Chaos, coherence and turbulence

TL;DR

The paper argues that turbulence can be understood in terms of coherent structures that decouple, at least partially, from the rest of the flow, especially in energy-input ranges driven by shear. It surveys free-shear and wall-bounded flows to illustrate how linear and quasi-linear mechanisms underpin large-scale coherence, while acknowledging that much of turbulence remains incoherent and multiscale. Looking forward, it highlights data-driven analytics and causality as powerful tools to tackle open questions, but cautions about their limits and the need to connect them to mechanistic understanding. The work emphasizes both the conceptual payoff of coherence-based views and their practical relevance for control, with an eye toward integrating new computational approaches while maintaining physics-based interpretation.

Abstract

This paper is a personal overview of the efforts over the last half century to understand fluid turbulence in terms of simpler coherent units. The consequences of chaos and the concept of coherence are first reviewed, using examples from free-shear and wall-bounded shear flows, and including how the simplifications due to coherent structures have been useful in the conceptualization and control of turbulence. It is remarked that, even if this approach has revolutionized our understanding of the flow, most of turbulence cannot yet be described by structures. This includes cascades, both direct and inverse, and possibly junk turbulence, whose role, if any, is currently unknown. This part of the paper is mostly a catalog of questions, some of them answered and others still open. A second part of the paper examines which new techniques can be expected to help in attacking the open questions, and which, in the opinion of the author, are the strengths and limitations of current approaches, such as data-driven science and causal inference.

Figures (6)

• Figure 1: (a) Plane turbulent shear layer at high Reynolds number between two streams of different gases. Reproduced with permission from brownr. The Reynolds number based on the velocity difference and on the maximum visual thickness is $Re\approx 2\times 10^5$. (b) Initial development of a low-Reynolds-number velocity discontinuity. $Re\approx 7,500$. Reproduced with permission from frey66.
• Figure 2: (a) Sketch of a turbulent shear layer. (b) Growth rate, $\kappa c_i$, of the unstable modes for two model velocity profiles of the shear layer. , $U=\tanh(y)$; , $U={\rm erf}(y)$.
• Figure 3: (a) Energy spectra of the transverse velocity component at the centerline of a free shear layer, plotted against the wavenumber normalized with the local momentum thickness delville. The dashed vertical line is the limit of stability in Eq. (\ref{['eq:oster']}), $\kappa\theta=0.45$. The two spectra correspond to different positions along the layer. , $\Delta U\theta/\nu = 3.9\times 10^3$; , $1.2\times 10^4$. (b) Growth of the momentum thickness of shear layers forced at the exit of the splitter plate with perturbations of frequency $\lambda/U_c$. Data from various experimenters and velocity ratios brow87. The dashed horizontal line is Eq. (\ref{['eq:oster']}), and the dotted diagonal is the growth rate of an unforced layer.
• Figure 4: Structures of a turbulent channel in a small computational box, $(L_x\times L_z)=(\pi/2\times \pi/4)h$, $Re_\tau=940$. The four panels are the same flow field, flowing from bottom left to top right. (a,b) For $yu_\tau/\nu\le 70$. (c,d) $yu_\tau/\nu\le 350$. The translucent red objects in all panels are low-velocity streaks, $u<-1.3\, u'(y)$. The yellow objects in (a,c) are vortices defined as $|\omega|<1.75\, |\omega|'(y)$. The green and blue objects in (b,d) are, respectively, positive and negative $v$-structures, defined as $|v|<1.75\, v'(y)$. Panels (c,d) include the full wall-parallel domain, while (a,b) only include one quarter of the plane, approximately centered spanwise on the streak in (c,d) at $x=0$. Axes are in wall units.
• Figure 5: Two-point correlations of: , streamwise velocity; , wall-normal velocity. Contour is $C_{**}=0.1$, and correlations are centered at $y/h=0.1,\,0.3$ and 0.5. (a) Channel at $Re_\tau=2003$hoy:jim:06. (b) Boundary layer at $Re_\tau\approx 1530$sil:jim:mos:14.