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A hybrid conservative finite difference/finite element scheme is proposed for the solution of the unsteady incompressible Navier-Stokes equations. Using velocity-pressure variables on a non-staggered grid system, the solution is obtained with a projection method based on the resolution of a pressure Poisson equation.The new proposed scheme is derived from the finite element spatial discretization using the Galerkin method with piecewise bilinear polynomial basis functions defined on quadrilateral elements. It is applied to the pressure gradient term and to the non-linear convection term as in the so-called group finite element method. It ensures strong coupling between spatial directions, inhibiting the development of oscillations during long-term computations, as demonstrated by the validation studies.Two-and three-dimensional unsteady separated flows with open boundaries have been simulated with the proposed method using Cartesian uniform mesh grids. Several examples of calculations on the backward-facing step configuration are reported and the results obtained are compared with those given by other methods. # 1997 by John Wiley & Sons, Ltd. Int. j. numer. methods fluids 24: 833-861, 1997.

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In this paper, a computational fluid dynamic prediction method is proposed, based on the resolution of the full unsteady incompressible N avier-Stokes equations. An original numerical method is elaborated, corresponding to the threedimensionnal cartesian version of the PEGASE code for DNS and LES of incompressible flows. A new improved subgrid-scale model, the Mixed Scale Model, is proposed. This model is based on both the largest resolved scale gradients and the smallest resolved scale kinetic energy. The problem of turbulent flow past a backward facing step is used in the present study to evaluate the potentiality of LES for the prediction and the analysis of separated flows. Numerical results obtained by LES at a Reynolds number equal to 11 200 are compared with experimental data of Eaton & Johnston [3].