Under consideration for publication inJ. Fluid Mech. 1 Large-eddy simulation of large-scale structures in long channel flow 0 1 0 D. CHUNG AND B. J. McKEON† 2 Graduate Aerospace Laboratories, California Instituteof Technology, Pasadena, CA 91125, n USA a [email protected] J 8 (Received ?? and in revised form ??) 1 We investigate statistics of large-scale structures from large-eddy simulation (LES) of ] turbulent channel flow at friction Reynolds numbers Reτ = 2k and 200k. To properly n capture the behaviour of large-scale structures, the channel length is chosen to be 96 y timesthechannelhalf-height.Inagreementwithexperiments,theselarge-scalestructures d - arefoundtogiverisetoanapparentamplitude modulationoftheunderlyingsmall-scale u fluctuations.Thiseffectisexplainedintermsofthephaserelationshipbetweenthelarge- l f and small-scale activity. The shape of the dominant large-scale structure is investigated . s by conditionalaveragesbasedonthe large-scalevelocity,determined using afilter width c equaltothechannelhalf-height.Theconditionedfielddemonstratescoherenceonascale i s of severaltimes the filter width, and the small-scale–large-scalerelative phase difference y increases away from the wall, passing through π/2 in the overlap region of the mean h p velocity before approaching π further from the wall. We also found that, near the wall, [ the convection velocity of the large-scales departs slightly, but unequivocally, from the mean velocity. 1 v 5 1 9 2 1. Introduction . 1 Recentstudies(Kim & Adrian1999;Morrison, McKeon, Jiang & Smits2004;Guala, Hommema & Adrian 0 2006;Monty, Stewart, Williams & Chong2007;Hutchins & Marusic2007b,a;Mathis, Hutchins & Marusic 0 2009)haveconfirmedearlierobservations(Favre, Gaviglio & Dumas1967;Kovasznay,Kibens & Blackwelder 1 1970) of very long large-scale structures in the wall region of boundary layers, channels : v and pipes. These structures are marked by streamwise-elongated, alternating low- and i X high-momentum,meanderingstreaks,withlength10δ(Kim & Adrian1999;Morrison et al. 2004),8–16δ(Guala et al.2006),25δ(Monty et al.2007),6δ(Hutchins & Marusic2007b), r a 20δ (Hutchins & Marusic2007a)andwidth 0.3–0.5δ (Mathis et al. 2009),where δ is the boundary layer thickness, channel half-height or pipe diameter. See Monty et al. (2009) foradescriptionofthedifferencesbetweenthecharacteristicsoftheselargestructuresin the different canonical flows. The bursting period of these structures, 6δ/U, where U(z) is the mean velocity, was already noted some decades ago, along with the long tails of the streamwisevelocityauto-correlations,see reviewby Cantwell (1981). The dynamical significanceoftheselarge-scalestructurescanbeseeninascaledecompositionofrelative energycontent,asmeasuredbythepremultipliedone-dimensionallongitudinalspectrum κ E plotted against the log streamwise wavelength, logλ , where λ = 2π/κ (equal x uu x x x area under the curve implies equal energy contribution). For a boundary layer at fric- tion Reynolds number Re = 7.3k (Hutchins & Marusic 2007b), the signature of these τ † Email address for correspondence: [email protected] 2 D. Chung and B. J. McKeon structures are relatedto the outer peak in κ E found at (z/δ,λ /δ)=(0.06,6).It has x uu x been proposed that the wall-normallocation of this peak is located at the middle of the log layer, z+ ∝ Re1/2 or Re3/4 (Mathis et al. 2009) (the choice of scaling depends on τ τ whether the lower limit of the log law is Reynolds number dependent), where z is the height from the wall and the superscript + indicates scaling in wall units: the friction velocityu andkinematicviscosityν.Howeverthescalingremainssomewhatambiguous. τ The large-scale structures were found (Bandyopadhyay & Hussain 1984; Mathis et al. 2009) to modulate the amplitudes of superimposed small-scale fluctuations. To test this