### 1995-28

#### Direct Galerkin-Approximation of the Plane-Parallel-Couette Flow by Stokes Eigenfunctions - New Results

Series: 1995-28, Preprints

MSC:
76F10 Shear flows
35Q30 ~Navier-Stokes equations
42C15 General harmonic expansions, frames
47A75 Eigenvalue problems
76H05 Transonic flows

Abstract:
We investigate the Couette flow of an Newtonian fluid within a domain
between two parallel walls moved in opposite directions.
We demand constant velocities of the walls and suppose nonslip
conditions of the fluid at the walls of the plane channel.
We formulate the initial-boundary value problem for the velocity of the
fluid by the Navier-Stokes equations for the unbounded domain between the
walls in ${\bf R}^{3}$.
We obtain by transition to non-dimensionalized quantities and equations a system
of the Navier-Stokes equations , where the physical properties of the
movement are included in a parameter $R$ (the half of the Reynolds number
$Re$ ).
The restriction of the domain on an open
bounded rectangular parallelepiped of ${\bf R}^{3}$ supplemented
with periodical
conditions for the sought velocity field in
the former unbounded directions and the decomposition of the velocity field
in two parts - the laminar flow fulfilling the nonzero boundary conditions on
the walls and the remaining velocity with homogeneous Dirichlet conditions
on the walls provide equations for the determination of the
remaining velocity.
We choose a fix period $2l{ }= 2\times 2,69$ for the first investigations .
By the use of the Galerkin method we obtain the Galerkin equations of
the weak solution of these
equations as an autonomous system of ordinary differential equations for the
coefficients of the eigenfunctions of the Stokes operator as the basic
elements of
the Galerkin-approximation space.\\
We regard 356 Stokes eigenfunctions
to involve at least all the Stokes eigenfunctions to eigenvalues $\lambda \le 4\pi^{2}$ in our considerations .
We solve the corresponding system of ordinary differential equations for several
values of the parameter $R$ and a set of initial values.
We use the kinetic energy of the approximated remaining velocity as a measure
of turbulence .\\
Our numerical investigations provide good agreements with
experimental results in the vicinity of the critical Reynolds number
at the study of the transition from laminar to turbulent flows and we
obtain satisfactory results for the mean velocities of the turbulent flow.

Keywords:
Navier-Stokes equations, eigenfunctions of the Stokes operator, Galerkin method