# Laminar Boundary Layer Theory

### Flat Plate Flow

From the previous section the boundary layer equations reduce to the following
when the boundary layer is assumed to be steady:

This is especially simple in the case of incompressible flow with no pressure
gradient. This laminar, flat plate, boundary layer flow satisfies:

These equations are easily solved by introducing the variable η:

also introducing the function:

From continuity, then:

and the boundary layer equation becomes:

f''' + ff'' = 0

This ordinary differential equation is accompanied by the boundary conditions
which state that the velocity right at the surface is 0 and that far away
the velocity approaches the specified U_{e}. These B.C.'s are written:

f = f' = 0 at h = 0

f' = 2 at h = infinity

The problem was first formulated in this way by Blasius in 1908.

The equation seems simple, but it is nonlinear and no closed form solution
is known. The problem can be solved by assuming a series approximation for
f.

When all is said and done, the following relations are found:

Note that C_{f} is about twice the momentum thickness for a flat plate. Is
this what you would expect?

### Thwaites Method

Thwaites method is used for computing the boundary layer characteristics
in laminar flow with pressure gradients. It is based on the steady boundary
layer equations with a specified external pressure gradient, and gives an
approximate solution.

Starting with the Navier-Stokes Equations we make the usual boundary layer
assumptions, that the flow is steady, incompressible, and we ignore higher
order terms (See Kuethe and Chow pg. 461 or 330 or 314)

In the x-direction:

This may be rewritten in terms of the boundary layer variables, θ, H,
U_{e}, and C_{fl}:

where θ is the momentum thickness:

H is the shape factor:

and C_{fl} is the local skin friction coefficient:

The basic idea behind many laminar boundary layer methods method is to assume
a particular form for the variation of u with y: i.e. u/u_{e} = f(y). Pohlhausen's
method is based on a quartic polynomial for u(y). Thwaites' method is not
quite so direct. It uses results from exact solutions to certain laminar
boundary layers to obtain an approximate relationship between H, C_{f}, Re,
and θ.

Substituting these results for θ and H into the expression above Thwaites
obtained:

This is easily integrated for θ(x).