Hostname: page-component-68c7f8b79f-tw422 Total loading time: 0 Render date: 2025-12-23T23:55:32.365Z Has data issue: false hasContentIssue false
  • English
  • Français

Scale-by-scale kinetic energy budgets in multiphase turbulence

Published online by Cambridge University Press:  23 December 2025

Fabien Thiesset Open the ORCID record for Fabien Thiesset [Opens in a new window]  and
Jonathan Vahé
Fabien Thiesset*
Affiliation:
CNRS, CORIA, UMR 6614, UNIROUEN, INSA Rouen, Normandy Univ., 76000 Rouen, France
Jonathan Vahé
Affiliation:
CNRS, CORIA, UMR 6614, UNIROUEN, INSA Rouen, Normandy Univ., 76000 Rouen, France
*
Corresponding author: Fabien Thiesset, fabien.thiesset@cnrs.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

The present work aims at exploring the scale-by-scale kinetic energy exchanges in multiphase turbulence. For this purpose, we derive the Kármán–Howarth–Monin equation which accounts for the variations of density and viscosity across the two phases together with the effect of surface tension. We consider both conventional and phase conditional averaging operators. This framework is applied to numerical data from detailed simulations of forced homogeneous and isotropic turbulence covering different values for the liquid volume fraction, the liquid–gas density ratio, the Reynolds number and the Weber number. We confirm the existence of an additional transfer term due to surface tension. Part of the kinetic energy injected at large scales is transferred into kinetic energy at smaller scales by classical nonlinear transport while another part is transferred to surface energy before being released back into kinetic energy, but at smaller scales. The overall kinetic energy transfer rate is larger than in single-phase flows. Kinetic energy budgets conditioned in a given phase show that the scale-by-scale transport of turbulent kinetic energy due to pressure is a gain (loss) of kinetic energy for the lighter (heavier) phase. Its contribution can be dominant when the gas volume fraction becomes small or when the density ratio increases. Building on previous work, we hypothesise the existence of a pivotal scale above which kinetic energy is stored into surface deformation and below which the kinetic energy is released by interface restoration. Some phenomenological predictions for this scale are discussed.

JFM classification

Turbulent Flows: Turbulent Flows Turbulent Flows: Turbulence theory Multiphase and Particle-laden Flows: Multiphase flow

Information

Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1025 , 25 December 2025 , A63
Check for updates
Copyright
© The Author(s), 2025. 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.)

Article purchase

Temporarily unavailable

References

Aithal, A.B., Tipirneni, M. & Ferrante, A. 2023 Temporal accuracy of FastRK3. J. Comput. Phys. 475, 111853.10.1016/j.jcp.2022.111853CrossRefGoogle Scholar
Aluie, H. 2011 Compressible turbulence: the cascade and its locality. Phys. Rev. Lett. 106 (17), 174502.10.1103/PhysRevLett.106.174502CrossRefGoogle ScholarPubMed
Aluie, H. 2013 Scale decomposition in compressible turbulence. Physica D 247 (1), 54–65.10.1016/j.physd.2012.12.009CrossRefGoogle Scholar
Antonia, R.A. & Burattini, P. 2006 Approach to the 4/5 law in homogeneous isotropic turbulence. J. Fluid Mech. 550, 175–184.10.1017/S0022112005008438CrossRefGoogle Scholar
Antonia, R.A., Zhou, T., Danaila, L. & Anselmet, F. 2000 Streamwise inhomogeneity of decaying grid turbulence. Phys. Fluids 12 (11), 3086–3089.10.1063/1.1314336CrossRefGoogle Scholar
Arun, S., Sameen, A., Srinivasan, B. & Girimaji, S.S. 2021 Scale-space energy density function transport equation for compressible inhomogeneous turbulent flows. J. Fluid Mech. 920, A31.10.1017/jfm.2021.468CrossRefGoogle Scholar
Balachandar, S. & Eaton, J.K. 2010 Turbulent dispersed multiphase flow. Annu. Rev. Fluid Mech. 42, 111–133.10.1146/annurev.fluid.010908.165243CrossRefGoogle Scholar
Banerjee, S. & Galtier, S. 2013 Exact relation with two-point correlation functions and phenomenological approach for compressible magnetohydrodynamic turbulence. Phys. Rev. E 87 (1), 013019.10.1103/PhysRevE.87.013019CrossRefGoogle ScholarPubMed
Boukharfane, R., Er-Raiy, A., Parsani, M. & Chakraborty, N. 2021 Structure and dynamics of small-scale turbulence in vaporizing two-phase flows. Sci. Rep. UK 11 (1), 15242.10.1038/s41598-021-94334-xCrossRefGoogle ScholarPubMed
Brahami, Y. 2020 Active Scalar Mixing in Turbulent Jet Flows. PhD theses, Normandie Université, Rouen.Google Scholar
Bunner, B. & Tryggvason, G. 2002 Dynamics of homogeneous bubbly flows Part 2. Velocity fluctuations. J. Fluid Mech. 466, 53–84.10.1017/S0022112002001180CrossRefGoogle Scholar
Cannon, I., Soligo, G. & Rosti, M.E. 2024 Morphology of clean and surfactant-laden droplets in homogeneous isotropic turbulence. J. Fluid Mech. 987, A31.10.1017/jfm.2024.380CrossRefGoogle Scholar
Crialesi-Esposito, M., Boffetta, G., Brandt, L., Chibbaro, S. & Musacchio, S. 2023 a Intermittency in turbulent emulsions. J. Fluid Mech. 972, A37.10.1017/jfm.2023.628CrossRefGoogle Scholar
Crialesi-Esposito, M., Chibbaro, S. & Brandt, L. 2023 b The interaction of droplet dynamics and turbulence cascade. Commun. Phys. 6 (1), 5.10.1038/s42005-022-01122-8CrossRefGoogle Scholar
Crialesi-Esposito, M., Rosti, M.E., Chibbaro, S. & Brandt, L. 2022 Modulation of homogeneous and isotropic turbulence in emulsions. J. Fluid Mech. 940, A19.10.1017/jfm.2022.179CrossRefGoogle Scholar
Danaila, L., Antonia, R.A. & Burattini, P. 2012 a Comparison between kinetic energy and passive scalar energy transfer in locally homogeneous isotropic turbulence. Physica D 241 (3), 224–231.10.1016/j.physd.2011.10.008CrossRefGoogle Scholar
Danaila, L., Krawczynski, J.F., Thiesset, F. & Renou, B. 2012 b Yaglom-like equation in axisymmetric anisotropic context. Physica D 241, 216–223.10.1016/j.physd.2011.08.011CrossRefGoogle Scholar
Davidson, P.A. & Pearson, B.R. 2005 Identifying turbulent energy distributions in real, rather than Fourier, space. Phys. Rev. Lett. 95 (21), 214501.10.1103/PhysRevLett.95.214501CrossRefGoogle ScholarPubMed
Dodd, M.S. & Ferrante, A. 2016 On the interaction of Taylor length scale size droplets and isotropic turbulence. J. Fluid Mech. 806, 356–412.10.1017/jfm.2016.550CrossRefGoogle 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, 93–105.10.1016/j.ijmultiphaseflow.2011.11.014CrossRefGoogle Scholar
