Project 4
Aim: To compare discretization schemes in OpenFoam.
The report of the same has been written in LaTex and presented here in the form of code. The .pdf file of the same is attached herewith along with supporting files.
The LaTex code:
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\title{\textbf{Project-4: To compare discretization schemes in {\fontfamily{qcr}\selectfont OpenFoam} using a customised test case {\fontfamily{qcr}\selectfont scalarTransport}}}
\author{Kavendra Ashokrao Thakare}
\begin{document}
\maketitle
\begin{abstract}
In this project we built a customised test case {\fontfamily{qcr}\selectfont scalarTransport}, using the existing tutorial test cases {\fontfamily{qcr}\selectfont \$FOAM\_TUTORIALS/compressible/sonicFoam/laminar/shockTube} and \newline {\fontfamily{qcr}\selectfont \$FOAM\_TUTORIALS/basic/scalarTransportFoam/pitzDaily}.
This case has been solved for pure diffusion with different values of diffusion coefficients to study the diffusion. Similarly the case has been solved for pure convection with different divergence schemes, to compare the results obtained from different schemes and effects of numerical diffusion.
The results are discussed in depth in the later section. Also the causes of numerical diffusion and methods to minimise numerical diffusion are explained here in this report.
\end{abstract}
\section{Introduction to the solver {\fontfamily{qcr}\selectfont scalarTransportFoam}}
The {\fontfamily{qcr}\selectfont scalarTransportFoam} solver solves the incompressible flow for scalar transport equations for any {\fontfamily{qcr}\selectfont volScalarField} $\phi$. Here $\phi$ is T.
The Scalar Transport equation for temperature is as:
\begin{equation}
\frac{\partial T}{\partial t} + (U.\nabla) T = \nabla.(DT \nabla T)
\end{equation}
where \textit{DT} is the heat diffusion coefficient,
\newline When the velocity is set to zero then the above scalar transport equation reduces to pure diffusion equation as,
\begin{equation}
\frac{\partial T}{\partial t} = \nabla.(DT \nabla T)
\end{equation}
With different values of heat diffusion coefficient \textit{DT}, the pure diffusion is solved and results are obtained and investigated.
\newline When the diffusion coefficient \textit{DT}, is set to zero then the above scalar transport equation reduces to pure convection (Advection) equation as,
\begin{equation}
\frac{\partial T}{\partial t} + (U.\nabla) T = 0
\end{equation}
With different divergence schemes the pure convection is solved and results are examined.
\section{Steps in the creation of the customised test case
\newline {\fontfamily{qcr}\selectfont scalarTransport}}
\subsection{To copy the existing tutorial case {\fontfamily{qcr}\selectfont ~/sonicFoam/laminar/shockTube} into the {\fontfamily{qcr}\selectfont \$FOAM\_RUN} directory and to modify it to {\fontfamily{qcr}\selectfont scalarTransport}}
Copy the existing tutorial case
\newline {\fontfamily{qcr}\selectfont \$FOAM\_TUTORIALS/compressible/sonicFoam/laminar/shockTube} to {\fontfamily{qcr} \selectfont \$FOAM\_RUN} directory, using copy command as follows:
\newline {\fontfamily{qcr} \selectfont cp -r \$FOAM\_TUTORIALS/compressible/sonicFoam/laminar/shockTube scalarTransport}
\newline Get into the {\fontfamily{qcr} \selectfont scalarTransport} case. Check the contents and remove the unwanted files and directories as follows:
\newline {\fontfamily{qcr} \selectfont cd scalarTransport}
\newline {\fontfamily{qcr} \selectfont ls *}
\newline The case contents are as below:
\begin{verbatim}
Allclean Allrun
0.orig:
T U p
constant:
thermophysicalProperties turbulenceProperties
system:
blockMeshDict controlDict fvSchemes fvSolution sample setFieldsDict
\end{verbatim}
Remove the {\fontfamily{qcr} \selectfont Allrun} and {\fontfamily{qcr} \selectfont Allclean} scripts using command \newline {\fontfamily{qcr} \selectfont rm All*}
\newline Rename the {\fontfamily{qcr}\selectfont 0.orig} directory to the {\fontfamily{qcr}\selectfont 0} by using rename command and remove the initial pressure file {\fontfamily{qcr} \selectfont p} from it. Commands are as follows:
\newline {\fontfamily{qcr} \selectfont mv 0.orig 0}
\newline {\fontfamily{qcr} \selectfont rm 0/p}
\newline Add the files {\fontfamily{qcr} \selectfont transportProperties} into {\fontfamily{qcr} \selectfont constant} directory. \newline Remove the existing files {\fontfamily{qcr} \selectfont thermophysicalProperties} and {\fontfamily{qcr} \selectfont turbulenceProperties}.\newline Commands are as follows:
\newline {\fontfamily{qcr} \selectfont cp constant/turbulenceProperties constant/transportProperties}
\newline {\fontfamily{qcr} \selectfont rm constant/thermophysicalProperties}
\newline {\fontfamily{qcr} \selectfont rm constant/turbulenceProperties}
\newline Change the file contents of {\fontfamily{qcr} \selectfont transportProperties} as it is copied from some other file, to the content given below using the text editor.
\begin{verbatim}
FoamFile
{
version 2.0;
format ascii;
class dictionary;
location "constant";
object transportProperties;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
DT 0.0; //The value will be changed as required for pure diffusion case.
// ************************************************************************* //
\end{verbatim}
Now get the into the {\fontfamily{qcr}\selectfont system} directory and Keep the existing file {\fontfamily{qcr}\selectfont blockMeshDict} as it is.
\newline The file {\fontfamily{qcr}\selectfont setFieldsDict} is kept as it is for time being as it is required with slight modifications.
\newline Remove the existing files {\fontfamily{qcr}\selectfont controlDict , fvSchemes , fvSolution} and {\fontfamily{qcr}\selectfont sample} as these will be replaced by corresponding files from the tutorial case, {\fontfamily{qcr}\selectfont \$FOAM\_TUTORIALS/basic/scalarTransportFoam/pitzDaily}.
