In this paper, we investigate the streamer discharge process in transformer oil under positive nanosecond pulse voltage through developing a two-dimensional axially symmetric fluid model and simulating the physics of discharge inception and propagation. The streamer discharge profile and distributions of electric field and space charge density are obtained under different conditions such as the amplitude of applied voltage, rise time and gap distance. Simulation results show that space charges enhance the front field of streamer head, which is conducive to the longer propagation of discharge channel, therefore "ionization wave" is formed. The magnitude and rise time of applied voltage have evident influences on the average speed of streamer propagation. It can be observed that the higher the applied impulse voltage, the faster the streamer propagates, and the steeper the rise time of applied impulse, when streamer arrives at the same position, the larger the discharging radius will be and the smaller maximal electric field will be. The cases of different gap distances indicate that longer gap distance corresponds to a faster average speed of streamer. It is considered that field-dependent molecular ionization predominates the charge generation mechanism of streamer discharge process in transformer oil, and space charge effect contributes to further developing ionization until the whole gap eventually breakdowns. The study is dedicated to the better understanding of the process from inception to breakdown of discharging in transformer oil, as well as ionization mechanism in liquid dielectric.