**B.E.S. Rayleigh-Benard **

I Introduction

II Theory

II 1 Unconditional
stability

II 2 Conditional stability

III
Determination of the stability threshold

III 1 Methode

III 2 explotation of results

IV Influence
of meshing

V Influence
of the order discretization scheme

VI
Influences of the initials conditions

VI 1 Presentation

VI 2 making of the
initial condition

VI 3 Results

VII To conclude

When a spatially extended, nonlinear system is driven out of equilibrium, it often displays complex spatio-temporal patterns. Many systems from Biology to Geophysics show similar behavior and one hopes to find universal features in these phenomena.

Convective systems, in particular, are intesively studied. Convection is one of the most common types of fluid flow observed in nature. It can be found in systems as widely varied as stars, the weather, and the oceans.

Rayleigh-Benard convection is significant because of comparatively easy-to-replicate phenomenon, it continues to provide insight into understanding how heat energy moves through a flow system. 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.

The experimental system that we are concerned with for this report 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.

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:

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

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.

We have a copetition betwen

- the Archimede one the one hand

- the drag forces and the thermal diffusion one the other hand

The adimentional number useful in this study are :

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

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

**Prandtl Number**: Pr = nu/a.

Where

**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,

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.

III Determination of the stability threshold

**A The meshing**

We use a grid 40*20 nodes showen here :

**Characteristics :
**Length 0.02 m

Height 0.01 m

Wall bounderies on the top and on the floor

**B Fluent initialization **

We initialize temperature at 300 K and velocity at 0 m/s. As we can see under 0 it's not a perfect initialization.

**C Unconditional Stability**

We can see above the result for the top plate warmed. It's a case of unconditional stability :

**D Conditional stabitilty**

As we see in the theoritical chapter the unconditional stability appear when the floor wall is hotter than the top wall.

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

from the stability ...

Dt = 0.1

Rayleigh = 803

We can note that the velocity is very low about 1e-9. but not equal to zero.

to the instability :

The differents Dt are 0.20, 0.21, 0.23, and 0.5

We can note differents things :

a/ The two dimensional projection of the pattern are approximately cosine functions of the form cos(k.x) where |k| = 2*pi/lambda and lambda is the wavelength.

In practice, of course, the patterns do not always emerge so uniformly. Different regions may have straight rolls oriented in different directions, for example, or the rolls may not be perfectly straight. In some cases, structures called defects or dislocations appear in the pattern. Defects correspond to places where a roll pair terminates. At defects, a wavenumber k cannot be uniquely defined.

b/ The determination of the stability threshold isn t accurate if we just see the result and don't explote them.

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 now plot Nusselt Number and Velocity to find the critical Rayleigh Number value.

We find the Nussel Number equal to one when convection
is smaller than conduction ( the case is stable). In the same time, velocity
are very "small" 1e-9 ~1e-7

When the Nusselt Number and the velocities increases we start do be in
an instables cases.

We find a Critical Rayleigh number around 1770. instead
of 1708. The pratical Rc is not exactly good. two things can explain that
because the dimention of the box aren't exactly made for a Rc of 1708

To test the influence of the meshing, we buid a new grid with 20*10 nodes

We find in this case a critical Rayleigh number around 1050 instade of 1708. The matter come from the meshing. We can conclude that the way of meshing influence the result.

V Influence of the order discretization scheme

After the study of the meshing influence, we are studying the order discretization scheme. For that we are using the worst meshing to see if an second-order scheme is able to improve the determination of the critical number of Rayleigh.

We see with theses curves, that the second-order scheme doesn't give us a better precision. We had try to explain this incoherent result in the fluent help.

As we can see Second-order accuracy is automatically used
for the viscous term.

When we use a second-order therm we juste apply it to ve velocity term
that a small when cases are stable. Viscosity and conduction phenomenon
are alredy discretize whit a second-order scheme. So, we dont change the
threshold by changing the discretization scheme in fluent.

Note :

We can think that we can improve the Nusselt and velocity curves in the
instable part of the graph.

VI Influences of the initials conditions

We have seen that with a null velocity initialization we have always two rollers. In theory, we can have a number of rollers different of two for this configuration.

VI-2 making of the initial condition

We have done a function (user1.c) with fluent to test the different cases, we have created a C function under Fluent. With this function, we have: X velocity equal null Y velocity equal A*sin(pi*y/ylast)*(cos(D*pi*x/xlast)+B*cos(C*p*x/xlast)). The syntax of the file is done in the initray.c (You must compile it on fluent) .

**Help for initialization:**

To make use of this function in **fluent/uns**, it
must first be compiled.

**Define **->**User Defined Function**s.

Use of the initialization function :

When the compilation is complete, you have to write in the Fluent/Uns window the following command :

(%chip-exec 'initialize)

We have test several initializations to try to understand how the initialization can influence the solution.

If the constant A is equal to 0, Y velocity and X velocity is null. We are in the same cases than before.

We have try a lot of cases of initialization with differents Raleigh number and differents numbers of rollers.... for exemple sixteen :

For example, we initialize at for rollers (A 1.0*e-04, B 0.0, C 1.0 and D 8)with Ra=4018, and we resolve unsteadies equations. We obtain three rollers. The following graphs show the passage from for rollers to three rollers.

The numerical simulation is able to solve problems which can be solve analytically. It is also cheaper than an experience. Furthermore we have to be careful. A lot of parameters are used (the boundaries conditions, the meshing, the scheme, the initialization, the underrelaxation.... ).

The user of an industrial workshop must know both the physical and the numerical problem.