high-reynolds-number eﬀects in turbulent channel ﬂow

Under consideration for publication in J. Fluid Mech. 1

High-Reynolds-number effects in turbulentchannel flow: evidence from DNS

MATTEO BERNARDINI, SERGIO PIROZZOLIand PAOLO ORLANDI

Dipartimento di Ingegneria Meccanica e Aerospaziale, Universita di Roma ‘La Sapienza’Via Eudossiana 18, 00184 Roma, Italy

(Received 20 May 2013)

The high-Reynolds-number behavior of the canonical incompressible turbulent channelflow is investigated through large-scale direct numerical simulation (DNS). A Reynoldsnumber is achieved (Reτ = h/δv ≈ 4000, where h is the channel half-height, and δv isthe viscous length scale) at which theory predicts the onset of phenomena typical of theasymptotic Reynolds number regime, namely a sensible layer with logarithmic variation ofthe mean velocity profile, and Kolmogorov scaling of the velocity spectra. Although higherReynolds numbers can be achieved in experiments, the main advantage of the presentDNS study is full control of the flow statistics, without possible experimental uncertaintiescaused by limited probe resolution. Less than a decade of nearly-logarithmic variationis observed in the mean velocity profiles, with log-law constants k ≈ 0.386, C ≈ 4.30. Alog layer is also observed in the spanwise velocity variance, as predicted by Townsend’sattached eddy hypothesis, whereas the streamwise variance seems to exhibit a plateau,perhaps being still affected by low-Reynolds-number effects. Comparison with previousDNS data at lower Reynolds number suggests strong enhancement of the imprinting effectof outer-layer eddies onto the near-wall region. This mechanisms is shown to be associatedwith excess turbulence kinetic energy production in the outer layer, and it clearly reflectsin flow visualizations and in the streamwise velocity spectra, which exhibit near-tonalbehavior in the outer layer. Associated with the outer energy production site, we findevidence of a Kolmogorov-like inertial range, limited to the spanwise spectral density ofu, whereas power laws with different exponents are found for the other spectra. Finally,arguments are given to explain the ‘odd’ scaling of the streamwise velocity variances,based on the analysis of the kinetic energy production term.

1. Introduction

Wall-bounded flows at high Reynolds number are a topic of obvious importance inengineering applications, but also have a great intrinsic academic interest. Early stud-ies showed that the structure of wall turbulence is relatively universal when representedin wall units (namely, the friction velocity uτ =

√

τw/ρ, and the viscous length-scaleδv = ν/uτ ). However, it is a relatively new finding that, as the Reynolds number in-creases (in terms of, e.g. the friction Reynolds number, Reτ = δ/δv), significant devia-tions from the universal structure emerge. One of the most evident effects is the variationof the near-wall peak of the streamwise turbulence intensity, whose amplitude is foundto grow logarithmically with the friction Reynolds number (DeGraaff & Eaton 2000).This behavior has been linked to the influence of outer-scaled eddies, which mainly con-tain wall-parallel motions (Kim & Adrian 1999; Guala et al. 2006; Hutchins & Marusic

2 M. Bernardini, S. Pirozzoli, P. Orlandi

2007), and which correspond to the inactive eddies postulated by Townsend (Townsend1961). The imprinting effects of the outer-layer eddies become more evident as Reτ isincreased, eventually leading to significant violation of the pure wall scaling. The interestof the community for these newly discovered (or re-discovered) physical effects, collec-tively referred to as high-Reynolds-number effects, is testified by the large number ofrecent dedicated reviews (Marusic et al. 2010; Smits et al. 2011; Jimenez 2012).Is is a common notion that, to be able to probe high-Reynolds-number effects, friction

Reynolds numbers of the order of a few thousands should be explored, the consensusthreshold to observe an order-of-magnitude logarithmic variation of the mean velocitybeing Reτ ≈ 4000 (Jimenez & Moser 2007; Hutchins & Marusic 2007). Studying wall-bounded turbulent flows (boundary layers, channels, pipes) in the high-Reynolds-numberregime is then quite a challenging task in experiments, since, as the Reynolds number isincreased, outer and inner scales become very disparate, thus placing stringent require-ments for accurate measurement of the velocity signal (Hultmark et al. 2010). Further,information on the velocity spectra is typically limited to the streamwise direction (uponuse of Taylor’s hypothesis), whereas (as we will show later) the spanwise spectra betterreflect the change of flow dynamics which occurs at high Reτ .Numerical simulations offer the advantage of closer control on resolution effects, thus