\newline List of commands is as given below:
\newline {\fontfamily{qcr} \selectfont cd system/}
\newline {\fontfamily{qcr} \selectfont rm controlDict}
\newline {\fontfamily{qcr} \selectfont rm fvSchemes}
\newline {\fontfamily{qcr} \selectfont rm fvSolution}
\newline {\fontfamily{qcr} \selectfont rm sample}
\newline {\fontfamily{qcr}\selectfont cp \$FOAM\_TUTORIALS/basic/scalarTransportFoam/pitzDaily/system/controlDict .}
\newline {\fontfamily{qcr}\selectfont cp \$FOAM\_TUTORIALS/basic/scalarTransportFoam/pitzDaily/system/fvSchemes .}
\newline {\fontfamily{qcr}\selectfont cp \$FOAM\_TUTORIALS/basic/scalarTransportFoam/pitzDaily/system/fvSolution .}
\newline
\subsection{To modify the file {\fontfamily{qcr}\selectfont system/setFieldsDict}}
Modify the contents of file {\fontfamily{qcr}\selectfont system/setFieldsDict}.
Change the content such that the temperature is in the box at the centre of the tube is set higher than the rest of the tube.
\newline The changed content of the file {\fontfamily{qcr}\selectfont system/setFieldsDict} is as follows:
\begin{verbatim}
\*---------------------------------------------------------------------------*/
FoamFile
{
version 2.0;
format ascii;
class dictionary;
location "system";
object setFieldsDict;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
defaultFieldValues
(
volScalarFieldValue T 0.0
);
regions
(
boxToCell
{
box (-0.5 -1 -1) (0.5 1 1);
fieldValues
(
volScalarFieldValue T 1.0
);
}
);
// ************************************************************************* //
\end{verbatim}
\subsection{To modify the file {\fontfamily{qcr}\selectfont system/controlDict}}
The {\fontfamily{qcr}\selectfont controlDict} file is modified by changing the application (solver) name to {\fontfamily{qcr}\selectfont scalarTransportFoam}.
The settings of {\fontfamily{qcr}\selectfont startTime, endTime, deltaT \& writeInterval } are modified as required.
The modified {\fontfamily{qcr}\selectfont controlDict} file is as follows:
\begin{verbatim}
FoamFile
{
version 2.0;
format ascii;
class dictionary;
location "system";
object controlDict;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
application scalarTransportFoam;
startFrom startTime;
startTime 0;
stopAt endTime;
endTime 5;
deltaT 0.05;
writeControl timeStep;
writeInterval 10;
purgeWrite 0;
writeFormat ascii;
writePrecision 6;
writeCompression off;
timeFormat general;
timePrecision 6;
runTimeModifiable true;
// ************************************************************************* //
\end{verbatim}
\subsection{To modify the file {\fontfamily{qcr}\selectfont system/fvSchemes}}
The {\fontfamily{qcr}\selectfont fvSchemes} file is modified by adding the gradient and divergence scheme for the Temperature i.e. schemes for {\fontfamily{qcr}\selectfont grad(T) \& div(phi,T)} under respective schemes.
The modified {\fontfamily{qcr}\selectfont controlDict} file is as follows:
\begin{verbatim}
FoamFile
{
version 2.0;
format ascii;
class dictionary;
location "system";
object fvSchemes;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
ddtSchemes
{
default Euler;
}
gradSchemes
{
default Gauss linear;
grad(p) Gauss linear;
grad(T) Gauss linear;
}
divSchemes
{
default none;
div(phi,U) Gauss linear;
div(phi,T) Gauss upwind;
}
laplacianSchemes
{
default Gauss linear orthogonal;
}
interpolationSchemes
{
default linear;
}
snGradSchemes
{
default orthogonal;
}
// ************************************************************************* //
\end{verbatim}
\subsection{To modify the file {\fontfamily{qcr}\selectfont system/fvSolution}}
The {\fontfamily{qcr}\selectfont fvSolution} file is modified by adding the solver for the Temperature and {\fontfamily{qcr}\selectfont SIMPLE} algorithm settings.
The modified {\fontfamily{qcr}\selectfont fvSolution} file is as follows:
\begin{verbatim}
FoamFile
{
version 2.0;
format ascii;
class dictionary;
location "system";
object fvSolution;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
solvers
{
U
{
solver smoothSolver;
smoother symGaussSeidel;
tolerance 1e-05;
relTol 0;
}
T
{
solver PBiCG;
preconditioner DILU;
tolerance 1e-06;
relTol 0;
}
}
SIMPLE
{
nNonOrthogonalCorrectors 0;
consistent yes;
}
// ************************************************************************* //
\end{verbatim}
\subsection{Content of the unchanged {\fontfamily{qcr}\selectfont system/blockMeshDict} file.}
The {\fontfamily{qcr}\selectfont blockMeshDict} file as shown below:
\begin{verbatim}
FoamFile
{
version 2.0;
format ascii;
class dictionary;
object blockMeshDict;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
scale 1;
vertices
(
(-5 -1 -1)
(5 -1 -1)
(5 1 -1)
(-5 1 -1)
(-5 -1 1)
(5 -1 1)
(5 1 1)
(-5 1 1)
);
blocks
(
hex (0 1 2 3 4 5 6 7) (1000 1 1) simpleGrading (1 1 1)
);
edges
(
);
boundary
(
sides
{
type patch;
faces
(
(1 2 6 5)
(0 4 7 3)
);
}
empty
{
type empty;
faces
(
(0 1 5 4)
(5 6 7 4)
(3 7 6 2)
(0 3 2 1)
);
}
);
mergePatchPairs
(
);
// ************************************************************************* //
\end{verbatim}
\subsection{Content of unchanged {\fontfamily{qcr}\selectfont 0/T} and {\fontfamily{qcr}\selectfont 0/U} files}
The content of the{\fontfamily{qcr}\selectfont 0/T} file is as given below:
\begin{verbatim}
FoamFile
{
version 2.0;
format ascii;
class volScalarField;
object T;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
dimensions [0 0 0 1 0 0 0];
internalField uniform 0;
boundaryField
{
sides
{
type zeroGradient;
}
empty
{
type empty;
}
}
// ************************************************************************* //
\end{verbatim}
The content of the{\fontfamily{qcr}\selectfont 0/U} file is as given below:
\begin{verbatim}
FoamFile
{
version 2.0;
format ascii;
class volVectorField;
location "0";
object U;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
dimensions [0 1 -1 0 0 0 0];
internalField uniform (0 0 0);
boundaryField
{
sides
{
type zeroGradient;
}
empty
{
type empty;
}
}
// ************************************************************************* //
\end{verbatim}
The list of content of case {\fontfamily{qcr}\selectfont scalarTransport} after all modifications is as follows:
\begin{verbatim}
0:
T U
constant:
transportProperties
system:
blockMeshDict controlDict fvSchemes fvSolution setFieldsDict
\end{verbatim}
\newpage
\section{Steps to solve the pure diffusion on customised case \newline {\fontfamily{qcr}\selectfont scalarTransport} with {\fontfamily{qcr}\selectfont ScalarTransportFoam} solver}
\subsection{To make the copies of the case {\fontfamily{qcr}\selectfont scalarTransport} with different values of diffusion coefficient \textit{DT} into {\fontfamily{qcr}\selectfont constant/transportProperties} file}
Four copies the case {\fontfamily{qcr}\selectfont scalarTransport} are made and modified as given in the table below:
\newline
\setlength{\arrayrulewidth}{0.5mm}
\setlength{\tabcolsep}{18pt}
\renewcommand{\arraystretch}{1.5}
\begin{table}[h!]
