The Joule heating is commonly used to melt materials in various industries. This is a particular case in glass industry for which electric melting is mainly employed for production of potentially volatile, polluting glasses and also for wool insulation products. In such processes, electrodes are introduced in the bath and the heating is due to the Joule dissipation. The raw materials are introduced from above floating at the surface of the bath forming a cohesive batch leading to a high level of thermal insulation.

The maximum of temperature is observed inside the bath which can be a source of thermal instabilities. In order to study these instabilities, we develop a numerical tool coupling the thermoconvection problem written in the framework of Boussinesq approximation with electric potential equation. The numerical method is based on the discontinuous Galerkin finite element approach. This choice is motivated by the fact that the molten silicate giving a glass presents a high Prandtl number. Consequently, a strong impact of the convective process in the thermal transfer is expected. The investigated domain is a 2D-enclosure with two electrodes corresponding to a portion of the two vertical walls.

The flow and thermal fields are characterized by the aspect ratio of the enclosure a set at 2 in our numerical simulations, the electrode length Le and the Rayleigh Ra and the Prandtl Pr numbers. We first study the case of the electrodes corresponding to the entire of the vertical walls. In a such case, the flow appears only when the Rayleigh number is larger than a critical value equal to Racr=1700 when a=2. The flow intensity measured by a Péclet number behaves like a square root of the difference between Ra-Racr as expected in supercritical bifurcation when Ra>Racr. The thermal efficiency can be quantified by computing the average temperature over the horizontal direction of the cavity. The inverse of the average temperature corresponds to the Nusselt number. Numerically, we observe that the Nusselt number which is constant below the critical Rayleigh number increases linearly with Ra-Racr.

We perform numerical simulations when electrodes are equal to 2/3 of the top of the vertical walls. In this situation, the flow appears without threshold. Two regimes emerge: at low Rayleigh number (below to 1e3), the Péclet number is proportional to the Rayleigh number and above Ra=1e3 the Péclet number is a function of the square root of Ra. These behaviors are completely independent of the Prandtl number.

The flow and thermal solution becomes periodic in time above a critical Rayleigh number which is function of the Prandtl number. The occurrence of this instability is mainly due to the creation of two counter-rotating cells forming a jet in the middle of the cavity which can not stay stable. At low Prandtl number (equal to one), the critical Rayleigh above which the flow becomes periodic is larger to 1.6e5 while when the Prandtl number is larger than 10 the critical Rayleigh number decreases to 2.5e4. For a larger Prandtl number, the critical Rayleigh number does not change significantly.

Finally, the periodic regime is analyzed by determining the frequency and amplitude of flow oscillations in order to characterize the nature of the instability.