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UBC Theses and Dissertations

Modal finite element method for the navier-stokes equations Savor, Zlatko


A modal finite element method is presented for the steady state and transient analyses of the plane flow of incompressible Newtonian fluid. The governing restricted functional is discretized with a high precision triangular stream function finite element. Eigenvalue analysis is carried out on the resulting discretized problem, under the assumption that the nonlinear convective term is equal to zero. After truncating at various levels of approximation to obtain a reduced number of modes, the transformation to the new vector space, spanned by these modes is performed. Advantage is taken of the ..symmetric and the antisymmetric properties of the modes in order to simplify the calculations. The Lagrange multipliers technique is employed to {incorporate the nonhomo-geneous boundary conditions. The steady state analysis is carried out by utilizing the Newton-Raphson iterative procedure. The algorithm for transient analysis is based upon backward finite differences in time. Numerical results are presented for the fully developed plane Poiseuille flow, the flow in a square cavity, and the flow over a circular cylinder problems. These resultscfor the steady state are compared with the results obtained by direct finite element approach on the same grids and the results obtained by finite differences technique. It is concluded that the number of modes, which are to be retained in the analysis in order to achieve reasonable results, increases with the refinement of the finite element grid. Furthermore, the choice of modes to be used depends on the problem. Finally it is established, that this new modal method yields good results in the range of moderate Reynolds numbers with about 50% or less of the modes of the problem. This, in turn, means that the time integrations can be performed on a greatly reduced number of equations and hence potential savings in computer time are significant.

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