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Methodology for determining coefficients of turbulent mixing model

Published online by Cambridge University Press:  30 October 2020

You-sheng Zhang Open the ORCID record for You-sheng Zhang [Opens in a new window] ,
Zhi-wei He ,
Han-song Xie ,
Meng-Juan Xiao Open the ORCID record for Meng-Juan Xiao [Opens in a new window]  and
Bao-lin Tian
You-sheng Zhang
Affiliation:
Institute of Applied Physics and Computational Mathematics, Beijing100094, PR China Center for Applied Physics and Technology, HEDPS, and College of Engineering, Peking University, Beijing100871, PR China
Zhi-wei He
Affiliation:
Institute of Applied Physics and Computational Mathematics, Beijing100094, PR China
Han-song Xie*
Affiliation:
Institute of Applied Physics and Computational Mathematics, Beijing100094, PR China
Meng-Juan Xiao*
Affiliation:
Institute of Applied Physics and Computational Mathematics, Beijing100094, PR China
Bao-lin Tian
Affiliation:
Institute of Applied Physics and Computational Mathematics, Beijing100094, PR China Center for Applied Physics and Technology, HEDPS, and College of Engineering, Peking University, Beijing100871, PR China
*
†Email addresses for correspondence: xiehansong19@gscaep.ac.cn; xiao_mengjuan@163.com
†Email addresses for correspondence: xiehansong19@gscaep.ac.cn; xiao_mengjuan@163.com
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Abstract

The accurate prediction of turbulent mixing induced by Rayleigh–Taylor (R–T), Richtmyer–Meshkov (R–M) and Kelvin–Helmholtz (K–H) instabilities is very important in understanding natural phenomena and improving engineering applications. In applications, the prediction of mixing with the Reynolds-averaged Navier–Stokes (RANS) equation remains the most widely used method. The RANS method involves two aspects, i.e. physical modelling and model coefficients. Generally, the latter is determined empirically; thus, there is a lack of universality. In this paper, inspired by the well-known Reynolds decomposition, we propose a methodology to determine the model coefficients with the following three steps: (i) preset a set of analytical RANS solutions by fully using the knowledge of mixing evolutions; (ii) simplify the differential RANS equations to algebraic equations by imposing the preset solutions to RANS equations; (iii) solve the algebraic equations approximately to give the values of the entire model coefficients. The specific application of this methodology in the widely used K–L mixing model shows that, using the same set of model coefficients determined from the current methodology, the K–L model successfully predicts the mixing evolutions in terms of different physical quantities (e.g. temporal scalings and spatial profiles), density ratios and problems (e.g. R–T, R–M, K–H and reshocked R–M mixings). It is possible to extend this methodology to other turbulence models characterised with self-similar evolutions, such as K-$\epsilon$ mixing models.

JFM classification

Mixing: Turbulent mixing Turbulent Flows: Turbulence modelling
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 905 , 25 December 2020 , A26
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