Estivalezes, J.-L., et al. 2022 A phase inversion benchmark for multiscale multiphase flows. J. Comput. Phys. 450, 110810.10.1016/j.jcp.2021.110810CrossRefGoogle Scholar
Eswaran, V. & Pope, S.B. 1988 An examination of forcing in direct numerical simulations of turbulence. Comput. Fluids 16 (3), 257–278.10.1016/0045-7930(88)90013-8CrossRefGoogle Scholar
Eyink, G.L. & Drivas, T.D. 2018 Cascades and dissipative anomalies in compressible fluid turbulence. Phys. Rev. X 8 (1), 011022.Google Scholar
Fang, L. & Bos, W. 2023 An EDQNM study of the dissipation rate in isotropic non-equilibrium turbulence. J. Turbul. 24 (6-7), 217–234.10.1080/14685248.2023.2189731CrossRefGoogle Scholar
Federrath, C., Klessen, R.S., Iapichino, L. & Beattie, J.R. 2021 The sonic scale of interstellar turbulence. Nat. Astron. 5 (4), 365–371.10.1038/s41550-020-01282-zCrossRefGoogle Scholar
Federrath, C., Roman-Duval, J., Klessen, R.S., Schmidt, W. & Mac Low, M.-M. 2010 Comparing the statistics of interstellar turbulence in simulations and observations-solenoidal versus compressive turbulence forcing. Astron. Astrophys. 512, A81.10.1051/0004-6361/200912437CrossRefGoogle 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), 457–492.10.1006/jcph.1999.6236CrossRefGoogle Scholar
Ferrand, R., Galtier, S., Sahraoui, F. & Federrath, C. 2020 Compressible turbulence in the interstellar medium: new insights from a high-resolution supersonic turbulence simulation. Astrophys. J. 904 (2), 160.10.3847/1538-4357/abb76eCrossRefGoogle Scholar
Freund, A. & Ferrante, A. 2019 Wavelet-spectral analysis of droplet-laden isotropic turbulence. J. Fluid Mech. 875, 914–928.10.1017/jfm.2019.515CrossRefGoogle Scholar
Frisch, H.L. & Stillinger, F.H. 1963 Contribution to the statistical geometric basis of radiation scattering. J. Chem. Phys. 38 (9), 2200–2207.10.1063/1.1733950CrossRefGoogle Scholar
Galtier, S. & Banerjee, S. 2011 Exact relation for correlation functions in compressible isothermal turbulence. Phys. Rev. Lett. 107 (13), 134501.10.1103/PhysRevLett.107.134501CrossRefGoogle ScholarPubMed
Gauding, M., Bode, M., Brahami, Y., Varea, É. & Danaila, L. 2021 Self-similarity of turbulent jet flows with internal and external intermittency. J. Fluid Mech. 919, A41.10.1017/jfm.2021.399CrossRefGoogle 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, 143–150.10.1016/j.ijheatfluidflow.2018.05.013CrossRefGoogle Scholar
Gauding, M., Thiesset, F., Varea, E. & Danaila, L. 2022 Structure of iso-scalar sets. J. Fluid Mech. 942, A14.10.1017/jfm.2022.367CrossRefGoogle Scholar
Goto, S. & Vassilicos, J.-C. 2015 Energy dissipation and flux laws for unsteady turbulence. Phys. Lett. A 379 (16-17), 1144–1148.10.1016/j.physleta.2015.02.025CrossRefGoogle Scholar
Hamba, F. 2015 Turbulent energy density and its transport equation in scale space. Phys. Fluids 27 (8), 085108.10.1063/1.4928698CrossRefGoogle Scholar
Hamba, F. 2018 Turbulent energy density in scale space for inhomogeneous turbulence. J. Fluid Mech. 842, 532–553.10.1017/jfm.2018.155CrossRefGoogle Scholar
Hellinger, P., Papini, E., Verdini, A., Landi, S., Franci, L., Matteini, L. & Montagud-Camps, V. 2021 a Spectral transfer and Kármán–Howarth–Monin equations for compressible hall magnetohydrodynamics. Astrophys. J. 917 (2), 101.10.3847/1538-4357/ac088fCrossRefGoogle Scholar
Hellinger, P., Verdini, A., Landi, S., Papini, E., Franci, L. & Matteini, L. 2021 b Scale dependence and cross-scale transfer of kinetic energy in compressible hydrodynamic turbulence at moderate Reynolds numbers. Phys. Rev. Fluids 6 (4), 044607.10.1103/PhysRevFluids.6.044607CrossRefGoogle Scholar
