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Lagrangian turbulence in the woods

Published online by Cambridge University Press:  06 April 2021

Laurent Chevillard*
Univ. Lyon, ENS de Lyon, Univ. Claude Bernard, CNRS, Laboratoire de Physique, 46 allée d'Italie, F-69342Lyon, France
Email address for correspondence:


A recent statistical analysis, proposed by Shnapp (J. Fluid Mech., vol. 913, 2021, R2), of Lagrangian velocity measurements in a wind tunnel in the presence of a canopy (a forest or urban morphology), using three-dimensional particle tracking velocimetry (Shnapp et al., Sci. Rep., vol. 9, issue 1, 2019, pp. 1–13), is a great read. In this strongly anisotropic situation, despite the additional roughness induced by the canopy, it is shown that fluctuations of Lagrangian velocity increments over small time scales display very similar behaviour as those observed in homogeneous and isotropic turbulent flows. This is all the more true when focussing on the non-Gaussian and intermittent nature of these fluctuations. At much larger time scales, of the order and greater than the characteristic turnover time scale of the flow, anisotropies implied by the presence of the canopy are quantified using averages of the fluctuating kinetic energy conditioned upon the direction of Lagrangian velocity with respect to the mean Eulerian flow. Shnapp (2021) evidences that, indeed, the canopy modifies the velocity along the trajectories at large scales, in particular its variance, but leaves unchanged its local regularity, as it is pinpointed by the power-law exponents of the structure functions.

Focus on Fluids
© The Author(s), 2021. Published by Cambridge University Press

1. Introduction

The Lagrangian investigation of laboratory and numerical turbulent flows has been intensively developed over the last thirty years, as reviewed in Yeung (Reference Yeung2002), Toschi & Bodenschatz (Reference Toschi and Bodenschatz2009) and Pinton & Sawford (Reference Pinton and Sawford2012), following an intense and vast effort aimed at characterizing with precision the statistical behaviour of the Eulerian velocity field (Frisch Reference Frisch1995). Following velocity along the path of fluid particles is not only important from a fundamental point of view (Monin & Yaglom Reference Monin and Yaglom1971; Tennekes & Lumley Reference Tennekes and Lumley1972), it is also an appropriate way to describe the mixing and dispersion properties of emitted tracers in geophysical situations (LaCasce Reference LaCasce2008).

Fluctuations of velocity along trajectories were initially observed in direct numerical simulations (DNSs) of the Navier–Stokes equations in controlled situations aimed at investigating some aspects of homogeneous and isotropic turbulence (Yeung & Pope Reference Yeung and Pope1989; Chevillard et al. Reference Chevillard, Roux, Lévêque, Mordant, Pinton and Arneodo2003; Biferale et al. Reference Biferale, Boffetta, Celani, Devenish, Lanotte and Toschi2004; Bentkamp, Lalescu & Wilczek Reference Bentkamp, Lalescu and Wilczek2019), and soon after in von Kármán swirling flows (Voth, Satyanarayan & Bodenschatz Reference Voth, Satyanarayan and Bodenschatz1998; Mordant et al. Reference Mordant, Metz, Michel and Pinton2001). Despite the artificial nature of the boundary conditions of DNSs, and the strong anisotropy of the experimental realisation of such flows driven by propellers, a remarkable agreement was observed in the statistical properties of the Lagrangian velocity in both situations (Arneodo et al. Reference Arneodo2008). The Johns Hopkins Turbulence Database (Yu et al. Reference Yu, Kanov, Perlman, Graham, Frederix, Burns, Szalay, Eyink and Meneveau2012) could, for instance, be used to further confirm the picture which follows.

We recall in a few words some of the key ingredients of the phenomenology of homogeneous and isotropic turbulence in the Lagrangian framework, as reviewed by Chevillard et al. (Reference Chevillard, Castaing, Arneodo, Lévêque, Pinton and Roux2012). Using the notation of Shnapp (Reference Shnapp2021), the statistical and multiscale nature of a Lagrangian velocity component $v_i(t)$, with $i\in \{1,2,3\}$, is well captured by the following probabilistic ansatz:

