alternate ways of defining an internal turbulent. Reynolds number, at least in the case of a closed flow. Among possible non-local, non-intrusive measurements.
J.
Phys.
II
FYance
7
(1997)
Characterization
#cole
J.-F.
Lyon (**),
de
1996,
November
29
revised
Hydrodynamic Turbulent
flows,
PACS.47.32.-y
Rotational
flow
Abstract.
characterize
to on
fluid. of
the
driving
the
We
the
numbers.
then
show
velocity
rms
We
of
state
torque
and
that
Lyon,
June
1997.
France
accepted
and
convection
flow
July 1997)
16
heat
transfer
vorticity
and
the
fluctuations
flow
closed
show
in
that
flow
the
and
that
reveals
it
Finally, we show that the transition once the calculation of fundamental global quantities allows the same the velocity fluctuations, the effective integral length as rms and Kolmogorov's dissipation length ~. That these quantities devices removed from the bulk of the flow is of importance complex geometries and for using corrosive fluids. the
Flow
Chillit
69364
J6
1729
between coaxial rotating disks, at moderate contra global ii. e. spatially averaged) quantities can be used the flow and its degree of turbulence. We first report measurements show how it depends on the is imparted to the momentum manner boundary provide a good estimate at the flow measurements pressure
investigate
We
high Reynolds
Closed
a
PAGE
stability
PACS.47.20.~k
PACS.47.27.-I
to
in
(*) and F.
Pinton
Sup6rieure
Normale
(Received
Turbulence
of
Mordant,
N.
NOVEMBER1997,
1729-1742
volume.
the
has
transition
to
occurred,
the
turbulence
for
characteristics
L*, Taylor's
scale may
turbulence
be
study of fluid
the
of
such
microscale
from
obtained
in
knowledge
A
measuring motion
in
Introduction
1.
turbulence The properties of flows at high Reynolds numbers are usually described experimentally in terms of the scaling properties of the local velocity field, as measured by a small probe documented and has produced placed inside the flow [1-3]. This technique is widely most of turbulent flows. Unfortunately the method is complex, local the existing experimental data on and intrusive; in addition, depending on the flow conditions, it is not always possible to use an probe m situ. Simple examples are flows in which the temperature is very high or anemometer liquids, flows in liquid metals [4], etc. The question then arises corrosive not uniform, flows in of
the
experimental
measureInents.
the
intensity of the
of
characterization
The
problem
is to
turbulence.
always
if
turbulent
a
determine
present.
We
flow
whether note
in
the
that
the
flow is
the
absence
turbulent
knowledge
local
of
and
of the
to
anemometry characterize
integral Reynolds by the values of
here integral determined that it is means diameter experiment (for example, the flow rate and tube in determined flow characteristics if one Turbulent compute an may only be can a pipe flow). internal Reynolds number~ I.e. based on internal flow parameters only. This is the turbulent number
external
is
not
parameters
sufficient
fixed
in
(*) Author for correspondence (**) CNRS URA 1325
@
Les
(ditions
de
Physique
the
(e~mail:
1997
pintonl3physique.ens-lyon.fr)
When
case
local
anemometry
velocity fluctuations rms global measurements to
defining
of
Among
global
are
turbulent
non-intrusive
measurements,
acoustic ary; measurements
by
which
the
where
quantity
V
volume is
average function
an
(fast-varying) g(t) j dg
in
let
f(Re)
r~J
In the
an
of
the
is
of
case
limit
known
in
the
disk
of
radius
a
on
way that the
fact
of
is
the
to
value
Dimensional
is
recovered
integral Reynolds dependent and
the
expressed
internal
quantities
describe
present
surements
in
be
into
section
next
results
our
can
inserted
the
on
3.I; for the
fluid.
number
combined
to
in
as
fl) thus
terms
of
can
be
of the
flow;
our
experimental
e.g.
the
transition
propose
of
f
to
the
that
[5]. We
the
the flow
internal
more
a
from
come
whereas
expected
energetic meaningful
f u)~~. e. g. global that do measurements not urm~[bulk] prm~[at wall].
only;
variables
r~J
r~J
techniques.
measurement
show
and
turbulence
in
velocity
rms
expression
an
related
boundaries
to
le. g.
parameters
that
f depends
that
stress
behavior
and
set-up
observed
been
just
we
measurements,
pressure
proceed with the deduction of internal flow dissipation, Taylor microscale, etc.) and we
via
has
r~J
in
flow
internal
accessed
bulk
characterize
its
be
can
and
via
=
slightly) rugose disks f(Re) Re° behavior.
