J. Fluid Mech.
(2013),
vol
. 734,
pp.
275–316.
c
©
Cambridge University Press 2013
275
doi:10.1017/jfm.2013.457
Model-based scaling of the streamwise energy
density in high-Reynolds-number turbulent
channels
Rashad Moarref
1
,
†
, Ati S. Sharma
2
, Joel A. Tropp
3
and Beverley J. McKeon
1
1
Graduate Aerospace Laboratories, California Institute of Technology, CA 91125, USA
2
Engineering and the Environment, University of Southampton, Southampton SO17 1BJ, UK
3
Computing & Mathematical Sciences, California Institute of Technology, CA 91125, USA
(Received 8 February 2013; revised 20 August 2013; accepted 24 August 2013;
first published online 9 October 2013)
We study the Reynolds-number scaling and the geometric self-similarity of a gain-
based, low-rank approximation to turbulent channel flows, determined by the resolvent
formulation of McKeon & Sharma (
J. Fluid Mech.
, vol. 658, 2010, pp. 336–382),
in order to obtain a description of the streamwise turbulence intensity from direct
consideration of the Navier–Stokes equations. Under this formulation, the velocity
field is decomposed into propagating waves (with single streamwise and spanwise
wavelengths and wave speed) whose wall-normal shapes are determined from the
principal singular function of the corresponding resolvent operator. Using the accepted
scalings of the mean velocity in wall-bounded turbulent flows, we establish that the
resolvent operator admits three classes of wave parameters that induce universal
behaviour with Reynolds number in the low-rank model, and which are consistent
with scalings proposed throughout the wall turbulence literature. In addition, it is
shown that a necessary condition for geometrically self-similar resolvent modes is
the presence of a logarithmic turbulent mean velocity. Under the practical assumption
that the mean velocity consists of a logarithmic region, we identify the scalings
that constitute hierarchies of self-similar modes that are parameterized by the critical
wall-normal location where the speed of the mode equals the local turbulent mean
velocity. For the rank-1 model subject to broadband forcing, the integrated streamwise
energy density takes a universal form which is consistent with the dominant near-wall
turbulent motions. When the shape of the forcing is optimized to enforce matching
with results from direct numerical simulations at low turbulent Reynolds numbers,
further similarity appears. Representation of these weight functions using similarity
laws enables prediction of the Reynolds number and wall-normal variations of the
streamwise energy intensity at high Reynolds numbers (
Re
τ
≈
10
3
–10
10
). Results
from this low-rank model of the Navier–Stokes equations compare favourably with
experimental results in the literature.
Key words:
mathematical foundations, Navier–Stokes equations, turbulent boundary layers
† Email address for correspondence: rashad@caltech.edu
276
R. Moarref, A. S. Sharma, J. A. Tropp and B. J. McKeon
1. Introduction
Understanding the behaviour of wall-bounded turbulent flows at high Reynolds
numbers has tremendous technological implications, for example, in air and water
transportation. This problem has received significant attention over the last two
decades especially in the light of full-field flow information revealed by direct
numerical simulations (DNS) at relatively small Reynolds numbers and high-Reynolds-
number experiments. Notwithstanding the recent developments, the highest Reynolds
numbers that are considered in DNS are an order of magnitude smaller than
experiments, which are in turn conducted at Reynolds numbers that are typically
two orders of magnitude smaller than most applications. This creates a critical demand
for model-based approaches that describe and predict the behaviour of turbulent flows
at technologically relevant Reynolds numbers.
Wall turbulence has been the topic of several reviews; see, for example, Robinson
(1991) and Adrian (2007) for structure of coherent motions, Gad-El-Hak &
Bandyopadhyay (1994) for turbulence statistics and scaling issues, Panton (2001) for
self-sustaining turbulence mechanisms, and Klewicki (2010), Marusic
et al.
(2010
c
)
and Smits, McKeon & Marusic (2011) for the latest findings and main challenges
in examining high-Reynolds-number wall turbulence. In the present study, special
attention is paid to scaling, universality, and geometric self-similarity of the turbulent
energy spectra at high Reynolds numbers. We also note that the energy spectra exhibit
clear signatures of coherent turbulent motions such as the near-wall streaks, the large-
scale motions (LSMs), and the very large-scale motions (VLSMs).
1.1.
Overview of dominant coherent motions
In the interests of giving a brief overview of the energetically dominant coherent
motions in wall turbulence, we will review three classes of structure. The near-wall
system of quasi-streamwise streaks and counter-rotating vortices with streamwise
length and spanwise spacing of approximately 1000 and 100 inner (viscous) units,
centred at approximately 15 inner units above the wall, has been well-studied. These
ubiquitous features of wall turbulence are responsible for large production of turbulent
kinetic energy (Kline
et al.
1967; Smith & Metzler 1983).
Another commonly observed feature of turbulent flows is the hairpin vortex. In
low-Reynolds-number flows, at least, packets of hairpin vortices have been observed to
extend from the wall to the edge of the boundary layer and constitute LSMs (Head
& Bandyopadhyay 1981; Adrian, Meinhart & Tomkins 2000; Adrian 2007), with
streamwise extent approximately 2–3 outer units (channel half-height, pipe radius, or
the boundary layer thickness).
VLSMs have been observed to reside in the logarithmic region of the turbulent
mean velocity, with lengths of approximately 10–15 outer units in boundary layers
and up to 30 outer units in channels and pipes (see, for example, Kim & Adrian
1999; Balakumar & Adrian 2007; Monty
et al.
2007). The emergence of VLSMs was
originally attributed to alignment of LSMs (Kim & Adrian 1999). However, Smits
et al.
(2011) concluded that this is unlikely since the detached LSMs are located
at a farther distance from the wall than the VLSMs and the attached LSMs have
much smaller width than VLSMs and are convected at different speeds. Recently, the
correlation between the envelope of small-scale activity and the large-scale velocity
signal (identified via filtering in spectral space), which has been interpreted as
an amplitude modulation of the small scales, has been investigated in detail, see
e.g. Hutchins & Marusic (2007
b
), Mathis, Hutchins & Marusic (2009
a
), Mathis
et al.
(2009
b
), Chung & McKeon (2010) and Hutchins
et al.
(2011).
Model-based scaling of the streamwise energy density
277
1.2.
