We have the governing equations
for a three-dimensional problem,
the Navier-Stokes equations,
on the equation for the vorticity field.
Now it's time to be specific and talk
about the problem we're actually going to solve,
which is the flow past an infinite cylinder.
Let me start by drawing the cylinder.
Here's a cross-section of our infinite cylinder.
It's infinite in this direction
coming out of the light board,
this cross-section of the circular cylinder
has a radius of
R. That's our flow
is going to be coming into the cylinder.
If we draw the freestream velocity,
it will look like this one.
Let me draw the velocity field in red.
It's coming in this direction we choose.
This is the freestream velocity.
Now at this point we can draw a coordinate system.
We want to draw the x-axis in
the direction of the freestream velocity.
We draw the x-axis in this direction.
This will be x.
And we want to draw the y-axis
then perpendicular to the freestream velocity.
This is our y-axis.
The freestream velocity then is in the x-direction.
We can write the freestream velocity u_f,
which is a vector,
is equal to a constant U times
the unit vector in the x-direction,
because that's how we've defined the coordinate system.
This is our flow geometry.
For non turbulent flow,
the velocity field will never
go in the direction of the z-axis,
it will always stay in this plane.
The velocity field won't point in the z-direction,
but it also won't depend on the z-direction.
Every cross-section of the cylinder
will have exactly the same velocity field.
That's for a non turbulent problem.
If the fluid becomes turbulent,
then you get random motion and you can have motion in
z and the flow field itself can depend on z.
But we're going to be doing
non turbulent flows in this class.
In that case, then we can write the velocity field
u as a component in the x-direction,
which I'll called scalar u,
that depends only on x and y.
It may also depend on time for unsteady flows.
Here I'll leave the time-dependence out.
And this is in the x-direction.
Plus the velocity field in the y-direction, we call V,
which is a function of x and y,
and that's in the y-direction.
So j is the unit vector in the y-direction.
This is our velocity field.
Then there are two cases here.
There's the steady flow where
the velocity field will be independent of
time and there's the unsteady flow
where the velocity fields will depend on time.
For the steady flows,
we can say something else about these velocity fields.
If I draw how we expect
the velocity field to behave as it passes the cylinder,
we expect the velocity field to come
in and then avoid the cylinder,
so it's flow around the cylinder.
This is in the positive y-plane.
And in the negative y-plane,
it will come in and avoid the cylinder, like so.
This would be our expectation of the velocity field.
Here I've shifted a little bit,
but you get the idea.
What does that mean in terms of u and v?
Well, if we look at the velocity field at some fixed x,
but in the lower half of the plane,
if y is positive at negative y,
the x component is going to be identical whether you're
above a distance y or below a distance minus y.
So this is an even function of y,
so this will be u of x, y.
What about the vertical component of the velocity?
If we look at v at negative y.
Here it's going down and up.
Here it's going up and down.
So it changes sign.
This is equal to negative v of x and y.
So the y component of
the velocity is an odd function of y.
We will make use of this symmetry
to improve our computational time of
our program because then we only need to
solve for 1/2 of the plane.
The bottom is just satisfying this even odd symmetry.
Let me summarize.
We're going to start now to
specialize to our particular fluid problem,
the flow around an infinite circular cylinder.
We're going to look at a cross-section of the cylinder.
Consider only non turbulent flows,
at which point the velocity field is independent of
z and the vector
itself has a component
in either the x-direction or the y-direction.
There is no flow going in or out of the board.
I introduced the idea of a freestream velocity.
We're far away from the cylinder,
the velocity is flowing then with my definition of the x
coordinate only in
the x-direction with constant velocity.
That'll be true far in front of the cylinder,
far behind the cylinder,
far above the cylinder,
and far below the cylinder.
The velocity will satisfy the freestream condition.
I'm Jeff Chasnov.
Thanks for watching,
and I'll see you in the next video.