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fluids

Article

Turbulence Intensity Scaling: A Fugue

Nils T. Basse

Elsas väg 23, 423 38 Torslanda, Sweden; nils.basse@npb.dk

Received: 05 July 2019; Accepted: 30 September 2019; Published: 9 October 2019

Abstract:

We study streamwise turbulence intensity deﬁnitions using smooth- and rough-wall

pipe ﬂow measurements made in the Princeton Superpipe. Scaling of turbulence intensity with

the bulk (and friction) Reynolds number is provided for the deﬁnitions. The turbulence intensity

scales with the friction factor for both smooth- and rough-wall pipe ﬂow. Turbulence intensity

deﬁnitions providing the best description of the measurements are identiﬁed. A procedure to

calculate the turbulence intensity based on the bulk Reynolds number (and the sand-grain roughness

for rough-wall pipe ﬂow) is outlined.

Keywords:

streamwise turbulence intensity deﬁnitions; Princeton Superpipe measurements; smooth-

and rough-wall pipe ﬂow; friction factor; computational ﬂuid dynamics boundary conditions

1. Introduction

The turbulence intensity (TI) is of great importance in, e.g., industrial ﬂuid mechanics, where it

can be used for computational ﬂuid dynamics (CFD) simulations as a boundary condition [

1

]. The TI

is at the center of the fruitful junction between fundamental and industrial ﬂuid mechanics.

This paper contains an extension of the TI scaling research in [

2

] (smooth-wall pipe ﬂow) and [

3

]

(smooth- and rough-wall pipe ﬂow). As in those papers, we treat streamwise velocity measurements

from the Princeton Superpipe [

4

,

5

]. The measurements were done at low speed in compressed air

with a pipe radius of about 65 mm. Details on, e.g., the bulk Reynolds number range and uncertainty

estimates can be found in [

4

]. Other published measurements including additional velocity components

can be found in [6] (smooth-wall pipe ﬂow) and [7,8] (smooth- and rough-wall pipe ﬂow).

Our approach to streamwise TI scaling is global averaging; physical mechanisms include separate

inner- and outer-region phenomena and interactions between those [

9

]. Here, the inner (outer) region

is close to the pipe wall (axis), respectively.

The local TI deﬁnition (see, e.g., Figure 9 in [10]) is:

I(r) = vRMS(r)

v(r), (1)

where

r

is the radius (

r=

0 is the pipe axis and

r=R

is the pipe wall),

v(r)

is the local mean streamwise

ﬂow velocity, and

vRMS(r)

is the local root-mean-square (RMS) of the turbulent streamwise velocity

ﬂuctuations. Using the radial coordinate

r

means that outer scaling is employed for the position.

As was done in [

2

,

3

], we use

v

as the streamwise velocity (in much of the literature,

u

is used for the

streamwise velocity).

The measurements in [

10

] were on turbulent ﬂow in a two-dimensional channel and similar work

for pipe ﬂow was published in [11].

In this paper, we study TI deﬁned using a global (radial) averaging of the streamwise velocity

ﬂuctuations. The mean ﬂow is either included in the global averaging or as a reference velocity.

This covers the majority of the standard TI deﬁnitions.

There is a plethora of TI deﬁnitions, which is why we use the term fugue in the title. This is

inspired by [12], where Frank Herbert’s Dune novels [13] are interpreted as “an ecological fugue”.

Fluids 2019,11, 5842; doi:10.3390/ﬂuids11205842 www.mdpi.com/journal/ﬂuids

Fluids 2019,11, 5842 2 of 13

The ultimate purpose of our work is to be able to present a robust and well-researched formulation

of the TI; an equivalent TI in the presence of shear ﬂow. Our work is not adding signiﬁcant knowledge

of the fundamental processes [

14

–

16

], but we need to understand them in order to use them as a

foundation for the scaling expressions.

The main contributions of this paper compared to [3] are:

•The introduction of additional deﬁnitions of the TI

•Log-law ﬁts in addition to power-law ﬁts

•

New ﬁndings on the rough pipe friction factor behaviour of the Princeton Superpipe

measurements.

Furthermore, we include a discussion on the link between the TI and the friction factor [

3

,

17

] in

the light of the Fukagata–Iwamoto–Kasagi (FIK) identity [18].

