Lagrangian differencing dynamics for incompressible flows

Abstract

A Lagrangian meshless method is introduced for numerical simulation of flows of incompressible fluids with free surface. Poisson Pressure Equation (PPE) formulation of Navier-Stokes equations is discretised in Lagrangian context using recently introduced spatial operators based on finite differences. The advection adjusts to moving or deforming boundaries, and volume-conservation is enforced by solving a set of geometrical constraints. A technique for detecting the edge or boundary of a point cloud is introduced, which is based on the Laplacian properties for detecting jumps. Since accurate and fast discrete versions of the operator are not straightforward to implement in meshless context, the technique exploits favourable properties of a discrete Laplacian that handles highly irregular arrangements of points. A geometrical scan-sphere test is optionally performed to verify that points are on the point–cloud boundary. The boundary–detection technique is validated on significantly deformed point clouds by artificial numerical experiments, and by simulating unsteady incompressible flows. The method was validated by simulating lid-driven cavity flow, dam-break and water-entry problems, and comparing the kinematics of flow and hydrodynamic loading during the impact. The method reproduced complex free surface shapes, violent fluid-structure interaction, and pressure fields with second-order accuracy.