A methodology to evaluate statistical errors in DNS ...

Computers and Fluids 130 (2016) 1–7

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A methodology to evaluate statistical errors in DNS data of plane channel flows Roney L. Thompson a,∗, Luiz Eduardo B. Sampaio b, Felipe A.V. de Bragança Alves a, Laurent Thais c, Gilmar Mompean c a b c

LMTA, Department of Mechanical Engineering (PGMEC), Universidade Federal Fluminense, Rua Passo da Patria 156, 24210-240 Niteroi, RJ, Brasil Institute for Computational and Mathematical Engineering, Stanford University, CA, USA Université Lille Nord de France, Lille I, Polytech’Lille, LML, CNRS UMR 8107, Cité Scientifique, F-59655 Villeneuve d’Ascq, France

a r t i c l e

i n f o

Article history: Received 5 September 2015 Revised 18 January 2016 Accepted 22 January 2016 Available online 19 February 2016 Keywords: Direct numerical simulation Turbulence Statistic errors Plane channel flow RANS modeling

a b s t r a c t Direct numerical simulations (DNS) provide useful information for the understanding and the modeling of turbulence phenomena. In particular, new methodologies recently allowed the achievement of high Reynolds number in DNS of the benchmark plane channel flow. In this scenario, estimating the statistical errors associated with DNS is a difficult but necessary task. Here, we present a methodology to evaluate the statistical errors of the second-moment DNS data. In this methodology, the momentum balance equation is used to calculate the mean velocity profile by considering the Reynolds stress tensor provided by DNS. This error evaluation was applied to different plane channel flow databases available in the literature. We show that using the Reynolds stress statistics obtained from standard DNS can lead to significant discrepancies for turbulence modeling. One interesting consequence of this approach is that we are able to compute the Reynolds shear stress from the converged first order statistic. This information can be used, for instance, to extract a more accurate turbulent viscosity for turbulence modeling purposes. Moreover, the new methodology seems to be a promising path to formulate a convergence criterion for future plane channel DNS. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Since the pioneering work of Kim et al. [12], direct numerical simulation of the Navier–Stokes equations applied to plane channel flows has been an intensive topic of research. The statistics obtained from the DNS constitute an important source for understanding sheared turbulence. In these simulations, the physics of the flow is captured without any turbulence model. Hence, theoretical predictions can be confirmed or invalidated, and it is even not uncommon to use DNS data as reference to check the accuracy of experimental data (see Schultz and Flack [23]). Obtaining experimental data able to provide the usual statistics necessary to describe the flow is a hard task, especially near the wall, where the probe length interferes on the measurements. This fact imposes to DNS the responsibility to provide reliable data (Moin and Mahesh [19]). In this context, evolving towards higher and higher Reynolds number is a necessity and stimulates different groups to move further in this direction. Recently, Bernardini et al. [4], Lee and Moser

∗

Corresponding author. Tel.: +55 21 26295576; fax: +55 21 26295588. E-mail address: [email protected], [email protected] (R.L. Thompson).

http://dx.doi.org/10.1016/j.compfluid.2016.01.014 0045-7930/© 2016 Elsevier Ltd. All rights reserved.

[13], Lozano-Durán and Jiménez [14] were able to exceed a friction Reynolds number Reτ = 40 0 0 in DNS of the Newtonian plane channel flow. They discussed theoretical predictions and specific findings associated with Reynolds numbers beyond that limit. Direct numerical simulations are frequently employed as a reference for the error assessment of other numerical approaches like Large Eddy Simulations (LES) and Reynolds Average Navier–Stokes (RANS) and even to evaluate alternative numerical strategies in DNS (Bauer et al. [3]). In LES, different error measurements can be evaluated (Meyers et al. [18]) and one can decouple the error effect in numerical and subgrid modeling contributions (Meyers et al. [17]). DNS also provides a mean for calculating the terms of the Reynolds transport equation. There are terms that need to be modeled but are difficult to be measured experimentally, like the pressure-strain correlation tensor (Moin and Mahesh [19]). An useful application is to employ DNS statistics to evaluate second order closures of turbulence, as in explicit RANS models, e.g., Pope [22]. A classical adopted procedure to evaluate the second order models is known as a priori tests. An emblematic example is found by Durbin [6], where the v2 − f model was compared with classical two-equation models. This procedure consists of inserting the DNS statistics in a turbulent model and verify the

