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Liquid transport in scale space

Published online by Cambridge University Press:  08 January 2020

F. Thiesset Open the ORCID record for F. Thiesset [Opens in a new window] ,
B. Duret ,
T. Ménard ,
C. Dumouchel ,
J. Reveillon  and
F. X. Demoulin
F. Thiesset*
Affiliation:
CNRS, Normandy Univ., UNIROUEN, INSA Rouen, CORIA, 76000Rouen, France
B. Duret
Affiliation:
CNRS, Normandy Univ., UNIROUEN, INSA Rouen, CORIA, 76000Rouen, France
T. Ménard
Affiliation:
CNRS, Normandy Univ., UNIROUEN, INSA Rouen, CORIA, 76000Rouen, France
C. Dumouchel
Affiliation:
CNRS, Normandy Univ., UNIROUEN, INSA Rouen, CORIA, 76000Rouen, France
J. Reveillon
Affiliation:
CNRS, Normandy Univ., UNIROUEN, INSA Rouen, CORIA, 76000Rouen, France
F. X. Demoulin
Affiliation:
CNRS, Normandy Univ., UNIROUEN, INSA Rouen, CORIA, 76000Rouen, France
*
†Email address for correspondence: thiesset@coria.fr
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Abstract

When a liquid stream is injected into a gaseous atmosphere, it destabilizes and continuously passes through different states characterized by different morphologies. Throughout this process, the flow dynamics may be different depending on the region of the flow and the scales of the involved liquid structures. Exploring this multi-scale, multi-dimensional phenomenon requires some new theoretical tools, some of which need yet to be elaborated. Here, a new analytical framework is proposed on the basis of two-point statistical equations of the liquid volume fraction. This tool, which originates from single phase turbulence, allows us notably to decompose the fluxes of liquid in flow–position space and scale space. Direct numerical simulations of liquid–gas turbulence decaying in a triply periodic domain are then used to characterize the time and scale evolution of the liquid volume fraction. It is emphasized that two-point statistics of the liquid volume fraction depend explicitly on the geometrical properties of the liquid–gas interface and in particular its surface density. The stretch rate of the liquid–gas interface is further shown to be the equivalent for the liquid volume fraction (a non-diffusive scalar) of the scalar dissipation rate. Finally, a decomposition of the transport of liquid in scale space highlights that non-local interactions between non-adjacent scales play a significant role.

JFM classification

Multiphase and Particle-laden Flows: Gas/liquid flow Drops and Bubbles: Breakup/coalescence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 886 , 10 March 2020 , A4
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Anez, J., Ahmed, A., Hecht, N., Duret, B., Reveillon, J. & Demoulin, F. X. 2019 Eulerian–Lagrangian spray atomization model coupled with interface capturing method for diesel injectors. Intl J. Multiphase Flow 113, 325–342.CrossRefGoogle Scholar
Antonia, R. A. & Burattini, P. 2006 Approach to the 4/5 law in homogeneous isotropic turbulence. J. Fluid Mech. 550, 175–184.CrossRefGoogle Scholar
Antonia, R. A. & Mi, J. 1998 Approach towards self-preservation of turbulent cylinder and screen wakes. Exp. Therm. Fluid Sci. 17, 277–284.CrossRefGoogle Scholar
Antonia, R. A. & Orlandi, P. 2004 Similarity of decaying isotropic turbulence with a passive scalar. J. Fluid Mech. 505, 123–151.CrossRefGoogle Scholar
