Evolution of the Surface of Hill's Vortex Subjected to a Small Three-Dimensional Disturbance for the Cases of<i><i>m</i>=<b>0</b></i>, 2, 3 and 4

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  • Rozi Tashpulat
    Department of Mathematical Sciences, College of Science and Engineering, Tokyo Denki University, Hatoyama–machi, Hiki–gun, Saitama–ken 350–0394

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  • Evolution of the Surface of Hill's Vortex Subjected to a Small Three-Dimensional Disturbance for the Cases of m=0, 2, 3 and 4.

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The purpose of this study is to examine the response of Hill's vortex to small three-dimensional disturbances of azimuthal numbers m=0, 2, 3 and 4. Specifically, we focus our attention on the temporal evolution of its surface shape. Moffatt and Moore (1978) used the streamfunction for the formulation of the problem for an axisymmetric disturbance (m=0), and obtained an approximate analytical solution for surface shape. To deal with three-dimensional modes, we must solve the Euler equation numerically inside the core. First, comparison is satisfactorily made, in the axisymmetric case (m=0), between the approximate analytical solution of Moffatt and Moore and the formulation using velocity potential in the entire space. Numerical computation for the disturbance with azimuthal number m=2 is performed. It is found that as time proceeds, double spike-like structure is formed around the rear stagnation point. We point out that some instability occurring near the front stagnation point, found in the previous paper, is due to the lack of number of terms in the expansion of the velocity field. Further numerical results for m=3 and 4 show that a peak of initial surface elevation moves downstream toward the rear stagnation point forming a m spike-like structure. A rough estimate suggests that the spikes grow with time t approximately as eα m t, with α being dependent on initial condition, implying a fast development of fine-scale structure around the rear.

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