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compatibility with the Navier–Stokes prediction given by the DNS. In this approach, epistemic uncertainties with respect to RANS models can also be evaluated (Emory et al. [7], Margheri et al. [16]). Alternatively, one can use the DNS data in order to extract target coefficients of a certain model constructed by a projection of the Reynolds stress tensor into a set of mean kinematic quantities (Thompson et al. [26]). Estimating uncertainties with respect to the different entities obtained from the statistics of DNS is not straightforward. As discussed by Ghosal [8] classical approaches for error analysis cannot be applied to strongly nonlinear problems, such as turbulence, due to the presence of a simultaneous continuum of space and time scales. Since statistical quantities are often provided as outputs of a DNS, the uncertainties associated to these statistics come from discretization errors and sampling errors. Recently, a discussion on this matter was conducted by Oliver et al. [21], where it was stated that is not common in the DNS literature to estimate the uncertainties of the statistical quantities. One of the main difficulties is related to the fact that samples used to generate DNS statistics are originated from a time history and spatial field that are generally not independent and the procedure used to mitigate this fact, namely taking samples far apart in time and space, does not eliminate it. The usual DNS parameters, such as: the box size, averaging time, mesh size, number of points, etc., have heuristically chosen values. Oliver et al. [21] proposed a Bayesian extension of Richardson extrapolation and they applied the method to a plane channel DNS flow for a low friction Reynolds number, Reτ = 180. This Reynolds number was chosen because it is sufficiently small so that it was feasible to run a simulation with twice the resolution of the simulation with heuristically chosen values. Recently, Vreman and Kuerten [27], 28] stressed the importance of high resolution DNS (HR-DNS). The main argument that support this approach is the convergence of statistics of higher order than the ones needed to compute the Reynolds stress tensor. These statistics are present in the dissipation and enstrophy budgets, which can be considered fingerprints of a turbulent flow (see Donzis et al. [5], Hamlington et al. [9], Jin et al. [11]). The approach presented here shows a different and simple method to estimate the statistical error associated with the Reynolds stress tensor provided by DNS of the plane channel flow. As it will be shown, this error impacts RANS modeling that uses this quantity as target. One direct consequence of the methodology here employed is that a new error evaluation, that can be used in a convergence criterion, rises naturally. The remainder of the paper is organized as follows. In Section 2, we present the simulation parameters of the database used in the present work, provided by different groups that have performed DNS of the plane channel. The methodology for the error calculation is presented in Section 3. In Section 4 results for errors associated to the Reynolds stress shear component and the mean velocity profile are shown for the different groups. In Section 5 the conclusions of the present work are provided. 2. The different DNS results for the benchmark plane channel flow In this work we have used databases of DNS for plane channel flows coming from different research teams: (i) Álamo et al. [2]–Hoyas and Jiménez [10]–Lozano-Durán and Jiménez [14]; (ii) Bernardini et al. [4]; (iii) Moser et al. [20]–Lee and Moser [13]; (iv) Thais et al. [24]; and (v) Vreman and Kuerten [27], 28]. In these works, the numerical simulation of fully developed turbulent channel flow at different Reynolds numbers are presented, and we can find data from Reτ = 180 up to Reτ = 5200, where Reτ denotes the friction Reynolds number based on the friction velocity uτ and on the channel half-gap h. It is known that the

Table 1 Parameters used in the DNS conducted by Jiménez and co-workers. [1]: Reτ = 547 [2]: Reτ = 934. [10]: Reτ = 2003. [14]: Reτ = 4179 (data on-line at http://torroja. dmt.upm.es/ftp/channels). Reτ

(Lx × Lz )/h

Nx × Ny × Nz

dx+

dy+ max

dz+

Tuτ /h

547 934 2003 4179

8π 8π 8π 2π

1536 × 1536 × 257 3072 × 2304 × 385 6144 × 633 × 4608 ? × 1081 × ?