simply
not
are
motion
It
moment
R~fl Iv. The the experiwith
Re also
varies
a
differences
The
the
torque
observes
rather
one
bulk.
the
in
when T is
point is that such require a probe to be
a
fluctuations
and
j,
thus
j2)
number
the
into
velocity
flow
the
Our
then
of
Section
in
injected
is
energy
related
behavior
We
point
this
to
global
a
high Reynolds numbers. For speed fl angular at constant analysis shows that the mean
very
rotating
R
v.
externally fixed parameters (such (such as urm~). The fluid is set into
variables cost
detail
in
return
the
such
,
Reynolds
itself
is
that
stress
pR5n2fjRe)
smooth, highly polished disks,
of
case
will
function
unknown
f
These
of the flow; it is
surface the condition of the disk. Indeed it geometry, e-gconst., in the limit of very high Reynolds number for (even so
ment's
f(Re)
is
form
We
mean
=
functional
of it.
realization
characteristics
f where
bound-
etc.
11)
large part
a
density p and kinematic viscosity required to drive the disks is given by:
torque
flow
driving mechanism, the ~'spatially averaged" mean
studied.
be
consider
us
the
at
pressure
are:
ways
flow.
closed
a
of
use
alternate
propose
of
case
the
flow
instantaneous
of
fluid
a
we
we
in the
justify
gix,t)d~x,
or
the
whose
time
scaling of j
the
cases
some
example,
of
and
using the
defined
is
=
flow
entire
space
over
of
must
=
In
the
covers
number
article~
this
least
at
of
we
/
gin)
In
measurements
term
N°11
fluctuations
consumption
power
ones,
A.
Reynolds number,
turbulent
non-local,
possible
microscale
the
II
Reynolds
turbulent
a
the
characterize
internal
an
available; Taylor
is
and
PHYSIQUE
DE
JOURNAL
1730
how
the
the
flow.
mea-
We
fluctuations,
for the
We
two
then energy
Reynolds
turbulent
number.
2.
2.I.
Experimental FLow
between
walls different
two
Set-Up
GEOMETRY. coaxial
surround
fluids,
We
use
counter~rotating
the
flow
volume.
air
and
water.
The
the disks
von
of
K6rm6n radius
experimental
R,
a
swirling
flow [6]
variable
distance
set-up is
sketched
in
produced H
apart.
Figure
I.
in
the
gap
Cylindrical We
use
two
CHARACTERIZATION
N°11
OF
TURBULENCE
CLOSED
IN A
FLOW
1731
MOTOR
H
PRESSURE TRANSDUCER
Q2 MOTOR
Fig. I. rotating wall, is
of the experimental set-up in water. equal frequencies. H 20 cm, R 9 cm. located in the mid plane between the disks.
Sketch
at
the
In
.
=
10
=
perpendicular
order
in
of air, R
case
blades, motors,
to
increase
the
rotation
the
In
used
in
order
torque
with smooth
disks
The
controlled
rotation
water
are
or
has
rugosity is
larger a
measured
The
disks
disks
disks
fitted
are
height hb
with are
of
radius of
cm
2
"
by
driven
=
9
=
flR~ Iv
cm,
and
thus is
H
in the
range
set
a
and 45 Hz
450
and
the
to
of 8
vertical
thickness
independent cm
counter-
flush
0.5
cm,
DC
watt
controlled
diameter.
in
The
10~.
=
and increase
0
20
two
40 Hz.
apart,
cm
are
enclosed
in
cm
in
Parvex-Alsthom The
Reynolds
RS420 numbers
motors
achieved
10~.
Experiments are performed with disks whose surface Calibrated waterproof sandpaper is used and rugosity. independently with an optical profilometer Iii. than
a
height. Light disks are cut-off of the the high frequency with a regulated constant water
19.8
maintained
independently by
rotated
with
cm
disks
the
mounted
=
diameter
temperature
frequencies
Re
R
inner
effects
inertia
are
about
by
motion
adjustable from 0 to are enclosed in a cylinder 23.2
are
thus
are
The
measurements.
circulation. at
reduce
to
the
and
cm
surfaces,
into
transducer,
pressure
frequencies of which
of water, disks container, 19.3
case
cylindrical
40
=
disks
entrainment.
by a feed-back loop. The largest Reynolds numbers
.
H
cm,
flow is set
The
=
the
to
The
controlled
is its
emphasize that in both cases, the forcing of the flow is such that the rotation rate of is kept through feed-back that other such loop. This quantities constant a means the by applied the disks fluctuating An the quantities in input time. torque as or energy are experimental study of these fluctuations has recently been done [8]. We
disks
the
2. 2. on
a
MEASUREMENT National
anti-aliasing
TECHNIQUES.
