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Structure of iso-scalar sets

Published online by Cambridge University Press:  17 May 2022

M. Gauding ,
F. Thiesset Open the ORCID record for F. Thiesset [Opens in a new window] ,
E. Varea  and
L. Danaila
M. Gauding
Affiliation:
CORIA, UMR 6614, CNRS, Normandy Univ., UNIROUEN, INSA Rouen, 76000 Rouen, France
F. Thiesset*
Affiliation:
CORIA, UMR 6614, CNRS, Normandy Univ., UNIROUEN, INSA Rouen, 76000 Rouen, France
E. Varea
Affiliation:
CORIA, UMR 6614, CNRS, Normandy Univ., UNIROUEN, INSA Rouen, 76000 Rouen, France
L. Danaila
Affiliation:
University of Rouen Normandy, CNRS, M2C, 76000 Rouen, France
*
†Email address for correspondence: thiesset@coria.fr
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Abstract

An analytical framework is proposed to explore the structure and kinematics of iso-scalar fields. It is based on a two-point statistical analysis of the phase indicator field which is used to track a given iso-scalar volume. The displacement speed of the iso-surface, i.e. the interface velocity relative to the fluid velocity, is explicitly accounted for, thereby generalizing previous two-point equations dedicated to the phase indicator in two-phase flows. Although this framework applies to many transported quantities, we here focus on passive scalar mixing. Particular attention is paid to the effect of Reynolds (the Taylor based Reynolds number is varied from 88 to 530) and Schmidt numbers (in the range 0.1 to 1), together with the influence of flow and scalar forcing. It is first found that diffusion in the iso-surface tangential direction is predominant, emphasizing the primordial influence of curvature on the displacement speed. Second, the appropriate normalizing scales for the two-point statistics at either large, intermediate and small scales are revealed and appear to be related to the radius of gyration, the surface density and the standard deviation of mean curvature, respectively. Third, the onset of an intermediate ‘scaling range’ for the two-point statistics of the phase indicator at sufficiently large Reynolds numbers is observed. The scaling exponent complies with a fractal dimension of 8/3. A scaling range is also observed for the transfer of iso-scalar fields in scale space whose exponent can be estimated by simple scaling arguments and a recent closure of the Corrsin equation. Fourth, the effects of Reynolds and Schmidt numbers together with flow or scalar forcing on the different terms of the two-point budget are highlighted.

JFM classification

Mathematical Foundations: General fluid mechanics Turbulent Flows: Turbulence theory Mixing: Turbulent mixing
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 942 , 10 July 2022 , A14
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

REFERENCES

Adler, P.M., Jacquin, C.G. & Quiblier, J.A. 1990 Flow in simulated porous media. Intl J. Multiphase Flow 16 (4), 691–712.CrossRefGoogle Scholar
Antonia, R.A. & Orlandi, P. 2003 Effect of Schmidt number on small-scale passive scalar turbulence. Appl. Mech. Rev. 56 (6), 615–632.CrossRefGoogle Scholar
Arns, C.H., Knackstedt, M.A. & Mecke, K.R. 2004 Characterisation of irregular spatial structures by parallel sets and integral geometric measures. Colloids Surf. A 241 (1–3), 351–372.CrossRefGoogle Scholar
Ashgriz, N. 2011 Handbook of Atomization and Sprays: Theory and Applications. Springer Science & Business Media.CrossRefGoogle Scholar
Batchelor, G.K. 1951 Pressure fluctuations in isotropic turbulence. Math. Proc. Camb. Philos. Soc. 47, 359–374.CrossRefGoogle Scholar
Berryman, J.G. 1987 Relationship between specific surface area and spatial correlation functions for anisotropic porous media. J. Math. Phys. 28 (1), 244–245.CrossRefGoogle Scholar
