Comptes Rendus
Structures and statistics of fluid turbulence / Structures et statistiques de la turbulence des fluides
Geometrical statistics of fluid deformation: Restricted Euler approximation and the effects of pressure
Comptes Rendus. Physique, Volume 13 (2012) no. 9-10, pp. 878-888.

The geometrical statistics of fluid deformation are analyzed theoretically within the framework of the restricted Euler approximation, and numerically using direct numerical simulations. The restricted Euler analysis predicts that asymptotically a material line element becomes an eigenvector of the velocity gradient regardless its initial orientation. The asymptotic stretching rate equals the intermediate eigenvalue of the strain rate tensor. Analyses of numerical data show that the pressure Hessian is the leading cause to destroy the alignment between the longest axis of the material element and the strongest stretching eigendirection of the strain rate. It also facilitates the alignment between the longest axis of the element and the intermediate eigendirection of the strain rate during initial evolution, but tends to oppose the alignment later.

Les statistiques de la géométrie de la déformation sont analysées dans le cadre théorique de lʼapproximation dʼEuler restreint, et numériquement en utilisant une simulation numérique directe des équations de Navier–Stokes. Sous lʼapproximation dʼEuler restreint, il est prédit quʼune ligne matérielle devienne asymptotiquement un vecteur propre du tenseur des gradients, quʼelle que soit son orientation initiale. De plus, le taux dʼétirement devient égal à la valeur propre intermédiaire du tenseur de déformation. Lʼanalyse des simulations numériques montre que le hessien de pression est la cause principale de la détérioration de lʼalignement du plus grand axe de lʼélément matériel avec la direction principale de plus grande déformation. Le hessien favorise aussi lʼalignement de ce plus grand axe avec la direction principale intermédiaire de la déformation lors de lʼévolution aux temps courts, mais a tendance à sʼy opposer par la suite.

Published online:
DOI: 10.1016/j.crhy.2012.09.003
Keywords: Turbulence, Geometrical statistics, Restricted Euler approximation, Modeling
Mot clés : Turbulence, Statistiques de la géométrie, Approximation dʼEuler restreint, Modélisation

Yi Li 1

1 School of Mathematics and Statistics, University of Sheffield, Sheffield S3 7RH, UK
@article{CRPHYS_2012__13_9-10_878_0,
     author = {Yi Li},
     title = {Geometrical statistics of fluid deformation: {Restricted} {Euler} approximation and the effects of pressure},
     journal = {Comptes Rendus. Physique},
     pages = {878--888},
     publisher = {Elsevier},
     volume = {13},
     number = {9-10},
     year = {2012},
     doi = {10.1016/j.crhy.2012.09.003},
     language = {en},
}
TY  - JOUR
AU  - Yi Li
TI  - Geometrical statistics of fluid deformation: Restricted Euler approximation and the effects of pressure
JO  - Comptes Rendus. Physique
PY  - 2012
SP  - 878
EP  - 888
VL  - 13
IS  - 9-10
PB  - Elsevier
DO  - 10.1016/j.crhy.2012.09.003
LA  - en
ID  - CRPHYS_2012__13_9-10_878_0
ER  - 
%0 Journal Article
%A Yi Li
%T Geometrical statistics of fluid deformation: Restricted Euler approximation and the effects of pressure
%J Comptes Rendus. Physique
%D 2012
%P 878-888
%V 13
%N 9-10
%I Elsevier
%R 10.1016/j.crhy.2012.09.003
%G en
%F CRPHYS_2012__13_9-10_878_0
Yi Li. Geometrical statistics of fluid deformation: Restricted Euler approximation and the effects of pressure. Comptes Rendus. Physique, Volume 13 (2012) no. 9-10, pp. 878-888. doi : 10.1016/j.crhy.2012.09.003. https://comptes-rendus.academie-sciences.fr/physique/articles/10.1016/j.crhy.2012.09.003/

[1] G.K. Batchelor The effects of homogeneous turbulence on material lines and surfaces, Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci., Volume 213 (1952), pp. 349-366

[2] W.J. Cocke Turbulent hydrodynamic line stretching: consequences of isotropy, Phys. Fluids, Volume 12 (1969), pp. 2488-2492

[3] S.A. Orszag Comments on “Turbulent hydrodynamic line stretching: consequences of isotropy”, Phys. Fluids, Volume 13 (1970), pp. 2203-2204

