# Space-time discontinuous Galerkin finite element method for two-fluid flows

W.E.H. Sollie

### Abstract

Multifluid and multiphase flows involve combinations of fluids and interfaces which separate these. These flows are of importance in many natural and industrial processes including fluidized beds and bubble columns. Often the interface is not static but moves with the fluid flow velocity. Also, interface topological changes due to breakup and coalescence processes may occur. Solutions typically have a discontinuous character at the interface between different fluids because of curvature and surface tension effects. In addition, the density and pressure differences across the interface can be very high, like in the case of liquid-gas flows. Also, the existence of shock or contact waves can introduce additional discontinuities into the problem. The aim of this research project was to develop a discontinuous Galerkin method for two-fluid flows, which is accurate, versatile and can alleviate some of the problems commonly encountered with existing methods. A novel numerical method for two-fluid flow computations is presented, which combines the space-time discontinuous Galerkin finite element discretization with the level set method and cut-cell based interface tracking. The space-time discontinuous Galerkin (STDG) finite element method offers high accuracy, an inherent ability to handle discontinuities and a very local stencil, making it relatively easy to combine with local {\it hp}-refinement. A front tracking approach is chosen because these methods ensure a sharp interface between the fluids are capable of high accuracy. The front tracking is incorporated by means of cut-cell mesh refinement, because this type of refinement is very local in nature and hence combines well with the STGD. To compute the interface dynamics the level set method (LSM) is chosen, because of its ability to deal with merging and breakup, since it was expected that the LSM combines well with the cut-cell mesh refinement and also because the LSM is easy to extend to higher dimensions. The small cell problem caused by the cut-cell refinement is solved by using a merging procedure involving bounding box elements, which improves stability and performance of the method. The interface conditions are incorporated in the numerical flux at the interface and the STDG discretization ensures that the scheme is conservative as long as the numerical fluxes are conservative. All possible cuts the 0-level set can make with square and cube shaped background elements are identified and for each cut an element refinement is defined explicitly. To ensure connectivity of the refined mesh, the $dim$-dimensional face refinements are defined equal to the $dim-1$-dimensional element refinements. It is expected that this scheme can accurately solve smaller scale problems where the interface shape is of importance and where complex interface physics are involved. To investigate the numerical properties and performance of the numerical algorithm it is applied to a number of one and two dimensional single and two-fluid test problems, including a magma - ideal gas shocktube and a helium cylinder - shock wave interaction problem. To remove oscillations in the flow field near the interface a novel interface flux is presented, which is based on the HLLC flux for a contact discontinuity and can compensate for small errors in the interface position by allowing for a small mass loss. Slope limiting was found to reduce spikes in the solution at the cost of a decrease in accuracy. It was found that the level set deformation restricted the simulation lengths. This problem can be solved by adding a level set reinitialization procedure. To improve the efficiency and stability of the two-fluid numerical algorithm it is advised to incorporate {\it hp}-refinement and a multigrid algorithm. Next, the Object Oriented Programming (OOP) design and implementation of the two-fluid method were discussed. The choice for the OOP language C++ was motivated by the general advantages of OOP such as reusability, reliability, robustness, extensibility and maintainability. In addition the use of OOP allowed for a strong connection between the numerical method and its implementation. In addition, {\it hp}GEM, an OOP package for DG methods was presented. The use of {\it hp}GEM allowed for a reduction of the development time and provided quality control and a coding standard which benefitted the sharing and maintenance of the codes.
Original language Undefined University of Twente Bokhove, Onno, Advisorvan der Vegt, Jacobus J.W., Supervisor 16 Apr 2010 Enschede University of Twente 978-90-365-3008-8 https://doi.org/10.3990/1.9789036530088 Published - 16 Apr 2010

### Keywords

• EWI-17844
• METIS-270806
• Space-time discontinuous Galerkin
• Front tracking
• Two fluid flow
• IR-70820
• Cut-cell
• Level set

