A nonlocal expression for the turbulent Reynolds stress $\overline{u^{\prime }v^{\prime }}$ 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
| \[ \overline{u^{\prime }v^{\prime }}=-\tilde{v}\ell \partial _y u \] |
with $\ell $ a mean free path but, by contrast with Prandtl, with $\tilde{v}$ a 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 $(10\pi )^{-1}\approx 0.031$ 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 Kármán constant is found to be $\kappa =6^{-1/2}\approx 0.41$.
Une expression non locale de la contrainte de Reynolds turbulente $\overline{u^{\prime }v^{\prime }}$ 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
| \[ \overline{u^{\prime }v^{\prime }}=-\tilde{v}\ell \partial _y u \] |
avec $\ell $ un libre parcours moyen mais, contrairement à Prandtl, avec $\tilde{v}$ une vitesse de transfert 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 $(10\pi )^{-1}\approx 0,{\hspace{-0.55542pt}}031$ 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 Kármán est prédite comme valant $\kappa =6^{-1/2}\approx 0,{\hspace{-0.55542pt}}41$.
Accepted:
Published online:
Mots-clés : Turbulence, contrainte de Reynolds, fermetures
Emmanuel Villermaux 1
CC-BY 4.0
@article{CRMECA_2025__353_G1_879_0,
author = {Emmanuel Villermaux},
title = {The nonlocal nature of the {Reynolds} stress},
journal = {Comptes Rendus. M\'ecanique},
pages = {879--899},
year = {2025},
publisher = {Acad\'emie des sciences, Paris},
volume = {353},
doi = {10.5802/crmeca.316},
language = {en},
}
Emmanuel Villermaux. The nonlocal nature of the Reynolds stress. Comptes Rendus. Mécanique, Volume 353 (2025), pp. 879-899. doi: 10.5802/crmeca.316
[1] On the dynamical theory of incompressible viscous fluids and the determination of the criterion, Philos. Trans. R. Soc. Lond., Ser. A, Volume 186 (1895), pp. 123-164 | Zbl
[2] Turbulent flows, Cambridge University Press, 2000, xxxiv+771 pages | DOI | MR | Zbl
[3] Homogeneous turbulence dynamics, Springer, 2018, xxix+897 pages | MR | DOI | Zbl
[4] Bericht über Untersuchungen zur ausgebildeten Turbulenz, Z. Angew. Math. Mech., Volume 5 (1925), pp. 136-139 | DOI | Zbl
[5] The transport of vorticity and heat through fluids in turbulent motion, Proc. R. Soc. Lond., Ser. A, Volume 135 (1932), pp. 685-705 | Zbl
[6] Momentum, vorticity & scalar transport in turbulence: the Taylor–Prandtl controversy (2025) (To appear in J. Fluid Mech.)
[7] Mechanische Ähnlichkeit und Turbulenz, Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl., Volume 1930 (1930), pp. 58-76 | Zbl
[8] Boundary-layer theory, McGraw-Hill Series in Mechanical Engineering, McGraw-Hill, 1979, xxii+817 pages | Zbl | MR
[9] About Boussinesq’s turbulent viscosity hypothesis: historical remarks and a direct evaluation of its validity, C. R. Méc., Volume 335 (2007), pp. 617-627 | Zbl | DOI
[10] Fluid mechanics, Course of Theoretical Physics, 6, Pergamon Press, 1987, xiii+539 pages | Zbl
[11] Some measurements in the self-preserving jet, J. Fluid Mech., Volume 38 (1969) no. 3, pp. 577-612 | DOI
[12] Bemerkungen zur Theorie der freien Turbulenz, Z. Angew. Math. Mech., Volume 22 (1942), pp. 241-243 | MR | Zbl | DOI
[13] The two-dimensional mixing region, J. Fluid Mech., Volume 41 (1970) no. 2, pp. 327-361 | DOI
[14] Limitations of gradient transport models in random walks and in turbulence, Adv. Geophys., Volume 18 (1975) no. A, pp. 25-60 | DOI
[15] Essai sur la théorie des eaux courantes, Mémoires présentés par divers savants à l’Académie des Sciences de l’Institut de France, XXIII, Imprimerie nationale, 1877, 60 pages | Zbl
[16] Momentum transfer across an open-channel, turbulent flow, J. Fluid Mech., Volume 981 (2024), A24 | DOI | MR | Zbl
[17] Atomic physics, Dover Books on Physics, Dover Publications, 1969, 495 pages
[18] Concentration fluctuations in a round turbulent free jet, Chem. Eng. Sci., Volume 26 (1971), pp. 95-107 | DOI
[19] An integral closure model for the vertical turbulent flux of a scalar in a mixed layer, J. Atmos. Sci., Volume 41 (1984) no. 4, pp. 674-680 | DOI
[20] Statistical nonlocality of dynamically coherent structures, J. Fluid Mech., Volume 966 (2023), A44 | DOI | MR | Zbl
[21] Scaling laws in turbulence, Chaos, Volume 30 (2020), 073137 | DOI | MR | Zbl
