Commit 65a7c9ea authored by Olivier's avatar Olivier

Exercises 7: general clean-up

parent 1a6be0d7
......@@ -22,7 +22,7 @@
F_\text{sphere} &=& 3 \pi \mu U D \ztag{\ref{eq_drag_creeping_sphere}}
\end{IEEEeqnarray}
\Cref{fig_viscosities} quantifies the viscosity of various fluids.
\Cref{fig_viscosities} quantifies the viscosity of various fluids as a function of temperature.
\end{boiboite}
\begin{figure}[h]
......@@ -31,7 +31,7 @@
\includegraphics[width=0.9\textwidth]{images/viscosities_horizontal.jpg}
\vspace{-0.5cm}
\end{center}
\supercaption{Viscosity of various fluids at a pressure of \SI{1}{\bar} (in practice viscosity is almost independent of pressure).}{Figure \copyright\xspace White, 2011, \textit{Fluid Mechanics}, 7th ed. pub. McGraw-Hill}
\supercaption{Viscosity of various fluids at a pressure of \SI{1}{\bar} (in practice viscosity is almost independent of pressure).}{Figure \copyright\xspace White 2008 \cite{white2008}}
\label{fig_viscosities}
\end{figure}
......
......@@ -194,10 +194,10 @@
Based on this work, it can be shown that for a laminar boundary layer flowing along a smooth wall, the four parameters about which we are interested are solely function of the distance-based Reynolds number $\rex$:
\begin{IEEEeqnarray}{rCl}
\frac{\delta}{x} &=& \frac{\num{4,91}}{\sqrt{\rex}} \IEEEyessubnumber\\\nonumber\\
\frac{\delta^*}{x} &=& \frac{\num{1,72}}{\sqrt{\rex}} \IEEEyessubnumber\\\nonumber\\
\frac{\theta}{x} &=& \frac{\num{0,664}}{\sqrt{\rex}} \IEEEyessubnumber\\\nonumber\\
c_{f_{(x)}} &=& \frac{\num{0,664}}{\sqrt{\rex}} \IEEEyessubnumber
\frac{\delta}{x} &=& \frac{\num{4,91}}{\sqrt{\rex}} \IEEEyessubnumber\label{eq_delta_lam}\\\nonumber\\
\frac{\delta^*}{x} &=& \frac{\num{1,72}}{\sqrt{\rex}} \IEEEyessubnumber\label{eq_deltastar_lam}\\\nonumber\\
\frac{\theta}{x} &=& \frac{\num{0,664}}{\sqrt{\rex}} \IEEEyessubnumber\label{eq_deltastarstar_lam}\\\nonumber\\
c_{f_{(x)}} &=& \frac{\num{0,664}}{\sqrt{\rex}} \IEEEyessubnumber\label{eq_cf_lam}
\end{IEEEeqnarray}
%%%%%%%%%%%%%%%%%
......@@ -291,10 +291,10 @@
In the same way that we have worked with the laminar boundary layer profiles, we can derive models for our characteristics of interest from this velocity profile:
\begin{IEEEeqnarray}{rCl}
\frac{\delta}{x} &\approx& \frac{\num{0,16}}{\rex^{\frac{1}{7}}} \IEEEyessubnumber\\\nonumber\\
\frac{\delta^*}{x} &\approx& \frac{\num{0,02}}{\rex^{\frac{1}{7}}} \IEEEyessubnumber\\\nonumber\\
\frac{\theta}{x} &\approx& \frac{\num{0,016}}{\rex^{\frac{1}{7}}} \IEEEyessubnumber\\\nonumber\\
c_{f_{(x)}} &\approx& \frac{\num{0,027}}{\rex^{\frac{1}{7}}} \IEEEyessubnumber
\frac{\delta}{x} &\approx& \frac{\num{0,16}}{\rex^{\frac{1}{7}}} \IEEEyessubnumber\label{eq_delta_turb}\\\nonumber\\
\frac{\delta^*}{x} &\approx& \frac{\num{0,02}}{\rex^{\frac{1}{7}}} \IEEEyessubnumber\label{eq_deltastar_turb}\\\nonumber\\
\frac{\theta}{x} &\approx& \frac{\num{0,016}}{\rex^{\frac{1}{7}}} \IEEEyessubnumber\label{eq_deltastarstar_turb}\\\nonumber\\
c_{f_{(x)}} &\approx& \frac{\num{0,027}}{\rex^{\frac{1}{7}}} \IEEEyessubnumber\label{eq_cf_turb}
\end{IEEEeqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Separation}
......
