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.
Mot clés : Turbulence, Statistiques de la géométrie, Approximation dʼEuler restreint, Modélisation
Yi Li 1
@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}, }
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] 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] Turbulent hydrodynamic line stretching: consequences of isotropy, Phys. Fluids, Volume 12 (1969), pp. 2488-2492
[3] Comments on “Turbulent hydrodynamic line stretching: consequences of isotropy”, Phys. Fluids, Volume 13 (1970), pp. 2203-2204
[4] Material element deformation in isotropic turbulence, J. Fluid Mech., Volume 220 (1990), pp. 427-458
[5] The kinematics of stretching and alignment of material elements in general flow fields, J. Fluid Mech., Volume 236 (1991), pp. 415-444
[6] Correlations of vorticity and material line elements with strain in decaying turbulence, Phys. Fluids, Volume 8 (1996), pp. 2203-2214
[7] 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] Line statistics: Stretching rate of passive lines in turbulence, Phys. Fluids, Volume 14 (2002), pp. 352-361
[9] Numerical study of comparison of vorticity and passive vectors in turbulence and inviscid flows, Phys. Rev. E, Volume 65 (2002), p. 046304
[10] On the evolution of material lines and vorticity in homogeneous turbulence, J. Fluid Mech., Volume 533 (2005), pp. 339-359
[11] Reynolds-number dependence of line and surface stretching in turbulence: folding effects, J. Fluid. Mech., Volume 586 (2007), pp. 59-81
[12] Lagrangian tetrad dynamics and the phenomenology of turbulence, Phys. Fluids, Volume 11 (1999), pp. 2394-2410
[13] Velocity-gradient dynamics in turbulence: effect of viscosity and forcing, Theoret. Comput. Fluid Dyn., Volume 16 (2003), pp. 421-432
[14] Origin of non-Gaussian statistics in hydrodynamic turbulence, Phys. Rev. Lett., Volume 95 (2005), p. 164502
[15] Lagrangian dynamics and statistical geometric structure of turbulence, Phys. Rev. Lett., Volume 97 (2006), p. 174501
[16] Matrix exponential-based closures for the turbulent subgrid-scale stress tensor, Phys. Rev. E, Volume 79 (2009), p. 016305
[17] Exploratory numerical experiments on the differences between genuine and “passive” turbulence, Phys. Fluids, Volume 15 (2003), p. 3514
[18] Local interaction between vorticity and shear in a perfect incompressible fluid, J. Phys., Volume 43 (1982), pp. 837-842
[19] Internal motion of a small element of fluid in an inviscid flow, Physica A, Volume 125 (1984), pp. 150-162
[20] Exact solution of a restricted Euler equation for the velocity gradient tensor, Phys. Fluids A, Volume 4 (1992), pp. 782-793
[21] Material deformation in a restricted Euler model for turbulent flows: analytic solution and numerical tests, Phys. Fluids, Volume 19 (2007), p. 015104
[22] 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] Eigenvalue problems in three-dimensional Euler flows, Phys. Fluids A, Volume 5 (1993), pp. 2570-2572
[24] Nonlocal nature of vortex stretching in an inviscid fluid, Phys. Fluids, Volume 7 (1995), pp. 411-421
[25] On the finite-time singularities of the 3D incompressible Euler equations, Comm. Pure Appl. Math., Volume 60 (2007), pp. 597-617
[26] Spectral dynamics of the velocity gradient field in restricted flows, Commun. Math. Phys., Volume 228 (2002), pp. 435-466
[27] Turbulent Flows, Cambridge University Press, Cambridge, 2000
[28] Orientation statistics of small particles in turbulence, New J. Phys., Volume 13 (2011), p. 093030
Cited by Sources:
Comments - Policy