Geoscience Reference
In-Depth Information
4
Rotation Effects on Wall-Bounded Flows:
Some Laboratory Experiments
P. Henrik Alfredsson
1
and Rebecca J. Lingwood
1
,
2
4.1. INTRODUCTION
dynamically important (however, it may be problematic
from a practical experimental point of view), whereas
the Coriolis term 2
Rotation of a solid boundary with respect to a fluid or
rotation of a fluid system occurs in both constructed and
naturally occurring systems. An example of the former is
rotating machinery, e.g., rotating compressors, turbines,
propellers, and centrifuges, and examples of the latter are
flows in the atmosphere and the oceans.
When studying such systems, one may choose from two
principally different approaches, namely to view the sys-
tem either from a nonrotating frame of reference or from
the rotating system itself. Depending on the flow system
under investigation, one approach may be more appro-
priate than the other. In the former case the rotation
enters through rotating boundary conditions, in the other
through the body forces set up by the rotation. In the latter
case, the rotation is felt through two well-known physi-
cal effects, namely the centrifugal and Coriolis forces. In
the rotating frame of reference, the continuity equation is
unchanged from its nonrotating equivalent and is given as
u
may completely change and
dominate the flow behavior compared with a nonrotating
counterpart.
×
4.1.1. Categorization of Different Rotating Flow
Systems
One may categorize rotating flows in many different
ways, but here we restrict ourselves to some cases where
there have been laboratory experiments of canonical flows
reported in the literature. For the present undertaking, we
focus on three different categories (see also Figure 4.1):
(a) System rotation vector parallel to mean-flow
vorticity
(b) Flows set up by the rotation of one or more
boundaries
(c) System rotation aligned with the mean-flow
direction
System rotation of shear flows in category (a) occurs in
many types of rotating machinery as well as in geophysical
flows. A canonical flow of this type is flow in the
x
direc-
tion through a two-dimensional channel with rotation
along the
z
axis (
z
). In this case the Coriolis acceleration
gives rise to a body force that can be “unstably stratified”
and, if it is large enough (compared with restoring viscous
forces), can therefore destabilize the flow. So, for exam-
ple, for pressure-driven Poiseuille flow, half of the channel
will be stabilized and half of the channel destabilized,
whereas for wall-driven Couette flow, the full channel will
be either stabilized (cyclonic rotation, i.e., the vorticity
of the mean flow is in the same direction as the rotation
vector) or destabilized (anticyclonic rotation). Analo-
gously, for flow over a flat plate rotating about a span-
wise axis, the boundary layer on the leading side will be
∇·
u
= 0,
(4.1)
where
u
is the velocity vector in the rotating system,
whereas the Navier-Stokes equation becomes
∂
u
∂t
+
(
u
1
ρ
∇
2
u
·∇
)
u
=
−
p
+
ν
∇
−
×
(
×
r
)
−
2
×
u
,
(4.2)
where
is the system rotation rate, which is assumed
to be constant, and the boundary conditions are cast in
the rotating frame. The centrifugal term
r
)
can be incorporated into the pressure term and is not
×
(
×
1
Royal Institute of Technology, Linné FLOW Centre, KTH
Mechanics, Stockholm, Sweden.
2
Institute of Continuing Education, University of Cambridge,
Cambridge, United Kingdom.
