1302.1594v2.pdf

Submitted to J. Fluid Mech.

Model-based scaling of the streamwise

energy density in high-Reynolds number

turbulent channels

By Rashad Moarref

1

, Ati S. Sharma

2

Joel A. Tropp

3

A N D

Beverley J. McKeon

1

Graduate Aerospace Laboratories, California Institute of Technology, CA 91125, USA

Engineering and the Environment, University of Southampton, SO17 1BJ, UK

Computing & Mathematical Sciences, California Institute of Technology, CA 91125, USA

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 (2010), in order to obtain a description of the stream-

wise 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 opera-

tor. 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 behavior with Reynolds number on the low-rank model, and which are

consistent with scalings proposed throughout the wall turbulence literature. In addition,

it was 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 (

≈

−

). Results from this low-rank model of the

Navier-Stokes equations compare favorably with experimental results in the literature.

1. Introduction

Understanding the behavior 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. Notwith-

standing 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

arXiv:1302.1594v2 [physics.flu-dyn] 3 Oct 2013

R. Moarref, A. S. Sharma, J. A. Tropp, & B. J. McKeon

at Reynolds numbers that are typically two orders of magnitude smaller than most ap-

plications. This creates a critical demand for model-based approaches that describe and

predict the behavior of turbulent flows at technologically relevant Reynolds numbers.

Wall turbulence has been the topic of several reviews; see, for example, Robinson

(1991); Adrian (2007) for structure of coherent motions, Gad-El-Hak & Bandyopadhyay

(1994) for turbulence statistics and scaling issues, Panton (2001) for self-sustaining tur-

bulence mechanisms, and Klewicki (2010); Marusic

et al.

(2010

); Smits

et al.

(2011) for

the latest findings and main challenges in examining high-Reynolds number wall turbu-

lence. 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, centered 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 & Bandy-

opadhyay 1981; Adrian

et al.

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

); Mathis

et al.

(2009

);

Chung & McKeon (2010); Hutchins

et al.

(2011).

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 spa-

tial 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,

≈O

(10

), with a notable exception of the atmospheric surface layer measurements

of, e.g., Metzger & Klewicki (2001) (

≈ O

(10

)) that are in turn generally contami-

Model-based scaling of the streamwise energy density

nated 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 behavior 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

;

Marusic

et al.

2010

). It is well-known that a region of the streamwise wavenumber

spectrum scales with inner units; Marusic

et al.

(2010

) 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

) 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

). Using data

from experiments of canonical wall-bounded turbulent flows, Alfredsson

et al.

(2012)

proposed a composite profile for the streamwise turbulence intensity and showed the

possibility for emergence 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

Theoretical approaches also offer insight into the scaling of the spectrum with in-

creasing 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 log-

arithmic 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 with the distance from the wall. The logarithmic behavior of the mean ve-

locity 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 be-

yond the logarithmic region; Marusic & Kunkel (2003) used empirical scaling arguments

concerning the effective forcing of the outer turbulence on the viscous region to propose a

similarity expression for the streamwise energy intensity that is valid throughout the zero

pressure boundary layer. Recently, Marusic

et al.

(2010

) 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 seemly 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 modeling, of the Navier-Stokes

R. Moarref, A. S. Sharma, J. A. Tropp, & B. J. McKeon

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 tur-

bulent flows was highlighted in direct numerical simulations 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

et al.

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 travel-

ing waves and periodic orbits, see e.g. Waleffe (2003); 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 even-

tual decay to zero (Gustavsson 1991; Klingmann 1992; Butler & Farrell 1992; Schmid

& Henningson 1994; Reddy & Henningson 1993). In addition, the high sensitivity is re-

sponsible 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 distur-

bances (Farrell & Ioannou 1993

; Bamieh & Dahleh 2001; Jovanovi ́c & Bamieh 2005).

The dominant structures that emerge from the above transient growth and energy am-

plification 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

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 addition to the

outer-scaled dominant structures, Butler & Farrell (1993); Farrell & Ioannou (1993

)

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 aug-

ment 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 amplifica-

tion (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

Model-based scaling of the streamwise energy density

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 analyze 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 systematical 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 behavior 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 analyzing the turbulent flow dynamics (Bourguignon

et al.

2013). This formulation

has also proven useful for pre- and post-diction of experimental observations in turbulent

pipe flow (Sharma & McKeon 2013; McKeon

et al.

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 behavior (independence) with Reynolds number are identified in

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

R. Moarref, A. S. Sharma, J. A. Tropp, & B. J. McKeon

(a)

(b)

Figure 1.

(a) Pressure driven channel flow. (b) Schematic of a two-dimensional propagating

wave with streamwise and spanwise wavelengths

and

and streamwise speed

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.

