Model-based scaling of the streamwise energy density in high-Reynolds number turbulent channels
Rashad Moarref, Ati S. Sharma, Joel A. Tropp, Beverley J. McKeon
SSubmitted to J. Fluid Mech. Model-based scaling of the streamwiseenergy density in high-Reynolds numberturbulent channels
By Rashad Moarref , Ati S. Sharma ,Joel A. Tropp A N D
Beverley J. McKeon 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 resolventformulation of McKeon & Sharma (2010), in order to obtain a description of the stream-wise turbulence intensity from direct consideration of the Navier-Stokes equations. Underthis formulation, the velocity field is decomposed into propagating waves (with singlestreamwise and spanwise wavelengths and wave speed) whose wall-normal shapes aredetermined 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 thatinduce universal behavior with Reynolds number on the low-rank model, and which areconsistent with scalings proposed throughout the wall turbulence literature. In addition,it was shown that a necessary condition for geometrically self-similar resolvent modes isthe presence of a logarithmic turbulent mean velocity. Under the practical assumptionthat the mean velocity consists of a logarithmic region, we identify the scalings thatconstitute 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 energydensity takes a universal form which is consistent with the dominant near-wall turbulentmotions. When the shape of the forcing is optimized to enforce matching with resultsfrom direct numerical simulations at low turbulent Reynolds numbers, further similarityappears. Representation of these weight functions using similarity laws enables predictionof the Reynolds number and wall-normal variations of the streamwise energy intensityat high Reynolds numbers ( Re τ ≈ − ). Results from this low-rank model of theNavier-Stokes equations compare favorably with experimental results in the literature.
1. Introduction
Understanding the behavior of wall-bounded turbulent flows at high Reynolds numbershas tremendous technological implications, for example, in air and water transportation.This problem has received significant attention over the last two decades especially inthe 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 inDNS are an order of magnitude smaller than experiments, which are in turn conducted a r X i v : . [ phy s i c s . f l u - dyn ] O c t 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 andpredict 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 c ); Smits et al. (2011) forthe 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 geometricself-similarity of the turbulent energy spectra at high Reynolds numbers. We also notethat the energy spectra exhibit clear signatures of coherent turbulent motions such asthe 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 motionsin 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 spanwisespacing of approximately 1000 and 100 inner (viscous) units, centered at approximately15 inner units above the wall, has been well-studied. These ubiquitous features of wallturbulence are responsible for large production of turbulent kinetic energy (Kline et al. et al. et al. et al. (2011) concludedthat this is unlikely since the detached LSMs are located at a farther distance from thewall than the VLSMs and the attached LSMs have much smaller width than VLSMsand are convected at different speeds. Recently, the correlation between the envelope ofsmall scale activity and the large scale velocity signal (identified via filtering in spectralspace), which has been interpreted as an amplitude modulation of the small scales, hasbeen investigated in detail, see e.g. Hutchins & Marusic (2007 b ); Mathis et al. (2009 a , b );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 Reynoldsnumber scaling of the turbulent energy spectra and the energy intensities remains anopen area of research. The main experimental obstacle is maintaining the necessary spa-tial resolution for measurement accuracy while achieving the high Reynolds numbersrequired 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 ), with a notable exception of the atmospheric surface layer measurementsof, e.g., Metzger & Klewicki (2001) ( Re τ ≈ O (10 )) that are in turn generally contami- odel-based scaling of the streamwise energy density et al. a ;Marusic et al. a ). It is well-known that a region of the streamwise wavenumberspectrum scales with inner units; Marusic et al. (2010 a ) showed by filtering that thecontribution of such scales to the streamwise energy intensity, and therefore by extensionalso the streamwise spectrum, is universal, i.e. independent of Reynolds number. Onthe 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 streamwiseintensity occurs close to the geometric mean of the limits of the logarithmic region in theturbulent mean velocity. The amplitude of this energetic peak increases with Reynoldsnumber and has a footprint down to the wall (Hutchins & Marusic 2007 b ). Using datafrom experiments of canonical wall-bounded turbulent flows, Alfredsson et al. (2012)proposed a composite profile for the streamwise turbulence intensity and showed thepossibility for emergence of an outer peak at high Reynolds numbers. Note however, thatavailable data are not sufficiently well-resolved to determine unequivocally the Reynoldsnumber scaling of either the inner or outer peaks of the streamwise energy intensity (see,for example, Marusic et al. a ).Theoretical approaches also offer insight into the scaling of the spectrum with in-creasing Reynolds numbers, originating with the attached eddy concepts described byTownsend (1976). These eddies are attached in the sense that their height scales withtheir distance from the wall, and they are geometrically self-similar since their wall-parallel length scales are proportional to their height. Perry & Chong (1982) developedthese ideas to include hierarchies of geometrically self-similar attached eddies in the log-arithmic region of the turbulent mean velocity. They systematically predicted that ifthe population density of the attached eddies inversely decreases with their height, boththe turbulent mean velocity and the wall-parallel energy intensities exhibit logarithmicdependence with the distance from the wall. The logarithmic behavior of the mean ve-locity and the streamwise energy intensity was recently confirmed using high-Reynoldsnumber experiments (Marusic et al. et al. (2010 b ) outlined an observationally-based, predictive formulation for the variation of the streamwise turbulent intensity upto the geometric mean of the logarithmic region based on consideration of the correlationbetween large and small scales. Most recently, Mizuno & Jim´enez (2013) used DNS toshow that self-similarity of the velocity fluctuations is sufficient and seemly importantfor reproducing a logarithmic profile in the mean velocity. They also observed that thelogarithmic 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 importantfeatures 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). Inaddition, it was shown that nonlinearity plays an important role in regenerating thenear-wall region of turbulent shear flows through a self-sustaining process (Hamilton et al. exact solutions of the NSE, such as travel-ing waves and periodic orbits, see e.g. Waleffe (2003); Wedin & Kerswell (2004) and thereview paper by Kerswell (2005).It is understood that high sensitivity of the laminar flow to disturbances providesalternative paths to transition that bypass linear instability; see, for example, Schmid &Henningson (2001). Trefethen et al. (1993) showed that the high flow sensitivity is relatedto 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-varyingfluctuations. Physically, as originally explained by Landahl (1975), a large streamwisedisturbance 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 achievea large variance at the steady state for the flow subject to zero-mean stochastic distur-bances (Farrell & Ioannou 1993 b ; 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 stagesof transition to turbulence (Matsubara & Alfredsson 2001). They are characterized byinfinitely long spanwise-periodic regions of high and low streamwise velocity associatedwith pairs of counter-rotating streamwise vortices that are separated by approximately3 . 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 infinitelylong 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 stochasticdisturbances over a coherence time of 90 inner units (Farrell & Ioannou 1998).Reynolds & Hussain (1972) put forward a modified linear model to account for theeffect 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 maintainthe mean velocity. This model yields two local optima for the structures with largesttransient growth (del ´Alamo & Jim´enez 2006; Pujals et al. odel-based scaling of the streamwise energy density et al. et al. Paper outline
In this paper, we identify the Reynolds number scaling of a low-rank approximationto turbulent channel flow and utilize it for predicting the streamwise energy intensityat high Reynolds numbers. Our development is outlined as follows: In §
2, we brieflyreview the resolvent formulation, highlight its low-rank nature, and show that a rank-1approximation captures the characteristics of the most energetic modes of real turbulentchannels. The stage is set for studying the energy density of fluctuations using a minimumnumber of assumptions by considering a rank-1 model in the wall-normal direction subjectto broadband forcing in the wall-parallel directions and time. Furthermore, a summaryof the computational approach for determining the rank-1 model is provided.Three classes of wave parameters for which the low-rank approximation of the resolventexhibits universal behavior (independence) with Reynolds number are identified in § 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 propagatingwave with streamwise and spanwise wavelengths λ x and λ z and streamwise speed c . broadband forcing, we reveal the universal streamwise energy densities, and show that thepeaks 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 broadbandforcing can be optimally weighted as a function of wave speed to match the intensity ofsimulations at low turbulent Reynolds numbers. The weight functions are then formulatedusing similarity laws which, in conjunction with the universal energy densities, enableprediction of the streamwise energy intensity at high Reynolds numbers. The paper isconcluded in §
2. Low-rank approximation to channel flow
An overview of the rationale for considering a low-rank approximation to turbulentchannel 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 channelsand highlighting the new observations.The pressure-driven flow of an incompressible Newtonian fluid is governed by thenondimensional NSE and the continuity constraint u t + ( u · ∇ ) u + ∇ P = (1 /Re τ )∆ 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 gradientoperator, and ∆ = ∇ · ∇ is the Laplacian. The streamwise and spanwise directions, x and z , are infinitely long, the wall-normal direction is finite, 0 ≤ y ≤
2, and t denotestime; 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 = (cid:112) τ w /ρ , where τ w is the shearstress at the wall, and ρ is the density. Velocity is normalized by u τ , spatial variablesby h , time by h/u τ , and pressure by ρu τ . The spatial variables are denoted by + whennormalized by the viscous length scale ν/u τ , e.g. y + = Re τ y .2.1. Decomposition in homogenous directions
The velocity is decomposed using the Fourier transform in the homogenous directionsand time u ( x, y, z, t ) = (cid:90) (cid:90) (cid:90) ∞−∞ ˆ u ( y ; κ x , κ z , ω ) e i( κ x x + κ z z − ωt ) d κ x d κ z d ω, (2.2) odel-based scaling of the streamwise energy density κ x , κ z , ω ) isthe streamwise and spanwise wavenumbers and the temporal (angular) frequency. TheFourier basis is optimal in the homogeneous wall-parallel directions. It is also an appro-priate basis in time under stationary conditions. For any ( κ x , κ z , ω ) (cid:54) = 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) foran 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 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 classesof propagating waves that are universal with Reynolds number.The turbulent mean velocity U ( y ) = [ U ( y ) 0 0 ] T = ˆ u ( y ; 0 , ,
0) corresponds to( κ x , κ z , ω ) = 0 and is assumed to be known. Note that our main results, i.e. the identifiedscalings 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 closureproblem for the mean velocity by using U ( y ) obtained in pipe flow experiments, but notethat the resolvent formulation could be used to determine the mean velocity profile, atopic of ongoing work (see McKeon et al. U ( y ) U ( y ) = Re τ (cid:90) y − ξ ν T ( ξ ) d ξ,ν T ( y ) = 12 (cid:40) (cid:32) κRe τ (cid:0) y − y (cid:1) (cid:0) − y + 2 y (cid:1) (cid:26) − e (cid:0) | y − |− (cid:1) Re τ α (cid:27) (cid:33) (cid:41) / − , (2.3)where ν T is normalized by ν , and the parameters α and κ appear in the van Driest’swall law and the von K´arm´an log law. These parameters are obtained by minimizing thedeviation 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 thatboth of these values converge for large Re τ . We take α = 25 . κ = 0 .