(1.1)\begin{equation} \Delta_\tau v_i = \mathcal{B}\left(\frac{\tau}{T_L}\right) \Delta_{T_L} v_i, \end{equation}

where the velocity increment $\Delta _\tau v_i(t)=v_i(t+\tau )-v(t)$ is introduced. Equation (1.1) relates an equality in probability law between the random variable $\Delta _\tau v_i$, made up of the instances of the increments along the trajectories at a given time scale $\tau$, and its instances $\Delta _{T_L} v_i$, at the large integral time scale of the flow $T_L$ at which velocity decorrelates, weighted by a random scale-dependent multiplier $\mathcal {B}$. At large scales $\tau \gg T_L$, $\mathcal {B}$ tends to the deterministic value $1$, meaning that the increment is statistically equal to $\Delta _{T_L} v_i$, usually taken to be a Gaussian random variable of zero average and of variance $2\langle v_i^2\rangle$, as dictated by observations. In the inertial range $\tau _\eta \ll \tau \ll T_L$, where $\tau _\eta$ is the Kolmogorov dissipative time scale, $\mathcal {B}$ fluctuates in the same way as $(\tau /T_L)^h$, the randomness being encoded in the exponent $h$. Dimensional arguments, mostly based on the irrelevance of viscosity at these scales (Tennekes & Lumley Reference Tennekes and Lumley1972), suggest that $\langle h \rangle \approx {1}/{2}$, at any scale $\tau$. Using the language of the multifractal formalism (Frisch Reference Frisch1995), in this statistically averaged sense, we can say that the Lagrangian velocity shares the same local regularity as that of the Brownian motion. Further analyses of experimental and numerical data (Chevillard et al. Reference Chevillard, Roux, Lévêque, Mordant, Pinton and Arneodo2003; Arneodo et al. Reference Arneodo2008) indicate that, indeed, $h$ fluctuates around its mean value, independently of both the Reynolds number and the geometry of the flow, which is known as intermittency. The level of Lagrangian intermittency is observed in the right proportion compared to that measured in the Eulerian framework, consistently with the elegant theory of Borgas (Reference Borgas1993).

2. Overview

Much more could be said on the statistical behaviour of the multiplier $\mathcal {B}$ entering in (1.1), in particular on the rich and predictive physics that has been developed to include the differential action of viscosity at small scales $\tau \ll \tau _\eta$, where fluctuations of the velocity increment are similar to those of acceleration. Let us keep in mind that this ansatz is well posed and closed from a probabilistic point of view if we furthermore assume that $h$ and $\Delta _{T_L} v_i$ are statistically independent. It is then possible to derive explicit predictions for the probability density function (p.d.f.) of $\Delta _{\tau } v_i$ and its moments (i.e. the structure functions), at any scale $\tau$, for a given Reynolds number and a prescribed level of intermittency.

The novelty of the analysis of Shnapp (Reference Shnapp2021) is to show that this aforementioned Lagrangian phenomenology, initially designed for isotropic turbulent flows, gives a fair account of the fluctuations of velocity along the trajectories obtained in his wind tunnel (Shnapp et al. Reference Shnapp, Shapira, Peri, Bohbot-Raviv, Fattal and Liberzon2019). Despite the presence of a strong mean flow, and a model canopy laid out on the bottom of the tunnel, Shnapp (Reference Shnapp2021) evidences that the subset of particles flying just above this rough surface exhibits a statistical behaviour in quantitative agreement with the theoretical predictions for the velocity increment p.d.f.s and moments obtained within this formalism, with furthermore the same level of intermittency. Noticing that the characteristic height of the canopy is of the order of $U_{\infty }T_L$, where $U_{\infty }$ is the wind mean velocity, this important observation further illustrates that Lagrangian fluctuations at small scales are universal and decoupled from the large scale flow. Nonetheless, signatures of the anisotropic nature of the canopy are evidenced when comparing the variance and correlation time scales of the different components $v_i$, which impact $\Delta _{T_L} v_i$, changing weakly the distribution of $\mathcal {B}$, similarly to what was observed by Ouellette et al. (Reference Ouellette, Xu, Bourgoin and Bodenschatz2006) and Huck, Machicoane & Volk (Reference Huck, Machicoane and Volk2019).

To characterize more precisely the anisotropic nature of the large scale flow from a Lagrangian perspective, Shnapp (Reference Shnapp2021) decomposes the set of the trajectories according to four quadrants which represent different directions of Lagrangian velocity with respect to the mean Eulerian flow. This original method of classification allows him to analyse the distribution of kinetic energy depending of the amplitude of the streamwise component, and makes some connections with the fluctuating nature of the drag induced by the canopy.

3. Future

The results of Shnapp (Reference Shnapp2021) remarkably show that Lagrangian tracking experiments are an original and fertile characterization of realistic turbulent flows, such as wind tunnels (Ayyalasomayajula et al. Reference Ayyalasomayajula, Gylfason, Collins, Bodenschatz and Warhaft2006; Shnapp et al. Reference Shnapp, Shapira, Peri, Bohbot-Raviv, Fattal and Liberzon2019), jets (Poulain et al. Reference Poulain, Mazellier, Gervais, Gagne and Baudet2004; Viggiano et al. Reference Viggiano, Basset, Solovitz, Barois, Gibert, Mordant, Chevillard, Volk, Bourgoin and Cal2021) and channel flows (Stelzenmuller et al. Reference Stelzenmuller, Polanco, Vignal, Vinkovic and Mordant2017). These newly developed techniques shed new light on Eulerian measurements and simulations of modelled canopies (Bai, Katz & Meneveau Reference Bai, Katz and Meneveau2015; Glick et al. Reference Glick, Ali, Bossuyt, Calaf and Cal2020), and their consequences for the intermittency phenomenon (Katul et al. Reference Katul, Porporato, Cava and Siqueira2006; Dupont et al. Reference Dupont, Argoul, Gerasimova-Chechkina, Irvine and Arneodo2020).

