Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-06-06T12:46:32.161Z Has data issue: false hasContentIssue false
  • English
  • Français

Lagrangian stochastic modelling of acceleration in turbulent wall-bounded flows

Published online by Cambridge University Press:  14 April 2020

Alessio Innocenti Open the ORCID record for Alessio Innocenti [Opens in a new window] ,
Nicolas Mordant Open the ORCID record for Nicolas Mordant [Opens in a new window] ,
Nick Stelzenmuller  and
Sergio Chibbaro Open the ORCID record for Sergio Chibbaro [Opens in a new window]
Alessio Innocenti
Affiliation:
Sorbonne Université, CNRS, UMR 7190, Institut Jean Le Rond d’Alembert, F-75005Paris, France
Nicolas Mordant
Affiliation:
Laboratoire des Ecoulements Géophysiques et Industriels, Université Grenoble Alpes, CNRS, Grenoble-INP, F-38000Grenoble, France
Nick Stelzenmuller
Affiliation:
Laboratoire des Ecoulements Géophysiques et Industriels, Université Grenoble Alpes, CNRS, Grenoble-INP, F-38000Grenoble, France
Sergio Chibbaro*
Affiliation:
Sorbonne Université, CNRS, UMR 7190, Institut Jean Le Rond d’Alembert, F-75005Paris, France
*
†Email address for correspondence: sergio.chibbaro@sorbonne-universite.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

The Lagrangian approach is natural for studying issues of turbulent dispersion and mixing. We propose in this work a general Lagrangian stochastic model for inhomogeneous turbulent flows, using velocity and acceleration as dynamical variables. The model takes the form of a diffusion process, and the coefficients of the model are determined via Kolmogorov theory and the requirement of consistency with velocity-based models. We show that this model generalises both the acceleration-based models for homogeneous flows as well as velocity-based generalised Langevin models. The resulting closed model is applied to a channel flow at high Reynolds number, and compared to experiments as well as direct numerical simulations. A hybrid approach coupling the stochastic model with a Reynolds-averaged Navier–Stokes model is used to obtain a self-consistent model, as is commonly used in probability density function methods. Results highlight that most of the acceleration features are well represented, notably the anisotropy between streamwise and wall-normal components and the strong intermittency. These results are valuable, since the model improves on velocity-based models for boundary layers while remaining relatively simple. Our model also sheds some light on the statistical mechanisms at play in the near-wall region.

JFM classification

Turbulent Flows: Turbulence modelling Turbulent Flows: Turbulent boundary layers
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 892 , 10 June 2020 , A38
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

