%~Mouliné par MaN_auto v.0.34.0 (b0f8651d) 2025-07-16 12:27:36 \documentclass[CRMECA,Unicode,biblatex,published]{cedram} \addbibresource{CRMECA_Villermaux_20250501.bib} \newcommand*{\dd}{\mathrm{d}} \newcommand*{\Reynolds}{\mathrm{Re}} \DeclareMathOperator{\erf}{erf} %%%--------------------------------------------------------------------------- %%% DELIMITERS \usepackage{mathtools} \DeclarePairedDelimiter{\parens}{\lparen}{\rparen} \DeclarePairedDelimiter{\braces}{\{}{\}} \DeclarePairedDelimiter{\abs}{\lvert}{\rvert} \DeclarePairedDelimiter{\bracks}{[}{]} %%%--------------------------------------------------------------------------- %%% SYMBOLS %%% \lesssim \newcommand{\lesssim}{\mathrel{\raisebox{-2.5pt}{$\stackrel{\raisebox{-1pt}{\scalebox{1}[.8]{<}}}{\sim}$}}} %%% \gtrsim \newcommand{\gtrsim}{\mathrel{\raisebox{-2.5pt}{$\stackrel{\raisebox{-1pt}{\scalebox{1}[.8]{>}}}{\sim}$}}} %%%--------------------------------------------------------------------------- %%% VERTICAL BAR %%% ''Evaluated at'' macro \newcommand\restr[2]{{ \left.\kern-\nulldelimiterspace #1 \vphantom{\big\vert} \right\vert_{#2} }} %%%--------------------------------------------------------------------------- \graphicspath{{./figures/}} \newcommand*{\mk}{\mkern -1mu} \newcommand*{\Mk}{\mkern -2mu} \newcommand*{\mK}{\mkern 1mu} \newcommand*{\MK}{\mkern 2mu} \hypersetup{urlcolor=purple, linkcolor=blue, citecolor=red} \newcommand*{\relabel}{\renewcommand{\labelenumi}{(\theenumi)}} \newcommand*{\romanenumi}{\renewcommand*{\theenumi}{\roman{enumi}}\relabel} \newcommand*{\Romanenumi}{\renewcommand*{\theenumi}{\Roman{enumi}}\relabel} \newcommand*{\alphenumi}{\renewcommand*{\theenumi}{\alph{enumi}}\relabel} \newcommand*{\Alphenumi}{\renewcommand*{\theenumi}{\Alph{enumi}}\relabel} \let\oldtilde\tilde \renewcommand*{\tilde}[1]{\mathchoice{\widetilde{#1}}{\widetilde{#1}}{\oldtilde{#1}}{\oldtilde{#1}}} \title{The nonlocal nature of the Reynolds stress} \alttitle{La nature non locale des contraintes de Reynolds} \author{\firstname{Emmanuel} \lastname{Villermaux}} \address{Aix Marseille Universit\'e, CNRS, Centrale Marseille, IRPHE, Marseille, France} \email{emmanuel.villermaux@univ-amu.fr} \thanks{This project has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement no.~956457} \CDRGrant[European Union's Horizon 2020 research and innovation program]{956457} \begin{abstract} A nonlocal expression for the turbulent Reynolds stress $\overline{u'v'}$ solves in a unified manner for the velocity profiles in jets, shear layers, wakes, and boundary layers, offering predictions consistent with known measurements in these canonical flows. The framework is the random walk inspired closure of Prandtl (1925) since we write \[ \overline{u'v'}=-\tilde{v}\ell\partial_y u \] with $\ell$ a mean free path but, by contrast with Prandtl, with $\tilde{v}$ a \emph{nonlocal} transfer velocity resulting from an integration over an appropriate portion of space of the mean velocity profile $u(y)$. The status of $\ell$ is discussed and its value, a fraction $(10\pi)^{-1}\approx0.031$ of the integral scale, is computed in opened shear flows. The closure is adapted to the special case of boundary layers, where the value of the von K\'arm\'an constant is found to be $\kappa=6^{-1/2}\approx 0.41$. \end{abstract} \begin{altabstract} Une expression non locale de la contrainte de Reynolds turbulente $\overline{u'v'}$ permet de résoudre de manière unifiée les profils de vitesse turbulents dans les jets, les couches de cisaillement, les sillages et les couches limites. Les prédictions sont cohérentes avec les mesures connues dans ces écoulements canoniques. Le cadre du raisonnement est inspiré de la fermeture de marche aléatoire de Prandtl (1925) puisque nous écrivons \[ \overline{u'v'}=-\tilde{v}\ell\partial_y u \] avec $\ell$ un libre parcours moyen mais, contrairement à Prandtl, avec $\tilde{v}$ une vitesse de transfert \emph{non locale} résultant d'une intégration sur une portion appropriée de l'espace du profil de vitesse moyen $u(y)$. Le statut de $\ell$ est discuté et sa valeur, une fraction $(10\pi)^{-1}\approx0,{\mkern -1mu}031$ de l'échelle intégrale, est calculée dans les écoulements cisaillés ouverts. La fermeture est adaptée au cas particulier des couches limites, où