J. Fluid Mech. (2004), vol. 501, pp. 135–147.
c
2004 Cambridge University Press
DOI: 10.1017/S0022112003007304 Printed in the United Kingdom
135
Further observations on the mean velocity
distribution in fully developed pipe flow
By B. J. M c KEON, J. LI,† W. JIANG‡,
J. F. MORRISON¶
AND A. J. SMITS
Department of Mechanical and Aerospace Engineering, Princeton University,
Princeton, NJ 08544-0710, USA
(Received 7 October 2002 and in revised form 19 September 2003)
The measurements by Zagarola & Smits (1998) of mean velocity profiles in fully
developed turbulent pipe flow are repeated using a smaller Pitot probe to reduce the
uncertainties due to velocity gradient corrections. A new static pressure correction
(McKeon & Smits 2002) is used in analysing all data and leads to significant
differences from the Zagarola & Smits conclusions. The results confirm the presence
of a power-law region near the wall and, for Reynolds numbers greater than 230 ×10
3
(R
+
> 5 ×10
3
), a logarithmic region further out, but the limits of these regions and
some of the constants differ from those reported by Zagarola & Smits. In particular,
the log law is found for 600 <y
+
< 0.12R
+
(instead of 600 <y
+
< 0.07R
+
), and the
von K
´
arm
´
an constant κ, the additive constant B for the log law using inner flow
scaling, and the additive constant B
∗
for the log law using outer scaling are found
to be 0.421 ± 0.002, 5.60 ± 0.08 and 1.20 ± 0.10, respectively, with 95% confidence
level (compared with 0.436 ±0.002, 6.15 ±0.08, and 1.51 ±0.03 found by Zagarola &
Smits). The data also confirm that the pipe flow data for Re
D
6 13.6 ×10
6
(as a
minimum) are not affected by surface roughness.
1. Introduction
1.1. Background and previous work
Zagarola & Smits (1998) (referred to hereinafter as ZS) presented measurements in
fully developed pipe flow for Reynolds numbers in the range 31 ×10
3
to 35 ×10
6
to study the scaling of the mean velocity profile. They found two overlap regions: a
power law for 60 <y
+
< 500, and, for Reynolds numbers greater than 400 ×10
3
,a
log law in the region 600 <y
+
< 0.07R
+
. Here, y
+
= yu
τ
/ν, y is the distance from the
wall, u
τ
=
√
τ
w
/ρ, ν is the kinematic viscosity, τ
w
is the shear stress at the wall, and
ρ is the fluid density. Also, R
+
= Ru
τ
/ν where R is the radius of the pipe (=D/2).
These findings were supported by a new scaling argument based on dimensional
analysis. For high Reynolds numbers, the scaling for the mean velocity profile in fully
developed turbulent pipe flow may be expressed in terms of an inner-layer scaling
given by
U = f
(y,u
i
,ν,R) (1.1)
† Permanent address: School of the Built Environment, Victoria University of Technology, PO
Box 14428, MCMC, Melbourne, Australia.
‡ Permanent address: CARDC, PO Box 211 Mianyang, Sichuan 621000, P. R. China.
¶ Permanent address: Department of Aeronautics, Imperial College, London SW7 2BY, UK.
136 B. J. McKeon, J. Li, W. Jiang, J. F. Morrison and A. J. Smits
and an outer-layer scaling given by
U
c
−U = g
(
y,u
0
,ν,R
)
(1.2)
where U is the mean velocity, U
c
is the centreline velocity, and f
and g
denote
a functional dependence. The inner velocity scale u
i
is always taken to be u
τ
,but
choosing the outer velocity scale u
0
is more controversial, as will be seen below.
Non-dimensionalizing equations (1.1) and (1.2) gives, respectively,
U
+
= f (y
+
,R
+
), (1.3)
U
c
−U
u
0
= g(η, R
+
), (1.4)
where U
+
= U/u
τ
and η = y/R. ZS showed that the velocity scaling in the overlap
region (where y
+
1andη 1) can be of two types: complete similarity where u
0
/u
τ
is independent of Reynolds number, and incomplete similarity where u
0
/u
τ
continues
to depend on Reynolds number. In the case of complete similarity, matching of the
velocity gradients in the overlap region leads to a logarithmic velocity profile, which
can be expressed in inner-layer variables as
U
+
=
1
κ
ln y
+
+ B (1.5)
and in outer-layer variables as
U
c
−U
u
τ
= −
1
κ
ln η + B
∗
(1.6)
where κ (usually called the von K
´
arm
´
an constant), B and B
∗
are constants ind-
ependent of Reynolds number.
Alternatively, in the case of incomplete similarity, matching the velocities as well
as the velocity gradients in the overlap region yields a power-law dependence which
in terms of inner-layer variables may be written as
U
+
= Cy
+γ
(1.7)
where the coefficient C and the exponent γ are independent of Reynolds number.
