DIRECTIONAL DO-NOTHING CONDITION FOR THE NAVIER-STOKES EQUATIONS

The numerical solution of flow problems usually requires bounded domains although the physical problem may take place in an unbounded or substantially larger domain. In this case, artificial boundaries are necessary. A well established artificial boundary condition for the Navier-Stokes equations di...

Full description

Saved in:
Bibliographic Details
Published inJournal of computational mathematics Vol. 32; no. 5; pp. 507 - 521
Main Authors Braack, Malte, Mucha, Piotr Boguslaw
Format Journal Article
LanguageEnglish
Published Chinese Academy of Mathematices and System Sciences (AMSS) Chinese Academy of Sciences 01.01.2014
Subjects
Online AccessGet full text
ISSN0254-9409
1991-7139
1991-7139
DOI10.4208/jcm.1405-m4347

Cover

More Information
Summary:The numerical solution of flow problems usually requires bounded domains although the physical problem may take place in an unbounded or substantially larger domain. In this case, artificial boundaries are necessary. A well established artificial boundary condition for the Navier-Stokes equations diseretized by finite elements is the “do-nothing” condition. The reason for this is the fact that this condition appears automatically in the variational formulation after partial integration of the viscous term and the pressure gradient. This condition is one of the most established outflow conditions for Navier-Stokes but there are very few analytical insight into this boundary condition. We address the question of existence and stability of weak solutions for the Navier-Stokes equations with a “directional do-nothing” condition. In contrast to the usual “do-nothing” condition this boundary condition has enhanced stability properties. In the case of pure outflow, the condition is equivalent to the original one, whereas in the case of inflow a dissipative effect appears. We show existence of weak solutions and illustrate the effect of this boundary condition by computation of steady and non-steady flows.
Bibliography:Boundary conditions Navier-Stokes, Outflow condition, Existence.
11-2126/O1
The numerical solution of flow problems usually requires bounded domains although the physical problem may take place in an unbounded or substantially larger domain. In this case, artificial boundaries are necessary. A well established artificial boundary condition for the Navier-Stokes equations diseretized by finite elements is the “do-nothing” condition. The reason for this is the fact that this condition appears automatically in the variational formulation after partial integration of the viscous term and the pressure gradient. This condition is one of the most established outflow conditions for Navier-Stokes but there are very few analytical insight into this boundary condition. We address the question of existence and stability of weak solutions for the Navier-Stokes equations with a “directional do-nothing” condition. In contrast to the usual “do-nothing” condition this boundary condition has enhanced stability properties. In the case of pure outflow, the condition is equivalent to the original one, whereas in the case of inflow a dissipative effect appears. We show existence of weak solutions and illustrate the effect of this boundary condition by computation of steady and non-steady flows.
Malte Braack, Piotr Boguslaw Mucha(1 Mathematisches Seminar, Christian-Albrechts-Universitat zu Kiel, Ludewig-Meyn-Str. 4, D-24098 Kiel, Germany;2 Institute of Applied Mathematics and Mechanics, University of Warsaw, ul Banacha 2, 02-097 Warszawa, Poland)
ISSN:0254-9409
1991-7139
1991-7139
DOI:10.4208/jcm.1405-m4347