Merge pull request #14239 from rmcdermo/master

Manuals: fix issues with \ct{ }

Author: rmcdermo

Manuals/FDS_Technical_Reference_Guide/Aerosol_Chapter.tex

@@ -215,7 +215,7 @@ \subsection{Gas Condensation}
 
 The total mass evaporated or condensed is added to the bulk mass source terms, $\dot{m}_{{\rm b},\alpha}^{\ppp}$, for the condensed and vapor phase species. The bulk energy source term, $\dot{q}_{\rm b}^{\ppp}$, is taken as the mass source term multiplied by the heat of vaporization, $h_v$ where condensation results in positive energy source term and evaporation results in a negative energy source term.
 
-The contribution to divergence is added to the {\ct D\_SOURCE} term following Eq~(\ref{eq:D_SOURCE_vap}). Since the condensed and vapor phases have the same molecular weight, the first term in the equation is zero.
+The contribution to divergence is added to the \ct{D_SOURCE} term following Eq~(\ref{eq:D_SOURCE_vap}). Since the condensed and vapor phases have the same molecular weight, the first term in the equation is zero.
 
 For the condensed phase, the radiation absorption is computed following Sec.~\ref{droplet-radiation}.
 
@@ -239,5 +239,5 @@ \subsection{Wall Condensation}
 \be
 \dot{q}_{{\rm dep},\alpha}^{\pp} = \dot{m}_{{\rm dep},\alpha}^{\pp} (h_v(T_w)+h_{s,\alpha}(T_g)-h_{s,\alpha}(T_w))
 \ee
-For the {\ct D\_SOURCE} term only the first term in Eq~(\ref{eq:D_SOURCE_vap}) applies since all the phase change energy comes from the wall cell.
+For the \ct{D\_SOURCE} term only the first term in Eq~(\ref{eq:D_SOURCE_vap}) applies since all the phase change energy comes from the wall cell.
 

Manuals/FDS_Technical_Reference_Guide/Combustion_Chapter.tex

@@ -121,7 +121,7 @@ \subsection{Default Hydrocarbon Combustion Chemistry}
 \end{tabular}
 \end{center}
 
-\noindent The preceding table shows that the addition of carbon monoxide and soot increases the number of primitive species in the reaction from five to seven. The number of lumped species, however, remains at three---the composition of Products has changed to include to the two additional species.  Note that FDS prints the $A$ matrix in the {\ct CHID.out} file so that the user can double check the reaction system.
+\noindent The preceding table shows that the addition of carbon monoxide and soot increases the number of primitive species in the reaction from five to seven. The number of lumped species, however, remains at three---the composition of Products has changed to include to the two additional species.  Note that FDS prints the $A$ matrix in the \ct{CHID.out} file so that the user can double check the reaction system.
 
 
 \clearpage
@@ -463,25 +463,25 @@ \subsection{Critical Flame Temperature}
 \subsection{Extinction Based Mainly on Oxygen Concentration}
 \label{o2_based_model}
 
