97-32

Direct Galerkin Approximation of Plane-Parallel-Couette and Channel Flows by Stokes Eigenfuctions

by Noske, A.; Rummler, B.

 

Preprint series: 97-32, Preprints

The paper is published: Not. Num. Fl. Mech., Vol. 64 (ed. by Friedrich, R. and \rBontroux, P.), Vieweg, 03 - 19

MSC:
76F10 Shear flows
35Q30 Stokes and Navier-Stokes equations, See also {76D05, 76D07, 76N10}
42C15 Series of general orthogonal functions, generalized Fourier expansions, nonorthogonal expansions
76H05 Transonic flows

 

Abstract: The plane parallel Couette flow and Poiseuille flow of an incompressible Newtonian fluid within an unbounded layer in $ {\bf R}^{3} $ between the parallel plates of distance $h$ are studied using Galerkin approximations based on Stokes eigenfunctions on an open bounded rectangular parallelepiped in $ {\bf R}^{3} $ furnished with periodical conditions. For the Galerkin method we utilize the first 356 Stokes eigenfunctions and a fixed period $ 2l{ }= 2\times 2,69 $. From the dimensionless Navier-Stokes equations for the difference $\bu $ between the velocity and the laminar velocity we get an autonomous system of ordinary differential equations for the time-dependent coefficients of the Stokes eigenfunctions. We apply the kinetic energy of $\bu$ as a measure of turbulence. The numerical calculations yield satisfactory results in comparison with measurements keeping in mind the small dimension of our approximation spaces.

Keywords: Navier-Stokes equations, Stokes eigenfunctions, Galerkin approximations


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