Commit 86ae97c0 authored by Daniele de Grazia's avatar Daniele de Grazia
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Changes to ADRSolver.tex

parent d6cf6a4e
......@@ -49,7 +49,6 @@ $\dfrac{\partial u}{\partial t} + u\nabla u = 0$
\label{t:ADR1}
\end{table}
\subsection{Usage}
ADRSolver session.xml
......@@ -88,22 +87,22 @@ UnstedayInviscidBurger & \checkmark & & &\\
\label{t:ADR2}
\end{table}
\vspace{-1 cm}
\item \textbf{Projection}: The Galerkin projection used may be either
\item \textbf{Projection}: The Galerkin projection used may be either:
\begin{itemize}
\item Continuous for a C0-continuous Galerkin (CG) projection.
\item Discontinuous for a discontinous Galerkin (DG) projection.
\end{itemize}
\item \textbf{DiffusionAdvancement}: This specifies how to treat the diffusion term. This will be restricted by the choice of time integration scheme.
\item \textbf{DiffusionAdvancement}: This specifies how to treat the diffusion term. This will be restricted by the choice of time integration scheme:
\begin{itemize}
\item Explicit: Requires the use of an explicit time integration scheme.
\item Implcit: Requires the use of a diagonally implicit, IMEX or Implicit scheme.
\end{itemize}
\item \textbf{AdvectionAdvancement}: This specifies how to treat the advection term. This will be restricted by the choice of time integration scheme
\item \textbf{AdvectionAdvancement}: This specifies how to treat the advection term. This will be restricted by the choice of time integration scheme:
\begin{itemize}
\item Explicit: Requires the use of an explicit or IMEX time integration scheme.
\item Implicit: Not supported at present.
\end{itemize}
\item \textbf{AdvectionType}: Specifies the type of advection
\item \textbf{AdvectionType}: Specifies the type of advection:
\begin{itemize}
\item NonConservative (for CG only).
\item WeakDG (for DG only).
......@@ -125,10 +124,10 @@ The following parameters can be specified in the PARAMETERS section of the sessi
\item \textbf{epsilon}: sets the diffusion coefficient $\epsilon$.\\
\textit{Can be used} in: SteadyDiffusionReaction, SteadyAdvectionDiffusionReaction, UnsteadyDiffusion, UnsteadyAdvectionDiffusion. \\
\textit{Default value}: 0.
\item \textbf{d00, d11, d22}: sets the diagonal entries of the diffusion tensor formula $D$. \\
\item \textbf{d00, d11, d22}: sets the diagonal entries of the diffusion tensor $D$. \\
\textit{Can be used in}: UnsteadyDiffusion \\
\textit{Default value}: All set to 1 (i.e. identity matrix).
\item \textbf{lambda}: sets the reaction coefficient formula $\lambda$. \\
\item \textbf{lambda}: sets the reaction coefficient $\lambda$. \\
\textit{Can be used in}: SteadyDiffusionReaction, Helmholtz, SteadyAdvectionDiffusionReaction\\
\textit{Default value}: 0.
\end{itemize}
......@@ -138,13 +137,12 @@ The following parameters can be specified in the PARAMETERS section of the sessi
The following functions can be specified inside the CONDITIONS section of the session file:
\begin{itemize}
\item \textbf{AdvectionVelocity}: specifies the advection velocity formula.
\item \textbf{AdvectionVelocity}: specifies the advection velocity $\mathbf{V}$.
\item \textbf{InitialConditions}: specifies the initial condition for unsteady problems.
\item \textbf{Forcing}: specifies the forcing function formula
\item \textbf{Forcing}: specifies the forcing function f.
\end{itemize}
\subsection{Examples}
\subsubsection{1D Advection equation}
\subsection{1D Advection equation}
In this example, it will be demonstrated how the Advection equation can be solved on a one-dimensional domain.
This problem is a particular case of the Advection-Diffusion-Reaction Solver. \\
......@@ -276,9 +274,7 @@ FldToTecplot Advection1D.xml Advection1D.fld \\
FldToVtk Advection1D.xml Advection1D.fld
\subsubsection{2D Helmholtz Problem}
\subsection{2D Helmholtz Problem}
In this example, it will be demonstrated how the Helmholtz equation can be solved on a two-dimensional domain.
This problem is a particular case of the Advection-Diffusion-Reaction Solver.