From a theoretical perspective, the Lagrangian framework naturally calls for the stochastic modelling of the trajectories using random walks, as they were developed for isotropic flows (Sawford Reference Sawford1991; Pope Reference Pope2002; Viggiano et al. Reference Viggiano, Friedrich, Volk, Bourgoin, Cal and Chevillard2020). In the spirit of recent propositions made by Innocenti et al. (Reference Innocenti, Mordant, Stelzenmuller and Chibbaro2020) and Shnapp et al. (Reference Shnapp, Bohbot-Raviv, Liberzon and Fattal2020), generalizing these approaches to anisotropic situations sounds like a fantastic perspective.

Declaration of interests

The author reports no conflict of interest.



Arneodo, A., et al. 2008 Universal intermittent properties of particle trajectories in highly turbulent flows. Phys. Rev. Lett. 100, 254504.CrossRefGoogle ScholarPubMed
Ayyalasomayajula, S., Gylfason, A., Collins, L.R., Bodenschatz, E. & Warhaft, Z. 2006 Lagrangian measurements of inertial particle accelerations in grid generated wind tunnel turbulence. Phys. Rev. Lett. 97 (14), 144507.CrossRefGoogle ScholarPubMed
Bai, K., Katz, J. & Meneveau, C. 2015 Turbulent flow structure inside a canopy with complex multi-scale elements. Boundary-Layer Meteorol. 155 (3), 435457.CrossRefGoogle Scholar
Bentkamp, L., Lalescu, C.C. & Wilczek, M. 2019 Persistent accelerations disentangle Lagrangian turbulence. Nat. Commun. 10, 3550.CrossRefGoogle ScholarPubMed
Biferale, L., Boffetta, G., Celani, A., Devenish, B.J., Lanotte, A. & Toschi, F. 2004 Multifractal statistics of Lagrangian velocity and acceleration in turbulence. Phys. Rev. Lett. 93 (6), 064502.CrossRefGoogle ScholarPubMed
Borgas, M.S. 1993 The multifractal Lagrangian nature of turbulence. Phil. Trans. R. Soc. Lond. A 342, 379.Google Scholar
Chevillard, L., Castaing, B., Arneodo, A., Lévêque, E., Pinton, J.-F. & Roux, S. 2012 A phenomenological theory of Eulerian and Lagrangian velocity fluctuations in turbulent flows. C.R. Phys. 13, 899.CrossRefGoogle Scholar
Chevillard, L., Roux, S.G., Lévêque, E., Mordant, N., Pinton, J.-F. & Arneodo, A. 2003 Lagrangian velocity statistics in turbulent flows: effects of dissipation. Phys. Rev. Lett. 91, 214502.CrossRefGoogle ScholarPubMed
Dupont, S., Argoul, F., Gerasimova-Chechkina, E., Irvine, M.R. & Arneodo, A. 2020 Experimental evidence of a phase transition in the multifractal spectra of turbulent temperature fluctuations at a forest canopy top. J. Fluid Mech. 896, A15.CrossRefGoogle Scholar
Frisch, U. 1995 Turbulence, The Legacy of A.N. Kolmogorov. Cambridge University Press.CrossRefGoogle Scholar
Glick, A., Ali, N., Bossuyt, J., Calaf, M. & Cal, R.B. 2020 Utility-scale solar PV performance enhancements through system-level modifications. Sci. Rep. 10 (1), 19.CrossRefGoogle ScholarPubMed
Huck, P.D., Machicoane, N. & Volk, R. 2019 Lagrangian acceleration timescales in anisotropic turbulence. Phys. Rev. Fluids 4 (6), 064606.CrossRefGoogle Scholar
Innocenti, A., Mordant, N., Stelzenmuller, N. & Chibbaro, S. 2020 Lagrangian stochastic modelling of acceleration in turbulent wall-bounded flows. J. Fluid Mech. 892, A38.CrossRefGoogle Scholar
Katul, G., Porporato, A., Cava, D. & Siqueira, M. 2006 An analysis of intermittency, scaling, and surface renewal in atmospheric surface layer turbulence. Physica D 215 (2), 117126.CrossRefGoogle Scholar
LaCasce, J.H. 2008 Lagrangian statistics from oceanic and atmospheric observations. In Transport and Mixing in Geophysical Flows, Lecture Notes in Physics (ed. J.B. Weiss & A. Provenzale), pp. 165–218. Springer.CrossRefGoogle Scholar
