[MUSIC]
We now have our full governing equations for our two dimensional problem.
An equation for the stream function that depends on the scale of vorticity and
an equation for the scale of vorticity with the time derivative in it.
That depends on the stream function.
I've replaced the velocity here by the derivatives of the stream function.
These equations have a parameter mew,
which is the kinematic viscosity that has units.
And in addition we have the external parameters
that define the flow, the free stream velocity,
U, capital U, and the radius of the cylinder.
So we have these three parameters.
So what one usually does before one does a numerical calculation is to non
dimensionalize the equations to reduce the number of free parameters.
There is a theorem called the Buckingham pie theorem that says we can
reduce the number of free parameters by
the number of independent dimensions in the problem.
The interdependent dimensions in this problem are a length and a time.
So let's see how we do the dimensional
analysis, dimensional analysis.
Okay, so the way to proceed is first to figure out
the dimensions of all the terms in the equation.
So we have time derivative.
I'm going to use this bracket notation to denote the dimension.
So that means the dimension of the time derivative is a 1 over time and
the dimension of a spatial derivative.
Then there's 1 over length, l to the -1.
We have this Laplacian, which is the second,
the square, the second derivative.
So we have the dimension of del squared is the length to the -2.
We have, what else do we have?
We have r that's a dimension of length, right?
The angle theta is a dimensionless variable.
Theta doesn't have any dimension.
And then what else can we figure out here?
So if we look at this second equation, there's an omega on every term,
so the dimensions of omega will cancel.
So we have a time derivative here, so we have a 1 over time and
then we have a del squared here.
So we have a 1 over length squared.
So the viscosity then will have to be a length squared divided by
time in order to get the dimensions of the equation right.
Length square, time to the -1.
Remember we're working under the physical principle that every term in
a valid equation has to have exactly the same dimension.
Okay, so that's the viscosity.
What else can we figure out here?
We can figure out the stream function from this second equation.
So we have the scale of vorticity cancels here.
We have a 1/r and a derivative, so 1 over length squared and a 1 over time.
So we bring the length squared on the left, we have a length squared over time.
So the stream function has to have units of length squared over time.
Length square t to the -1.
Okay, that gives us one more quantity, which is the scale of vorticity.
Now we can go to the first equation.
So the stream function is length squared over time.
The del squared is 1 over length squared.
So this is a 1 over time.
So the vorticity then has units of 1 over time.
Okay, so that's the dimensional analysis of every term in this equation.
Then we have two more parameters.
We have R which is the radius of the cylinder.
The dimension of [R] is length.
And then we have the free stream velocity U, which is a velocity.
That's a length per unit time or a lt to the -1.
Okay, so we can choose units of length and units of time.
So for length will choose to measure length in the radius
of the cylinder and units of the radius of the cylinder.
So in our dimensionless equations then the radius of the cylinder will be
1 because we've measured length in terms of the radius of the cylinder.
We can measure time using [U] and l.
So if we take R and divided by U we get a time.
So we can measure time in units of radius
of the cylinder provided by the free stream velocity.
Okay, so that means we take our equations,
any R's in this equation.
We non dimensionalize using capital R.
Any time in this equation we non dimensionalize using R/U,
you work through all the equation.
Everything will cancel except that this mew then will become a non
dimensional parameter.
So this mew.
Well, when we go through the non dimensionalization procedure,
this mew, which is a length squared over time.
We'll go to a mew and normalized so that the length squared over time cancels.
So we get a length, to get a length squared over time.
We have a U times an R, right, a length over time,
times of length is a length squared over time.
And this then becomes our non dimensional parameter.
So this is related to the Reynolds number.
So we define a Reynolds number.
This is Re for Reynolds number.
And traditionally in this problem of flow around the cylinder,
you define the Reynolds number in terms of the diameter of the cross-section,
not the radius.
So the non dimensional quantity which becomes
the Reynolds number will be 2, twice the radius.
So 2UR divided by the viscosity.
And that's the traditional definition for Reynolds number.
Okay, so then what become, what is our governing equations?
Well, the first equation remains the same.
Del squared Psi equals minus omega except that now del
squared is a dimensionless differential operator.
Psi is a dimensionless stream function and
omega is a dimensionless scale of vorticity.
The second equation becomes d omega/ dt.
Now dimensionless plus 1/r
times the same factor here,
d Psi /d theta of the omega
dr- d Psi/ dr d omega / d theta.
And the right hand side instead of the viscosity now we have the Reynolds number.
But so now mew goes to mew over UR, which is two times r,
which is 1 over the Reynolds number.
Except that we have to cancel the factor of 2.
So this becomes a 2 over the Reynolds number times del squared omega,
where del squared and omega are now dimensionless variables.
So the non dimensionalization of these equations is relatively straightforward.
You figure out what all of the dimensions of the variables are.
And then we have the two extra parameters.
We have the radius of the cylinder and we have the free stream velocity.
So we make use of the radius of the cylinder for
the length and the radius divided by the free stream velocity for the time.
We non dimensionalize all the variables in terms of our length scale and
our time scale.
And what that means is this viscosity here gets converted into
2 divided by the Reynolds number.
So now we have our full non dimensional equations.
This plus the radius of the cylinder is 1.
The free stream velocity is 1 in the X direction.
I'm Jeff Khazanov.
Thanks for watching and I'll see you in the next video.