[MUSIC]
So we have our Vorticity equation in three dimensions.
Now we want to write the vorticity equation in two dimensions,
which is the equation will be working with for our flow geometry.
So we have our cylinder in our coordinate system.
So this is x, this is y.
We're putting the cylinder in the middle of the coordinate system.
And right now I might as well introduce the polar coordinates that
we'll be modifying later and using log polar coordinates.
So if we have a point here, the polar coordinates then has an angle theta and
a distance r from the origin, which is the center of the cylinder.
And has two unit vectors, has a unit vector in the r direction,
which we call r hat, and a unit vector in increasing theta direction,
which we call theta hat, okay?
So, we have a two dimensional problem and
I want to write that problem in polar coordinates.
So, we'll have our velocity field then we'll have a component in the r direction.
U sub r, r theta, r hat plus a component in the theta direction,
which I will call U sub theta, which depends on r and
theta and that will be in the theta hat direction.
So, our velocity field in polar coordinates then has two
components, U sub r and U sub theta, okay?
What does the vorticity field look like in polar coordinates?
So the vorticity field,
omega is equal to the curl of the velocity field.
If you go back to your vector calculus notes,
we have the curl of a vector field written in polar coordinates.
I don't need to re derive that here.
This is given by one over r, one over the radius
times d/dr of r,U theta minus dur/d theta and
there's only a single component and
this is in the direction perpendicular to
the plane, so it's in the k direction.
So the vorticity field in a two dimensional flow has
only a single non zero component which is in the K direction.
So I'm going to be calling that the scale of vorticity omega times k, okay?
So, omega here is the scalar vorticity.
So now let's look at our full vorticity equation and
see how it simplifies then in two dimensions.
If we look at this vortex stretching term,
this omega dot del U term because the vorticity
field has only a single component in the Z direction,
this becomes the scale of vorticity omega times dU vector dz,
but the velocity field is independent of the Z coordinate.
So, this is strictly zero, okay?
So this term then, which is so crucial for
turbulence is zero in the two dimensional flow.
Okay, so what else do we need to do?
Well, we would like to write the gradient operator in polar coordinates.
So, the gradient in polar coordinates
is r hat d/dr plus theta hat one
over r, d/d theta, okay?
And then we need the laplacian in polar coordinates.
So the laplacean in polar
coordinates is del squared
is equal to one over r squared
times r d/dr, r d/dr.
So this is one of the many ways you can write the laplacian
plus d squared, d theta squared, okay?
So that's the laplacian in polar coordinates.
Again, you can go back to my vector calculus course and
lecture notes and I derived this form of the laplacian.
Okay, now we're ready to write the vorticity equation.
There's only the Z component.
So, the vorticity equation becomes d omega/ dt.
So, this is now the scale of vorticity and then the z component here.
So, it's going to be U in polar coordinates dotted into
the gradient in polar coordinates.
So r hat dot r hat is one.
r hat dot theta hat is zero.
Theta hat dot r hat is zero.
And theta hat dot theta hat is one.
So we have a U dot del.
So, we have a U sub r,
U dot del, d/dr of omega.
So, d omega dr plus
U sub theta divided by r.
So plus the use of feta divided by r times d omega/ d theta.
So, that's the U dot del omega equation.
And that's equal to new del squared omega.
So, it's a new over r squared.
And then we have the laplacian, r d/dr,
r d/dr plus d squared d theta squared.
All operating on the scalar vorticity.
And then this becomes our single equation.
So, the three component vorticity equation for
three dimensional turbulence in a two dimensional flow
reduces to a single component equation for the scalar vorticity.
A great simplification when you can change three equations into one equation.
So let me summarize, we're trying to write the vorticity
equation in three dimensions using polar coordinates and
a velocity field that doesn't depend on Z.
And it also doesn't have any vector component in the Z direction.
The three dimensional vorticity field reduces to a scale of vorticity
omega that is only in the Z direction, in the perpendicular direction.
So, then we need to only write this equation,
the Z component of this equation,
we make use of the gradient in polar coordinates and
we make use of the laplacian in polar coordinates.
I'm Jeff Chazanov.
Thanks for watching and I'll see you in the next video.