ruling out some of the uncertainties incurred in experiments. Given the proportionalitybetween the viscous length scale and the Kolmogorov scale (Pope 2000), it is generallyaccepted that all the energetically relevant scales of motion are resolved provided thespacings in the wall-parallel directions are ∆x+ ≈ 10, ∆z+ ≈ 5 (the superscript +is hereafter used to denote quantities made nondimensional with uτ and δv), and, tobe able to resolve the viscous sublayer, the first point off the wall must be placed at∆y+ . 1. This implies that the total number of points in well-resolved boundary layersimulations must grow roughly as Re3τ . This restriction makes the numerical simulation ofwall-bounded flows at high Reynolds number (in the operational sense given above) quitechallenging. To date, the highest Reynolds number attained in incompressible boundarylayer DNS is Reτ ≈ 2000 (Sillero et al. 2011), for channel flows Reτ ≈ 2000 (Hoyas &Jimenez 2006), and for pipe flows Reτ ≈ 1100 (Wu & Moin 2008). In a recent work bythe present authors (Pirozzoli & Bernardini 2013), for the first time a Reynolds numberReτ ≈ 4000 was achieved in a compressible boundary layer DNS, at which typical high-Reynolds-number effects start to manifest themselves.

In this paper we present novel data from incompressible channel flow DNS, whichextends the Reynolds number envelope to Reτ ≈ 4000, thus meeting the constraints forthe flow to be considered genuinely high-Reynolds. The numerical methodology used forthe purpose is explained in §2, and the flow statistics are presented in §3, together witha discussion of the results. Final comments are given in §4.

2. Numerical setup

We solve the Navier-Stokes equations for a divergence-free velocity field, by enforcinga time-varying pressure gradient to maintain a constant flow rate. The equations arediscretized in a Cartesian coordinate system (x, y, z denote the streamwise, wall-normaland spanwise directions) using staggered central second-order finite-difference approxi-mations, which guarantee that kinetic energy is globally conserved in the limit of inviscidflow. Time advancement is achieved by a hybrid third-order low-storage Runge-Kutta al-gorithm (Bernardini & Pirozzoli 2009) coupled with the second-order Crank-Nicolsonscheme, combined in the fractional-step procedure, whereby the convective and diffusiveterms are treated explicitly and implicitly, respectively. The Poisson equation for the

3

Flow case Line style Reb Reτ Nx Ny Nz

CH1 Dashed 20063 546 1152 192 576CH2 Dash-dot 39600 999 2048 384 1024CH3 Dash-dot-dot 87067 2012 4096 768 2048CH4 Solid 191333 4080 8192 1024 4096

Table 1: List of parameters for turbulent channel flow cases. Reb = hub/ν is the bulkReynolds number, and Reτ = huτ/ν is the friction Reynolds number. Nx, Ny, Nz arethe number of grid points in the streamwise, wall-normal, and spanwise directions.

pressure field, stemming from the incompressibility condition, is efficiently solved usinga direct solver based on Fourier transform methods, as described by Kim & Moin (1985).A full description of the numerical method is provided in Orlandi (2000).

It is worth noting that the computations are performed in a convective reference framefor which the bulk velocity is zero, i.e. in which the net streamwise mass flux is zero.In addition to allowing a larger computational time-step, this expedient minimizes thedispersion errors associated with the finite-difference discretization, leading to nearlyidentical results as obtained with spectral methods (Bernardini et al. 2013).

The DNS have been carried out in a (Lx×Ly×Lz) = (6πh×2h×2πh) computationalbox (h is the channel half-height), which is large enough to accommodate the largestouter-layer flow structures (Flores & Jimenez 2010), and the mesh spacing in the wall-parallel directions has been kept (nearly) constant in wall units for all simulations, namely∆x+ ≈ 9, ∆z+ ≈ 6. A cosine stretching function has been used to cluster points in thewall-normal direction. Information on the flow parameters for the DNS are reported intable 1. The simulations have been initiated with a laminar parabolic Poiseuille velocityprofile, with maximum velocity up at the centerline, and bulk velocity ub = 2/3up. Thecalculations (labeled as CH1-4) have then been time-advanced by forcing constant massflux through the channel, in such a way that ub does not vary with time. After an initialtransient the pressure gradient starts to fluctuate about a nearly constant value. Afterthe end of this transient, flow statistics are collected by averaging in the wall-paralleldirections and in time. Comparison of the flow statistics (including the velocity spectra)with the data of del Alamo & Jimenez (2003) up to Reτ = 2000, gave nearly identicalresults.

