12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

WALL TURBULENCE AT HIGH FRICTION REYNOLDS NUMBERS

Sergio Hoyas
IUMPA, Universitat Politècnica de València,
València, Spain, serhocal@mot.upv.es

Martin Oberlack
TU Darmstadt, Chair of Fluid Dynamics,
Darmstadt, Germany, oberlack@fdy.tu-darmstadt.de

ABSTRACT

Hoyas et al. (2022).
This friction Reynolds number is less than the largest realisation of the flow obtained by experimental means, see Samieet al. (2018) and discussion inside. In particular, Samie et al.
(2018) reached a value of 20, 000 with very good spatial resolution. The main advantage, however, is that DNS allows thecomputation of any conceivable quantity over the entire domain, including derivatives close to walls. In addition to contributing to discussions on some of the issues that have arisenover the past few years, with this simulation we are providing
community data that hopefully will help better model turbulence.
In the present article we will restrict ourselves to present
the data and the kinematics of the flow, referring the interested reader to Oberlack et al. (2022) for scaling laws about thestreamwise mean velocity. This work is organized as follows.
Section two describes the numerical method and the validationof the data. Section three discusses the one-point statistics ofthe flow, including mean flow and intensities. The energy turbulent budgets are discussed in section four. Finally, section
five contains the conclusions of this work.
The database containing the mean flow, intensities, and turbulent budgets can be downloaded from the TUdatalib Repository of TU Darmstadt at https://doi.org/10.48328/tudatalib658

A new direct numerical simulation of Poiseuille channel
flow with a frictional Reynolds number of 10, 000 was performed using the LISO pseudospectral code. The mean flowpresents a long logarithmic layer ranging from 400 to 2500
wall units, longer than expected. The maximum intensity of
the flow velocity increases with the Reynolds number. The
elusive second maximum of this intensity is also yet to appear.
If it existed, its position would be around y+ ≈ 120, extrapolating the friction Reynolds number to around 13500. For
several Reynolds numbers, this slight difference in the nearwall gradient of the intensity is associated with a scaling failure of dissipation, confirming this hypothesis. The scaling of
the turbulent users in the middle of the channel is almost perfect above 1000 wall units. Finally, the peak pressure intensity
grows with increasing Reynolds number and does not scale inwall units.

INTRODUCTION
After almost 140 years of the publication of O. Reynolds’s
first work, turbulence is still an open problem. Research of turbulent flows has been dominated by experimental techniques
until the eighties of the last century, where supercomputers
started to be powerful enough to solve the equations of turbulent flows. In fact, Direct Numerical Simulation (DNS) has become one of the main tools in studying wall turbulence. However, DNSs are extremely expensive in terms of computational
resources. Thus, the most studied configurations have been
turbulent boundary layers and channels. Here we study thelater, where the fluid is confined between two parallel plates
and the flow is driven by pressure.
The friction Reynolds number, defined as Reτ = uτ h/ν, isthe main√ control parameter in wall bounded turbulence. Hereuτ = τw /ρ is the friction velocity, ν is the kinematic viscosity, ρ is the density, and τw is the friction at the wall. his the semi-height of the channel and is equivalent to the radius in pipes and δ99 in boundary layers. Since the seminal work of Kim, Moin and Moser Kim et al. (1987), theReτ has steadily increased from 180 in 1987 to 8000 in 2018
(Kim et al., 1987; Hoyas & Jiménez, 2006; Lozano-Durán &Jiménez, 2014; Bernardini et al., 2014; Yamamoto & Tsuji,2018). A simulation reaching the Reτ = 10000 frontier is presented here. This simulation has been very recently published,and some more details are given here: Oberlack et al. (2022);