Monin, A.S. & Yaglom, A.M. 1971 Statistical Fluid Mechanics, vols. 1 & 2. MIT Press.Google Scholar
Mordant, N., Metz, P., Michel, O. & Pinton, J.-F. 2001 Measurement of Lagrangian velocity in fully developed turbulence. Phys. Rev. Lett. 87 (21), 214501.CrossRefGoogle ScholarPubMed
Ouellette, N.T., Xu, H., Bourgoin, M. & Bodenschatz, E. 2006 Small-scale anisotropy in Lagrangian turbulence. New J. Phys. 8 (6), 102.CrossRefGoogle Scholar
Pinton, J.-F. & Sawford, B.L. 2012 A Lagrangian view of turbulent dispersion and mixing. In Ten Chapters in Turbulence (ed. P.A. Davidson, Y. Kaneda & K.R. Sreenivasan), pp. 132–175. Cambridge University Press.CrossRefGoogle Scholar
Pope, S.B 2002 A stochastic Lagrangian model for acceleration in turbulent flows. Phys. Fluids 14 (7), 23602375.CrossRefGoogle Scholar
Poulain, C., Mazellier, N., Gervais, P., Gagne, Y. & Baudet, C. 2004 Spectral vorticity and Lagrangian velocity measurements in turbulent jets. Flow Turbul. Combust. 72 (2), 245271.CrossRefGoogle Scholar
Sawford, B.L. 1991 Reynolds number effects in Lagrangian stochastic models of turbulent dispersion. Phys. Fluids A 3 (6), 15771586.CrossRefGoogle Scholar
Shnapp, R. 2021 On small-scale and large-scale intermittency of Lagrangian statistics in canopy flow. J. Fluid Mech. 913, R2.CrossRefGoogle Scholar
Shnapp, R., Bohbot-Raviv, Y., Liberzon, A. & Fattal, E. 2020 Turbulence-obstacle interactions in the Lagrangian framework: applications for stochastic modeling in canopy flows. Phys. Rev. Fluids 5 (9), 094601.CrossRefGoogle Scholar
Shnapp, R., Shapira, E., Peri, D., Bohbot-Raviv, Y., Fattal, E. & Liberzon, A. 2019 Extended 3D-PTV for direct measurements of Lagrangian statistics of canopy turbulence in a wind tunnel. Sci. Rep. 9 (1), 113.CrossRefGoogle Scholar
Stelzenmuller, N., Polanco, J.I., Vignal, L., Vinkovic, I. & Mordant, N. 2017 Lagrangian acceleration statistics in a turbulent channel flow. Phys. Rev. Fluids 2 (5), 054602.CrossRefGoogle Scholar
Tennekes, H. & Lumley, J.L. 1972 A First Course in Turbulence. MIT Press.CrossRefGoogle Scholar
Toschi, F. & Bodenschatz, E. 2009 Lagrangian properties of particles in turbulence. Annu. Rev. Fluid Mech. 41, 375.CrossRefGoogle Scholar
Viggiano, B., Basset, T., Solovitz, S., Barois, T., Gibert, M., Mordant, N., Chevillard, L., Volk, R., Bourgoin, M. & Cal, R.B. 2021 Lagrangian diffusion properties of a free shear turbulent jet. arXiv:2102.08333.Google Scholar
Viggiano, B., Friedrich, J., Volk, R., Bourgoin, M., Cal, R.B. & Chevillard, L. 2020 Modelling Lagrangian velocity and acceleration in turbulent flows as infinitely differentiable stochastic processes. J. Fluid Mech. 900, A27.CrossRefGoogle Scholar
Voth, G.A., Satyanarayan, K. & Bodenschatz, E. 1998 Lagrangian acceleration measurements at large Reynolds numbers. Phys. Fluids 10, 2268.CrossRefGoogle Scholar
Yeung, P.K. 2002 Lagrangian investigations of turbulence. Annu. Rev. Fluid Mech. 34 (1), 115142.CrossRefGoogle Scholar
Yeung, P.K. & Pope, S.B. 1989 Lagrangian statistics from direct numerical simulations of isotropic turbulence. J. Fluid Mech. 207, 531.CrossRefGoogle Scholar
Yu, H., Kanov, K., Perlman, E., Graham, J., Frederix, E., Burns, R., Szalay, A., Eyink, G. & Meneveau, C. 2012 Studying Lagrangian dynamics of turbulence using on-demand fluid particle tracking in a public turbulence database. J. Turbul. 13, N12.CrossRefGoogle Scholar