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chapter_01_3_exp_results.md

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Preliminary results

\begin{figure}[h!] \centerline{ \includegraphics[width=4.14cm]{paper_05_milestone_issf/Figures/exp21/vh_400.pdf} \includegraphics[width=3.44cm]{paper_05_milestone_issf/Figures/exp21/vh_655.pdf} \includegraphics[width=4.72cm]{paper_05_milestone_issf/Figures/exp21/vh_890.pdf}} \vspace{0mm} \centerline{ \includegraphics[width=4.16cm]{paper_05_milestone_issf/Figures/exp21/vv_890.pdf} \includegraphics[width=3.48cm]{paper_05_milestone_issf/Figures/exp21/vv_655.pdf} \includegraphics[width=4.63cm]{paper_05_milestone_issf/Figures/exp21/vv_400.pdf} } \vspace{-2mm} \caption{Instantaneous horizontal large fields (top, $z=40$~cm) and vertical fields (bottom) for $t/T = [1.31,~1.5,~1.68]$, $F_{hc} = 0.1$ and $\mathcal{R}_c=450$. Background colors indicate the norm of the 2D field normalized by $U_c$, which is $(u_x^2 + u_y^2)/U_c$ and $(u_x^2 + u_z^2)/U_c$, respectively.} \label{fig:field} \end{figure}

The experiment was conducted as two campaigns in the years 2016 and 2017, hitherto referred to with the prefixes M16 and M17. The two experimental setups are nearly identical. They differ mainly on the precise location of the density and temperature probes, the number and location of the cameras used for PIV, and the stroke pattern of the oscillating comb. Here we highlight some of the important results reported in @campagne2016 and ongoing post-experiment analysis.

The experiment with parameters $N = 0.8$ rad/s, $U_c = 6$ cm/s, $F_{hc} = 0.1$ and $\R_c = 450$ from M16 campaign is described here. The oscillating comb was observed to inject kinetic energy $u^2 \approx 0.08 U_c^2$. The emergence of layered structures with vertical length scale of \order{1} cm can be visually observed in the velocity fields plotted in @fig:field, thereby conforming to the prediction $l_v \sim u / N$. Note that, there is a time delay when similar dynamics are captured in the horizontal (top row) and vertical (bottom row) PIV fields.

\begin{figure}[ht!] \centerline{ \includegraphics[width=0.7\textwidth]{paper_05_milestone_issf/Figures/exp28/normalized_S2_exp28.pdf}} \caption{Normalized second-order structure $S_h$ function as a function of $r/M$ for $F_{hc} = 0.1$ and $\mathcal{R}_c=450$. The arrow represents the progress of time as kinetic energy decays.} \label{fig:S2} \end{figure}

\begin{figure}[hb!] \centerline{ \includegraphics[width=0.65\textwidth]{_paper_06_milestone/1st/tmp/fig_energy_pot_vs_time}} \centerline{ \includegraphics[width=0.75\textwidth]{_paper_06_milestone/1st/tmp/fig_dt_pot_energy} } \caption{Evolution of potential energy normalized by linear stratification for experiment M17-21 (top) and normalized mixing coefficient $\eps_P / (3\times10^{-3} {U_c}^3/D_c)$ for some MILESTONE 17 experiments (bottom).}% \label{fig:dt:pot:energy}

\end{figure}

In @fig:S2, we plot the normalized second-order structure function, defined as \begin{eqnarray} S_h(r) = (S_{xx} + S_{yy} + S_{xy} + S_{yx})/2, \end{eqnarray} with $S_{ij} = \langle\overline{(u_i({\bf x}+r{\bf e}_j)-u_i({\bf x}))^2}\rangle$, and $\overline{}$ as time average when the carriage is moving. For fully developed turbulence undergoing forward cascade we expect second order structure functions to scale as $S_h(r) = C (\varepsilon r)^{2/3}$ [@Lindborg2006;@AugierBillantChomaz2015]. In @fig:S2 the normalized structure function $S_h(rU_c^3/M)^{-2/3}$ is plotted for different times after one stroke of the carriage. As can be seen, there is a range where the curves are flat consistent with the prediction $S_h \sim r^{2/3}$.

The evolution of the background potential energy is shown in the left plot of @fig:dt:pot:energy for an experiment from M17-21 with parameters $N = 0.55$ rad/s, $U_c = 12$ cm/s, $F_{hc} = 0.436$ and $\R_c = 11425$. The value -1 would correspond to a linear stratification with the same initial \text{Brunt-V"ais"al"a} frequency. Since the profiles at the beginning of the experiment are already eroded from previous experiments, the initial normalized potential energy is slightly smaller than -0.6. We see that it increases approximately linearly with time while the fluid is stirred and stabilize at a constant value after the stop of the carriage. From this curve, we can evaluate the mean rate of increase of the average background potential energy $\eps_P$ during an experiment. This quantity $\eps_P$ is normalized by an estimation of the dissipation of kinetic energy $3\times10^{-3} {U_c}^3/D_c$ where $D_c$ is the diameter of the cylinder and is approximately proportional to the mixing coefficient $\Gamma$. It is plotted as a function of the Froude number in figure \ref{fig:dt:pot:energy}. The colors represent the buoyancy Reynolds number such that the yellow / light coloured points correspond to $\R > 15$. We see that, these points fall on a power law fit of $F_h^{-1.2}$, while there are more variations for smaller values of buoyancy Reynolds number. The points at high $F_h$ are more consistent with a ${F_h}^{-2}$ scaling law observed and predicted with @maffioli_mixing_2016.

At this stage, these results seem promising but should be taken with a grain of salt as they are not final. Since we observe too much spread in some results and some values are larger than what has been theoretically and numerically observed, there might be overlooked errors in the interpretation of the results. A third series of experiments might be needed to confirm the findings.