idea, these authors first split the streamwise velocity into large- and small-scale com- ponents via a filter at λ /δ = U(z)/(fδ), and then used either a filtered and rectified x small-scale signal or the Hilbert-transform to determine the envelope for the small-scale fluctuations, finally forming the correlation coefficient between the large-scale fluctua- tions and the low-passfiltered envelope of the small-scalefluctuations. They found that, near the wall,large-scalehigh-speedregionscarryintense superimposedsmall-scalefluc- tuations,butthiscorrelationisreversedaboveaheightthatdecreasesinouterunitswith Re . We shall attempt to reproduce these features presently. τ The footprint of structures centred far from the wall provides an obvious challenge in terms of determining appropriate convection velocities across the range of turbulent scales, with particular importance for obtaining the correct wavenumber spectra from temporal frequency spectra obtained by, for example, hot-wire anemometry. It has been known for some time that convectionvelocities deviate from the local mean in the near- wall region, (e.g. Krogstadet al. 1998). The common practice is to use Taylor’s frozen- turbulence hypothesis to map from the frequency to the wavenumberdomain, that is to use the assumption that all structures at a given wall distance z convect at the same scale-independent mean velocity U(z). It was shown from a particle image velocimetry (PIV) experiment (Dennis & Nickels 2008) that this is indeed good approximation at z/δ =0.16for a Re =4.7kboundary layer,atleastfor scalessmallerthantheir fieldof θ view, 3.2δ in space and 6.3δ/U in time. However note that this wall-normal distance is sufficiently far fromthe wallthat it is beyondthe large-scaleenergy peak,suchthat any convectionvelocityquestionsarelikelyinsignificantbecauseofthelowshearintheouter region.With afieldofview largerthan20δ×20δ/U andheightdownto z/δ =0.049,we revisit the question of whether the footprint of the large-scalestructures, having centres further from the wall, still convect at the local mean velocity near the wall. To properly assess the dynamics of these long structures, reported to reach up to 25δ (Monty et al. 2007), we use large-eddy simulation (LES) coupled with a wall model (Chung & Pullin 2009). This investigation is ideally suited to the present wall modelled LES since its cost depends only on the number of ‘large eddies’, which, for a channel, is Reynoldsnumberindependent.Incontrast,thefullyresolveddirectnumericalsimulation (DNS) is prohibitively expensive. For reference, the most ambitious DNS of a channel flow to date is the Re =2k simulation (Hoyas & Jim´enez 2006) in an L /δ =8π≈25 τ x channel, where L is the streamwise length; a DNS investigation at higher Re and x τ larger L of these large-scale structures is not yet possible. Of course, the use of LES x comesatthecostofsubgrid-scale(SGS)modelling,wallmodellingandnumericalerrors, but LES is much faster (hours for simulation, minutes for post-processing) than DNS and experiments; we hope that a controlled application of the present LES combined with experience in the subject may shed some light on the physics of these large-scale structures. Details of the simulations are given in §2 and discussion of observations are found in §3 before we conclude in §4. LES of large-scale structures in long channel flow 3 Run Reτ Lx/δ Ly/δ h+0 ∆x/δ ∆tuτ/δ Nx Ny Nz Nt TUc/Lx G1 2k 95 7.9 15 0.17 0.006 576 48 48 72000 110 H3 200k 96 8.0 750 0.083 0.002 1152 96 96 15800 12 G1b 2k 95 7.9 15 0.17 0.006 576 48 48 22000 34 Table 1. LES parameters for long channelflows. Channel-transit times based on data-recording period T and centreline velocity Uc. Figure 1. Comparison between DNS data of Hoyas & Jim´enez (2006) and present Re = 2k channel flow LES, run G1b (table 1). Spectra at z/δ =0.049,0.090,0.17,0.26⇔z+ =98,180,350,510 (ordered in decreasing energy). 