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This work concerns the modeling of electrohydrodynamic phenomena in a flat duct where a corona discharge is generated by a single or multiple fine-wire electrodes. The objective is to disturb the air flow and to increase the convective heat transfer coefficient on the down wall. The coupled equations of Navier-Stokes, energy, electrodynamics and Ohm's law are modeled by the finite element method. An original numerical procedure based on a minimization algorithm is implemented to determine the charge density on the discharge electrodes surfaces. The developed numerical model makes it possible to undertake complete parametric studies taking into account the main couplings. We validate our approach in the case of a wire-plate configuration without cross-flow. Then, we study different steady state configurations for Reynolds numbers ranging from 3846 to 12820 using a standard k-epsiv model. This model enables to analyze velocity field in the channel and the heat transfer by convection between the air and the grounded electrode. The results indicate that a three-fold increase of heat transfer may be locally obtained when the air enters at the lower Reynolds number in the channel. A 'barrier' effect is also highlighted in the case of multiple wire electrodes.