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Turbulence intensity in the transition from hydraulically

smooth to fully rough pipe ﬂow

Nils T. Bassea

aToftehøj 23, Høruphav, 6470 Sydals, Denmark

January 4, 2017

Abstract

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

ﬂow measurements made in the Princeton Superpipe. The proﬁle develop-

ment 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 propose a

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 turbulence in smooth and rough pipes have been carried

out in the Princeton Superpipe [1] [2] [3]. We have treated the smooth pipe

measurements as a part of [4]; here, we focused on the turbulence intensity

(TI) Iwhich is deﬁned as:

I=vRMS

v,(1)

where vis the mean ﬂow velocity and vRMS is the RMS of the turbulent

velocity ﬂuctuations. Only streamwise components are considered.

In this paper, we add the rough pipe measurements to our previous anal-

ysis. The smooth (rough) pipe had a radius Rof 64.68 (64.92) mm and an

Email address: nils.basse@npb.dk (Nils T. Basse)

1

RMS roughness of 0.15 (5) µm, respectively. The corresponding sand-grain

roughness is 0.45 (8) µm [5].

The smooth pipe is hydraulically smooth for all Reynolds numbers Re

covered. The rough pipe evolves from hydraulically smooth through transi-

tionally rough to fully rough with increasing Re.

Our paper is structured as follows: In Section 2, we (i) study how the

TI proﬁles change over the transition from smooth to rough pipe ﬂow and

(ii) ﬁt the proﬁles to an analytical expression. Thereafter we present the

resulting scaling of the TI with Re in Section 3. A discussion of the possible

correspondence between the friction factor and the TI is contained in Section

4. Finally, we conclude in Section 5.

2. Turbulence intensity proﬁles

2.1. Post-processed measurements

We have constructed the TI proﬁles for the measurements available, see

Fig. 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.

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

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

Figure 1: Turbulence intensity as a function of pipe radius, left: Smooth pipe, right:

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. Further, we use a normalized pipe radius

rnto account for the diﬀerence in smooth and rough pipe radii. The result

is a comparison of the TI proﬁles at four Re, see Fig. 2. As Re increases, we

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

2

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

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

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

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

Figure 2: Comparison of smooth and rough pipe TI proﬁles for the four Re values where

the rough pipe measurements are done.

3

To make the comparison more quantitative, we deﬁne the turbulence

intensity ratio (TIR):

rI,Rough/Smooth =IRough

ISmooth

=vRMS,Rough

vRMS,Smooth ×vSmooth

vRough

(2)

The TIR is shown in Fig. 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 Re, 1.98 ×106.

The events close to the wall are most clearly seen in the right-hand plot

of Fig. 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. So the 0.13 mm closest to the wall are not considered.

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

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

Figure 3: Turbulence intensity ratio, left: All radii, right: Zoom to outer 5%.

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

ﬁxed rnvs. Re, see Fig. 4. From this plot we observed that the peak close

to the wall (rn= 0.99) increases linearly with Re.

2.2. Fits

As we have done for the smooth pipe measurements in [4], we can also ﬁt

the rough pipe measurements to this function:

4

0 1 2 3 4 5 6

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

Figure 4: Turbulence intensity ratios for ﬁxed rn.

5

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

= [α+β×rγ

n] + [δ× |ln(1 −rn)|ε],(3)

where α,β,γ,δand εare ﬁt parameters. A comparison of ﬁt parameters

found for the smooth- and rough-pipe measurements is shown in Fig. 5.

Overall, we can state that the ﬁt parameters for the smooth and rough pipes

are in a similar range for 106< Re < 6×106.

104105106107108

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)

104105106107108

Re

0

1

2

3

4

5

Fit exponents [a.u.]

(Smooth pipe)

(Smooth pipe)

(Rough pipe)

(Rough pipe)

Figure 5: Comparison of smooth- and rough-pipe ﬁt parameters.

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

Fig. 6.

104105106107108

Re

-15

-10

-5

0

5

10

15

Deviation of fit [%]

Smooth pipe

Mean

Min

Max

104105106107108

Re

-15

-10

-5

0

5

10

15

Deviation of fit [%]

Rough pipe

Mean

Min

Max

Figure 6: Deviation of ﬁts to measurements, left: Smooth pipe, right: Rough pipe.

The core and wall ﬁts for the smooth and rough pipe ﬁts are compared

in Fig. 7. Both the core and wall TI increase for the largest Re.

6

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)

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)

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)

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)

Figure 7: Comparison of smooth and rough pipe core and wall ﬁts.

7

The position where the core and wall TI levels are equal is shown in Fig.

8. 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 8: Normalised pipe radius where the core and wall TI levels are equal.

