Commit 300b9849 authored by Olivier's avatar Olivier

Changed order of chapters 8 and 9

parent e79c01c2
../../11/images/F5_tornado_Elie_Manitoba_2007.jpg
\ No newline at end of file
......@@ -53,7 +53,7 @@ Welcome to the Fluid Mechanics course of the \textit{Chemical and Energy Enginee
This course script is the only document you need to work through and succeed in this course. I highly recommend that you print it out (at 200 pages, a copy at the University print shop costs less than 9\ €). If you can’t afford this, print at least the exercise sheets and the first page of each chapter.
Note that this year, one chapter is missing. I will complete the all-new \chaptereight a few weeks after the course has started.
Note that this year, one chapter is missing. I will complete the all-new \chapternine a few weeks after the course has started.
In order to receive weekly updates about the course, send an email to fluidmech@ovgu.de from your university email account, indicating your name and matriculation number.
......
......@@ -440,7 +440,7 @@
\item [Turbulence] One last characteristic that we systematically attempt to identify in fluid flows is \vocab{turbulence} (or its opposite, \vocab{laminarity}). While laminar flows are generally very smooth and steady, turbulent flows feature multiple, chaotic velocity field variations in time and space.
We shall first approach the concept of turbulence in \chapterseven, and study it more formally in \chaptereight. In the meantime, we may assess whether a flow will become turbulent or not using a non-dimensional parameter named the \vocab{Reynolds number}, noted $\re$:
We shall first approach the concept of turbulence in \chapterseven, and study it more formally in \chapternine. In the meantime, we may assess whether a flow will become turbulent or not using a non-dimensional parameter named the \vocab{Reynolds number}, noted $\re$:
\begin{IEEEeqnarray}{rCl}
\re &\equiv& \frac{\rho V L}{\mu} \label{eq_def_reynolds_number}
\end{IEEEeqnarray}
......
......@@ -638,7 +638,7 @@
\item $\re < \num{e4}$ ? Likely turbulent\pause
\end{itemize}
{\small (ok it’s not really magic $\to$ chapter 9)}
{\small (ok it’s not really magic $\to$ chapter 8)}
\end{description}
\end{frame}
......
......@@ -256,7 +256,7 @@
\section{Boundary layer transition}
\label{ch_bl_transition}
After it has traveled a certain length along the wall, the boundary layer becomes very unstable and it transits rapidly from a laminar to a turbulent regime (\cref{fig_bl_transition}). We have already described the characteristics of turbulence in broadly in \chapterseven and more extensively in \chaptereight; they apply to turbulence within the boundary layer. It is worth reminding ourselves that the boundary layer may be turbulent in a globally laminar flow (e.g.\ around an aircraft in flight, the boundary layer is turbulent, but the main flow is laminar). Here, we refer to the regime of the boundary layer only, not the outer flow.
After it has traveled a certain length along the wall, the boundary layer becomes very unstable and it transits rapidly from a laminar to a turbulent regime (\cref{fig_bl_transition}). We have already described the characteristics of turbulence in broadly in \chapterseven and more extensively in \chapternine; they apply to turbulence within the boundary layer. It is worth reminding ourselves that the boundary layer may be turbulent in a globally laminar flow (e.g.\ around an aircraft in flight, the boundary layer is turbulent, but the main flow is laminar). Here, we refer to the regime of the boundary layer only, not the outer flow.
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=\textwidth]{boundary_layer_transition.png}
......@@ -277,7 +277,7 @@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Turbulent boundary layers}
The extensive description of turbulent flows remains an unsolved problem. As we have seen in \chaptereight, by contrast with laminar counterparts turbulent flows result in
The extensive description of turbulent flows remains an unsolved problem. As we have seen in \chapternine, by contrast with laminar counterparts turbulent flows result in
\begin{itemize}
\item increased mass, energy and momentum exchange;
\item increased losses to friction;
......
