The stability of elliptical vortices in an external straining flow
The stability of elliptical vortices in an external straining flow
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DOI：
10.1017/s0022112090001276
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发表时间：
1990-01
影响因子：
3.7
通讯作者：
D. Dritschel
中科院分区：
文献类型：
--
作者：
D. Dritschel
在均匀应变的作用下，粘性、不可压缩的二维流体中的涡旋椭圆斑块通常会旋转或章动并延伸或压缩，同时保持精确的椭圆形状（Kida 解）。这个结果很有趣，因为均匀应变理想化了具有许多涡流的流动中远处涡流的主阶畸变影响。由于远处涡流的不稳定运动，应变速率和应变轴的旋转速率通常随时间变化。在应变速率和旋转速率稳定的特殊情况下，当应变速率不太大时，椭圆涡旋的周期运动是可能的。较大的应变率导致涡流无限延伸。然而，均匀应变只能近似模拟远处涡流的影响。涡旋周围应变场的局部变化会扰乱涡旋，使其无法保持简单的椭圆形形状。这些干扰可能会因不稳定而放大。在本文中，我们首先通过线性 Floquet 理论，然后通过直接、高分辨率、非线性数值积分，研究了在稳定、均匀应变和旋转速率的情况下，周期性椭圆运动对小边界扰动的稳定性。人们发现周期解的很大一部分是线性不稳定的。即使应变率任意小且基本运动任意接近圆形，也可能发生不稳定。扩展的非线性计算在某些情况下会出现复发，而在另一些情况下则会由于重复的波放大、陡峭和破裂而导致涡流的磨损。
Subject to uniform strain, an elliptical patch of vorticity in an in viscid, incompressible, two-dimensional fluid generally rotates or nutates and extends or compresses while retaining a precisely elliptical shape (the Kida solutions). This result is of interest because the uniform strain idealizes the leading-order distortional influence of distant vortices in a flow with many vortices. Because of the unsteady motion of the distant vortices, both the strain rate and the rotation rate of the strain axes typically vary with time. In the special case that the strain rate and rotation rate are steady, and when the strain rate is not too large, periodic motion of an elliptical vortex is possible. Larger strain rates lead to indefinite extension of the vortex. Uniform strain, however, only approximately mimics the effect of distant vortices. The local variations- in the strain field around a vortex disturb the vortex, preventing it from retaining a simple, elliptical shape. These disturbances may amplify because of instabilities. In this paper, we examine the stability of periodic elliptical motion to small boundary disturbances, for the case of steady, uniform strain and rotation rate, first by linear Floquet theory and then by direct, high-resolution, nonlinear numerical integrations. It is discovered that a significant portion of the periodic solutions are linearly unstable. Instability can occur even when the strain rate is arbitrarily small and the basic motion arbitrarily close to circular. Extended nonlinear calculations exhibit recurrence, in some cases, and attrition of the vortex by repeated wave amplification, steepening, and breaking in others.