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Local transition indicator and modelling of turbulent mixing based on the mixing state

Published online by Cambridge University Press:  23 December 2024

Meng-Juan Xiao Open the ORCID record for Meng-Juan Xiao [Opens in a new window] ,
Han Qi Open the ORCID record for Han Qi [Opens in a new window]  and
You-Sheng Zhang Open the ORCID record for You-Sheng Zhang [Opens in a new window]
Meng-Juan Xiao
Affiliation:
Institute of Applied Physics and Computational Mathematics, Beijing 100094, PR China National Key Laboratory of Computational Physics, Beijing 100088, PR China
Han Qi
Affiliation:
Institute of Applied Physics and Computational Mathematics, Beijing 100094, PR China
You-Sheng Zhang*
Affiliation:
Institute of Applied Physics and Computational Mathematics, Beijing 100094, PR China National Key Laboratory of Computational Physics, Beijing 100088, PR China Center for Applied Physics and Technology, HEDPS, and College of Engineering, Peking University, Beijing 100871, PR China
*
†Email address for correspondence: [email protected]
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Abstract

Turbulent mixing is a pivotal phenomenon in fusion research with profound implications for energy gain. A Reynolds-averaged Navier–Stokes model capable of predicting realistic mixing transition processes is of significant importance for fusion applications, yet such a model remains elusive. This work addresses the limitations of prevalent global transition criteria, proposing a new idea to quantify local transition characteristics based on the mixing state, recognizing its direct relevance to fusion reaction rates. We delve into the description and analysis of the spatiotemporal evolution of the mixing state and its interplay with the transition process. Then, a local transition indicator is developed and compared with conventional global criteria using the large-eddy simulation (LES) of Rayleigh–Taylor turbulent mixing. Building upon this foundation, we introduce a novel eddy viscosity model based on the local transition indicator. A posterior assessment using LES data validates that it significantly outperforms standard gradient transport models during the transition stage. Consequently, we integrate this new eddy viscosity model with the Besnard–Harlow–Rauenzahn model to construct a comprehensive transition model, which demonstrates reasonably good performance in comparison with LES results. This work paves the way for future research in developing advanced modelling strategies that can effectively address the complexities of transitional flows in fusion engineering applications.

JFM classification

Mixing: Turbulent mixing Turbulent Flows: Turbulent transition Instability: Transition to turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1002 , 10 January 2025 , A4
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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References