......@@ -34,7 +34,7 @@
\str\ \partialderivative{\vec V^*}{t^*} + [1]\ \vec V^* \cdot \gradient^* \vec V^* & = & \frac{1}{\fr^2}\ \vec g^* - \eu\ \gradient^* p^* + \frac{1}{\re}\ \gradient^{*2} \vec V^*\nonumber\label{eq_ns_nondim_chapeight}\\
\end{IEEEeqnarray}
We saw in \chapternine that we could compare the relative weight of terms: when the Reynolds number $\re$ is very large, the last term becomes negligible relative to the other four. Thus, our governing equation can be reduced as follows:
We saw in \chaptereight that we could compare the relative weight of terms: when the Reynolds number $\re$ is very large, the last term becomes negligible relative to the other four. Thus, our governing equation can be reduced as follows:
\begin{IEEEeqnarray}{rCl}
\str\ \partialderivative{\vec V^*}{t^*} + [1]\ \vec V^* \cdot \gradient^* \vec V^* & \approx & \frac{1}{\fr^2}\ \vec g^* - \eu\ \gradient^* p^* \label{eq_euler_nondim}
\end{IEEEeqnarray}
......@@ -49,12 +49,12 @@
\subsection{Investigation of inviscid flows}
From what we studied in \chapternine, we recognize immediately that flows governed by eq.~\ref{eq_euler_equation} are troublesome: the absence of viscous effects facilitates the occurrence of turbulence and makes for much more chaotic behaviors. Although the removal of shear from the Navier-Stokes equation simplifies the governing equation, the \emph{solutions} to this new equation become even harder to find and describe.
From what we studied in \chaptereight, we recognize immediately that flows governed by eq.~\ref{eq_euler_equation} are troublesome: the absence of viscous effects facilitates the occurrence of turbulence and makes for much more chaotic behaviors. Although the removal of shear from the Navier-Stokes equation simplifies the governing equation, the \emph{solutions} to this new equation become even harder to find and describe.
What can be done in the other two branches of fluid mechanics?
\begin{itemize}
\item Large-scale flows are difficult to investigate experimentally. As we have seen in \chapternine, scaling down a flow (e.g. so it may fit inside a laboratory) while maintaining constant $\re$ requires increasing velocity by a corresponding factor.
\item Large-scale flows are also difficult to investigate numerically. At high $\re$, the occurrence of turbulence makes for either an exponential increase in computing power (we saw in \chaptereightshort that direct numerical simulation computing power increases with $\re^{\num{3,5}}$), or for increased reliance on hard-to-calibrate turbulence models (in Reynolds-averaged simulations).
\item Large-scale flows are difficult to investigate experimentally. As we have seen in \chaptereight, scaling down a flow (e.g. so it may fit inside a laboratory) while maintaining constant $\re$ requires increasing velocity by a corresponding factor.
\item Large-scale flows are also difficult to investigate numerically. At high $\re$, the occurrence of turbulence makes for either an exponential increase in computing power (we saw in \chapternineshort that direct numerical simulation computing power increases with $\re^{\num{3,5}}$), or for increased reliance on hard-to-calibrate turbulence models (in Reynolds-averaged simulations).
\end{itemize}
All three branches of fluid mechanics, therefore, struggle with large-scale flows, because of turbulence.
......
......@@ -477,7 +477,7 @@
Today indeed, 150 years after it was first written, no general expression has been found for velocity or pressure fields that would solve this vector equation in the general case. Nevertheless, in this course we will use it directly:
\begin{itemize}
\item to understand and quantify the importance of key fluid flow parameters, in \chapternine;
\item to understand and quantify the importance of key fluid flow parameters, in \chaptereight;
\item to find analytical solutions to flows in a few selected cases, in the other remaining chapters.
\end{itemize}
......
......@@ -67,7 +67,7 @@
\begin{enumerate}
\item What is the “correct” answer?\\\pause
\small $\to$ chapters 6, 7, 8
\small $\to$ chapters 8, 10, 11
\end{enumerate}
\end{frame}
......
