01-14
Some Strange Numerical Solutions of the Non-stationary Navier-Stokes Equations in Pipes
by Rummler, B.
Preprint series: 01-14, Preprints
- MSC:
- 76F99 None of the above but in this section
- 34A34 Nonlinear equations and systems, general
- 35Q30 Stokes and Navier-Stokes equations, See also {76D05, 76D07, 76N10}
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 coefficients 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
Notes: The right MSC 2000 are: 76F06 and 34A34, 35Q30, 65M60, 76F65
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