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Turbulence intensity in the transition from hydraulically smooth to fully rough pipe flow

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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 propose a correspondence between turbulence intensity and the friction factor.
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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
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
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
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
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
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
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
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
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
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
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
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 ×Re0.0407
ISmooth pipe area = 0.227 ×Re0.100 (4)
We note that the Blasius smooth pipe (Darcy) friction factor [6] is also
expressed as an Re power-law:
λBlasius = 0.3164 ×Re0.25 (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
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
m4I2
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=pτwis 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) R3P=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
R2ZR
0
vRMS(r)
v(r)rdr(A.1)
The AD leads to a somewhat diﬀerent scaling for the smooth pipe mea-
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, AD (Smooth pipe fit)
Pipe axis (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 ×Re0.110 (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
(A.3)
We can now deﬁne the AD version of the average velocity of the turbulent
ﬂuctuations:
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:
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
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
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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.
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19
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