......@@ -298,7 +298,7 @@
\item Pipe flow is turbulent for $\reD \gtrsim \num{4000}$.
\end{itemize}
The significance of the Reynolds number extends far beyond pipe flow; we shall explore this in \chapternine.
The significance of the Reynolds number extends far beyond pipe flow; we shall explore this in \chaptereight.
\subsection{Characteristics of turbulent flow}
......
......@@ -618,7 +618,7 @@
\item[\textbf{B}] He stole the secret from the soviets in an incredibly daring Bond-like operation;\pause
% \item[\textbf{B}] Luke, you switched off your targeting computer! What's wrong?\pause
\item[\textbf{C}] It’s just instinct that fluid dynamicists have about things;\pause
\item[\textbf{D}] He figured it out using material from Chapter~9.
\item[\textbf{D}] He figured it out using material from Chapter~8.
\end{itemize}
\end{frame}
......@@ -631,7 +631,7 @@
~
\begin{centering}
\small (magic, until we get to chapter 9)
\small (magic, until we get to chapter 8)
\end{centering}
\end{frame}
......
This diff is collapsed.
\renewcommand{\lastedityear}{2018}
\renewcommand{\lasteditmonth}{06}
\renewcommand{\lasteditday}{12}
\renewcommand{\lastedityear}{2019}
\renewcommand{\lasteditmonth}{08}
\renewcommand{\lasteditday}{13}
\renewcommand{\numberofthischapter}{8}
\renewcommand{\titleofthischapter}{\namechaptereight}
\atstartofexercises
\fluidmechexercisestitle
\mecafluboxen
\mecafluexboxen
\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}}\\
&& \frac{t_\eta}{t_\Lambda} &=& \re_\Lambda^{-1/2}\ztag{\ref{eq_kolmogorov_time}}
\end{IEEEeqnarray}
\begin{IEEEeqnarray}{rCl}
\eta &=& \left(\frac{\mu^3}{\rho^3} \frac{1}{\epsilon} \right)^{\frac{1}{4}}\ztag{\ref{eq_kolmogorov_epsilon_size}}\\
u_\eta &=& \left(\frac{\mu}{\rho} \frac{1}{\epsilon} \right)^{\frac{1}{4}}\ztag{\ref{eq_kolmogorov_epsilon_vel}}\\
t_\eta &=& \left(\frac{\mu}{\rho} \frac{1}{\epsilon} \right)^{\frac{1}{2}}\ztag{\ref{eq_kolmogorov_epsilon_time}}
\end{IEEEeqnarray}
Non-dimensional incompressible Navier-Stokes equation:
\begin{IEEEeqnarray}{rCl}
\str\ \partialderivative{\vec V^*}{t^*} + [1]\ \vec V^* \cdot \gradient^* \vec V^* & = & \frac{1}{\fr^2}\ \vec g^* - \eu\ \gradient^* p^* + \frac{1}{\re}\ \gradient^{*2} \vec V^*\ztag{\ref{eq_ns_nondim}}
\end{IEEEeqnarray}
\begin{equationterms}
\item in which $\str \equiv \frac{f\ L}{V}$, \ \ $\eu \equiv \frac{p_0 - p_\infty}{\rho\ V^2}$, \ \ $\fr \equiv \frac{V}{\sqrt{g\ L}}$ and \ \ $\re \equiv \frac{\rho\ V\ L}{\mu}$.