\renewcommand{\lastedityear}{2015}
\renewcommand{\lasteditmonth}{03}
\renewcommand{\lasteditday}{12}
\renewcommand{\lasteditmonth}{06}
\renewcommand{\lasteditday}{10}
\renewcommand\coursnumber{7}
\renewcommand\courstitle{Boundary layer}
\atstartofexercises
......@@ -9,39 +9,40 @@
\mecafluboxen
Laminar boundary layer along a smooth surface, exact solutions:
\begin{IEEEeqnarray}{rCl}
\frac{\delta}{x} &=& \frac{\num{4,91}}{\sqrt{\text{[Re]}_x}} \\\nonumber\\
\frac{\delta^*}{x} &=& \frac{\num{1,72}}{\sqrt{\text{[Re]}_x}} \\\nonumber\\
\frac{\theta}{x} &=& \frac{\num{0,664}}{\sqrt{\text{[Re]}_x}} \\\nonumber\\
c_{f_{(x)}} &=& \frac{\num{0,664}}{\sqrt{\text{[Re]}_x}}
\end{IEEEeqnarray}
Transition occurs around $\rex \approx \num{5e5}$.\\
Turbulent boundary layer along a smooth surface, approximate values:
\begin{IEEEeqnarray}{rCl}
\frac{\delta}{x} &\approx& \frac{\num{0,16}}{\text{[Re]}_x^{\frac{1}{7}}} \\\nonumber\\
\frac{\delta^*}{x} &\approx& \frac{\num{0,02}}{\text{[Re]}_x^{\frac{1}{7}}} \\\nonumber\\
\frac{\theta}{x} &\approx& \frac{\num{0,016}}{\text{[Re]}_x^{\frac{1}{7}}} \\\nonumber\\
c_{f_{(x)}} &\approx& \frac{\num{0,027}}{\text{[Re]}_x^{\frac{1}{7}}}
\end{IEEEeqnarray}
\mecafluexboxen
\begin{boiboite}
Exact solutions to the laminar boundary layer along a smooth surface yield:
\begin{align}
\frac{\delta}{x} &= \frac{\num{4,91}}{\sqrt{\rex}}
&\frac{\delta^*}{x} &= \frac{\num{1,72}}{\sqrt{\rex}} \tag{\ref{eq_deltastar_lam}}\\
\frac{\theta}{x} &= \frac{\num{0,664}}{\sqrt{\rex}}
&c_{f_{(x)}} &= \frac{\num{0,664}}{\sqrt{\rex}} \tag{\ref{eq_cf_lam}}
\end{align}
On a flat surface, transition occurs around $\rex \approx \num{5e5}$.
Solutions to the turbulent boundary layer along a smooth surface yield the following time-averaged characteristics:
\begin{align}
\frac{\delta}{x} &\approx \frac{\num{0,16}}{\rex^{\frac{1}{7}}}
&\frac{\delta^*}{x} &\approx \frac{\num{0,02}}{\rex^{\frac{1}{7}}} \tag{\ref{eq_deltastar_turb}}\\
\frac{\theta}{x} &\approx \frac{\num{0,016}}{\rex^{\frac{1}{7}}}
&c_{f_{(x)}} &\approx \frac{\num{0,027}}{\rex^{\frac{1}{7}}} \tag{\ref{eq_cf_turb}}
\end{align}
\Cref{fig_viscosities} quantifies the viscosity of various fluids as a function of temperature.
\end{boiboite}
\begin{figure}[h]
\begin{center}
%\vspace{-0.5cm}
\includegraphics[width=0.9\textwidth]{images/viscosities_horizontal.jpg}
\vspace{-0.5cm}
\end{center}
\supercaption{Viscosity of various fluids at a pressure of \SI{1}{\bar} (in practice viscosity is almost independent of pressure).}{Figure \copyright\xspace White 2008 \cite{white2008}}
\label{fig_viscosities}
\end{figure}
\begin{comment}
exos:
calcul delta, delta*, theta
calcul de Cf avec cf
calcul plus précis séparation (modèle dans White)
\end{comment}
\clearpage
\subsubsection{Water and air flow}
\wherefrom{White \smallcite{white2008} E7.2}
......@@ -51,13 +52,16 @@ Turbulent boundary layer along a smooth surface, approximate values:
\item in water of temperature \SI{20}{\degreeCelsius}?
\end{enumerate}
\subsubsection{Boundary layer sketches}
\wherefrom{\cczero \oc}
Sketch the velocity profile formed by a fluid flowing along a straight wall, at the leading edge, in a laminar regime, and in a turbulent regime.\\
Draw a few streamlines and the thickness $\delta^*$.
How can the transition to turbulent regime be triggered, or delayed?
\subsubsection{Shear force}
\wherefrom{White \smallcite{white2008} E7.3}
......@@ -67,49 +71,74 @@ Turbulent boundary layer along a smooth surface, approximate values:
How will these shear efforts evolve with the plate is tilted with an angle to the flow of about \SI{10}{\degree}?
\subsubsection{Wright Flyer I}
\wherefrom{non-examinable, \cczero \oc}
The \we{Wright Flyer I}, the first aircraft flown into powered control flight (1903), was a biplane with a \SI{12}{\metre} wingspan and \SI{47}{\metre\squared} wing surface. The wing profile was extremely thin and it could only fly at very low angles of attack. Its flight speed was approximately \SI{40}{\kilo\metre\per\hour}.
