涡度是流体力学中的一个重要概念，用来描述流体微元的局部旋转强度。数学上定义为速度场的旋度，记作omega等于nabla叉乘v。为了直观理解涡度，我们可以观察一个刚体旋转的流场，其中流体粒子围绕中心轴以恒定角速度旋转。
涡度的数学定义基于速度场的偏导数。在二维情况下，涡度的z分量等于v_y对x的偏导数减去v_x对y的偏导数。三维情况下，涡度是一个矢量，每个分量都由对应的偏导数差值构成。图中展示的是一个简单的剪切流场，虽然流体没有明显的旋转运动，但由于速度梯度的存在，仍然具有非零的涡度。
涡度的物理意义是流体微元角速度的两倍。当涡度为正值时，表示流体微元逆时针旋转；为负值时表示顺时针旋转；为零时表示流体微元没有旋转，可能是平移运动或纯变形。图中展示的是典型的涡旋流场，流体围绕中心点做圆周运动，具有很高的涡度值。
涡度在许多领域都有重要应用。在气象学中，它用于分析大气涡旋和龙卷风的形成机制；在海洋学中，帮助研究海洋环流和涡流结构；在航空航天领域，用于分析机翼绕流和尾涡现象；在工程流体中，指导泵和压缩机的设计优化；在湍流研究中，是理解复杂流动结构的关键参数。总的来说，涡度是理解和分析各种流动现象的重要工具，在理论研究和工程实践中都有广泛的应用价值。
The mathematical definition of vorticity is based on partial derivatives of the velocity field. In two dimensions, the z-component of vorticity equals the partial derivative of v_y with respect to x minus the partial derivative of v_x with respect to y. In three dimensions, vorticity is a vector with each component defined by corresponding partial derivative differences. The figure shows a simple shear flow where fluid moves horizontally with velocity proportional to height, resulting in constant non-zero vorticity despite no apparent rotational motion.
The physical meaning of vorticity is twice the local angular velocity of fluid elements. Positive vorticity indicates counterclockwise rotation, negative vorticity indicates clockwise rotation, and zero vorticity means no rotation. The figure shows a typical vortex flow where fluid moves in circular paths around a center point, demonstrating high vorticity values. The red particle traces a circular path, illustrating how fluid elements rotate in such flows.
Vorticity behaves differently in various flow types. In uniform flow, all fluid moves with the same velocity, resulting in zero vorticity everywhere. In solid body rotation, the entire fluid rotates as a rigid body with constant angular velocity, producing uniform non-zero vorticity. In shear flow, velocity varies linearly with position, creating spatially varying vorticity. These examples demonstrate how vorticity quantifies the local rotation characteristics of different flow patterns.
The vorticity transport equation describes how vorticity evolves in fluid motion. It includes convection by the flow, vorticity stretching in three dimensions, and viscous diffusion. In two-dimensional incompressible flow, the equation simplifies significantly. The visualization shows viscous diffusion of a vortex, where vorticity spreads outward over time due to molecular viscosity, causing the vortex to weaken and expand. This equation is fundamental for understanding vorticity dynamics in turbulent flows and engineering applications.