### Cite this

Sollie, W.E.H.. / Space-time discontinuous Galerkin finite element method for two-fluid flows. Enschede : University of Twente, 2010. 150 p.
title = "Space-time discontinuous Galerkin finite element method for two-fluid flows",
abstract = "Multifluid and multiphase flows involve combinations of fluids and interfaces which separate these. These flows are of importance in many natural and industrial processes including fluidized beds and bubble columns. Often the interface is not static but moves with the fluid flow velocity. Also, interface topological changes due to breakup and coalescence processes may occur. Solutions typically have a discontinuous character at the interface between different fluids because of curvature and surface tension effects. In addition, the density and pressure differences across the interface can be very high, like in the case of liquid-gas flows. Also, the existence of shock or contact waves can introduce additional discontinuities into the problem. The aim of this research project was to develop a discontinuous Galerkin method for two-fluid flows, which is accurate, versatile and can alleviate some of the problems commonly encountered with existing methods. A novel numerical method for two-fluid flow computations is presented, which combines the space-time discontinuous Galerkin finite element discretization with the level set method and cut-cell based interface tracking. The space-time discontinuous Galerkin (STDG) finite element method offers high accuracy, an inherent ability to handle discontinuities and a very local stencil, making it relatively easy to combine with local {\it hp}-refinement. A front tracking approach is chosen because these methods ensure a sharp interface between the fluids are capable of high accuracy. The front tracking is incorporated by means of cut-cell mesh refinement, because this type of refinement is very local in nature and hence combines well with the STGD. To compute the interface dynamics the level set method (LSM) is chosen, because of its ability to deal with merging and breakup, since it was expected that the LSM combines well with the cut-cell mesh refinement and also because the LSM is easy to extend to higher dimensions. The small cell problem caused by the cut-cell refinement is solved by using a merging procedure involving bounding box elements, which improves stability and performance of the method. The interface conditions are incorporated in the numerical flux at the interface and the STDG discretization ensures that the scheme is conservative as long as the numerical fluxes are conservative. All possible cuts the 0-level set can make with square and cube shaped background elements are identified and for each cut an element refinement is defined explicitly. To ensure connectivity of the refined mesh, the $dim$-dimensional face refinements are defined equal to the $dim-1$-dimensional element refinements. It is expected that this scheme can accurately solve smaller scale problems where the interface shape is of importance and where complex interface physics are involved. To investigate the numerical properties and performance of the numerical algorithm it is applied to a number of one and two dimensional single and two-fluid test problems, including a magma - ideal gas shocktube and a helium cylinder - shock wave interaction problem. To remove oscillations in the flow field near the interface a novel interface flux is presented, which is based on the HLLC flux for a contact discontinuity and can compensate for small errors in the interface position by allowing for a small mass loss. Slope limiting was found to reduce spikes in the solution at the cost of a decrease in accuracy. It was found that the level set deformation restricted the simulation lengths. This problem can be solved by adding a level set reinitialization procedure. To improve the efficiency and stability of the two-fluid numerical algorithm it is advised to incorporate {\it hp}-refinement and a multigrid algorithm. Next, the Object Oriented Programming (OOP) design and implementation of the two-fluid method were discussed. The choice for the OOP language C++ was motivated by the general advantages of OOP such as reusability, reliability, robustness, extensibility and maintainability. In addition the use of OOP allowed for a strong connection between the numerical method and its implementation. In addition, {\it hp}GEM, an OOP package for DG methods was presented. The use of {\it hp}GEM allowed for a reduction of the development time and provided quality control and a coding standard which benefitted the sharing and maintenance of the codes.",
keywords = "EWI-17844, METIS-270806, Space-time discontinuous Galerkin, Front tracking, Two fluid flow, IR-70820, Cut-cell, Level set",
author = "W.E.H. Sollie",
note = "10.3990/1.9789036530088",
year = "2010",
month = "4",
day = "16",
doi = "10.3990/1.9789036530088",
language = "Undefined",
isbn = "978-90-365-3008-8",
publisher = "University of Twente",
school = "University of Twente",

}

Enschede : University of Twente, 2010. 150 p.

TY - THES

T1 - Space-time discontinuous Galerkin finite element method for two-fluid flows

AU - Sollie, W.E.H.

N1 - 10.3990/1.9789036530088

PY - 2010/4/16

Y1 - 2010/4/16

N2 - Multifluid and multiphase flows involve combinations of fluids and interfaces which separate these. These flows are of importance in many natural and industrial processes including fluidized beds and bubble columns. Often the interface is not static but moves with the fluid flow velocity. Also, interface topological changes due to breakup and coalescence processes may occur. Solutions typically have a discontinuous character at the interface between different fluids because of curvature and surface tension effects. In addition, the density and pressure differences across the interface can be very high, like in the case of liquid-gas flows. Also, the existence of shock or contact waves can introduce additional discontinuities into the problem. The aim of this research project was to develop a discontinuous Galerkin method for two-fluid flows, which is accurate, versatile and can alleviate some of the problems commonly encountered with existing methods. A novel numerical method for two-fluid flow computations is presented, which combines the space-time discontinuous Galerkin finite element discretization with the level set method and cut-cell based interface tracking. The space-time discontinuous Galerkin (STDG) finite element method offers high accuracy, an inherent ability to handle discontinuities and a very local stencil, making it relatively easy to combine with local {\it hp}-refinement. A front tracking approach is chosen because these methods ensure a sharp interface between the fluids are capable of high accuracy. The front tracking is incorporated by means of cut-cell mesh refinement, because this type of refinement is very local in nature and hence combines well with the STGD. To compute the interface dynamics the level set method (LSM) is chosen, because of its ability to deal with merging and breakup, since it was expected that the LSM combines well with the cut-cell mesh refinement and also because the LSM is easy to extend to higher dimensions. The small cell problem caused by the cut-cell refinement is solved by using a merging procedure involving bounding box elements, which improves stability and performance of the method. The interface conditions are incorporated in the numerical flux at the interface and the STDG discretization ensures that the scheme is conservative as long as the numerical fluxes are conservative. All possible cuts the 0-level set can make with square and cube shaped background elements are identified and for each cut an element refinement is defined explicitly. To ensure connectivity of the refined mesh, the $dim$-dimensional face refinements are defined equal to the $dim-1$-dimensional element refinements. It is expected that this scheme can accurately solve smaller scale problems where the interface shape is of importance and where complex interface physics are involved. To investigate the numerical properties and performance of the numerical algorithm it is applied to a number of one and two dimensional single and two-fluid test problems, including a magma - ideal gas shocktube and a helium cylinder - shock wave interaction problem. To remove oscillations in the flow field near the interface a novel interface flux is presented, which is based on the HLLC flux for a contact discontinuity and can compensate for small errors in the interface position by allowing for a small mass loss. Slope limiting was found to reduce spikes in the solution at the cost of a decrease in accuracy. It was found that the level set deformation restricted the simulation lengths. This problem can be solved by adding a level set reinitialization procedure. To improve the efficiency and stability of the two-fluid numerical algorithm it is advised to incorporate {\it hp}-refinement and a multigrid algorithm. Next, the Object Oriented Programming (OOP) design and implementation of the two-fluid method were discussed. The choice for the OOP language C++ was motivated by the general advantages of OOP such as reusability, reliability, robustness, extensibility and maintainability. In addition the use of OOP allowed for a strong connection between the numerical method and its implementation. In addition, {\it hp}GEM, an OOP package for DG methods was presented. The use of {\it hp}GEM allowed for a reduction of the development time and provided quality control and a coding standard which benefitted the sharing and maintenance of the codes.

