# problem with cavity flow

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 August 20, 2012, 05:02 problem with cavity flow #1 New Member   Mark sperc Join Date: Jul 2012 Posts: 10 Rep Power: 5 Hi All, I'm trying to simulate flow inside a cavity using stream function-vorticity formulation and AF scheme. I've written a code that runs fine for Re<2000 but it strongly depends on the initial guessed values for vorticity, is this normal? For Re>2000 (e.g. 4000) the absolute monitored residuals of vorticity on 61x61 grid gets down to 0.005 after 3000 iteration (seems high) but oscillate about it so that it never reaches to steady-state solution. That said,I need to make sure up to what Re we can expect steady-state solution? I'm really stuck on what the problem is and need some help? thanks.

 August 20, 2012, 13:59 #2 New Member   Mark sperc Join Date: Jul 2012 Posts: 10 Rep Power: 5 any ideas?

August 20, 2012, 14:07
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 Originally Posted by dersi Hi All, I'm trying to simulate flow inside a cavity using stream function-vorticity formulation and AF scheme. I've written a code that runs fine for Re<2000 but it strongly depends on the initial guessed values for vorticity, is this normal?
Hi Mark,

if the solution is steady then it should not depend on the initial condition.

Quote:
 so that it never reaches to steady-state solution. That said,I need to make sure up to what Re we can expect steady-state solution?
you should work with the lidd driven cavity poblem because a lot of results and bechmarks are available on this configuration.

For Re=5000 the solution is steady so you can run your code with this example and check.

 August 20, 2012, 16:49 #4 New Member   Mark sperc Join Date: Jul 2012 Posts: 10 Rep Power: 5 Thanks leflix for the reply, actually the problem is lid driven with square geometry, but the weird part is it converges for Re<2000 but not greater than that which goes oscillating. Generally, how many iteration cycle is needed to reach a steady state solution for e.g. Re=5000?

August 25, 2012, 05:49
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 Originally Posted by dersi Generally, how many iteration cycle is needed to reach a steady state solution for e.g. Re=5000?
Hi Mark,

Sorry for this delay.
I have made a run for Re=5000 to check. The grid is coarse 50x50.
The code is fully transient and it took about 25000 time iterations with a time step of 0.01 to reach the steady state.
With a steady state computation process it would have taken cetainly much less.

 August 25, 2012, 06:33 #6 Senior Member   Filippo Maria Denaro Join Date: Jul 2010 Posts: 1,780 Rep Power: 23 Hi, I did several tests many years ago ... I tried to repeat the range of Re covered by the well-known paper of Ghia&Ghia with my time-dependent FV solver based on the stream-function/vorticity formulation. I found steady solutions only up to Re=3200 (and, as leflix stated, the solution is independent from the initial guess), then my code started to have an unsteady behaviour. I remember that such behaviour was found also by some authors...

August 25, 2012, 11:59
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 Originally Posted by FMDenaro Hi, I did several tests many years ago ... I tried to repeat the range of Re covered by the well-known paper of Ghia&Ghia with my time-dependent FV solver based on the stream-function/vorticity formulation. I found steady solutions only up to Re=3200 (and, as leflix stated, the solution is independent from the initial guess), then my code started to have an unsteady behaviour. I remember that such behaviour was found also by some authors...
Hi Filippo,

Here are the solutions obtained at Re=5000 with two different codes that I developped , one in primitive variables and the other one with vorticity-stream function. The grid is now 100x100 because I thought that with a too coarse grid(50x50) the solution may be inaccurate and thus leads too a fake steady solution.

As you will notice it, the solution is still steady and there is a perfect agreement between the results. The both codes are able to capture the tertiary vortice. Check my results with the figure 3 of Ertuk et al: ref 1)

In the following references you will find two studies with a complete bibliography on this subject which reports steady state for Re=5000.
The authors found also in their study that for Re=5000 the solution was steady. The results have been obtained with different computational strategies.

In the youtube link, you will find a computation obtained with LBM method and it gives the same steady behaviour for Re=5000
Finaly in the very good book of Griebel et al those reference is given below, computations at Re=5000 have been undertaken and similar results, as far as I remember, have been found.

