Opus Heat

This test case has been proposed by Annabelle Le-Hyaric and Michel Fouquembergh formerly at AIRBUS.

We consider a 2D model representative of the neighboring of an electronic component submitted to a cooling air flow. It is described by four geometrical domains in $\mathbb{R}^2$ named $\Omega_i,i=1,2,3,4$, see figure. We suppose the velocity $\vec{v}$ is known in each domain --- for instance in $\Omega_4$ it is the solution of previous Navier-Stokes computations. --- The temperature $T$ of the domain $\Omega = \cup_{i=1}^4 \Omega_i$ is then solution of heat transfer equation :

where $t$ is the time and in each sub-domain $\Omega_i$, $\rho C_i$ is the volumic thermal capacity, $k_i$ is the thermal conductivity. $k_1$ and $k_2$ are parameters of the model.

ICs dissipate heat, so the volumic heat dissipated $Q_1$ and $Q_2$ are also parameters of the model, while $Q_3=Q_4=0$.

One should notice that the convection term in heat transfer equation may lead to spatial oscillations which can be overcome by Petrov-Galerkin type or continuous interior penalty stabilization techniques.

Integrated circuits (ICs) (domains $\Omega_1$ and $\Omega_2$ ) are respectively soldered on PCB at $\mathbf{x1}=(e_{Pcb}, h_1)$ and $\mathbf{x_2}=(e_{Pcb}, h_2)$. They are considered as rectangles with width $e_{IC}$ and height $h_{IC}$. The printed circuit board (PCB) is a rectangle $\Omega_3$ of width $e_{PCB}$ and height $h_{PCB}$. The air(Air) is flowing along the PCB in domain $\Omega_4$. Speed in the air channel $\Omega_4$ is supposed to have a parabolic profile function of $x$ coordinate. Its expression is simplified as follows (notice that $\vec{v}$ is normally solution of Navier-Stokes equations; the resulting temperature and velocity fields should be quite different from that simplified model), we have for all $0 \leq y \leq h_{PCB}$

where V is a parameter of the model.

The medium velocity $\vec{v}_i = \vec{0}, i=1,2,3$ in the solid domains $\Omega_i, i=1,2,3$.