8.9 9.2 8.2 12.8

6.7 7.6 8.9 10.7

4.5 3.8 4.1 6.4

12 8.5 10.3 15

× × × ×

4π 3π 3π

π

Table 2 Parameters used in the DNS conducted by [4]. In this work the dymax is nondimensionalized with the Kolmogorov scale (η) (data on-line at http://newton.dma. uniroma1.it/channel). Reτ

(Lx × Lz )/h

Nx × Ny × Nz

dx+

(dy/η)max

dz+

Tuτ /h

550 999 2022 4079

6π 6π 6π 6π

1024 × 256 × 512 2048 × 384 × 1024 4096 × 768 × 2048 8192 × 1024 × 4096

10.0 9.2 9.3 9.4

1.84 1.84 1.84 1.84

6.7 6.1 6.2 6.2

36.3 26.9 14.9 8.54

× × × ×

2π 2π 2π 2π

Table 3 + Parameters used in the DNS conducted by [13]. The quantities dy+ w and dyc are grid spacing at wall and center line, respectively. (data on-line at http://turbulence.ices. utexas.edu). Reτ

(Lx × Lz )/h

Nx × Ny × Nz

dx+

dy+ w

dy+ c

dz+

Tuτ /h

544 10 0 0 5186

8π × 3π 8π × 3π 8π × 3π

? × 384 × ? ? × 512 × ? 10240 × 1536 × 7680

8.9 10.9 12.7

0.019 0.019 0.498

4.5 6.2 10.3

5.0 4.6 6.4

13.6 12.5 7.80

Table 4 Parameters used in the DNS conducted by [24]. (data on-line at http://lml. univ-lille1.fr/channeldata). Reτ

(Lx × Lz )/h

Nx × Ny × Nz

dx+

dy+ max

dz+

Tuτ /h

590 10 0 0

8π × 1.5π 6π × 1.5π

1536 × 257 × 512 1536 × 513 × 768

9.6 12.3

10.4 8.4

5.4 6.1

19.9 7.1

Table 5 Parameters used in the DNS at Reτ = 590: MKM conducted by [20] and VK conducted by [28]. Group

Reτ

(Lx × Lz )/h

Nx × Ny × Nz

dx+

dy+ max

dz+

Tuτ /h

MKM VK

587.19 590

2π × π 2π × π

384 × 257 × 384 768 × 385 × 768

9.7 4.8

7.2 4.8

4.8 2.4

? 100

two main sources of errors in DNS calculations arise from: numerical discretization of the Navier–Stokes equations and statistics being evaluated over a finite statistical sampling time. The averaging time for statistics is an important parameter related with the size of the domain, as pointed out by Lozano-Durán and Jiménez [14]. Tables 1–5 show the main parameters of the numerical simulations for different Reynolds numbers: domain size, number of grid points, mesh resolution and the averaging times. The different DNS data used in this work are briefly summarized below. Jiménez and co-workers use the spatial numerical discretization originally proposed by Kim et al. [12], combined with a third order temporal discretization found by Moser et al. [20]. For the present analysis we have chosen the data at Reτ = 547 and 934 from Álamo et al. [2] (Series 1, long periodicities Lx and Lz , L550 and L950); together with Reτ = 2003 from Hoyas and Jiménez [10], and recently completed with Reτ = 4179 from Lozano-Durán and Jiménez [14]. The details of the spatial discretization and sampling times are summarized in Table 1. Bernardini et al. [4], see Table 2, present results of DNS turbulent channel flow. From its database, we have chosen the following Reynolds numbers : Reτ = 550, 999, 2022, 4079. A staggered

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central second-order finite-different was used to discretize the Navier–Stokes equations; and the time advancement was achieved with an hybrid third-order algorithm. Lee and Moser [13], see Table 3, made use of a Fourier–Galerkin method in the streamwise and spanwise directions, while the wall-normal direction was discretized using a B-spline collocation method. The Navier–Stokes equations were solved using the method of Kim et al. [12], in which equations for the wall-normal vorticity and the Laplacian of the wall-normal velocity are timeadvanced. An implicit-explicit scheme based on third-order Runge– Kutta for the non-linear terms and Crank–Nicolson for the viscous terms was used to advance in time. Thais et al. [24] present results of DNS of Newtonian and viscoelastic flows for different Reynolds numbers, from Reτ = 180 up to 30 0 0. In the present work we only consider the Newtonian flow at Reτ = 590 and 10 0 0. Table 4 presents the numerical domain, the grid used and the averaging times. The spatial discretization is hybrid, making use of two-dimensional fast Fourier transforms in the streamwise and spanwise directions and 6th-order finite differences in the wall-normal direction. A detailed discussion of the numerical procedure used to solve the Navier–Stokes equations can be found by Thais et al. [25]. Table 5 summarizes two independent simulations at Reτ = 590 by Moser et al. [20] and Vreman and Kuerten [28]. These simulations share in common a small computational box of 2π × π . However, Vreman and Kuerten [28] produced statistics from a high-resolution numerical scheme. Their numerical method is fully spectral, Fourier in the streamwise and spanwise directions, and Chebyshev-tau in the wall-normal direction. Their method, based on the equations for the wall-normal vorticity, is an independent implementation of the method described by Moser et al. [20]. The time integration scheme is a three-stage Runge–Kutta method with implicit treatment of the viscous terms. The statistics were collected during an extremely long period of 100 h/uτ . 3. Methodology Following Pope [22], we know that in steady-state fullydeveloped channel flow, the total shear stress, τ , is a linear function of wall distance y. We also have that τ is constrained by the boundary conditions τ = τw at the wall (y = 0) and τ = 0 at the half-channel (y = h) At the same time we can split the total shear stress, τ , as the sum of the molecular shear stress and of the Reynolds shear stress Ryx = −u v . Hence, we get