Hill, R.J. 2002 Exact second-order structure-function relationships. J. Fluid Mech. 468, 317–326.10.1017/S0022112002001696CrossRefGoogle Scholar
Hinze, J.O. 1955 Fundamentals of the hydrodynamic mechanism of splitting in dispersion processes. AIChE J. 1 (3), 289–295.10.1002/aic.690010303CrossRefGoogle Scholar
Innocenti, A., Jaccod, A., Popinet, S. & Chibbaro, S. 2021 Direct numerical simulation of bubble-induced turbulence. J. Fluid Mech. 918, A23.10.1017/jfm.2021.288CrossRefGoogle Scholar
Kida, S. & Orszag, S.A. 1990 Energy and spectral dynamics in forced compressible turbulence. J. Sci. Comput. 5, 85–125.10.1007/BF01065580CrossRefGoogle Scholar
Kirste, R. & Porod, G. 1962 Röntgenkleinwinkelstreuung an kolloiden systemen asymptotisches verhalten der streukurven. Kolloid-Z. Z. Polym. 184 (1), 1–7.10.1007/BF01840132CrossRefGoogle Scholar
Kolla, H., Hawkes, E.R., Kerstein, A.R., Swaminathan, N. & Chen, J.H. 2014 On velocity and reactive scalar spectra in turbulent premixed flames. J Fluid Mech. 754, 456–487.10.1017/jfm.2014.392CrossRefGoogle Scholar
Kolmogorov, A.N. 1949 On the breakage of drops in a turbulent flow. Dokl. Akad. Nauk SSSR 66, 825–828.Google Scholar
Kritsuk, A.G., Wagner, R. & Norman, M.L. 2013 Energy cascade and scaling in supersonic isothermal turbulence. J. Fluid Mech. 729, R1.10.1017/jfm.2013.342CrossRefGoogle Scholar
Lai, C.C.K., Charonko, J.J. & Prestridge, K. 2018 A Kármán–Howarth–Monin equation for variable-density turbulence. J. Fluid Mech. 843, 382–418.10.1017/jfm.2018.125CrossRefGoogle Scholar
Lance, M. & Bataille, J. 1991 Turbulence in the liquid phase of a uniform bubbly air–water flow. J. Fluid Mech. 222, 95–118.10.1017/S0022112091001015CrossRefGoogle Scholar
Ling, Y., Fuster, D., Tryggvason, G. & Zaleski, S. 2019 A two-phase mixing layer between parallel gas and liquid streams: multiphase turbulence statistics and influence of interfacial instability. J. Fluid Mech. 859, 268–307.10.1017/jfm.2018.825CrossRefGoogle Scholar
Loisy, A. & Naso, A. 2017 Interaction between a large buoyant bubble and turbulence. Phys. Rev. Fluids 2 (1), 014606.10.1103/PhysRevFluids.2.014606CrossRefGoogle Scholar
Lu, J. & Tryggvason, G. 2018 Direct numerical simulations of multifluid flows in a vertical channel undergoing topology changes. Phys. Rev. Fluids 3 (8), 084401.10.1103/PhysRevFluids.3.084401CrossRefGoogle Scholar
Lu, J. & Tryggvason, G. 2019 Multifluid flows in a vertical channel undergoing topology changes: effect of void fraction. Phys. Rev. Fluids 4 (8), 084301.10.1103/PhysRevFluids.4.084301CrossRefGoogle Scholar
Lucci, F., Ferrante, A. & Elghobashi, S. 2010 Modulation of isotropic turbulence by particles of Taylor length-scale size. J. Fluid Mech. 650, 5–55.10.1017/S0022112009994022CrossRefGoogle Scholar
Lundgren, T.S. 2003 Linearly Forced Isotropic Turbulence. Center for Turbulence Research Annual Research Briefs 2003.Google Scholar
McCaslin, J.O. & Desjardins, O. 2014 Theoretical and computational modeling of turbulence/interface interactions. In Proceedings of the Summer Program, pp. 79. Center for Turbulence Research, Stanford University.Google 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), 510–524.10.1016/j.ijmultiphaseflow.2006.11.001CrossRefGoogle Scholar
Mukherjee, S., Safdari, A., Shardt, O., Kenjereš, S. & Van den Akker, H.E.A. 2019 Droplet–turbulence interactions and quasi-equilibrium dynamics in turbulent emulsions. J. Fluid Mech. 878, 221–276.10.1017/jfm.2019.654CrossRefGoogle Scholar
Narula, H., Pandey, V., Mitra, D. & Perlekar, P. 2025 Scale-by-scale energy transfers in bubbly flows. arXiv preprint arXiv:2506.07693.Google Scholar