REFERENCES

Akula, B. & Ranjan, D. 2016 Dynamics of buoyancy-driven flows at moderately high Atwood numbers. J. Fluid Mech. 795, 313–355.CrossRefGoogle Scholar
Anderews, M. J. & Spalding, D. B. 1990 A simple experiment to investigate 2-dimensional mixing by Rayleigh–Taylor instability. Phys. Fluids 2 (6), 922–927.CrossRefGoogle Scholar
Banerjee, A. & Andrews, M. J. 2009 3D simulations to investigate initial condition effects on the growth of Rayleigh–Taylor mixing. Intl J. Heat Mass Transfer 52 (17), 3906–3917.CrossRefGoogle Scholar
Bell, J. H. & Mehta, R. D. 1990 Development of a two-stream mixing layer from tripped and untripped boundary layers. AIAA J. 28 (12), 2034–2042.CrossRefGoogle Scholar
Brown, G. L. & Roshko, A. 1974 On density effects and large structure in turbulent mixing layers. J. Fluid Mech. 64 (4), 775–816.CrossRefGoogle Scholar
Burrows, A. 2000 Supernova explosions in the universe. Nature 403 (6771), 727–733.CrossRefGoogle ScholarPubMed
Cabot, W. & Zhou, Ye 2013 Statistical measurements of scaling and anisotropy of turbulent flows induced by Rayleigh–Taylor instability. Phys. Fluids 25 (1), 15107.CrossRefGoogle Scholar
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
Chiravalle, V. P. 2006 The k-L turbulence model for describing buoyancy-driven fluid instabilities. Laser Part. Beams 24 (3), 381–394.CrossRefGoogle Scholar
Clark, T. T. & Zhou, Y. 2003 Self-similarity of two flows induced by instabilities. Phys. Rev. E 68 (6), 066305.CrossRefGoogle ScholarPubMed
Cook, A. W. & Cabot, W. H. 2006 Reynolds number effects on Rayleigh–Taylor instability with possible implications for type Ia supernovae. Nat. Phys. 2 (8), 562–568.Google Scholar
Dimonte, G. 2000 Spanwise homogeneous buoyancy-drag model for Rayleigh–Taylor mixing and experimental evaluation. Phys. Plasmas 7 (6), 2255–2269.CrossRefGoogle Scholar
Dimonte, G. 2004 Dependence of turbulent Rayleigh–Taylor instability on initial perturbations. Phys. Rev. E 69 (5), 056305.Google ScholarPubMed
Dimonte, G. & Schneider, M. 2000 Density ratio dependence of Rayleigh–Taylor mixing for sustained and impulsive acceleration histories. Phys. Fluids 12 (2), 304–321.Google 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), 085101.CrossRefGoogle Scholar
Dimonte, G., Youngs, D. L., Dimits, A., Weber, S., Marinak, M., Wunsch, S., Garasi, C., Robinson, A., Andrews, M. J., Ramaprabhu, P., et al. 2004 A comparative study of the turbulent Rayleigh–Taylor instability using high-resolution three-dimensional numerical simulations: the alpha-group collaboration. Phys. Fluids 16 (5), 1668–1693.CrossRefGoogle Scholar
Gao, F., He, Z., Zhang, Y., Li, L. & Tian, B. 2017 The characteristic of turbulent mixing at late stage of the Richtmyer–Meshkov instability. AIP Adv. 7 (7), 075020.CrossRefGoogle Scholar
Gao, F., Zhang, Y., He, Z. & Tian, B. 2016 Formula for growth rate of mixing width applied to Richtmyer–Meshkov instability. Phys. Fluids 28 (11), 114101.CrossRefGoogle Scholar
Harten, A. 1997 High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 135 (2), 260–278.Google Scholar
Harten, A., Lax, P. D. & Leer, B. V. 1983 On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25 (1), 35–61.Google Scholar
Helmholtz, V. 1868 On discontinuous movements of fluids. Lond. Edinb. Dubl. Phil. Mag. J. Sci. 36 (244), 337–346.CrossRefGoogle Scholar
Kelvin, L. 1871 Hydrokinetic solutions and observations. Lond. Edinb. Dubl. Phil. Mag. J. Sci. 42 (281), 362–377.Google Scholar
Kim, K. H. & Kim, C. 2005 a Accurate, efficient and monotonic numerical methods for multi-dimensional compressible flows. Part I: spatial discretization. J. Comput. Phys. 208 (2), 527–569.Google Scholar
