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Article

Turbulence Intensity and the Friction Factor for

Smooth- and Rough-Wall Pipe Flow

Nils T. Basse

Toftehøj 23, Høruphav, 6470 Sydals, Denmark; nils.basse@npb.dk

Academic Editor: William Layton

Received: 24 April 2017; Accepted: 8 June 2017; Published: 10 June 2017

Abstract:

Turbulence intensity proﬁles are compared for smooth- and rough-wall pipe ﬂow

measurements made in the Princeton Superpipe. The proﬁle development in the transition from

hydraulically smooth to fully rough ﬂow displays a propagating sequence from the pipe wall towards

the pipe axis. The scaling of turbulence intensity with Reynolds number shows that the smooth-

and rough-wall level deviates with increasing Reynolds number. We quantify the correspondence

between turbulence intensity and the friction factor.

Keywords:

turbulence intensity; Princeton Superpipe measurements; ﬂow in smooth- and rough-wall

pipes; friction factor

1. Introduction

Measurements of streamwise turbulence [

1

] in smooth and rough pipes have been carried out in

the Princeton Superpipe [

2

–

4

] (Note that the author of this paper did not participate in making the

Princeton Superpipe measurements.). We have treated the smooth pipe measurements as a part of [

5

].

In this paper, we add the rough pipe measurements to our previous analysis. The smooth (rough)

pipe had a radius

R

of 64.68 (64.92) mm and a root mean square (RMS) roughness of 0.15 (5)

µ

m,

respectively. The corresponding sand-grain roughness is 0.45 (8) µm [6].

The smooth pipe is hydraulically smooth for all Reynolds numbers

Re

covered. The rough pipe

evolves from hydraulically smooth through transitionally rough to fully rough with increasing

Re

.

Throughout this paper, Re means the bulk Re deﬁned using the pipe diameter D.

We deﬁne the turbulence intensity (TI) Ias:

I(r) = vRMS(r)

v(r), (1)

where

v

is the mean ﬂow velocity,

vRMS

is the RMS of the turbulent velocity ﬂuctuations and

r

is the

radius (r=0 is the pipe axis, r=Ris the pipe wall).

An overview of past research on turbulent ﬂows over rough walls can be found in the pioneering

work by Nikuradse [7] and a more recent review by Jiménez [8].

The development of predictive drag models has previously been carried out using both

measurements [

9

] and direct numerical simulations (DNS) [

10

]. This work covered the transitionally

and fully rough regimes and a variety of rough surface geometries.

The aim of this paper is to provide the ﬂuid mechanics community with a scaling of the TI with

Re

,

both for smooth- and rough-wall pipe ﬂow. An application example is computational ﬂuid dynamics

(CFD) simulations, where the TI at an opening can be speciﬁed. A scaling expression of TI with

Re

is

provided as Equation (6.62) in [

11

]. However, this formula does not appear to be documented, i.e.,

no reference is provided.

Fluids 2017,2, 30; doi:10.3390/ﬂuids2020030 www.mdpi.com/journal/ﬂuids

Fluids 2017,2, 30 2 of 13

Our paper is structured as follows: in Section 2, we study how the TI proﬁles change over the

transition from smooth to rough pipe ﬂow. Thereafter, we present the resulting scaling of the TI with

Re

in Section 3. Quantiﬁcation of the correspondence between the friction factor and the TI is contained

in Section 4, and we discuss our ﬁndings in Section 5. Finally, we conclude in Section 6.

2. Turbulence Intensity Proﬁles

We have constructed the TI proﬁles for the measurements available (see Figure 1). Nine proﬁles are

available for the smooth pipe and four for the rough pipe. In terms of

Re

, the rough pipe measurements

are a subset of the smooth pipe measurements. Corresponding friction Reynolds numbers can be

found in Table 1 in [3].

0 0.01 0.02 0.03 0.04 0.05 0.06

r [m]

10-2

10-1

100

Turbulence intensity

Smooth pipe

Re = 8.13e+04

Re = 1.46e+05

Re = 2.47e+05

Re = 5.13e+05

Re = 1.06e+06

Re = 2.08e+06

Re = 3.95e+06

Re = 4.00e+06

Re = 5.98e+06

(a)

0 0.01 0.02 0.03 0.04 0.05 0.06

r [m]

10-2

10-1

100

Turbulence intensity

Rough pipe

Re = 9.94e+05

Re = 1.98e+06

Re = 3.83e+06

Re = 5.63e+06

(b)

Figure 1. Turbulence intensity as a function of pipe radius, (a): smooth pipe; (b): rough pipe.

