Energy Transport

Solve a multi-physics problem that couples the Navier–Stokes equation with a heat equation:

Here is the vector-valued velocity field, is the pressure and the identity matrix. and are the density and dynamic viscosity, respectively. is the heat capacity of the liquid and the temperature.

Specify parameters and a rectangular region with cutouts.

Visualize the region.

Specify a Prandtl number and a Rayleigh number .

Set up a viscous Navier–Stokes equation that is coupled to a heat equation making use of a Boussinesq approximation. Use the material parameters specified.

Set up no-slip boundary conditions for the velocities on all boundary walls.

Set up a reference pressure point.

Specify a temperature difference between the left and right walls.

Replace parameters in the boundary conditions.

Set up initial conditions such that the system is at rest.

Monitor the time-integration progress and the total time it takes to solve the PDE while using a specified mesh spacing and interpolating the velocities and and the temperature with second order and the pressure with first order.

Construct a visualization of the boundary of the region.

Visualize the pressure distribution.

Visualize the temperature distribution.

Visualize the velocity field.

Animate the change in temperature and the velocity streamlines.

For the curious, investigate the different velocity field produced by simply swapping the cutout positions.