\end{equationterms}
\end{boiboiboite}
The force coefficient $C_F$ and power coefficient $C_\P$ are defined as:
\begin{align}
C_F &\equiv \frac{F}{\frac{1}{2} \rho S V^2}
&C_\P &\equiv \frac{\dot W}{\frac{1}{2} \rho S V^3}\tag{\ref{def_power_coefficient}}
\end{align}
\clearpage
\subsubsection{Hypothetical flow}
\label{exo_hypothetical_flow}
\wherefrom{Non-examinable. From De Nevers \cite{denevers2004} Ex 18.1}
We imagine a turbulent flow described at some point with the equations (in \si{\metre\per\second})
The speed of sound $c$ in air is modeled as:
\begin{IEEEeqnarray}{rCl}
u &=& 10 + \sin t\\
v &=& 0\\
w &=& 0
c &=& \sqrt{\gamma R T} \ztag{\ref{eq_speed_sound_perfect_gas}}
\end{IEEEeqnarray}
(No real turbulent flow can be described by equations this simple — but this is a nice first basis for practice)
What are the values of $\overline{u}$, $u'$, $\overline{u'}$, $I_x$, $I$, and $k$?
Hint: $\int \sin^2 x \diff x = \frac{1}{2}\left(x + \frac{\sin 2x}{2} \right) + b$
\subsubsection{Turbulent channel flow}
\label{exo_turbulent_channel_flow}
\wherefrom{Non-examinable. From De Nevers \cite{denevers2004} Ex 18.2}
\Cref{fig_viscosities_various_fluids_two} quantifies the viscosity of various fluids as a function of temperature.
\end{boiboiboite}
A wind tunnel carries air through a channel which is \SI{1}{\metre} wide and \SI{0.24}{\metre} high. The average velocity is \SI{0.82}{\metre\per\second}. The pressure drop caused by both friction on the walls and turbulent dissipation is measured at \SI{-0,0286}{\pascal\per\metre}.
\begin{enumerate}
\item What is the non-turbulent kinetic energy per unit mass of the flow?
\item At what average rate does this kinetic energy degrade into heat?
\item If there was no heat transfer, what would be the rate of temperature increase of the air?
\end{enumerate}
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{figure}
\begin{center}
\vspace{-0.5cm}
\includegraphics[width=8cm]{t_measurements}
\vspace{-0.5cm}
\vspace{-1.5cm}%handmade
\includegraphics[width=\textwidth]{viscosities_horizontal}
\vspace{-1.1cm}
\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}\vspace{-0.5cm}
\supercaption{Viscosity of various fluids at a pressure of \SI{1}{\bar} (in practice viscosity is almost independent of pressure).}{Figure repeated from \cref{fig_viscosities_various_fluids} p.\pageref{fig_viscosities_various_fluids}; \copyright\xspace White 2008~\cite{white2008}}
\label{fig_viscosities_various_fluids_two}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Scaling with the Reynolds number}
\wherefrom{\cczero \oc}
\label{exo_scaling_re}
\begin{enumerate}
\shift{3}
\item What is the value of $k$ at a point \SI{2}{\centi\metre} from the wall?
\item Give the definition of the Reynolds number, indicating the \textsc{si} units for each term.
\item What is the consequence on the velocity field of having a low Reynolds number? A high Reynolds number?
\item Give an example of a high Reynolds number flow, and of a low Reynolds number~flow.
\end{enumerate}
\begin{comment}
A computational fluid dynamics (\cfd) simulation of the flow is carried out, in which $C_\mu = \num{0,09}$. It predicts that the value of $\epsilon$ at the point where $k$ was calculated above is $\epsilon = \SI{0,025}{\metre\squared\per\second\cubed}$.
The standard golf ball has a diameter of \SI{42,67}{\milli\metre} and a mass of \SI{45,93}{\gram}. A typical maximum velocity for such balls is \SI{200}{\kilo\metre\per\hour}.
A student wishes to investigate the flow over a golf ball using a model in a wind tunnel. S/he prints an enlarged model with a diameter of \SI{50}{\centi\metre}.
\begin{enumerate}
\shift{3}
\item What is the value of the turbulent viscosity at the point where $k$ was measured above, and how does it compare to the value of the viscosity of the air?
\item If the atmospheric conditions are identical, what flow velocity needs to be generated during the experiment in order to reproduce the flow patterns around the real ball?