3. Results

The mean streamwise velocity profiles for the CH1-4 simulations are shown in Fig. 1,together with recent experimental data (Schultz & Flack 2013), at Reynolds numbersvery close to those of the CH2-4 simulations. Figure 1 highlights the onset of a layerwith nearly logarithmic velocity variation, whose extent visually increases with Reτ , withexcellent agreement with experiments. More quantitative information can be gained frominspection of the log-law diagnostic function

Ξ = y+ du+/dy+, (3.1)


(a)

100 101 102 103 1040

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25

30

y+

u+

(b)

100 101 102 103 1042

2.2

2.4

2.6

2.8

3

3.2

3.4

y+

y+du+/dy+

Figure 1: Distribution of (a) mean velocity profile and (b) log-law diagnostic function,as defined in equation (3.1). Symbols indicate experimental data from Schultz & Flack(2013), at Reτ = 1010 (squares), Reτ = 1956 (diamonds), Reτ = 4048 (circles). In panel(b) the thick grey lines correspond to the defect-layer profiles defined in equation (3.3), forReτ corresponding to the CH2-4 DNS, and the horizontal red line denotes the reference1/k log-law value (k = 0.386). See table 1 for nomenclature of DNS data.

whose constancy would imply the presence of a logarithmic layer in the mean velocityprofile. For reference, in figure 1(b) we also show the trends of the defect-layer velocityprofile predicted by Townsend (1976, p. 147), under the assumption of uniform eddyviscosity

u+

CL − u+ =1

2Rs(1− y/h)2, (3.2)

which implies

Ξ = Rsy/h(1− y/h), (3.3)

where uCL is the mean velocity at the channel centerline, and Rs is a constant to bedetermined empirically. Specifically, the curves drawn in figure 1 were obtained using

5

104 105 106

0.003

0.004

0.005

0.006

0.007

Reb

Cf

Figure 2: Comparison of skin friction coefficient with correlations and experiments.Square symbols denote DNS from CH1-4 datasets, and circles experimental data (Schultz& Flack 2013). The solid line indicates the friction law (3.4) with k = 0.386, C = 4.30;the dashed line indicates the friction law (3.4) with k = 0.37, C = 3.7; the dot-dashedline indicates Deans’ friction law (equation (3.5)).

Rs = 12.7. The figure clearly demonstrates universality of the mean velocity in inner unitsup to y+ ≈ 50, and it also supports convergence to the outer-layer scaling given by (3.3)for y/h & 0.25. Between those two extreme regions the distribution varies significantlywith the Reynolds number. While for Reτ . 2000 the diagnostic function connects theinner-layer minimum with the outer-layer maximum monotonically, at Reτ ≈ 4000 aclear plateau forms in the range 300 6 y+ 6 700, which is the hint of the formation ofa log layer in u+. Based on the CH4 data it is then possible to estimate the value ofthe von Karman constant, since in the presence of a logarithmic velocity distribution,u+ = 1/k log y++C, one has Ξ = 1/k. The value that we obtain, k ≈ 0.386, is not too far,but still sensibly different than the value k = 0.37 quoted by Nagib & Chauhan (2008) tobe appropriate for channel flow. However, it is very close to the value (k = 0.39) recentlyclaimed to be ‘universal’ for boundary layers and pipe flows by Marusic et al. (2013).It is noteworthy that the experimental data highlight significant scatter in this type ofrepresentation, and estimates of the von Karman constant are thus inevitably affected bysome form of uncertainty, and also to some extent by flow three-dimensionality (Monty2005). Of course, at this stage we must rely on the results of a single DNS exhibitinggenuine logarithmic behavior, but certainly DNS at higher Reynolds number are mostwelcome to confirm or refute our finding. The value of the additive constant in the loglaw was also estimated by fitting the velocity distribution from the CH4 flow case inthe range 300 6 y+ 6 700, with the result C = 4.30, again sensibly different than thepreviously quoted value of C = 4.17 (Nagib & Chauhan 2008). We must stress that thesedifferences are not appreciable in the classical inner semi-logarithmic representation. Toverify possible alternatives to the logarithmic law, we have also considered the diagnosticfunction for power-law behavior, namely y+/u+(du+/dy+), whose constancy would implypower-law variation of the mean velocity. The resulting distributions, not shown, haveno significant plateau, which leads us to believe that the logarithmic law is more robustthan the power law, in agreement with most current literature on the subject (Marusicet al. 2010).


Strictly related to the behavior of the mean velocity profile is the friction law. Earlystudies proposed approximations based on either log-law representations of the wholemean velocity profile, such as the classical Prandtl’s smooth flow formula (Durand 1935)

√

2

Cf=

1

klog

(

Reb

2

√

Cf

2

)

+ C −1

k, (3.4)

and power-law representations, such as Dean’s (Dean 1978)

Cf = 0.073Re−0.25b , (3.5)

where Cf = 2τw/(ρu2b), Reb = hub/ν, and ub =

∫ 2h

0u dy/(2h) is the bulk velocity.

