Abstracts - KTH Mechanics

Global stability of the rotating disk boundary layer and the
effects of suction and injection
Christopher Davies ∗ and Christian Thomas ∗
The rotating disk boundary-layer can be shown to be absolutely unstable, using
an analysis that deploys the usual ‘parallel-flow’ approximation, where the base flow
is simplified by taking it to be homogeneous along the radial direction1 . But for the
genuine radially inhomogeneous base flow, numerical simulations indicate that the
absolute instability does not give rise to any unstable linear global mode. 2 . This is
despite the fact that the temporal growth rates for the absolute instability display a
marked increase with radius.
The apparent disparity between the radially increasing strength of the absolute
instability and the absence of any global instability can be understood by considering
analogous behaviour in solutions of the linearized complex Ginzburg-Landau equation3
. These solutions show that detuning, arising from the radial variation of the
temporal frequency of the absolute instability, may be enough to globally stabilize
disturbances. Depending on the precise balance between the radial increase in the
growth rates and the corresponding shifts in the frequencies, it is possible for an
absolutely unstable flow to remain globally stable.
Similar behaviour has been identified in more recent numerical simulations, where
mass injection was introduced at the disk surface but the modified flow still appeared
to be globally stable. More interestingly, it was found that globally unstable behaviour
was promoted when suction was applied. This is illustrated in the figure,
which displays numerical simulation results for the impulse response of a range of inhomogeneous
base flows with varying degrees of mass injection and suction. The plots
show the evolution of locally computed temporal growth rates at the critical point for
the onset of absolute instability. The temporal variation of the negative-valued growth
rates for the cases with injection is associated with convective propagation behaviour.
But when suction is applied the disturbance eventually displays an increasingly rapid
growth at the radial position of the impulse, albeit without any selection of a dominant
frequency, as would be more usual for an unstable global mode.
∗School of Mathematics, Cardiff University, Cardiff, CF24 4YH, UK
1Lingwood, J. Fluid Mech. 299, 17 (1995).
2Davies and Carpenter, J. Fluid Mech. 486, 287 (2003).
3Hunt and Crighton, Proc. R. Soc. Lond. A 435, 109 (1991).
Growth rates
0.1
0
−0.1
−0.2
−0.3
−0.4
−0.5
a = −1.0
a = −0.5
a = 0.0
a = 0.5
a = 1.0
Suction
Injection
0.2 0.4 0.6
t/T
0.8 1
13