la valeur de la constante de von K\'arm\'an est prédite comme valant $\kappa=6^{-1/2}\approx 0,{\mkern -1mu}41$. \end{altabstract} \keywords{\kwd{Turbulence} \kwd{Reynolds stress} \kwd{closures}} \altkeywords{\kwd{Turbulence} \kwd{contrainte de Reynolds} \kwd{fermetures}} \dateposted{2025-07-18} \begin{document} \maketitle \section{Introduction: closures for the Reynolds stress}\label{Closure} One possible way to represent a turbulent velocity field $U_i=u_i+u_i'$ is to split it into its mean $u_i$ and fluctuating parts $u'_i$, a method inaugurated by Reynolds~\cite{Reynolds_95}. The interest of this approach is that in large Reynolds number (viscous term discarded), pressure gradient free flows (unconstrained opened flows like jets, wakes, etc.), the Navier--Stokes equations \begin{equation} \partial_tU_i+U_j\partial_jU_i=\partial_j\parens[\big]{-(p/\rho)\delta_{ij}+\nu\bigl[\partial_jU_i+\partial_iU_j\bigr]}\quad \text{with} \ \ \partial_iU_i=0, \end{equation} when time-averaged as $\overline{(\,\cdot\,)}=\lim_{T \rightarrow +\infty}\frac{1}{T}\int_0^T(\,\cdot\,)\, \dd t$, reduce to a form \begin{equation} u_j\partial_ju_i=-\partial_j\overline{u_i'u_j'} \end{equation} interestingly singling-out the correlation of the fluctuations only. The so-called Reynolds stress tensor $\overline{u_i'u_j'}$ which encodes all the information carried by the erratic motions has been the subject of several interpretations~\cite{Pope00,sagaut_18}, starting with the one of~\cite{Prandtl25}. Because it was a question of computing the value of a correlation between random variables, the temptation was great to make an analogy with the kinetic theory of gases, which Prandtl did. The construction is well known: one considers a turbulent flow as a medium equipped with a diffusivity $\mathcal{D}$ linking the stress tensor to the time-averaged velocity gradient (in analogy with gases and laminar flows) as $\overline{u_i'u_j'}=-\mathcal{D}(\partial_ju_i+\partial_iu_j)$, this effective coefficient being (by contrast) a function of the mean flow structure itself, namely of $u_i$ and/or of its spatial derivatives $\partial_ju_i$. This description assumes that momentum transport is mediated by ``liquid balls'' (\emph{Fl\"ussigkeitsballen}) of mean free path $\ell$ (\emph{Mischungswegl\"ange}) with short-time and space correlation as compared to those of the flow at scale $L$, an assumption which is questionable, in general. While not questioning the random walk construction \emph{per se}, Taylor~\cite{Taylor_32} suggested that, in two dimensional turbulent flows at least, it is not the momentum of the ``eddies'' which is conserved from one step to the other (unlike in the Reynolds--Prandtl theory), but their vorticity; we discuss this point of view and its relation to scalar transport in jets elsewhere~\cite{Rotily_24}. For instance, if $U_i=\{U,V\}=\braces[\big]{u(x,y)+u'(x,y,t),v(x,y)+v'(x,y,t)}$ in a plane jet, wake or boundary layer, or $\braces[\big]{u(x,r)+u'(x,r,t),v(x,r)+v'(x,r,t)}$ as in an axisymmetrical round jet, Prandtl's closure starts with \begin{equation} \overline{u'v'}=-\mathcal{D}\partial_yu \quad \text{or} \quad -\mathcal{D}\partial_ru, \quad \text{with} \ \ \mathcal{D}=\overline{v'\ell} \end{equation} where $\mathcal{D}$ remains to be interpreted. Since, as opposed to thermally activated transport (which defines the fluid viscosity $\nu$), there is no obvious intrinsic length scale in turbulence (see however Section~\ref{meanfreep}), it is assumed that $\ell$ is, while smaller, proportional to the dimension of the flow $L$ where the shear producing turbulence is sustained. In jets, $L$ is the jet radius and $\ell\sim L$ thus depends on the axial distance $x$ only (and not on $y$ or $r$), but in boundary layers $\ell\sim y$ is taken as proportional to the distance to the wall~$y$. It was even suggested by von K\'arm\'an~\cite{vonKarman_30} that $\ell$ could be related to the spatial structure of the mean flow $u$ itself by $\ell\sim \partial_yu/\partial^2_y u$ in pipe flow. A number of \emph{ad hoc} formulae have been proposed for $\mathcal{D}$, applied opportunistically (see~\cite{Schlichting87} and~\cite{Schmitt_07} for a historical perspective) to different types of configurations (boundary