426 for allReynolds numbers and note that these values are optimized for Re τ = 2003 (del ´Alamo& Jim´enez 2006; Pujals et al. f = − ( u · ∇ ) u that drives the velocity fluctuations, see also figure 2.For any ( κ x , κ z , ω ) (cid:54) = 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 usingthe orthonormality of the complex exponential functions − i ω ˆ u + ( U · ∇ )ˆ u + (ˆ u · ∇ ) U + ∇ ˆ p − (1 /Re τ )∆ˆ u = ˆ f , ∇ · ˆ u = 0 . (2.4)Here, ∇ = [ i κ x ∂ y i κ z ] T , ∆ = ∂ yy − κ with κ = κ x + κ z , and R. Moarref, A. S. Sharma, J. A. Tropp, & B. J. McKeon ˆ f ( y ; κ x , κ z , ω ) = [ ˆ f ˆ f ˆ f ] T = − (cid:90) (cid:90) (cid:90) ( κ (cid:48) x , κ (cid:48) z , ω (cid:48) ) (cid:54) = (0 , , κ (cid:48) x , κ (cid:48) z , ω (cid:48) ) (cid:54) = ( κ x , κ z , ω ) (ˆ u ( y ; κ (cid:48) x , κ (cid:48) z , ω (cid:48) ) · ∇ ) ˆ u ( y ; κ x − κ (cid:48) x , κ z − κ (cid:48) z , ω − ω (cid:48) ) d κ (cid:48) x d κ (cid:48) z d ω (cid:48) . (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 v and wall-normal vorticityˆ 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 pressureterm and the continuity constraint from (2.4) and obtain − (i ωI + 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 ) , ˆ u ( y ; κ x , κ z , ω ) = C ( κ x , κ z ) ˆ ζ ( y ; κ x , κ z , ω ) . (2.6)Here, A is the state operator, C maps the state vector to the velocity vector, and theadjoint of C (denoted by C † ) maps the forcing vector to the state vector. A , C , and C † are operators in y and parameterized by κ x and κ z A = (cid:20) ∆ − (cid:0) (1 /Re τ )∆ + i κ x ( U (cid:48)(cid:48) − U ∆) (cid:1) − i κ z U (cid:48) (1 /Re τ )∆ − i κ x U (cid:21) ,C = 1 κ i κ x ∂ y − i κ z κ κ z ∂ y i κ x , C † = (cid:20) − i κ x ∆ − ∂ y κ ∆ − − i κ z ∆ − ∂ y i κ z − i κ x (cid:21) , (2.7)where ∆ = ∂ yyyy − κ ∂ yy + κ , and the prime denotes differentiation in y , e.g. U (cid:48) ( y ) =d U/ d y . The input-output relationship between ˆ f and ˆ u is obtained upon elimination ofˆ 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 ) ,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)where R A ( κ x , κ z , ω ) = − (i ωI + A ( κ x , κ z )) − is the resolvent of AR A = (cid:20) ∆ − (cid:0) i κ x (( U − c )∆ − U (cid:48)(cid:48) ) − (1 /Re τ )∆ (cid:1) κ z U (cid:48) i κ x ( U − c ) − (1 /Re τ )∆ (cid:21) − . (2.9)As illustrated in figure 2, the only source of coupling between propagating waves withdifferent wavenumbers is the quadratic dependence of f ( x, y, z, t ) on u ( x, y, z, t ). For anywavenumber 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 theinput-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- odel-based scaling of the streamwise energy density Figure 2.
For any triplet ( κ x , κ z , ω ), the operator H ( κ x , κ z , ω ) maps the forcing ˆ f to theresponse 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 trans-form, respectively. The input-output map (shown with the dashed rectangle) is the main focusof the present study. 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 )= ∞ (cid:88) 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 ) ,a j ( κ x , κ z , ω ) = (cid:90) − ˆ φ ∗ j ( y ; κ x , κ z , ω ) ˆ f ( y ; κ x , κ z , ω ) d y, (2.10)where σ ≥ σ ≥ · · · > H , and the singular functions ˆ φ j =[ ˆ f j ˆ f j ˆ f j ] T and ˆ ψ j = [ ˆ u j ˆ v j ˆ w j ] T are respectively the forcing and response directionscorresponding to σ j . In principle, there are infinite number of singular values/modesbecause the wall-normal coordinate is continuous. For the discretized equation, the totalnumber of singular values/modes is twice the number of grid points in y since the resolventoperator 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 area limited number of relatively highly-amplified modes within this total number of modes.Throughout this paper, we consistently refer to ˆ ψ j by the resolvent mode , and distinguishit from the real turbulent flow that, under stationary conditions, can be represented by aweighted sum of the resolvent modes. The latter is denoted by the weighted mode . Notethat the resolvent modes were denoted by response modes in McKeon & Sharma (2010);McKeon et al. (2013); Sharma & McKeon (2013).While the singular values of H are unique, additional treatment is necessary to obtainunique singular functions. Unlike in a pipe, the singular values come in pairs due to thewall-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 legitimatesingular function. For example, if the symmetric and anti-symmetric modes are denotedby ψ s and ψ a where | ψ s | = | ψ a | , the singular function given by ψ d = ψ s − ψ a is zero in onehalf of the channel and twice ψ s in the other half. Clearly, ψ d is also a singular functionof the transfer function with the same singular value as ψ s and ψ a . Physically, this meansthat the modes with lengths and widths smaller than the channel half-height exhibit thepotential to independently evolve in either halves of the channel provided that they areforced with a forcing (e.g. disturbance) that is present only in one half of the channel.0 R. Moarref, A. S. Sharma, J. A. Tropp, & B. J. McKeon u u (color); v , w (arrows) y + z + x + z + Figure 3.
The principal velocity response ψ ( x, y, z, t ; κ x , κ z , c ) = [ u v w ] T for λ + x = 700, λ + z = 100, c = 10, and Re τ = 10000 at t = 0. (Left) The isosurfaces of the streamwise velocityat 60% of its maximum; (Right) The streamwise velocity (contours) and the spanwise andwall-normal velocity (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.
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 oranti-symmetric in the opposite halves of the channel. Physically, these modes representconvective global phenomena meaning that they cannot take place independently in theopposite halves of the channel. They convect with the same magnitude in the oppositehalves 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 (cid:90) − ˆ φ ∗ j ( y ; κ x , κ z , ω ) ˆ φ k ( y ; κ x , κ z , ω ) d y = (cid:90) − 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 areequal, we impose a symmetry/anti-symmetry constraint on the singular functions in ad-dition to the above orthonormality constraint. In other words, the corresponding singularfunctions assume the same magnitude throughout the channel while being in phase inone 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 theabsolute 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 spanwisedirections 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 (cid:0) ˆ u j ( y ; κ x , κ z , ω ) e i( κ x x − ωt ) (cid:1) ,v j ( x, y, z, t ; κ x , κ z , ω ) = 4 cos( κ z z ) Re (cid:0) ˆ v j ( y ; κ x , κ z , ω ) e i( κ x x − ωt ) (cid:1) ,w j ( x, y, z, t ; κ x , κ z , ω ) = − κ z z ) Im (cid:0) ˆ w j ( y ; κ x , κ z , ω ) e i( κ x x − ωt ) (cid:1) , 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 similarexpressions. odel-based scaling of the streamwise energy density φ j direction with unit energy, the responseis aligned in the ˆ ψ j direction with energy σ j . Consequently, the forcing and responsedirections with the largest gain correspond to the principal singular functions ˆ φ and ˆ ψ .For any ( κ x , κ z , ω ), the singular functions of H should be thought of as propagating wavesin the physical domain. In the rest of the paper, the resolvent modes are characterizedby 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); McKeon et al. (2013) are obtained for channel flows. For example, the principalsingular function ψ ( x, y, z, t ; κ x , κ z , ω ) = [ u v w ] T for the propagating wave corre-sponding to the energetic near-wall cycle ( λ + x = 700, λ + z = 100, c = U ( y + = 15) = 10)for Re τ = 10000 is shown in figure 3. The streamwise component of these structurescontains regions of fast- and slow- moving fluids that are aligned in the streamwise direc-tion, slightly inclined to the wall, and are sandwiched between counter-rotating vorticalmotions 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 significantportion of its response to a broadband forcing in y is captured by projection on the firstfew 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 thatthe largest pair of singular values is approximately one order of magnitude larger thanthe other singular values.The energetic contribution of the k -th direction ˆ ψ k to the total response in the modelsubject to broadband forcing in y with fixed λ x , λ z , and c is quantified by σ k / ( (cid:80) ∞ j =1 σ j ).Figures 4(b)-4(d) highlight the low-rank nature of H by showing that the first two princi-pal response directions ˆ ψ and ˆ ψ contribute to more than 80% of the total response overa large range of wall-parallel wavelengths (red region) for wave speeds c = U ( y + = 15), U ( y + = 100), U ( y = 0 . Re τ = 2003. The relevance of studying the low-rank ap-proximation of H is further emphasized by noting that the most energetic wavenumbersfrom the DNS of Hoyas & Jim´enez (2006) (contours) coincide with the wavenumbersand critical wave speeds for which H is low-rank. We note that the streamwise velocityhas 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 thestreamwise 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 valuesis 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 energeticforcing and response directions corresponding to σ and show that significant under-standing of the scaling of wall turbulence can be obtained using this simple model. Thisis motivated by the observation in § H is essentially a directionalamplifier. In other words, we expect to see the principal singular response of H in realturbulent flows provided that the principal forcing direction is present in the nonlinearforcing term. Even though the resolvent modes corresponding to σ and σ comparablycontribute to the total response, cf. § R. Moarref, A. S. Sharma, J. A. Tropp, & B. J. McKeon σ j ( σ + σ ) / ( (cid:80) ∞ j =1 σ j )( a ) ( b ) λ + z j λ + x ( σ + σ ) / ( (cid:80) ∞ j =1 σ j ) ( σ + σ ) / ( (cid:80) ∞ j =1 σ j )( c ) ( d ) λ + z λ + z λ + x λ + x Figure 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 color plots show the energy that is contained in the largest tworesponse modes relative to the total response for different streamwise and spanwise wavelengthsand (b) c = U ( y + = 15); (c) c = U ( y + = 100); and (d) c = U ( y = 0 . y + = 15; (c) y + = 100; and (d) y = 0 .
