Geoscience Reference
In-Depth Information
so-called Ekman spiral. Although boundary layer velocity
profiles that approximate the Ekman layer occur in the
atmospheric boundary layer and in wind-driven surface
layers of the ocean, turbulence always plays a role in atmo-
spheric and oceanic boundary layers because of the high
Reynolds numbers involved. Unsteadiness of the mean-
flow and thermal effects may also be important.
From Ekman's Ph.D. work, the flow induced when the
boundary and the fluid at a large distance from the bound-
ary approach the same rotation rate has become known
as the Ekman layer. Later,
Bödewadt
[1940] studied the
boundary layer flow created near a stationary plate in a
body of fluid rotating with constant rotation rate at large
distances from the plate (think of the boundary layer at
the bottom of a cup of stirred tea!). For the Bödewadt
layer the centrifugal and radial pressure gradient forces
are in equilibrium in the fluid rotating at large distances
above the plate but the centrifugal forces are reduced
within the boundary layer, due to viscous action, and
the axially independent radial pressure gradient causes a
radial flow that is predominantly inward (that is why the
tea leaves settle in the center of the cup) and a consequent
(from continuity arguments) upward axial flow.
The
vonKármán
[1921] layer is in some senses the reverse
of the Bödewadt layer. It is induced by a rotating bound-
ary in otherwise quiescent fluid and sets up a flow where
fluid is thrown radially outward at the boundary due to
the action of centrifugal forces and is replaced by a down-
ward axial flow. The BEK boundary layers are particular
examples of a family of rotating flows and are sketched
in Figure 4.3. In addition to the BEK family, one can
envisage other rotation-induced flows, such as boundary
layer flows over rotating cones or spheres, with or without
added coflow (see, e.g.,
Kohama
[2000];
Garrett and Peake
[2007];
Hussain et al.
[2011]).
In the laboratory, the Bödewadt and von Kármán flows
are set up on plates, that are stationary and rotating,
respectively, in the laboratory frame. The Ekman flow is
usually set up between two disks with slightly different
rotation rates (
z
=
0
±
), where boundary lay-
ers are formed on both disks and the flow spirals radially
inward on the slower moving disk and vice versa on the
faster moving disk. In the region between the disks and
outside the boundary layers, the fluid rotates with
0
and
there is an axial flow exchange between the two boundary
layers.
In category (c) we have, for example, the pressure-driven
axially rotating pipe or channel flows. For an axially
rotating pipe, i.e., with the rotation axis parallel to the
mean-flow direction, the rotation has no direct influence
on the axial laminar mean flow and the basic parabolic
axial mean-flow profile of the nonrotating case is retained.
Early linear stability analysis and experiments have shown
that rotation has a destabilizing effect [
Pedley
, 1969;
Imao
et al.
, 1992]. In the turbulent case, the azimuthal reflection
symmetry is broken and the cross-flow Reynolds stress
term (which is zero for the nonrotating case) gives rise
to a deviation from solid-body rotation (first observed by
Kikuyama et al.
[1983]). The streamwise velocity distribu-
tion becomes less full in this case. Only one experiment of
channel flow with streamwise rotation has been reported,
where
Recktenwaldetal.
[2007] measured the velocity field
with particle image velocimetry (PIV). Comparisons were
made with direct numerical simulations (DNSs) and in
that case, which is more complicated than the rotating pipe
flow, a secondary flow is set up perpendicular to the main
flow direction, and the mean-flow profiles were found to
become fuller than for the nonrotating case.
4.1.2. Layout of Chapter
The flows in the different categories above differ with
respect to their geometry but, more importantly, in how
rotation affects them. In the following, we focus on three
different flows that are relatively amenable to laboratory
investigation, one from each category described above:
One is plane Couette flow undergoing system rotation
about an axis normal to the mean flow, another is the von
Kármán boundary layer flow, and the third is axially rotat-
ing pipe flow. These flows are associated with interesting
flow features that may be relevant to both technical and
naturally occurring fluid systems. In Section 4.2, we define
the important nondimensional parameters that govern
them and discuss some of their interesting flow features
in various parameter ranges. Various experimental real-
izations of the three different flow systems are described
in Section 4.3 and considerations and limitations regard-
ing the laboratory systems are discussed. Some intriguing
results are described in Section 4.4, and in Section 4.5
Ω
Ω
Von Kármán disk flow
Bödewadt flow
Ω+ΔΩ
Ω-ΔΩ
Ekman boundary layer flow
Figure 4.3.
Three different examples of the BEK family of flows.