Antonia, R. A., Satyaprakash, B. R. & Hussain, A. K. M. F. 1980 Measurements of dissipation rate and some other characteristics of turbulent plane and circular jets. Phys. Fluids 23 (4), 695–700.CrossRefGoogle Scholar
Antonia, R. A., Smalley, R. J., Zhou, T., Anselmet, F. & Danaila, L. 2003 Similarity of energy structure functions in decaying homogeneous isotropic turbulence. J. Fluid Mech. 487, 245–269.CrossRefGoogle Scholar
Antonia, R. A., Smalley, R. J., Zhou, T., Anselmet, F. & Danaila, L. 2004 Similarity solution of temperature structure functions in decaying homogeneous isotropic turbulence. Phys. Rev. E 69, 1–11.Google ScholarPubMed
Bevilaqua, P. M. & Lykoudis, P. S. 1978 Turbulence memory in self-preserving wakes. J. Fluid Mech. 89 (3), 589–606.CrossRefGoogle Scholar
Bradbury, L. J. S. 1965 The structure of a self-preserving turbulent plane jet. J. Fluid Mech. 23 (1), 31–64.CrossRefGoogle Scholar
Bradshaw, P. 1966 The effect of initial conditions on the development of a free shear layer. J. Fluid Mech. 26 (2), 225–236.CrossRefGoogle Scholar
Brown, G. L. & Roshko, A. 1974 On density effects and large structure in turbulent mixing layers. J. Fluid Mech. 64, 775–816.CrossRefGoogle Scholar
Burattini, P., Antonia, R. A. & Danaila, L. 2005a Scale-by-scale energy budget on the axis of a turbulent round jet. J. Turbul. 6, 1–11.CrossRefGoogle Scholar
Burattini, P., Antonia, R. A. & Danaila, L. 2005b Similarity in the far field of a turbulent round jet. Phys. Fluids 17, 025101.CrossRefGoogle Scholar
Campagne, A., Gallet, B., Moisy, F. & Cortet, P. P. 2014 Direct and inverse energy cascades in a forced rotating turbulence experiment. Phys. Fluids 26 (12), 125112.CrossRefGoogle Scholar
Canu, R., Puggelli, S., Essadki, M., Duret, B., Menard, T., Massot, M., Reveillon, J. & Demoulin, F. X. 2018 Where does the droplet size distribution come from? Intl J. Multiphase Flow 107, 230–245.CrossRefGoogle Scholar
Casciola, C. M., Gualtieri, P., Benzi, R. & Piva, R. 2003 Scale-by-scale budget and similarity laws for shear turbulence. J. Fluid Mech. 476, 105–114.CrossRefGoogle Scholar
Chasnov, J. R. 1994 Similarity states of passive scalar transport in isotropic turbulence. Phys. Fluids 6 (2), 1036–1051.CrossRefGoogle Scholar
Chesnel, J., Menard, T., Reveillon, J. & Demoulin, F. X. 2011 Subgrid analysis of liquid jet atomization. Atomiz. Sprays 21 (1), 41–67.CrossRefGoogle Scholar
Cimarelli, A. & De Angelis, E. 2011 Analysis of the Kolmogorov equation for filtered wall-turbulent flows. J. Fluid Mech. 676, 376–395.CrossRefGoogle Scholar
Cimarelli, A., De Angelis, E. & Casciola, C. M. 2013 Paths of energy in turbulent channel flows. J. Fluid Mech. 715, 436–451.CrossRefGoogle Scholar
Cimarelli, A., De Angelis, E., Schlatter, P., Brethouwer, G., Talamelli, A. & Casciola, C. M. 2015 Sources and fluxes of scale energy in the overlap layer of wall turbulence. J. Fluid Mech. 771, 407–423.CrossRefGoogle Scholar
Cui, G. X., Xu, C.-X., Fang, L., Shao, L. & Zhang, Z. S. 2007 A new subgrid eddy-viscosity model for large-eddy simulation of anisotropic turbulence. J. Fluid Mech. 582, 377–397.CrossRefGoogle Scholar
Danaila, L., Anselmet, F., Zhou, T. & Antonia, R. A. 1999 A generalization of Yaglom’s equation which accounts for the large-scale forcing in heated decaying turbulence. J. Fluid Mech. 391, 359–372.CrossRefGoogle Scholar
Danaila, L., Anselmet, F., Zhou, T. & Antonia, R. A. 2001 Turbulent energy scale budget equations in a fully developed channel flow. J. Fluid Mech. 430, 87–109.CrossRefGoogle Scholar
