what we want to do here is once we have
made a miniature or an enlarged version
of some kind of fluid flow and we have
done so by scaling the main flow
parameters the Strouhal number the Froude
number the Euler number and the Reynolds
number once we have done this and we
make measurements on this tiny or
enlarged version of the flow how do we
translate those into the real flow
and in general what we want to do is to
save money the question is of course as
I explained before you have a very large
machine let's say an Airbus a380 this is
80 meters wide and 80 meters long 600
tons weight so it's pretty pretty large
beefy machine these are humans standing
below the wing of this airplane this is
the wing and here are the engines you
can see how small humans are in
comparison to the machine you make
modifications to this machine to make it
more efficient what do you do do you
make a new airplane every time you want
to try out something or do you try to
save the 200 million euros this machine
costs and build a model instead and test
on the model and then carry out the
modifications on the production machine
once you're certain is going to work
this applies all kinds of situation you
may take a look at a submarine instead
of building building a new
billion-dollar submarine you want to
just test it off using a model inside
the water channel and once you do that
when you have measurements force and
power measurements on this model how do
you carry them out to have the
equivalent on a real full-size machine
yeah so the question is this if I
reproduce the real flow on my 1 to 100
model car then how small or how large
will the drag force be in reality ok so
let's take a look at that the answer to
this to this problem is this in two
physically similar experiments in
experiments where you have carried out
the Reynolds number the Froude number the
Euler number
and the Strouhal number carry out the flow
parameters and those two physically
similar experiments then the force and
the power coefficients are the same so
let's take a look what is the force and
the power coefficient to figure out a
force coefficient we have to go back to
basics and we have steady flow you draw
a potato around your fluid flow and you
want to quantify the force the force is
the net sum of each term that is gonna
be the mass flow multiplied by the
velocity of the fluid the momentum of
the fluid and this is density times
velocity times area multiplied by
density so you could say that the net
force is proportional or grows together
where density times area times the
square of velocity and this is the basis
for figuring out for coming up with a
definition which is the definition of
the force coefficient the force
coefficient is the force the fluid
induced force that you are measuring
that you are interested in divided by
one-half of Rho times some reference
area times V squared okay so really
this relates the magnitude of F to some
amount of fluid momentum that is in the
path of the object and there are a few
remarks to make about us
the first remark is that this is a
definition there is not much to argue
about about this and you just have to
accept that it's being quantified this
way historically in fluid mechanics
second this F this with force is any
force that is generated by the fluid on
the object this could be the drag which
is typically in the direction of the
flow or the lift which is perpendicular
to the flow or any force at any angle so
the force in which you are interested
that is generated by the fluid on the
object s the area is arbitrary but it
must be representative so if you're
looking at the car you could take the
frontal area or the top area or the side
area you could take an area of the wheel
you
could take the area of a mirror it doesn't
matter as long as it is something that's
going to grow and shrink together with
the object you are looking at as you
scale it up and down
typically in industries there are some
conventions for which area to take so if
you take your car for example it is going
to be the front area if you take an
airplane it's going to be the top view of
the area of the wings and so on and so
forth and this is just by convention
again this is a definition this is not
pure physics this is just some
engineering scale parameter that we like
to quantify and again V like s is
somewhat arbitrary and typically we take
the velocity far away from the object so
the incoming velocity far away from
there from the object so say in the
pipe we could take the average velocity
and so every time in every industry or
in every area of fluid mechanics you are
going to find new conventions for which
parameters to put in there this one
half that you see at the bottom of the
fraction here this one have is just
arbitrary and conventional it could have
been one third it could have been PI
it's just a bad convention and we just
stick with it
okay there are other coefficients then
the force coefficient yeah we're gonna
talk about the power coefficient again
if you go back to basics you draw a
control volume around your flow you're
going to find that the net power
involved in a fluid flow is the sum the
net sum of incoming and outgoing mass
flow multiplied by the sum of specific
energies and so in this mass flow you're
gonna have velocity if this is going
good to be equal to Rho VA and in here
you have the square of velocity so it's
fair enough to say that power over here
it grows together it scales up and down
together with Rho VA yes and V squared
and so this is the basis again for
coming up with the definition of the
power coefficient which is power divided
by one-half or Rho SV cube now again
this is a definition there is nothing
much to argue about about this this is