Overview of scaling issues
In spite of recent advances in understanding the structure of wall turbulence, the
Reynolds-number scaling of the turbulent energy spectra and the energy intensities
remains an open area of research. The main experimental obstacle is maintaining
the necessary spatial resolution for measurement accuracy while achieving the high
Reynolds numbers required for large separation between the small and large turbulent
scales. For example, the available experiments are performed at relatively small
friction Reynolds numbers,
Re
τ
≈
O
(
10
4
)
, with a notable exception of the atmospheric
surface layer measurements of e.g. Metzger & Klewicki (2001) (
Re
τ
≈
O
(
10
6
)
) that
are in turn generally contaminated by surface roughness effects. Most high-Reynolds-
number experiments suffer from spatial resolution issues in the inner region (see, for
example, Hutchins
et al.
2009).
Significant experimental effort has been devoted to determining the behaviour of
the streamwise energy intensity at high Reynolds numbers since it dominates the
turbulent kinetic energy and is easier to measure relative to the wall-normal and
spanwise velocities. It is understood that both small and large scales contribute
to the streamwise energy intensity (Metzger & Klewicki 2001; Marusic & Kunkel
2003; Hutchins & Marusic 2007
a
; Marusic, Mathis & Hutchins 2010
a
). It is well-
known that a region of the streamwise wavenumber spectrum scales with inner units;
Marusic
et al.
(2010
a
) showed by filtering that the contribution of such scales to
the streamwise energy intensity, and therefore by extension also the streamwise
spectrum, is universal, i.e. independent of Reynolds number. On the other hand,
the large motions have been shown to scale in outer units (Kim & Adrian 1999);
Mathis
et al.
(2009
a
) proposed that the corresponding peak in streamwise intensity
occurs close to the geometric mean of the limits of the logarithmic region in the
turbulent mean velocity. The amplitude of this energetic peak increases with Reynolds
number and has a footprint down to the wall (Hutchins & Marusic 2007
b
). Using
data from experiments of canonical wall-bounded turbulent flows, Alfredsson,
̈
Orl
̈
u &
Segalini (2012) proposed a composite profile for the streamwise turbulence intensity
and showed the possibility of an outer peak at high Reynolds numbers. Note however,
that available data are not sufficiently well-resolved to determine unequivocally the
Reynolds-number scaling of either the inner or outer peaks of the streamwise energy
intensity (see, for example, Marusic
et al.
2010
a
).
Theoretical approaches also offer insight into the scaling of the spectrum with
increasing Reynolds numbers, originating with the attached-eddy concepts described
by Townsend (1976). These eddies are attached in the sense that their height scales
with their distance from the wall, and they are geometrically self-similar since their
wall-parallel length scales are proportional to their height. Perry & Chong (1982)
developed these ideas to include hierarchies of geometrically self-similar attached
eddies in the logarithmic region of the turbulent mean velocity. They systematically
predicted that if the population density of the attached eddies inversely decreases with
their height, both the turbulent mean velocity and the wall-parallel energy intensities
exhibit logarithmic dependence on the distance from the wall. The logarithmic
behaviour of the mean velocity and the streamwise energy intensity was recently
confirmed using high-Reynolds-number experiments (Marusic
et al.
2013). However,
the attached-eddy hypothesis does not predict the exact shape of the eddies or their
evolution in time.
Subsequent works by Perry and co-authors extended the attached-eddy formulation
beyond the logarithmic region; Marusic & Kunkel (2003) used empirical scaling
arguments concerning the effective forcing of the outer turbulence on the viscous
278
R. Moarref, A. S. Sharma, J. A. Tropp and B. J. McKeon
region to propose a similarity expression for the streamwise energy intensity that
is valid throughout the zero-pressure boundary layer. Recently, Marusic, Mathis &
Hutchins (2010
b
) outlined an observationally based, predictive formulation for the
variation of the streamwise turbulent intensity up to the geometric mean of the
logarithmic region based on consideration of the correlation between large and small
scales. Most recently, Mizuno & Jim
́
enez (2013) used DNS to show that self-similarity
of the velocity fluctuations is sufficient and seemingly important for reproducing a
logarithmic profile in the mean velocity. They also observed that the logarithmic
region can be maintained independent of the near-wall dynamics.
1.3.
Review of previous model-based approaches
We seek in this work a description of the streamwise turbulence intensity for
all wall-normal locations arising from direct consideration, and modelling, of the
Navier–Stokes equations (NSE). There has been much work in this vein, highlighting
several important features of the NSE. We provide a brief review of the most relevant
literature here.
The critical role of linear amplification mechanism in promoting and maintaining
turbulent flows was highlighted in DNS of Kim & Lim (2000). In addition, it was
shown that nonlinearity plays an important role in regenerating the near-wall region
of turbulent shear flows through a self-sustaining process (Hamilton, Kim & Waleffe
1995; Waleffe 1997; Schoppa & Hussain 2002). More recently, significant effort has
been directed at identification and analysis of exact solutions of the NSE, such as
travelling waves and periodic orbits, see e.g. Waleffe (2003) and Wedin & Kerswell
(2004) and the review paper by Kerswell (2005).
It is understood that high sensitivity of the laminar flow to disturbances provides
alternative paths to transition that bypass linear instability; see, for example, Schmid
& Henningson (2001). Trefethen
et al.
(1993) showed that the high flow sensitivity
is related to non-normality of the coupled Orr–Sommerfeld and Squire operators; see
also Schmid (2007). These operators are coupled in the presence of mean shear and
spanwise-varying fluctuations. Physically, as originally explained by Landahl (1975), a
large streamwise disturbance is induced on the flow in response to lift-up of a fluid
particle by the wall-normal velocity such that its wall-parallel momentum is conserved.
Even in linearly stable flows, the high sensitivity can result in large transient
responses, meaning that the energy of certain initial perturbations significantly grows
before eventual decay to zero (Gustavsson 1991; Butler & Farrell 1992; Klingmann
1992; Reddy & Henningson 1993; Schmid & Henningson 1994). In addition, the high
sensitivity is responsible for high energy amplification, meaning that the velocity
fluctuations achieve a large variance at the steady state for the flow subject to
zero-mean stochastic disturbances (Farrell & Ioannou 1993
b
; Bamieh & Dahleh
2001; Jovanovi
́
c & Bamieh 2005). The dominant structures that emerge from the
above transient growth and energy amplification analyses are reminiscent of the
streamwise streaks observed at the early stages of transition to turbulence (Matsubara
& Alfredsson 2001). They are characterized by infinitely long spanwise-periodic
regions of high and low streamwise velocity associated with pairs of counter-rotating
streamwise vortices that are separated by approximately 3
.
5 outer units.
