Commit 23f88977 authored by Olivier's avatar Olivier
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Exercises 6: expanded tornado problem, proof-read

The problem remains non-examinable this year, but should be
interesting enough to be part of the main problem sheet now
parent 1cc9f1d6
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\atstartofexercises
......@@ -10,12 +10,12 @@
\mecafluexboxen
\begin{boiboite}
\begin{boiboiboite}
In a highly-viscous (creeping) steady flow, the drag~$F_\D$ exerted on a spherical body of diameter~$D$ at by flow at velocity~$U_\infty$ is quantified as:
\begin{IEEEeqnarray}{rCl}
F_\text{D sphere} &=& 3 \pi \mu U_\infty D \ztag{\ref{eq_drag_creeping_sphere}}
\end{IEEEeqnarray}
\end{boiboite}
\end{boiboiboite}
%%%%%%
\subsubsection{Volcanic ash from the Eyjafjallajökull}
......@@ -28,7 +28,7 @@
\begin{enumerate}
\item What is the terminal velocity of the particle?
\item Will this terminal velocity increase or decrease as the particle progresses towards the ground? (give a brief, qualitative answer)
\item Will this terminal velocity increase or decrease as the particle progresses towards the ground? (briefly justify your answer, e.g.\ in 30 words or less)
\end{enumerate}
......@@ -138,6 +138,93 @@
\end{figure}
%%%%%%
\clearpage
\subsubsection{Flow field of a tornado}
\wherefrom{Non-examinable. From Çengel \& al. \smallcite{cengelcimbala2010} E9-5, E9-14 \& E10-3}
\label{exo_tornado}
In this problem, we attempt to model a very large-scale flow: that of a tornado (\cref{fig_tornado}). We begin by pretending the tornado is one perfectly straight, stationary structure. We divide the flow into two regions: a core cylinder that rotates almost like a solid body, and an outer region where flow spins in an irrotational matter. This model is called the \we{Rankine vortex} (displayed in \cref{fig_rankine_vortex})and is used widely as a simple, first approximation to model flows as large as a hurricane and as small as turbulence-induced vortices.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.5\textwidth]{F5_tornado_Elie_Manitoba_2007}
\end{center}\vspace{-0.5cm}%handmade
\supercaption{Photo of an approaching tornado in Manitoba, Canada}{\wcfile{RankineVortex.svg}{Figure} \ccbysa by \weu{Justin1569}}
\label{fig_tornado}
\end{figure}
\begin{figure}[h]
\begin{center}\vspace{-1cm}%handmade
\includegraphics[width=0.5\textwidth]{RankineVortex}
\end{center}\vspace{-0.5cm}%handmade
\supercaption{Modeled angular velocity in a vortex, according to the \we{Rankine vortex} model.}{\wcfile{F5 tornado Elie Manitoba 2007.jpg}{Photo} \ccbysa by \wcu{Grhu}}
\label{fig_rankine_vortex}
\end{figure}
We are first interested in the outer region of the tornado flow field. We model the flow as being steady, two-dimensional (neglecting any movement in the vertical, $z$-direction), and having a rotational velocity $v_\theta$ such that:
\begin{equation}
v_\theta = \frac{\Gamma}{2 \pi r}
\end{equation}
\begin{equationterms}
\item in which $\Gamma$ is the \vocab{circulation} (measured in \si{\per\second}) and remains constant and uniform.
\end{equationterms}
\begin{enumerate}
\item The mass balance equation for incompressible flow (eq.~\ref{eq_continuity_der_inc} p.\pageref{eq_continuity_der_inc}) is developed in cylindrical coordinates as follows:
\begin{IEEEeqnarray}{rCl}
\frac{1}{r} \partialderivative{r v_r}{r} + \frac{1}{r} \partialderivative{v_\theta}{\theta} + \partialderivative{v_z}{z} = 0
\end{IEEEeqnarray}
According to this mass balance equation, what form must the radial velocity $v_r$ take?