Blakeley, B.C., Wang, W. & Riley, J.J. 2019 On the kinematics of scalar iso-surfaces in decaying homogeneous, isotropic turbulence. J. Turbul. 20 (10), 661–680.CrossRefGoogle Scholar
Bray, K.N.C. & Moss, J.B. 1977 A unified statistical model of the premixed turbulent flame. Acta Astronaut. 4 (3-4), 291–319.CrossRefGoogle Scholar
Candel, S. & Poinsot, T. 1990 Flame stretch and the balance equation for the flame area. Combust. Sci. Technol. 70 (1–3), 1–15.CrossRefGoogle Scholar
Catrakis, H.J. & Dimotakis, P.E. 1996 Mixing in turbulent jets: scalar measures and isosurface geometry. J. Fluid Mech. 317, 369–406.CrossRefGoogle Scholar
Constantin, P. 1994 a Geometric and analytic studies in turbulence. In Trends and Perspectives in Applied Mathematics (ed. L. Sirovich), pp. 21–54. Springer.CrossRefGoogle Scholar
Constantin, P. 1994 b Geometric statistics in turbulence. SIAM Rev. 36 (1), 73–98.CrossRefGoogle Scholar
Constantin, P., Procaccia, I. & Sreenivasan, K.R. 1991 Fractal geometry of isoscalar surfaces in turbulence: theory and experiments. Phys. Rev. Lett. 67 (13), 1739.CrossRefGoogle ScholarPubMed
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
De Gennes, P.-G., Brochard-Wyart, F. & Quéré, D. 2013 Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves. Springer Science & Business Media.Google Scholar
Di Battista, R., Bermejo-Moreno, I., Ménard, T., de Chaisemartin, S. & Massot, M. 2019 Post-processing of two-phase DNS simulations exploiting geometrical features and topological invariants to extract flow statistics: application to canonical objects and the collision of two droplets. In 10th International Conference on Multiphase Flow.Google Scholar
Dimotakis, P.E. & Catrakis, H.J. 1999 Turbulence, fractals, and mixing. In Mixing (ed. H. Chaté, E. Villermaux & J.M. Chomaz), pp. 59–143. Springer.CrossRefGoogle Scholar
de Divitiis, N. 2014 Finite scale Lyapunov analysis of temperature fluctuations in homogeneous isotropic turbulence. Appl. Math. Model. 38 (21–22), 5279–5297.CrossRefGoogle Scholar
de Divitiis, N. 2016 von Kármán–Howarth and Corrsin equations closure based on Lagrangian description of the fluid motion. Ann. Phys. 368, 296–309.CrossRefGoogle Scholar
de Divitiis, N. 2020 von Kármán–Howarth and Corrsin equations closures through Liouville theorem. Results Phys. 16, 102979.CrossRefGoogle Scholar
Donzis, D.A., Sreenivasan, K.R. & Yeung, P.K. 2005 Scalar dissipation rate and dissipative anomaly in isotropic turbulence. J. Fluid Mech. 532, 199–216.CrossRefGoogle Scholar
Drew, D.A. 1990 Evolution of geometric statistics. SIAM J. Appl. Maths 50 (3), 649–666.CrossRefGoogle Scholar
Dumouchel, C. 2008 On the experimental investigation on primary atomization of liquid streams. Exp. Fluids 45 (3), 371–422.CrossRefGoogle Scholar
Dumouchel, C. & Grout, S. 2009 Application of the scale entropy diffusion model to describe a liquid atomization process. Intl J. Multiphase Flow 35 (10), 952–962.CrossRefGoogle Scholar
Dumouchel, C., Thiesset, F. & Ménard, T 2022 Morphology of contorted liquid structures. Intl J. Multiphase flow 152, 104055.CrossRefGoogle Scholar
Elsas, J.H., Szalay, A.S. & Meneveau, C. 2018 Geometry and scaling laws of excursion and iso-sets of enstrophy and dissipation in isotropic turbulence. J. Turbul. 19 (4), 297–321.CrossRefGoogle Scholar
Essadki, M., Drui, F., Larat, A., Ménard, T. & Massot, M. 2019 Statistical modeling of the gas–liquid interface using geometrical variables: toward a unified description of the disperse and separated phase flows. Intl J. Multiphase Flow 120, 103084.Google Scholar
Eswaran, V. & Pope, S.B. 1988 An examination of forcing in direct numerical simulations of turbulence. Comput. Fluids 16 (3), 257–278.CrossRefGoogle Scholar
Federer, H. 1959 Curvature measures. Trans. Am. Math. Soc. 93 (3), 418–491.CrossRefGoogle Scholar
