On the relationship of the injected power between constant torque forcing
and constant velocity forcing of fully turbulent flow
Olivier Cadot
a)
Unite ´ de Me ´canique de l’Ecole Nationale Supe ´rieure de Techniques Avance ´es, Chemin de la Hunie `re,
92170 Palaiseau Cedex, France
Jean Hugues Titon
Laboratoire de Me ´canique, Universite ´ du Havre, 25 Rue Philippe Lebon, 76058 Le Havre Cedex, France
Received 9 December 2003; accepted 27 February 2004; published online 4 May 2004
DOI: 10.1063/1.1714666
In a recent experimental work, Titon and Cadot
1
consid-
ered the turbulent flow that is either forced with a constant
angular velocity mode, or with a constant torque
mode. For each case, the power injected in the turbulent
flow is given by
P t = P
eff
t -
1
2
I
1
d
1
t
2
dt
-
1
2
I
2
d
2
t
2
dt
, 1
where P
eff
(t)=
1
(t)
1
eff
(t)+
2
(t)
2
eff
(t) with
i
( t ) and
i
eff
(t)
being, respectively, the instantaneous angular velocity and
the torque applied on the forcing device i. For the mode,
there are not any angular velocity fluctuations and the power
injected in the flow is
P
t =
0
t . 2
We denote by
0
=
1
=-
2
the counter-rotation angular
velocity, and by ( t ) =
1
( t ) -
2
( t ) the total torque that the
turbulent drag exerts on both devices. For the mode, the
torques on both forcing devices are fixed in order to have
0
=
1
eff
=-
2
eff
. We denote by ( t ) =
1
( t ) -
2
( t ) the to-
tal velocity. Thus the power injected in the flow becomes
P
t =
0
t -
1
2
I
1
d
1
t
2
dt
-
1
2
I
2
d
2
t
2
dt
. 3
In this case the power depends strongly on the inertia.
In this Comment, we show that the nonlinear changing
of forcing variables between both forcing modes allows us to
retrieve completely the differences in the shape of the in-
jected power statistics measured by Titon and Cadot.
1
We
start by assuming that the inertia of the forcing devices is
zero, in this case the injected power fluctuations result from
the angular velocity fluctuations of the forcing devices only.
With this assumption, the power injected for the mode is
P
t =
0
t . 4
The angular velocity fluctuations can be seen as a reversed
mirror of the drag torque fluctuations due to the flow: those
that are directly measured in the case of the mode. Thus,
to maintain a constant torque for the mode, the angular
velocity increases respectively, decreases when the torque
decreases respectively, increases. The relationship between
the velocity and the drag torque is a property of turbulence.
We will assume it to be simply
t =a
2
t . 5
The relation is dimensional and has been experimentally
checked for the relationship between time averaged torque
and angular velocity in this experimental setup Ref. 1 and
references therein. Hence, due to a turbulent torque fluctua-
tion of magnitude ( t ), the forcing device has to adjust its
velocity to
t = - t / 2 a t , 6
in order to keep the drag torque constant.
We denote by P
0
, the mean power injected for the
mode, from the measurements of Titon and Cadot:
1
P
0
= P
=a
0
3
. 7a
It is worth noticing that Titon and Cadot
1
did not measure
any differences between the power injected for both modes,
they also have for the mode
P
=a
0
3
,
0
=a
0
2
where
0
=
1
=-
2
. 7b
According to Eqs. 5, 7a, and 7b we can construct a
relation between both powers defined in Eqs. 2 and 4:
P
( t ) = P
O
P
( t ). This variable change takes into account
the nonlinearity between torque and velocity but not the re-
versed behavior in the fluctuations and the fact that both
averaged powers are identical. It is then natural to propose
the simplest nonlinear relationship between the variables as
P
t = P
0
+
P
O
P
- P
O
P
t . 8
Their probability density functions PDFs, denoted by f
*
for the variable P
and by f
for the variable P
, should
then verify from Eq. 8
a
Electronic mail: cadot@ensta.fr
PHYSICS OF FLUIDS JUNE 2004 VOLUME 16, NUMBER 6
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