results

tex/appendix-figures.tex

@@ -9,6 +9,13 @@
   \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}
+
+\begin{figure}[ht]
+  \centering
+  \tikzinput{14-u--2-N4-gT}
+  \caption{Development of $\abs{∇T}$ over time at $\Rey=100$ and $N = 4$ (26496 cells). We see no stabilisation. Therefore this value is not reliable. For $N=1$ and $N=2$ the curves look different but not stabilising either.}%
+  \label{fig:gradTNotStable}
+\end{figure}
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tex/results.tex

@@ -30,7 +30,21 @@ 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 $3900$.
 The values for $100$ and $1000$ are inserted into table \ref{tab:values} for comparison.
 
-\textcite[4.1.1 The laminar flow][5]{same} reports that at $\Rey = 100$ a steady state is reached after \SI{15}{\second}. This was not confirmed by my experiments: the visual inspection indicated a stabilisation around \SIrange{60}{80}{\second}. The development of the drag coefficiant over time indicates stabilisation at around \SI{}{\second}.?? % check at end
+\paragraph{Low Reynolds number}
+\textcite[4.1.1 The laminar flow][5]{same} reports that at $\Rey = 100$ a steady state is reached after \SI{15}{\second}. This was not confirmed by my experiments: the visual inspection indicated a stabilisation around \SIrange{60}{80}{\second}. The development of the drag coefficiant over time indicates stabilisation at around \SI{130}{\second}, for $N=4$ even later. Therefore this experiment was run until $t=\SI{150}{\second}$.
+The $\abs{∇T}$ did not stabilise within \SI{150}{\second} though as visible in figure{fig:gardTNotStable}.
+
+\paragraph{Medium Reynolds number}
+\textcite[4.1.2 The turbulent flow][9]{same} noted that for $\Rey = 1000$ the necessary time steps need to be very small and therefore the maximal time length could not be long.
+They observed periodical fluctuations of the drag coefficiant stabilising at around \SI{140}{\second}.
+My experiment did not do the same but decreases within the time range of \SI{150}{\second} and did not stabilise during this time. Therefore the simulation time was increased to \SI{230}{\second} where it was getting closer to a stable value for $N=1$ but showed signs of either divergence or fluctuation for $N=2$.
+
+\textcite{same} introduced an additional pertubation that should decrease the time the simulation needs to reach an interesting state. This was not done in this experiment.
+The time step for $N=2$ to guarantee a maximal Courant number of 1 was about $0.05$ with about 5 time steps calculated per (real-world) second.
+There was not enough time to calculate the case for $N=4$.
+
+\paragraph{High Reynolds number}
+Due to the long running time of simulations with $\Rey \in \{\num{e4}, \num{e5}\}$ only one experiment with $N=1$, $\Rey = \num{e4}$ and $t_{\max} = \SI{20}{\second}$ was calculated. The development of drag coeffciant and $\abs{∇T}$ looked as if it stabilised but due to the short running time this is uncertain.
 
 \section{Measurements}
 The drag coefficiant was calculated with the OpenFOAM intrinsic function
@@ -43,30 +57,33 @@ 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. ref-$\Nus$ is the Nusselt number according to the Churchill-Bernstein equation.}
+  \caption{Scalar results of the numerical experiments. ref-$\Nus$ is the Nusselt number according to the Churchill-Bernstein equation. Reference drag coefficiants are taken from \textcite[Table 1][14]{same}. Value with (*) have an high uncertainty because they have not stabilised yet.}
   \label{tab:values}
   \begin{tabular}{cccccccccc}
-    $\Rey$ & $u_{in}$ & \# cells & $Δt$ & $t_{\max}$ $C_D$ & $\vec n · ∇T\vert_{\text{cyl}}$ & ref-$C_D$ & $\Nus$ & ref-$\Nus$\\
-    \num{e2} & \num{0.05} & 1664 & &  & & 5.1%56
+    $\Rey$ & $u_{in}$ & \# cells & $Δt$ & $t_{\max}$ & $C_D$ & $\vec n · ∇T\vert_{\text{cyl}}$ & $\Nus$ & ref-$C_D$  & ref-$\Nus$\\
+    \num{e2} & \num{0.05} & 1656 & \SI{0.2}{\second} & \SI{150}{\second} & \num{8.44e-2} & \num{2.84e2} & 5.68 & 1.245 & 5.1%56
     \\
-    \num{e2} & \num{0.05} & 6656 & & & & 5.1%56
+    \num{e2} & \num{0.05} & 6624 &  \SI{0.2}{\second} & \SI{150}{\second} & \num{7.18e-2} & \num{2.69e2} & 5.38 & 1.245 & 5.1%56
     \\
-    \num{e2} & \num{0.05} & 26624 & & & & 5.1%56
+    \num{e2} & \num{0.05} & 26496 & \SI{0.2}{\second} & \SI{150}{\second} & \num{5.66e-2} & \num{2.3e2} &4.6 & 1.245 & 5.1%56
     \\
-    \num{e3} & \num{0.5} & & & & & 15%.930
+    \num{e3} & \num{0.5} & 1656 & \SI{0.12}{\second} & \SI{230}{\second} & \num{4.84} & \num{7.9e2} & \num{15.8} & 0.995 & 15%.930
     \\
-    \num{e4} & \num{5} & & & & & 53%.328
+    \num{e3} & \num{0.5} & 6624 & \SI{0.2}{\second} & \SI{230}{\second} & \num{6.39} & \num{7.5e2}(*) & \num{15} & 0.995 & 15%.930
     \\
-    \num{e5} & \num{50} & & & & & 214%.126
+    \num{e4} & \num{5} & 1656 & \SI{0.009}{\second} & \SI{20}{\second} & \num{4.58} & \num{2.58e3} & \num{51.6} & --- & 15%.930
   \end{tabular}
 \end{table}
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+\section{Comparison and conclusion}
+For the relatively low Reynolds number of $\Rey = 100$ this setup produced results that did not fit to the values in the literature: the drag coefficiant is off by one to two orders of magnitude, it did not stabilise within the same time frame. Only the Nusselt number was in the same order of magnitude as predicted by the Churchill-Bernstein equation but the value varied greatly between different mesh densities and did not stabilise during run time. So the setup is not suitable.
+
+For $\Rey = 1000$ the simulation was not able to exhibit the regular fluctuations observed by \textcite{same} and the drag coefficiant was off by a factor of five (closer than for $\Rey = 100$ but the Nusselt number was again close to the Churchill-Bernstein equation value.
+
+For higher Reynold numbers $\Rey \in \{\num{e4}, \num{e5}\})$ the computation became unfeasible slow because the Courant numbers dictate time steps at the order of magnitude of  $0.008$ ($N=1$, $\Rey=\num{e4}$) to $0.0008$ ($\Rey=\num{e5}$) and the stabilisation times dictate running times of at least \SI{30}{\second} (according to turbulent case in \textcite{same}). Therefore those cases could not even be compared to the literature.
+
+For all Reynold numbers the different meshes showed quite different results. That gives no confidence for the reliability of the coarse meshes.
+This setup for numerical computation for unsteady flow around a cylinder in 2 dimensions is suitable to calculate the rough estimates on the Nusselt number but does not give reliable results for drag coefficiants or fluctuation frequencies and does not scale to denser meshes or higher turbulence, indicated by higher Reynold numbers.
+The author cannot tell if that is due to an inferior mesh, wrong boundary conditions, unsuitable time stepping or solver and scheme choice. On each of those variables except boundary conditions some variations were tried without noticeable improvement.
 
 \docEnd