Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-28T12:37:53.515Z Has data issue: false hasContentIssue false
  • English
  • Français

Global linear instability analysis of thermal convective flow using the linearized lattice Boltzmann method

Published online by Cambridge University Press:  30 June 2022

Hao-Kui Jiang ,
Kang Luo Open the ORCID record for Kang Luo [Opens in a new window] ,
Zi-Yao Zhang ,
Jian Wu Open the ORCID record for Jian Wu [Opens in a new window]  and
Hong-Liang Yi Open the ORCID record for Hong-Liang Yi [Opens in a new window]
Hao-Kui Jiang
Affiliation:
School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, PR China
Kang Luo*
Affiliation:
School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, PR China
Zi-Yao Zhang
Affiliation:
School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, PR China
Jian Wu
Affiliation:
School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, PR China
Hong-Liang Yi*
Affiliation:
School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, PR China
*
† Email addresses for correspondence: luokang@hit.edu.cn, yihongliang@hit.edu.cn
† Email addresses for correspondence: luokang@hit.edu.cn, yihongliang@hit.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

Modal global linear stability analysis of thermal convection is performed with the linearized lattice Boltzmann method (LLBM). The onset of Rayleigh–Bénard convection in rectangular cavities with conducting and adiabatic sidewalls and the instability of two-dimensional (2-D) and three-dimensional (3-D) natural convection in cavities are studied. The method of linearizing the local equilibrium probability distribution function that was first proposed by Pérez et al. (Theor. Comp. Fluid Dyn., vol. 31, 2017, pp. 643–664) is extended to solve the coupled linear Navier–Stokes equations together with the linear energy equation in this work. A multiscale analysis is also performed to recover the macroscopic linear Navier–Stokes equations from the discrete lattice Boltzmann equations for both the single and multiple relaxation time models. The present LLBM is implemented in the framework of the Palabos library. It is validated by calculating the linear critical value of 2-D natural convection that the LLBM with the multiple relaxation time model has an error less than 1 % compared with the spectral method. The instability mechanism of the flow is explained by kinetic energy transfer analysis. It is shown that the buoyancy mechanism and inertial mechanism tend to stabilize the Hopf bifurcation of the 2-D natural convection at Pr < 0.08 and Pr > 1, respectively. For 3-D natural convection, subcritical bifurcation of the Hopf type is found for low-Prandtl-number fluids (Pr < 0.1).

JFM classification

Convection: Convection in cavities
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 944 , 10 August 2022 , A31
Check for updates
Copyright
© The Author(s), 2022. 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.)