Kim, K. H. & Kim, C. 2005 b Accurate, efficient and monotonic numerical methods for multi-dimensional compressible flows. Part II: multi-dimensional limiting process. J. Comput. Phys. 208 (2), 570–615.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.Google 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
Krivets, V. V., Ferguson, K. J. & Jacobs, J. W. 2017 Turbulent mixing induced by Richtmyer–Meshkov instability. AIP Conf. Proc. 1793 (1), 1–4.Google Scholar
Li, H., He, Z. W., Zhang, Y. S. & Tian, B. L. 2019 a Growth law of reshocked Richtmyer–Meshkov mixing width. J. Fluid Mech. (submitted).Google Scholar
Li, H., He, Z. W., Zhang, Y. S. & Tian, B. L. 2019 b On the role of rarefaction/compression waves in Richtmyer–Meshkov instability with reshock. Phys. Fluids 31 (5), 54102.Google Scholar
Liu, H. & Xiao, Z. 2016 Scale-to-scale energy transfer in mixing flow induced by the Richtmyer–Meshkov instability. Phys. Rev. E 93 (5), 053112.CrossRefGoogle ScholarPubMed
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 ScholarPubMed
Livescu, D., Wei, T. & Petersen, M. R. 2011 Direct numerical simulations of Rayleigh–Taylor instability. J. Phys.: Conf. Ser. 318 (8), 082007.Google Scholar
Meshkov, E. E. 1969 Instability of the interface of two gases accelerated by a shock wave. Fluid Dyn. 4 (5), 101–104.Google Scholar
Moran-Lopez, J. T. & Schilling, O. 2013 Multicomponent Reynolds-averaged Navier–Stokes simulations of reshocked Richtmyer–Meshkov instability-induced mixing. High Energ. Dens. Phys. 9 (1), 112–121.CrossRefGoogle Scholar
Morgan, B. E. 2018 Response to ‘Comment on “Large-eddy simulation and unsteady RANS simulations of a shock-accelerated heavy gas cylinder” by B. E. Morgan, J. Greenough’. Shock Waves 28 (6), 1301–1302.Google Scholar
Morgan, B. E. & Greenough, J. A. 2016 Large-eddy and unsteady RANS simulations of a shock-accelerated heavy gas cylinder. Shock Waves 26, 355–383.Google Scholar
Morgan, B. E, Schilling, O. & Hartland, T. A. 2018 Two-length-scale turbulence model for self-similar buoyancy-, shock-, and shear-driven mixing. Phys. Rev. E 97 (1), 013104.CrossRefGoogle ScholarPubMed
Mueschke, N. J. & Schilling, O. 2009 Investigation of Rayleigh–Taylor turbulence and mixing using direct numerical simulation with experimentally measured initial conditions. I. Comparison to experimental data. Phys. Fluids 21 (1), 14106.Google Scholar
Olson, D. H. & Jacobs, J. W. 2009 Experimental study of Rayleigh–Taylor instability with a complex initial perturbation. Phys. Fluids 21 (3), 34103.CrossRefGoogle Scholar
Poggi, F., Thorembey, M.-H. & Rodriguez, G. 1998 Velocity measurements in turbulent gaseous mixtures induced by Richtmyer–Meshkov instability. Phys. Fluids 10 (11), 2698–2700.CrossRefGoogle Scholar
Ramaprabhu, P., Dimonte, G. & Andrews, M. J. 2005 A numerical study of the influence of initial perturbations on the turbulent Rayleigh–Taylor instability. J. Fluid Mech. 536, 285–319.CrossRefGoogle Scholar
Rayleigh, Lord 1882 Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proc. Lond. Math. Soc. 201 (1), 170–177.CrossRefGoogle Scholar
Read, K. I. 1984 Experimental investigation of turbulent mixing by Rayleigh–Taylor instability. Physica D 12 (1–3), 45–58.Google Scholar
Richtmyer, R. D. 1960 Taylor instability in shock acceleration of compressible fluids. Commun. Pure. Appl. Maths 13 (2), 297–319.CrossRefGoogle Scholar
Roberts, M. S. & Jacobs, J. W. 2016 The effects of forced small-wavelength, finite-bandwidth initial perturbations and miscibility on the turbulent Rayleigh–Taylor instability. J. Fluid Mech. 787, 50–83.CrossRefGoogle Scholar