layer, wake, jet, mixing layer\dots): \begin{quote} ``\dots agreement with experimental results can be obtained with various expressions for the turbulent viscosity'', \end{quote} as noted by Landau \& Lifshitz~\cite{Land87}. Among them, two well-known proposals: \begin{enumerate} \item The ``mixing length'' theory of Prandtl~\cite{Prandtl25} for which $v'\sim u'\sim \ell\partial_yu$ in a plane jet ($\ell\partial_ru$ in an axisymmetrical jet) giving \begin{equation}\label{DPrandtl} \mathcal{D}\sim\ell^2\abs{\partial_y u} \quad \text{or}\quad \ell^2\abs{\partial_r u}. \end{equation} This expression was originally designed for and successfully applied to turbulent boundary layers (the famous ``Log-law of the wall'') but was also recognized as ``unsatisfactory''~\cite{Schlichting87} for jets because $\mathcal{D}$ vanishes where the velocity gradient is zero that is, in particular, at the center of the jet in $y=0$ where, on the contrary, the fluctuations intensities $\overline{u'^2}$ and $\overline{v'^2}$ are the largest but $\partial_y u=0$~\cite{WygnanskyFiedler}. That flaw was circumvented by Prandtl himself~\cite{Prandtl_42} who proposed to use in that case \begin{equation}\label{Dconst} \mathcal{D}\sim\Delta u\,\ell, \end{equation} a diffusivity taken as constant across the jet, proportional to the net velocity difference absorbed by the turbulent shear layer $\Delta u=u(x,0)-u(x,\infty)$. Prandtl's closure was, on the other hand, better suited to the plane mixing layer geometry where fluctuations and $\partial_y u$ are both the largest in $y=0$, where $\partial_y^2 u=0$~\cite{WygnanskyFiedler70}, although its solution for that flow using~\eqref{DPrandtl} suffers from an ``esthetical deficiency''~\cite{Schlichting87}, to say the least. This lack of unity has certainly contributed to moderate the enthusiasm for the ``gradient type'' approach to turbulent dispersion, of which the quote above is a typical reflection (see also the critical appraisal by~\cite{Corrsin_75}). \item The question was approached earlier in very similar terms by Boussinesq~\cite{Bouss77} who proposed, for the flow in tubes and channels (see~\cite{Lajeunesse_24} for a recent study), an expression identical to~\eqref{Dconst} with $\ell$ being the tube radius. His idea, later reiterated by~\cite{Reynolds_95} is that ``eddy agitations'' (\emph{agitation tourbillonaire}) mediate momentum transfer. One may push Boussinesq's logic one step further by applying the argument locally, stating that in a flow with ``tumultuous movements'', viscosity should be proportional to the intensity of the ``mean agitation'' of which $u(x,y)$ (or $u(x,r)$) is a measure, like in the kinetic theory of gases where temperature sets the thermal velocity (Maxwell formula, see e.g.~\cite{Born69}). In that perspective, $v'\sim u$ giving \begin{equation}\label{bouss} \mathcal{D}\sim u\ell \end{equation} where $u$ actually means $u(x,r)-u(x,\infty)$ for Galilean invariance. This formulation is reminiscent of the $\mathrm{k}-\epsilon$ theory (with $\mathrm{k}\sim u^2$, $\epsilon\sim u^3/\ell$ and $\mathcal{D}\sim \mathrm{k}^2/\epsilon$), first proposed to model concentration fluctuations in jets~\cite{Spalding_71}. But nothing guarantees that~\eqref{bouss} should be universally true, in every flow. \end{enumerate} The different proposals for $\mathcal{D}$ listed above have different levels of ``locality'': \eqref{DPrandtl} is local because of the spatial derivative of $u$ while~\eqref{Dconst} is comparatively more ``integral'' in the sense that it involves the total velocity difference across the whole flow; \eqref{bouss}, where $\mathcal{D}$ is proportional to the local velocity rather than to its spatial derivative, is intermediate. Turbulent flows in nature are generated by differences of velocity, density, temperature etc.