J. Fluid Mech. (2004), vol. 508, pp. 99–131.
c
2004 Cambridge University Press
DOI: 10.1017/S0022112004008985 Printed in the United Kingdom
99
Scaling of the streamwise velocity component
in turbulent pipe flow
By J. F. MORRISON
1
,B.J.McKEON
2
, W. JIANG
3
AND A. J. SMITS
2
1
Department of Aeronautics, Imperial College, London SW7 2AZ, UK
2
Department of Mechanical & Aerospace Engineering, Princeton University,
Princeton, NJ 08544-0710, USA
3
CARDC, PO Box 211 Mianyang, Sichuan 621 000, People’s Republic of China
(Received 7 March 2003 and in revised form 30 January 2004)
Statistics of the streamwise velocity component in fully developed pipe flow are
examined for Reynolds numbers in the range 5.5 ×10
4
6 Re
D
6 5.7 ×10
6
. Probability
density functions and their moments (up to sixth order) are presented and their
scaling with Reynolds number is assessed. The second moment exhibits two maxima:
the one in the viscous sublayer is Reynolds-number dependent while the other, near
the lower edge of the log region, follows approximately the peak in Reynolds shear
stress. Its locus has an approximate (R
+
)
0.5
dependence. This peak shows no sign of
‘saturation’, increasing indefinitely with Reynolds number. Scalings of the moments
with wall friction velocity and (U
cl
− U) are examined and the latter is shown to be
a better velocity scale for the outer region, y/R >0.35, but in two distinct Reynolds-
number ranges, one when Re
D
< 6 ×10
4
, the other when Re
D
> 7 ×10
4
. Probability
density functions do not show any universal behaviour, their higher moments showing
small variations with distance from the wall outside the viscous sublayer. They are
most nearly Gaussian in the overlap region. Their departures from Gaussian are
assessed by examining the behaviour of the higher moments as functions of the lower
ones. Spectra and the second moment are compared with empirical and theoretical
scaling laws and some anomalies are apparent. In particular, even at the highest
Reynolds number, the spectrum does not show a self-similar range of wavenumbers
in which the spectral density is proportional to the inverse streamwise wavenumber.
Thus such a range does not attract any special significance and does not involve a
universal constant.
1. Introduction
It has long been accepted that the motion in the viscous sublayer (y
+
= yu
τ
/ν < 30,
where u
τ
=
√
τ
w
/ρ, y is the wall-normal distance and τ
w
is the wall shear stress)
is directly affected by viscosity, ν. Either a conventional overlap analysis that
uses asymptotic matching (Millikan 1938; Wosnik, Castillo & George 2000) or a
dimensional analysis that yields the logarithmic law for the mean velocity suggests
that the motion of the log region is independent of viscosity. However, such analyses
do not specify directly the range in y over which the log law applies, this being the
subject of much experimental investigation. Zagarola & Smits (1998) have shown
that the mean velocity exhibits a log-law dependence for 600 <y
+
< 0.07R
+
only
when R
+
> 9 ×10
3
(R
+
is the K
´
arm
´
an number based on pipe radius) and that
100 J. F. Morrison, B. J. McKeon, W. Jiang and A. J. Smits
for 60 <y
+
< 500, there is a power law when R
+
> 9000 which expresses the direct
influence of viscosity out to y
+
≈0.15R
+
when R
+
< 9000 (see also Zagarola, Perry
& Smits 1997). Using an improved data analysis, McKeon et al. (2004) have recently
confirmed that complete similarity in the form of a log law (with slightly modified
constants) occurs for R
+
> 5 ×10
3
only, in the range 600 <y
+
< 0.12R
+
. These scalings
for the first moment therefore raise questions concerning the most appropriate choice
of velocity scale for the higher moments, particularly because the fluctuating motion
at a point comprises a range of scales, or equivalently, motion at a given wavenumber
receives contributions from the entire physical domain. Moreover, near walls much
of the turbulence information resides in the smaller scales which, in addition to
inhomogeneity and anisotropy, show significant departures from a Gaussian velocity
distribution. For example, close to the wall, the flatness of the wall-normal velocity
component becomes very large (Eggels et al. 1994) and can be attributed to the
spatially alternating behaviour of ejections and sweeps. Statistically, this suggests
inter-connection between many possible degrees of freedom, so contravening one of
the requirements of the central limit theorem. The behaviour of the higher moments
at very high Reynolds numbers is therefore very interesting.