References

Abu-Nada, E., Masoud, Z. & Hijazi, A. 2008 Natural convection heat transfer enhancement in horizontal concentric annuli using nanofluids. Intl Commun. Heat Mass Transfer 35, 657–665.10.1016/j.icheatmasstransfer.2007.11.004CrossRefGoogle Scholar
Albensoeder, S., Kuhlmann, H.C. & Rath, H.J. 2001 Three-dimensional centrifugal-flow instabilities in the lid-driven-cavity problem. Phys. Fluids 13, 121–135.10.1063/1.1329908CrossRefGoogle Scholar
Barkley, D. 2006 Linear analysis of cylinder wake mean flow. Europhys. Lett. 75 (5), 750–756.10.1209/epl/i2006-10168-7CrossRefGoogle Scholar
Bergé, P. & Dubois, M. 1984 Rayleigh-bénard convection. Contemp. Phys. 25, 535–582.10.1080/00107518408210730CrossRefGoogle Scholar
Bilgen, E. & Oztop, H. 2005 Natural convection heat transfer in partially open inclined square cavities. Intl J. Heat Mass Transfer 48, 1470–1479.10.1016/j.ijheatmasstransfer.2004.10.020CrossRefGoogle Scholar
Blackburn, H.M. & Sherwin, S.J. 2004 Formulation of a Galerkin spectral element-Fourier method for three-dimensional incompressible flows in cylindrical geometries. J. Comput. Phys. 197, 759–778.10.1016/j.jcp.2004.02.013CrossRefGoogle Scholar
Bouzidi, M., d'Humieres, D., Lallemand, P. & Luo, L.S. 2001 Lattice Boltzmann equation on a two-dimensional rectangular grid. J. Comput. Phys. 172, 704–717.10.1006/jcph.2001.6850CrossRefGoogle Scholar
Chandrasekhar, S. 1981 Hydrodynamic and Hydromagnetic Stability. Dover Publications.Google Scholar
Chang, H.-W., Hong, P.-Y., Lin, L.-S. & Lin, C.-A. 2013 Simulations of flow instability in three dimensional deep cavities with multi relaxation time lattice Boltzmann method on graphic processing units. Comput. Fluids 88, 866–871.10.1016/j.compfluid.2013.08.019CrossRefGoogle Scholar
Chapman, S. & Cowling, T.G. 1953 The mathematical theory of non-uniform gases. 2nd ed.. Cambridge University Press.Google Scholar
Chen, S., Doolen, & Gary, D. 1998 Lattice Boltzmann method for fluid flows. Rev. Fluid Mech. 30, 329–364.10.1146/annurev.fluid.30.1.329CrossRefGoogle Scholar
Davis, S.H. 1967 Convection in a box: linear theory. J. Fluid Mech. 30, 465–478.10.1017/S0022112067001545CrossRefGoogle Scholar
Dijkstra, H.A. 1992 On the structure of cellular solutions in Rayleigh-Benard-Marangoni flows in small-aspect-ratio containers. J. Fluid Mech. 243, 73–102.10.1017/S0022112092002647CrossRefGoogle Scholar
Ehrenstein, U. 1996 On the linear stability of channel flow over riblets. Phys. Fluids 8, 3194–3196.10.1063/1.869091CrossRefGoogle Scholar
Fischer, P., Kerkemeier, S. & Peplinski, A. 2008 nek5000 home page. Available at: https://nek5000.mcs.anl.gov/.Google Scholar
Gelfgat, A.Y. 1999 Different modes of Rayleigh–Bénard Instability in two- and three-dimensional rectangular enclosures. J. Comput. Phys. 156, 300–324.10.1006/jcph.1999.6363CrossRefGoogle Scholar
Gelfgat, A.Y., Bar-Yoseph, P.Z. & Solan, A. 2000 Axisymmetry breaking instabilities of natural convection in a vertical bridgman growth configuration. J. Cryst. Growth 220, 316–325.10.1016/S0022-0248(00)00842-3CrossRefGoogle Scholar
Gelfgat, A.Y. & Tanasawa, I. 1994 Numerical analysis of oscillatory instability of buoyancy convection with the Galerkin spectral method. Numer. Heat Transfer A: Appl. 25, 627–648.10.1080/10407789408955970CrossRefGoogle Scholar
Gill, A.E. 1966 The boundary-layer regime for convection in a rectangular cavity. J. Fluid Mech. 26, 515–536.10.1017/S0022112066001368CrossRefGoogle Scholar
Goldhirsch, L., Orszag, S.A. & Maulik, B.K. 1987 An efficient method for computing leading eigenvalues and eigenvectors of large asymmetric matrices. J. Sci. Comput. 2, 33–58.10.1007/BF01061511CrossRefGoogle Scholar
Gomez, F., Gomez, R. & Theofilis, V. 2014 On three-dimensional global linear instability analysis of flows with standard aerodynamics codes. Aerosp. Sci. Technol. 32, 223–234.10.1016/j.ast.2013.10.006CrossRefGoogle Scholar
Gomez, F., Pérez, J.M., Blackburn, H.M. & Theofilis, V. 2015 On the use of matrix-free shift-invert strategies for global flow instability analysis. Aerosp. Sci. Technol. 44, 69–76.10.1016/j.ast.2014.11.003CrossRefGoogle Scholar
Guo, Z., Shi, B. & Zheng, C. 2002 a A coupled lattice BGK model for the Boussinesq equations. Intl J. Numer. Meth. Fluids 39, 325–342.10.1002/fld.337CrossRefGoogle Scholar
Guo, Z., Zheng, C. & Shi, B. 2002 b Discrete lattice effects on the forcing term in the lattice Boltzmann method. Phys. Rev. E 65, 046308.10.1103/PhysRevE.65.046308CrossRefGoogle ScholarPubMed
Guo, Z., Zheng, C. & Shi, B. 2002 c An extrapolation method for boundary conditions in lattice Boltzmann method. Phys. Fluids 14, 2007–2010.10.1063/1.1471914CrossRefGoogle Scholar
Haghshenas, A., Nasr, M.R. & Rahimian, M.H. 2010 Numerical simulation of natural convection in an open-ended square cavity filled with porous medium by lattice Boltzmann method. Heat Mass Transfer 37, 1513–1519.10.1016/j.icheatmasstransfer.2010.08.006CrossRefGoogle Scholar
