1. Reference problem#

1.1. Geometry#

_images/1000000000000A1F0000058B33E289E31DB72F88.png

We consider the plane thermal problem of a square cavity (with a side equal to 1) where heat propagates:

  • by convection (i.e. the particles constituting the middle of the cavity move at a speed \(u\) assumed to be constant here); the speed \(u\) is supposed to make an angle of \(67.5°\) with the axis \(x\),

  • by conduction.

1.2. Material properties#

We take \(\rho {C}_{P}=1.\) \(\lambda ={10}^{-6}\)

Hence a diffusivity \(\alpha =\frac{\lambda }{\rho {C}_{P}}={10}^{-6}\)

and as we take \(\parallel u\parallel =1\), we have the Peclet number \({p}_{e}=\frac{\parallel u\parallel \mathrm{.}L}{\alpha }={10}^{6}\) (\(L\) is the characteristic length, here \(L=1.\)).

1.3. Boundary conditions and loads#

On segments \(\mathrm{AB}\) and \(\mathrm{BC}\), a temperature \(T=1.\) is imposed

On segment \(\mathrm{AE}\), we impose a temperature \(T=0.\)

On the other 2 sides, we have the condition by default, namely, we have zero flow.