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Model for the dynamics of micro-bubbles in high-Reynolds-number flows

Published online by Cambridge University Press:  01 October 2019

Zhentong Zhang ,
Dominique Legendre Open the ORCID record for Dominique Legendre [Opens in a new window]  and
Rémi Zamansky Open the ORCID record for Rémi Zamansky [Opens in a new window]
Zhentong Zhang
Affiliation:
Institut de Mécanique des Fluides de Toulouse (IMFT), Université de Toulouse, CNRS, INPT, UPS, Toulouse, France
Dominique Legendre
Affiliation:
Institut de Mécanique des Fluides de Toulouse (IMFT), Université de Toulouse, CNRS, INPT, UPS, Toulouse, France
Rémi Zamansky*
Affiliation:
Institut de Mécanique des Fluides de Toulouse (IMFT), Université de Toulouse, CNRS, INPT, UPS, Toulouse, France
*
†Email address for correspondence: [email protected]
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Abstract

We propose a model for the acceleration of micro-bubbles (smaller than the dissipative scale of the flow) subjected to the drag and fluid inertia forces in a homogeneous and isotropic turbulent flow. This model, that depends on the Stokes number, Reynolds number and the density ratio, reproduces the evolution of the acceleration variance as well as the relative importance and alignment of the two forces as observed from direct numerical simulations (DNS). We also report that the bubble acceleration statistics conditioned on the local kinetic energy dissipation rate are invariant with the Stokes number and the dissipation rate. Based on this observation, we propose a stochastic model for the instantaneous bubble acceleration vector accounting for the small-scale intermittency of the turbulent flows. The norm of the bubble acceleration is obtained by modelling the dissipation rate along the bubble trajectory from a log-normal stochastic process, whereas its orientation is given by two coupled random walks on a unit sphere in order to model the evolution of the joint orientation of the drag and inertia forces acting on the bubble. Furthermore, the proposed stochastic model for the bubble acceleration is used in the context of large eddy simulations (LES) of turbulent flows laden with small bubbles. To account for the turbulent motion at scales smaller than the mesh resolution, we decompose the instantaneous bubble acceleration in its resolved and residual parts. The first part is given by the drag and fluid inertia forces computed from the resolved velocity field, and the second term refers to the random contribution of small unresolved turbulent scales and is estimated with the stochastic model proposed in the paper. Comparisons with DNS and standard LES, show that the proposed model improves significantly the statistics of the bubbly phase.

JFM classification

Drops and Bubbles: Bubble dynamics Multiphase and Particle-laden Flows: Particle/fluid flow Turbulent Flows: Isotropic turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 879 , 25 November 2019 , pp. 554 - 578
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Copyright
© 2019 Cambridge University Press 

