Velocity Averaging and Mean Value Property

Point Vortex Dynamics

For our point vortex case, we have:

$$-\Delta \psi = \omega \implies \psi = - \Delta ^{-1}\omega$$

Where $\omega = \sum_{k=1}^{N} \Gamma_{k}\delta_{k}$ where $\delta_{k}$ is a delta function at the position of the $k^{th}$ point vortex. And as $\Delta$ is a linear operator, we have:

$$\psi_{1} = - \Delta ^{-1}\delta_{k_{1}}, \quad \psi_{2} = - \Delta ^{-1}\delta_{k_{2}}, \quad\dots$$ $$\therefore \psi = \sum_{k=1}^{N} \psi_{k}$$

That means on a small circular patch around point vortex $k_{1}$ we can write the global stream function as:

$$\psi = \psi_{1} + \sum_{k=2}^{N}\psi_{k}$$

$\psi_{1}$ is singular and blows up at the center of point vortex. But $h = \sum_{k=2}^{N}\psi_{k}$ is a smooth harmonic scalar function on this small patch.

$$\therefore \psi = \psi_{1} + h \implies \boxed{h = \psi - \psi_{s}} \tag{1}$$

where $\psi_{s}$ is the singular stream function part of a point vortex inside the patch $B_{r}$ centered at $a$ (position of the point vortex).

In the classical term, Eq. $(2)$ can be interpreted as $h$ is the actual stream function value obtained as a combination of Robin’s function and pairwise interaction of Green’s function (excluding the self interaction).

So for correct Advection, we want to get rid of the $\psi_{s}$ part and then have the velocity at the point vortex (point $a$) as:

$$\vec{u}(a) = J \nabla h(a)$$

For this we need the $\nabla h(a)$.

Mean Value Property of the Harmonic Function

For a harmonic, function $h$ the value at a point $x_{0}$ is given as:

$$h(x_{0}) = \frac{1}{\lvert B_{r} \rvert }\int_{B_{r(x_{0})}}h(x)\,dx$$

where $B_{r}$ is a ball of radius $r$ around the point $x_{0}$. Similarly this holds for the boundary of the ball as well.

$$h(x_{0}) = \frac{1}{\lvert \partial B_{r} \rvert}\int_{\partial B_{r(x_{0})}}h\,dS$$

In this case we are integrating over a sphere or a circle.

For 2D case, integral over the circle:

$$h(x_{0}) = \frac{1}{2\pi}\int_{0}^{2\pi}h(x_{0} + r(\cos\theta, \sin\theta))\,d\theta$$

Consequently, we can take gradient of $h$ in terms of $x$ as:

$$\nabla h(x_{0}) = \frac{1}{2\pi}\int_{0}^{2\pi} \nabla h(x_{0} + r(\cos\theta, \sin\theta))\,d\theta$$

Equivalently,

$$\boxed{\nabla h(a) = \frac{1}{\lvert \partial B_{r} \rvert }\oint_{\partial B_{r}}\nabla h(x)\,dx} \tag{2}$$

where $B_{r}$ is a small circular patch around point $a$ with radius $r$.

Velocity Averaging

Now, from Eq. $(1)$ and $(2)$:

$$\nabla h(a) = \frac{1}{\lvert \partial B_{r} \rvert }\oint_{\partial B_{r}}\nabla h(x)dx$$ $$= \frac{1}{\lvert \partial B_{r} \rvert }\oint_{\partial B_{r}}\nabla(\psi - \psi_{s})\,dx$$ $$\nabla h(a)= \frac{1}{\lvert \partial B_{r} \rvert }\oint_{\partial B_{r}}\nabla\psi\,dx - \frac{1}{\lvert \partial B_{r} \rvert }\oint_{\partial B_{r}}\nabla\psi_{s}\,dx \tag{3}$$

One important thing about $\psi_{s}$ we know is that it’s radially symmetrical around a point $a$. For a small enough patch, in continuum case it’s equal to $\psi_{s} = -\frac{\Gamma}{2\pi}\log \lvert x - a \rvert$.

Given that $B_{r}$ is a circular patch with center $a$, $\psi_{s}$ is constant over $\partial B_{r}$ i.e. it’s a contour line, and $\psi_{s}(r)$ is constant for any value of $r\leq R$ where $R$ is the radius of patch $B_{r}$. Therefore, magnitude of $\nabla \psi_{s}$ is constant over $\partial B_{r}$ and direction is pointing radially outwards in the direction of normal $\vec{n}$. Therefore,

$$\oint_{\partial B_{r}}\nabla \psi_{s}\,dx = \oint_{\partial B_{r}} C\vec{n}\,dx = C \oint_{\partial B_{r}}\vec{n}\,dx = 0$$ $$\therefore \boxed{\oint_{\partial B_{r}} \nabla \psi_{s}\,dx = 0} \tag{4}$$

Now, Eq. $(3)$ becomes:

$$\nabla h(a) = \frac{1}{\lvert \partial B_{r} \rvert }\oint_{\partial B_{r}}\nabla \psi\,dx$$ $$\boxed{\vec{u}(a) = J\nabla h(a) = \frac{1}{\lvert \partial B_{r} \rvert }\oint_{\partial B_{r}} J\nabla \psi\,dx} \tag{5}$$