Discontinuous Galerkin Finite Element Methods for the Navier-Stokes Equations in Entropy Variable Formulation

L. Pesch

Research output: ThesisPhD Thesis - Research UT, graduation UTAcademic

Abstract

Many numerical methods for fluid dynamics are suitable only for a single, idealized type of fluid. Most prominently, algorithms for compressible flow are often tailored to ideal gases and another class of schemes is designed for incompressible media. This dissertation targets a numerical method for the simulation of the flow of fluids with differing thermodynamical properties, in particular both compressible and incompressible fluids. By considering the Navier--Stokes equations in terms of certain generalized variable sets, the singularity of the incompressible limit, which arises when using conservation variables, can be avoided. Especially entropy variables, which stem from symmetrization theory, can ensure desirable properties like the fulfillment of the second law of thermodynamics. To obtain a discretization that allows local adaptation and high geometric flexibility, a space-time discontinuous Galerkin finite element method is developed. Particular attention is paid to the examination which components of the numerical method have to be adapted when using specific thermodynamical models. Numerical results for a diverse range of compressible and incompressible test cases underline the applicability of the method for various fluids and conditions. Apart from the numerical algorithms, the project also involved the further development of the object-oriented general-purpose discontinuous Galerkin finite element software framework hpGEM. This thesis presents aspects of its philosophy and design, and discusses examples of how the components can be applied. The formulation in terms of general sets of variables, a mathematical method that accommodates geometric flexibility and adaptivity, and a software environment for the implementation of numerical methods are important parts for the construction of finite element methods for gas-liquid multiphase flows.
LanguageEnglish
Awarding Institution
  • University of Twente
Supervisors/Advisors
  • van der Vegt, Jacobus J.W., Supervisor
Thesis sponsors
Award date28 Sep 2007
Place of PublicationEnschede
Publisher
Print ISBNs978-90-365-2545-9
Publication statusPublished - 28 Sep 2007

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Discontinuous Galerkin Finite Element Method
Navier-Stokes Equations
Numerical Methods
Entropy
Formulation
Fluid
Flexibility
Incompressible Limit
Second Law of Thermodynamics
Software
Symmetrization
Multiphase Flow
Liquid Flow
Ideal Gas
Discontinuous Galerkin
Compressible Fluid
Adaptivity
Compressible Flow
Fluid Dynamics
Incompressible Fluid

Keywords

  • METIS-241960
  • EWI-11168
  • IR-57928

Cite this

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title = "Discontinuous Galerkin Finite Element Methods for the Navier-Stokes Equations in Entropy Variable Formulation",
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Discontinuous Galerkin Finite Element Methods for the Navier-Stokes Equations in Entropy Variable Formulation. / Pesch, L.

Enschede : University of Twente, 2007. 174 p.

Research output: ThesisPhD Thesis - Research UT, graduation UTAcademic

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AB - Many numerical methods for fluid dynamics are suitable only for a single, idealized type of fluid. Most prominently, algorithms for compressible flow are often tailored to ideal gases and another class of schemes is designed for incompressible media. This dissertation targets a numerical method for the simulation of the flow of fluids with differing thermodynamical properties, in particular both compressible and incompressible fluids. By considering the Navier--Stokes equations in terms of certain generalized variable sets, the singularity of the incompressible limit, which arises when using conservation variables, can be avoided. Especially entropy variables, which stem from symmetrization theory, can ensure desirable properties like the fulfillment of the second law of thermodynamics. To obtain a discretization that allows local adaptation and high geometric flexibility, a space-time discontinuous Galerkin finite element method is developed. Particular attention is paid to the examination which components of the numerical method have to be adapted when using specific thermodynamical models. Numerical results for a diverse range of compressible and incompressible test cases underline the applicability of the method for various fluids and conditions. Apart from the numerical algorithms, the project also involved the further development of the object-oriented general-purpose discontinuous Galerkin finite element software framework hpGEM. This thesis presents aspects of its philosophy and design, and discusses examples of how the components can be applied. The formulation in terms of general sets of variables, a mathematical method that accommodates geometric flexibility and adaptivity, and a software environment for the implementation of numerical methods are important parts for the construction of finite element methods for gas-liquid multiphase flows.

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