Numerical method
In this work we present the results of a DNS of a pressuredriven (Poiseuille) channel flow at a nominal Reτ = 10000. Superscript (+) indicates that the quantities have been normalizedby uτ and ν. This simulation has been performed in a computational box of sizes Lx = 2πh, Ly = 2h and Lz = πh. For
Poiseuille flow, Lozano-Durán & Jiménez (2014) noticed thateven relatively small computational boxes of stream- and spanwise sizes of only 2πh × πh can satisfactorily recover the onepoint statistics of the flow. The streamwise, wall-normal, andspanwise coordinates are x, y, and z, respectively. The corresponding velocity components are U, V andW or, using index
notation, Ui . Statistically averaged quantities in time, x and z
are denoted by an overbar, U, whereas fluctuating quantities
are denoted by lowercase letters, i. e., U = U + u. Primes arereserved for intensities, u′ = uu1/2 .
The Navier-Stokes equations have been solved using the
LISO code, which has successfully been employed to run some1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
Case

Line

Reτ

Reb

Lx

Lz

∆x+

∆z+

Tuτ /h

HJ02

⋯⋯

2000

43650

8πh

3πh

12.3

6.1

11

LJ04

4000

98302

2πh

πh

12.8

6.4

15

LM05

5200

125000

8πh

3πh

8.2

4.1

7.80

HO10

10000

261000

2πh

πh

15.3

7.6

19.8

Table 1: Parameters of the simulations. Reb is the bulk Reynolds number, Reb = Ub h/ν, where Ub is the bulk velocity. ∆+xand ∆+z are in terms of dealiased Fourier modes. The last column is the total simulation time in terms of eddy turnovers.

of the largest simulations of turbulence (Hoyas & Jiménez,
2006; Avsarkisov et al., 2014a,b; Kraheberger et al., 2018;
Alcántara-Ávila et al., 2021). Briefly, the code uses the same
strategy than Kim et al. (1987), but using a seven-point compact finite differences in y direction with fourth-order consistency and extended spectral-like resolution (Lele, 1992). The
temporal discretization is a third-order semi-implicit RungeKutta scheme (Spalart et al., 1991). The wall-normal grid
spacing is adjusted to keep the resolution at ∆y = 1.5η, i.e.,
approximately constant in terms of the local isotropic Kolmogorov scale η = (ν 3 /ε)1/4 . In wall units, ∆y+ varies from
0.3 at the wall, up to ∆y+ ≃ 12 at the centerline. The resolution
in x and z is similar to the largest simulations of turbulence,
see table 1. A code similar to the one used presently, including
the energy equation, is explained in Lluesma-Rodrı́guez et al.
(2021).
The initial file of this simulation was taken from a smaller
Reynolds number simulation. To accelerate the compilation of
statistics, three initial files were prepared and thus three simulations were run at the same time. In every case, the code was
run until some transition phase had passed and the flow had
adjusted to the new set of parameters. Once the flow was ina statistically steady state, statistics were compiled. The running times to compile statistics are shown in terms of eddyturnovers in the rightmost column of table 1. The transitions
until the simulations reached a statistically steady state, whichwere very time consuming, are not contemplated in this table.
Table 1 also shows the parameters of the simulations HJ02
(Hoyas & Jiménez, 2006), LJ04 (Lozano-Durán & Jiménez,2014), and LM05 (Lee & Moser, 2015). These simulations
will be used in the paper in the colour code described in thesecond column of table 1. As it is said above, this code has
already proved its worth, but to further validate the statistics,
figure 1 shows the error in the momentum equation,

Figure 1: Error in the computation of momentum equa-tion (1)+profile of U is indeed logarithmic, using the Lie-symmetry
theory applied to turbulence. Furthermore, it this article it isshown that in the log-region we haveU1+ =ω(n−1)(U1n )+ =Cn (y+ )(2)− Bn , for n ≥ 2,

(3)

Cn =αeβ n , Bn = α̃eβ̃ n , for n ≥ 2.

