Energy Inequalities and Outflow Boundary Conditions for the Navier–Stokes Equations

Abstract

Artificial boundary conditions play important roles in numerical simulation of real-world flow problems. A typical example is a class of outflow boundary conditions for blood flow simulations in large arteries. The common outflow boundary conditions are a prescribed constant pressure or traction, a prescribed velocity profiles, and a free-traction (do-nothing) conditions. However, the flow distribution and pressure field are unknown and cannot be prescribed at the outflow boundary in many simulations. Moreover, with those boundary conditions, we are unable to obtain energy inequalities. This disadvantage may cause numerical instability in unstationary 3D simulations. In this paper, we examine some outflow boundary conditions for the Navier–Stokes equations from the viewpoint of energy inequalities. Further, we propose an energy-preserving unilateral condition and review mathematical results including the well-posedness, variational inequality formulations, and finite element approximations.