-The first of two optional extinction models (referred to as {\ct 'EXTINCTION 1'} in the FDS input file) linearizes Eq.~(\ref{YO2eq}) to form a limiting oxygen concentration\footnote{The extinction model is written in terms of the oxygen {\em volume} fraction, $X_\OTWO$, because usually experimental results are reported as such. However, within the numerical algorithm, all values are converted into mass fraction, $Y_\OTWO$.} that is a piecewise-linear function of the cell bulk temperature, $T_{ijk}$ (see Fig.~\ref{extinction_1_sketch}):
+The first of two optional extinction models (referred to as \ct{'EXTINCTION 1'} in the FDS input file) linearizes Eq.~(\ref{YO2eq}) to form a limiting oxygen concentration\footnote{The extinction model is written in terms of the oxygen {\em volume} fraction, $X_\OTWO$, because usually experimental results are reported as such. However, within the numerical algorithm, all values are converted into mass fraction, $Y_\OTWO$.} that is a piecewise-linear function of the cell bulk temperature, $T_{ijk}$ (see Fig.~\ref{extinction_1_sketch}):
 \be
    X_{\OTWO,\lim}(T_{ijk}) = \left\{ \begin{array}{c@{\quad \quad}l} X_{\OI} \, \left( \frac{T_{\OI}-T_{ijk}}{T_{\OI}-T_\infty} \right) & T_{ijk}<T_{\rm fb} \\[.1in]  0 &  T_{ijk} \ge T_{\rm fb} \end{array} \right.  \label{extinction_model}
 \ee
 \begin{figure}
 \centering
 \includegraphics[width=4.5in]{FIGURES/extinction_1_sketch}
 \vskip-.2cm
-\caption{Extinction criteria for the {\ct 'EXTINCTION 1'} model.}
+\caption{Extinction criteria for the \ct{'EXTINCTION 1'} model.}
 \label{extinction_1_sketch}
 \end{figure}
 If $X_{\OTWO,ijk}<X_{\OTWO,\lim}$, local extinction is assumed and $\dot{m}_\alpha'''=0$ and $\dot{q}'''=0$ for that grid cell at that time step. At an ambient temperature of 20~$^\circ$C, the default limiting oxygen volume fraction is 0.135.  This value is consistent with the measurements of Morehart et al.~\cite{Morehart:1991}, who measured the oxygen concentration near self-extinguishing flames. They found that flames self-extinguished at oxygen volume fractions of 12.4~\% to 14.3~\%. Note that their results are expressed as volume, not mass, fractions. Beyler's chapter in the SFPE Handbook references other researchers who measured oxygen concentrations at extinction ranging from 12~\% to 15~\%.
 
-The {\ct 'EXTINCTION 1'} model is intended for relatively coarse fire simulations where the grid cell cannot resolve details of the flame structure or capture flame temperatures. The ``free-burn'' temperature, $T_{\rm fb}$, in Eq.~(\ref{extinction_model}) is needed for simulations in which the characteristic grid cell size, $\dx$, is much larger than 1~cm. In such cases, the combustion occurs within a fraction of the grid cell and its energy cannot raise the cell bulk temperature to the critical value. Its default value is 600~$^\circ$C. Measurements of Pitts~\cite{Pitts:1995}, Bundy~\cite{Bundy:1}, and others, have shown that the upper layer oxygen concentration drops to zero in flashover compartment fire experiments when the temperature increases above approximately 600~$^\circ$C.
+The \ct{'EXTINCTION 1'} model is intended for relatively coarse fire simulations where the grid cell cannot resolve details of the flame structure or capture flame temperatures. The ``free-burn'' temperature, $T_{\rm fb}$, in Eq.~(\ref{extinction_model}) is needed for simulations in which the characteristic grid cell size, $\dx$, is much larger than 1~cm. In such cases, the combustion occurs within a fraction of the grid cell and its energy cannot raise the cell bulk temperature to the critical value. Its default value is 600~$^\circ$C. Measurements of Pitts~\cite{Pitts:1995}, Bundy~\cite{Bundy:1}, and others, have shown that the upper layer oxygen concentration drops to zero in flashover compartment fire experiments when the temperature increases above approximately 600~$^\circ$C.
 
