LBM-MHFEM section - updated to refer to the synthetic turbulence generator in the appendix

For simplicity, we assume that $\Omega_2$ is completely immersed in the domain $\Omega_1$ (i.e., none of the domain boundaries coincide: $\partial\Omega_1 \cap \partial\Omega_2 = \emptyset$).

The kinematic viscosity $\nu = \SI{15.52e-6}{\metre\squared\per\second}$ and the diffusion coefficient $D_0 = \SI{25.52e-6}{\metre\squared\per\second}$ are set the same as in \cref{chapter:vapor transport} (see \cref{tab:physical parameters}).

\begin{figure}[tb]

    \centering

    \includegraphics[width=0.8\textwidth]{figures/lbm-mhfem/ADE2_domain_annotated.png}

    \caption{Schematic configuration of the computational domains $\Omega_1$ and $\Omega_2$ (2D cross-section of a 3D cuboidal channel along the plane $y = 0$).}

    \label{fig:lbm-mhfem:domain}

\end{figure}

Both \cref{eq:lbm-mhfem:ns,eq:ADE} must be supplemented by suitable initial and boundary conditions.

For simplicity, the velocity field is initialized by zero ($\vec v(\vec x, 0) = \vec 0$) and the variable $\phi$ is initialized as $\phi(\vec x, 0) = \phi_0$, where $\phi_0$ is assumed to be constant.

For simplicity, the velocity field is initialized by zero ($\vec v(\vec x, 0) = \vec 0$ for $x \in \Omega_1$) and the variable $\phi$ is initialized by one ($\phi(\vec x, 0) = 1$ for $\vec x \in \Omega_2$).

The boundary conditions on $\partial \Omega_1$ are posed as follows:

\begin{itemize}

    \item

        A Neumann-type condition is used to prescribe zero gradient in the normal direction on all remaining sides of $\Omega_2$.

        %($\frac{\partial \phi}{\partial \vec x} \cdot \vec n = 0$)

\end{itemize}

The concrete inflow boundary profiles for both $\vec v$ and $\phi$ will be specified later in \cref{sec:lbm-mhfem:numerical analysis} where the benchmark problem will be studied numerically.

\begin{figure}[tb]

    \centering

    \includegraphics[width=0.8\textwidth]{figures/lbm-mhfem/ADE2_domain_annotated.png}

    \caption{Schematic configuration of the computational domains $\Omega_1$ and $\Omega_2$ (2D cross-section of a 3D cuboidal channel along the plane $y = 0$).}

    \label{fig:lbm-mhfem:domain}

\end{figure}

In order to study the differences between the conservative and non-conservative formulations, the boundary conditions are specified with the following profiles.

Turbulent flow in $\Omega_1$ is induced by prescribing a fluctuating velocity profile on the inflow boundary of $\Omega_1$ using the algorithm described in \cref{appendix:turbgen} with the following parameters: mean velocity $\overline{\vec v}_{\mathrm{in}} = [1, 0, 0]^T$~\si{\metre\per\second}, integral length scale $\mathcal L_{\mathrm{int}} = \SI{0.05}{\metre}$, turbulent kinetic energy $k = \SI{e-2}{\metre\squared\per\second\squared}$, and number of discrete modes $N_{\mathrm{modes}} = 3000$.

On the inflow boundary of $\Omega_2$ ($x = \SI{0.5}{\metre}$), we prescribe a fixed value $\phi_{\mathrm{in}} = 1$.

Given a velocity field $\vec v(\vec x,t)$ satisfying the divergence-free condition \cref{eq:lbm-mhfem:ns:mass}, this initial--boundary--value problem has a trivial analytical solution $\phi(\vec x, t) = 1$ for all $\vec x \in \Omega_2$ and $t > 0$.

\section{Computational algorithm and time adaptivity}

\label{sec:lbm-mhfem:algorithm}

        \begin{algsteps}

            \item %If $i \pmod{1000} = 0$, %% NOTE: i is undefined...

                After every 1000 iterations, recompute inflow velocity fluctuations that will be used in the next 1000 iterations.

                \todo{Velocity fluctuations were not introduced yet...}

                See \cref{appendix:turbgen} for details.

            \item Perform one iteration of LBM on the lattice $\mathcal L_{\overline{\Omega}_1}$ (i.e., perform the steps \ref{algitem:forcing} to \ref{algitem:output} from \cref{alg:LBM} on \cpageref{alg:LBM}).

            \item Set $t_L := t_L + \delta_t$.

            \item \label{step:mesh-comp}

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}.

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}.

In order to study the differences between the conservative and non-conservative formulations, the initial and boundary conditions outlined in \cref{sec:lbm-mhfem:problem formulation} are specified with the following profiles.

Initially, we set $\phi(\vec x, 0) = \phi_0 = 1$ for all $\vec x \in \Omega_2$.

On the inflow boundary of $\Omega_2$ ($x = \SI{0.5}{\metre}$), we prescribe a fixed value $\phi_{\mathrm{in}} = 1$.

Turbulent flow in $\Omega_1$ is induced using the unsteady (time-varying) inflow boundary condition described in \cref{sec:inflow:fluctuations}\todo{move the section to this chapter to avoid a forward reference} with the mean inflow velocity magnitude $v_{\max} = \SI{1}{\metre\per\second}$.\todo{the symbol $v_{\max}$ is used in the power-law profile, not in the fluctuations -- change it to $\bar{\vec v}_{\mathrm{in}}$?}

Given a divergence-free velocity field due to \cref{eq:lbm-mhfem:ns:mass}, this initial-boundary-value problem has a trivial analytical solution $\phi(\vec x, t) = 1$ for all $\vec x \in \Omega_2$ and $t > 0$.

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

Each variant was computed in three resolutions denoted as RES-A1, RES-A2, and RES-A3, see \cref{tab:ADE2:resolutions}.

Note that single precision was used in the LBM part for fluid flow and double precision was used in the MHFEM part.