Let me use this video to summarize
now all of our boundary conditions.
Let me draw the rectangle.
This is our radial direction, cosine,
and this is our polar angle direction,
this is theta,
sub j. Cosign variable goes from
zero to it's
maximum value theta n grid points in cosine,
this goes to cosine sub n. The theta variable goes
from zero on the mid line behind the cylinder,
to Pi on the mid line in front of the cylinder.
What are the boundary conditions?
When theta equals zero,
this is on the mid line,
we know that the stream function and
vorticity are zero on the mid line.
This boundary condition on this side of
the rectangle would be psi.
We have the first grid point in
theta and all the grid points in cosine.
That means i_1,
and that's supposed to be zero,
but also true for the vorticity,
the vorticity i_1 is also zero.
This is the mid-line behind the cylinder.
The top side of
the rectangle is the mid-line in front of the cylinder,
and again the fields are zero.
This is psi i.
Then we have m points in the theta direction,
this is Psi im equals omega im also equal to zero.
Those are the two mid lines of the flow.
What about this left side of the rectangle?
This corresponds to psi equal to zero.
This is the cylinder boundary condition.
We know that the stream function
is zero on the cylinder,
so that means the stream function on the cylinder means
the first grid point for all angle j has to be zero.
What about the vorticity on the cylinder?
That was this complicated boundary condition
which we derived.
We have the vorticity on the cylinder,
one comma j is equal to one over two times h squared,
so h is the grid spacing times
the stream function two points away from the cylinder,
minus eight times the stream function
one point away from the cylinder.
This is the most sophisticated boundary condition
which generates
the vorticity on the cylinder by
computing the difference between these stream functions,
one and two grid points away from the cylinder.
You have a division by h squared here,
h is very small that
they'll be balanced with side being relatively small.
We know psi is zero on the cylinder.
This is a crucial boundary condition.
Finally, on the right side of the rectangle,
this is the maximum value of cosine.
With the farthest distance away from the cylinder,
that's the free stream boundary condition.
The free stream boundary condition on the stream function
will be psi n comma j,
because we're at the last grid point,
and the free stream boundary condition is e to
the psi and times sine theta j.
It depends on the polar angle.
But we're at the farthest distance
away from the cylinder.
We take the simple boundary condition on
the vorticity in the freestream to be zero.
These are all of the boundary conditions.
We have a boundary condition on the stream function,
and vorticity on the mid line of the flow,
right behind the flow,
in front of the flow and on the cylinder,
and in the free stream.
This is what you need to implement in your code.
I'm Jeff Chasnov.
Thanks for watching,
and I'll see you in the next video.