What is a boundary layer?
We talk about boundary layers when we are discussing internal or external, and some
ways they are exactly the same, and in other ways they differ.
So the overall description is its this thin region.
It is on the surface of a body and it is were viscous affects are important.
So our outside boundary layer we can consider the flow in-viscid.
So how does this occur?
Well you have a fluid and we will consider this internal flow, and it enters with a uniform
velocity.
As it moves through viscus affects are going to cause it to hit the wall.
The reason this is, because of the no slip condition.
So let me try to draw a pipe with this.
So lets say here you are entering a pipe, or any kind of internal flow.
As you go along the pipe here the velocity profile is changing because of this growing
boundary layer.
So lets call this the boundary layer.
Remember this is a circular pipe.
In the boundary layer we have viscous affect that are important.
However in here we have in in-viscid core, and one of the reasons that could be consider
important is in fact that the flow there is in-viscid implies that we can actually use
bernoulli.
When finding these boundary layers to completely fill the pipe we get our velocity profile.
So when this happens we get what is known as fully developed flow.
So what that means is a flow does not change in the direction of the flow.
Again the best way to put that is mathematically.
The change in U with respect to X equals 0 or the change in U with respect to z is equal
to 0.
The change in U to R however is not 0.
You can tell that from the velocity profile.
So until it becomes fully developed that is what is known as the entrance or the entry
length, entrance region.
For the rest of the pipe now.
It is fully developed.
So how do we determine what this entry length is.
Well this entrance length or dimensionless entrance length divided by D is a function
of the Re regardless of whether the flow is laminar or turbulent.
It has been found if the flow is laminar this l over D equals 0.06 times the Re.
However if the flow is turbulent.
This entry length divided by D equals 4.4Re to the 1/6.
So they were looking at a pipe.
Lets say instead we look at a flat plate.
So this is what we would consider external flow.
Again for a flat plate the purpose of the boundary layer is allow the fluid to change
its velocity from its upstream value, which is this uniform velocity profile, and we will
call this U, and we will call this U 0 at the surface.
So what happens is you get from the edge.
You get this boundary layer that grows, and then it becomes turbulent.
It jumps and grows some more, and we defined this height of this boundary layer as del.
Del is considered the y at which our velocity little u, equals 0.99 percent of our free
stream velocity.
So above this are we still have beyond the boundary layer.
We have our uniform or up stream velocity.
So the question is how do we find this height of our boundary layer.
So for laminar flow.
It has been determined that our boundary layer length or width equals 5 times the square
root of the kinematic viscosity times x over our free stream velocity.
This allows us to do is find the width of the boundary layer at any x on the flat plate.
We also have a similar boundary layer for turbulent flow, and here this del divided
by x equals 0.370 divided by Re subs x to the 1/5.
So again we are able to find the width or the height of the boundary layer at any x
regardless of whether we are laminar or turbulent flow.
So sometimes we would like to know what the velocity is.
So we want to find out what that u is.
In order to determine that the navier-stokes equations or simplified.
I am taking into account of properties such that v is
very much less then u, and d/dx is very less then d/dy, and that is because the boundary
layer is thin the component of velocity normal to the plate is going to be smaller then that
is parallel and the rate of change of any perimeter around that boundary layer should
be much greater then that along the direction of the flow.
So we simplify the Navier-Stokes equations to make this the boundary layer equations.
These cannot be solved analytically.
However they have been solved numerically, and we therefore have solutions based on a
similarity variable.
So the similarity variable takes into consideration not only, and this is the kinematic viscosity,
the x direction.
It also takes into account the y direction, and we use this similarity variable, and again
remember this was figured out numerically.
To define this derivative of this function of our similarities variables.
Which equals u, little u divided by the big U.
So what that allows us to do is at any x and any y find our similarities variable.
Then using a table.
Which is laminar flow along a flat plate.
This is also known as the Blasius solution.
We have a table that uses eta, and from that you can find your little u over big U.
For example, if you have an eta of 2.8, which you calculated based on your free stream velocity
in your x and your y.
You would find that little u over your free stream velocity would be 0.8115, and clearly
these numbers are not going to be greater then 1, because your velocity at any point
is going to be greater then the free stream velocity.
We can then use this number such that u equals 0.8115 times our free stream velocity.