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References

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/S002211200500844XGoogle Scholar
Bec, J., Biferale, L., Cencini, M., Lanotte, A. & Toschi, F. 2010 Intermittency in the velocity distribution of heavy particles in turbulence. J. Fluid Mech. 646, 527–536.10.1017/S0022112010000029Google Scholar
Berrouk, A. S., Laurence, D., Riley, J. J. & Stock, D. E. 2007 Stochastic modelling of inertial particle dispersion by subgrid motion for les of high Reynolds number pipe flow. J. Turbul. 8, N50.10.1080/14685240701615952Google Scholar
Bos, W. J. T. & Zamansky, R. 2019 Power fluctuations in turbulence. Phys. Rev. Lett. 122 (12), 124504.10.1103/PhysRevLett.122.124504Google Scholar
Breuer, M. & Hoppe, F. 2017 Influence of a cost–efficient Langevin subgrid-scale model on the dispersed phase of large–eddy simulations of turbulent bubble–laden and particle–laden flows. Intl J. Multiphase Flow 89, 23–44.10.1016/j.ijmultiphaseflow.2016.10.007Google Scholar
Burton, G. C. & Dahm, W. J. A. 2005a Multifractal subgrid-scale modeling for large-eddy simulation. I. Model development and a priori testing. Phys. Fluids 17, 075111.Google Scholar
Burton, G. C. & Dahm, W. J. A. 2005b Multifractal subgrid-scale modeling for large-eddy simulation. II. Backscatter limiting and a posteriori evaluation. Phys. Fluids 17, 075112.Google Scholar
Calzavarini, E., Kerscher, M., Lohse, D. & Toschi, F. 2008 Dimensionality and morphology of particle and bubble clusters in turbulent flow. J. Fluid Mech. 607, 13–24.10.1017/S0022112008001936Google Scholar
Calzavarini, E., Volk, R., Bourgoin, M., Lévêque, E., Pinton, J.-F. & Toschi, F. 2009 Acceleration statistics of finite-sized particles in turbulent flow: the role of Faxén forces. J. Fluid Mech. 630, 179–189.10.1017/S0022112009006880Google Scholar
Castaing, B. 1996 The temperature of turbulent flows. J. Phys. II 6, 105–114.Google Scholar
Castaing, B., Gagne, Y. & Hopfinger, E. J. 1990 Velocity probability density functions of high Reynolds number turbulence. Physica D 46 (2), 177–200.10.1016/0167-2789(90)90035-NGoogle Scholar
Chen, S., Doolen, G. D., Kraichnan, R. H. & She, Z.-S. 1993 On statistical correlations between velocity increments and locally averaged dissipation in homogeneous turbulence. Phys. Fluids 5 (2), 458–463.10.1063/1.858897Google Scholar
Climent, E. & Magnaudet, J. 1999 Large-scale simulations of bubble-induced convection in a liquid layer. Phys. Rev. Lett. 82, 4827–4830.10.1103/PhysRevLett.82.4827Google Scholar
Dhotre, M. T., Deen, N. G., Niceno, B., Khan, Z. & Joshi, J. B. 2013 Large eddy simulation for dispersed bubbly flows: a review. Intl J. Chem. Engng 2013, 343276.Google Scholar
Fede, P. & Simonin, O. 2006 Numerical study of the subgrid fluid turbulence effects on the statistics of heavy colliding particles. Phys. Fluids 18 (4), 045103.10.1063/1.2189288Google Scholar
Gatignol, R. 1983 The Faxén formulae for a rigid particle in an unsteady non-uniform Stokes flow. J. Méc. Théor. Appl. 1, 143–160.Google Scholar
Ghate, A. S. & Lele, S. K. 2017 Subfilter-scale enrichment of planetary boundary layer large eddy simulation using discrete fourier–gabor modes. J. Fluid Mech. 819, 494–539.10.1017/jfm.2017.187Google Scholar
Gorokhovski, M. & Zamansky, R. 2018 Modeling the effects of small turbulent scales on the drag force for particles below and above the Kolmogorov scale. Phys. Rev. Fluids 3 (3), 1–23.10.1103/PhysRevFluids.3.034602Google Scholar
van den Hengel, E. I. V., Deen, N. G. & Kuipers, J. A. M. 2005 Application of coalescence and breakup models in a discrete bubble model for bubble columns. Ind. Engng Chem. Res. 44 (14), 5233–5245.10.1021/ie0492449Google Scholar
Hinze, J. O. 1975 Turbulence, 2nd edn. McGraw-Hill.Google Scholar
Hu, G. & Celik, I. 2008 Eulerian–Lagrangian based large-eddy simulation of a partially aerated flat bubble column. Chem. Engng Sci. 63 (1), 253–271.10.1016/j.ces.2007.09.015Google Scholar
Johnson, P. L. & Meneveau, C. 2017 Predicting viscous-range velocity gradient dynamics in large-eddy simulations of turbulence. J. Fluid Mech. 837, 80–114.10.1017/jfm.2017.838Google Scholar