\centering
\begin{tabular}{ |p{4cm}|p{3cm}|p{4cm}| }
\hline
\multicolumn{3}{|c|}{List of changes in the different files for Pure Diffusion case} \\
\hline
Case Name (Copy of {\fontfamily{qcr}\selectfont scalarTransport}) & {\fontfamily{qcr}\selectfont 0/U} & {\fontfamily{qcr}\selectfont constant/\newline transportProperties} \\
\hline
{\fontfamily{qcr}\selectfont scalarTransportD0} & {\fontfamily{qcr}\selectfont U (0 0 0) } & {\fontfamily{qcr}\selectfont DT 0.0 } \\
{\fontfamily{qcr}\selectfont scalarTransportD0001} & {\fontfamily{qcr}\selectfont U (0 0 0) } & {\fontfamily{qcr}\selectfont DT 0.0001 } \\
{\fontfamily{qcr}\selectfont scalarTransportD01} & {\fontfamily{qcr}\selectfont U (0 0 0) } & {\fontfamily{qcr}\selectfont DT 0.01 } \\
{\fontfamily{qcr}\selectfont scalarTransportD1} & {\fontfamily{qcr}\selectfont U (0 0 0) } & {\fontfamily{qcr}\selectfont DT 1.0 } \\
\hline
\end{tabular}
\caption{List of changes in the different copies of case {\fontfamily{qcr}\selectfont scalarTransport} for Pure Diffusion}
\label{table:1}
\end{table}
\subsection {To solve the respective cases for pure diffusion using {\fontfamily{qcr}\selectfont scalarTransportFoam} solver}
By getting into the respective pure diffusion cases the following list of commands is executed to run the case with {\fontfamily{qcr}\selectfont scalarTransportFoam} solver.
\newline {\fontfamily{qcr} \selectfont blockMesh}
\newline {\fontfamily{qcr} \selectfont checkMesh}
\newline {\fontfamily{qcr} \selectfont setFields}
\newline {\fontfamily{qcr} \selectfont scalarTransportFoam}
\newline
\newline To generate mesh.
\newline To check mesh.
\newline To set the fields.
\newline To solve the case.
\newline The respective {\fontfamily{qcr} \selectfont .foam} files are saved. The results are post-processed in the {\fontfamily{qcr} \selectfont ParaView}.
\newpage
\subsection{The Results of the case {\fontfamily{qcr}\selectfont scalarTransportD0}}
The initial temperature contours at {\fontfamily{qcr}\selectfont t=0} and the final temperature contours at {\fontfamily{qcr}\selectfont t=5} for diffusion coefficient {\fontfamily{qcr}\selectfont DT=0.0} are observed as follows:
\begin{figure}[h]
\centering
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Contour DT0(t=0).png}
\caption{Initial Temperature Contours DT=0}
\label{fig:3}
\end{minipage}
\hfill
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Contour DT0(t=5).png}
\caption{Final Temperature Contours DT=0}
\label{fig:4}
\end{minipage}
\end{figure}
The initial temperature graph at {\fontfamily{qcr}\selectfont t=0} and the final temperature graph at {\fontfamily{qcr}\selectfont t=5} with respect to X-Coordinate, for diffusion coefficient {\fontfamily{qcr}\selectfont DT=0.0} are as shown below:
\begin{figure}[h]
\centering
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Profile DT0(t=0).png}
\caption{Initial Temperature graph DT=0}
\label{fig:3}
\end{minipage}
\hfill
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Profile DT0(t=5).png}
\caption{Final Temperature graph DT=0}
\label{fig:4}
\end{minipage}
\end{figure}
No difference has been observed there in initial and final contours as well as temperature graphs as diffusion coefficient is set to 0, i.e.\textit{DT=0}.
\newpage
\subsection{The Results of the case {\fontfamily{qcr}\selectfont scalarTransportD0001}}
The initial temperature contours at {\fontfamily{qcr}\selectfont t=0} and the final temperature contours at {\fontfamily{qcr}\selectfont t=5} for diffusion coefficient {\fontfamily{qcr}\selectfont DT=0.0001} are observed as follows:
\begin{figure}[h]
\centering
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Contour DT0.0001(t=0).png}
\caption{Initial Temperature Contours DT=1e-4}
\label{fig:3}
\end{minipage}
\hfill
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Contour DT0.0001(t=5).png}
\caption{Final Temperature Contours DT=1e-4}
\label{fig:4}
\end{minipage}
\end{figure}
The initial temperature graph at {\fontfamily{qcr}\selectfont t=0} and the final temperature graph at {\fontfamily{qcr}\selectfont t=5} with respect to X-Coordinate, for diffusion coefficient {\fontfamily{qcr}\selectfont DT=0.0001} are as shown below:
\begin{figure}[h]
\centering
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Profile DT0.0001(t=0).png}
\caption{Initial Temperature graph DT=1e-4}
\label{fig:3}
\end{minipage}
\hfill
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Profile DT0.0001(t=5).png}
\caption{Final Temperature graph DT=1e-4}
\label{fig:4}
\end{minipage}
\end{figure}
The slight difference has been observed in initial and final temperature contours as the sign of temperature diffusion. The Final temperature contours are slightly diffused at the boundaries as diffusion coefficient \textit{DT} is set to 0.0001 which is very low value as compared to scale of geometry.
