in the forcing-response
gain or (
c
) a change in mode structure leading to a reduction
in the Reynolds stress contribution from highly amplified resolvent modes. The present
analysis will identify the effectiveness of the frequency-tuned surfaces as a control scheme
through mechanisms (
b
)and(
c
) on a linear mode-by-mode basis noting that mechanism
(
a
) requires knowledge of the
non
lin
ear interactions, via the weighting factors (McKeon,
Sharma & Jacobi
2013
) or statistical estimation methods (Towne, Lozano-Durán & Yang
2019
), which themselves require data from experiment or simulation. While a more
complete model would require knowledge of the
non
lin
ear interactions and the coupling
between the resolvent modes, previous studies have shown that analysis of the resolvent
modes alone can provide valuable insight into the turbulent flow structure and can
approximate the response of the full
non
lin
ear system to control. Despite the considered
simplifications in the present approach, it can determine control-induced drag reduction in
trends which agree with direct numerical simulation (DNS) results
, as shown by Toedtli
et al.
(
2019
). In addition, the present approach based on the mode-by-mode analysis
provides valuable knowledge for optimal design of surfaces that have specific spatial
periodicity that is tuned to specific frequencies, i.e. frequency-tuned surfaces, and are not
aimed for overall drag reduction. Hence, a pattern search is adopted in the present approach
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A. Jafari, B.J. McKeon and M. Arjomandi
to find the wall impedance which favourably affects the turbulent flow structures with a
focus on suppression of the resolvent modes resembling the near-wall cycle and VLSMs.
Favourable is defined as: (
a
) a reduction in forcing-response amplification (
σ
k
) relative
to the uncontrolled flow, and (
b
) a reduction in the channel-integrated Reynolds stress
contribution from the resolvent mode (
RS
), defined as (Luhar
et al.
2015
)
RS
k
=
2
0
σ
2
k
Re
(
u
∗
k
v
k
)(
y
−
1
)
d
y
.
(2.6)
The weighted channel-integrated Reynolds stress in (
2.6
) is proportional to the turbulent
component of friction coefficient in the turbulent channel flow.
2.2.
Impedance boundary condition
The frequency-tuned surface is modelled by a generalised complex impedance. We
implement the generalised impedance proposed by Gabard (
2020
) that correlates the forces
inserted onto the surface by the fluid to the velocity vector via Cauchy stresses. This
generalised complex impedance is defined
as
̄
Z
=
̄
Z
tt
̄
Z
tn
̄
Z
nt
̄
Z
nn
,
(2.7)
and
̄
Z
.
̄
u
t
̄
u
n
=
− ̄
τ
nt
̄
p
− ̄
τ
nn
.
(2.8)
Here,
̄
represents dimensional variables,
t
is
a unit vector tangent to the surface and
n
is the wall-normal unit vector pointing into the surface
;
̄
Z
nn
represents the classical
acoustic impedance known as the inverse of admittance, i.e.
̄
Z
nn
correlates the pressure
at the surface to wall-normal velocity and if
̄
Z
nt
=
0, then:
̄
Z
nn
= ̄
p
/
̄
v
. This form of
complex impedance in combination with a dynamic boundary condition to account for
wall movements was previously used by Luhar
et al.
(
2015
) to simulate a compliant wall
(note that the impedance surfaces in the current study do not allow any wall deformation
or displacements). In (
2.8
),
̄
Z
tt
incorporates the effect of streamwise wall shear stress;
̄
Z
tn
and
̄
Z
nt
are related to the tangential force generated by the wall-normal velocity component
and the normal force created by the streamwise velocity component, respectively. The
non-diagonal components of the impedance tensor will be non-zero as, for instance for
a perforated surface with the perforations made at an angle to the
wall
nor
mal (Gabard
2020
).
We apply the concept of generalised impedance for modelling the frequency-tuned
surfaces and consider that the surface impedance affects the forces inserted on the flow at
the wall boundary by permitting slip and/or transpiration. The effect of surface impedance
is introduced as boundary conditions relating the fluctuating pressure and streamwise
wall shear stress to the fluctuating streamwise and wall-normal velocities. Hence, the
impedance boundary conditions at the bottom wall (
y
=
0) are expressed
as
−
Z
xx
u
k
(
0
)
+
Z
xy
v
k
(
0
)
=
1
Re
τ
∂
u
∂
y
(
0
),
(2.9)
Z
yx
u
k
(
0
)
−
Z
yy
v
k
(
0
)
=
p
k
(
0
).
(2.10)
Note that here the wall-normal vector (
y
axis) is pointing outward of the wall and all
parameters are normalised with friction velocity (specifically
Z
=
̄
Z
/ρ
u
τ
). We define the
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Frequency-tuned surfaces for control of wall turbulence
boundary
con
di
tions such that a passive surface with a positive solely real
Z
yy
allows
transpiration into the wall at high pressure regions. Accordingly, it is ensured that the wall
surface receives more energy than it provides to the fluid. Similarly, a passive surface with
a positive solely real
Z
xx
is defined to allow a negative slip velocity when at high shear
stresses. Equations (
2.9
)and(
2.10
) together with the no-slip condition for the spanwise
velocity (
w
k
=
0) are applied within the resolvent before computing the SVD to introduce
the effect of the frequency-tuned surface at the bottom wall.