[4] S.S. Girimaji; S.B. Pope Material element deformation in isotropic turbulence, J. Fluid Mech., Volume 220 (1990), pp. 427-458

[5] E. Dresselhaus; M. Tabor The kinematics of stretching and alignment of material elements in general flow fields, J. Fluid Mech., Volume 236 (1991), pp. 415-444

[6] M.J. Huang Correlations of vorticity and material line elements with strain in decaying turbulence, Phys. Fluids, Volume 8 (1996), pp. 2203-2214

[7] J. Duplat; E. Villermaux Persistency of material element deformation in isotropic flows and growth rate of lines and surfaces, Eur. Phys. J. B, Volume 18 (2000), pp. 353-361

[8] S. Kida; S. Goto Line statistics: Stretching rate of passive lines in turbulence, Phys. Fluids, Volume 14 (2002), pp. 352-361

[9] K. Ohkitani Numerical study of comparison of vorticity and passive vectors in turbulence and inviscid flows, Phys. Rev. E, Volume 65 (2002), p. 046304

[10] M. Guala; B. Lüthi; A. Liberzon; A. Tsinober; W. Kinzelbach On the evolution of material lines and vorticity in homogeneous turbulence, J. Fluid Mech., Volume 533 (2005), pp. 339-359

[11] S. Goto; S. Kida Reynolds-number dependence of line and surface stretching in turbulence: folding effects, J. Fluid. Mech., Volume 586 (2007), pp. 59-81

[12] M. Chertkov; A. Pumir; B.I. Shraiman Lagrangian tetrad dynamics and the phenomenology of turbulence, Phys. Fluids, Volume 11 (1999), pp. 2394-2410

[13] E. Jeong; S.S. Girimaji Velocity-gradient dynamics in turbulence: effect of viscosity and forcing, Theoret. Comput. Fluid Dyn., Volume 16 (2003), pp. 421-432

[14] Y. Li; C. Meneveau Origin of non-Gaussian statistics in hydrodynamic turbulence, Phys. Rev. Lett., Volume 95 (2005), p. 164502

[15] L. Chevillard; C. Meneveau Lagrangian dynamics and statistical geometric structure of turbulence, Phys. Rev. Lett., Volume 97 (2006), p. 174501

[16] Y. Li; L. Chevillard; G. Eyink; C. Meneveau Matrix exponential-based closures for the turbulent subgrid-scale stress tensor, Phys. Rev. E, Volume 79 (2009), p. 016305

[17] A. Tsinober; B. Galanti Exploratory numerical experiments on the differences between genuine and “passive” turbulence, Phys. Fluids, Volume 15 (2003), p. 3514

[18] P. Vieillefosse Local interaction between vorticity and shear in a perfect incompressible fluid, J. Phys., Volume 43 (1982), pp. 837-842

[19] P. Vieillefosse Internal motion of a small element of fluid in an inviscid flow, Physica A, Volume 125 (1984), pp. 150-162

[20] B.J. Cantwell Exact solution of a restricted Euler equation for the velocity gradient tensor, Phys. Fluids A, Volume 4 (1992), pp. 782-793

[21] Y. Li; C. Meneveau Material deformation in a restricted Euler model for turbulent flows: analytic solution and numerical tests, Phys. Fluids, Volume 19 (2007), p. 015104

[22] L. Chevillard; C. Meneveau; L. Biferale; F. Toschi Modeling the pressure Hessian and viscous Laplacian in turbulence: comparisons with DNS and implications on velocity gradient dynamics, Phys. Fluids, Volume 20 (2008), p. 101504

[23] K. Ohkitani Eigenvalue problems in three-dimensional Euler flows, Phys. Fluids A, Volume 5 (1993), pp. 2570-2572

[24] K. Ohkitani; S. Kishiba Nonlocal nature of vortex stretching in an inviscid fluid, Phys. Fluids, Volume 7 (1995), pp. 411-421

[25] D. Chae On the finite-time singularities of the 3D incompressible Euler equations, Comm. Pure Appl. Math., Volume 60 (2007), pp. 597-617

[26] H. Liu; E. Tadmor Spectral dynamics of the velocity gradient field in restricted flows, Commun. Math. Phys., Volume 228 (2002), pp. 435-466

[27] S.B. Pope Turbulent Flows, Cambridge University Press, Cambridge, 2000

[28] A. Pumir; M. Wilkinson Orientation statistics of small particles in turbulence, New J. Phys., Volume 13 (2011), p. 093030

Cited by Sources:

Comments - Policy