Steady - Unsteady







      Every process begins with an unsteady phase before reaching the steady state. The unsteady state is caracterized by an evolution of all the parameters with time. As soon as the parameters describing the process do not depend anymore on time, the steady state is reached. These two states can be modelled separately with FLUENT. But we did not know what happened for the transition between these two states. Indeed, is there a difference between the results of calculations  with a steady state  model and the results of calculations with an unsteady state model,whose calculation time is long enough for the steady state to be reached. In other words, does the unsteady state model tend to the steady state model, when time increases?
 
 

1) Results of the Steady State Case
        The geometry of the problem is the same as cases' second chapter. The difference of temperature is 0.5 K. To compare the steady results with unsteady results, we have chosen several parameters: the velocity and the temperature profile at x = 0.5 cm (middle of one of the two rolls), the highest velocity and the heat flux throw the fluid.

Velocity vectors:











Velocity profile at the middle of the rolls:








Temperature profile at the middle of the rolls:








Other parameters:
 

Vmax =  3.02 e-4 m/s
= 1.354 W
2) Results of the Unsteady State Case
    The unsteady calculations have exactly the same initial data than the steady one's.
 

To predict the time needed to reach the steady state, we considered that the steady state might be reached for a time equal to  avec:

o / 

The Rayleigh number is:    Ra = 6145.    Since the critical Rayleigh number is 1708, we have:

=2.6

In the other hand:

                with

a    : layer thickness (m)
   : thermal diffusivity (sq.m / s)
Pr  : Prandt unitless number
 
    In this case,
a = 0.01 m
=1.43 e -7 sq.m/s                                        ==> o = 35 s
Pr = 7


So = 13.5 s

 We have run several calculations o after o. To know if the steady state was reached, we have diplayed the Vmax versus time with Vmax(steady state) = 3e-4 m/s.
 
 

    From this graph, the steady state seems to be reached at =255 s.Then we picked some values up to compare with the first case:
 

  Vmax = 3.1e-4 m/s
= 1.356 W


We also displayed the velocity profile and the temperature profile for x = 0.5 cm and x = 1.5 cm:

Velocity profile:








Temperature:


3) Comparison and conclusion
     We first notice that the Vmax, the heat flux and the shape of the rolls are the same for the steady state reached by the unsteady calculations and the steady state reached by the steady state calculations. Besides if we compare the velocity profile and the temperature profile at several values of x, they are quite the same. Here are the velocity profile got by both models at two different positions in the cell (at the middle of each roll):


 

So we can conclude that the unsteady state of FLUENT tends to its steady state.
      The theoritical time to reach the steady state was around 20 seconds whereas the experimental determined in this study is around 250 seconds, so 10 times longer. This difference makes us think that the time  as it is defined theoritically, represent the transition time between two steady states and not the transition time between the conduction state (V = 0 m/s) and a steady state. Manuel Velarde and Christiane Normand [1] explain that the experimental discovery of the instabilities of Rayleigh-Benard was long because for a long time after the beginning of the experience, nothing occurs. And as soon as the force of Archimede and the dissipation forces are equal (Ra = Rac), rolls appear.
 
Reference
 [1]        M. Velarde, C. Normand, La convection