Kerstein, A. R. 1999 One-dimensional turbulence: model formulation and application to homogeneous turbulence, shear flows, and buoyant stratified flows. J. Fluid Mech. 392, 277–334.10.1017/S0022112099005376Google Scholar
Kolmogorov, A. N. 1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Dokl. Akad. Nauk SSSR 434, 9–13, translation by V. Levin 1991 Phil. Trans. R. Soc. Lond. A 434, 9–13.Google Scholar
Kolmogorov, A. N. 1962 A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J. Fluid Mech. 13, 82–85.10.1017/S0022112062000518Google Scholar
Lalescu, C. C. & Wilczek, M. 2018 Acceleration statistics of tracer particles in filtered turbulent fields. J. Fluid Mech. 847, R2.10.1017/jfm.2018.381Google Scholar
Lanotte, A., Calzavarini, E., Toschi, F., Bec, J., Biferale, L. & Cencini, M.2011 Heavy particles in turbulent flows. International CFD Database, RM-2007-GRAD-2048.St0. iCFDdatabase.Google Scholar
Legendre, D. & Magnaudet, J. 1997 A note on the lift force on a spherical bubble or drop in a low-Reynolds-number shear flow. Phys. Fluids 9 (11), 3572–3574.10.1063/1.869466Google Scholar
Legendre, D. & Magnaudet, J. 1998 The lift force on a spherical bubble in a viscous linear shear flow. J. Fluid Mech. 368, 81–126.10.1017/S0022112098001621Google Scholar
Loisy, A. & Naso, A. 2017 Interaction between a large buoyant bubble and turbulence. Phys. Rev. Fluids 2, 014606.10.1103/PhysRevFluids.2.014606Google Scholar
Lukassen, L. J. & Wilczek, M. 2017 Lagrangian intermittency based on an ensemble of Gaussian velocity time series. In Progress in Turbulence VII (ed. Örlü, R., Talamelli, A., Oberlack, M. & Peinke, J.), pp. 23–29. Springer International Publishing.10.1007/978-3-319-57934-4_4Google Scholar
Magnaudet, J. & Eames, I. 2000 The motion of high-Reynolds-number bubbles in inhomogeneous flows. Annu. Rev. Fluid Mech. 32, 659–708.10.1146/annurev.fluid.32.1.659Google Scholar
Mathai, V., Calzavarini, E., Brons, J., Sun, C. & Lohse, D. 2016 Microbubbles and microparticles are not faithful tracers of turbulent acceleration. Phys. Rev. Lett. 117, 024501.10.1103/PhysRevLett.117.024501Google Scholar
Maxey, M. R. & Riley, J. J. 1983 Equation of motion for a small rigid sphere in a nonuniform flow. Phys. Fluids 26 (4), 883–889.10.1063/1.864230Google Scholar
Mazzitelli, I. M. & Lohse, D. 2004 Lagrangian statistics for fluid particles and bubbles in turbulence. New J. Phys. 6 (1), 203.10.1088/1367-2630/6/1/203Google Scholar
Mazzitelli, I. M., Lohse, D. & Toschi, F. 2003 On the relevance of the lift force in bubbly turbulence. J. Fluid Mech. 488, 283–313.10.1017/S0022112003004877Google Scholar
Meneveau, C. & Katz, J. 2000 Scale-invariance and turbulence models for large-eddy simulation. Annu. Rev. Fluid Mech. 32, 1–32.10.1146/annurev.fluid.32.1.1Google Scholar
Minier, J.-P., Chibbaro, S. & Pope, S. B. 2014 Guidelines for the formulation of Llagrangian stochastic models for particle simulations of single-phase and dispersed two-phase turbulent flows. Phys. Fluids 26, 113303.10.1063/1.4901315Google Scholar
Mordant, N., Crawford, A. M. & Bodenschatz, E. 2004 Three-dimensional structure of the Lagrangian acceleration in turbulent flows. Phys. Rev. Lett. 93 (21), 214501.10.1103/PhysRevLett.93.214501Google Scholar
Mordant, N., Metz, P. & Michel, O. 2001 Measurement of Lagrangian velocity in fully developed turbulence. Phys. Rev. Lett. 21, 214501.Google Scholar
Park, G. I., Bassenne, M., Urzay, J. & Moin, P. 2017 A simple dynamic subgrid-scale model for les of particle-laden turbulence. Phys. Rev. Fluids 2 (4), 044301.10.1103/PhysRevFluids.2.044301Google Scholar
Pereira, R. M., Moriconi, L. & Chevillard, L. 2018 A multifractal model for the velocity gradient dynamics in turbulent flows. J. Fluid Mech. 839, 430–467.10.1017/jfm.2018.12Google Scholar
Pope, S. B. 1990 Lagrangian microscales in turbulence. Phil. Trans. R. Soc. Lond. A 333 (1631), 309–319.Google Scholar
Pope, S. B. & Chen, Y. L. 1990 The velocity-dissipation probability density function model for turbulent flows. Phys. Fluids 2 (8), 1437–1449.10.1063/1.857592Google Scholar