......@@ -370,7 +366,7 @@ This section defines the polynomial expansions used on each composites.
\textbf{\footnotesize{Conditions definition}}
This sections defines the problem solved. In this example formula and the Continuous Galerkin Method
This sections defines the problem solved. In this example $\lambda = 1$ and the Continuous Galerkin Method
is used as projection scheme to solve the Helmholtz equation.
\begin{lstlisting}[style=XMLStyle]
......@@ -455,7 +451,7 @@ Simulation results are written in the file Test\_Helmholtz2D\_modal.fld. \nekpp
\normalsize
provides the file Test\_Helmholtz2D\_modal\_u.pos in the Gmsh format which gives the following image.
\begin{figure}[h!]
\begin{figure}[h!]
\begin{center}
\includegraphics[width=6cm]{Figures/Helmholtz2D}
\caption{Solution of the 2D Helmholtz Problem.}
......@@ -464,4 +460,175 @@ provides the file Test\_Helmholtz2D\_modal\_u.pos in the Gmsh format which gives
By writing FldToTecplot or FldToVtk instead of FldToGmsh in the previous command, Tecplot or Paraview can be used to visualize the results.
\subsection{Advection dominated mass transport in a pipe}
The following example demonstrates the application of the ADRsolver for modelling advection dominated mass transport in a straight pipe.
Such a transport regime is encountered frequently when modelling mass transport in arteries. This is because the diffusion
coefficient of small blood borne molecules, for example oxygen or adenosine triphosphate, is very small $O(10^{-10})$.
\textbf{Background}
The governing equation for modelling mass transport is the unsteady advection diffusion equation:
\begin{equation}
\dfrac{\partial u}{\partial t} + v\nabla u + \epsilon \nabla^2 u = 0
\end{equation}
For small diffusion coefficient, $\epsilon$, the transport is dominated by advection and this leads to a very fine boundary
layer adjacent to the surface which must be captured in order to get a realistic representation of the wall mass transfer processes.
This creates problems not only from a meshing perspective, but also numerically where classical oscillations
are observed in the solution due to under-resolution of the boundary layer.\\
The Graetz-Nusselt solution is an analytical solution of a developing mass (or heat) transfer boundary layer in a pipe.
Previously this solution has been used as a benchmark for the accuracy of numerical methods to capture the fine
boundary layer which develops for high Peclet number transport (the ratio of advection to diffusion).
The solution is derived based on the assumption that the velocity field within the mass transfer boundary layer
is linear i.e. the Schmidt number (the relative thickness of the momentum to mass transfer boundary layer) is sufficiently large.
The analytical solution for the non-dimensional mass transfer at the wall is given by:
\begin{equation}
S h(z) = \dfrac{2^{4/3}(Pe R/z)^{1/3}}{g^{1/3}\Gamma(4/3)} ,
\end{equation}
where $z$ is the streamwise coordinate, $R$ the pipe radius, $\Gamma(4/3)$ an incomplete
Gamma function and $Pe$ the Peclet number given by:
\begin{equation}
Pe = \dfrac{2 U R}{\epsilon}
\end{equation}
In the following we will numerically solver mass transport in a pipe and compare the calculated mass transfer
at the wall with the Graetz-Nusselt solution. The Peclet number of the transport regime under consideration is
750000, which is physiologically relevant.
\textbf{Geometry}
The geometry under consideration is a pipe of radius, $R = 0.5$ and length $l = 0.5$
\begin{figure}[h!]
\begin{center}
\includegraphics[width=6cm]{Figures/pipe}
\caption{Pipe.}
\end{center}
\end{figure}
Since the mass transport boundary layer will be confined to a very small layer adjacent to the wall we do not need to mesh
the interior region, hence the mesh consists of a layer of ten prismatic elements over a thickness of 0.036R.
The elements progressively grow over the thickness of domain.
\textbf{Input parameters}
\textbf{\footnotesize{Expansion}}
In this example we utilise heterogeneous polynomial order, in which the polynomial order normal to the wall is
higher so that we avoid unphysical oscillations, and hence the incorrect solution, in the mass transport boundary layer.
