Von Karman Boundary Layer Thickness Video Lecture | Fluid Mechanics for Mechanical Engineering

position position position position
position position position
the next thing that von Karman did after
coming up with the quadratic velocity
profile in his momentum integral
technique was he went ahead and
evaluated the boundary layer thickness
and he used the relations that he got
out of the control volume analysis for
both the wall friction as well as the
drag on the plate and so we combined
those two together and enabled him to
get the boundary layer thickness that's
we're gonna take a look at in this
segment
so what von-karman did is he had this
relationship for the momentum thickness
and this one for his quadratic velocity
profile subject to the boundary
condition as we talked about in the last
segment he plugged that profile into a
momentum thickness and was able to come
up with this relationship here and we
will refer to this as being equation a
and he also had a relationship for the
wall shear stress
and from that he could obtain and this
for the wall shear stress and that would
have been using the velocity profile
that he had obtained now also from the
control volume analysis
and the following and so using the
expression from a
and this is equation D and then he
combines those two
and we will call this equation e now
what he did is he had this expression
for shear stress and if we look back
here he had this expression for shear
stress so he equated those two so
rearranging he was able to come up with
this relationship here and what he then
did is he integrated that
and knowing that the boundary layer
thickness was zero at x equals zero
so he used the boundary condition in
order to get the constant of integration
and rearranging
and taking the square root
he obtained this relationship here where
that is under the square root sign and
that was the relationship that he
obtained for the thickness of the
boundary layer as a function of X and
one thing that we can look at is this
term in here and that is a slightly
familiar term to us and that is
basically one over the Reynolds number
because we have the kinematic viscosity
and the numerator there so that can also
be written as Delta over X so the
boundary layer thickness divided by
stream-wise position along the plate is
approximately equal to five point five
divided by our e X remember that's way
that we write the boundary or the
Reynolds number for a boundary layer
raised to the power one half and that's
another way of writing the expression of
von-karman obtained for the thickness of
the boundary layer so that was a very
powerful advancement that had occurred
in fluid mechanics when he derived this
however I should make the caveat that
there this was done is subject to a
assumed velocity profile the quadratic
profile and consequently this is not a
perfect solution but it was one that
helped them quite a bit and it was
actually pretty close to the actual
value layer thickness growth rate for a
laminar boundary layer so what we're
going to do in the next segment is we're
going to take a look at the skin
friction now that's another thing that
von Karman was able to calculate using
his momentum integral technique