[22] Kinetic theory of plastic flow in soft glassy materials, Phys. Rev. Lett., Volume 103 (2009), 036001 | DOI
[23] Nonlocal rheology of granular flows across yield conditions, Phys. Rev. Lett., Volume 111 (2013), 238301 | DOI
[24] Vortices enhance diffusion in dense granular flows, Phys. Rev. Lett., Volume 119 (2017), 178001 | DOI
[25] Impuls- und Warmeaustausch in freier Turbulenz, Z. Angew. Math. Mech., Volume 24 (1944) no. 5, pp. 268-272 | DOI | MR | Zbl
[26] On the dynamical theory of gases, Phil. Mag. (4), Volume CLVII (1867) no. I, pp. 49-88 | DOI
[27] Turbulent transfer and entrainment in a low-density jet, J. Fluid Mech., Volume 968 (2023), A27 | Zbl | DOI | MR
[28] Laminare Strahlausbreitung, Z. Angew. Math. Mech., Volume 13 (1933) no. 4, pp. 260-263 | DOI | Zbl
[29] A new exact solution of the Navier–Stokes equations, C. R. (Dokl.) Acad. Sci. URSS, n. Ser., Volume 43 (1944), pp. 286-288 | MR | Zbl
[30] Studies in turbulent mixing I, Chem. Eng. Sci., Volume 16 (1961), pp. 267-277 | DOI
[31] On similarity solutions for turbulent and heated round jets, Z. Angew. Math. Phys., Volume 37 (1986), pp. 624-631 | Zbl
[32] A simple analytical model for turbulent kinetic energy dissipation for self-similar round turbulent jets, J. Fluid Mech., Volume 983 (2024), A44 | DOI | MR | Zbl
[33] Entrainment, diffusion and effective compressibility in a self-similar turbulent jet, J. Fluid Mech., Volume 947 (2022), A29 | DOI | MR
[34] Berechnung turbulenter Ausbreitungsvorgange, Z. Angew. Math. Mech., Volume 6 (1926) no. 6, pp. 468-478 | DOI | Zbl
[35] Hydrodynamic instabilities in open flows, Hydrodynamics and nonlinear instabilities (C. Godrèche; P. Manneville, eds.) (Collection Aléa-Saclay: Monographs and Texts in Statistical Physics), Volume 3, Cambridge University Press, 1998, pp. 81-294 | DOI | Zbl
[36] Fragmentation versus cohesion, J. Fluid Mech., Volume 898 (2020), P1, 121 pages | DOI | MR | Zbl
[37] On the stability, or instability of certain fluid motion, Proc. Lond. Math. Soc., Volume 11 (1880), pp. 57-72 | MR | Zbl
[38] The mixing layer and its coherence examined from the point of view of two-dimensional turbulence, J. Fluid Mech., Volume 192 (1988), pp. 511-534 | DOI | MR
[39] Stability and transition in shear flows, Applied Mathematical Sciences, 142, Springer, 2001, xiv+556 pages | DOI | MR | Zbl
[40] On density effects and large scale structures in turbulent mixing layers, J. Fluid Mech., Volume 64 (1974) no. 4, pp. 775-815 | DOI
[41] Stability of a shear layer between parallel streams, Phys. Fluids, Volume 6 (1963), pp. 1391-1395 | DOI | Zbl
[42] On the role of viscosity in shear instabilities, Phys. Fluids, Volume 10 (1998), pp. 368-373 | DOI | MR | Zbl
[43] The theory of homogeneous turbulence, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, 1959, xi+197 pages | MR
[44] Structure in turbulent mixing layers and wakes using a chemical reaction, J. Fluid Mech., Volume 109 (1981), pp. 1-24 | DOI
[45] The mixing transition in turbulent flows, J. Fluid Mech., Volume 409 (2000), pp. 69-98 | DOI | MR | Zbl
[46] Local isotropy in turbulent boundary layers at high Reynolds number, J. Fluid Mech., Volume 268 (1994), pp. 333-372 | DOI
[47] Coarse grained scale of turbulent mixtures, Phys. Rev. Lett., Volume 97 (2006), 144506 | DOI
[48] The structure of turbulent shear flow, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, 1976, xi+429 pages | Zbl | MR
[49] Hierarchical random additive process and logarithmic scaling of generalized high order, two-point correlations in turbulent boundary layer flow, Phys. Rev. Fluids, Volume 1 (2016), 024402 | DOI
[50] On the mathematical classification of physical quantities, Proc. Lond. Math. Soc., Volume III (1871) no. 34, pp. 224-233 | MR | Zbl
[51] Turbulent diffusion in the environment, Geophysics and Astrophysics Monographs, 3, Reidel Publishing Company, 1973, xii+249 pages | DOI
[52] Mathematical biology, Biomathematics (Berlin), 19, Springer, 1993, xiv+767 pages | MR | DOI | Zbl
[53] Log at first sight, J. Fluid Mech., Volume 1000 (2024), F6, 5 pages | DOI | MR
[54] Natural logarithms, J. Fluid Mech., Volume 718 (2013), pp. 1-4 | Zbl | DOI
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