Commit 89e08df8 authored by Olivier's avatar Olivier

Chapter 9: updated slides

parent 9b89e5d8
9/images/pressure_wave_0.png

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9/images/pressure_wave_0.png
9/images/pressure_wave_0.png
9/images/pressure_wave_0.png
9/images/pressure_wave_0.png
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......@@ -3,9 +3,9 @@
\renewcommand{\documentnumber}{9}
\renewcommand{\titleofthisdocument}{Compressible flow}
\renewcommand{\keywordsofthisdocument}{}
\renewcommand{\lastedityear}{2017}
\renewcommand{\lasteditmonth}{07}
\renewcommand{\lasteditday}{29}
\renewcommand{\lastedityear}{2018}
\renewcommand{\lasteditmonth}{06}
\renewcommand{\lasteditday}{21}
% Syntax for single-image slides:
% (the first argument (number) being the maximum fraction of the
......@@ -132,7 +132,7 @@
In control volume, mass conservation means: \pause
\begin{IEEEeqnarray}{rCl}
\rho A a &=& (\rho + \diff \rho) A (a - \diff V)
\rho A c &=& (\rho + \diff \rho) A (c - \diff V)
\end{IEEEeqnarray} \pause
\footnotesize $\to$ the density change is caused by the velocity change from one side of the wave to the other.
......@@ -142,12 +142,12 @@
re-arrange:
\begin{IEEEeqnarray*}{rCl}
\rho a &=& \rho a - \rho \diff V + a \diff \rho - \diff \rho \diff V \pause
\rho c &=& \rho c - \rho \diff V + c \diff \rho - \diff \rho \diff V \pause
\end{IEEEeqnarray*}
from which we drop $\diff \rho \diff V$: \pause
\begin{IEEEeqnarray}{rCl}
\rho \diff V &=& a \diff \rho \label{eq_tmpsoundone} \pause
\rho \diff V &=& c \diff \rho \label{eq_tmpsoundone} \pause
\end{IEEEeqnarray}
\footnotesize let’s eliminate this $\diff V$!
......@@ -159,9 +159,9 @@
\small
\begin{IEEEeqnarray}{rCl}
F_\net &=& -\dot m_\inn V_\inn + \dot m_\out V_\out \nonumber\\ \pause
p A - (p + \diff p) A &=& - \rho a A a + (\rho - \diff \rho) (a - \diff V) A (a - \diff V) \nonumber\\\pause
-\diff p A &=& \rho a A \left[-a + (a - \diff V)\right] \nonumber\\\pause
\frac{\diff p}{a} &=& \rho \diff V \label{eq_tmpsoundtwo}
p A - (p + \diff p) A &=& - \rho c A c + (\rho - \diff \rho) (c - \diff V) A (c - \diff V) \nonumber\\\pause
-\diff p A &=& \rho c A \left[-c + (c - \diff V)\right] \nonumber\\\pause
\frac{\diff p}{c} &=& \rho \diff V \label{eq_tmpsoundtwo}
\end{IEEEeqnarray}
\end{frame}
......@@ -170,17 +170,17 @@ p A - (p + \diff p) A &=& - \rho a A a + (\rho - \diff \rho) (a - \diff V) A (a
\small
\begin{IEEEeqnarray*}{rCl}
\rho \diff V = a \diff \rho
\rho \diff V = c \diff \rho
\end{IEEEeqnarray*}
and
\begin{IEEEeqnarray*}{rCl}
\frac{\diff p}{a} = \rho \diff V
\frac{\diff p}{c} = \rho \diff V
\end{IEEEeqnarray*}
\normalsize
combine to obtain: \pause
\begin{IEEEeqnarray}{rCl}
a &=& \sqrt{\frac{\diff p }{\diff \rho}}
c &=& \sqrt{\frac{\diff p }{\diff \rho}}
\end{IEEEeqnarray}
\end{frame}
......@@ -188,30 +188,35 @@ p A - (p + \diff p) A &=& - \rho a A a + (\rho - \diff \rho) (a - \diff V) A (a
\begin{frame}
Generalizing for any kind of wave: \pause
\begin{IEEEeqnarray}{rCl}
a &=& \sqrt{\left.\partialderivative{p}{\rho}\right|_{s=\cst}} \label{eq_def_speed_sound} \pause
c &=& \sqrt{\left.\partialderivative{p}{\rho}\right|_{s=\cst}} \label{eq_def_speed_sound} \pause
\end{IEEEeqnarray}
\small “the speed of sound is the square root of the partial derivative of pressure with respect to density at constant entropy”.\\\pause
“…which happens when the integral of the infinitesimal heat transfers divided by the temperature at which they occur along a reversible path between the two end states is equal to zero.”
