Who told you that we dont'worry about that? The CFL is one of the conditions to fulfill for the stability of explicit time integration methods

CFL number specifies how solution is progressing in numerical domain with respect to physical domain. If these two domain does not match then solution becomes unstable or sometimes erroneous. Thus we have to ensure that this condition should be satisfied for solution to be accurate and also stable.

That’s right, just to add that the time-accuracy in URANS has a quite different meaning since the time-averaging filters out all high wavenumber modes.

This a gross mistake and I think there is a huge misunderstanding. Sometimes fed by commercial packages. This is my personal experience and as such it might be completely wrong. Implicit and Explicit schemes are used to march the solution in time. However, when we have complex physics with different time scales things are difficult to asses. For example, Imagine we have a multi-species with mass transfer problem. Since the time scale is so small we always use implicit schemes. However, if you abuse with large dt (based on your implicit formulation) your solution will remain stable but the final solution will be garbage. I remember one case where I saw that the final temperature field had temperatures unbounded in the order of 3000 kelvins, whereas the BC was in the order of 300-500 kelvins. You will be surprised that the solution was provided by a support engineer from a commercial CFD company. Of course, I was not happy at all, and I solved the case by my self. Our results demonstrated that delta t needed to keep the solution bounded was smaller than the one obtained with the CFL and sometimes in the order of the Von Neumann delta t.

Please take the following with a pinch of salt since is based on my own conclusion. Most of the stability studies (CFL and Von Neumann limits) are based for hyperbolic, Parabolic and elliptic problems. However, the NS is a combination of them all and extrapolating it explicitly is not easy. As such, we have to keep looking at every single direction when we try to draw a conclusion and take them fro granted from such studies.

Thank you all,

I think, as Julio said, we shouldn't solve each problem in the same way. In some problems, we should use very small time step sizes to have accurate physical results but by using larger time step sizes in that problem, we can also get a numerically converged solution but NOT a physical solution. So in spite of using Implicit methods, we have to pay attention to physical results and not convinced that numerical stability will lead to accurate physical results.

Dear Farid Sepehrianazar

I think that the focus should be first on the type of solution of the PDE you want to solve. Implicit or explicit methods of the same accuracy order, formally have somehow same impact on the solution at equal CFL number. The problem arises when we think that implicit method are unconditionally stable (a fact that is not true) and we use them at high values of the CFL. That can make some sense if you are using a transient method only to get a steady state and you want a fast solution.

In your case, URANS is expected to have a transient solution (but you should be careful on its meaning, I suggest to read this discussion https://www.researchgate.net/post/URANS_what_is_the_meaning_for_statistically_steady_flows_and_what_compared_to_LES) and the accuracy is a constraint to be considered. When we consider CFL