3. Turbulence intensity scaling

Scaling of the TI with Re for smooth pipe ﬂow has been covered in [4];

measurements for rough pipe ﬂow are added in Fig. 9. For Re = 106, 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.

We have not made ﬁts to the rough wall pipe measurements because of the

limited number of datapoints.

To quantify the diﬀerences, we interpolate the smooth pipe measurements

to the Re where the rough pipe measurements were made. Then we construct

8

104105106107108

Re

0

0.02

0.04

0.06

0.08

0.1

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 9: Turbulence intensity for smooth and rough pipe ﬂow.

9

the TIR, both for TI on the pipe axis and for TI averaged over the pipe area,

see Fig. 10. The pipe axis TIR only shows a change for the largest Re (same

as in Fig. 4), whereas the pipe area TIR increases for all Re.

0 1 2 3 4 5 6

Re 106

0.95

1

1.05

1.1

1.15

1.2

Turbulence intensity ratio (rough/smooth)

Pipe axis

Pipe area

Figure 10: Turbulence intensity ratios for pipe axis and pipe area.

4. Friction factor

The ﬁts shown in Fig. 9 have been derived in [4] and are repeated here:

ISmooth pipe axis = 0.0550 ×Re−0.0407

ISmooth pipe area = 0.227 ×Re−0.100 (4)

We note that the Blasius smooth pipe (Darcy) friction factor [6] is also

expressed as an Re power-law:

λBlasius = 0.3164 ×Re−0.25 (5)

10

The Blasius friction factor matches measurements best for Re < 105; the

friction factor by e.g. Gersten (Eq. (1.77) in [7]) is preferable for larger

Re. The Blasius and Gersten friction factors are compared in Fig. 11. The

deviation between the smooth and rough pipe Gersten friction factors above

Re = 105is qualitatively similar to the deviation between the smooth and

rough pipe area TI in Fig. 9. For the Gersten friction factors, we have used

the measured pipe roughnesses.

104105106107108

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 11: Friction factor.

For the smooth pipe, we can combine Eqs. (4) and (5) to link the pipe

area TI and the Blasius friction factor:

ISmooth pipe area = 0.360 ×λ0.4

Blasius

λBlasius = 12.89 ×I2.5

Smooth pipe area

(6)

The TI and Blasius friction factor scaling is shown in Fig. 12.

The relationship between the pipe area TI and a generalized friction factor

can be derived by assuming these scalings:

11

0 0.02 0.04 0.06 0.08 0.1

Turbulence intensity (pipe area)

0

0.01

0.02

0.03

0.04

0.05

Blasius

Smooth pipe

Figure 12: Relationship between pipe area turbulence intensity and the Blasius friction

factor.

12

IPipe area =a×Reb

λ=c×Red,(7)

where a,b,cand dare ﬁt parameters. In principle, this can be used for both

smooth- and rough-wall pipes. So we can express the pipe area TI and the

friction factor as:

IPipe area =a×λ

cb/d

λ=c×IPipe area

ad/b (8)

From Eq. (6) we propose a general approximation:

IPipe area ∼√λ/2 (9)

Eq. (9) can be combined with Eq. (1.1) in [8]:

λ=4τw

1

2ρv2

m

=−(∆P/L)D

1

2ρv2

m

= 8 ×v2

τ

v2

m∼4I2

Pipe area,(10)

where τwis the wall shear stress, ρis the ﬂuid density, vmis the mean ﬂow

velocity averaged over the pipe area, ∆Pis the pressure loss, Lis the pipe

length, Dis the pipe diameter and vτ=pτw/ρ is the friction velocity.

For axisymmetric ﬂow in the streamwise direction, we have:

vm=2

R2×ZR

0

v(r)rdr(11)

We have deﬁned the TI averaged over the pipe area as:

IPipe area =1

RZR

0

vRMS(r)

v(r)dr(12)

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

ﬂuctuations:

hvRMSi=vmIPipe area =2

R3ZR

0

v(r)rdrZR

0

vRMS(r)

v(r)dr(13)

13

From Eqs. (10) and (13), we ﬁnd that the average turbulent velocity

ﬂuctuations are proportional to the friction velocity:

hvRMSi ∼ √2vτ(14)

The relationship in Eq. (14) is illustrated in Fig. 13.

In Appendix A, we provide results using an alternative deﬁnition of the

TI averaged over the pipe area.

0 0.2 0.4 0.6 0.8

v [m/s]

0

0.2

0.4

0.6

0.8

1

vRMS [m/s]

Smooth pipe

Rough pipe

Eq. (14)

Figure 13: Relationship between friction velocity and the average velocity of the turbulent

ﬂuctuations.