Different authors mentioned a hop bifurcation which turns the flow to become unsteady around Re=7500.

so we are forced to conclude that your code might have a bugg

1) Numerical Solutions of 2-D Steady Incompressible Driven Cavity Flow at High Reynolds
Numbers: (Erturk, Corke, Gokçol)
http://arxiv.org/pdf/cs/0411047.pdf

2) The 2D lid-driven cavity problem revisited (Charles-Henri Bruneau , Mazen Saad)

Attached Images
 primitive_variable.jpg (99.0 KB, 6 views) steam_vorticity.jpg (98.8 KB, 4 views)

August 25, 2012, 12:15
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Hi, many thanks for your details, I am not aware about the recent references on lid-driven cavity flows since I stopped working on 2d problems many years ago, therefore I just remember some discussions about the Hopf bifurcation after Re= 3200.
If you are interested, this what I experienced in a past study (G. de Felice, F.M. Denaro, C. Meola, Multidimensional Single Step Vector Upwinded Schemes for Highly Convective Transport Problems. Numerical Heat Transfer. Part B Vol 23, pp. 425-460, 1993.), I can say that Re <= 3200 are for sure steady cases and Re=10^4 is for sure a not-steady case (despite the steady solution reported by Ghia & Ghia), for Re = 3200 - 7500 I just remember some debates about bifurcation. Some authors supposed the difference in the results were due to different treatment of the corner points that are singular. But you are much more aware than me on this topic

August 25, 2012, 12:28
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 Originally Posted by FMDenaro If you are interested, this what I experienced in a past study (G. de Felice, F.M. Denaro, C. Meola, Multidimensional Single Step Vector Upwinded Schemes for Highly Convective Transport Problems. Numerical Heat Transfer. Part B Vol 23, pp. 425-460, 1993.)
Thanx for the reference, Iwill have a look...

Quote:
 I can say that Re <= 3200 are for sure steady cases and Re=10^4 is for sure a not-steady case (despite the steady solution reported by Ghia & Ghia),
and now you can reduce a bit the range. for Re<=5000 it is definitely steady.
I'm very confident in this result.

Quote:
 Some authors supposed the difference in the results were due to different treatment of the corner points that are singular.
In the two codes (one in FV the other one inFD) there is no special treatment concerning the corner point as it is never involved in the discretization, nor in the computation.
But however the codes are able to capture the tertiary vortice.

Quote:
 But you are much more aware than me on this topic
No I'm sure that you are much more sharp than me on this subject.
I just use Lid driven cavity as a very good validation test case when I developp codes as it is well documented in the litterature (Ghia, Erturk,....), easy boundary conditions, steady states, ...

 August 25, 2012, 12:39 #10 Senior Member   Filippo Maria Denaro Join Date: Jul 2010 Posts: 1,780 Rep Power: 23 good to know new results, I am quite surprised that the 2D cavity flow is still studied deeply... I wonder why the 3D lid-driven is not so ... yes, for standard central finite-difference the corner-points are never involved, however I used multidimensional stencil in my finite-volume flux reconstruction and I remember that I had hard time to treat them in consistent way ...

August 25, 2012, 12:53
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 Originally Posted by FMDenaro I am quite surprised that the 2D cavity flow is still studied deeply... I wonder why the 3D lid-driven is not so ...
Because not every one has a 3D code available and also because it requires
more computational ressources that not every one has on his desk.
To capture tertiary vortice you need at least 100x100 mesh in 2D. If you use a 3D code you would need 1 million points and if your code is not paralellized or optimized it could require at least 1 day of computation according the number of time step you want to simulate. And it' s too much for people.
Paralelizing a 3D code is absolutely necessary and it is a hard task...at least for me
And I should underline that 1 million grid nodes for 3D computations is really not enough to be taken seriously (it 's like 20x20 in 2D computations )

August 25, 2012, 12:56
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Filippo Maria Denaro
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 Originally Posted by leflix Because not every one has a 3D code available and also because it requires more computational ressources that not every one has on his desk. To capture tertiary vortice you need at least 100x100 mesh in 2D. If you use a 3D code you would need 1 million points and if your code is not paralellized or optimized it could require at least 1 day of computation according the number of time step you want to simulate. And it' s too much for people. Paralelizing a 3D code is absolutely necessary and it is a hard task...at least for me And I should underline that 1 million grid nodes for 3D computations is really not enough to be taken seriously (it 's like 20x20 in 2D computations )

yes, but now is quite common to have LES and DNS with several tenth of millions of unknown ... but the 3D lid driven cavity is still poorly explored...

August 25, 2012, 13:03
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 Originally Posted by FMDenaro yes, but now is quite common to have LES and DNS with several tenth of millions of unknown ... but the 3D lid driven cavity is still poorly explored...
yes it's not false !! But I guess studying hypersonic exhaust jet in a gas turbine sounds much more sexy than "Lid driven cavity" !!

August 25, 2012, 13:15
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Filippo Maria Denaro
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 Originally Posted by leflix yes it's not false !! But I guess studying hypersonic exhaust jet in a gas turbine sounds much more sexy than "Lid driven cavity" !!
yes, definitely!

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