Blasius Equation

 

The Blasius equation describes the nondimensional velocity distribution in the laminar boundary layer over a flat plate.  It describes the similarity solution of the fluid flow influenced by viscous effects [33].  The Navier Stokes equation and the boundary conditions on the problem are

 

                                                                                           (3.1)

                                                                                    (3.2)

 

where u and v are the velocity in the x and y direction.  The problem is described by Figure 3.1 below where U is the free stream velocity.  The pressure is assumed constant and is not solved in this simplification.

 

Figure 3.1 Laminar flow over a flat plate

 

 

Using a stream function the continuity equation, the first equation in Equation 3.1, is satisfied.  Introducing the dimensionless coordinate η, and a dimensionless stream function f [33], the second equation in Equation (3.1) is reduced to

 

                                                                                                  (3.3)

 

The boundary conditions then translate to

 

                                                                          (3.4)

 

 

This equation, termed as the Blasius equation, is a third order nonlinear ordinary differential equation with boundary conditions at two points.  Published solutions indicate that the infinity boundary condition is easily met at η = 6. The analytical solution was obtained by Blasius by a series expansion at η = 0 and asymptotic expansion for large η.  It was solved numerically for the first time by Toepfer in 1912. A better solution was obtained by Howarth [34].  Howarth’s solution can now be reproduced using a numerical boundary value problem solver.

           

The Blasius problem is part of the set of problems defined by the Falkner-Skan equations, also appears to be a bench mark for comparing the different techniques for the solution to the NLBVP, particularly analytical ones.  Reference 35 transforms the BVP into a pair of initial value problems using transformation groups.  The interval is broken into 17 domains with a different power series in each domain.  They are able to obtain good agreement with Howarth’s solution.  Reference 36 applies the Adomain decomposition to obtain the transformation of the Blasius equation to Banach space and then solves for the missing initial condition necessary to convert it to an initial value problem.  A standard numerical technique is then used to obtain the solution.  The procedure is strongly problem dependent.  Reference 37 applies to problems that have asymptotic boundary conditions, which is true of the Blasius equation.  It uses a special numerical technique to obtain the solution.  The original differential equations are normalized and rewritten over a set of intervals.  A perturbation system is developed to solve the normalized equation and the whole process is iterated.  Reference 38 uses a direct perturbation technique by imbedding a small parameter into the Blasius equation.  This parameter value is then iterated until desired accuracy is reached.  Reference 39 uses perturbation and calculus of variations to establish the analytic solution of the Blasius equation, which resembles a power series, but has imbedded in it a factor that is similar to a basis definition.  Reference 40 uses the δ- perturbation scheme which involves iteration on δ.   The accuracy demonstrated in the paper is quite weak as only a couple of iterations are shown.

 

3.1.1 The Bezier Function Formulation

The problem in Equation (3.3, 3.4) is solved using a Bezier function. Find the Bezier function [x, y] or the vertices of the parametric curve {P1, P2,..,Pm+1 } or [aj ,bj], j = 1, .., m,     that

 

Minimizes

                                                            (3.5)

Subject to:       a1=0,    b1=0,    []

                         b2=0,   []                                                                               (3.6)

                       

                        am+1 = 6.

 

where i is point on the curve.  The number of points (np) is 100.  Every vertex of the parametric curve, Pj, is represented by its coordinate values as [aj, bj].  In the optimization problem, aj, j = 1 m+1, and bj, j = 1,m+1 are the design variables.  The objective function F is the sum of the residuals along the trajectory.  The constraints are the boundary condition on the curve.  The explicit condition on the vertices can also be used to reduce the number of design variables.  This is not done here to keep the formulation consistent.   The optimizer from MATLAB allows us to explicitly prescribe linear equality and inequality constraints among the design variables.

 

The problem in Equation (3.5, 3.6) is solved using fmincon from the MATLAB optimization toolbox.  The version used is MATLAB v.7.04. 

 

3.1.2 The Bezier Solution

The solution to the Blasius equation is presented in Figure 3.2.  There are five vertices used to define the curve.  A fourth order Bezier function is used to generate the curve.  The minimum value of the objective function in Equation (3.5) is 7.4e-05.

Figure 3.2 Final Solution

 

Figure 3.3 is the comparison of the Bezier solution and the analytical solution (obtained numerically through the BVP solver).  The axes are interchanged with respect to Figure 3.2 as η represents the y-variable. The figure illustrates that the Bezier function is can match the solution produced by the BVP solver available in the MATLAB package. 

 

Figure 3.3 Solution to the Blasius equation

 

 

3.1.3 Explicit Bezier Solution

The explicit Bezier solution is

           

                                         (3.7)

 

The analytical solution from Blasius was not used here because of convenience.  This means that Figure 3.3 remains the major source of comparison.  Since the residuals are computed exactly, its low value for the final iteration should provide reasonable confidence that the Bezier function provides an excellent solution.