\end{frame}
\begin{frame}
\begin{IEEEeqnarray}{rCl}
a &=& \sqrt{\left.\partialderivative{p}{\rho}\right|_{s=\cst}} \label{eq_def_speed_sound}
c &=& \sqrt{\left.\partialderivative{p}{\rho}\right|_{s=\cst}} \label{eq_def_speed_sound}
\end{IEEEeqnarray}
\small “the speed of sound is the square root of the change of pressure with density when the medium is squished gently.”
\small “the speed of sound is the square root of the change of pressure with density {\color{niceblue}when the medium is squished gently}.”
\end{frame}
\begin{frame}
\begin{IEEEeqnarray*}{rCl}
a &=& \sqrt{\left.\partialderivative{p}{\rho}\right|_{s=\cst}} \label{eq_def_speed_sound}
c &=& \sqrt{\left.\partialderivative{p}{\rho}\right|_{s=\cst}} \label{eq_def_speed_sound}
\end{IEEEeqnarray*}
\small “the speed of sound is the square root of the hardness of the fluid when you nudge it gently”.
\small “the speed of sound is the square root of {\color{niceblue}the hardness of the fluid when you nudge it gently}”.
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\skipinprint{
\section{Thermodynamics of isentropic flow of perfect gases}
\setcounter{section}{2}
}
\sectionwithsubtitle{Thermodynamics of isentropic flow of perfect gases}{Harry Potter and the adiabatic and reversible compressions and expansions of the Perfect Gas}
\sectionwithsubtitle{Thermodynamics of isentropic flow for a perfect gas}{Harry Potter and the adiabatic and reversible compressions and expansions of the Perfect Gas}
\begin{frame}
......@@ -267,9 +272,9 @@ p A - (p + \diff p) A &=& - \rho a A a + (\rho - \diff \rho) (a - \diff V) A (a
\end{frame}
\begin{frame}
Take \ref{eq_hor_three} ($p v^\gamma = \text{cst.}$), then re-write $a$ as: \pause
Take \ref{eq_hor_three} ($p v^\gamma = k$), then re-write $c$ as: \pause
\begin{IEEEeqnarray*}{rCl}
a &=& \sqrt{\left.\partialderivative{p}{\rho}\right|_{s=\cst}} \pause
c &=& \sqrt{\left.\partialderivative{p}{\rho}\right|_{s=\cst}} \pause
= \sqrt{k \gamma \rho^{\gamma - 1}} = \sqrt{p \rho^{-\gamma} \gamma \rho^{\gamma - 1}} \nonumber\\ \pause
&=& \sqrt{\gamma p \rho^{-1}}
\end{IEEEeqnarray*}
......@@ -278,7 +283,7 @@ p A - (p + \diff p) A &=& - \rho a A a + (\rho - \diff \rho) (a - \diff V) A (a
\begin{frame}
So, in a perfect gas, the speed of sound is: \pause
\begin{IEEEeqnarray}{rCl}
a &=& \sqrt{\gamma R T}\\\nonumber\\ \pause
c &=& \sqrt{\gamma R T}\\\nonumber\\ \pause
\ma &=& \frac{V}{\sqrt{\gamma R T}} \label{eq_speed_sound_pg}
\end{IEEEeqnarray}
\end{frame}
......@@ -298,7 +303,7 @@ p A - (p + \diff p) A &=& - \rho a A a + (\rho - \diff \rho) (a - \diff V) A (a
Plug the Mach number into eq. (\ref{eq_stag_temp}):\pause
\begin{IEEEeqnarray*}{rCl}
\frac{T_0}{T} &=& 1 + \frac{1}{c_p T} \frac{1}{2} V^2 = 1 + \frac{1}{\frac{\gamma R}{\gamma - 1} T} \frac{1}{2} V^2\\ \pause