Frisch, H.L. & Stillinger, F.H. 1963 Contribution to the statistical geometric basis of radiation scattering. J. Chem. Phys. 38 (9), 2200–2207.CrossRefGoogle Scholar
Garcia-Ruiz, J.-M., Louis, E., Meakin, P. & Sander, L.M. 2012 Growth Patterns in Physical Sciences and Biology, vol. 304. Springer Science & Business Media.Google Scholar
Gauding, M., Danaila, L. & Varea, E. 2017 High-order structure functions for passive scalar fed by a mean gradient. Intl J. Heat Fluid Flow 67, 86–93.CrossRefGoogle Scholar
Gauding, M., Goebbert, J.H., Hasse, C. & Peters, N. 2015 Line segments in homogeneous scalar turbulence. Phys. Fluids 27 (9), 095102.CrossRefGoogle Scholar
Gauding, M., Wang, L., Goebbert, J.H., Bode, M., Danaila, L. & Varea, E. 2019 On the self-similarity of line segments in decaying homogeneous isotropic turbulence. Comput. Fluids 180, 206–217.CrossRefGoogle Scholar
Giannakopoulos, G.K., Frouzakis, C.E., Mohan, S., Tomboulides, A.G. & Matalon, M. 2019 Consumption and displacement speeds of stretched premixed flames-theory and simulations. Combust. Flame 208, 164–181.CrossRefGoogle Scholar
Gibson, C.H. 1968 Fine structure of scalar fields mixed by turbulence. I. Zero-gradient points and minimal gradient. Phys. Fluids 11 (11), 2305–2315.CrossRefGoogle Scholar
Girimaji, S.S. & Pope, S.B. 1992 Propagating surfaces in isotropic turbulence. J. Fluid Mech. 234, 247–277.CrossRefGoogle Scholar
Gouldin, F.C. 1987 An application of fractals to modeling premixed turbulent flames. Combust. Flame 68 (3), 249–266.CrossRefGoogle Scholar
Gouldin, F.C., Hilton, S.M. & Lamb, T. 1989 Experimental evaluation of the fractal geometry of flamelets. In Symposium (International) on Combustion, vol. 22, pp. 541–550. Elsevier.CrossRefGoogle Scholar
Gran, I.R., Echekki, T. & Chen, J.H. 1996 Negative flame speed in an unsteady 2-D premixed flame: A computational study. In Symposium (International) on Combustion, vol. 26, pp. 323–329. Elsevier.CrossRefGoogle Scholar
Grossmann, S. & Lohse, D. 1994 Fractal-dimension crossovers in turbulent passive scalar signals. Europhys. Lett. 27 (5), 347.CrossRefGoogle Scholar
Grout, S., Dumouchel, C., Cousin, J. & Nuglisch, H. 2007 Fractal analysis of atomizing liquid flows. Intl J. Multiphase Flow 33 (9), 1023–1044.CrossRefGoogle Scholar
Hawkes, E.R., Chatakonda, O., Kolla, H., Kerstein, A.R. & Chen, J.H. 2012 A petascale direct numerical simulation study of the modelling of flame wrinkling for large-eddy simulations in intense turbulence. Combust. Flame 159 (8), 2690–2703.CrossRefGoogle Scholar
Hentschel, H.G.E. & Procaccia, I. 1984 Relative diffusion in turbulent media: the fractal dimension of clouds. Phys. Rev. A 29 (3), 1461.CrossRefGoogle 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
Hou, T.Y. & Li, R. 2007 Computing nearly singular solutions using pseudo-spectral methods. J. Comput. Phys. 226 (1), 379–397.CrossRefGoogle Scholar
Iyer, K.P., Schumacher, J., Sreenivasan, K.R. & Yeung, P.K. 2020 Fractal iso-level sets in high-Reynolds-number scalar turbulence. Phys. Rev. Fluids 5 (4), 044501.CrossRefGoogle Scholar
Jay, S., Lacas, F. & Candel, S. 2006 Combined surface density concepts for dense spray combustion. Combust. Flame 144 (3), 558–577.CrossRefGoogle Scholar
Kirste, R. & Porod, G. 1962 Röntgenkleinwinkelstreuung an kolloiden systemen asymptotisches verhalten der streukurven. Kolloid-Z. Z. Polym. 184 (1), 1–7.CrossRefGoogle Scholar
Krug, D., Holzner, M., Lüthi, B., Wolf, M., Kinzelbach, W. & Tsinober, A. 2015 The turbulent/non-turbulent interface in an inclined dense gravity current. J. Fluid Mech. 765, 303–324.CrossRefGoogle Scholar
Krug, D., Holzner, M., Marusic, I. & van Reeuwijk, M. 2017 Fractal scaling of the turbulence interface in gravity currents. J. Fluid Mech. 820, R3.CrossRefGoogle Scholar
