Energy-stable boundary conditions based on a quadratic form: Applications to outflow/open-boundary problems in incompressible flows

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.

On a simple and effective thermal open boundary condition for convective heat transfer problems

“…Open boundary conditions for the incompressible Navier-Stokes equations have been studied extensively in a number of previous works (see e.g. [33,54,8,21,23,19,48], among others). In this paper, for the Navier-Stokes equations, we will employ the open boundary condition developed in [19].…”

“…In particular, it contains an inertial term (time derivative of temperature) and an extra nonlinear term combining the velocity and the temperature, apart from the temperature directional derivative at the boundary. The nonlinear term in the thermal open boundary condition can be analogized to a term in those open boundary conditions for the incompressible Navier-Stokes equations [19,23,48,7], and it also bears a similarity to the conditions considered in [51,47,9].…”

Numerical considerations for advection‐diffusion problems in cardiovascular hemodynamics

“…Backflow stabilization : Neumann conditions have typically been prescribed for outlet boundaries in cardiovascular flows, either through direct imposition of a known traction (ie, zero or constant pressure condition) 1,2 or, more recently, through the coupling of reduced order models (ie, lumped parameter networks) of the distal vasculature, which ultimately results in the specification of a time‐varying weak traction on the outlet face 3 . However, Neumann conditions in boundaries exhibiting partial or complete inflow are known to lead to numerical divergence 4‐16 . Specifically, prescribing a diffusive flux fails to guarantee stable energy estimates due to the unknown velocity profile at these boundaries 17 .…”

“…Specifically, prescribing a diffusive flux fails to guarantee stable energy estimates due to the unknown velocity profile at these boundaries 17 . To mitigate these difficulties associated with flow modeling, several strategies have been proposed including adding a backflow stabilization term to the boundary nodes, 7‐18 constraining the velocity to be normal to the outlet, 5 or using Lagrange multipliers to constrain the velocity profile at all or some of the outlets 4 . A comparison of these strategies determined that backflow stabilization was the most robust approach with the least impact on both the solution and computational cost 5…”

A gPAV-based unconditionally energy-stable scheme for incompressible flows with outflow/open boundaries

“…[34,17,50,11,22,1,15,36]. In the presence of outflow/open boundaries, the numerical methods employed in [16,13,20,14,21,40] also belong to the semiimplicit type schemes. The unconditionally energy-stable schemes (see e.g.…”