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

nondimensional NSE and the continuity constraint

+ (

·∇

)

∇

= (1

/Re

)∆

∇·

= 0

(2.1)

where

(

x,y,z,t

) is the velocity vector,

(

x,y,z,t

) is the pressure,

∇

is the gradient

operator, and ∆ =

∇·∇

is the Laplacian. The streamwise and spanwise directions,

and

, are infinitely long, the wall-normal direction is finite, 0

≤

2, and

denotes

time; see figure 1(a) for the geometry. The subscript

represents temporal derivative, e.g.

∂

/∂t

. The Reynolds number

h/ν

is defined based on the channel half-

height

, kinematic viscosity

, and friction velocity

√

/ρ

, where

is the shear

stress at the wall, and

is the density. Velocity is normalized by

, spatial variables

, time by

h/u

, and pressure by

ρu

. The spatial variables are denoted by

when

normalized by the viscous length scale

ν/u

, e.g.

2.1.

Decomposition in homogenous directions

The velocity is decomposed using the Fourier transform in the homogenous directions

and time

(

x,y,z,t

) =

∫∫∫

∞

−∞

(

;

,κ

,ω

) e

−

ωt

)

ω,

(2.2)

Model-based scaling of the streamwise energy density

where ˆ denotes a variable in the transformed domain, and the triplet (

,κ

,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 appro-

priate basis in time under stationary conditions. For any (

,κ

,ω

)

= 0,

(

;

,κ

,ω

)

represents a propagating wave with streamwise and spanwise wavelengths

= 2

π/κ

and

= 2

π/κ

and speed

ω/κ

in the streamwise direction; see figure 1(b) for

an illustration. Some special cases include standing waves (

= 0), infinitely long waves

(

= 0), and infinitely wide waves (

= 0). In this study, we emphasize the emi-

nent 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

(

) = [

(

) 0 0 ]

(

; 0

0) corresponds to

(

,κ

,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

. McKeon & Sharma (2010) avoided the closure

problem for the mean velocity by using

(

) 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

(

)

(

) =

∫

−

1 +

(

)

ξ,

(

) =

{

1 +

(

κRe

(

−

)(

−

+ 2

)

{

−

(

−

|−

)

}

)

}

−

(2.3)

where

is normalized by

, and the parameters

and

appear in the van Driest’s

wall law and the von K ́arm ́an log law. These parameters are obtained by minimizing the

deviation between

(

) in (2.3) and the DNS-based turbulent mean velocity profile. The

and

obtained for

= 186, 547, and 934 (Moarref & Jovanovi ́c 2012) suggest that

both of these values converge for large

. We take

= 25

4 and

= 0

426 for all

Reynolds numbers and note that these values are optimized for

= 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

−

(

·∇

)

that drives the velocity fluctuations, see also figure 2.

For any (

,κ

,ω

)

= 0, an equation for velocity fluctuations

(

;

,κ

,ω

) = [ ˆ

]

around the turbulent mean velocity is obtained by substituting (2.2) in (2.1), and using

the orthonormality of the complex exponential functions

−

+ (

·∇

)

+ (

·∇

)

∇

−

/Re

)∆

∇·

= 0

(2.4)

Here,

∇

= [ i

∂

]

, ∆ =

∂

−

with

, and

R. Moarref, A. S. Sharma, J. A. Tropp, & B. J. McKeon

(

;

,κ

,ω

) = [

]

−

∫∫∫

(

′

,κ

′

,ω

′

)

= (0

(

′

,κ

′

,ω

′

)

= (

,κ

,ω

)

(

;

′

,κ

′

,ω

′

)

·∇

)

(

;

−

′

,κ

−

′

,ω

−

′

) d

′

(2.5)

McKeon & Sharma (2010) implicitly accounted for the continuity constraint by pro-

jecting the velocity field onto the divergence-free basis of Meseguer & Trefethen (2003).

Here, we use a standard choice of wall-normal velocity ˆ

and wall-normal vorticity

= i

−

as the state variables,

(

;

,κ

,ω

) = [ ˆ

]

, to eliminate the pressure

term and the continuity constraint from (2.4) and obtain

−

ωI

(

,κ

))

(

;

,κ

,ω

) =

†

(

,κ

)

(

;

,κ

,ω

)

(

;

,κ

,ω

) =

(

,κ

)

(

;

,κ

,ω

)

(2.6)

Here,

is the state operator,

maps the state vector to the velocity vector, and the

adjoint of

(denoted by

†

) maps the forcing vector to the state vector.

, and

†

are operators in

and parameterized by

and

[

∆

−

(

/Re

)∆

+ i

(

′′

−

∆)

)

−

′

/Re

)∆

−

]





∂

−

∂





, C

†

[

−

∆

−

∂

∆

−

∆

−

∂

−

]

(2.7)

where ∆

∂

yyyy

−

∂

, and the prime denotes differentiation in

, e.g.