\ maintained between distant boundaries. There is no reason to discard \emph{a priori} the possibility that small scale features of the flow would depend on the large scale structure of its forcing; or at least that some aspects of the stress tensor may be sensitive to the general organisation of the flow, like the overall shape of the velocity profile, and not only to its local value or gradient. Nonlocal closures for fluxes have besides already been considered in sheared layers~\cite{Fiedler_84}, in more general multi-scale flows~\cite{Souza_23}, including turbulent flows explicitly~\cite{Pomeau_20}. In other sectors of condensed matter physics concerned with jammed, glassy materials or dense granular flows, this idea is commonplace: the nonlocality of the motion in sheared dense colloids and granular media has been recognized for some time~\cite{Bocquet_09,Andreotti_13} where the existence of large scale structures like vortices has been documented~\cite{Rognon_17}. Newton's force chains, which mediate the stress in granular materials, mediate pressure in condensed normal fluids; for this reason, incompressible media are subjected to a long-range order. Our approach, taking advantage of the qualitative remarks on the nonlocality in turbulence made above, first discusses the general form the Reynolds $\overline{u'v'}$ should have, introduces an effective, nonlocal transfer velocity $\tilde {v}$ and computes, from there, the mean velocity profiles of some canonical flows of fluid mechanics. We describe the degree of nonlocality for each flow (jets in Sections~\ref{RJ} and~\ref{CJ}, mixing layers in Section~\ref{ML}, wakes in Section~\ref{W}), including for the particular problem of the boundary layer (Section~\ref{BL}) a more ``local'' configuration than the one in open flows. Our analysis also offers an interpretation of the mean free path $\ell$ (Section~\ref{meanfreep}), as well as a value of the von K\'arm\'an constant in Section~\ref{BL}. \section{Relationships in jets}\label{RJ} For stationary free (zero pressure gradient), plane or axisymmetrical jets at large Reynolds number, we note as in Section~\ref{Closure}, $\{U,V\}=\{u+u',v+v'\}$. In practice, jets are slender (the opening angle of a round jet is $10^{\circ}$) and the axial gradient of the stress $\overline{u'v'}$ is negligible in front of the transverse one. \subsection{Plane jet} The representation above translates into $U\partial_x U+V\partial_yU=0$ and $\partial_x U+\partial_yV=0$, that is \begin{align} &u\partial_x u+v\partial_yu=-\partial_y\overline{u'v'}, \label{planejet} \\ &\partial_xu+\partial_yv=0, \label{incompp} \end{align} valid up to corrections of order $\mathcal{O}\parens{\partial_x\overline{u'^2}}$ which we neglect owing to slenderness. Far downstream from the injection region, the jet transverse size $L$ broadens proportionally to the downstream distance $x$, and axial momentum flux $\sim u^2 x$ (per unit span-wise length and mass) conservation dictates that $u=\mathcal{O}(x^{-1/2})$, and $\mathcal{D}=\mathcal{O}(u\, x)\sim x^{1/2}$, see e.g.~\cite{Land87,Schlichting87,Pope00}. Introducing a stream function $\psi(\eta)$ consistent with the incompressibility condition in~\eqref{incompp}, we have \begin{equation}\label{vits} \begin{aligned} &\psi\sim x^{1/2}\mathcal{F}(\eta), \quad \text{with} \ \ \eta=\frac{y}{x},\\ \text{providing} \ \ &u=\partial_y\psi\sim x^{-1/2}\mathcal{F}', \quad \text{and} \ \ v=-\partial_x\psi\sim x^{-1/2}\parens[\Bigg]{\eta\mathcal{F}'-\frac{1}{2}\mathcal{F}}. \end{aligned} \end{equation} A useful reformulation of Euler's equation was introduced by von Reichardt~\cite{Reichardt_44}. It amounts to extract a relationship between the Reynolds stress $\overline{u'v'}$ and the mean fields $u$ and $v$. From~\eqref{planejet} \&~\eqref{vits} we find $ \overline{u'v'}=u\parens{\eta\, u-v}$, leading to \begin{equation}\label{Reynoldstress} \overline{u'v'} \sim\frac{1}{2x}\mathcal{F}\mathcal{F'}. \end{equation} \subsection{Round jet} The above results are readily transposed to a round jet for which the balance equations are \begin{align}\label{roundjet} &u\partial_x u+v\partial_ru=-\partial_r\parens{r\,\overline{u'v'}}/r,\\ &\partial_xu+\partial_r\parens{r\,v}/r=0. \end{align} Far downstream from the injection region, the jet radius $L$ broadens proportionally to the downstream distance $x$, and axial momentum flux $\sim u^2 x^2$ (per unit mass) conservation dictates that $u=\mathcal{O}(x^{-1})$, and $\mathcal{D}=\mathcal{O}(u\, x)=\mathrm{C^{st}}$. The stream function $\psi(\eta)$ is now \begin{equation}\label{vitsr} \begin{aligned} &\psi\sim x \mathcal{F}(\eta), \quad \text{with} \ \ \eta=\frac{r}{x},\\ \text{which provides} \ \ &u=\partial_r\psi/r\sim\frac{\mathcal{F}'}{x\eta}, \quad \text{and} \ \ v=-\partial_x\psi/r\sim \frac{-\mathcal{F}+\eta\mathcal{F}'}{x\eta}. \end{aligned} \end{equation} A similar von Reichardt's type of calculation now provides $\overline{u'v'}=u\parens[\big]{\eta\, u-v}$ as for a plane jet, that~is \begin{equation}\label{stressround} \overline{u'v'} \sim\frac{1}{x^2}\frac{\mathcal{F}\mathcal{F}'}{\eta^2}. \end{equation} \section{The closure and its formulation in jets}\label{CJ} Let us start by discussing the plane jet for which~\eqref{Reynoldstress} has shown that \begin{equation}\label{euler} \overline{u'v'}\sim\frac{1}{2x}\mathcal{F}\mathcal{F}'. \end{equation} The function $\mathcal{F}$ is defined from the stream function $\psi$, which itself arises from incompressibility; its dependence on the sole variable $\eta=y/x$ reflects the flow self-similarity, which is a separate issue (a flow may well be incompressible while non self-similar). The fact that the stress tensor is proportional to $\mathcal{F}$, that is to the integral over space (i.e.\ on $\eta$) of the velocity ($u\sim\mathcal{F}'$) shows that it incorporates ingredients which are necessarily \emph{nonlocal}. This is not an artifact of a particular model since~\eqref{Reynoldstress} is nothing but a reformulation of the basic conservation equations. This nonlocality of the stress, which arises from the dynamical equations of the flow, does not necessarily imply that, in general, the closure for $\overline{u'v'}$ should be nonlocal itself: when usual molecular, thermally activated viscosity $\nu$ is at play, the stress $-\nu(\partial_ju_i+\partial_iu_j)$ is \emph{very local} since it involves the spatial derivatives of $u_i$. But here, viscous stresses are not extrinsic (i.e.\ thermally activated), but intrinsic; the effective viscosity $\mathcal{D}$ is the result of the flow itself, and since stresses in the flow are ruled by nonlocal effects, so should be $\mathcal{D}$. Therefore, in looking for a model for the stress tensor in a flow with mean velocity profile $u(x,y)$, we need to have in mind that $\overline{u'v'}$ must: \begin{enumerate} \item be Galilean invariant: when we write $u(x,y)$, we actually mean $u(x,y)-u(x,\infty)$, where $u(x,\infty)$ is an arbitrary velocity shift, taken here as zero; \item vanish when $\partial_yu=0$, another way to express that fluxes are mediated by velocity \emph{differences} only; \item be sensitive to the large-scale organization of the flow i.e.\ be \emph{nonlocal} by involving, for example, an integral $\int u\, \dd y$ between bounds which need to be discussed for each flow. \end{enumerate} A possible form satisfying the constraints above may be derived from an extension of the familiar momentum transport model used by~\cite{Prandtl25} in analogy with the kinetic theory of gases~\cite{maxwell_67,Born69}. In the direction of the velocity gradient $\partial_yu$, the net axial momentum flux (per unit mass) crossing the plane $y$ through instantaneous transverse fluctuations with mean free path $\ell$ at velocity $v'$ is \begin{equation} \overline{u'v'}=\overline{v'\bracks[\Bigg]{u\parens[\Big]{y+\frac{\ell}{2}}-u\parens[\Big]{y-\frac{\ell}{2}}}}=\overline{v'\ell}\partial_yu. \end{equation} In a turbulent flow, both $\ell$ and $v'$ are broadly distributed but may have a loose level of inter-correlation defining, in this construction, the intensity of the momentum transfer. Our model is the following: in shear flows like jets, fluctuations are driven by a sustained large-scale shear, namely by the velocity difference between the core of the jet, and the outside; this is, in essence, the nature of the Kelvin--Helmholtz instability. The correlation length of the motion is the jet radius itself and thus $\ell$ should on average scale with the jet radius $L=\alpha x$. For the same reason, fluctuations of any extensity (momentum, vorticity, concentration) are likely to be transported from the rich core of the jet to any off-center depleted location $y$. The velocity $v'$ driving the transfer in $y0$ (or larger $\abs{y}$ in two-dimensions) for jets, and from large $\abs{y}$ towards $y=0$ in mixing layers and wakes, explaining the sign of $\tilde{v}$ and the integration interval of its definition ($\int_0^yu\, \dd y$ for jets and $\int_y^\infty u\, \dd y$ for mixing layers and wakes).