copy-edit - typos and wording

The chapter is organized as follows.

\Cref{sec:lbm-mhfem:problem formulation} formulates the general problem and its special case with an analytical solution for numerical analysis.

Then, \cref{sec:lbm-mhfem:algorithm,sec:lbm-mhfem:interpolation,sec:lbm-mhfem:decomposition} provide details related to the coupled computational approach and its implementation.

The final \cref{sec:lbm-mhfem:numerical analysis} describes the results of the numerical analysis using the benchmark problem from the first section.

The final \cref{sec:lbm-mhfem:numerical example} describes the results of the numerical analysis using the benchmark problem from the first section.

\section{Problem formulation}

\label{sec:lbm-mhfem:problem formulation}

The surrounding lattice points $\hat{\vec x} \in \mathcal L_{\overline{\Omega}_1}$ can be easily found and the linear or cubic interpolation in $\mathbb R^3$ can be used to obtain the velocity at $\vec x$ from the velocities at $\hat{\vec x}$.

Note that linear interpolation can be implemented more efficiently than cubic interpolation as it uses fewer input data points.

Our implementation of the cubic interpolation is not efficient and may cause the whole solver to run multiple times slower compared to the linear interpolation.

The impact of using the linear or cubic interpolation on the accuracy of the numerical solution is investigated in \cref{sec:lbm-mhfem:numerical analysis}.

The impact of using the linear or cubic interpolation on the accuracy of the numerical solution is investigated in \cref{sec:lbm-mhfem:numerical example}.

The finite element space used by MHFEM imposes requirements on the interpolation of the velocity field.

In this work, we use the Raviart--Thomas--Nédélec space of the lowest order $\mathbf{RTN}_0(\mathcal K_h)$ for the finite element--approximation of the velocity field.

Finally, it is important to note that the MHFEM schemes for \cref{eq:ADE:non-conservative,eq:ADE:conservative} do not behave equivalently with a general discrete velocity field interpolated to the mesh.

This is because the discrete velocity field computed by LBM may not satisfy \cref{eq:lbm-mhfem:ns:mass} exactly and even if it did, the interpolation scheme combines values from different locations in the flow field on a single element.

Hence, the field interpolated to the unstructured mesh may be locally non-conservative, i.e., the discrete approximation of the velocity divergence $\sum\limits_{E \in \mathcal E_K} \hat{\vec v}_E \cdot \vec n_{K,E}$ on element $K \in \mathcal K_h$ may be non-zero.

The accuracy of the numerical scheme applied to the conservative form of \cref{eq:ADE:conservative} and non-conservative form of \cref{eq:ADE:non-conservative} is investigated in \cref{sec:lbm-mhfem:numerical analysis} on a benchmark problem.

The accuracy of the numerical scheme applied to the conservative form of \cref{eq:ADE:conservative} and non-conservative form of \cref{eq:ADE:non-conservative} is investigated in \cref{sec:lbm-mhfem:numerical example} on a benchmark problem.

\section{Domain decomposition for overlapped lattice and mesh}

\label{sec:lbm-mhfem:decomposition}

The result of this decomposition procedure is illustrated in \cref{fig:lbm-mhfem:non-uniform decomposition}.

Overall, the decomposition algorithm optimizes the computational cost and memory requirements of each MPI rank at the cost of increased communication due to increased number of lattice subdomains.

\section{Numerical analysis}

\label{sec:lbm-mhfem:numerical analysis}

\section{Experimental convergence analysis}

\label{sec:lbm-mhfem:numerical example}

In this section, we study numerically the convergence of the coupled LBM-MHFEM scheme using an artificial benchmark problem described in \cref{sec:lbm-mhfem:problem formulation}.

In this section, we study the convergence of the coupled LBM-MHFEM scheme using a numerical experiment based on an artificial benchmark problem described in \cref{sec:lbm-mhfem:problem formulation}.

The aim of this section is to study the differences between the conservative and non-conservative formulations of the advection--diffusion equation \eqref{eq:ADE}.

Several variants of the MHFEM scheme from \cref{chapter:MHFEM} were used, namely explicit or implicit upwind (see \cref{sec:mhfem:upwind}), and linear or cubic interpolation of the velocity field (see \cref{sec:lbm-mhfem:interpolation}).

\begin{table}[tb]

    \centering

    \caption{Characteristics of lattice and grid resolutions used for the numerical analysis.}

    \caption{Characteristics of lattice and grid resolutions used for the experimental convergence analysis.}

    \label{tab:ADE2:resolutions}

    \begin{tabular}{lrrr}

        \toprule

\begin{table}[tb]

    \centering

    \caption{Results of the numerical analysis for different formulations and variants of the MHFEM scheme.}

    \caption{Results of the experimental convergence analysis for different formulations and variants of the MHFEM scheme.}

    \label{tab:ADE2:norms}

    \input{data/lbm-mhfem/norms_concentration.tex}

\end{table}

The presented results suggest several key areas where future experimental efforts could be improved, allowing the analysis of this model's performance to be extended and further explored.