\end{enumerate}
Among all the possibilities for $v_r$, we choose the simplest form, so that now, we model radial velocity as:
\begin{IEEEeqnarray}{rCl}
v_r &=& 0
\end{IEEEeqnarray}
\begin{enumerate}
\shift{1}
\item The momentum balance equation for incompressible flow (eq.~\ref{eq_navierstokes} p.\pageref{eq_navierstokes}) is developed in cylindrical coordinates are as follows:
\begin{IEEEeqnarray}{lr}
\rho \left[ \partialtimederivative{v_r} + v_r \partialderivative{v_r}{r} + \frac{v_\theta}{r} \partialderivative{v_r}{\theta} - \frac{v_\theta^2}{r} + v_z \partialderivative{v_r}{z} \right] &\nonumber\\
\ \ \ \ \ = \ \rho g_r - \partialderivative{p}{r} + \mu \left[ \frac{1}{r} \partialderivative{}{r} \left(r \partialderivative{v_r}{r} \right) - \frac{v_r}{r^2} + \frac{1}{r^2} \secondpartialderivative{v_r}{\theta} -\frac{2}{r^2} \partialderivative{v_\theta}{\theta} + \secondpartialderivative{v_r}{z} \right] \nonumber \\
\\
\rho \left[ \partialtimederivative{v_\theta} + v_r \partialderivative{v_\theta}{r} + \frac{v_\theta}{r} \partialderivative{v_\theta}{\theta} + \frac{v_r v_\theta}{r} + v_z \partialderivative{v_\theta}{z} \right] &\nonumber\\
\ \ \ \ \ = \ \rho g_\theta - \frac{1}{r} \partialderivative{p}{\theta} + \mu \left[ \frac{1}{r} \partialderivative{}{r} \left(r \partialderivative{v_\theta}{r} \right) - \frac{v_\theta}{r^2} + \frac{1}{r^2} \secondpartialderivative{v_\theta}{\theta} + \frac{2}{r^2} \partialderivative{v_r}{\theta} + \secondpartialderivative{v_\theta}{z} \right] \nonumber \\
\\
\rho \left[ \partialtimederivative{v_z} + v_r \partialderivative{v_z}{r} + \frac{v_\theta}{r} \partialderivative{v_r}{\theta} + v_z \partialderivative{v_z}{z} \right] &\nonumber\\
\ \ \ \ \ = \ \rho g_z - \partialderivative{p}{z} + \mu \left[ \frac{1}{r} \partialderivative{}{r} \left(r \partialderivative{v_z}{r} \right) + \frac{1}{r^2} \secondpartialderivative{v_z}{\theta} + \secondpartialderivative{v_z}{z} \right] \nonumber \\
\end{IEEEeqnarray}
Starting from those equations, show that the pressure distribution in the outer region of the tornado can be expressed as:
\begin{IEEEeqnarray}{rCl}
p &=& p_\infty -\frac{1}{2} \rho \Gamma^2 \frac{1}{r^2}
\end{IEEEeqnarray}
\begin{equationterms}
\item where $p_\infty$ is the atmospheric pressure far away from the tornado.
\end{equationterms}
\end{enumerate}
We now turn to the core of the tornado, which we model as if it were a rotating solid (a \vocab{vortex core}).
\begin{enumerate}
\shift{2}
\item What is the radial velocity $v_\theta$ distribution?
\item What is the pressure field within the rotational core of the tornado?\\
(hint: you may start directly from an energy balance equation, eq.~\ref{eq_sfee} p.\pageref{eq_sfee}, without having to use the Navier-Stokes equations above).
\item Make a simple, qualitative sketch (i.e. without numerical data) of the pressure as a function of radius throughout the entire tornado flow field.
\end{enumerate}
It is finally time to calibrate and exploit our model. We estimate the tornado diameter to be \SI{50}{\metre} and the maximum wind velocity at \SI{180}{\kilo\metre\per\hour}.
\begin{enumerate}
\shift{5}
\item According to the model, what is the lowest pressure attained by the air?
\item According to the model, at what distance from the core are winds lower than \SI{50}{\kilo\metre\per\hour}?
\end{enumerate}
(curious students may play with the above model by adding a non-zero radial velocity, and look up the phenomenon of \vocab{vortex stretching})
%%%%%%
\subsubsection{Lift on a symmetrical object}
\wherefrom{non-examinable}
......@@ -181,69 +268,10 @@
On the bottom surface, the speed is measured as being constant ($u=U$) to within experimental error.
What is the lift coefficient on the airfoil?
%%%%%%
\clearpage
\subsubsection{Flow field of a tornado}
\wherefrom{From Çengel \& al. \smallcite{cengelcimbala2010} E9-5, E9-14 \& E10-3}
\label{exo_tornado}
What is the lift coefficient of the airfoil?