To do this we specify explicitly the expansion type, points type and distribution in each direction as follows:
\begin{lstlisting}[style=XMLStyle]
<EXPANSIONS>
<E COMPOSITE="C[0]"
NUMMODES="3,5,3"
BASISTYPE="Modified_A,Modified_A,Modified_B"
NUMPOINTS="4,6,3"
POINTSTYPE="GaussLobattoLegendre,GaussLobattoLegendre,GaussRadauMAlpha1Beta0"
FIELDS="u" />
</EXPANSIONS>
\end{lstlisting}
The above represents a quadratic polynomial order in the azimuthal and streamwise direction and
4th order polynomial normal to the wall for a prismatic element.
\textbf{\footnotesize{Solver information}}
\begin{lstlisting}[style=XMLStyle]
<SOLVERINFO>
<I PROPERTY="EQTYPE" VALUE="UnsteadyAdvectionDiffusion" />
<I PROPERTY="Projection" VALUE="Continuous" />
<I PROPERTY="DiffusionAdvancement" VALUE="Implicit" />
<I PROPERTY="AdvectionAdvancement" VALUE="Explicit" />
<I PROPERTY="TimeIntegrationMethod" VALUE="IMEXOrder1" />
<I PROPERTY="GlobalSysSoln" VALUE="IterativeStaticCond" />
</SOLVERINFO>
\end{lstlisting}
\textbf{\footnotesize{Parameters}}
\begin{lstlisting}[style=XMLStyle]
<PARAMETERS>
<P> TimeStep = 0.0005 </P>
<P> FinalTime = 30 </P>
<P> NumSteps = FinalTime/TimeStep </P>
<P> IO_CheckSteps = 1000 </P>
<P> IO_InfoSteps = 200 </P>
<P> epsilon = 1.33333e-6 </P>
</PARAMETERS>
\end{lstlisting}
The value of $\epsilon$ is $\epsilon = 1/Pe$.
\textbf{\footnotesize{Boundary conditions}}
The analytical solution represents a developing mass transfer boundary layer in a pipe. In order to
reproduce this numerically we assume that the inlet concentration is a uniform value and the outer
wall concentration is zero; this will lead to the development of the mass transport boundary layer along the length
of the pipe. Since we do not model explicitly the mass transfer in the interior region of the pipe we assume that
the inner wall surface concentration is the same as the inlet concentration; this assumption is valid based on the large
Peclet number meaning the concentration boundary layer is confined to the region in the immediate vicinity of the wall.
The boundary conditions are specified as follows in the input file:
\begin{lstlisting}[style=XMLStyle]
<BOUNDARYREGIONS>
<B ID="0"> C[3] </B> <!-- inlet -->
<B ID="1"> C[4] </B> <!-- outlet -->
<B ID="2"> C[2] </B> <!-- outer surface -->
<B ID="3"> C[5] </B> <!-- inner surface -->
</BOUNDARYREGIONS>
<BOUNDARYCONDITIONS>
<REGION REF="0">
<D VAR="u" VALUE="1" />
</REGION>
<REGION REF="1">
<N VAR="u" VALUE="0" />
</REGION>
<REGION REF="2">
<D VAR="u" VALUE="0" />
</REGION>
<REGION REF="3">
<D VAR="u" VALUE="1" />
</REGION>
</BOUNDARYCONDITIONS>
\end{lstlisting}
\textbf{\footnotesize{Functions}}
The velocity field within the domain is fully developed pipe flow (Poiseuille flow), hence we
can define this through an analytical function as follows:
\begin{lstlisting}[style=XMLStyle]
<FUNCTION NAME="AdvectionVelocity">
<E VAR="Vx" VALUE="0" />
<E VAR="Vy" VALUE="0" />
<E VAR="Vz" VALUE="2.0*(1-(x*x+y*y)/0.25)" />
</FUNCTION>
\end{lstlisting}
We assume that the initial domain concentration is uniform everywhere and the same as the inlet. This is defined by,
\begin{lstlisting}[style=XMLStyle]
<FUNCTION NAME="InitialConditions">
<E VAR="u" VALUE="1" />
</FUNCTION>
\end{lstlisting}
\textbf{Results}
To compare with the analytical expression we numerically calculate the concentration gradient at the surface of the pipe.
This is then plotted against the analytical solution by extracting the solution along a line in the streamwise direction.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=7cm]{Figures/graetz-nusselt}
\caption{Concentration gradient at the surface of the pipe.}
\end{center}
\end{figure}
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