For example, extending measurements with flow characteristics in the transverse direction (e.g., $v_y$, RMS$_y$, $\overline{v'_x v'_y}$, $\overline{v'_y v'_z}$) would allow us to compare the turbulent kinetic energy and improve the fluctuating inflow velocity condition for the simulations.

Another possible improvement is to arrange measurements in horizontal profiles in regions behind the plants, which would allow us to study the convergence of the numerical method (i.e, the effect of mesh resolution) by comparing the horizontal location of the vortical structures.

Another possible improvement is to arrange measurements in horizontal profiles in regions behind the plants, which would allow us to study the convergence of the numerical method (i.e., the effect of mesh resolution) by comparing the horizontal location of the vortical structures.

Last but not least, the applicability of the measured evaporative mass flux to the close spacing scenario EX-1 should be investigated.

Improving the methodology for measuring the evaporation from the plants would allow for prescribing more appropriate boundary conditions.

\label{sec:WT:model}

The air flow in the free space above the soil surface is governed by the Navier--Stokes equations.

As the model targets low Mach number situations ($M\!a \approx 0.003$ in the wind tunnel), the fluid is considered to be incompressible \cite{lions1998,desjardins1999}.

As the model targets low Mach number situations ($\mathrm{Ma} \approx 0.003$ in the wind tunnel), the fluid is considered to be incompressible \cite{lions1998,desjardins1999}.

For convenience, we copy \cref{eq:lbm-mhfem:ns} here from \cref{chapter:LBM-MHFEM}.

The momentum and mass conservation laws for the air are therefore represented by

\begin{subequations}\label{eq:ns}

In this chapter, we use conforming unstructured cuboidal meshes that are refined around the synthetic plants immersed in the domain.

The configuration of the solver is similar or the same as described in the previous chapter.

Based on the results from \cref{sec:lbm-mhfem:numerical analysis}, we use the linear interpolation of the velocity field and the explicit upwind scheme in MHFEM.

Based on the results from \cref{sec:lbm-mhfem:numerical example}, we use the linear interpolation of the velocity field and the explicit upwind scheme in MHFEM.

Discrete boundary conditions are applied based on the continuous description from \cref{sec:WT:boundary conditions}.

In the MHFEM scheme, Dirichlet-type conditions are used to prescribe fixed values of relative humidity and the Neumann-type condition for the diffusive flux is used to prescribe zero gradient in the normal direction, see \cref{sec:mhfem:boundary conditions} for details.

The following approaches are used the LBM part.

        \midrule

        Lattice space step & \SI{7.88}{\mm} & \SI{3.94}{\mm} & \SI{1.97}{\mm} \\

        Lattice dimensions & $496 \times 96 \times 144$ & $991 \times 224 \times 288$ & $1981 \times 480 \times 575$ \\

        No. of lattice sites & approx. $7 \cdot 10^6$ & approx. $64 \cdot 10^6$ & approx. $547 \cdot 10^6$ \\

        No. of mesh cells & approx. $1.5 \cdot 10^6$ & approx. $12 \cdot 10^6$ & approx. $96 \cdot 10^6$ \\

        No. of lattice sites & approx. $7 \times 10^6$ & approx. $64 \times 10^6$ & approx. $547 \times 10^6$ \\

        No. of mesh cells & approx. $1.5 \times 10^6$ & approx. $12 \times 10^6$ & approx. $96 \times 10^6$ \\

        Total memory & 2.5 GiB & 24 GiB & 200 GiB \\

        Base time step $\Delta t$ & \SI{1.33e-3}{\second} & \SI{3.33e-4}{\second} & \SI{8.32e-5}{\second} \\

        \multicolumn{1}{p{12em}}{Average no. of LBM iters per MHFEM step ($\floor{C_{\max} / C}$)} & 1 & 2 & 4 \\

\noindent

It is my pleasure to thank all giants on whose shoulders this thesis stands and everyone who supported me during my Ph.D. studies.

First of all, I thank my supervisors, Tomáš Oberhuber and Radek Fučík, for their expert guidance that helped to shape my research work.

I also thank Tissa Illangasekare and Andrew Trautz for their invaluable advices, ideas and repeated opportunities to visit their institutions.

I also thank Tissa Illangasekare and Andrew Trautz for their invaluable advice, ideas and repeated opportunities to visit their institutions.

Finally, I thank all members of the Mathematical Modeling Group formed around Michal Beneš for providing me with a comfortable working environment with a friendly atmosphere.

\medskip