I. Introduction

The goal of this report is to study an example of hydrodynamic instability: the instability of Rayleigh-Benard. At first, we present the problem studied.
Then, we will compare the steady and unsteady option for a same problem. Finally, we try to determinate the critical Rayleigh number with several different initial conditions.
 

1. The problem

One considers a domain between two walls which contains a fluid. One imposes a difference of temperature between this two walls.
When the higher temperature is on top wall, the solution of the thermic equations is a stable gradient of temperature.

Nevertheless, when the higher temperature is on the lower wall, different solutions can appear.
Indeed, it exists a critical value for the difference of temperature.
Under this value, the solution is always a stable gradient of temperature.
Up to this value, an instability appears: 2D stationnary contrarotative rolls of Rayleigh-Benard.

2. Theory

This instability is due to the competition of two types of forces: Force of stabilization and force of unstabilization.
The first one is the drag force and the heat diffusion and the other is the Archimed force.
When Archimed force is preponderant compared to the drag one, the instability can appear.
A critical number of Rayleigh caracterises the limit between stabilities and unstabilities.
 

Stabilizative forces	Unstabilizative forces

 
 

3. The mesh

One studies a domain which dimensions are 1 cm for the height and 2 cm for the width.
The mesh had been created with Prebfc.
 
 

We decided to generate a structured mesh because we can genarate unstructured mesh which zoom on high gradient zones. Indeed, these zones are depending on the number of rolls which appears.

During the simulation with Fluent, the fluid used was water with the following properties:

Moreover, during all simulations, we used the option Presto! of Fluent which is applied to pressure. You choose this option in the following menu: