On the use of Laplace's equation for pressure and a mesh-free method for 3D simulation of nonlinear sloshing in tanks

• A proof for validity of pressure Laplace's equation for incompressible fluids
• A meshless method for 3D incompressible-inviscid fluids with moving boundaries
• The use of exponential basis functions for non-linear sloshing in 3D tanks
• A precise geometry updating by a simple implicit time marching algorithm.
• Highly accurate results in 3D by few boundary nodes in tanks with various shapes

In this paper, it is first proven that instead of Poisson's equation one can use Laplace’s equation for the pressure, which is much simpler to solve, in Lagrangian simulation of incompressible inviscid Newtonian fluid flow problems starting from a divergence-free initial acceleration condition. When Laplace’s equation for the pressure is used in Newmark time integration scheme it guarantees mass conservation with O(Δt3)O(Δt3) accuracy. Next in this paper a consistent 3D mesh-free method for the solution of free surface sloshing in tanks is presented. In this method a linear summation of exponential basis functions (EBFs) is assumed as an approximation to the solution. The coefficients of the series are determined by a collocation technique used on a set of boundary nodes. These coefficients and the surface boundary nodes are updated through a time marching algorithm. Linear/non-linear 3D sloshing problems are solved in both rectangular and cylindrical basins. It is shown that the method may be used as an effective tool for 3D simulation of tanks with various shapes without the need for a huge number of domain/boundary elements for the discretization.