Corona discharges on conductors are unavoidable in high-voltage direct current (HVDC) transmission lines. The cross-section geometry of the conductor is an important factor which significantly affects the positive corona discharges of dc transmission lines. Here, an improved fluid model is proposed to simulate the dc positive corona discharges on conductors with different cross-section geometries. The evolution of electric field and plasma sheath of circular conductor and aluminum conductor steel reinforced (ACSR) has been demonstrated. It is shown that the cross-section geometry of the conductor mainly affects the corona characteristics at the start stage of the discharge, whereas it has little influence with the development of corona discharges. Moreover, the effects of cross-section geometries of the conductors on plasma sheath characteristics are also discussed. The results would provide guidance for the selection and design of conductor section in high-voltage overhead transmission lines, which may find wide applications in the diagnosis of corona effects on the surface of conductors.