}\label{Figcas} \end{figure} \subsection{The round jet} In a round jet, our closure, identically to~\eqref{closure}, is $\overline{u'v'}=-\tilde{v}\ell \partial_r u$ with $\ell =ax$ except that now the velocity $\tilde{v}$ must be understood as an average over the jet cross section since there are two degrees of freedom for radial transport, and involves a surface integration ($\, \dd x_j\, \dd x_k=r\, \dd \theta \, \dd r$, $\theta\in[0,2\pi]$, see Figure~\ref{fig1} and~\eqref{general}) \begin{equation}\label{closround} \tilde{v}=\frac{1}{\pi r^2}\int_0^r 2\pi r' u(r')\, \dd r'=\frac{2}{x}\frac{\mathcal{F}}{\eta^2}. \end{equation} Since $\partial_ru=(\mathcal{F'}/\eta)'/x$ and that, from Euler equation in~\eqref{stressround} \begin{equation} \overline{u'v'}=\frac{1}{x^2}\frac{\mathcal{F F'}}{\eta^2}, \end{equation} one sees that $\mathcal{F}$ must solve (with $\mathcal{F}(0)=0$ and $\mathcal{F'}/\eta\xrightarrow[\eta\to 0]{}1$) \begin{equation}\label{rounj} \begin{aligned} 2a\parens[\Bigg]{\frac{\mathcal{F}'}{\eta}}'&=-\mathcal{F}'\\ \text{giving} \ \ \mathcal{F}&=2a\parens{1-e^{-\frac{\eta^2}{4a}}}, \quad \text{that is} \ \ u\sim\mathcal{F}'/\eta=e^{-\frac{\eta^2}{4a}}. \end{aligned} \end{equation} The velocity profile is also Gaussian, as again experiments show (Figure~\ref{Figcas}). \subsection{The dispersion coefficients}\label{dispcoefs} Rather than being at the source of the computation providing the form of $u$, the dispersion coefficient $\mathcal{D}$ is a consequence of it. We have not conjectured a particular form of $\mathcal{D}$ (various forms are compared in Figure~\ref{fig2}), we have instead discussed what features the stress $\overline{u'v'}$ should have, and obtained $u$ right away. We can now compute $\mathcal{D}$. It is true that we have argued in~\eqref{closure} why $\overline{u'v'}$ should be proportional to the velocity gradient and this implies, assuming a gradient type form for the flux \begin{equation} \begin{aligned} \overline{u'v'}&=-\mathcal{D}\partial_yu \ \ \text{or} \ \ -\mathcal{D}\partial_ru,\\ \text{that} \ \ \mathcal{D}&=\tilde{v}\ell, \end{aligned} \end{equation} but that was not our starting point. In a plane jet, we have $\overline{u'v'}=-\mathcal{D}\partial_yu$ so that \begin{equation} \begin{aligned} \mathcal{D}&\sim -x^{1/2}\frac{\mathcal{F F}'}{2\mathcal{F''}}\\ &=a\sqrt{\pi x}\,\frac{\erf\parens[\Big]{\frac{\eta}{2\sqrt{a}}}}{\eta/\sqrt{a}}. \end{aligned} \end{equation} The dispersion coefficient increases with downstream distance as $x^{1/2}$, is constant close to the jet center for $\eta\ll 1$ and decays far away from it like $\eta^{-1}$ for $\eta\gg 1$. In a round jet, where $\overline{u'v'}=-\mathcal{D}\partial_ru$ we have \begin{equation}\label{Dround} \begin{aligned} \mathcal{D}&\sim\frac{\mathcal{F F'}}{\mathcal{F'-\eta F''}}\\ &=a \frac{1-e^{-\frac{\eta^2}{4a}}}{(\eta/2\sqrt{a})^2}, \end{aligned} \end{equation} showing that $\mathcal{D}$ is independent of $x$, and again that it is constant close to the jet center for $\eta\ll 1$ and decays far away from it like $\eta^{-2}$ this time, for $\eta\gg 1$. \begin{figure} \includegraphics[width=0.7\textwidth]{Fig2} \caption{Different models of the turbulent dispersion coefficient $\mathcal{D}$ for a turbulent round jet (Section~\ref{Closure}). The closure $\mathcal{D}\sim \Delta u\ell$ of Boussinesq~\cite{Bouss77} where $\Delta u$ is the amplitude of the velocity profile $u$ across the jet, its local version $\mathcal{D}\sim u\ell$, the Prandtl~\cite{Prandtl25} version $\mathcal{D}\sim u\ell^2\partial_ru$, initially conceived for boundary layers, clearly at odd for velocity profiles with a maximum (see also~\cite{Prandtl_42}), and the one derived from the present conjecture of the stress tensor in~\eqref{closure}, leading to a Gaussian velocity profile $u\sim e^{-\frac{\eta^2}{4a}}$, which is intermediate between the Boussinesq local, and constant forms (see~\eqref{Dround}).