Our paper is organised as follows: In Section 2, we introduce the velocity deﬁnitions. These are

used in Section 3to deﬁne various TI expressions. In Section 4we present scaling laws using the

presented deﬁnitions. We discuss our ﬁndings in Section 5and conclude in Section 6.

2. Velocity Deﬁnitions

The friction velocity is:

vτ=pτw/ρ, (2)

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

The area-averaged (AA) velocity of the turbulent ﬂuctuations is:

hvRMSiAA =2

R2×ZR

0vRMS(r)rdr(3)

The ﬁt between vτand hvRMSiAA is shown in Figure 1:

hvRMSiAA =1.7277 ×vτ(4)

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 AA [m/s]

Smooth pipe

Rough pipe

Fit

Figure 1. Relationship between vτand hvRMS iAA.

The velocity on the pipe axis is the centerline (CL) velocity:

vCL =v(r=0)(5)

Fluids 2019,11, 5842 3 of 13

The (area-averaged) mean velocity is given by:

vm=2

R2×ZR

0v(r)rdr(6)

The difference between the centerline and the mean velocity scales with the friction velocity.

The corresponding ﬁt is shown in Figure 2:

vCL −vm=4.4441 ×vτ, (7)

where the ﬁt constant is close to the value of 4.28 [

19

] found using earlier Princeton Superpipe

measurements [20].

0 0.2 0.4 0.6 0.8

v [m/s]

0

1

2

3

4

vCL-vm [m/s]

Smooth pipe

Rough pipe

Fit

Figure 2. Relationship between vτand vCL −vm.

3. Turbulence Intensity Deﬁnitions

3.1. Local Velocity Deﬁnitions

The arithmetic mean (AM) deﬁnition is:

IPipe area, AM =1

RZR

0

vRMS(r)

v(r)dr(8)

The area-averaged deﬁnition is:

IPipe area, AA =2

R2ZR

0

vRMS(r)

v(r)rdr(9)

In [

3

] (Equation (9)),

hvRMSi

was deﬁned as the product of

vm

and

IPipe area, AA

. Comparing the

resulting Equations (11) and (12) in [

3

] to the current Equation (4), we ﬁnd a difference of less than 5%

(9/5 compared to 1.7277).

Finally, the volume-averaged (VA) deﬁnition (inspired by the FIK identity) is:

IPipe area, VA =3

R3ZR

0

vRMS(r)

v(r)r2dr(10)

Fluids 2019,11, 5842 4 of 13

3.2. Reference Velocity Deﬁnitions

As mentioned in the Introduction, we use outer scaling for the radial position

r

. For the TI,

we separate the treatment to inner and outer scaling below, see, e.g., [15].

3.2.1. Inner Scaling

For inner scaling, we deﬁne the TI using vτas the reference velocity:

Iτ=hvRMSiAA

vτ

=1.7277, (11)

where the ﬁnal equation is found using Equation (4).

The square of the local version of this,

I2

τ(r) = vRMS (r)2

v2

τ

, is often used as the TI in the literature,

see also Section 5.1 in [

3

]. This is the normal streamwise Reynolds stress normalised by the friction

velocity squared.

Iτis shown in Figure 3; no scaling is observed with ReD, the bulk Reynolds number:

ReD=Dvm

νkin , (12)

where D=2Ris the pipe diameter and νkin is the kinematic viscosity.

104105106107108

ReD

1

1.5

2

2.5

3

I

Smooth pipe

Rough pipe

1.7277

Figure 3. Iτas a function of ReD.

3.2.2. Outer Scaling

For outer scaling, we use either vmor vCL as the reference velocity.

The TI using vmas the reference velocity is:

Im=hvRMSiAA

vm(13)

Finally, the TI using vCL as the reference velocity [15] is:

ICL =hvRMSiAA

vCL (14)

Fluids 2019,11, 5842 5 of 13

4. Turbulence Intensity Scaling Laws

The Princeton Superpipe measurements and the TI deﬁnitions in Section 3are used to create the

TI data points. Thereafter we ﬁt the points using the power-law ﬁt:

QPower−law ﬁt(x) = a×xb, (15)

and the log-law ﬁt:

QLog−law ﬁt(x) = c×ln(x) + d(16)

Here,

a

,

b

,

c

, and

d

are constants.