It is believed that the NSE linearized around the turbulent mean velocity are stable
for all Reynolds numbers (Malkus 1956; Reynolds & Tiederman 1967). Early model-
based approaches extended the aforementioned sensitivity analyses of the laminar flow
to the turbulent channel flow and found dominance of streamwise streaks that are
spaced by 3 outer units, which is approximately the same as in the laminar flow. In
Model-based scaling of the streamwise energy density
279
addition to the outer-scaled dominant structures, Butler & Farrell (1993) and Farrell
& Ioannou (1993
a
) showed that the largest transient response over an eddy turnover
time of 80 inner units, associated with the near-wall cycle, is obtained for initial
perturbations that are infinitely long and have the same spanwise spacing as the
near-wall streaks, i.e. 100 inner units. The same streamwise and spanwise lengths were
obtained in flows subject to stochastic disturbances over a coherence time of 90 inner
units (Farrell & Ioannou 1998).
Reynolds & Hussain (1972) put forward a modified linear model to account for the
effect of background Reynolds stresses on the velocity fluctuations. They proposed to
augment the molecular viscosity by the turbulent eddy viscosity that is required to
maintain the mean velocity. This model yields two local optima for the structures with
largest transient growth (del
́
Alamo & Jim
́
enez 2006; Pujals
et al.
2009) and energy
amplification (Hwang & Cossu 2010) without the need for confining the optimization
time. These peaks correspond to streamwise-elongated structures with a spacing of 80
inner units and 3–4 outer units and are in fair agreement with the spacing of near-
wall streaks and the very large-scale motions in real turbulent flows. The geometric
similarity of the optimal transient response to initial perturbations and the optimal
responses to harmonic and stochastic forcings was highlighted by Hwang & Cossu
(2010) using the linearized NSE with turbulent eddy viscosity. These authors found
that the streamwise-constant optimal responses scale with the spanwise wavelength
in the wall-normal direction for spanwise wavelengths between the inner- and outer-
scaled regions.
An exact representation of the NSE was introduced by McKeon & Sharma (2010)
in which: (i) a set of linear sub-systems describe extraction of energy from the mean
velocity at individual wavenumbers/frequencies; and (ii) the only source of coupling
between these sub-systems is the conservative nonlinear interaction of their outputs,
that determines both the input to the sub-systems and the turbulent mean velocity.
At its heart is the ability to analyse the flow of energy from the mean velocity to
all the velocity scales and identify the essential linear amplification and nonlinear
redistribution mechanisms that drive the turbulent flow. The input–output relationship
of the linear sub-systems can be described by transfer functions whose low-rank nature
in the wall-normal direction enables significant simplification of their analysis.
One of the main differences between the formulation of McKeon & Sharma (2010)
and other input–output analyses of laminar and turbulent flows (see, for example,
Jovanovi
́
c & Bamieh 2005; Hwang & Cossu 2010) is parameterization of the waves
with wave speed rather than temporal frequency. The latter approaches showed
that the globally optimal transient growth and energy amplification takes place for
zero streamwise wavenumber and temporal frequency. Selecting the wave speed, as
emphasized by McKeon & Sharma (2010): (i) enables a systematic search for both
locally (in wall-normal direction) and globally optimal wave shapes and parameters;
(ii) removes the ambiguity about the wave speed corresponding to the globally optimal
waves by determining the limit of the ratio between zero streamwise wavenumber
and temporal frequency; and (iii) distinguishes between non-normality and critical
behaviour as the main linear amplification mechanisms.
McKeon & Sharma (2010) showed that the principal forcing and response directions
associated with the linear sub-systems are consistent with the dominant response
shapes in real turbulent pipe flows. In addition, the low-dimensional and sparse feature
of the resulting model enables development and utilization of compressive sampling
techniques for analysing the turbulent flow dynamics (Bourguignon
et al.
2013).
This formulation has also proven useful for pre- and post-diction of experimental
280
R. Moarref, A. S. Sharma, J. A. Tropp and B. J. McKeon
observations in turbulent pipe flow (McKeon, Sharma & Jacobi 2013; Sharma &
McKeon 2013).
1.4.
Paper outline
In this paper, we identify the Reynolds-number scaling of a low-rank approximation to
turbulent channel flow and utilize it for predicting the streamwise energy intensity at
high Reynolds numbers. Our development is outlined as follows. In § 2, we briefly
review the resolvent formulation, highlight its low-rank nature, and show that a
rank-1 approximation captures the characteristics of the most energetic modes of
real turbulent channels. The stage is set for studying the energy density of fluctuations
using a minimum number of assumptions by considering a rank-1 model in the wall-
normal direction subject to broadband forcing in the wall-parallel directions and time.
Furthermore, a summary of the computational approach for determining the rank-1
model is provided.
Three classes of wave parameters for which the low-rank approximation of the
resolvent exhibits universal behaviour (independence) with Reynolds number are
identified in § 3. The requirement for universality highlights the role of wave speed
in distinguishing these classes. Each class of waves is characterized by a unique range
of wave speeds and a unique spatial scaling that emerge from the resolvent. For the
rank-1 model subject to broadband forcing, we reveal the universal streamwise energy
densities, and show that the peaks of these energy densities roughly agree with the
most energetic turbulent motions, i.e. the near-wall streaks, the VLSMs, and the LSMs.
In § 4, we show that the streamwise energy density of the rank-1 model with
broadband forcing can be optimally weighted as a function of wave speed to match
the intensity of simulations at low turbulent Reynolds numbers. The weight functions
are then formulated using similarity laws which, in conjunction with the universal
energy densities, enable prediction of the streamwise energy intensity at high Reynolds
numbers. The paper is concluded in § 5 and limitations and several future directions
are discussed.
2. Low-rank approximation to channel flow
An overview of the rationale for considering a low-rank approximation to turbulent
channel flow is presented in this section. We follow the development of McKeon &
Sharma (2010) for turbulent pipe flow, showing that equivalent results are obtained for
channels and highlighting the new observations.
The pressure-driven flow of an incompressible Newtonian fluid is governed by the
non-dimensional NSE and the continuity constraint
u
t
+
(
u
·∇
)
u
+
∇
P
=
(
1
/
Re
τ
)1
u
,
∇·
u
=
0
,
}
(2.1)
where
u
(
x
,
y
,
z
,
t
)
is the velocity vector,
P
(
x
,
y
,
z
,
t
)
is the pressure,
∇
is the gradient
operator, and
1
=
∇·∇
is the Laplacian. The streamwise and spanwise directions,
x
and
z
, are infinitely long, the wall-normal direction is finite, 0
6
y
6
2, and
t
denotes
time; see figure 1(
a
) for the geometry. The subscript
t
represents temporal derivative,
e.g.
u
t
=
∂
u
/∂
t
. The Reynolds number
Re
τ
=
u
τ
h
/ν
is defined based on the channel
half-height
h
, kinematic viscosity
ν
, and friction velocity
u
τ
=
√
τ
w
/ρ
, where
τ
w
is the
shear stress at the wall, and
ρ
is the density. Velocity is normalized by
u
τ
, spatial
variables by
h
, time by
h
/
u
τ
, and pressure by
ρ
u
2
τ
. The spatial variables are denoted by
+
when normalized by the viscous length scale
ν/
u
τ
, e.g.
y
+
=
Re
τ
y
.
