### 2001-14

#### Some Strange Numerical Solutions of the Non-stationary Navier-Stokes Equations in Pipes

by Rummler, B..

Series: 2001-14, Preprints

MSC:
76F99 None of the above, but in this section
34A34 Nonlinear equations and systems, general
35Q30 ~Navier-Stokes equations

Abstract:
A general class of boundary-pressure-driven flows of
incompressible Newtonian fluids in three-dimensional pipes
with known steady laminar realizations is investigated.
Considering the laminar velocity as a 3D-vector-function
of the cross-section-circle arguments, we fix the scale
for the velocity by the $L_{2}$-norm of the laminar velocity.
The usual new variables are introduced to get
dimension-free Navier-Stokes equations. The
characteristic physical and geometrical quantities are subsumed in
the energetic Reynolds number $Re$ and a parameter $\psi$,
which involves the energetic ratio and the directions of
the boundary-driven part and the pressure-driven part
of the laminar flow.
The solution of non-stationary dimension-free Navier-Stokes
equations is sought in the form
$\underline{{\bf u}}\,=\,{\bf u}_{L}\,+\,{\bf u}$,
where ${\bf u}_{L}$ is the scaled laminar velocity and
periodical conditions in center-line-direction
are prescribed for $\bf u$. An autonomous system (S) of
ordinary differential equations for the time-dependent
coeff\/icients of the spatial Stokes eigenfunction is got by
application of the Galerkin-method
to the dimension-free Navier-Stokes equations for $\bf u$.
The finite-dimensional approximations
${\bf u}_{N(\lambda)}$ of ${\bf u}$ are defined
in the usual way. A class of timely periodical solutions
near to the laminar velocities
but different from them was found by parameter studies
for the numerical solution
of finite-dimensional subsystems of (S).
This class of timely periodical strange numerical solutions
seems to stay for one of the first links in the bifurcation chain to
turbulence.

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