Limitations on time the step for lid driven cavity flow

ja0335 · October 21, 2016, 10:48

I'm following these slides

http://www3.nd.edu/~gtryggva/CFD-Cou...-Lecture-5.pdf

They formulate N-S 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!

FMDenaro · October 21, 2016, 10:58

Quote:

Originally Posted by ja0335

I'm following these slides

http://www3.nd.edu/~gtryggva/CFD-Cou...-Lecture-5.pdf

They formulate N-S 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!

It stems from the numerical stability analysis for the explicit Forward Time Central Space (FTCS) applied on a 2D convection-diffusion 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.

ja0335 · October 21, 2016, 11:11

Quote:

Originally Posted by FMDenaro

It stems from the numerical stability analysis for the explicit Forward Time Central Space (FTCS) applied on a 2D convection-diffusion 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.

I'm curious about your answer, because as I know it is a Von Newman stability analysis, however this problem is not linear so the Von Newman analysis can not be applied here. I'm missing something?

FMDenaro · October 21, 2016, 11:17

Quote:

Originally Posted by ja0335

I'm curious about your answer, because as I know it is a Von Newman stability analysis, however this problem is not linear so the Von Newman analysis can not be applied here. I'm missing something?

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.

FMDenaro · October 21, 2016, 11:18

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 wiggles-free solution is to work with the Reh<2 condition.

ja0335 · October 21, 2016, 11:20

Quote:

Originally Posted by FMDenaro

P.S.:the second constraint must have the ratio to the mesh size h

sorry to be annoying, but do you know any paper or article where this specific analysis is done?

sbaffini · October 21, 2016, 11:26

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...-3-642-85952-6

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 ( $\Delta x = \Delta y$ ):

$\Delta t \le \frac{\Delta x^2}{4 \nu}$

$\Delta t \le \frac{4 \nu}{\left(|u|+|v|\right)^2}$

ja0335 · October 21, 2016, 11:57

Quote:

Originally Posted by sbaffini

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...-3-642-85952-6

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 ( $\Delta x = \Delta y$ ):

$\Delta t \le \frac{\Delta x^2}{4 \nu}$

$\Delta t \le \frac{4 \nu}{\left(|u|+|v|\right)^2}$

Thanks a lot!

FMDenaro · October 21, 2016, 12:05

you can read any good textbook of both CFD and numerical anlysis...
as example, http://dl.iranidata.com/book/daneshg...idata.com).pdf

ja0335 · October 21, 2016, 12:32

Quote:

Originally Posted by sbaffini

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...-3-642-85952-6

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 ( $\Delta x = \Delta y$ ):

$\Delta t \le \frac{\Delta x^2}{4 \nu}$

$\Delta t \le \frac{4 \nu}{\left(|u|+|v|\right)^2}$

Sbafiny, I've checked the books you mentioned but again I don't see they do any analysis to arrive to the time step condition.

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?

sbaffini · October 21, 2016, 12:55

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.

FMDenaro · October 21, 2016, 18:28

I strongly suggest to plot the amplification factor for the value = 1 in the (Reh, cfl) plane.

FMDenaro · October 21, 2016, 19:13

This issue is not as simple as it would appear...

https://estudogeral.sib.uc.pt/bitstr...20analysis.pdf

October 21, 2016, 10:48	Limitations on time the step for lid driven cavity flow	#1
ja0335 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/CFD-Cou...-Lecture-5.pdf They formulate N-S 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, 11:26		#7
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,152 Blog Entries: 29 Rep Power: 39	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...-3-642-85952-6 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 ( $\Delta x = \Delta y$ ): $\Delta t \le \frac{\Delta x^2}{4 \nu}$ $\Delta t \le \frac{4 \nu}{\left(\|u\|+\|v\|\right)^2}$ Last edited by sbaffini; October 21, 2016 at 11:45. Reason: Added the actual formula from Fletcher

October 21, 2016, 19:13		#13
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,764 Rep Power: 71	This issue is not as simple as it would appear... https://estudogeral.sib.uc.pt/bitstr...20analysis.pdf sbaffini likes this. Last edited by FMDenaro; October 22, 2016 at 10:19.

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October 21, 2016, 11:18		#5
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,764 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 wiggles-free solution is to work with the Reh<2 condition.

October 21, 2016, 12:05		#9
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,764 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, 18:28		#12
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,764 Rep Power: 71	I strongly suggest to plot the amplification factor for the value = 1 in the (Reh, cfl) plane.