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Hierarchical parcel-swapping representation of turbulent mixing. Part 4. Extension to the viscous range and to mixing of scalars with non-unity Schmidt numbers

Published online by Cambridge University Press:  30 September 2025

Masoomeh Behrang ,
Tommy Starick ,
Isaac Wheeler ,
Heiko Schmidt ,
Alan Kerstein Open the ORCID record for Alan Kerstein [Opens in a new window]  and
David Lignell Open the ORCID record for David Lignell [Opens in a new window]
Masoomeh Behrang
Affiliation:
330 EB, Brigham Young University, Provo, UT 84602, USA
Tommy Starick
Affiliation:
Brandenburg University of Technology Cottbus-Senftenberg, Cottbus, Germany
Isaac Wheeler
Affiliation:
330 EB, Brigham Young University, Provo, UT 84602, USA
Heiko Schmidt
Affiliation:
Brandenburg University of Technology Cottbus-Senftenberg, Cottbus, Germany
Alan Kerstein
Affiliation:
Consultant, 72 Lomitas Road, Danville, CA 94526, USA
David Lignell*
Affiliation:
330 EB, Brigham Young University, Provo, UT 84602, USA
*
Corresponding author: David Lignell, davidlignell@byu.edu
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Abstract

Hierarchical parcel swapping (HiPS) is a multiscale stochastic model of turbulent mixing based on a binary tree. Length scales decrease geometrically with increasing tree level, and corresponding time scales follow inertial range scaling. Turbulent eddies are represented by swapping subtrees. Lowest-level swaps change fluid parcel pairings, with new pairings instantly mixed. This formulation suitable for unity Schmidt number $Sc$ is extended to non-unity $Sc$. For high $Sc$, the tree is extended to the Batchelor level, assigning the same time scale (governing the rate of swap occurrences) to the added levels as the time scale at the base of the $Sc=3$ tree. For low $Sc$, a swap at the Obukhov–Corrsin level mixes all parcels within corresponding subtrees. Well-defined model analogues of turbulent diffusivity, and mean scalar-variance production and dissipation rates are identified. Simulations idealising stationary homogeneous turbulence with an imposed scalar gradient reproduce various statistical properties of viscous-range and inertial-range pair dispersion, and of the scalar power spectrum in the inertial-advective, inertial-diffusive and viscous-advective regimes. The viscous-range probability density functions of pair separation and scalar dissipation agree with applicable theory, including the stretched-exponential tail shape associated with viscous-range scalar intermittency. Previous observation of that tail shape for $Sc=1$, heretofore not modelled or explained, is reproduced. Comparisons to direct numerical simulation allow evaluation of empirical coefficients, facilitating quantitative applications. Parcel-pair mixing is a common mixing treatment, e.g. in subgrid closures for coarse-grained flow simulation, so HiPS can improve model physics simply by smarter (yet nearly cost-free) selection of pairs to be mixed.

JFM classification

Turbulent Flows: Turbulence modelling Mass Transport: Coupled diffusion and flow Mass Transport: Dispersion

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Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1020 , 10 October 2025 , A13
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© The Author(s), 2025. Published by Cambridge University Press

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References

Batchelor, G.K. 1959 Small-scale variation of convected quantities like temperature in a turbulent fluid. Part 1. General discussion and the case of small conductivity. J. Fluid Mech. 5, 113–133.10.1017/S002211205900009XCrossRefGoogle Scholar
Bos, W.T.J. 2014 On the anisotropy of the turbulent passive scalar in the presence of a mean scalar gradient. J. Fluid Mech. 744, 38–64.CrossRefGoogle Scholar
Briard, A. & Gomez, T. 2017 Prandtl number effects in decaying homogeneous isotropic turbulence with a mean scalar gradient. J. Turbul. 18, 418–442.CrossRefGoogle Scholar
Chasnov, J. 1991 Simulation of the inertial-conductive subrange. Phys. Fluids A 3, 1164–1168.10.1063/1.858044CrossRefGoogle Scholar
Chasnov, J. & Rogallo, R.S. 1988 Turbulence spectrum of a passive temperature field: results of a numerical simulation. Phys. Fluids 31, 2065–2067.10.1063/1.867013CrossRefGoogle Scholar
Chertkov, M., Falkovich, G. & Kolokolov, I. 1998 Intermittent dissipation of a passive scalar in turbulence. Phys. Rev. Lett. 80, 2121–2124.CrossRefGoogle Scholar
Crawford, F.W., Ho, L.S.T. & Suchard, M.A. 2018 Computational methods for birth-death processes. WIREs Comput. Stat. 10, e1423.10.1002/wics.1423CrossRefGoogle ScholarPubMed
Curl, R.L. 1963 Dispersed phase mixing: I. Theory and effects in simple reactors. AIChE J. 9, 175–181.CrossRefGoogle Scholar
Donzis, D.A., Sreenivasan, K.R. & Yeung, P.K. 2010 The Batchelor spectrum for mixing of passive scalars in isotropic turbulence. Flow Turbul. Combust. 85, 549–566.10.1007/s10494-010-9271-6CrossRefGoogle Scholar
Elsinga, G., Ishihara, T. & Hunt, J. 2022 Non-local dispersion and the reassessment of Richardson’s $t^3$ -scaling law. J. Fluid Mech. 932, A17.CrossRefGoogle Scholar
Fox, R.O. 2003 Computational Models for Turbulent Reacting Flows. Cambridge University Press.CrossRefGoogle Scholar
Girimaji, S.S. & Pope, S.B. 1990 Material-element deformation in isotropic turbulence. J. Fluid Mech. 220, 427–458.CrossRefGoogle Scholar
Gotoh, T. & Watanabe, T. 2012 Scalar flux in a mean scalar gradient in homogeneous isotropic steady turbulence. Physica D 241, 141–148.10.1016/j.physd.2010.12.009CrossRefGoogle Scholar
Gotoh, T. & Yeung, P.K. 2013 Passive scalar transport in turbulence: a computational perspective. In Ten Chapters in Turbulence (ed. P.A. Davidson, Y. Kaneda & K. Sreenivasan), pp. 87–131. Cambridge University Press.Google Scholar
Gurvich, A.S. & Yaglom, A.M. 1967 Breakdown of eddies and probability distributions for small-scale turbulence. Phys. Fluids 10, 59–65.CrossRefGoogle Scholar
Hill, H.J. 1978 Models of the scalar spectrum for turbulent advection. J. Fluid Mech. 88, 541–562.CrossRefGoogle Scholar
Holzer, M. & Siggia, E.D. 1994 Turbulent mixing of a passive scalar. Phys. Fluids 6, pp. 1820–1837.CrossRefGoogle Scholar
Kalda, J. & Morozenko, A. 2008 Turbulent mixing: the roots of intermittency. New J. Phys. 10, 093003.10.1088/1367-2630/10/9/093003CrossRefGoogle Scholar
Kerstein, A.R. 1991 a Scaling properties of the viscous-convective scalar spectral subrange in turbulent jets. Phys. Fluids A 3, 1832–1834.10.1063/1.857911CrossRefGoogle Scholar
Kerstein, A.R. 1991 b Linear-eddy modelling of turbulent transport. Part 6. Microstructure of diffusive scalar mixing fields. J. Fluid Mech. 231, 361–394.CrossRefGoogle Scholar
Kerstein, A.R. 2013 Hierarchical parcel-swapping representation of turbulent mixing. Part 1. Formulation and scaling properties. J. Stat. Phys. 153, 142–161.10.1007/s10955-013-0811-zCrossRefGoogle Scholar
Kerstein, A.R. 2014 Hierarchical parcel-swapping representation of turbulent mixing. Part 2. Application to channel flow. J. Fluid Mech. 750, 421–463.CrossRefGoogle Scholar