 
 \subsection{Extinction Based on Both Fuel and Oxygen}
 
-The second optional extinction model in FDS, referred to as {\ct 'EXTINCTION 2'}, considers both the oxygen and the fuel content of a given grid cell at the start of a time step. If the potential heat release from the reactants cannot raise the temperature of the cell above the empirically determined critical flame temperature, $T_{\rm CFT}$, combustion is suppressed.  Consider the simple reaction $\mbox{Fuel} + \mbox{Air} \rightarrow \mbox{Products}$. The mass fractions of lumped species Fuel, Air, and Products in the mixed portion of the grid cell at the beginning and the end of the reaction part of the time step are $[Z_{\rm F}^0, Z_{\rm A}^0, Z_{\rm P}^0]$ and $[Z_{\rm F}, Z_{\rm A}, Z_{\rm P}]$, respectively. The Products include the products of combustion as well as diluents like argon or water vapor from droplet evaporation. Define a modified form of the equivalence ratio:
+The second optional extinction model in FDS, referred to as \ct{'EXTINCTION 2'}, considers both the oxygen and the fuel content of a given grid cell at the start of a time step. If the potential heat release from the reactants cannot raise the temperature of the cell above the empirically determined critical flame temperature, $T_{\rm CFT}$, combustion is suppressed.  Consider the simple reaction $\mbox{Fuel} + \mbox{Air} \rightarrow \mbox{Products}$. The mass fractions of lumped species Fuel, Air, and Products in the mixed portion of the grid cell at the beginning and the end of the reaction part of the time step are $[Z_{\rm F}^0, Z_{\rm A}^0, Z_{\rm P}^0]$ and $[Z_{\rm F}, Z_{\rm A}, Z_{\rm P}]$, respectively. The Products include the products of combustion as well as diluents like argon or water vapor from droplet evaporation. Define a modified form of the equivalence ratio:
 \be
    \label{eq:dza}
    \tilde{\phi} \equiv  \min \left( \, 1 \, , \, \frac{s \, Z_{\rm F}^0}{Z_{\rm A}^0} \, \right) = \frac{Z_{\rm A}^0-Z_{\rm A}}{Z_{\rm A}^0}

Manuals/FDS_Technical_Reference_Guide/Complex_Geometry_Chapter.tex

@@ -16,7 +16,7 @@ \chapter{Unstructured Geometry (Beta)}
 
 Unstructured geometry is treated using a ``cut-cell'' (CC) method, following the seminal work of Michael Aftosmis and Marsha Berger \cite{Berger:2012,Berger:2017,May:2017}.  The term cut-cell refers to the gas phase region of a Cartesian grid cell that has been ``cut'' by the unstructured geometry partitioning the cell into gas and solid sub-volumes.  Faces of the cut-cell that have also been carved from cartesian faces and connected to other cut-cells are called \emph{gasphase cut-faces} (these are gas-gas interfaces).  The faces on the boundary of the solid in a cut-cell are \emph{in-boundary cut-faces} (these are gas-solid interfaces).  There are also faces of a cut-cell which are not cut and which connect the cell to a neighboring regular cartesian gas phase cell; these faces are referred to as \emph{regular faces} (also a gas-gas interface).
 
-The introduction of cut-cells surrounding an embedded boundary generates the need for an unstructured solution algorithm.  In particular, the pressure equation is unstructured and so fast trigonometric solvers are not viable (i.e. the {\ct FFT} Poisson solver).  Instead, an unstructured local matrix ({\ct ULMAT}) solver has been developed using the Pardiso and Sparse Cluster Solvers available in the Intel Math Kernel Library (MKL).
+The introduction of cut-cells surrounding an embedded boundary generates the need for an unstructured solution algorithm.  In particular, the pressure equation is unstructured and so fast trigonometric solvers are not viable (i.e. the \ct{FFT} Poisson solver).  Instead, an unstructured local matrix (\ct{ULMAT}) solver has been developed using the Pardiso and Sparse Cluster Solvers available in the Intel Math Kernel Library (MKL).
 
 The following sections describe the finite volume methods used to discretize the governing equations in the unstructured cut-cells surrounding embedded boundaries.
 
@@ -324,7 +324,7 @@ \subsection{Unsteady Evolution: Explicit Time Integration for Scalars} \label{se
 In general, there will arise \texttt{GASPHASE} cut-cells whose small-size will severely penalize
 the time step. We recall that each cell on the gas phase, including cut-cells, needs to meet CFL and Von Neuman stability constraints. Several different ways have been proposed in the literature to deal with this problem, i.e., cell merging, mixing or linking methods. In general, these lead to ad hoc selection procedures for surrounding cells, having to deal with many special cases, and in some cases potential solution deterioration close to the boundary.
 
