In this paper we present the numerical results of the behavior of the horizontal vortex generated by ejecting a liquid vertically upward from an orifice into the bulk fluid above the orifice. The numerical calculation has been performed for the axi-symmetric Navier-Stokes equation. A simple flow-visualization experiment was also conducted to qualitatively verify the numerical solutions. Three cases of the flow configurations studied in this paper are; firstly, the vortex was generated without any background rotation, secondly, the vortex was made under a full background rotation, and thirdly, the vortex was made during the spin-up process such that only the region adjacent to the side wall was set into motion viewed in the inertial frame of reference. It was shown that the swirl flow at the inlet boundary affects considerably the formation and development of the vortex for the second case. In the third case, it was remarkable to see that the vortex cannot penetrate into the region near to the side wall of the tank, because of the strong swirl flow and corresponding high pressure gradient in the region.