Besnard, D., Harlow, F.H., Rauenzahn, R.M. & Zemach, C. 1992 Turbulence transport equations for variable-density turbulence and their relationship to two-field models. Tech. Rep. LA-12303-MS; ON: DE92017292.CrossRefGoogle Scholar
Burrows, A. 2000 Supernova explosions in the universe. Nature 403 (17), 727–733.CrossRefGoogle ScholarPubMed
Cabot, W.H. & Cook, A.W. 2006 Reynolds number effects on Rayleigh–Taylor instability with possible implications for type-Ia supernovae. Nat. Phys. 2 (8), 562–568.CrossRefGoogle Scholar
Cook, A.W., Cabot, W. & Miller, P.L. 2004 The mixing transition in Rayleigh–Taylor instability. J. Fluid Mech. 511, 333–362.CrossRefGoogle Scholar
Denissen, N.A., Rollin, B., Reisner, J.M. & Andrews, M.J. 2014 The tilted rocket rig: a Rayleigh–Taylor test case for RANS models. Trans. ASME J. Fluids Engng 136 (9), 091301.CrossRefGoogle Scholar
Dimonte, G. & Tipton, R. 2006 K-L turbulence model for the self-similar growth of the Rayleigh–Taylor and Richtmyer–Meshkov instabilities. Phys. Fluids 18 (8), 85101.CrossRefGoogle Scholar
Dimotakis, P.E. 2000 The mixing transition in turbulent flows. J. Fluid Mech. 409, 69–98.CrossRefGoogle Scholar
Griffond, J., Soulard, O. & Gréa, B.-J. 2023 A modified dissipation equation for Reynolds stress turbulent mixing models. Trans. ASME J. Fluids Engng 145 (2), 021502.CrossRefGoogle Scholar
Grinstein, F.F. 2017 Initial conditions and modeling for simulations of shock driven turbulent material mixing. Comput. Fluids 151, 58–72.CrossRefGoogle Scholar
Grinstein, F.F., Rauenzahn, R., Saenz, J. & Francois, M. 2017 Coarse grained simulation of shock-driven turbulent mixing. In Proceedings of the ASME Fluids Engineering Division Summer Meeting, vol. 1b. ASME.CrossRefGoogle Scholar
Haines, B.M., Grinstein, F.F. & Schwarzkopf, J.D. 2013 Reynolds–averaged Navier–Stokes initialization and benchmarking in shock-driven turbulent mixing. J. Turbul. 14 (2), 46–70.CrossRefGoogle Scholar
Helmholtz, V. 1868 On discontinuous movements of fluids. Lond. Edinb. Dublin Phil. Mag. J. Sci. 36 (244), 337–346.CrossRefGoogle Scholar
Kelvin, L. 1871 Hydrokinetic solutions and observations. Lond. Edinb. Dublin Phil. Mag. J. Sci. 42 (281), 362–377.Google Scholar
Kokkinakis, I.W., Drikakis, D. & Youngs, D.L. 2019 Modeling of Rayleigh–Taylor mixing using single-fluid models. Phys. Rev. E 99 (1), 013104.CrossRefGoogle ScholarPubMed
Kokkinakis, I.W., Drikakis, D., Youngs, D.L. & Williams, R.J.R. 2015 Two-equation and multi fluid turbulence models for Rayleigh–Taylor mixing. Intl J. Heat Fluid Flow 56, 233–250.CrossRefGoogle Scholar
Livescu, D. 2013 Numerical simulations of two-fluid turbulent mixing at large density ratios and applications to the Rayleigh–Taylor instability. Phil. Trans. R. Soc. Lond. A 371 (2003), 20120185.Google Scholar
Livescu, D., Ristorcelli, J.R., Gore, R.A., Dean, S.H., Cabot, W.H. & Cook, A.W. 2009 High-Reynolds number Rayleigh–Taylor turbulence. J. Turbul. 10, N13.CrossRefGoogle Scholar
Livescu, D., Wei, T. & Petersen, M.R. 2011 Direct numerical simulations of Rayleigh–Taylor instability. J. Phys.: Conf. Ser. 318 (8), 082007.Google Scholar
Lombardini, M., Pullin, D.I. & Meiron, D.I. 2012 Transition to turbulence in shock-driven mixing: a Mach number study. J. Fluid Mech. 690, 203–226.CrossRefGoogle Scholar
Meshkov, E.E. 1969 Instability of the interface of two gases accelerated by a shock wave. Fluid Dyn. 4 (5), 101–104.CrossRefGoogle Scholar
Mohaghar, M., Carter, J., Pathikonda, G. & Ranjan, D. 2019 The transition to turbulence in shock-driven mixing: effects of Mach number and initial conditions. J. Fluid Mech. 871, 595–635.CrossRefGoogle Scholar
Morgan, B.E. & Black, W.J. 2019 Parametric investigation of the transition to turbulence in Rayleigh–Taylor mixing. Phys. D: Nonlinear Phenom. 402, 132223.CrossRefGoogle Scholar
Morgan, B.E. & Wickett, M.E. 2015 Three-equation model for the self-similar growth of Rayleigh–Taylor and Richtmyer–Meskov instabilities. Phys. Rev. E 91 (4), 043002.CrossRefGoogle ScholarPubMed
