Download Lecture: Boundary Value Problem Boundary Value Problem 1 The

Lecture: Boundary Value Problem
Boundary Value Problem 1
The Blasius equation describes the non-dimensional velocity distribution in the laminar boundary layer over a flat plate.
It describes the similarity solution of the fluid flow influenced by viscous effects . The Navier Stokes equation and the
boundary conditions on the problem are
u v

 0;
x y
u
u
u
 2u
 v  2
x
y
y
(1)
y  0 : u  v  0; y   : u  U
(2)
where u and v are the velocity in the x and y direction. The problem is described by the Figure below where U is the free
stream velocity. The pressure is assumed constant and is not solved in this simplification.
U
y
U
u (y)

x
Figure: Laminar flow over a flat plate
Using a stream function the continuity equation, the first equation in Equation (1), is satisfied. Introducing the
dimensionless coordinate η, and a dimensionless stream function f , the second equation in Equation ((1) is reduced to
d3 f
d2 f

0.5
f
0
d 3
d 2
The boundary conditions then translate to
f (0)  0;
df
(0)  0;
d
df
(6)  1;
d
(3)
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 in 1912. A better solution was obtained by Howarth. 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.
Boundary Value Problem 2
This example is also a fluid flow problem. The original problem definition is available in Schlichting [Boundary Layer
Theory]. Figure below illustrates the features of the problem. A disk of radius R is rotating with the angular speed ω in
still fluid. The flow is steady, incompressible, has constant property, and is axisymmetric. The fluid at the disk has to
satisfy the no slip condition. The centrifugal effects cause the fluid to leave the disk radially near the disk. The flow
above the disk must replace this airflow through a downward spiraling flow. A cylindrical coordinate system (r, θ, z) is
used for description. Vr, Vθ, Vz, are the velocity components. p is the pressure, ν, the dynamic viscosity. The continuity
and the Navier-Stokes equations are
z
Vz
r


Vr
R
V
Figure: Flow over a rotating disk
Vr Vr Vz
 
0
r
r
z
V V 2
V
Vr r    Vz z
r
r
z

Vr
V VrV
V

 Vz  
r
r
z
Vr
Vz
r
 Vz
Vz
z
  2V
1 p
 V
   2r   r
 r

r r

r


  2V
 
2
 r

  V

r  r
  2Vr 

2 
 z 
  2V 

2 
 z 
  2V
1 p
1 Vz  2Vz 
   2z 
 2 
 z
r r
z 
 r
(4
The boundary conditions are
z  0 : Vr  0; V  r ; Vz  0;
z   : Vr  0; V  0;
(5)
In Schlicting, it is derived that using a differential volume over the disk an approximate value for the boundary layer
thickness can be found. This is used for scaling the z- coordinate. Next, scaling for the velocity and pressure are
introduced so that dependence on r and z can be separated. This is an essential technique for reducing the PDE to ODE.
The scaling relations for the various terms are

z
 ; Vr  r F (Z ); V  r G(Z );


Vz   H (Z ); p( z )   P(Z );
Zz
(6)
Substituting in Equations (4, 5), and simplifying with Z being the independent variable and primes representing
derivatives with respect to Z,
2F  H 
0
2
2
F  F H  G  F   0
2 FG  HG   G   0
P  HH   H 
0
(7)
The transformed boundary conditions are
Z  0; F  0; G  1; H  0; P  0
Z   ( 6); F  0; G  0
(8)
The original Navier-Stokes equations and boundary conditions are now reduced to the set of ODE and corresponding
boundary conditions in Equations (7, 8). Usually, the last differential equation for pressure is not solved with the first
three. It can be obtained from the solution of F, G, and H.
The solution to Equation (7, 8) was first obtained using a power series around Z =0, and an asymptotic series for large
values of Z. The numerical solution was obtained by Sparrow and Gregg . Numerical solutions are currently accepted as
alternate analytic solutions for this problem today. MATLAB is used to generate the numerical solutions used for
comparison. Once again the final value of the residuals should provide reasonable confidence that the Bezier functions
provide an excellent solution.