Finite element method for incompressible viscous flows with adaptive hybrid grids

A new adaptive finite element numerical method has been developed for the unsteady Navier-Stokes equations of incompressible flow in two dimensions. The momentum equations combined with a pressure correction equation are solved employing a nonstaggered grid. The solution is advanced in time with an explicit/implicit marching scheme. An adaptive algorithm has been implemented, which refines the grid locally to resolve detected flow features. A combination of quadrilateral as well as triangular cells provides a stable and accurate numerical treatment of grid interfaces that are located within regions of high gradients. Applications of the developed adaptive algorithm include both steady and unsteady flows, with low and high Reynolds numbers. Comparisons with analytical as well as experimental data evaluate accuracy and robustness of the method. I. Introduction I NCOMPRESSIBLE flows are frequently encountered in engineering applications. During the past two decades a significant number of numerical algorithms have been developed for solution of the incompressible Navier-Stokes equations.1 liie lack of pressure term in the continuity equation makes solution of the momentum equations with the divergence-free constraint more difficult. In the case of incompressible flows, the conservation of mass acts as a constraint condition that the velocity field must satisfy, whereas in compressible flows, the conservation of mass is given through a partial differential equation for the temporal variation of density. The infinite speed of sound in the incompressible case requires an implicit treatment of the pressure. Furthermore, spatial discretization for pressure and velocity may produce oscillatory solutions. One approach followed is to formulate the equations in terms of a stream function and a vorticity. Extension of this method to three dimensions is not possible. Different formulations have been used in three dimensions, such as the vorticity-velocity approach. Another approach is to use the compressible flow equations and solve them for low Mach numbers. The required time step for such computations is very small because the speed of sound approaches infinity at the incompressible limit. A method that uses compressible-like governing equations is the artificial compressibility approach.2"^ A time derivative of the pressure is added to the continuity equation, and the incompressible flowfield is treated as compressible during the transient stage. Time accuracy of the simulation is usually not preserved. Another class of algorithms uses a special Poisson equation for the pressure field.5"8 The usual computational procedure is to assume an initial pressure field, and then an iterative process is defined until the continuity equation is satisfied. A major issue of the corresponding pressure and velocity spatial discretization is oscillations in the pressure field. To reject these modes, staggered grids have been employed by several of these algorithms.9'10 On the other hand, employment of nonstaggered grids11"13 requires dissipation in the algorithms. Stability of both approaches with high-Reynolds-number flows is an important issue. A review of numerical methods for incompressible flows, as well as references