Nonreflecting outlet boundary conditions for incompressible flows using SPH

In this paper we implement a simple strategy, based on Jin and Braza's method, to deal with nonreflecting outlet boundary conditions for incompressible Navier-Stokes flows using the method of smoothed particle hydrodynamics (SPH). The outflow boundary conditions are implemented using an outflow zone downstream of the outlet, where particles are moved using an outgoing wave equation for the velocity field so that feedback noise from the outlet boundary is greatly reduced. For unidirectional flow across the outlet, this condition reduces to Orlanski's wave equation. The performance of the method is demonstrated through several two-dimensional test problems, including unsteady, plane Poiseuille flow, flow between two inclined plates, the Kelvin-Helmholtz instability in a channel, and flow in a constricted conduit, and in three-dimensions for turbulent flow in a $90^{\circ}$ section of a curved square pipe. The results show that spurious waves incident from the outlet are effectively absorbed and that steady-state laminar flows can be maintained for much longer times compared to periodic boundary conditions. In addition, time-dependent anisotropies in the flow, like fluid recirculations, are convected across the outlet in a very stable and accurate manner.