As described in \cref{chapter:LBM}, the lattice Boltzmann method is an effective tool for numerical fluid flow simulations and its coupling with other methods is still subject of intensive research in order to develop solvers for complex multi-physics models \cite{Dapelo2021,Gaedtke2018,Haussmann2021,fucik2019,Maier2017,Mink2020,Mink2022}.
As described in \cref{chapter:LBM}, the lattice Boltzmann method is an effective tool for numerical fluid flow simulations and its coupling with other methods is still subject of intensive research in order to develop solvers for complex multiphysics models \cite{Dapelo2021,Gaedtke2018,Haussmann2021,fucik2019,Maier2017,Mink2020,Mink2022}.
In this work, we investigate a novel computational approach based on the coupling of LBM with the \emph{NumDwarf} scheme described in \cref{chapter:MHFEM}, which is based on the mixed-hybrid finite element method.
As the initial step towards the development of a flexible multi-physics solver, a rather simple model coupling the Navier--Stokes equations with a linear advection--diffusion equation is considered.
As the initial step towards the development of a flexible multiphysics solver, a rather simple model coupling the Navier--Stokes equations with a linear advection--diffusion equation is considered.
The content of this chapter deals with numerical details of the coupled approach based on the paper \cite{klinkovsky2022:WT} and represents original work of the author.
%An application of the developed approach to the mathematical modeling of vapor transport in air is described in the next chapter.
An application of the developed approach is described in the next chapter.
@@ -178,7 +178,7 @@ PETSc \cite{petsc-web-page,petsc-user-ref,petsc-efficient} is an open-source lib
Unlike Trilinos, PETSc is a monolithic library developed in the C language.
Its \ic{KSP} module provides many parallel and sequential, direct and iterative solvers for linear systems and the \ic{PC} module provides preconditioners such as stationary methods, ILU factorizations, algebraic multigrid, or BDDC.
OpenFOAM \cite{openfoam:8.0,jasak:2007openfoam} is a large open-source multi-physics package based on the finite volume method.
OpenFOAM \cite{openfoam:8.0,jasak:2007openfoam} is a large open-source multiphysics package based on the finite volume method.
It also provides its own implementation of various iterative methods and preconditioners for linear systems, such as stationary methods, incomplete factorizations, CG and BiCGstab methods, and geometric agglomerated algebraic multigrid.
OpenFOAM itself does not support GPU acceleration, though the possibilities are being explored via external extensions \cite{bna2020petsc4foam,posey:2019,martineau:2020}.
\inline{experiments, setup of the simulations, validation results}
In this chapter, we use the coupled LBM-MHFEM computational approach developed in \cref{chapter:LBM-MHFEM} to simulate vapor transport in air.
The content is based on the paper \cite{klinkovsky2022:WT}.
The author developed the mathematical model and computational methodology, performed all simulations and compared the results with experimental data.
The experimental methodology was developed by Andrew Trautz and Tissa Illangasekare, the analysis of the results and overall integration of mathematical modeling and experimental work were performed collectively by all co-authors of the paper.
The chapter is organized as follows.
First, the motivation and introduction to the mathematical modeling of vapor transport in air is described in \cref{sec:WT:introduction}.
@@ -12,7 +14,7 @@ Finally, the achieved results and future work are summarized in \cref{sec:WT:con
\label{sec:WT:introduction}
Numerous proprietary or open-source computational tools are available for solving partial differential equations originating from mathematical modeling of various biological, environmental, or industrial problems.
In particular, computational software such as deal.II~\cite{bangerth:2007deal.II}, DUNE~\cite{bastian:2006DUNE}, OpenFOAM~\cite{jasak:2007openfoam}, TOUGH2~\cite{pruess:1999TOUGH2}, MFiX~\cite{syamlal:1993}, ANSYS Fluent~\cite{ansys-fluent:2009} or COMSOL Multiphysics~\cite{COMSOLMultiphysics1998} are suitable for complex multi-physics simulations involving multiphase or compositional flows.
In particular, computational software such as deal.II~\cite{bangerth:2007deal.II}, DUNE~\cite{bastian:2006DUNE}, OpenFOAM~\cite{jasak:2007openfoam}, TOUGH2~\cite{pruess:1999TOUGH2}, MFiX~\cite{syamlal:1993}, ANSYS Fluent~\cite{ansys-fluent:2009} or COMSOL Multiphysics~\cite{COMSOLMultiphysics1998} are suitable for complex multiphysics simulations involving multiphase or compositional flows.
%However, there is no universal software package that would fit all practical scenarios. %% RF: this sentence is too controversial
However, each software has limitations: the underlying numerical methods may restrict the applicability of the software; the approach for code execution may cause limited or no advantage of using high-performance architectures, in particular graphical processing units (GPUs), for the acceleration of computations; and the software design in general might make it difficult to combine different tools for solving coupled problems.
Furthermore, extending large software packages such as the aforementioned ones with novel mathematical approaches and numerical methods is challenging and unfeasible for most external users.
author={Askar, Ahmad H. and Illangasekare, Tissa H. and Trautz, Andrew and Solovský, Jakub and Zhang, Ye and Fučík, Radek},
author={Askar, Ahmad H. and Illangasekare, Tissa H. and Trautz, Andrew C. and Solovský, Jakub and Zhang, Ye and Fučík, Radek},
journal={Water Resources Research},
title={Exploring the impacts of source condition uncertainties on far-field brine leakage plume predictions in geologic storage of {CO2}: integrating intermediate-scale laboratory testing with numerical modeling},