Model-based scaling of the streamwise energy density
281
y
xu
z
w
(
a
)(
b
)
F
IGURE
1. (Colour online) (
a
) Pressure-driven channel flow. (
b
) Schematic of a two-
dimensional propagating wave with streamwise and spanwise wavelengths
λ
x
and
λ
z
and
streamwise speed
c
.
2.1.
Decomposition in homogeneous directions
The velocity is decomposed using the Fourier transform in the homogeneous directions
and time
u
(
x
,
y
,
z
,
t
)
=
∫∫∫
∞
−∞
ˆ
u
(
y
;
κ
x
,κ
z
,ω)
e
i
(κ
x
x
+
κ
z
z
−
ω
t
)
d
κ
x
d
κ
z
d
ω,
(2.2)
where the hat denotes a variable in the transformed domain, and the triplet
(κ
x
,κ
z
,ω)
is the streamwise and spanwise wavenumbers and the temporal (angular) frequency.
The Fourier basis is optimal in the homogeneous wall-parallel directions. It is
also an appropriate basis in time under stationary conditions. For any
(κ
x
,κ
z
,ω)
6=
0,
ˆ
u
(
y
;
κ
x
,κ
z
,ω)
represents a propagating wave with streamwise and spanwise
wavelengths
λ
x
=
2
π
/κ
x
and
λ
z
=
2
π
/κ
z
and speed
c
=
ω/κ
x
in the streamwise
direction; see figure 1(
b
) for an illustration. Some special cases include standing
waves (
c
=
0), infinitely long waves (
κ
x
=
0), and infinitely wide waves (
κ
z
=
0).
In this study, we emphasize the eminent role of wave speed, a factor that was
highlighted by McKeon & Sharma (2010) while being predominantly neglected in
the previous studies, in determining the classes of propagating waves that are universal
with Reynolds number.
The turbulent mean velocity
U
(
y
)
= [
U
(
y
)
0 0
]
T
=
ˆ
u
(
y
;
0
,
0
,
0
)
corresponds to
(κ
x
,κ
z
,ω)
=
0 and is assumed to be known. Note that our main results, i.e. the
identified scalings in § 3, rely on the accepted scales of the turbulent mean velocity
and, otherwise, do not depend on the exact shape of
U
. McKeon & Sharma (2010)
avoided the closure problem for the mean velocity by using
U
(
y
)
obtained in pipe flow
experiments, but note that the resolvent formulation could be used to determine the
mean velocity profile, a topic of ongoing work (see McKeon
et al.
2013). Here,
we use a semi-empirical turbulent viscosity model, originally proposed for pipe
flow (Malkus 1956; Cess 1958) and extended to channel flow (Reynolds & Tiederman
1967), to determine
U
(
y
)
:
U
(
y
)
=
Re
τ
∫
y
0
1
−
ξ
1
+
ν
T
(ξ)
d
ξ,
(2.3
a
)
ν
T
(
y
)
=
1
2
{
1
+
(
κ
Re
τ
3
(
2
y
−
y
2
)(
3
−
4
y
+
2
y
2
)
{
1
−
e
(
|
y
−
1
|−
1
)
Re
τ
/α
}
)
2
}
1
/
2
−
1
2
,
(2.3
b
)
where
ν
T
is normalized by
ν
, and the parameters
α
and
κ
appear in the van Driest
wall law and the von K
́
arm
́
an log law. These parameters are obtained by minimizing
282
R. Moarref, A. S. Sharma, J. A. Tropp and B. J. McKeon
Nonlinearity
coupling
FT
IFT
F
IGURE
2. For any triplet
(κ
x
,κ
z
,ω)
, the operator
H
(κ
x
,κ
z
,ω)
maps the forcing
ˆ
f
to the
response
ˆ
u
. The different wavenumbers are coupled via the quadratic relationship between
f
(
x
,
y
,
z
,
t
)
and
u
(
x
,
y
,
z
,
t
)
. FT and IFT stand for Fourier transform and inverse Fourier
transform, respectively. The input–output map (shown with the dashed rectangle) is the main
focus of the present study.
the deviation between
U
(
y
)
in (2.3) and the DNS-based turbulent mean velocity profile.
The
α
and
κ
obtained for
Re
τ
=
186, 547 and 934 (Moarref & Jovanovi
́
c 2012)
suggest that both of these values converge for large
Re
τ
. We take
α
=
25
.
4 and
κ
=
0
.
426 for all Reynolds numbers and note that these values are optimized for
Re
τ
=
2003 (del
́
Alamo & Jim
́
enez 2006; Pujals
et al.
2009).
Following McKeon & Sharma (2010), the convective nonlinearity in (2.1) is
considered as a forcing term
f
=−
(
u
·∇
)
u
that drives the velocity fluctuations,
see also figure 2. For any
(κ
x
,κ
z
,ω)
6=
0, an equation for velocity fluctuations
ˆ
u
(
y
;
κ
x
,κ
z
,ω)
= [ˆ
u
ˆ
v
ˆ
w
]
T
around the turbulent mean velocity is obtained by
substituting (2.2) in (2.1), and using the orthonormality of the complex exponential
functions:
−
i
ω
ˆ
u
+
(
U
·∇
)
ˆ
u
+
(
ˆ
u
·∇
)
U
+
∇
ˆ
p
−
(
1
/
Re
τ
)1
ˆ
u
=
ˆ
f
,
(2.4
a
)
∇·
ˆ
u
=
0
.
(2.4
b
)
Here,
∇
=[
i
κ
x
∂
y
i
κ
z
]
T
,
1
=
∂
yy
−
κ
2
with
κ
2
=
κ
2
x
+
κ
2
z
, and
ˆ
f
(
y
;
κ
x
,κ
z
,ω)
=[
ˆ
f
1
ˆ
f
2
ˆ
f
3
]
T
=
∫∫∫
(κ
′
x
,κ
′
z
,ω
′
)
6=
(
0
,
0
,
0
)
(κ
′
x
,κ
′
z
,ω
′
)
6=
(κ
x
,κ
z
,ω)
(
ˆ
u
(
y
;
κ
′
x
,κ
′
z
,ω
′
)
·∇
)
×
ˆ
u
(
y
;
κ
x
−
κ
′
x
,κ
z
−
κ
′
z
,ω
−
ω
′
)
d
κ
′
x
d
κ
′
z
d
ω
′
.