The study by ZS showed that the power-law constants were given by C =8.70 and
γ =0.137, and the log-law constants were given by κ =0.436 ±0.002, B =6.15 ±0.08,
and B
∗
=1.51 ± 0.03. These values of the log-law constants are different from the
commonly accepted values of about 0.41, 5.0 and 0.8, respectively. for the so-called
‘standard log law’ discussed below.
ZS also proposed a new velocity scale for the outer region, u
0
= U
c
−
¯
U.They
argued that u
0
= U
c
−
¯
U was a more representative outer velocity scale than the
friction velocity, which describes the inner flow and is impressed upon the outer
region. Since the ratio (U
c
−
¯
U)/u
τ
becomes constant at high Reynolds number, there
is no change to the overlap analysis leading to a log law at these Reynolds numbers. At
low Reynolds numbers (high enough that an overlap region still exists), the variation
of (U
c
−
¯
U)/u
τ
with Reynolds number means that matching velocity gradients does
not lead to a Reynolds-number-independent condition. However matching velocities
and velocity gradients leads to the power law of equation (1.7).
ZS obtained their measurements using a round Pitot probe of 0.9 mm OD. A
number of corrections were made to the data, including corrections to the Pitot tube
pressure for the effects of turbulence, viscosity, and velocity gradient, and corrections
to the static pressure for viscous effects. Subsequent examination of the ZS data
Mean velocity in fully developed pipe flow 137
Figure 1. Velocity profiles for the 0.9 mm (ZS) data at Re
D
=3×10
6
(solid lines) and 10 ×10
6
(dashed lines): (a) results using Shaw’s (1960) correction as reported by ZS; (b) results using
correction according to McKeon & Smits (2002).
revealed slight deviations between the velocity profiles in inner scaling (figure 1a).
The deviations (also called ‘steps’) appear as departures from the logarithmic region
for Re
D
> 3 ×10
6
and increase with Reynolds number.
In ZS, the comparison of the data with the log law and the power law was done
by fitting the scaling laws to the mean velocity profile. As pointed out by W. George
(private communication), the mean velocity itself is not a very sensitive quantity in
distinguishing a log law from a power law because the differences are small. This is
especially true at low Reynolds number. Fractional differences were therefore used, as
suggested by Zagarola, Perry & Smits (1997). The fractional difference E = U
+
/U
+
between a fitted curve and the measured data is defined as
E =
U
+
U
+
=1−
U
+
fit
U
+
measured
. (1.8)
Zagarola et al. suggested that a good fit to the experimental data should result in
random relative errors that are less than the mean experimental error.
Traditionally, dimensional analysis using u
0
= u
τ
has yielded a log law (Millikan
1938). The so-called ‘standard log law’ arose from the conclusions of several studies
in low-Reynolds-number boundary layers, including that by Bradshaw (1976), that
the overlap region was best represented by a log law with κ ≈0.41 and B ≈5.0
(Schlichting 1979).
¨
Osterlund et al. (2000) in a more recent study of boundary layers found a log law
with κ =0.38 and B =4.1 in the range 200/δ
+
<y/δ<0.15 for Re
θ
> 6000. Although
this log law appears over a rather short region in y
+
(200 to about 1000), we should
note that this includes virtually all boundary layer studies, where few studies have
a large enough Reynolds number range to investigate the overlap region with an
upper extent where y
+
> 1000. This includes the work by
¨
Osterlund et al. where the
maximum value of Re
θ
was 27 000.
In a medium-Reynolds-number channel flow (Re
h
< 1.2 ×10
5
based on half-channel
height and mean velocity, 1000 <Re
τ
< 5000), Zanoun et al. (2002) found a similar
value of the von K
´
arm
´
an constant, κ =0.379, with B =4.05, for Re
τ
> 2000 within
the limits y
+
> 100 and y/h < 0.15. However, they also suggested that a log law may
138 B. J. McKeon, J. Li, W. Jiang, J. F. Morrison and A. J. Smits
be valid for limits reaching y
+
=50 and y/h =0.9, which presumably yields different
log-law constants.
1.2. Corrections to the measurements
Perry, Hafez & Chong (2001) attributed the steps in the data (which were exacerbated
in their analysis by a different choice of κ and B) to roughness effects combined
with inappropriate Pitot tube corrections. Recent experiments at Princeton, however,
have demonstrated that these deviations were in fact due to inaccurate static pressure
corrections (McKeon & Smits 2002) and the absence of a wall term in the Pitot
correction. ZS used the static pressure correction proposed by Shaw (1960), who
suggested that the error, p , depends only on d
+
t
and reaches an asymptotic value
of approximately 3.0τ
w
for d
+
t
> 1000, where d
+
t
= u
τ
d
t
/ν and d
t
is the hole diameter.
