Blasius Equation: Some Explorations Part 2

Solution of Blasius Equation in Matlab

A direct attack on the Blasius equation requires some kind of iteration such as a shooting method, because it is a two-point boundary value problem. Fortunately, there is a reformulation of the problem that avoids an iteration. We begin this reformulation by introducing a new dependent variable :

Where is a positive parameter to be determined. As one may show, the equation for g is then the same as that for f:

From the conditions on f , we have g(0) = 0, and



as

 

Instead of imposing the condition on g at infinity , we impose



Then the problem for g is an initial value problem with initial conditions

In order to carry out this integration numerically using matlab, we need to convert the 3^rd order differential equation into a system of three 1st order equations.

We have the following equivalent system of equations

   1:  >> blas =@(t,f) [f(2); f(3); -0.5*f(1)*f(3)];

   2:  >> options = odeset('RelTol', 1e-10, 'AbsTol', 1e-12);

   3:  >> sol=ode45(blas, [0,15], [0.0 0.0 1.0], options);

   4:  >> plot(sol.x, sol.y(2,:), 'LineWidth', 2.5)

   5:  >> xlabel('\eta'); ylabel('g(\eta)')

   6:  >> plot(sol.x, sol.y(2,:), 'LineWidth', 2.5)

   7:  >> xlabel('\eta'); ylabel('g(\eta)')

Blasius Equation: Some Explorations - Part 1

 

What is Blasius Equation:

Blasuis Equation describes the flow of a fluid over a flat plate.

The x-y coordinate system is chosen so that x is along the plate, and y is perpendicular to the plate. The leading edge of the plate is at x = 0, y = 0. The velocity components - u in the x-direction and v in the y-direction -- are expressed in terms of a stream function (x,y):

The fundamental equation which determines y is the x-component of the momentum equation in the boundary layer approximation. For the flat plate there is no pressure gradient, and the boundary layer approximation to the x-momentum equation takes the form

Where is the kinematic viscosity. By substituting the first eqaution into the second, we get for the partial differential equation

At the wall (y = 0) both velocity components must vanish, and far away from the plate, the horizontal velocity must approach the given free stream velocity . These conditions translate into the following conditions on :

The absence of a length scale (the plate is semi-infinite in length) suggests a similarity solution, as originally used by Blasius. The solution has the form:

The scale is comparable with the boundary layer thickness. This substitution into the equation for  leads to the following nonlinear ordinary differential equation for :

This equation is called the Blasius equation. We will solve it numerically in the next part. Once f is known, the velocity components may be computed as

Very good freeware editor for ANSYS APDL

 

I was trying to find an APDL freeware editor, with auto-completion feature. After a while of searching, I found a very good one that is based on the freeware PSpad editor. you can find it in the following location:

http://www.2doworld.com/software/ansys-editor-10.html 

It supports 1500 ANSYS command lines for user. Each text editor supports instant command help for APDL commands by locating the cursor on the line containing the command of interest. I failed to make the context help command work with new ANSYS help system based on java, which is not mentioned in the site, probably they developed the editor for the old help system. But you still can activate the auto-completion  by CTRL+SPACE it will give you a short description for the command. It is portable, standalone application.

Another interesting aspect of the above mentioned site is the available tutorials for ANSYS products. Check it out it is informative.