steady state verkackt

tex/main.tex

@@ -7,6 +7,8 @@
 % use \textinput as described in /header and _TEMPLATE
 \textinput{mesh}
 \textinput{reynold}
+\section{Results}
+\textinput{steady-state}
 \textinput{diagrams}
 % \begin{appendices}
   % \textinput{some-appendix-section}

tex/mesh.tex

@@ -11,14 +11,20 @@ One subdomain consists of the \SI{1.8}{\metre} on the side of the outlet without
 On this part (called \enquote{pipe}) the grading is chosen in a way that the cells closest to the cylinder are smaller than those at the outlet and almost of the same size as the neigboring cells in the neigboring subdomain.
 
 The area around the cylinder is divided into four quarters, each with a quarter of the cylinder and one boundary part as boundaries.
-The grading is chosen in a way that the cells closest to the cylinder are smallest and all cells are quite close to squares. (|Max-aspect-ratio = 2.4|)
+% The grading is chosen in a way that the cells closest to the cylinder are smallest and all cells are quite close to squares. (|Max-aspect-ratio = 2.4|)
+There is no grading since there is a lot of change close to the cylinder and the outer walls, so we need smaller cells on the outside than on the inside of the blocks which is not possible with |simpleGrading|.
 Since each of the four quarters dictates the number and grading of the cells in radius direction to its neighbors, all four quarters have the same number in this direction even though the cylinder is not perfectly in the center between the walls.
 In the corners of the area around the cylinder the angle of the outermost cells are \SI{45}{°}. I could not find a way to improve this, but |checkMesh| calls the mesh \enquote{OK}.
 
 \begin{figure}[htpb]
   \centering
-  \includegraphics[width=0.8\linewidth]{.maindir/zeichnungen/meshN8}
-  \caption{The mesh for $N = 8$. With $2$ correctors, $N=8$ does not work (Courant number blows up), but with $4$ correctors, it does work.}%
+  \includegraphics[width=0.8\linewidth]{.maindir/zeichnungen/mesh}
+  \caption{The mesh for $N = 6$.% With $2$ correctors, $N=8$ does not work (Courant number blows up), but with $4$ correctors, it does work.
+  }%
   \label{fig:mesh}
 \end{figure}
+
+The mesh has disadvantages.
+The cells in the pipe are not shaped into the flow direction because they have to fit with the area around the pipe.
+The same holds for the area in front and behind the cylinder but since there is also vertical flow it might actually be OK.
 \docEnd

tex/reynold.tex

@@ -9,7 +9,7 @@ consider the definition of the Reynolds number:
 \begin{align*}
   \Rey = \frac{\bar u d}{ν}
 \end{align*}
-where $d = \SI{0.1}{\metre}$ is the diameter of the cylinder and $ν = \SI{10e-3}{\square\metre\per\second}$ is the kinematic viscosity\footnote{\url{https://en.wikipedia.org/wiki/Viscosity#Kinematic_viscosity}}.
+where $d = \SI{0.1}{\metre}$ is the diameter of the cylinder and $ν = \SI{10e-3}{\square\metre\per\second}$ is the kinematic viscosity\footnote{\url{https://en.wikipedia.org/wiki/Viscosity\#Kinematic_viscosity}}.
 Therefore
 \begin{align*}
   \bar u = \frac{\Rey ν}{d} = \frac{\Rey · \SI{10e-3}{\square\metre\per\second}}{\SI{0.1}{\metre}} = \frac{\Rey}{100} \si{\metre\per\second} \\

tex/steady-state.tex

+%! TEX program = lualatex
+
+\input{.maindir/tex/header/preamble-section}
+% inputs the preamble only if necessary
+\docStart
+\subsection{Steady state}
+The configuration with $\Rey = 20$ and $\bar u = \SI{0.2}{\metre\per\second}$ results in a steady state.
+After a very short time (\SI{0.6}{\second}) of adjusting, the flow reaches a situation that (almost) does not change anymore. Hence only the first $2$ instead of $20$ seconds were calculated.
+
+The calculations were performed with $N = 6$ ($1728$ cells), $6$ correctors, $2$ NonOrthogonalCorrectors and a time step of $0.02$. After a starting period the max Courant number remained at $1.058$.
+
+\begin{figure}[htpb]
+  \centering
+  \newcommand{\picwidth}{0.95\linewidth}
+  \includegraphics[width=\picwidth]{.maindir/zeichnungen/20_u_x}
+  \includegraphics[width=\picwidth]{.maindir/zeichnungen/20_u_y}
+  \includegraphics[width=\picwidth]{.maindir/zeichnungen/20_p}
+  \caption{The steady state is reached at time \SI{0.6}{\second}.
+    From top to bottom we have the velocity in flow direction,
+    the velocity perpendicular to it and the pressure.%
+  }
+  \label{fig:steadystate}
+\end{figure}
+In figure \ref{fig:steadystate} we can see the recirculation zone with a length of approximately \SIrange{15}{20}{\centi\metre}.
+Here I take the area where the flow is disturbed, hence not a straight flow as the recirculation zone.
+In the literature the length of the recirculation is zone is about \SI{8}{\centimetre},
+about half of my value.
+But the paper does not define what this zone is, hence those values are not comparable.
+
+The pressure values in front and behind the cylinder and the coefficiants are plotted in  the figures \ref{fig:steadyP}, \ref{fig:steadyCd} and \ref{fig:steadyCl}.
+The captions include the comparison with the literature.
+\begin{figure}[ht]
+  \centering
+  \tikzinput{bothpressures}
+  \caption{Pressure in front and behind the cylinder and the difference.
+    The pressure difference quickly steadies at \SI{0.5}{\square\metre\per\square\second}.
+    In the literature, the pressure difference is in most cases about \num{0.1}{\square\metre\per\second} which is a fifth of my value.
+  }
+\label{fig:steadyP}
+\end{figure}
+\begin{figure}[ht]
+  \centering
+  \tikzinput{Cd}
+  \caption{The drag coefficiant over time. It steadies at \num{33.77}.
+    In the literature $C_D$ is about \num{5.5} which is about one sixth of my value.
+  }
+  \label{fig:Cd}
+\end{figure}
+\begin{figure}[ht]
+  \centering
+  \tikzinput{Cl}
+  \caption{The lift coefficiant over time. It steadies at \num{7.59}.
+    In the literature $C_L$ is about \num{0.01} which is a 800th of my value.
+  }
+  \label{fig:Cl}
+\end{figure}
+The comparisons to the literature indicate that there is a major flaw in the setup.
+\docEnd