Ruan, Y. C., Zhang, Y. S., Tian, B. L. & Zhang, X. T. 2019 Density-ratio-invariant mean-species-profile of classical Rayleigh–Taylor mixing. Phys. Rev. Fluids (submitted).Google Scholar
Slessor, M. D., Zhuang, M. & Dimotakis, P. E. 2000 Turbulent shear-layer mixing: growth-rate compressibility scaling. J. Fluid Mech. 414, 35–45.CrossRefGoogle Scholar
Sweby, P. K. 1984 High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal. 21 (5), 995–1011.Google Scholar
Taylor, G. 1950 The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I. Proc. R. Soc. Lond. A 201 (1065), 192.Google Scholar
Thomas, V. A. & Kares, R. J. 2012 Drive asymmetry and the origin of turbulence in an ICF implosion. Phys. Rev. Lett. 109 (7), 075004.Google Scholar
Thornber, B., Drikakis, D., Williams, R. J. R. & Youngs, D. 2008 a On entropy generation and dissipation of kinetic energy in high-resolution shock-capturing schemes. J. Comput. Phys. 227 (10), 4853–4872.CrossRefGoogle Scholar
Thornber, B., Drikakis, D., Youngs, D. L. & Williams, R. J. R. 2010 The influence of initial conditions on turbulent mixing due to Richtmyer–Meshkov instability. J. Fluid Mech. 654, 99–139.Google Scholar
Thornber, B., Groom, M. & Youngs, D. 2018 A five-equation model for the simulation of miscible and viscous compressible fluids. J. Comput. Phys. 372, 256–280.CrossRefGoogle Scholar
Thornber, B., Mosedale, A., Drikakis, D., Youngs, D. & Williams, R. J. R. 2008 b An improved reconstruction method for compressible flows with low Mach number features. J. Comput. Phys. 227 (10), 4873–4894.Google Scholar
Toro, E. F., Spruce, M. & Speares, W. 1994 Restoration of the contact surface in the HLL-Riemann solver. Shock Waves 4 (1), 25–34.CrossRefGoogle Scholar
Tritschler, V. K., Zubel, M., Hickel, S. & Adams, N. A. 2014 Evolution of length scales and statistics of Richtmyer–Meshkov instability from direct numerical simulations. Phys. Rev. E 90 (6), 063001.CrossRefGoogle ScholarPubMed
Vetter, M. & Sturtevant, B. 1995 Experiments on the Richtmyer–Meshkov instability of an air/SF6 interface. Shock Waves 4 (5), 247–252.CrossRefGoogle Scholar
Xiao, M., Zhang, Y. & Tian, B. 2020 Unified prediction of reshocked Richtmyer–Meshkov mixing with K-L model. Phys. Fluids 32 (3), 032107.Google Scholar
Youngs, D. L. 1989 Modelling turbulent mixing by Rayleigh–Taylor instability. Physica D 37 (1–3), 270–287.Google 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
Youngs, D. L. 2017 Rayleigh–Taylor mixing: direct numerical simulation and implicit large eddy simulation. Phys. Scr. 92 (7), 074006.CrossRefGoogle Scholar
Zhang, Y.-S. 2018 Comment on ‘Large-eddy and unsteady RANS simulations of a shock-accelerated heavy gas cylinder’ by B. E. Morgan, J. Greenough. Shock Waves 28 (6), 1299–1300.CrossRefGoogle Scholar
Zhang, Y.-S., He, Z.-W., Gao, F.-J., Li, X.-L. & Tian, B.-L. 2016 Evolution of mixing width induced by general Rayleigh–Taylor instability. Phys. Rev. E 93 (6), 063102.CrossRefGoogle ScholarPubMed
Zhang, Y.-S., Ruan, Y.-C., Xie, H.-S. & Tian, B.-L. 2020 Mixed mass of classical Rayleigh–Taylor mixing at arbitrary density ratios. Phys. Fluids 32 (1), 011702.Google 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., Cabot, W. H. & Thornber, B. 2016 Asymptotic behavior of the mixed mass in Rayleigh–Taylor and Richtmyer–Meshkov instability induced flows. Phys. Plasmas 23 (5), GO5.012.CrossRefGoogle Scholar
Zhou, Z.-R., Zhang, Y.-S. & Tian, B.-L. 2018 Dynamic evolution of Rayleigh–Taylor bubbles from sinusoidal, w-shaped, and random perturbations. Phys. Rev. E 97 (3), 033108.Google ScholarPubMed