Instruments filters.
NBA2150F
In
each
16-bits
case
the
digitizing
transducers signal from the are card which incorporates the
recorded necessary
PHYSIQUE
DE
JOURNAL
1732
II
N°11
(local) are performed in air using a TSI Velocity subminiature hot-film measurements probe with a sensing element 10 ~m thick and I mm long. Velocities deduced from are voltage the validity using the usual King's law fi a)16, of which measurements (e~ has been checked, and the coefficients a and b obtained from in a calibrated measurements wind tunnel. The position of the probe is adjustable. We have checked that the presence of the probe does not affect the flow by performing identical measurements measurements with the probe support different angles. inclined at
.
=
Pressure PCBHl12A21 piezoelectric transin water done with a 5 mm measurements are ducer, mounted flush with the lateral wall, in the mid-plane the disks. It is between acceleration-compensated and has a low frequency cut-off at -3 dB equal to 50 mHz; its
.
is I
rise-time
flush
mounted
Torque
.
the
EG&G
3.
performed driving
amplifier
read
to
and
its
rise-time
also mm,
is 25 ~s.
experimental apparatus using liquids. One of (strain by a calibrated Lebow torque sensor Signal-to-noise ratio is improved by the use of an gage bridge output. in the
Inotor
1102-3.53Nm).
Model
lock-in
Hz
0.05
is
the
to
piezoelectric PCB103A02, transducer mide-plane. Its active diameter is 2.I
a
and in the
wall
-5%
at
are
connected
is
shaft,
gage
cut-off
by
measured
are
lateral
measurements
disks
air
in
the
with
frequency
low
its
ms.
fluctuations
Pressure
the
Results
We
first
with
the
at
report
which
to
recast
turbulence
to
TORQUE
3.I.
the
sure
The
torque applied of
case
blades,
disk
a
is
and
a
of the
flow.
PRESSURE
AND
driving torque the
is
boundary
flow
they permit transition
the driving torque and show how it depends on the on manner imparted to the fluid. We then show that measurements pressure provide a good estimate of the rms velocity fluctuations in the flow and the torque data the In particular it reveals onto a very simple form.
measurements
moInentuln
on
one
fitted
MEASUREMENTS: of the
blades
with
just equal to very simple
disks
what
of
needed
is
referring
that
the
authors The
thickness
sion
depends that
cannot
In
fluid is
done
the
tribution
in
be
disks
is
on
whether
flow is
the
equation (2) f(Re) observed
turbulent law
in
our
lit&. ~w
experiment
regime Schlichting's
yields
a
boundary layer
or
thickness
c~
The as
turbulent. has
been rotation
using e
fl~.
This
is
~w
the
fluid is
of the
entrainment
the
argument
f
and
at
case
This
where
the
between
in
also
the
when
case
rugosity hb is larger than the boundary high rotation rates as observed by many
whose
decreases
which
laminar
situation,
of fluid
slices
mea-
rates.
'
dependent.
viscosity
boundary layer,
-of the
e
T
the
subtle:
more
layers and is thus
We
equal
at
hb
const.
~w
disks
uia
VARIABLES.
the
motion
into
set
~2~2
~
equation (2), f(Re)
to
smooth
of
case
boundary
the
so
"
the
is
FLow
opposite directions simplest one: in that
in
rotate
to:
sublayer. This is eventually always the [5,9j and again in Figure 2 (asterisks).
viscous
disks
back of the
entrainment
to
leads
calculation
y so
INTERNAL
they height hb
when
skin
Re
increases;
In the
verified rate
in
addition
laminar
regime
(see [10j
of the
empirical
(v/fl)~/~;
through
done
pv(0vjj /0z)
friction
this
for
a
its e
the
involves expres-
/fl
~w
review),
but
disks is always to high.
(-th-power velocity gives f(Re)
c~
dis-
Re~~/~
CHARACTERIZATION
N°11
OF
IN A
TURBULENCE
CLOSED
o.7
(~
FLOW
1733
(a)
o,
QJ
ii~
§ c
~i
~
5
0
10
Rotation
Qi
slope
[Hz]
Q2
io
°
lo
=
35
30
25
20
15
frequency
(~)
2.01
£ E
j~_
~o
.
~ ~
~
.
~
~~-
)
-2
lo
o ~
>
io°
lo'
Fig. of
Torque
2.
smooth
Our
(o)
in
measurement
and
(*, b
rugose
measurements
=
~ with
a
We
1.82.
