Hi,

I would like have some suggestion from those who has performed DNS for turbulent flows.

(1). Which spatial discretization method is suitable and of what order?

I looked into the 6th order compact schemes but it seems quite expensive to obtain all the derivatives after solving huge matrices. So, I'm making my mind to use a 6-th order Central-Difference scheme on non-uniform grids. However, I am not sure how should I derive all those 6-th order expressions for various derivatives. Any efficient idea?

Also, if I use 6th order central-difference, I need to extrapolate the variables for three imaginary cells on the boundary. Would it bring any significant inaccuracy?

(2). Same is true with the time-marching scheme. I'm currently using a special 3rd order Runge-Kutta scheme for Differential Algebraic Equations (DAE) proposed by M.Arnold: "Half-Explicit Runge-Kutta Methods with Explicit Stages for Differential-Algebraic Systems of Index 2". This is the link to download it: http://sim.mathematik.uni-halle.de/~...s/1998/A98b.ps

I'm planning to change the time-marching scheme also, either with the general Runge-Kutta schemes for ODEs or if there is any other better and efficient scheme.

Please suggest me with your opinions.

Thank you very much in anticipation!

The best way would be to use explicit central difference scheme of 6th order/ Spectral methods along the periodic direction and non uniform 6th order compact scheme in the wall normal direction. Do not use coordinate transformation in the wall normal direction as it would introduce numerical errors which might introduce numerical errorsn. Refer to the non uniform compact scheme paper by Gamet et.al in journal of computational physics. They performed DNS of compressible channel flow using non uniform compact scheme. For time marching use Runga-Kutta method implemented in low-storage form. This would help in the reduction of memory requirements by a factor of 2 or 3.

Hope this helps.

Thanks for your reply. I looked at Gamet's paper in "International Journal for Numerical Methods in Fluids" with title "Compact Finite Difference Schemes in Non-Uniform Meshes. Application to Direct Numerical Simulations of Compressible Flows". However, I couldn't find his paper in JCP. Could you please be more specific about it.

In the Gamet's paper I mentioned, he has described the compact scheme of 4th-order on non-uniform mesh using 5-point stencils resulting in a tri-diagonal system. That looks attractive.

However, if I want to use 6th-order non-uniform mesh using 3-point stencils, "Chu & Fan" have proposed some schemes on non-uniform grids. But, the problem with using that is the resulting system is not a band-diagonal system. Instead, it seems to be a general sparse matrix which is computationally very expensive.

If I carry on Gamet's idea, I think, I need to use a 7-point stencils for 6th-order scheme. It is same as a 6th order CD-scheme. So, I still have to figure out which one to use

Please suggest your opinions! It will be very helpful.

I think, I don't need to use 7-point stencil in order to get 6th-order compact scheme from Gamet's idea. 5-point stencil is sufficient.

Also, I read that Compact Schemes are much more accurate that ordinary CD-schemes. Perhaps, that's the reason why people prefer compact schemes over CD-schemes in DNS.

I think that i gave the wrong journal. It was published in IJNMF. Compact schemes resolve higher range of numerical wavenumbers compared to ordinary CD schemes and hence are more accurate. The problems with using implicit scheme if you have periodic directions is that the tridiagonal solver becomes much more expensive than a non periodic tridiagonal solver. It would be best to use Upwind compact scheme of Zhong or DRP scheme of Tam and Web in the periodic direction to make the computations faster. Also explicit schemes in certain directions will make it easier to parallelize the code in slices.

Thank you very much for the valuable information. Now, I can understand why not to use the compact scheme in periodic direction.

I have one final question though: I am using the classic MAC method by Harlow and Welch for my computation. The grid is staggered i.e. velocity vectors are defined on the face centers and the scalars are on the cell-centers. The compact schemes what I have come across until now or what you have suggested, all are for collocated grids, not for the staggered grids. If u have any suggestion in this regard or any reference, could u please let me know? It will indeed be very helpful to me.

Thank you very much once again.