Rotation Effects on Wall-Bounded Flows: Some Laboratory Experiments - Modeling Atmospheric and Oceanic Flows - page 90

are aware) the first of any boundary layer flow found
to exhibit an absolute instability [ Lingwood , 1995, 1996],
and the first example to link the onset of nonlinearity
and transition to turbulence to such a well-defined critical
Reynolds number, explaining the observed highly repro-
ducible transition Reynolds number. Subsequently, other
members of the BEK family have been shown to exhibit
absolute instability as well [ Lingwood , 1997]. The BEK
flows are “semiclosed”in the sense that a disturbance that
grows in time at a fixed radial position while continuing
to convect azimuthally will not convect out of the domain
of interest but cycle round and build on itself. It is per-
tinent to note that it is the azimuthal periodicity of the
rotating disk configuration, namely its semiclosed nature,
that means the unidirectional (radial) absolute instabil-
ity is particularly relevant to the physical phenomena
observed.
In a similar three-dimensional boundary layer without
azimuthal periodicity, e.g., a swept-wing boundary layer,
a disturbance that grows in time at a fixed chordwise posi-
tion while continuing to convect in the spanwise direction
(i.e., as would apply analogously to a unidirectional abso-
lute instability) could well pass out of the domain of
interest, e.g., off the end of the wing, before triggering
nonlinearity.
The onset of transition for the rotating disk flow was
identified by Lingwood [1995] as being highly repro-
ducible, which was the motivation for looking for absolute
instability of the boundary layer flow. The flow was stud-
ied using linear stability theory assuming a locally spa-
tially invariant boundary layer and found to be locally
absolutely unstable above a Reynolds number of 507. She
hypothesized that the growth in time at a fixed radial
position would lead to nonlinearity and the onset of tran-
sition and corroborated this experimentally [ Lingwood ,
1996]. However, for a complete and mathematically con-
sistent theoretical stability analysis, it is necessary to
include the spatial variation of the flow (i.e., variations
in Reynolds number with radius), and therefore it is the
global stability characteristics that are relevant to the
real flow. If there is a global instability associated with
laminar-turbulent transition, it would imply that the onset
of transition should be highly repeatable across differ-
ent experimental facilities. While it has previously been
shown that local absolute instability does not necessar-
ily lead to linear global instability, Pier [2003] showed
that finite-amplitude traveling waves are subject to a sec-
ondary instability leading to a steep-fronted nonlinear
global mode situated at the onset of primary local abso-
lute instability. Furthermore, recently Healey [2010] has
shown, using the linearized complex Ginzburg-Landau
equation, that if the finite nature of the low domain (finite
radius) is accounted for, then local absolute instability
can give rise to linear global instability, leading directly

to a nonlinear global mode located at the convective-
absolute transition boundary. Imayama et al. [2012b]
investigated experimentally the effect of the disk edge
on transition and compared with others' results for the
onset of nonlinearity/transition, which is shown to lie at
approximately 500-520. They found that there is in fact
even greater reproducibility in the transition Reynolds
number for smooth/clean disks than might have pre-
viously been thought when a consistent definition for
transition is applied to others' results. This supports
the hypothesis of Lingwood [1995, 1996] that the local
absolute instability is the trigger for transition as well
as the suggestion of Healey [2010] that it is the finite
disk (present, of course, in all experiments) that allows
the absolute instability to lead to (supercritical) linear
global instability (and then to a nonlinear steep-fronted
global mode).
The turbulent rotating disk boundary layer flow is
arguably more relevant to geophysical flows than the lam-
inar one; however, it is yet relatively underinvestigated.
Recent experimental results by Imayama [2012] for the
turbulent boundary layer compare the mean azimuthal
velocity and variance, as well as higher moments, with
results obtained for a (nonrotating) two-dimensional tur-
bulent flat-plate boundary layer. The mean azimuthal
velocity profile shows a strong similarity to the mean
velocity for the two-dimensional flat-plate boundary layer,
especially in the near-wall region and in the logarithmic
region. However, the so-called wake region, i.e., the region
where the flow deviates from the logarithmic distribution,
is clearly less pronounced for the turbulent rotating disk
flow. Imayama [2012] has been able to determine the skin
friction (or friction velocity) accurately by direct measure-
ment of the velocity distribution close to the disk and
shows that the turbulence intensities in the rotating disk
experiment are qualitatively similar, although they have
lower values near the wall compared to measurements
in two-dimensional boundary layers. This can partly be
explained by the spatial averaging of the hot-wire probe,
which for the experiments of Imayama [2012] has a length
of almost 30 ∗

despite the fact that the sensing length
is only 0.3 mm long. While the higher moments, i.e., the
skewness and flatness factors, also show similar behav-
ior in the near-wall and logarithmic regions as compared
to the two-dimensional boundary layer, differences are
apparent in the outer region.
An intriguing aspect of Imayama's [2012] experimental
results as compared to two-dimensional flat-plate bound-
ary layers [ Örlü and Schlatter , 2013] is the difference
between the spectral maps of the azimuthal and stream-
wise turbulence fluctuations for the two cases. First, the
spectral peak corresponding to the maximum in the rms
distribution near the wall is located at higher wave num-
bers (smaller wavelengths) for the rotating disk boundary