\item Would the Mach number for the real ball then be reproduced?
\item Would this velocity need to be adjusted if the experiment was run on a very hot summer day?
\item If the model was made of the same material as the real ball, how heavy would it~be?
\end{enumerate}
\end{comment}
\subsubsection{Cumulus cloud}
\label{exo_cumulus_cloud}
\wherefrom{Non-examinable. From Tennekes \& Lumley \cite{tennekesetal1972} P1.1}
A cumulus cloud (one of those “fluffy” summer clouds, figure~\ref{fig_fluffy_cloud}) has roughly the size of a sphere of diameter $D = \SI{50}{\meter}$. To a good approximation, it features isotropic, homogeneous, fully-developed turbulence. The largest-scale air currents in the cloud reach a maximum velocity $V = \SI{3}{\metre\per\second}$.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.3\textwidth]{Bluesky.jpg}
\end{center}
\supercaption{Summer clouds (\textit{Cumulus humilis}) form when hot moist air convected from the ground is cooled down when it rises.}{\wcfile{Bluesky.jpg}{Photo} \ccbysa by \weu{Dwindrim}}
\label{fig_fluffy_cloud}
\end{figure}
\begin{enumerate}
\item What is approximately the size of the smallest eddies in the cloud?
\item What is approximately the dissipation power, per unit mass of air and for the entire cloud?
\item What will those three values become once the cloud has grown to a diameter of $D_2 = \SI{100}{\meter}$?
\end{enumerate}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Fluid mechanics of a giant airliner}
\wherefrom{non-examinable, \cczero \oc}
%homemade
\label{exo_scale_airbus}
\subsubsection{Reactor tank}
\label{exo_reactor_tank}
\wherefrom{Non-examinable. \cczero \oc}
In this exercise, we consider that that viscosity is entirely independent from pressure and density (this is a reasonable first approximation).
A tank used to store chemical reactants has roughly the size of a cube of side length $L = \SI{2}{\metre}$ (figure~\ref{fig_tank_agitator}). The tank is filled with a water-like liquid and vigorously stirred with a large agitator propeller for a prolonged amount of time. The propeller induces a maximum fluid velocity of \SI{1,5}{\metre\per\second}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.3\textwidth]{tank_agitator}
\end{center}
\supercaption{Schematic drawing of a cubic tank stirred with an agitator propeller}{Figure \cczero \oc}
\label{fig_tank_agitator}
\end{figure}
\begin{enumerate}
\item What is approximately the size of the smallest eddies in the tank?
\item What is approximately the specific dissipation power?
\end{enumerate}
A full-scale simulation (\textsc{dns}) of the flow was carried out, which required 500 hours of computing time on a supercomputer. Now the same simulation is to be carried out again, for the same flow, but with a fluid whose viscosity is half that of water.
\begin{enumerate}
\shift{2}
\item What do you expect the new computation time to be?
\end{enumerate}
An Airbus A380 airliner has an \SI{80}{\metre}-wingspan and is designed to cruise at $\ma = \num{0,85}$ where the air temperature and pressure are \SI{-40}{\degreeCelsius} and \SI{0,25}{\bar}.
A group of students wishes to study the flow field around the aircraft, with a wind tunnel whose test section has a diameter of~\SI{1}{\metre} (in which, obviously, the model has to fit!).
\clearpage
\subsubsection*{Answers}
\startofanswers
\begin{enumerate}
\item p.~\pageref{exo_hypothetical_flow}
\begin{enumerate}
\item $\overline u = \SI{10}{\metre\per\second}$
\item $u' = \sin (t)$
\item $\overline u' = \SI{0}{\metre\per\second}$ (as always)
\item $I_x = \SI{7,07}{\percent}$
\item $I = \SI{4,08}{\percent}$
\item $k = \SI{0,25}{\joule\per\kilogram}$
\end{enumerate}
\item p.~\pageref{exo_turbulent_channel_flow}
\begin{enumerate}
\item $\dot e_{m \text{main}} = \SI{0,336}{\joule\per\kilogram}$
\item $\epsilon = \SI{0,0191}{\watt\per\kilogram}$
\item $\dot T = \SI{0,02}{\milli\kelvin\per\second}$
\item $k = \SI{5,76}{\milli\joule\per\kilogram}$
\item If the temperature inside the tunnel is maintained at $T_\text{tunnel} = \SI{20}{\degreeCelsius}$, propose a combination of wind tunnel velocity and pressure which would enable the team to adequately model the effects of viscosity.
