 ### 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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