~w
in
note
We
rotation
not
that
flow
the
disks, the
forcing
the
of the
measurements
on
geometry flow that
in
many
result;
Figure
see
p/fl2
~'~'
~~~
~
2
(circles)
the
gap
between
of the
probe the
the
geometrical container, etc.
bulk
but
the
of the
we
observe:
~~.
observed scalings are in boundary layers are always turbulent.
(power input),
where
'
the
other
case
~
asymptotic
these
rates:
Linear
that
that so
flow
rotation varying equal and opposite coordinates. and 16) logarithmic
with
(a)
this
Re~°
f(Re)
find
2
rates
depends
the
with
~~
~
disks
disks.
~w
Figure
explored Inean
thus
for
water
37 ~Jm)
agreement
in
are
Qi
equency
otation
state
flow.
disks
is
itself
characteristics, The
of the
torque fluid
turbulent. such
as
gives only motion
can
the
entire
This
Indeed
the
in
the
separation
some
only
range state
of
between
indication
be
of does
turn
obtained
on
the
from
JOURNAL
1734
PHYSIQUE
DE
N°11
II
(a) w
~~
~
o
o
Cid
~
-
o
oo
w o
~
£
~
~
o~
~
o~
w~ ~o
~°oooo°°~~~ ~0
Qi
frequency
Rotafion
25
20
15
10
5
~o
o~
~»
wwwwww~
o
o~
~
35
30
[Hz]
Q2
=
(b)
p
slope
2.06
~
~
P
»
/
w
»
g
, -
,
cz C~~
g
~*
fl
w
~ ~
~
slope 1.76
o
~
~W w »
o
o
~w ~
»
o o
~o w
o°
» o
~
o
o
~
smooth
We
of the
variance
asymptotic
regime
Evolution
3.
the
In
water.
lo). (a)
disks
show
that
the
measurements
at
now
pressure
Linear
just
measurement;
recall
of prrrs
for
rw
intensity the
disks
rugose coordinates.
[Hz]
Q2
=
fluctuations
pressure
~~
(b) logarithmic
and
that
the
10~
Qi
frequency
Rotation
in
o
10'
10°
Fig.
o
the
in
(*,
b
=
fluctuations in the velocity wall, in the mid-plane- We equation for the pressure: of
lateral the
AP
~Pv'
~
ll~v)~l
"
~P~j)fl~
mid-plane, 37 ~Jm)
bulk
and
can
that
stress
flow
the
at
~~
prrrs
be
wall, for
~~
~w
obtained
this
is
a
uia
global
13) ,
j
is
a
Poisson
(see [11j Figure varied.
for 3
It is
the
one, an
shows
the
readily
of
term
source
extended
which
involves
the
velocity
level
prms
gradients
of
disks
rotation
the
flow
review).
evolution observed
of the
that,
fluctuation
pressure
contrary
to
torque
measurements,
as
the the
pressure
rate
fluctuations
is
CHARACTERIZATION
N°11
OF
TURBULENCE
CLOSED
IN A
1o°
FLOW
1735
(a)
io.'
-8
-6
2
0
-4
4
(p-P)/P,~ns 0
(b)
~
»
~i
o~
o°°~
»
o
~
~°o@~
o o
0.3
Ul
*
°°
°
.
~j
~
O
O
~z
~
nJ
~'
j~
o
,
w
-0.5
uJ
°o~
*
o
°~o
ww
o
*.
-0.6
o
w
».*
o
-0.8 ~
~
~~
-0.9 ~~
5
0
10
Fig.
Evolution
4.
water,
in
high
and with
disks
Qi
"
Ll2
35
[Hz]
fluctuations. Probability Density of the Measurement Function pressure mid-plane. (a) Comparison of low rotation 2 Hz) rate (dotted line, Q (solid line, ~ 34 Hz). (b) Comparison of the variation of the PDFS' skewness frequency; (o): smooth disks and (*); disks with b 37 ~Jm rugosity.
in
rate
rotation
change
30
25
20
of the
wall
rotation
the
display
the
at
15
frequency
Rotation
the
=
=
=
when going from slow to fast disk At low fl, prms rotation. remains asymptotically as level, while in the limit of large rotation rates it scales turbulent flow [12,13]. Note, that however again the scaling a power law, as expected in a once depends on the between entrainment mechanism, as illustrated by the difference of behavior at
an
smooth This pressure
a
almost
and
of
behavior
constant
rugose
change
of
fluctuations
disks. behavior
is
also
visible
on
at
low fl
(see Fig. 4a):
regime they display the well behavior in the PDFS shape remains rates, the skewness regime. We note that the
known may
about transition
be
seems
tails
by
while
constant to
be
PDFS
the
exponential measured
Density Functions (PDFS) of the Gaussian, while in the turbulent are towards low values [14]. This change of skewness (see Fig. 4b). At low rotation
Probability
the
its
it more
decreases
abrupt
almost when
linearly
rugose
disks
in are
the
turbulent
being
used.