\item Which flow conditions would be required in a \emph{water} tunnel with the same dimensions?
\end{enumerate}
\item p.~\pageref{exo_cumulus_cloud}
Shocked by their answers to the above questions, the group of students decides instead to study the effect of compressibility (while accepting that viscous effects may not be adequately modeled).
\begin{enumerate}
\item Since $\re_\Lambda \equiv \num{6,1e3}$, $\eta \approx \SI{0,2}{\milli\metre}$
\item $\epsilon \approx \SI{1,147}{\watt\per\kilogram}$ \& $\dot W_\epsilon \approx \SI{92}{\kilo\watt}$
\item $\eta_2 = \num{0,6} \eta_1$, $\epsilon_2 = 8 \epsilon_1$, $\dot W_{\epsilon 2} = \num{64} \dot W_{\epsilon 1}$
\shift{2}
\item If the maximum velocity attainable in the wind tunnel is \SI{80}{\metre\per\second}, which air temperature is required for compressibility effects to be modeled?
\end{enumerate}
\item p.\pageref{exo_reactor_tank}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\clearpage \clearfloats %handmade, fucking figure placement
\subsubsection{Scale effects on a dragonfly}
%homemade
\label{exo_scale_dragonfly}
A dragonfly (sketched in \cref{fig_dragonfly}) has a \SI{10}{\centi\metre} wingspan, a mass of \SI{80}{\milli\gram}, and cruises at~\SI{4}{\metre\per\second}, beating its four wings \num{20} times per second.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.4\textwidth]{dragonfly}
\vspace{-0.5cm}%handmade
\end{center}
\supercaption{Plan view of a dragonfly}{\wcfile{Dragonfly Calopteryx maculata -wingveins (2).svg}{Figure} by \oc, A. Plank, Drury, Dru, Westwood, J. O. (\cczero)}
\label{fig_dragonfly}
\end{figure}
You are tasked with the investigation of the flight performance of the dragonfly, and have access to a wind tunnel with a test section of diameter \SI{1}{\metre}.
\begin{enumerate}
\item $\eta \approx \SI{0,023}{\milli\metre}$ (width of human hair)
\item $\epsilon \approx \SI{3,4}{\watt\per\kilogram}$
\item Increase resolution in all three directions according to $\eta$, and the time resolution according to $t_\eta$: the computation time increases by a factor \num{6,7}.
\item Which model size and flow velocity would you use for this experiment?
\item How many wing beats per second would then be required on the model?
\item What would be the lift developed by the model during the experiment?
\item How much mechanical power would the model require, compared to the real dragonfly?
\end{enumerate}
\end{enumerate}
\subsubsection{Formula One testing}
%homemade
\label{exo_formula_one}
You are leading the Aerodynamics team in a successful Formula One racing team. Your team races a car that is \SI{5,1}{\metre} long, \SI{1,8}{\metre} wide, has a~\SI{610}{\kilo\watt} power plant, and a mass of~\SI{750}{\kilogram}. The car can reach a top speed of \SI{310}{\kilo\metre\per\hour}, at which speed the drag force is~\SI{7,1}{\kilo\newton}.
In order to test different aerodynamic configurations for the car, your team invests in a 50-million euro wind tunnel in which you will run tests on a model. You currently have the choice between two possible model sizes: a \SI{60}{\percent} model and a (smaller) \SI{50}{\percent} model.
\begin{enumerate}
\item What would be the frontal area of each model, in proportion to the frontal area of the real car?