In figure 2 we show the DNS data together with experimental data from Schultz &Flack (2013), compared with theoretical predictions. The DNS data are in overall goodagreement with experiments, albeit with systematic deviations of about 1%, which arehowever within the experimental uncertainty (Schultz & Flack 2013). It is also clear thatDean’s power-law prediction rapidly loses accuracy at high Reynolds number, and once asensible log layer is formed, Prandtl’s formula is clearly superior. We have tested Prandtl’sformula (as given in equation (3.4), with two different sets of log-law coefficients, thosesuggested by Nagib & Chauhan (2008), and those derived from fitting the present DNSdata. While the discrepancies are minor at low Reynolds number (Reb . 2 · 105), whereNagib’s set of coefficients seems to be in closer agreement with the available data, thereversal behavior is found at higher Reb, although no clear preference can be given toany set of constants.

The second-order velocity fluctuations statistics are shown in inner coordinates in fig-ure 3. The main impression gained from the figure is that the trends observed at lowerReynolds number (Hoyas & Jimenez 2006) continue to hold. Specifically, the longitudinalstress (a) shows evident lack of universality in inner scaling, with fluctuation amplitudeswhich increase logarithmically with Reτ (see figure inset). This behavior has been at-tributed (DeGraaff & Eaton 2000) to the increasing influence of inactive outer layermodes on the near-wall dynamics. At this Reynolds number, no clear range with log-arithmic variation of the streamwise velocity variance is observed, as predicted by theattached eddy hypothesis (Townsend 1961; Perry & Li 1990), and for which experimentsin boundary layers and pipes at much higher Reynolds number suggest (Marusic et al.

2013)

u′2/u2τ ≈ B1 −A1 log(y/h), (3.6)

with A1 ≈ 1.26, which we have tentatively sketched in the figure. The spanwise velocityfluctuations (b) have a similar behavior, with substantial dependence of the inner-scaledintensities on Reτ . In this case, the formation of an extended logarithmic layer is muchmore evident, and we find evidence for a universal scaling

w′2/u2τ ≈ B3 −A3 log(y/h), (3.7)

with A3 ≈ 0.456, B3 ≈ 0.886. The vertical velocity fluctuations (c) exhibit universalityin inner scaling in a wider part of the near-wall region, but their peak still grows withReτ , even though the growth is sub-logarithmic, and perhaps v′2 tends to saturate asu′v′ approaches the unit value (d). Furthermore, no evident plateau of v′2 has formed atthis Reynolds number, as requested by the attached-eddy hypothesis. Overall, it shouldbe noted that the agreement of the observed trends with experiments is excellent, exceptperhaps for the peak of the wall-normal velocity, which is by the way notoriously difficultto measure.

7

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102 103 1047

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9

10

y+

u′2/u2 τ

Reτ

(b)

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3

102 103 1041.5

2

2.5

3

y+

Reτ

w′2/u2 τ

(c)

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102 103 1041

1.1

1.2

1.3

y+

Reτ

v′2/u2 τ

(d)

100 101 102 103 1040

0.2

0.4

0.6

0.8

1

y+

u′v′/u2 τ

Figure 3: Distribution of Reynolds stress components across the channel. Symbols in-dicate experimental data from Schultz & Flack (2013), at Reτ = 1010 (squares),Reτ = 1956 (diamonds), Reτ = 4048 (circles). The dashed diagonal line in (a) denotesthe distribution given in equation (3.6) for the CH4 flow case, and the dashed diagonallines in (b) denote the distribution given in equation (3.6), with A3 ≈ 0.456, B3 ≈ 0.886.The insets in panels (a-c) highlight the trend of the velocity variance peaks with Reτ .Refer to table 1 for nomenclature of the DNS data.

Velocity variances are shown in outer scaling (uτ , h) in figure 4 to highlight outer-layer trends. Clear collapse of the outer-scaled distributions is observed for the spanwisevelocity fluctuations, whereas violation of the uτ scaling is observed for the streamwisevelocity fluctuations. This discrepancy was first noticed by DeGraaff & Eaton (2000),who attributed it to the effect of large-scale motions scaling with the velocity at theedge of the wall layer (in this case, the mean centerline velocity). Later (del Alamoet al. 2004), it became clear that the increase of u′ at fixed y/h is associated withh-scaled outer-layer modes which are superposed onto the wall-attached modes, whichscale on uτ . In any case, consistent with the findings of DeGraaff & Eaton (2000), it isfound that scaling u′ by uMIX = (uτuCL)

1/2 effectively removes this dependency, andcollapse of the streamwise velocity is observed for y/h & 0.2. Further explanation for theobserved scaling are provided later on. More ambiguous is the behavior of v′. Inspectionof figure 4(c-d) shows than none of the two velocity scalings is able to remove the Reτdependency. However, as also found for the inner layer, the uτ scaling seems to yield aflat asymptotic behavior at the highest Reτ available, which leads us to suspect that lackof universality is but a low-Reynolds-number effect.