2. Simulation details As full details of the LES, including the numerical method and SGS model, are given by Chung & Pullin (2009), we only highlight the important points here. We solve the filtered Navier–Stokes equations for the LES velocity field u using the stretched-spiral vortexSGSmodel(Misra & Pullin1997;Voelkl et al.2000).Tocircumventtheinhibitive cost of resolving the near-wall region (Chapman 1979), z < h , we use a wall model 0 (Chung & Pullin 2009) that supplies off-wall slip-velocity boundary conditions at h to 0 the interior LES, operating in h < z < 2δ −h , where z = 0,2δ locates the walls. 0 0 Presently, we fix h = 0.18∆ , and the slip velocity is calculated using a wall model 0 z comprising 1) an evolution equation for the wall shear stress derived from assuming local inner scaling for the streamwise momentum equation and 2) an extended form of the stretched-vortex SGS model that provides a local log relation along with a dynamic estimate for the local K´arm´an constant. The parameters for the three LES runs are given in table 1. The grid is uniform, ∆ = ∆ = 4∆ , throughout the simulation domain. To capture the physics of long x y z large-scalestructures,weuselongachannel,L /δ ≈96,andthestatisticsaretakenover x T channel-transit times. It was shown (Chung & Pullin 2009) that the LES-predicted statistics from the Re = 2k case, including means, turbulent intensities and spectra, compare reasonably well with the DNSofHoyas & Jim´enez(2006).We showthe root-mean-square(r.m.s.)ofthe streamwise velocity fluctuations and the LES-resolved spectra in figure 1. The points 4 D. Chung and B. J. McKeon in 1(a) correspond to actual discretization points. Even though the total (subgrid plus resolved) r.m.s. is within 90% of the DNS result, its spectra plotted in energy-content form overpredicts the DNS spectra by about 20%. (Note, however, that the comparison between experimentalchannel flow data and DNS has revealeda similar trend, with the experimental premultiplied spectra at large wavelengths in the overlap layer being up to order 10% larger than the equivalent DNS values (Monty & Chong 2009).) As such, the results presented here should be viewed as approximate, despite capturing energy at significantly larger wavelengths.On the other hand, the physics reported here can be seen as robust features of wall turbulence if they are also observed elsewhere. We note that the peak value, (κ E )+ ≈ 1 at z/δ = 0.090, λ/δ ≈ 6 (figure 1) is within the x uu rangeofthe peak values fromboundary layerspectra for Re =1.0–7.3k(see figure 9of τ Hutchins & Marusic (2007a)). We interpret the LES results as a model of the real flow, and emphasize that Re =200k is far out of the reach of current DNS approaches. τ Inordertocomputecorrelationsbasedontemporalaveraging,anumericalrakeinruns G1 and H3, fixed in streamwise–spanwise location, is set up to record the LES velocity u and its modelled subgrid fluctuations T (≡ uu−uu) at the wall-normal locations xx z =n ∆ (n =0,1,...,N ) andtime steps t=n ∆ (n =0,1,...,N −1). Analogous z z z z t t t t correlationsbasedonspatialaveragesarealsocomputedfromtheserunsfromasnapshot in time. The only difference between runs G1 and G1b is the recorded data. For the latter, the three-dimensional data set, u(n ∆ ,y,n ∆ ,n ∆ ), is recorded at fixed y, for n = x x z z t t x 0,1,...,575,n =0,1,...,48andn =0,1,...,21999,where∆ /δ =0.17,∆ /δ =0.041 z t x z and ∆ u /δ =0.006. t τ Presently, x, y and z respectively denote the streamwise, spanwise and wall-normal directions; the velocity components, u, v and w, are defined accordingly. 3. Discussion of observations We present observations of the LES velocity fields, with emphasis on the large scales. 