Abstracts - KTH Mechanics

Continuous Spectrum Growth and Modal Instability in
Swirling Duct Flow
C. J. Heaton ∗ ,N.Peake ∗
We discuss the acoustics and stability of inviscid ducted swirling flow. Swirling
flow arises in many problems, from aeroacoustics to vortex and jet stability. The
vorticity in the mean flow complicates the usual acoustic spectrum: there is no simple
scalar velocity potential and the unsteady pressure and vorticity are coupled. The
spectrum typically contains two distinct families of modes, respectively analogous to
the acoustic and vorticity waves found in irrotational flow 1 . In particular a family
of so-called nearly-convected modes can be present in the spectrum, often having an
infinite accumulation of eigenvalues in the complex plane and we analytically classify
this, and describe in which cases the family contains instability modes. We also
treat the continuous spectrum of the acoustic-vorticity waves and find that it can be
responsible for a new convective instability. The growth rate for this instability is
algebraic (rather than exponential), with an exponent that depends on the mean flow
and which is given in the analysis.
The stability of swirling inviscid flow has been considered by many previous authors,
from Rayleigh’s famous stability criterion for axisymmetric perturbations to
incompressible rotating fluid to many modern numerical and asymptotic studies. The
family of unstable nearly-convected modes we identify (see figure 1 for an example,
which shows the complex wavenumber plane for a fixed temporal frequency calculation)
is shown to correspond to existing large wavenumber WKB theory 2 and among
our results is an extension of their results to finite azimuthal orders. The non-modal
instability we identify and investigate is due to the continuous spectrum of the linearised
Euler equations, and is in addition to any modal growth that may or may not
be present. The continuous spectrum of parallel shear flow can lead to linear growth
of the flow energy 3 , but here we find that taking a fully three dimensional mean flow
leads to a growth rate which can take any value.
∗ DAMTP, Wilberforce Road, Cambridge CB3 0WA,UK
1 Golubev and Atassi, J. Sound Vib. 209, 203 (1998).
2 Leibovich and Stewartson, J. Fluid Mech. 126, 335 (1983).
3 Landahl, J. Fluid Mech. 98, 243 (1980).
0.5
0
−0.5
13 13.2 13.4 13.6 13.8
Figure 1: Unstable nearly convected modes (+ signs) and the continuous spectrum
(thick grey line) for a swirling flow. These numerics agree very well the the asymptotic
analysis to be presented.
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