Commit cdffde56 authored by Olivier's avatar Olivier

Re-organization of chapters. Work in progress.

parent 40130fe5
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\renewcommand{\lastedityear}{2018}
\renewcommand{\lasteditmonth}{04}
\renewcommand{\lasteditday}{01}
\renewcommand{\numberofthischapter}{0}
\renewcommand{\titleofthischapter}{Important concepts}
\atstartofexercises
\fluidmechexercisestitle
\mecafluexboxen
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Compressibility effects}
%homemade
\label{exo_compressibility_effects}
An aircraft is flying in air with density \SI{0,9}{\kilogram\per\metre\cubed} and temperature \SI{-5}{\degreeCelsius}. Above which flight speed would you expect the air flow over the wings to become compressible?
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Pressure-induced force}
%homemade
\label{exo_pressure_induced_force}
A \SI{2}{\metre} by \SI{2}{\metre} flat panel is used as the wall of a swimming pool (\cref{fig_pressure_distribution_plate}). On the left side, the pressure is uniform at \SI{1}{\bar}.
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=0.6\textwidth]{pressure_distribution_plate}
\end{center}
\supercaption{Pressure distribution on a flat plate}{\wcfile{Pressure distribution on a flat plate.svg}{Figure} \cczero \oc}
\label{fig_pressure_distribution_plate}
\end{figure}
\begin{enumerate}
\item What is the pressure force exerted on the left side of the plate?
\end{enumerate}
On the right side of the plate, the water exerts a pressure which is not uniform: it increases with depth. The relation, expressed in \si{pascals}, is:
\begin{IEEEeqnarray}{rCl}
p_\text{water} &=& \num{1,3e5} - \num{9,81e3} \times z
\end{IEEEeqnarray}
\begin{enumerate}
\shift{1}
\item What is the pressure force exerted on the right side of the plate?\\
\textit{[Hint: we will explore the required expression in chapter~1 as eq.~\ref{eq_pressure_force_scalar} p.\pageref{eq_pressure_force_scalar}]}
\end{enumerate}
\clearpage %handmade, fucking figure float won’t work
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Shear-induced force}
%homemade
\label{exo_shear_induced_force}
A fluid flows over a \SI{3}{\metre} by \SI{3}{\metre} flat horizontal plate, in the $x$-direction as shown in \cref{fig_shear_force_plate}. Because of this flow, the plate is subjected to uniform shear $\tau_{zx} = \SI{1,65}{\pascal}$.
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=0.6\textwidth]{shear_force_plate}
\end{center}
\supercaption{Shear force exerting on a plate}{\wcfile{Shear force on a plate}{Figure} \cczero \oc}
\label{fig_shear_force_plate}
\end{figure}\vspace{-1cm}%handmade
\begin{enumerate}
\item What is the shear force applying on the plate?
\item What would be the shear force if the shear was not uniform, but instead was a function of $x$ expressed (in \si{pascals}) as $\tau_{zx} = \num{1,65} - \num{0,01} \times x^2$?\\
\textit{[Hint: we will explore the required expression in chapter~2 as eq.~\ref{eq_shear_force_twod_integration_general} p.\pageref{eq_shear_force_twod_integration_general}]}
\end{enumerate}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Speed of sound}
\wherefrom{White \smallcite{white2008} P1.87}
\label{exo_speed_sound_newton}
Isaac Newton measured the speed of sound by timing the interval between observing smoke produced by a cannon blast and the hearing of the detonation. The cannon is shot~\SI{8,4}{\kilo\metre} away from Newton. What is the air temperature if the measured interval is~\SI{24,2}{\second}? What is the temperature if the interval is~\SI{25,1}{\second}?
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Power lost to drag}
%homemade
\label{exo_power_lost_to_drag}
A truck moves with constant speed $\vec V$ on a road, with $V = \SI{100}{\kilo\metre\per\hour}$. Because it experiences cross-wind, it is subjected to a drag $\vec F_D$ with $F_D = \SI{5}{\kilo\newton}$ at an angle $\theta = \SI{20}{\degree}$, as shown in \cref{fig_truck_drag_power}.
\begin{figure}
\begin{center}
\includegraphics[width=0.6\textwidth]{truck_drag_power}
\end{center}
\supercaption{Top view of a truck traveling at velocity $\vec V$ and subject to a drag force $\vec F_D$}{\wcfile{Force and velocity.svg}{Figure} \cczero \oc}
\label{fig_truck_drag_power}
\end{figure}
\begin{enumerate}
\item What is the power required to overcome drag?
\end{enumerate}
The drag force $\vec F_D$ is applying at a distance \SI{0,8}{\metre} behind the center of gravity of the truck.
\begin{enumerate}
\shift{1}
\item What are the magnitude and the direction of the moment exerted by the drag $\vec F_D$ about the center of gravity?
