Modeling and simulation of transitional Rayleigh–Taylor flow with partially averaged Navier–Stokes equations

Pereira, Filipe Miguel Soares; Grinstein, Fernando F.; Israel, Daniel M.; Rauenzahn, Rick Meier; Girimaji, Sharath S.

doi:10.1063/5.0067552

Title: Modeling and simulation of transitional Rayleigh–Taylor flow with partially averaged Navier–Stokes equations

Journal Article · Mon Nov 15 00:00:00 EST 2021 · Physics of Fluids

DOI:https://doi.org/10.1063/5.0067552· OSTI ID:1844161

^[1];

^[1]; Girimaji, Sharath S. ^[2]

Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Texas A & M Univ., College Station, TX (United States)

In this work, the partially averaged Navier–Stokes (PANS) equations are used to predict the variable-density Rayleigh–Taylor (RT) flow at Atwood number 0.5 and maximum Reynolds number 500. This is a prototypical problem of material mixing, featuring laminar, transitional, and turbulent flow, instabilities and coherent structures, density fluctuations, and production of turbulence kinetic energy by both shear and buoyancy mechanisms. These features pose numerous challenges to modeling and simulation, making the RT flow ideal to develop the validation space of the recently proposed PANS Besnard–Harlow–Rauenzahn-linear eddy viscosity model closure. The numerical simulations are conducted at different levels of physical resolution and test three approaches to set the parameters $$f_\phi$$ defining the range of physically resolved scales. The computations demonstrate the efficiency (accuracy vs cost) of the PANS model predicting the spatiotemporal development of the RT flow. Results comparable to large-eddy simulations and direct numerical simulations are obtained at significantly lower physical resolution without the limitations of the Reynolds-averaged Navier–Stokes equations in these transitional flows. The data also illustrate the importance of appropriate selection of the physical resolution and the resolved fraction of each dependent quantity $$\phi$$ of the turbulent closure, $$f_\phi$$. These two aspects determine the ability of the model to resolve the flow phenomena not amenable to modeling by the closure and, as such, the computations’ fidelity.

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Research Organization:: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: 89233218CNA000001

OSTI ID:: 1844161

Report Number(s):: LA-UR-21-28227; TRN: US2302576

Journal Information:: Physics of Fluids, Vol. 33, Issue 11; ISSN 1070-6631

Publisher:: American Institute of Physics (AIP)Copyright Statement

Country of Publication:: United States

Language:: English

References (81)