Pozorski, J. & Apte, S. V. 2009 Filtered particle tracking in isotropic turbulence and stochastic modeling of subgrid-scale dispersion. Intl J. Multiphase Flow 35 (2), 118–128.10.1016/j.ijmultiphaseflow.2008.10.005Google Scholar
Prakash, V. N.2013 Light particles in turbulence. PhD thesis, University of Twente, Enschede.Google Scholar
Prakash, V. N., Tagawa, Y., Calzavarini, E., Mercado, J. M., Toschi, F., Lohse, D. & Sun, C. 2012 How gravity and size affect the acceleration statistics of bubbles in turbulence. New J. Phys. 14 (10), 105017.Google Scholar
Sabelnikov, V., Barge, A. & Gorokhovski, M. 2019 Stochastic modeling of fluid acceleration on residual scales and dynamics of suspended inertial particles in turbulence. Phys. Rev. Fluids 4 (4), 044301.10.1103/PhysRevFluids.4.044301Google Scholar
Sabel’nikov, V., Chtab, A. & Gorokhovski, M. 2007 The coupled LES – sub-grid stochastic acceleration model (LES-SSAM) of a high Reynolds number flows. In Advances in Turbulence XI, vol. 117, pp. 209–211; 11th EUROMECH European Turbulence Conference, June 25–28, 2007, Porto, Portugal: Springer Proceedings in Physics.10.1007/978-3-540-72604-3_66Google Scholar
Sabel’nikov, V., Chtab-Desportes, A. & Gorokhovski, M. 2011 New sub-grid stochastic acceleration model in LES of high-Reynolds-number flows. Eur. Phys. J. B 80 (2), 177–187.10.1140/epjb/e2011-10455-1Google Scholar
Sagaut, P. 2002 Large Eddy Simulation for Incompressible Flows: An Introduction, 2nd edn. Springer.10.1007/978-3-662-04695-1Google Scholar
Sawford, B. L. 1991 Reynolds number effects in Lagrangian stochastic models of turbulent dispersion. Phys. Fluids A 3, 1577–1588.10.1063/1.857937Google Scholar
Sawford, B. L. & Guest, F. M. 1991 Lagrangian statistical simulation of the turbulent motion of heavy particles. Boundary-Layer Meteorol. 54 (1–2), 147–166.10.1007/BF00119417Google Scholar
Sawford, B. L. & Yeung, P. K. 2011 Kolmogorov similarity scaling for one-particle Lagrangian statistics. Phys. Fluids 23 (9), 091704.10.1063/1.3643852Google Scholar
Tagawa, Y., Mercado, J. M., Prakash, V. N., Calzavarini, E., Sun, C. & Lohse, D. 2012 Three-dimensional Lagrangian Voronoï analysis for clustering of particles and bubbles in turbulence. J. Fluid Mech. 693, 201–215.10.1017/jfm.2011.510Google Scholar
Tagawa, Y., Roghair, I., Prakash, V. N., van Sint Annaland, M., Kuipers, H., Sun, C. & Lohse, D. 2013 The clustering morphology of freely rising deformable bubbles. J. Fluid Mech. 721, R2.10.1017/jfm.2013.100Google Scholar
Tchen, C. M.1947 Mean value and correlation problems connected with the motion of small particles suspended in a turbulent fluid. PhD thesis, Delft University, Netherlands.Google Scholar
Tennekes, H. & Lumley, J. L. 1972 A First Course in Turbulence. MIT Press.10.7551/mitpress/3014.001.0001Google Scholar
Volk, R., Calzavarini, E., Verhille, G., Lohse, D., Mordant, N., Pinton, J.-F. & Toschi, F. 2008a Acceleration of heavy and light particles in turbulence: comparison between experiments and direct numerical simulations. Physica D 237 (14–17), 2084–2089.10.1016/j.physd.2008.01.016Google Scholar
Volk, R., Mordant, N., Verhille, G. & Pinton, J.-F. 2008b Laser doppler measurement of inertial particle and bubble accelerations in turbulence. Eur. Phys. Lett. 81, 34002.Google Scholar
Yeung, P. K. 2001 Lagrangian characteristics of turbulence and scalar transport in direct numerical simulations. J. Fluid Mech. 427, 241–274.10.1017/S0022112000002391Google 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, 065103.10.1063/1.2204053Google Scholar
Zaichik, L. I. & Alipchenkov, V. M. 2011 A model for predicting the acceleration variance of arbitrary-density finite-size particles in isotropic turbulence. Intl J. Multiphase Flow 37 (3), 236–240.10.1016/j.ijmultiphaseflow.2010.11.003Google Scholar
Zamansky, R., Coletti, F., Massot, M. & Mani, A. 2016 Turbulent thermal convection driven by heated inertial particles. J. Fluid Mech. 809, 390–437.10.1017/jfm.2016.630Google Scholar
Zamansky, R., Vinkovic, I. & Gorokhovski, M. 2013 Acceleration in turbulent channel flow: universalities in statistics, subgrid stochastic models and an application. J. Fluid Mech. 721, 627–668.10.1017/jfm.2013.48Google Scholar