Kerstein, A.R. 2021 Hierarchical parcel-swapping representation of turbulent mixing. Part 3. Origins of correlation patterns observed in turbulent boundary layers. Phys. Rev. Fluids 6, 044611.CrossRefGoogle Scholar
Kerstein, A.R. 2022 Reduced numerical modeling of turbulent flow with fully resolved time advancement. Part 1. Theory and Physical Interpretation.. Fluids 7, 76.CrossRefGoogle Scholar
Kolmogorov, A.N. 1962 A refinement of previous hypotheses concerning the local structure of turbulence in a viscous in-compressible fluid at high Reynolds number. J. Fluid Mech. 13, 82–85.CrossRefGoogle Scholar
Kraichnan, R.H. 1959 The structure of isotropic turbulence at very high Reynolds number. J. Fluid Mech. 5, 497–543.10.1017/S0022112059000362CrossRefGoogle Scholar
Kraichnan, R.H. 1962 Turbulent thermal convection at arbitrary Prandtl number. Phys. Fluids 5, 1374–1389.CrossRefGoogle Scholar
Kraichnan, R.H. 1974 Convection of a passive scalar by a quasi-uniform random straining field. J. Fluid Mech. 64, 737–762.10.1017/S0022112074001881CrossRefGoogle Scholar
Lignell, D.O., Fredline, G.C. & Lewis, A.D. 2015 Comparison of one-dimensional turbulence and direct numerical simulations of soot formation and transport in a nonpremixed ethylene jet flame. Proc. Combust. Inst. 35, 1199–1206.CrossRefGoogle Scholar
Lundgren, T.S. 1981 Turbulent pair dispersion and scalar diffusion. J. Fluid Mech. 111, 27–57.10.1017/S0022112081002280CrossRefGoogle Scholar
Obukhov, A.M. 1941 Spectral energy distribution in turbulent flow. Izv. Akad. Nauk: SSSR. Ser. Matematicheskaya 5, 453–566.Google Scholar
Obukhov, A.M. 1962 Some specific features of atmospheric turbulence. J. Fluid Mech. 13, 77–81.10.1017/S0022112062000506CrossRefGoogle Scholar
O’Gorman, P.A. & Pullin, D.I. 2005 Effect of Schmidt number on the velocity-scalar cospectrum in isotropic turbulence with a mean scalar gradient. J. Fluid Mech. 532, 111–140.CrossRefGoogle Scholar
Pope, S.B. 2000 Turbulent Flows. Cambridge University Press.Google Scholar
Richardson, L. 1926 Atmospheric diffusion shown on a distance-neighbour graph. Proc. R. Soc. Lond. A 110, 709–737.Google Scholar
Salazar, J.P.L.C. & Collins, L.R. 2009 Two-particle dispersion in isotropic turbulent flows. Annu. Rev. Fluid Mech.. 41, 405–432.CrossRefGoogle Scholar
Scatamacchia, R., Biferale, L. & Toschi, F. 2012 Extreme events in the dispersion of two neighboring particles under the influence of fluid turbulence. Phys. Rev. Lett. 45, 14501.Google Scholar
Shete, K.P., Boucher, D.J., Riley, J.J. & de Bruyn Kops, S.M. 2022 Effect of viscous-convective subrange on passive scalar statistics at high Reynolds number. Phys. Rev. Fluids 7, 024601.CrossRefGoogle Scholar
Shraiman, B.I. & Siggia, E.D. 2000 Scalar turbulence. Nature 405, 639–646.10.1038/35015000CrossRefGoogle ScholarPubMed
Su, L.K. & Clemens, N.T. 2003 The structure of fine-scale scalar mixing in gas-phase planar turbulent jets. J. Fluid Mech. 488, 1–29.CrossRefGoogle Scholar
Subramaniam, S. & Pope, S.B. 1998 A mixing model for turbulent reactive flows based on Euclidean minimum spanning trees. Combust. Flame 115, 487–514.10.1016/S0010-2180(98)00023-6CrossRefGoogle Scholar
Touchette, H. 2009 The large deviation approach to statistical mechanics. Phys. Rep. 478, 1–69.CrossRefGoogle Scholar
Warhaft, Z. 2000 Passive scalars in turbulent flows. Annu. Rev. Fluid Mech. 32, 203–240.CrossRefGoogle Scholar
Watanabe, T. & Gotoh, T. 2004 Statistics of a passive scalar in homogeneous turbulence. New J. Phys. 6, 40.10.1088/1367-2630/6/1/040CrossRefGoogle Scholar
Yeung, P.K. & Sreenivasan, K.R. 2013 Spectrum of passive scalars of high molecular diffusivity in turbulent mixing. J. Fluid Mech. 716, R14.CrossRefGoogle Scholar
Yeung, P.K. & Sreenivasan, K.R. 2014 Direct numerical simulation of turbulent mixing at very low Schmidt number with a uniform mean gradient. Phys. Fluids 26, 015107.CrossRefGoogle Scholar