A First Course in Turbulence Tennekes, Henk; Lumley, John L. MIT Press https://doi.org/10.7551/mitpress/3014.001.0001		January 1972
Rayleigh–Taylor and Richtmyer–Meshkov instability induced flow, turbulence, and mixing. I Zhou, Ye Physics Reports, Vol. 720-722 https://doi.org/10.1016/j.physrep.2017.07.005	journal	December 2017
Restoration of the contact surface in the HLL-Riemann solver Toro, E. F.; Spruce, M.; Speares, W. Shock Waves, Vol. 4, Issue 1 https://doi.org/10.1007/BF01414629	journal	July 1994
Rayleigh–Taylor and Richtmyer–Meshkov instability induced flow, turbulence, and mixing. II Zhou, Ye Physics Reports, Vol. 723-725 https://doi.org/10.1016/j.physrep.2017.07.008	journal	December 2017
Dynamics of laboratory diapir and plume models Whitehead, John A.; Luther, Douglas S. Journal of Geophysical Research, Vol. 80, Issue 5 https://doi.org/10.1029/JB080i005p00705	journal	February 1975
Development and validation of a turbulent-mix model for variable-density and compressible flows Banerjee, Arindam; Gore, Robert A.; Andrews, Malcolm J. Physical Review E, Vol. 82, Issue 4 https://doi.org/10.1103/PhysRevE.82.046309	journal	October 2010
General Circulation Experiments with the Primitive Equations: i. the Basic Experiment* Smagorinsky, J. Monthly Weather Review, Vol. 91, Issue 3 https://doi.org/10.1175/1520-0493(1963)091<0099:GCEWTP>2.3.CO;2	journal	March 1963
On the simulation of the flow around a circular cylinder at $R e = 140, 000$ Pereira, Filipe S.; Eça, Luís; Vaz, Guilherme International Journal of Heat and Fluid Flow, Vol. 76 https://doi.org/10.1016/j.ijheatfluidflow.2019.01.007	journal	April 2019
Effects of Diffusion on Interface Instability between Gases Duff, R. E.; Harlow, F. H.; Hirt, C. W. Physics of Fluids, Vol. 5, Issue 4 https://doi.org/10.1063/1.1706634	journal	January 1962
Reactive Rayleigh–Taylor turbulence Chertkov, M.; Lebedev, V.; Vladimirova, N. Journal of Fluid Mechanics, Vol. 633 https://doi.org/10.1017/S0022112009007666	journal	August 2009
Application of a second-moment closure model to mixing processes involving multicomponent miscible fluids Schwarzkopf, John D.; Livescu, Daniel; Gore, Robert A. Journal of Turbulence, Vol. 12 https://doi.org/10.1080/14685248.2011.633084	journal	January 2011
Partially averaged Navier-Stokes closure modeling for variable-density turbulent flow Pereira, F. S.; Grinstein, F. F.; Israel, D. M. Physical Review Fluids, Vol. 6, Issue 8 https://doi.org/10.1103/PhysRevFluids.6.084602	journal	August 2021
On the identification of a vortex Jeong, Jinhee; Hussain, Fazle Journal of Fluid Mechanics, Vol. 285, Issue -1 https://doi.org/10.1017/S0022112095000462	journal	February 1995
Toward Predictive RANS and SRS Computations of Turbulent External Flows of Practical Interest Pereira, Filipe S.; Eça, Luís; Vaz, Guilherme Archives of Computational Methods in Engineering https://doi.org/10.1007/s11831-021-09563-0	journal	February 2021
Towards the Ultimate Conservative Difference Scheme van Leer, Bram Journal of Computational Physics, Vol. 135, Issue 2 https://doi.org/10.1006/jcph.1997.5704	journal	August 1997
Two-equation and multi-fluid turbulence models for Rayleigh–Taylor mixing Kokkinakis, I. W.; Drikakis, D.; Youngs, D. L. International Journal of Heat and Fluid Flow, Vol. 56 https://doi.org/10.1016/j.ijheatfluidflow.2015.07.017	journal	December 2015
The mechanics of an organized wave in turbulent shear flow Hussain, A. K. M. F.; Reynolds, W. C. Journal of Fluid Mechanics, Vol. 41, Issue 2 https://doi.org/10.1017/S0022112070000605	journal	April 1970
