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Commit f119a320 authored by Olivier's avatar Olivier

Chapter 7: Major re-work + proof-read

The chapter is still unsatisfying in my eyes. I would like to see
a more systematic method for quantifying pressure difference
in pipes (mixing height, expansion/contraction, local losses,
and wall friction losses).
I would also like to see a good systematic (even if approximate)
exploration of the dependency between the main parameters at hand.

* Delta p_friction is now uniformly refered to as Delta p_loss
* Ditched entrance effects, focus in on fully-developed flow and
  on methodology
* Summary added at end to better picture relevance of chapter in
  view of the entire course
parent efec80a1
This diff is collapsed.
\subsubsection{Couette flow}
\wherefrom{\cczero \oc}
\label{exo_couette_flow}
We consider laminar flow of a fluid between two parallel plates (named \vocab{Couette flow}), as shown in \cref{fig_twoplates_exo}.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.6\textwidth]{velocity_distribution_couette_flow.png}
\end{center}
\supercaption{Two-dimensional laminar flow between two plates, also called \vocab{Couette flow}.}{\wcfile{Couette flow flat plate laminar velocity distributions.svg}{Figure} \cczero \oc}
\label{fig_twoplates_exo}\vspace{-0.5cm}%handmade
\end{figure}
\begin{enumerate}
\item Starting from the Navier-Stokes equations for incompressible, two-dimensional flow,
\begin{IEEEeqnarray}{cCc}
\rho \left[ \partialtimederivative{u} + u \partialderivative{u}{x} + v \partialderivative{u}{y} \right] & = & \rho g_x - \partialderivative{p}{x} + \mu \left[ \secondpartialderivative{u}{x} + \secondpartialderivative{u}{y} \right] \\%\ztag{\ref{eq_ns_twodone}}\\
\rho \left[ \partialtimederivative{v} + u \partialderivative{v}{x} + v \partialderivative{v}{y} \right] & = & \rho g_y - \partialderivative{p}{y} + \mu \left[ \secondpartialderivative{v}{x} + \secondpartialderivative{v}{y} \right] \\%\ztag{\ref{eq_ns_twodtwo}}
\end{IEEEeqnarray}
show that the velocity profile in a horizontal, laminar, steady, fully-developed flow between two horizontal plates separated by a gap of height~$2H$ is:
\begin{IEEEeqnarray}{rCl}
u &=& \frac{1}{2 \mu} \left(\partialderivative{p}{x} \right) (y^2 - H^2)\ztag{\ref{eq_tmp5}}
\end{IEEEeqnarray}
\item Why would this equation fail to describe turbulent flow? (Briefly justify your answer, e.g.\ in 30 words or less)
\end{enumerate}
\subsubsection{Design of a wind tunnel}
\wherefrom{non-examinable}
\label{exo_wind_tunnel}
%homemade
Describe the main characteristics of a wind tunnel that could be installed and operated in the room you are standing in.
In order to do this:
\begin{itemize}
\item Start by proposing key characteristics for the test section;
\item From these dimensions, draw approximately an air circuit to feed the test section (while attempting to minimize the size of the fan, whose cost increases exponentially with diameter).
\item Quantify the static and stagnation pressures along the air circuit, by estimating the losses generated by wall shear and in the bends (you may use data from \cref{fig_loss_coefficients_stator_vanes});
\item Quantify the minimum power required to generate your chosen test section flow characteristics.
\end{itemize}
\begin{figure}
\begin{center}
\includegraphics[width=0.5\textwidth]{wind_tunnel_guide_vanes_barlow1999.png}\\
Filter screen: $\eta = \num{0,05}$
\end{center}
\supercaption{Loss coefficients $K_L$ (here noted $\eta$) generated by the use of various components within wind tunnel ducts.}{Figure \copyright\xspace Barlow, Rae \& Pope 1999~\cite{barlowraepope1999}}
\label{fig_loss_coefficients_stator_vanes}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%
\item [\ref{exo_couette_flow}]%
\tab The structure is given in the derivation of equation~\ref{eq_tmp5} p.~\pageref{eq_tmp5}, and more details about the math are given in the derivation of the (very similar) equation~\ref{eq_u_lam} p.~\pageref{eq_u_lam}.
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