Commit 65a7c9ea by 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

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 ... ... @@ -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}