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# Turbulence Intensity and the Friction Factor for Smooth- and Rough-Wall Pipe Flow

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• Independent Scientist

## 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 quantify the correspondence between turbulence intensity and the friction factor.
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fluids
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
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)
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 ×108×Re +1.0161. (3)
Fluids 2017,2, 30 3 of 13
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
(a)
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
(b)
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
(c)
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
(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
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
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 ×Re0.0407,
ISmooth pipe area =0.317 ×Re0.110.(5)
The Blasius smooth pipe (Darcy) friction factor [12] is also expressed as an Re power-law:
λBlasius =0.3164 ×Re0.25. (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.
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.
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
m200
81 ×I2
Pipe area, (13)
where Pis the pressure loss and Lis the pipe length. This can be reformulated as:
IPipe area 9
102×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)R3P
=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
=B1A1ln Rr
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
τ9
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
102×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(1rn)|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
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
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
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
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
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
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 ×Re0.100. (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×vτr2
3×hvRMS i. (A7)
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
τ2
3×9
52
=2.16. (A9)
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2.
Hultmark, M.; Vallikivi, M.; Bailey, S.C.C.; Smits, A.J. Turbulent pipe ﬂow at extreme Reynolds numbers.
Phys. Rev. Lett. 2012,108, 094501.
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Hultmark, M.; Vallikivi, M.; Bailey, S.C.C.; Smits, A.J. Logarithmic scaling of turbulence in smooth- and
rough-wall pipe ﬂow. J. Fluid Mech. 2013,728, 376–395.
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Princeton Superpipe. 2017. Available online: https://smits.princeton.edu/superpipe-turbulence-data/
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c
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Chapter
Turbulent free shear flows occur if there are no walls directly at the flow. Figure 22.1 shows some examples: a free jet, a buoyant jet, a mixing layer with the free jet–boundary flow as a special case, and a wake flow. The corresponding laminar flows are treated in Sects. 7.2, 7.5, 10.5.4 and 12.1.5. The flow of a turbulent wall jet, which is a jet bounded on one side by a wall, is treated in Sect. 22.8 (the laminar wall jet is discussed in Sect. 7.2.7).
Article
The streamwise mean velocity profile in a turbulent boundary layer is classically described as the sum of a log law extending all the way to the edge of the boundary layer and a wake function. While there is theoretical support for the log law, the wake function, defined as the deviation of the measured velocity profile from the log law, is essentially an empirical fit and has no real physical underpinning. Here, we present a new physically motivated formulation of the velocity profile in the outer region, and hence for the wake function. In our approach, the entire flow is represented by a two-state model consisting of an inertial self-similar region designated as ‘pure wall flow state’ (featuring a log-law velocity distribution) and a free stream state, which results in a jump in velocity at the interface separating the two. We show that the model provides excellent agreement with the available high Reynolds number mean velocity profiles if this interface is assumed to fluctuate randomly about a mean position with a Gaussian distribution. The new concept can also be extended to internal geometries in the same form, again confirmed by the data. Furthermore, adopting the same interface distribution in a two-state model for the streamwise turbulent intensities, with unchanged parameters, also yields a reliable and consistent prediction for the decline in the outer region of these profiles in all geometries considered. Finally, we discuss differences between our model interface and the turbulent/non-turbulent interface (TNTI) in turbulent boundary layers. We physically interpret the two-state model as lumping the effects of internal shear layers and the TNTI into a single discontinuity at the interface.
Chapter
Incompressible pipe flows are fully developed if the velocities are independent of the axial coordinate. The theory of turbulent pipe flows at high Reynolds numbers leads to analytical expressions for the velocity profile and the friction factor, which contain free constants. One of the latter is the well-known Krmn constant. The free constants can be determined from the best available experimental data. Final formulae for the velocity distribution and the friction factor are derived. Further effects on these laws are also considered, such as wall roughness and low Reynolds number effects.