Estimate the power necessary to compensate the shear exerted by the airflow on the wings during flight. What other forms of drag would also be found on the aircraft?
\subsubsection{Shear friction on a fuselage}
\wherefrom{\cczero \oc}
An Airbus A340-600 is cruising at $M=\num{0,82}$ at an altitude of \SI{10 000}{\metre} (viscosity \SI{1,457e-5}{\newton\second\per\metre\squared}, temperature \SI{220}{\kelvin}, density \SI{0,4}{\kilogram\per\metre\cubed}). Estimate the power dissipated to friction on the cylindrical part of the fuselage (diameter~\SI{5,6}{\metre}, length~\SI{65}{\metre}).
An \we{Airbus A340-600} is cruising at $M=\num{0,82}$ at an altitude of \SI{10 000}{\metre} (viscosity \SI{1,457e-5}{\newton\second\per\metre\squared}, temperature \SI{220}{\kelvin}, density \SI{0,4}{\kilogram\per\metre\cubed}). Estimate the power dissipated to friction on the cylindrical part of the fuselage (diameter~\SI{5,6}{\metre}, length~\SI{65}{\metre}).
In practice, in which circumstances could flow separation occur on the fuselage skin?
\subsubsection{Separation according to Pohlhausen}
\wherefrom{non-examinable, based on Richecœur 2012~\cite{richecoeur2012}}
Air at \SI{1}{\bar} and \SI{20}{\degreeCelsius} flows along a smooth surface, and decelerates slowly, with a constant rate of $\SI{-0,25}{\metre\per\second\per\metre}$.
According to the Pohlhausen model, at which distance downstream will separation occur? Is the boundary layer still laminar then?
How could one generate such a deceleration in practice?
\subsubsection{Separation mechanism}
\wherefrom{non-examinable, \cczero \oc}
Sketch the velocity profile of a laminar or turbulent boundary layer shortly upstream of, and at a separation point.
The two equations below describe flow in laminar boundary layer:
\begin{IEEEeqnarray*}{rCl}
\begin{IEEEeqnarray}{rCl}
u \partialderivative{u}{x} + v \partialderivative{u}{y} & = & U \frac{\diff U}{\diff x} + \frac{\mu}{\rho} \secondpartialderivative{u}{y} \ztag{\ref{eq_ns_bl_lam_un}}\\
\partialderivative{u}{x} + \partialderivative{v}{y} & = & 0 \ztag{\ref{eq_ns_bl_lam_deux}}
\end{IEEEeqnarray*}
\end{IEEEeqnarray}
Identify these two equations, list the conditions in which they apply, and explain why a boundary layer cannot separate when a favorable pressure gradient is applied along the~wall.
\subsubsection{Laminar wing profile}
\wherefrom{non-examinable, based on a diagram from Bertin et al. 2010~\cite{bertincummings2010}}
The characteristics of a so-called “laminar” wing profile are compared in \cref{fig_laminar_profile_bertin_one,fig_laminar_profile_bertin_two,fig_laminar_profile_bertin_three} with those of an ordinary profile.
\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{images/bertin_laminar_profile_1.png}
\end{center}
\supercaption{Comparison of the thickness distribution of two uncambered wing profiles: an ordinary medium-speed \textsc{naca} 0009 profile, and a “laminar” \textsc{naca} 66-009 profile.}{Figure \copyright\xspace Bertin \& Cummings 2010~\cite{bertincummings2010}}
\label{fig_laminar_profile_bertin_one}
\end{figure}
The characteristics of a so-called “laminar” wing profile are compared in \cref{fig_lam} below with those of an ordinary profile.
\begin{figure}
\begin{center}
\includegraphics[width=0.85\textwidth]{images/bertin_laminar_profile_2.png}
\end{center}
\supercaption{Static pressure distribution (represented as a the local non-dimensional \vocab{pressure coefficient} $C_p \equiv \frac{p -p_\infty}{\frac{1}{2} \rho V^2}$) as a function of distance $x$ (non-dimensionalized with the chord $c$) over the surface of the two airfoils shown in \cref{fig_laminar_profile_bertin_one}.}{Figure \copyright\xspace Bertin \& Cummings 2010~\cite{bertincummings2010}}
\label{fig_laminar_profile_bertin_two}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.75\textwidth]{images/bertin1.jpg}
\includegraphics[width=0.85\textwidth]{images/bertin2.jpg}
\includegraphics[width=0.75\textwidth]{images/bertin_laminar_profile_3.png}
\end{center}
\supercaption{Characteristics of a laminar wing profile compared to those of an ordinary medium-speed profile.}{Fig. \copyright\xspace Bertin \& Cummings 2010~\cite{bertincummings2010}}
\label{fig_lam}
\supercaption{Values of the section drag coefficient $C_d \equiv \frac{d}{\frac{1}{2} c \rho V^2}$ as a function of the section lift coefficient $C_l \equiv \frac{l}{\frac{1}{2} c \rho V^2}$ for both airfoils presented in \cref{fig_laminar_profile_bertin_one}.}{Figure \copyright\xspace Bertin \& Cummings 2010~\cite{bertincummings2010}, based on data by Abott \& Von Doenhoff 1949~\cite{abbottvondoenhoff1959}}
\label{fig_laminar_profile_bertin_three}
\end{figure}
On the graph representing the pressure coefficient~$C_p \equiv \frac{p -p_\infty}{\frac{1}{2} \rho V^2}$, identify the curve corresponding to each profile.