To make a direct comparison of the smooth and rough pipe measurements, we interpolate the

smooth pipe measurements to the four

Re

values where the rough pipe measurements are done.

Furthermore, we use a normalized pipe radius

rn=r/R

to account for the difference in smooth and

rough pipe radii. The result is a comparison of the TI proﬁles at four

Re

(see Figure 2). As

Re

increases,

we observe that the rough pipe TI becomes larger than the smooth pipe TI.

To make the comparison more quantitative, we deﬁne the turbulence intensity ratio (TIR):

rI,Rough/Smooth(rn) = IRough(rn)

ISmooth(rn)=vRMS,Rough (rn)

vRMS,Smooth(rn)×vSmooth (rn)

vRough(rn). (2)

The TIR is shown in Figure 3. The left-hand plot shows all radii; prominent features are:

•The TIR on the axis is roughly one except for the highest Re, where it exceeds 1.1.

•

In the intermediate region between the axis and the wall, an increase is already visible for the

second-lowest Re, 1.98 ×106.

The events close to the wall are most clearly seen in the right-hand plot of Figure 3. A local peak

of TIR is observed for all Re; the magnitude of the peak increases with Re. Note that we only analyse

data to 99.8% of the pipe radius. Thus, the 0.13 mm closest to the wall is not considered.

The TIR information can also be represented by studying the TIR at ﬁxed

rn

vs.

Re

(see Figure 4).

From this plot, we ﬁnd that the magnitude of the peak close to the wall (

rn=

0.99) increases linearly

with Re:

rI,Rough/Smooth(rn=0.99) = 2.5137 ×10−8×Re +1.0161. (3)

Fluids 2017,2, 30 3 of 13

0 0.2 0.4 0.6 0.8 1

Normalized pipe radius

10-2

10-1

100

Turbulence intensity

Re = 9.94e+05

Smooth pipe

Rough pipe

(a)

0 0.2 0.4 0.6 0.8 1

Normalized pipe radius

10-2

10-1

100

Turbulence intensity

Re = 1.98e+06

Smooth pipe

Rough pipe

(b)

0 0.2 0.4 0.6 0.8 1

Normalized pipe radius

10-2

10-1

100

Turbulence intensity

Re = 3.83e+06

Smooth pipe

Rough pipe

(c)

0 0.2 0.4 0.6 0.8 1

Normalized pipe radius

10-2

10-1

100

Turbulence intensity

Re = 5.63e+06

Smooth pipe

Rough pipe

(d)

Figure 2.

Comparison of smooth and rough pipe turbulence intensity (TI) proﬁles for the four

Re

values where the rough pipe measurements are done, (

a

): Re = 9.94e + 05; (

b

): Re = 1.98e + 06;

(c): Re = 3.83e + 06; (d): Re = 5.63e + 06.

0 0.2 0.4 0.6 0.8 1

Normalized pipe radius

0.8

0.9

1

1.1

1.2

Turbulence intensity ratio (rough/smooth)

Re = 9.94e+05

Re = 1.98e+06

Re = 3.83e+06

Re = 5.63e+06

(a)

0.95 0.96 0.97 0.98 0.99 1

Normalized pipe radius

0.95

1

1.05

1.1

1.15

1.2

Turbulence intensity ratio (rough/smooth)

Re = 9.94e+05

Re = 1.98e+06

Re = 3.83e+06

Re = 5.63e+06

(b)

Figure 3. Turbulence intensity ratio (TIR); (a): all radii; (b): zoom to outer 5%.

Information on ﬁts of the TI proﬁles to analytical expressions can be found in Appendix A.

Fluids 2017,2, 30 4 of 13

Based on uncertainties in Table 2 in [

3

], the uncertainty of TI for the smooth (rough) pipe is 2.9%

(3.5%), respectively. Note that we have used 4.4% instead of 4.7% for the uncertainty of

v2

RMS/v2

τ

to

derive the rough pipe uncertainty. The resulting TIR uncertainty is 4.5%.

