Commit 7374e502 authored by Olivier's avatar Olivier

Chapter 8: two figures

* Replace figure with energy and dissipation distributions (from
  Leschnziner) with self-drawn, simpler alternative
* Add figure to illustrate instantaneous/average decomposition
parent 04cd3d27
\renewcommand{\lastedityear}{2019}
\renewcommand{\lasteditmonth}{06}
\renewcommand{\lasteditday}{08}
\renewcommand{\lasteditday}{11}
\renewcommand{\numberofthischapter}{8}
\renewcommand{\titleofthischapter}{\namechaptereight}
......@@ -107,11 +107,19 @@
\subsection{Average and fluctuation}
For the purpose of quantifying turbulence, we distinguish, in a given flow, between the average velocity and the “turbulent part” of velocity. We thus decompose the velocity field into two components: one is the \vocab{average} flow $\left(\overline u, \overline v, \overline w\right)$, and the other the \vocab{instantaneous fluctuation} flow $\left(u', v', w'\right)$:
For the purpose of quantifying turbulence, we distinguish, in a given flow, between the average velocity and the “turbulent part” of velocity, as illustrated in figure~\ref{fig_instantaneous_average}. We thus decompose the velocity field into two components: one is the \vocab{average} flow $\left(\overline u, \overline v, \overline w\right)$, and the other the \vocab{instantaneous fluctuation} flow $\left(u', v', w'\right)$:
\begin{IEEEeqnarray}{rCl}
u_i &\equiv& \overline u_i + u_i'\label{eq_def_average_u}\\
\overline{u_i'} &\equiv& 0
\end{IEEEeqnarray}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.9\textwidth]{instantaneous_average}
\end{center}
\supercaption{An example of the separation between instantaneous and average values, here for temperature. The instantaneous temperature $T$ is decomposed as the sum of the time-averaged temperature $\overline T$ (blue curve) and the fluctuation $T'$, whose average $\overline{T'}$ is zero (red curve).}{\wcfile{Average and instantaneous values.svg}{Figure} \cczero \oc}
\label{fig_instantaneous_average}
\end{figure}
\subsection{Turbulence intensity}
\coveredin{De Nevers \cite{denevers2004}}
......@@ -205,9 +213,9 @@
These two integral equations are only useful to understand the meaning of figure~\ref{fig_cascade_k_epsilon}, where the distribution of $k$ and $\epsilon$ across the scales of eddies is plotted.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=\textwidth]{spectra}
\includegraphics[width=0.9\textwidth]{spectra}
\end{center}
\supercaption{Distribution of turbulent kinetic energy (left) and turbulent dissipation rate (right) in fully-developed homogeneous isotropic turbulence. In those diagrams, the $x$-axis displays the \vocab{wavelength} $\kappa \equiv 2\pi/l$, so that the small-scale eddies are on the right side, with large values of $\kappa$.}{Figure extracted from Leschziner~\cite{leschziner2015}}
\supercaption{Distribution of turbulent kinetic energy (left) and of turbulent dissipation rate (right) in fully-developed homogeneous isotropic turbulence. The top diagrams are in linear scale, while the bottom diagrams are in logarithmic scale. In those diagrams, the horizontal axis displays $1/l$, so that the small-scale eddies are on the right side, and large-scale eddies are on the left side. \\Those energy and dissipation distributions are for the simplest occurrences of turbulence; their features (in particular, the curves’ slopes and the ratios between $\Lambda$ and $\epsilon$) are used as reference cases in the study of more complex cases.}{Figure \ccbysa by \olivier}
\label{fig_cascade_k_epsilon}
\end{figure}
......
8/images/spectra.png

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8/images/spectra.png

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8/images/spectra.png
8/images/spectra.png
8/images/spectra.png
8/images/spectra.png
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