2. Thecontours represent 10% to 90% of the maximum energy spectrum at each wall-normal locationwith increments of 20%. resolvent modes are symmetric/anti-symmetric counterparts of each other and have thesame magnitude. Therefore, accounting for both resolvent modes yields the same resultas accounting for one resolvent mode.It is well-known that the streamwise energy spectrum can be divided into regionsthat scale in inner and outer variables (see, for example, Morrison et al. Re τ . We focuson the streamwise velocity because it dominates the kinetic energy density in turbulentflows. Similarly, the principal singular responses of H that result in the largest energyamplification are dominated by their streamwise component, such that the proposed gain-based decomposition yields the streamwise velocity most accurately. This is in agreementwith previous linear analyses of the global optimal responses, e.g. del ´Alamo & Jim´enez odel-based scaling of the streamwise energy density f equals the principal forcing direction ˆ φ . Consequently, the forcing has unit energy forall wave parameters, meaning that it is broadband in κ x , κ z , and c . For the rank-1 modelwith broadband forcing, we define the premultiplied streamwise energy density of theprincipal response of H by E uu ( y ; κ x , κ z , c ) = κ x κ z ( σ ( κ x , κ z , c ) | u | ( y ; κ x , κ z , c )) , (2.12)such that the premultiplied one-dimensional energy densities and the energy intensityare 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 ) = (cid:90) (cid:90) S E uu ( y ; κ x , κ z , c ) d log( κ x ) d log( κ z ) ,E uu ( y ) = (cid:90) (cid:90) (cid:90) S E uu ( y ; κ x , κ z , c ) d log( κ x ) d log( κ z ) d c, (2.13)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 § Re τ for properly selected subsets ofwave parameters. It is further shown that the emerging scales are consistent with thoseobserved in experiments. In addition, the scales of energetically dominant waves roughlyagree 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 theMatlab Differentiation Matrix Suite developed by Weideman & Reddy (2000). Table 1summarizes the selected range of wave parameters and their respective resolution innumerical computations. It has been verified that the excluded wave parameters are notenergetically important and therefore do not change the results of the present study.An efficient randomized scheme developed by Halko et al. (2011) is utilized to computethe principal singular directions of H for different Reynolds numbers and wave parame-ters. The accuracy and computation time depend on the decay of the singular values; afaster decay results in high accuracy or equivalently less computation time to reach thesame accuracy. In addition, if the singular values are not well separated, the problemof computing the associated singular functions is badly conditioned, meaning that it ishard for any method to determine them very accurately. In this study, the above schemeapproximately halves the total computation time relative to Matlab’s svds algorithm.This becomes increasingly important considering the three-dimensional wave parameterspace 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 makesit especially suitable for large-scale computations. Even though we have not used thisfeature in the present study, it may find use in designing turbulent flow control strategies,e.g. by means of spatially or temporally periodic actuations.4 R. Moarref, A. S. Sharma, J. A. Tropp, & 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 .
07 10 100 22 . × .
15 10 50 24 . × .
18 10 30 25 . . . . × . . Table 1.
Summary of the selected parameters in numerical computations at different Reynoldsnumbers. 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 linearlyspaced wave speeds are chosen between c min = 2 and c max = U cl .
3. Universal behavior of the resolvent
The formulation of § κ x , κ z , c ) to the streamwise energy density. For the rank-1 model with broadbandforcing, the energy density of each wave is determined from the principal singular valuesand singular functions of the transfer function H ; see (2.12). In this section, we identifyunique classes of wave parameters for which E uu ( y ; κ x , κ z , c ) exhibits either universalbehavior with Re τ or geometrically self-similar behavior with distance from the wall.Each class is characterized by a unique range of wave speeds and a unique scaling of thewall-normal coordinate and the wall-parallel wavelengths. These classes are inherent tothe linear mechanisms in the NSE and are rigorously identified by analysis of the transferfunction. 3.1. Requirement for universality of the resolvent modes
We start by showing that a requirement for universal behavior is the wall-normallocality 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 andthe wave speed, U ( y ) − c , and its wall-normal derivatives, U (cid:48) ( y ) and U (cid:48)(cid:48) ( y ), appear asspatially-varying coefficients in H . Since the turbulent mean velocity scales differentlywith Re τ in different wall-normal locations, only the resolvent modes that are sufficientlynarrow in y have the potential to be universal. This is because such resolvent modes arepurely 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 significantmodes are in fact localized. As summarized by LeHew et al. (2011), the energetic con-tribution of structures with convection velocities less than 10 u τ and larger than thecenterline velocity U cl = U ( y = 1) is negligible in real turbulent flows. However, we areinterested in determining the effect of a broader range of wave speeds on the energydensity. Note that small values of c result in small amplification because the correspond-ing singular values are small. In fact, it is shown in § c (cid:46) S , denoted by S e ,that includes all wall-parallel wavenumbers and the energetically important wave speeds S e = { ( κ x , κ z , c ) | ≤ c ≤ U cl } . (3.1)Figure 5 shows the one-dimensional energy density as a function of wave speed E uu ( y, c ) odel-based scaling of the streamwise energy density ( a ) ( b ) c c Uy + y + Figure 5. (Color online) (a) The one-dimensional energy density Re − 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 fixedvalues of c . The colors are in logarithmic scale. The turbulent mean velocity is shown by theblack curve in (b). 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 colors are given in logarithmicscale. The localization is highlighted in figure 5(b) where E uu ( y, c ) is normalized by itsmaximum value over y for fixed values of c . We see that the largest energy amplificationtakes place in the vicinity of the critical wall-normal location where the turbulent meanvelocity (thick black curve) equals the wave speed. McKeon & Sharma (2010) argued thatemergence of critical layers is one of the three means of maximizing the Hilbert-Schmidtnorm of H (sum of squares of the singular values), i.e. by locally minimizing the term U ( y ) − c that appears in the resolvent operator R A given in (2.9).According to Taylor’s frozen turbulence hypothesis (Taylor 1938), the flow structuresin boundary layers propagate downstream with a speed close to the local mean velocity.Consistent with this hypothesis, figure 5(b) shows that among all the waves with arbitrarystreamwise and spanwise wavelengths at the wall-normal location y , the ones with criticalspeed c = U ( y ) are the most highly amplified. This provides strong evidence for theimportance of critical layers in amplification of flow disturbances. In addition, figure 5(b)shows that the scatter in the energetic wave speeds increases as the peak of energy densityapproaches the wall. This agrees with the practical observation that Taylor’s hypothesisyields inaccurate energy spectra close to the wall; see, for example, Kim & Hussain (1993);Monty & Chong (2009); del ´Alamo & Jim´enez (2009); LeHew et al. (2011).3.2. Requirement for geometric self-similarity of the resolvent modes
We show that a necessary condition for existence of geometrically self-similar resol-vent modes is the presence of a logarithmic region in the turbulent mean velocity. Theboundary conditions in the inhomogeneous direction y , the wall-normal symmetry rela-tive to the center plane, and the presence of y -dependent coefficients, e.g. U ( y ) − c , in theresolvent pose limitations on wall-normal scaling of the transfer function. As discussedlater in § § U ( y ) − c , U (cid:48) ( y ), and U (cid:48)(cid:48) ( y ) in the resolvent, cf. (2.9), and reduces to identifying the necessary6 R. Moarref, A. S. Sharma, J. A. Tropp, & B. J. McKeon conditions under which U ( y ) − c = g ( y/y c ) , (3.2)for some functions U ( y ) and g ( y ) and some scale y c to be determined. Let the relation-ship between c and y c be governed by c = g ( y c ). Then, we seek the functions U , g , g ,and the scale y c such that U ( y ) − g ( y c ) = g ( y/y c ) . (3.3)It follows from (3.3) that g ( y ) = U ( y ) − g (1), g ( y ) = U ( y ) − g (1), and g (1) = U (1) − g (1). Therefore, (3.3) can be rewritten as U ( y ) − (cid:0) U ( y c ) − g (1) (cid:1) = U ( y/y c ) − (cid:0) U (1) − g (1) (cid:1) , or U ( y ) − U ( y c ) = U ( y/y c ) − U (1). The only functions that satisfy thisconstraint are the constant function and the logarithmic function and we have U ( y ) = d + d log d ( y ) , c = U ( d y c ) , (3.4)where d to d are constants. The wall-normal scale corresponds to the wall-normallocation where c = U ( d y c ). The constant d is arbitrary since it enters as a coefficientin front of the scale y c . We select d = 1 such that y c is the critical wall-normal locationcorresponding to the wave speed c . Therefore, in the presence of a logarithmic meanvelocity, the height of the resolvent modes scales with y c .3.3. Universal modes and self-similar modes
We start by reviewing the universal behavior of the turbulent mean velocity. This isa prerequisite to studying the universality of the principal propagating waves since thelatter holds for critical modes only, as discussed in § U = B ( y + ) + (1 /κ ) log( y + ) + (2Π /κ ) W ( y ) , (3.5)where B is the inner-scaled wall function, Π is the wake factor, W is the outer-scaledwake function, and κ is the von K´arm´an’s constant also appearing in (2.3). Consequently, U − c is universal with Re τ for certain intervals of wave speed and appropriate wall-normalscales; see figure 6. Figure 7(a) shows that U ( y + ) − c is universal for y + (cid:46)
100 and fixed c (cid:46)
16 (inner region). As shown in figure 7(b), the function U ( y ) − c is universal for y (cid:38) . (cid:46) U cl − c (cid:46) .
15, with U cl = U ( y = 1) (outer region).The gap between the inner and outer regions of the turbulent mean velocity is bridgedby a middle region between y + = 100 and y = 0 .
1. There is an abundance of numericaland experimental evidence that support the presence of a logarithmic turbulent meanvelocity in this region (see, for a recent summary, Smits et al. ≤ U ≤ U cl − .
15, and note that recent experiments suggest that the lower bound on the logarithmicregion depends on Reynolds number: y + ∼ Re / τ , see e.g. Marusic et al. (2013).The existence, at least approximately, of a logarithmic region in U satisfies the neces-sary conditions in § ≤ c ≤ U cl − .
15 areat least one decade away from the walls and the center plane and the boundary effectsare negligible. This eliminates the first two limitations for presence of self-similar modes,cf. § d = 5 . d = 1 /κ , and d = e in U given by (3.4) are obtainedupon direct comparison with (3.5).Associated with each region of the mean velocity, there is a class of wave parametersfor which the low-rank approximation of H exhibits either universal behavior with Re τ odel-based scaling of the streamwise energy density Figure 6.
