In this chapter, the computational approach for solving \cref{eq:ns,eq:transport} based on the combination of the lattice Boltzmann and mixed-hybrid finite element methods is described.
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}.
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.
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.
The chapter is organized as follows.
\todo{TODO}
Both methods are introduced and their coupling in a time-adaptive manner is explained.
Finally, various implementation details are described.
\inline{define domains $\Omega_1$ and $\Omega_2$}
\section{Problem formulation}
\label{sec:lbm-mhfem:problem formulation}
\cref{eq:ns,eq:transport}
\inline{equations: copy Navier--Stokes here, add a general advection-diffusion equation (without physical interpretation)}
\inline{define domains $\Omega_1$ and $\Omega_2$ -- use the problem from the appendix in \cite{klinkovsky2022:WT}}
\section{Computational algorithm and time adaptivity}
\label{sec:algorithm}
\label{sec:lbm-mhfem:algorithm}
The previous two sections described the numerical methods used for the discretizations of \cref{eq:ns,eq:transport}, respectively.
In this section, we describe the main computational algorithm of the coupled scheme focusing on its adaptive time control.
abstract={The design and optimization of photobioreactor(s) (PBR) benefit from the development of robust and quantitatively accurate computational fluid dynamics (CFD) models, which incorporate the complex interplay of fundamental phenomena. In the present work, we propose a comprehensive computational model for tubular photobioreactors equipped with glass sponges. The simulation model requires a minimum of at least three submodels for hydrodynamics, light supply, and biomass kinetics, respectively. First, by modeling the hydrodynamics, the light–dark cycles can be detected and the mixing characteristics of the flow (besides the mass transport) can be analyzed. Second, the radiative transport model is deployed to predict the local light intensities according to the wavelength of the light and scattering characteristics of the culture. The third submodel implements the biomass growth kinetic by coupling the local light intensities to hydrodynamic information of the CO2 concentration, which allows to predict the algal growth. In combination, the novel mesoscopic simulation model is applied to a tubular PBR with transparent walls and an internal sponge structure. We showcase the coupled simulation results and validate specific submodel outcomes by comparing the experiments. The overall flow velocity, light distribution, and light intensities for individual algae trajectories are extracted and discussed. Conclusively, such insights into complex hydrodynamics and homogeneous illumination are very promising for CFD-based optimization of PBR.},
abstract={The design and optimization of photobioreactor(s) (PBR) benefit from the development of robust and quantitatively accurate computational fluid dynamics (CFD) models, which incorporate the complex interplay of fundamental phenomena. In the present work, we propose a comprehensive computational model for tubular photobioreactors equipped with glass sponges. The simulation model requires a minimum of at least three submodels for hydrodynamics, light supply, and biomass kinetics, respectively. First, by modeling the hydrodynamics, the light--dark cycles can be detected and the mixing characteristics of the flow (besides the mass transport) can be analyzed. Second, the radiative transport model is deployed to predict the local light intensities according to the wavelength of the light and scattering characteristics of the culture. The third submodel implements the biomass growth kinetic by coupling the local light intensities to hydrodynamic information of the CO2 concentration, which allows to predict the algal growth. In combination, the novel mesoscopic simulation model is applied to a tubular PBR with transparent walls and an internal sponge structure. We showcase the coupled simulation results and validate specific submodel outcomes by comparing the experiments. The overall flow velocity, light distribution, and light intensities for individual algae trajectories are extracted and discussed. Conclusively, such insights into complex hydrodynamics and homogeneous illumination are very promising for CFD-based optimization of PBR.},
doi={10.3390/en15207671},
}
@@ -1537,6 +1537,46 @@
doi={10.1007/s10334-020-00837-5},
}