3.1. Convection velocities from spatio-temporal spectra We beginbyusingthesomewhatuniquecombinationofspatialandtemporaldataavail- able in this study to investigatethe validity ofTaylor’shypothesis.Giventhe autocorre- lation of the streamwise velocity fluctuations, ′ ′ R(ρ,τ)=hu(x,t)u(x+ρ,t+τ)i, (3.1) ′ where ρ is the streamwise separation; τ is the time delay; and u ≡ U +u such that hu′i = 0, we define the spatio-temporal spectrum Ψ(κ,ω) as the Fourier transform of R(ρ,τ). That is, together they form the Fourier transform pair, given by ∞ ∞ 1 Ψ(κ,ω)= R(ρ,τ)e−i(κρ−ωτ)dρdτ, (2π)2 −∞ −∞ Z Z ∞ ∞ R(ρ,τ)= Ψ(κ,ω)ei(κρ−ωτ)dκdω. −∞ −∞ Z Z The wavenumber spectrum Θ(κ) and the frequency spectrum Φ(ω) are both related to Ψ via ∞ ∞ Θ(κ)= Ψ(κ,ω)dω, Φ(ω)= Ψ(κ,ω)dκ, (3.2) −∞ −∞ Z Z LES of large-scale structures in long channel flow 5 whence the mean-square of u fluctuations can be recoveredfrom ∞ ∞ ∞ ∞ hu′2i= Ψ(κ,ω)dκdω = Φ(ω)dω = Θ(κ)dκ. −∞ −∞ −∞ −∞ Z Z Z Z Theseareeven,Φ(ω)=Φ(−ω)andΘ(κ)=Θ(−κ).SinceΦ(ω)canbemeasureddirectly using hot wires, the one-dimensional longitudinal spectrum, ′ Θ(κ)≡ω (κ)Φ(ω(κ)), (3.3) strictly defined in terms of a dispersion relation ω(κ) and associated group velocity ω′(κ)≡dω/dκ,isoftenapproximatedusingTaylor’sfrozen-turbulencehypothesis.Under thisassumption,thedispersionrelationcanbeformallywrittenasω (κ)=Uκand,from T (3.3), Θ(κ) = UΦ(Uκ). This is another interpretation to Taylor’s hypothesis based on the Fourier decomposition: the group velocity ω′ (κ)=U and phase velocity ω /κ=U T T of all eddies contributing to the turbulent kinetic energy at a particular wall-normal location are constants independent of wavenumber and equal to the mean velocity at that location, an approximationaccurate for sufficiently small eddies. To obtain Θ from Φ,thisstrictinterpretationofTaylor’shypothesiscanberelaxedtoaccountforenergetic eddieswhichtravelatU±∆U providedΨ(κ,ω)issymmetricwithrespecttotheω =Uκ line, that is Ψ(κ,ω)=Ψ(ω/U,Uκ), (3.4) because (3.4) then relates the two definitions in (3.2): ∞ ∞ ∞ ′ ′ Θ(κ)≡ Ψ(κ,ω)dω = Ψ(ω/U,Uκ)dω=U Ψ(κ,Uκ)dκ ≡UΦ(Uκ). −∞ −∞ −∞ Z Z Z For the purpose of obtaining wavenumber spectrum from frequency spectrum, a test of thevalidityofTaylor’shypothesiscanberecastasaquestionofthesymmetryofΨ(κ,ω) with respect to the line ω = Uκ. When this symmetry is broken, a different dispersion relation ω (κ) is necessary to relate Φ to Θ. Since we have access to both Φ and Θ, we c compute the ω (κ) directly by finding the inverse to the monotonic function K c t ∞ ∞ ω (κ)=K−1(K (κ)), K (κ)≡ Θ(κ′)dκ′, K (ω)≡ Φ(ω′)dω′, c t x x t Zκ Zω where K (0) = K (0) = hu′2i/2. The analogy to the convection velocity of individual x t eddies is complex. To first order, ω (κ) describes the apparent passing frequency of the c most energetic eddies with wavenumber κ, although this is an integral effect over the range of energetic spanwise wavenumbers. When computing Ψ from the present LES simulation, a normalised Hann window in time, 2/3[1−cos(2πn /M )](M ∆ isthetemporalwindowsize),isappliedtoubefore t t t t taking the discrete Fourier transform because the velocity is not periodic in time. From p run G1b (table 1) where N = 22000 and M = 576, the spectrum is averaged across t t ⌊22000/(576/2)−1⌋ = 75 half-overlapping windows. No windowing is necessary in the periodic streamwise direction. We plot in figure 2 contours of the premultiplied spectrum, κ κ Ψ/u2 versus logλ x t τ x and logλ , where λ = 2π/κ , λ = 2π/κ and κ = ω/U. When Taylor’s hypothesis t x x t t t is valid, contours of κ κ Ψ/u2 should be symmetrical about the λ = λ line. Observe x t τ t x from figure 2 that Taylor’s hypothesis is indeed a good approximation, except near the wall,z/δ =0.041and0.083,andforthelargescales,λ /δ,λ /δ >10.NotethattheLES t x formulationdoes notpermit effective examinationofsmaller scalesor locationscloser