### 1999-35

#### Investigation of the Nonstationary Navier-Stokes Equations in special Domains - Transition to Turbulence

by Rummler, B..

Series: 1999-35, Preprints

MSC:
34A34 Nonlinear equations and systems, general
35Q30 ~Navier-Stokes equations

Abstract:
We regard a general class of boundary-pressure-driven flows of
incompressible Newtonian fluids in unbounded layers and in unbounded pipes
in ${\bf R}^{3}$ with thickness $2R$ or radius $R$, which marginal cases are
e.g. plane Couette flows and Poiseuille flows in channels,
respectively.
We avail the incompressible nonstationary Navier-Stokes equations as
description of the physical process. \\
We def\/ine energetic Reynolds numbers. Using Galerkin approximations
based on Stokes eigenfunctions on open bounded subdomains
in ${\bf R}^{3}$ furnished with periodical conditions in the at first
unbounded spatial directions, we get an autonomous system of
ordinary differential equations for the time-dependent coeff\/icients
of the Stokes eigenfunctions from the dimensionless
Navier-Stokes equations for the difference $\bu$
between the velocity and the laminar velocity.
For the Galerkin method we utilize f\/ixed periods $2l$ and the f\/irst
$N(l)$ 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
and hope raising results for investigations of bifurcations.

Keywords:
Navier-Stokes equations, Stokes eigenfunctions, Galerkin methods, transition to turbulence