(4)

where κ, B, Bn , Cn , α, β , α̃ and β̃ are constants that need tobe fitted with experiments. For several of the flows analysed,
ω = 0.10.
For every case, the first local minimum of Γ is reachedaround y+ ≈ 70, which more or less coincide with the classic
starting point of the logarithmic layer (Pope, 2000). However,
the indicator function is not flat until y+ ≈ 400, so this could bea new starting point. The logarithmic layer extends to aroundy+ ≈ 2500 or y/h = 0.25, above the usual value of y/h = 0.2.
To obtain the values of κ and B, we have restricted ourselvesto the region where the indicator function is flattest, i.e., fromy+ = 400 to y/h = 0.25 (figure 2a), obtaining κ = 0.394 andB = 4.61. This value of κ is similar to the experimental one ofMarusic et al. (2013), and only 0.010 and 0.007 units larger
than the one given by Lee & Moser (2015) and Yamamoto &Tsuji (2018). Abe & Antonia (2016) also obtained this value
in their study of global energy. Studying the finite Reynolds
number effects on the flow, Luchini (2017), and Spalart & Abe(2021), give a similar value.
It is worth mentioning that some other authors, with dif-+dU
− uv+ = 1 − y+ .
dy+1ln (y+ ) + B,κ(1)

The difference between both sides of this equation is below 2×10−3 , similar to the other tree simulations utilized here.
Thus, it has been considered that enough statistical information was obtained.

One point statistics
The mean velocity profile is shown in figure 2a in terms+of the indicator function, Γ = y+ ∂y+ U . This function should
show a plateau if the classical scaling for the logarithmic layer+
U = κ −1 log(y+ ) + B holds, where κ is the von Kármán constant. Moreover, in Oberlack et al. (2022) it is shown that the212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

Figure 2: Lines as in table 1. (a) Indicator function, showing a log layer in the range y+ = 400 − 2500. (b) Budgets for

Reynolds stresses in wall units. Production ∎, dissipation ⧫, viscous diffusion ∗ , pressure-strain ▼, pressure diffusion ▲,turbulent diffusion •.

Figure 3: Lines as in table 1. (a) u′+ . Box: dy+ u′+ evaluated at the wall. (b) dy+ u′+ , close to the possible second maximum.

Box: maximum value of dy+ u′+
ferent tools, have obtained values above 0.40. McKeon McKeon et al. (2004) in high Reynolds numbers involving pipes
got 0.42. More recently, Monkewitz (2021) has developed an+algorithm to model U for very large Reynolds numbers. The
value of κ is similar to the one obtained by us. However, in
Monkewitz (2021) the actual logarithmic profile for Reτ = 105
would start around 103 wall units for a final value κ = 0.42. As
a DNS reaching this Reτ is approximately 3500 times morecostly than the one presented here, this is probably an open
problem for the next decade or more.
The intensity of the streamwise velocity, u′+ , is shown infigure 3a. The well known scaling failure in the buffer layer
is still present (Hoyas & Jiménez, 2006), and the maximum of
the intensity is u′+ = 3.07. About the open question of a possible second maximum of u′+ the situation is shown in figure 3b.
If it exists, this maximum would be located around y+ ≈ 120.
However, the derivative of u′+ is still not zero. Fitting the data
to a logarithmic grow law, we obtain dy u′+
0 = 0.29 log Reτ −2.7,
and a approximate critical value of Reτ = 13500.

Hoyas & Jiménez (2008)
Bi j ≡ Dui u j /Dt = Pi j + εi j + Ti j + Πsi j + Πdij +Vi j .

(5)

The terms in the right hand side of equation (5) are referred as production, dissipation, turbulent diffusion, viscous
diffusion, pressure-strain, and pressure-diffusion.

Pi j = −ui uk ∂kU j − u j uk ∂kUi ,Ti j = ∂k ui u j uk ,
Πsi j = p(∂ j ui + ∂ j ui ),εi j = −2ν∂k ui ∂k u j ,Vi j = ν∂kk ui u jΠdij = ∂k (pui δ jk + pu j δik ),where δi j is Kronecker’s delta. The splitting of the pressure in
two different terms is not unique, but this one offers more information in the B12 and B22 terms (Hoyas & Jiménez, 2008).
Finally, in channels Bi j ≡ 0. In the viscous and buffer layers,
3budgets should scale in wall units, B+
i j = Bi j ν/uτ . The B11 budget is shown in figure 3b, using this scale. Except those terms
that are identically zero, all are active. The well-known scaling failure (Hoyas & Jiménez, 2008) of the dissipation at thewall for B11 is still present. As expected all terms collapse for