Mosedale, A. & Drikakis, D. 2007 Assessment of very high order of accuracy in implicit LES models. Trans. ASME J. Fluids Engng 129 (12), 1497–1503.CrossRefGoogle Scholar
Petrasso, R.D. 1994 Inertial fusion -Rayleigh's challenge endures. Nature 367, 217–218.CrossRefGoogle Scholar
Rayleigh, Lord 1883 Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proc. Lond. Math. Soc. 201 (1), 170–177.Google Scholar
Richtmyer, R.D. 1960 Taylor instability in shock acceleration of compressible fluids. Commun. Pure Appl. Maths 13 (2), 297–319.CrossRefGoogle Scholar
Ristorcelli, J.R. 2017 Exact statistical results for binary mixing and reaction in variable density turbulence. Phys. Fluids 29, 020705.CrossRefGoogle Scholar
Robey, H.F., Zhou, Y., Buckingham, A.C., Keiter, P., Remington, B.A. & Drake, R.P. 2003 The time scale for the transition to turbulence in a high Reynolds number, accelerated flow. Phys. Plasmas 10, P614.CrossRefGoogle Scholar
Rollin, B., Denissen, N.A., Reisner, J.M. & Andrews, M.J. 2012 Simulations of the tilted rig experiment using the xrage and flag hydrocodes. ASME Intl Mech. Engng Cong. Exposition 7, 1353–1361.Google Scholar
Schwarzkopf, J.D., Livescu, D., Gore, R.A., Rauenzahn, R.M. & Ristorcelli, J.R. 2011 Application of a second-moment closure model to mixing processes involving multicomponent miscible fluids. J. Turbul. 12, 1–35.CrossRefGoogle Scholar
Singh, A.P., Duraisamy, K. & Morgan, B.E. 2019 Data-augmented modeling of transition to turbulence in Rayleigh–Taylor mixing layers. LLNL-TR-767683.CrossRefGoogle Scholar
Taylor, G. 1950 The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I. Proc. R. Soc. Lond. A Math. Phys. Sci. 201 (1065), 192.Google Scholar
Wang, R., Song, Y., Ma, Z.Q., Zhang, C.B., Shi, X.F., Wang, L.L. & Wang, P. 2022 Transitional model for rarefaction-driven Rayleigh–Taylor mixing on the diffuse interface. Phys. Fluids 34, 075136.CrossRefGoogle Scholar
Xiao, M.J., Hu, Z.X., Dai, Z.H. & Zhang, Y.S. 2022 Experimentally consistent large-eddy simulation of re-shocked Richtmyer–Meshkov turbulent mixing. Phys. Fluids 34, 125125.CrossRefGoogle Scholar
Xie, H.S., Xiao, M.J. & Zhang, Y.S. 2021 Unified prediction of turbulent mixing induced by interfacial instabilities via Besnard–Harlow–Rauenzahn-2 model. Phys. Fluids 33 (10), 105123.CrossRefGoogle Scholar
Youngs, D.L. 2013 The density ratio dependence of self-similar Rayleigh–Taylor mixing. Phil. Trans. R. Soc. Lond. A 371 (2003), 20120173.Google ScholarPubMed
Zhang, Y.S., He, Z.W., Xie, H.S., Xiao, M.J. & Tian, B.L. 2020 a Methodology for determining the coefficients of turbulent mixing model. J. Fluid Mech. 905, A26.CrossRefGoogle Scholar
Zhang, Y.S., Ni, W.D., Ruan, Y.C. & Xie, H.S. 2020 b Quantifying mixing of Rayleigh–Taylor turbulence. Phys. Rev. Fluids 5, 104501.CrossRefGoogle Scholar
Zhang, Y.-S., Ruan, Y.-C., Xie, H.-S. & Tian, B.-L. 2020 c Mixed mass of classical Rayleigh–Taylor mixing at arbitrary density ratios. Phys. Fluids 32 (1), 011702.CrossRefGoogle Scholar
Zhou, Y. 2007 Unification and extension of the similarity scaling criteria and mixing transition for studying astrophysics using high energy density laboratory experiments or numerical simulations. Phys. Plasmas 14, 082701.CrossRefGoogle Scholar
Zhou, Y. 2017 a Rayleigh–Taylor and Richtmyer–Meshkov instability induced flow, turbulence, and mixing. I. Phys. Rep. 720–722 (C), 1–136.Google Scholar
Zhou, Y. 2017 b Rayleigh–Taylor and Richtmyer–Meshkov instability induced flow, turbulence, and mixing. II. Phys. Rep. 723–725, 1–160.Google Scholar
Zhou, Y., Clark, T.T., Clark, D.S., Glendinning, S.G., Skinner, M.A., Huntington, C.M., Hurricane, O.A., Dimits, A.M. & Remington, B.A. 2019 Turbulent mixing and transition criteria of flows induced by hydrodynamic instabilities. Phys. Plasmas 26 (8), 080901.CrossRefGoogle Scholar
Zhou, Y., Robey, H.F. & Buckingham, A.C. 2003 Onset of turbulence in accelerated high-Reynolds-number flow. Phys. Rev. E 67, 056305.CrossRefGoogle ScholarPubMed