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| Titel |
Three-dimensional radiative instabilities in stratified compressible flows |
| VerfasserIn |
Julien Candelier, Stéphane Le Dizès, Christophe Millet |
| Konferenz |
EGU General Assembly 2010
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| Medientyp |
Artikel
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| Sprache |
Englisch
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| Digitales Dokument |
PDF |
| Erschienen |
In: GRA - Volume 12 (2010) |
| Datensatznummer |
250034522
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| Zusammenfassung |
| We investigate the three-dimensional stability of two-dimensional flows Ux(z) such as plane
Bickley jet or boundary layer in a stably stratified fluid of constant Brunt-Väisälä frequency
N. The angle θ between the shear x-z plane and the vertical is considered as a control
parameter. For θ = 0, the gravitational acceleration is along the (-z)-axis so the shear plane
is vertical and for θ = Ï-2, along the y-axis thus orthogonal to the stratification.
Following the parallel flow approximation, the instability wave solution is sought in the
form of a normal mode in the x and y directions: p ~Ëp  exp(iÏt - ikxx - ikyy).
The eigenvalue problem for Ï is solved with a spectral collocation method. For
radiative instability waves, an appropriate contour deformation in the complex z-plane
is developed in order to apply correctly the condition of radiation in numerical
codes.
It emerges that, as a wave propagates downstream, its transverse behaviour may be
dispersive. This phenomenon depends strongly on the stratification of the mean flow, through
the parameter θ. For θ = 0, Miles (1961) and Howard (1961) found a sufficient condition
for stability, in terms of Richardson number. The case associated with θ = Ï-2
for a Bickley jet was studied recently by Deloncle et al (2007) under Boussinesq
approximation. But the way such results may be extended into a wider context, for a given θ,
cannot be simply deduced from their analysis. We draw attention to a mechanism
whereby gravity waves may be generated by the mean flow without necessarily
invoking Kelvin-Helmholtz instabilities. The radiating modes found in our study are
similar to the “unstable gravity waves” mentioned by Satomura (1980) in shallow
water or “supersonic instability waves” described by Tam and Hu (1988,1989)
for high-speed jets. From a mathematical point of view, it can be established that
the modes with a real part of frequency satisfying kxN--k2--+-k2
x y < Ïr < N,
where N is the Brunt-Väisälä frequency, give rise to gravity waves in the transverse
direction.
The boundary layer flow is studied under fully compressible framework. The radiative
instability still occurs and is analysed by using different asymptotic approaches, for large
wavenumbers kx and ky or for small Brunt-Väisälä frequency. In both cases, the most
unstable modes properties are captured in those limits. |
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