&=& 1 + \left(\frac{\gamma - 1}{2}\right) \frac{V^2}{a^2} = 1 + \left(\frac{\gamma - 1}{2}\right) \ma^2\nonumber\\
&=& 1 + \left(\frac{\gamma - 1}{2}\right) \frac{V^2}{c^2} = 1 + \left(\frac{\gamma - 1}{2}\right) \ma^2\nonumber\\
\end{IEEEeqnarray*}
\end{frame}
......@@ -330,7 +335,11 @@ p A - (p + \diff p) A &=& - \rho a A a + (\rho - \diff \rho) (a - \diff V) A (a
Back to squishy fluid mechanics…
\end{frame}
\skipinprint{
\pictureframe{1}{C8RVgNOXcAATFtp}{}{}
\pictureframe{0.9}{C8K20vBVYAAJxya}{}{}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
......@@ -375,7 +384,7 @@ p A - (p + \diff p) A &=& - \rho a A a + (\rho - \diff \rho) (a - \diff V) A (a
which we differentiate to obtain:
\begin{IEEEeqnarray}{rCl}
\diff h + V \diff V &=& 0\\\nonumber \pause
v \diff p + V \diff V &=& 0\\ \pause
\frac{1}{\rho} \diff p + V \diff V &=& 0\\ \pause
\frac{1}{\rho} &=& -\frac{1}{\diff p} V \diff V \label{eq_tmp_rho}
\end{IEEEeqnarray}
\end{frame}
......@@ -383,9 +392,9 @@ p A - (p + \diff p) A &=& - \rho a A a + (\rho - \diff \rho) (a - \diff V) A (a
\begin{comment}
\begin{frame}
Side note: we remember from \ref{eq_def_speed_sound} that the speed of sound $a$ is expressed~as: \pause
Side note: we remember from \ref{eq_def_speed_sound} that the speed of sound $c$ is expressed~as: \pause
\begin{IEEEeqnarray*}{rCl}
a^2 = \left(\partialderivative{p}{\rho}\right)_s \label{eq_a_derivative}
c^2 = \left(\partialderivative{p}{\rho}\right)_s \label{eq_a_derivative}
\end{IEEEeqnarray*}
\end{frame}
\end{comment}
......@@ -397,7 +406,7 @@ p A - (p + \diff p) A &=& - \rho a A a + (\rho - \diff \rho) (a - \diff V) A (a
\frac{\diff \rho}{\rho} + \frac{\diff A}{A} + \frac{\diff V}{V} &=& 0\\ \pause
- \derivative{\rho}{p} V \diff V + \frac{\diff A}{A} + \frac{\diff V}{V} &=& 0\\ \pause
\frac{\diff V}{V} \left[1 - \derivative{\rho}{p} V^2\right] + \frac{\diff A}{A} &=& 0\\ \pause
\frac{\diff V}{V} \left[1 - \frac{1}{a^2} V^2\right] + \frac{\diff A}{A} &=& 0\\ \pause
\frac{\diff V}{V} \left[1 - \frac{1}{c^2} V^2\right] + \frac{\diff A}{A} &=& 0\\ \pause
\frac{\diff V}{V} \left[1 - \ma^2\right] + \frac{\diff A}{A} &=& 0
\end{IEEEeqnarray*}
\footnotesize cool!
......@@ -684,7 +693,7 @@ p A - (p + \diff p) A &=& - \rho a A a + (\rho - \diff \rho) (a - \diff V) A (a
Now, we compare conditions upstream and downstream:
\begin{IEEEeqnarray}{rCl}
\rho_1 V_1 &=& \rho_2 V_2 \nonumber\\ \pause
\frac{\rho_1}{\rho_2} &=& \frac{V_2}{V_1} = \frac{\ma_2 a_2}{\ma_1 a_1}\nonumber\\ \pause
\frac{\rho_1}{\rho_2} &=& \frac{V_2}{V_1} = \frac{\ma_2 c_2}{\ma_1 c_1}\nonumber\\ \pause
&=& \frac{\ma_2 \sqrt{\gamma R T_2}}{\ma_1 \sqrt{\gamma R T_1}}\nonumber\\ \pause
\frac{\ma_1}{\ma_2} &=& \frac{p_2}{p_1} \left(\frac{T_1}{T_2}\right)^\frac{1}{2} \pause
\end{IEEEeqnarray}
......
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