I am looking for information on whether it is more favourable to use finite volume method or spectral method for a direct numerical simulation code a channel flow. Can anyone help me with that?

Guus Jacobs

FVM can deal common problems. SM is more accurate, however it has restriction in geometry. You can find more in some books on spectral method.

Hi THere,

Spectral methods are good to resolve the fine structure of the flow and it has little or no numerical dissipation.

THe main disadvantage with it is that it can not easily deal with shocks and supersonic flow, since the discountinuous variables (density, velocity,..) cannot be represent by a finite series. Basically you get two-point oscillations and the solution is quickly poluted by that.

The other thing that you need to take into consideration is that when you use the spectral methods, you need to treat the boundary conditions on the charactersitics of the flow and not on the primitive variables. You also need not to overspecify the boundary conditions, otherwise the method is explosively unstable. This just means that the method is so accurate that it does not tollearate boundary conditions which are not correct mathematically (other methods of low order accuracy have enough numerical dissipation to avoid the instability, but they also do not conserve enerby, momentum, etc...).

You can use the SM with a spectral filter to cut-off high frequencies.

You have also to introduce an hyperviscosity to simulate the dissipation of the energy in the smalest scale.

In the periodic dimension of domain (for example an angle in non-cartesian coordinate system) you can use expansion of periodic series. Usually one uses the Fourier series and makes use of FFT.

In the non-periodic case one uses non-periodic polynomial such as the Chebushev one. IN this case fast cosine transforms, making use again of FFT, can be used.

If you could tell about the symmetry of the channel flow and all the rest, like boundary, boundary conditions, initial conditions, etc.. I could say if it will be easy to use SM.

I am myself using spectral methods for about 10 years or so, in non-cartesian geometry, for 2D rotating, viscous, compressible, turbulent flows (Fourier-Chebyshev expansion).

Cheers,

Patrick

Thank you for your answers. I am left with a few questions though. I would appreciate if you could answer these too:

Can spectral methods be used for parallel programming?

Can you recommend literature in fluid mechanics on the spectral methods besides the Rogallo (1981) and Spalart (1991) paper?

Greetings,

Guus Jacobs,

Spectral Methods that I mentioned use FFT, and FFT has been implemented on massively parallel machine. The grid, however, cannot - as far as I know - be cut into portions for each processor.I believed that using parallel FFT and clever coding can lead to some good results (I know some people who ported their SM code on a parallel machine very sucessfully).

ref: books:

Voigt, Gottlieb, Hussaini, 1984, Spectral Methods for Partial Differential Equations (PHiladelphia: SIAM-CBMS)

Canuto, Hussaini, Quarteroni and Zang, 1988 Spectral Methods in Fluid Dynamics, New York Springer. I recomand this one particularly.

See also all kind of papers with these names, Gottlieb, Orszag, Canuto, Deville, Husaini, etc...

a paper:

She, Jackson, Orszag, 1991, Proceeding of the Royal Society of London A 434 (1890), 101.

The book I recomand is a very good start indeed.

Patrick

Hi,

Here is some more on parallel spectral stuff.

Olus Boratav ************************************************** *******

Pelz, R.B., ``Parallel Fouier spectral methods on ensemble architectures,''

Computer Methods in Applied Mechanics and Engineering, 89, pp.

529-542, (1991).

Pelz, R.B., ``The parallel Fourier pseudospectral method,'' Journal of

Computational Physics, 92, pp. 296-312, (1991).

Pelz, R.B. ``Pseudospectral methods on massively parallel computers,''

Computer Methods in Applied Mechanics and Engineering, 80, pp.

493-503, (1990).

[sasank]

Im Sasank doing my masters in IIT kgp,Im in very much need of "Spectral Methods in Fluid Dynamics" by Caunuto,1988 as my project is pertained to it..Im just a beginner If u hav the access to that book plz send to me..I will be highly thankful to U..Plz suggest me few gud books on the spectral methods..
My email ID:sasank.komarla@gmail.com
awaiting for ur response!!!
Thanks