Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-19T10:03:15.218Z Has data issue: false hasContentIssue false
  • English
  • Français

Modelling Lagrangian velocity and acceleration in turbulent flows as infinitely differentiable stochastic processes

Published online by Cambridge University Press:  11 August 2020

Bianca Viggiano Open the ORCID record for Bianca Viggiano [Opens in a new window] ,
Jan Friedrich ,
Romain Volk ,
Mickael Bourgoin ,
Raúl Bayoán Cal  and
Laurent Chevillard Open the ORCID record for Laurent Chevillard [Opens in a new window]
Bianca Viggiano
Affiliation:
Laboratoire de Physique, Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, 46 allée d'Italie, F-69342Lyon, France Department of Mechanical and Materials Engineering, Portland State University, Portland, OR97201, USA
Jan Friedrich
Affiliation:
Laboratoire de Physique, Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, 46 allée d'Italie, F-69342Lyon, France
Romain Volk
Affiliation:
Laboratoire de Physique, Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, 46 allée d'Italie, F-69342Lyon, France
Mickael Bourgoin
Affiliation:
Laboratoire de Physique, Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, 46 allée d'Italie, F-69342Lyon, France
Raúl Bayoán Cal
Affiliation:
Laboratoire de Physique, Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, 46 allée d'Italie, F-69342Lyon, France Department of Mechanical and Materials Engineering, Portland State University, Portland, OR97201, USA
Laurent Chevillard*
Affiliation:
Laboratoire de Physique, Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, 46 allée d'Italie, F-69342Lyon, France
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We develop a stochastic model for Lagrangian velocity as it is observed in experimental and numerical fully developed turbulent flows. We define it as the unique statistically stationary solution of a causal dynamics, given by a stochastic differential equation. In comparison with previously proposed stochastic models, the obtained process is infinitely differentiable at a given finite Reynolds number, and its second-order statistical properties converge to those of an Ornstein–Uhlenbeck process in the infinite Reynolds number limit. In this limit, it exhibits furthermore intermittent scaling properties, as they can be quantified using higher-order statistics. To achieve this, we begin with generalizing the two-layered embedded stochastic process of Sawford (Phys. Fluids A, vol. 3 (6), 1991, pp. 1577–1586) by considering an infinite number of layers. We then study, both theoretically and numerically, the convergence towards a smooth (i.e. infinitely differentiable) Gaussian process. To include intermittent corrections, we follow similar considerations as for the multifractal random walk of Bacry et al. (Phys. Rev. E, vol. 64, 2001, 026103). We derive in an exact manner the statistical properties of this process, and compare them with those estimated from Lagrangian trajectories extracted from numerically simulated turbulent flows. Key predictions of the multifractal formalism regarding the acceleration correlation function and high-order structure functions are also derived. Through these predictions, we understand phenomenologically peculiar behaviours of the fluctuations in the dissipative range, that are not reproduced by our stochastic process. The proposed theoretical method regarding the modelling of infinitely differentiability opens the route to the full stochastic modelling of velocity, including the peculiar action of viscosity on the very fine scales.

JFM classification

Turbulent Flows: Homogeneous turbulence Turbulent Flows: Isotropic turbulence Turbulent Flows: Turbulence theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 900 , 10 October 2020 , A27