Nguyen, M.Q., Delache, A., Simoëns, S., Bos, W. & El Hajem, M. 2016 Small scale dynamics of isotropic viscoelastic turbulence. Phys. Rev. Fluids 1 (8), 083301.10.1103/PhysRevFluids.1.083301CrossRefGoogle Scholar
Ni, R. 2024 Deformation and breakup of bubbles and drops in turbulence. Annu. Rev. Fluid Mech. 56 (1), 319–347.10.1146/annurev-fluid-121021-034541CrossRefGoogle Scholar
Nie, Q. & Tanveer, S. 1999 A note on third–order structure functions in turbulence. Proc. R. Soc. 455 (1985), 1615–1635.10.1098/rspa.1999.0374CrossRefGoogle Scholar
Pan, N. & Banerjee, S. 2022 Exact relations for energy transfer in simple and active binary fluid turbulence. Phys. Rev. E 106 (2), 025104.10.1103/PhysRevE.106.025104CrossRefGoogle ScholarPubMed
Pan, N. & Banerjee, S. 2025 Universal energy cascade in homogeneous binary fluid turbulence: a direct comparison of different exact relations. Phys. Rev. Fluids 10 (6), 064615.10.1103/pfwf-cp3pCrossRefGoogle Scholar
Pandey, V., Mitra, D. & Perlekar, P. 2023 Kolmogorov turbulence coexists with pseudo-turbulence in buoyancy-driven bubbly flows. Phys. Rev. Lett. 131 (11), 114002.10.1103/PhysRevLett.131.114002CrossRefGoogle ScholarPubMed
Pandey, V., Ramadugu, R. & Perlekar, P. 2020 Liquid velocity fluctuations and energy spectra in three-dimensional buoyancy-driven bubbly flows. J. Fluid Mech. 884, R6.10.1017/jfm.2019.991CrossRefGoogle Scholar
Perlekar, P. 2019 Kinetic energy spectra and flux in turbulent phase-separating symmetric binary-fluid mixtures. J. Fluid Mech. 873, 459–474.10.1017/jfm.2019.425CrossRefGoogle Scholar
Perlekar, P., Biferale, L., Sbragaglia, M., Srivastava, S. & Toschi, F. 2012 Droplet size distribution in homogeneous isotropic turbulence. Phys. Fluids 24 (6), 065101.10.1063/1.4719144CrossRefGoogle Scholar
Ramirez, G., Burlot, A., Zamansky, R., Bois, G. & Risso, F. 2024 Spectral analysis of dispersed multiphase flows in the presence of fluid interfaces. Intl J. Multiphase Flow 177, 104860.10.1016/j.ijmultiphaseflow.2024.104860CrossRefGoogle Scholar
Richardson, L.F. 1922 Weather Prediction by Numerical Process. Cambridge University Press.Google Scholar
Risso, F. 2018 Agitation, mixing, and transfers induced by bubbles. Annu. Rev. Fluid Mech. 50, 25–48.10.1146/annurev-fluid-122316-045003CrossRefGoogle Scholar
Riviere, A., Mostert, W., Perrard, S. & Deike, L. 2021 Sub-Hinze scale bubble production in turbulent bubble break-up. J. Fluid Mech. 917, A40.10.1017/jfm.2021.243CrossRefGoogle Scholar
Rosales, C. & Meneveau, C. 2005 Linear forcing in numerical simulations of isotropic turbulence: physical space implementations and convergence properties. Phys Fluids 17 (9), 095106.10.1063/1.2047568CrossRefGoogle Scholar
Rosti, M.E., Ge, Z., Jain, S.S., Dodd, M.S. & Brandt, L. 2019 Droplets in homogeneous shear turbulence. J. Fluid Mech. 876, 962–984.10.1017/jfm.2019.581CrossRefGoogle Scholar
Rudman, M. 1998 A volume-tracking method for incompressible multifluid flows with large density variations. Intl J. Numer. Meth. Fluids 28 (2), 357–378.10.1002/(SICI)1097-0363(19980815)28:2<357::AID-FLD750>3.0.CO;2-D3.0.CO;2-D>CrossRefGoogle Scholar
Sabelnikov, V.A., Lipatnikov, A.N., Nishiki, S. & Hasegawa, T. 2019 a Application of conditioned structure functions to exploring influence of premixed combustion on two-point turbulence statistics. Proc. Combust. Inst. 37 (2), 2433–2441.10.1016/j.proci.2018.08.029CrossRefGoogle Scholar
Sabelnikov, V.A., Lipatnikov, A.N., Nishiki, S. & Hasegawa, T. 2019 b Investigation of the influence of combustion-induced thermal expansion on two-point turbulence statistics using conditioned structure functions. J. Fluid Mech. 867, 45–76.10.1017/jfm.2019.128CrossRefGoogle 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), 469–505.10.1016/j.jcp.2006.06.020CrossRefGoogle Scholar