PHYSIQUE
DE
JOURNAL
1736
N°11
II
~~ ~
~~ (
i ~
.35
-)------
1.3
15
25
20
Velocity
5.
tuations
the
at
m/s). (o):
in
height,
at
r
thus
We
prms
cm
observe
now
fluctuations,
urms,
regime, the simplest
turbulent
the
in
g(Re)
where
case
=
wall
a
g(Re)
~
pressure
fluc-
(both
scaled
velocity
recorded
the
at
same
cm.
the
as
of
transition
flow
the
of the
measurement
On
grounds, one Reynolds number.
the
of Re
cc,
~
that
so
pr~s
to
expects the
In
pu)~~.
c~
a
of
variance
dimensional
function const.
from
anemometry
calculated
and
disk
reveals
flow.
arbitrary
an
is
3
[Hz]
hotwire
meaningful
of the
bulk
is
flow
yields
prms
using
lower r
Q2
=
50
45
blades)
with
the
and
cm
the
at
that
fitted
above
11
=
Qi
measurements
cm
h
pressure
show
pu)~~ g(Re)
=
the
the 5
=
(*):
(disks
air
to
h
at
axis.
the
that
We
state.
velocity
the
measured from
in
compared
wall
pressure
5
=
turbulent
fluctuations
rms
lateral
40
35
frequency
Rotation
Fig.
30
It
quasi-Gaussian approximation [15,16], so that the pressure case fluctuations mostly governed by the velocity fluctuations at the integral scale although the are involves the in velocity gradients. its equation term source We have directly tested the proportionality of u)~~ and prms in the experimental set-up in air, simultaneous where and velocity possible. Velocity measurements measurements pressure are made hotwire the flow. different radial and vertical within using positions anemometry, at are (0.22 + 0.03)udisk throughout the 2grRfl where udwk We obtain urms volume, measurement /p (0.32 0.02) when the is the disk rim velocity. On the other hand, we observe + udisk prms has
been
shown
the
to be
the
in
~
=
=
of
recorded
is
pressure
the
at
the
Altogether,
disks.
of the
wall
probe
anemometer
either
and
in the
of
the
of the
center
prms/p
gives
it
(1.45
=
measurement
transducers,
pressure
(see Fig. 5). torque scale (an approach
then
We
reconsider
velocity shown in Figure 6,
teristic as
in
our
where
we
for
fi
using
measurements
volume
already used in [17] ). have plotted f/Vprms us.
The
fi
all
disks
of the
frequencies
rotation
of
Rfl)
behavior
is
radically
the
one
locations
(instead V is
to
nearer
or
0.07)urms, independent
+
charac-
a
as
changed
volume
of
fluid
a
clear
motion). (I)
a
sharp
transition
transition
in
for
values
scaling
can
flow,
be
observed
which
at
becomes
a
critical
value
turbulent
when
p)~~, giving evidence the
velocity
fluctuations
for
exceed
threshold.
certain
(it)
the
of the
range
is
rotation
verified
rate as
soon
such as
that
the
prms
entire
>
flow
p)~~,
the
becomes
scales torque turbulent.
as
prms.
This
a
N°11
CHARACTERIZATION
TURBULENCE
OF
0.12
~
~
)~~ ~
$~'~
~~
CLOSED
~~~l ~ § ~
+~+
X~
IN A
~
~
~
~
~
~
FLOW
1737
*
~~
/#
.~
#
~'
t
0 08
4'
j
i
~
~j~
* w
$$t
# ~
~
+
i~
~
~~
~
#
~ +
~ +
~
#~ w +
#
~
+
~
~
~
~
1°
sq"( Fig. flow,
6.
that
Note
all
threshold
the
for
two
to
access
forcing intensity) the
the
performed
measurements
to
torque,
rotation
power
input (P
the
quantities:
amplitude
of the
quantities
These
velocity allow
integral length scale length 1/.
the
motion:
dissipation
L*
of
shape,
internal
flow
in
the
of the 37
pm.
showing variables
(of
state
course
gives the
and
flow. and
rate
fl,
r
=
of the the
fluctuations
pressure
characteristic
prms/p,
calculation
(*)
pm,
experiment)
fluctuations
the
same
of
a
the
on
23.7
turbulent
of the
the
fluid).