Schematic of the different regions of the mean velocity and the associated classesof induced scales on the propagating waves. The mean velocity is denoted by U cl in the centerplane and by U m in the geometric mean of the middle region. The inner, self-similar, and outerclasses of modes are denoted by S i , S h , and S o , respectively. See also table 2 and figures 7 and 8. or self-similar behavior with distance from the wall; see tables 2 and 3 for a summary.As illustrated in figure 6, these classes are primarily distinguished by the wave speed.The identified scales represent inherent features of the linear mechanisms in the NSEand are not arbitrary: (i) The wall-normal length scale is inherited from the turbulentmean velocity at the critical layer, and (ii) the streamwise and spanwise length scalesare determined from the balance between the viscous dissipation term, (1 /Re τ )∆, andthe mean advection terms, e.g. i κ x ( U − c ), in the resolvent in (2.9). In addition, themagnitude of the singular values and singular functions scale uniquely in each class ofwave parameters, which induces unique scales on the premultiplied streamwise energydensity E uu ( y ; κ x , κ z , c ). Next, we separately discuss each class and refer the reader toAppendices A, B, and C for detailed derivation of the scales.3.3.1. The universal inner class S i For wave speeds in the inner region of the turbulent mean velocity, universality of H requires constant λ + x , y + , λ + z , and c ; cf. Appendix A, table 2, and figure 7(c). As aresult, the time T c = λ x /c over which the wave convects downstream for one wavelengthrelative to the wall reduces with Re τ and the convective frequency ω c = 2 π/T c increaseswith Re τ . In other words, a truly inner scale is induced on the length, height, width, andconvective time of the waves that correspond to the principal resolvent modes. Therefore,the wall-normal support of the resolvent modes in outer units linearly decreases with Re τ ,and the unit energy constraint on the resolvent modes requires that the magnitude of theresolvent modes increase with Re / τ . The number of these waves per unit wall-parallelarea and time increases with Re τ as their length, width, and convective time decreasewith Re τ . Since the singular values of H linearly decrease with Re τ , the overall result isthat E uu ( y ; κ x , κ z , c ) increases with Re τ ; cf. table 3.3.3.2. The universal outer class S o For wave speeds close to the centerline, universality of H requires constant λ x /Re τ , y , λ z , and U cl − c , such that an aspect ratio constraint λ x /λ z (cid:38) γRe τ /Re τ, min is satisfied(a conservative value for γ is √ T cl = λ x / ( U cl − c ) over which the wave convects upstream for onewavelength relative to an observer with speed U cl increases with Re τ and the convectivefrequency ω cl = 2 π/T cl decreases with Re τ . In addition, the aspect ratio λ x /λ z of theuniversal waves increases as Re τ . This explains why universality for this class holds for8 R. Moarref, A. S. Sharma, J. A. Tropp, & B. J. McKeon
U U c l − U y + y ( a ) ( b ) S i S o c U c l − c λ + x λ + z λ x /Re τ λ z ( c ) ( d ) Figure 7. (Color online) (a) The turbulent mean velocity U ( y + ) and (b) the defect velocity U cl − U ( y ) relative to the centerline, for Re τ = 3333 (blue), Re τ = 10000 (red), and Re τ = 30000(black). The arrows indicate increase in the Reynolds number. Notice that the shaded regionsare invariant with Re τ . (c) The inner class S i and (d) the outer class S o of wave parameters( λ x , λ z , c ) that induce universal behavior with Re τ on the low-rank approximation of H . S o isobtained for Re τ > Re τ, min = 3333.Class x -scale y -scale z -scale t -scale Subset of wave parametersInner λ + x y + λ + z T + c S i : 2 (cid:46) c (cid:46) λ x Re τ y λ z Re τ T cl S o : (cid:40) (cid:46) U cl − c (cid:46) . λ x /λ z (cid:38) γRe τ /Re τ, min Self-similar λ x y + c y c yy c λ z y c − S h : (cid:46) c (cid:46) U cl − . c = U ( y c ) = B + (1 /κ ) log( y + c ) λ x /λ z (cid:38) γ Middle λ x (cid:112) yy + (cid:112) λ z λ + z T m S m : (cid:40) | U m − c | (cid:46) dλ x /λ z (cid:38) γ (cid:112) Re τ /Re τ, min Table 2.
Summary of the length scales and wave speeds for the universal modes of thetransfer function H . See also figure 6. odel-based scaling of the streamwise energy density Class Subset κ x (d / d y ) κ z ω i,o,m σ u E uu Inner S i Re τ Re τ Re τ Re τ Re − τ Re / τ Re τ Outer S o Re − τ Re − τ Re τ Re τ Self-similar S h ( y + c y c ) − y − c y − c − ( y + c ) y c y − / c Re τ Middle S m Re / τ Re / τ Re / τ Re / τ Re τ Table 3.
Summary of the growth/decay rates (with respect to Re τ or y c ) of the wall-parallelwavenumbers, the wall-normal derivative, the convective frequency, the principal singular valueand the principal streamwise singular function of H , and the premultiplied three-dimensionalstreamwise energy density for the classes of universal waves outlined in table 2. the waves with aspect ratios larger than a threshold: As the aspect ratio of the resolventmodes increases with Reynolds number, the Laplacian operator in the resolvent becomesindependent of κ x . Therefore, the necessary condition for the Laplacian to be universalwith Re τ is that κ z dominates κ x even for the smallest Reynolds number Re τ, min thatis considered. This poses the above-mentioned aspect ratio constraint on the universalwaves. The magnitude of resolvent modes is independent of Re τ since the resolvent modesscale with outer units in the wall-normal direction. The number of waves per unit areaand time decreases with Re τ since their length and convective time increase with Re τ .The singular values increase with Re τ and the overall result is that the energy density E uu ( y ; κ x , κ z , c ) increases with Re τ ; cf. table 3.The waves in the outer class asymptotically approach the streamwise constant fluctu-ations, i.e. κ x = 0, as Re τ increases. These infinitely long fluctuations exhibit the largestlinear transient growth in response to initial perturbations in laminar (Gustavsson 1991;Butler & Farrell 1992; Reddy & Henningson 1993) and turbulent (Butler & Farrell 1993;del ´Alamo & Jim´enez 2006; Pujals et al. b ; Bamieh& Dahleh 2001; Jovanovi´c & Bamieh 2005) and turbulent (Hwang & Cossu 2010) flowssubject to stochastic disturbances.The effect of Reynolds number on the streamwise constant fluctuations has been stud-ied in laminar flows. For example, Gustavsson (1991) showed that the peak of lineartransient growth scales with the square of centerline Reynolds number Re cl = U cl h/ν .For the flow subject to harmonic disturbances, Jovanovi´c & Bamieh (2005) showed thatthe singular values of H increase as Re cl when the temporal frequency ω linearly de-creases with Re cl . No other scales for the singular values were found since the laminarmean velocity U/U cl = 2 y − y is universal with Reynolds number throughout the chan-nel. Our study shows that the singular values in the turbulent flow increase quadraticallywith Re τ for the waves with defect speeds, U cl − c (cid:46) .
15, and streamwise wavelengthsthat linearly increase with Re τ , i.e. λ x (cid:38) γ λ z Re τ /Re τ, min ; cf. table 3.3.3.3. The geometrically self-similar class S h The logarithmic region of the turbulent mean velocity yields a hierarchy of geometri-cally self-similar resolvent modes that are uniquely parameterized by the critical wall-normal distance y c , i.e. c = U ( y c ); see Appendix C for derivation. As summarized intable 2, the height and width of the self-similar modes scale with y c and their length with y + c y c . In addition, the self-similar modes satisfy an aspect ratio constraint, λ x /λ z (cid:38) γ ,where a conservative value for γ is √
10. This agrees with the observation of Hwang &0
R. Moarref, A. S. Sharma, J. A. Tropp, & B. J. McKeon
Cossu (2010) that the optimal responses were approximately similar for κ x (cid:28) κ z . Noticethat the difference between the streamwise scaling of the self-similar resolvent modes λ x ∼ y + c y c and the scaling λ x ∼ y chosen in original developments of the attached-eddyhypothesis (Townsend 1976; Perry & Chong 1982) does not contradict the philosophy ofself-similar attached eddies, i.e. the resolvent modes are still self-similar.Any hierarchy is a subset of S and can be described by a representative mode with λ x,r , λ z,r , and c r = U ( y c r ) S h ( λ x,r , λ z,r , c r ) = (cid:40) ( λ x , λ z , c ) | λ x = λ x,r (cid:16) y + c y c y + c r y c r (cid:17) ,λ z = λ z,r (cid:16) y c y c r (cid:17) ,c = B + (1 /κ ) log y + c , Re τ ≤ y c ≤ y c ≤ . (cid:41) . (3.6)Here, y c is the critical wall-normal location associated with the smallest wave speed c above which the aspect ratio constraint is satisfied, see Appendix C.The concept of hierarchies is illustrated in figure 8(a) where the loci of wave parametersthat belong to three demonstrative hierarchies with representative modes marked byopen circles are shown. The mode with κ x,r = 1, κ z,r = 10, and c r = (2 / U cl ( h ,black) is representative of the very large-scale motions (McKeon & Sharma 2010). Therepresentative modes for the other hierarchies have the same wavenumbers but differentspeeds, i.e. c r = 16 ( h , blue) and U cl − .
15 ( h , red), corresponding to the mean velocityat the upper limit of the inner region and the lower limit of the outer region. Each locusconstitutes a vertical line after normalizing the length, width, and height of the modesaccording to the scales in (3.6) obtained from the resolvent. In fact, the resolvent modesare self-similar along any vertical line as long as λ x /λ z > γ . The aspect ratio constraintrequires that the wave parameters lie above the shaded threshold plane λ x /λ z = γ infigure 8(a). For example, the waves corresponding to the dashed segment of the hierarchywith c r = U cl − .
15 do not belong to any hierarchy.Owing to the self-similar behavior, the principal singular values and singular functionsof H for all the modes in a given hierarchy can be determined from its representativemode. The principal singular value σ corresponding to the waves that belong to thehierarchies in figure 8(a) are shown in figure 8(b). The singular values grow with ( y + c ) ( y c )as theoretically predicted, cf. table 3. Figure 8(c) shows the principal streamwise resolventmode u corresponding to the hierarchy with κ x,r = 1, κ z,r = 10, and c r = (2 / U cl for 100 /Re τ ≤ y c ≤ .
1. The arrow shows the direction of increasing y c . Normalizingand scaling the resolvent modes according to table 2 collapses the resolvent modes fordifferent wave speeds; see black curves marked as h in figure 8(d). This figure also showsthe scaled resolvent modes corresponding to the hierarchies with κ x,r = 1, κ z,r = 10,and c r = 16 ( h , blue) and U cl − .
15 ( h , red). We see that the normalized and scaledresolvent modes lie on the top of each other for the hierarchy with c r = 16. For thehierarchy with c r = U cl − .
15, the resolvent modes for large y c collapse on each otherwhile the resolvent modes for small y c are considerably different. This is expected sincethe aspect ratios of the modes fall below γ as y c decreases. Notice that for this hierarchy,the modes corresponding to small y c lie below the threshold plane in figure 8(a).3.3.4. The universal middle class S m The Reynolds number scaling of the self-similar class depends on the wave speed andis consistent with the inner and outer classes of resolvent modes, cf. tables 2 and 3. For odel-based scaling of the streamwise energy density ( a ) ( b ) y c c = U ( y c ) σ λ x / ( y + c y c ) λ z /y c y c ( c ) ( d ) | u | √ y c | u | y c y/y c Figure 8. (Color online) (a) The vertical lines are the loci of wave parameters that belong tothe hierarchies with representative modes (open circles) κ x,r = 1, κ z,r = 10, and c r = U cl − . h , red), (2 / U cl ( h , black), and 16 ( h , blue) for Re τ = 10000. The shaded threshold planecorresponds to the wavenumbers with aspect ratio λ x /λ z = √
10. The modes below this plane donot belong to any hierarchy. (b) The principal singular values along the hierarchies in figure 8(a).(c) The principal streamwise resolvent modes that belong to the hierarchy with c r = (2 / U cl ( h , black) in figure 8(a). (d) The normalized and scaled (according to table 2) principal resolventmodes along the hierarchies in figure 8(a). The arrows show the direction of increasing y c with100 /Re τ ≤ y c ≤ . example, when the wave speed is fixed as Re τ changes, y + c remains constant and the innerscale is recovered. When the defect wave speed U cl − c is fixed, y c remains constant and theouter scale is recovered. Consequently, the energy density corresponding to the completerange of wave speeds in the self-similar region, 16 < c < U cl − .
