Study of an horizontal gradient of temperature

In this example, we want to see which are the effects of an horizontal gradient of temperature.
In our case, the density is dependent on the temperature, so the isochore lines are vertical. Moreover, the isobar lines are  horizontal, due to the hydrostatic configuration. Thus, the theory explains us that this configuration is not stable. Indeed, as shown on the draw, the gravity center (G) of a fluid element is not the same that the center (P) due to the pressure forces, so there is apparition of a momentum which creates a local rotation.

The configuration of the simulation is the following :

- Initial conditions
. the fluid velocity is equal to zero everywhere,
. the temperature is given by T(x) = 301.5-1.5 cos(50 Pi x), that is to say a difference of 3° C between the left and right sides and a temperature equal for all points placed on a vertical line.

- Boundary conditions
the temperature on the both walls is equal to the initial temperature

The results obtained are the following :

We notice that the fluid is rolling in the whole domain which is given to it. This allows a heat transfer by convection. This one is stopped in the first top half of the domain when there is a steady-state flow.