标题： 流体微元旋转和内在涡度分解的一般运动学理论 显示英文标题

标题： A general kinematic theory of fluid-element rotation and intrinsic vorticity decompositions

摘要： 本研究提出了一种通用的运动学理论，用于流体元旋转和内在涡量分解，该理论是在涡量和涡旋动力学的背景下提出的。 材料线和表面元的角速度包括由体积元旋转驱动的经典贡献（等于局部涡量的一半），以及由应变率引起的特定角速度。 然后，构建了两种方向依赖的涡量分解（DVD），揭示了涡量的刚性旋转和自转模式。 我们推导了正交线-表面元对的内在耦合关系，阐明了它们互补的运动学和几何作用。 值得注意的是，我们严格证明了基于表面元的DVD中的自转模式与广义Caswell公式中的相对涡量相同，从而准确地考虑了牛顿流体中的表面剪切应力。 接下来，在场论框架下，利用微分几何提出了基于流线和流面的涡量分解。 通过与DVD、Caswell公式和Helmholtz-Hodge分解的统一分析，明确了特征代数描述中六个旋转不变量的物理作用（包括速度梯度张量（VGT）的正常-幂零分解（NND）及其产生的不变涡量分解（IVD））。 ... 发现物理允许的DVD涡量模式在相空间中必须被IVD模式所限制，而强制最小化原理自然得出Liutex公式。 最后，该理论的有效性在各种流动中得到了验证，从简单的到复杂的流动。 结果表明，耦合的IVD-DVD分析可以在代数和场论框架下增强对复杂涡旋流动的物理理解。 显示英文摘要

摘要： The present study proposes a general kinematic theory for fluid-element rotation and intrinsic vorticity decompositions within the context of vorticity and vortex dynamics. Both the angular velocities of material line and surface elements comprise a classical contribution driven by volume-element rotation (equal to half the local vorticity), and a strain-rate-induced specific angular velocity. Then, two direction-dependent vorticity decompositions (DVDs) are constructed, revealing the rigid rotation and spin modes of vorticity. We derive intrinsic coupling relations for orthogonal line-surface element pair, elucidating their complementary kinematic and geometric roles. Notably, we rigorously prove that the spin mode (in the surface-element-based DVD) is identical to the relative vorticity in the generalized Caswell formula, thereby faithfully accounting for surface shear stress in Newtonian fluids. Next, within a field-theoretic framework, vorticity decompositions are proposed based on streamline and streamsurface using differential geometry. The physical roles of the six rotational invariants in the characteristic algebraic description (including the the normal-nilpotent decomposition (NND) of the velocity gradient tensor (VGT) and the resulting invariant vorticity decomposition (IVD)) are clarified through unified analysis with the DVD, Caswell formula, and Helmholtz-Hodge decomposition. ... It is found that physical admissible DVD vorticity modes must be bounded by IVD modes in phase space, whereas enforcing a minimization principle naturally yields the Liutex formula. Finally, the effectiveness of theory is validated across diverse flows, from simple to complex. Results show that a coupled IVD-DVD analysis could enhance physical understanding of complex vortical flows under both algebraic and field-theoretic frameworks.