Kolmogorov
geometry
the
term
flow
(#)
pm,
on
in
The
turbulent
of the
scales
global
11.3
recast
of the
the
on
(x)
disks) collapse
transition
depends
pm,
intensity of velocity fluctuations by Vprrrs, where V is the volume
CHARACTERISTICS. and
of the
response
the
8.6
when
torque
of the it
(+)
rugose
or
driving
with
adimensionalized
fundamental
TURBULENCE
3.2.
give
and
universal,
not
scaling
appropriate
smooth
of the
Pr~nS
disk
one
on
is
rugosities:
disks
determination
a
is
torque
(with
behavior
allows
it
The
various
curves
simple
very
alone;
for
applied
torque
the
fi.
by
Measurements
fluid.
the
of
Variation
measured
as
30
25
20
15
of the
characteristic
characteristic
flow, Taylor
of
length
microscale
I
Integral Scale L*. In most experiments L* is identified with a charactersitic scale L experimental set-up (e.g. the grid mesh size in grid-turbulence). It is a fixed parameter, independent of the forcing of the flow, even though its physical meaning is to give the size of the studies which containing eddies. Its evaluation is made more precise in numerical energy define it through the velocity power spectrum: 3.2.1.
of the
f kE(k)dk fE(k)dk
2gr
fi We the
to
propose energy
change of
keep
input the
into
kinetic
the
spirit of this the
definition
Equating
flow.
of
energy
~
structures
)
and
the
power
of size
L*,
~w
~
L* ~w
calculate
L*
consumption one
~~
~
P
to
as
a
of the
characteristic
size of
flow
rate
to
the
of
obtains:
_j f P
3/2
(4)
PHYSIQUE
DE
JOURNAL
1738
N°11
II
0.2 0.18
~ -
0.1
~
0
Fig. 7. consumption
The
correspond calculate
to
Taylor
3.2.2.
Its
flow
This
rates
constant
to
to
that
2
cm
the
energy
It
decreases
at the
transition
7.
We
note
that
disks
and
that
it
This
confirms
value.
set-up.
from
Figure
in
about
of the
diameter
length
inner
R~ which arrangement.
the
in
the
does
not
need
the
to
is
scale
obtained
it
filaments
local
from
j
hotwire
)
maximum
turbulent
the
addition,
In
of enstrophy. Together with urms it motion. intensity of the turbulence, independently has been proposed [17,18] that I is the observed in swirling flows. Traditionally, of the
characteristic
of the
characterizes
vorticity
of the
scale
scale
the
is
number
experimental length characteristic microscale the Taylor
of
I.
gives the
Renolds
a
calculated
displayed
is
experimental
the
in
scale L*, boundary.
rotation
the
than
[Hz]
Q2
"
40
35
measurements.
Microscale
measurement
yields
length
particular
any
smaller
times
at low
remains
it
the
at
frequency
rotation
(20 cm)
threshold,
the
length
fluctuations
disks
the
Integral
disks.
pressure
of the fluid
L* is 10
from
it
smooth the
with
Above
regime
turbulent
and
height
the
from
turbulence.
to
flow
of L*
variation
rapidly
with
Measurement
of the
Lli
frequency
Rotation
30
25
20
15
10
5
using:
anemometry,
(VU)~,
~
lL~~~
computation, the Taylor hypothesis is used to spatial velocity profile [19-21]. We propose of I, but to the velocity gradients from the dissipation, estimate input and the viscous dissipation rate: power and
a
in
the
fixed
present
location
to
the
P
pV The from the are
numerical
values
corresponding comparable to R~
@ ~w
Figure
scattering
ultrasound
behavior
see
~jp~~2
turbulent the is
one
~'~'
~
8a
are
measurements
Reynolds measured
observed
in
the
'~)mS
~2
fl/pV
~
~
close to the
one
turbulent
anemometry
regime.
equating
above
the
time
averaged
j~~
obtained
one
in
[17].
size
and
at
definition
Prms
directly
=
local
measurements
the
retain
fl/V
~
of the filaments core number R~ urmsl/v;
from
time
relate to
observes that
the
similar
conditions
Figure 8b that usual
the
shows values
asymptotic
CHARACTERIZATION
N°11
OF
TURBULENCE
IN A
CLOSED
FLOW
1739
5
i
15
10
Qi
Q2
=
35
30
25
20
frequency
Rotation
[Hz] (b)
12
o
~y~ ~
~
~
Ct~
o
o
i
a~
o
ooooooooooooooo°°
°o~ oo
~5
15
10
Fig.