Q

is the quantity to ﬁt and

x

is a corresponding variable. We ﬁrst

apply the two ﬁts using Q=Iand x=ReD.

The log-law ﬁt is obtained by taking the (natural) logarithm of the power-law ﬁt. The reason we

use these two ﬁts is that they have been discussed in the literature [

21

,

22

] as likely scaling candidates.

We apply the two ﬁts to the measurements and calculate the resulting root-mean-square deviations

(RMSD) between the ﬁts and the measurements. A small RMSD means that the ﬁt is closer to

the measurements.

Note that the smooth pipe

ReD

measurement range is much larger than the rough pipe

ReD

measurement range: a factor of 74 (9 points) compared to a factor of 6 (4 points). The consequence is a

major uncertainty in the rough pipe results, e.g., (i) ﬁts and (ii) extrapolation.

It is also important to be aware that we only have two sets of measurements with the following

sand-grain roughnesses ks:

•Smooth pipe: ks=0.45 µm [23]

•Rough pipe: ks=8µm [24] (see the related discussion in Section 5.1)

The results are presented in Figures 4and 5and Tables 1–4.

We do not discuss the quality of ﬁts to the rough pipe, since there are only 4 measurements for a

single ks. Thus, these values are provided as a reference.

For the smooth pipe, the power-law ﬁts perform slightly better than the log-law ﬁts, except for

the CL deﬁnition. The best ﬁt is using the power-law ﬁt to the AA deﬁnition of the TI:

IPipe area, AA =0.3173 ×Re−0.1095

D, (17)

which is the same as Equation (5) in [3].

Table 1. Power-law ﬁt constants, smooth pipe.

TI Deﬁnition a b RMSD

IPipe area, AM 0.2274 −0.1004 4.0563 ×10−4

IPipe area, AA 0.3173 −0.1095 3.5932 ×10−4

IPipe area, VA 0.3758 −0.1134 3.6210 ×10−4

Im0.2657 −0.1000 5.2031 ×10−4

ICL 0.1811 −0.0837 6.9690 ×10−4

Table 2. Log-law ﬁt constants, smooth pipe.

TI Deﬁnition c d RMSD

IPipe area, AM −0.0059 0.1391 6.7748 ×10−4

IPipe area, AA −0.0080 0.1808 8.8173 ×10−4

IPipe area, VA −0.0093 0.2074 1.1018 ×10−3

Im−0.0069 0.1634 5.8899 ×10−4

ICL −0.0049 0.1257 5.4179 ×10−4

Fluids 2019,11, 5842 6 of 13

Table 3. Power-law ﬁt constants, rough pipe.

TI Deﬁnition a b RMSD

IPipe area, AM 0.1172 −0.0522 4.0830 ×10−4

IPipe area, AA 0.1702 −0.0638 3.5784 ×10−4

IPipe area, VA 0.1989 −0.0667 3.7697 ×10−4

Im0.1568 −0.0610 3.4902 ×10−4

ICL 0.1177 −0.0519 3.4317 ×10−4

Table 4. Log-law ﬁt constants, rough pipe.

TI Deﬁnition c d RMSD

IPipe area, AM −0.0028 0.0960 4.3111 ×10−4

IPipe area, AA −0.0042 0.1291 4.0028 ×10−4

IPipe area, VA −0.0050 0.1477 4.2938 ×10−4

Im−0.0039 0.1213 3.8575 ×10−4

ICL −0.0028 0.0967 3.6542 ×10−4

104105106107108

ReD

0.04

0.06

0.08

0.1

0.12

0.14

Turbulence intensity

Power-law fit, smooth pipe

Pipe area, VA

Pipe area, VA (fit)

Pipe area, AA

Pipe area, AA (fit)

Pipe area, AM

Pipe area, AM (fit)

Mean

Mean (fit)

CL

CL (fit)

104105106107108

ReD

0.04

0.06

0.08

0.1

0.12

0.14

Turbulence intensity

Log-law fit, smooth pipe

Pipe area, VA

Pipe area, VA (fit)

Pipe area, AA

Pipe area, AA (fit)

Pipe area, AM

Pipe area, AM (fit)

Mean

Mean (fit)

CL

CL (fit)

Figure 4.

Smooth pipe turbulence intensity as a function of

ReD

, left: Power-law ﬁt, right: Log-law ﬁt.