In contrast to the mean velocity, it has been established for some time that the
Reynolds stresses near the wall scale neither with ‘wall’ (or ‘inner’) variables (ν/u
τ
,u
τ
),
nor with ‘outer’ variables (R, u
o
,whereu
o
is an outer velocity scale left undefined
at present). Of particular importance is the difference between the behaviour of
the wall-parallel (u, w) and wall-normal (v) components where the impermeability
constraint, which affects eddies out to a distance from the wall that is of the order
of the eddy size (‘blocking’ or ‘splatting’), is responsible for an increase in the wall-
parallel components at the expense of the wall-normal one. Thus the behaviour of the
statistics for the u-andw-components is different from those for the v-component,
and in the case of pipe flow, the streamwise and azimuthal components are subject to
specialized homogeneous boundary conditions. Failure of the u-andw-components
to scale on wall variables was explained by Townsend (1961, 1976) as the influence of
‘inactive’ motion (Bradshaw 1967; Morrison, Subramanian & Bradshaw 1992), that
of the large eddies inducing a ‘meandering or swirling’ on the near-wall motion. Being
largely confined to the (x, z)-plane and scaling on outer variables, the inactive motion
does not, to a first order, contribute to either the ρ
v
2
normal stress, or to the shear
stress, −ρ
uv, the ‘active’ component of near-wall motion. It is therefore supposed that
the two modes do not interact, the active one being modulated by an ‘irrotational
free stream’ (Bradshaw 1967), the result of both large-scale vorticity and irrotational
pressure fluctuations.
For ‘high’ Reynolds numbers, when y
+
1andy/R 1, an overlap region for
the first moment only of the streamwise component becomes apparent. Even then,
there are many measurements that show that the active component of the Reynolds
stresses does not scale on inner variables either. In particular, the ‘constant-stress’
region (−
uv ≈u
2
τ
) does not hold, except in the limit of very high Reynolds number.
But it is by no means clear just how high a Reynolds number is required: this
is particularly noteworthy since the constant-stress region can be deduced by the
same dimensional arguments that lead to the log law, but it has yet to be determined
whether or not the constant-stress region emerges at about the same Reynolds number
as that at which the log law does (Zagarola & Smits 1998; McKeon et al. 2004).
Morrison et al. (1992) show that the Reynolds-stress-bearing motion in a boundary
layer at Re
θ
≈1.5 ×10
4
(equivalent to R
+
≈5000) is associated with a Kolmogorov
scaling and this should be recognized as evidence of the failure of universal inner
Scaling of the streamwise velocity component in turbulent pipe flow 101
scaling, that is, u
τ
= − (uv)
1/2
with lengthscale y, y
+
1. See also Antonia & Kim
(1994) and Wei & Willmarth (1989). These studies were performed at much lower
Reynolds numbers than those studied in the present investigation.
Nevertheless, in considering the Reynolds-number dependence of near-wall
turbulence, it is important to distinguish direct viscous influence (physically, the
eruptions of low-momentum fluid from the sublayer) from that of the outer-region
motion (large-scale inrushes that produce splats near the wall). The ratio of the two
relevant lengthscales is, of course, R
+
. Not only is the outer-scaling influence of
inactive motion more apparent at high Reynolds numbers, it is also more prevalent
in boundary layers in which the influence of inactive motion is larger than in internal
flows. This raises the question of the extent to which the active and inactive modes
interact, and indeed whether such a delineation is meaningful. In this context, it is
important to remember that Townsend’s original distinction was based on the basic,
but conceptual element of near-wall structure, the ‘attached wall eddy’. Recently, the
precise nature of so-called inactive motion in high-Reynolds-number boundary layers
has been examined by Hunt & Morrison (2000) who suggest that ‘top-down’, outer-
layer interactions are important to the dynamics in the near-wall region. Note that
any interaction between these two modes implies that, even at very high Reynolds
numbers when direct viscous effects are small, wall motion cannot be universal in
any meaningful sense. Outer-layer influences may also have ramifications for the
self-similarity of the mean velocity: both Bradshaw (1967) and Townsend (1976) use
a simple linear analysis to show that the effect of inactive motion is to make the von
K
´
arm
´
an constant, κ, Reynolds-number dependent. These considerations lead also to
the conclusion that simple arguments concerning the overlap of scales for the higher
moments in a particular region of wall turbulence are inappropriate. Crucial to the
understanding of these issues is the realization that the wall affects wall-normal and
wall-parallel components differently.
