From 7fa59c368ec0d033dc3c9f90003b021c29b9087a Mon Sep 17 00:00:00 2001
From: Felix <feh7@hi.is>
Date: Sun, 25 Apr 2021 13:06:57 +0000
Subject: [PATCH] some more writing of my problems

---
 tex/appendix-figures.tex     |  21 +++++++++++++++++
 tex/bibliography.bib         |   9 ++++++++
 tex/results.tex              |  43 +++++++++++++++++++++++++++++------
 tex/setup.tex                |  12 ++++++----
 tex/timestep.tex             |  38 +++++++++++++++++--------------
 zeichnungen/R3-N2-Drag.png   | Bin 0 -> 40245 bytes
 zeichnungen/code/diagrams.py |  14 ++++++++----
 7 files changed, 103 insertions(+), 34 deletions(-)
 create mode 100644 tex/appendix-figures.tex
 create mode 100644 zeichnungen/R3-N2-Drag.png

diff --git a/tex/appendix-figures.tex b/tex/appendix-figures.tex
new file mode 100644
index 0000000..f5781d1
--- /dev/null
+++ b/tex/appendix-figures.tex
@@ -0,0 +1,21 @@
+%! TEX program = lualatex
+
+\input{.maindir/tex/header/preamble-section}
+% inputs the preamble only if necessary
+\docStart
+\begin{figure}[ht]
+  \centering
+  \includegraphics[width=0.9\linewidth]{.maindir/zeichnungen/R3-N2-Drag}
+  \caption{The drag coefficiant at a Reynolds number of \num{e3} ($u = \SI{5}{\metre\per\second}$) and $N=2$ (6624 cells). The Courant number rose to about 3.4 and it can be seen that this resulted in highly inaccurate and useless results. For higher $\Rey$ this became worse (also for $N=1$).}%
+  \label{fig:R3N2Drag-fluctuation}
+\end{figure}
+% do not use \input nor \begin{document nor \end{document}
+
+% write the text here
+% for including a tikz picture, place it in the directory tikz and
+% use \tikzinput{tikzpicture}
+% for including another tex (text) file, place it in the directory
+% tex and use \textinput{textfile}
+% for including some other file use \maininput{file/path/relative/to/main/repodir}
+
+\docEnd
diff --git a/tex/bibliography.bib b/tex/bibliography.bib
index e69de29..8f38481 100644
--- a/tex/bibliography.bib
+++ b/tex/bibliography.bib
@@ -0,0 +1,9 @@
+@article{same,
+  title={Numerical investigation of unsteady flow past a circular cylinder using 2-D finite volume method},
+  author={Bakzuzan Nakbas and Baokut Wanib and Abdul Allm},
+  url={https://www.banglajol.info/index.php/JNAME/article/download/914/980/},
+  journal={Journal of Naval Architecture and Marine Engineering},
+  publisher={Aname Publication},
+  year={2007},
+  month={June}
+}
diff --git a/tex/results.tex b/tex/results.tex
index 913775c..37e08c0 100644
--- a/tex/results.tex
+++ b/tex/results.tex
@@ -3,6 +3,32 @@
 \input{.maindir/tex/header/preamble-section}
 % inputs the preamble only if necessary
 \docStart
+\section{Literature}
+Since the studied geometry is pretty simple, it has been studied numerous times before.
+For the Nusselt number Churchill and Berstein found the following formula in terms of $\Rey$ and $\Pr$ with an accuracy of about \SI{20}{\percent}:%
+\footnote{\url{https://en.wikipedia.org/wiki/Churchill\%E2\%80\%93Bernstein_equation}}
+\begin{equation*}
+\overline{\Nus} =
+0.3 + \frac{
+  0.62 \sqrt{\Rey} \sqrt[3]{\Pr}
+}{
+  \sqrt[4]{1 +
+           \left(
+             \frac{0.4}{\Pr}
+           \right)^{\frac23}
+          }
+}
+· \left(
+    1 +
+    \left(
+      \frac{\Rey}{282000}
+    \right)^{\frac58}
+  \right)^{\frac45}
+\end{equation*}
+The results of this equation are listed in table \ref{tab:values}.
+
+For reference values for the drag coefficiant we can refer to the paper \textcite{same} which solved the same problem for $\Rey = 100, 1000$ and $3900$.
+
 \section{Measurements}
 The drag coefficiant was calculated with the OpenFOAM intrinsic function
 |forceCoeffsIncompressible| with the reference area |Aref| of
@@ -14,14 +40,17 @@ We need $\abs{n·∇T}$ instead of $\abs{∇T}$ but in this case those two value
 The results are summerized in table \ref{tab:values}.
 \begin{table}[htpb]
   \centering
-  \caption{Scalar results of the numerical experiments}
+  \caption{Scalar results of the numerical experiments. ref-$\Nus$ is the Nusselt number according to the Churchill-Bernstein equation.}
   \label{tab:values}
-  \begin{tabular}{cccccc}
-    $\Rey$ & $u_{in}$ & \# cells & $C_D$ & $\vec n · ∇T\vert_{\text{cyl}}$ & $\Nus$ \\
-    \num{e2} & \num{0.05} & \\
-    \num{e3} & \num{0.5} & \\
-    \num{e4} & \num{5} & \\
-    \num{e5} & \num{50} &
+  \begin{tabular}{ccccccc}
+    $\Rey$ & $u_{in}$ & \# cells & $C_D$ & $\vec n · ∇T\vert_{\text{cyl}}$ & $\Nus$ & ref-$\Nus$\\
+    \num{e2} & \num{0.05} & & & & & 5.1%56
+    \\
+    \num{e3} & \num{0.5} & & & & & 15%.930
+    \\
+    \num{e4} & \num{5} & & & & & 53%.328
+    \\
+    \num{e5} & \num{50} & & & & & 214%.126
   \end{tabular}
 \end{table}
 % do not use \input nor \begin{document nor \end{document}
diff --git a/tex/setup.tex b/tex/setup.tex
index 0156a6b..94382e2 100644
--- a/tex/setup.tex
+++ b/tex/setup.tex
@@ -9,8 +9,13 @@ The mesh has two zones: one around the cylinder and a wider one for the remainde
 The zone around the cylinder is a square with arked edges.
 Those arks distribute the angles about equally around the corners of the square and distribute the grading partially into the outer zone.
 
