96-29
Direct Galerkin-Approximation of the Channel Flow by Stokes Eigenfunctions
Preprint series: 96-29, Preprints
- MSC:
- 76F10 Shear flows
- 35Q30 Stokes and Navier-Stokes equations, See also {76D05, 76D07, 76N10}
- 42C15 Series of general orthogonal functions, generalized Fourier expansions, nonorthogonal expansions
- 47A75 Eigenvalue problems, See also {49Rxx}
- 76H05 Transonic flows
Abstract: We explore the plane parallel channel flows of incompressible Newtonian fluidsin an infinite layer of R 3 with a thickness of h. We presume non-slip conditionsof the fluid at the walls. We write the task as initial-boundary value problemof the Navier-Stokes equations. Now we transform these equations to a non-dimensionalized version. The limitation of the domain on an open bounded rect-angular parallelepiped in R 3 furnished with periodical conditions for the soughtvelocity field in the antecedent unbounded directions and the decomposition ofthe velocity field in two parts affords equations for the determination of theremaining velocity. The laminar velocity is fulfilling homogenous Dirichlet con-ditions. The mainspring of flow is a constant pressure gradient. The applicationof Galerkin method precipitates a corresponding system of ordinary differentialequations. The features of this system depend on one parameter Re (Reynoldsnumber based on the laminar velocity on the midline of the channel). For theGalerkin method we utilize Stokes eigenfunctions and a fix period 2l = 2 \Theta 2; 69.For our first considerations we use 356 eigenfunctions (to attain all eigenvaluesless then or equal to 4 2 ). For solving our problem in time we exert a Runge-Kutta-Fehlberg method. We define a kinetic energy as an essential quantity forthe valuation of our solution. The results of computations lead to the conclu-sions that some features of the laminar turbulent transition are describable byour Galerkin system.
Keywords: Navier-Stokes equations, plane channel flows, Galerkin method
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