Taguelmimt, N., Danaila, L. & Hadjadj, A. 2016 Effects of viscosity variations in temporal mixing layer. Flow Turbul. Combust. 96, 163–181.10.1007/s10494-015-9649-6CrossRefGoogle Scholar
Taylor, M.A., Kurien, S. & Eyink, G.L. 2003 Recovering isotropic statistics in turbulence simulations: the Kolmogorov 4/5th law. Phys. Rev. E 68 (2), 026310.10.1103/PhysRevE.68.026310CrossRefGoogle ScholarPubMed
Terrington, S.J., Hourigan, K. & Thompson, M.C. 2022 Vorticity generation and conservation on generalised interfaces in three-dimensional flows. J. Fluid Mech. 936, A44.10.1017/jfm.2022.91CrossRefGoogle Scholar
Thiesset, F., Duret, B., Ménard, T., Dumouchel, C., Reveillon, J. & Demoulin, F.-X. 2020 Liquid transport in scale space. J. Fluid Mech. 886, A4.10.1017/jfm.2019.1056CrossRefGoogle Scholar
Thiesset, F. & Federrath, C. 2023 Structure of iso-density sets in supersonic isothermal turbulence. Astron. Astrophys. 676, A12.10.1051/0004-6361/202346575CrossRefGoogle Scholar
Thiesset, F., Ménard, T. & Dumouchel, C. 2021 Space-scale-time dynamics of liquid–gas shear flow. J. Fluid Mech. 912, A39.10.1017/jfm.2020.1152CrossRefGoogle 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. https://hal.science/hal-02614565v1 Google Scholar
Torquato, S. 2002 Random Heterogeneous Materials. Microstructure and Macroscopic Properties. Springer-Verlag New York.10.1007/978-1-4757-6355-3CrossRefGoogle Scholar
Trautner, E., Klein, M., Bräuer, F. & Hasslberger, J. 2021 Conditional and unconditional second-order structure functions in bubbly channel flows of power-law fluids. Phys. Fluids 33 (5), 055121.10.1063/5.0049799CrossRefGoogle Scholar
Trefftz-Posada, P. & Ferrante, A. 2023 On the interaction of Taylor length-scale size droplets and homogeneous shear turbulence. J. Fluid Mech. 972, A9.10.1017/jfm.2023.647CrossRefGoogle Scholar
Valente, P.C., Da Silva, C.B. & Pinho, F.T. 2014 The effect of viscoelasticity on the turbulent kinetic energy cascade. J. Fluid Mech. 760, 39–62.10.1017/jfm.2014.585CrossRefGoogle 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, 204–216.10.1016/j.compfluid.2017.04.023CrossRefGoogle Scholar
Wagner, R., Falkovich, G., Kritsuk, A.G. & Norman, M.L. 2012 Flux correlations in supersonic isothermal turbulence. J. Fluid Mech. 713, 482–490.10.1017/jfm.2012.470CrossRefGoogle Scholar
Wang, J., Wan, M., Chen, S. & Chen, S. 2018 Kinetic energy transfer in compressible isotropic turbulence. J. Fluid Mech. 841, 581–613.10.1017/jfm.2018.23CrossRefGoogle Scholar
Whitman, S.H.R., Towery, C.A.Z., Poludnenko, A.Y. & Hamlington, P.E. 2019 Scaling and collapse of conditional velocity structure functions in turbulent premixed flames. Proc. Combust. Inst. 37 (2), 2527–2535.10.1016/j.proci.2018.07.010CrossRefGoogle Scholar
Wilcox, D.C. 1998 Turbulence Modeling for CFD, vol. 2. DCW Industries.Google Scholar
Yao, H., Alves-Portela, F. & Papadakis, G. 2020 Evolution of conditionally averaged second-order structure functions in a transitional boundary layer. Phys. Rev. Fluids 5 (9), 093902.10.1103/PhysRevFluids.5.093902CrossRefGoogle Scholar
Yao, H. & Papadakis, G. 2023 On the role of the laminar/turbulent interface in energy transfer between scales in bypass transition. J. Fluid Mech. 960, A24.10.1017/jfm.2023.194CrossRefGoogle Scholar
Zamansky, R., De Bonneville, F.L.R. & Risso, F. 2024 Turbulence induced by a swarm of rising bubbles from coarse-grained simulations. J. Fluid Mech. 984, A68.10.1017/jfm.2024.230CrossRefGoogle 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