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

TL;DR

This paper introduces new energy-stable open boundary conditions for incompressible flow simulations that prevent backflow instability and are implementable with existing numerical algorithms, validated through extensive 2D and 3D tests.

Contribution

The authors develop a quadratic form-based framework for energy-stable boundary conditions that effectively handle backflow in incompressible flow simulations, improving stability and accuracy.

Findings

Successfully prevent backflow instability at moderate and high Reynolds numbers.

Implementable with existing splitting-type algorithms.

Validated with extensive 2D and 3D numerical experiments.

Abstract

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.