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We present a set of new energy-stable open boundary conditions for tacklingthe backflow instability in simulations of outflow/open boundary problems forincompressible flows. These boundary conditions are developed through twosteps: (i) devise a general form of boundary conditions that ensure the energystability by re-formulating the boundary contribution into a quadratic form interms of a symmetric matrix and computing an associated eigen problem; and (ii)require that, upon imposing the boundary conditions from the previous step, thescale of boundary dissipation should match a physical scale. These openboundary conditions can be re-cast into the form of a traction-type condition,and therefore they can be implemented numerically using the splitting-typealgorithm from a previous work. The current boundary conditions can effectivelyovercome the backflow instability typically encountered at moderate and highReynolds numbers. These boundary conditions in general give rise to a non-zerotraction on the entire open boundary, unlike previous related methods whichonly take effect in the backflow regions of the boundary. Extensive numericalexperiments in two and three dimensions are presented to test the effectivenessand performance of the presented methods, and simulation results are comparedwith the available experimental data to demonstrate their accuracy.

Energy-Stable Boundary Conditions Based on a Quadratic Form: Applications to Outflow/Open-Boundary Problems in Incompressible Flows