3D Simulations to investigate initial condition effects on the growth of Rayleigh–Taylor mixing Banerjee, Arindam; Andrews, Malcolm J. International Journal of Heat and Mass Transfer, Vol. 52, Issue 17-18 https://doi.org/10.1016/j.ijheatmasstransfer.2009.03.032	journal	August 2009
High-Reynolds number Rayleigh–Taylor turbulence Livescu, D.; Ristorcelli, J. R.; Gore, R. A. Journal of Turbulence, Vol. 10 https://doi.org/10.1080/14685240902870448	journal	January 2009
Chemically Driven Hydrodynamic Instabilities Almarcha, C.; Trevelyan, P. M. J.; Grosfils, P. Physical Review Letters, Vol. 104, Issue 4 https://doi.org/10.1103/PhysRevLett.104.044501	journal	January 2010
Initial conditions and modeling for simulations of shock driven turbulent material mixing Grinstein, Fernando F. Computers & Fluids, Vol. 151 https://doi.org/10.1016/j.compfluid.2016.11.003	journal	June 2017
An improved reconstruction method for compressible flows with low Mach number features Thornber, B.; Mosedale, A.; Drikakis, D. Journal of Computational Physics, Vol. 227, Issue 10 https://doi.org/10.1016/j.jcp.2008.01.036	journal	May 2008
Families of salt Domes in the gulf Coastal Province Selig, Franz; Wermund, E. G. GEOPHYSICS, Vol. 31, Issue 4 https://doi.org/10.1190/1.1439806	journal	August 1966
Self-similar solutions of asymmetric Rayleigh-Taylor mixing Hillier, A. Physics of Fluids, Vol. 32, Issue 1 https://doi.org/10.1063/1.5130893	journal	January 2020
Evaluation of RANS and SRS methods for simulation of the flow around a circular cylinder in the sub-critical regime Pereira, Filipe S.; Vaz, Guilherme; Eça, Luís Ocean Engineering, Vol. 186 https://doi.org/10.1016/j.oceaneng.2019.05.049	journal	August 2019
Scrutinizing URANS in Shedding Flows: The Case of Cylinder in Cross-Flow in the Subcritical Regime Palkin, E.; Mullyadzhanov, R.; Hadžiabdić, M. Flow, Turbulence and Combustion, Vol. 97, Issue 4 https://doi.org/10.1007/s10494-016-9772-z	journal	October 2016
A K–L model with improved realizability for turbulent mixing Xiao, Mengjuan; Zhang, Yousheng; Tian, Baolin Physics of Fluids, Vol. 33, Issue 2 https://doi.org/10.1063/5.0038212	journal	February 2021
Partially-Averaged Navier Stokes Model for Turbulence: Implementation and Validation Girimaji, Sharath; Abdol-Hamid, Khaled 43rd AIAA Aerospace Sciences Meeting and Exhibit https://doi.org/10.2514/6.2005-502	conference	June 2005
The scaling of the turbulent/non-turbulent interface at high Reynolds numbers Silva, Tiago S.; Zecchetto, Marco; da Silva, Carlos B. Journal of Fluid Mechanics, Vol. 843 https://doi.org/10.1017/jfm.2018.143	journal	March 2018
Effects of Atwood and Reynolds numbers on the evolution of buoyancy-driven homogeneous variable-density turbulence Aslangil, Denis; Livescu, Daniel; Banerjee, Arindam Journal of Fluid Mechanics, Vol. 895 https://doi.org/10.1017/jfm.2020.268	journal	May 2020
Turbulent Flows Pope, Stephen B. https://doi.org/10.1017/CBO9780511840531	book	July 2012
Effect of the numerical discretization scheme in Shock-Driven turbulent mixing simulations Pereira, F. S.; Grinstein, F. F.; Israel, D. Computers & Fluids, Vol. 201 https://doi.org/10.1016/j.compfluid.2020.104487	journal	April 2020
K-L turbulence model for the self-similar growth of the Rayleigh-Taylor and Richtmyer-Meshkov instabilities Dimonte, Guy; Tipton, Robert Physics of Fluids, Vol. 18, Issue 8 https://doi.org/10.1063/1.2219768	journal	August 2006
The RAGE radiation-hydrodynamic code Gittings, Michael; Weaver, Robert; Clover, Michael Computational Science & Discovery, Vol. 1, Issue 1 https://doi.org/10.1088/1749-4699/1/1/015005	journal	October 2008
