  We present an unconditionally energy-stable scheme for approximating the
incompressible Navier-Stokes equations on domains with outflow/open boundaries.
The scheme combines the generalized Positive Auxiliary Variable (gPAV) approach
and a rotational velocity-correction type strategy, and the adoption of the
auxiliary variable simplifies the numerical treatment for the open boundary
conditions. The discrete energy stability of the proposed scheme has been
proven, irrespective of the time step sizes. Within each time step the scheme
entails the computation of two velocity fields and two pressure fields, by
solving an individual de-coupled Helmholtz (including Poisson) type equation
with a constant pre-computable coefficient matrix for each of these field
variables. The auxiliary variable, being a scalar number, is given by a
well-defined explicit formula within a time step, which ensures the positivity
of its computed values. Extensive numerical experiments with several flows
involving outflow/open boundaries in regimes where the backflow instability
becomes severe have been presented to test the performance of the proposed
method and to demonstrate its stability at large time step sizes.