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∼ −(∆P/L)D/4 = τw,(15)

with V=LπR2so we have:

14

hEkin,RMSi=1

2mhvRMSi2∼ −(π/2) R3∆P=τwV, (16)

where mis the ﬂuid mass. The pressure loss corresponds to an increase of

the turbulent kinetic energy.

5. Conclusions

We have compared TI proﬁles for smooth- and rough-wall pipe ﬂow mea-

surements made in the Princeton Superpipe.

The change of the TI proﬁle 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. We show that the rough pipe TI can be

ﬁtted to a function consisting of a core- and a wall-term.

The scaling of TI with Re shows that the smooth- and rough-wall level

deviates with increasing Reynolds number. We speculate that IPipe area ∼

√λ/2. 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.

Acknowledgement

We thank Professor A.J.Smits for making the Superpipe data publicly

available [3].

15

Appendix A. Alternative deﬁnition of turbulence intensity aver-

aged over the pipe area

In the main paper, we have deﬁned the TI over the pipe area in Eq. (12);

this constitutes the arithmetic mean. An alternative is to deﬁne it similar to

the mean ﬂow velocity averaged over the pipe area (Eq. (11)). This takes

into account that the area increases with r. We will name this the alternative

deﬁnition (AD):

IPipe area,AD =2

R2ZR

0

vRMS(r)

v(r)rdr(A.1)

The AD leads to a somewhat diﬀerent scaling for the smooth pipe mea-

surements which is illustrated in Fig. A.14. See also Fig. 9.

104105106107108

Re

0

0.02

0.04

0.06

0.08

0.1

Turbulence intensity

IPipe area, AD

Pipe axis (Smooth pipe)

Pipe axis (Smooth pipe fit)

Pipe area, AD (Smooth pipe)

Pipe area, AD (Smooth pipe fit)

Pipe axis (Rough pipe)

Pipe area, AD (Rough pipe)

Figure A.14: Turbulence intensity for smooth and rough pipe ﬂow. The AD is used for

the pipe area TI.

The modiﬁed scaling is:

16

ISmooth pipe area,AD = 0.317 ×Re−0.110 (A.2)

The revised scaling also has implications for the relationship with the

Blasius friction factor scaling (Eq. (6)):

ISmooth pipe area,AD = 0.526 ×λ0.44

Blasius

λBlasius = 4.307 ×I2.27

Smooth pipe area,AD

(A.3)

We can now deﬁne the AD version of the average velocity of the turbulent

ﬂuctuations:

hvRMSiAD =vmIPipe area,AD =4

R4ZR

0

v(r)rdrZR

0

vRMS(r)

v(r)rdr(A.4)

The AD deﬁnition can be considered as a second order moment equation

for vRMS, whereas the deﬁnition in Eq. (13) is a ﬁrst order moment equation.

Again, we ﬁnd that the AD average turbulent velocity ﬂuctuations are

proportional to the friction velocity. However, the constant of proportionality

is diﬀerent than the one in Eq. (14), see Fig. A.15. The AD case can be

approximated as:

hvRMSiAD ∼(5/4) ×√2vτ(A.5)

17

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 AD [m/s]

IPipe area, AD

Smooth pipe

Rough pipe

Eq. (14)

Eq. (A.5)

Figure A.15: Relationship between friction velocity and the AD average velocity of the

turbulent ﬂuctuations.

18

References

[1] Hultmark M, Vallikivi M, Bailey SCC, Smits AJ. Turbulent pipe ﬂow

at extreme Reynolds numbers. Phys Rev Lett 2012;108:094501.

[2] Hultmark M, Vallikivi M, Bailey SCC, Smits AJ. Logarithmic scal-

ing of turbulence in smooth- and rough-wall pipe ﬂow. J Fluid Mech

2013;728:376-395.

[3] Princeton Superpipe; 2016. [Online]

<https://smits.princeton.edu/superpipe-turbulence-data/>.

[4] Russo F, Basse NT. Scaling of turbulence intensity for low-speed ﬂow

in smooth pipes. Flow Meas Instrum 2016;52:101-114.

[5] Langelandsvik LI, Kunkel GJ, Smits AJ. Flow in a commercial steel

pipe. J Fluid Mech 2008;595:323-339.

[6] Blasius H, Das ¨

Ahnlichkeitsgesetz bei Reibungsvorg¨angen in

Fl¨ussigkeiten. Forschg. Arb. Ing. 1913; VDI Heft 131:1-40.

[7] Gersten K, Fully developed turbulent pipe ﬂow, in: Merzkirch W (Ed.)

Fluid Mechanics of Flow Metering, Springer, Berlin, Germany, 2005.

[8] McKeon BJ, Zagarola MV, Smits AJ. A new friction factor relationship

for fully developed pipe ﬂow. J Fluid Mech 2005;538;429-443.

19