\newline The final temperature graph is slightly deformed at the top with same magnitude as initial temperature which justifies the diffusion with low value of diffusion coefficient.
\newpage
\subsection{The Results of the case {\fontfamily{qcr}\selectfont scalarTransportD01}}
The initial temperature contours at {\fontfamily{qcr}\selectfont t=0} and the final temperature contours at {\fontfamily{qcr}\selectfont t=5} for diffusion coefficient {\fontfamily{qcr}\selectfont DT=0.01} are observed as follows:
\begin{figure}[h]
\centering
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Contour DT0.01(t=0).png}
\caption{Initial Temperature Contours DT=1e-2}
\label{fig:3}
\end{minipage}
\hfill
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Contour DT0.01(t=5).png}
\caption{Final Temperature Contours DT=1e-2}
\label{fig:4}
\end{minipage}
\end{figure}
The initial temperature graph at {\fontfamily{qcr}\selectfont t=0} and the final temperature graph at {\fontfamily{qcr}\selectfont t=5} with respect to X-Coordinate, for diffusion coefficient {\fontfamily{qcr}\selectfont DT=0.01} are as shown below:
\begin{figure}[h]
\centering
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Profile DT0.01(t=0).png}
\caption{Initial Temperature graph DT=1e-2}
\label{fig:3}
\end{minipage}
\hfill
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Profile DT0.01(t=5).png}
\caption{Final Temperature Contours DT=1e-2}
\label{fig:4}
\end{minipage}
\end{figure}
The major differences has been observed in initial and final temperature contours as the sign of high temperature diffusion. The Final temperature contours are more diffused at the boundaries as diffusion coefficient \textit{DT} is set to 0.01 which is comparatively high value as compared to scale of geometry.
\newline The final temperature graph is deformed and smeared with less magnitude as compared to initial temperature magnitude which justifies the more diffusion with comparatively high value of diffusion coefficient.
\newpage
\subsection{The Results of the case {\fontfamily{qcr}\selectfont scalarTransportD1}}
The initial temperature contours at {\fontfamily{qcr}\selectfont t=0} and the final temperature contours at {\fontfamily{qcr}\selectfont t=5} for diffusion coefficient {\fontfamily{qcr}\selectfont DT=1.0} are observed as follows:
\begin{figure}[h]
\centering
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Contour DT1(t=0).png}
\caption{Initial Temperature Contours DT=1}
\label{fig:3}
\end{minipage}
\hfill
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Contour DT1(t=5).png}
\caption{Final Temperature Contours DT=1}
\label{fig:4}
\end{minipage}
\end{figure}
The initial temperature graph at {\fontfamily{qcr}\selectfont t=0} and the final temperature graph at {\fontfamily{qcr}\selectfont t=5} with respect to X-Coordinate, for diffusion coefficient {\fontfamily{qcr}\selectfont DT=1.0} are as shown below:
\begin{figure}[h]
\centering
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Profile DT1(t=0).png}
\caption{Initial Temperature graph DT=1}
\label{fig:3}
\end{minipage}
\hfill
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Profile DT1(t=5).png}
\caption{Final Temperature graph DT=1}
\label{fig:4}
\end{minipage}
\end{figure}
The Initial and final temperature contours show that temperature diffusion rate is so high that, the Final temperature contours are fully diffused at the boundaries as diffusion coefficient \textit{DT} is set to 1 which is comparatively higher value as compared to scale of geometry.
\newline The final temperature graph is so deformed and smeared to the lower magnitude as compared to initial temperature magnitude which justifies the higher diffusion with comparatively higher value of diffusion coefficient.
\newpage
\subsection{The combined results of all the pure diffusion cases }
The initial temperature contours at {\fontfamily{qcr}\selectfont t=0} and the final temperature contours at {\fontfamily{qcr}\selectfont t=5} for different diffusion coefficients are observed in the following Figures.
\newline In the figure of contours the same sequence of pure diffusion cases is followed from top to bottom as mentioned in the Table 1. i.e.
\newline Temperature contours of {\fontfamily{qcr}\selectfont scalarTransportD0} at the top
\newline Temperature contours of {\fontfamily{qcr}\selectfont scalarTransportD0001} second from the top.
\newline Temperature contours of {\fontfamily{qcr}\selectfont scalarTransportD01} third from the top.
\newline Temperature contours of {\fontfamily{qcr}\selectfont scalarTransportD1} at the bottom.
\begin{figure}[h]
\centering
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Contour (t=0)-Combined.png}
\caption{Initial Temp Contours combined}
\label{fig:3}
\end{minipage}
\hfill
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Contour (t=5)-Combined.png}
\caption{Final Temp Contours combined}
\label{fig:4}
\end{minipage}
\end{figure}
The initial temperature graph at {\fontfamily{qcr}\selectfont t=0} and the final temperature graph at {\fontfamily{qcr}\selectfont t=5} with respect to X-Coordinate, for different diffusion coefficients are as shown below:
\begin{figure}[h]
\centering
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Profile (t=0)-Combined.png}
\caption{Combined Initial Temperature graph}
\label{fig:3}
\end{minipage}
\hfill
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Profile (t=5)-Combined.png}
\caption{Combined Final Temperature graph}
\label{fig:4}
\end{minipage}
\end{figure}
The initial temperature contours are same setFields conditions were applied but the final temperature contours show the comparative strength of diffusion with respect to scale of geometry.
The study of pure diffusion and the results implies that, as the value of diffusion coefficients increases the rate of diffusion also increases which is true.
\newpage
\section{Steps to solve the pure convection (advection) on customised case {\fontfamily{qcr}\selectfont scalarTransport} with {\fontfamily{qcr}\selectfont ScalarTransportFoam} solver}
\subsection{To make the copies of the case {\fontfamily{qcr}\selectfont scalarTransport} for Pure convection study with different divergence schemes for Temperature i.e. {\fontfamily{qcr}\selectfont div(phi,T)}}
Six copies the case {\fontfamily{qcr}\selectfont scalarTransport} are made with different divergence schemes and modified for Pure convection study, as given in the table below:
\setlength{\arrayrulewidth}{0.5mm}
\setlength{\tabcolsep}{10pt}
\renewcommand{\arraystretch}{1.5}
\begin{table}[h!]
