Stream function and Vorticity relationship with Streamlines

Question

For a two-dimensional flow, the velocity is given by $\textbf u= (u(x,y,t),v(x,y,t),0)$. Define the stream function $ψ (x,y,t)$.

Evaluate $(\textbf u·∇) ψ$ and deduce a relationship between the stream function and streamlines.

For a two-dimensional flow, the vorticity is given by $ω = (0,0, \xi (x,y,t))$.

We now consider an inviscid steady two-dimensional flow. Evaluate $(\textbf u·∇)\xi$ and deduce a relationship between the vorticity $\xi(x,y)$ and the streamlines.

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I am stuck on the bold parts. I am pretty sure that $$(\textbf u·∇) ψ=u\frac{\partial ψ}{\partial x}+v\frac{\partial ψ}{\partial y} $$ and $(\textbf u·∇)\xi=0$. But I don't know the relationships.

For the first bold part, is it just that the streamlines can be given by $ψ=\text{constant}$ for different constants because the stream function is constant on streamlines?

For the second bold bit, I have no idea.

Answer 1

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

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

Using the incompressiblity condition $\nabla \cdot \mathbb{u} = 0$ 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,

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

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

We can relate the stream function and vorticity, using the definition $\mathbb{\omega} = \nabla \times \mathbb{u}$ by

$$\xi = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} = \frac{\partial }{\partial x}\left(-\frac{\partial \psi}{\partial x}\right) - \frac{\partial }{\partial y}\left(\frac{\partial \psi}{\partial y}\right) = -\nabla^2 \psi.$$

In steady inviscid flow we have the Euler equation

$$\mathbb{u} \cdot \nabla \mathbb{u} = -\frac{1}{\rho} \nabla p.$$

Taking the curl of both sides we find

$$\mathbb{u} \cdot \nabla \mathbb{\mathbb{\omega}} + \mathbb{\omega} \cdot \nabla \mathbb{\mathbb{u}} = 0.$$

Since $\mathbb{\omega}$ has only the non-zero $z-$ component $\xi$ this reduces to

$$\mathbb{u} \cdot \nabla \xi = 0,$$

and, by the same argument for the stream function, te vorticity must be constant along a streamline.