ON THE CONNECTION BETWEEN THIN VORTEX LAYERS AND VORTEX SHEETS
ON THE CONNECTION BETWEEN THIN VORTEX LAYERS AND VORTEX SHEETS
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DOI：
10.1017/s0022112090002609
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发表时间：
1990-06-01
影响因子：
3.7
通讯作者：
SHELLEY, MJ
中科院分区：
文献类型：
--
作者：
BAKER, GR;SHELLEY, MJ
本文在小厚度的极限下，研究了不可压缩无粘流体中均匀涡量层的二维运动方程。在适当的情况下，极限是涡面，其强度是涡量乘以层的局部厚度。然而，涡面可以在有限时间内发展出奇点，其后续性质是一个悬而未决的问题。另一方面，涡旋层虽然可能在边界上发展出奇点，但始终有运动。幸运的是，层中的材质曲线始终存在。在某些假设下，它的极限运动又是涡面，因此它的行为可以表明在奇点时刻之后涡面的性质和可能存在。对于摩尔（1978）定义的中心曲线的极限行为，也得到了类似的渐近结果。通过考察一系列层的行为，对涡面曲率奇异性的形成有了一些物理上的理解。应变流，部分地由片的周期性延伸引起，导致涡量被平流输送到片上的某个点，速度快到足以形成奇点。然而，由于不可压缩性，涡层只是简单地向外凸出，随后形成一个带拖曳臂的涡核，拖曳臂环绕涡核。有证据表明，在涡层的边界曲线上不会形成奇点。在涡面的奇异时间之外，涡层的极限行为是不均匀的。远离涡核，各层会聚成一条光滑的曲线，看起来像一个双分支螺旋。当核心周围的环流消失时，涡面强度的近似值变得无界，这表明一个复杂的局部结构，其确切性质仍然无法确定。
The equations for the two-dimensional motion of a layer of uniform vorticity in an incompressible, inviscid fluid are examined in the limit of small thickness. Under the right circumstances, the limit is a vortex sheet whose strength is the vorticity multiplied by the local thickness of the layer. However, vortex sheets can develop singularities in finite time, and their subsequent nature is an open question. Vortex layers, on the other hand, have motions for all time, though they may develop singularities on their boundaries. Fortunately, a material curve within the layer does exist for all time. Under certain assumptions, its limiting motion is again the vortex sheet, and thus its behaviour may indicate the nature and possible existence of the vortex sheet after the singularity time. Similar asymptotic results are obtained also for the limiting behaviour of the centre curve as defined by Moore (1978). By examining the behaviour of a sequence of layers, some physical understanding of the formation of the curvature singularity for a vortex sheet is gained. A strain flow, induced partly by the periodic extension of the sheet, causes vorticity to be advected to a certain point on the sheet rapidly enough to form the singularity. A vortex layer, however, simply bulges outwards as a consequence of incompressibility and subsequently forms a core with trailing arms that wrap around it. The evidence indicates that no singularities form on the boundary curves of the layer. Beyond the singularity time of the vortex sheet, the limiting behaviour of the vortex layers is non-uniform. Away from the vortex core, the layers converge to a smooth curve which has the appearance of a doubly branched spiral. While the circulation around the core vanishes, approximations to the vortex sheet strength become unbounded, indicating a complex, local structure whose precise nature remains undetermined.