  y dU τ = 1 − τw = ρν + ρ Ryx , h

dy

(1)

where U is the mean velocity in the streamwise x-direction, y is the wall direction, ρ is the mass density, ν is the kinematic vis√ cosity. Using uτ = τw /ρ as characteristic velocity and δv = ν /uτ as characteristic length, we can rewrite Eq. (1) as



1−

y+ Reτ



=

dU + + R+ yx dy+

(2)

where the variables with a superscript +, are expressed in wall units and Reτ ≡ uτ h/ν is the usual friction Reynolds number. Probably, the two most obvious statistics provided in a plane channel DNS are the mean velocity field and the Reynolds stress tensor associated to the fully developed steady-state. They are both determined by a hybrid averaging process: averaging is performed spatially in wall-parallel slabs and in time, using as many flow snapshots as available. Therefore, both are subject to discretization and sampling errors. The numerical experience shows that the Reynolds shear stress yx-component, R+ yx , besides being a second-order statistical quantity, is the most difficult component to converge.

3

The methodology employed here is to use Eq. (2) as means to discuss the consequences of the numerical imbalance of the fully developed momentum equation. We employ the symbol (ˆ) to label any variable that is provided by the DNS hybrid space-time average. The symbol  ( ) is used to label a variable that is computed from a conservative equation such as Eq. (2). With this convention, we can write the following two equations + ˆ+ + (y+ ) = 1 − y − dU , R yx Reτ dy+  y+ +2   + (y+ ) = y+ − y U − Rˆ+ yx (y )dy . 2Reτ 0

(3)

(4)

Eq. (3) expresses the yx-component of the Reynolds stress tensor that balances the momentum equation in the x-direction using the axial mean velocity gradient provided by DNS as input data. On the other hand, Eq. (4) expresses the mean velocity profile that would balance the momentum equation using the yx-component of the Reynolds stress tensor available from DNS as input data. Eqs. (3) and (4) can be used to define two residuals: ER (y+ ) = + (y+ ) − Rˆ+ (y+ ) and E (y+ ) = U + (y+ ) − Uˆ + (y+ ). Using Eq. (3), R U yx yx + ER (y ) can be written as + ˆ+ + (y+ ) − Rˆ+ (y+ ) = 1 − y − dU − Rˆ+ (y+ ) ER ( y+ ) = R yx yx yx Reτ dy+

(5)

and can be interpreted as the residual of the momentum balance with respect to steady state fully developed channel flow. To guarantee steady state fully developed flow, ER (y+ ) computed from DNS data ought to be as small as possible across the channel width. Using Eq. (4), the second residual EU (y+ ) can be written as

+ (y+ ) − Uˆ + (y+ ) = EU (y+ ) = U



y+ 0

ER (y )dy ,

(6)

i.e., EU (y+ ) is originated from the cumulative error associated with the residual of the momentum equation. Hence, introducing the y+ mean residual ER (y+ ) = (1/y+ ) 0 ER (y )dy , we can re-write,

EU (y+ ) = y+ ER (y+ ).