}\label{fig2} \end{figure} It would have been, admittedly, difficult to intuit these relatively complicated formulae for $\mathcal{D}$ in order to obtain the ubiquitous Gaussian form for $u$. Although qualitatively similar in plane and round jets, the detailed form of $\mathcal{D}$ is besides different in each case. This illustrates how guessing~$\mathcal{D}$ to infer $u$ is a risky exercise, if not an \emph{a priori} lost battle. Although the velocity profiles are both Gaussian in plane and round jets, the dispersion coefficients are different in each case. This is at contrast with standard molecular diffusion where the dispersion coefficient is intrinsic, because thermally activated (the viscosity $\nu$ in fluids); the originality of turbulence is, on the contrary, that the dispersion coefficient depends on the structure of the flow itself and has therefore no reason to be universal; the velocity profiles, interestingly, are. \subsection{Relation to experiments} These Gaussian mean velocity profiles fit well the observed ones, including on the more stringent log-scale compared to the lin-lin scaling routinely used in this context~\cite{Amielh_23}, in any case better than those derived by Schlichting~\cite{Schlichting_33,Schlichting87} under the assumption of a constant, and uniform dispersion coefficient $\mathcal{D}$ across the jet, as suitable for a laminar jet~\cite{landau_44,Land87}. Indeed, \eqref{planj} \&~\eqref{rounj} compare favorably well with experiments, unlike the corresponding predictions of Schlichting, namely \begin{equation} \begin{aligned} u&\sim 1-\tanh^2\eta \quad \text{for a plane jet},\\ \text{and}\quad u&\sim\parens[\Bigg]{1+\frac{\eta^2}{2}}^{-2} \quad \text{for a round jet}, \end{aligned} \end{equation} which both present much too heavy tails, obvious in logarithmic units (see~\cite{Rotily_24}). We have, on purpose, mentioned in Figure~\ref{Figcas} the early seminal measurements reported in~\cite{Schlichting87} only, but the empirical adequacy of the Gaussian fit in jets has been since then reported consistently~\cite{Danckwerts_61,So_86,Rimbert_24}. In the present representation, $\mathcal{D}$ decays (slowly) on the edge of the jet, while dispersion is intense close to the center of the jet~\cite{Bourgoin_22}. This results in a more ``localized'' profile around the jet centerline than those assuming a constant $\mathcal{D}$ in the transverse direction (a fact true for both plane and round jets). Prandtl's parametrization in~\eqref{DPrandtl} has also a vanishing $\mathcal{D}$ at the jet edge (where the velocity gradient weakens), but dramatically fails on the centerline, predicting oddly that $\mathcal{D}=0$ there (see~\cite{Tollmien_26} for an application of Prandtl's mixing length theory to the turbulent round jet). For the diffusive description of the phenomenon to be consistent, the mean free path $\ell=ax$ in~\eqref{closure} must be appreciably smaller than the jet radius $L=\alpha x$, implying that $a\ll \alpha$. In a round jet, $\alpha=1/6\approx 0.16$ and $a\approx 0.004$ fit well the experimental profiles, thus fulfilling the condition of a short mean free path, when seen at the scale of the jet radius. \section{Mixing layers}\label{ML} \subsection{The temporal mixing layer} The mixing layer has for a long time been considered as the \emph{Drosophila} of turbulent shear flows: a benchmark to exercise new ideas, methods and concepts. We work out the closure in~\eqref{closure} in a two-dimensional, temporally growing mixing layer between two streams of opposite velocities $\pm\Delta u/2$ initially separated by a sharp discontinuity. The flow is uniform along $x$, and the layer grows in the $y$-direction so that from the Euler equations $\partial_tU+\partial_xU^2+\partial_y(UV)=0$ with $u(y,t)$ and $v=0$, we have \begin{equation}\label{mixl} \partial_tu=-\partial_y\overline{u'v'} \end{equation} where now the average $\overline{(\,\cdot\,)}$ must be taken over a time-period $T$ large compared to the correlation time of $u'$ and $v'$ (taken as zero in the pure brownian limit), but small compared to the characteristic time of the flow evolution $u/\dot{u}$. The only length scale in this problem is $\Delta u\, t$, we thus look for a solution of the form \begin{equation}\label{uml} u=\Delta u\,\mathcal{F}(\eta) \quad \text{with} \ \ \eta=\frac{y}{\Delta u\, t}. \end{equation} As usual we compute the stress $\overline{u'v'}$ as a function of $u$ and we have from~\eqref{mixl}, making use of $\partial_tu=-\Delta