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benchmark_thin_layer_entrainment.tex
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benchmark_thin_layer_entrainment.tex
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\begin{flushright} {\tiny {\color{gray} benchmark\_thin\_layer\_entrainment.tex}} \end{flushright}
\vspace{0.5cm}
\begin{flushright}
Data pertaining to this section are to be found at:
\url{https://github.com/cedrict/fieldstone/tree/master/images/benchmark_thinlayer}
\end{flushright}
\vspace{0.5cm}
The problem is a simulation to study the
amount of entrainment by thermal convection of a dense,
thin layer at the bottom of the model \cite{vaks97}.
To the author's knowledge only two other publications (Tackley \& King (2003) \cite{taki03},
van Thienen (2007) \cite{vant07})
have presented results pertaining to this benchmark.
The results shown here after are obtained with my \elefant code using
the particle-in-cell technique and originate in the \elefant paper \cite{thie14}.
The box is $2\times 1$, and contains two fluids:
\begin{center}
\includegraphics[width=0.45\linewidth]{images/benchmark_thinlayer/temperature_init}
\includegraphics[width=0.45\linewidth]{images/benchmark_thinlayer/mat_init}
\end{center}
Fluid 1 has a density $\rho_1=1$ and a viscosity $\eta=1$.
Fluid 2 is heavier ($\rho_2=\rho_1 + \Delta \rho$)
but has the same viscosity.
Both fluids have a thermal expansion coefficient $\alpha=10^{-10}$, a
thermal conductivity $k=1$, and a heat capacity coefficient $c_p=1$.
Fluid 2 is placed at the bottom of the box ($0\leq y \leq 0.025$).
This experiment is parameterised by the thermal Rayleigh number $\Ranb=300,000$ and
and the compositional Rayleigh number $\Ranb_c=450,000$ which are defined as follows:
\begin{eqnarray}
\Ranb_T&=&\frac{\alpha \rho g \Delta T L_y^3}{\kappa \eta}
= \frac{\alpha \rho^2 g \Delta T L_y^3 c_p}{k \eta}
= \alpha g \\
\Ranb_c&=&\frac{ \Delta \rho g L_y^3}{\kappa \eta}
= \frac{ \rho \Delta \rho g L_y^3 c_p}{k \eta}
= \Delta \rho g
\end{eqnarray}
where I have used the relationship $\kappa=k/\rho c_p$.
$B$ is defined as $B=\Ranb_T/\Ranb_c$ so
The gravity acceleration is therefore set to $g=\Ranb/\alpha$ and
this yields $\Delta \rho=\Ranb_c/g=B \Ranb_T/g = B\times \alpha$.
Free-slip boundary conditions are imposed on all sides of the domain.
Temperature boundary conditions are $T(x,y=0)=1$ and
$T(x,y=1)=0$. The analytical initial temperature field is given by
\begin{equation}
T(x,y)=T_u(x,y)+T_l(x,y)+T_r(x,y)+T_s(x,y)-\frac{3}{2}
\end{equation}
where
\begin{eqnarray}
T_u(x,y) &=& \frac{1}{2} {\rm erf} \left( \frac{1-y}{2} \sqrt{\frac{u_0}{x}} \right) \nonumber\\
T_l(x,y) &=& 1-\frac{1}{2} {\rm erf} \left( \frac{y}{2} \sqrt{\frac{u_0}{L_x-x}} \right) \nonumber\\
T_r(x,y) &=& \frac{1}{2} + \frac{Q}{2\sqrt{\pi}} \sqrt{\frac{u_0}{y+1}} \exp \left( -\frac{x^2u_0}{4y+4} \right) \nonumber\\
T_s(x,y) &=& \frac{1}{2} - \frac{Q}{2\sqrt{\pi}} \sqrt{\frac{u_0}{2-y}} \exp \left( -\frac{(L_x-x)^2u_0}{8-4y} \right)
\end{eqnarray}
with
\begin{equation}
u_0=\frac{L_x^{7/3}}{(1+L_x^4)^{2/3}} \left(\frac{\Ranb}{2\sqrt{\pi}} \right)^{2/3}
\quad\quad
Q=2\sqrt{\frac{L_x}{\pi u_0}}
\end{equation}
Using $L_x=2$, $\Ranb=3\times10^5$, one gets
$u_0 \simeq 1469.315 $ and $Q\simeq 0.0416305$.
