A pressure correction scheme for generalized form of energy-stable open boundary conditions for incompressible flows

We present a generalized form of open boundary conditions, and an associated numerical algorithm, for simulating incompressible flows involving open or outflow boundaries. The generalized form represents a family of open boundary conditions, which all ensure the energy stability of the system, even in situations where strong vortices or backflows occur at the open/outflow boundaries. Our numerical algorithm for treating these open boundary conditions is based on a rotational pressure correction-type strategy, with a formulation suitable for $C^0$ spectral-element spatial discretizations. We have introduced a discrete equation and associated boundary conditions for an auxiliary variable. The algorithm contains constructions that prevent a numerical locking at the open/outflow boundary. In addition, we have also developed a scheme with a provable unconditional stability for a sub-class of the open boundary conditions. Extensive numerical experiments have been presented to demonstrate the performance of our method for several flow problems involving open/outflow boundaries. We compare simulation results with the experimental data to demonstrate the accuracy of our algorithm. Long-time simulations have been performed for a range of Reynolds numbers at which strong vortices or backflows occur at the open/outflow boundaries. We show that the open boundary conditions and the numerical algorithm developed herein produce stable simulations in such situations.