\end{enumerate}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Go-faster exhaust pipe}
%homemade
\label{exo_go_faster_exhaust_pipe}
The engine exhaust gases of a student’s hot-rod car are flowing quasi-steadily in a cylindrical outlet pipe, whose outlet is slanted at an angle $\theta = \SI{25}{\degree}$ to improve the good looks of the car and provide the opportunity for an exercise.
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=0.7\textwidth]{go_faster_pipe_photo}\vspace{0.5cm}
\includegraphics[width=0.7\textwidth]{go_faster_pipe}
\end{center}
\supercaption{Exhaust gas pipe of a car. The outlet cross-section is at an angle $\theta$ relative to the axis of the pipe.}{\wcfile{Go-faster tailpipe.svg}{Figure} \cczero \oc\\ \wcfile{Classic Car (1) (3495188372).jpg}{Photo} cropped, mirrored and edited from an \flickrfile{kazandrew2/3495188372/}{original} \ccbysa by \flickrname{Kaz Andrew}{kazandrew2}}
\label{fig_go_faster_pipe}
\end{figure}
The outlet velocity is measured at \SI{15}{\metre\per\second}, and the exhaust gas density is \SI{1,1}{\kilogram\per\metre\cubed}. The slanted outlet section area $A$ is \SI{420}{\centi\metre\squared}.
\begin{enumerate}
\item What is the mass flow $\dot m$ through the pipe?
\item What is the volume flow $\dot \vol$ of exhaust gases?
\end{enumerate}
Because of the shear within the exhaust gases, the flow through the pipe induces a pressure loss of \SI{21}{\pascal} (we will learn to quantify this in chapter~5). In these conditions, the specific heat capacity of the exhaust gases is $c_{p \text{gases}} = \SI{1100}{\joule\per\kilogram\per\kelvin}$.
\begin{enumerate}
\shift{2}
\item What is the power required to carry the exhaust gases through the pipe?
\item What is the gas temperature increase due to the shear in the flow?
\end{enumerate}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Acceleration of a particle}
%homemade
\label{acceleration_fluid_particle}
Inside a complex, turbulent water flow, we are studying the trajectory of a cubic fluid particle of width \SI{0,1}{\milli\metre}. The particle is accelerating at a rate of \SI{2,5}{\metre\per\second\squared}.
\begin{enumerate}
\item What is the net force applying to the particle?
\item In practice, which types of forces could cause it to accelerate?
\end{enumerate}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Flow classifications}
%homemade
\label{exo_flow_classifications}
\begin{enumerate}
\item Can an incompressible flow also be unsteady?
\item Can a very viscous fluid flow in a turbulent manner?
\item \textit{[more difficult]} Can a compressible flow also be isothermal?
\item Give an example of an isothermal flow, of an unsteady flow, of a compressible flow, and of an incompressible flow.
\end{enumerate}
\clearpage
\subsubsection*{Answers}
\NumTabs{2}
\begin{description}
\item [\ref{exo_compressibility_effects}]%
\tab If you adopt $\ma = \num{0,6}$ as an upper limit, you will obtain $V_\max = \SI{709}{\kilo\metre\per\hour}$ (eqs.~\ref{eq_def_ma} \& \ref{eq_speed_sound_perfect_gas} p.\pageref{eq_speed_sound_perfect_gas}). Note that propellers, fan blades etc. will meet compressiblity effects far sooner.
\item [\ref{exo_pressure_induced_force}]%
\tab 1) $F_\text{left} = \SI{400}{\kilo\newton}$ (eq.~\ref{eq_first_def_pressure} p.\pageref{eq_first_def_pressure});
\tab 2) $F_\text{right} = \SI{480}{\kilo\newton}$ (eq.~\ref{eq_pressure_force_scalar} p.\pageref{eq_pressure_force_scalar}).
\item [\ref{exo_shear_induced_force}]%
\tab 1) $F_1 = \SI{14,85}{\newton}$ (eq.~\ref{eq_first_def_shear} p.\pageref{eq_first_def_shear});
\tab 2) $F_2 = \SI{14,58}{\newton}$ (eq.~\ref{eq_shear_force_twod_integration_general} p.\pageref{eq_shear_force_twod_integration_general}).
\item [\ref{exo_speed_sound_newton}]%
\tab \SI{26,7}{\degreeCelsius} \& \SI{5,6}{\degreeCelsius}.
\item [\ref{exo_power_lost_to_drag}]%
\tab 1) $\dot W = \vec F_\text{drag} \cdot \vec V_\text{truck} = \SI{130,5}{\kilo\watt}$;
\tab \tab 2) $M = || \vec r \wedge \vec F_\text{drag}|| = \SI{1368}{\newton\metre}$, $\vec M = \left(\begin{array}{c} 0\\ 0\\ \num{-1368}\end{array}\right)$ (points vertically upwards).
