Blasius Equation Discretization:

john_t · April 16, 2011, 12:59

Ive been trying the discretization of Blasius equation: f'''+1/2 ff''=0 using the center scheme (finite difference) but im not able to do it correctly. Any help or suggestions? Thanks

nickvinn · April 17, 2011, 06:18

I solved it with such discretizations:
f''''=(f(i+1)-3f(i)+3f(i-1)-f(i-2))/h^3 (take central differences for second derivatives in (i-1) and i, totally centered at (i-1/2)),
f''=(f(i-2)-2f(i-1)+f(i))/h^2 (centered at (i-1)),
f=f(i-1),
resulting in equation:
f(i+1)=3f(i)-3f(i-1)+f(i-2)-f(i-1)(f(i-2)-2f(i-1)+f(i))h/2
This variant is a little bit biased, but the performance is OK.At least when you have to solve it only before main computations in order to obtain initial laminar boundary layer solution.

john_t · April 17, 2011, 16:39

Quote:

Originally Posted by nickvinn

I solved it with such discretizations:
f''''=(f(i+1)-3f(i)+3f(i-1)-f(i-2))/h^3 (take central differences for second derivatives in (i-1) and i, totally centered at (i-1/2)),
f''=(f(i-2)-2f(i-1)+f(i))/h^2 (centered at (i-1)),
f=f(i-1),
resulting in equation:
f(i+1)=3f(i)-3f(i-1)+f(i-2)-f(i-1)(f(i-2)-2f(i-1)+f(i))h/2
This variant is a little bit biased, but the performance is OK.At least when you have to solve it only before main computations in order to obtain initial laminar boundary layer solution.

can this be solved using C++ or Fortran ??
Any idea about Tri Diagonal matrix discretization ??

nickvinn · April 18, 2011, 03:22

Well, I wrote my code in Fortran. It is quite simple.
There is no tridiagonal matrix here. The discretization is explicit: if you know values in 3 neighbouring points e.g. f(0), f(h) and f(2h), then you obtain f(3h) according to above formula. Then you get f(4h), etc. Of course, you need to know values in 3 points at the left border to start the process. You have the boundary conditions f(0)=f'(0)=0. So, f(0)=f(h)=0. Since this is the 3-rd order equation, you need one more initial condition for Cauchy problem. So, you can set some value for the second derivative f''(0) and thus obtain f(2h). Then, you get the solution for all points and find out if the boundary condition at the right boundary f'(infinity)=1 is OK. Of course, it is not, and you adjust the value of f''(0), and try once more. This is called the shooting method, you can find it in any elementary computational methods textbook.

April 16, 2011, 12:59	Blasius Equation Discretization:	#1
john_t New Member Join Date: Apr 2011 Posts: 2 Rep Power: 0	Ive been trying the discretization of Blasius equation: f'''+1/2 ff''=0 using the center scheme (finite difference) but im not able to do it correctly. Any help or suggestions? Thanks

Similar Threads
Thread	Thread Starter	Forum	Replies	Last Post
mass flow in is not equal to mass flow out	saii	CFX	12	March 19, 2018 06:21
Calculation of the Governing Equations	Mihail	CFX	7	September 7, 2014 07:27
Blasius equation help :(	johnt447	Main CFD Forum	0	March 18, 2011 18:57
Constant velocity of the material	Sas	CFX	15	July 13, 2010 09:56
Space and time discretization of Euler equation	Hooman	Main CFD Forum	2	June 6, 2010 09:30

April 17, 2011, 06:18		#2
nickvinn New Member Nikolay Vinnichenko Join Date: Nov 2010 Posts: 14 Rep Power: 16	I solved it with such discretizations: f''''=(f(i+1)-3f(i)+3f(i-1)-f(i-2))/h^3 (take central differences for second derivatives in (i-1) and i, totally centered at (i-1/2)), f''=(f(i-2)-2f(i-1)+f(i))/h^2 (centered at (i-1)), f=f(i-1), resulting in equation: f(i+1)=3f(i)-3f(i-1)+f(i-2)-f(i-1)(f(i-2)-2f(i-1)+f(i))h/2 This variant is a little bit biased, but the performance is OK.At least when you have to solve it only before main computations in order to obtain initial laminar boundary layer solution.

April 18, 2011, 03:22		#4
nickvinn New Member Nikolay Vinnichenko Join Date: Nov 2010 Posts: 14 Rep Power: 16	Well, I wrote my code in Fortran. It is quite simple. There is no tridiagonal matrix here. The discretization is explicit: if you know values in 3 neighbouring points e.g. f(0), f(h) and f(2h), then you obtain f(3h) according to above formula. Then you get f(4h), etc. Of course, you need to know values in 3 points at the left border to start the process. You have the boundary conditions f(0)=f'(0)=0. So, f(0)=f(h)=0. Since this is the 3-rd order equation, you need one more initial condition for Cauchy problem. So, you can set some value for the second derivative f''(0) and thus obtain f(2h). Then, you get the solution for all points and find out if the boundary condition at the right boundary f'(infinity)=1 is OK. Of course, it is not, and you adjust the value of f''(0), and try once more. This is called the shooting method, you can find it in any elementary computational methods textbook.