Consider these two cases.
There is fluid flowing over a sphere and the fluid can either flow at a slow speed, or
it can flow at a fast speed, and in the slow speed it is nice laminar flow profile.
As a high speed there is turbulent regime in the back.
Lets say you wanted to calculate the force of drag in both cases.
Now there is no way to characterize the force of drag for this turbulent flow.
There is no way to integrate the Novier-Stokes equations and calculate the drag force.
What you could do is actually run some experiments and measure to force of drag in both cases.
Thus you might have some way to measure the fluid flow, the velocity from left to right,
and you might note that things like the density and the viscosity of the fluid.
In that case you can just measure the forces of drag as a function of these other parameters.
What you could do is make a graph of the drag force as a function of velocity.
So you would run the experiment at a low velocity and run it at a high velocity.
Pleasurably you would get some curve force of 0 at a low velocity.
You might see some increasing curve like this.
What you would do is fit a line to this data or curve of some sort and you might do this
with a sphere with a diameter of 1 cm.
Then what you would want to do is explore what would happen if I change it to a 10 cm
sphere and you would get another curve.
So in this case these data points represent 10 cm, and you fit a line to these, and you
might will want to do something at 1 mm.
Here is what the data might look like for a smaller sphere.
So presumably you might get a bigger sphere and you would expect to have a higher drag
force.
Then what you might be interested in is not the force as a function of velocity.
You are looking at a bunch of different fluids.
Lets see how the force varies with the viscosity of the fluid.
So we will do an experiment.
Lets say we have a fixed diameter, which we are going to say 1 cm sphere, and we are going
to collect data as a function of viscosity.
We are going to do it for a couple of different experiments.
What you might find is some trend, and use a velocity of 10 cm/s.
You draw a curve and so here is a force as a function of viscosity.
So what happens if I have a higher velocity or a lower velocity in the same diameter.
So presumably what we would see.
Is at a small velocity the drag force would be small and again you increase the velocity
and if we increase the viscosity I would anticipate that the force of drag would be greater.
So we get more experimental data.
Notice though that all of these experimental points are only for 1 diameter sphere.
In this case it was 1 cm.
So we are doing a whole lot of experiments, and we want to figure out if what happens
if the diameter now is 10 cm.
Well in this case you have to go back and measure all of these velocities again using
a sphere that was 10 times bigger.
You would calculate the force in that case as a function of viscosity.
So we are doing a whole lot of experiments just to get the drag force around a simple
geometry like a sphere.
Then you might say well not only is the viscosity significant.
What happens if we look at the force as a function of the density of the fluid.
It is hard to say of the back what this relationship might be.
Does a denser fluid have a bigger drag, a smaller density.
If we look at this now how does the density compare of we had given fluid with a certain
density and a higher viscosity or high density and a low viscosity.
You can imagine you would have 100s and 100s of experiments that you could possibly run
to explore the interactions between all of these variables.
As it terms out we can combine all of those data based on viscosity, Re number, and diameter
of the sphere, and the density of the fluid.
We could make a graph of what is known as the coefficient of drag as a function of what
is known as the Re number.
Both of these are dimensionless parameters they don't have dimensions like meters or
kilograms.
The coefficient of drag is equal to the force of drag divided by 1/2 times the area of the
sphere times the density times the velocity of the fluid squared.
So this area of the sphere is the cross sectional area that is exposed to the fluid.
So the area in this case is pi over 4 times the diameter squared.
In the Re number is equal to the density of the fluid, times the velocity, times the diameter
of the sphere, divided by the viscosity of the fluid.
So lets say you did a whole lot of experiments and in each experiment you measured the force
of drag and you look at a particular diameter sphere, particular cross-sectional area, particular
density of the fluid, and a particular velocity.
So if you know all of these parameters you can calculate the coefficient of drag.
You also calculate for a given experiment you know the density of the fluid, its velocity,
its diameter of the sphere, the viscosity of the fluid.
You could calculate what is known as the Re number.
We make a graph of this and it turns out for a sphere.
What you would absorb is some decrease in the coefficient of drag and in some independent
region where the coefficient of drag does not change much with the Re numbers.
Then it decreases due to the changes in the fluid flow behavior around the sphere, and
then it increases again.
So lets say we did that, and say that the blue would indicate a certain diameter, and
if you do this again we will double the diameter.
So now we are looking at a bigger sphere.
We are going to vary the velocity the density of the fluids, and different types pf fluids.
If you graph it this way.
What it turns out these data points fall right on top of each other.
They are very, very close to one another.
You see this exact same trend.
This might be a smaller sphere or it might have a more viscous fluid what have you, but
you calculate the coefficient of drag and the Re number, and you do it again.
We will say alright here an even bigger sphere and in this case we are going to do it in
a very thick fluid, and in this case it turns out the data points all fall on top of each
other.
What you are able to do is collapse all of these different experiments into a single
curve.
What that allows you to do is extrapolate now.
Lets say we have a diameter sphere that we have not tested, and we have tested 1 cm,
10 cm, 100 cm, well lets say we have a 50 cm diameter sphere.
Some strange fluid with a different viscosity.
You haven't actually done any experiments.
You just know the diameter of the sphere, the velocity, the density, and the viscosity
of the fluid.
What we ultimately want to know is the drag force on this sphere.
Lets say the Re number is in this region.
You might be something like 100,000 for this Re number.
So I will draw a line.
What that means is in this case there is a particular coefficient of drag associated
with that, and this coefficient of drag might be 0.6.
So if you know the velocity of all these other fluid diameters.
You have not even done the experiment yet, but you can calculate the force of drag.