Axisymmetric turbulent statistics of long slender circular cylinders
Stephen A. Jordan
a)
Naval Undersea Warfare Center, Newport, Rhode Island 02841, USA
(Received 17 November 2010; accepted 1 June 2011; published online 21 July 2011)
The experimental evidence leads us to believe that long slender circular cylinders have similar
axisymmetric turbulent statistics along most of their axial length. The respective boundary layer
reaches a maximum thickness (d) with no further downstream net growth. Despite their small
radius (a), these long cylinders still own high radius-based Reynolds numbers (Re
a
) as well as
transverse curvatures (d/a). The influence of these flow conditions (and others) on the turbulent
statistics is still chiefly unknown. The present effort begins an investigation that targets axial
similarity (or homogeneity) of the long thin cylinder statistics. The database is a collection of
previous experimental measurements and observations as well as the present computational results
by the large-eddy simulation methodology. Interestingly, this investigation shows that reaching
axial homogeneity is reliant essentially on Re
a
with lesser influence by the transverse curvature.
But the Re
a
value depends on the turbulent statistic of interest. Likewise, this same result was
found for spotting the radial location of the respective statistical peak. Axial homogeneity starts
near the cylinder wall then migrates outward radially with increasing Re
a
until full saturation
through the turbulent intermediate layer. [doi:10.1063/1.3609272]
I. INTRODUCTION
In spite of their simple geometry, long slender circular
cylinders own rather complex turbulent flow characteristics.
On one hand, their axial turbulent boundary layers (ATBL)
under zero pressure gradients will appear planar-like if the
respective thickness is on the same order as the cylinder ra-
dius. But conversely, a much thicker boundary layer will
spawn turbulent statistics unique only to the thin cylinder
itself. According to the experimental evidence,
1,2
the ATBL
will quickly reach a maximum thickness generally on one to
two orders of magnitude greater than the cylinder radius.
Although fluctuating, this thickness will remain consistent
along the remaining portions of the long cylinder’s extent.
This unique observation exposes an axial similarity that
offers possible recognition of certain turbulent statistics
which are effectively germane along most of the cylinder
length. From a practical perspective, long slender cylinders
serve well as acute undersea listening devices because their
extensive length supports a wide acoustic aperture. Tow
arrays (or streamers) are excellent examples that Navy scien-
tists and oceanographers use for undersea defense detection
as well as mapping the seafloor topography and stratification
of the sub-seafloor sediments.
Due to the experimental difficulties of measuring the
outer layers (Rao and Keshaven
3
), earliest studies of the long
slender cylinder were restricted to thin boundary layers. But
as noted earlier, the ATBL statistics become flat-plate-like as
the boundary layer thickness (d) approaches the cylinder ra-
dius (a). The dependence of the inner and outer turbulent
regions on the transverse curvature (d/a) essentially disap-
pears from the thin cylinder similarity laws. This observation
was reported by Willmarth et al.
4
(and later emphasized by
Lueptow et al.
2
) where the wall-normal statistical distribu-
tions (turbulent velocity means, intensities, Reynolds
stresses, etc.) were found independent of the local wall
length scale u
s
/ (u
s
is the local friction velocity and is the
kinematic viscosity). Moreover, the log layer of the mean
axial velocity in wall units possessed the same slope as von
Karman’s constant (j¼ 0.4) for turbulent flat plates.
As the transverse curvature d/a !1, both experimental
and numerical studies show that the ATBL statistics depend
on the coupling effects between the local wall length scale
(u
s
/) and the transverse curvature. In particular, Luxton
et al.
5
noted the importance of the additional length scale (a)
toward discovering useful similarity laws in terms of a
þ
(a
þ
¼ au
s
/) and Re
a
(Re
a
¼ aU
0
/ where U
o
is the free-
stream velocity). Both a
þ
and Re
a
are Reynolds numbers
where the former utilizes the wall friction velocity instead of
the freestream. Under small a
þ
, Reid and Wilson
6
and Rao
7
separately converted the near-wall momentum equilibrium
expression by Glauert and Lighthill
8
into a relationship for
the mean axial velocity (V
x
) in wall units; u
þ
¼ a
þ
ln
[1þ y
þ
/a
þ
] where u
þ
¼ V
x
/u
s
and y
þ
¼ yu
s
/. This law-of-
the-wall relationship for the slender cylinder clearly recog-
nizes the influence of the transverse curvature on the inner
flow physics. If both parameters a
þ
and Re
a
are low, the
transverse curvature strongly impacts both the inner and
outer regions of the ATBL (Ref. 5). The log region exhibits
a variable slope that eventually vanishes as Re
a
becomes
decreasing small. Under this latter condition, the outer region
transforms into a wake-like flow. Notably, the ATBL statis-
tics will then scale on the outer quantities and indicate little
dependence on Re
a
in the absence of the log layer. Given
these same parameters, Lueptow et al.
2
quantified the de-
pendence of d/a on the log layer slope (and intercept). They
found an asymptotic approach of the mean axial velocity (in
a)
Author to whom correspondence should be addressed. Electronic mail:
stephen.jordan@navy.mil.
1070-6631/2011/23(7)/075105/11/$30.00 23, 075105-1
PHYSICS OF FLUIDS 23, 075105 (2011)