### 1997-32

by Noske, A., Rummler, B..

**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.

\\

**Keywords:**

Navier-Stokes equations, Stokes eigenfunctions, Galerkin approximations

**This paper was published in:**

Not. Num. Fl. Mech., Vo