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\title{\fontsize{10pt}{12pt}{\sffamily{\textbf{EG501 COMPUTATIONAL FLUID DYNAMICS REPORT}}}}
\author{\fontsize{10}{15}{\rmfamily{James Long (51550519)}}}
\date{\vspace{-5ex}}
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\maketitle
\begin{abstract}
    \rmfamily{
    \fontsize{8pt}{12pt}{
    A hot cylinder cross flow is a common flow geometry encountered in any process plant when shell and tube heat exchangers are used. Heat exchanged in these methods is significantly influenced by the degree of mixing and boundary layer formation, making turbulence models an invaluable tool to investigate and quantify heat exchanged between fluids without experiments.
    This report investigates the turbulent behaviours of a cylinder in cross flow for 3 separate Reynolds' number as well as the heat transfer behaviour. The aims of these simulations were to successfully verify and validate the models and simulations against literature, and analyse the results to inspect the turbulent mixing and how it affect the heat transfer from the cylinder to the fluid.
    It was found that the k-$\varepsilon$ model did not give an accurate result for the drag co-efficient of the cylinder in cross flow, giving an under-estimation. The Transition SST model gave a result that was more accurate and had excellent agreement with the values given in literature detailed in the document. Vortex shedding occurred in the simulation for $Re = 10,000$.   }
}
\end{abstract}

\section*{\textbf{INTRODUCTION}}

\section*{\textbf{SIMULATION SET UP}}
\subsection*{Geometry}
Figure \ref{fig:geometry} shows the geometry set up used for all simulations.
\begin{figure}[h]
\centering
    \resizebox{0.6\textwidth}{0.5\textwidth}{
    \begin{tikzpicture}
    \node[anchor=south west,inner sep=0] at (0,0) {\includegraphics{geometry.png}};
    \draw[>=stealth, ultra thick, <->] (7.3,8.79) -- (8.5,8.79);
    \draw[ultra thick] (7.3,8.79) -- (8.5,8.79) -- node[above] {\Large{D = 2.5cm}} ++(2.5,0);
    \draw[dashed, thick] (6,8.79) -- (8.5,8.79);
    \draw[>=stealth, ultra thick, <->] (1.9,7.5) -- node[above] {\Large{L\textsubscript{1} = 11.5cm}} (7.9,7.5);
    \draw[>=stealth, ultra thick, <->] (7,8.79) -- node[right] {\Large{H\textsubscript{1} = 12.5cm}} (7,15.25);
    \draw[>=stealth, ultra thick, <->] (15.5,2.3) -- node[right] {\Large{H = 25cm}} (15.5,15.25);
    \draw[dashed, thick] (7.9,11) -- (7.9,7);
    \draw[>=stealth, ultra thick, <->] (1.9,5) -- node[above] {\Large{L = 31.5cm}} (18.3,5);
    \end{tikzpicture}
    }
    \caption{An image exported from the geometry file used in Ansys. Annotated with the dimensions used to make the geometry file. Dashed lines go through the centre of the cylinder circle. The shaded grey area is the flow domain.}
    \label{fig:geometry}
\end{figure}

\subsection*{Mesh}
The meshes used in the simulations were all unstructured, made with a triangular method. the mesh tightness around the cylinder and the mesh tightness in the bulk of the flow domain will both affect the solution from the simulation. For this reason, refinement was used around cylinder. This way, decreasing the element size of the mesh will increase the tightness around the cylinder and the bulk of the flow domain, as the refinement method used in Ansys meshing will use element sizes smaller than that of the over-all mesh element size.
\subsection*{Simulation Set Up}
\subsubsection*{Initial and Boundary Conditions}
The boundary conditions of the upper and lower walls of the geometry was set to a wall with no shear. The inlet boundary condition was set to as a velocity inlet, with a uniform flow velocity profile. The outlet broundary condition was set as a pressure outlet.