Danaila, L., Antonia, R. A. & Burattini, P. 2004 Progress in studying small-scale turbulence using exact two-point equations. New J. Phys. 6 (1), 128.CrossRefGoogle Scholar
Danaila, L., Krawczynski, J. F., Thiesset, F. & Renou, B. 2012 Yaglom-like equation in axisymmetric anisotropic context. Phys. D 241, 216–223.Google Scholar
Danaila, L. & Mydlarski, L. 2001 Effect of gradient production on scalar fluctuations in decaying grid turbulence. Phys. Rev. E 64 (1), 016316.Google ScholarPubMed
Danaila, L., Voivenel, L. & Varea, E. 2017 Self-similarity criteria in anisotropic flows with viscosity stratification. Phys. Fluids 29 (2), 020716.CrossRefGoogle Scholar
Debye, P., Anderson, H. R. Jr & Brumberger, H. 1957 Scattering by an inhomogeneous solid. II. The correlation function and its application. J. Appl. Phys. 28 (6), 679–683.CrossRefGoogle Scholar
Debye, P. & Bueche, A. M. 1949 Scattering by an inhomogeneous solid. J. Appl. Phys. 20 (6), 518–525.CrossRefGoogle Scholar
Dumouchel, C. 2009 The maximum entropy formalism and the prediction of liquid spray drop-size distribution. Entropy 11 (4), 713–747.CrossRefGoogle Scholar
Dumouchel, C., Aniszewski, W., Vu, T.-T. & Ménard, T. 2017 Multi-scale analysis of simulated capillary instability. Intl J. Multiphase Flow 92, 181–192.CrossRefGoogle Scholar
Dumouchel, C., Cousin, J. & Grout, S. 2008 Analysis of two-dimensional liquid spray images: the surface-based scale distribution. J. Flow Vis. Image Process. 15 (1), 59–83.CrossRefGoogle Scholar
Duret, B., Luret, G., Reveillon, J., Ménard, T., Berlemont, A. & Demoulin, F. X. 2012 DNS analysis of turbulent mixing in two-phase flows. Intl J. Multiphase Flow 40, 93–105.CrossRefGoogle Scholar
Duret, B., Reveillon, J., Menard, T. & Demoulin, F. X. 2013 Improving primary atomization modeling through DNS of two-phase flows. Intl J. Multiphase Flow 55, 130–137.CrossRefGoogle Scholar
Ewing, D., Frohnapfel, B., George, W. K., Pedersen, J. M. & Westerweel, J. 2007a Two-point similarity in the round jet. J. Fluid Mech. 577 (1), 309–330.CrossRefGoogle Scholar
Ewing, D., George, W. K., Rogers, M. M. & Moser, R. D. 2007b Two-point similarity in temporally evolving plane wakes. J. Fluid Mech. 577, 287–307.CrossRefGoogle Scholar
Fedkiw, R. P., Aslam, T., Merriman, B. & Osher, S. 1999 A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J. Comput. Phys. 152 (2), 457–492.CrossRefGoogle Scholar
Fiedler, H. E. 1975 On turbulence structure and mixing mechanism in free turbulent shear flows. In Turbulent Mixing in Nonreactive and Reactive Flows, pp. 381–409. Springer.CrossRefGoogle Scholar
Fitzhugh, R. 1983 Statistical properties of the asymmetric random telegraph signal, with applications to single-channel analysis. Math. Biosci. 64 (1), 75–89.CrossRefGoogle Scholar
Gauding, M., Danaila, L. & Varea, E. 2018 One-point and two-point statistics of homogeneous isotropic decaying turbulence with variable viscosity. Intl J. Heat Fluid Flow 72, 143–150.CrossRefGoogle Scholar
Gauding, M., Wick, A., Pitsch, H. & Peters, N. 2014 Generalised scale-by-scale energy-budget equations and large-eddy simulations of anisotropic scalar turbulence at various Schmidt numbers. J. Turbul. 15 (12), 857–882.CrossRefGoogle Scholar
George, W. K. 1989 The self-preservation of turbulent flows and its relation to initial conditions and coherent structures. In Advances in Turbulence, pp. 39–74.Google Scholar
