The example of Rayleigh-Benard convection. Pattern-forming instabilities: The example of Rayleigh-Benard convection.

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Presentation transcript:

The example of Rayleigh-Benard convection

Pattern-forming instabilities: The example of Rayleigh-Benard convection

Rayleigh-Benard convection. Boussinesq approximation, incompressible flow

Rayleigh-Benard convection. Boussinesq approximation, incompressible flow

Rayleigh-Benard convection. Boussinesq approximation, incompressible flow

Rayleigh-Benard convection. Boussinesq approximation, incompressible flow

Rayleigh-Benard convection: static state

Rayleigh-Benard convection

Rayleigh-Benard convection: static state with pure conduction Fixed temperature b.c.

Rayleigh-Benard convection. Boussinesq approximation, incompressible flow

Rayleigh-Benard convection: non dimensional formulation

Rayleigh-Benard convection Important parameters: R = g D 3 T 2 -T 1 ) / = a = L D

Rayleigh-Benard convection. If R < R crit conduction T(x,y,z,t)=T cond (z)=T 2 - z (u,v,w)=(0,0,0) If R > R crit convection T= T cond + (u,v,w) non zero

Rayleigh-Benard convection. Linear stability analysis

2D Rayleigh-Benard convection (non dimensional formulation)

2D Rayleigh-Benard convection

2D Rayleigh-Benard convection: Linearization around the static state

2D Rayleigh-Benard convection: Linearization around the static state

2D Rayleigh-Benard convection: Linearization around the static state

2D Rayleigh-Benard convection: Linearization around the static state

2D Rayleigh-Benard convection: Linearization around the static state

Rayleigh-Benard convection. Linear stability analysis

2D Rayleigh-Benard convection: Threshold to convection

Rayleigh-Benard convection: above R crit, convective motion occurs. This takes the form of parallel rolls

The convective rolls saturate the instability

Amplitude expansion

Amplitude expansion: first order

Amplitude expansion: second order

Amplitude expansion: third order

Amplitude expansion: third order The Fredhom alternative: eliminate the secular term and get a solvability condition: The Landau equation

Amplitude expansion: third order The Landau equation with real coefficients

Pitchfork bifurcation at R 2 =0

Stability of the rolls: Busse balloon

Stability of the rolls

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