Commit c1c5dfc8 authored by Olivier's avatar Olivier

Ch.8 (turbulence): fit draft content into mold

parent cac61be3
\documentclass[12pt,a4paper]{article}
\usepackage{magdeburg1} % from https://git.framasoft.org/u/olivier/sensible-styles
\renewcommand{\authorofthisdocument}{Olivier Cleynen}
\renewcommand{\titleofthisdocument}{(Draft) Dealing with turbulence}
\renewcommand{\keywordsofthisdocument}{}
\usepackage{magdeburg2}
\renewcommand{\lastedityear}{2019}
\renewcommand{\lasteditmonth}{03}
\renewcommand{\lasteditday}{18}
\renewcommand{\numberofthischapter}{8}
\renewcommand{\titleofthischapter}{Dealing with turbulence}
\newgeometry{inner=3cm, outer=3cm}
\fluidmechchaptertitle
\begin{document}
\mecafluboxen
\mecafluboxtmp
\begin{center}
~\vspace{2cm}\par
{\LARGE \titleofthisdocument\par}
\vspace{1em}
{\footnotesize Last modified \today\\
Olivier Cleynen -- olivier.cleynen@ovgu.de\\}
\vspace{2em}
\end{center}
\vspace{3cm}
\section{Introduction}
\section{Motivation}
Most flows of interest to engineers and scientists are turbulent. Fluid flow in industrial and domestic piping, in engines and in turbomachinery, is turbulent. Flow close to solid surfaces and in the wake of objects is turbulent at all but the slowest speeds. Blood flow in our largest veins and arteries, and air flow in our nostrils and tracheae, are turbulent. River flows, ocean currents, and all but the calmest winds are turbulent.
......@@ -146,72 +134,4 @@
The difference between eqs.~\ref{eq_rans} and~\ref{eq_ns_cartone} can perhaps be expressed differently: the time-average of a real flow cannot be calculated by solving for the time-average velocities. Or, more bluntly: \emph{the average of the solution cannot be obtained with only the average of the flow}. This is a tremendous burden in computational fluid dynamics, where limits on the available computational power prevent us in practice from solving for these fluctuations. In the overwhelming majority of computations, the Reynolds stress has to be approximated in bulk with schemes named \vocab{turbulence models}.
\section{Exercises}
\subsection{Hypothetical flow}
We imagine a turbulent flow described at some point with the equations (in \si{\metre\per\second})
\begin{IEEEeqnarray}{rCl}
V_x &=& 10 + \sin t\\
V_y &=& 0\\
V_z &=& 0
\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 $\overline{V_x}$, $v_x$, $\overline{v_x}$, $T_x$, $T$, and $k$?
\subsection{Turbulent channel flow}
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 XX.
\begin{enumerate}
\shift{2}
\item What is the value of $k$ at a point \SI{2}{\centi\metre} from the wall?
\end{enumerate}
A computational fluid dynamics 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 $\epsilon = \SI{0,025}{\metre\squared\per\second\cubed}$.
\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?
\end{enumerate}
\subsection{Cumulus cloud}
A cumulus cloud (one of those “fluffy” summer clouds) 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{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}
\subsection{Reactor tank}
A tank used to store chemical reactants has roughly the size of a cube of side length $L = \SI{2}{\metre}$. The tank is filled 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{enumerate}
\item What is approximately the size of the smallest eddies in the tank?
\item What is approximately the specific dissipation power?
\end{enumerate}
The agitator propeller is stopped and removed from the tank. The fluid motion is left to itself and the turbulence slowly decays. The turbulence subsists until the Reynolds number of the largest vortices reaches a value of approximately \num{10}.
\begin{enumerate}
\shift{2}
\item At what rate does the turbulence intensity initially approximately decay?
\item How much time approximately is required before the turbulence has died down?
\end{enumerate}
\end{document}
\atendofchapternotes
\renewcommand{\lastedityear}{2018}
\renewcommand{\lasteditmonth}{03}
\renewcommand{\lasteditday}{18}
\renewcommand{\numberofthischapter}{8}
\renewcommand{\titleofthischapter}{Dealing with turbulence}
\atstartofexercises
\fluidmechexercisestitle
\mecafluboxen
\mecafluboxtmp
\mecafluexboxen
\begin{boiboite}
todo %TODO
\end{boiboite}
\subsubsection{Hypothetical flow}
We imagine a turbulent flow described at some point with the equations (in \si{\metre\per\second})
\begin{IEEEeqnarray}{rCl}
V_x &=& 10 + \sin t\\
V_y &=& 0\\
V_z &=& 0
\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 $\overline{V_x}$, $v_x$, $\overline{v_x}$, $T_x$, $T$, and $k$?
\subsubsection{Turbulent channel flow}
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 XX.
\begin{enumerate}
\shift{2}
\item What is the value of $k$ at a point \SI{2}{\centi\metre} from the wall?
\end{enumerate}
A computational fluid dynamics 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 $\epsilon = \SI{0,025}{\metre\squared\per\second\cubed}$.
\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?
\end{enumerate}
\subsubsection{Cumulus cloud}
A cumulus cloud (one of those “fluffy” summer clouds) 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{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{Reactor tank}
A tank used to store chemical reactants has roughly the size of a cube of side length $L = \SI{2}{\metre}$. The tank is filled 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{enumerate}
\item What is approximately the size of the smallest eddies in the tank?
\item What is approximately the specific dissipation power?
\end{enumerate}
The agitator propeller is stopped and removed from the tank. The fluid motion is left to itself and the turbulence slowly decays. The turbulence subsists until the Reynolds number of the largest vortices reaches a value of approximately \num{10}.
\begin{enumerate}
\shift{2}
\item At what rate does the turbulence intensity initially approximately decay?
\item How much time approximately is required before the turbulence has died down?
\end{enumerate}
\atendofexercises
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