George, W. K. 1992a The decay of homogeneous isotropic turbulence. Phys. Fluids 4, 1492–1509.CrossRefGoogle Scholar
George, W. K. 1992b Self-preservation of temperature fluctuations in isotropic turbulence. In Studies in Turbulence, pp. 514–528. Springer.CrossRefGoogle Scholar
George, W. K. & Gibson, M. M. 1992 The self-preservation of homogeneous shear flow turbulence. Exp. Fluids 13 (4), 229–238.CrossRefGoogle Scholar
Gomes-Fernandes, R., Ganapathisubramani, B. & Vassilicos, J. C. 2015 The energy cascade in near-field non-homogeneous non-isotropic turbulence. J. Fluid Mech. 771, 676–705.CrossRefGoogle Scholar
Gonzalez, M. & Fall, A. 1998 The approach to self-preservation of scalar fluctuations decay in isotropic turbulence. Phys. Fluids 10 (3), 654–661.CrossRefGoogle Scholar
Gorokhovski, M. A. & Saveliev, V. L. 2003 Analyses of Kolmogorov’s model of breakup and its application into Lagrangian computation of liquid sprays under air-blast atomization. Phys. Fluids 15 (1), 184–192.CrossRefGoogle Scholar
Gotoh, T., Fukayama, D. & Nakano, T. 2002 Velocity field statistics in homogeneous steady turbulence obtained using a high-resolution direct numerical simulation. Phys. Fluids 14 (3), 1065–1081.CrossRefGoogle Scholar
Gualtieri, P., Casciola, C. M., Benzi, R. & Piva, R. 2007 Preservation of statistical properties in large-eddy simulation of shear turbulence. J. Fluid Mech. 592, 471–494.CrossRefGoogle Scholar
Herrmann, M. & Gorokhovski, M. 2008 An outline of a LES subgrid model for liquid/gas phase interface dynamics. In Proceedings of the Summer Program. Center for Turbulence Research, pp. 171–181.Google Scholar
Hill, R. J. 2002 Exact second-order structure-function relationships. J. Fluid Mech. 468, 317–326.CrossRefGoogle Scholar
Hirt, C. W. & Nichols, B. D. 1981 Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys. 39 (1), 201–225.CrossRefGoogle Scholar
Hussain, A. K. M. F. & Zedan, M. F. 1978a Effects of the initial condition on the axisymmetric free shear layer: effect of the initial fluctuation level. Phys. Fluids 21 (9), 1475–1481.CrossRefGoogle Scholar
Hussain, A. K. M. F. & Zedan, M. F. 1978b Effects of the initial condition on the axisymmetric free shear layer: effects of the initial momentum thickness. Phys. Fluids 21, 1100–1112.CrossRefGoogle Scholar
von Kármán, T. & Howarth, L. 1938 On the statistical theory of isotropic turbulence. Proc. R. Soc. Lond. A 164 (917), 192–215.CrossRefGoogle Scholar
Kolmogorov, A. 1941a Dissipation of energy in the locally isotropic turbulence. Dokl. Akad. Nauk SSSR 125, 15–17.Google Scholar
Kolmogorov, A. N. 1941b On the log-normal law of distribution of particles during fragmentation. Dokl. Akad. Nauk SSSR 125, 99–101.Google Scholar
Lai, C. C. K., Charonko, J. J. & Prestridge, K. 2018 A Kármán–Howarth–Monin equation for variable-density turbulence. J. Fluid Mech. 843, 382–418.CrossRefGoogle Scholar
Lamriben, C., Cortet, P. P. & Moisy, F. 2011 Direct measurements of anisotropic energy transfers in a rotating turbulence experiment. Phys. Rev. Lett. 107 (2), 024503.CrossRefGoogle Scholar
Lebas, R., Menard, T., Beau, P. A., Berlemont, A. & Demoulin, F. X. 2009 Numerical simulation of primary break-up and atomization: DNS and modelling study. Intl J. Multiphase Flow 35 (3), 247–260.CrossRefGoogle Scholar
Lesieur, M. & Metais, O. 1996 New trends in large-eddy simulations of turbulence. Annu. Rev. Fluid Mech. 28 (1), 45–82.CrossRefGoogle Scholar