\centering
\begin{tabular}{ |p{3.7cm}|p{2cm}|p{2cm}|p{4cm}| }
\hline
\multicolumn{4}{|c|}{List of changes in the different files for Pure Convection case} \\
\hline
Case Name \newline (Copy of {\fontfamily{qcr}\selectfont scalarTransport}) & {\fontfamily{qcr}\selectfont 0/U} & {\fontfamily{qcr}\selectfont constant/\newline{transport-}\newline{Properties}} & {\fontfamily{qcr}\selectfont system/\newline fvSchemes} \\
\hline
{\fontfamily{qcr}\selectfont scalarTransportCUp} & {\fontfamily{qcr}\selectfont U (1 0 0) } & {\fontfamily{qcr}\selectfont DT 0.0 } & {\fontfamily{qcr}\selectfont div(phi,T) upwind } \\
{\fontfamily{qcr}\selectfont scalarTransportCLU} & {\fontfamily{qcr}\selectfont U (1 0 0) } & {\fontfamily{qcr}\selectfont DT 0.0 } & {\fontfamily{qcr}\selectfont div(phi,T) linearUpwind grad(T)} \\
{\fontfamily{qcr}\selectfont scalarTransportCQd} & {\fontfamily{qcr}\selectfont U (1 0 0) } & {\fontfamily{qcr}\selectfont DT 0.0 } & {\fontfamily{qcr}\selectfont div(phi,T) quadraticFit 1 } \\
{\fontfamily{qcr}\selectfont scalarTransportCCb} & {\fontfamily{qcr}\selectfont U (1 0 0) } & {\fontfamily{qcr}\selectfont DT 0.0 } & {\fontfamily{qcr}\selectfont div(phi,T) cubic } \\
{\fontfamily{qcr}\selectfont scalarTransportCVL} & {\fontfamily{qcr}\selectfont U (1 0 0) } & {\fontfamily{qcr}\selectfont DT 0.0 } & {\fontfamily{qcr}\selectfont div(phi,T) vanLeer } \\
{\fontfamily{qcr}\selectfont scalarTransportCQK} & {\fontfamily{qcr}\selectfont U (1 0 0) } & {\fontfamily{qcr}\selectfont DT 0.0 } & {\fontfamily{qcr}\selectfont div(phi,T) QUICK } \\
\hline
\end{tabular}
\caption{List of changes in the different copies of case {\fontfamily{qcr}\selectfont scalarTransport} for Pure Convection}
\label{table:2}
\end{table}
\subsection {To solve the respective cases for pure convection using {\fontfamily{qcr}\selectfont scalarTransportFoam} solver}
By getting into the respective pure convection cases the following list of commands is executed
\newline to run the case with {\fontfamily{qcr}\selectfont scalarTransportFoam} solver.
\newline {\fontfamily{qcr} \selectfont blockMesh}
\newline {\fontfamily{qcr} \selectfont checkMesh}
\newline {\fontfamily{qcr} \selectfont setFields}
\newline {\fontfamily{qcr} \selectfont scalarTransportFoam}
\newline
\newline To generate mesh.
\newline To check mesh.
\newline To set the fields.
\newline To solve the case.
\newline The respective {\fontfamily{qcr} \selectfont .foam} files are saved. The results are post-processed in the {\fontfamily{qcr} \selectfont ParaView}.
\newpage
\subsection{The Results of the case {\fontfamily{qcr}\selectfont scalarTransportCUp}}
The initial temperature contours at {\fontfamily{qcr}\selectfont t=0} and the final temperature contours at {\fontfamily{qcr}\selectfont t=5} for pure convection case with uniform x-velocity of 1 m/s with the {\fontfamily{qcr}\selectfont upwind} divergence scheme for temperature, are observed as follows:
\begin{figure}[h]
\centering
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Contour Pure Convection(t=0)Upwind.png}
\caption{Initial Temp Contours-upwind}
\label{fig:3}
\end{minipage}
\hfill
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Contour Pure Convection(t=5)Upwind.png}
\caption{Final Temp Contours-upwind}
\label{fig:4}
\end{minipage}
\end{figure}
The initial temperature graph at {\fontfamily{qcr}\selectfont t=0} and the final temperature graph at {\fontfamily{qcr}\selectfont t=5} with respect to X-Coordinate,for pure convection case with uniform x-velocity of 1 m/s with the {\fontfamily{qcr}\selectfont upwind} divergence scheme for temperature, are as shown below:
\begin{figure}[h]
\centering
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Profile Pure Convection(t=0)Upwind.png}
\caption{Initial Temp graph-upwind}
\label{fig:3}
\end{minipage}
\hfill
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Profile Pure Convection(t=5)Upwind.png}
\caption{Final Temp graph-upwind}
\label{fig:4}
\end{minipage}
\end{figure}
The initial temperature contours are as per the setFileds conditions.The final temperature contours are shifted 5 meters along x-axis as this is pure convection case with x-velocity 1m/s and simulated for 5 seconds of time.The final temperature contours are smeared and magnitude has also been dropped significantly though diffusion coefficient is set to 0. This is due to the numerical diffusion or artificial diffusion which arises due to the numerical schemes used for interpolations.Thus the {\fontfamily{qcr}\selectfont upwind} divergence scheme produces significant amount of numerical diffusion this may lead to wrong interpretation of the final results. This numerical diffusion needs to be countered for the use of the scheme and good results.