0123456

Re 106

0.95

1

1.05

1.1

1.15

1.2

Turbulence intensity ratio (rough/smooth)

rn=0.0

rn=1/3

rn=2/3

rn=0.99

rn=0.99 (fit)

Figure 4. Turbulence intensity ratios for ﬁxed rn.

3. Turbulence Intensity Scaling

We deﬁne the TI averaged over the pipe area as:

IPipe area =2

R2ZR

0

vRMS(r)

v(r)rdr. (4)

In [

5

], another deﬁnition was used for the TI averaged over the pipe area. Analysis presented in

Sections 3and 4is repeated using that deﬁnition in Appendix B.

Scaling of the TI with

Re

for smooth- and rough-wall pipe ﬂow is shown in Figure 5. For

Re =

10

6

,

the smooth and rough pipe values are almost the same. However, when

Re

increases, the TI of the

rough pipe increases compared to the smooth pipe; this increase is to a large extent caused by the TI

increase in the intermediate region between the pipe axis and the pipe wall (see Figures 3and 4). We

have not made ﬁts to the rough wall pipe measurements because of the limited number of datapoints.

10410 510610 7108

Re

0

0.05

0.1

0.15

Turbulence intensity

Pipe axis (Smooth pipe)

Pipe axis (Smooth pipe fit)

Pipe area (Smooth pipe)

Pipe area (Smooth pipe fit)

Pipe axis (Rough pipe)

Pipe area (Rough pipe)

Figure 5. Turbulence intensity for smooth and rough pipe ﬂow.

Fluids 2017,2, 30 5 of 13

4. Friction Factor

The ﬁts shown in Figure 5are:

ISmooth pipe axis =0.0550 ×Re−0.0407,

ISmooth pipe area =0.317 ×Re−0.110.(5)

The Blasius smooth pipe (Darcy) friction factor [12] is also expressed as an Re power-law:

λBlasius =0.3164 ×Re−0.25. (6)

The Blasius friction factor matches measurements best for

Re <

10

5

; the friction factor by e.g.,

Gersten (Equation (1.77) in [

13

]) is preferable for larger

Re

. The Blasius and Gersten friction factors

are compared in Figure 6. The deviation between the smooth and rough pipe Gersten friction factors

above

Re =

10

5

is qualitatively similar to the deviation between the smooth and rough pipe area TI in

Figure 5. For the Gersten friction factors, we have used the measured pipe roughnesses.

10410 510610 7108

Re

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Smooth pipe (Gersten)

Rough pipe (Gersten)

Smooth pipe (Blasius)

Figure 6. Friction factor.

For the smooth pipe, we can combine Equations (5) and (6) to relate the pipe area TI to the Blasius

friction factor: ISmooth pipe area =0.526 ×λ0.44

Blasius,

λBlasius =4.307 ×I2.27

Smooth pipe area.(7)

The TI and Blasius friction factor scaling is shown in Figure 7.

For axisymmetric ﬂow in the streamwise direction, the mean ﬂow velocity averaged over the pipe

area is:

vm=2

R2×ZR

0v(r)rdr. (8)

Now, we are in a position to deﬁne an average velocity of the turbulent ﬂuctuations:

hvRMSi=vmIPipe area =4

R4ZR

0v(r)rdrZR

0

vRMS(r)

v(r)rdr. (9)

The friction velocity is:

vτ=pτw/ρ, (10)

where τwis the wall shear stress and ρis the ﬂuid density.

Fluids 2017,2, 30 6 of 13

The relationship between hvRMSiand vτis illustrated in Figure 8. From the ﬁt, we have:

hvRMSi=1.8079 ×vτ, (11)

which we approximate as:

hvRMSi ∼ 9

5×vτ. (12)

Equations (11) and (12) above correspond to the usage of the friction velocity as a proxy for the

velocity of the turbulent ﬂuctuations [

14

]. We note that the rough wall velocities are higher than for

the smooth wall.

0 0.05 0.1 0.15

Turbulence intensity (pipe area)

0

0.01

0.02

0.03

0.04

0.05

0.06

Blasius

Smooth pipe

Figure 7. Relationship between pipe area turbulence intensity and the Blasius friction factor.

0 0.2 0.4 0.6 0.8

v [m/s]

0

0.2

0.4

0.6

0.8

1

1.2

vRMS [m/s]

Smooth pipe

Rough pipe

Fit

Approximation

Figure 8.