-The grading around the cylinder is chosen in a way that maximizes the grading while keeping the cell size ratio at \SI{105}{\percent}.
-So the grading increases with higher number of cells.
+In the simulations with a previous version with 20 cells in the radial direction,
+the heat around the cylinder could not be resolved well at Reynold number of \num{e4} and \num{e5}.
+Therefore this radial resolution was increased to 40 (times $N$).
+
+In an earlier version, the grading around the cylinder is chosen in a way that maximizes the grading while keeping the cell size ratio at \SI{105}{\percent}.
+So the grading increasd with higher number of cells but this got too extreme leading to extremely high Courant numbers and bad cell size ratios at the square ark edges.
+So the grading was fixed to $45$.
 
 Since the cells close to the cylinder should be smaller but also dictate the size in one dimension over, under, left and right of the square part, outer cells in those four directions get very long and thin.
 
@@ -24,9 +29,6 @@ The mesh at $N=1$ is displayed in figure \ref{fig:mesh}.
 
 % In order to keep the cell sizes similar at the corners of the square, the number of cells in the tangential direction needs to be quite high, creating quite rectangular (in radial direction) cells at the cylinder wall. It would be good to have a good idea to avoid this.
 
-In the simulations with a previous version with 20 cells in the radial direction,
-the heat around the cylinder could not be resolved well at Reynold number of \num{e4} and \num{e5}.
-Therefore this radial resolution was increased to 40 (times $N$).
 \begin{figure}[htpb]
   \centering
   \includegraphics[width=0.95\linewidth]{.maindir/zeichnungen/mesh}  % check at end
diff --git a/tex/timestep.tex b/tex/timestep.tex
index 8e45622..3e10b3d 100644
--- a/tex/timestep.tex
+++ b/tex/timestep.tex
@@ -5,24 +5,28 @@
 \docStart
 \section{Time stepping}
 The timesteps were adjusted automatically because for moderate to higher velocities the Courant numbers were very high (over 40) at time steps of one second but stable.
-This resulted in time steps listed in table \ref{fig:timesteps}.
-As it is visible there, the combination of high Reynolds number ($\num{e5}$) with higher cell number $N > 2$) is not viable to run.
+This resulted in time steps from \SIrange{0.0008}{1.3}{\second}.% listed in table \ref{fig:timesteps}.
+My computer with parallel execution was able to run about 8 timesteps for second.
+Therefore is was not feasible for Reynold number \num{e4} and \num{e5}.
+% As it is visible there, the combination of high Reynolds number ($\num{e5}$) with higher cell number $N > 2$) is not viable to run.
 