104105106107108

ReD

0.04

0.06

0.08

0.1

0.12

0.14

Turbulence intensity

Power-law fit, rough pipe

Pipe area, VA

Pipe area, VA (fit)

Pipe area, AA

Pipe area, AA (fit)

Pipe area, AM

Pipe area, AM (fit)

Mean

Mean (fit)

CL

CL (fit)

104105106107108

ReD

0.04

0.06

0.08

0.1

0.12

0.14

Turbulence intensity

Log-law fit, rough pipe

Pipe area, VA

Pipe area, VA (fit)

Pipe area, AA

Pipe area, AA (fit)

Pipe area, AM

Pipe area, AM (fit)

Mean

Mean (fit)

CL

CL (fit)

Figure 5. Rough pipe turbulence intensity as a function of ReD, left: Power-law ﬁt, right: Log-law ﬁt.

Instead of ReD, one can also express the TI ﬁts using the friction Reynolds number [4]:

Reτ=Rvτ

νkin =vτ

2vm×ReD(18)

The relationship between

ReD

and

Reτ

can ﬁtted using Equation (15) where

Q=Reτ

and

x=ReD

,

see Figure 6and Table 5. A log-law ﬁt was also performed but resulted in a bad ﬁt, i.e., a RMSD which

was between one and two orders of magnitude larger than for the power-law ﬁt. As mentioned above,

Fluids 2019,11, 5842 7 of 13

the rough pipe ﬁt is only provided as a reference. For channel ﬂow, it has been found that

a=

0.09 and

b=0.88, see Figure 7.11 in [25] and associated text.

We choose to focus on the bulk Reynolds number since it is possible to determine for applications

where the friction velocity is unknown.

104105106107108

ReD

102

104

106

Re

Power-law fit

Smooth pipe

Smooth pipe fit

Rough pipe

Rough pipe fit

Figure 6. Relationship between ReDand Reτ.

Table 5. Bulk and friction Reynolds number ﬁts.

Case a b

Smooth 0.0621 0.9148

Rough 0.0297 0.9675

5. Discussion

5.1. Friction Factor

In [

26

], the following expression for the smooth pipe friction factor has been derived based on

Princeton Superpipe measurements:

1

√λSmooth

=1.930 log10 ReDpλSmooth−0.537 (19)

A corresponding rough pipe friction factor has been proposed in [27]:

1

qλRough

=−2 log10

ks

3.7D+2.51

ReDqλRough

(20)

The friction factor can also be expressed using the friction velocity and the mean velocity,

see Equation (1.1) in [26]:

λ=8×v2

τ

v2

m

(21)

or:

vτ

vm

=rλ

8(22)

Fluids 2019,11, 5842 8 of 13

The equations for the smooth- and rough-wall pipe ﬂow friction factors are shown in Figure 7,

along with Princeton Superpipe measurements.

We have included additional smooth pipe measurements [

20

,

28

]. Both sets agree with

Equation (19).

Additional rough pipe measurements can be found in Table 2 (and Figure 3) in [

24

]. Here, it was

found that

ks=

8

µ

m. For our main data set [

4

,

5

],

ks=

8

µ

m does not match the measurements;

instead we get

ks=

3

µ

m for a ﬁt to those measurements using Equation (20). It is the same pipe;

the reason for the discrepancy is not clear [

29

]. However, the difference is within the experimental

uncertainty of 5% stated in [24].

For the rough pipe friction factor, we use ks=3µm in the remainder of this paper.

104105106107108

Re D

0.005

0.01

0.015

0.02 Smooth pipe

Meas. (Hultmark et al. 2013)

Meas. (McKeon et al. 2004)

Eq. (McKeon et al. 2005)

104105106107108

Re D

0.005

0.01

0.015

0.02 Rough pipe

Meas. (Hultmark et al. 2013)

Meas. (Langelandsvik et al. 2008)

Eq. (Colebrook 1939, ks=8 m)

Eq. (Colebrook 1939, ks=3 m)

Figure 7. Friction factors, left: Smooth pipe, right: Rough pipe.

5.2. Turbulence Intensity Aspects

5.2.1. Importance for Flow

The TI is an important quantity for many physical phenomena [30], e.g.:

•The critical Reynolds number for the drag of a sphere [31]

•The laminar–turbulent transition [32]

•Development of the turbulent boundary layer [33]

•The position of ﬂow separation [34]

•Heat transfer [35]

•Wind farms [36]

•Wind tunnels [37].