At the top wall (
y
=
2), the boundary conditions are expressed with a sign change to
account for the change of wall-normal direction opposite to the channel
y
axis, and are
given
as
−
Z
xx
u
k
(
2
)
−
Z
xy
v
k
(
2
)
=−
1
Re
τ
∂
u
∂
y
(
2
),
(2.11)
Z
yx
u
k
(
2
)
+
Z
yy
v
k
(
2
)
=
p
k
(
2
).
(2.12)
As described in §
2.1
, to determine the effect of surface impedance, the resolvent modes
for the channel with impedance boundary conditions are compared with the uncontrolled
flow with the standard no-slip boundary conditions (
u
k
=
v
k
=
w
k
=
0) at the lower and
upper walls.
2.3.
Numerical implementation
A MATLAB code based on the resolvent code of a turbulent channel flow by Toedtli
et al.
(
2019
) is developed and employed in the present study. The resolvent operator is discretised
in the wall-normal direction (
y
) using a spectral collocation method on Chebyshev points.
The mean velocity profile
U
(
y
)
needed for the resolvent operator is computed from the
eddy viscosity model given by Reynolds & Tiederman (
1967
). It is assumed that the
surface impedance does not alter the mean velocity profile and
the
same mean profile
is applied to the case of the frequency-tuned surface. As discussed by Toedtli
et al.
(
2019
),
a sufficient estimation for the response of the
non
lin
ear system to control can be obtained
by using the canonical mean velocity profile.
For the present study, a grid resolution study was conducted which showed that
for
N
400, the singular values converged to within
O
(
10
−
7
)
and
O
(
10
−
4
)
for the
uncontrolled and controlled flow cases, respectively. Similarly, the Reynolds stress
contribution,
RS
k
, was found to converge to within
O
(
10
−
4
)
for
N
400 for both
uncontrolled and controlled cases. Therefore,
N
=
400 was used in this study, and
N
=
800 was used only for the plots showing the wall-normal profiles (such as
figure 9
).
2.4.
Modelling verification and comparison with previous simulations
In order to verify the predictions of resolvent analysis with the impedance boundary
conditions, the developed model is used for estimation of flow response to compliant and
porous walls, and the results are compared with DNS results from three different studies.
It is important to note that the current model does not reproduce the entire flow field
and the results are presented for one (or individual) resolvent modes. The model does not
consider the feedback to the mean flow and assumes broadband forcing. In addition, the
impedance formulation does not consider movement of the interface between the flow
and the subsurface as opposed to the compliant walls in DNS simulations that move
and deform. Considering these differences between the resolvent modelling approach and
DNSs, lack of quantitative agreement and precise match of profiles is to be expected.
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A. Jafari, B.J. McKeon and M. Arjomandi
0.2
0.4
0.6
0.8
–(
u
v
)
0
25
50
75
100
125
150
175
200
y
+
0
2
4
σ
2
Re(
u
∗
v
)
0
10
20
30
40
50
60
70
80
10
–2
10
0
10
2
Nondimensional damping
0.5
0.6
0.7
0.8
0.9
1.0
Drag ratio
(
a
)(
b
)(
c
)
Figure 1. Resolvent analysis predictions for the passive streamwise-shear-driven impedance study by Józsa
et al.
(
2019
): (
a
) Reynolds shear stress profiles from DNS results by Józsa
et al.
(
2019
), and (
b
) Reynolds shear
stress contribution of the resolvent mode representing the
near
-
wall cycle. The black solid lines show the base
flow and the dashed blue lines correspond to the control case. (
c
) The ratio of drag for control to base flow for
different damping ratios obtained for the near-wall resolvent mode in comparison with total drag determined
from DNS. The filled blue circles show the predictions by resolvent analysis and the unfilled black circles show
the DNS results by Józsa
et al.
(
2019
).
However, it is demonstrated that the developed model is able to predict the impacts of
surface porosity and compliance on the key structural features of turbulence (in terms of
variations in resolvent modes).
First, a passive compliant wall driven by streamwise shear (Józsa
et al.
2019
)is
considered. As found by the DNS of Józsa
et al.
(
2019
), a passive shear-driven compliant
wall control reduces Reynolds shear stress specifically at its peak which is associated with
the near-wall cycle (
figure 1
a
). We employ resolvent analysis to evaluate the effect of
the compliant wall by focusing only on the resolvent mode representing the near-wall
cycle using a surface impedance tensor. For this compliant wall,
Z
yy
=
Z
yx
=
Z
xy
=
0
and
Z
xx
is calculated using a
mass–spring–damper model. The wall properties are used
to calculate its mechanical admittance
c
p
=
i
ω/(ω
2
Λ
m
+
i
ωΛ
d
−
Λ
s
)
(Landahl
1962
)
with
Λ
m
,
Λ
d
and
Λ
s
representing the normalised mass, damping and spring coefficients
(which as shown by Nagy & Paál (
2019
) can also be applied for shear-driven impedance).
