A Convective Boundary Condition for the Navier-Stokes Equations: Existence Analysis and Numerical Implementations

TL;DR

This paper introduces a convective boundary condition for the Navier-Stokes equations that ensures solution existence and better captures fluid outflow, improving upon the traditional do-nothing boundary condition.

Contribution

It establishes existence and uniqueness results for Navier-Stokes systems with the new convective boundary condition and compares it numerically to the do-nothing condition.

Findings

Convective boundary condition guarantees existence of solutions.

Numerical examples show improved outflow modeling.

Comparison indicates advantages over do-nothing condition.

Abstract

Due to computational complexity, fluid flow problems are mostly defined on a bounded domain. Hence, capturing fluid outflow calls for imposing an appropriate condition on the boundary where the said outflow is prescribed. Usually, the Neumann-type boundary condition called do-nothing condition is the go-to description for such outflow phenomenon However, such condition does not ensure an energy estimate for the Navier--Stokes equations - let alone establish the existence of solutions. In this paper, we analyze a convective boundary condition that will capture outflow and establish the existence of solutions to the governing equation. We shall show existence and uniqueness results for systems with mixed boundary conditions - Dirichlet condition and the convective boundary condition. The first system is a stationary equation where the Dirichlet condition is purely homogeneous, the other is where an input function is prescribed, and lastly a dynamic system with prescribed input function. We end by showing numerical examples to illustrate the difference between the current outflow condition and the usual do-nothing condition.