Incompressible Rayleigh–Taylor Turbulence Boffetta, Guido; Mazzino, Andrea Annual Review of Fluid Mechanics, Vol. 49, Issue 1 https://doi.org/10.1146/annurev-fluid-010816-060111	journal	January 2017
Direct numerical simulations of incompressible Rayleigh–Taylor instabilities at low and medium Atwood numbers Hamzehloo, Arash; Bartholomew, Paul; Laizet, Sylvain Physics of Fluids, Vol. 33, Issue 5 https://doi.org/10.1063/5.0049867	journal	May 2021
Theory of the Rayleigh-Taylor instability Kull, H. J. Physics Reports, Vol. 206, Issue 5 https://doi.org/10.1016/0370-1573(91)90153-D	journal	August 1991
Growth rates of the ablative Rayleigh–Taylor instability in inertial confinement fusion Betti, R.; Goncharov, V. N.; McCrory, R. L. Physics of Plasmas, Vol. 5, Issue 5 https://doi.org/10.1063/1.872802	journal	May 1998
Gradient and counter-gradient scalar transport in turbulent premixed flames Veynante, D.; Trouvé, A.; Bray, K. N. C. Journal of Fluid Mechanics, Vol. 332 https://doi.org/10.1017/S0022112096004065	journal	February 1997
Vortex Dynamics in the Cylinder Wake Williamson, C. H. K. Annual Review of Fluid Mechanics, Vol. 28, Issue 1 https://doi.org/10.1146/annurev.fl.28.010196.002401	journal	January 1996
New phenomena in variable-density Rayleigh–Taylor turbulence Livescu, D.; Ristorcelli, J. R.; Petersen, M. R. Physica Scripta, Vol. T142 https://doi.org/10.1088/0031-8949/2010/T142/014015	journal	December 2010
Density ratio dependence of Rayleigh–Taylor mixing for sustained and impulsive acceleration histories Dimonte, Guy; Schneider, Marilyn Physics of Fluids, Vol. 12, Issue 2 https://doi.org/10.1063/1.870309	journal	February 2000
Magnetic field generation in Rayleigh-Taylor unstable inertial confinement fusion plasmas Srinivasan, Bhuvana; Dimonte, Guy; Tang, Xian-Zhu Physical Review Letters, Vol. 108, Issue 16 https://doi.org/10.1103/PhysRevLett.108.165002	journal	April 2012
Time-dependent study of anisotropy in Rayleigh-Taylor instability induced turbulent flows with a variety of density ratios Zhou, Ye; Cabot, William H. Physics of Fluids, Vol. 31, Issue 8 https://doi.org/10.1063/1.5110914	journal	August 2019
An overview of Rayleigh-Taylor instability Sharp, D. H. Physica D: Nonlinear Phenomena, Vol. 12, Issue 1-3 https://doi.org/10.1016/0167-2789(84)90510-4	journal	July 1984
Impact of numerical hydrodynamics in turbulent mixing transition simulations Grinstein, F. F.; Pereira, F. S. Physics of Fluids, Vol. 33, Issue 3 https://doi.org/10.1063/5.0034983	journal	March 2021
Variable-density buoyancy-driven turbulence with asymmetric initial density distribution Aslangil, Denis; Livescu, Daniel; Banerjee, Arindam Physica D: Nonlinear Phenomena, Vol. 406 https://doi.org/10.1016/j.physd.2020.132444	journal	May 2020
The dynamical destruction of shocked gas clouds Nittmann, J.; Falle, S. A. E. G.; Gaskell, P. H. Monthly Notices of the Royal Astronomical Society, Vol. 201, Issue 4 https://doi.org/10.1093/mnras/201.4.833	journal	December 1982
Three-dimensional simulation strategy to determine the effects of turbulent mixing on inertial-confinement-fusion capsule performance Haines, Brian M.; Grinstein, Fernando F.; Fincke, James R. Physical Review E, Vol. 89, Issue 5 https://doi.org/10.1103/PhysRevE.89.053302	journal	May 2014
A simple experiment to investigate two‐dimensional mixing by Rayleigh–Taylor instability Andrews, M. J.; Spalding, D. B. Physics of Fluids A: Fluid Dynamics, Vol. 2, Issue 6 https://doi.org/10.1063/1.857652	journal	June 1990
Effects of operator splitting and low Mach-number correction in turbulent mixing transition simulations Grinstein, F. F.; Saenz, J. A.; Dolence, J. C. Computers & Mathematics with Applications, Vol. 78, Issue 2 https://doi.org/10.1016/j.camwa.2018.05.008	journal	July 2019
