Contents

Introduction

Critical Rayleigh

Velocity study

Unsteady case

Conclusion

Introduction




Presentation of the study



fig. 1

This study aims at simulate the Rayleigh-Bernard instabilities. To do this, we'll use the two softwares FLUENT and PHOENICS. The case, we have chosen, is a rectangular box (1cmx2cm) composed by 2 plates (see fig. 1) maintained at two different temperatures. This box was filled of water. All of the computation will be made in the hypothesis of laminar flow.


The Rayleigh-Bernard instabilities

The instabilities are determined by the differences of temperature, T1-T2, between the two walls (see fig. 1).
    • 1sr case: T1>T2; no instabilities: U=0 and T(y)=T2+(T1-T2)y/a
    • 2nd case: T1<T2; the former mathematics solution exists, but it isn't sure that it's realised. For T2-T1=delta tc

, steady rolls appears.





The rolls appearance mechanics is linked to the lack of balance between the Archimede force and the drag of the fluid particules. A dimensionless number caracterizing this, is the Rayleigh number (Ra).

Ra=g*beta*delta(t)*d3/(nu*ath)

So for Ra>Rac (where Rac is the Rayleigh number for T1-T2= delta t), we have instabilities.


The equations of this problem

We have to solve:
    • the continuity equation
    • the Navier-Stockes equations
    • the energy equation
    • and the state equation

Moreover, we apply the Boussinesq approximation: (where is constant) in the Navier-Stockes equation where the expression of is replaced by . is the density at T0, and alpha is the thermal expenssion coefficient.

All of this hypothesis have been computed in FLUENT and PHOENICS. Our study can start. As a first step, we chose to determined the critical Ra (Rac) and the meshing influence on this number. In a second step, we'll study the evolution of the velocity in function of the Rayleigh number. And at least, we'll make a brief study in a hypothesis of a unsteady flow.