涡量方程的一种简化形式及涡量守恒定理向三维的推广
A simplification of the vorticity equation and an extension of the vorticity persistence theorem to three dimensions
AI总结：
本研究通过对任意坐标系下的涡量方程进行数学简化，将涡量守恒定理推广至三维，定义了不依赖坐标系的时间导数，完善了三维Euler流的涡量相关理论。
摘要：
一个多世纪以来，学界已知在二维（平面）Euler流中，涡量沿流线守恒。但在三维情形下，尚无此类结论，这主要是由运动方程中的涡拉伸项导致的。本文将涡量守恒定理推广至三维情形：对于存在流线坐标系、且流线速度分量与某一横坐标无关的Euler流，涡量张量中垂直于流线方向的所有分量均沿流线守恒。这一推广是通过对任意坐标系下推导得到的涡量方程进行数学简化实现的。经过数学简化后，涡量方程中的非线性对流项剩余部分为涡量张量相对于流体速度的Lie导数。本文还定义了一种不依赖坐标系的时间导数，当该导数取零时，根据其输入变量的不同，可分别表示连续性方程或（不含粘性项的）涡量方程。
英文摘要：
It has been known for more than a century that in two-dimensional (planar) Euler flow, vorticity is conserved along streamlines. In three-dimensions, however, no such result has been established, and this is primarily due to the vortex stretching term in the equation of motion. The vorticity persistence theorem is herein extended to three dimensions. It states that all components of the vorticity tensor normal to the streamline direction are conserved along streamlines in Euler flows admitting streamline coordinate systems in which the streamline velocity component is independent of a transverse coordinate. This extension is accomplished with the aid of a mathematical simplification of the vorticity equation derived for arbitrary coordinate systems. What remains of the nonlinear convective terms in the vorticity equation, after the mathematical simplification, is the Lie derivative of the vorticity tensor with respect to fluid velocity. A coordinate-independent temporal derivative is defined which, when set to zero, expresses either the continuity or vorticity equation (excluding the viscous term), depending upon the argument supplied to it.
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