I am wondering the capabilities of 2nd order finite difference formula in space and first order formula in time to discretize the vorticity-streamfunction equation to solve Taylor Green Vortex Problem. I am mostly concerned about capturing the vortex interaction using these formulas. Do I need to use high order finite difference schemes both in space and time to capture vortex interaction?
In theory yes! but you may have to reduce the mesh size and time step to a very small number which hits practical limits
If you are considering a second order central discretization with a first order explicit time integration the only chance to get a stable solution is in fulfilling the constraint Re_h=1 everywhere in the domain. The time step must be of the same order of the mesh size.
However, this is not also a guarantee that the solution is sufficiently accurate as the action of the local truncation error can introduce some spurious superimposition.
Generally, unsteady flow problems such as this one require at least a second order temporal accuracy
Filippo Maria Denaro , Thank you very much professor. Is RK2 in time with second order central in space will be able to capture the vortex interaction?
Do I need to reduce the mesh size significantly in that case?
Professor Tapan K. Sengupta
, Thank you very much for your comment. I'd be glad to take a look at your notes.On a side note, What can be the relatively easier finite difference scheme that you would recommend me to try first? I'd be more than happy to take your advises.
If centered finite differences are used for space semi-discretization of advection, the resulting ODE system has purely imaginary eigenvalues. Therefore, in a method of lines time discretization approach, an ODE method must be used which includes at least a small portion of the imaginary axis in its stability region.
Runge Kutta like methods of order lower than 3 do not have this property, therefore for pure advection problems they yield unconditionally unstable discretizations. On the other hand, as remarked by Filippo Maria Denaro ,
adding the diffusion term can partially cure this problem, as long as the time step
is sufficiently small. A nice explanation can be found in this paper
Article Vorticity Boundary Condition and Related Issues for Finite D...
Despite the inaccuracy of the FTCS scheme, I personally consider very instructive for a student to convince himself by using such a scheme for solving the viscous Burgers equation. In particular starting from the initial sin x condition (k=1) with periodic conditions, the student can experiment the quality of the solution by analyzing the resulting spectra. This is a good experiment to check both stability and accuracy of FTCS, FTUS and higher order discretizations.
The modified equation analysis applied to the linear convection -diffusion equation discretized by the FTCS scheme shows that if the cell Reynolds number is below 2 you have a chance to get a stable solution (provided a correct value of the time step). That can be analysed also by the signs of the coefficients
Dear Tapan K. Sengupta
yes, your results seem in accord to my computed stability region in the (cfl, Reh) plane I attach here. There is a very restricted stability region for Reh
The cell Peclet number (or the cell Reynolds number) is reported in the x-axis, going from 0 to 160. The y-axis has the cfl number. This region is computed from the spectral analysis, by computing the modulus of the amplification factor for any pair (cfl, Reh) and for any phase. What is different in your CAF paper is the Pe expression, you defined it by means of the time step (alpha*dt/h^2). Actually, this is not a Peclet number (or Reynolds number). I used the definitions: cfl=c*dt/h and Reh=c*h/alpha. This way, your Peclet definition is cfl/Reh in my terminology.
Dr. Sengupta, thank you very much for your suggestions and notes. I will try to use your advice quickly to see the results.
Dr. Denaro,
I didn't understand the initial condition for
the viscous burger's equation.
I'd like to know a little more about the problem.
Thank you.
Tapan K. Sengupta
yes, professor. I have already read this article.Indeed, this article helped me a lot.
Mahdi Hasan
for the Burgers problems have a look at slides 16 and 17 here
Presentation Lecture on LES - Part I and II
Tapan K. Sengupta
and Mahdi HasanTapan is correct! I made a mistake. look at this problem from "nonlinear dynamics" point of view. First derivative can capture all changes except in Biforcations. As first derivative is either zero or way smaller than higher order terms (H.O.T.). That is the simple reason you cannot capture [very] nonlinear phenomena with low-order-methods. See discussion of Steven Strogatz in his book "Nonlinear Dynamics and Chaos" on that.
Thanks for reminding that Tapan K. Sengupta
Hi Mahdi, I saw that professor Tapan and Denaro answered the question. Who am I to disagree with them? Nevertheless, I will my two cents. If I am mistaken, please correct me because I am willing to learn.
There is nothing preventing you from getting such interaction with low order schemes. At least theoretically, you can demonstrate that all schemes tend to the same solution as dt and dx decrease. At the end of the day, this is what Roache's work is based on! You can converge to the same solution faster with a high order scheme. For example, refer to the Taylor Expansion series. The thing is the fidelity of your solution. In addition, after a brief literature review, you will see that most of the current research is carried out using High order schemes. Here is the catch, a second order scheme can be more ACCURATE than a 6th or 7th order scheme. For example, refer to the DRP.
Having said that, you can obtain your interaction using first order in time and space, but the temporal and spatial discretization will be extremely small. For example, if your Re_cell = 1 ( I would say < 1) You round off and discretization error will be smaller than your cutoff (Nyquist). However, to obtain such resolution, provided that you are reproducing professor's Tapan's paper, the mesh will be huge. Here is the point I want to make: You can obtain your interaction, but your solution will not be accurate. I am not surprised at all, and Professor Tapan gave a pristine answer for that.
Agree professor, exa-scale computing is something that fascinates me but, my humble opinion is that the CFD field is not ready yet. Unlike other fields, I’m seeing that CFD hasn't embraced that yet.
Professor I am looking forward to see the proceedings from CISM summer school soon. Any other event like that in the future?
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