Relationship between friction velocity and the average velocity of the turbulent ﬂuctuations.

Equations (9) and (12) can be combined with Equation (1.1) in [15]:

λ=4τw

1

2ρv2

m

=−(∆P/L)D

1

2ρv2

m

=8×v2

τ

v2

m∼200

81 ×I2

Pipe area, (13)

where ∆Pis the pressure loss and Lis the pipe length. This can be reformulated as:

IPipe area ∼9

10√2×√λ. (14)

Fluids 2017,2, 30 7 of 13

We show how well this approximation works in Figure 9. Overall, the agreement is within 15%.

We proceed to deﬁne the average kinetic energy of the turbulent velocity ﬂuctuations

hEkin,RMSi

(per pipe volume V) as:

hEkin,RMSi/V=1

2ρhvRMSi2∼ −81

50 ×(∆P/L)D/4

=81

50 ×τw=81

50 ×v2

τρ,(15)

with V=LπR2, so we have:

hEkin,RMSi=1

2mhvRMSi2∼ −81

50 ×(π/2)R3∆P

=81

50 ×τwV=81

50 ×v2

τm,(16)

where

m

is the ﬂuid mass. The pressure loss corresponds to an increase of the turbulent kinetic energy.

The turbulent kinetic energy can also be expressed in terms of the mean ﬂow velocity and the TI or the

friction factor:

hvRMSi2=v2

mI2

Pipe area ∼81

200 ×v2

mλ. (17)

10410 510610 7108

Re

0

0.05

0.1

0.15

Turbulence intensity

Figure 9.

Turbulence intensity for smooth and rough pipe ﬂow. The approximation in Equation (14) is

included for comparison.

5. Discussion

5.1. The Attached Eddy Hypothesis

Our quantiﬁcation of the ratio

hvRMSi/vτ

as a constant can be placed in the context of the

attached eddy hypothesis by Townsend [16,17]. Our results are for quantities averaged over the pipe

radius, whereas the attached eddy hypothesis provides a local scaling with distance from the wall.

By proposing an overlap region (see Figure 1 in [

18

]) between the inner and outer scaling [

19

], it can be

deduced that

hvRMSi/vτ

is a constant in this overlap region [

20

,

21

]. Such an overlap region has been

shown to exist in [

2

,

20

]. The attached eddy hypothesis has provided the basis for theoretical work on

e.g., the streamwise turbulent velocity ﬂuctuations in ﬂat-plate [

22

] and pipe ﬂow [

23

] boundary layers.

Work on the law of the wake in wall turbulence also makes use of the attached eddy hypothesis [24].

As a consistency check for our results, we can compare the constant 9

/

5 in Equation (12) to the

prediction by Townsend:

vRMS,Townsend(r)2

v2

τ

=B1−A1ln R−r

r, (18)

where ﬁts have provided the constants

B1=

1.5 and

A1=

1.25. Here,

A1

is a universal constant,

whereas

B1

is not expected to be a constant for different wall-bounded ﬂows [

25

]. The constants are

Fluids 2017,2, 30 8 of 13

averages of ﬁts presented in [

3

] to the smooth- and rough-wall Princeton Superpipe measurements.

The Townsend-Perry constant A1was found to be 1.26 in [25]. Performing the area averaging yields:

hvRMS,Townsendi2

v2

τ

=B1+3

2×A1=3.38. (19)

Our ﬁnding is:

hvRMSi2

v2

τ∼9

52

=3.24, (20)

which is within 5% of the result in Equation (19). The reason that our result is smaller is that

Equation (18) is overpredicting the turbulence level close to the wall and close to the pipe axis.

Equation (18) as an upper bound has also been discussed in [26].

5.2. The Friction Factor and Turbulent Velocity Fluctuations

The proportionality between the average kinetic energy of the turbulent velocity ﬂuctuations

and the friction velocity squared has been identiﬁed in [

27

] for

Re >

10

5

. This corresponds to our

Equation (16).

A correspondence between the wall-normal Reynolds stress and the friction factor has been

shown in [

28

]. Those results were found using DNS. The main difference between the cases is that we

use the streamwise Reynolds stress. However, for an eddy rotating in the streamwise direction, both a

wall-normal and a streamwise component should exist which connects the two observations.

5.3. The Turbulence Intensity and the Diagnostic Plot

Other related work can be found beginning with [

29

] where the diagnostic plot was introduced.

