99-35
Investigation of the Nonstationary Navier-Stokes Equations in special Domains - Transition to Turbulence
by Rummler, B.
Preprint series: 99-35, Preprints
- MSC:
- 34A34 Nonlinear equations and systems, general
- 35Q30 Stokes and Navier-Stokes equations, See also {76D05, 76D07, 76N10}
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 define 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 coefficients 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 fixed periods $ 2l $ and the first $N(l)$ 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
Notes: MSC 2000: 34A34, 35Q30, 65M60, 76F06, 76F65
The author(s) agree, that this abstract may be stored asfull text and distributed as such by abstracting services.