(2.5)
McKeon & Sharma (2010) implicitly accounted for the continuity constraint by
projecting the velocity field onto the divergence-free basis of Meseguer & Trefethen
(2003). Here, we use a standard choice of wall-normal velocity
ˆ
v
and wall-normal
vorticity
ˆ
η
=
i
κ
z
ˆ
u
−
i
κ
x
ˆ
w
as the state variables,
ˆ
ζ
(
y
;
κ
x
,κ
z
,ω)
=[ˆ
v
ˆ
η
]
T
, to eliminate the
pressure term and the continuity constraint from (2.4) and obtain
−
(
i
ω
I
+
A
(κ
x
,κ
z
)
)
ˆ
ζ
(
y
;
κ
x
,κ
z
,ω)
=
C
†
(κ
x
,κ
z
)
ˆ
f
(
y
;
κ
x
,κ
z
,ω),
(2.6
a
)
ˆ
u
(
y
;
κ
x
,κ
z
,ω)
=
C
(κ
x
,κ
z
)
ˆ
ζ
(
y
;
κ
x
,κ
z
,ω).
(2.6
b
)
Here,
A
is the state operator,
C
maps the state vector to the velocity vector, and the
adjoint of
C
(denoted by
C
†
) maps the forcing vector to the state vector.
A
,
C
, and
C
†
Model-based scaling of the streamwise energy density
283
are operators in
y
and parameterized by
κ
x
and
κ
z
:
A
=
[
1
−
1
(
(
1
/
Re
τ
)1
2
+
i
κ
x
(
U
′′
−
U
1)
)
0
−
i
κ
z
U
′
(
1
/
Re
τ
)1
−
i
κ
x
U
]
,
(2.7
a
)
C
=
1
κ
2
i
κ
x
∂
y
−
i
κ
z
κ
2
0
i
κ
z
∂
y
i
κ
x
,
C
†
=
[
−
i
κ
x
1
−
1
∂
y
κ
2
1
−
1
−
i
κ
z
1
−
1
∂
y
i
κ
z
0
−
i
κ
x
]
,
(2.7
b
)
where
1
2
=
∂
yyyy
−
2
κ
2
∂
yy
+
κ
4
, and the prime denotes differentiation in
y
, e.g.
U
′
(
y
)
=
d
U
/
d
y
. The input–output relationship between
ˆ
f
and
ˆ
u
is obtained upon
elimination of
ˆ
ζ
from (2.6):
ˆ
u
(
y
;
κ
x
,κ
z
,ω)
=
H
(κ
x
,κ
z
,ω)
ˆ
f
(
y
;
κ
x
,κ
z
,ω),
(2.8
a
)
H
(κ
x
,κ
z
,ω)
=
C
(κ
x
,κ
z
)
R
A
(κ
x
,κ
z
,ω)
C
†
(κ
x
,κ
z
),
(2.8
b
)
where
R
A
(κ
x
,κ
z
,ω)
=−
(
i
ω
I
+
A
(κ
x
,κ
z
))
−
1
is the resolvent of
A
:
R
A
=
[
1
−
1
(
i
κ
x
((
U
−
c
)1
−
U
′′
)
−
(
1
/
Re
τ
)1
2
)
0
i
κ
z
U
′
i
κ
x
(
U
−
c
)
−
(
1
/
Re
τ
)1
]
−
1
.
(2.9)
As illustrated in figure 2, the only source of coupling between propagating waves with
different wavenumbers is the quadratic dependence of
f
(
x
,
y
,
z
,
t
)
on
u
(
x
,
y
,
z
,
t
)
. For
any wavenumber triplet, the input–output map from
ˆ
f
to
ˆ
u
(shown by the dashed
rectangle) represents a sub-system of the full NSE.
2.2.
Decomposition in the wall-normal direction
The transfer function
H
(κ
x
,κ
z
,ω)
provides a large amount of information about the
input–output relationship between
ˆ
f
and
ˆ
u
. Following the gain analysis of McKeon
& Sharma (2010), we use the Schmidt (singular value) decomposition to provide a
wall-normal basis based on the most highly amplified forcing and response directions:
ˆ
u
(
y
;
κ
x
,κ
z
,ω)
=
H
(κ
x
,κ
z
,ω)
ˆ
f
(
y
;
κ
x
,κ
z
,ω)
=
∞
∑
j
=
1
σ
j
(κ
x
,κ
z
,ω)
a
j
(κ
x
,κ
z
,ω)
ˆ
ψ
j
(
y
;
κ
x
,κ
z
,ω),
(2.10
a
)
a
j
(κ
x
,κ
z
,ω)
=
∫
1
−
1
ˆ
φ
∗
j
(
y
;
κ
x
,κ
z
,ω)
ˆ
f
(
y
;
κ
x
,κ
z
,ω)
d
y
,
(2.10
b
)
where
σ
1
>
σ
2
>
···
>
0 denote the singular values of
H
, and the singular functions
ˆ
φ
j
= [
ˆ
f
1
j
ˆ
f
2
j
ˆ
f
3
j
]
T
and
ˆ
ψ
j
= [ˆ
u
j
ˆ
v
j
ˆ
w
j
]
T
are respectively the forcing and response
directions corresponding to
σ
j
. In principle, there is an infinite number of singular
values/modes because the wall-normal coordinate is continuous. For the discretized
equation, the total number of singular values/modes is twice the number of grid points
in
y
since the resolvent operator
R
A
in (2.9) acts on a vector of two functions in
y
. As
highlighted by McKeon & Sharma (2010), the singular value decomposition effectively
demonstrates that there is a limited number of relatively highly amplified modes within
this total number of modes. Throughout this paper, we consistently refer to
ˆ
ψ
j
as
the
resolvent mode
, and distinguish it from the real turbulent flow that, under stationary
284
R. Moarref, A. S. Sharma, J. A. Tropp and B. J. McKeon
conditions, can be represented by a weighted sum of the resolvent modes. The latter
is denoted
the weighted mode
. Note that the resolvent modes were denoted response
modes in McKeon & Sharma (2010), McKeon
et al.
(2013) and Sharma & McKeon
(2013).