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Arneodo, A., Bacry, E. & Muzy, J.-F. 1998 Random cascades on wavelet dyadic tree. J. Math. Phys. 39, 4142–4164.10.1063/1.532489CrossRefGoogle Scholar
Arneodo, A., Benzi, R., Berg, J., Biferale, L., Bodenschatz, E., Busse, A., Calzavarini, E., Castaing, B., Cencini, M., Chevillard, L., et al. . 2008 Universal intermittent properties of particle trajectories in highly turbulent flows. Phys. Rev. Lett. 100, 254504.10.1103/PhysRevLett.100.254504CrossRefGoogle ScholarPubMed
Arratia, A., Cabana, A. & Cabana, E. M. 2014 Modeling stationary data by a class of generalized Ornstein–Uhlenbeck processes: the Gaussian case. In International Symposium on Intelligent Data Analysis, pp. 13–24. Springer.10.1007/978-3-319-12571-8_2CrossRefGoogle Scholar
Bacry, E., Delour, J. & Muzy, J.-F. 2001 Multifractal random walk. Phys. Rev. E 64, 026103.10.1103/PhysRevE.64.026103CrossRefGoogle ScholarPubMed
Bacry, E. & Muzy, J.-F. 2003 Log-infinitely divisible multifractal processes. Commun. Math. Phys. 236 (3), 449–475.10.1007/s00220-003-0827-3CrossRefGoogle Scholar
Batchelor, G. K. 1951 Pressure fluctuations in isotropic turbulence. In Mathematical Proceedings of the Cambridge Philosophical Society, vol. 47, pp. 359–374. Cambridge University Press.10.1017/S0305004100026712CrossRefGoogle Scholar
Bec, J., Biferale, L., Boffetta, G., Celani, A., Cencini, M., Lanotte, A., Musacchio, S. & Toschi, F. 2006 Acceleration statistics of heavy particles in turbulence. J. Fluid Mech. 550, 349–358.10.1017/S002211200500844XCrossRefGoogle Scholar
Bec, J., Biferale, L., Cencini, M., Lanotte, A. S. & Toschi, F. 2011 Spatial and velocity statistics of inertial particles in turbulent flows. J. Phys.: Conf. Ser. 333, 012003.Google Scholar
Beck, C. 2003 Lagrangian acceleration statistics in turbulent flows. Europhys. Lett. 64 (2), 151–157.10.1209/epl/i2003-00498-4CrossRefGoogle Scholar
Bentkamp, L., Lalescu, C. C. & Wilczek, M. 2019 Persistent accelerations disentangle Lagrangian turbulence. Nat. Commun. 10, 1–810.1038/s41467-019-11060-9CrossRefGoogle ScholarPubMed
Benzi, R., Biferale, L., Crisanti, A., Paladin, G., Vergassola, M. & Vulpiani, A. 1993 A random process for the construction of multiaffine fields. Physica D 65, 352–358.10.1016/0167-2789(93)90060-ECrossRefGoogle Scholar
Benzi, R., Biferale, L., Fisher, R., Lamb, D. Q. & Toschi, F. 2010 Inertial range Eulerian and Lagrangian statistics from numerical simulations of isotropic turbulence. J. Fluid Mech. 653, 221–244.10.1017/S002211201000056XCrossRefGoogle Scholar
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), 064502.10.1103/PhysRevLett.93.064502CrossRefGoogle ScholarPubMed
Borgas, M. S. 1993 The multifractal Lagrangian nature of turbulence. Phil. Trans. R. Soc. Lond. A 342, 379–411.Google Scholar
Borgas, M. S. & Sawford, B. L. 1994 Stochastic equations with multifractal random increments for modeling turbulent dispersion. Phys. Fluids 6 (2), 618–633.10.1063/1.868358CrossRefGoogle Scholar
Bos, W. J. T., Chevillard, L., Scott, J. F. & Rubinstein, R. 2012 Reynolds number effect on the velocity increment skewness in isotropic turbulence. Phys. Fluids 24 (1), 015108.10.1063/1.3678338CrossRefGoogle Scholar
Buaria, D., Pumir, A., Bodenschatz, E. & Yeung, P. K. 2019 Extreme velocity gradients in turbulent flows. New J. Phys. 21 (4), 043004.10.1088/1367-2630/ab0756CrossRefGoogle Scholar
Castaing, B., Gagne, Y. & Hopfinger, E. 1990 Velocity probability density functions of high Reynolds number turbulence. Physica D 46, 177–200.10.1016/0167-2789(90)90035-NCrossRefGoogle Scholar
Chevillard, L. 2017 Regularized fractional ornstein-uhlenbeck processes and their relevance to the modeling of fluid turbulence. Phys. Rev. E 96 (3), 033111.10.1103/PhysRevE.96.033111CrossRefGoogle ScholarPubMed
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, 899–928.10.1016/j.crhy.2012.09.002CrossRefGoogle Scholar