15, is centered around thegeometric mean of the middle region of the turbulent mean velocity, i.e. y m = (cid:112) /Re τ .The self-similar class is primarily concerned with geometric self-similarity of the resolventmodes. We next construct a middle class of modes S m , a subset of the self-similar class S h , with unique Reynolds number scalings.The wave speeds in the middle class are confined to | U m − c | < d , with d denoting aradius around U m = U ( y m ). For the resolvent modes in the middle class, universality of H requires constant λ x , (cid:112) y + y , (cid:112) λ + z λ z , and U m − c such that the aspect ratio constraint λ x /λ z (cid:38) γ (cid:112) Re τ /Re τ, min is satisfied; cf. table 2. These scales are equal to the geometric2 R. Moarref, A. S. Sharma, J. A. Tropp, & B. J. McKeon mean of the scales in the inner and outer classes, and can also be recovered from the scalesof the self-similar class for fixed (cid:112) y + c y c as Re τ changes. When only one Reynolds numberis considered, we have Re τ, min = Re τ and the aspect ratio constraints in the self-similarand middle classes are equivalent, i.e. the constraint λ x /λ z (cid:38) γ (cid:112) Re τ /Re τ, min = γ inthe middle class is the same as λ x /λ z (cid:38) γ in the self-similar class. When the self-similarmodes are compared across more than one Reynolds number, the aspect ratio constrainton the middle class is more restrictive. This is because, the modes remain self-similar asthe Reynolds number increases. However, they do not remain independent of Re τ unlesstheir aspect ratio is larger than γ even for the smallest Reynolds number Re τ, min thatis considered, resulting in the modified aspect ratio constraint λ x /λ z (cid:38) γ (cid:112) Re τ /Re τ, min for the middle class. This constraint can be obtained similarly to the constraint for theouter class, cf. § T m = λ x / | U m − c | over which a wave in the middle class convects away forone wavelength relative to an observer with speed U m remains independent of Re τ andsame does the convective frequency ω m = 2 π/T m . These waves have the same scales asthe structures in the meso-layer; see, for example, Long & Chen (1981); Afzal (1984);Sreenivasan & Sahay (1997); Wei et al. (2005). The aspect ratio constraint follows fromsimilar arguments to those discussed for the outer class. The magnitude of the corre-sponding resolvent modes increases with Re / τ because of the unit energy constraint.The number of waves per unit area and time increases with Re / τ . In addition, the prin-cipal singular values of the waves in S m increase with Re / τ . The above scales result ingrowth of E uu ( y ; κ x , κ z , c ) with Re τ ; cf. table 3.There is a direct relationship between S m and S h : The union of the middle class ofmodes equals the union of the geometrically self-similar modes with speeds | U m − c | < d and aspect ratio constraint λ x /λ z (cid:38) γ (cid:112) Re τ /Re τ, min . Since the difference between themiddle class and the self-similar class becomes larger as Re τ increases, our ongoingresearch is focused on analytical developments using the scalings of the self-similar classthat bridges the gap between the inner and outer classes.3.4. Universality of the streamwise energy density
We compute the streamwise energy density of the rank-1 model with broadband forcingand illustrate its universal behavior with Re τ . These computations build the basis forprediction of the streamwise energy intensity at the technologically relevant values of Re τ in §
4. Because of the unique scales in the inner, middle, and outer classes of waveparameters, we distinguish the corresponding intervals of wave speeds by expanding thepremultiplied energy density into the following three integrals: E uu ( y, κ x , κ z ) = (cid:90) E uu ( y, κ x , κ z , c ) d c + (cid:90) U cl − . E uu ( y, κ x , κ z , c ) d c + (cid:90) U cl U cl − . E uu ( y, κ x , κ z , c ) d c. (3.7)Similar expansions can be written for E uu ( y, κ x ), E uu ( y, κ z ), and E uu ( y ). In spite of thedifferent behavior of singular values and singular functions, the energy density increaseswith Re τ in all three classes of wave parameters. Figure 9 shows the premultiplied one-dimensional energy densities and the energy intensity confined to each class of waveparameters and normalized by Re τ . The same contour levels are used for all Reynoldsnumbers, Re τ = 3333 (blue), 10000 (red), and 30000 (black), in figures 9(a)-9(f). Noticethat confining the wavenumbers to S i , S m , and S o yields a universal energy density assummarized in tables 2 and 3. odel-based scaling of the streamwise energy density S i S m S o λ + x λ x λ x / R e τ y + (cid:112) yy + y ( a ) ( b ) ( c ) λ + z (cid:112) λ z λ + z λ z y + (cid:112) yy + y ( d ) ( e ) ( f ) y + (cid:112) yy + y ( g ) ( h ) ( i ) Figure 9. (Color online) The premultiplied one-dimensional streamwise energy density Re − τ E uu ( y, λ x ) in (a)-(c); and Re − τ E uu ( y, λ z ) in (d)-(f); and the streamwise energy inten-sity Re − τ E uu ( y ), dashed curves in (g)-(i) for the rank-1 model with broadband forcing. Thewave parameters are confined to S = S i in (a), (d), (g); S = S m in (b), (e), (h); and S = S o in(c), (f), (i). The Reynolds numbers are Re τ = 3333 (blue), Re τ = 10000 (red), and Re τ = 30000(black). The contour levels decrease by 0 .
05 from their maximum value of 0 .
25 (a); 0 .
15 (b);and 0 .
25 (c), and by 0 . . . . S = S e . The arrows indicateincrease in the Reynolds number. The inner peak of streamwise energy density in the rank-1 model with broadbandforcing occurs at y + ≈ λ + x ≈ λ + z ≈
44; see figures 9(a) and 9(d). The locationof the above wall-normal peak represents an integral effect over all wave parameters in S i and corresponds to the critical speed c ≈ .
5. The inner peak is comparable withthe location, length, and spacing of the most energetic structures associated with thenear-wall cycle, i.e. y + ≈ λ + x ≈ − λ + z ≈
100 (see, for example, Hoyas &Jim´enez 2006). Figures 9(c) and 9(f) show that the outer peak takes place at y ≈ . U cl − c ≈
2) for λ x /Re τ ≈ .
1, and λ z ≈ R. Moarref, A. S. Sharma, J. A. Tropp, & B. J. McKeon
This peak points to much longer structures relative to the LSM structures observed inexperiments, i.e. λ x ≈ et al. et al. et al. (cid:112) yy + ≈ . − . c ≈ U m ), λ x ≈ −
16, and (cid:112) λ z λ + z ≈
20; see figures 9(b) and 9(e). It has the samestreamwise and wall-normal scalings as the VLSMs and its location is comparable withthe most energetic VLSMs located at (cid:112) yy + ≈ . λ x ≈ et al. a ) and at λ x ≈ −
20 in pipes and channels (see,for example, Monty et al. y , see e.g. Monty et al. (2007); Bailey et al. (2008). The self-similar scales of theresolvent modes in the middle region are consistent with these results. The Reynoldsnumber scaling of the spanwise wavelength appears to still be under investigation. Sincethe spanwise peak of the one-dimensional spectrum is obtained by including a range ofmodes with speeds | U m − c | < d (instead of focusing on one mode), the Reynolds numberscaling of the spanwise peak is similar to the wall-normal scaling of the modes. Theorganization of the self-similar coherent motions in the logarithmic layer of real turbulentflows has been studied by many authors, e.g. see Tomkins & Adrian (2003); del ´Alamo et al. (2006); Flores & Jim´enez (2010). In addition, Hwang & Cossu (2011) addressed theself-sustaining mechanisms of these coherent motions. Studying the implications of theidentified scalings of the resolvent modes on these structures is a topic of future research.In making the above comparisons, it is important to note the distinction betweenthe resolvent modes and the real turbulent flow that can be represented by a weightedsuperposition of the resolvent modes. The agreement between the admitted scales of theprincipal resolvent modes and the scalings observed in real flows is striking considering thesimplicity of the rank-1 model subject to broadband forcing. This agreement emphasizesthe role of linear mechanisms and critical layers in determining the scaling of turbulentflows. In addition, the differences between the scalings highlights the role of nonlinearityin shaping the weights of the resolvent modes. We also note that the experimentallyobtained outer peak in the two-dimensional spectrum and the wavelengths associatedwith VLSMs and LSMs may be contaminated by use of the Taylor’s hypothesis and lackof sufficient scale separation at relatively low Re τ .The one-dimensional energy densities can be integrated in the remaining wall-paralleldirection to obtain the streamwise energy intensity E uu ( y ) for the rank-1 model withbroadband forcing. The dashed curves in figures 9(g)-9(i) are the energy intensities nor-malized by Re τ obtained by confining the wave parameters to S i , S m , and S o , respectively.As expected, the energy intensities are independent of Reynolds number when confinedto the universal classes of wave parameters. The solid curves are obtained by integratingthe energy density over all wavenumbers and wave speeds 2 ≤ c ≤ U cl , i.e. by confiningthe wave parameters to S e . These figures highlight the selection of two local peaks by thelinear amplification mechanism where the inner and outer peaks dominate the middlepeak. The inner peak takes place close to the inner peak of the streamwise intensity inreal turbulent flows. While the energy intensity of real flows exhibits outer scales nearthe center of the channel, there is no strong evidence for presence of an outer peak evenfor high Re τ .As evident from figure 9(g), the universal inner waves contribute to more than 96%of the total energy intensity for y + <
20 for all Reynolds numbers. On the other hand,figure 9(i) shows that the universal outer waves capture a smaller amount of the total odel-based scaling of the streamwise energy density y = 0 .
45 as Re τ increases; 95% for Re τ = 3333 vs. 86% for Re τ = 30000.This is because the aspect ratio constraint in S o excludes more wavenumbers from S e as Re τ increases. The excluded waves are not universal with Re τ and their contribution tothe energy intensity is not completely negligible. A similar reasoning explains why theuniversal middle scale captures 82% of the total energy intensity at (cid:112) yy + = √
10 for Re τ = 3333 vs. 72% for Re τ = 30000; cf. figure 9(h).At the end of this section, we recall that the streamwise energy densities and intensitiesthus far were obtained for the model with broadband forcing in λ x , λ z , and c . In §
4, weconsider a non-broadband forcing by introducing an optimally shaped energy density.
4. Predicting the streamwise energy intensity
In this section, we introduce a model for predicting the energy intensity of real tur-bulent flows by considering a non-broadband forcing in wave speed. This is done byincorporating a positive weight function W ( c ) that amplifies or attenuates the energydensity E uu ( y, c ) of the rank-1 model with broadband forcing. Even though W ( c ) differsfrom a true forcing spectrum (that also depends on the wall-parallel wavelengths), itprovides the model with sufficient degrees of freedom for predicting the energy intensity.In addition, since each wave speed is associated with a certain class of wavelengths, W ( c )affects different classes of wavelengths as the wave speed changes.First, we show that W ( c ) can be optimally shaped such that the model-based stream-wise energy intensity, E uu,W ( y ) = (cid:90) U cl W ( c ) E uu ( y, c ) d c, (4.1)matches the intensity of real flows at low Reynolds numbers. Then, we estimate similaritylaws to approximate the optimal weight functions at high values of Re τ . These weightfunctions in conjunction with the energy density of the rank-1 model with broadbandforcing enable prediction of the streamwise energy intensity at technologically relevantReynolds numbers. 4.1. Optimal weights for small Reynolds numbers
The weight function W ( c ) is determined by minimizing the deviation between E uu,W ( y )in (4.1) and the streamwise energy intensity obtained from DNS, E uu, DNS ( y ), in the in-terval y + ≥ y ≤ .
