### 1997-32

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

Series: 1997-32, Preprints

MSC:
76F10 Shear flows
35Q30 ~Navier-Stokes equations
42C15 General harmonic expansions, frames
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
stu\-died
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 f\/irst 356 Stokes eigenfunctions
and a f\/ixed 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 coeff\/icients
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.
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Keywords:
Navier-Stokes equations, Stokes eigenfunctions, Galerkin approximations

This paper was published in:
Not. Num. Fl. Mech., Vo