Chevillard, L., Castaing, B. & Lévêque, E. 2005 On the rapid increase of intermittency in the near-dissipation range of fully developed turbulence. Eur. Phys. J. B 45, 561–567.CrossRefGoogle Scholar
Chevillard, L., Castaing, B., Lévêque, E. & Arneodo, A. 2006 Unified multifractal description of velocity increments statistics in turbulence: intermittency and skewness. Physica D 218, 77–82.10.1016/j.physd.2006.04.011CrossRefGoogle Scholar
Chevillard, L., Garban, C., Rhodes, R. & Vargas, V. 2019 On a skewed and multifractal unidimensional random field, as a probabilistic representation of Kolmogorov's views on turbulence. Ann. Henri Poincaré 20 (11), 3693–3741.10.1007/s00023-019-00842-yCrossRefGoogle 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
Dubrulle, B. 2019 Beyond Kolmogorov cascades. J. Fluid Mech. 867, P1.10.1017/jfm.2019.98CrossRefGoogle Scholar
Friedrich, R. 2003 Statistics of Lagrangian velocities in turbulent flows. Phys. Rev. Lett. 90 (8), 084501.10.1103/PhysRevLett.90.084501CrossRefGoogle ScholarPubMed
Frisch, U. 1995 Turbulence, The Legacy of A.N. Kolmogorov. Cambridge University Press.CrossRefGoogle Scholar
Hill, R. J. 2002 Scaling of acceleration in locally isotropic turbulence. J. Fluid Mech. 452, 361–370.CrossRefGoogle Scholar
Huck, P. D., Machicoane, N. & Volk, R. 2019 Lagrangian acceleration timescales in anisotropic turbulence. Phys. Rev. Fluids 4 (6), 064606.10.1103/PhysRevFluids.4.064606CrossRefGoogle Scholar
Ishihara, T., Kaneda, Y., Yokokawa, M., Itakura, K. & Uno, A. 2007 Small-scale statistics in high-resolution direct numerical simulation of turbulence: Reynolds number dependence of one-point velocity gradient statistics. J. Fluid Mech. 592, 335–366.10.1017/S0022112007008531CrossRefGoogle Scholar
La Porta, A., Voth, G. A., Crawford, A. M., Alexander, J. & Bodenschatz, E. 2001 Fluid particle accelerations in fully developed turbulence. Nature 409, 1017–1019.10.1038/35059027CrossRefGoogle ScholarPubMed
Lamorgese, A. G., Pope, S. B., Yeung, P. K. & Sawford, B. L. 2007 A conditionally cubic-Gaussian stochastic Lagrangian model for acceleration in isotropic turbulence. J. Fluid Mech. 582, 423–448.CrossRefGoogle Scholar
Li, Y., Perlman, E., Wan, M., Yang, Y., Burns, R., Meneveau, C., Burns, R., Chen, S., Szalay, A. & Eyink, G. 2008 A public turbulence database cluster and applications to study Lagrangian evolution of velocity increments in turbulence. J. Turbul. 9, 31.CrossRefGoogle Scholar
Meneveau, C. 1996 On the transition between viscous and inertial-range scaling of turbulence structure functions. Phys. Rev. E 54, 3657–3663.10.1103/PhysRevE.54.3657CrossRefGoogle ScholarPubMed
Meneveau, C. & Sreenivasan, K. R. 1987 Simple multifractal cascade model for fully developed turbulence. Phys. Rev. Lett. 59, 1424–1427.10.1103/PhysRevLett.59.1424CrossRefGoogle ScholarPubMed
Minier, J.-P., Chibbaro, S. & Pope, S. B. 2014 Guidelines for the formulation of Lagrangian stochastic models for particle simulations of single-phase and dispersed two-phase turbulent flows. Phys. Fluids 26 (11), 113303.CrossRefGoogle Scholar
Monin, A. S. & Yaglom, A. M. 1971 Statistical Fluid Mechanics, vol. 1 and 2. MIT.Google Scholar
Mordant, N., Delour, J., Léveque, E., Arnéodo, A. & Pinton, J.-F. 2002 Long time correlations in Lagrangian dynamics: a key to intermittency in turbulence. Phys. Rev. Lett. 89 (25), 254502.CrossRefGoogle ScholarPubMed
Mordant, N., Delour, J., Lévêque, E., Michel, O., Arneodo, A. & Pinton, J.-F. 2003 Lagrangian velocity fluctuations in fully developed turbulence: scaling, intermittency and dynamics. J. Stat. Phys. 112, 701–717.10.1023/A:1027352417365CrossRefGoogle 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
Nelkin, M. 1990 Multifractal scaling of velocity derivatives in turbulence. Phys. Rev. A 42 (12), 7226–7229.CrossRefGoogle ScholarPubMed
Nualart, D. 2000 The Malliavin Calculus and Related Topics. Springer.Google Scholar
