No abstract

An efficient strategy is developed for building suitable collision operators, to be used in a simplified version of the lattice gas Boltzmann equation. The resulting numerical scheme is shown to be linearly stable. The method is applied to the computation of the flow in a channel containing a periodic array of obstacles.

An alternative simulation procedure is proposed for lattice hydrodynamics, based on the lattice Boltzmann equation instead of on the microdynamical evolution. The averaging step, used by the latter method to derive macroscopic quantities, is suppressed, as well as the associated fluctuations. The collision operator is expressed in terms of its linearized part, and condensed into a few parameters, which can be selected, independently of a particular collision rule, to decrease viscosity as much as desired.

The hydraulic jump appearing in the viscous laminar flow of a thin liquid layer over a finite horizontal plate is studied using the boundary-layer approximation for the flow in and around the jump. The position and structure of the jump are determined by numerically solving the resulting problem with a boundary condition at the edge of the plate that expresses the matching of the layer with the shorter region where the liquid turns around and falls under the action of gravity. When the Froude number of the flow ahead of the jump is very large, the jump is much shorter than the horizontal extent of the layer, though still much longer than its depth. An asymptotic description of the inner structure of such a jump is given, building upon the analysis of Bowles & Smith for the short interaction region at the leading end of the jump. This structure consists of a fast moving separated flow in the upper part of the layer that progressively slows down by ingesting new fluid across its lower boundary, until the hydrostatically generated adverse pressure gradient makes it recirculate in the lower part of the layer. The effects of the surface tension and the cross-stream pressure variation owing to the curvature of the streamlines are taken into account in the jump and in the flow approaching the edge of the plate, showing that they can lead to quantitative and also qualitative changes of the jump structure, including a local breakdown of the boundary-layer approximation.

It is shown that the lattice Boltzmann equation (LBE) for a lattice gas provides a viable numerical method for the study of three-dimensional flows in complex geometries. Numerical results for low Reynolds number flows in a three-dimensional random medium are reported. The Darcy's law is recovered and a preliminary estimation of the permeability presented.