@Article{Gaedtke2018,
author={Gaedtke, Maximilian and Wachter, Simon and Rädle, Matthias and Nirschl, Hermann and Krause, Mathias J.},
journal={Computers \& Mathematics with Applications},
title={Application of a lattice {B}oltzmann method combined with a {S}magorinsky turbulence model to spatially resolved heat flux inside a refrigerated vehicle},
year={2018},
issn={0898-1221},
number={10},
pages={2315--2329},
volume={76},
abstract={In this work the simulation of velocity and temperature distributions inside a refrigerated vehicle is evaluated. For this purpose a 3D double distribution lattice Boltzmann method (LBM) with the Bhatnagar-Gross-Krook (BGK) collision operator is coupled by the buoyancy force calculated with the Boussinesq approximation. This LBM is extended by a Smagorinsky subgrid method, which numerically stabilizes the BGK scheme for low resolutions and high Reynolds and Rayleigh numbers. Besides validation against the two benchmark cases porous plate and natural convection in a square cavity evaluated at resolutions of y+≈2 for Ra numbers between 103 and 1010, the method and its implementation are tested via comparison with experimental data for a refrigerated vehicle at Re≈53000. The aim of the investigation is to provide a deeper understanding of the refrigerated vehicle’s insulation processes including its thermal performance under turbulent flow conditions. Therefore, we extend this method by the half lattice division scheme for conjugate heat transfer to simulate in the geometry of a refrigerated vehicle including its insulation walls. This newly developed method combination enables us to accurately predict velocity and temperature distributions inside the cooled loading area, while spatially resolving the heat flux through the insulation walls. We simulate the time dependent heating process of the open door test and validate against measurements at four characteristic velocity and 13 temperature positions in the truck.},
doi={10.1016/j.camwa.2018.08.018},
}
@Article{Haussmann2021,
author={Haussmann, Marc and Reinshaus, Peter and Simonis, Stephan and Nirschl, Hermann and Krause, Mathias J.},
journal={Fluids},
title={Fluid-structure interaction simulation of a {C}oriolis mass flowmeter using a lattice {B}oltzmann method},
year={2021},
issn={2311-5521},
number={4},
volume={6},
abstract={In this paper, we use a fluid-structure interaction (FSI) approach to simulate a Coriolis mass flowmeter (CMF). The fluid dynamics is calculated by the open-source framework OpenLB, based on the lattice Boltzmann method (LBM). For the structural dynamics we employ the open-source software Elmer, an implementation of the finite element method (FEM). A staggered coupling approach between the two software packages is presented. The finite element mesh is created by the mesh generator Gmsh to ensure a complete open source workflow. The Eigenmodes of the CMF, which are calculated by modal analysis, are compared with measurement data. Using the estimated excitation frequency, a fully coupled, partitioned, FSI simulation is applied to simulate the phase shift of the investigated CMF design. The calculated phase shift values are in good agreement to the measurement data and verify the suitability of the model to numerically describe the working principle of a CMF.},
doi={10.3390/fluids6040167},
}
@Article{Maier2017,
author={Maier, Marie-Luise and Henn, Thomas and Thaeter, Gudrun and Nirschl, Hermann and Krause, Mathias J.},
journal={Chemical Engineering Technology},
title={Multiscale simulation with a two-way coupled lattice {B}oltzmann method and discrete element method},
year={2017},
issn={0930-7516},
month=sep,
number={9},
pages={1591--1598},
volume={40},
abstract={Abstract Simulations are helpful to better understand the dynamics and interactions of large numbers of particles and fluid in processes that occur in chemical or process engineering. Depending on whether the suspension is dense, dilute or semi-dilute, the particles and fluid can be mutually affected. Here, a lattice Boltzmann method for the fluid is combined with a discrete element method for the particles which were treated as point particles. Both are two-way coupled by drag forces, based on momentum exchange. Single-particle sedimentation is chosen as a first validation example for one- and two-way coupling. For dense suspensions, contact forces are necessary and a scenario for two colliding particles is verified before the simulation of a multiparticle block is performed.},