(7)

It is reasonable to expect that the criteria available for considering ER small should be based on the maximum residual in the domain. However, Eq. (7) shows that using the residual of the integrated momentum equation on its own can be responsible for poor predictions of the mean velocity profile far from the wall. A direct consequence of Eq. (7) is that EU (Reτ ) = Reτ ER (Reτ ). This rationale indicates that if the same criterion adopted for a low Reτ simulation were used for a high Reτ simulation, one would probably obtain a worst prediction for U + far from the wall at high Reynolds number. It is important to notice at this point is that non-small oscillatory local residual ER (y+ ) could possibly lead to low integral residual EU in the event where positive and negative values of ER would compensate each other across the channel. The present methodology requires Eq. (4) to be numerically integrated using DNS data at co-located grid points, and the velocity gradient to be evaluated in Eq. (3) for those DNS databases which do not provide this statistic. We performed the integration with the trapezoidal rule, and the gradient was computed with the second-order midpoint rule. Although different high-order numerical schemes were tried, we did not notice any significant difference in the results presented below. As a final check of accuracy, we ∗ from Eq. (4), in which the Reynolds computed the velocity field U + evaluated from Eq. (3) was used in replacement of the stress R yx DNS-averaged Reynolds stress Rˆ+ yx . We observed a maximum dif∗ so computed and the original ference well below 10−3 between U DNS velocity field Uˆ + , which proved the accuracy of the numerics to be enough.

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R.L. Thompson et al. / Computers and Fluids 130 (2016) 1–7

(a) Reτ = 550, 590

(b) Reτ = 550, 590

(c) Reτ = 950, 1000

(d) Reτ = 950, 1000

(e) Reτ = 2000

(f) Reτ = 2000

(g) Reτ > 4000

(h) Reτ > 4000

+ , and DNS data, Rˆ+ yx-component of the Reynolds stress tensor for different values of Reτ (left column). The right column shows Fig. 1. Comparison between calculated, R yx yx + and Rˆ+ . AJZM [2]; BPO [4]; HJ [10]; LDJ [14]; LM [13]; MKM [15]; TGM [24]; VK [28] . the residual ER (y+ ), i.e., the difference between R yx yx

R.L. Thompson et al. / Computers and Fluids 130 (2016) 1–7

(a) Reτ = 550, 590

(b) Reτ = 550, 590

(c) Reτ = 950, 1000

(d) Reτ = 950, 1000

(e) Reτ = 2000

(f) Reτ = 2000

(g) Reτ > 4000

5

(h) Reτ > 4000

+ , and provided by DNS data, Uˆ + mean velocity profiles. The right column shows the residual EU (y+ ), i.e., the difference between Fig. 2. Comparison between calculated, U + and Uˆ + . Symbols are the same as in Fig. 1. U

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R.L. Thompson et al. / Computers and Fluids 130 (2016) 1–7

4. Results

5. Conclusions

The results are organized along similar Reτ selecting the flow cases where Reτ ≥ 550 which are more critical. The different flow cases were divided in four categories: (1) Reτ = 550, 590; (2) Reτ = 950, 10 0 0; (3) Reτ = 20 0 0; 4) Reτ > 40 0 0. Fig. 1 shows the comparison between the Reynolds stress com+ , obtained from Eq. (3), and the one available as part ponent, R yx of the DNS statistics, Rˆ+ yx . The right column of the figure shows + (y+ ) − Rˆ+ (y+ ). The relative error is not conthe residual E = R

The present work investigates the effects of the imbalance in the momentum equation in DNS of the plane channel flow. The methodology employed here can be directly computed by the agents that are running the simulation, since it uses their own database and does not require additional expensive numerical calculations nor other databases. The impact of the residual of the momentum equation on the velocity profile is cumulative in y+ , and, therefore, can be significant far from the wall in high Reynolds number simulations. This affects turbulence modeling, especially in a RANS approach, where the Reynolds stress tensor is a target. The present analysis puts in light the intrinsic velocity statistical error (Eq. 6) associated with the DNS Reynolds stress being assumed as the “correct” value. A model that replicates exactly the Reynolds stress tensor is doomed to obtain the same discrepancy between DNS and predicted velocity fields obtained in the present work. This result needs to be taken into account when one conducts an uncertainty quantification analysis. The mean velocity Uˆ provided by DNS data is a first order statistic, whereas the Reynolds stress Rˆ+ yx is a second order statistic which is, as a consequence, less converged. The present methodology evaluates the impact of computing the mean velocity from the less converged statistic by comparing it with the more converged value. Besides that, Eq. (3) provides the shear component of the Reynolds stress tensor from first order statistic. Hence, one can use this more accurate quantity for modeling purposes. The difference among the different groups on the first order statistic (mean velocity profile) is low, what indicates that the discretization error is small, since different schemes and box sizes were used by each group. The high resolution DNS statistics of Vreman and Kuerten [28] recovers the most accurate mean velocity profile from the Reynolds shear stress. This result shows that the error evaluated is a strictly related to the sampling error, since the averaging time employed was one order of magnitude higher. The present approach was able to provide a quantitative information about the statistical error associated and a relative comparison with the other available data. It should be pointed out that the performance of Vreman and Kuerten [28] was made possible not only by taking very long time-averaging but also by using a small computational box at a relatively modest Reynolds number. Owing to presently available computer power, it is not conceivable to reproduce this at higher Reynolds numbers in a large computational box. Therefore, a procedure able to provide more accurate data for high Reynolds number plane channel flows is still a requirement. Convergence criteria can be formulated based on the velocity residual EU (y+ ) instead of the stress residual ER (y+ ). These criteria would provide a more accurate Reynolds stress tensor and not only a more accurate shear component as described above. The present methodology provides a new path for statistical error evaluation that can be useful in future DNS of plane channel flows and, by consequence, provide more reliable targets for modeling purposes.

