hydrodynamic example : Rayleigh Benard intabilities

This year, we have studied an example of hydrodinamic instability, The Rayleigh Benard one.

The fluid is assumed to be Boussinesq. This means, essentially, that the density is assumed to be only a function of the temperature and that the parameters for the fluid such as viscosity and thermal diffusivity do not vary over the volume of the fluid. The system is governed by the Boussinesq equations.

Producing R-B convection involves isolating a liquid in a tiny enclosed cylindrical or rectangular cell and creating a temperature difference (gradient) between the bottom and top layers. It's the rough equivalent of heating a covered pan of water, though the experimental arrangement allows for precise control of fluid properties and the temperature gradient between the cylinder top and bottom, in addition to offering a safe view of the heated fluid's surface.

There's two possible boundary conditions :

- Free, in this case Rc=(27/4).Pi^4 and
Kc=Pi/2^0.5 this case can be theoreticaly
study

- Wall, in this case Rc=1707 and Kc=3.1
this case must
be experimentaly or numericaly study.it's the one we have stdudy.

The experimental system that we are concerned with for this part involves two flat parallel plates with separation d. The plates are arranged horizontally so that gravity acts perpindicular to the plane of the plates. The bottom plate is warmed and the top plate is cooled so that the system has a constant temperature gradient across it.

depending if we heat from the bottom or from above two case apear:

1 Unconditional stability

This case occure when fluid is heated from above and cooled from below

2 Conditional stability

Rayleigh-Benard convection occurs when fluid is heated from below and cooled from above. When the temperature difference between the top and bottom is large enough, an instability occurs and the fluid begins to convect. The colder, denser fluid sinks while the warmer and less dense fluid rises. This state of flow is called convection. It is known that when the Rayleigh number Ra exceeds a certain critical value, Rc = 1708 , convection occurs with a roll pattern described by a horizontal wavenumber of kc=3.117.

nu is the viscosity of the fluid, a is the thermal diffusivity, alpha is thermal expansion coefficient. DT is the temperature difference across the plates, v is the velocity field,

Nusselt Number : Nu = ( flow / flow without conduction )

Rayleigh Number : Ra = alpha*g*d^3*dT/(nu*a).

Prandtl Number: Pr = nu/a.

We can visualise here computational result for differents temperature of the top wall :

3 explotation of results

Fluent give us the Velocity and the exchange flow. Furthermore with the temperature differenc we can find The RAyleigh number and the conduction flow. We can plot Nusselt Number and Velocity to find the critical Rayleigh Number value.

there's other stable possibilities depending of the different boundary and initial conditions. For example, we initialize at for rollers (A 1.0*e-04, B 0.0, C 1.0 and D 8)with Ra=0.5, and we resolve unsteadies equations. We obtain three rollers. The following graphs show the passage from for rollers to three rollers.