Shayan Shahidi - Essay for Eas3

Using these three boundary conditions the solution of the governing equation may be obtained by the use of power series solution and shown in the table below. The important things to note are the points corresponding to the edge of the boundary layer. Since u  U, f  1, (we choose the value of .9915, since  is defined at u = .99U). Thus from the table below:

Using the alternate definition, , we get

Now,

Therefore, the wall shear stress, w, may be written as

We define Skin Friction Coefficient as the non-dimensional wall shear stress, given by:

In this case both and are claimed to be exact solution of steady, laminar boundary layer over a flat plate oriented along the x-axis.

We notice from the above expressions that both (x) and Cf change along the plate. While (x) increases (boundary layer grows) with , Cf  0 as x . Both quantities depend on the variable Reynolds number, Rex . If the plate length is not infinite, how do we obtain the shear force on it? We may do this by integrating directly or, the use of the concept of "overall Skin Friction Coefficient". For example, for a finite length, L, of the plate, the shear force

where, .

but:

Therefore the may be substituted above and Fyx obtained by integration.

Alternately, define = Overall Skin Effect Coefficient . Thus the is nothing but "length-averaged" friction coefficient. Unlike Cf(x), is a constant value for the whole plate. Similarly the average shear stress for the plate may be defined as . Finally, the shear force on the plate may be written as the product of and the plate area.

Approximate Solution Method

Unlike the Blasius solution, which is exact, approximate solution method assumes an approximate shape of the velocity profile. This velocity profile is then utilized to evaluate quantities related to the governing differential equation, given below by Karman and Pohlhausen. This method, which is called the momentum integral method, changes the two equations given by Prandtl into a single differential equation. This equation over a flat plate may be written as:

where, x = Wall shear stress

 = Density of fluid

U = Free stream velocity

and,  = Momentum thickness of the boundary layer

=

The above equation is applicable only when the pressure gradient term is zero. For the case of non-zero pressure gradients you should use

Velocity Profiles

Since the Karman-Pohlhausen method requires an assumed velocity profile, let us explore some velocity profiles and their characteristics (see example problem 1). For example, suppose we assume the velocity profile to be a second order polynomial where A, B, and C are constants.

To evaluate velocity profile constants A, B, and C, we must use boundary conditions. The following three conditions may be used:

1) No-slip: y = 0, u = 0

2) B.L. Edge Velocity: y = , u = U

3) B.L. Edge Shear: y = , = 0

Note that at the edge of the defined edge of the boundary layer u = .99U and . However we approximate them with the rounded values. This is the reason the solution method by Momentum Integral Method is considered an approximate one.

With the above profile,

1)  0 =

2)  U = [ ]

3) 

Subtracting the second condition from the third,

Using this in the second condition,

or,  Parabolic Profile (see plot in the example)

Remember the use of the boundary layer velocity profile is only meaningful when . The use of this velocity profile may now be made to obtain  and w

or,

Note that defining a new variable makes the evaluation much easier.

Similiarly, [ in boundary layer]

for the parabolic profile

Using the above results for  and w in the momentum integral equation for a flat plate gives

Separating the variables  and x, and integrating

To express the boxed equation in a non-dimensional form divide both sides by x2,

, where Rex = is the Reynolds number based upon the variable x.

Compare this result with the earlier exact solution obtained under the Blasius method.

We therefore see the popularity of the parabolic velocity profile. Although the solution by Karman-Pohlhausen method is approximate it gives less than 10% error when compared with the exact solution is laminar flows over a flat plate.

Now that we have obtained (x), the shear stress, w, and skin friction coefficient, Cf, may