2D convection

Result

Temperature and speed:

Temperature
Speed

Govern equations

Natural convection equation :

$\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u}\cdot \nabla \mathbf{u}) + \frac{1}{\rho_0}\nabla p = \nu \nabla^2 \mathbf{u} + \beta(T- T_0) \mathbf{g}$
$\frac{\partial T}{\partial t} = \kappa \nabla^2 T - \left(\mathbf{u}\cdot\nabla\right)T$
$\nabla\cdot \mathbf{u} = 0$
where $\mathbf{g} = (0,-g)$

Domain

Compuational domain:

Boundary condtion:

$(a),(b)\quad \mathbf{u} =0,\quad \frac{\partial p}{\partial n} = 0, \quad \frac{\partial T}{\partial n} = 0$
$(c)\quad \mathbf{u} =0,\quad \frac{\partial p}{\partial n} = 0, \quad T = T_1\quad (T_1 > T_2)$
$(d)\quad \mathbf{u} =0,\quad \frac{\partial p}{\partial n} = 0, \quad T = T_2\quad (T_1 > T_2)$