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benchmark_blbc89.tex
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benchmark_blbc89.tex
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\begin{flushright} {\tiny {\color{gray} benchmark\_blbc89.tex}} \end{flushright}
%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The abstract of the original publication by Blankenbach \etal (1989) \cite{blbc89} reads:
\begin{center}
{\it
We have carried out a comparison study for a set of benchmark problems
which are relevant for convection in the Earth's mantle. The cases comprise
steady isoviscous convection, variable viscosity convection and time-dependent
convection with internal heating. We compare Nusselt numbers, velocity,
temperature, heat-flow , topography and geoid data. Among the applied codes
are finite-difference, finite-element and spectral methods. In a synthesis
we give best estimates of the `true' solutions and ranges of uncertainty. We
recommend these data for the validation of convection codes in the future.
}
\end{center}
The temperature is fixed to zero on top and to $\Delta T$ at the bottom,
with reflecting symmetry at the sidewalls (i.e. $\partial_x T=0$)
and there are no internal heat sources.
Free-slip conditions are implemented on all boundaries.
The Rayleigh number is given by
\[
\Ranb = \frac{\alpha g_y \Delta T h^3 }{\kappa \nu}
=\frac{\alpha g_y \Delta T h^3 \rho^2 c_p}{k \mu}
\]
The initial temperature field is given by
\[
T(x,y)=(1-y) - 0.01\cos(\pi x) \sin(\pi x)
\]
The perturbation in the initial temperature fields leads to
a perturbation of the density field and sets the fluid in motion.
Depending on the initial Rayleigh number, the system ultimately reaches a
steady state after some time.
\begin{center}
a)\includegraphics[width=4.5cm]{images/benchmark_blbc89/temp1a}
b)\includegraphics[width=4.5cm]{images/benchmark_blbc89/temp1b}
c)\includegraphics[width=4.5cm]{images/benchmark_blbc89/temp1c}\\
{\captionfont Temperature fields at steady-state for
$\Ranb=10^4$ (a), $\Ranb=10^5$ (b), $\Ranb=10^6$ (c).
Obtained with \elefant code \cite{thie14}.}
\end{center}
\begin{center}
a)\includegraphics[width=13cm]{images/benchmark_blbc89/krhb12a}
b)\includegraphics[width=13cm]{images/benchmark_blbc89/krhb12b}\\
{\captionfont
a) Results for the 2-D benchmark problem with uniform mesh refinement.
\# DoFs indicates the number of degrees of freedom.
Reference results from Blankenbach \etal (1989).
b) Results with adaptive mesh refinement. The number of degrees of freedom (\# DoFs) for
each finest mesh size $h$ varies between time steps;
the indicated numbers provide a typical range.}
\end{center}
%\begin{center}
%\begin{tabular}{llcccc}
%\hline
% & & Blankenbach \etal (1989) & Tackley (1994) & King (2009) & \\
% & & \cite{blbc89} & \cite{tack94} & \cite{king09} & \elefant \\
%\hline
%\hline
%$\Ranb=10^4$ & $\upnu_{rms}$ & $42.864947 \pm 0.000020$ & 42.775 (0.2\%) & 42.867 (0.005\%) & 42.867 (0.01\%) \\
% & $\Nunb$ & $4.884409 \pm 0.000010$ & 4.878 (0.1\%) & 4.885 (0.02\%) & 4.882 (0.05\%)\\
%$\Ranb=10^5$ & $\upnu_{rms}$ & $193.21454 \pm 0.00010 $ & 193.11 (0.05\%)& 193.248 (0.02\%) & 193.255 (0.02\%)\\
% & $\Nunb$ & $10.534095 \pm 0.000010$ & 10.531 (0.03\%)& 10.536 (0.02\%) & 10.507 (0.26\%) \\
%$\Ranb=10^6$ & $\upnu_{rms}$ & $833.98977 \pm 0.00020 $ & 833.55 (0.05\%)& 834.353 (0.04\%) & 834.712 (0.08\%)\\
% & $\Nunb$ & $21.972465 \pm 0.000020$ & 21.998 (0.1\%)& 21.981 (0.04\%) & 21.695 (1.2\%)\\
%\hline
%\end{tabular} \\
%{\captionfont Steady state Nusselt number and Vrms measurements as reported in the literature and
%obtained with \elefant on a $200\times 200$ grid.}
%\end{center}
\begin{center}
\begin{tabular}{llll}
\hline
& & $\upnu_{rms}$ & $\Nunb$ \\
\hline
Blankenbach \etal (1989) \cite{blbc89} & $\Ranb=10^4$ & $42.864947 \pm 0.000020$ & $4.884409 \pm 0.000010$ \\
& $\Ranb=10^5$ & $193.21454 \pm 0.00010 $ & $10.534095 \pm 0.000010$ \\
& $\Ranb=10^6$ & $833.98977 \pm 0.00020 $ & $21.972465 \pm 0.000020$ \\
\hline
Tackley (1994) \cite{tack94} & $\Ranb=10^4$ & 42.775 & 4.878 \\
& $\Ranb=10^5$ & 193.11 & 10.531 \\
& $\Ranb=10^6$ & 833.55 & 21.998 \\
\hline
King (2009) \cite{king09} & $\Ranb=10^4$ & 42.867 & 4.885 \\
& $\Ranb=10^5$ & 193.248 & 10.536 \\
& $\Ranb=10^6$ & 834.353 & 21.981 \\
\hline
Thieulot (2014) \cite{thie14} & $\Ranb=10^4$ & 42.867 & 4.882 \\
& $\Ranb=10^5$ & 193.255 & 10.507 \\
& $\Ranb=10^6$ & 834.712 & 21.695 \\
\hline
\aspect \cite{aspectmanual} & $\Ranb=10^4$ & & \\
& $\Ranb=10^5$ & & \\
& $\Ranb=10^6$ & & \\
\hline
\end{tabular}\\
{\captionfont Steady state Nusselt number and Vrms measurements as reported in the literature and
obtained with \elefant on a $200\times 200$ grid. King (2009) results on 200x200 grid with ConMan.}
\end{center}
\Literature:
\textcite{trab90} (1990),
\textcite{ogaw93} (1993),
\textcite{trha98} (1998),
\textcite{auha99} (1999),
\textcite{chgs02} (2002),
\textcite{chhl08} (2008),
\textcite{kaks05} (2005),
\textcite{king09} (2009),
\textcite{bepo10} (2010),
\textcite{lezh11} (2011),
\textcite{dawk11} (2011),
\textcite{vyrc13} (2013),
\textcite{trbs21} (2021),
\textcite{dakg22} (2002),
\textcite{siwi20} (2020),
\stone 3.