A visual impression of the effect of outer-scaled eddies on the near-wall layer can begained from Fig. 5, where we show velocity fluctuation contours in wall-parallel planesat y+ ≈ 15, which corresponds to the position of the peak of streamwise velocity fluctu-ations, and in the outer layer, at y/h = 0.3. A two-scale organization is clearly visible at


(a)

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10u′2/u2 τ

(b)

0.2 0.4 0.6 0.8 10

0.05

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0.25

0.3

0.35

0.4

u′2/(u

τuCL)

(c)

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0.5

1

1.5

v′2/u2 τ

(d)

0 0.2 0.4 0.6 0.8 10

0.01

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v′2/(u

τuCL)

(e)

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0.5

1

1.5

2

2.5

3

y/h

w′2/u2 τ

(f)

0 0.2 0.4 0.6 0.8 10

0.04

0.08

0.12

y/h

w′2/(u

τuCL)

Figure 4: Distribution of Reynolds stress components in outer scaling (a,c,e) and mixedscaling (b,d,f). See table 1 for nomenclature of the DNS data.

y+ = 15, with high- and low-momentum streaks having size of the order of 100 wall units,which are part of the inner-layer turbulence sustainment cycle, and superposed on themmuch larger streaks, having size of the order of h. While this effect was previously noticedin DNS at lower Reynolds number (Hutchins & Marusic 2007), the scale separation andimprinting effects here are much more prominent. The flow organization in cross-streamplanes is monitored in figure 6, where we show the contours of velocity fluctuations. Itappears that the core flow is mainly organized into ’towering’ eddies which are attachedto the wall, and which occupy the entire half-channel in which they are generated. Signif-icant correlation of u′ and v′ events is observed (negative in the lower half of the channel,positive in the upper half), which points to a non-negligible contribution of these eddiesto the Reynolds stress far from the walls. Less clear is the organization of w′. In theclassical scenario (del Alamo & Jimenez 2006), outer-layer streaks (i.e. u′ events) areassociated with rollers, hence spanwise velocity fluctuations should have a quadrupolardistribution around a positive/negative u′ event. However, this pattern is not easy todiscern in the flow visualizations.

Spectral densities of the velocity fluctuations are shown in figure 7 for the inner layer,and in figure 8 for the outer layer. Note that wavelengths (λi = 2π/ki) are shown on the

9

(a)

(b)

Figure 5: Instantaneous streamwise velocity field in x− z planes at y+ = 15 (a), y/h =0.3, for the CH4 dataset. Contour levels are shown for −2.3 6 u′/uτ 6 2.3 (a), and−1.56 6 u′/ub 6 1.56 (b), from dark to light shades. The figure inset in (a) shows azoom of a (1500+ × 1000+) box, to highlight viscous length scales.

horizontal axis, rather than the corresponding wavenumbers, to more clearly highlightthe eddies length scales. Further, the spectra are pre-multiplied by the wavenumber, insuch a way that equal areas correspond to equal energies in a logarithmic plot. Con-sistent with expectations, inner scaling yields collapse of the spectra at the smallestresolved scales of motion across the range of Reynolds numbers, whereas outer scalingyields (approximate) collapse of the large scales. The inner-layer spectra clearly showthe presence of an energetic inner site corresponding to the near-wall streak regenerationcycle. The typical length scales in the spanwise direction remain very roughly universal,with λu

z ≈ λwz ≈ 100δv, λ

vz ≈ 50δv, which in the classical interpretation (Hamilton et al.

1995), correspond to the near-wall streaks/streamwise vortices system. Additional en-ergy appears at large wavelengths in the u and w spectra as Reτ increases, as a result ofouter-layer imprinting processes. The u spectra probably exhibit the onset a k−1

z spec-tral range (which amounts to a plateau of the pre-multiplied spectra) at increasing Reτ ,whose presence can be deduced from dimensional arguments as one of the consequencesof the attached eddy hypothesis (Nickels et al. 2005). The spanwise outer-layer spectra(figure 8) exhibit a qualitatively similar organization as the inner-layer spectra, but ona different scale. Specifically, the typical length scales of the eddies are λu

z ≈ λwz ≈ h,

λvz ≈ h/2, which suggests that the large-scale organization of the flow still consists of

streaks coupled with rollers. It is noteworthy that, while the spanwise spectra of v andw are relatively broad-banded, and tend to be weakly affected by Reτ , those of u exhibit


(a)

(b)

(c)

Figure 6: Instantaneous cross-stream visualizations of u′ (a), v′ (b), w′ (c) for the CH4dataset. Contour levels below −0.5uτ are shown in black, and contour levels above 0.5uτ

are shown in grey.

a sharp peak, and apparently a tonal spectrum arises as Reτ increases. This feature isdifficult to observe in experiments, which typically can only access streamwise spectraupon use of Taylor’s hypothesis, and it is likely caused by the activation of the outer-layermodes through a transient growth mechanism (Hutchins & Marusic 2007; del Alamo &Jimenez 2006). This observation is further corroborated from the uv co-spectra, shown infigure 8(d), which can be interpreted as spectra of turbulence kinetic energy production(at a given off-wall distance), and which exhibit the same spikes as the u spectra.

