Gas engines often utilize a small-volume pre-chamber in which fuel is injected at near stoichiometric condition to produce a hot turbulent jet which then ignites the lean mixture in the main chamber. Hot jet ignition has several advantages over traditional spark ignition, e.g., more reliable ignition of extra-lean mixtures and more surface area for ignition resulting in faster burning and improved combustion efficiency. Our previous experimental results show that supersonic jets could extend the lean flammability limit of fuel/air mixtures in the main chamber in comparison to subsonic jets. The present paper investigated the characteristics of supersonic hot jets generated by combustion of stoichiometric H2/air in a pre-chamber to understand the ignition mechanism of ultra-lean mixtures by supersonic hot jets. Numerical simulations were carried out to examine the transient hot jets issued from six different nozzles (two straight nozzles, one converging nozzle and three converging-diverging (C-D) nozzles) using a detailed H2/air chemistry. The detailed flame propagation process inside the pre-chamber was investigated. Then the characteristics of the hot jets from six nozzles were compared, including the spatial and temporal distribution of velocity, vorticity, pressure, turbulence quantities, temperature, shock structures, and species concentrations. The results show that supersonic jets exhibit shock diamond structures. The static temperature rises after each shock and a significant temperature rise occur after the final shock. The location of this high-temperature zone is consistent with the experimental observations where ignition was initiated. The profile of Damköhler numbers based on the local flow properties was determined. A critical Damköhler number was found to be 11, below which the main chamber ignition would unlikely to occur. Additionally, the Damköhler number profiles help to explain why the two C-D nozzles with an area ratio of 4 and 9 could extend the flammability limit, whereas the C-D nozzle with an area ratio of 16 failed to do so.