Three-Dimensional Hydrodynamical Simulations of the Supernovae-Driven gas loss in the Dwarf Spheroidal Galaxy ursa Minor Caproni, A.; Lanfranchi, G. A.; Silva, A. Luiz da The Astrophysical Journal, Vol. 805, Issue 2 https://doi.org/10.1088/0004-637X/805/2/109	journal	May 2015
Mixing enhancement by expansion waves in supersonic flows of different densities Li, C.; Kailasanath, K.; Book, D. L. Physics of Fluids A: Fluid Dynamics, Vol. 3, Issue 5 https://doi.org/10.1063/1.858066	journal	May 1991
Rayleigh-Taylor instabilities in inertial-confinement fusion targets Freeman, J. R.; Clauser, M. J.; Thompson, S. L. Nuclear Fusion, Vol. 17, Issue 2 https://doi.org/10.1088/0029-5515/17/2/005	journal	April 1977
Effect of Initial Conditions on the Development of Rayleigh–Taylor Instability Rozanov, V. B.; Kuchugov, P. A.; Zmitrenko, N. V. Journal of Russian Laser Research, Vol. 36, Issue 2 https://doi.org/10.1007/s10946-015-9487-y	journal	March 2015
Modelling turbulent mixing by Rayleigh-Taylor instability Youngs, David L. Physica D: Nonlinear Phenomena, Vol. 37, Issue 1-3 https://doi.org/10.1016/0167-2789(89)90135-8	journal	July 1989
Partially-Averaged Navier-Stokes Model for Turbulence: A Reynolds-Averaged Navier-Stokes to Direct Numerical Simulation Bridging Method Girimaji, Sharath S. Journal of Applied Mechanics, Vol. 73, Issue 3 https://doi.org/10.1115/1.2151207	journal	November 2005
Vertical Mixing, Energy, and the General Circulation of the Oceans Wunsch, Carl; Ferrari, Raffaele Annual Review of Fluid Mechanics, Vol. 36, Issue 1 https://doi.org/10.1146/annurev.fluid.36.050802.122121	journal	January 2004
Modeling and simulation of transitional Taylor-Green vortex flow with partially averaged Navier-Stokes equations Pereira, F. S.; Grinstein, F. F.; Israel, D. M. Physical Review Fluids, Vol. 6, Issue 5 https://doi.org/10.1103/PhysRevFluids.6.054611	journal	May 2021
Multiple-time-scale modeling of turbulent flows in one-point closures Schiestel, Roland Physics of Fluids, Vol. 30, Issue 3 https://doi.org/10.1063/1.866322	journal	January 1987
Experimental investigation of turbulent mixing by Rayleigh-Taylor instability Read, K. I. Physica D: Nonlinear Phenomena, Vol. 12, Issue 1-3 https://doi.org/10.1016/0167-2789(84)90513-X	journal	July 1984
Modeling of Rayleigh-Taylor mixing using single-fluid models Kokkinakis, Ioannis W.; Drikakis, Dimitris; Youngs, David L. Physical Review E, Vol. 99, Issue 1 https://doi.org/10.1103/PhysRevE.99.013104	journal	January 2019
Filamentary structure on the Sun from the magnetic Rayleigh–Taylor instability Isobe, Hiroaki; Miyagoshi, Takehiro; Shibata, Kazunari Nature, Vol. 434, Issue 7032 https://doi.org/10.1038/nature03399	journal	March 2005
A comparative study of the turbulent Rayleigh–Taylor instability using high-resolution three-dimensional numerical simulations: The Alpha-Group collaboration Dimonte, Guy; Youngs, D. L.; Dimits, A. Physics of Fluids, Vol. 16, Issue 5 https://doi.org/10.1063/1.1688328	journal	May 2004
Rayleigh–Taylor turbulence: self-similar analysis and direct numerical simulations Ristorcelli, J. R.; Clark, T. T. Journal of Fluid Mechanics, Vol. 507 https://doi.org/10.1017/S0022112004008286	journal	May 2004
Numerical study of particle-induced Rayleigh-Taylor instability: Effects of particle settling and entrainment Chou, Yi-Ju; Shao, Yun-Chuan Physics of Fluids, Vol. 28, Issue 4 https://doi.org/10.1063/1.4945652	journal	April 2016