\newpage
\subsection{The Results of the case {\fontfamily{qcr}\selectfont scalarTransportCLU}}
The initial temperature contours at {\fontfamily{qcr}\selectfont t=0} and the final temperature contours at {\fontfamily{qcr}\selectfont t=5} for pure convection case with uniform x-velocity of 1 m/s with the {\fontfamily{qcr}\selectfont linearUpwind} divergence scheme for temperature, are observed as follows:
\begin{figure}[h]
\centering
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Contour Pure Convection(t=0)LinearUpwind.png}
\caption{Initial Temp Contours-linearUpwind}
\label{fig:3}
\end{minipage}
\hfill
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Contour Pure Convection(t=5)LinearUpwind.png}
\caption{Final Temp Contours-linearUpwind}
\label{fig:4}
\end{minipage}
\end{figure}
The initial temperature graph at {\fontfamily{qcr}\selectfont t=0} and the final temperature graph at {\fontfamily{qcr}\selectfont t=5} with respect to X-Coordinate, for pure convection case with uniform x-velocity of 1 m/s with the {\fontfamily{qcr}\selectfont linearUpwind} divergence scheme for temperature, are as shown below:
\begin{figure}[h]
\centering
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Profile Pure Convection(t=0)LinearUpwind.png}
\caption{Initial Temp graph-linearUpwind}
\label{fig:3}
\end{minipage}
\hfill
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Profile Pure Convection(t=5)LinearUpwind.png}
\caption{Final Temp graph-linearUpwind}
\label{fig:4}
\end{minipage}
\end{figure}
The initial temperature contours are as per the setFileds conditions.The final temperature contours are shifted 5 meters along x-axis as this is pure convection case with x-velocity 1m/s and simulated for 5 seconds of time.The final temperature contours are smeared and magnitude has also been dropped significantly though diffusion coefficient is set to 0. As discussed in earlier case, this is due to the numerical diffusion or artificial diffusion which arises due to the numerical schemes used for interpolations.Thus the {\fontfamily{qcr}\selectfont linearUpwind} divergence scheme also produces significant amount of numerical diffusion this may lead to wrong interpretation of the final results. This numerical diffusion needs to be countered for the use of the scheme and good results.
\newpage
\subsection{The Results of the case {\fontfamily{qcr}\selectfont scalarTransportCQd}}
The initial temperature contours at {\fontfamily{qcr}\selectfont t=0} and the final temperature contours at {\fontfamily{qcr}\selectfont t=5} for pure convection case with uniform x-velocity of 1 m/s with the {\fontfamily{qcr}\selectfont quadraticFit} divergence scheme for temperature, are observed as follows:
\begin{figure}[h]
\centering
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Contour Pure Convection(t=0)QuadraticFit.png}
\caption{Initial Temp Contours-quadratic}
\label{fig:3}
\end{minipage}
\hfill
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Contour Pure Convection(t=5)QuadraticFit.png}
\caption{Final Temp Contours-quadratic}
\label{fig:4}
\end{minipage}
\end{figure}
The initial temperature graph at {\fontfamily{qcr}\selectfont t=0} and the final temperature graph at {\fontfamily{qcr}\selectfont t=5} with respect to X-Coordinate,for pure convection case with uniform x-velocity of 1 m/s with the {\fontfamily{qcr}\selectfont quadraticFit} divergence scheme for temperature, are as shown below:
\begin{figure}[h]
\centering
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Profile Pure Convection(t=0)QuadraticFit.png}
\caption{Initial Temp graph-quadratic}
\label{fig:3}
\end{minipage}
\hfill
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Profile Pure Convection(t=5)QuadraticFit.png}
\caption{Final Temp graph-quadratic}
\label{fig:4}
\end{minipage}
\end{figure}
The initial temperature contours are as per the setFileds conditions.The final temperature contours are shifted 5 meters along x-axis as this is pure convection case with x-velocity 1m/s and simulated for 5 seconds of time.The final temperature contours are less smeared and magnitude has also nearly same as that of initial.Though there is numerical diffusion the {\fontfamily{qcr}\selectfont quadraticFit} divergence scheme. It performs comparatively good in minimising the numerical diffusion effects.But the peak with magnitude higher than original initial temperature is observed near the inlet of tube which are the numerical oscillation. Although numerical diffusion in {\fontfamily{qcr}\selectfont quadraticFit} scheme is comparatively low but it produces high numerical oscillations which may in-stabilise the solution and give rise to unrealistic results.
\newpage
\subsection{The Results of the case {\fontfamily{qcr}\selectfont scalarTransportCCb}}
The initial temperature contours at {\fontfamily{qcr}\selectfont t=0} and the final temperature contours at {\fontfamily{qcr}\selectfont t=5} for pure convection case with uniform x-velocity of 1 m/s with the {\fontfamily{qcr}\selectfont cubic} divergence scheme for temperature, are observed as follows:
\begin{figure}[h]
\centering
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Contour Pure Convection(t=0)cubic.png}
\caption{Initial Temp Contours-cubic}
\label{fig:3}
\end{minipage}
\hfill
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Contour Pure Convection(t=5)cubic.png}
\caption{Final Temp Contours-cubic}
\label{fig:4}
\end{minipage}
\end{figure}
The initial temperature graph at {\fontfamily{qcr}\selectfont t=0} and the final temperature graph at {\fontfamily{qcr}\selectfont t=5} with respect to X-Coordinate,for pure convection case with uniform x-velocity of 1 m/s with the {\fontfamily{qcr}\selectfont cubic} divergence scheme for temperature, are as shown below:
\begin{figure}[h]
\centering
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Profile Pure Convection(t=0)Cubic.png}
\caption{Initial Temp graph-cubic}
\label{fig:3}
\end{minipage}
\hfill
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Profile Pure Convection(t=5)Cubic.png}
\caption{Final Temp graph-cubic}
\label{fig:4}
\end{minipage}
\end{figure}
The initial temperature contours are as per the setFileds conditions.The final temperature contours are shifted 5 meters along x-axis as this is pure convection case with x-velocity 1m/s and simulated for 5 seconds of time.The final temperature contours are smeared and magnitude has also dropped significantly as that of initial.This is due to numerical diffusion produced by the {\fontfamily{qcr}\selectfont cubic} divergence scheme.But the peak with considerable magnitude of temperature is observed near the inlet of tube which are the numerical oscillation. The {\fontfamily{qcr}\selectfont cubic} divergence scheme produces comparatively high numerical diffusion and considerable oscillations which may in-stabilise the solution and give rise to unrealistic results.
