# Vorticity constant along a streamline

In the case of a 2D incompressible flow $$\omega = - \nabla^2 \psi$$ where $$\psi$$ is the stream function.

From the vorticity equation of motion I am able to find the following: $$\frac{\mathrm{d}\omega}{\mathrm{d}t} = \frac{\partial\omega}{\partial t} + (u\cdot\nabla) \omega = 0$$ Is this meant to show that in the case of steady flow $$\nabla^2 \psi$$ is constant on a streamline?

How does this show this? I am very confused.

Presumably, this is inviscid flow since the viscous terms are absent in your vorticity equation. Since the flow is two-dimensional, there is only one non-zero vorticity component $$\omega$$ aligned in the direction normal to the plane of flow. Hence, there is no vortex-stretching term $$\bf{\omega} \cdot \nabla\mathbf{u}$$.

In steady flow, the vorticity equation reduces to $$\mathbf{u}\cdot \nabla \omega = 0$$.

The gradient $$\nabla \omega$$ is always oriented in a direction normal to a level curve $$\omega = \text{constant}.$$ The condition $$\mathbf{u} \cdot \nabla \omega = 0$$ implies that the velocity at a point is tangential to such a curve. Hence, level curves of $$\omega$$ correspond to streamlines.

Velocity field is tangential to streamlines (level curves of streamfunction)

In two-dimensional, incompressible flow we can define a stream function $$\psi$$ by a line integral along any path $$C$$ joining $$(0,0)$$ to $$(x,y)$$

$$\psi(x,y) = \int_C \mathbf{u} \cdot \mathbf{n} dl = \int_C u \, dy - v \, dx.$$

Using the incompressiblity condition $$\nabla \cdot \mathbf{u} = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0$$ and Green's theorem it follows that the line integral is independent of path and $$\psi$$ is a well-defined function which is related to the velocity field by

$$u = \frac{\partial \psi}{\partial y}, \,\,\,\ v = -\frac{\partial \psi}{\partial x}.$$

Hence,

$$\mathbb{u} \cdot \nabla \psi = u \frac{\partial \psi}{\partial x} + v \frac{\partial \psi}{\partial y}= u(-v) + v(u) = 0.$$

• I understand it now. A good way to think of it is: 𝐮⋅∇𝜔=0 implies the directional derivative of 𝜔 in the direction of 𝐮 is zero. This implies that 𝜔 must be constant along a streamline. Commented Apr 17, 2019 at 20:12