This mechanical admittance is correlated to the normalised impedance as:
Z
xx
=
1
/
Re
τ
c
p
.
For the compliant wall of Józsa
et al.
(
2019
), the mass, damping and spring coefficients
normalised by the bulk channel velocity were
Λ
m
=
4,
Λ
d
=
1
and
Λ
s
=
0
.
5, which when
normalised by friction velocity translate into
Λ
m
=
0
.
252,
Λ
d
=
0
.
063
and
Λ
s
=
0
.
0315.
Using these values at
Re
τ
=
180 and for the resolvent mode representing the near-wall
cycle
(κ
x
,κ
z
,
c
+
)
=
(
1
,
11
,
10
)
, it is found that
Z
xx
=−
0
.
0004
+
0
.
014i (with a negative
sign applied to account for conversion of coordinates).
Figure 1
(
b
) presents the predictions of resolvent analysis for
Z
xx
=−
0
.
0004
+
0
.
014i on
the turbulent Reynolds shear stress of the near-wall resolvent mode, and
figure 1
(
c
)shows
the effect of the damping of the passive wall on drag reduction (in terms of ratio of drag of
the controlled flow to drag of base flow which for the model is calculated from (
2.6
)). Note
that we are comparing results predicted by resolvent analysis for a single resolvent mode
(the near-wall mode) with the full DNS results. While the Reynolds shear stress profile (of
base and controlled flows) and the drag reductions do not describe the full flow, it is shown
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Frequency-tuned surfaces for control of wall turbulence
246810
k
x
4
6
8
10
12
14
c
+
246810
k
x
4
6
8
10
12
14
10
0
10
1
(
a
)(
b
)
Figure 2. Gain ratios for spanwise-constant modes (
κ
z
=
0) over a range of streamwise wavenumbers and wave
speeds for the passive pressure-driven compliant wall by Kim & Choi (
2014
) predicted by resolvent analysis:
(
a
) considering only the wall impedance, and (
b
) considering both impedance and the wall motion. The
+
symbol represents the two-dimensional waves observed in DNS of Kim & Choi (
2014
).
that the model is able to predict the response of the flow to shear-driven control and drag
reduction in trends that agree with DNS results.
The second comparison is made for a pressure-driven compliant wall simulated by
Kim & Choi (
2014
), in which large-amplitude two-dimensional waves were found to
emerge. To model this compliant wall (case II in the study of Kim & Choi
2014
), we
determine
Z
yy
from the mechanical admittance formulation using the
mass–spring–damper
model (
Z
yy
=
1
/
c
p
)and
Z
xx
=
Z
xy
=
Z
yx
=
0. Here, as given by Kim & Choi (
2014
),
Λ
m
=
2
and
the spring and damper coefficients normalised with the bulk velocity are
1 and 0.5, which translate into 440 and 10.5, respectively, when normalised with friction
velocity.
Figure 2
shows the gain ratios of
two
-
dimen
sional resolvent modes (
κ
z
=
0) over
a range of streamwise wavenumbers,
κ
x
, and wave speeds,
c
+
for this wall impedance at
Re
τ
=
140. Gain ratio is defined as the ratio of singular value of the compliant wall to
the base flow for each mode. As shown in
figure 2
(
a
), the predictions of the resolvent
analysis show a region of high amplification at
(κ
x
,
c
+
)
≈
(
4
.
5
,
6
.
5
)
to
(κ
x
,
c
+
)
≈
(
8
,
5
)
.
However, these modes correspond to a larger streamwise wavelength compared
with that
found in the DNS results with
λ
x
=
2
.
4
h
(
κ
x
=
2
.
6). The main reason for this difference
is lack of wall movement in the current model. This is demonstrated in
figure 2
(
b
), in
which
, in addition to the impedance boundary condition, wall movement is incorporated
in the boundary conditions using a linearised approximation by the equation derived
by Luhar
et al.
(
2015
)(
v
k
(
0
)
=−
i
ωη
k
, where
η
k
is the Fourier coefficient for wall
displacement). With consideration of wall movement, the amplified modes correspond
closely to those predicted in the DNS study. This comparison also suggests that the
deteriorating mechanisms observed over compliant walls are closely correlated with the
wall movement.
Finally, an analogy between surface impedance and permeability is adopted and the
resolvent predictions are compared with the results of a previous DNS study of a
zero-pressure-gradient boundary layer over a Darcy-type porous wall (Jiménez
et al.
2001
). Surface permeability,
K
, is interpreted as the inverse of pressure-driven impedance
resistance, i.e. Re
(
Z
yy
)
, and for a surface with
Z
yy
=
K
−
1
(noting that permeability can
be interpreted as a time-averaged impedance). The developed model is used to predict
the
two
-
dimen
sional resolvent modes (
κ
z
=
0) for a wall with
Z
yy
=
18
.
52 (and
Z
xy
=
Z
yx
=
Z
xx
=
0) representing the porous wall of Jiménez
et al.
(
2001
).
Figure 3
(
a
)shows
the ratio of singular values for the porous wall over an impermeable wall (base flow) at
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