Clearer control on the structural flow changes with Reτ is obtained from figure 9,where we show the distribution of the production-to-dissipation ratio, P/ε (where P =−(du/dy)u′v′, and ε = 2νs′ijs

′

ij), and of the effective Taylor-scale Reynolds number, de-

fined as Reλ = λq/ωrms, with λ = q/ωrms (where q2 = u′

iu′

i is the velocity fluctuation

11

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0.6kzE

uu/u2 τ

(b)

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0.005

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0.015

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0.04

kzE

vv/u2 τ

(c)

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kzE

ww/u2 τ

λ+z

(d)

101 102 103 104 1050

0.02

0.04

0.06

0.08

λ+z

kzE

uv/u2 τ

Figure 7: Pre-multiplied spanwise spectral densities of velocity fluctuations at y+ = 15.(a) u-spectra; (b) v-spectra; (c) w-spectra (d) uv co-spectra.

(a)

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kzE

uu/u2 τ

(b)

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kzE

vv/u2 τ

(c)

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λz/h

kzE

ww/u2 τ

(d)

10-2 10-1 100 1010

0.02

0.04

0.06

0.08

0.1

λz/h

kzE

uv/u2 τ

Figure 8: Pre-multiplied spanwise spectral densities of velocity fluctuations at y/h = 0.3.(a) u-spectra; (b) v-spectra; (c) w-spectra (d) uv co-spectra.

variance, and ωrms is the root-mean-square vorticity). The observed insensitivity of P/εon Reτ further supports the universality of the inner-layer dynamics. However, a slow(but steady) tendency for P/ε to increase with Reτ is observed in the outer layer, whereit attains a plateau. Eventually, production exceeds dissipation up to y/h ≈ 0.5, by upto 10% in the CH4 simulation. This non-marginal effect was first speculated by Brad-


(a)

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1.5

2

y+

P/ε

(b)

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20

40

60

80

100

120

140

160

y+

Reλ

Figure 9: Distribution of the production/dissipation ratio (a), and of the Taylor-scaleReynolds number (b).

shaw (1967), and it implies that the excess turbulence kinetic energy is transferred tothe underlying layers through turbulent diffusion (as we have verified), hence initiatingan energy leakage process. At the same time, the Taylor Reynolds number (figure 9b)increases in the outer layer, attaining values which make it possible the formation ofinertial ranges in the velocity spectra. The presence of power-law ranges in the velocityspectra is verified in figure 10, where data are shown in the Kolmogorov representation, aty/h = 0.3, which roughly corresponds to the peak value of Reλ. The spectra are reportedin compensated form, in such a way that a plateau corresponds to a power-law range,and compared with forced isotropic turbulence data at Reλ = 142 (Jimenez et al. 1993),to verify the accuracy in the prediction of the small scales of turbulence. The streamwisespectrum of u highlights the presence of a decade with k−1.46 variation, which is signifi-cantly less steep than Kolmogorov’s k−5/3 inertial spectrum. Similar deviations from anideal behavior were observed in experiments of grid turbulence (Mydlarski & Warhaft1996), and homogeneously sheared flow (Ferchichi & Tavoularis 2000), and attributed tolow-Reynolds number effects. In particular, Mydlarski & Warhaft (1996) proposed a fitfor the scaling exponent of the longitudinal spectra,

E(k) ∼ k5/3−p, p = 5.25Re−2/3λ . (3.8)

For Reλ ≈ 140 (which corresponds to the outer-layer value of the CH4 DNS), the re-sulting exponent is comparable with our observations. Similar conclusions also hold forthe transverse spectra (v and w), which exhibit power-law behavior with yet lower slopethan the longitudinal ones. On the other hand, the spanwise spectra of u show the forma-tion of a narrow k−5/3 spectral band (limited to the CH4 case) between the large-scalepeak and the dissipative range, whereas power-law ranges with exponents different than−5/3 are observed for v and w. This behavior can be tentatively ascribed to the factthat energy is mainly pumped into the streamwise velocity mode through a tonal forcingacting in the spanwise direction, and which is reflected in the sharp spectral peak atλz ≈ h. Such forcing is not observed either in the other two velocity components, norin any streamwise spectra. Since Kolmogorov’s theory is based on the assumption thatenergy if fed into the system only at the largest scales of motion, and on the assumptionof scale separation with the dissipative scales, it is then natural that Kolmogorov spectraare more easily observed in the spanwise spectra of u, where production is concentrated.Again, we point out that this type of analysis is difficult in experiments, and as a matterof fact, clear Kolmogorov spectra were not observed in longitudinal spectra, even at muchhigher Reynolds number (Saddoughi & Veeravalli 1994; McKeon & Morrison 2007). Inthis sense, the use of DNS at high Reynolds number can provide significant contributions.