8. We do not enforce matching for y ≥ . W ( c ) for wave speeds close to U cl . This is because E uu ( y, c )is considerably smaller and more localized near the centerline compared to other loca-tions and results in sensitivity of W ( c ); see, for example, figure 5(a) for y + > y > . Re τ = 30000. Note that the main amplification mechanismsfor waves with speeds close to U cl is critical behavior of the resolvent modes since thenon-normality effect is small as the mean shear approaches zero. This results in smallgains and resolvent modes that are localized in the wall-normal direction.Since E uu ( y, c ) scales with Re τ (cf. table 3) while E uu, DNS ( y ) does not, we find thenormalized weight function W ( c ) = Re τ W ( c ) that solves the following optimizationproblem minimize: (cid:107) E uu, DNS ( y ) − E uu,W ( y ) (cid:107) e (cid:107) E uu, DNS ( y ) (cid:107) e + γ w (cid:107) W ( c ) (cid:107) w , subject to: W ( c ) > , ≤ c ≤ U cl . (4.2)6 R. Moarref, A. S. Sharma, J. A. Tropp, & B. J. McKeon
Here, (cid:107) g ( y ) (cid:107) e is defined as (note integration in log y + ) (cid:107) g ( y ) (cid:107) e = (cid:90) log(0 . Re τ )0 g (log y + ) d log y + , to equally penalize the deviation of energy intensities near the inner peak as well as inthe channel core. The second term in the objective function, (cid:107) W ( c ) (cid:107) w = 1 U cl − (cid:90) U cl W ( c ) d c, provides the weight function with smoothness by penalizing the magnitude of W , and γ w ≥ Re τ = 934 (del ´Alamo et al. Re τ = 2003 (Hoyas & Jim´enez2006). Even though these are orders of magnitude smaller than the Reynolds numbers forwhich we predict the energy intensity in § γ w = 0 . (cid:107) E uu, DNS ( y ) − E uu,W ( y ) (cid:107) e and smoothness of W ( c ). The optimizationproblem is robust with respect to the choice of γ w . For example, changing γ w by afactor of two has negligible effect on the matching error and the optimal weights for7 (cid:46) c (cid:46) U cl − W ( c ) elsewhere.Figures 10(a) and 10(b) show the optimal weights as a function of c and U cl − c for Re τ = 934 and 2003. These weight functions match E uu, DNS and E uu,W with a relativeerror of approximately 0 . W is qualitatively similar for c (cid:46)
16 and U cl − c (cid:46) .
15 since both the model-based and DNS-based intensities exhibitinner and outer scaling in the respective regions. Figure 10(b) shows that W ( U cl − c )approximately coincides for Re τ = 934 and 2003 for U cl − c ≤ .
9. We denote thisuniversal function by W ( U cl − c ). For simplicity, the weights are approximated by linearfunctions in the self-similar region for 16 ≤ c ≤ U cl − .
9. These lines are denoted by L (black) and intersect for L ( c = −
2) = 19 .
88 and L ( U cl − c = 3 .
9) = 2 .
54. This gives ananalytical expression for L as a function of wave speed and Reynolds number since U cl varies with Re τ L ( c ; Re τ ) = 2 .
54 + 17 . (cid:18) U cl − c − . U cl − . (cid:19) . (4.3)As the Reynolds number increases, W ( c ) is shifted upward for c ≤
16 by the kick thatit receives from the self-similar region. This is expected since the DNS-based energyintensity increases with Re τ in the inner region while E uu ( y + , c ) remains constant. Morediscussion about the relationship between the weights in the self-similar and inner regionsare provided in § W ( c ; Re τ ) = W ( c ) + (cid:0) L ( c ; Re τ ) − L ( c ) (cid:1) , ≤ c ≤ ,L ( c ; Re τ ) , < c < U cl − . ,W ( U cl − c ) , U cl − . ≤ c ≤ U cl , (4.4)that consists of three segments: A universal outer segment represented by W ( U cl − c )for U cl − . ≤ c ≤ U cl ; a Reynolds number dependent linear segment L ( c ; Re τ ) that is odel-based scaling of the streamwise energy density c U cl − c ( a ) ( b ) W W y + c ( c ) ( d ) E uu , W ; E uu , D N S W Figure 10. (Color online) (a)-(b): The optimal weight functions W as a function of c in (a) and U cl − c in (b) for Re τ = 934 (orange) and 2003 (green). The tangent lines, L , to W ( U cl − c ) at U cl − c = 3 . Re τ are shown in black. (c) The model-based streamwise energy intensity E uu, W ( y + ) for Re τ = 934 (orange) and 2003 (green) and the DNS-based intensity E uu, DNS ( y + )(black) are optimally matched by solving (4.2) for each Re τ . The respective curves lie on the topof each other. (d) The optimal weight functions W for Re τ = 934 (orange) and 2003 (green) arecompared with the weight functions obtained using the similarity law (4.4) (black dots). Thearrows indicate increase in the Reynolds number. analytically determined by (4.3); and an inner segment composed of a universal function W ( c ) for 2 ≤ c ≤
16 superposed by a linear function L ( c ; Re τ ) − L ( c ) where L ( c ) = L ( c ; Re τ = 934). Figure 10(d) shows that the optimal weights computed by solving (4.2)are well-captured by the weights formulated using the similarity law (4.4). We notethat more complex approximations could be used if more than two DNS datasets wereavailable. Efforts to determine these weights analytically are ongoing.4.2. Predictions at high Reynolds numbers
The similarity law in (4.3)-(4.4) is used to predict the weight functions, and consequently,the streamwise energy intensity at high Re τ using (4.1). Figures 11(a)-11(c) show thepredicted weights and energy intensities for Re τ = 934, 2003, 3333, 10000, and 30000.An approximately logarithmic dependence of the energy intensity on the distance from8 R. Moarref, A. S. Sharma, J. A. Tropp, & B. J. McKeon c U cl − c ( a ) ( b ) W W y + y + ( c ) ( d ) E uu , W E uu , W y + y + ( e ) ( f ) E uu , W E uu , W Figure 11. (Color online) (a)-(b): The optimal weight functions W obtained using the similaritylaw (4.4) as a function of c in (a) and U cl − c in (b). (c)-(f): The model-based streamwise energyintensity E uu, W ( y + ) for Re τ = 934 (orange), 2003 (green), 3333 (blue), 10000 (red), and 30000(black) in (c); Re τ = 3165 (green dotted), 4000 (blue solid), and 5813 (red dashed) in (d); and Re τ = 5813 (red dashed), 13490 (blue solid), and 23013 (green dotted) in (e)-(f). The arrows in(a)-(c) indicate increase in the Reynolds number and the line in (c) shows logarithmic scaling.The symbols in (d) are experimental data from channel flows for Re τ = 3165; (cid:5) and 4000; (cid:79) (Monty 2005), and for Re τ = 4000; ◦ and 6000; × (Schultz & Flack 2013). The symbols in(e) are experimental data from boundary layers for Re τ = 5813; × , 13490; + (De Graaff &Eaton 2000), and 23013; (cid:63) (Fernholz et al. odel-based scaling of the streamwise energy density et al. et al. Re τ = 3165, 4000, and 6000; especially note thecomparison at Re τ = 4000 with the data of Schultz & Flack (2013) that maintains asufficient spatial resolution down to the wall-normal location of the inner peak. Notethat the data of Monty (2005) and the data of Schultz & Flack (2013) at Re τ = 6000are not fully spatially resolved near the wall. In the absence of channel flow data athigher Reynolds numbers, figure 11(e) compares the model-based streamwise intensitieswith the data from boundary layer experiments for Re τ = 5813 and 13490 (De Graaff& Eaton 2000) and Re τ = 23013 (Fernholz et al. et al. (2009) showedthat the behavior of boundary layers, pipes, and channels is similar in the near-wallregion in spite of the differences between channels/pipes and boundary layers furtheraway from the wall. The experimental measurements are not accurate near the wallas they suffer from spatial resolution issues (see, for example, Hutchins et al. et al. Re τ . Alterna-tively, the universal behavior of E uu ( y ; κ x , κ z , c ) can be used to avoid these computations.In the present study, we employ the universality (invariance with Re τ ) of E uu ( y ; κ x , κ z , c )for S i to predict the inner peak of the streamwise intensity at arbitrary high Re τ . Ex-panding the weighted energy density according to the wave speed and substituting forthe weight function using the similarity law (4.4) yields E uu,W ( y ) = (cid:90) W ( c ) (cid:0) Re − τ E uu ( y, c ) (cid:1)(cid:124) (cid:123)(cid:122) (cid:125) universal (cid:18) L ( c ; Re τ ) − L ( c ) W ( c ) (cid:19) d c + (cid:90) U cl − . L ( c ; Re τ ) (cid:0) Re − τ E uu ( y, c ) (cid:1) d c + (cid:90) U cl U cl − . W ( c ) (cid:0) Re − τ E uu ( y, c ) (cid:1) d c. (4.5)The first integral, corresponding to the inner class of wave parameters S i , contains auniversal function multiplied by a coefficient L ( c ; Re τ ) that also appears in the secondintegral for the faster and larger waves in the self-similar region. It represents the contri-bution from the inner class of wave parameters that are coupled with and amplified bythe large scales in the self-similar region. This is similar to the model that Marusic et al. (2010 b ) proposed to capture the influence of the large scales u L (close to the geometricmean of the middle region of U ) on the small scales u S close to the inner peak of theenergy intensity u S = u ∗ (1 + β u L ) + α u L . (4.6)For the purpose of the present study, (4.6) implies that the small structures are deter-0 R. Moarref, A. S. Sharma, J. A. Tropp, & B. J. McKeon y + y + (a) (b) Figure 12. (Color online) (a) The red solid curve is the model-based energy intensity E uu,W ( y )for Re τ = 10 . The blue dashed curve is the contribution from the universal function in (4.5),and the black dotted curve is the contribution from the inner class of wave parameters S i . (b)The contribution to the energy intensity from the inner class of wave parameters for Re τ = 10 ,10 , and 10 . mined by a universal inner-scaled function u ∗ multiplied by a coefficient 1 + β u L thatincreases with the energy of the large structures. Physically, the first term in (4.6) de-scribes the amplitude modulation of small scales by the large scales and the second termrepresents the direct superimposition of the large scales on the inner-scaled near-wallpeak (Marusic et al. b ).The blue dashed curve in figure 12(a) shows the contribution of the universal functionin (4.5) to the energy intensity. This is equal to the contribution of the inner class ofwave parameters to the energy intensity for Re τ = 934, i.e. for L = L . In other words,the inner class of wave parameters is not influenced by the large scales in the middleregion for Re τ = 934. This is expected since at Re τ ≈ c = 16: i.e. innerscales for c <
16 and outer scales for c > U cl − . ≈
16. Therefore, Re τ ≈ Re τ ≈ Re τ = 10 . Notice that the large scales from the self-similarregion increase the inner peak by amplifying the universal function through the coefficient1 + ( L − L ) /W . The red solid curve is the total intensity obtained by integratingthe contribution of all wave parameters S e . The inner peak is captured by the firstintegral and the direct superimposition of the large scales on the inner peak is negligible.Therefore, the first integral readily yields the behavior of the streamwise intensity nearthe inner peak. For example, figure 12(b) illustrates how the more energetic large scalesat Re τ = 10 and 10 further increase the inner peak relative to Re τ = 10 by amplifyingthe universal function W .Figure 13 is adapted from figure 8 in Marusic et al. (2010 a ) where the DNS and ex- odel-based scaling of the streamwise energy density Figure 13. (Color online) Variation of the inner ( y + = 15) and outer ( y + = 3 . Re / τ ) peaksof the streamwise energy intensity with Reynolds number. The figure is adapted from Marusic et al. (2010 a ). The black open symbols are experimental and simulation data from channels andboundary layers, see Hutchins & Marusic (2007 a ) for a full list of references. The open trianglesare from large-eddy simulations of boundary layers (Inoue et al. perimental data from channels and boundary layers are summarized (open and filledblack symbols). The black filled squares and circles, respectively, show the magnitude ofthe inner ( y + = 15) and outer ( y + = 3 . Re / τ ) peaks in recent boundary layer exper-iments (Marusic et al. a ). Using these data, the authors proposed two possibilitiesfor the behavior of the inner peak at high Reynolds numbers. The first possibility is toextrapolate following the trend suggested by the filled black squares (line 1). The secondpossibility, motivated by the fact that the large scales increase the energy of the smallscales, is to