What is Velocity potential; circulation? — Physics

Definition 2.14 University Physics — Year 2 · Chapter 2 — Fluid Kinematics

In an irrotational flow the velocity derives from a velocity potential: $\vect v = \operatorname{\vect{grad}}\varphi$ (the circulation of $\vect v$ along a path then depends only on its ends, $\int_A^B\vect v \cdot\dd\vect l = \varphi(B) - \varphi(A)$ ). If the flow is also incompressible, $\Delta\varphi = \operatorname{div}\operatorname{\vect{grad}} \varphi = 0$ : the potential obeys Laplace’s equation, exactly like the electrostatic potential in a charge-free region — a potential flow. The circulation of the velocity along a closed curve $C$ is $\Gamma = \oint_C\vect v\cdot\dd\vect l$ .

Three plane flows. Left: the stagnation flow, irrotational, with hyperbolic streamlines. Middle: the point vortex — circular streamlines, yet zero vorticity away from the centre. Right: the Rankine vortex profile, solid rotation in the core and a point vortex outside.

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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