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Tornadoes are one type of violent flow phenomenon and occur in many places in the world. There are many research methods that aim to reduce the loss of human lives and material damage caused by tornadoes. One effective method is numerical simulation such as that in Ishihara et al. (J. Wind Engng Ind. Aerodyn., vol. 99, 2011, pp. 239-248). The swirling structure of the Navier-Stokes flow is significant for both the mathematical analysis and numerical simulations of tornadoes. In this paper, we try to clarify the swirling structure. More precisely, we performed numerical computations on axisymmetric Navier-Stokes flows with a no-slip flat boundary. We compared a hyperbolic flow with swirl and one without swirl, and observed that the following phenomenon occurs only in the swirl case: the distance between the point with the maximum magnitude of velocity |v| and the z-axis changed drastically at a specific time (which we call the turning point). Besides, an 'increasing velocity phenomenon' occurred near the boundary, and the maximum value of |v| was obtained near the axis of symmetry and the boundary when the time was close to the turning point in the swirl case. [ABSTRACT FROM AUTHOR]