Blasius Solution for Boundary Layer Thickness

The flat plate laminar boundary layer functions
that were developed by Blasius can be used to find a number of different characteristics
of a boundary layer. In this screencast, I’ll show you how we use this table to find the
boundary layer thickness at any point along a flat plate and demonstrate it with an example.
By definition, the thickness of the boundary layer, which we call del, is the value of
Y where the velocity in the boundary, u, is 99%, or 0.99, of the free stream velocity.
In other words, U divided by U infinity equals 0.99. So let’s look at this table here, and
if you notice, between these two numbers is 0.99, which means that our eta is going to
equal 5.0. That means that 5.0, if we look at what this column means, equals del, because
that’s where our y is, times the square root of our free stream velocity divided by the
kinematic viscosity of the fluid times x, where x is the length along the plate. Let’s
write this in terms of del. So we’re going to multiply the top and bottom of this by
x. So we get 5.0 times x, divided by the square root of U infinity times x squared, divided
by our kinematic viscosity times x, or we can write this as 5.0 times x divided by the
square root of (U infinity)(x) divided by the kinematic viscosity. And hopefully we
recognize this quantity in the denominator under the square root as the Reynolds Number.
That means, at any x, if we know the fluid property, in particular the kinematic viscosity
and the free stream velocity, we can find the boundary layer thickness, or del. Let’s
look at an example. Let’s say we have water – it’s at 300 K – and our free stream velocity
is 2 m/s. And what we want to do is find the boundary layer thickness at different points
along the plate. So the fist thing that we’re going to need is the kinematic viscosity of
water at 300 K, and that’s equal to 8.6 x 10^-4 m^2/s. Next we find the Reynolds Number,
which is defined as the free stream velocity times x, divided by our kinematic viscosity.
Our Reynolds Number in terms of x equals 2326 m^-1 times x. So then when we put that into
our equation that we found up here for del, we end up with del, which equals 5.0 times
x divided by the square root of the Reynolds Number, or 2326 square root of x, and this
equals 0.104x to the 1/2. Let’s look at this at different points on the plate. So let’s
say we look at 10 mm down our flat plate. Our boundary layer thickness is going to be
0.0104 m. We look at 100 mm and our boundary layer is 0.033 m. And as we expect, as we
go down further on the plate, our boundary layer thickness is gong to increase and in
fact, at 1 m, our boundary layer thickness is 0.104 m. In fact, if we look at a plot
of the boundary layer thickness as we go along the plate, we see the tendency for this thickness
to increase sharply at the beginning, and then plateauing as we continue along the plate.

Bernard Jenkins

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