\item [\ref{exo_go_faster_exhaust_pipe}]%
\tab 1) $\dot m = \SI{0,2929}{\kilogram\per\second}$ (eq.~\ref{eq_basic_mass_flow} p.\pageref{eq_basic_mass_flow});
\tab 2) $\dot \vol = \SI{266,2}{\liter\per\second}$ (eq.~\ref{eq_basic_volume_flow} p.\pageref{eq_basic_volume_flow});
\tab 3) $\dot W = \SI{5,59}{\watt}$ (eq.~\ref{eq_power_deltap} p.\pageref{eq_power_deltap});
\tab 4) $\Delta T = \SI{+0,0174}{\kelvin}$ (eq.~\ref{eq_power_heat} p.\pageref{eq_power_heat}), an illustration of remarks made in \S\ref{ch_temperature_distribution} p.\pageref{ch_temperature_distribution} regarding temperature distribution.
\item [\ref{acceleration_fluid_particle}]%
\tab 1) $F_\net = \SI{2,5e-9}{\newton}$ (eq~\ref{eq_secondlaw} p.\pageref{eq_secondlaw}), such are the orders of magnitude involved in \textsc{cfd} calculations!
\tab 2) Only three kinds: forces due pressure, shear, and gravity.
\item [\ref{exo_flow_classifications}]%
\tab 1) yes, 2) yes if $\re$ is high enough, 3) yes (in very specific cases such as high pressure changes combined with high heat transfer or high irreversibility, therefore generally no), 4) open the cap of a water bottle and turn it upside down: you have an isothermal, unsteady, incompressible flow. An example of compressible flow could be the expansion in a jet engine nozzle.
\end{description}
\atendofexercises
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\usepackage{fluidmechslides} % from https://git.framasoft.org/u/olivier/sensible-styles
\renewcommand{\documentnumber}{0}
\renewcommand{\titleofthisdocument}{Important concepts}
\renewcommand{\keywordsofthisdocument}{}
\renewcommand{\lastedityear}{2017}
\renewcommand{\lasteditmonth}{04}
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% Syntax for single-image slides:
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% \figureframe{1}{filename}{Title}{Attribution}
% Do I want the print version? (no \pause, larger peamble)
\printversion
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\begin{document}
\mainslidesbegin
\begin{frame}{Take-away slide for Chapter 0}
\begin{enumerate}\pause
\item Fluid particles instead of molecules\pause
\item Pressure (perpendicular)\\
Shear (parallel)\pause
\item Not Everything Will Always Be Known\pause
%\begin{itemize}
% \item Theory vs. CFD vs. Experiment\pause
% \item Classify flows to simplify problems
%\end{itemize}
\end{enumerate}
\end{frame}
\section{Concept of a fluid}
\begin{frame}{}
\vocab{fluid}:\\\pause
matter that is continuously deformable,\\\pause
occupies all of the space\\ made available to it.
\end{frame}
\section{Fluid mechanics}
\subsection{Solution of a flow}
\begin{frame}{Typical problem}\pause
What is the fluid flow around (or through) an object?
\end{frame}
\begin{frame}{}
\begin{centering}
\vocab{Solution}\pause
the entire set of velocities of fluid particles.\\
{\small (sets of discrete values, or functions)}
\end{centering}\pause
~
From a known solution, we calculate forces and moments on object
\end{frame}
\subsection{Modeling of fluids}
\begin{frame}{}
What \emph{is} a fluid?
\end{frame}
\begin{frame}{}
fluid = matter\pause~= molecules? \pause
…not in fluid mechanics!
\end{frame}
\begin{frame}{The macroscopic scale}
In fluid mechanics, we treat fluids like a \vocab{continuum}\\\pause
(all physical properties continuously differentiable)
\end{frame}
\begin{frame}{}
\begin{centering}
~
1 “empty” bottle of air\pause
=\pause
\num{2e22} molecules\pause
at \SI{1000}{\kilo\metre\per\hour}.\pause
~
\textit{uh-oh}
\end{centering}
\end{frame}
\begin{frame}{}
\begin{centering}
{\Large 20000000000000000000000}\\
equations\pause
with\\\pause
{\Large 20000000000000000000000}\\
unknowns\pause
\end{centering}
~
Result: $\vec V \left(\begin{array}{c}u\\ v\\ w\end{array}\right) = f(x,y,z,t)$\\\pause
but $\vec V_\text{average} = \vec 0$ !
\end{frame}
\begin{frame}{}
The \vocab{continuum assumption} treats groups of millions of molecules as patches\pause
1 patch = a \vocab{fluid particle}\pause
$\approx \SI{1}{\micro\metre\cubed}$ for complex flow\\\pause
$\approx \SI{e3}{\metre\cubed}$ for upper atmosphere
\end{frame}
\figureframe{1}{property_shrinking_volume}{}{\wcfile{Macroscopic microscopic property_2.svg}{Figure} \ccbysa \oc}
\begin{frame}{}
A fluid: not “marbles”,
instead, “continuous expanding dough”.
\end{frame}
\subsection{Theory, numerics, experiment}
\begin{frame}{Analytical fluid mechanics}
\begin{itemize}\pause
\item First useful results in mid-1930s\pause
\item Able to provide \textbf{insight} over complex flows\pause
\item Provides solutions for simple flows\\\pause