Commit 2a1956fc authored by Olivier's avatar Olivier

Chapter 8: minor code cleanup/formatting

parent ae315a1b
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\mecafluboxen
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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.
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In answering those, we will keep in mind that at our level, turbulence characterization is not an exact science: we expect that our models may be off by a factor of 10 or so. An imprecise or inexact understanding is better than none at all, and, at the very least, we are developing some familiarity with turbulence that will be useful in further studies, especially when exploring computational fluid dynamics or experimental fluid dynamics.
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\section{Recognizing turbulence}
\coveredin{Tennekes \& Lumley \cite{tennekesetal1972}}
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\subsection{A brief definition}
We define \vocab{turbulence} as fluid motion in multiple scales which is chaotic and dissipative.
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\subsection{Chaos, not randomness}
\coveredin{Leschziner \cite{leschziner2015}}
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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.
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\subsection{Growth and decay}
Turbulence occurs because of shear (sometimes also pressure) applied non-uniformly in a fluid. Typical sources are a sudden turn at the downstream side of an obstacle, and shear alongside solid walls.
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In this sense, turbulence is not magic: it occurs in circumstances that are well understood and documented. Turbulence takes energy out of the main flow (this is usually measurable as a streamwise pressure drop) and ultimately transforms it into heat.
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\subsection{A cascade of vortices}
\coveredin{Davidson \cite{davidson2015}}
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A turbulent flow thus consists of a \vocab{cascade} of structures of decreasing size. Energy is always passed down from larger into smaller structures. Smaller vortices have lower diameter and feature lower velocities: they are then more strongly affected by viscosity. Ultimately, the smallest vortices disappear, their energy dissipated down into heat.
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\subsection{Not turbulence}
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.
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Although more formal definitions of turbulence exist (see e.g. \cite{pope2000,mathieuscott2000,davidson2015}), we are satisfied with focusing on just three requisites: a flow is turbulent if the motion contains many scales, is chaotic, and is dissipative.
\clearpage
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\section{The effects of turbulence}
\coveredin{Tennekes \& Lumley \cite{tennekesetal1972}}
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\subsection{Dissipation (losses)}
Turbulence extracts energy from the main flow and dissipates it at a molecular level. This energy is never recovered. Turbulence therefore causes additional losses in the flow, that would not be present if it were to remain laminar. Typically, friction losses are increased by a factor 2 to 10.
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In some special cases, turbulence leads to a reduction in losses. We shall study one such occurrence in \chapterten, where we sometimes intentionally create \vocab{turbulent boundary layers}. This is not important at the moment.
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\subsection{Main flow patterns}
In any given flow, the laminar and turbulent regimes result in markedly different flow patterns. Turbulent flows always lead to wider shear zones and very unsteady patterns, which, when averaged over time, have more uniform velocity profiles.
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When the flow is clearly turbulent, increases in the Reynolds number do not lead to main flow pattern changes anymore. For example, jet plumes at $\red = \num{e5}$ and \num{e7} have the nearly identical spread, length, and time-averaged velocity distribution. Some properties of turbulence still change with $\re$, as we will see below.
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\subsection{Mixing}
Turbulence tremendously increases mixing. The large range of the scales of motion (i.e.\ the many different sizes of vortices which occur simultaneously) increases the contact surface between two mixing fluids, for example. This makes turbulence a desired property in many chemical reactors, or in cases where pollutants have to be dissipated (e.g.\ for exhaust gases).
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The increased mixing also affects exchange of momentum, with increase in the interaction between slower and faster fluid particles. This tends to widen areas of interaction between fluids of different velocities, such as plumes, exhaust stacks and shear layers.
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\section{Quantifying turbulence}
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\subsection{Average and fluctuation}
For the purpose of quantifying turbulence, we distinguish, in a given flow, between the average velocity and the “turbulent part” of velocity, as illustrated in figure~\ref{fig_instantaneous_average}. We thus decompose the velocity field into two components: one is the \vocab{average} flow $\left(\overline u, \overline v, \overline w\right)$, and the other the \vocab{instantaneous fluctuation} flow $\left(u', v', w'\right)$:
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\label{fig_instantaneous_average}
\end{figure}
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\subsection{Turbulence intensity}
\coveredin{De Nevers \cite{denevers2004}}
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Typically, a low-turbulence channel flow would feature $I = \SI{3}{\percent}$, while a highly-turbulent channel flow could display for example $I = \SI{80}{\percent}$.
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\subsection{The size of eddies}
\coveredin{Leschziner \cite{leschziner2015}}
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\subsection{Turbulent kinetic energy and dissipation rate}
\coveredin{De Nevers \cite{denevers2004}}
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\subsection{Turbulence anisotropy and inhomogeneity}
The relationships between largest and smallest scales derived by Kolmogorov only hold true for very “straightforward” turbulence. In most flows of interest, turbulence turbulence is being created and dissipated in more complex ways. In such cases, which are outside the scope of this course, important properties to quantify are:
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In most of those cases, the basic models studied above serve as reference cases, against which comparisons can be made.
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\section{Computing turbulent flow}
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\subsection{Basic premise}
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.
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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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\subsection{Accounting for turbulence}
\coveredin{Versteeg et al.\ \cite{versteegetal2007}, Leschziner \cite{leschziner2015}, and Wilkes \cite{wilkes2006}}
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\pictureframe{1}{Kea_0334_(24833991).jpeg}{Transition to turbulence}{\wcfile{Kea_0334_(24833991).jpeg}{Photo} \ccby by \flickrcodename{49503154413@N01}{THOR}}
\pictureframe{1}{Smoke_Series_(4130758692).jpg}{Transition to turbulence}{\wcfile{Smoke_Series_(4130758692).jpg}{Photo} \ccby by \attlink{https://500px.com/rafaespada}{Rafa Espada} }
\pictureframe{1}{Smoke_Series_(4130758692).jpg}{Transition to turbulence}{\wcfile{Smoke_Series_(4130758692).jpg}{Photo} \ccby by \attlink{https://500px.com/rafaespada}{Rafa Espada}}
\begin{frame}
\begin{IEEEeqnarray}{rCl}
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