Commit 339174db authored by Olivier's avatar Olivier
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Chapter 8: added illustrations

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\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.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=\textwidth]{Argentina.TMOA2003041_lrg.jpg}
\end{center}
\supercaption{Plankton blooming in the Atlantic ocean reveals the complexity of the flow passing over the coast of Argentina. The scale of the image is so that the height covers approximately \SI{500}{\kilo\metre} in this take.}{\attlink{https://frama.link/southatlanticphytoplankton}{Image} by Jacques Descloitres, MODIS Rapid Response Team NASA GSFC (\pd)}
\label{fig_plankton_argentina}
\end{figure}
Turbulence may be ubiquitous, but it remains an incredibly complex phenomenon, and describing it accurately requires either extraordinarily powerful numerical computations, or advanced mathematics. Neither of those is available in this course.
......@@ -34,9 +41,18 @@
\begin{IEEEeqnarray}{rCl}
\re &\equiv& \frac{\rho V L}{\mu}
\end{IEEEeqnarray}
Typically, when $\re$ exceeds \num{1000}, the flow is very likely to be or become turbulent. We will see in \chapternine that this is because the magnitude of the Laplacian of the velocity field is \num{1000} times smaller than the magnitude of its advective. (Dampening factors other than viscosity sometimes also exist, such as density gradients or interaction with soft solid surfaces: in those cases, other non-dimensional parameters are used).
Typically, when $\re$ exceeds \num{1000}, the flow is very likely to be or become turbulent (figure~\ref{fig_rising_smoke}). We will see in \chapternine that this is because the magnitude of the Laplacian of the velocity field is \num{1000} times smaller than the magnitude of its advective. (Dampening factors other than viscosity sometimes also exist, such as density gradients or interaction with soft solid surfaces: in those cases, other non-dimensional parameters are used).
This instability is what gives turbulence its chaotic characteristic. If a rider-less bicycle is rolled forward and left to itself, it will continue rolling and eventually fall to the side. Which side, left or right, depends on the initial conditions: even a minute modification to the start position is likely to influence the result. The fall is deterministic, and can be calculated very precisely, but with a very strong dependence on the initial conditions.\\
\vimeothumb{84518319}{simulation of two miscible fluids of different densities layered one on top of the other (color representing density). The “perfect” uniform initial situation is unstable and leads to chaotic (hard to predict) patterns whose details will depend strongly on minute changes in the initial conditions. (A \textsc{2d} \dns simulation performed with MicroHH)}{Chiel van Heerwaarden (\ccby)}
This instability is what gives turbulence its chaotic characteristic. If a rider-less bicycle is rolled forward and left to itself, it will continue rolling and eventually fall to the side. Which side, left or right, depends on the initial conditions: even a minute modification to the start position is likely to influence the result. The fall is deterministic, and can be calculated very precisely, but with a very strong dependence on the initial conditions.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.5\textwidth]{Smoke_Series_(4130758692).jpg}
\end{center}
\supercaption{A column of hot air from burning incense rising through cold air is an unstable situation. After a certain length, the flow breaks down into chaotic patterns. The occurrence is predicted across all fluids, plume diameters, and velocities: every time, the Reynolds number is the determining parameter.}{\wcfile{Smoke_Series_(4130758692).jpg}{Photo} \ccby by \attlink{https://500px.com/rafaespada}{Rafa Espada}}
\label{fig_rising_smoke}
\end{figure}
Turbulent motion in fluids has the same properties. Fluid motion follows laws which are fully deterministic (the Navier-Stokes equation, eq.~\ref{eq_navierstokes}), but the exact patterns in situations where the Reynolds number is high cannot be predicted because, much like the for the bicycle above, they depend very minutely on the initial configuration.
Thus, in two identical turbulent flow experiments, the details of the flow will be different. In this sense, turbulent flow is \vocab{chaotic} (depending extremely sensitively on initial conditions) but not \vocab{random}: it remains predictable, governed by well-known deterministic laws in which chance does not play a role. The effective engineer will determine 1) what general characteristics of turbulence \emph{do} remain identical in both flows, and 2) how they affect the main, global flow characteristics.
......@@ -63,9 +79,17 @@
\subsection{Not turbulence}
Not all unsteady flows are turbulent. Well-known patterns such as a \wed{von Karman vortex street}{von Kármán vortex street} or a series of \wed{Wave cloud}{wave clouds}, for example, are not turbulent. Those oscillations occur at a single recognizable frequency and scale, and once the phenomena has begun, their evolution is easily predictable.
Not all unsteady flows are turbulent. Well-known patterns such as a \wed{von Karman vortex street}{von Kármán vortex street} (figure~\ref{fig_not_turbulence}) or a series of \wed{Wave cloud}{wave clouds}, for example, are not turbulent. Those oscillations occur at a single recognizable frequency and scale, and once the phenomena has begun, their evolution is easily predictable.
