In this problem we are interested in the
boundary layer that is developing above
an advertising board that is placed
above a car by fluid dynamics teacher who
wants to advertise for their Fluid Dynamics course and we'd like to qualify is
two things one is whether the boundary
layer transits and which means whether
it becomes turbulent starting laminar
and if so where and the second question
which we’re try to answer is how thick does the
boundary layer become above the board so
as it develops over the board over
here how thick will it become overall
so perhaps if you represent the
situation as seen from the top above the
car you will get something that looks
like this this will be the board like so
and on each side of the board you will
have a boundary layer developing and the
boundary layer starts at zero and it is
laminar and then it will transit at some
point and then becomes turbulent at
which point it becomes thicker and it
grows faster whoop this should not be in
a straight line like so the question
that we are trying to answer really is
two things one is if it transits at
which distance the X transition the
longitudinal distance at which is
transiting and the second question we
try to answer is what is the maximum
thickness that the boundary layer
reaches over the board so let's take a
look the transition point is guided by a
general principle which says that the
Reynoldss number based on distance at
which a boundary layer transits in
general is approximately half a million
five times 10 to the power 5 like so and
based on this we can find which which
position the transition occurs at for
the given flow conditions so let's
expand this we have Rho V or usually u u
times xx transition in this case here
divided by mu the viscosity this has to
be equal to the Reynolds number
of the transition this is not a fixed
absolute rule for from physics from the
universe this is just a general
guideline a general rule depending on
the conditions on the board you will
have transition a bit earlier or a bit
later so it's a isolate here X
transition it's equal to
the Reynolds number X transition
multiplied by the viscosity multiplied
by 1 over density times the main
velocity and if we now put numbers into
this then we're going to get here is our
half million 5 times 10 to the power 5
approximately the viscosity is 1.5 times
10 to the power minus 5 for air in this
case at this temperature and then we
have 1 over the density of air which is
one point two two five kilograms per
meter cube and the velocity of the car
which in this case happens to be I
forgot
it happens to be ten meters per
second like so and so we get you plug
this into your calculator you get
something like zero point six one two
and this is a distance that we calculate
in this so this is a distance in
meters the position of the transition
point like so it means that if you look
back at the board over the car you're
gonna have a laminar boundary layer
right up until approximately this point
here and so the X transition here will
be 0.6 meters 60 centimeters like this
like so yes so for a little bit of the
board we're gonna have a laminar
boundary layer then it transits and
then becomes turbulent for the rest of
the board so this point is this position
here is 0.6 meters okay what about the
thickness what about the maximum
thickness that the flow will will reach
well the thickness is quantified in
boundary layers for both the laminar and
turbulent case
and in this case what we want to have is
the equation for thickness inside the
turbulent boundary layer which is which is
occurring here what we're gonna do is to
quantify Delta Max using the distance
based Reynolds number at the trailing edge of the board so for this
we apply the model for boundary layer
thickness in a turbulent boundary layer
and this model is like so it says that
this I have to go back to the formula
sheet it says that Delta the thickness
of the boundary layer divided by X this
happens to be modeled approximately
approximately as 0.16 as it would
turn out divided by the reynolds number
based on distance to the power of 1 over
7 and this is a general model for how
thick a boundary layer becomes as it as
it grows as it as it develops over a
flat surface it's again not a basic law
of physics it is what is generally
observed look at the videos for what a
turbulent boundary the boundary layer
looks like in the lecture notes and you
will see that it's very difficult to
define you really to see what's going on
with the thickness and how how thick
exactly or even on average a boundary
layer would be when it's turbulent so
keep this as a rule of thumb but don't
tattoo it on your arm it's not worth it and
people certainly will argue about
whether this is 1 over 7 or 1 over 6 and
a half and so on so forth so anyway what
we only have here is Delta Max and so
for this we can put all the values at
the x max position like so and so what
we're gonna do is to expand this and
have it like this so Delta max ya as as
it turns out in our case 0.16 I
multiplied by x max multiplied by the
Reynolds number
based on distance it's a maximum value
^ -1 over seven like so and if we expand
this further we get 0.16 x x max
multiplied by rho u x max here to the
power minus 1 over 7 and then we have
the mu which is trailing over there
which is to the power 1 over 7 like so
and so I'm now putting in numbers we are
looking at x max x max is the maximum
distance that we have traveled along the
boundary layer so we're looking at this
point here and this point here is at the
trailing edge of the board so it's 3
meters is the maximum length of the of
the board and so we have a 0.16 and it
turns out to be 3 meters for x max and
then we put in the values for density of
the air 1.2 to 5 kilograms per meter
cube multiplied by the free stream
velocity 10 meters per second multiplied
by again x max which is the maximum
distance along which we have traveled
with a boundary layer all of this to the
power of minus 1 over 7 and then I
multiply this by the value of the
density 1.5 times 10 to the power minus
5 like so minus 5 and this is to the
power 1 over 7 there's a little bit
tedious to type into your calculator so
always be very careful when you do this
during the rules for priorities in how
you type in the numbers and this should
get you 0.5 8 7 and this is a distance
that were calculating it's a distance
away from the wall the thickness so I'm
sorry I mistyped this I misread my
own notes this is 5.8 7 times 10 to the
power minus 2 meters and so this becomes
now 5 [point] 8 7
centimetres and this is the maximum
thickness that we have over the board
over here so let's go back to the to the
diagram here so we have a sense of the
scale of this flow this board over all
the width of this board sorry the length
of this board is three meters and above
that we have a boundary layer that's
going to become five to six centimeters
thick so that you see that this diagram
here in this diagram the vertical scale
is a very much exaggerated compared to
the longitudinal scale if the car were
to drive faster
and then this boundary layer thickness
would reduce and it would become smaller
and smaller but in this case with ten
meters per second you're gonna get a
boundary layer that's approximately six
centimeters again I insist very much
this is the rule of thumb for the ideal
case and so do not put too much focus on
the very last digits of your calculation
because all of those equations that were
using here the transition points and the
thickness model are rules of thumbs for
modeling the thickness of the boundary
layers and they're not fundamental
physical laws
