Hi, I have to work numerically in a channel, but without streamwise periodic conditions ( non-established regime), and a priori unbounded. Is it possible to use Chebyshev's Polynomials for it, and, if one can, how? (Up to here, I had thought to imposing some lenght for the channel and map it into the interval [-1,1], but that seems to me excesive as approach, since a priori I dont know the lenght necessary for the analysis). Thanks in advance,

No matter whether you know the domain length a priori or not, you must choose one for your computation. Generally, you can choose a length which seems large enough based on your knowledge on the flow field and based on the different types of flow. An appropriate length should be that whose enlargment should not change the numerical result.

-Paul

And you will need to impose the boundary conditions properlly, because the chebyshev method is very sensitive to wrong boundary conditions. You will have to impose the BC on the 'invariants' of the flow, see for example treatment on non-reflective BC. ALso since your channel is 'infinite', it means that waves can actually 'escape' to infinity, i.e. not be reflected at the boundary.

For propblems with viscosity on an infinite domain it may be best to first try a spectral method with polynomials that are localized, i.e., polynomials that satisfy the fact that long away from the domain of interest the viscosity will damp everything to zero (or pretty close to it).

Peter

Thank you very much for all your suggestions.

I am going to try with the first suggestion (to fix a lenght a priori).

Concerning the second answer (suitable BC, the condition of zero normal derivative; or "do nothing" BC, is this one a "good" condition?

Finally, the use of the localized polynomyals seems interesting. Can You give me some bibliographical references?

Thank you once again

Carlos

There are two things that you need to consider when imposing the BC. First what are the 'physical' BC you want to impose, in this case you mean for example zero derivative (I am not sure what you mean by do nothing BC). The best is to impose kind of fixed BC, BC that are not a results of the simulations themselves. The second stage is that you imposed these conditions on the variables through the characteristics of the flow, in 1D the characteristics are the Rieman invariants. It would be very long to explain all that here, the best would be to look at some paper on non-reflective BC for example. Or look at the details on the imposition of BC in spectral methods (in books or in some papers in a journal).

For non reflective BC, see for example:

Givoli, D., 1991, Journal of Computational PHysics, volume 94, page 1.

For treatment and imposition of BC in spectral methods, see for example:

Abarbanel et al. 1991, Journal of Fluid Mechanics, 225, 557.

Gottlieb, Gunzburger, Turkel, 1982, SIAM J. Numer. Anal. 19, 671.

You might also want to have a look at the book:

Spectral Methods for partial differential equations, SIAM-CBMS, 1984, by Voigt, Gottlieb, Hussaini (Philadelphia, PA).