J. Liou

A new strategy for finite element computations involving moving boundaries and interfaces—the deforming-spatial-domain/space-time procedure. I: The concept and the preliminary numerical tests

Abstract A new strategy based on the stabilized space-time finite element formulation is proposed for computations involving moving boundaries and interfaces. In the deforming-spatial-domain/space-time (DSD/ST) procedure the variational formulation of a problem is written over its space-time domain, and therefore the deformation of the spatial domain with respect to time is taken into account automatically. Because the space-time mesh is generated over the space-time domain of the problem, within each time step, the boundary (or interface) nodes move with the boundary (or interface). Whether the motion of the boundary is specified or not, the strategy is nearly the same. If the motion of the boundary is unknown, then the boundary nodes move as defined by the other unknowns at the boundary (such as the velocity or the displacement). At the end of each time step a new spatial mesh covers the new spatial domain. For computational feasibility, the finite element interpolation functions are chosen to be discontinuous in time, and the fully discretized equations are solved one space-time slab at a time.

INCOMPRESSIBLE FLOW COMPUTATIONS BASED ON THE VORTICITY-STREAM FUNCTION AND VELOCITY-PRESSURE FORMULATIONS

Abstract Finite element procedures and computations based on the velocity-pressure and vorticitystream function formulations of incompressible flows are presented. Two new multi-step velocity-pressure formulations are proposed and are compared with the vorticity-stream function and one-step formulations. The example problems chosen are the standing vortex problem and flow past a circular cylinder. Benchmark quality computations are performed for the cylinder problem. The numerical results indicate that the vorticity-stream function formulation and one of the two new multi-step formulations involve much less numerical dissipation than the one-step formulation.

Grouped element-by-element iteration schemes for incompressible flow computations

Abstract Grouped element-by-element (GEBE) iteration schemes for incompressible flows are presented in the context of vorticity- stream function formulation. The GEBE procedure is a variation of the EBE procedure and is based on arrangement of the elements into groups with no inter-element coupling within each group. With the GEBE approach, vectorization and parallel implementation of the EBE method becomes more clear. The savings in storage and CPU time are demonstrated with two unsteady flow problems.

On the downstream boundary conditions for the vorticity-stream function formulation of two-dimensional incompressible flows

Downstream boundary conditions equivalent to the homogeneous form of the natural boundary conditions associated with the velocity-pressure formulation of the Navier-Stokes equations are derived for the vorticity-stream function formulation of two-dimensional incompressible flows. Of particular interest are the zero normal and shear stress conditions at a downstream boundary.

Adaptive implicit-explicit finite element algorithms for fluid mechanics problems

Abstract The adaptive implicit-explicit (AIE) approach is presented for the finite element solution of various problems in computational fluid mechanics. In the AIE approach the elements are dynamically (adaptively) arranged into differently treated groups. The differences in treatment could be based on considerations such as the cost efficiency, the type of spatial or temporal discretization employed, the choice of field equations, etc. Several numerical tests are performed to demonstrate that with this approach substantial savings in the CPU time and memory can be achieved.

Computation of spatially periodic flows based on the vorticity-stream function formulation

Abstract Finite element solution strategies are presented for the two-dimensional, spatially periodic, viscous and inviscid, incompressible flows governed by the verticity-stream function formulation. These strategies are successfully tested on various uniperiodic and biperiodic viscous flow problems involving arrays of cylinders with Reynolds number 0 and 100. It is shown that in all cases for Reynolds number 100 the solution becomes unsteady and ceases to satisfy the symmetry conditions along the horizontal centerlines of the cylinders.