chapter on vapor transport - copy-edit

Verification of the developed approach against exact solutions is not possible, since there are no known exact analytical solutions for a coupled system of Navier--Stokes and transport equations.

We therefore turn to controlled experimental data for model validation purposes.

The data used for validation in this paper\todo{thesis} include velocity and relative humidity profiles measured in a low speed, climate controlled wind tunnel where 3D turbulent flow above a soil surface with cuboidal bluff bodies were investigated in \cite{trautz2017role}.

The data used for validation in this thesis include velocity and relative humidity profiles measured in a low speed, climate controlled wind tunnel where 3D turbulent flow above a soil surface with cuboidal bluff bodies were investigated in \cite{trautz2017role}.

The wind tunnel used for these measurements was designed specifically for the investigation of coupled soil–plant–atmosphere processes as soil-test beds containing any desired soil(s) and vegetation can be interfaced at a sufficiently large scale along a \SI{7.4}{\metre} long test-section.

Validating the numerical solver using experimental data obtained in controlled environments with simplified geometry is an important first step towards the development of a more advanced solver that could be used to guide sensitivity analysis for measurements in regions with high uncertainty, inform sampling in future experimental efforts, or even to predict the behavior in regions where measurements are not possible.

\subsection{Experimental setup and methodology}

\label{sec:experimental setup}

All experimental data used in this paper\todo{thesis (or cite the paper)} were generated in the Center for Experimental Study of Subsurface Environmental Processes (CESEP) wind tunnel--porous media test facility now located at the US Army Engineer Research and Development Center (ERDC) Synthetic Environment for Near-Surface Sensing and Experimentation (SENSE) Research Facility.

All experimental data used in this thesis and the paper \cite{klinkovsky2022:WT} were generated in the Center for Experimental Study of Subsurface Environmental Processes (CESEP) wind tunnel--porous media test facility now located at the US Army Engineer Research and Development Center (ERDC) Synthetic Environment for Near-Surface Sensing and Experimentation (SENSE) Research Facility.

The facility is centered around a closed-circuit, climate-controlled, low-speed wind tunnel that can be interfaced with soil test-beds of varying size.

The test facility was designed specifically for the investigation of coupled soil--plant--atmosphere processes, including air flow and heat and mass transport at a 1:1 scale; the wind tunnel meets similarity criteria and is therefore suitable for momentum scaling studies as well.

A brief description of the components relevant to this paper\todo{thesis (or cite the paper)} is presented below for the convenience of the reader.

A brief description of the components relevant to this work is presented below for the convenience of the reader.

Details concerning the test facility can be found in~\cite{trautz2017development,trautz2018experimental}.

The wind tunnel was interfaced with a \SI{7.15}{\metre} long and \SI{0.3}{\metre} wide soil test-bed along the centerline of its \SI{7.4}{\metre} long, \SI{1}{\metre} wide, and \SI{1}{\metre} tall test-section.

\end{table}

The measurements generated by \cite{trautz2017development,trautz2017role} are available as a public dataset~\cite{trautz:dataset}.

Note that only a subset of these measurements is relevant for the purpose of this paper,\todo{thesis (or cite the paper)} namely the airflow properties and relative humidity above the soil surface.

Note that only a subset of these measurements is relevant for the purpose of this work, namely the airflow properties and relative humidity above the soil surface.

The airflow and relative humidity measurement locations were varied between the three configurations based on the spacing distance between the synthetic plants and the resulting flow regime created.

The measurement locations are highlighted in the figures in the last section.

As in \cref{eq:ns:mass}, no sources/sinks are considered in \cref{eq:transport:rho_alpha}.

The diffusive flux is given by the Fick's law \cite{bird:2002} as

\begin{equation} \label{eq:transport:Fick law}

    \vec J_\alpha = - \rho D \nabla \omega_\alpha,

    \vec J_\alpha = - \rho D_\alpha \nabla \omega_\alpha,

\end{equation}

where $D$\todo{$D$ was used to denote the spatial dimension in previous chapters} [\si{\metre\squared\per\second}] is the molecular diffusivity coefficient and $\omega_\alpha = \rho_\alpha / \rho$ [-] is the mass fraction of the component $\alpha$ in the mixture.

where $D_\alpha$ [\si{\metre\squared\per\second}] denotes the molecular diffusivity coefficient of the component $\alpha$ in the mixture and $\omega_\alpha = \rho_\alpha / \rho$ [-] is the mass fraction of the component $\alpha$ in the mixture.

