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Corner eddies

Corner eddies

\[\newcommand{\p}{\partial}\]

Next we want to describe a behavior of creeping flow that occurs when one encounters a (concave) corner; below some threshold angle, eddies start to appear. Mathematically, there are an infinite family of increasingly smaller nested eddies as one approaches the corner; of couse, physically these will of course get cut off by microphysics, and at any rate the eddies rapidly become too small to be observed.

Setup: the vorticity equations

We consider 2-dimensional creeping flow inside a wedge open at some angle \(2 \alpha\), symmetric about \(\theta = 0\) in the \(r-\theta\) plane. We leave aside what is driving the flow – in one experiment, it is a rotating cylinder just outside of the wedge – and assume the wedge extends out to infinity. Thus the equations are

\[\begin{split} & - \frac{1}{\rho} \p_r p + \nu \nabla^2 u = 0 \\ & {\vec\nabla}\cdot{\vec u} = 0\\ & u_r \Big|_{\theta = \pm \alpha} = 0\\ & u_{\theta} \Big|_{\theta = \pm \alpha} = 0 \end{split}\]

Because we are working in two dimensions, the incompressibility condition in a simply connected domain means that we can write, in Cartesian coordinates, the velocity in terms of the stream function \(\psi\):

\[\begin{split} & u_x = \p_y \psi\\ & u_y = - \p_x \psi \end{split}\]

In polar coordinates we can write:

\[\begin{split} & u_r = \frac{1}{r} \p_{\theta} \psi\\ & u_{\theta} = - \p_r \psi \end{split}\]

The vorticity in two dimensions is defined as

\[ \omega = \p_x u_y - \p_y u_x = - \nabla^2 \psi \]

If we imagined this fluid was trapped on the \(x-y\) plane, we wold have \({\vec\nabla} \times {\vec u} = -\nabla^2 \psi\ {\hat z}\).

We can eliminate the pressure in this case by taking the curl of the creeping flow equations to find:

\[\begin{split} - \nu \nabla^2 \nabla^2 \psi & = - \nu \left(\frac{1}{r}\frac{\p}{\p r}\left(r\frac{\p}{\p r}\right) + \frac{1}{r^2}\frac{\p^2}{\p\theta^2}\right)^2 \psi\\ & = -\nu \left(\frac{\p^2}{\p r^2} + \frac{1}{r}\frac{\p}{\p r} + \frac{1}{r^2}\frac{\p^2}{\p\theta^2}\right)^2 \psi = 0 \end{split}\]

So now we have one scalar equation for \(\psi\).

General Solution

Since every term in this equation scales the same under rescalings of \(r\), we will consider solutions of the form \(\psi = r^{\ell} f(\theta)\). This is a guess, it makes sense because in the equation above, every term will be proportional to \(r^{\ell - 4}\). Plugging this in and dropping the \(-\nu\) prefactor, we find:

\[ \left((\ell - 2)^2 + \p_{\theta}^2\right)\left(\ell^2 + \p_{\theta}^2\right) f(\theta) r^{\ell - 4} \]

This means that either \((\ell^2 + \p_{\theta}^2)f = 0\) or \(((\ell - 2)^2 + \p_{\theta}^2) f = 0\), so we have the following general solution for fixed \(\ell\):

\[ \psi = r^{\ell} \left(A \cos\ell\theta + B \sin\ell\theta + C \cos (\ell - 2)\theta + D \sin(\ell - 2)\theta\right) \]

Here we have four unknowns, reflecting the fact that we have set up a fourth order differential equation. Two of them will be fixed by the two boundary conditions. We will fix two more by considering flows in which \(u_r\) changes sign under \(\theta \to - \theta\), which eliminates the sine terms (recall \(u_r = - \frac{1}{r} \p_{\theta} \psi\).) This will ensure the solutions inherit he symmetry of the boundary forcing. Thus we set \(B = D = 0\). Finally, we demand \(u_r = u_{\theta} = 0\) at \(\theta = \pm \alpha\), which translates to:

\[\begin{split} & u_r = - \frac{1}{r} \p_{\theta} \psi\Big|_{\theta = \pm\alpha} = \pm r^{\ell} \left(A \ell \sin \ell \alpha + C (\ell - 2) \sin (\ell - 2) \alpha\right) = 0\\ & u_{\theta}\Big|_{\theta = \pm\alpha} = - \frac{\p}{\p r} \psi = \ell r^{\ell -1} \left(A \cos \ell \alpha + C \cos(\ell - 2)\alpha\right) = 0 \end{split}\]

Moving the terms proportional to \(C\) to the othe side of the equation, and then dividing one equation by the other, we find:

\[ \ell \tan\ell \alpha = (\ell - 2) \tan (\ell - 2) \alpha \]

Next, we combine half angle formulae:

\[ \tan x = \frac{\sin 2x}{1 + \cos 2x} = \frac{1 - \cos 2x}{\sin 2x} \]

with addition formulae

\[\begin{split} \cos(2(\ell - 2)\alpha) & = \cos (2(\ell - 1)\alpha - 2\alpha) = \cos 2(\ell - 1)\alpha \cos 2\alpha + \sin 2(\ell - 1)\alpha \sin 2\alpha\\ \sin(2(\ell - 2)\alpha) & = \sin (2(\ell - 1)\alpha - 2\alpha) = \sin 2(\ell - 1)\alpha \cos 2\alpha - \cos 2(\ell - 1)\alpha \sin 2\alpha \end{split}\]

to find:

\[ \frac{sin 2(\ell - 1) \alpha}{2(\ell -1)\alpha} = - \frac{\sin 2\alpha}{2\alpha} \]

which is a condition on \(\ell\).

When do corner eddies form?

Now in this equation, \(\ell\) can be real or imaginary. If it is real, solutions will flow simply in once side of the wedge and ou the other.

If it is imaginary, the linearity of the differential equation \(\nabla^4 \psi = 0\) means that we can take the real or imaginary part of the solution and still get a solution. For a solution proportional to \(r^{a + i b}\), we can write this as

\[ \psi = r^a e^{i b \ln r} f(\theta) = r^a \left(\cos b \ln r + i b \sin \ln r\right) f(\theta) \]

Consider \(u_{\theta}\) at \(\theta = 0\) generated by the real part of the above: clearly this oscillates with increasing rapidity as \(r \to 0\), indicating an infinite sequence of increasingly smaller counterrotating eddies.

The condition on \(\ell\) is prohibitive to solve analytically. We can, however, ask when real solutions no longer become available. The question of whether there are simultaneous solutions comes by first plotting \(\sin y/y\) as a function of \(y\). For \(y = 2\alpha\), \(0 \leq \alpha \leq \pi\), we find the value of \(\sin y/y\) and mark the vertical axis at value \(Q\). Mark it also at \(-Q\); if this is larger than the minimum of \(\sin y/y\), there will be one or more points \(y_i\) on the curve \(\sin y/y\) with this value, we can just solve \(2(\ell -1) \alpha = y_i\) for \(\ell\).

However, \(\sin y/y\) has a maximum of \(1\) at the origin and a minimum of \(\sim - .217\). Thus there will be no real solutions if \(\sin 2\alpha/2\alpha > .217\), which happens at \(2 \alpha \sim 146.3^{\circ}\). Thus there is a threshold angle below which we form “corner eddies”.

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General properties of creeping flow