To know the nonlinear characteristics of pipe flow, we simulate the flow as a superposition of many vortex rings. As a first step, we investigated the nonlinear interaction among vortex rings whose number is three at most. The pipe wall is replaced by many bound vortices. A free vortex ring moves forward or backward according to the radius, and that of a particular radius keeps its initial position. Two vortices of equal radii always show an overtaking process repeatedly, i.e., the backward ring increases the velocity due to the forward ring while the forward one is decreased by the induced velocity due to the vortex behind it. In the case of three vortices, there appears a wide variety of behavior. For certain combinations of the radii and the axial spaces among them, the motion seems to be very complex. It is expected that our result will give a clear sight for seeking the nonlinear behavior of many vortices.

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The correspondence of the coherent structure on the surface of a rotating cylinder to the circumferential distortion of a vortex ring was investigated by using the flow visualization technique. When a circular cylinder rotates suddenly about its axis, some mushroom type eddies which surround the rotating cylinder are regularly generated along its axis. The relation between the spacing of mushroom eddies, p/d, and the Reynolds number, Rer, is shown as p/d = 64·Rer^-0.77. The circulation of the vortex ring which is made from a vortex filament is given by Γ= 2R·V₀, where 2R is the diameter of vortex ring and V₀ the induced velocity at the center of vortex ring. This simple relation is also applicable to the real vortex ring. The vortex in a vortex ring can be replaced by the Rankine vortex model which has the same circulation as real vortex ring, when the diameter of vortex core is 1.2 times the half width of the vorticity distribution of the real vortex ring and the constant vorticity in vortex core is equal to that of the vortex center. It is assumed that the core diameter of vortex ring which is made from the Rankine vortex is equivalent to the diameter of a rotating cylinder. Then, the relation between the spacing of the circumferential distortion and the Reynolds number for the vortex ring well corresponds to the above relation for the rotating cylinder. This fact suggests that the distortion of vortex ring is caused by the centrifugal instability that generates mushroom eddies on the surface of rotating cylinder.

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