The Boussinesq-Approximation

For the discretization of the energy equation, a first order Boussinesq approximation is used, which takes into account buoyancy effects induced by temperature differences. Here one assumes that the density, viscosity and the Prandtl number do not depend on the temperature. Furthermore, dissipation terms like $\Delta {\bf u}\cdot {\bf u}$ are not considered for the energy equation. The temperature only affects the forcing term ${\bf g}$, i.e. if there is a volume force ${\bf g}_0$, then the resulting forcing term in () reads

$${\bf g}=(1-\beta (T-T_{ref})){\bf g}_0~.$$ (3.9)

$\beta$ is the volume expansion coefficient and $T_{ref}$ is a certain reference temperature, like the mean temperature in $\Omega$ or so.