Dear Tapan K. Sengupta

no, I separetely consider accuracy and stability, let me consider the case of the implicit Crank Nicolson scheme. Since from their original paper the appearence of an unstable mode was highlighted. The von Neumann stability analysis showed that such unstable mode can be dumped. But, first of all, that is true for infinite precision representation (no round-off) and for linear PDE. When we work at very small space and time steps, the magnitude of the discretization error (as well as the local truncation error) can reduce until to the point that the round-off error evolution (that does not obey to the same governing equation as we already discussed) can strongly affect the stability. Furthermore, historically the CN method was already analysed under some circumstances showing the appearence of instability (see for example https://watermark.silverchair.com/9-1-110.pdf?token=AQECAHi208BE49Ooan9kkhW_Ercy7Dm3ZL_9Cf3qfKAc485ysgAAAnAwggJsBgkqhkiG9w0BBwagggJdMIICWQIBADCCAlIGCSqGSIb3DQEHATAeBglghkgBZQMEAS4wEQQMDvKlAZZhcouyCuP-AgEQgIICI4S9VBd6a3-VXezP-vcUW5jApaufEvyyLMdoLcGgR-zAX7x3ZHGB6JihwRj6QtjMDgG9rt8rlMCej7mTq4Ev5m3H_GdW0X752u8tIGMY87VPsIEhRDBduej_65YczJUoMgroVzlBVFb_mX5kG186rO7oc3d9M68wtlcZIVEG0G1D5_gcLyuChwDzMwuTqwbPTFjdUryWFsQG6tYbs40Y7wa5A6szLoNoZ1Pbbd3cABgzDnHQ2BbS7QBXAZW5NtGNIL2AfG7Po4xPFe-5GuGCJagDbEDuPTi7XCAjNjbf7Yp_NmwJ0BybzgAWuxjPBNXABM9Rt0acygtvafMEtf3O7M3S1nfeqVdwNWByjxQuHizLeOsLAeP9kBDRnEPdYRhdu-c-5CQEUGYWI_9E6F4nXywdOCjI9PPRa7JCqWTuViLr3sMYnOMuCwt2UNoTRHGiJONQ9UoKQFGYziZO9lIsq48-9a76rAGAo1rxPnLtj_3hY2OyKAWrDU04T1O_p991d5d1ccsIk4fLAvA45U0Ssww-kPlOP3s_wct_NCH7N1yEWxu2NrKPy7fz2OXqFWA9o7N4l4RbwvCJr4PB87hpQImjexFnx6kiZdzoMO3bLUqMtHxiYZfB1bnJsjXiGYnAEI0nyfJxpW5Pku0i-JVAbJcDlkBNi3UV_dh5mRFukN3hUIbeZp8ANf0vtMsS4_EOZEpvNPljhZmE_9f0zmPQN32Qv8c)

You can find also a warning in the textbook of Hirsch (page 446):

"It is also seen that not all the implicit methods (θ =0) are A-stable, showing that implicitness does not ensure always that the schemes will be unconditionally stable."

Dear Filippo Maria Denaro,

thank you for your valuable comments,

what about rotating computational domains?

in some papers (including rotating zones, e.g. wind turbines), time step sizes are selected due to rotational degree and the solution is converged after proper number of rotations.

In a computational domain, including stationary and rotating zones, CFL number for rotating zone is greater than that of stationary zone. In these papers, by reducing time step sizes, there are no changes in results even for CFL>1.

Dear Farid Sepehrianazar

I am not sure to understand your questions.

If you have an external time-depending forcing term having some periodical behavior your time step must be a fraction of the period to describe the cycle.

I am not sure about your statement " solution is converged after proper number of rotations". For "convergence" do you mean that a fully correlated turbulent field is obtained after some rotations?

Then, your last sentence has to do with the greater effects of the magnitude in the turbulence model rather than the local truncation error.

Dear Tapan K. Sengupta

Other than the paper of Mitchell I posted before, there are old analyses with criticisms to the stability behavior of the CN method. For example, the non-linear convection equation is analysed here

https://pdfs.semanticscholar.org/157f/cec799c12022129cde3197502b9f151e913c.pdf

The classical stability analysis for linear equation syas CN is unconditionally stable and it highlights the existence of one unstable mode in the CN scheme. In case of the diffusion equation it is dumped while the advection equation could behave differently as the correct equation for the evolution of the round-off error should have relevance in the analysis.

Thank you Tapan and Filippo,

about "solution convergence" :

I mean,

as i mentioned, in papers about wind turbine, the writers plot Cm in terms of time (Cm: Coefficient of momentum) with different time step sizes for the last revolution of wind turbine. They show that by reducing time step size less than 0.1 degree of revolution (time needed for 0.1 degree revolution of wind turbine), changes in results are not significant although for that time steps CFL is greater than 1 (CFL>1) and the results are in agreement with experimental data.

Hi Farid, that surprises me. Few years ago, when I started my PhD I did some work on Wind and Marine Turbines. Please, be careful with my experience since that was at the of my PhD, thus I was building my experience in such physics. Having said that, I worked on LES and we needed to use dt that leaded to CFL smaller than 1 or sometimes close to one. But, I dont recall > or >>1.

Here I want to add few extra taughta that I remeber from that experience. Firstly, the stability of your scheme depends on the order of your time scheme. Not surprise ! My scheme was not bigh order

Implicit solution do not need to meet CFL