Biferale, L., Bodenschatz, E., Cencini, M., Lanotte, A. S., Ouellette, N. T., Toschi, F. & Xu, H. 2008 Lagrangian structure functions in turbulence: a quantitative comparison between experiment and direct numerical simulation. Phys. Fluids 20 (6), 065103.CrossRefGoogle Scholar
Chibbaro, S. & Minier, J.-P. 2011 A note on the consistency of hybrid Eulerian/Lagrangian approach to multiphase flows. Intl J. Multiphase Flow 37 (3), 293–297.CrossRefGoogle Scholar
Chibbaro, S. & Minier, J.-P. 2014 Stochastic Methods in Fluid Mechanics. Springer.CrossRefGoogle Scholar
Dreeben, T. D. & Pope, S. B. 1997 Probability density function and Reynolds-stress modeling of new near-wall turbulent flows. Phys. Fluids 9, 154–163.CrossRefGoogle Scholar
Dreeben, T. D. & Pope, S. B. 1998 Probability density function/Monte Carlo simulation of near-wall turbulent flows. J. Fluid Mech. 357, 141–166.CrossRefGoogle Scholar
Durbin, P. A. 1991 Near-wall turbulence closure modeling without damping functions. Theor. Comput. Fluid Dyn. 3 (1), 1–13.Google Scholar
Durbin, P. A. 1993 A Reynolds stress model for near-wall turbulence. J. Fluid Mech. 249, 465–498.CrossRefGoogle Scholar
Fox, R. O. 2003 Computational Models for Turbulent Reacting Flows. Cambridge University Press.CrossRefGoogle Scholar
Gardiner, C. W. 1990 Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Springer.Google Scholar
Gotoh, T. & Kraichnan, R. H. 2004 Turbulence and Tsallis statistics. Physica D 193 (1–4), 231–244.Google Scholar
Hockney, R. W. & Eastwood, J. W. 1988 Computer Simulation Using Particles. CRC Press.CrossRefGoogle Scholar
Jenny, P., Pope, S. B., Muradoglu, M. & Caughey, D. A. 2001 A hybrid algorithm for the joint pdf equation of turbulent reactive flows. J. Comput. Phys. 166 (2), 218–252.CrossRefGoogle Scholar
Kloeden, P. E. & Platen, E. 1992 Numerical Solution of Stochastic Differential Equations. Springer.CrossRefGoogle Scholar
Krasnoff, E. & Peskin, R. L. 1971 The Langevin model for turbulent diffusion. Geophys. Astrophys. Fluid Dyn. 2 (1), 123–146.CrossRefGoogle Scholar
La Porta, A., Voth, G. A., Crawford, A. M., Alexander, J. & Bodenschatz, E. 2001 Fluid particle accelerations in fully developed turbulence. Nature 409 (6823), 1017–1019.CrossRefGoogle 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
Launder, B. E., Reece, G. Jr & Rodi, W. 1975 Progress in the development of a Reynolds-stress turbulence closure. J. Fluid Mech. 68 (3), 537–566.CrossRefGoogle Scholar
Lee, C., Yeo, K. & Choi, J.-I. 2004 Intermittent nature of acceleration in near wall turbulence. Phys. Rev. Lett. 92 (14), 144502.CrossRefGoogle ScholarPubMed
Lundgren, T. S. 1969 Model equation for nonhomogeneous turbulence. Phys. Fluids 12 (3), 485–497.CrossRefGoogle Scholar
Marconi, U. M. B., Puglisi, A., Rondoni, L. & Vulpiani, A. 2008 Fluctuation–dissipation: response theory in statistical physics. Phys. Rep. 461 (4–6), 111–195.Google Scholar
Meneveau, C. 2011 Lagrangian dynamics and models of the velocity gradient tensor in turbulent flows. Annu. Rev. Fluid Mech. 43, 219–245.CrossRefGoogle Scholar
Minier, J.-P. 2016 Statistical descriptions of polydisperse turbulent two-phase flows. Phys. Rep. 665, 1–122.Google Scholar
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
Minier, J.-P. & Peirano, E. 2001 The PDF approach to turbulent and polydispersed two-phase flows. Phys. Rep. 352 (1–3), 1–214.Google Scholar
Minier, J.-P., Peirano, E. & Chibbaro, S. 2003 Weak first-and second-order numerical schemes for stochastic differential equations appearing in Lagrangian two-phase flow modeling. Monte Carlo Meth. Appl. 9 (2), 93–133.CrossRefGoogle Scholar
Monin, A. S. & Yaglom, A. M. 2013 Statistical Fluid Mechanics: Mechanics of Turbulence. Dover.Google Scholar
Mordant, N., Crawford, A. M. & Bodenschatz, E. 2004a Experimental Lagrangian acceleration probability density function measurement. Physica D 193 (14), 245–251.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., Lévêque, E. & Pinton, J.-F. 2004b Experimental and numerical study of the Lagrangian dynamics of high Reynolds turbulence. New J. Phys. 6 (1), 116–160.CrossRefGoogle 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