......@@ -117,7 +146,7 @@ Turbulent boundary layer along a smooth surface, approximate values:
What advantages and disadvantages does the laminar wing profile have, and how can they be explained? In which applications will it be most useful?
\subsubsection{Separation model}
\wherefrom{White \smallcite{white2008} E7.5}
\wherefrom{non-examinable, from White \smallcite{white2008} E7.5}
In 1949, Bryan Thwaites explored the limits of the Pohlhausen separation model. He proposed a different model to describe the laminar boundary layer velocity profile, which is articulated upon an expression for the momentum thickness $\theta$ and is more accurate than the Reynolds-number based descriptions that we have studied. Thwaites proposed the~model:
\begin{IEEEeqnarray}{rCl}
......@@ -135,12 +164,4 @@ Turbulent boundary layer along a smooth surface, approximate values:
A classical model for flow deceleration is the Howarth longitudinal velocity profile $U_{(x)} = U_0 \left(1 - \frac{x}{L}\right)$, in which $L$ is a reference length of choice. In this velocity distribution with linear deceleration, what is the distance at which the Thwaites model predicts the boundary layer separation?
\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{images/viscosite_rotated.jpg}
\vspace{-1cm}
\end{center}
\supercaption{Viscosity of various fluids at a pressure of \SI{1}{\bar}}{Figure \copyright\xspace White 2008~\cite{white2008}}
\end{figure}
\atendofexercises
......@@ -31,10 +31,10 @@
\end{frame}
\begin{frame}{Take-home slide for lecture 7}
\begin{itemize}%\pause
\item BL is viscosity-dominated; rest of flow (almost) inviscid%\pause
\item \vocab{Transition}: from laminar (low-drag, fragile) to turbulent (high-drag, resistant)%\pause
\item \vocab{Separation}: streamlines diverge from surface,\\%\pause
\begin{itemize}\pause
\item BL is viscosity-dominated; rest of flow (almost) inviscid\pause
\item \vocab{Transition}: from laminar (low-drag, fragile) to turbulent (high-drag, resistant)\pause
\item \vocab{Separation}: streamlines diverge from surface,\\\pause
happens when $\tau_\text{wall} = 0$
\end{itemize}
\end{frame}
......@@ -47,8 +47,8 @@
\end{frame}
\begin{frame}{Objectives:}
\begin{itemize}%\pause
\item How can we describe flows close to the wall?%\pause
\begin{itemize}\pause
\item How can we describe flows close to the wall?\pause
\item How can we describe and predict separation?
\end{itemize}
\end{frame}
......@@ -60,16 +60,16 @@
\begin{frame}
In 1904, Ludwig Prandtl observed that in most flows,
viscous effects are concentrated in a zone very close to the wall.%\pause
viscous effects are concentrated in a zone very close to the wall.\pause
He called this zone the \vocab{boundary layer}.
\end{frame}
\begin{frame}
Boundary layer: a \emph{concept}%\pause
Boundary layer: a \emph{concept}\pause
Zone between wall and point where velocity is \SI{99}{\percent} of external velocity%\pause
Zone between wall and point where velocity is \SI{99}{\percent} of external velocity\pause
A blurry and sometimes un-definable boundary!
\end{frame}
......@@ -83,7 +83,7 @@
\begin{frame}
Boundary layer thickness depends strongly of flow conditions.%\pause
Boundary layer thickness depends strongly of flow conditions.\pause
It \emph{decreases} when speed is increased or when viscosity is decreased(!)
\end{frame}
......@@ -92,15 +92,15 @@
\begin{frame}
Flow within a boundary layer is laminar up to a certain point:\\%\pause
$\to$ \vocab{transition point}. %\pause
Flow within a boundary layer is laminar up to a certain point:\\\pause
$\to$ \vocab{transition point}. \pause
Beyond this, the layer becomes turbulent: greater thickness, larger growth, more dissipation.
\end{frame}
\begin{frame}
We may have laminar boundary layer in a turbulent flow;%\pause
We may have laminar boundary layer in a turbulent flow;\pause
and turbulent boundary layers are commonplace in laminar flows!