While the singular values of
H
are unique, additional treatment is necessary to
obtain unique singular functions. Unlike in a pipe, the singular values come in pairs
due to the wall-normal symmetry in the channel (which reflects itself in the resolvent
operator); see, for example, figure 4(
a
). For the modes with smaller streamwise
and spanwise wavelengths than the channel half-height, the singular values come in
equal pairs. Therefore, any linear combination of the corresponding singular functions
represents a legitimate singular function. For example, if the symmetric and anti-
symmetric modes are denoted by
ψ
s
and
ψ
a
where
|
ψ
s
|=|
ψ
a
|
, the singular function
given by
ψ
d
=
ψ
s
−
ψ
a
is zero in one half of the channel and twice
ψ
s
in the other
half. Clearly,
ψ
d
is also a singular function of the transfer function with the same
singular value as
ψ
s
and
ψ
a
. Physically, this means that the modes with lengths
and widths smaller than the channel half-height exhibit the potential to independently
evolve in either half of the channel provided that they are forced with a forcing (e.g.
disturbance) that is present only in one half of the channel. On the other hand, for
the modes with larger wavelengths than the channel half-height, the paired singular
values are different and the singular modes are either symmetric or anti-symmetric in
the opposite halves of the channel. Physically, these modes represent convective global
phenomena, meaning that they cannot take place independently in the opposite halves
of the channel. They convect with the same magnitude in the opposite halves of the
channel even though they can be of the same or opposite phase.
When the paired singular values are different, we obtain unique singular functions,
modulo a complex multiplicative constant of unit magnitude, by imposing an
orthonormality constraint on them:
∫
1
−
1
ˆ
φ
∗
j
(
y
;
κ
x
,κ
z
,ω)
ˆ
φ
k
(
y
;
κ
x
,κ
z
,ω)
d
y
=
∫
1
−
1
ˆ
ψ
∗
j
(
y
;
κ
x
,κ
z
,ω)
ˆ
ψ
k
(
y
;
κ
x
,κ
z
,ω)
d
y
=
δ
jk
,
(2.11)
where
δ
denotes the Kronecker delta. In the case where the paired singular values
are equal, we impose a symmetry/anti-symmetry constraint on the singular functions
in addition to the above orthonormality constraint. In other words, the corresponding
singular functions assume the same magnitude throughout the channel while being in
phase in one half of the channel and out of phase in the other half.
In
this
study,
we
select
the
unknown
multiplicative
constant
(after
orthonormalization) such that
ˆ
u
j
(
y
max
;
κ
x
,κ
z
,ω)
is a real number at the wall-normal
location
y
max
where the absolute value of
ˆ
u
j
is the largest. This choice places the
maximum of
u
j
(
x
,
y
,
z
,
t
;
κ
x
,κ
z
,ω)
at the origin
x
=
z
=
t
=
0. The channel symmetries
in the streamwise and spanwise directions can be used to obtain
u
j
,
v
j
, and
w
j
in the
physical domain:
u
j
(
x
,
y
,
z
,
t
;
κ
x
,κ
z
,ω)
=
4 cos
(κ
z
z
)
Re
(
ˆ
u
j
(
y
;
κ
x
,κ
z
,ω)
e
i
(κ
x
x
−
ω
t
)
)
,
(2.12
a
)
v
j
(
x
,
y
,
z
,
t
;
κ
x
,κ
z
,ω)
=
4 cos
(κ
z
z
)
Re
(
ˆ
v
j
(
y
;
κ
x
,κ
z
,ω)
e
i
(κ
x
x
−
ω
t
)
)
,
(2.12
b
)
w
j
(
x
,
y
,
z
,
t
;
κ
x
,κ
z
,ω)
=−
4 sin
(κ
z
z
)
Im
(
ˆ
w
j
(
y
;
κ
x
,κ
z
,ω)
e
i
(κ
x
x
−
ω
t
)
)
,
(2.12
c
)
Model-based scaling of the streamwise energy density
285
80
60
40
20
c
50
–50
0
–200200
0
80
60
40
20
0
–25025
(
a
)(
b
)
–5050
F
IGURE
3. (Colour online) The principal velocity response
ψ
1
(
x
,
y
,
z
,
t
;
κ
x
,κ
z
,
c
)
=
[
u
1
v
1
w
1
]
T
for
λ
+
x
=
700,
λ
+
z
=
100,
c
=
10, and
Re
τ
=
10 000 at
t
=
0. (
a
) The isosurfaces
of the streamwise velocity,
u
1
, at 60 % of its maximum. (
b
) The streamwise velocity (
u
1
,
contours) and the spanwise and wall-normal velocity (
v
1
,
w
1
, arrows) at
x
+
=
λ
+
x
/
2. The
contours in (
b
) represent positive (thick solid) and negative (thin dashed) values from 3 to 15
with increments of 3.
where Re and Im denote the real and imaginary parts of a complex number. The
representation of the forcing directions in the physical domain is obtained using
similar expressions.
From the singular value decomposition (2.10) and the orthonormality constraints
(2.11) it follows that if the forcing is aligned in the
ˆ
φ
j
-direction with unit energy, the
response is aligned in the
ˆ
ψ
j
-direction with energy
σ
2
j
. Consequently, the forcing and
response directions with the largest gain correspond to the principal singular functions
ˆ
φ
1
and
ˆ
ψ
1
. For any
(κ
x
,κ
z
,ω)
, the singular functions of
H
should be thought of
as propagating waves in the physical domain. In the rest of the paper, the resolvent
modes are characterized by
c
instead of
ω
and we note that prescribing any two of
κ
x
,
ω
, and
c
leads to the other.
Equivalent near-wall structures to those reported for pipe flows by McKeon &
Sharma (2010) and McKeon
et al.
(2013) are obtained for channel flows. For
example, the principal singular function
ψ
1
(
x
,
y
,
z
,
t
;
κ
x
,κ
z
,ω)
=[
u
1
v
1
w
1
]
T
for the
propagating wave corresponding to the energetic near-wall cycle (
λ
+
x
=
700,
λ
+
z
=
100,
c
=
U
(
y
+
=
15
)
=
10) for
Re
τ
=
10 000 is shown in figure 3. The streamwise
component of these structures contains regions of fast- and slow-moving fluids that are
aligned in the streamwise direction, slightly inclined to the wall, and are sandwiched
between counter-rotating vortical motions in the cross-stream plane.
2.3.
Low-rank nature of
H
The operator
H
, acting on functions of
y
, can be described as low rank if a significant
portion of its response to a broadband forcing in
y
is captured by projection on
the first few response directions. McKeon & Sharma (2010) highlighted the low-rank
nature of
H
for turbulent pipe flow. Figure 4(
a
) shows the first twenty singular values
of
H
for
λ
+
x
=
700,
λ
+
z
=
100, and
c
=
10 in turbulent channel flow with
Re
τ
=
2003.
