The theory of short-gap breakdown of needle point-plane gaps in air using finite-difference and finite-element methods

Numerical results are presented for the formation of breakdown streamers which bridge a 1 mm gap between a positive 50 µm radius hyperboloid point and a plane, when a dc voltage is applied. The results show that, for such gaps, no streamers form at voltages lower than or equal to 2.5 kV and that streamers bridge the gap at higher voltages. The streamer speed and radial dimensions of the streamer are found to linearly increase with the applied voltage and this agrees with what is predicted by existing two-dimensional models. The electric field in the streamer channel behind the streamer head is initially found to be much lower than that for longer gaps and becomes comparable only later in the development of the streamer. The results are obtained using a pre-existing finite-difference code and a new finite-element code developed using a new finite-element flux-corrected transport (FE-FCT) method. The finite-element results are shown to be almost identical to the finite-difference results. The finite-element method, however, through the use of unstructured grids, reduces significantly the number of unknowns and makes the modelling of streamers and arbitrarily shaped electrodes in two dimensions a feasible task.