to the wall. The dispersion relation line computed from Ψ, λ = λ ω/ω , appearing below t x c 6 D. Chung and B. J. McKeon Figure 2. Premultiplied spatio-temporal spectra of streamwise velocity fluctuations, κxκtΨ/u2τ = 0.1,0.2,0.3,0.4, at various heights from Reτ = 2k channel flow LES, run G1b (table 1): , dispersion relation computed from Ψ, λt =λxω/ωc; , Taylor’s hypothe- sis λt =λx. LES of large-scale structures in long channel flow 7 (to the right) of Taylor’s hypothesis, λ = λ ω/ω = λ · 1, implies that the phase t x T x velocity U ≡ω /κ is larger than the mean velocity U. c c Despite overwhelmingsimilarities,thephasevelocityU isnotstrictlythe sameasthe c convection velocity defined by (2.7) from Wills (1964), where it is related to the ridge of Ψ. Recall from (3.3) that by construction U (κ) is defined such that the wavenum- c ber spectrum Θ can be recovered from the frequency spectrum Φ, that is Θ(κ) ≡ Φ(κU (κ))d(κU (κ))/dκ. Put another way,it is the mapping from wavenumber space to c c frequency space such that the energy observedin wavenumberspace at the wavenumber κ,Θ(κ)dκ,isequaltothatobservedinfrequencyspaceΦ(κU (κ))d(κU (κ))atfrequency c c ω =κU (κ)(seealsodiscussioninMonty & Chong(2009)).Inpractice,seefigure2,this c integral-based definition often traces out the ridge of Ψ, that is the definition in Wills (1964). Because it can be measured readily, the ridge-based definition is often taken as the surrogatefor U . Energy transmissionoccurs at neither U nor the velocity given by c c Wills (1964), but the group velocity d(κU (κ))/dκ, that is the velocity of the energy of c wavepacketswithwavenumbernearκ.Summarising,toobtainthewavenumberspectrum from the frequency spectrum, we require both the phase velocity and groupvelocity, see (3.3):theformerappearinginthe argumentofΦasκU (κ)to rescalethefrequency;and c the latter appearing as a factor of Φ as d(κU (κ))/dκ to rescale the rate of change in c frequency. FollowingDennis & Nickels(2008),wecanalsotestthe validityofTaylor’shypothesis in physical space by plotting the autocorrelationR defined by (3.1). Taylor’s hypothesis is valid, or more precisely there is a straightforward conversion from the temporal to the spatial domain, where the contours of R are symmetrical about the ρ = ρ line, x t whereρ =τU.Likefigure2,Rinfigure3showsanunequivocaldeparturefromTaylor’s t hypothesisnearthewall,z/δ =0.041and0.083andforthelargescales,ρ /δ,ρ /δ >10. t x There also appears to be a slight discrepancy of the opposite sign for the large scales at z/δ = 0.5, which is more marked in the autocorrelation than our presentation of the spectrum.TheboundarylayerPIVexperimentperformedbyDennis & Nickels(2008)at Re ≈ 4.7k reported that Taylor’s hypothesis is still valid at the height z/δ = 0.16 for θ the field of view ρ/δ < 0.29m/0.09m = 3.2 and τU/δ < 1s×0.57ms−1/0.09m = 6.3. This is consistentwith the presentLESdata since atz/δ =0.17,figure 3(c), R is indeed symmetrical about the ρ =ρ line, even up to very large scales ρ , ρ =20δ. x t t x Consideration of spatio-temporal spectra permits some speculation about the convec- tion velocities of the energetic structures and the error associated with identifying the footprint of eddies with a particular streamwise scale on the near-wall region from tem- poral data. The LES velocity fields unequivocally indicate that the most energetic large structuresconvectfasterthanthelocalmeanvelocityclosetothewall,withthedeviation growingclosetothewall.Conversely,theselargestructuresconvectslowerthanthelocal mean velocity nearthe channelcentre,figure 3(e, f). This suggeststhat these eddies are “local”toaregionintheoverlaplayer,inthesensethatthemeanvelocitymatchestheir convective velocity somewhere in the log region. We speculate that the location of this velocity matching correspondsto the locationof the large-scalestreamwise energy peak, whichis consistentwith the approximatemagnitude ofthe difference betweenω/ω and T ω/ω for large wavelength.This suggests that the departure from Taylor’s hypothesis at c thelargescalesshouldstrengthenwithincreasingReynoldsnumberduetotheincreasing shear near the wall. This is an area of current experimental study. 