\subsubsection*{Flow past a cylinder}
For the steady simulations at Re = 10, the laminar viscous model in Fluent was used. For Re = 100, the laminar viscous model was used, but the simulation was switched from steady to transient, with 10 iterations per time step. The number of time steps taken was 500, which made sure that the flow in the simulation would reach a pseudo steady state. 2 models were used for Re = 10,000, the k-$\varepsilon$ and the transition shear stress transport (tranistion-SST) model. These were both run as transient simulation, and were allowed to run for 1000 time steps to ensure that the flow reached a pseudo-steady state.
\subsubsection*{Flow past a heated cylinder}
The fluid used was air at 75$^\circ$C, the properties of air were evaluated at this temperature using \citep{perrys}

\section*{\textbf{RESULTS AND ANALYSIS}}

\subsection*{Task 2.2.2}
\subsubsection*{Verification}
For the laminar model, the error in the velocity profile 5 cm down stream of the cylinder after 500 of the same time steps (and therefor the velocity profiles at the same time)  was used as the variable of interest to evaluate mesh independence. The error between solution from different mesh sizes is calculated as:
\begin{equation}
    \varepsilon = \frac{1}{N}\large{\sum_{j = 1}^N} \left| \frac{u_{x,j}^{i+1} - u_{x,j}^{i}}{u_{x,j}^{i}} \right|
\end{equation}
For higher Reynolds numbers, $Re = 10,000$, it was found that calculating the error using velocity profiles was too difficult due to the chaotic nature of the turbulent mixing after the cylinder. The variable of interest in this case was the drag co-efficient. This was used to evaluate mesh independence instead of velocity profiles. Since it was seen that the drag co-efficient and nusselt number oscillated with time in the simulation, the time averages of these values were calculated as:
\begin{equation}
    \overline{C_D} = \frac{\displaystyle{\int_{t_1}^{t_2}C_D\,dt}}{t_2 - t_1}\qquad \overline{Nu} = \frac{\displaystyle{\int_{t_1}^{t_2}Nu\,dt}}{t_2 - t_1}
\end{equation}
Where the integral was calculated numerically from the exported $C_D\times$T and $Nu\times$T data using the simpsons rule in a Python3 script.
The verification results are tabulated in table \ref{tab:verification1}, where it is shown the error between the solutions provided by the mesh are reasonably low, enough to say that the results have mesh independance.
\begin{table}[h]
    \centering
    \refstepcounter{table} \label{tab:verification1}
    {\renewcommand{\arraystretch}{1.3}%
    \begin{tabular}{|c c c c c |}
    \hline
    &&&&\\
    \multicolumn{5}{|l|}{\shortstack[l]{\footnotesize{\textbf{Table \ref{tab:verification1}. Tabulated errors between variables of interest for different mesh sizes at different Reynolds's numbers. }} \\
     \footnotesize{\textbf{The underlined mesh sizes are the selected and used mes resolutions.}}}} \\
     \hline
         Re &  Mesh Size & $\overline{C_D}$ & Literature $\overline{C_D}$ & $\varepsilon$\\
         10  & 50,000 cells & 3.57 & 2.83 & 0.2615 \\
         100  & 42,900 cells &  1.52 & 1.41 & 0.0780 \\
         10,000 (K-$\varepsilon$) & 71,808 cells & 0.88 & 1.09 & 0.1927 \\
         10,000 (Transition-SST model) & 115,197 cells & 1.12 & 1.09 & 0.0275 \\
         \hline
    \end{tabular}
    }
\end{table}
\begin{figure}[h]
    \centering
    \begin{tikzpicture}
        \begin{axis}[
                    height = 0.5\textwidth,
                    width  = 0.7\textwidth,
                    ylabel = {Pressure co-efficient, $C_p$},
                    xlabel = {Angle about cylinder, rad},
                    xmin = 0, xmax = 6.282914,
                    xtick={0,1.5708,3.14159,4.7124,6.282914},
                    xticklabels={$0$,$\frac{\pi}{2}$,$\pi$,$\frac{3\,\pi}{2}$,$2\pi$},
                    no markers
                    ]
                \addplot[red,thick] table [x=x, y=Pc, col sep=comma] {Re10000Cp.csv};
                \addlegendentry{$Re = 10,000$}%
                 \addplot[blue,thick] table [x=x, y=Pc, col sep=comma] {Re100Cp.csv};
                 \addlegendentry{$Re = 100$}
                \addplot[black,thick] table [x=x, y=Pc, col sep=comma] {Re10Cp.csv};
                \addlegendentry{$Re = 10$}
                \node[circle,fill,blue, inner sep=1.5pt,label = below:{\color{blue}$\theta_s = 1.61$ rad}, mark size = 1pt,text=blue] at (1.53,-1.427) {};
                \node[circle,fill,blue, inner sep=1.5pt,label = right:{\color{blue}$\theta_s = 5.04$ rad}, mark size = 1pt] at (4.94,-1.02) {};
                \node[circle,fill,red, inner sep=1.5pt,label = below:{\color{red}$\theta_s = 1.18$ rad}, mark size = 1pt,text=blue] at (1.4,-2.2) {};
                \node[circle,fill,red, inner sep=1.5pt,label = below:{\color{red}$\theta_s = 5.04$ rad}, mark size = 1pt] at (5.05,-1.63) {};
        \end{axis}
    \end{tikzpicture}
    \caption{Pressure co-efficient around the cylinder wall. The origin of the horizontal axis begins at the front of the cylinder to the flow, going clockwise around it. The Result for Re = 10,000 is taken from the transition-SST simulation and not the k-$\varepsilon$ model as it gave more accurate results.}
    \label{fig:my_label}
\end{figure}

