CFD Online Logo CFD Online URL
Home > Forums > Main CFD Forum

Exact 3D solutions

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

LinkBack Thread Tools Display Modes
Old   February 4, 2009, 02:11
Default Exact 3D solutions
Posts: n/a
It is interesting to know the exact (analytical) solutions of the incompressible Navier-Stokes equations for 3-D rectangular channel (steady state or transient). The solution as an infinite series probably. Need for debugging the codes.

Thank you.

  Reply With Quote

Old   February 4, 2009, 05:37
Default Re: Exact 3D solutions
Paolo Lampitella
Posts: n/a
It could be a very challenging task to write here the series solution for this case. You can find it on the White's book "Viscous fluid Flow".

However, if you can't go through it, consider that for unsteady laminar incompressible flows in straight ducts with any kind of costant cross section the Navier-Stokes equations will semplify in the 2D unsteady heat equation for the axial velocity component, the only non zero velocity component.

In this case the pressure gradient has to be considered as a general, known, unsteady forcing term constant in the cross section of the duct, that is:

du/dt = nu ( d2u/dy2 + d2u/dz2 ) + F(t)

where F is at most a function of time and, of course, the only source of unsteadyness. A lot of solution for this problem are also available through the Carslaw and Jaeger book. However, the solution of this problem is very simple if a rectangular cross section is considered.

If you don't want to go through the solution process, also considering that the solution will not effectively be a 3D one, you can consider the following:

Another option is to consider the 2D taylor vortex solution (google it) and apply a simmetry boundary condition in the third direction, actually obtaining a 2D unsteady solution constant in the third direction.

I suggest to follow the second path because the series summation is a very time consuming task...a non necessary one.
  Reply With Quote

Old   February 12, 2009, 18:53
Default Re: Exact 3D solutions
Posts: n/a

This book seems to contain such a solution --->
  Reply With Quote


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
Exact solution of Burgers equation mcaro Main CFD Forum 3 January 25, 2011 07:46
Analytical flow solutions Antonio Main CFD Forum 0 December 14, 2005 16:43
Analytical flow solutions Antonio Main CFD Forum 0 November 15, 2005 18:47
Exact solutions zhor Main CFD Forum 4 May 13, 2001 11:26
EXACT SOLUTIONS OF NON-NEWTONIAN FREE SURFACE FLOWS Valdemir G. Ferreira Main CFD Forum 0 December 7, 1999 13:25

All times are GMT -4. The time now is 17:41.