Given the small thickness of the bottom layer, it seems quite legitimate to
investigate the influence of grid resolution on the simulation.
I have therefore looked at the initial root mean square velocity measurement
as a function of the element diagonal value (a proxy for the average resolution
in the case where elements are not square).
Results are confirm that
the element size plays a non negligible role at startup on the dynamics of the system.
Superimposed on the figure are the measurements provided by Prof. van Keken (black squares
in the gray box).
They agree well with my measurements but also indicate that
none of the authors in the original study ran the experiment at a high-enough resolution
to start with (their results were therefore most likely resolution dependent).
We see that the number of markers per element at startup is critical at
(very) low resolution but that it does not lead to
significant velocity variations at high resolution.
\begin{center}
\includegraphics[width=0.65\linewidth]{images/benchmark_thinlayer/vrms/vrmszero.pdf}\\
{\captionfont
Thin layer entrainment experiment: root mean square velocity measurements at
$t=0$ as a function of the element diagonal size.
The red square points correspond to resolutions where the number of elements in each direction
is a multiple of 40 (i.e. $L_y/d$), so that no element would contain a mix of fluids 1 and 2.
Pink points correspond to cases wherethe number of markers within each element was varied between 4 and 500
(random spatial distribution). Taken from \elefant paper \cite{thie14}}
\end{center}
Looking at the root mean square velocity measurements, we see that
the measurements done with \elefant agree nicely with those presented in van Keken \etal \cite{vaks97}.
Past $t\sim0.015$, the curves diverge clearly across all codes and authors,
so I only need to focus the comparison for times $t <0.015$.
For the three tested resolutions,measurements agree well and fall within the grey curves
representing all results of van Keken \etal.
Additional tests have been carried out concerning the value of the
Courant number (0.1 to 0.25) and the initial number of markers per element (100 or 200)
and these parameters led to extremely similar results.
\begin{center}
\includegraphics[width=0.5\linewidth]{images/benchmark_thinlayer/vrms/vrms.pdf}\\
{\captionfont Thin layer entrainment experiment. Root mean square velocity as a function of time.
All results presented in van Keken \etal (1997) are collapsed in black dashed lines.
All simulations were run with an initial marker density of 100 markers per element and with
a Courant numberof 0.25. Taken from \elefant paper \cite{thie14}.}
\end{center}
As observed in van Keken \etal, the dense layer is first swept
into the lower left corner. Thermal instabilities then further develop in an asymmetrical way
and entrain the dense material. Past $t\simeq 0.015$ the system becomes more
and more chaotic with markers being randomly mixed in the system in a non-orderly fashion.
\begin{center}
\includegraphics[width=0.23\linewidth]{images/benchmark_thinlayer/maarkers0000}
\includegraphics[width=0.23\linewidth]{images/benchmark_thinlayer/maarkers0025}
\includegraphics[width=0.23\linewidth]{images/benchmark_thinlayer/maarkers0050}
\includegraphics[width=0.23\linewidth]{images/benchmark_thinlayer/maarkers0075}\\
\includegraphics[width=0.23\linewidth]{images/benchmark_thinlayer/maarkers0100}
\includegraphics[width=0.23\linewidth]{images/benchmark_thinlayer/maarkers0125}
\includegraphics[width=0.23\linewidth]{images/benchmark_thinlayer/maarkers0150}
\includegraphics[width=0.23\linewidth]{images/benchmark_thinlayer/maarkers0175}\\
\includegraphics[width=0.23\linewidth]{images/benchmark_thinlayer/maarkers0200}
\includegraphics[width=0.23\linewidth]{images/benchmark_thinlayer/maarkers0225}
\includegraphics[width=0.23\linewidth]{images/benchmark_thinlayer/maarkers0250}
\includegraphics[width=0.23\linewidth]{images/benchmark_thinlayer/maarkers0275}\\
{\captionfont Marker distribution as obtained with \elefant for
grid 240x120, init\_marker\_density=7, \\
random distribution,
CFL=0.25, rkmethod=2, m\_to\_q=2. (unpublished).}
\end{center}
\Literature: Trim \etal (2020) \cite{trlb20}.