We mention these examples to illustrate the importance of the TI for real world applications.

5.2.2. Scaling with the Friction Factor

The wall-normal [

17

] and streamwise (this paper and [

3

]) Reynolds stress have both been shown

to be linked to the friction factor

λ=

4

Cf

, where

Cf

is the skin friction coefﬁcient. These observations

can be interpreted as manifestations of the FIK identity [

18

], where an equation for

Cf

is derived based

on the streamwise momentum equation:

Cf

is proportional to the integral over the Reynolds shear

stress weighted by the quadratic distance from the pipe axis.

An alternative formulation for

Cf

based on the streamwise kinetic energy is derived in [

38

]: here,

Cf

is proportional to the integral over the Reynolds shear stress multiplied by the streamwise mean

velocity gradient weighted by the distance from the pipe axis. It is concluded that the logarithmic

region dominates friction generation for high Reynolds number ﬂow. The dominance of the logarithmic

region has been conﬁrmed in [39].

Fluids 2019,11, 5842 9 of 13

The seeming equivalence between Reynolds stress (shear or normal) and the friction factor leads

us to propose that the TI scales with

vτ/vm

. Therefore we ﬁt to Equations (15) and (16) using

Q=I

and x=vτ/vm, see Figure 8and Tables 6and 7.

In this case, the power-law ﬁts are best for the local velocity deﬁnitions and the log-law ﬁts are

best for the reference velocity deﬁnitions. Overall, the best ﬁt is the power-law ﬁt using the AM

deﬁnition of the TI:

IPipe area, AM =0.6577 ×λ0.5531 , (23)

which is a modiﬁcation of Equation (14) in [3].

0 0.02 0.04 0.06

v /vm

0

0.05

0.1

0.15

Turbulence intensity

Power-law fit

Pipe area, VA

Pipe area, VA (fit)

Pipe area, AA

Pipe area, AA (fit)

Pipe area, AM

Pipe area, AM (fit)

Mean

Mean (fit)

CL

CL (fit)

0 0.02 0.04 0.06

v /vm

0

0.05

0.1

0.15

Turbulence intensity

Log-law fit

Pipe area, VA

Pipe area, VA (fit)

Pipe area, AA

Pipe area, AA (fit)

Pipe area, AM

Pipe area, AM (fit)

Mean

Mean (fit)

CL

CL (fit)

Figure 8. TI as a function of vτ/vm.

Table 6. Power-law ﬁt constants.

TI Deﬁnition a b RMSD

IPipe area, AM 2.0776 1.1062 4.7133 ×10−4

IPipe area, AA 3.5702 1.2088 5.0224 ×10−4

IPipe area, VA 4.6211 1.2530 4.9536 ×10−4

Im2.4238 1.1039 6.8412 ×10−4

ICL 1.1586 0.9260 7.5876 ×10−4

Table 7. Log-law ﬁt constants.

TI Deﬁnition c d RMSD

IPipe area, AM 0.0658 0.2715 5.5459 ×10−4

IPipe area, AA 0.0890 0.3602 6.6955 ×10−4

IPipe area, VA 0.1036 0.4162 8.0775 ×10−4

Im0.0775 0.3195 5.5182 ×10−4

ICL 0.0551 0.2367 5.8536 ×10−4

Combining Equation (22) with Equations (15) and (16) leads to:

IPower−law ﬁt(λ) = a×λ

8b/2

(24)

ILog−law ﬁt(λ) = c

2×ln λ

8+d(25)

The predicted TI for the best case (power-law AM) is shown in Figure 9; one reason for the better

match to measurements compared to Figure 9 in [

3

] is that we use

ks=

3

µ

m instead of

ks=

8

µ

m

for the rough pipe. The correspondence between the TI and the friction factor means that the TI will

Fluids 2019,11, 5842 10 of 13

approach a constant value for rough-wall pipe ﬂow at large

ReD

(fully rough regime). It also means

that a larger ksleads to a higher TI.

104105106107108

ReD

0.04

0.06

0.08

0.1

0.12

0.14

Turbulence intensity

Figure 9. AM deﬁnition of TI as a function of ReDfor smooth- and rough-wall pipe ﬂow.

