Rayleigh-Benard instability

Principle of Rayleigh-Benard instability
Mathematical theory : determination of the critical Rayleigh number
Manifestations of Rayleigh-Benard instability

Principle of Rayleigh-Benard instability

    This problem of Rayleigh-Benard instability is a problem of convection inside a fluid.

    This instability occurs inside a fluid confined between two infinite horizontal planes at different temperatures $ T_{1}$ and $ T_{2}$.

Schemes of the domain for Rayleigh-Benard instability 

    If $ T_{2}>T_{1}$ then the system is stable and stays stratified in temperature. But, if $ T_{2}<T_{1}$ and if a perturbation is introduced in the system, then for a critical value of the difference of temperature ($ T_{1}-T_{2}=dT_{C}$) it can appear movements inside the fluid : the system is unstable and movements are organized in periodic contrarotative rolls. These rolls, or also called cells of Rayleigh-Benard, appear when there is a coupling between the dynamic field and the thermal field.

Rolls of Rayleigh-Benard 

    The principle of the instability is simple. Consider a drop of fluid near the lower plane. The drop is heated so its density decreases and it goes up inside the fluid due to the Archimede force. When the drop reaches the upper plane, more cold, it is cooled so its density increases and it can goes down inside the fluid. Rolls of Rayleigh-Benard instability are caused by this mechanism.

    The Archimede force must surpass the viscous drag force and the heat diffusion to allow convection inside the fluid. So there is a notion of threshold.

    Effectively, this instability occurs following the value of the Rayleigh number which is:

$\displaystyle Ra=\frac{\alpha \Delta T g d^{3}}{\nu a} $
with g denotes the acceleration due to gravity, d the depth of the layer, $ \alpha$ the coefficient of volume expansion, $ \Delta T$ the difference of temperature between the two planes $ \Delta T=T_{1}-T_{2}$$ \nu$ the kinematic viscosity and a the thermometric conductivity.
Enthalpie field for Ra=1468  Enthalpie field for Ra=4405 

    The critical value of the Rayleigh number for apparition of Rayleigh-Benard instability is 1707. For this critical value we can observe 2 rolls inside the fluid. This state of the fluid with two rolls is the most stable. But if initial conditions are well chosen, if Rayleigh number increases, number of rolls inside the fluid increases too.

              Display of rolls of Rayleigh-Benard following Ra 


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Mathematical theory : determination of the critical Rayleigh number

    To study this problem, we have to solve classic equations of continuity, motion and heat conduction.

For this problem, Boussinesq approximation is used and we have following equations for incompressible fluid:

$\displaystyle \vec{div}\vec{U}=0 $
$\displaystyle \frac{d\vec{U}}{dt}=-\vec{grad} \Pi+\alpha g \theta \vec{z}+\nu \Delta \vec{U} $
$\displaystyle \frac{d\theta}{dt}=-\Gamma w+\kappa \Delta\theta $

$\displaystyle \Pi=\frac{P}{\rho_{0}}+gz-\alpha g(T_{1}-T_{2})z-\frac{1}{2}\alpha g \Gamma z^{2} $
$\displaystyle \Gamma = \frac{T_{1}-T_{2}}{d}= \frac{q}{\kappa} $
$\displaystyle \kappa= \frac{k}{\rho_{0}C_{v}} $
$\displaystyle \nu = \frac{\mu_{n}}{\rho_{0}} $
$\displaystyle T(\vec{x},t)=T_{c}(z)+\theta(\vec{x},t) $

    The velocity limit conditions are $ u=v=w=0$ for z=0 and z=d. For temperature we can take constant temperatures or constant gradients of temperature for z=0 and z=d.

    The equations become, in thermal adimensionnal form ($ [L]=d$,$ [\tau]=\frac{d^{2}}{\nu}$ and $ [\theta]=d\Gamma$), and for the 2D case: 

$\displaystyle \frac{\partial u}{\partial x}+\frac{\partial w}{\partial z}=0 $
$\displaystyle \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+w\frac{\partial u}{\partial z}=-\frac{\partial \Pi}{\partial x}+Pr \Delta u $
$\displaystyle \frac{\partial w}{\partial t}+u\frac{\partial w}{\partial x}+w\fr......rtial w}{\partial z}=-\frac{\partial \Pi}{\partial z}+Ra Pr \theta+Pr \Delta w $
$\displaystyle \frac{\partial \theta}{\partial t}+u\frac{\partial \theta}{\partial x}+w\frac{\partial \theta}{\partial z}=-w+\Delta \theta $

    Velocity limit conditions become : $ u=w=0$ for z=0 and z=1.

Determination of the critical Rayleigh number :

    The fonction $ \psi$ is introduced : $ u=-\frac{\partial\psi}{\partial z}$ and $ v=\frac{\partial \psi}{\partial x}$. The problem become after linearisation :

$\displaystyle \frac{\partial}{\partial t}\Delta \psi = -Ra Pr \frac{\partial \theta}{\partial x} + Pr \Delta^{2} \psi $
$\displaystyle \frac{\partial \theta}{\partial t} = - \frac{\partial \psi}{\partial x} + \Delta \theta $

    If we look for solutions with the following form$ [\psi(x,z,t),\theta(x,z,t)]=[\Psi(z),\Theta(z)]e^{ik_{1}x+\lambda t}$, the system of equations gives :

$\displaystyle \lambda (D^{2}-k_{1}^{2})\Psi=-ik_{1}Ra Pr \Theta+Pr(D^{2}-k_{1}^{2})\Psi $
$\displaystyle \lambda \Theta=-ik_{1}\Psi+(D^{2}-k_{1}^{2})\Theta $
                                          with $ D=\frac{d}{dz}$

    Limit conditions for z=0 and z=1 are : 

$\displaystyle \Theta=0$
$\displaystyle \Psi=0$
$\displaystyle D\Psi=0$

    By solving this sytem we can find the value of critical Rayleigh number et critical value of k :

$\displaystyle Ra_{c}=1707,762 $
$\displaystyle k_{c}=3,117$

Graphic of stability 


    We have studied the case where there are two planes under and above the fluid. Then, velocity limit conditions are two rigid conditions :  for z=0 and z=d.

    Two others cases exist following the type of boundary. Fluid can be placed on a heated plane but its upper face can be a free surface.

Case where there is one free surface 

    In this case we have a rigid condition for the lower boundary
( for z=0 ) and a free condition for the upper boundary ( and w=0 for z=d) . So critical values to see instability change :

    The last case is the case where the fluid is confined between two free surfaces :  and w=0 for z=0 and z=d .

Case where there is two free surfaces 

    The critical values are now :

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Manifestations of Rayleigh-Benard instability
Rayleigh-Benard instability visualization  3D Rayleigh-Benard instability 

Click here for animation (.mpg)

    This phenomenon of Rayleigh-Benard instability is very important in the industrial domain. Examples are numerous : cooling of nuclear plants, heating of building, exchanges between atmosphere and oceans..... But manifestations of Rayleigh-Benard instability are not always visible to our eyes that's why we have not a lot of pictures of it.

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