Buoyancy-Driven Natural Convection

Mathematical Model

Consider the Boussinesq approximation for buoyancy-driven natural convection:

\[\begin{split} \begin{align} \frac{\partial \mathbf{v}}{\partial t} + (\mathbf{v} \cdot \nabla)\mathbf{v} &= -\frac{1}{\rho_0}\nabla p + \nu \nabla^2 \mathbf{v} + \beta g (T - T_0)\mathbf{k} \\ \frac{\partial T}{\partial t} + \mathbf{v} \cdot \nabla T &= \alpha \nabla^2 T \end{align} \end{split}\]

Boundary Conditions

• Vertical walls: \(\mathbf{v} \cdot \mathbf{t} = 0\) and \(\frac{\partial T}{\partial n} = 0\).

• Inlet and outlet: Specify velocity or temperature conditions.

• Top and bottom: No-slip and adiabatic conditions.

Initial Conditions

Start with an initial velocity and temperature distribution.

Weak Formulation

Find \(\mathbf{v} \in H_0^1(\Omega)\) and \(T \in H^1(\Omega)\) such that:

\[\begin{split} \begin{align} &\int_{\Omega} \left(\frac{\partial \mathbf{v}}{\partial t} + (\mathbf{v} \cdot \nabla)\mathbf{v}\right) \cdot \mathbf{w} \, d\Omega + \int_{\Omega} \frac{1}{\rho_0}\nabla p \cdot \mathbf{w} \, d\Omega \\ &+ \int_{\Omega} \nu \nabla \mathbf{v} : \nabla \mathbf{w} \, d\Omega - \beta g (T - T_0)\mathbf{k} \cdot \mathbf{w} \, d\Omega \\ &+ \int_{\Omega} \left(\frac{\partial T}{\partial t} + \mathbf{v} \cdot \nabla T - \alpha \nabla^2 T\right) \psi \, d\Omega \\ &= 0 \end{align} \end{split}\]

for all test functions \(\mathbf{w}, \psi \in H_0^1(\Omega)\).