
[Sponsors] 
Limitations on time the step for lid driven cavity flow 

LinkBack  Thread Tools  Search this Thread  Display Modes 
October 21, 2016, 10:48 
Limitations on time the step for lid driven cavity flow

#1 
New Member
Juan Camilo Acosta Arango
Join Date: Mar 2016
Posts: 5
Rep Power: 10 
I'm following these slides
http://www3.nd.edu/~gtryggva/CFDCou...Lecture5.pdf They formulate NS equations usin stream, vorticity functions, I'm want to know what kind of analysis they do to conclude the time step restriction showed in the page 5. Its very important for me to know how it is done, thanks!
__________________
Juan Camilo Acosta Arango 

October 21, 2016, 10:58 

#2  
Senior Member
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,793
Rep Power: 71 
Quote:
It stems from the numerical stability analysis for the explicit Forward Time Central Space (FTCS) applied on a 2D convectiondiffusion equation. Note that the first constraint is exact only for the pure diffusive case (no convection). The method is unconditionally unstable for the pure convective case. In the mixed case the constraint involves a more complex functional law in the (cfl, Reh) plane. You can find details in any good CFD textbook. 

October 21, 2016, 11:11 

#3  
New Member
Juan Camilo Acosta Arango
Join Date: Mar 2016
Posts: 5
Rep Power: 10 
Quote:
__________________
Juan Camilo Acosta Arango 

October 21, 2016, 11:17 

#4 
Senior Member
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,793
Rep Power: 71 
Yes. The Von Neumann analysis is applied for the linear case (as well as the matrix analysis you can perform by using the Gershgorin locus method). The constraint in those slides are valid for the linear equation. Usually, they are roughly used also for the non linear equation.


October 21, 2016, 11:18 

#5 
Senior Member
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,793
Rep Power: 71 
P.S.:the second constraint must have the ratio to the mesh size h but, as I wrote, the case for the pure convective case is unstable. The key for a wigglesfree solution is to work with the Reh<2 condition.


October 21, 2016, 11:20 

#6 
New Member
Juan Camilo Acosta Arango
Join Date: Mar 2016
Posts: 5
Rep Power: 10 
sorry to be annoying, but do you know any paper or article where this specific analysis is done?
__________________
Juan Camilo Acosta Arango 

October 21, 2016, 11:26 

#7 
Senior Member

Fletcher: Computational Techniques for Fluid Dynamics, Vol. 2, pp. 335 cites a more general condition for the 2D equations. He actually cites this:
http://link.springer.com/book/10.100...3642859526 pp. 183 as original source. If you don't trust them, I swear, I did it by myself (straightforward Neumann on the equivalent scalar equation) and it is correct . It is (): Last edited by sbaffini; October 21, 2016 at 11:45. Reason: Added the actual formula from Fletcher 

October 21, 2016, 11:57 

#8  
New Member
Juan Camilo Acosta Arango
Join Date: Mar 2016
Posts: 5
Rep Power: 10 
Quote:
__________________
Juan Camilo Acosta Arango 

October 21, 2016, 12:05 

#9 
Senior Member
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,793
Rep Power: 71 
you can read any good textbook of both CFD and numerical anlysis...
as example, http://dl.iranidata.com/book/daneshg...idata.com).pdf 

October 21, 2016, 12:32 

#10  
New Member
Juan Camilo Acosta Arango
Join Date: Mar 2016
Posts: 5
Rep Power: 10 
Quote:
You mentioned something about "straightforward Neumann on the equivalent scalar equation", may you point me to some source about these formulation and the analysis done over it?
__________________
Juan Camilo Acosta Arango 

October 21, 2016, 12:55 

#11 
Senior Member

That's what i meant by all those "cites" and "swear". If you can handle the italian language between the 14 equations on 2 pages (i guess so), here it is attached my old analysis. As you can see, it is pretty straightforward Neumann analysis.


October 21, 2016, 18:28 

#12 
Senior Member
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,793
Rep Power: 71 
I strongly suggest to plot the amplification factor for the value = 1 in the (Reh, cfl) plane.


October 21, 2016, 19:13 

#13 
Senior Member
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,793
Rep Power: 71 
This issue is not as simple as it would appear...
https://estudogeral.sib.uc.pt/bitstr...20analysis.pdf Last edited by FMDenaro; October 22, 2016 at 10:19. 

Tags 
time step size 
Thread Tools  Search this Thread 
Display Modes  


Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Transient simulation not converging  skabilan  OpenFOAM Running, Solving & CFD  14  December 16, 2019 23:12 
Stuck in a Rut interDyMFoam!  xoitx  OpenFOAM Running, Solving & CFD  14  March 25, 2016 07:09 
time step directories naming issue  Andrea_85  OpenFOAM  3  April 3, 2014 08:38 
dynamic Mesh is faster than MRF????  sharonyue  OpenFOAM Running, Solving & CFD  14  August 26, 2013 07:47 
Orifice Plate with a fully developed flow  Problems with convergence  jonmec  OpenFOAM Running, Solving & CFD  3  July 28, 2011 05:24 