Paladin, G. & Vulpiani, A. 1987 Degrees of freedom of turbulence. Phys. Rev. A 35, 1971–1973.CrossRefGoogle ScholarPubMed
Papoulis, A. 1991 Probability, Random Variables and Stochastic Processes, 3rd edn. McGraw-Hill International Editions.Google Scholar
Pereira, R. M., Garban, C. & Chevillard, L. 2016 A dissipative random velocity field for fully developed fluid turbulence. J. Fluid Mech. 794, 369–408.CrossRefGoogle Scholar
Pereira, R. M., Moriconi, L. & Chevillard, L. 2018 A multifractal model for the velocity gradient dynamics in turbulent flows. J. Fluid Mech. 839, 430–467.CrossRefGoogle Scholar
Pinton, J.-F. & Sawford, B. L. 2012 A Lagrangian view of turbulent dispersion and mixing. In Ten Chapters in Turbulence (ed. Davidson, P. A., Kaneda, Y. & Sreenivasan, K. R.), pp. 132–175. Cambridge University Press.Google Scholar
Pope, S. B. 1990 Lagrangian microscales in turbulence. Phil. Trans. R. Soc. Lond. A 333 (1631), 309–319.Google Scholar
Pope, S. B. 2002 A stochastic Lagrangian model for acceleration in turbulent flows. Phys. Fluids 14 (7), 2360–2375.CrossRefGoogle Scholar
Pope, S. B. & Chen, Y. L. 1990 The velocity-dissipation probability density function model for turbulent flows. Phys. Fluids A 2 (8), 1437–1449.CrossRefGoogle Scholar
Reynolds, A. M. 2003 Superstatistical mechanics of tracer-particle motions in turbulence. Phys. Rev. Lett. 91 (8), 084503.CrossRefGoogle ScholarPubMed
Reynolds, A. M., Mordant, N., Crawford, A. M. & Bodenschatz, E. 2005 On the distribution of Lagrangian accelerations in turbulent flows. New J. Phys. 7 (1), 58.10.1088/1367-2630/7/1/058CrossRefGoogle Scholar
Rhodes, R. & Vargas, V. 2014 Gaussian multiplicative chaos and applications: a review. Probab. Surveys 11, 315–392.CrossRefGoogle Scholar
Robert, R. & Vargas, V. 2010 Gaussian multiplicative chaos revisited. Ann. Probab. 38 (2), 605–631.CrossRefGoogle Scholar
Sawford, B. L. 1991 Reynolds number effects in Lagrangian stochastic models of turbulent dispersion. Phys. Fluids A 3 (6), 1577–1586.10.1063/1.857937CrossRefGoogle Scholar
Sawford, B. L., Yeung, P. K., Borgas, M. S., Vedula, P., La Porta, A., Crawford, A. M. & Bodenschatz, E. 2003 Conditional and unconditional acceleration statistics in turbulence. Phys. Fluids 15 (11), 3478–3489.CrossRefGoogle Scholar
Schmitt, F. & Marsan, D. 2001 Stochastic equations generating continuous multiplicative cascades. Eur. Phys. J. B 20 (1), 3–6.Google Scholar
Tennekes, H. & Lumley, J. L. 1972 A First Course in Turbulence. MIT.CrossRefGoogle Scholar
Toschi, F. & Bodenschatz, E. 2009 Lagrangian properties of particles in turbulence. Ann. Rev. Fluid Mech. 41, 375–404.CrossRefGoogle Scholar
Voth, G. A., Satyanarayan, K. & Bodenschatz, E. 1998 Lagrangian acceleration measurements at large Reynolds numbers. Phys. Fluids 10, 2268–2280.CrossRefGoogle Scholar
Wilson, J. D. & Sawford, B. L. 1996 Review of Lagrangian stochastic models for trajectories in the turbulent atmosphere. Boundary-Layer Meteorol. 78, 191–120.CrossRefGoogle Scholar
Yakhot, V. & Sreenivasan, K. R. 2005 Anomalous scaling of structure functions and dynamic constraints on turbulence simulations. J. Stat. Phys. 121 (5-6), 823–841.CrossRefGoogle Scholar
Yeung, P. K. & Pope, S. B. 1989 Lagrangian statistics from direct numerical simulations of isotropic turbulence. J. Fluid Mech. 207, 531–586.CrossRefGoogle Scholar
Yeung, P. K., Pope, S. B., Kurth, E. A. & Lamorgese, A. G. 2007 Lagrangian conditional statistics, acceleration and local relative motion in numerically simulated isotropic turbulence. J. Fluid Mech. 582, 399–422.CrossRefGoogle Scholar
Yeung, P. K., Pope, S. B., Lamorgese, A. G. & Donzis, D. A. 2006 Acceleration and dissipation statistics of numerically simulated isotropic turbulence. Phys. Fluids 18 (6), 065103.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, N12.10.1080/14685248.2012.674643CrossRefGoogle Scholar