Lévêque, E., Toschi, F., Shao, L. & Bertoglio, J.-P. 2007 Shear-improved Smagorinsky model for large-eddy simulation of wall-bounded turbulent flows. J. Fluid Mech. 570, 491–502.CrossRefGoogle Scholar
Lucci, F., Ferrante, A. & Elghobashi, S. 2010 Modulation of isotropic turbulence by particles of Taylor length-scale size. J. Fluid Mech. 650, 5–55.CrossRefGoogle Scholar
Lundgren, T. S. 1982 Strained spiral vortex model for turbulent fine structure. Phys. Fluids 25 (12), 2193–2203.CrossRefGoogle Scholar
Machlup, S. 1954 Noise in semiconductors: spectrum of a two-parameter random signal. J. Appl. Phys. 25 (3), 341–343.CrossRefGoogle Scholar
Marati, N., Casciola, C. M. & Piva, R. 2004 Energy cascade and spatial fluxes in wall turbulence. J. Fluid Mech. 521, 191–215.CrossRefGoogle Scholar
Marmottant, P. & Villermaux, E. 2004 On spray formation. J. Fluid Mech. 498, 73–111.CrossRefGoogle Scholar
Ménard, T., Tanguy, S. & Berlemont, A. 2007 Coupling level set/VOF/ghost fluid methods: validation and application to 3D simulation of the primary break-up of a liquid jet. Intl J. Multiphase Flow 33 (5), 510–524.CrossRefGoogle Scholar
Moisy, F., Tabeling, P. & Willaime, H. 1999 Kolmogorov equation in a fully developed turbulence experiment. Phys. Rev. Lett. 82 (20), 3994.CrossRefGoogle Scholar
Mollicone, J.-P., Battista, F., Gualtieri, P. & Casciola, C. M. 2018 Turbulence dynamics in separated flows: the generalised Kolmogorov equation for inhomogeneous anisotropic conditions. J. Fluid Mech. 841, 1012–1039.CrossRefGoogle Scholar
Monin, A. S. & Yaglom, A. M. 1975 Statistical Fluid Mechanics, Volume 2: Mechanics of Turbulence. Courier Corporation.Google Scholar
Moser, R. D., Moser, R. M. & Ewing, D. W. 1998 Self-similarity of time-evolving plane wakes. J. Fluid Mech. 367, 255–289.CrossRefGoogle Scholar
Nie, Q. & Tanveer, S. 1999 A note on third–order structure functions in turbulence. Proc. R. Soc. Lond. A 455 (1985), 1615–1635.CrossRefGoogle Scholar
Orlandi, P. & Antonia, R. A. 2002 Dependence of the non-stationary form of Yaglom’s equation on the Schmidt number. J. Fluid Mech. 451, 99–108.CrossRefGoogle Scholar
Peters, N. 1992 A spectral closure for premixed turbulent combustion in the flamelet regime. J. Fluid Mech. 242, 611–629.CrossRefGoogle Scholar
Portela, F. A., Papadakis, G. & Vassilicos, J. C. 2017 The turbulence cascade in the near wake of a square prism. J. Fluid Mech. 825, 315–352.CrossRefGoogle Scholar
Richardson, L. F. 1920 The supply of energy from and to atmospheric eddies. Proc. R. Soc. Lond. A 97 (686), 354–373.CrossRefGoogle Scholar
Rosti, M. E., Ge, Z., Jain, S. S., Dodd, M. S. & Brandt, L. 2018 Emulsions in homogeneous shear turbulence. In Proceedings of the Summer Program. Center for Turbulence Research.Google Scholar
Rosti, M. E., Ge, Z., Jain, S. S., Dodd, M. S. & Brandt, L. 2019 Droplets in homogeneous shear turbulence. J. Fluid Mech. 876, 962–984.CrossRefGoogle Scholar
Rudman, M. 1998 A volume-tracking method for incompressible multifluid flows with large density variations. Intl J. Numer. Meth. Fluids 28 (2), 357–378.3.0.CO;2-D>CrossRefGoogle Scholar
Saikrishnan, N., De Angelis, E., Longmire, E. K., Marusic, I., Casciola, C. M. & Piva, R. 2012 Reynolds number effects on scale energy balance in wall turbulence. Phys. Fluids 24 (1), 015101.CrossRefGoogle Scholar
