Generalized Rayleigh criterion for non-axisymmetric centrifugal - PowerPoint PPT Presentation

Generalized Rayleigh criterion for non-axisymmetric centrifugal instabilities Paul Billant 1 and François Gallaire 2 1 LadHyX, Ecole Polytechnique-CNRS, Palaiseau, France 2 Lab. J.-A. Dieudonné- UNSA-CNRS, Nice, France

Axisymmetric centrifugal instabilities cyclone anticyclone Taylor-Couette Counter-rotating vortex pair Van Dyke, Album of Fluid motion in a rotating fluid

Rayleigh criterion Pressure Centrifugal force gradient Angular velocity Equilibrium unstable if: Axial vorticity Necessary (Rayleigh 1916) and sufficient (Synge 1933) condition for instability of axisymmetric perturbations Typically the case of isolated vortices: r r Stability of non-axisymmetric modes ?

Stability equations for an inviscid fluid Basic state: axisymmetric vortex Perturbation: ⇒ Boundary conditions: WKB Asymptotic analysis for k >>1

WKB analysis m=0 B 0 (r) Oscillatory Evanescent Evanescent r 2 r 1 r r 0

WKB Dispersion relation for m=0 similar to Le Dizès & Lacaze (2004) for B 0 (r) Kelvin waves Parabolic approximation: r 1 r 2 r 0 identical to Bayly (1988), Sipp & Jacquin (2000)

Carton & McWilliams vortex profiles r r

Results: axisymmetric mode Carton & McWilliams vortex profile with α =2 numerics WKB dispersion relation WKB+ parabolic approximation Numerics : shooting and Tchebitscheff collocation methods

m ≠ 0: B 0 is complex: Stokes Lines Turning point r 2 r 0 B 0 (r 2 )=0 Turning point r 1 B 0 (r 1 )=0

WKB dispersion relation for non-axisymmetric modes Parabolic approximation:

Results Carton & McWilliams vortex profile with α =4 m=1 m=2 shooting shooting σ parabolic approximation parabolic σ approximation WKB WKB k k

Generalized Rayleigh criterion for non-axisymmetric modes An azimuthal mode m is centrifugally unstable if Re ( ) >0 where r 0 is defined by α =4 Re( σ 0 ) 1 0,8 0,6 0,4 0,2 6 m 0 0 1 2 3 4 5 ⇑ Cutoff

Competition between 2D shear/centrifugal instabilities 3D centrifugal instability 2D shear instability α =4 Re( σ 0 ) 1 σ max ̃ √ α 0,8 m s ̃1/shear thickness̃ α 0,6 0,4 σ max ̃ α 0,2 6 m 0 0 1 2 3 4 5 ⇑ Cutoff

Taylor-Couette flow B 0 (r) R* r R 1 R 2 Linear approximation: In constrast with isolated vortices, the growth rate is independent of m !

Generalisation to rotating and stratified fluids Same WKB dispersion relation: Except that Background rotation Brunt-Väisälä frequency The generalized rayleigh criterion is independent of the stratification

Conclusion An azimuthal mode m is centrifugally unstable if Re ( ) >0 where r 0 is defined by