X. Exact Solutions of the Incompressible Boundary Layer Equations

	In this chapter, we seek exact solutions to 2-D, steady, incompressible boundary layer equations, given by

(1)
	Here n is the kinematic viscosity and equals m/r . Observe that we are using dp/dx, rather than ∂p/∂x to denote the pressure gradient. This is because ∂p/∂y = 0 across the boundary layer.

	The description "exact solution" is a somewhat rough description of what we are about to do. We will attempt to reduce the above system of PDEs into a single ODE, which may be numerically solved. This was the preferred approach during the first half of the 20th century, when digital computers were not available. Today, it is possible to solve the above system of PDEs on a PC class machine. (A typical numerical solution of the PDE requires about 5 minutes from airfoil nose to trailing edge).

	We seek exact solution for the class of flows, where the edge velocity distribution ue(x) may be described as

(2)
	Many potential flows obey the above description of velocity. For example, for the potential flow over a flat plate we can set the exponent m to be zero, giving ue = A, a constant. In the vicinity of the stagnation point at the nose of an airfoil, potential flow theory states that ue linearly increases with x giving m= 1; here x is the distance along the airfoil surface measured from the front stagnation point. Flows within 2-D convergent and divergent channels may also be described using the above description.

	The flow over a flat plate was analyzed by Blasius, one of Prandtl's students in 1908, shortly after the boundary layer equations were developed. The stagnation point flow was studied by Hiemenz in 1911. For the more general case where m is not known a priori, exact solutions (i.e. ODEs which must be subsequently solved numerically) were developed by Falkner and Skan in 1931.

Solution of Blasius Flow:

	Since the Blasius problem (m=0) is a little easier to analyze than the more general Falkner-Skan flow, we study this problem first.

	Our starting point is to choose the stream function y(x,y) as the primary unknown, rather than the primitive variables u and v. This reduces the unknowns from 2 to 1. Using y also means continuity is automatically satisfied.

(3)
	The next step is to assume a functional form for y. This form will have two parts: a dimensional part, with the dimensions of y; and a nondimensional part. From the derivation of boundary layer equations, we expect that the square root of kinematic viscosity n must appear in this form, because several quantities such as boundary layer thickness d and the v- component of velocity vary as the square root of Reynolds number. We therefore assume

(4)
	We keep the first part F as simple as possible, pushing all the complexity into the second function f.  Guided by dimensional analysis, Blasius chose F1 to be . (Check that this quantity has the dimensions of y).

	The second part f is nondimensional. We group its arguments (x,y,u• and n) into nondimensional forms. Since we are seeking ODEs, it is desirable that the function f has only one independent variable. Blasius defined an independent variable h as follows:

(5)
	With this variable, the stream function y(x,y,n) takes the form:

(6)

We can express ∂u/∂x, ∂u/∂y and the higher derivatives from (6) as follows:

(7)
	We can also express dp/dx that appears in the u- momentum equation in terms of the edge velocity ue from Bernoulli's equation as follows:

(8)
	In the present case, of course, ∂p/∂x=0.

	When the dust settles, for our flows defined by equation (2), the u- momentum equation becomes:

(9)

	Equation (9) is a third order ODE, for f. We are, in particular, interested in the u- velocity profile, where u = ue fh/2. If we can solve this ODE then, in effect, we know the velocity field at all x- stations, at all y- stations, and for all n values! We need to solve this ODE just once. This implies that the velocity profiles at different x- and y- locations for different n values are similar, and may be computed from a general universal solution. For this reason, equations (5) and (6) are called similarilty transformations, and the flows satisfied by equation (9) are called similar solutions.

	Equation (9) requires three boundary conditions. At the solid surface (y=0, thus h=0) we require u=v=0. At boundary layer edge (large h), we require u = ue. Then, the appropriate boundary conditions for equation (9) are:

(10)
	Do not be alarmed by the appearance of the symbol • in equation (10). It is just a figurative way of saying that at the boundary layer edge, we are far away from the solid surface.

	Equation (9), known as the Blasius equation, may be solved on PC class computers using an iterative process to be described. 

Falkner-Skan Solution

	Falkner and Skan generalized the special case that Blasius considered (m=0) and the stagnation point solution that Hiemenz considered (m=1) to a more general class of edge velocity distributions. Their approach is quite similar to what Blasius did. The similarity transformations they used are:

(10)
Or,

(11)
and

(12)
	when this transformation was used in the boundary layer equations, the following ODE resulted.

(13)
	The above equation may be solved using a procedure identical to that used for the Blasius equation, discussed in detail later.