\end{frame}
......@@ -108,7 +108,7 @@
\subsection{Why study the boundary layer?}
\begin{frame}
Why this frenzy about a puny half-millimeter?%\pause
Why this frenzy about a puny half-millimeter?\pause
\
......@@ -117,7 +117,7 @@
\begin{frame}
\begin{enumerate}
\item we \textbf{avoid dealing with the full Navier-Stokes equations}!\\%\pause
\item we \textbf{avoid dealing with the full Navier-Stokes equations}!\\\pause
~\\
\emph{Outside} of the boundary layer and wake areas, viscous effects are negligible.
In that case, $\rho \totaltimederivative{\vec V} = - \gradient{p}$: this is surmountable.\\
......@@ -130,7 +130,7 @@
\begin{frame}
\begin{enumerate}
\shift{1}
\item We \textbf{quantify shear forces}.\\%\pause
\item We \textbf{quantify shear forces}.\\\pause
~\\
A good resolution of the boundary layer = a quantification of friction
\end{enumerate}
......@@ -139,7 +139,7 @@
\begin{frame}
\begin{enumerate}
\shift{2}
\item We predict \textbf{flow separation}.\\%\pause
\item We predict \textbf{flow separation}.\\\pause
~\\
Boundary layer control is key to imparting a given trajectory on a fluid!
\end{enumerate}
......@@ -151,13 +151,13 @@
\subsection{Characterization of the boundary layer}
\begin{frame}
Three parameters to quantify boundary layer thickness%\pause
Three parameters to quantify boundary layer thickness\pause
\begin{enumerate}
\item The \vocab{thickness} $\delta$,\\%\pause
\item The \vocab{thickness} $\delta$,\\\pause
\begin{IEEEeqnarray}{rCl}
\delta &\equiv& y_{u=\num{0,99}U}
\end{IEEEeqnarray}%\pause
\end{IEEEeqnarray}\pause
$\to$ distance at which velocity $u$ is \SI{99}{\percent} of~$U$.
\end{enumerate}
......@@ -166,9 +166,9 @@
\begin{frame}
\begin{enumerate}
\shift{1}
\item The \vocab{displacement thickness} $\delta^*$ :%\pause
\item The \vocab{displacement thickness} $\delta^*$ :\pause
What is the distance by which the flow streamlines are shifted away from the wall?%\pause
What is the distance by which the flow streamlines are shifted away from the wall?\pause
\begin{IEEEeqnarray}{rCl}
\delta^* &\equiv& \int_0^\infty \left( 1 - \frac{u}{U}\right) \diff y
......@@ -182,9 +182,9 @@
\begin{frame}
\begin{enumerate}
\shift{2}
\item The \vocab{momentum thickness} $\theta$. %\pause
\item The \vocab{momentum thickness} $\theta$. \pause
How thick is the fluid layer that we would need to remove in order to generate the same drag as the boundary layer?%\pause
How thick is the fluid layer that we would need to remove in order to generate the same drag as the boundary layer?\pause
\begin{IEEEeqnarray}{rCl}
\theta &\equiv& \int_0^\delta \frac{u}{U} \left( 1 - \frac{u}{U}\right) \diff y
......@@ -196,29 +196,29 @@
\begin{frame}
What about the shear $\tau_\text{wall}$ ?%\pause
What about the shear $\tau_\text{wall}$ ?\pause
Simply dependent on $u_{(y)}$ :%\pause
Simply dependent on $u_{(y)}$ :\pause
\begin{IEEEeqnarray}{rCl}
\tau_\text{wall} &=& \mu \frac{\partial u}{\partial y}
\end{IEEEeqnarray}%\pause
this expression is a function of $x$ \\%\pause
\end{IEEEeqnarray}\pause
this expression is a function of $x$ \\\pause
\small and typically $\tau_\text{wall}$ decreases with distance in a laminar boundary layer.
\end{frame}
\begin{frame}
Useful parameter: the \vocab{shear coefficient},%\pause
Useful parameter: the \vocab{shear coefficient},\pause
\begin{IEEEeqnarray}{rCl}
c_{f_{(x)}} &\equiv& \frac{\tau_\text{wall}}{\frac{1}{2} \rho U^2}
\end{IEEEeqnarray}%\pause
\end{IEEEeqnarray}\pause
\small (also a function of $x$)
\end{frame}
\begin{frame}
Shear force is then obtained by integration of shear over the surface $S$ of interest:%\pause
Shear force is then obtained by integration of shear over the surface $S$ of interest:\pause
\begin{IEEEeqnarray}{rCl}
F_\text{shear} &=& \int_S \tau_\text{wall} \diff x \diff z
\end{IEEEeqnarray}%\pause
\end{IEEEeqnarray}\pause
\end{frame}
......@@ -230,7 +230,7 @@
\begin{frame}
What happens in a laminar steady boundary layer?%\pause
What happens in a laminar steady boundary layer?\pause
$\to$ To the Navier-Stokes!
