Channel flow - Galerkin method
I want to apply a GLOBAL Galerkin method to the developing flow in a straight channel; I am NOT interested in the stability of the channel flow. I prescribe a plug-like uniform velocity profile at the inlet (u=1, v=0) and I expect to find the well-known parabolic Poiseuille velocity profile further downstream. The no-slip boundary condition is imposed on the channel walls. I want to carry out the computation in a streamfunction-vorticity formulation (psi, omega) rather than in primitive variables (u,v,p). Can anybody give me a hint if this has already been done (and if so by whom) and what type of global trial functions I should use (Fourier, Chebyshev, Legendre)? I guess Sine/Cosine series are not possible because the problem is non-periodic? Any comments are highly appreciated!
Thank you very much in advance,
Re: Channel flow - Galerkin method
I do not know what the Galerkin method is but probably you can use a usual pseudo-spectral channel code for this problem with Chebyshev in the wall-normal direction and Fourier in the spanwise and streamwise direction. The only thing you have to add is a kind of fringe technique in the streamwise direction that forces the flow to the desired inflow profile. This technique has been used to simulate developing boundary layers, pipe flows and so on. Articles about this can be found in J. Fluid Mech. and Phys. Fluids. I think.
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