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

by Rummler, B.; Noske, A.


Preprint series: 95-28, Preprints

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 investigate the Couette flow of an Newtonian fluid within a domainbetween two parallel walls moved in opposite directions.We demand constant velocities of the walls and suppose nonslipconditions of the fluid at the walls of the plane channel.We formulate the initial-boundary value problem for the velocity of thefluid by the Navier-Stokes equations for the unbounded domain between thewalls in $ {\bf R}^{3} $.We obtain by transition to non-dimensionalized quantities and equations a systemof the Navier-Stokes equations , where the physical properties of themovement are included in a parameter $ R $ (the half of the Reynolds number$ Re $ ).The restriction of the domain on an openbounded rectangular parallelepiped of $ {\bf R}^{3} $ supplementedwith periodicalconditions for the sought velocity field inthe former unbounded directions and the decomposition of the velocity fieldin two parts - the laminar flow fulfilling the nonzero boundary conditions onthe walls and the remaining velocity with homogeneous Dirichlet conditionson the walls provide equations for the determination of theremaining 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 ofthe weak solution of theseequations as an autonomous system of ordinary differential equations for thecoefficients of the eigenfunctions of the Stokes operator as the basicelements ofthe Galerkin-approximation space.\\We regard 356 Stokes eigenfunctionsto 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 severalvalues of the parameter $ R $ and a set of initial values.We use the kinetic energy of the approximated remaining velocity as a measureof turbulence .\\Our numerical investigations provide good agreements withexperimental results in the vicinity of the critical Reynolds numberat the study of the transition from laminar to turbulent flows and weobtain satisfactory results for the mean velocities of the turbulent flow.

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

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