\end{frame}
......@@ -238,7 +238,7 @@
\begin{frame}
What happens in a laminar steady boundary layer?%\pause
What happens in a laminar steady boundary layer?\pause
\small
\begin{IEEEeqnarray}{cCc}
......@@ -248,7 +248,7 @@
\end{frame}
\begin{frame}
Let us apply three simplifications (hypothesis based on observation):%\pause
Let us apply three simplifications (hypothesis based on observation):\pause
\begin{enumerate}
\item Gravity plays a negligible role;
......@@ -259,7 +259,7 @@
\begin{enumerate}
\shift{1}
\item The component of speed perpendicular to the wall $v$ is very small ($v \ll u$). \\
Thus, its space variations can be neglected:\\%\pause
Thus, its space variations can be neglected:\\\pause
$\partialderivative{v}{x} \approx 0$ and $\secondpartialderivative{v}{x} \approx 0$.\\
$\partialderivative{v}{y} \approx 0$ and $\secondpartialderivative{v}{y} \approx 0$.
\end{enumerate}
......@@ -280,18 +280,18 @@
\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]
\end{IEEEeqnarray*}
becomes:%\pause
becomes:\pause
\begin{IEEEeqnarray}{rCl}
\frac{\partial p}{\partial y} & \approx & 0
\end{IEEEeqnarray}%\pause
\end{IEEEeqnarray}\pause
yippee!
\end{frame}
\begin{frame}
So $\frac{\partial p}{\partial y} \approx 0$. What does that mean to us?%\pause
So $\frac{\partial p}{\partial y} \approx 0$. What does that mean to us?\pause
Pressure is a function of $x$ only\\
$\partialderivative{p}{x} = \derivative{p}{x}$
......@@ -307,14 +307,14 @@
\begin{frame}
And what does this pressure depend on?%\pause
And what does this pressure depend on?\pause
\small
\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]
\end{IEEEeqnarray*}
at the point where $u=U$, becomes:%\pause
at the point where $u=U$, becomes:\pause
\begin{IEEEeqnarray}{rCl}
\derivative{p}{x} &=& - \rho U \frac{\diff U}{\diff x}
......@@ -323,7 +323,7 @@
\begin{frame}
And now, can we find the velocity profile?%\pause
And now, can we find the velocity profile?\pause
\small
\begin{IEEEeqnarray*}{cCc}
......@@ -332,7 +332,7 @@
becomes
\begin{IEEEeqnarray}{cCc}
u \partialderivative{u}{x} + v \partialderivative{u}{y} & = & -\frac{1}{\rho} \partialderivative{p}{x} + \frac{\mu}{\rho} \secondpartialderivative{u}{y}\nonumber\\%\pause
u \partialderivative{u}{x} + v \partialderivative{u}{y} & = & -\frac{1}{\rho} \partialderivative{p}{x} + \frac{\mu}{\rho} \secondpartialderivative{u}{y}\nonumber\\\pause
&=& U \derivative{U}{x} + \frac{\mu}{\rho} \secondpartialderivative{u}{y}
\end{IEEEeqnarray}
\end{frame}
......@@ -340,14 +340,14 @@
\begin{frame}
Voilà.%\pause
Voilà.\pause
The velocity field $\vec V = (u ; v) = f(x,y)$ in the laminar boundary layer must be such that:%\pause
The velocity field $\vec V = (u ; v) = f(x,y)$ in the laminar boundary layer must be such that:\pause
\begin{IEEEeqnarray}{rCl}
u \partialderivative{u}{x} + v \partialderivative{u}{y} & = & U \frac{\diff U}{\diff x} + \frac{\mu}{\rho} \secondpartialderivative{u}{y} \label{eq_ns_bl_lam_un}\\%\pause
u \partialderivative{u}{x} + v \partialderivative{u}{y} & = & U \frac{\diff U}{\diff x} + \frac{\mu}{\rho} \secondpartialderivative{u}{y} \label{eq_ns_bl_lam_un}\\\pause
\partialderivative{u}{x} + \partialderivative{v}{y} & = & 0 \label{eq_ns_bl_lam_deux}
\end{IEEEeqnarray}%\pause
\end{IEEEeqnarray}\pause
no luck! We can’t find the analytical solution!
\end{frame}
......@@ -363,9 +363,9 @@
\end{frame}
\begin{frame}
Superb intuition and simplification:%\pause
Superb intuition and simplification:\pause
The geometry of the velocity profile in the boundary layer is \emph{always the same}.%\pause
The geometry of the velocity profile in the boundary layer is \emph{always the same}.\pause
$u$ can be simply expressed as a function of $\eta$:
\begin{IEEEeqnarray}{rCl}
......@@ -376,11 +376,11 @@
\end{frame}
\begin{frame}
TLDR:%\pause
TLDR:\pause
$u$ is a function such that $\frac{u}{U} = f'_{(\eta)}$ and $f''' + \frac{1}{2} f f'' = 0$.%\pause
$u$ is a function such that $\frac{u}{U} = f'_{(\eta)}$ and $f''' + \frac{1}{2} f f'' = 0$.\pause
Dag-nagit! there is no simple analytical solution!\\%\pause
Dag-nagit! there is no simple analytical solution!\\\pause
We find numerical values of $f'$ corresponding to values of $y$.
