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

▶
generalized rayleigh criterion for non axisymmetric
SMART_READER_LITE
LIVE PREVIEW

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

Mar 04, 2023 1.19k likes •1.37k views

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

slide-1
SLIDE 1

Generalized Rayleigh criterion for non-axisymmetric centrifugal instabilities

Paul Billant1 and François Gallaire2

1LadHyX, Ecole Polytechnique-CNRS, Palaiseau, France

  • 2Lab. J.-A. Dieudonné- UNSA-CNRS, Nice, France
slide-2
SLIDE 2

Counter-rotating vortex pair in a rotating fluid Taylor-Couette

Van Dyke, Album of Fluid motion

Axisymmetric centrifugal instabilities

cyclone anticyclone

slide-3
SLIDE 3

Rayleigh criterion

Pressure gradient Centrifugal force

Equilibrium unstable if:

Angular velocity Axial vorticity Necessary (Rayleigh 1916) and sufficient (Synge 1933) condition for instability

  • f axisymmetric perturbations

Typically the case

  • f isolated vortices:

r r

Stability of non-axisymmetric modes ?

slide-4
SLIDE 4

Stability equations for an inviscid fluid

WKB Asymptotic analysis for k >>1

⇒

Basic state: axisymmetric vortex Perturbation: Boundary conditions:

slide-5
SLIDE 5

WKB analysis m=0

Oscillatory Evanescent Evanescent B0(r) r r1 r2 r0

slide-6
SLIDE 6

WKB Dispersion relation for m=0

identical to Bayly (1988), Sipp & Jacquin (2000)

Parabolic approximation:

B0(r)

r1 r2 r0

similar to Le Dizès & Lacaze (2004) for Kelvin waves

slide-7
SLIDE 7

Carton & McWilliams vortex profiles

r r

slide-8
SLIDE 8

Results: axisymmetric mode

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

slide-9
SLIDE 9

m≠0: B0 is complex:

r0 Stokes Lines Turning point r2 B0(r2)=0 Turning point r1 B0(r1)=0

slide-10
SLIDE 10

Parabolic approximation:

WKB dispersion relation for non-axisymmetric modes

slide-11
SLIDE 11

Results

k shooting parabolic approximation WKB m=2 m=1 shooting parabolic approximation WKB k

σ σ

Carton & McWilliams vortex profile with α=4

slide-12
SLIDE 12

Generalized Rayleigh criterion for non-axisymmetric modes

An azimuthal mode m is centrifugally unstable if where r0 is defined by Re ( ) >0

0,2 0,4 0,6 0,8 1 1 2 3 4 5 6 m

α =4 Re(σ0)

⇑ Cutoff

slide-13
SLIDE 13

Competition between 2D shear/centrifugal instabilities

0,2 0,4 0,6 0,8 1 1 2 3 4 5 6 m

α =4 Re(σ0)

⇑ Cutoff

ms̃1/shear thickness̃ α σmax ̃ α σmax ̃ √α

3D centrifugal instability 2D shear instability

slide-14
SLIDE 14

Taylor-Couette flow

B0(r)

r R1 R2 R*

Linear approximation: In constrast with isolated vortices, the growth rate is independent of m !

slide-15
SLIDE 15

Generalisation to rotating and stratified fluids

Same WKB dispersion relation: Except that

Brunt-Väisälä frequency Background rotation

The generalized rayleigh criterion is independent of the stratification

slide-16
SLIDE 16

Conclusion

An azimuthal mode m is centrifugally unstable if where r0 is defined by Re ( ) >0