\inline{to deal with LBM compressibility ($\nabla \cdot \vec v \ne 0$), we would likely have to solve the general equation (\cref{eq:transport:rho_alpha_2} below) or do different manipulations with it}

\inline{the continuity eq is $\nabla \cdot \vec v = 0$ here, but LBM solves $\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec v) = 0$ (according to LBMAT)}

\todo{The following is not necessary to arrive to \cref{eq:transport}.}

Using \cref{eq:transport:Fick law} and $\rho_\alpha = \omega_\alpha \rho$ in \cref{eq:transport:rho_alpha} gives

\begin{equation} \label{eq:transport:rho_alpha_2}

    \frac{\partial (\rho \omega_\alpha)}{\partial t} + \nabla \cdot \left( \rho \omega_\alpha \vec v - \rho D \nabla \omega_\alpha \right) = f_{\rho_\alpha}.

    \frac{\partial (\rho \omega_\alpha)}{\partial t} + \nabla \cdot \left( \rho \omega_\alpha \vec v - \rho D_\alpha \nabla \omega_\alpha \right) = f_{\rho_\alpha}.

\end{equation}

Applying the product rule in \cref{eq:transport:rho_alpha_2} and using the continuity equation $\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec v) = 0$ \todo{But we have \cref{eq:ns:mass}!?!}, we obtain

\begin{equation} \label{eq:transport:omega_alpha:general}

    \rho \left( \frac{\partial \omega_\alpha}{\partial t} + \nabla \cdot \left( \omega_\alpha \vec v \right) \right)

    - \nabla \cdot \left( \rho D \nabla \omega_\alpha \right) = f_{\rho_\alpha}.

    - \nabla \cdot \left( \rho D_\alpha \nabla \omega_\alpha \right) = f_{\rho_\alpha}.

\end{equation}

Assuming a mixture with a constant density $\rho$, \cref{eq:transport:omega_alpha:general} can be rewritten as

\begin{equation} \label{eq:transport:omega_alpha}

    \frac{\partial \omega_\alpha}{\partial t} + \nabla \cdot \left( \omega_\alpha \vec v - D \nabla \omega_\alpha \right) = f_{\omega_\alpha},

    \frac{\partial \omega_\alpha}{\partial t} + \nabla \cdot \left( \omega_\alpha \vec v - D_\alpha \nabla \omega_\alpha \right) = f_{\omega_\alpha},

\end{equation}

where $f_{\omega_\alpha} = f_{\rho_\alpha} / \rho$.

\inline{in the end we got a conservative equation for $\omega_\alpha$, but without the $\rho \vec v$ term -> problem for LBM}

Using $\rho_\alpha = \phi \rho^*_\alpha$, \cref{eq:transport:Fick law} and the assumption of constant density $\rho$, \cref{eq:transport:rho_alpha} transforms to the conservative transport equation

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

    \begin{equation}\label{eq:transport:conservative}

        \frac{\partial \phi}{\partial t} + \nabla \cdot \left( \phi \vec v - D \nabla \phi \right) = 0. %f_\phi.

        \frac{\partial \phi}{\partial t} + \nabla \cdot \left( \phi \vec v - D_{H_2O} \nabla \phi \right) = 0. %f_\phi.

    \end{equation}

    Combining \cref{eq:transport:conservative,eq:ns:mass} leads to the non-conservative form

    \begin{equation}\label{eq:transport:non-conservative}

        \frac{\partial \phi}{\partial t} + \vec v \cdot \nabla \phi - \nabla \cdot \left( D \nabla \phi \right) = 0. %f_\phi,

        \frac{\partial \phi}{\partial t} + \vec v \cdot \nabla \phi - \nabla \cdot \left( D_{H_2O} \nabla \phi \right) = 0. %f_\phi,

    \end{equation}

\end{subequations}

%where $f_\phi = f_{\rho_\alpha} / \rho^*_\alpha$.