\newpage
\subsection{The Results of the case {\fontfamily{qcr}\selectfont scalarTransportCVL}}
The initial temperature contours at {\fontfamily{qcr}\selectfont t=0} and the final temperature contours at {\fontfamily{qcr}\selectfont t=5} for pure convection case with uniform x-velocity of 1 m/s with the {\fontfamily{qcr}\selectfont vanLeer} divergence scheme for temperature, are observed as follows:
\begin{figure}[h]
\centering
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Contour Pure Convection(t=0)Vanleer.png}
\caption{Initial Temp Contours-vanleer}
\label{fig:3}
\end{minipage}
\hfill
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Contour Pure Convection(t=5)Vanleer.png}
\caption{Final Temp Contours-vanleer}
\label{fig:4}
\end{minipage}
\end{figure}
The initial temperature graph at {\fontfamily{qcr}\selectfont t=0} and the final temperature graph at {\fontfamily{qcr}\selectfont t=5} with respect to X-Coordinate,for pure convection case with uniform x-velocity of 1 m/s with the {\fontfamily{qcr}\selectfont vanLeer} divergence scheme for temperature, are as shown below:
\begin{figure}[h]
\centering
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Profile Pure Convection(t=0)VanLeer.png}
\caption{Initial Temp graph-vanleer}
\label{fig:3}
\end{minipage}
\hfill
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Profile Pure Convection(t=5)VanLeer.png}
\caption{Final Temp graph-vanleer}
\label{fig:4}
\end{minipage}
\end{figure}
The initial temperature contours are as per the setFileds conditions.The final temperature contours are shifted 5 meters along x-axis as this is pure convection case with x-velocity 1m/s and simulated for 5 seconds of time.The final temperature contours are somewhat smoothened and magnitude is roughly same as that of initial.Smearing and minute change in magnitude is due to numerical diffusion produced by the {\fontfamily{qcr}\selectfont vanLeer} divergence scheme.No signs of the numerical oscillations are observed here. The {\fontfamily{qcr}\selectfont vanLeer} divergence scheme produces less amount of numerical diffusion and no numerical oscillations which is the sign of better choice for the divergence scheme to used.
\newpage
\subsection{The Results of the case {\fontfamily{qcr}\selectfont scalarTransportCQK}}
The initial temperature contours at {\fontfamily{qcr}\selectfont t=0} and the final temperature contours at {\fontfamily{qcr}\selectfont t=5} for pure convection case with uniform x-velocity of 1 m/s with the {\fontfamily{qcr}\selectfont QUICK} divergence scheme for temperature, are observed as follows:
\begin{figure}[h]
\centering
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Contour Pure Convection(t=0)QUICK.png}
\caption{Initial Temp Contours-QUICK}
\label{fig:3}
\end{minipage}
\hfill
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Contour Pure Convection(t=5)QUICK.png}
\caption{Final Temp Contours-QUICK}
\label{fig:4}
\end{minipage}
\end{figure}
The initial temperature graph at {\fontfamily{qcr}\selectfont t=0} and the final temperature graph at {\fontfamily{qcr}\selectfont t=5} with respect to X-Coordinate,for pure convection case with uniform x-velocity of 1 m/s with the {\fontfamily{qcr}\selectfont QUICK} divergence scheme for temperature, are as shown below:
\begin{figure}[h]
\centering
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Profile Pure Convection(t=0)QUICK.png}
\caption{Initial Temp graph-QUICK}
\label{fig:3}
\end{minipage}
\hfill
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Profile Pure Convection(t=5)QUICK.png}
\caption{Final Temp graph-QUICK}
\label{fig:4}
\end{minipage}
\end{figure}
The initial temperature contours are as per the setFileds conditions.The final temperature contours are shifted 5 meters along x-axis as this is pure convection case with x-velocity 1m/s and simulated for 5 seconds of time.The final temperature contours are somewhat smoothened and magnitude is roughly same as that of initial.Smearing and minute change in magnitude is due to numerical diffusion produced by the {\fontfamily{qcr}\selectfont QUICK} divergence scheme.No signs of the numerical oscillations are observed here. The {\fontfamily{qcr}\selectfont QUICK} divergence scheme produces less amount of numerical diffusion and no numerical oscillations which is the sign of better choice for the divergence scheme to used.
\newpage
\subsection{The combined results of all the pure convection cases }
The initial temperature contours at {\fontfamily{qcr}\selectfont t=0} and the final temperature contours at {\fontfamily{qcr}\selectfont t=5} for pure convection with different divergence schemes for Temperature, are observed as follows:
\newline In the figure of contours the same sequence of pure convection cases is followed from top to bottom as mentioned in the Table 2. i.e.
\newline Temperature contours of {\fontfamily{qcr}\selectfont scalarTransportCUp} at the top
\newline Temperature contours of {\fontfamily{qcr}\selectfont scalarTransportCLU} second from the top.
\newline Temperature contours of {\fontfamily{qcr}\selectfont scalarTransportCQd} third from the top.
\newline Temperature contours of {\fontfamily{qcr}\selectfont scalarTransportCCb} fourth from the top
\newline Temperature contours of {\fontfamily{qcr}\selectfont scalarTransportCVL} fifth from the top
\newline Temperature contours of {\fontfamily{qcr}\selectfont scalarTransportCQK} at the bottom.
\begin{figure}[h]
\centering
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Contour Pure Convection(t=0)Combined.png}
\caption{Initial Temp Contours combined}
\label{fig:3}
\end{minipage}
\hfill
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Contour Pure Convection(t=5)Combined.png}
\caption{Final Temp Contours combined}
\label{fig:4}
\end{minipage}
\end{figure}
The initial temperature graph at {\fontfamily{qcr}\selectfont t=0} and the final temperature graph at {\fontfamily{qcr}\selectfont t=5} with respect to X-Coordinate, for pure convection with different divergence schemes for Temperature, are as shown below:
\begin{figure}[h]
\centering
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Profile Pure Convection(t=0)Combined.png}
\caption{Combined Initial Temperature graph}
\label{fig:3}
\end{minipage}
\hfill
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=1\textwidth]{Temperature Profile Pure Convection(t=5)Combined.png}
\caption{Combined Final Temperature graph}
\label{fig:4}
\end{minipage}
\end{figure}
From the above results it is clear that the divergence schemes {\fontfamily{qcr}\selectfont upwind,linearUpwind and \newline cubic}, produce comparatively more numerical diffusions than {\fontfamily{qcr}\selectfont quadraticFit,vanLeer and QUICK} schemes. While it can be observed that the divergence schemes like {\fontfamily{qcr}\selectfont quadraticFit and cubic}
\newline produce significant numerical oscillations which instabilise the solution. The peaks near the inlet with different magnitude are the amplified numerical oscillations. Thus the divergence schemes, {\fontfamily{qcr}\selectfont vanLeer and QUICK} are better choice to minimise the numerical diffusion issue as they have better accuracy than others.
