In this paper, the penetration of a long bubble through a Newtonian fluid in a horizontal circular tube is studied by numerical simulation. We simulate the evolution of the asymptotic bubble profiles with an interface-tracking method by conservative level set method. The ratio of the asymptotic bubble width to the radius of the circular tube. i.e. λ. dominates not only the type of the flow patterns but also the location of the stagnation point. Three flow patterns suggested by Taylor in front of the bubble, namely the complete bypass flow, the transient flow and the recirculation flow are shown graphically. They make a good agreement with the study of Giavedoni and Hsu respectively. The stagnation point moves downstream as λ increases. The stagnation point moves here downstream with an almost constant rate of λ while 0.72≦λ≧ 0.9.