\subsubsection*{Validation}
The results for the calculated drag co-efficient were compared to literature values which are all tabulated in table \ref{tab:validation1}.

It is seen that the transition shear stress transport model (sometimes known as gamma-Retheta SST) model gives a more accurate result than the k-$\varepsilon$ model. The k-$\varepsilon$ model is a 2 equation Reynold's averaged Navier Stokes model that is known to have a weak response to adverse pressure gradients and consequently under-predict boundary layer separation \citep{menter}. Both of these phenomena are significant for this flow geometry at this high Reynolds number. The transition-SST model on the other hand is a 4 equation model, that gives better treatment to the boundary layer in adverse pressure gradients \citep{argy}. This is likely the reason for the deference.

\begin{figure}[h]
\centering
\subfigure[\sffamily{\textbf{Vorticity contour plot for Re = 10}}]{
\includegraphics[height = 0.45\textwidth, width = 0.48\textwidth]{vortRe10.png}
}
\subfigure[\sffamily{\textbf{Vorticity contour plot for Re = 10,000}}]{
\includegraphics[height = 0.45\textwidth, width = 0.48\textwidth]{vortRe10000.png}
}
\caption{Vorticity contour plots for different Reynolds numbers.}
\label{fig:vort}
\end{figure}

\begin{figure}[h]
\centering
\subfigure[\sffamily{\textbf{Stream line plot for Re = 10}}]{
\includegraphics[height = 0.45\textwidth, width = 0.45\textwidth]{streamlineRe10.png}
}
\subfigure[\sffamily{\textbf{Stream line plot for Re = 10,000}}]{
\includegraphics[height = 0.45\textwidth, width = 0.45\textwidth]{streamlineRe10000.png}
}
\caption{Streamline plots for different Reynolds numbers.}
\label{fig:streamline}
\end{figure}

\subsection*{Task 2.2.2 part c)}

It is seen that for the laminar case, where Re = 10, the streamline profiles and pressure co-efficient all follow a profile expected from \citep{schlicht2}. That it, the streamlines for this case diverge around the cylinder and re-converge around the cylinder, returning to a similar distribution as before the cylinder, and there is no separation of the boundary layer from the cylinder body. The pressure co-efficient around the cylinder is symmetric, matching the limiting solution profile in figure 1.13 from \citep{schlicht2}.  As the Reynolds number increases, the pressure co-efficient profile minima come closer together, indicating the angle of separation is decreasing (this angle being, the angle between the front of the cylinder to the flow and the point on the cylinder shoulder where the boundary layer separates, a satisfactory rendition is shown in figure 1.14 of \citep{schlicht2}), indicating that the flow separation occurs earlier along the cylinder with increasing Reynolds number.