In this paper, we report hot-wire measurements of the streamwise velocity
component in the Reynolds-number range (based on pipe diameter, D and mean
velocity,
U)5.5 ×10
4
6 Re
D
6 5.7 ×10
6
. Statistics up to the sixth moment are
calculated as the moments of a probability density function (p.d.f.). Equivalent spectra
as a function of streamwise wavenumber, φ(k
1
), are also presented. The scaling of
both is investigated. It is becoming increasingly apparent that there are significant
differences between different flows of the same species: thus in the present context, we
distinguish between not only external and internal flows, but also between pipe and
channel flows (Nieuwstadt & Bradshaw 1997). Therefore reference to channel flows is
not made unless the results are specifically relevant to the present work. Early work
on pipe flow includes that of Laufer (1954) and Sandborn (1955) and more recently,
Durst, Jovanovi
´
c & Sender (1995), Eggels et al. (1994) and Fontaine & Deutsch
(1995). However, all of these studies are limited in terms of the range of Reynolds
numbers over which data were obtained. The main influence, pervading much of the
work reported here, comes from Townsend’s seminal work concerning the self-similar
structure of attached wall eddies, which forms the basis of the supposed self-similarity
of the spectra for the surface-parallel velocities and the functional forms for the
normal stresses. This stimulated further work, principally that of Perry & Abell (1975,
1977) and Perry, Henbest & Chong (1986) (pipes) and Perry & Li (1990), Perry &
Marusic (1995), Marusic & Perry (1995), Marusic, Uddin & Perry (1997) and Jones,
Marusic & Perry (2001) (boundary layers). Marusic & Kunkel (2003) have recently
extended these ideas. In later sections, we interpret the data using the concept of
inactive motion. But, owing to the very significant potential benefits accruing from a
102 J. F. Morrison, B. J. McKeon, W. Jiang and A. J. Smits
self-similar description of wall turbulence, it seems prudent first to examine the exact
requirements for this to be so.
2. Similarity considerations
Zagarola & Smits (1998) show that asymptotic matching of the mean velocity
gradients in the overlap region leads to complete similarity (in the form of the log
law) for 600 <y
+
< 0.07R
+
when the Reynolds number is sufficiently high and when
the velocity scales for the inner and outer regions are the same, that is, given by
u
τ
. (See also Zagarola et al. 1997; McKeon, Li, Jiang, Morrison & Smits 2004) At
smaller y
+
, they argue that the ratio of inner to outer velocity scales, u
τ
/u
o
,isa
function of R
+
, and that simultaneous matching of both the velocities and velocity
gradients leads to a power law. The appearance of a power law may be regarded as
a form of incomplete similarity and is supported by the data of both Zagarola &
Smits (1998) for 60 <y
+
< 500 and McKeon et al. (2004) for 60 <y
+
< 300. The term
‘complete (or self-) similarity’ means that, first, the lengthscale used to normalize the
independent variable, y, in the log argument may be freely chosen, and second, the
von K
´
arm
´
an constant is universal. For this reason the log law is valid using inner
or outer scaling, or even using a rough-surface lengthscale. In fact, demonstration of
complete similarity requires simultaneous collapse using both inner and outer scaling.
Owing to the difficulties in scaling the Reynolds stresses near the wall, there have
been several attempts at finding a more suitable inner velocity scale. Prominent among
the alternatives to u
τ
is so-called ‘mixed’ scaling (where the velocity scale is (u
τ
U
cl
)
1/2
,
see for example, DeGraaff & Eaton 2000) for the horizontal stresses. Zagarola &
Smits (1998) have suggested that a true outer velocity scale is U
cl
− U , where U
cl
is the centreline velocity. For Re
D
> 2 ×10
5
, they show that (U
cl
− U )/u
τ
−→ 4.34
although, with more data and a revised analysis, McKeon et al. (2004) suggest that
the constant is 4.28. The use of both mixed scaling and (U
cl
− U) as a velocity scale
is considered in §5.
Since publication of Townsend’s seminal work, considerable attention has been
devoted to the deduction of spectral forms associated with the self-similar nature of
attached wall eddies. Such self-similarity manifests itself at ‘high’ Reynolds numbers
as a range of streamwise wavenumber, k
1
, in which the spectrum φ
11
∝u
2
τ
k
−1
1
.
