CFD Online Logo CFD Online URL
www.cfd-online.com
[Sponsors]
Home > Forums > Main CFD Forum

Numerical solution to the rotating disk problem?

Register Blogs Members List Search Today's Posts Mark Forums Read

Reply
 
LinkBack Thread Tools Display Modes
Old   September 2, 2005, 01:24
Default Numerical solution to the rotating disk problem?
  #1
johny
Guest
 
Posts: n/a
Hi, all:

I am long harassed by finding a numerical method to the problem of rotating disk.

Here, the name "rotating disk" problem refer to a set of four ordinary differential equations modelled from the axis-symmetric flow from a rotating disk (see P163 ~ 168 of the book <<Viscous Fluid Flow>> by Frank M. White):

H' = -2F F'' = -G^2 + F^2 + F'H G'' = 2FG + HG' P' = 2FH - 2F'

with boundary conditions F(0)=H(0)=P(0)=0, G(0)=1, and F(infinity)=G(infinity)=0.

I know that to solve the equations numerically, we must find out the needed inital value of F'(0), and G'(0), to let F and G vanish at the infinite far end.

But for me, it seems a stiff system, and shooting just can't converge.

Any suggestions?

Btw, is there any paper that is on this topic? After all, it is a textbook problem, and should be solved numrically long before.

Thanks,
  Reply With Quote

Old   September 2, 2005, 04:20
Default Re: Numerical solution to the rotating disk proble
  #2
ramp
Guest
 
Posts: n/a
The following two refs, are for stationary disk, may be of some use to you.....

S. Nitin and R.P. Chhabra, Wall effects in two-dimensional axisymmetric flow over a circular disk oriented normal to flow in a cylindrical tube, Canadian Journal of Chemical Engineering, 83, 450-457 (2005).

S. Nitin and R.P. Chhabra, Sedimentation of a circular disk in power law fluids, Journal of Colloid and Interface Science, In Press, Available online 1 September 2005,
  Reply With Quote

Old   September 2, 2005, 04:37
Default Re: Numerical solution to the rotating disk proble
  #3
johny
Guest
 
Posts: n/a
I don't know why the book's name can't be displayed.

The problem is described in Frank M. White's book "Viscous Fluid Flow" in Page 163 ~ 168.

It can be reduced to solve the four ordinary differential equations numerically:

H' = -2F F'' = -G^2 + F^2 + F'H G'' = 2FG + HG' P' = 2FH - 2F'

subject to F(0)=H(0)=P(0)=0, G(0)=1, and F(infinity)=G(infinity)=0

The key point is to find the two initial values F'(0) and G'(0) to let F(inf) and G(inf) vanish.

Up to this point, it is a pure math problem. And maybe it can be solved with shooting method.

The difficulty is: there are two valued to be shooted, and this may be quite difficult due to the system's stiffness.

Since this is quite an old textbook problem, I don't think these 2 new papers can be of help. If it is solved, it should be long before.

Anyway, thanks for your attention,
  Reply With Quote

Old   September 2, 2005, 04:38
Default Re: Numerical solution to the rotating disk proble
  #4
Tom
Guest
 
Posts: n/a
The easiest way to solve this equation is to write the equations in finite difference form and Newton iterate. Otherwise try the book "Two point boundary value problems" by H.B. Keller which I vaguely recall discusses the related Falkner-Skan problem. You could also try Schlicting or any other book on boundary-layers (your problem is usually referred to as the von Karman similarity solution).
  Reply With Quote

Old   September 2, 2005, 05:24
Default Re: Numerical solution to the rotating disk proble
  #5
johny
Guest
 
Posts: n/a
Yes, I have consulted that book. However, in Falkner-Skan problem, there is only one value to be shooted, so that's relatively easy.

I even solved Blasius equation numerically only by linear shooting.

But for this rotating disk problem, two values are to be determined.

Schlicting's classical book also didn't talk of how to numerically get F'(0) and G'(0), it mentioned that those two valued were obtained by the method of series expansion and match. But today this method is clearly two complicated and outdated. With the power of modern computer, it should not be that difficult to solve it numerically.

Actally, I have tried to mail Prof Keller about this problem, coz he is an established expert in boundary-value problem of ODE; but i got no reply.
  Reply With Quote

Old   September 2, 2005, 07:44
Default Re: Numerical solution to the rotating disk proble
  #6
Tom
Guest
 
Posts: n/a
Have you tried my ealier suggestion of Newton iteration on the finite difference version of the equations - there is no need to know either F'(0) or G'(0) in this case?

I think you're being a bit unfair on the series solution method - have a look in either the Journal of Fluid Mechanics or the European Journal of Mechanics B/Fluids to see that these methods/ideas are still used. Also have a look at

http://anziamj.austms.org.au/V42/CTAC99/Kels/home.html

I suspect Keller didn't reply to your e-mail because he's retired (he's about 80 years old now!)
  Reply With Quote

Old   September 4, 2005, 22:30
Default Re: Numerical solution to the rotating disk proble
  #7
johny
Guest
 
Posts: n/a
Hi, Tom:

Sorry I didn't pay attention to the Finite-Difference method you methioned in last post.

And it didn't occure to me that series method is still in large use these days.

Thanks for your opinions and that URL - it helps.
  Reply With Quote

Old   September 5, 2005, 05:53
Default Re: Numerical solution to the rotating disk proble
  #8
zxaar
Guest
 
Posts: n/a
I am not sure how much it is relevant to your work, but if you have not, have a look at numerical recipe book, might find something useful , here is the link:

http://www.library.cornell.edu/nr/cbookcpdf.html

  Reply With Quote

Reply

Thread Tools
Display Modes

Posting Rules
You may not post new threads
You may not post replies
You may not post attachments
You may not edit your posts

BB code is On
Smilies are On
[IMG] code is On
HTML code is Off
Trackbacks are On
Pingbacks are On
Refbacks are On


Similar Threads
Thread Thread Starter Forum Replies Last Post
Convergence Centurion2011 FLUENT 24 May 9, 2015 08:02
Problem with the 2D rotating blade-CFX-12.1 Araz CFX 5 September 3, 2011 07:54
Naca 0012 (compressible and inviscid) flow convergence problem bipulsaha FLUENT 1 July 6, 2011 07:51
IcoFoam parallel woes msrinath80 OpenFOAM Running, Solving & CFD 9 July 22, 2007 02:58
Wall functions Abhijit Tilak Main CFD Forum 6 February 5, 1999 02:16


All times are GMT -4. The time now is 17:48.