\newpage
\section{Details on Numerical diffusion and numerical oscillations observed into the results of pure convection cases.}
\subsection{Numerical diffusion and Numerical oscillations}
\subsubsection{Numerical diffusion}
The diffusion which occurs mainly due to the numerical schemes implemented for the numerical solution is called as numerical diffusion.
\newline Here this has been observed in case of pure convection but it also appears in combined diffusion convection problem.Thus needed to be minimised for the accuracy of the solution.
\subsubsection{Numerical oscillations}
The sudden occurrence comparatively high and low value of numerical solution due to the nature of numerical scheme used are known as numerical oscillations.
\newline The numerical oscillations should be eliminated for the converged and stable solution.
\subsection{The Numerical oscillations observed at different time steps}
Here for pure convection cases the origin of the numerical oscillation has been investigated by the plotting the negative error in Temperature of magnitude upto 1e-3 on the temperature contours. The contours of the same for various time steps are presented as follows:
\begin{figure}[h]
\centering
\includegraphics[scale=0.275]{Temperature Contour Error(t=0.5)Combined.png}{ \label{fig:}}
\caption{Numerical oscillations at time (t=0.5)}
\end{figure}
Above figure for time t=0.5 shows comparative numerical diffusion and the origin of numerical oscillations for divergence schemes like {\fontfamily{qcr}\selectfont linearUpwind,quadraticFit and cubic}. These numerical oscillations are observed at left side of the higher temperature zone.
\newpage
\begin{figure}[h]
\centering
\includegraphics[scale=0.275]{Temperature Contour Error(t=1)Combined.png}{ \label{fig:}}
\caption{Numerical oscillations at time (t=1)}
\end{figure}
Above figure for time t=1 shows comparative numerical diffusion and the propagation of numerical oscillations for divergence schemes like {\fontfamily{qcr}\selectfont quadraticFit and cubic} in the opposite direction of flow velocity. Whereas the numerical oscillations in the scheme {\fontfamily{qcr}\selectfont linearUpwind} have been diminished.
\begin{figure}[h]
\centering
\includegraphics[scale=0.275]{Temperature Contour Error(t=2)Combined.png}{ \label{fig:}}
\caption{Numerical oscillations at time (t=2)}
\end{figure}
Above figure for time t=2 also shows comparative numerical diffusion and the propagation of numerical oscillations for divergence schemes like {\fontfamily{qcr}\selectfont quadraticFit and cubic} in the opposite direction of flow velocity.
\newpage
\begin{figure}[h]
\centering
\includegraphics[scale=0.275]{Temperature Contour Error(t=3)Combined.png}{ \label{fig:}}
\caption{Numerical oscillations at time (t=3)}
\end{figure}
Above figure for time t=3 also shows comparative numerical diffusion and the propagation of numerical oscillations for divergence schemes like {\fontfamily{qcr}\selectfont quadraticFit and cubic} in the opposite direction of flow velocity.The numerical oscillations are reflected the inlet boundary due to {\fontfamily{qcr}\selectfont zeroGradient} boundary condition at inlet and suddenly increased in magnitude which is substantial.After reflection the amplified oscillations are propagating in same direction as that of flow velocity.
\begin{figure}[h]
\centering
\includegraphics[scale=0.275]{Temperature Contour Error(t=4)Combined.png}{ \label{fig:}}
\caption{Numerical oscillations at time (t=4)}
\end{figure}
Above figure for time t=4 also shows comparative numerical diffusion and the propagation of amplified numerical oscillations for divergence schemes like {\fontfamily{qcr}\selectfont quadraticFit and cubic} in the same direction of flow velocity after striking back to inlet boundary.
\newpage
\begin{figure}[h]
\centering
\includegraphics[scale=0.275]{Temperature Contour Error(t=5)Combined.png}{ \label{fig:}}
\caption{Numerical oscillations at time (t=5)}
\end{figure}
Above figure for time t=5 also shows comparative numerical diffusion and the propagation of amplified numerical oscillations for divergence schemes like {\fontfamily{qcr}\selectfont quadraticFit and cubic} in the same direction of flow velocity after striking back to inlet boundary.
\subsection{Conclusions on Numerical diffusion and Numerical oscillations}
\begin{enumerate}
\item The numerical oscillations for divergence schemes like {\fontfamily{qcr}\selectfont quadraticFit and cubic} are observed to be propagating in opposite direction of flow velocity initially. Thus these schemes are prone to numerical oscillations.
\item The numerical oscillations for divergence scheme {\fontfamily{qcr}\selectfont linearUpwind} are diminished quickly.This is the advantage of this scheme.
\item The numerical oscillations for divergence schemes like {\fontfamily{qcr}\selectfont quadraticFit and cubic} are observed to be reflected back from inlet boundary due to {\fontfamily{qcr}\selectfont zeroGradinet} boundary condition at inlet boundary and got amplified. Thus the type of boundary conditions used are influence the numerical oscillations in the solution.
\end{enumerate}
\newpage
\section{Observations and Conclusions}
\begin{enumerate}
\item The study of pure diffusion cases show the increased strength of diffusion with increase in diffusion coefficient.The strength of diffusion can be controlled by selecting appropriate {\fontfamily{qcr}\selectfont laplacian schemes} in the {\fontfamily{qcr}\selectfont fvSchemes} file.
\item The study of pure convection cases show deviations in final results with initial conditions.These deviations are due to the numerical diffusion produced by divergence schemes itself.
\item Along with numerical diffusion some schemes show the numerical oscillations which result in unstable solution if increased beyond limit.
\item The divergence schemes like {\fontfamily{qcr}\selectfont vanLeer and QUICK} show better and comparable accuracy and stability than other schemes in the list. But {\fontfamily{qcr}\selectfont QUICK} has slightly better accuracy than {\fontfamily{qcr}\selectfont vanLeer} thus it is preferred for solution to the convection problems.
\item The use of appropriate boundary conditions is also utmost important for stability and convergence of numerical solution.
\end{enumerate}
\end{document}