Geoscience Reference
In-Depth Information
thickness (proportional to
√
ν/
z
) and is therefore par-
allel in the physical sense (
z
is the coordinate normal to
the disk surface). In the von Kármán flow, there is only
one nondimensional parameter, namely a Reynolds num-
ber based on the angular rotational speed, the radius, and
the viscos
ity. U
sually the Reynolds number is taken as
Re =
r
√
z
/ν
,where
r
is the (dimensional) radius and
hence Re increases linearly with
r
.
One of the most interesting features of the rotating
disk boundary layer flow is that the onset of transition
to turbulence is observed experimentally within an unusu-
ally limited range of Reynolds numbers (500
<
Re
<
520).
Reported transition Reynolds numbers slightly outside
of this range can be largely attributed to the use of dif-
ferent definitions for the onset of transition rather than
fundamental variations in the physical phenomena; see
Imayama et al.
[2012b]. This degree of reproducibility
across different experimental facilities contrasts with the
transition behavior observed for most other boundary
layer flows. This was first pointed out by
Lingwood
[1995],
and she also offered a hypothesis that this is due to the
rotating disk boundary layer being absolutely unstable.
Figure 4.7 shows a visualization from
Cederholm and
Lundell
[1998] of the temperature of a heated disk, where
the transition to turbulence is seen as a distinct change
in temperature at a specific radial position (i.e., specific
Reynolds number).
The von Kármán flow is a three-dimensional bound-
ary layer flow that is susceptible to an inviscid cross-flow
instability (often referred to as type 1 or B instability) due
to an inflectional mean radial velocity component, giv-
ing rise to so-called cross-flow vortices. Figure 4.7 shows
Re
Stable tertiary solutions exist
100
75
50
Roll cells (2
D
flow)
25
Pure Couette flow
0
0
2
4
6
8
10
12
Ω
Figure 4.6.
Predicted nonlinear steady states by
Nagata
[1998]
(upper) and experimental realization by
Tsukahara et al.
[2010a]
(lower) at Re = 100,
= 8.7. Copyright © Cambridge Univer-
sity Press, 2011. Photograph reprinted with permission.
[2010a] (see Figure 4.6). For even higher Reynolds num-
bers, the roll cells themselves become turbulent.
For the nonrotating case, plane Couette flow (PCF)
starts to develop turbulent spots around Re = 325 to
Re = 360 [
Tillmark and Alfredsson
, 1991, 1992;
Daviaud
et al.
, 1992] and turbulent stripes [
Prigent et al.
, 2002;
Barkley and Tuckerman
, 2005, 2007] are observed at inter-
mediate Reynolds numbers in both experiments and sim-
ulations. For cyclonic rotation, formation of turbulent
spots is suppressed until higher Reynolds numbers and are
followed by turbulent stripes if the Reynolds number is
further increased, as observed both in experiments and in
DNS [
Tsukahara et al.
, 2010b]. The first DNSs of a few
cases of cyclonic rotation at low Reynolds numbers were
reported already by
Komminaho et al.
[1996] and showed
the stabilization effect. At sufficiently high Re the flow
becomes fully turbulent, as can be seen in the mapping
of the parameter space done by
Tsukahara et al.
[2010a]
(see their Figure 2).
Figure 4.7.
Temperature distribution of a heated rotating disk
shown by temperature-sensitive floating crystals [
Cederholm
and Lundell
, 1998]. The photograph is obtained as a long-time
exposure using stroboscopic light. Copyright © Cederholm &
Lundell, 1998. Reprinted with permission. For color detail,
please see color plate section.
4.2.2. von Kármán Boundary Layer Flow
The laminar von Kármán boundary layer flow can be
expressed as an exact solution of the Navier-Stokes equa-
tions, which is an attractive feature for theoretical anal-
yses. The laminar boundary layer also has a constant