McKeon & Smits (2002) have shown instead that the error continues to grow with
increasing Reynolds number as long as the ratio d
t
/D is small, and reaches a value
of greater than 7τ
w
at the highest Reynolds numbers obtained in the pipe, where
d
+
t
≈6500. Although the new correction corresponds to a change of less than 1% in
the maximum dynamic pressure for all cases presented here, the new static pressure
correction removes the deviations observed in the ZS profiles to within the limits of
experimental uncertainty (see figure 1b).
To understand the effect of Pitot probe corrections on the similarity scaling and
the value of the log-law constants, a further study was undertaken by McKeon
et al. (2003) using four different sized Pitot tubes, measuring 0.3 mm to 1.83 mm OD.
A new correction scheme was suggested that collapses the data markedly better than
other wall corrections methods, including that suggested by MacMillan (1956), for
y<2d,whered is the outer diameter of the Pitot tube, and that agrees well with
MacMillan’s and Chue’s (1975) methods over the entire Reynolds number range for
y > 2d.
1.3. Current study
ZS measured 28 velocity profiles over a range of Reynolds numbers from 31 ×10
3
to 36 ×10
6
using a 0.9 mm Pitot probe. We use these data, in addition to new
measurements encompassing 21 profiles obtained using a Pitot probe of 0.3 mm
replicating the ZS Reynolds numbers over the range from 74 ×10
3
to 35 ×10
6
.Both
sets of data included measurements of the pressure drop to determine the friction
factor. The Reynolds numbers and friction factors for the mean velocity measurements
analysed in this work are shown in table 1, along with the symbols that will be used
in the figures that follow.
Since an appropriate Pitot correction must surely collapse data from probes of all
diameters, the pipe flow mean velocity measurements were analysed in two different
ways to ensure the conclusions were not affected by the Pitot correction. First,
all points taken within 2d of the wall, that is, where the Pitot probe corrections
disagreed, were removed from the data set. The remaining data points were therefore
independent of the particular Pitot tube corrections used. To be consistent with ZS the
Chue (1975) correction was applied to the data, although any of the other corrections,
such as MacMillan (1956), give the same results. This data set (y > 2d) was used in
the analysis of the log-law shown here. Second, to investigate the near-wall behaviour,
all the data points for y > d were used. In this case, the data were corrected using
the method suggested by McKeon et al. (2003), which gives good agreement between
data from probes of different diameters for y > d. It should be noted that analysis of
Mean velocity in fully developed pipe flow 139
Symbol Re
D
λ Symbol Re
D
λ
M 75 × 10
3
0.0193 H 2.3 × 10
6
0.0103
150 × 10
3
0.0167 F 3.1 ×10
6
0.0099
N 230 × 10
3
0.0153 C 4.4 × 10
6
0.0094
310 × 10
3
0.0147 | 6.1 ×10
6
0.0090
410 × 10
3
0.0138 J 7.7 × 10
6
0.0086
◦ 540 × 10
3
0.0132 P 10.2 ×10
6
0.0082
750 × 10
3
0.0125 B 13.6 × 10
6
0.0080
• 1.0 × 10
6
0.0118 18.2 ×10
6
0.0077
O 1.3 × 10
6
0.0113 I 35.3 × 10
6
0.0071
X1.7 × 10
6
0.0108
Tab l e 1. Nomenclature and friction factor data (0.9 mm Pitot probe).
this second data set confirmed all the conclusions derived using the first data set (the
set that was independent of the Pitot tube correction used).
1.4. Experimental considerations
In other respects, the experimental apparatus and techniques used for the new
dataset were virtually the same as those used by ZS, and their error analysis applies
almost unchanged to the measurements reported here. The maximum error in velocity
measurement remained 0.35%. However, to improve the accuracy of the wall distance
measurement, a linear encoder with an accuracy of 5 µm per count was used to
determine the wall distance (ZS reported an accuracy of 25 µm). In addition, the
starting position for the Pitot probe was determined by detecting the electrical contact
between the Pitot probe and the pipe surface to an accuracy of 5 µm, compared to
50 µm for ZS. The accumulated position error of running the present traverse system
forward and backward once over a distance of 71 mm was generally less than 30 µm,
compared with 50 µm accuracy for ZS. The position accuracy was estimated to be
±1.7% for measuring points close to the pipe surface and ±0.05% for data points
taken near the centre of the pipe. Across the pipe radius, 57 data points were taken
with logarithmically uniform spacing. The first point was taken with the Pitot probe
touching the pipe surface, the 56th point was located at the centre of the pipe, and the
57th point was on the other side of the pipe centre. The symmetry of the flow across
the pipe, first noted by Zagarola (1996) was confirmed in the present experiment.
The data were sampled using a new PC-based data acquisition system (National
Instrument data acquisition boards driven by Labview software) at 500 Hz over a two
minute period.
2. Mean velocity results
Figure 2 shows sample mean velocity profiles from the present work (i.e. under the
new corrections) for the ZS data.
Both datasets were processed in three steps. First, the value of κ was determined
from the friction factor data by fitting
1
√
λ
= C
1
log(Re
√
λ)+C
2
(2.1)