Measurements
8.
velocity
fluctuations
Reynolds
number
R>
smooth
Iv
itrrrsl
=
/$,
cc
of the
the
smallest
transfer
possible
consumption
power
Length
kinetic
of the
structures
of the
the
Calculation
the
variations
3.2.4.
one
of q with the recovers
Comparison
with
of the
Taylor
microscale
from
the
Taylor-based
flow.
This
of the
in
energy the flow.
scale is
characteristic
turbulent
In
the
number
Re =
rms
turbulent
R~~ Iv.
In
motion
regime
turbulent
of the heat.
to
it is
It
friction,
viscous
gives the
estimated
of
size
from
the
flow:
~3 /PV~
The
lHz]
lb) Evolution of the integral (experimental) Reynolds in
1/.
~
regime,
Q2
expected.
as
Kolmogorov Dissipation
3.2.3. I.e.
with
la)
disks.
dissipation
energy
regime R>
turbulent
the
with
and
Dim
35
30
25
20
frequency
Rotation
the usual Hot-
disks
~~~
~
rotation
relationship Wire
1/4
frequency
)
~w
Measurements.
shown
are
Re~@, To
as
in
observed
establish
Figure 9a. In the in Figure 9b.
further
global
the
calculation
turbulent
of tur-
compared the results we with traditional hot-wire the fluid. anemometry, in the working The dissias (0u/0z)~ pation is estimated from the local velocity C~v where u measurements < >, as EL is the local flow velocity and a localized Taylor hypothesis [21] is used to relate temporal and bulence
small
scales
characteristics
from
have
measurements, setup using air
~w
JOURNAL
1740
PHYSIQUE
DE
N°11
II
40
(a)
~
35
G
o
30 a -
~i
~
~
o
25
o
~
o
o
~
°°o
20
°o~
o
~ooo
o~
15
o~~
~~5
frequency
Rotation
30
25
20
15
IO
Qi
Ooo
35
Q2
"
lHz]
~
o(b) o°
5
2 rn~
°
oo
o
4
go
oo°
~o°
coo°
o
a
il~
~
'~
i~
XXXXXXXXXXrXX
°
~
OXX OX
~
~X
~/
o~ X
R w
10
5
Fig.
Measurement
9.
input
energy
in
the
with
(experimental) Reynolds spatial
derivatives.
sured
dissipation Kolmogorov's
The
The
Re
constant
of the
center
near
number
is also
=
Ce
calculated
with
the
considered
is
disks,
the
from
the
Q2
"
motors
regime
equal
be
to
1/
15
to
as
function
from
of the
the
integral
L*Re~~/~
cc
velocity u is meabe homogeneous.
the
since
consumption
power
calculated
~,
a
expected
flow is
the
length
L*,
scale
turbulent
where
[Hz]
dissipation
integral length
R~~ Iv. In the
between
gap
(a) Kolmogorov
disks.
smooth
(b) Scaling of1/
flow.
Qi
35
30
25
20
15
frequency
Rotation
to
eG
as
"
(Pm~~hanicai) /M.
dissipation scale is then calculated as 1/G,L (v~ leG,L)~@ for several values of frequencies. The results, displayed in Figure 10, show that both methods yield the disk rotation LRe~~@ the correct order of magnitude for 1/. However, the Reynolds number dependence1/ It shows that it is very difficult to obtain a good is only observed for the global measurements. level is fluctuation of EL using local velocimetry in closed flows where the velocity estimate of EL, such for the calculation other alternatives very high (about 35% here). We have tried Karman-Howarth relationship, again the order of magnitude is correct, but not using the as of the Taylor microscale dependence. It also affects the the Reynolds number measurements ~
~w
I
when same
as
it is
estimated
when
derived
expected We the
I
~w
thus turbulence
the
from
from
the
velocity
local
mechanical
Re~~/~ scaling. observe small
that scale
the
global
measurements
characteristics
in
a
the
measurements:
power
input
in the
yield
closed
flow.
a
flow,
correct
order but
and
of
it
magnitude
does
coherent
not
is
exhibit estimation
the
the of
CHARACTERIZATION
N°11
x
OF
TURBULENCE
CLOSED
IN A
10~~
FLOW
1741
~
9
8 ~.73
S
7 ~'~
~'~
~'~
~ -
~
f
5
~ ~
4
~
~ '
T
3 15
10.