13

(a)

10-3 10-2 10-1 100 1010

0.5

1

1.5

2

kxη

(kxη)1

.46E

uu(εν5)1

/4

(b)

10-3 10-2 10-1 100 1010

0.5

1

1.5

2

kzη

(kzη)1

.66E

uu(εν5)1

/4

(c)

10-3 10-2 10-1 100 1010

1

2

3

4

5

kxη

(kxη)1

.12E

vv(εν5)1

/4

(d)

10-3 10-2 10-1 100 1010

1

2

3

4

5

6

kzη

(kzη)1

.24E

vv(εν5)1

/4

(e)

10-3 10-2 10-1 100 1010

1

2

3

4

5

6

kxη

(kxη)1

.12E

ww(εν5)1

/4

(f)

10-3 10-2 10-1 100 1010

0.2

0.4

0.6

0.8

1

kzη

(kzη)1

.58E

ww(εν5)1

/4

Figure 10: Compensated spectral densities of velocity fluctuations at y/h = 0.3 in thestreamwise direction (left column) and in the spanwise direction (right column). Theexponent for compensation (p) was determined for each case as the value for whicha wider plateau was observed for the CH4 flow case. The symbols indicate isotropicturbulence data at Reλ = 142 (Jimenez et al. 1993). See table 1 for nomenclature ofDNS data.

Further insight into the mechanisms of turbulence kinetic energy production can begained by considering the following decomposition (Orlandi 1997)

P = −u′v′du

dy= u

(w′ω′

y − v′ω′

z

)

︸︷︷︸

PR

−d

dy

(uu′v′

)

︸︷︷︸

PC

, (3.9)

whereby the production term is split into a term (PR) containing the streamwise compo-nent of the fluctuating Lamb vector (p = −ω

′×u′), where ω′ = ∇×u′, and which is anindicator of the rate of energy transfer to small scales (Rogers & Moin 1987), and a termin divergence form (PC), which thus represents redistribution of kinetic energy across thechannel. The distributions of the budget terms of P are shown in figure 11 in inner andouter scaling. For that purpose, we note that the obvious inner normalization for P is


(a)

10-1 100 101 102 103

-0.4

-0.2

0

0.2

0.4

0.6

0.8

y+

P+

P

PC

PR

(b)

0 0.2 0.4 0.6 0.8 1-40

-20

0

20

40

P

PC

PR

y/h

Ph/u3 τ

(c)

0 0.2 0.4 0.6 0.8 1

-1

0

1

P

PC

PR

y/h

Ph/(u

2 τuCL)

Figure 11: Contributions to kinetic energy production in inner units (a), and in outercoordinates, with standard outer normalization (b), and with ‘mixed’ normalization (c).

by u3τ/δv. On the other hand, the natural outer normalization is by u3

τ/h, assuming thatuτ is the only legitimate outer-layer velocity scale. The near-wall distributions, shown infigure 11(a), highlight close universality in wall units, and confirm the pattern found inpipe flow (Orlandi 1997), with accumulation of kinetic energy production associated withPC , which is partly balanced by the energy flux from the large to the small scales. Theopposite scenario holds away from the wall, where PR is positive, and PC is negative, thecrossover occurring near the peak position of the turbulent shear stress. However, in thiscase the natural outer scaling is not capable of collapsing the distributions across theReτ range. We then consider an alternative scaling, which should hold for PC , PR andP . We focus on PR, under the following assumptions: i) v′ ∼ w′ ∼ uτ , as confirmed fromthe analysis of the velocity variances; ii) ω′

y ∼ ω′

z ∼ uτ/δv, which is the widely acceptedscaling for the vorticity fluctuations; iii) u ∼ uCL in the outer layer. We then conclude

P ∼ PC ∼ PR ∼ u2τuCL/h. (3.10)

The scaling 3.10 is tested in figure 11(c), and shown to be much more accurate that

15

the standard outer scaling for y/h & 0.2. A consequence of this finding is that thestreamwise velocity fluctuations should scale as u′2 ∼ P/τ , where τ = h/uτ is the typicaleddy turnover time in the outer layer (Simens et al. 2009). Hence, it follows that

u′2 ∼ uτuCL, (3.11)

which is precisely the mixed scaling proposed by DeGraaff & Eaton (2000).