3.2. Large-scale–small-scale interaction With the differences between the spatial and the temporal decompositions in mind, we now describe the correlation that characterizes the interaction between the large-scales 8 D. Chung and B. J. McKeon Figure 3. Spatio-temporal correlations of streamwise velocity fluctuations, R/u2τ = 0.3,0.6,0.9,1.2, at various heights from Reτ = 2k channel flow LES, run G1b (table1): ,dispersionrelationcomputedfromΨ,ρt=ρxω/ωc; ,Taylor’shypothesis ρt =ρx. LES of large-scale structures in long channel flow 9 andthesmall-scales.Wefirstextractthelarge-scalefluctuationsu byapplyingasliding- L window top-hat time average,centred at t, to u: 1 t+τ/2 ′ ′ u (t)= u(t)dt, (3.5) L τ Zt−τ/2 whereτ isthe widthofthe slidingwindow.Alow-passfilter,(3.5),dampens fluctuations withfrequencieshigherthan1/τ.Inspectralspace,(3.5)isequivalenttoamultiplication by the filter sin(τω/2)/(ω/2), where ω is the angular frequency. For clarity, we have suppressed the x-dependence of u in this part of the discussion since x is held constant. The small-scale fluctuations are defined to be the remaining part of the motion, u = S u−u . The small-scale intensity can be measured by its local r.m.s., L 1/2 1 t+τ/2 u (t)= u2(t′)dt′ . (3.6) S τ Zt−τ/2 S ! e Physically, u measures the local envelope or intensity of small-scale fluctuations. For S example, if u is normally distributed, 95% of its amplitude is estimated to lie within S 2u . If u is not normal, u still measures the spreador envelope of u . In any case, u2 S S e S S S appearsintheequationsgoverningu ,obtainedbyapplyingthefilter(3.5)totheNavier– L Stokesequations(seeReynolds & Hussain(1972)forarelatedtwo-scaledecomposition), e e e which is another way to interpret u . Note that equivalent approaches have been used S by Bandyopadhyay& Hussain (1984) and Guala, Metzger & McKeon (2009) to obtain similar results in a range of flows including a laboratory turbulent boundary layer and e the near-wall region of the near neutrally stable atmospheric surface layer, respectively. An elegant alternative to obtain the envelope of u is via the Hilbert transform S (Mathis et al.2009),andthisapproachhasledtoasignificantadvanceinunderstanding of the large-small scale interactions. However it is harder to relate the results to the governing equations of turbulence. Whencalculatinganr.m.s.definedlocally,(3.6),onehastocontendwiththeinevitabil- ity that large-scales (low frequencies) have been aliased into the small-scale signature. Perhaps a better alternative is to use a tapered window in (3.6), alleviating some but not all of the aliasing. We have tried this and found some minor changes, but the gen- eral picture is unaltered, and so we decided to keep the simple definition in (3.6). The Hilbert transform bypasses this aliasing difficulty at the enveloping stage, but the issue reappears when one filters the envelope signal. We note that even if a perfect decompo- sition can be found, nature herself does not permit it, that is the two peaks in κ E x uu (Hutchins & Marusic 2007b) are never completely isolated, at least in Fourier space. In terms of LES quantities, we can write (3.5) and (3.6) as 1 t+τ/2 ′ ′ u (t)= u(t)dt, (3.7a) L τ Zt−τ/2 1/2 1 t+τ/2 u (t)= u2(t′)+T (t′) dt′ , (3.7b) S τ Zt−τ/2 S xx ! (cid:2) (cid:3) e whereuistheresolvedvelocity;u =u−u ;andT isthemodelledsubgridfluctuations S L xx associated with time scales smaller than the numerical discretization ∆ . Using (3.7), t we now