The scaling failure of the dissipation
The budget equation for the component ui u j of the
Reynolds-stress tensor is written as Mansour et al. (1988);

312th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
y+ > 10. However, below this more or less arbitrary limit, the++absolute values of ε11
and V11
increase with the Reynolds number. This scaling failure can be linked to the growing of the firstmaximum of u′+ . At the wall (Hoyas & Jiménez, 2008),
V11 ∣y=0 = ν∂yy u2 ∣y=0 = 2ν (∂y u)2 ∣y=0 = −ε11Avsarkisov, V., Oberlack, M. & Hoyas, S. 2014b New scalinglaws for turbulent Poiseuille flow with wall transpiration.
Journal of Fluid Mechanics 746, 99–122.
Bernardini, M., Pirozzoli, S. & Orlandi, P. 2014 Velocity
statistics in turbulent channel flow up to Reτ = 4000. Journal of Fluid Mechanics 758, 327–343.
Hoyas, Sergio & Jiménez, Javier 2006 Scaling of the velocity
fluctuations in turbulent channels up to Reτ = 2003. Physics
of Fluids 18 (1), 011702.
Hoyas, S. & Jiménez, J. 2008 Reynolds number effects on the
Reynolds-stress budgets in turbulent channels. Physics of
Fluids 20 (10), 101511.
Hoyas, Sergio, Oberlack, Martin, Alcántara-Ávila, Francisco,
Kraheberger, Stefanie V. & Laux, Jonathan 2022 Wall turbulence at high friction reynolds numbers. Phys. Rev. Fluids
7, 014602.
Kim, J., Moin, P. & Moser, R. 1987 Turbulence statistics infully developed channels flows at low Reynolds numbers.
Journal of Fluid Mechanics 177, 133–166.
Kraheberger, S., Hoyas, S. & Oberlack, M. 2018 Dns of aturbulent couette flow at constant wall transpiration up toReτ = 1000. Journal of Fluid Mechanics 835, 421–443.
Lee, M. & Moser, R. 2015 Direct numerical simulation of turbulent channel flow up to Reτ ≈ 5200. Journal of Fluid Mechanics 774, 395–415.
Lele, S. K. 1992 Compact finite difference schemes with
spectral-like resolution. Journal of Computational Physics
103 (1), 16–42.
Lluesma-Rodrı́guez, F., Álcantara Ávila, F., Pérez-Quiles,
M.J. & Hoyas, S. 2021 A code for simulating heat transfer in turbulent channel flow. Mathematics 9 (7).
Lozano-Durán, A. & Jiménez, J. 2014 Effect of the computational domain on direct simulations of turbulent channels
up to Reτ = 4200. Physics of Fluids 26 (1), 011702.
Luchini, P. 2017 Universality of the turbulent velocity profile.
Physical Review Letters 118 (22), cited By 26.
Mansour, N. N., Kim, J. & Moin, P. 1988 Reynolds-stress anddissipation-rate budgets in a turbulent channel flow. Journalof Fluid Mechanics 194, 15.
Marusic, Ivan, Monty, Jason P., Hultmark, Marcus & Smits,
Alexander J. 2013 On the logarithmic region in wall turbulence. Journal of Fluid Mechanics 716, R3.
McKeon, B. J., Li, J., Jiang, W., Morrison, J. F. & Smits, A. J.
2004 Further observations on the mean velocity distribution
in fully developed pipe flow. Journal of Fluid Mechanics
501, 135–147.
Monkewitz, P.A. 2021 The late start of the mean velocity overlap log law at-a generic feature of turbulent wall layers inducts. Journal of Fluid Mechanics 910.
Oberlack, Martin, Hoyas, Sergio, Kraheberger, Stefanie V.,
Alcántara-Ávila, Francisco & Laux, Jonathan 2022 Turbulence statistics of arbitrary moments of wall-bounded shear
flows: A symmetry approach. Phys. Rev. Lett. 128, 024502.
Pope, S. B. 2000 Turbulent flows. Cambridge University Press.
Samie, M., Marusic, I., Hutchins, N., Fu, M.K., Fan, Y., Hultmark, M. & Smits, A.J. 2018 Fully resolved measurementsof turbulent boundary layer flows up to Reτ = 20000. Journal of Fluid Mechanics 851, 391–415.
Spalart, P.R. & Abe, H. 2021 Empirical scaling laws for wallbounded turbulence deduced from direct numerical simulations. Physical Review Fluids 6 (4), cited By 0.
Spalart, Philippe R, Moser, Robert D & Rogers, Michael M
1991 Spectral methods for the Navier-Stokes equations with
one infinite and two periodic directions. Journal of Computational Physics 96 (2), 297–324.