Kulkarni, T., Buttay, R., Kasbaoui, M.H., Attili, A. & Bisetti, F. 2021 Reynolds number scaling of burning rates in spherical turbulent premixed flames. J. Fluid Mech. 906, A2.CrossRefGoogle Scholar
Le Moyne, L., Freire, V. & Queiros-Condé, D. 2008 Fractal dimension and scale entropy applications in a spray. Chaos, Solitons Fractals 38 (3), 696–704.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
Libby, P.A. & Bray, K.N.C. 1980 Implications of the laminar flamelet model in premixed turbulent combustion. Combust. Flame 39 (1), 33–41.CrossRefGoogle Scholar
Liepmann, H.W. 1949 Die anwendung eines satzes über die nullstellen stochastischer funktionen auf turbulenzmessungen. Helv. Phys. Acta 22 (2), 119–126.Google Scholar
Liss, P.S. & Johnson, M.T. 2014 Ocean-Atmosphere Interactions of Gases and Particles. Springer Nature.CrossRefGoogle Scholar
Lovejoy, S. 1982 Area-perimeter relation for rain and cloud areas. Science 216 (4542), 185–187.CrossRefGoogle ScholarPubMed
Lu, J. & Tryggvason, G. 2018 Direct numerical simulations of multifluid flows in a vertical channel undergoing topology changes. Phys. Rev. Fluids 3 (8), 084401.CrossRefGoogle Scholar
Lu, J. & Tryggvason, G. 2019 Multifluid flows in a vertical channel undergoing topology changes: effect of void fraction. Phys. Rev. Fluids 4 (8), 084301.CrossRefGoogle Scholar
Ma, Z. & Torquato, S. 2018 Precise algorithms to compute surface correlation functions of two-phase heterogeneous media and their applications. Phys. Rev. E 98 (1), 013307.CrossRefGoogle ScholarPubMed
Mandelbrot, B.B. 1975 On the geometry of homogeneous turbulence, with stress on the fractal dimension of the iso-surfaces of scalars. J. Fluid Mech. 72 (3), 401–416.CrossRefGoogle Scholar
Mazellier, N. & Vassilicos, J.C. 2008 The turbulence dissipation constant is not universal because of its universal dependence on large-scale flow topology. Phys. Fluids 20 (1), 015101.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
Morán, J., Fuentes, A., Liu, F. & Yon, J. 2019 FracVAL: an improved tunable algorithm of cluster-cluster aggregation for generation of fractal structures formed by polydisperse primary particles. Comput. Phys. Commun. 239, 225–237.CrossRefGoogle Scholar
Peters, N. 1988 Laminar flamelet concepts in turbulent combustion. In Symposium (International) on Combustion, vol. 21, pp. 1231–1250. Elsevier.CrossRefGoogle Scholar
Peters, N. 1992 A spectral closure for premixed turbulent combustion in the flamelet regime. J. Fluid Mech. 242, 611–629.CrossRefGoogle Scholar
Peters, N., Terhoeven, P., Chen, J.H. & Echekki, T. 1998 Statistics of flame displacement speeds from computations of 2-D unsteady methane-air flames. In Symposium (International) on Combustion, vol. 27, pp. 833–839. Elsevier.CrossRefGoogle Scholar
Pope, S.B. 1988 The evolution of surfaces in turbulence. Intl J. Engng Sci. 26 (5), 445–469.CrossRefGoogle Scholar
Pope, S.B., Yeung, P.K. & Girimaji, S.S. 1989 The curvature of material surfaces in isotropic turbulence. Phys. Fluids 1 (12), 2010–2018.CrossRefGoogle Scholar
Seaton, N.A. & Glandt, E.D. 1986 Spatial correlation functions from computer simulations. J. Chem. Phys. 85 (9), 5262–5268.CrossRefGoogle Scholar
Shete, K.P. & de Bruyn Kops, S.M. 2020 Area of scalar isosurfaces in homogeneous isotropic turbulence as a function of Reynolds and Schmidt numbers. J. Fluid Mech. 883, A38.CrossRefGoogle Scholar
da Silva, C.B., Hunt, J.C.R., Eames, I. & Westerweel, J. 2014 Interfacial layers between regions of different turbulence intensity. Annu. Rev. Fluid Mech. 46, 567–590.CrossRefGoogle Scholar
de Silva, C.M., Philip, J., Chauhan, K., Meneveau, C. & Marusic, I. 2013 Multiscale geometry and scaling of the turbulent-nonturbulent interface in high Reynolds number boundary layers. Phys. Rev. Lett. 111 (4), 044501.CrossRefGoogle ScholarPubMed