Numerical simulations of two-fluid turbulent mixing at large density ratios and applications to the Rayleigh–Taylor instability Livescu, D. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 371, Issue 2003 https://doi.org/10.1098/rsta.2012.0185	journal	November 2013
Self-similar Reynolds-averaged mechanical–scalar turbulence models for Rayleigh–Taylor, Richtmyer–Meshkov, and Kelvin–Helmholtz instability-induced mixing in the small Atwood number limit Schilling, Oleg Physics of Fluids, Vol. 33, Issue 8 https://doi.org/10.1063/5.0055193	journal	August 2021
Molecular viscosity and diffusivity effects in transitional and shock-driven mixing flows Pereira, F. S.; Grinstein, F. F.; Israel, D. M. Physical Review E, Vol. 103, Issue 1 https://doi.org/10.1103/PhysRevE.103.013106	journal	January 2021
Self-similarity and universality in Rayleigh–Taylor, Boussinesq turbulence Vladimirova, Natalia; Chertkov, Michael Physics of Fluids, Vol. 21, Issue 1 https://doi.org/10.1063/1.3054152	journal	January 2009
Dependence of turbulent Rayleigh-Taylor instability on initial perturbations Dimonte, Guy Physical Review E, Vol. 69, Issue 5 https://doi.org/10.1103/PhysRevE.69.056305	journal	May 2004
The Piecewise Parabolic Method (PPM) for gas-dynamical simulations Colella, Phillip; Woodward, Paul R. Journal of Computational Physics, Vol. 54, Issue 1 https://doi.org/10.1016/0021-9991(84)90143-8	journal	April 1984
Reynolds number effects on Rayleigh–Taylor instability with possible implications for type Ia supernovae Cabot, William H.; Cook, Andrew W. Nature Physics, Vol. 2, Issue 8 https://doi.org/10.1038/nphys361	journal	July 2006
The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I Taylor, Geoffrey Ingram Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, Vol. 201, Issue 1065, p. 192-196 https://doi.org/10.1098/rspa.1950.0052	journal	March 1950
Challenges in Scale-Resolving Simulations of turbulent wake flows with coherent structures Pereira, Filipe S.; Eça, Luís; Vaz, Guilherme Journal of Computational Physics, Vol. 363 https://doi.org/10.1016/j.jcp.2018.02.038	journal	June 2018
Transition stages of Rayleigh–Taylor instability between miscible fluids Cook, Andrew W.; Dimotakis, Paul E. Journal of Fluid Mechanics, Vol. 443 https://doi.org/10.1017/S0022112001005377	journal	September 2001
Simulations of Richtmyer–Meshkov instabilities in planar shock-tube experiments Grinstein, F. F.; Gowardhan, A. A.; Wachtor, A. J. Physics of Fluids, Vol. 23, Issue 3 https://doi.org/10.1063/1.3555635	journal	March 2011
Rayleigh–Taylor mixing: direct numerical simulation and implicit large eddy simulation Youngs, David L. Physica Scripta, Vol. 92, Issue 7 https://doi.org/10.1088/1402-4896/aa732b	journal	June 2017
Rayleigh–Taylor instability with gravity reversal Livescu, D.; Wei, T.; Brady, P. T. Physica D: Nonlinear Phenomena, Vol. 417 https://doi.org/10.1016/j.physd.2020.132832	journal	March 2021
A new approach for the calculation of the cut-off resolution parameter in bridging methods for turbulent flow simulation Basara, Branislav; Pavlovic, Zoran; Girimaji, Sharath International Journal of Heat and Fluid Flow, Vol. 74 https://doi.org/10.1016/j.ijheatfluidflow.2018.09.011	journal	December 2018
Investigation of the Character of the Equilibrium of an Incompressible Heavy Fluid of Variable Density Proceedings of the London Mathematical Society, Vol. s1-14, Issue 1 https://doi.org/10.1112/plms/s1-14.1.170	journal	November 1882

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Title: Modeling and simulation of transitional Rayleigh–Taylor flow with partially averaged Navier–Stokes equations

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