R

yx

yx

sidered since R+ yx is expected to vanish at the wall and at the centerline. In every DNS considered the residual remains reasonably small, with peak values all below 0.03. However, looking more into the details, we can identify considerable differences. In the Reynolds number range 500 (Fig. 1(b)) the high resolution results obtained by Vreman and Kuerten [28] is distinguished from the others with an extremely tiny residual never exceeding 0.002 across the channel. Singular patterns can be identified for each dataset, irrespectively of the Reynolds number. The results of Bernardini et al. [4] always present a peak at y+ ≈ 30. Then ER monotonically decreases (except at Reτ = 4079) towards the center of the channel, with negative values beyond y+ ≈ 170. Moser et al. [20] and Thais et al. [24] present increasing ER (y+ ) values until y+ ≈ 175 and decreasing values towards the center of the channel. Hoyas and Jiménez [10] present small oscillating ER (y+ ) values over the entire domain. Lee and Moser [13] shows an even smaller oscillating behavior until y+ ≈ 250, and after that a monotonic decreasing behavior towards the channel center. The results displayed in Fig. 1(d) and (f) for the stress residual at Reτ ≈ 10 0 0 and Reτ = 20 0 0, respectively, also exhibit typical patterns for each dataset. The ER (y+ ) behavior obtained by Jimenez and co-authors as well as Lee and Moser [13] are very small over the entire domain. Bernardini et al. [4] data present a high peak and a monotonic decreasing behavior, while Thais et al. [24] data present an intermediate smooth curve. For Reτ > 40 0 0 (see Fig 1(h)), only Lee and Moser [13] manage to reproduce relatively low ER (y+ ) across the channel as observed at lower Reynolds numbers. The results of Lozano-Durán and Jiménez [14] and Bernardini et al. [4] have a valley at y+ ≈ 160, however the former was characterized by smaller stress residual and smoother behavior. Fig. 2 shows the comparison between calculated and DNS mean velocities, (left column) together with the velocity residual EU (y+ ) (right column). + , differ significantly from The calculated velocity profiles, U the available DNS data, especially away from the wall, with the noticeable exception of Vreman and Kuerten [28] dataset which is the only one exhibiting a negligible velocity residual (see Fig. 2(b)). The velocity residual obtained by Hoyas and Jiménez [10] at Reτ = 20 0 0 (see Fig. 2(f)) is particularly small when compared to other (non HR-DNS) results at lower Reτ . Not surprisingly, the velocity residuals obtained for Reτ > 40 0 0 were large for all datasets. The velocity residual can go above 10 wall units at the channel center line which shows the extreme difficulty in obtaining converged statistics at such high Reynolds numbers. Also remarkable is the particular behavior of the computed velocity profile for the Lozano-Durán and Jiménez [14] dataset: a + /dy+ straight horizontal line after y+ ≈ 200, indicating that dU in negligible in this range. It is worth mentioning that a new procedure was proposed to obtain these data where a smaller box was used.

Acknowledgments We would like to thank the support given by the Brazilian funding agency CNPq - Conselho Nacional de Desenvolvimento Científico e Tecnológico by means of the Science without boarders program through grant 402394/2012-7. References [1] Álamo JCD, Jiménez J. Spectra of the very large anisotropic scales in turbulent channels. Phys Fluids 2003;15(1):L41–4.

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