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

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Abstract and Figures

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
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 ×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
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
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
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 ×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
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
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