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layer. Second, the energy distribution across the boundary
layer differs. It is suggested that the differences observed
between these two canonical turbulent boundary layer
flows may reflect the influence of rotation on the inclina-
tion and dynamics of the large-scale turbulent structures.
4.5. CONCLUSIONS
It is clear that rotation plays an important, if not dom-
inant, role in many geophysical situations, whether it be
in the atmosphere or the oceans. However, for these flows
there are also other competing effects, such as temperature
and density stratification. In this review, we have high-
lighted some cases where rotation drastically changes or
is responsible for setting up the flow field and where the
resulting flow fields are preferably studied through labora-
tory experiments. The examples reviewed here do not have
in mind any specific geophysical analogies (or for that mat-
ter technical applications) but are intended, through some
general flow cases, to describe various aspects of rotation
and how it affects stability, transition, and turbulence.
At this point, one may wish to consider the continu-
ing reasons for undertaking laboratory experiments when
simulations are becoming more and more powerful, espe-
cially in the case of system rotation where it is easy to
add the Coriolis term to a working simulation of the cor-
responding nonrotating case (to obtain a solution in the
rotating frame). Some simulations of RPCF and axially
rotating pipe flow have been performed, but they are still
limited to a few parameter values and low Reynolds num-
bers. In the case of RPCF,
Tsukahara et al.
[2010a] were
able to explore the parameter plane consisting of more
than 400 combinations of Re and
, a task that would
be too time consuming to contemplate using numerical
simulations. Also, for both RPCF and axially rotating
pipe flow, simulations still cannot (and maybe never will)
fully resolve small-scale turbulence at high Reynolds num-
bers. Such explorations would therefore require the use of
experimental facilities for the foreseeable future.
For the single-rotating-disk case, full nonlinear simu-
lations have not been reported so far. Here, one would
expect the treatment of the boundary conditions to be cru-
cial given the role upstream (i.e., radially inwardly) travel-
ing modes have in absolute instability. An advantage that
rotating disk simulations have over some other flow cases
is that there is only one nondimensional parameter that
defines the flow. In the Linné FLOW Centre, simulations
of the rotating disk boundary layer flow are presently
underway and we hope will give interesting results in the
future to compare directly with our experimental results
(e.g.,
Imayama et al.
[2012a, 2012b];
Imayama
[2012]).
Moreover, our interest is also directed toward other types
of rotating body induced flows, for instance, the bound-
ary layer flows over a rotating sphere and rotating cones
where simulations would be even harder to carry out due
to geometric complications.
4.4.3. Axially Rotating Pipe Flow
One of the most surprising results for the mean-flow
field of a rotating pipe flow is the deviation from solid-
body rotation. In a cylindrical coordinate system the equa-
tion for the mean circumferential velocity can be reduced
to (for a derivation see
Wallin and Johansson
[2000])
ν
d
2
V
dr
2
=
d
+
1
r
dV
dr
−
V
r
2
dr
(vw)
+2
vw
r
.
(4.8)
Equation (4.8) can be integrated twice, first from 0 to
r
and thereafter from
r
to
R
,togive
R
r
R
−
r
ν
vw
r
dr
.
V(r)
=
V
w
(4.9)
r
The first term on the right-hand side gives the solid-body
rotation, whereas the sec
on
d term gives a contribution
from the Reynolds stress
vw
that results in the deviation
from the solid-body rotation. As was discussed earlier and
sketched in Figure 4.8, the mean circumferential velocity
distribution can be described accurately by a parabola,
which then can
be
used to obtain an expression from
equation (4.9) of
vw
such that
vw
U
b
=
2
S
Re
r
R
,
(4.10)
i.e., the distribution is a straight line. As can be seen, the
resulting cross-stream Reynolds stress is obtained through
a delicate balance between viscous and rotation effectsand
is due to the fact that, by rotating the pipe, the azimuthal
reflection symmetry of the flow is broken. The fact that
rotation also influences the mean streamwise velocity such
that the wall shear stress decreases and maximum velocity
in the center of the pipe increases is also rather perplexing.
Finally, the theoretical modeling of the turbulent
flow in axially rotating pipes by
Oberlack
[1999] should
be mentioned. He used a Lie group approach to the
Reynolds-averaged Navier-Stokes equations and was able
to derive new scaling laws for both the azimuthal and
streamwise mean velocities for the rotating pipe flow. For
instance, the theory gives the azimuthal mean velocity as
V
V
w
=
ξ
r
R
ψ
,
which with
ξ
=1and
ψ
= 2 corresponds to the parabolic
velocity distribution observed in experiments and simu-
lations. Some other results from that analysis are also
substantiated by the experiments of
Facciolo et al.
[2007].
Acknowledgments.
This review is based on both the
authors'own work, and that of several former and present
Ph.D. students and postdoctoral researchers. They are all