\end{frame}
......@@ -388,10 +388,10 @@
\begin{frame}
It can thus be shown that for a laminar boundary layer along a smooth surface, we have:
\begin{IEEEeqnarray}{rCl}%\pause
\frac{\delta}{x} &=& \frac{\num{4,91}}{\sqrt{\rex}} \IEEEyessubnumber\\\nonumber\\%\pause
\frac{\delta^*}{x} &=& \frac{\num{1,72}}{\sqrt{\rex}} \IEEEyessubnumber\\\nonumber\\%\pause
\frac{\theta}{x} &=& \frac{\num{0,664}}{\sqrt{\rex}} \IEEEyessubnumber\\\nonumber\\%\pause
\begin{IEEEeqnarray}{rCl}\pause
\frac{\delta}{x} &=& \frac{\num{4,91}}{\sqrt{\rex}} \IEEEyessubnumber\\\nonumber\\\pause
\frac{\delta^*}{x} &=& \frac{\num{1,72}}{\sqrt{\rex}} \IEEEyessubnumber\\\nonumber\\\pause
\frac{\theta}{x} &=& \frac{\num{0,664}}{\sqrt{\rex}} \IEEEyessubnumber\\\nonumber\\\pause
c_{f_{(x)}} &=& \frac{\num{0,664}}{\sqrt{\rex}} \IEEEyessubnumber
\end{IEEEeqnarray}
\end{frame}
......@@ -401,18 +401,18 @@
\begin{frame}
Pohlhausen started from the other end…
what if the solution was simple and beautiful?%\pause
what if the solution was simple and beautiful?\pause
\begin{IEEEeqnarray}{rCl}
\frac{u}{U} = g_{(Y)} &=& a Y + b Y^2 + c Y^3 + d Y^4 \nonumber\\\label{eq_idee_Pohlhausen}
\end{IEEEeqnarray}%\pause
\end{IEEEeqnarray}\pause
where $Y \equiv y/\delta$.%\pause
where $Y \equiv y/\delta$.\pause
Now: what are $a$, $b$, $c$ and $d$?
\end{frame}
\begin{frame}
Pohlhausen then simply calibrated his model on the known boundary conditions:%\pause
Pohlhausen then simply calibrated his model on the known boundary conditions:\pause
\begin{enumerate}
\item for $y=0$ (meaning $Y = 0$), we have both $u=0$ and $v=0$. Eq. (\ref{eq_ns_bl_lam_un}) becomes:
......@@ -425,7 +425,7 @@
\begin{frame}
\begin{enumerate}
\shift{1}
\item for $y=\delta$ (meaning $Y = 1$) we have $u=U$, $\partial u/\partial y = 0$ and $\partial^2 u/(\partial y)^2 = 0$.\\%\pause
\item for $y=\delta$ (meaning $Y = 1$) we have $u=U$, $\partial u/\partial y = 0$ and $\partial^2 u/(\partial y)^2 = 0$.\\\pause
Therefore we know of $g$ that:
\begin{IEEEeqnarray}{rCCCl}
u &=& U &=& U g_{(1)}\\
......@@ -436,24 +436,24 @@
\end{frame}
\begin{frame}
$\to$ a system of four equations%\pause
$\to$ a system of four equations\pause
We introduce variable $\Lambda$ :%\pause
We introduce variable $\Lambda$ :\pause
\begin{IEEEeqnarray}{rCl}
\Lambda &\equiv& \delta^2 \frac{\rho}{\mu} \frac{\diff U}{\diff x}
\end{IEEEeqnarray}%\pause
\end{IEEEeqnarray}\pause
\small a non-dimensionalized measure of the boundary layer thickness
\end{frame}
\begin{frame}
And now:%\pause
And now:\pause
\begin{IEEEeqnarray}{rCl}
\frac{u}{U} &=& a Y + b Y^2 + c Y^3 + d Y^4 \nonumber\\%\pause
\frac{u}{U} &=& a Y + b Y^2 + c Y^3 + d Y^4 \nonumber\\\pause
\frac{u}{U} &=& 1 - (1+Y)(1-Y)^3 + \Lambda \frac{Y}{6} (1 - Y^3) \nonumber\\\label{eq_modele_Pohlhausen}
\end{IEEEeqnarray}%\pause%\pause
\end{IEEEeqnarray}\pause\pause
In practice, a close match to that of Blasius (but we have the full equation!)