Calculation
(o),
fluctuations The
inset
and
data
the
shows
Kolmogorov
of
from
local
dissipation
hotwire
Qi
from
(*)
50
45
[Hz]
Q2
"
length,
measurements
logarithmic
in
~
40
35
frequency
Rotation Fig.
30
25
20
~
consumption position h 11
power
probe
=
and cm,
pressure r
=
3
cm.
coordinates.
Conclusion
4.
behavior of a closed flow investigate the dynamical using global ii. e. spatially averaged) measurements only. These sophisticated simple that probes be and do require measurements not are very introduced bulk of the flow. physical quantities such as the The scaling of relevant in the applied torque or power consumption in terms of internal flow variables (e.g. urms) reveals the turbulent transition regime. It does so much more clearly than the corresponding to the rate). Furthermore, variations with the experimental control rotation parameters (e.g. disks the has occurred, the turbulence be characterized with the knowledge of transition once can of the global Indeed, the and of quantities. the input measurements same pressure power We
have
at
moderate
shown
fluctuations the
as
of
measuring motion
in
wall
it is
possible
sizes
the
in
removed
complex
or
flow,
from
geometries
calculate
to
the of
intensity of the
overall
devices
sufficient
are
to
numbers
fluctuations
motion
the
measures
the
at
velocity
rms
interval
that
here
high Reynolds
to
the
turbulence
fundamental
typical length L* IA yields an
L*, I and
scales estimate
of the
characteristics
1/. inertial
obtained
study of fluid
range,
turbulence.
That
these
quantities
may
be
bulk
flow is of
importance
for
the
of the
and/or using
such
[L*,1/] is the
Then
while
Rj from
fluids.
corrosive
Acknowledgments We
acknowledge
helpful
assisted
The
experimental
set-up
could
expertise of Marc
Moulin
and
us
in
the
discussions
who
measurement not
Franck
Stephan
with
of the have
sand
been
Vittoz.
Many thanks to Sergio Ciliberto rugosity with his optical profilometer.
Fauve. paper
modified
so
many
times
without
the
(patient)
JOURNAL
1742
DE
PHYSIQUE
N°11
II
References
[3]
Lumley J-L-, A Nelkin M., Adu. Phys. 43 (1994) (Cambridge Turbulence Frish U.,
[4]
Magnetohydrodynamics
[1]
[2]
Tennekes
[6] [7] [8] [9]
[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
and
flows
are
first
course
in
turbulence
(The
MIT
Press, 1971).
143-181. U. of
Press.
1995).
importance
in
a
wide
range
of
ranging from areas, understanding of the
cooling circuits study of heat transfers in nuclear reactors dynamo effect that generates the magnetic field of planets. Nagata S., Mixing, chap-I, (J. Wiley & Sons, 1975). so-called swirling flows", see for instance, Zandbergen these "von Karman For reviews on Ann. Mech. and Dijkstra D., Reu. Fluid P-J19 (1987) 465-491. Commitn. Arecchi F-T-, Bertani D. and Ciliberto S., Opt. 31 (1979) 263. LabbA R., Pinton J.-F. and Fauve S., J. Phys II France 6 (1996) 1099-1100. J.-F. and Fauve S., Phys. Flitids 8 (1996) 914-922. LabbA R., Pinton Schlichting H., Boundary-layer theory (McGraw-Hill, 1979). J.-F., to appear in Non Linear Pinton Dernoncourt B.. LabbA R. and Fauve S., Abry P., Science Today, Beheringer, Ed. (Springer-Verlag, 1996). Abry P., Fauve S., Flandrin P. and Laroche C., J. Phys. II France 4 (1994) 725-733. Cadot O., Douady S. and Couder Y., Phys. Flitids 7 (1995) 630-646. Fauve S., Laroche C. and Castaing B., J. Phys. II France 3 (1993) 271-278. Batchelor G-K-, Proc. Cambridge Phil. Soc. 47 (1951) 359-374. George W-K-, Beuther P-D- and Arndt R-E-A-, J. Flitid Mech. 148 (1984) 155-191. Dernoncourt B., Pinton J.-F. and Fauve S., submitted to Physica D. Conference, Turbulence Hernandez R. and Baudet C., in Proceedings of the Vth European Kluwer, (1996). Taylor G-I-, Proc. Roy. Soc. A. 164 (1928) 476. Fisher M-J- and Davies P-O-A-L-, J. Flitid Mech. 18 (1964) 97-116. Pinton J-F- and LabbA, R., J. Phys. II France 4 (1994) 1461-1468.
the
[5]
H.
to the