\item What would be the volume of each model, compared to the volume of the real car?
\item How much less weight would the \SI{50}{\percent} model have than the \SI{60}{\percent} model?
\end{enumerate}
In the end, the regulations for the Formula One racing change, and your team is forced to opt for a \SI{50}{\percent} car model.
\begin{enumerate}
\shift{3}
\item If the ambient atmospheric conditions cannot be changed, which flow speed in the wind tunnel is required, so that the air flow around the real car is reproduced around the model?
\end{enumerate}
In practice the regulations change once again, and the maximum speed authorized in wind tunnel tests is now \SI{50}{\metre\per\second}.
\begin{enumerate}
\shift{4}
\item Your team considers modifying the air temperature to compensate for the limit in the air speed. If the temperature in the tunnel can be controlled between \SI{-10}{\degreeCelsius} and \SI{40}{\degreeCelsius}, but the pressure remains atmospheric, what is the maximum race-track speed that can be reproduced in the wind tunnel?
\item In that case, by which factor should the model drag force measurements be multiplied in order to correspond to the real car?
\item \textit{[non-examinable question]} The wind tunnel has a \SI{2}{\metre} by \SI{2}{\metre} square test cross-section. What is the power required to bring atmospheric air ($T_\atm = \SI{15}{\degreeCelsius}$, $V_\atm = \SI{0}{\metre\per\second}$) to the desired speed and temperature in the test section?
\end{enumerate}
{\small \textit{NB: this exercise is inspired by \href{https://www.youtube.com/watch?v=KC0E0wU6inU}{an informative and entertaining video by the Sauber F1 team about their wind tunnel testing}, which the reader is encouraged to watch at\\ \href{https://www.youtube.com/watch?v=KC0E0wU6inU}{https://www.youtube.com/watch?v=KC0E0wU6inU}.}}
\clearpage
\subsubsection*{Answers}
\NumTabs{2}
\begin{description}
\item [\ref{exo_scaling_re}]%
\tab 1) See equation~\ref{eq_def_re} and subsequent comments;
\tab 2) See \S\ref{ch_inertial_viscous_effects}, in practice this can be summarized in two or three sentences;
\tab 3) High-Re: air flow around an airliner in cruise; Low-Re: air flow around a dust particle falling to the ground.
\tab 4) Match $\re$: $V_2 = \SI{4,74}{\metre\per\second}$
\tab 5) No, but it is very low anyway (no compressibility effects to be reproduced)
\tab 6) Yes! (discuss in class)
\tab 7) $m_2 = \SI{73,9}{\kilogram}$
\item [\ref{exo_scale_airbus}]%
\tab 1) With a half-aircraft model of half-length $\frac{L_2}{2} = \SI{80}{\centi\metre}$, at identical speed, we would need an air density $\rho_2 = \SI{24,9}{\kilogram\per\metre\cubed}$. At ambient temperature, this requires a pressure $p_2 = \SI{20,9}{\bar}$!
\tab 2) In water, the density cannot be reasonably controlled, and we need a velocity $V_3 = \SI{418}{\metre\per\second}$!
\tab 3) In air, at $V_4 = \SI{80}{\metre\per\second}$, Mach number can be reproduced at $T_4 = \SI{-251}{\degreeCelsius}$ (although the Reynolds number is off). Wind tunnels used to investigate compressible flow around aircraft have very powerful coolers.
\item [\ref{exo_scale_dragonfly}]%
\tab 1) With e.g. a model of span $\SI{60}{\centi\metre}$, match the Reynolds number: $V_2 = \SI{0,67}{\metre\per\second}$;
\tab 2) Match the Strouhal number: $f_2 = \SI{0,56}{\hertz}$. Mach, Froude and Euler numbers will have no effect here;
\tab 3) and 4) are left as a surprise for the student.