As the Reynolds number increases, the wake behind the cylinder becomes unstable.
These instabilities cause the wake to oscillate as does the velocity with time, moving down stream from the cylinder. This shown in the figures \ref{} and \ref{}, called Von K\'arm\'an vortex streets \citep{dudeman}. For Reynolds number (but Re < 3$\times$10$^{5}$), the boundary layer remains laminar, which seen to be the case as shown in figure \ref{} where the streamlines close to the cylinder are seen to be parallel in the boundary region before separation, although the wake becomes turbulent \citep{dudeman}. A separation bubble forms behind the cylinder where the boundary has separated, in which eddies form. For regions where 20 < Re < 50, a pair of symmetric, stable eddies form in the separation bubble that do not shed. As the Reynolds number increases further, the flow field becomes unstable. The instabilities formed in the wake, close to the cylinder cause these vortexes behind the cylinder to shed, id est, they cause the onset of the Von K\'arm\'an vortex street.


\subsection*{Task 2.3.1}
\subsubsection{Time averaged Nusselt Number}
The time averaged Nusselt number for the simulation using the transition-SST model was 51.854, and the average Reynolds number around the cylinder was found to be 8,774, which is significantly different to the inlet Reynolds number of 10,000. Though the correlations used here from literature are insensitive to the Reynolds number, the difference here is large enough to make the assumption that the inlet Reynold's being equal to the Reynold's number around the cylinder invalid.  Thus the the averaged Reynolds number around the cylinder is used in the correlations used.
For the k-$\varepsilon$ the time averaged Nusselt number was 87.081 and the averaged Reynold's number around the cylinder was 10736.3. This is compared to the literature values in section. To use the functions provided from \citep{bernie}, \citep{Zman}, \citep{holman} and \citep{hilpi}, The prandlt number of the film, surface and bulk of the flow past cylinder had to be calculated.
The Prandlt number for the bulk of the flow was calculated using the inputed values for the transport and thermodynamic properties (specific heat $c_p$, dynamic viscosity $\mu$ and thermal conductivity $k$) into the simulation, which were calculated using chapter one from \citep{perrys}. The same book chapter used to calculate $c_p$, $\mu$ and $k$ at he cylinder surface and film temperature. The calculations for the Prandlt numbers went as follows:
{\renewcommand{\arraystretch}{1.3}%
\begin{gather*}
    Pr = \frac{c_p \, \mu}{k}\\
    Pr = \frac{1003.70 \left[\sfrac{J}{kg\,K} \right] \times 2.0742  \times 10^{-5}\left[ \sfrac{kg}{m\,s} \right] }{0.0295 \left[ \sfrac{W}{m\,K} \right]} = 0.706 \\
    Pr_s =\frac{1118.91 \left[\sfrac{J}{kg\,K} \right] \times 2.9413  \times 10^{-5}\left[ \sfrac{kg}{m\,s} \right] }{ 0.0439 \left[ \sfrac{W}{m\,K} \right]} = 0.750 \\
    Pr_f =\frac{ 1017.34 \left[\sfrac{J}{kg\,K} \right] \times 2.5375  \times 10^{-5}\left[ \sfrac{kg}{m\,s} \right] }{ 0.0369 \left[ \sfrac{W}{m\,K} \right]} = 0.699
\end{gather*}
}
The film temperature was estimated as the average of the wall and bulk temperature (as conjectured by \citep{bergman}) T$_f$ = 188$^\circ$C. Of course, in the simulation, the thermodynamic properties and transport properties (except from density) are held constant as temperature changes, thus the film and surface Prandlt numbers in the simulation are the same