′

(

) =

. The input-output relationship between

and

is obtained upon elimination of

from (2.6)

(

;

,κ

,ω

) =

(

,κ

,ω

)

(

;

,κ

,ω

)

(

,κ

,ω

) =

(

,κ

)

(

,κ

,ω

)

†

(

,κ

)

(2.8)

where

(

,κ

,ω

) =

−

ωI

(

,κ

))

−

is the resolvent of

[

∆

−

(

((

−

)∆

−

′′

)

−

/Re

)∆

)

′

(

−

)

−

/Re

)∆

]

−

(2.9)

As illustrated in figure 2, the only source of coupling between propagating waves with

different wavenumbers is the quadratic dependence of

(

x,y,z,t

) on

(

x,y,z,t

). For any

wavenumber triplet, the input-output map from

(shown by the dashed rectangle)

represents a sub-system of the full NSE.

2.2.

Decomposition in the wall-normal direction

The transfer function

(

,κ

,ω

) provides a large amount of information about the

input-output relationship between

and

. Following the gain analysis of McKeon &

Sharma (2010), we use the Schmidt (singular value) decomposition to provide a wall-

Model-based scaling of the streamwise energy density

Figure 2.

For any triplet (

,κ

,ω

), the operator

(

,κ

,ω

) maps the forcing

to the

response

. The different wavenumbers are coupled via the quadratic relationship between

(

x,y,z,t

) and

(

x,y,z,t

). FT and IFT stand for Fourier transform and inverse Fourier trans-

form, respectively. The input-output map (shown with the dashed rectangle) is the main focus

of the present study.

normal basis based on the most highly amplified forcing and response directions:

(

;

,κ

,ω

) =

(

,κ

,ω

)

(

;

,κ

,ω

)

∞

∑

(

,κ

,ω

)

(

,κ

,ω

)

(

;

,κ

,ω

)

(

,κ

,ω

) =

∫

−

∗

(

;

,κ

,ω

)

(

;

,κ

,ω

) d

(2.10)

where

≥

≥···

0 denote the singular values of

, and the singular functions

[

]

and

= [ ˆ

]

are respectively the forcing and response directions

corresponding to

. In principle, there are 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

since the resolvent

operator

in (2.9) acts on a vector of two functions in

. As highlighted by McKeon &

Sharma (2010), the singular value decomposition effectively demonstrates that there are

a limited number of relatively highly-amplified modes within this total number of modes.

Throughout this paper, we consistently refer to

the resolvent mode

, and distinguish

it from the real turbulent flow that, under stationary conditions, can be represented by a

weighted sum of the resolvent modes. The latter is denoted by

the weighted mode

. Note

that the resolvent modes were denoted by response modes in McKeon & Sharma (2010);

McKeon

et al.

(2013); Sharma & McKeon (2013).

While the singular values of

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 wave-

lengths 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

and

where

, the singular function given by

−

is zero in one

half of the channel and twice

in the other half. Clearly,

is also a singular function

of the transfer function with the same singular value as

and

. Physically, this means

that the modes with lengths and widths smaller than the channel half-height exhibit the

potential to independently evolve in either halves of the channel provided that they are

forced with a forcing (e.g. disturbance) that is present only in one half of the channel.

R. Moarref, A. S. Sharma, J. A. Tropp, & B. J. McKeon

(color);

(arrows)

Figure 3.

The principal velocity response

ψ

(

x,y,z,t

;

,κ

) = [

]

for

= 700,

= 100,

= 10, and

= 10000 at

= 0. (Left) The isosurfaces of the streamwise velocity

at 60% of its maximum; (Right) The streamwise velocity (contours) and the spanwise and

wall-normal velocity (arrows) at

2. The contours in (b) represent positive (thick solid)

and negative (thin dashed) values from 3 to 15 with increments of 3.

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 phases.

When the paired singular values are different, we obtain unique singular functions,

modulo a complex multiplicative constant of unit magnitude, by imposing an orthonor-

mality constraint on them

∫

−

∗

(

;

,κ

,ω

)

(

;

,κ

,ω

) d

∫

−

∗

(

;

,κ

,ω

)

(

;

,κ

,ω

) d

(2.11)

where

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 ad-

dition 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

(

max

;

,κ

,ω

) is a real number at the wall-normal location

max

where the

absolute value of ˆ

is the largest. This choice places the maximum of

(

x,y,z,t

;

,κ

,ω

)

at the origin

= 0. The channel symmetries in the streamwise and spanwise

directions can be used to obtain

, and

in the physical domain

(

x,y,z,t

;

,κ

,ω

) = 4 cos(

) Re

(

;

,κ

,ω

) e

−

ωt

)

(

x,y,z,t

;

,κ

,ω

) = 4 cos(

) Re

(

;

,κ

,ω

) e

−

ωt

)

(

x,y,z,t

;

,κ

,ω

) =

−

4 sin(

) Im

(

;

,κ

,ω

) e

−

ωt

)

where Re and Im denote the real and imaginary parts of a complex number. The rep-

resentation of the forcing directions in the physical domain is obtained using similar

expressions.