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Eroding dipoles and vorticity growth for Euler flows in R 3 : Axisymmetric flow without swirl

机译:R 3中Euler流的腐蚀偶极和涡度增长:无涡旋的轴对称流

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摘要

A review of analyses based upon anti-parallel vortex structures suggests that structurally stableuddipoles with eroding circulation may offer a path to the study of vorticity growth in solutions ofudEuler’s equations in R3ud. We examine here the possible formation of such a structure in axisymmetricudflow without swirl, leading to maximal growth of vorticity as tud4/3ud. Our study suggests that theudoptimizing flow giving the tud4/3 growth mimics an exact solution of Euler’s equations representingudan eroding toroidal vortex dipole which locally conserves kinetic energy. The dipole cross-sectionudis a perturbation of the classical Sadovskii dipole having piecewise constant vorticity, whichudbreaks the symmetry of closed streamlines. The structure of this perturbed Sadovskii dipole isudanalyzed asymptotically at large times, and its predicted properties are verified numerically. Weudalso show numerically that if mirror symmetry of the dipole is not imposed but axial symmetryudmaintained, an instability leads to breakup into smaller vortical structures.
机译:对基于反平行涡结构的分析的回顾表明,具有稳定环流的结构稳定的偶极子可能为研究R3 ud中 udEuler方程解的涡度增长提供一条途径。我们在这里检查了在没有涡旋的轴对称 udflow中这种结构的可能形成,从而导致涡度最大增长为t ud4 / 3 ud。我们的研究表明,使t ud4 / 3增长的 u最优化流模拟了Euler方程的精确解,该方程表示 udan腐蚀的环形涡旋偶极子,该方程局部保存了动能。偶极子的横截面反映出具有分段恒定涡度的经典萨多夫斯基偶极子的摄动,这打破了闭合流线的对称性。对该动静的萨多夫斯基偶极子的结构进行了大幅度渐近分析,并对其数值进行了验证。我们还从数值上表明,如果不施加偶极子的镜像对称性但保持轴向对称性,则不稳定性会导致分裂成较小的涡旋结构。

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