AB - Multifluid and multiphase flows involve combinations of fluids and interfaces which separate these. These flows are of importance in many natural and industrial processes including fluidized beds and bubble columns. Often the interface is not static but moves with the fluid flow velocity. Also, interface topological changes due to breakup and coalescence processes may occur. Solutions typically have a discontinuous character at the interface between different fluids because of curvature and surface tension effects. In addition, the density and pressure differences across the interface can be very high, like in the case of liquid-gas flows. Also, the existence of shock or contact waves can introduce additional discontinuities into the problem. The aim of this research project was to develop a discontinuous Galerkin method for two-fluid flows, which is accurate, versatile and can alleviate some of the problems commonly encountered with existing methods. A novel numerical method for two-fluid flow computations is presented, which combines the space-time discontinuous Galerkin finite element discretization with the level set method and cut-cell based interface tracking. The space-time discontinuous Galerkin (STDG) finite element method offers high accuracy, an inherent ability to handle discontinuities and a very local stencil, making it relatively easy to combine with local {\it hp}-refinement. A front tracking approach is chosen because these methods ensure a sharp interface between the fluids are capable of high accuracy. The front tracking is incorporated by means of cut-cell mesh refinement, because this type of refinement is very local in nature and hence combines well with the STGD. To compute the interface dynamics the level set method (LSM) is chosen, because of its ability to deal with merging and breakup, since it was expected that the LSM combines well with the cut-cell mesh refinement and also because the LSM is easy to extend to higher dimensions. The small cell problem caused by the cut-cell refinement is solved by using a merging procedure involving bounding box elements, which improves stability and performance of the method. The interface conditions are incorporated in the numerical flux at the interface and the STDG discretization ensures that the scheme is conservative as long as the numerical fluxes are conservative. All possible cuts the 0-level set can make with square and cube shaped background elements are identified and for each cut an element refinement is defined explicitly. To ensure connectivity of the refined mesh, the $dim$-dimensional face refinements are defined equal to the $dim-1$-dimensional element refinements. It is expected that this scheme can accurately solve smaller scale problems where the interface shape is of importance and where complex interface physics are involved. To investigate the numerical properties and performance of the numerical algorithm it is applied to a number of one and two dimensional single and two-fluid test problems, including a magma - ideal gas shocktube and a helium cylinder - shock wave interaction problem. To remove oscillations in the flow field near the interface a novel interface flux is presented, which is based on the HLLC flux for a contact discontinuity and can compensate for small errors in the interface position by allowing for a small mass loss. Slope limiting was found to reduce spikes in the solution at the cost of a decrease in accuracy. It was found that the level set deformation restricted the simulation lengths. This problem can be solved by adding a level set reinitialization procedure. To improve the efficiency and stability of the two-fluid numerical algorithm it is advised to incorporate {\it hp}-refinement and a multigrid algorithm. Next, the Object Oriented Programming (OOP) design and implementation of the two-fluid method were discussed. The choice for the OOP language C++ was motivated by the general advantages of OOP such as reusability, reliability, robustness, extensibility and maintainability. In addition the use of OOP allowed for a strong connection between the numerical method and its implementation. In addition, {\it hp}GEM, an OOP package for DG methods was presented. The use of {\it hp}GEM allowed for a reduction of the development time and provided quality control and a coding standard which benefitted the sharing and maintenance of the codes.

KW - EWI-17844

KW - METIS-270806

KW - Space-time discontinuous Galerkin

KW - Front tracking

KW - Two fluid flow

KW - IR-70820

KW - Cut-cell

KW - Level set

U2 - 10.3990/1.9789036530088

DO - 10.3990/1.9789036530088

M3 - PhD Thesis - Research UT, graduation UT

SN - 978-90-365-3008-8

PB - University of Twente

CY - Enschede

ER -