%Assuming that water vapor is not formed or lost due to chemical reactions with other components of the air, we can set $f_\phi = 0$.

Although a slightly variable temperature distribution above the soil tank was observed during the experiments \cite{trautz2017role}, we assume its impact on the density, kinematic viscosity, molecular diffusivity, and relative humidity to be negligible compared to the sensor accuracies.

The isothermal model given by \cref{eq:ns,eq:transport} is used with constant parameters $\rho$, $\nu$, and $D$.

The isothermal model given by \cref{eq:ns,eq:transport} is used with constant parameters $\rho$, $\nu$, and $D_{H_2O}$.

Furthermore, the fluid density $\rho$ is assumed to be independent of the relative humidity $\phi$ and the effect of gravity is neglected due to the dimensions of the experimental facility.

Model parameters for air under standard atmospheric conditions are given in~\cref{tab:physical parameters}.

\begin{table}[!h]

        Density $\rho$ \cite{engeneeringtoolbox:air-viscosity} & \SI{1.184}{\kg\per\cubic\metre} \\

        %Dynamic viscosity $\mu$ \cite{engeneeringtoolbox:air-viscosity} & \SI{18.37e-6}{\kg\per\metre\per\second} \\

        Kinematic viscosity $\nu$ \cite{engeneeringtoolbox:air-viscosity} & \SI{15.52e-6}{\metre\squared\per\second} \\

        Molecular diffusivity $D$ of water vapor in air \cite{massman1998} & \SI{25.52e-6}{\metre\squared\per\second} \\

        Molecular diffusivity $D_{H_2O}$ of water vapor in air \cite{massman1998} & \SI{25.52e-6}{\metre\squared\per\second} \\

        \bottomrule

    \end{tabular}

    \caption{Model parameters for air under standard atmospheric conditions (\SI{25}{\degreeCelsius} and pressure of \SI{1}{bar}).}

Note that \cref{eq:ns} is solved in domain $\Omega_1$ and \cref{eq:transport} is solved in domain $\Omega_2$.

Since \cref{eq:ns,eq:transport} are coupled only via the velocity field $\vec v$, \cref{eq:ns} can be solved without \cref{eq:transport} and the latter can be solved only in a subdomain of $\Omega_1$, i.e., $\Omega_2 \subset \Omega_1$.

For the purpose of this paper,\todo{thesis (or cite the paper)} we are interested in simulating only the free flow region, where the soil-atmosphere and synthetic plant (bluff body)--atmosphere interfaces are treated using boundary conditions (described in \cref{sec:boundary conditions}).

For the purpose of this thesis we are interested in simulating only the free flow region, where the soil-atmosphere and synthetic plant (bluff body)--atmosphere interfaces are treated using boundary conditions (described in \cref{sec:boundary conditions}).

Hence, the computational domain is considered as shown in \cref{fig:domain}.

Based on the experimental setup described in \cref{sec:experimental setup}, the computational domain $\Omega_1$ is defined as an inset of the whole test-section starting at a downstream distance of \SI{-3.507}{\metre} from the test-section inlet.

The total dimensions of $\Omega_1$ are approximately\todo[color=gray]{Technically, the exact length and width depend on the resolution due to rounding.} $3.89 \times 1 \times 1.13$\si{~\cubic\metre}.

The total dimensions of $\Omega_1$ are approximately\todo[color=gray]{Technically, the exact length and width depend on the resolution due to rounding.} \qtyproduct[product-symbol = \ensuremath{\,\times\,}]{3.89 x 1 x 1.13}{\metre}.

The upper side of the domain $\Omega_1$ coincides with the inclined ceiling of the wind tunnel; the back and front sides coincide with the test-section walls.

The bottom boundaries of both domains $\Omega_1$ and $\Omega_2$ coincide with the soil surface in which the two synthetic plants were planted as illustrated in~\cref{fig:domain}.

The dimensions of the subdomain $\Omega_2$ are $2.94 \times 0.7 \times 0.5$\si{~\cubic\metre}.

The dimensions of the subdomain $\Omega_2$ are \qtyproduct[product-symbol = \ensuremath{\,\times\,}]{2.94 x 0.7 x 0.5}{\metre}.

\begin{figure}[!ht]

    \centering