In the following publications, a version of the diagnostic plot was brought forward where the local TI

is plotted as a function of the local streamwise velocity normalised by the free stream velocity [

30

–

32

].

Equation (3) in [31] corresponds to our ICore (see Equation (A1) in Appendix A).

5.4. Applicability of Turbulence Intensity Scaling with Friction Factor

The scaling of TI with the friction factor (Equation (14)) was found based on pipe ﬂow

measurements with two roughnesses. It is an open question whether our result holds in the fully

rough regime. For the fully rough regime, the friction factor becomes a constant for high

Re

. As a

consequence of our scaling expression, this should also be the case for the TI.

It is clear that the speciﬁc formula is not directly applicable for other wall-bounded ﬂows, since

B1

takes different values. However, the basic behaviour, i.e., that the TI scales with the square root of

the friction factor, may be universally valid.

6. Conclusions

We have compared TI proﬁles for smooth- and rough-wall pipe ﬂow measurements made in the

Princeton Superpipe.

The change of the TI proﬁle with increasing

Re

from hydraulically smooth to fully rough ﬂow

exhibits propagation from the pipe wall to the pipe axis. The TIR at rn=0.99 scales linearly with Re.

The scaling of TI with

Re

—on the pipe axis and averaged over the pipe area—shows that the

smooth- and rough-wall level deviates with increasing Reynolds number.

We ﬁnd that

IPipe area ∼9

10√2×√λ

. This relationship can be useful to calculate the TI given

a known

λ

, both for smooth and rough pipes. It follows that given a pressure loss in a pipe,

the turbulent kinetic energy increase can be estimated.

Acknowledgments: We thank Alexander J. Smits for making the Superpipe data publicly available.

Conﬂicts of Interest: The authors declare no conﬂict of interest.

Fluids 2017,2, 30 9 of 13

Appendix A. Fits to the Turbulence Intensity Proﬁle

As we have done for the smooth pipe measurements in [

5

], we can also ﬁt the rough pipe

measurements to this function:

I(rn) = ICore(rn) + IWall(rn)

=α+β×rγ

n+[δ×|ln(1−rn)|ε],(A1)

where

α

,

β

,

γ

,

δ

and

ε

are ﬁt parameters. A comparison of ﬁt parameters found for the smooth- and

rough-pipe measurements is shown in Figure A1. Overall, we can state that the ﬁt parameters for the

smooth and rough pipes are in a similar range for 106<Re <6×106.

10410 510610 7108

Re

0

0.02

0.04

0.06

0.08

0.1

Fit constant and multipliers [a.u.]

(Smooth pipe)

(Smooth pipe)

(Smooth pipe)

(Rough pipe)

(Rough pipe)

(Rough pipe)

(a)

10410 510610 7108

Re

0

1

2

3

4

5

Fit exponents [a.u.]

(Smooth pipe)

(Smooth pipe)

(Rough pipe)

(Rough pipe)

(b)

Figure A1.

Comparison of smooth- and rough-pipe ﬁt parameters, (

a

): ﬁt parameters

α

,

β

and

δ

;

(b): ﬁt parameters γand ε.

The min/max deviation of the rough pipe ﬁt from the measurements is below 10%; see the

comparison to the smooth wall ﬁt min/max deviation in Figure A2.

10410 510610 7108

Re

-15

-10

-5

0

5

10

15

Deviation of fit [%]

Smooth pipe

Mean

Min

Max

(a)

10410 510610 7108

Re

-15

-10

-5

0

5

10

15

Deviation of fit [%]

Rough pipe

Mean

Min

Max

(b)

Figure A2. Deviation of ﬁts to measurements; (a): smooth pipe, (b): rough pipe.

The core and wall ﬁts for the smooth and rough pipe ﬁts are compared in Figure A3.

