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 wh ich bridge a 1 mm gap between a positive 50 mu m radius hyperboloid point a nd a plane, when a de voltage is applied. The results show that, for such g aps, no streamers form at voltages lower than or equal to 2.5 kV and that s treamers bridge the gap at higher voltages. The streamer speed and radial d imensions of the streamer are found to linearly increase with the applied v oltage and this agrees with what is predicted by existing two-dimensional m odels. 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 c omparable only later in the development of the streamer. The results are ob tained 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 th e finite-difference results. The finite-element method, however, through th e use of unstructured grids, reduces significantly the number of unknowns a nd makes the modelling of streamers and arbitrarily shaped electrodes in tw o dimensions a feasible task.