Hydrogen will be used more from now in energy systems such as fuel cell systems, hydrogen tanks, chemical plants, rockets, and combustion engines. Hydrogen is supplied as gas jets for many cases of the energy systems. There are important experimental reports of penetration length and longitudinal coverage of various gas jets [Hamamoto et al. Trans. of Japan Society of Mechanical Engineers, 1987]. However, the mixing process of hydrogen gas and air (or oxygen) is still mysterious on penetration length and longitudinal coverage, i.e., diffusion speed to the direction normal to the jet axis for various conditions. This is because, unfortunately, numerical simulations at the level of large eddy simulation (LES) and direct numerical simulation (DNS) for gases of multi-components and their experimental visualizations are not performed very much, whereas studies for liquid sprays of fossil fuels have been done extremely frequently until now. It is difficult to visualize hydrogen gas jets experimentally. Thus, the purpose of this research is to reveal the turbulent mixing process of single high-speed hydrogen gas jet by performing unsteady three-dimensional computation based on an approach extended from the LES. Computation results indicate that the penetration lengths both for jets of hydrogen-air and nitrogen-air are nearly the same. Also, the longitudinal coverage of hydrogen is wider than that of nitrogen, although hydrogen has a relatively small density. The computational results for two types of jets of hydrogen and nitrogen agree well with the predictions based on the empirical relation of Hamamoto et al. [Trans. of Japan Society of Mechanical Engineers, 1987]. Computational results for nitrogen jets also agree well with the penetration lengths obtained by our experimental visualizations.