Muradoglu, M., Pope, S. B. & Caughey, D. A. 2001 The hybrid method for the pdf equations of turbulent reactive flows: consistency conditions and correction algorithms. J. Comput. Phys. 172 (2), 841–878.CrossRefGoogle Scholar
Onsager, L. & Machlup, S. 1953 Fluctuations and irreversible processes. Phys. Rev. 91 (6), 1505.CrossRefGoogle Scholar
Ouellette, N. T., Xu, H. & Bodenschatz, E. 2006 A quantitative study of three-dimensional Lagrangian particle tracking algorithms. Exp. Fluids 40, 301–313.CrossRefGoogle Scholar
Peirano, E., Chibbaro, S., Pozorski, J. & Minier, J.-P. 2006 Mean-field/PDF numerical approach for polydispersed turbulent two-phase flows. Prog. Energy Combust. Sci. 32 (3), 315–371.CrossRefGoogle Scholar
Pope, S. B. 1981 Transport equation for the joint probability density function of velocity and scalars in turbulent flow. Phys. Fluids 24 (4), 588–596.CrossRefGoogle Scholar
Pope, S. B. 1985 Pdf methods for turbulent reactive flows. Prog. Energy Combust. Sci. 11, 119–192.CrossRefGoogle Scholar
Pope, S. B. 1994a Lagrangian pdf methods for turbulent reactive flows. Annu. Rev. Fluid Mech. 26, 23–63.CrossRefGoogle Scholar
Pope, S. B. 1994b On the relationship between stochastic Lagrangian models of turbulence and second-order closures. Phys. Fluids 6 (2), 973–985.CrossRefGoogle Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Pope, S. B. 2002 Stochastic Lagrangian models of velocity in homogeneous turbulent shear flow. Phys. Fluids 14 (5), 1696–1702.CrossRefGoogle Scholar
Pope, S. B. 2014 The determination of turbulence-model statistics from the velocity–acceleration correlation. J. Fluid Mech. 757, R1.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. 2003a On the application of nonextensive statistics to Lagrangian turbulence. Phys. Fluids 15 (1), L1–L4.CrossRefGoogle Scholar
Reynolds, A. M. 2003b Third-order Lagrangian stochastic modeling. Phys. Fluids 15 (9), 2773–2777.CrossRefGoogle Scholar
Sawford, B. L. 1991 Reynolds number effects in Lagrangian stochastic models of turbulent dispersion. Phys. Fluids A 3 (6), 1577–1586.CrossRefGoogle Scholar
Sawford, B. L. & Guest, F. M. 1988 Uniqueness and universality of Lagrangian stochastic models of turbulent dispersion. In 8th Symposium on Turbulence and Diffusion, pp. 96–99. American Meteorological Society.Google Scholar
Stelzenmuller, N., Polanco, J. I., Vignal, L., Vinkovic, I. & Mordant, N. 2017 Lagrangian acceleration statistics in a turbulent channel flow. Phys. Rev. 2 (5), 054602.Google Scholar
Tavoularis, S. & Corrsin, S. 1981 Experiments in nearly homogenous turbulent shear flow with a uniform mean temperature gradient. Part 1. J. Fluid Mech. 104, 311–347.CrossRefGoogle Scholar
Voth, G. A., La Porta, A., Crawford, A. M., Alexander, J. & Bodenschatz, E. 2002 Measurement of particle accelerations in fully developed turbulence. J. Fluid Mech. 469, 121–160.CrossRefGoogle Scholar
Voth, G. A., Satyanarayan, K. & Bodenschatz, E. 1998 Lagrangian acceleration measurements at large Reynolds numbers. Phys. Fluids 10 (9), 2268–2280.CrossRefGoogle Scholar
Wacławczyk, M., Pozorski, J. & Minier, J.-P. 2004 Probability density function computation of turbulent flows with a new near-wall model. Phys. Fluids 16, 1410–1422.CrossRefGoogle Scholar
Watteaux, R., Sardina, G., Brandt, L. & Iudicone, D. 2019 On the time scales and structure of Lagrangian intermittency in homogeneous isotropic turbulence. J. Fluid Mech. 867, 438–481.CrossRefGoogle Scholar
Wilson, J. D. & Sawford, B. L. 1996 Review of Lagrangian stochastic models for trajectories in the turbulent atmosphere. Boundary-Layer Meteorol. 78, 191–210.CrossRefGoogle Scholar
Xu, J. & Pope, S. B. 1999 Assessment of numerical accuracy of pdf/Monte Carlo methods for turbulent reacting flows. J. Comput. Phys. 152 (1), 192–230.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
Zamansky, R., Vinkovic, I. & Gorokhovski, M. 2010 LES approach coupled with stochastic forcing of subgrid acceleration in a high-Reynolds-number channel flow. J. Turbul. 11, N30.CrossRefGoogle Scholar