Commit 6df20790 authored by Olivier's avatar Olivier

Exercises 8: expanded formula sheet

parent a1b2566b
......@@ -11,8 +11,18 @@
\mecafluexboxen
\begin{boiboite}
In homogeneous, isotropic, fully-developed turbulence, we accept the following relationships apply between the largest-scale and smallest-scale eddies:
\begin{boiboiboite}
Turbulence intensity $I$:
\begin{IEEEeqnarray}{rCl}
I &\equiv& \frac{1}{\overline V} \left[\frac{1}{3} \left[ \overline{\left(u'^2\right)} + \overline{\left(v'^2\right)} + \overline{\left(w'^2\right)} \right]\right]^\frac{1}{2} \ztag{\ref{eq_i}}
\end{IEEEeqnarray}
Turbulent kinetic energy $k$:
\begin{IEEEeqnarray}{rCl}
k &\equiv& \frac{1}{2} \left(\overline{\left(u'^2\right)} + \overline{\left(v'^2\right)} + \overline{\left(w'^2\right)} \right)\ztag{\ref{eq_k}}
\end{IEEEeqnarray}
In homogeneous, isotropic, fully-developed turbulence, the following relationships apply between the largest-scale and smallest-scale eddies:
\begin{IEEEeqnarray}{rCcCl}
\frac{L_\min}{L_\max} &=& \frac{\eta}{\Lambda} &=& \re_\Lambda^{-3/4}\ztag{\ref{eq_kolmogorov_size}}\\
&& \frac{u_\eta}{u_\Lambda} &=& \re_\Lambda^{-1/4}\ztag{\ref{eq_kolmogorov_speed}}\\
......@@ -24,9 +34,9 @@ In homogeneous, isotropic, fully-developed turbulence, we accept the following r
t_\eta &=& \left(\frac{\mu}{\rho} \frac{1}{\epsilon} \right)^{\frac{1}{2}}\ztag{\ref{eq_kolmogorov_epsilon_time}}
\end{IEEEeqnarray}
\end{boiboite}
\end{boiboiboite}
\clearpage
\subsubsection{Hypothetical flow}
\wherefrom{Non-examinable. From De Nevers \cite{denevers2004} Ex 18.1}
......@@ -42,7 +52,7 @@ In homogeneous, isotropic, fully-developed turbulence, we accept the following r
Hint: $\int \sin^2 x \diff x = \frac{1}{2}\left(x + \frac{\sin 2x}{2} \right) + b$
\clearpage
\subsubsection{Turbulent channel flow}
\wherefrom{Non-examinable. From De Nevers \cite{denevers2004} Ex 18.2}
......@@ -56,13 +66,15 @@ In homogeneous, isotropic, fully-developed turbulence, we accept the following r
Measurements are carried out to measure the turbulent intensity through the channel. Those are displayed in figure~\ref{fig_t_measurements}.
\begin{figure}[ht]
\begin{center}
\vspace{-0.5cm}
\includegraphics[width=8cm]{t_measurements}
\vspace{-0.5cm}
\end{center}
\supercaption{Measurements of turbulent intensity in $x$ and $y$ directions in a rectangular channel \SI{1}{\metre} wide and \SI{0.24}{\metre} wide, in which the centerline velocity $\overline v$ is \SI{1}{\metre\per\second}. Here $u'$ and $v'$ are written $v_x$ and $v_y$ respectively.}{Figure extracted from De Nevers \cite{denevers2004}, with source data from Reichardt 1938, \textit{Naturwissenschaften 26}:407}
\label{fig_t_measurements}
\label{fig_t_measurements}\vspace{-0.5cm}
\end{figure}
\begin{enumerate}
\shift{2}
\shift{3}
\item What is the value of $k$ at a point \SI{2}{\centi\metre} from the wall?
\end{enumerate}
......
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