\end{frame}
......@@ -470,11 +470,11 @@
\figureframe{}{images/boundary_layer_transition.png}{1}{figure \ccby \olivier}
\begin{frame}
$x_\text{transition}$ reduces when $U$ is increased, or $\mu$ is reduced. We accept:%\pause
$x_\text{transition}$ reduces when $U$ is increased, or $\mu$ is reduced. We accept:\pause
\begin{IEEEeqnarray}{rCl}
\text{[Re]}_{x\ \text{transition}} &\approx& \num{5e5}
\end{IEEEeqnarray}%\pause%\pause
\end{IEEEeqnarray}\pause\pause
\small Transition generated earlier on rough surfaces, with obstacles (turbulators, trip wires etc)\\
Transition delayed on very smooth surfaces and uniform, steady incoming flows.\par
......@@ -484,7 +484,7 @@
\begin{frame}
What a challenge!%\pause
What a challenge!\pause
\begin{itemize}
\item increased mass, energy and momentum exchange;
......@@ -496,11 +496,11 @@
\figureframe{$\text{[M]} = \num{0,8}$, $\text{[Re]}_{(\delta1)}=\num{2500}$}{Dns_schlierenimage.png}{1}{\wcfile{Dns schlierenimage.png}{figure} \ccbyde Andreas Babucke}
\begin{frame}
we satisfy ourselves with describing the average speed, $\overline{u}$. Widely-accepted model:%\pause
we satisfy ourselves with describing the average speed, $\overline{u}$. Widely-accepted model:\pause
\begin{IEEEeqnarray}{rCl}
\frac{\overline{u}}{U} &\approx& \left(\frac{y}{\delta}\right)^{\frac{1}{7}}
\end{IEEEeqnarray}%\pause
\end{IEEEeqnarray}\pause
\begin{description}
\item for flow over a smooth surface.
\end{description}
......@@ -510,11 +510,11 @@
\begin{frame}
From this, it can be shown that on a smooth surface, in a turbulent layer:%\pause
From this, it can be shown that on a smooth surface, in a turbulent layer:\pause
\begin{IEEEeqnarray}{rCl}
\frac{\delta}{x} &\approx& \frac{\num{0,16}}{\rex^{\frac{1}{7}}} \IEEEyessubnumber\\\nonumber\\%\pause
\frac{\delta^*}{x} &\approx& \frac{\num{0,02}}{\rex^{\frac{1}{7}}} \IEEEyessubnumber\\\nonumber\\%\pause
\frac{\theta}{x} &\approx& \frac{\num{0,016}}{\rex^{\frac{1}{7}}} \IEEEyessubnumber\\\nonumber\\%\pause
\frac{\delta}{x} &\approx& \frac{\num{0,16}}{\rex^{\frac{1}{7}}} \IEEEyessubnumber\\\nonumber\\\pause
\frac{\delta^*}{x} &\approx& \frac{\num{0,02}}{\rex^{\frac{1}{7}}} \IEEEyessubnumber\\\nonumber\\\pause
\frac{\theta}{x} &\approx& \frac{\num{0,016}}{\rex^{\frac{1}{7}}} \IEEEyessubnumber\\\nonumber\\\pause
c_{f_{(x)}} &\approx& \frac{\num{0,027}}{\rex^{\frac{1}{7}}} \IEEEyessubnumber
\end{IEEEeqnarray}
\end{frame}
......@@ -528,21 +528,21 @@
\figureframe{}{boundary_layer_separation.png}{1}{figure \ccby \olivier}
\begin{frame}
Yikes!%\pause
Yikes!\pause
Flow separates from the wall (streamlines diverge). The boundary layer disintegrates…%\pause
Flow separates from the wall (streamlines diverge). The boundary layer disintegrates…\pause
\emph{Separation is our failure to control the velocity of the fluid.}
\end{frame}
\begin{frame}
When $U = f_{(x)}$, the \emph{geometry} of the boundary layer changes:%\pause
When $U = f_{(x)}$, the \emph{geometry} of the boundary layer changes:\pause
\begin{itemize}
\item When speed increases ($\diff U/\diff x >0$), the layer is flattened;%\pause
\item When speed decreases ($\diff U/\diff x <0$), the boundary layer straightens up.\\%\pause
\item When speed increases ($\diff U/\diff x >0$), the layer is flattened;\pause
\item When speed decreases ($\diff U/\diff x <0$), the boundary layer straightens up.\\\pause
When the speed profile becomes vertical, streamlines separate!
\end{itemize}%\pause
\end{itemize}\pause
Can we predict separation mathematically?
......@@ -552,7 +552,7 @@
\begin{frame}
We need a robust model for $u$. Let us come back to fundamental equations, stating that \textbf{at the separation point, the shear on the wall is zero}:%\pause
We need a robust model for $u$. Let us come back to fundamental equations, stating that \textbf{at the separation point, the shear on the wall is zero}:\pause
\begin{IEEEeqnarray}{rCl}
\tau_\text{wall} = 0 &=& \mu \left(\frac{\partial u}{\partial y}\right)_{y=0}