The continuity and Navier-Stokes equations for incompressible flow written in cylindrical coordinates are as follows:
\begin{IEEEeqnarray}{lr}
\rho \left[ \partialtimederivative{v_r} + v_r \partialderivative{v_r}{r} + \frac{v_\theta}{r} \partialderivative{v_r}{\theta} - \frac{v_\theta^2}{r} + v_z \partialderivative{v_r}{z} \right] &\nonumber\\
\ \ \ \ \ = \ \rho g_r - \partialderivative{p}{r} + \mu \left[ \frac{1}{r} \partialderivative{}{r} \left(r \partialderivative{v_r}{r} \right) - \frac{v_r}{r^2} + \frac{1}{r^2} \secondpartialderivative{v_r}{\theta} -\frac{2}{r^2} \partialderivative{v_\theta}{\theta} + \secondpartialderivative{v_r}{z} \right] \nonumber \\
\\
\rho \left[ \partialtimederivative{v_\theta} + v_r \partialderivative{v_\theta}{r} + \frac{v_\theta}{r} \partialderivative{v_\theta}{\theta} + \frac{v_r v_\theta}{r} + v_z \partialderivative{v_\theta}{z} \right] &\nonumber\\
\ \ \ \ \ = \ \rho g_\theta - \frac{1}{r} \partialderivative{p}{\theta} + \mu \left[ \frac{1}{r} \partialderivative{}{r} \left(r \partialderivative{v_\theta}{r} \right) - \frac{v_\theta}{r^2} + \frac{1}{r^2} \secondpartialderivative{v_\theta}{\theta} + \frac{2}{r^2} \partialderivative{v_r}{\theta} + \secondpartialderivative{v_\theta}{z} \right] \nonumber \\
\\
\rho \left[ \partialtimederivative{v_z} + v_r \partialderivative{v_z}{r} + \frac{v_\theta}{r} \partialderivative{v_r}{\theta} + v_z \partialderivative{v_z}{z} \right] &\nonumber\\
\ \ \ \ \ = \ \rho g_z - \partialderivative{p}{z} + \mu \left[ \frac{1}{r} \partialderivative{}{r} \left(r \partialderivative{v_z}{r} \right) + \frac{1}{r^2} \secondpartialderivative{v_z}{\theta} + \secondpartialderivative{v_z}{z} \right] \nonumber \\
\\
\frac{1}{r} \partialderivative{r v_r}{r} + \frac{1}{r} \partialderivative{v_\theta}{\theta} + \partialderivative{v_z}{z} = 0
\end{IEEEeqnarray}
In this exercise, we are interested in solving the pressure field in a simplified description of a tornado. For this, we consider only a horizontal layer of the flow, and we consider that all properties are independent of the altitude $z$ and of the time $t$.% and of the angular position $\theta$.
We start by modeling the tornado as a vortex imparting an angular velocity such that:
\begin{equation}
v_\theta = \frac{\Gamma}{2 \pi r}
\end{equation}
\begin{equationterms}
\item in which $\Gamma$ is the \vocab{circulation} (measured in \si{\per\second}) and remains constant everywhere.
\end{equationterms}
\begin{enumerate}
\item What form must the radial velocity $v_r$ take in order to satisfy continuity?
\end{enumerate}
Among all the possibilities for $v_r$, we choose the simplest in our study, so that:
\begin{equation}
v_r = 0
\end{equation}
\begin{equationterms}
\item all throughout the tornado flow field.
\end{equationterms}
\begin{enumerate}
\shift{1}
\item What is the pressure field throughout the horizontal layer of the tornado?
\end{enumerate}
The flow field described above becomes unphysical in the very center of the tornado vortex. Indeed, our model for $v_\theta$ is typical of an \vocab{irrotational vortex}, which, like all irrotational flows, is constructed under the premise that the flow is inviscid. However, in the center of the vortex, we are confronted with high velocity gradients over very small distances: viscous effects can no longer be neglected and our model breaks down.
It is observed that in most such vortices, a \vocab{vortex core} region forms that rotates just like a solid cylindrical body. This flow is \vocab{rotational} and its governing equations result in a realistic pressure distribution.
\begin{enumerate}
\shift{2}
\item What is the pressure field within the rotational core of the tornado?\\
Make a simple sketch showing the pressure distribution as a function of radius throughout the entire tornado flow field.
\item \textit{[An opening question for curious students]}\\
What is the basic mechanism of \vocab{vortex stretching}? How would it modify the flow field described here?
\end{enumerate}
%%%%%%
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