Rotation Effects on Wall-Bounded Flows: Some Laboratory Experiments - Modeling Atmospheric and Oceanic Flows - page 84

(in the blue-green region) the footprints of such vortices,
which are stationary with respect to the rotating disk, vor-
tices that are predicted by linear stability theory to form
part of the convectively unstable cross-low mode. The ini-
tial discovery of the cross-flow instability was by Smith
[1947], was further substantiated by Gregory et al. [1955],
and was visualized in the now classical study of Kohama
[1984]. A second convectively unstable mode, which is vis-
cous in nature and often referred to as type 2 (or type A)
instability, that has a lower critical Reynolds number for
linear instability also exists. While the von Kármán flow is
induced by the disk rotation, acting like a centrifugal fan,
a similar three-dimensional boundary layer is established
in nonrotating configurations, e.g., on swept wings, where
cross-flow instability has been found to be of importance
for understanding transition (see the review by Saric et al.
[2003]).
In the 1980s and 1990s several studies of the linear insta-
bility of rotating disk flow were made to investigate the
type 1 and type 2 convectively unstable modes. However,
the theoretical discovery by Lingwood [1995] of an abso-
lute instability turned the research in a new direction. As
discussed by Huerre and Monkewitz [1990], the response
of a flow to impulsive forcing shows whether it is con-
vectively or absolutely unstable. If the response to the
transient disturbance grows with time at a fixed location
in space, then the flow is absolutely unstable. Lingwood
[1995] showed that above a critical Reynolds number there
is an absolute instability of the von Kármán boundary
layer produced by a coalescence of the inviscidly unstable
(type 1) mode and a third mode that is spatially damped
and inwardly propagating. The local absolute instability
was found to occur above Re = 507, and this value was
confirmed by Lingwood [1996] in a concurrent experimen-
tal study. Her work suggested that this instability mech-
anism is responsible for the onset of nonlinear behavior
and thereby possibly for the onset of laminar-turbulent
transition.
Davies and Carpenter [2003] performed direct numer-
ical simulations solving the linearized Navier-Stokes
equations and suggested that the convective behavior
eventually prevails even for strongly locally absolutely
unstable regions and concluded that the absolute insta-
bility does not produce a linear amplified global mode.
This work has been followed by several recent theoret-
ical (e.g., Pier [2003]; Healey [2010]) and experimental
(e.g., Corke et al. [2007]; Imayama et al. , [2012a, 2012b])
studies that have focused on detailed investigations of the
absolute instability and the transition to turbulence and
to determine whether Davies and Carpenter's [2003] find-
ing of linear global stability is a product of the linear
approximation and/or neglect of the inwardly traveling
disturbances from the outer radial boundary. Pier [2003]
proposed a subcritical mechanism where finite-amplitude

disturbances with secondary instability lead to a nonlinear
steep-fronted global mode at the location of the onset of
absolute instability. Healey [2010] proposed a supercritical
mechanism where inclusion of the finite radius of the disk
in the theoretical analysis creates global instability leading
directly to a steep-fronted global mode regardless of how
low the background disturbance level is.
Only relatively few studies have so far investigated the
turbulent disk flow [ Cham and Head , 1969; Erian and
Tong , 1971; Itoh and Hasegawa , 1994; Imayama , 2012], in
part because of experimental difficulties associated with
achieving spatially well-resolved measurements. The exist-
ing measurements show both similarities and dissimilari-
ties with a turbulent flat-plate boundary layer and will be
further discussed in Section 4.4.2.

4.2.3. Axially Rotating Pipe Flow

In the axially rotating pipe flow, two nondimensional
flow parameters arise, the Reynolds number based on the
average flow velocity and pipe radius ( R ) and the angu-
lar rotation speed of the pipe ( x ). The rotation rate can
be given as the azimuthal velocity of the pipe inner wall,
V w = R x , normalized with the bulk velocity such that
S = V w /U b ,where S is sometimes called the swirl rate.
As already mentioned, the fully developed laminar axial
mean flow is not affected by the axial rotation since the
fluid undergoes solid-body rotation. However, theoretical
studies show that rotation destabilizes the flow, so that it
is no longer linearly stable at all Reynolds numbers. Fur-
thermore, the study of rotating Hagen-Poiseuille flow by
Shresthaetal. [2013] shows the first experimental evidence
(via flow visualization) of absolute instability of a con-
fined spatially developing forward flow. Their results build
on theoretical linear stability analyses of the convective
versus absolute instability of the flow (e.g., Fernandez-
Feria and del Pino [2002]; del Pino et al. [2003]) and the
three-dimensional numerical simulations of Sanmiguel-
RojasandFernandez-Feria [2005] and show that the axially
rotating pipe flow can be absolutely unstable for mod-
erate values of Reynolds numbers and low values of
the swirl rate. Shrestha et al. 's [2013] flow visualizations
show transitions, with increased Reynolds number and
swirl rate, from stable cases, where the rotating bound-
ary layer develops with an axisymmetric conical shape
in the inlet region with no disturbance field downstream,
to convectively unstable cases, where the inlet region is
unchanged but with sinusoidal waveforms in the down-
stream region, to absolutely unstable cases, where the
azimuthal symmetry in the inlet region is broken by a
spiral structure overlaid on the conical boundary layer
development. Further analysis of nonlocal and nonlin-
ear effects is required to elucidate the laminar-turbulent
transition process.