Most surface waves on a body of water (e.g.\ waves in open sea) are not turbulence: although they may be partly chaotic, they dissipate very little energy (unless they crash on a shore) and propagate over very large distances.
Most surface waves on a body of water (e.g.\ waves in open sea, figure~\ref{fig_not_turbulence}) are not turbulence: although they may be partly chaotic, they dissipate very little energy (unless they crash on a shore) and propagate over very large distances.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.9\textwidth]{Clouds_over_the_Atlantic_Ocean_cropped} \vspace{0.05\textwidth}
\includegraphics[width=0.9\textwidth]{Karmansche_Wirbelstr_kleine_Re}
\end{center}
\supercaption{Not turbulence: surface waves on the ocean (top) are complex, but not fully-chaotic, and they feature very little dissipation. Well-known oscillatory patterns such as the \wed{von Karman vortex street}{von Kármán vortex street} feature one dominant frequency and one dominant vortex size: they are not turbulent either.}{\wcfile{Clouds_over_the_Atlantic_Ocean.jpg}{Sea wave photo} \ccbysa by \wcun{Tfioreze}{Tiago Fioreze} (cropped)\\{\wcfile{Karmansche_Wirbelstr_kleine_Re.JPG}{Wake photo} \ccbysa by Jürgen Wagner}}
\label{fig_not_turbulence}
\end{figure}
Molecular motion is, at the macroscopic scale, completely random, and will appear in measurements as Gaussian white noise with no distinguishable range of frequencies, and no dissipation phenomenon: it is not turbulence, either.
......@@ -153,6 +177,13 @@
\begin{IEEEeqnarray}{rCcCl}
\re_\eta &\equiv& \frac{\rho u_\eta \eta}{\mu} &\approx& 1
\end{IEEEeqnarray}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=\textwidth]{PIA22256_North Atlantic_Mar1}
\end{center}
\supercaption{Surface-relative vorticity in the Atlantic ocean. Blue color indicates clockwise rotation, and red color anticlockwise rotation. Rotating structures of many different sizes can be observed. In homogeneous isentropic turbulence (and in this flow case by approximation), the size of the largest and smallest vortices are related to one another through the Reynolds number.}{\doilink{10.1038/s41467-018-02983-w} \ccby by Z.\ Su, J.\ Wang, P.\ Klein, A.\ F.\ Thompson \& D.\ Menemenlis \cite{suetal2018ocean}}
\label{fig_vorticity_atlantic}
\end{figure}
Based on this postulate, when the turbulence has been given time and space enough to develop fully, is homogeneous, and isotropic (has identical properties in all three directions) —these are important restrictions—, Kolmogorov and his peers showed using dimensional analysis that
\begin{mdframed}
......@@ -242,6 +273,7 @@
We have seen in \S\ref{ch_cfd} that in principle, the flow of fluids can be computed for any given flow by solving for the change in time of the unknowns $u$, $v$ and $w$ in the Navier-Stokes equation. Such a formulation is called a \vocab{Direct Numerical Simulation} (\textsc{dns}); it allows solving for \emph{all} flows and will very well describe turbulent flows.
\youtubethumb{aR-hehP1pTk}{\dns simulation of air flow over an airfoil at relatively low speeds ($\re = \num{4e5}$, so $V \approx \SI{50}{\kilo\metre\per\hour}$). Because the complete details of the flow are solved, \num{35} million \textsc{cpu}-hours were needed for this calculation in 2015. On an ordinary desktop computer, this would take 500 years to complete.}{Linné FLOW Centre (\styl)}
Unfortunately, the reality is that the computational cost of \textsc{dns} is enormous, and precludes us from solving most flows of interest. Turbulence exacerbates the problem. With eqs.~\ref{eq_kolmogorov_size} \& \ref{eq_kolmogorov_time} we can see that as the Reynolds number increases, the spatial and temporal discretization of the computation must increase, too. Every decrease in the size of the grid cell and in the length of the time step increases the total number of equations to be solved by the algorithm. Halving each of $\delta x$, $\delta y$, $\delta z$ and $\delta t$ multiplies the total number of equations by \num{16}, so that soon enough the designer of the simulation will wish to know what maximum (coarsest) grid size is appropriate or tolerable. Furthermore, in many practical cases, we may not even be interested in an exhaustive description of the velocity field, and just wish to obtain a general, coarse description of the fluid flow.
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@article{suetal2018ocean,
title={Ocean submesoscales as a key component of the global heat budget},
author={Su, Zhan and Wang, Jinbo and Klein, Patrice and Thompson, Andrew F and Menemenlis, Dimitris},
journal={Nature communications},
volume={9},
number={1},
pages={775},
year={2018},
doi={10.1038/s41467-018-02983-w},
}
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