Scardovelli, R. & Zaleski, S. 1999 Direct numerical simulation of free-surface and interfacial flow. Annu. Rev. Fluid Mech. 31 (1), 567–603.CrossRefGoogle Scholar
Sellens, R. W. & Brzustowski, T. A. 1986 A simplified prediction of droplet velocity distributions in a spray. Combust. Flame 65 (3), 273–279.CrossRefGoogle Scholar
Speziale, C. G. & Bernard, P. S. 1992 The energy decay in self-preserving isotropic turbulence revisited. J. Fluid Mech. 241, 645–667.CrossRefGoogle Scholar
Stewart, R. W. & Townsend, A. A. 1951 Similarity and self-preservation in isotropic turbulence. Phil. Trans. R. Soc. Lond. A 243 (867), 359–386.CrossRefGoogle Scholar
Sussman, M., Smith, K. M., Hussaini, M. Y., Ohta, M. & Zhi-Wei, R. 2007 A sharp interface method for incompressible two-phase flows. J. Comput. Phys. 221 (2), 469–505.CrossRefGoogle Scholar
Thiesset, F., Antonia, R. A. & Danaila, L. 2013 Scale-by-scale turbulent energy budget in the intermediate wake of two-dimensional generators. Phys. Fluids 25 (11), 115105.CrossRefGoogle Scholar
Thiesset, F., Antonia, R. A. & Djenidi, L. 2014a Consequences of self-preservation on the axis of a turbulent round jet. J. Fluid Mech. 748, R2.CrossRefGoogle Scholar
Thiesset, F., Danaila, L. & Antonia, R. A. 2014b Dynamical interactions between the coherent motion and small scales in a cylinder wake. J. Fluid Mech. 749, 201–226.CrossRefGoogle Scholar
Thiesset, F., Dumouchel, C. & Ménard, T.2019a A new theoretical framework for characterizing the transport of liquid in turbulent two-phase flows. In ILASS-Europe, Paris.Google Scholar
Thiesset, F., Dumouchel, C., Ménard, T., Aniszewski, W., Vaudor, G. & Berlemont, A.2019b Probing liquid atomization using probability density functions, the volume-based scale distribution and differential geometry. In ILASS-Europe, Paris.Google Scholar
Togni, R., Cimarelli, A. & De Angelis, E. 2015 Physical and scale-by-scale analysis of Rayleigh–Bénard convection. J. Fluid Mech. 782, 380–404.CrossRefGoogle Scholar
Tsinober, A. 2009 An Informal Conceptual Introduction to Turbulence (Second Edition of An Informal Introduction to Turbulence). Springer Science & Business Media.CrossRefGoogle Scholar
Valente, P. C. & Vassilicos, J. C. 2015 The energy cascade in grid-generated non-equilibrium decaying turbulence. Phys. Fluids 27 (4), 045103.CrossRefGoogle Scholar
Vassilicos, J. C. 2015 Dissipation in turbulent flows. Annu. Rev. Fluid Mech. 47, 95–114.CrossRefGoogle Scholar
Vaudor, G., Ménard, T., Aniszewski, W., Doring, M. & Berlemont, A. 2017 A consistent mass and momentum flux computation method for two phase flows. Application to atomization process. Comput. Fluids 152, 204–216.CrossRefGoogle Scholar
Villermaux, E. 2007 Fragmentation. Annu. Rev. Fluid Mech. 39, 419–446.CrossRefGoogle Scholar
Villermaux, E., Marmottant, P. & Duplat, J. 2004 Ligament-mediated spray formation. Phys. Rev. Lett. 92 (7), 074501.CrossRefGoogle ScholarPubMed
Watanabe, T. & Gotoh, T. 2004 Statistics of a passive scalar in homogeneous turbulence. New J. Phys. 6 (1), 40.CrossRefGoogle Scholar
Yaglom, A. M. 1949 On the local structure of a temperature field in a turbulent flow. Dokl. Akad. Nauk SSSR 69, 743–746.Google Scholar
Yeong, C. L. Y. & Torquato, S. 1998 Reconstructing random media. Phys. Rev. E 57 (1), 495.Google Scholar
Zhang, J. 1996 Acceleration of five-point red-black Gauss-Seidel in multigrid for Poisson equation. Appl. Math. Comput. 80 (1), 73–93.Google Scholar