We see that the largest pair of singular values is approximately one order of magnitude
larger than the other singular values.
The energetic contribution of the
k
th-direction
ˆ
ψ
k
to the total response in the
model subject to broadband forcing in
y
with fixed
λ
x
,
λ
z
, and
c
is quantified by
286
R. Moarref, A. S. Sharma, J. A. Tropp and B. J. McKeon
10
–1
10
–2
0
5
10
15
20
10
4
10
3
10
2
10
5
10
4
10
3
10
2
10
1
1.0
0.8
0.6
0.4
0.2
0
10
4
10
3
10
2
10
5
10
4
10
3
10
2
1.0
0.8
0.6
0.4
0.2
0
10
5
10
4
10
3
10
2
10
1
10
5
10
4
10
3
10
2
10
1
10
1
1.0
0.8
0.6
0.4
0.2
0
j
10
5
10
1
(
a
)(
b
)
(
c
)(
d
)
10
5
10
1
F
IGURE
4. (
a
) The twenty largest singular values of
H
for
λ
+
x
=
700,
λ
+
z
=
100,
c
=
10, and
Re
τ
=
2003. (
b
–
d
) The energy that is contained in the largest two response modes relative to
the total response,
(σ
2
1
+
σ
2
2
)/(
∑
∞
j
=
1
σ
2
j
)
, for different streamwise and spanwise wavelengths
and: (
b
)
c
=
U
(
y
+
=
15
)
; (
c
)
c
=
U
(
y
+
=
100
)
; and (
d
)
c
=
U
(
y
=
0
.
2
)
. The black contours
show the turbulent kinetic energy spectrum from the DNS of Hoyas & Jim
́
enez (2006) at
the corresponding critical wall-normal locations: (
b
)
y
+
=
15; (
c
)
y
+
=
100; and (
d
)
y
=
0
.
2.
The contours represent 10 % to 90 % of the maximum energy spectrum at each wall-normal
location with increments of 20 %.
σ
2
k
/(
∑
∞
j
=
1
σ
2
j
)
. Figures 4(
b
)–4(
d
) highlight the low-rank nature of
H
by showing that
the first two principal response directions
ˆ
ψ
1
and
ˆ
ψ
2
contribute more than 80 % of the
total response over a large range of wall-parallel wavelengths (red region) for wave
speeds
c
=
U
(
y
+
=
15
)
,
U
(
y
+
=
100
)
,
U
(
y
=
0
.
2
)
, and
Re
τ
=
2003. The relevance of
studying the low-rank approximation of
H
is further emphasized by noting that the
most energetic wavenumbers from the DNS of Hoyas & Jim
́
enez (2006) (contours)
coincide with the wavenumbers and critical wave speeds for which
H
is low rank.
We note that the streamwise velocity has the largest contribution to the kinetic energy.
Even though the shapes of the two-dimensional wall-normal and spanwise spectra may
be significantly different from the streamwise spectrum, the contours corresponding
to 70 % of the maximum in all spectra (not shown) lie within the region where the
contribution of the largest two singular values is more than 50 %.
2.4.
Rank-1 model subject to broadband forcing
In the present study, we consider a rank-1 model by only keeping the most
energetic forcing and response directions corresponding to
σ
1
and show that significant
Model-based scaling of the streamwise energy density
287
understanding of the scaling of wall turbulence can be obtained using this simple
model. This is motivated by the observation in § 2.3 that the operator
H
is essentially
a directional amplifier. In other words, we expect to see the principal singular response
of
H
in real turbulent flows provided that the principal forcing direction is present
in the nonlinear forcing term. Even though the resolvent modes corresponding to
σ
1
and
σ
2
comparably contribute to the total response, cf. § 2.3, considering one of the
resolvent modes is sufficient for capturing the wall-normal shape of the energy density.
This is because the two resolvent modes are symmetric/anti-symmetric counterparts
of each other and have the same magnitude. Therefore, accounting for both resolvent
modes yields the same result as accounting for one resolvent mode.
It is well-known that the streamwise energy spectrum can be divided into regions
that scale in inner and outer variables (see, for example, Morrison
et al.
2004). Our
objective is to explore the Reynolds-number scaling of the streamwise energy density
and predict the behaviour of the streamwise turbulence intensity at high
Re
τ
. We
focus on the streamwise velocity because it dominates the kinetic energy density in
turbulent flows. Similarly, the principal singular responses of
H
that result in the
largest energy amplification are dominated by their streamwise component, such that
the proposed gain-based decomposition yields the streamwise velocity most accurately.
This is in agreement with previous linear analyses of the global optimal responses, e.g.
del
́
Alamo & Jim
́
enez (2006) and Hwang & Cossu (2010). We note that higher-order
resolvent modes may have comparable or larger wall-normal and spanwise components
relative to the streamwise velocity, the study of which is a subject of ongoing work.
In order to use the smallest number of assumptions, we consider the case where the
forcing
ˆ
f
equals the principal forcing direction
ˆ
φ
1
. Consequently, the forcing has unit
energy for all wave parameters, meaning that it is broadband in
κ
x
,
κ
z
, and
c
. For the
rank-1 model with broadband forcing, we define the premultiplied streamwise energy
density of the principal response of
H
by
E
uu
(
y
;
κ
x
,κ
z
,
c
)
=
κ
2
x
κ
z
(
σ
1
(κ
x
,κ
z
,
c
)
|
u
1
|
(
y
;
κ
x
,κ
z
,
c
)
)
2
,
(2.13)
such that the premultiplied one-dimensional energy densities and the energy intensity
are obtained by integrating
E
uu
(
y
;
κ
x
,κ
z
,
c
)
over the set of all wave parameters
S
, e.g.
E
uu
(
y
,
c
)
=
∫∫
S
E
uu
(
y
;
κ
x
,κ
z
,
c
)
d log
(κ
x
)
d log
(κ
z
),
(2.14
a
)
E
uu
(
y
)
=
∫∫∫
S
E
uu
(
y
;
κ
x
,κ
z
,
c
)
d log
(κ
x
)
d log
(κ
z
)
d
c
,
(2.14
b
)
and
E
uu
(
y
,κ
x
)
and
E
uu
(
y
,κ
z
)
are determined similarly.
