What is Vorticity? — Physics

Definition 2.12 University Physics — Year 2 · Chapter 2 — Fluid Kinematics

The vorticity of a flow is the vector field

\vect\omega = \operatorname{\vect{curl}}\vect v , \qquad \operatorname{\vect{curl}}\vect A = \Bigl(\frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z},\; \frac{\partial A_x}{\partial z} - \frac{\partial A_z} {\partial x},\; \frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y}\Bigr) .

A flow is irrotational where $\vect\omega = \vect 0$ .

Examples

Example 2.15 (Three plane flows)

(i) Stagnation flow $\vect v = k(x, -y)$ : $\operatorname{div}\vect v = 0$ , $\vect\omega = \vect 0$ , $\varphi = \tfrac12k(x^2 - y^2)$ ; streamlines $xy =$ const, hyperbolas — a jet hitting a wall, near the axis. (ii) Point vortex $\vect v = \dfrac{\Gamma}{2\pi r}\vect e_\theta$ (plane polar coordinates): incompressible and irrotational everywhere except at $r = 0$ , $\varphi = \Gamma\theta/2\pi$ (multivalued), and the circulation on any circle around the centre is $\Gamma$ — the vorticity is concentrated on the axis. (iii) Rankine vortex: a core $r < a$ in solid rotation $v_\theta = \Omega r$ (vorticity $2\Omega$ ) matched to the point vortex $v_\theta = \Omega a^2/r$ outside: the model of a tornado, a bathtub swirl or the eddy behind the bridge pier — the velocity peaks at the edge of the core.

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