\item [\ref{exo_formula_one}]
\tab 3) The third model would have \SI{42}{\percent} less mass than the second;
\tab 4) Maintaining $\re$ requires $V_3 = \SI{172,2}{\metre\per\second}$ (fast!);
\tab 5) Now the fastest track speed that can be studied is $\SI{27}{\metre\per\second}$ (assuming identical viscosity everywhere);
\tab 6) Multiply force measurements by \num{1,74} to scale up to reality;
\tab 7) One needs powerful coolers!
\end{description}
\atendofexercises
This diff is collapsed.
This diff is collapsed.
\renewcommand{\lastedityear}{2019}
\renewcommand{\lasteditmonth}{04}
\renewcommand{\lasteditday}{02}
\renewcommand{\lasteditmonth}{08}
\renewcommand{\lasteditday}{13}
\renewcommand{\numberofthischapter}{9}
\renewcommand{\titleofthischapter}{\namechapternine}
\atstartofexercises
\fluidmechexercisestitle
\mecafluboxen
\mecafluexboxen
\begin{boiboiboite}
Non-dimensional incompressible Navier-Stokes equation:
\begin{IEEEeqnarray}{rCl}
\str\ \partialderivative{\vec V^*}{t^*} + [1]\ \vec V^* \cdot \gradient^* \vec V^* & = & \frac{1}{\fr^2}\ \vec g^* - \eu\ \gradient^* p^* + \frac{1}{\re}\ \gradient^{*2} \vec V^*\ztag{\ref{eq_ns_nondim}}
\end{IEEEeqnarray}
\begin{equationterms}
\item in which $\str \equiv \frac{f\ L}{V}$, \ \ $\eu \equiv \frac{p_0 - p_\infty}{\rho\ V^2}$, \ \ $\fr \equiv \frac{V}{\sqrt{g\ L}}$ and \ \ $\re \equiv \frac{\rho\ V\ L}{\mu}$.
\end{equationterms}
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}}\\
&& \frac{t_\eta}{t_\Lambda} &=& \re_\Lambda^{-1/2}\ztag{\ref{eq_kolmogorov_time}}
\end{IEEEeqnarray}
\begin{IEEEeqnarray}{rCl}
\eta &=& \left(\frac{\mu^3}{\rho^3} \frac{1}{\epsilon} \right)^{\frac{1}{4}}\ztag{\ref{eq_kolmogorov_epsilon_size}}\\
u_\eta &=& \left(\frac{\mu}{\rho} \frac{1}{\epsilon} \right)^{\frac{1}{4}}\ztag{\ref{eq_kolmogorov_epsilon_vel}}\\
t_\eta &=& \left(\frac{\mu}{\rho} \frac{1}{\epsilon} \right)^{\frac{1}{2}}\ztag{\ref{eq_kolmogorov_epsilon_time}}
\end{IEEEeqnarray}
The force coefficient $C_F$ and power coefficient $C_\P$ are defined as:
\begin{align}
C_F &\equiv \frac{F}{\frac{1}{2} \rho S V^2}
&C_\P &\equiv \frac{\dot W}{\frac{1}{2} \rho S V^3}\tag{\ref{def_power_coefficient}}
\end{align}
\end{boiboiboite}
The speed of sound $c$ in air is modeled as:
\clearpage
\subsubsection{Hypothetical flow}
\label{exo_hypothetical_flow}
\wherefrom{Non-examinable. From De Nevers \cite{denevers2004} Ex 18.1}
We imagine a turbulent flow described at some point with the equations (in \si{\metre\per\second})
\begin{IEEEeqnarray}{rCl}
c &=& \sqrt{\gamma R T} \ztag{\ref{eq_speed_sound_perfect_gas}}
u &=& 10 + \sin t\\
v &=& 0\\
w &=& 0
\end{IEEEeqnarray}
(No real turbulent flow can be described by equations this simple — but this is a nice first basis for practice)