Commit 300b9849 authored by Olivier's avatar Olivier

Changed order of chapters 8 and 9

parent e79c01c2
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......@@ -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.
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......@@ -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}
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......@@ -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}
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......@@ -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;
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......@@ -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.
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......@@ -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}
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......@@ -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}
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......@@ -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}
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......@@ -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}
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