the way it's usually done in the
industry to quantify
and scale power in fluid flows there
are many other parameters that you're going to
see in fluid throws and these are we saw
some of these previously in studying
pipe flow this is the pressure loss
coefficient you have the friction factor
you have the shear coefficient you have
all kinds of coefficients for the
fluid dynamicists love non-dimensional fluid flow
coefficients to relate measurements that
they make to the incoming parameters of
the flow and so this is the recipe again
to save 200 million dollars to save
yourself from building one complete new
machine every time you make a
modification the difficult part is the
first the second part actually first
part is the math you quantify the
non-dimensional parameters inside the
non-dimensional navier-stokes equation
these are best Strouhal the Froude the Euler
the Reynolds and also sometimes in
compressible flow the Mach number yes
the difficult part is to make miniature
physics in the lab or maximized if you
want to enlarged version or the physics
inside the lab and these involves in
theory reproducing all five of those
numbers yes but in practice only
reproducing the most important of those
because it turns out reproducing all
five is extremely difficult if not
completely impossible so you select in
those the most representative the ones
that are most important for the
situation you trying to study most often
in in engineering this is the Reynolds
number yes and you reproduce them in the
miniature or an enlarged version of the
flow you're looking at once you have
done this the hard work is over and you
can just enjoy the fact that you don't
have to build one new airplane every
time you make a modification you change
a tiny thing on your tiny model you make
a measurement and then once you make the
measurement you know that the force and
power coefficients on your miniature
version is the same as on the larger
airplane and this allows you to scale up
and down the forces that you have on the
small objects or enlarged object
compared to the real flow yes and then
what you do with the savings well you
of course spend them on something meaningful and purposeful
yeah okay so to illustrate now a little
bit what this looks like this is before
we had a good this here on this side
this is before we had a good
understanding of scaling in fact in
fluid mechanics we would build airplanes
and test airplanes inside the wind
tunnels we would put the actual full
airplane inside a gigantic wind tunnel
to be able to test whether it was any
good or not and we would carry out
modifications on the real thing and so
you would have real size airplanes in
the 1940s put inside enormous machines
which move the air around the airplane
so we can make measurements on those
most comfortably and so the machines are
some of them are still standing today
this is a wind tunnel at NASA and you
can see over there is a car at the
entrance of the wind tunnel this is the
entrance of the of the air inside the
wind tunnel and this whole huge
structure over there we see in another
photo which is here and this is
the inside of the wind tunnel with those
gigantic doors and you can see there's
no human on this picture but you can see
there are rail guards here and a
ladder here which gives you a sense of
the scale of this enormous machine
that is needed to move the flow around
the whole airplane because we did not
understand completely exactly how to
scale flows up and down of course it
would need monstrously powerful machines
to move and cool and heat up the air
inside those machines so this is the fan
that is needed to move such a large
amount of air and now of course this is
over we have very good understanding of
the effects of scale on on models and so
when you develop an airplane when you
develop a submarine what you do is you
make a model of that and you test you
make measurement on the model and then
you scale those up and down now this is
I believe in f-35 and this is an f-16
both are American airplanes because NASA
publishes lots of photos in the public
domain that are available for people to
reuse and I like this picture pretty
well pretty much because we see pretty
well what the technique is to measure the
flow which is called particle image
velocimetry and this consists in
you seeding smoke into the wind tunnel
and then lighting up using laser sheets
which here come from the top so they cut
across the flow they come from the top
and they land on the bottom here those
laser sheets light up in a very bright
manner the smoke as it passes through
and above the through the tunnel and
above and around the model and then we
take videos or photos of this smoke
patterns and we try to reconstruct the
velocities inside the sheets that we see
through through those photos here so
very well-known a very well developed
technique to measure fluid flow velocity
inside of wind tunnels you may if
you're really into scaling you may make
models that are smaller and smaller and
smaller until you need you need optical
instruments to to observe their surface
but this this becomes harder and harder
as you do that because the treatment of
the surface becomes more and more
critical as you that this is a limit of
course to how much scale you can
safely carry out but again summing
summarizing the the main contents of
this you quantify the main flow
parameters and then you do the hard
things which is to reproduce the most
important of those flow parameters in
your miniature or enlarged version of
the flow yes
this is very hard and once you've done
this it pays off because the force in
power coefficients are the same and you
can spend all the money that's left over
doing other interesting things