(6)

as all other terms vanish. u′+ can be approximated by u′+ =2(bu y+ + cu y+ + ...) . Therefore, near the wall, u′+ ≈ bu y+ , and+V11≈ b2u . Thus, the reason why this term of the turbulent budget does not scale with the Reynolds number in the wall comesfrom the differences in the bu terms. This term represents theslope of u′+ near the wall. Looking at the box in figure 3a, one
can see that, effectively, the value of du′+ /dy+ at the wall does
not collapse, but slightly increase. Apart from our data, there
are evidences that the peak at y+ ≈ 15 keeps growing with respect to Reτ (Samie et al., 2018). Because the position of thepeak is constant in y+ , the slope of u′+ has to be higher forlarger Reτ . In other words, as long as the peak of u′+ increases+
with Reτ , bu will also increase and V11
cannot scale at the wall.

Conclusions
To conclude, we have simulated a Poiseuille turbulent
channel flow at a friction Reynolds number of Reτ = 10000.
This simulation was made in a small box of size (2πh, 2h, πh),large enough to accurately compute the statistics of the flow.
The profile of U shows a long log layer, extending fromy+ ≈ 400 to y+ ≈ 2500. The value of the von Kármán constantis κ = 0.394. The first maximum of the streamwise profile u′+continues growing, which is the cause of the scaling failure of
the dissipation at the wall. The second maximum of u′+ has
not appeared yet, and it is foreseen to appear at approximatelyReτ = 13500. On the other hand, The turbulent budgets show aalmost perfect scaling in the outer region with B∗i j = yBi j /u3τ .

Acknowledgements
The authors gratefully acknowledge providing computing time on the Gauss Centre for Supercomputing e.V. on
the GCS Supercomputer SuperMUC-NG at Leibniz Supercomputing Centre under project number pr92la, on the supercomputer Lichtenberg II at TU Darmstadt under project
number project00072 and on the supercomputer CLAIX-2018
at RWTH-Aachen, project bund0008. MO would like toacknowledge her funding by the German Research Foundation (DFG) through the project OB96/39-1 and OB96/481. SH was supported by contract RTI2018-102256-B-I00 of
MINECO/FEDER.

REFERENCES
Abe, H. & Antonia, R.A. 2016 Relationship between the energy dissipation function and the skin friction law in a turbulent channel flow. Journal of Fluid Mechanics 798, 140–
164.
Alcántara-Ávila, F., Hoyas, S. & Pérez-Quiles, M.J. 2021
Direct numerical simulation of thermal channel flow forReτ = 5000 and Prand = 0.71. Journal of Fluid Mechanics
916, A29.
Avsarkisov, V., Hoyas, S., Oberlack, M. & Garcı́a-Galache,
J.P. 2014a Turbulent plane Couette flow at moderately highreynolds number. Journal of Fluid Mechanics 751, R1.

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
Yamamoto, Yoshinobu & Tsuji, Yoshiyuki 2018 Numericalevidence of logarithmic regions in channel flow at Reτ =8000. Physical Review Fluids 3, 012602(R).

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