-We use a simple and robust procedure to address this problem. Within the scheme for numeration of scalar cell unknowns, a test is performed on cut-cells. If the cut-cell volume is less than the threshold volume $V_{thr}= ${\ct CCVOL\_LINK} $V_{cart}$, where $V_{cart}$ is the local Cartesian cell volume and {\ct CCVOL\_LINK}$<1$ is a threshold factor (default value 0.5), the unknown number this cell takes is the one of an adjacent cell which has a volume larger than $V_{thr}$. This mathematically defines a single control volume of the two linked cells. Cell volumes are added in building the mass matrix for the FV discretization, and fluxes and matrix terms are added with their corresponding signs. Note that, flux quantities corresponding to the common face of these two cells effectively cancel on the single equation for the set.
+We use a simple and robust procedure to address this problem. Within the scheme for numeration of scalar cell unknowns, a test is performed on cut-cells. If the cut-cell volume is less than the threshold volume $V_{thr}= $\ct{CCVOL\_LINK} $V_{cart}$, where $V_{cart}$ is the local Cartesian cell volume and \ct{CCVOL\_LINK}$<1$ is a threshold factor (default value 0.5), the unknown number this cell takes is the one of an adjacent cell which has a volume larger than $V_{thr}$. This mathematically defines a single control volume of the two linked cells. Cell volumes are added in building the mass matrix for the FV discretization, and fluxes and matrix terms are added with their corresponding signs. Note that, flux quantities corresponding to the common face of these two cells effectively cancel on the single equation for the set.
 Alternatively, if after a number of cell numbering iterations (default 2) an unlinked small cell persists in the mesh, this small cell is blocked and its corresponding cartesian cell tagged as solid. % Default 2 link numbering iterations.
 
 \section{Momentum Time Marching and Immersed Boundaries}
@@ -370,7 +370,7 @@ \subsection{Momentum-Pressure Coupling}
 
 
 The next component of the scheme involves approximating velocity boundary conditions for immersed solid boundaries. In FDS, velocity field components are staggered on cartesian or cut faces. 
-For complex geometry the projection scheme defined above is applied in the unstructured mesh composed by cut-cells and regular gas cells. Therefore, the Poisson equation is discretized in an unstructured grid and solved by matrix direct solvers, either by FDS mesh ({\ct ULMAT}) or globally ({\ct UGLMAT}). Currently the numerical scheme allows only one cut-cell/face cartesian per cell/face, due to the current need to define a single background pressure per cartesian cell. The blocking treatment of split cells is described in the Users guide. The projection step to obtain final velocities involves grid cell sizes in regular faces and centroid to centroid distance in cut-faces.
+For complex geometry the projection scheme defined above is applied in the unstructured mesh composed by cut-cells and regular gas cells. Therefore, the Poisson equation is discretized in an unstructured grid and solved by matrix direct solvers, either by FDS mesh (\ct{ULMAT}) or globally (\ct{UGLMAT}). Currently the numerical scheme allows only one cut-cell/face cartesian per cell/face, due to the current need to define a single background pressure per cartesian cell. The blocking treatment of split cells is described in the Users guide. The projection step to obtain final velocities involves grid cell sizes in regular faces and centroid to centroid distance in cut-faces.
 
 Cross velocities and tangent stresses and vorticity emanating from the immersed boundary are defined on cartesian faces being intersected by the geometry surface and surrounding cartesian edges. We use a wall modeled stress imposition method (STM) to this end described next.
 

Manuals/FDS_Technical_Reference_Guide/Equation_Chapter.tex

@@ -110,7 +110,7 @@ \subsection{Cell Edges}
 
 \end{tikzpicture}
 }
-\caption[Position of flow variables around a computational edge]{Position of flow variables around a computational edge along the $x$ axis.  The Index of the Edge Component ({\ct IEC}) is 1 in this case.  The orientations of the faces (defined by the directions normal to the faces) that connect to the edge are {\ct IOR} = $\pm2$, $\pm3$.  The vorticity around the edge is $\omega_x = \partial w/\partial y - \partial v/\partial z$.}
+\caption[Position of flow variables around a computational edge]{Position of flow variables around a computational edge along the $x$ axis.  The Index of the Edge Component (\ct{IEC}) is 1 in this case.  The orientations of the faces (defined by the directions normal to the faces) that connect to the edge are \ct{IOR} = $\pm2$, $\pm3$.  The vorticity around the edge is $\omega_x = \partial w/\partial y - \partial v/\partial z$.}
 \label{fig:edge_positions}
 \end{figure}