Fluids 2017,2, 30 10 of 13

0 0.2 0.4 0.6 0.8 1

Normalized pipe radius

10-2

10-1

100

Turbulence intensity

Re = 9.94e+05

Core fit (Smooth pipe)

Core fit (Rough pipe)

Wall fit (Smooth pipe)

Wall fit (Rough pipe)

(a)

0 0.2 0.4 0.6 0.8 1

Normalized pipe radius

10-2

10-1

100

Turbulence intensity

Re = 1.98e+06

Core fit (Smooth pipe)

Core fit (Rough pipe)

Wall fit (Smooth pipe)

Wall fit (Rough pipe)

(b)

0 0.2 0.4 0.6 0.8 1

Normalized pipe radius

10-2

10-1

100

Turbulence intensity

Re = 3.83e+06

Core fit (Smooth pipe)

Core fit (Rough pipe)

Wall fit (Smooth pipe)

Wall fit (Rough pipe)

(c)

0 0.2 0.4 0.6 0.8 1

Normalized pipe radius

10-2

10-1

100

Turbulence intensity

Re = 5.63e+06

Core fit (Smooth pipe)

Core fit (Rough pipe)

Wall fit (Smooth pipe)

Wall fit (Rough pipe)

(d)

Figure A3.

Comparison of smooth and rough pipe core and wall ﬁts, (

a

): Re = 9.94e + 05;

(b): Re = 1.98e + 06; (c): Re = 3.83e + 06; (d): Re = 5.63e + 06.

The position where the core and wall TI levels are equal is shown in Figure A4. This position

does not change signiﬁcantly for the rough pipe; however, the position does increase with

Re

for the

smooth pipe: this indicates that the wall term becomes less important relative to the core term.

0123456

Re 106

0.985

0.99

0.995

Normalized pipe radius

Equal core and wall turbulence intensity

Smooth pipe

Rough pipe

Figure A4. Normalised pipe radius where the core and wall TI levels are equal.

Fluids 2017,2, 30 11 of 13

Appendix B. Arithmetic Mean Deﬁnition of Turbulence Intensity Averaged Over the Pipe Area

In the main paper, we have deﬁned the TI over the pipe area in Equation (4). In [

5

], we used the

arithmetic mean (AM) instead:

IPipe area, AM =1

RZR

0

vRMS(r)

v(r)dr. (A2)

The AM leads to a somewhat different pipe area scaling for the smooth pipe measurements, which

is illustrated in Figure A5. Compare to Figure 5.

10410 510610 7108

Re

0

0.05

0.1

0.15

Turbulence intensity

IPipe area, AM

Pipe axis (Smooth pipe)

Pipe axis (Smooth pipe fit)

Pipe area, AM (Smooth pipe)

Pipe area, AM (Smooth pipe fit)

Pipe axis (Rough pipe)

Pipe area, AM (Rough pipe)

Figure A5.

Turbulence intensity for smooth and rough pipe ﬂow. The arithmetic mean (AM) is used

for the pipe area TI.

The scaling found in [5] using this deﬁnition is:

ISmooth pipe area, AM =0.227 ×Re−0.100. (A3)

The AM scaling also has implications for the relationship with the Blasius friction factor scaling

(Equation (7)):

ISmooth pipe area, AM =0.360 ×λ0.4

Blasius,

λBlasius =12.89 ×I2.5

Smooth pipe area, AM.(A4)

We can now deﬁne the AM version of the average velocity of the turbulent ﬂuctuations:

hvRMSiAM =vmIPipe area, AM =2

R3ZR

0v(r)rdrZR

0

vRMS(r)

v(r)dr. (A5)

The AM deﬁnition can be considered as a ﬁrst order moment equation for

vRMS

, whereas the

deﬁnition in Equation (9) is a second order moment equation.

Again, we ﬁnd that the AM average turbulent velocity ﬂuctuations are proportional to the

friction velocity. However, the constant of proportionality is different than the one in Equation (11)

(see Figure A6). The AM case can be ﬁtted as:

hvRMSiAM =1.4708 ×vτ, (A6)

which we approximate as:

Fluids 2017,2, 30 12 of 13

hvRMSiAM ∼r2

3×9

5×vτ∼r2

3×hvRMS i. (A7)

0 0.2 0.4 0.6 0.8

v [m/s]

0

0.2

0.4

0.6

0.8

1

vRMS AM [m/s]

IPipe area, AM

Smooth pipe

Rough pipe

Fit

Approximation

Figure A6.

Relationship between friction velocity and the AM average velocity of the

turbulent ﬂuctuations.

As we did in Section 5, we can perform the AM averaging of Equation (18) (also done in [26]):

hvRMS,Townsendi2

AM

v2

τ

=B1+A1=2.75, (A8)

where we ﬁnd: hvRMSi2

AM

v2

τ∼2

3×9

52

=2.16. (A9)

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