Commit cdffde56 by Olivier

### Re-organization of chapters. Work in progress.

parent 40130fe5
This diff is collapsed.
 \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

85.6 KB

 \documentclass[17pt]{beamer} \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} \renewcommand{\lasteditday}{06} % Syntax for single-image slides: % (the first argument (number) being the maximum fraction of the % slide width that the image is allowed to have.) % \figureframe{1}{filename}{Title}{Attribution} % Do I want the print version? (no \pause, larger peamble) \printversion %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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