extrapolate following line 3 which is parallel to line 2 that captures the vari-ation of the outer peak with Re τ . The data (open triangles) from large-eddy simulationsof boundary layers (Inoue et al. et al. (2010 b ) are shown for comparison. The current understanding, at least for relatively smallintervals of Reynolds numbers, suggests logarithmic growth of the inner peak. However,due to lack of sufficient spatial resolution close to the wall, the available experimentaldata conducted for different ranges of Re τ , predict different rates for the logarithmicgrowth, e.g. see Marusic et al. (2010 a ). Therefore, the available data is not sufficient forpredicting the exact behavior of the inner peak as Re τ increases.The diamonds in figure 13 show the model-based prediction of the inner peak of thestreamwise intensity up to Re τ = 10 . These predictions are made at no additional costusing the universal energy density for the inner class of wave parameters and the similaritylaw for the weight functions. These results are obtained for channels and are potentiallydifferent than boundary layers. In spite of an approximately logarithmic growth of thepredicted inner peak up to Re τ ∼ , a sub-logarithmic behavior becomes evidentwhen seven decades of Re τ are considered. As shown in equation (4.5), the linear partof the weight function, modeling the influence of large outer-scaled modes on the smallinner-scaled modes, affects the growth of the inner peak with Re τ . The sub-logarithmicgrowth of the predicted inner peak can be attributed to the decrease in the slope of L ( c ) as Re τ increases, cf. equation (4.3) and figure 10(a). Understanding the Reynoldsnumber dependence of L ( c ) is an essential part of our ongoing research which is focusedon analysis of the self-similar modes in the logarithmic region.2 R. Moarref, A. S. Sharma, J. A. Tropp, & B. J. McKeon
5. Concluding remarks
Starting from the NSE, we highlighted the low-rank nature of the resolvent, formulatedfor individual wall-parallel wavenumbers and frequencies, and illustrated its power byshowing that the most energetic motions of real turbulent flows correspond to wavenum-bers and frequencies whose resolvent is approximately rank-1 (in the wall-normal di-rection). Motivated by this observation, we studied the streamwise energy density ofthe rank-1 model subject to forcings in the wall-parallel directions and time that werebroadband and optimized, or “trained”, with respect to the available DNS data.Our analysis consists of two steps: firstly identifying the modes that are highly amplifiedby the NSE and their scaling (essentially an analysis of the resolvent operator), and thencalculating weighting functions (by matching to DNS results) which determine which ofthese modes will be sustained in the real flow (connecting the linear system of resolventoperators back to the full NSE).5.1.
Scaling of the most amplified resolvent modes
It was shown that the resolvent admits three classes of wavenumbers and wave speedswhere the corresponding principal singular values and singular functions exhibit universalbehavior with Reynolds number. These classes are directly related to the universal regionsof the turbulent mean velocity (which is assumed known a priori ) and thus are primarilydistinguished by the wave speed: (i) a truly inner-scaled class of waves with constantspeeds in the inner region of the turbulent mean velocity; (ii) a class of waves with outer-scaled height and width and constant defect speeds relative to the centerline; and (iii) aclass of waves with outer-scaled length and constant defect speeds relative to geometricmean of the middle region of the turbulent mean velocity. In addition, we showed thathierarchies of geometrically self-similar modes whose length and width respectively scalequadratically and linearly with their height are admitted by the resolvent in the presenceof a logarithmic mean velocity.The integral role of wave speed and critical layers in characterizing the classes ofuniversal modes with Reynolds number and the geometrically self-similar modes with thewall-normal distance is understood and emphasized for the first time. The conventionalunderstanding about the scales of turbulent flows comes from the time-averaged velocityspectra in DNS and experiments. Upon integration in time, the separated scales in the(temporal) frequency domain are overlaid, and distinction of different scales in the spatialspectra becomes difficult. Therefore, the identified scales have significant implications forunderstanding the scaling of wall turbulence. They are inherent features of the linearmechanisms in the NSE and, consequently, the energy extraction mechanisms from themean velocity. In both the universal and self-similar classes, the wall-normal length scaleis inherited from the turbulent mean velocity, and the wall-parallel length scales aredetermined from the balance between the viscous dissipation term, (1 /Re τ )∆, and themean advection terms in the resolvent, e.g. i κ x ( U − c ).The main results of the present paper, i.e. the identified scalings in § U . Therefore, the choice of eddy viscosity or the von K´arm´an’s constant κ doesnot change our main results. On the other hand, the debate on the universality and/orexact value of κ is ongoing, e.g. see Nagib & Chauhan (2008) and using the turbulentviscosity given in (2.3) can result in inaccuracies in the considered mean velocity. Thiscan affect the quantitative results of § §
4, e.g. the shape of the resolvent modesand the predicted growth rate of the inner peak. Characterizing these effects is a topicof future work, and the sensitivity is known to be highest in the region of highest shear,close to the wall. Since closing the feedback loop in figure 2 eventually generates the odel-based scaling of the streamwise energy density λ z ∼ λ x y c which is different from the trend λ z ∼ λ x y ob-served in the DNS-based two-dimensional streamwise spectrum (Jim´enez & Hoyas 2008).Understanding the scaling differences between the resolvent modes and the weightedmodes requires detailed scrutiny of the weights and the nonlinear effects, a topic of ongo-ing research. In addition, our results suggest that, owing to scale separation in frequency,there is a large benefit to obtaining and analyzing the scaling of three-dimensional time-resolved spectral measurements.5.2. Effect of nonlinearity
From a systems theory point of view, the nonlinear terms wrap a feedback loop aroundthe linear sub-systems in the NSE and redistribute the energy. They determine the wall-normal shape and the magnitude/phase of the driving force for an individual mode.Therefore, the real flow is obtained by superposing the resolvent response modes thatare weighted according to projection of the driving force on the resolvent forcing modes.We started by assuming that the nonlinear forcing is broadband in the wall-paralleldirections and time and aligned in the principal resolvent forcing modes. It was shownthat these simple assumptions can qualitatively produce different scaling regions of thestreamwise energy spectra. Therefore, the proposed analysis effectively narrows downthe scaling problem in wall-bounded turbulent flows to the problem of understandingthe influence of nonlinearity on the inevitable scales that are admitted by the linearmechanisms, i.e. determining which of those admitted modes will be required in realflows for the flow to be self-sustaining. A full description of the latter effects is beyondthe scope of the present paper, but the subject of ongoing work.A non-broadband forcing in time was accounted for by considering a weight functionin the wave speed. We showed that “training” the weights based on the wave speed canresult in streamwise energy intensities that quantitatively match DNS and experiments.As the Reynolds number increases, the optimal weight functions increase for wave speedsin the inner region of the mean velocity. Representation of the optimal weights usingsimilarity laws revealed that the amount of upward shift is linearly correlated with theweight function for wave speeds in the middle region of the mean velocity. In otherwords, the weight function increases with the energy intensity of the large scales andamplifies the universal inner-scaled energy density of the rank-1 model. Therefore, itimplicitly captures the well-known coupling of small scales with the large scales andtheir subsequent amplification in real turbulent flows.A consequence of the simplicity of the identified scaling in wavenumber-frequencydomain is the success of the simple weighting based on convection velocity in post- andpre-dicting the variation of the streamwise velocity fluctuations with Reynolds number.One of the main results of this study is that the rank-1 approximation, together withthe optimal weight functions and the (well-known) mean velocity profile, is sufficient forpredicting the streamwise energy intensity at high Reynolds numbers. Even though theweight function provides a rough intuition about the effect of nonlinearity, the explicitanalysis of the nonlinear feedback on the velocity field remains a subject of ongoingresearch.4
R. Moarref, A. S. Sharma, J. A. Tropp, & B. J. McKeon
Outlook of the present analysis as a predictive tool
The present study effectively divides the streamwise energy density of the rank-1 modelwith broadband forcing into inner- and outer-scaled universal regions with Reynoldsnumber and a geometrically self-similar region with distance from the wall that bridgesthe gap between the inner and outer regions. This enables scaling of the streamwiseenergy density to arbitrary large Reynolds numbers. It was shown that the most energeticwave parameters and the corresponding scales roughly agree with the dominant near-wallmotions in real turbulent flows.The identified self-similar resolvent modes facilitate analytical developments in thelogarithmic region of the turbulent mean velocity and can result in significant simplifica-tions in analysis of wall turbulence. In addition, the wall-normal locality of the self-similarmodes in a given hierarchy suggests that the linear sub-systems in the NSE impose adirect correspondence between wall-parallel scales and wall-normal locations in the log-arithmic region. In the classical cascade analogy, e.g. see the review paper by Jim´enez(2012), this is reminiscent of an inertial regime the study of which is a topic of ongoingresearch. Furthermore, ongoing research is focused on utilizing the identified scalings tobetter understand the structure and evolution of the hypothesized attached eddies.The available predictive models of wall turbulence, e.g. the attached eddy hypothe-sis (Townsend 1976; Perry & Chong 1982) and the model of Marusic & Kunkel (2003),rely on physical intuition that is gained from DNS and experiments. For example, themethod proposed by Marusic & Kunkel (2003) is based on an assumption about theinfluence of outer-layer modes on the near-wall modes (their equation (2)), where theunderlying functions are determined by empirical curve fits to the experimental data(their equations (3)-(5)). The present model is more fundamental as it directly uses theNSE for decomposing the flow into classes of modes that are uniquely scaled with theReynolds number and distance from the wall. Since the wall-normal shape of these modesis one of the model outputs, the contribution of the present work goes beyond reportingan empirical fit to the model-based data, namely by exploring the scaling of the modesadmitted by the NSE.In essence, this work supports the efficacy of the low-rank model of wall turbulence pro-posed by McKeon & Sharma (2010) by demonstrating that it can be used both to deter-mine self-similar mode scalings and to obtain a low-rank representation of the streamwiseintensity, given appropriate, self-similar weighting of the modes. Our ongoing research,to be reported elsewhere, is focused on analytical expression of the streamwise energydensity for wave speeds in the logarithmic region of the mean velocity and a priori deriva-tion of the weight functions. Addressing the limitations and implications of the low-rankmodel for predicting the wall-normal and spanwise energy spectra as well as the Reynoldsstresses is another topic of future research.
