The difference between eqs.~\ref{eq_rans} and~\ref{eq_ns_cartone} can perhaps be expressed differently: the time-average of a turbulent 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}. That is, a local value for $\rho\ \inlinepartialderivative{\overline{u_i' u_j'}}{j}$ is estimated everywhere, depending on the average values ($\overline{u_i}$). The most well-known method for doing this is the \mbox{\textit{$k$-epsilon}} turbulence model, which involves solving partly arbitrary transport equations for both $k$ and $\epsilon$. The delights, shortcomings and mysteries of that method and more are left for the reader to discover in a good hands-on course on the youngest and most promising area of this discipline, \cfd.

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\section{Commented bibliography}

Most books on turbulence spill into mathematical intricacies which are irrelevant to the engineer; it is regrettable that some of the very best documentation on turbulence is in the user manuals of \cfd software and experimental measurement devices. It is also regrettable that few books provide actual applied \emph{problems} to be solved quantitatively. A commented (and necessarily subjective) bibliography for learning about turbulence is proposed here:

\begin{itemize}

\item General understanding (encyclopedic knowledge)\\

The introductions of Mathieu \& Scott~\cite{mathieuscott2000}, of Tennekes \& Lumley~\cite{tennekesetal1972}, and of Davidson~\cite{davidson2015} will provide excellent information.

\item Reference works\\

Unfortunately, no book truly aimed at engineers is known to the author. The following books provide in-depth insight over the physics of turbulence:

\begin{itemize}

\item Tennekes \& Lumley~\cite{tennekesetal1972}: despite its age, an outstanding book, in particular for its first chapter.

\item Leschziner~\cite{leschziner2015}: despite its focus on \cfd, the book serves as a great step-by-step exploration of turbulence.

\item Mathieu \& Scott~\cite{mathieuscott2000} (reference book). An appropriate amount of comment is provided around mathematical expressions.

\item Pope~\cite{pope2000} (reference book). Terser than its counterparts above.

\end{itemize}

\item Works containing useful problems\\

De Nevers~\cite{denevers2004} (a single chapter within) is a very applied, down-to-earth and useful treatment; Tennekes \& Lumley~\cite{tennekesetal1972} also contains good examples in the first chapter.

\item Dealing with turbulence in \cfd\\

The best coverage is probably in Leschziner~\cite{leschziner2015}. Some useful (passing) information can be found in Tu et al.~\cite{tuetal2018computational} and Versteeg et al.~\cite{versteegetal2007}.

\item Books with parts useful for specific purposes include:

\begin{itemize}

\item Bernard \& Wallace~\cite{bernard2002turbulent}, for theory about experiments;

\item Libby~\cite{libby1996introduction}, for its introduction to applied statistics;

\item Sagaut~\cite{sagaut2006les}, for its coverage of the notion of spectra, and its excellent illustrations;

\item Cebeci~\cite{cebeci2004analysisturbulentflows}, for its rigorous introduction into basic metrics, and excellent fundamental data and diagrams.