The above formulation of the energy density is used in § 3 to identify the
contribution of confined subsets of wave parameters to the energy density. We
establish that the energy density exhibits universal behaviour with
Re
τ
for properly
selected subsets of wave parameters. It is further shown that the emerging scales are
consistent with those observed in experiments. In addition, the scales of energetically
dominant waves roughly agree with the scales of dominant near-wall motions in real
turbulent flows.
2.5.
Computational approach
A pseudo-spectral method is used to discretize the differential operators in the wall-
normal direction on a set of Chebyshev collocation points. This is implemented using
the Matlab differentiation matrix Suite developed by Weideman & Reddy (2000).
288
R. Moarref, A. S. Sharma, J. A. Tropp and B. J. McKeon
Re
τ
N
x
N
y
N
z
N
c
λ
+
x
,
min
λ
x
,
max
y
+
min
λ
+
z
,
min
λ
z
,
max
U
cl
934
64
251
32
100
10
10
6
0
.
07
10
100
22
.
39
2003
64
251
32
100
10
5
×
10
5
0
.
15
10
50
24
.
02
3333
64
301
32
100
10
3
×
10
5
0
.
18
10
30
25
.
22
10 000
64
401
32
100
10
10
5
0
.
3
10
10
27
.
81
30 000
80
601
40
100
10
3
.
3
×
10
6
0
.
4
10
33
30
.
39
T
ABLE
1. Summary of the selected parameters in numerical computations at different
Reynolds numbers. In the wall-normal direction,
N
y
Chebyshev collocation points are used
with
y
+
min
denoting the closest point to the wall. In the streamwise and spanwise directions,
N
x
and
N
z
logarithmically spaced wavelengths are used between
λ
+
min
and
λ
max
. In addition,
N
c
linearly spaced wave speeds are chosen between
c
min
=
2 and
c
max
=
U
cl
.
Table 1 summarizes the selected range of wave parameters and their respective
resolution in numerical computations. It has been verified that the excluded wave
parameters are not energetically important and therefore do not change the results of
the present study.
An efficient randomized scheme developed by Halko, Martinsson & Tropp (2011)
is utilized to compute the principal singular directions of
H
for different Reynolds
numbers and wave parameters. The accuracy and computation time depend on the
decay of the singular values; a faster decay results in high accuracy or equivalently
less computation time to reach the same accuracy. In addition, if the singular values
are not well separated, the problem of computing the associated singular functions
is badly conditioned, meaning that it is hard for any method to determine them
very accurately. In this study, the above scheme approximately halves the total
computation time relative to Matlab’s
svds
algorithm. This becomes increasingly
important considering the three-dimensional wave parameter space that we need to
explore and the large size of the discretized resolvent operator (twice the number of
collocation points in
y
) at high Reynolds numbers. In addition, the randomized nature
of this scheme enables its parallel implementation which makes it especially suitable
for large-scale computations. Even though we have not used this feature in the present
study, it may find use in designing turbulent flow control strategies, e.g. by means of
spatially or temporally periodic actuations.
3. Universal behaviour of the resolvent
The formulation of § 2 facilitates analysis of the contribution of different wave
parameters
(κ
x
,κ
z
,
c
)
to the streamwise energy density. For the rank-1 model with
broadband forcing, the energy density of each wave is determined from the principal
singular values and singular functions of the transfer function
H
; see (2.13). In this
section, we identify unique classes of wave parameters for which
E
uu
(
y
;
κ
x
,κ
z
,
c
)
exhibits either universal behaviour with
Re
τ
or geometrically self-similar behaviour
with distance from the wall. Each class is characterized by a unique range of wave
speeds and a unique scaling of the wall-normal coordinate and the wall-parallel
wavelengths. These classes are inherent to the linear mechanisms in the NSE and
are rigorously identified by analysis of the transfer function.
Model-based scaling of the streamwise energy density
289
20
15
10
5
10
0
10
1
10
2
10
3
–1
–2
–3
–4
–5
–6
–7
20
15
10
5
10
0
10
1
10
2
10
3
–1
–2
–3
–4
–5
0
–6
c
(
a
)(
b
)
U
F
IGURE
5. (Colour online) (
a
) The one-dimensional energy density
Re
−
2
τ
E
uu
(
y
,
c
)
for
S
=
S
e
and
Re
τ
=
2003; and (
b
) the energy density normalized by its maximum value
over all
y
for fixed values of
c
. The colours are in logarithmic scale. The turbulent mean
velocity is shown by the black curve in (
b
).
3.1.
Requirement for universality of the resolvent modes
We start by showing that a requirement for universal behaviour is the wall-normal
locality of the resolvent modes. This is done by examining the underlying operators
in
H
, cf. (2.7)–(2.9). We see that the difference between the turbulent mean velocity
and the wave speed,
U
(
y
)
−
c
, and its wall-normal derivatives,
U
′
(
y
)
and
U
′′
(
y
)
,
appear as spatially varying coefficients in
H
. Since the turbulent mean velocity scales
differently with
Re
τ
in different wall-normal locations, only the resolvent modes that
are sufficiently narrow in
y
have the potential to be universal. This is because such
resolvent modes are purely affected by a certain part of the mean velocity that scales
uniquely with
Re
τ
.
We next show that the resolvent modes corresponding to the energetically significant
modes are in fact localized. As summarized by LeHew, Guala & McKeon (2011),
the energetic contribution of structures with convection velocities less than 10
u
τ
and
larger than the centreline velocity
U
cl
=
U
(
y
=
1
)
is negligible in real turbulent flows.
However, we are interested in determining the effect of a broader range of wave
speeds on the energy density. Note that small values of
c
result in small amplification
because the corresponding singular values are small. In fact, it is shown in § 4 that
including the modes with
c
.
2 does not improve the matching error between the
model-based and DNS-based energy intensities. This motivates defining a conservative
subset of
S
, denoted by
S
e
, that includes all wall-parallel wavenumbers and the
energetically important wave speeds:
S
e
={
(κ
x
,κ
z
,
c
)
|
2
6
c
6
U
cl
}
.
(3.1)
Figure 5 shows the one-dimensional energy density as a function of wave speed
E
uu
(
y
,
c
)
for
Re
τ
=
2003 and
S
=
S
e
. As evident from figure 5(
a
), the energy density
for a fixed
c
is localized in a narrow wall-normal region; note that the colours
are given in logarithmic scale. The localization is highlighted in figure 5(
b
) where
E
uu
(
y
,
c
)
is normalized by its maximum value over
y
for fixed values of
c
. We see that
the largest energy amplification takes place in the vicinity of the critical wall-normal
location where the turbulent mean velocity (thick black curve) equals the wave speed.
McKeon & Sharma (2010) argued that emergence of critical layers is one of the three