Numerical modelling of flow through moving water-control gates by vortex method. Part I – problem formulation

A vortex method for simulating a flat flow within moving complex boundaries is presented. Thanks to the use of Lagrange variables (the trajectories of vortex particles) to determine the evolution of vorticity and velocity fields the method offers the possibility of modelling high Reynolds number flows. A procedure for formulating boundary conditions for flows confined by a moving impervious boundary, consisting in the superposition of three velocity fields: the first originating from the vortex particles, the second being a potential field satisfying the no-through-flow boundary condition and the third one resulting from the fact that the vortex sheet is modelled along the impervious boundaries and satisfying the no-slip-flow condition, is described. The original derivation of a formula for vortex sheet intensity, based on the single layer potential theory and leading to the formulation a second-kind Fredholm equation for vortex sheet intensity, is presented. This paper is the first part of a work covering the theoretical foundations and general description of the vortex method algorithms and boundary conditions. An example illustrating the computation of the vorticity and velocity fields of the flow through a moving hydraulic gate will be provided in the second part of the work.