u\,\eta\mathcal{F}'/t$, \begin{equation}\label{stressml} \begin{aligned} \overline{u'v'}&=\restr{\smash{\overline{u'v'}}}{0}-\int_0^y\partial_tu\, \dd y\\ &=\restr{\smash{\overline{u'v'}}}{0}+(\Delta u)^2\int_0^\eta\eta\mathcal{F}'\, \dd \eta\\ &=-(\Delta u)^2\int_\eta^\infty\eta\mathcal{F}'\, \dd \eta \end{aligned} \end{equation} since $\overline{u'v'}\xrightarrow[\eta\to\pm\infty]{}0$ and that therefore $\restr{\smash{\overline{u'v'}}}{0}=-(\Delta u)^2\int_0^\infty\eta\mathcal{F}'\, \dd \eta$. Now comes the closure: the mean free path $\ell=a\Delta u\,t$ is proportional to the layer thickness, $\partial_yu=\mathcal{F}'/t$ and~\eqref{closure} \&~\eqref{uml} give $\overline{u'v'}=-a\Delta u\tilde{v}\mathcal{F}'$, the minus sign reflecting the direction of the momentum transfer, from the (infinite) upper layer above the sheared region (in $y>0$, say, the profile is anti-symmetrical in $y<0$), towards the core of the layer in $y=0$. That direction is opposite to the one in inertial jets diluting their momentum in a quiescent environment, as those examined before. The momentum reservoir thickness $h$ being practically infinite, we have \begin{equation} \tilde{v}=\frac{1}{h}\int_y^{y+h\to\infty} u\, \dd y\approx\Delta u/2 \end{equation} since the integral extends to infinity. Then, from~\eqref{stressml}, we find \begin{equation}\label{mlf} \begin{aligned} \mathcal{F}'&=\frac{2}{a}\int_\eta^\infty\eta\mathcal{F}'\, \dd \eta \quad \text{or, differentiating,} \ \ \mathcal{F}''=-\frac{2}{a}\,\eta\mathcal{F}',\\ \text{finally providing} \ \ \mathcal{F}&=\frac{1}{2}\erf\parens[\Bigg]{\frac{\eta}{\sqrt{a}}}, \end{aligned} \end{equation} which fulfills the boundary conditions $u(\eta=\pm\infty)=\pm\Delta u/2$. The dispersion coefficient $\mathcal{D}$ in a turbulent mixing layer, such that $\overline{u'v'}=-\mathcal{D}\partial_yu$, or \begin{equation} \mathcal{D}=\frac{(\Delta u)^2\int_\eta^\infty\eta\mathcal{F}'\, \dd \eta}{\mathcal{F}'}\,t=\frac{a}{2}(\Delta u)^2\,t, \end{equation} is proportional to the product of the mean free path $\ell=a\Delta u\, t$ times the velocity difference $\Delta u$ sustained across the layer. It is independent of $y$, increases linearly in time in proportion of $\ell$, while the turbulent stresses $\overline{u'v'}\sim \sqrt{a}\,e^{-\eta^2/a}$ remain localized over a region, called the mixing layer, whose width increases accordingly. If one chooses the ``vorticity thickness'' $L=\restr{\Delta u/\partial_yu}{\max}$, equal to $\Delta u t/\mathcal{F}'(0)$ in the present theory (since the maximum of the shear rate $\partial_yu$ is in $y=0$) to quantify the layer thickness, we find that \begin{equation}\label{thicknes} L=\sqrt{\pi a}\,\Delta u t. \end{equation} As already recalled, the growth of the layer thickness (as well as the growth of jets widths) is driven by the Kelvin--Helmholtz instability of the sustained large-scale shear $\Delta u/L$. Since this instability is at the core of the phenomenon, one may expect to link our description, which still lacks a fundamental understanding of the---free at this stage---parameter $a$, with the fundamentals of the Kelvin--Helmholtz instability. The (temporal) growth rate of a disturbance amplitude with wavenumber $k$ along the layer is given by (see~\cite{HuerreRossi} for a comprehensive review, and~\cite[Appendix]{Vill_20} for a synthetic formulaire) \begin{equation} \frac{\Delta u}{L}\mathcal{J}(kL) \end{equation} where $\mathcal{J}(kL)$ is a function describing the amplification of mode $k$, which is tangent to $kL$ for $kL\to 0$ and falls to zero for $k_cL=\mathcal{O}(1)$. For instance, $\mathcal{J}(kL)=\frac{1}{2}\sqrt{e^{-2kL}-(1-kL)^2}$ for a broken-line profile absorbing a total velocity difference $\Delta u$ over a uniform crossover of thickness $L$, first studied by~\cite{Rayleigh_80}. In that generic case (the precise shape of $\mathcal{J}(kL)$ is very little sensitive to the detailed shape of $u(y)$, see~\cite{HuerreRossi}), the maximal amplification rate is \begin{equation}\label{gr} 0.2\frac{\Delta u}{L} \end{equation} obtained at $k_mL\approx 0.8$. How does this result connect with the linear growth of $L$ in~\eqref{thicknes}? The increase of $L$ reflects the amplification of a superposition of all the modes $k$ in the unstable range $0