In the vicinity of boundaries of all kinds, the kernel summation process in SPH is deficient and requires some form of correction to avoid spurious and physically incorrect flow phenomena being generated.  This is particularly the case for open boundaries where apart from the SPH particles leaving and entering the domain, waves within the solution must be allowed to leave the domain without generating numerical reflection.  Until now, the only way to implement open boundaries in SPH was to use periodic (or recirculating) boundaries which are very limited in scope and application.

To tackle the more general problem, I have utilised the Reproducing Kernel Particle Method (RKPM) technique developed by Wing-Kam Lui & others at Northwestern University to correct the SPH kernel along with their Essential Boundary Conditions (EBCs) method to enforce a specified boundary value.

1-D Density Wave on a Uniform Flow
This seemingly straightfoward test is difficult.  It involves a initial density profile on a uniform stream of particles travelling at 0.8 m/s.  The density wave splits into a d'Alembert-like solution with one wave travelling upstream against the flow and one wave downstream with the flow.  In the animation one can observe particles leaving the downstream boundary and new particles arriving at the upstream boundary with the correct local properties while the density waves propagate out of the domain unaltered (avi: 3.0MB).

The same test as above but in 2-D.  The enforcement of linear reproducing conditions of RKPM does not entirely capture second-order accurate effects which one can see in the v-velocity.  However, in this case, this turns out not to be important.  Slip lateral boundary conditions have also been implemented here using SPH-RKPM, see below (avi: 3.6MB)

The same technique of combining SPH with RKPM can be used to specify solid boundary conditions by ensuring we have velocity equal to zero normal to a solid object.  This includes modelling both slip and non-slip boundary conditions.

This is a variant of the open boundaries version, except that this time, the density wave reflects off solid walls.  The wave is eventually caused to decay due to viscosity (avi: 11MB).

A well known test to benchmark any scheme.  In addition to modelling the interior viscous flow, this tests the ability of the scheme to model both open boundaries and non-slip walls along the sides (avi: 26MB).

The following were my initial efforts to implement open boundary conditions before using some form of kernel correction.

This is a 2-D SPH simulation (5000 particles) avi movie for Steady Transcritical Flow over a hump with a stationary shock in an open channel including the analytical solution. 

This is benchmark test case for Shock-Capturing shallow water models and is a difficult test of SPH and open boundaries.

And here is an animation of 2-D flow past a submerged hydrofoil which is trying reproduce the experiments of Duncan (1981):