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## The example of Rayleigh-Benard convection

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### Title: PowerPoint Presentation Author: antonello Last modified by: antonello provenzale Created Date: 1/29/2007 11:26:31 AM Document presentation format – PowerPoint PPT presentation

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Title: The example of Rayleigh-Benard convection

1
The example of Rayleigh-Benard convection
2
Pattern-forming instabilities The example of
Rayleigh-Benard convection
3
Rayleigh-Benard convection. Boussinesq
approximation, incompressible flow
4
Rayleigh-Benard convection. Boussinesq
approximation, incompressible flow
5
Rayleigh-Benard convection. Boussinesq
approximation, incompressible flow
6
Rayleigh-Benard convection. Boussinesq
approximation, incompressible flow
7
Rayleigh-Benard convection static state
8
Rayleigh-Benard convection
9
Rayleigh-Benard convection static state with
pure conduction
Fixed temperature b.c.
10
Rayleigh-Benard convection. Boussinesq
approximation, incompressible flow
11
Rayleigh-Benard convection non dimensional
formulation
12
Rayleigh-Benard convection
Important parameters R ga D3(T2-T1) / nk s
n / k a L / D
13
Rayleigh-Benard convection.
If R lt Rcrit conduction T(x,y,z,t)Tcond
(z)T2 - b z (u,v,w)(0,0,0) If R gt Rcrit
convection T Tcond q (u,v,w) non zero
14
Rayleigh-Benard convection.
Linear stability analysis
15
2D Rayleigh-Benard convection (non dimensional
formulation)
16
2D Rayleigh-Benard convection
17
2D Rayleigh-Benard convection Linearization
around the static state
18
2D Rayleigh-Benard convection Linearization
around the static state
19
2D Rayleigh-Benard convection Linearization
around the static state
20
2D Rayleigh-Benard convection Linearization
around the static state
21
2D Rayleigh-Benard convection Linearization
around the static state
22
Rayleigh-Benard convection.
Linear stability analysis
23
2D Rayleigh-Benard convection Threshold to
convection
24
Rayleigh-Benard convection above Rcrit,
convective motion occurs. This takes the form of
parallel rolls
25
The convective rolls saturate the instability
26
Amplitude expansion
27
Amplitude expansion
28
Amplitude expansion first order
29
Amplitude expansion second order
30
Amplitude expansion third order
31
Amplitude expansion third order
32
Amplitude expansion third order The Fredhom
alternative eliminate the secular term and get a
solvability condition
The Landau equation
33
Amplitude expansion third order The Landau
equation with real coefficients
34
Amplitude expansion third order The Landau
equation with real coefficients
Pitchfork bifurcation at R20
35
Stability of the rolls Busse balloon
36
Stability of the rolls
37
Stability of the rolls