in this problem we try to reproduce the
effect of turbulence around a real car
which is represented here using a model
car which is represented there
this model car is twenty times smaller
and the question we try to answer is
at what speed should the air flow around
model car to be able to simulate what is
happening around the real car which is here
the answer to this is found using
the Reynolds number because the Reynolds
number dictates the amount of
turbulence that we're gonna have and it
describes the importance, the role of
viscosity in the flow so what we do is
we write the Reynolds number
for the real car and we set it equal to
the Reynolds number for the model car
and I'm going to call the real car A
and the model car B and I advise you to
do the same things to be very careful
with how you label different situations
this is because situations get
increasingly complex as you mix
velocities and temperatures and scales
as you compare different situations so
always be very rigorous in how you label
the situations well the Reynolds number
around the real car is Rho V L over
mu so the density around the real car
times the velocity coming into the real
car
times some representative length of the
real car / viscosity and this we set
equal to the same parameters but for the
model will be V b lb / ub and in here
what we want to calculate what we want
to isolate here is V B the velocity of
the model car so the question is what is
V B well this is not a difficult mathematical
problem we take V B and write it by
itself on the left side and then we have
to rearrange and we sort out all
different terms so VB is equal to VA
times a series of factors and
these factors are […]
[…]
this case we are having the same air
flow over both cars the real one and the
model one and so the density of the air
and the viscosity of air will be the
same and so we can simply replace VA by
its original value Rho a over Rho B is
going to be 1 and LA over LB
this is where the labeling is useful la
is the length of the car lb is the
length of the model so la over lb is 20
okay then mu B mu a that's 1 and so we
have V be the velocity that we need
around the tiny car to be able to
simulate the flow around the real car
here is 20 times VA now this is quite a
disappointing result because Theory
tells us it's very easy if you want if
you want to reproduce the flow around an
object with the model then all you need
to do is reproduce the same Reynolds number
and here you see that it's not that easy
if the real car down here if the real
car here goes at 100 kilometres per hour and you want to
reproduce that flow with a tiny model
with a 1 to 20 model you need to go 20
times faster 20 times faster is 2,000
kilometers per hour and clearly this is
not feasible so there are different ways
around this problem that fluid
dynamicists use one way is to use the
other parameters in this equation so you
can play a little bit with this fraction
here and that fraction in there if you
adjust the properties of the fluid
around the model car so in this case you
want this fraction to be smaller to
basically compensate the factor 20 and
so if you increase the density B
compared to density a then you're gonna
get away with some of that increasing
density B typically is done by
cooling down the fluid and so having
very dense fluid very dense air cooler
will help you compensate for this and
changing the temperature of the air will
also change the ratio of viscosity as
you cool down on the air its viscosity will increase and so the ratio of mu B
to mu a will also increase, bigger than
one nevertheless this is not enough to
compensate for a factor 20 maybe with
good quality very powerful
equipment you can cool down the air so
that you get a factor two or three here
using those two fractions so it's not
enough to compensate for a factor 20 and
so what fluid dynamicists use on top of this is
to change the model so that there is
more turbulence on the model than there
would be around the real car and doing
this
can be done with more turbulent
incoming air or can be done with rough
patches on the surface of the model so
that you artificially increase the
amount of turbulence around the small
model so you get away with this so
here's the lesson equate the Reynolds
number you squeeze down the size of
something means you have to increase the
velocity by the same amount
this means it may not be easily done in
practice.
