We now want to consider the unsteady flow problem.
When the flow becomes unsteady,
the symmetry between the upper half
and the lower half of the plane gets
broken and we have to consider
the full range of the polar angle.
To do that,
we still need boundary conditions on theta.
The boundary condition we're going to use is
called periodic boundary conditions.
Let me show you how that works.
If we draw our domain.
This is not necessarily the cylinder,
but this is just some circle
with the cylinder in the center.
I can illustrate the periodic boundary conditions.
This is X and this is Y.
To implement Periodic Boundary Conditions,
we know that the flow field is
periodic and the angle Theta,
because after Theta goes from zero to two Pi,
it repeats itself, so,
the boundary condition we can use is to start
one grid point below the zero angle.
This will be our first grid point
and then when we cross the x-axis here,
this will be our second grid point.
There are M grid points in total so we're going to go
around and finish on the x-axis after M grid points.
This first grid point will also be our m minus
1 grid point and
the second grid point will also be our m grid point.
We say that the first grid point and
the last grid point are what's called ghost points.
J equals 1 and j equals m,
are what we call ghost points because they're repeated.
The first grid point is
the same as the m minus 1 grid point.
The second grid point is the same as the m grid point.
These are our boundaries.
We have the polar angle sub j then is going to start
at minus h. It'll be j minus 2 times h,
when j is equal to 1,
meaning starting at minus h 2,
2 m. That will be our polar angle.
What are the boundary conditions?
We want a boundary condition on the stream function.
This is for all of
the radial coordinate points and
the first point in the Theta angle.
That's going to be equal to the stream function for
all the radial points and the m minus
1 point in the Theta angle.
This is our boundary condition on
the first value of
Theta and the same is true for the vorticity.
What about the last value of Theta?
Psi for all the radial point at the last value
of Theta is going to be the
same as the second value of Theta.
Psi of i comma two and the same for the vorticity.
Those are our boundary conditions
for the first Theta value,
and for the last Theta value.
What is our h here?
What is the grid spacing?
We know that we have m points in
Theta and then when we go around endpoints,
we should get to two Pi.
We know that two Pi then has to be equal to
m minus 2 times h. After starting one below,
when we go around endpoints,
we need to end up at two Pi.
That gives us our value of h.
Our grid spacing in this case is going to be two Pi.
Now, no longer Pi,
but two Pi divided by the number of
grid points we choose in Theta minus 2.
Let me summarize.
Because we're solving now the unsteady flow problem,
we have a broken symmetry
between the upper and the lower planes.
In fact, we're going to be simulating vortex shedding.
A vortex coming off of
the upper half of the cylinder will be
followed by a vortex shattered
off the lower half of the cylinder.
There's no symmetry that we can
exploit to reduce the computational domain by two,
so we have to solve for the whole computational domain.
To still have boundary conditions in the Theta angle,
we choose periodic boundary conditions.
The first grid point in Theta is going to be
the same value as the m minus 1 grid point in
Theta and the m grid point in Theta is going to
be the same value as the second grid point in Theta.
That gives us our four boundary conditions.
I'm Jeff Chasnov,
thanks for watching, and I'll see you in the next video.