The resulting Nusselt number from literature sources and the correlations used are tabulated in table \ref{tab:Verification2}. The results from the k-$\varepsilon$ model are neglected from table \ref{tab:Verification2} as it was clear that k-$\varepsilon$ result was significantly different to the literature values, and bared no value in comparison. The error tabulated in table \ref{tab:Verification2} is the error between the simulation using the transition-SST model and the literature value.
\begin{table}[h]
    \centering
    \refstepcounter{table} \label{tab:Verification2}
{\renewcommand{\arraystretch}{1.2}%
\begin{tabular}{|p{2.9cm} p{8.5cm} p{1.2cm} p{1.2cm}|}
     \hline
     &&&\\
     \multicolumn{4}{|l|}{\shortstack[l]{\footnotesize{\textbf{Table \ref{tab:Verification2}. Tabulated correlations, references and value found in calculating the average Nusselt}} \\
     \footnotesize{\textbf{ number as well as the error between the calculation and the simulation results. $Re = 10,000$.}}}} \\
     \hline
     Reference & Correlation & Nusselt number & Error (\%) \\
     \hline
     \citep{bernie} & \parbox{8.5cm}{
                                                    \begin{equation}\label{bernie}
                                                    \normalsize{
                                                    \overline{Nu} = 0.3 + \frac{0.62\,Re^{\sfrac{1}{2}}Pr^{\sfrac{1}{2}}}{\left[ 1 + \left( \sfrac{0.4}{Pr}\right)^{\sfrac{2}{3}}\right]^{\sfrac{1}{4}}} \left[ 1+ \left( \frac{Re}{282000} \right)^{\sfrac{1}{2}} \right]}
                                                    \end{equation}
                                                     } &
    43.70 & 03.4 \\
    \citep{hilpi} & \parbox{6cm}{\begin{equation}\label{hilpi}
                                    \overline{Nu} = 0.193\,Re^{0.618} Pr^{\sfrac{1}{3}}
                                   \end{equation}
                                                     } &
    39.91 & 29.9 \\
    \citep{Zman} & \parbox{6cm}{\begin{equation}\label{Zman}
                                    \overline{Nu} = 0.26\,Re^{0.6} Pr^{0.37}\left( \frac{Pr}{Pr_s}\right)^{\sfrac{1}{4}}
                                   \end{equation}
                                                     } &
    52.30 & 00.8 \\
    \citep{holman} & \parbox{6cm}{\begin{equation}\label{holman}
                                    \overline{Nu} = 0.25\,Re^{0.6} Pr^{0.38}\left( \frac{Pr_f}{Pr_s}\right)^{\sfrac{1}{4}}
                                   \end{equation}
                                                     } &
    49.98 & 03.7 \\
    \hline
\end{tabular}
}
\end{table}
It is seen the best agreement between the simulated Nusselt number and correlatoin are the results from equation \eqref{Zman}, \eqref{bernie} and \eqref{holman}. Although the errors associated with each correlation could not be retrieved from literature, it is known that these errors are typically in the range of 5\% to 15\%. Thus it can be said, that the Nusselt number predicted by the simulation has good agreement with the correlation proposed by \eqref{Zman}, \eqref{bernie} and \eqref{holman}.


\subsubsection*{Time averaged drag co-efficient}
$\overline{C_D}$ for the simulation using the k-$\varepsilon$ model was !!! and for the transition-SST model was !!!. Drag co-efficients for this turbelent flow for Re = 10,000 could not be sourced from literature. However, it is known that when a cylinder is heated, the drag co-efficient should increase be larger than that of a flow for the same Reynolds number \citep{Vman}. This is contrary to the results from both the transition-SST simulation and the k-$\varepsilon$ model.

\clearpage
\bibliography{bibli.bib}





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