  We present a set of new energy-stable open boundary conditions for tackling
the backflow instability in simulations of outflow/open boundary problems for
incompressible flows. These boundary conditions are developed through two
steps: (i) devise a general form of boundary conditions that ensure the energy
stability by re-formulating the boundary contribution into a quadratic form in
terms of a symmetric matrix and computing an associated eigen problem; and (ii)
require that, upon imposing the boundary conditions from the previous step, the
scale of boundary dissipation should match a physical scale. These open
boundary conditions can be re-cast into the form of a traction-type condition,
and therefore they can be implemented numerically using the splitting-type
algorithm from a previous work. The current boundary conditions can effectively
overcome the backflow instability typically encountered at moderate and high
Reynolds numbers. These boundary conditions in general give rise to a non-zero
traction on the entire open boundary, unlike previous related methods which
only take effect in the backflow regions of the boundary. Extensive numerical
experiments in two and three dimensions are presented to test the effectiveness
and performance of the presented methods, and simulation results are compared
with the available experimental data to demonstrate their accuracy.