construct the normalised large-scale–small-scale correlation based on temporal 10 D. Chung and B. J. McKeon filtering: h(u −U)(u −hu i)i L S S R (z)= , (3.8) τ h(u −U)2i1/2h(u −hu i)2i1/2 L S S where the global or ensemble averageis formallyegiveneby e e 1 T/2 ′ ′ hφi≡ lim φ(t)dt. (3.9) T→∞T Z−T/2 Note that the visual envelope of u , e.g. 2u , does not affect R because the constant S S τ factorcancelsoutinthenormalisedcorrelation(3.8).Inotherwords,R doesnotcontain τ amplitude information;itdoes,however,containphaseinformation,sinceitisthe cosine e of the angle (or phase) between u −U and u −hu i, using the inner product h i. In L S S practice,weobtainR (z)byreplacingtheintegralswithsumsandensuringtherecording τ period T is much larger than the largest physical time scale in the flow, see table 1. e e The spatial counterpart to (3.8), R (z), is defined analogously, with x and ρ respec- ρ tivelyreplacingtandτ,whileholdingothervariablesconstant.ForR ,theglobalaverage ρ (3.9) is replaced by an average over the wall-parallel plane with area L L . The spatial x y correlations are calculated from runs G1 and H3 at one snapshot in time. The physical meaning of the correlations R = R , R are as follows. If large-scale τ ρ higher-speed regions carry higher small-scale intensity (positively correlated, in phase), thenR≈1.Similarly,iflarge-scalehigher-speedregionscarrylowersmall-scaleintensity (negatively correlated, π out of phase), then R ≈−1. R ≈0 can occur either if there is no correlationbetween the large and small scales,or if they are π/2 out of phase, which is physically the more likely option given the strong correlation for small and large z/δ, as already demonstrated by Bandyopadhyay & Hussain (1984); Mathis et al. (2009). Although the r.m.s.-based correlation coefficient (3.8) is different from its Hilbert- transform-based counterpart in the boundary layer study of Mathis et al. (2009), we expect similar qualitative features if the large-scale–small-scale phase relationship is a universalaspect ofwall-boundedflows,namely channelsand boundarylayers.As we are interestedinlargescaleswithsizesofthe order10δ,weexpectthattheLESwillcapture this statistic satisfactorily since the premise of an LES is to directly simulate the large scales;presently,there are6gridpointsperδ inbothwall-paralleldirectionsand24grid points per δ in the wall-normaldirection. Figure 4 compares the correlations based on temporal filtering, R , and correlations τ based on spatial filtering, R , for Re = 2k and Re = 200k and different values of τ ρ τ τ and ρ. Observe that near the wall, u and u are positively correlated, up to R ≈ 0.4 L S (the maximum in the domain we resolve, although note that the maximum value likely increasesclosertothe wall),butabovea certaincrossingheight, z/δ ≈0.2forRe =2k e τ and z/δ ≈0.11 for Re =200k, they are negatively correlated, down to R≈−0.4. The τ trend of decreasing crossing height with increasing Reynolds number is also reported by Mathis et al. (2009) for the turbulent boundary layer, with z/δ ≈ 0.07 for Re = 2.8k τ andz/δ ≈0.03forRe =19k.Presently,thecorrelationsarelargelyindependentoffilter τ sizes 1.7<τU/δ,ρ/δ <12.3, although the deviation between R and R with the larger ρ τ filtersizesfromtheonewithρ/δ =1.7closetothewallisexacerbatedinthespatialplots, as would be expected from the arguments concerning Taylor’s hypothesis at the large scalesintheprecedingsection.Thekick-upofR relativetoR forcurvescorresponding ρ τ to ρ/δ = 7.0, 12.3 in the vicinity of z/δ = 0.45, see figure 4(a, b) is presumably related to the convectionvelocity effect demonstratedin figure 3(e). A small sensitivity to filter sizes is reported by Mathis et al. (2009). Perhaps a precise quantitative comparison is impossible owing to the different envelope-extraction techniques and the different type