Sorensen, C.M. 2001 Light scattering by fractal aggregates: a review. Aerosol Sci. Technol. 35 (2), 648–687.CrossRefGoogle Scholar
Sreenivasan, K.R. & Meneveau, C. 1986 The fractal facets of turbulence. J. Fluid Mech. 173, 357–386.CrossRefGoogle Scholar
Sreenivasan, K.R., Prabhu, A. & Narasimha, R. 1983 Zero-crossings in turbulent signals. J. Fluid Mech. 137, 251–272.CrossRefGoogle Scholar
Sreenivasan, K.R., Ramshankar, R. & Meneveau, C. 1989 Mixing, entrainment and fractal dimensions of surfaces in turbulent flows. Proc. R. Soc. Lond. A 421 (1860), 79–108.Google Scholar
Sussman, M., Smereka, P. & Osher, S. 1994 A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114 (1), 146–159.CrossRefGoogle Scholar
Teubner, M. 1990 Scattering from two-phase random media. J. Chem. Phys. 92 (7), 4501–4507.CrossRefGoogle Scholar
Thiesset, F., Duret, B., Ménard, T., Dumouchel, C., Reveillon, J. & Demoulin, F.-X. 2020 Liquid transport in scale space. J. Fluid Mech. 886, A4.CrossRefGoogle Scholar
Thiesset, F., Maurice, G., Halter, F., Mazellier, N., Chauveau, C. & Gökalp, I. 2016 a Geometrical properties of turbulent premixed flames and other corrugated interfaces. Phys. Rev. E 93 (1), 013116.CrossRefGoogle ScholarPubMed
Thiesset, F., Maurice, G., Halter, F., Mazellier, N., Chauveau, C. & Gökalp, I. 2016 b Modelling of the subgrid scale wrinkling factor for large eddy simulation of turbulent premixed combustion. Combust. Theor. Model. 20 (3), 393–409.CrossRefGoogle Scholar
Thiesset, F., Ménard, T. & Dumouchel, C. 2021 Space-scale-time dynamics of liquid–gas shear flow. J. Fluid Mech. 912, A39.CrossRefGoogle Scholar
Thiesset, F. & Poux, A. 2020 Numerical assessment of the two-point statistical equations in liquid/gas flows. Tech. Rep. CNRS, Normandy Univ., UNIROUEN, INSA Rouen, CORIA. Available at https://hal.archives-ouvertes.fr/hal-02614565.Google Scholar
Thiesset, F., Schaeffer, V., Djenidi, L. & Antonia, R.A. 2014 On self-preservation and log-similarity in a slightly heated axisymmetric mixing layer. Phys. Fluids 26 (7), 075106.CrossRefGoogle Scholar
Torquato, S. 2002 Random Heterogeneous Materials. Microstructure and Macroscopic Properties. Springer.CrossRefGoogle Scholar
Trouvé, A. & Poinsot, T. 1994 The evolution equation for the flame surface density in turbulent premixed combustion. J. Fluid Mech. 278, 1–31.CrossRefGoogle Scholar
Vassilicos, J.C. 1992 The multispiral model of turbulence and intermittency. In Topological Aspects of the Dynamics of Fluids and Plasmas (ed. H. K. Moffatt, G. M. Zaslavsky, P. Comte & M. Tabor), pp. 427–442. Springer.CrossRefGoogle Scholar
Vassilicos, J.C. & Hunt, J.C.R. 1991 Fractal dimensions and spectra of interfaces with application to turbulence. Proc. R. Soc. Lond. A 435 (1895), 505–534.Google Scholar
Vassilicos, J.C. & Hunt, J.C.R. 1996 Moving surfaces in turbulent flows. In Institute of Mathematics and its Applications Conference Series, vol. 56, pp. 127–154. Oxford University Press.Google Scholar
Villermaux, E. & Gagne, Y. 1994 Line dispersion in homogeneous turbulence: stretching, fractal dimensions, and micromixing. Phys. Rev. Lett. 73 (2), 252.CrossRefGoogle ScholarPubMed
Yeung, P.K., Girimaji, S.S. & Pope, S.B. 1990 Straining and scalar dissipation on material surfaces in turbulence: implications for flamelets. Combust. Flame 79 (3–4), 340–365.CrossRefGoogle Scholar
Yon, J., Morán, J., Ouf, F.-X., Mazur, M. & Mitchell, J.B. 2021 From monomers to agglomerates: a generalized model for characterizing the morphology of fractal-like clusters. J. Aerosol Sci. 151, 105628.CrossRefGoogle Scholar
Yu, R., Nilsson, T., Fureby, C. & Lipatnikov, A.N. 2021 Evolution equations for the decomposed components of displacement speed in a reactive scalar field. J. Fluid Mech. 911, A38.CrossRefGoogle Scholar