There are several derivations, the earliest provided by Tchen (1953), reappraised
by Hinze (1975), involving the balance between the spectral transfer of energy
by the mean shear and that by inertial interactions of the turbulence – a strong
interaction or ‘resonance’ condition. Tchen’s theory involves several assumptions that
are questionable in highly anisotropic wall turbulence, such as a constant strain rate
and a spherically symmetric eddy viscosity. As such, the individual components are
not distinguished. More pragmatically, it should be noted that a prescribed slope over
some region of wavenumber can usually be found in turbulence spectra on log–log
axes. The simpler theory for pipe flow was proposed by Perry & Abell (1977) and
Perry et al. (1986), but it is equally appropriate for boundary layers (see papers by
Perry and colleagues). The theory has been the subject of much attention, but it
appears that often the existence of self-similarity at practical Reynolds numbers is
taken for granted (Nikora 1999; H
¨
ogstr
¨
om, Hunt & Smedman 2002), or that its proof
is the result of ab initio assumptions (Kader & Yaglom 1991). Given the prominence
of the theory and, in terms of the Reynolds number, the uniqueness of the present
data, a careful reappraisal is clearly needed (see also Morrison et al. 2002a, b). In this,
Scaling of the streamwise velocity component in turbulent pipe flow 103
it would appear sensible to focus on that range of y in which the mean velocity is
known to exhibit self-similarity in the form of a log law.
‘Large’ scales (in which the direct effects of viscosity may be neglected) that
contribute to the streamwise velocity component may be scaled using either inner
or outer scales. Outer scaling suggests that y is not important and, taking u
τ
as the
appropriate velocity scale, dimensional analysis therefore yields
φ
11
(k
1
)
Ru
2
τ
=
φ
11
(k
1
R)
u
2
τ
= g
1
(k
1
R), (2.1)
while, alternatively, inner scaling suggests the exclusion of R as a relevant lengthscale
so that, at higher wavenumbers,
φ
11
(k
1
)
yu
2
τ
=
φ
11
(k
1
y)
u
2
τ
= g
2
(k
1
y). (2.2)
The veracity of these scalings is usually judged by the degree of collapse of the
spectra at wavenumbers lower than that at which spectral transfer (which at high
Reynolds numbers is given by the mean dissipation rate) becomes important. In the
range of wave numbers R
−1
<k
1
<y
−1
over which both (2.1) and (2.2) are valid (that
is collapse is evident with both scalings, as required by asymptotic matching), it then
follows that
φ
11
(k
1
)=Ru
2
τ
g
1
(k
1
R)=yu
2
τ
g
2
(k
1
y). (2.3)
Dimensional arguments and direct proportionality between g
1
and g
2
therefore imply
φ
11
(k
1
R)
u
2
τ
=
A
1
k
1
R
= g
1
(k
1
R), (2.4)
and
φ
11
(k
1
y)
u
2
τ
=
A
1
k
1
y
= g
2
(k
1
y), (2.5)
where A
1
is a universal constant. Collapse with both length scales therefore suggests
a self-similar structure such that φ
11
(k
1
) ∝ u
2
τ
k
−1
1
. We will therefore call this situation
‘complete similarity’. In this situation, the only relevant lengthscale is k
−1
1
itself, and,
owing to the nature of the Fourier transform and because the foregoing analysis is
equally valid for the spanwise velocity component, a self-similar structure would have
to be space-filling in (x, z)-planes parallel to the surface. Now, it is possible that, for
example, while y and u
τ
might form a complete parameter set to define the motion
in the range of wavenumbers over which collapse is apparent with (2.2), these wave
numbers might, in fact, be too high for collapse to be possible using R and u
τ
as
in (2.1). Thus simultaneous collapse is not possible. We shall refer to this situation
as ‘incomplete similarity’, in which case the constant A
1
in (2.4) and (2.5) cannot be
universal.
Note that this analysis is predicated on two principal assumptions. The first is that
the kinematic viscosity, ν, does not enter the problem. This requires that k
1
ν/u
τ
1.
In turn, this requires the Reynolds number to be sufficiently high, or equivalently that
y is sufficiently large, such the energy-containing scales are not affected directly by
viscosity. Taking the outer limit to the power-law region for the first moment to be
y
+
= 500, it would seem unlikely that higher moments would be free of direct viscous
effects below y
+
≈1000, as shown by the conditional sampling results of Morrison
et al. (1992). The second assumption is that u
τ
is the correct velocity scale for both
the inner and outer regions. In particular, in conformity with Townsend’s theory,
