Geoscience Reference
In-Depth Information
out all turbulence within less than one full revolution of
the channel.
The stabilizing effect on turbulence can be understood
by scrutinizing the turbulent energy equation. For unidi-
rectional flows (e.g., channel flows) there is no variation of
mean quantities in the
x
1
=
x
and
x
3
=
z
directions, and
the mean velocity components satisfy
U
2
=
U
3
=0.The
equations for the different components can be written as
∂
∂t
N
I
H
G
K
L
F
M
J
2
uv
dU
2
z
+
11
−
u
2
=
B
D
−
dy
−
11
+
D
11
, (4.4)
A
E
v
2
=
∂
∂t
−
4
uv
z
+
22
−
22
+
D
22
,
(4.5)
C
w
2
=
33
−
∂
∂t
33
+
D
33
,
(4.6)
Figure 4.11.
Rotating pipe apparatus used by
Facciolo et al.
[2007]: (A) throttle valve, (B) centrifugal fan, (C) valve regulated
bypass, (D) electrical heater, (E) distribution chamber, (F) honey-
comb, (G) stagnation chamber, (H) coupling between stationary
and rotating pipe, (I) honeycomb (J) DC motor, (K) ball bear-
ings, (L) rotating pipe, (M) circular and plate, (N) pipe outlet.
Adapted from
Ferro
[2012]. Copyright © Ferro, 2012. Reprinted
with permission.
dy
+2
u
2
v
2
z
+
12
−
∂
∂t
(
uv)
=
v
2
dU
12
+
D
21
.
(4.7)
Here,
ij
is the pressure transport,
ij
the dissipation,
and
D
ij
the viscous diffusion (for a detailed discussion of
these equations see
Johnston et al.
[1972]). Depending on
the sign of
z
, turbulent energy will be transferred to or
from the streamwise component, equation (4.4), and vice
versa for
th
e wall-normal one, equation (4.5), through the
term
−
−
rotating part of the honed steel pipe has a diameter of
60 mm and is 6000 mm long, giving a ratio
L/D
= 100.
After the settling chamber, there is a 1000 mm long nonro-
tating pipe that is connected to the rotating part through a
rotating connection. At the entrance of the pipe a 100 mm
long honeycomb is mounted to bring the air into rotation
where it can develop freely. The pipe is supported by four
ball bearings distributed along its length. The maximum
rotational speed obtainable was 900 rpm.
4
uv
z
, which appears with opposite sign in the two
equations. For anticyclonic rotation (note that the mea
n-
low vorticity
ω
z
f
or
PCF is equal to
ω
z
=
±
dU/dy
),
u
2
−
will decrease and
v
2
increase since
uv
is positive where
dU/dy
is positive and vice versa. It is not obvious what
the overall result will be on the turbulence, except that it
will tend towards an equalization of the two components.
On the ot
her
hand
, n
egative (cyclonic) rotation transfers
energy to
u
2
from
v
2
. The normal component is impor-
tan
it for turbulence production since it directly influences
uv
(see equation (4.7)) and it can be intuitively unde
rst
ood
that negative rotation will lead to a decrease in
−
4.4. SOME INTRIGUING RESULTS
IN ROTATING FLOWS
uv
and
hence in a decrease in the production of turbulent energy.
This type of reasoning can be generalized to other
types of shear flows and shows the competition between
the mean-flow vorticity and the background vorticity
imposed by system rotation. It is not hard to imagine that
similar reasoning may be applied to flows on a plane-
tary scale.
−
4.4.1. Rotating Plane Couette Flow
There are a number of interesting observations related
to RPCF. The theoretical discovery of stationary ter-
tiary states by
Nagata
[1998] shows that the Navier-Stokes
equations may have multiple steady states, and at least
some of these states seem to have been verified in experi-
ments [
Hiwatashietal.
, 2007;
Tsukaharaetal.
, 2010a]. The
flow also has a large number of possible flow states within
the parameter ranges spanned by experiments, revealing
various types of instabilities and also robust turbulent
structures. The stabilization of turbulence for cyclonic
rotation is also quite remarkable for anybody who has
observed it in the laboratory. Even a small rotation rate of
the order of one revolution in 20s can dramatically wipe
4.4.2. von Kármán Boundary Layer Flow
One of the most important and interesting aspects of
the rotating disk flow is the absolute instability of the
boundary layer above a critical Reynolds number (i.e.,
outward from a critical radial position). While the con-
vectively unstable nature of the rotating disk boundary
layer was well established, it was (as far as the authors
