Law of Similarity for Boundary Layer Flows
- It states that the u component of velocity with two velocity profiles of u(x,y) at different x locations differ only by scale factors in u and y .
- Therefore, the velocity profiles u(x,y) at all values of x can be made congruent if they are plotted in coordinates which have been made dimensionless with reference to the scale factors.
- The local free stream velocity U(x) at section x is an obvious scale factor for u, because the dimensionless u(x) varies between zero and unity with y at all sections.
- The scale factor for y , denoted by g(x) , is proportional to the local boundary layer thickness so that y itself varies between zero and unity.
- Velocity at two arbitrary x locations, namely x1 and x2 should satisfy the equation
- Now, for Blasius flow, it is possible to identify g(x) with the boundary layers thickness δ we know
Thus in terms of x we get
i.e.,
where   
or more precisely,
The stream function can now be obtained in terms of the velocity components as
or
 | (28.20) |
where D is a constant. Also  and the constant of integration is zero if the stream function at the solid surface is set equal to zero.
Now, the velocity components and their derivatives are:
or
Substituting (28.2) into (28.15), we have
or,
where
| (28.22) |
and
 | |
This is known as Blasius Equation .
- The boundary conditions as in Eg. (28.16), in combination with Eg. (28.21a) and (28.21b) become
at  , therefore  |
at  therefore 
| (28.23) |
Equation (28.22) is a third order nonlinear differential equation .
- Blasius obtained the solution of this equation in the form of series expansion through analytical techniques
- We shall not discuss this technique. However, we shall discuss a numerical technique to solve the aforesaid equation which can be understood rather easily.
- Note that the equation for
 does not contain  .
- Boundary conditions at
 and  merge into the condition  . This is the key feature of similarity solution.
- We can rewrite Eq. (28.22) as three first order differential equations in the following way
- Let us next consider the boundary conditions.
- The condition
 remains valid.
- The condition
 means that  .
- The condition
 gives us  .
Note that the equations for f and G have initial values. However, the value for H(0) is not known. Hence, we do not have a usual initial-value problem.
Shooting Technique
We handle this problem as an initial-value problem by choosing values of  and solving by numerical methods  , and  .
In general, the condition  will not be satisfied for the function  arising from the numerical solution.
We then choose other initial values of  so that eventually we find an  which results in  .
This method is called the shooting technique .
In Eq. (28.24), the primes refer to differentiation wrt. the similarity variable  . The integration steps following Runge-Kutta method are given below.
 | (28.25a) |
 | (28.25b) |
 | (28.25c) |
- One moves from
 to  . A fourth order accuracy is preserved if h is constant along the integration path, that is,  for all values of n . The values of k, l and m are as follows.
- For generality let the system of governing equations be
In a similar way K3, l3, m3 and k4, l4, m4 mare calculated following standard formulae for the Runge-Kutta integration. For example, K3 is given by The functions F1, F2and F3 are G, H , - f H / 2 respectively. Then at a distance  from the wall, we have
- As it has been mentioned earlier
 is unknown. It must be determined such that the condition  is satisfied.
The condition at infinity is usually approximated at a finite value of  (around  ). The process of obtaining  accurately involves iteration and may be calculated using the procedure described below.
- For this purpose, consider Fig. 28.2(a) where the solutions of
 versus  for two different values of  are plotted.
The values of  are estimated from the  curves and are plotted in Fig. 28.2(b).
- The value of
 now can be calculated by finding the value  at which the line 1-2 crosses the line  By using similar triangles, it can be said that  . By solving this, we get  .
- Next we repeat the same calculation as above by using
 and the better of the two initial values of  . Thus we get another improved value  . This process may continue, that is, we use  and  as a pair of values to find more improved values for  , and so forth. The better guess for H (0) can also be obtained by using the Newton Raphson Method. It should be always kept in mind that for each value of  , the curve  versus  is to be examined to get the proper value of  .
- The functions
 and  are plotted in Fig. 28.3.The velocity components, u and v inside the boundary layer can be computed from Eqs (28.21a) and (28.21b) respectively.
- A sample computer program in FORTRAN follows in order to explain the solution procedure in greater detail. The program uses Runge Kutta integration together with the Newton Raphson method
Fig 28.2 Correcting the initial guess for H(O)
Fig 28.3 f, G and H distribution in the boundary layer
- Measurements to test the accuracy of theoretical results were carried out by many scientists. In his experiments, J. Nikuradse, found excellent agreement with the theoretical results with respect to velocity distribution
 within the boundary layer of a stream of air on a flat plate.
- In the next slide we'll see some values of the velocity profile shape
 and  in tabular format.
|
