vapor transport chapter - updated velocity fluctuations to refer to the appendix

\subsubsection{Inflow: velocity fluctuations}

\label{sec:inflow:fluctuations}

\inline{Move to \cref{chapter:LBM} or \cref{chapter:LBM-MHFEM}?}

Direct numerical simulations (DNS) of turbulent flow in a regular domain require sufficiently high resolution and large domain size, otherwise the turbulent boundary layer may not fully develop and the simulation may give wrong results.

As will be seen in \cref{sec:results}, prescribing a time-constant velocity profile at the inflow boundary may lead to non-physical results, because the simulated flow field may remain laminar until it reaches the first obstacle placed in the domain.

An alternative is to induce turbulent flow by adding synthetic fluctuations to the prescribed velocity profile.

An alternative is to turn away from the DNS approach and add synthetic fluctuations to the prescribed velocity profile, which should help induce turbulent flow and let the boundary layer develop faster.

The inflow velocity $\vec v_{\mathrm{in}} = \vec v_{\mathrm{in}}(\vec x, t)$ is decomposed as

\begin{equation}

    \vec v_{\mathrm{in}}(\vec x, t) = \overline{\vec v}_{\mathrm{in}}(\vec x) + \vec v_{\mathrm{in}}'(\vec x, t),

\end{equation}

where $\overline{\vec v}_{\mathrm{in}}(\vec x)$ is the mean (time-averaged) value given in \subsubsecref{sec:inflow:time-constant}, and $\vec v_{\mathrm{in}}'(\vec x, t)$ is the velocity fluctuation.

The procedure for generating synthetic turbulent fluctuating velocity field $\vec v_{\mathrm{in}}'$ is based on \cite{davidson2006,davidson2007,davidson2008,davidson:lecture}.

In the simulation, inflow velocity fluctuations for a uniform grid with the spacing $\Delta x$ are computed at discrete time levels $t_n = n \Delta t$, where $n$ is an integer denoting the time level and $\Delta t$ is the time step.

Firstly, independent realizations of random fluctuations $\hat{\vec v}'_{\mathrm{in}}$ are generated for each time level for the specified length scale $\mathcal L_{\mathrm{int}}$ and energy spectrum of synthetic turbulence.

In this work, we set the turbulent length scale to one half of the inflow boundary layer $z_\delta$ used in \cref{eq:inflow vx}, i.e. $\mathcal L_{\mathrm{int}} = \SI{0.05}{\metre}$, and use the modified von~Kármán spectrum \cite{davidson2007,davidson:lecture} with the highest wave number $\kappa_{\mathrm{max}} = 2\pi/\Delta x$, smallest wave number $\kappa_{\mathrm{min}} = \kappa_e / 5$, 3000 discrete modes, turbulent kinetic energy $k_{\mathrm{in}} = \SI{e-2}{\metre\squared\per\second\squared}$, and kinematic viscosity $\nu$ given in \cref{tab:physical parameters}.

Then, time correlation between the realizations is introduced using an asymmetric time filter

\begin{equation}

    (\vec v'_{\mathrm{in}})^n = a (\vec v'_{\mathrm{in}})^{n-1} + b (\hat{\vec v}'_{\mathrm{in}})^n,

\end{equation}

where $\vec v'_{\mathrm{in}}$ denotes the time-correlated field, $\hat{\vec v}'_{\mathrm{in}}$ denotes the time-independent field, subscripts denote the time levels and the coefficients are chosen as $a = \exp(-\Delta t / \mathcal T_{\mathrm{in}})$ and $b = \sqrt{1 - a^2}$, where $\mathcal T_{\mathrm{int}} = \mathcal L_{\mathrm{int}} / \abs{v_{x,\mathrm{in,max}}}$.

The time filter ensures that $\mathcal T_{\mathrm{int}}$ corresponds to the turbulent integral time scale and that the variance of the generated fluctuations is preserved~\cite{davidson:lecture}.

where $\overline{\vec v}_{\mathrm{in}}(\vec x)$ is the mean (time-averaged) value given in \cref{sec:inflow:time-constant} above, and $\vec v_{\mathrm{in}}'(\vec x, t)$ is the velocity fluctuation.

The field $\vec v_{\mathrm{in}}'(\vec x, t)$ is generated using the algorithm described in \cref{appendix:turbgen} with the following parameters.

The turbulent length scale is set to one half of the inflow boundary layer $z_\delta$ used in \cref{eq:inflow vx}, i.e. $\mathcal L_{\mathrm{int}} = \SI{0.05}{\metre}$.

The number of discrete modes is $N_{\mathrm{modes}} = 3000$, the turbulent kinetic energy is $k_{\mathrm{in}} = \SI{e-2}{\metre\squared\per\second\squared}$, and the kinematic viscosity $\nu$ is given in \cref{tab:physical parameters}.

Time correlation is introduced with the turbulent integral time scale $\mathcal T_{\mathrm{int}} = \mathcal L_{\mathrm{int}} / v_{\max}$, where $v_{\max}$ is the mean free stream velocity specified in \cref{sec:inflow:time-constant}.

Note that the fluctuations generated with the aforementioned procedure are isotropic which is an acknowledged simplification of anisotropic real-world turbulence.

The procedure could be improved based on a specified anisotropic Reynolds stress tensor~\cite{davidson:lecture}, however, even using the inflow condition based on isotropic synthetic turbulence lead to improved results in this work.