5.2.3. CFD Deﬁnition

Let us now consider a typical CFD turbulence model, the standard

k−ε

model [

40

]. Here,

k

is the

turbulent kinetic energy (TKE) per unit mass and εis the rate of dissipation of TKE per unit mass.

As an example of a boundary condition, the user provides the TI (

Iuser

) and the turbulent viscosity

ratio

µt/µ

, where

µt

is the dynamic turbulent viscosity and

µ

is the dynamic viscosity. For a deﬁned

reference velocity vref,kcan then be calculated as:

k=3

2(vref Iuser)2(26)

As the next step, εis deﬁned as:

ε=ρCµk2

(µt/µ)µ, (27)

where ρis density and Cµ=0.09.

An example of default CFD settings is Iuser =0.01 (1%) and µt/µ=10.

The output from a CFD simulation is the total TKE, not the individual components. If we assume

that the turbulence is isotropic, the streamwise TI we are treating in this paper is proportional to the

square root of the TKE:

vRMS =r2

3k(28)

5.2.4. Proposed Procedure for CFD and an Example

A standard deﬁnition of TI for CFD is to use the free-stream velocity as the reference velocity, i.e.,

ICL

for pipe ﬂow. For

ICL

, we use the log-law version since this has the smallest RMSD, see Tables 6

and 7:

ICL =0.0276 ×ln(λ) + 0.1794 (29)

We note that this scaling is based on the Princeton Superpipe measurements; for industrial

applications, the TI may be much higher, so our scaling should be considered as a lower limit.

A procedure to calculate the TI, e.g., for use in CFD is:

Fluids 2019,11, 5842 11 of 13

1. Deﬁne ReD(and ksfor a rough pipe)

2.

Calculate the friction factor: Equation (19) for a smooth pipe and Equation (20) for a rough pipe

3. Use Equation (29) to calculate the TI.

As a concrete example, we consider incompressible (water) ﬂow through a 130 mm diameter pipe.

CFD boundary conditions can be velocity inlet and pressure outlet. The steps are:

1. Deﬁne the mean velocity, we use 10 m/s

2. Calculate ReD= 1.3 ×106

3. Use Equation (19) to calculate λSmooth =0.0114

4. Use Equation (29) to calculate ICL =0.0561.

For this example, we conclude that the minimum TI is 5.6%. A code with this example is available

as Supplementary Materials, a link is provided after the Conclusions.

5.2.5. Open Questions

It remains an open question to what extent the quality of the ﬁts (RMSD) impacts the outcome of

a CFD simulation. For the TI deﬁnitions used, the RMSD varies less than a factor of two for the ﬁts

of TI as a function of

vτ/vm

, see Tables 6and 7. For CFD, a single TI deﬁnition is used, so it is not

possible to switch TI models in CFD and compare simulations to measurements.

To continue our measurement-based research, we would need measurements of all velocity

components, i.e., wall-normal and spanwise, in addition to the available streamwise measurements.

As mentioned earlier [

2

], it would also be interesting to have measurements for higher Mach

numbers, where compressibility will play a larger role.

In addition to pipe ﬂow, other canonical ﬂows, such as zero-pressure gradient ﬂows [

14

], might

be suitable for analysis similar to what we have presented.

6. Conclusions

We have used Princeton Superpipe measurements of smooth- and rough-wall pipe ﬂow [

4

,

5

] to

study the properties of various TI deﬁnitions. The scaling of TI with

ReD

is provided for the deﬁnitions.

For scaling purposes, we recommend the AA deﬁnition and a power-law ﬁt: Equation (17). The TI

also scales with

vτ/vm

, where the best result is obtained with a power-law ﬁt and the AM deﬁnition.

This ﬁt implies that the turbulence level scales with the friction factor: Equation (23).

Scaling of TI with ReDand vτ/vmwas done using both power-law and log-law ﬁts.

A proposed procedure to calculate the TI, e.g., CFD is provided and exempliﬁed in Section 5.2.4.

Supplementary Materials:

The following is available online at https://www.researchgate.net/publication/

336374461_Python_code_to_calculate_turbulence_intensity_based_on_Reynolds_number_and_surface_

roughness, Supplementary Material: Python code to calculate turbulence intensity based on Reynolds number

and surface roughness.

Funding: This research received no external funding.

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

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

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