4. Conclusions

Flow statistics from DNS at computationally high Reynolds number have been col-lected and commented. At the Reynolds number of the largest simulation (Reτ ≈ 4000)some effects which are believed to be typical of the asymptotic high-Re regime startto manifest themselves. Specifically, for the first time in DNS we observe a range withlogarithmic behavior of the mean velocity, as indicated by a plateau in the log-law indica-tor function, and which smoothly connects the (quasi)-universal inner and core regions.Exploration of the high-Re regime also allows to verify other theoretical predictions, in-cluding the logarithmic decrement of the wall-parallel velocity variances with the walldistance (Townsend 1976). While clear and extended logarithmic ranges are observed forw′, the same does not hold for u′, which rather tends to form a plateau in the overlaplayer. This finding is perhaps due to the need for yet higher Reynolds numbers. It isconfirmed that, while v′, w′, and u′v′ scale with the friction velocity, as in the standardtheory (except for low-Re effects), u′ exhibits a mixed scaling, with (uτuCL)

1/2 (De-Graaff & Eaton 2000). Justifications for the observed scaling are proposed, based ondimensional analysis applied to the kinetic energy production term. A limited spectralrange with k−5/3 Kolmogorov scaling is only observed for the outer-layer spanwise spectraof u′, whereas all other longitudinal and transverse spectra exhibit power-law behaviorwith less steep slope. This behavior has been related to the presence of a tonal spectralforcing, associated with O(h) outer-layer modes.

Flow statistics are available at the web page http://reynolds.dma.uniroma1.it/channel/,with supporting documentation.

We acknowledge that some of the results reported in this paper have been achieved us-ing the PRACE Research Infrastructure resource JUGENE based at the Forschungszen-trum Julich (FZJ) in Julich, Germany.

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(a)

(b)

(c)

(d)

(e)

Figure 12: Instantaneous cross-stream visualizations of u′ at Reτ = 180 (a), 550 (b), 1000(c), 2000 (d), 4000 (e). Contour levels below −0.5uτ are shown in black, and contour levelsabove 0.5uτ are shown in grey.

19

(a)

(b)

(c)

(d)

(e)

Figure 13: Instantaneous cross-stream visualizations of v′ at Reτ = 180 (a), 550 (b), 1000(c), 2000 (d), 4000 (e). Contour levels below −0.5uτ are shown in black, and contour levelsabove 0.5uτ are shown in grey.


(a)

(b)

(c)

(d)

(e)

Figure 14: Instantaneous cross-stream visualizations of w′ at Reτ = 180 (a), 550 (b), 1000(c), 2000 (d), 4000 (e). Contour levels below −0.5uτ are shown in black, and contour levelsabove 0.5uτ are shown in grey.

21

(a)

(b)

(c)

(d)

(e)

Figure 15: Instantaneous side view of u′ at Reτ = 180 (a), 550 (b), 1000 (c), 2000 (d),4000 (e). Contour levels below −0.5uτ are shown in black, and contour levels above 0.5uτ

are shown in grey.


(a)

(b)

(c)

(d)

(e)

Figure 16: Instantaneous side view of v′ at Reτ = 180 (a), 550 (b), 1000 (c), 2000 (d),4000 (e). Contour levels below −0.5uτ are shown in black, and contour levels above 0.5uτ

are shown in grey.

23

(a)

(b)

(c)

(d)

(e)

Figure 17: Instantaneous side view of w′ at Reτ = 180 (a), 550 (b), 1000 (c), 2000 (d),4000 (e). Contour levels below −0.5uτ are shown in black, and contour levels above 0.5uτ

are shown in grey.


(a)

(b)

(c)

(d)

(e)

Figure 18: Instantaneous visualizations of u′ in a wall-parallel plane (y/h = 0.3) at Reτ =180 (a), 550 (b), 1000 (c), 2000 (d), 4000 (e). Contour levels below −0.5uτ are shown inblack, and contour levels above 0.5uτ are shown in grey.

25

(a)

(b)

(c)

(d)

(e)

Figure 19: Instantaneous visualizations of v′ in a wall-parallel plane (y/h = 0.3) at Reτ =180 (a), 550 (b), 1000 (c), 2000 (d), 4000 (e). Contour levels below −0.5uτ are shown inblack, and contour levels above 0.5uτ are shown in grey.


(a)

(b)

(c)

(d)

(e)

Figure 20: Instantaneous visualizations of w′ in a wall-parallel plane (y/h = 0.3) at Reτ =180 (a), 550 (b), 1000 (c), 2000 (d), 4000 (e). Contour levels below −0.5uτ are shown inblack, and contour levels above 0.5uτ are shown in grey.

high-reynolds-number eﬀects in turbulent channel ﬂow

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