 Since for low Reynold numbers, the time step became too big, a maximal time step of \num{0.1} was introduced.
 
-\begin{table}[htpb]
-  \centering
-  \caption{Time steps due to automatic adjustment with max courant number of \num{1}.}
-  \label{tab:timesteps}
-  \begin{tabular}{ccc}
-    Nr. of cells & Reynold number & time step \\
-    % 1016 = "N = 1"; 4064 = "N = 2"
-    4064 & \num{e2} & 1.3 \\ % todo: run again with smaller timestep
-    4064 & \num{e3} & 0.0795 \\
-    4064 & \num{e4} & 0.0065 \\
-    1016 & \num{e5} & 0.0017 \\
-    4065 & \num{e5} & 0.00051
-  \end{tabular}
-\end{table}
-
+When running with Courant numbers greater than one and a time step of $0.01$ it still took long times and the results were not useful as visible in diagram \ref{fig:R3N2Drag-fluctuation}.
+{
+  % \begin{table}[htpb]
+  %   \centering
+  %   \caption{Time steps due to automatic adjustment with max courant number of \num{1}.}
+  %   \label{tab:timesteps}
+  %   \begin{tabular}{ccc}
+  %     Nr. of cells & Reynold number & time step \\
+  %     % 1016 = "N = 1"; 4064 = "N = 2"
+  %     4064 & \num{e2} & 1.3 \\ % todo: run again with smaller timestep
+  %     4064 & \num{e3} & 0.0795 \\
+  %     4064 & \num{e4} & 0.0065 \\
+  %     1016 & \num{e5} & 0.0017 \\
+  %     4065 & \num{e5} & 0.00051
+  %   \end{tabular}
+  % \end{table}
+}
 \docEnd
diff --git a/zeichnungen/R3-N2-Drag.png b/zeichnungen/R3-N2-Drag.png
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HcmV?d00001

diff --git a/zeichnungen/code/diagrams.py b/zeichnungen/code/diagrams.py
index 1db2272..b25d5aa 100755
--- a/zeichnungen/code/diagrams.py
+++ b/zeichnungen/code/diagrams.py
@@ -13,6 +13,10 @@ SUBPATH_GRADT = "patchAverage(name=cylinder,mag(grad(T)))"
 PATH_DIAGRAMS = ".maindir/zeichnungen/generated/"
 DIAGRAM_WIDTH = r'\diagramwidth'
 
+def startindextake(alllength):
+    # return min(alllength//5, 20)
+    return alllength//5
+
 if __name__ == "__main__":
     if len(sys.argv) == 3:
         N = int(sys.argv[1])
@@ -29,8 +33,8 @@ if __name__ == "__main__":
             print(error)
             print("abort this case")
             continue
-        times = force[len(force)//3:, 0]
-        drag_coeffs = force[len(force)//3:, 1]
+        times = force[startindextake(len(force)):, 0]
+        drag_coeffs = force[startindextake(len(force)):, 1]
         if len(cases) == 1:
             print("times:", times)
             print("D:", drag_coeffs)
@@ -38,7 +42,7 @@ if __name__ == "__main__":
         axis = pyplot.axes()
         axis.set_xlabel('time')
         axis.set_ylabel('Drag coefficiant')
-        print("plot D_c for N={N}, u={u}")
+        print("plot D_c for N={N}, u={u}".format(N=N, u=u))
         axis.plot(times, drag_coeffs)
         tikzplotlib.save(figure=fig, filepath=(
                 PATH_DIAGRAMS + '{nr}-D.tex'.format(nr=filenumber)),
@@ -57,8 +61,8 @@ if __name__ == "__main__":
             print(error)
             print("abort this case")
             continue
-        times = gradT[len(gradT)//3:, 0]
-        dT = gradT[len(gradT)//3:, 1]
+        times = gradT[startindextake(len(gradT)):, 0]
+        dT = gradT[startindextake(len(gradT)):, 1]
         if len(cases) == 1:
             print("times:", times)
             print("dT:", dT)
-- 
GitLab