Acknowledgments
The support of Air Force Office of Scientific Research under grants FA 9550-09-1-0701 (P.M. John Schmisseur) and FA 9550-12-1-0469 (P.M. Doug Smith) is gratefullyacknowledged.
Appendix A. Derivation of the inner scalings
We show that the transfer function H admits universal behavior for the modes withspeeds c (cid:46)
16. For these modes, following equation (3.5), the y -dependent coefficients in odel-based scaling of the streamwise energy density H , are either independent of Re τ , e.g. U ( y + ) − c , or scale with Re τ ,e.g. U (cid:48) ( y + ). This allows for scaling the height of the resolvent modes with the viscousunit ν/u τ . In addition, the balance between the viscous dissipation term, (1 /Re τ )∆, andthe mean advection terms, e.g. i κ x ( U − c ), in the resolvent in (2.9) requires scaling of thewall-parallel wavelengths with the viscous unit ν/u τ λ + x = Re τ λ x , y + = Re τ y, λ + z = Re τ λ z . The differential operators in y and the wavenumber symbols in the inner coordinates are ∂/∂ y + = Re − τ ∂/∂ y , κ + x = Re − τ κ x , κ + z = Re − τ κ z , ∆ + = Re − τ ∆ . Consequently in the inner coordinates, the operators R A , C , and C † in (2.7) and (2.9)scale as R A = (cid:20) Re τ X Re τ X Re τ X (cid:21) − = (cid:20) Re − τ Y Y Re − τ Y (cid:21) ,C = C Re − τ C C C Re − τ C , C † = (cid:20) C † C † C † Re τ C † Re τ C † (cid:21) . (A 1)For given κ + x and κ + z , the operators C to C and their adjoints are independent of Re τ . On the other hand, the operators X to X and Y to Y contain spatially-varyingcoefficients, U − c and its first two derivatives, that depend on Re τ . As discussed in thebeginning of § U scales with y + and is independent of Re τ for y + (cid:46) κ + x , κ + z , and c (cid:46) U ( y + = 100) = 16, the operators X to X and Y to Y are independent of Re τ when acting on functions whose supports are inside the interval0 < y + (cid:46) c ), the resolventmodes are negligible outside y + (cid:46)
100 for c (cid:46)
16 and all of the aforementioned operatorsare effectively independent of Re τ . It follows from (A 1) that H = CR A C † = Re − τ H Re − τ H Re − τ H Re − τ H Re − τ H Re − τ H Re − τ H Re − τ H Re − τ H , where the operators H ij are effectively independent of Re τ when acting on their principalresolvent modes. Therefore, the principal singular value of H is proportional to Re − τ . Inaddition, the orthonormality constraints (2.11) on ˆ ψ and ˆ φ require that these functionsscale as Re / τ . This is because the supports of ˆ ψ and ˆ φ are independent of Re τ ininner units (hence, proportional to Re − τ in outer units). In other words, ˆ ψ ( y ) andˆ φ ( y ) become thinner and taller as Re τ increases. Finally, the streamwise energy density E uu = κ x κ z σ | u | scales with (cid:0) Re τ (cid:1) (cid:0) Re τ (cid:1) (cid:0) Re − τ (cid:1) (cid:0) Re / τ (cid:1) = Re τ . Appendix B. Derivation of the outer scalings
For the modes with defect speeds 0 (cid:46) U cl − c (cid:46) .
15, we show that the transferfunction H admits universal behavior with Reynolds number. For these modes, followingequation (3.5), the y -dependent coefficients in the transfer function H , e.g. U ( y ) − c , areindependent of Re τ . This allows for scaling the height of the resolvent modes with h .Furthermore, the balance between the viscous dissipation term, (1 /Re τ )∆, and the mean6 R. Moarref, A. S. Sharma, J. A. Tropp, & B. J. McKeon advection terms, e.g. i κ x ( U − c ), in the resolvent in (2.9) requires scaling of the spanwisecoordinate with h and the streamwise coordinate with hRe τ . Therefore, the streamwisewavenumber symbol in the outer coordinates is given by κ − x = Re τ κ x . The Laplacian∆ = ∂ yy − Re − τ ( κ − x ) − κ z , is independent of Re τ if κ z dominates Re − τ ( κ − x ) for all values of Re τ . For fixed κ − x and κ z , it suffices that λ − x λ z = κ z κ − x (cid:38) γRe τ, min . (B 1)In the outer coordinates, the operators R A , C , and C † in (2.7) and (2.9) scale as R A ≈ (cid:20) Re − τ ˜ X X Re − τ ˜ X (cid:21) − = (cid:20) Re τ ˜ Y Re τ ˜ Y Re τ ˜ Y (cid:21) ,C ≈ Re − τ ˜ C ˜ C ˜ C C Re − τ ˜ C , C † ≈ (cid:20) Re − τ ˜ C † ˜ C † ˜ C † ˜ C † Re − τ ˜ C † (cid:21) . (B 2)For given κ − x and κ z that satisfy the constraint (B 1), the operators ˜ C to ˜ C and theiradjoints are approximately independent of Re τ . In addition, the defect velocity U cl − U ( y )is independent of Re τ for y (cid:38) .
1. Therefore, for given κ − x , κ z , and U cl − c (cid:46) U cl − U ( y =0 .
1) = 6 .
15, the operators ˜ X to ˜ X and ˜ Y to ˜ Y are approximately independent of Re τ when acting on functions whose supports are inside the interval 0 . (cid:46) y (cid:46)
1. From (B 2),we have H = CR A C † ≈ Re τ ˜ H Re τ ˜ H Re τ ˜ H ˜ H Re τ ˜ H Re τ ˜ H ˜ H Re τ ˜ H Re τ ˜ H . (B 3)Owing to the locality of the principal resolvent modes around the critical layer, theoperators ˜ H ij are approximately independent of Re τ when acting on their principalresolvent modes. Therefore, the principal singular value of H is proportional to Re τ .Since ˆ ψ and ˆ φ scale in the outer length scale, the orthonormality constraints (2.11)require that these functions be independent of Re τ . Finally, the streamwise energy density E uu = κ x κ z σ | u | scales with (cid:0) Re − τ (cid:1) (cid:0) (cid:1) (cid:0) Re τ (cid:1) (cid:0) (cid:1) = Re τ . Appendix C. Derivation of the geometrically self-similar scalings
The transfer function H admits geometrically self-similar modes with speeds in thelogarithmic region of the turbulent mean velocity. In this region, it follows from thediscussion in § y -dependent coefficient in the transfer function H can beexpressed as U ( y ) − c = (1 /κ ) log( y/y c ), where y c is the critical wall-normal locationcorresponding to c , i.e. c = U ( y c ). Similarly, U (cid:48) and U (cid:48)(cid:48) are functions of y/y c . This allowsfor scaling the height of the resolvent modes with y c . Furthermore, the balance betweenthe viscous dissipation term, (1 /Re τ )∆, and the mean advection terms, e.g. i κ x ( U − c ),in the resolvent in (2.9) requires scaling of the spanwise wavelength with y c and thestreamwise wavelength with y + c y c ,¯ λ x = λ x / ( y + c y c ) , ¯ y = y/y c , ¯ λ z = λ z /y c . odel-based scaling of the streamwise energy density y and the wavenumber symbols in the y c -scaled coordinatesare ∂/∂ ¯ y = y c ( ∂/∂ y ) , ¯ κ x = ( y + c y c ) κ x , ¯ κ z = y c κ z . For given ¯ κ x and ¯ κ z , the Laplacian∆ = y − c (cid:0) ∂ ¯ y ¯ y − ( y + c ) − (¯ κ x ) − (¯ κ z ) (cid:1) , approximately scales with y − c if (¯ κ z ) dominates ( y + c ) − (¯ κ x ) , i.e. κ z /κ x = λ x /λ z = y + c (¯ λ x / ¯ λ z ) (cid:38) γ, (C 1)where a conservative value for γ is √
10. Since the aspect ratio λ x /λ z increases with y + c , the smallest value of y + c for which (C 1) is guaranteed is equal to y + c = γ (¯ λ z / ¯ λ x ).Therefore, the smallest wave speed that satisfies the aspect ratio constraint and lies abovethe inner region is given by c = max (cid:0) , B + (1 /κ ) log y + c (cid:1) . (C 2)Then, the operators R A , C , and C † in (2.7) and (2.9) scale as R A = (cid:34) (cid:0) y + c y c (cid:1) − ¯ X y − c ¯ X (cid:0) y + c y c (cid:1) − ¯ X (cid:35) − = (cid:20) (cid:0) y + c y c (cid:1) ¯ Y y + c ) ¯ Y (cid:0) y + c y c (cid:1) ¯ Y (cid:21) ,C = (1 /y + c ) ¯ C ( y c ) ¯ C ¯ C C (1 /Re τ ) ¯ C , C † = (cid:34) (1 /y + c ) ¯ C † ¯ C † ¯ C † (1 /y c ) ¯ C † (cid:0) y + c y c (cid:1) − ¯ C † (cid:35) . (C 3)For given ¯ κ x and ¯ κ z that satisfy the constraint (C 1), the operators ¯ C to ¯ C and theiradjoints are approximately independent of y c and Re τ . In addition, the operators ¯ X to¯ X and ¯ Y to ¯ Y are approximately independent of y c and Re τ when acting on functionswhose supports are localized in the interval 100 /Re τ ≤ y ≤ .
1. From (C 3), we have H = CR A C † = (cid:0) y + c y c (cid:1) ¯ H (cid:0) y + c (cid:1) ( y c ) ¯ H (cid:0) y + c (cid:1) ( y c ) ¯ H ( y c ) ¯ H (cid:0) y + c y c (cid:1) ¯ H (cid:0) y + c y c (cid:1) ¯ H ( y c ) ¯ H (cid:0) y + c y c (cid:1) ¯ H (cid:0) y + c y c (cid:1) ¯ H , where the operators ¯ H ij are effectively independent of y c and Re τ when acting on theirprincipal resolvent modes. Therefore, the principal singular value of H is proportional to( y + c ) ( y c ). In addition, the orthonormality constraints (2.11) on ˆ ψ and ˆ φ require thatthese functions scale with ( y c ) − / . This is because the supports of ˆ ψ and ˆ φ expandwith y c . Finally, the streamwise energy density E uu = κ x κ z σ | u | for the waves thatbelong to the same hierarchy scales with (cid:0) y + c y c (cid:1) − (cid:0) y c (cid:1) − (cid:0) ( y + c ) ( y c ) (cid:1) (cid:0) y c (cid:1) − = Re τ . REFERENCESAdrian, R. J.
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