Said GHALIMI and Jeremie SAGARRA

BES : INSTABILITY OF RAYLEIGH BENARD

## Introduction

Presentation of the problem :

We want to study the stability of a fluid situated between two plane and horizontal plates maintained at two different temperatures. Each plate is maintained at an uniform temperature and the fluid is considered as having a weight. We try to evaluate the conditions and the nature of the convective movements of the fluid.
In the case that the up plate has a temperature higher than the temperature of the down plate, the thermal stratification is stable. In this case, a particle of fluid moved to up will go in a zone where the volumic mass is less important. The strength of weight will thus play a role of stabilization since it will tend to bring back the particle to its initial position.

Stable case

If we consider the inverse situation, where the up plate is colder than the down plate, the fluid more heavy will be over the lighter fluid. The action of the weight will have an effect of destabilization on the system. The particles of the down will tend to go up under the effect of the forces of flottability and the particles of the up will tend to go down under the action of the weight. The thermic stratification will be then unstable.
When the gradient of temperature is over a critical value we can observe between the two plates the appearance of rolls of convection. This phenomenon is called the unsteadiness of Rayleigh-Bénard and it is quantified by the adimensional number called number of Rayleigh (Ra) :

Ra = (g.alpha.DT.h3) / (nu.a)

With :

g : gravity acceleration.

alpha : coefficient of thermic dilatation of the fluid.

DT : temperature gradient between the plates.

h : distance between the plates.

nu : cinematic viscosity of the fluid.

a : thermic diffusivity of the fluid.

In the case hich interest us of the liquid water, we have : Ra = 8068.33 * DT

According to the theory the critical value of the Rayleigh number is : 1708.

Aims of the study :

The aim of this study is to modelise the problem of Rayleigh – Bénard with the code of simulation FLUENT, in order to visualize the details of this phenomenon.

We define the model of the flow as laminar and we compute the flow and the heat flux of the fluid. The computations are made first in a two -dimension geometry, and in a second time in a three dimension circular domain and a three dimension rectangular domain.

Summary

## Study of a 2 - dimension case.

We have at first studied the Rayleigh - Benard problem with a two - dimension mesh. The dimensions of the domain taken into account are 10 mm in height and 20 mm in length. We have with this visualistion a clear view of the phenomenons in the fluid.

View of the mesh :

View of the velocity :

When the gradient of temperature is such that the Rayleigh number is inferior to the critical Rayleigh ( Ra < Rac ), the case is purely diffusive. There is no velocity on the fluid. (The values observed on the graphic are due to the numerical errors of the software).

When the Rayleigh increases and become superior to the critical Rayleigh ( Ra > Rac ), the heat transfer become convective :

Indeed, we observe the appearance of convective rolls in the fluid. The number of rolls depend of the initial condition put on the unstable system. In the majority of the cases we observe two rolls. This correspond to the most stable mode.

summary

View of the temperature :

In the purely convective case, the gradient of temperature is linear :

In the convective case, the movements of the fluid create a hotter zone where the fluid goes up and a colder zone where the fluid goes down :

We have seen that the velocity depends on the Rayleigh number. It can be interesting to plot the velocity in function of the Rayleigh number :

summary

Study of the Nusselt number :

The Nusselt number is representative of the heat exchange due to the convection. It is equal to the heat flux really exchanged quotiented by the heat flux exchanged in the pure diffusive case, which is equal in our case to : Phi = 1.2 * DT.

We notice, as it seems logical, that the heat flux increase with the convection.

Summary

## Study of a 3 - dimension case.

Circular geometry :

In order to have a more general point of view of the phenomenon, we have chosen to visualise it in a three dimension geometry. We have first generated a circular mesh. At the sides of the cylinder we have put a condition of symmetry.

Velocity for Ra > Rac :

Temperature for Ra > Rac :

In order to be able to see better the interior of the cylinder, we have made a cut by the plane x = 0.

Velocity :

Temperature :

Rectangular geometry :

We have generated a rectangular geometry to compare with the circular geometry :

Velocity :

Temperature in the plane x = 0 :

Velocity in the plane x = 0 :

We observe the same rolls that in two dimension. We have the same stable frequency of two rolls.

Like in the first study, we plot the velocity and the Nusselt number in function of the Rayleigh. The results are quite similar :

Velocity :

Nusselt :

Summary

## Conclusion

We could, during this study, visualise many theoretical results about the convection of Rayleigh Bénard. The number used to qualify this flow is the number of Rayleigh ; thus we have computed our different cases for different Ra. We observed that :

• Under the critical Rayleigh, (Rac=1708), the flow is purely diffusive. There is no velocity in the fluid and the temperature gradient is linear. The forces of stabilisation ( viscosity ) are more important than the forces of flottability.
• Over the critical Rayleigh, the flow began to be convective. We observe the appearance of rolls between the plates. In this case the destabilising forces are more important than the forces of viscosity. The system is unsteady and is destabilised by any perturbation.
• This study allowed us to discover the simulation code FLUENT. We could define with this software the hypothesis of the computation : laminar flow, computation of the velocity and heat flux equations.

Summary