Hammond, D.A. & Redekoop, L.G. 1997 Global dynamic of symmetric and asymmetric wake. J. Fluid Mech. 331, 231–260.10.1017/S0022112096003825CrossRefGoogle Scholar
Henkes, R. & Hoogendoorn, C. 1990 On the stability of the natural convection flow in a square cavity heated from the side. Appl. Sci. Res. 47, 195–220.10.1007/BF00418051CrossRefGoogle Scholar
Henkes, & Quere, P.L. 1996 Three-dimensional transition of natural-convection flows. J. Fluid Mech. 319, 281–303.10.1017/S0022112096007343CrossRefGoogle Scholar
Huang, R. & Wu, H. 2015 Phase interface effects in the total enthalpy-based lattice Boltzmann model for solid-liquid phase change. J. Comput. Phys. 294, 346–362.10.1016/j.jcp.2015.03.064CrossRefGoogle Scholar
Jackson, C.P. 1987 A finite-element study of the onset of vortex shedding in flow past variously shaped bodies. J. Fluid Mech. 182, 23–45.10.1017/S0022112087002234CrossRefGoogle Scholar
Jasak, H., Jemcov, A. & Tukovic, Z. 2007 OpenFOAM: a C++ library for complex physics simulations. In International Workshop on Coupled Methods in Numerical Dynamics (IUC Dubrovnik Croatia).Google Scholar
Jiang, M., Xu, Z.G. & Zhou, Z.P. 2021 Pore-scale investigation on reactive flow in porous media considering dissolution and precipitation by LBM. J. Petrol. Sci. Engng 204, 108712.10.1016/j.petrol.2021.108712CrossRefGoogle Scholar
Joseph, D.D. 2013 Stability of Fluid Motions I (Vol. 27). Springer Science & Business Media.Google Scholar
Ke, J., Williamson, N., Armfield, S.W., Mcbain, G.D. & Norris, S.E.J. 2019 Stability of a temporally evolving natural convection boundary layer on an isothermal wall. J. Fluid Mech. 877, 1163–1185.10.1017/jfm.2019.639CrossRefGoogle Scholar
Kooper, M.N., Vorst, H., Poedts, S. & Goedbloed, J.P. 1995 Application of the implicitly updated arnoldi method with a complex shift-and-invert strategy in MHD. J. Comput. Phys. 118, 320–328.10.1006/jcph.1995.1102CrossRefGoogle Scholar
Lallemand, P. & Luo, L.S. 2000 Theory of the lattice Boltzmann method: dispersion, dissipation, isotropy, galilean invariance, and stability. Phys. Rev. E 61, 6546–6562.10.1103/PhysRevE.61.6546CrossRefGoogle ScholarPubMed
Latt, J. & Chopard, B. 2006 Lattice Boltzmann method with regularized pre-collision distribution functions. Math. Comput. Simul. 72, 165–168.10.1016/j.matcom.2006.05.017CrossRefGoogle Scholar
Latt, J., et al. 2021 Palabos: parallel lattice Boltzmann solver. Comput. Math. Appl. 81, 334–350.10.1016/j.camwa.2020.03.022CrossRefGoogle Scholar
Lee, N.Y., Schultz, W.W. & Boyd, J.P. 1989 Stability of fluid in a rectangular enclosure by spectral method. Intl J. Heat Mass Transfer 32, 513–520.10.1016/0017-9310(89)90139-7CrossRefGoogle Scholar
Li, Q., He, Y.L., Wang, Y. & Tao, W.Q. 2007 Coupled double-distribution-function lattice Boltzmann method for the compressible Navier-Stokes equations. Phys. Rev. E 76, 056705.10.1103/PhysRevE.76.056705CrossRefGoogle ScholarPubMed
Lin, R.S. & Malik, M.R. 1997 On the stability of attachment-line boundary layers. Part 2. The effect of leading-edge curvature. J. Fluid Mech. 333, 125–137.10.1017/S0022112096004260CrossRefGoogle Scholar
Lockett, F.J. 1969 On Squire's Theorem for viscoelastic fluids. Intl J. Engng Sci. 7, 337–349.10.1016/0020-7225(69)90044-5CrossRefGoogle Scholar
Luijkx, J. & Platten, J. 1981 On the onset of free convection in a rectangular channel. J. Non-Equilib. Thermodyn. 6, 141–158.10.1515/jnet.1981.6.3.141CrossRefGoogle Scholar
Luo, K., Wu, J., Yi, H.-L. & Tan, H.-P. 2016 Lattice Boltzmann modelling of electro-thermo-convection in a planar layer of dielectric liquid subjected to unipolar injection and thermal gradient. Intl J. Heat Mass Transfer 103, 832–846.10.1016/j.ijheatmasstransfer.2016.07.108CrossRefGoogle Scholar
Luo, K., Yi, H.-L., Tan, H.-P. & Wu, J. 2017 Unified lattice Boltzmann method for electric field-space charge coupled problems in complex geometries and its applications to annular electroconvection. IEEE Trans. Ind. Applics. 53, 3995–4007.10.1109/TIA.2017.2691761CrossRefGoogle Scholar
Ma, D.J., Henry, D. & Ben Hadid, H. 2005 Three-dimensional numerical study of natural convection in vertical cylinders partially heated from the side. Phys. Fluids 17, 124101.10.1063/1.2141430CrossRefGoogle Scholar
Marquet, O., Sipp, D., Chomaz, J.M. & Jacquin, L. 2008 Amplifier and resonator dynamics of a low-Reynolds-number recirculation bubble in a global framework. J. Fluid Mech. 605, 429–443.10.1017/S0022112008000323CrossRefGoogle Scholar
Merle, X., Alizard, F. & Robinet, J.C. 2010 Finite difference methods for viscous incompressible global stability analysis. Comput. Fluids 39, 911–925.10.1016/j.compfluid.2009.12.002CrossRefGoogle Scholar
Metzener, P. 1986 The effect of rigid sidewalls on nonlinear two-dimensional Benard convection. Phys. Fluids 29, 1373–1377.10.1063/1.865703CrossRefGoogle Scholar
Morgan, R. 1996 On restarting the Arnoldi method for large nonsymmetric eigenvalue problems. Math. Comput. 65, 1213–1230.10.1090/S0025-5718-96-00745-4CrossRefGoogle Scholar
Morzynski, M. & Thiele, F. 2008 4th AIAA Flow Control Conference, Seattle, Washington, USA. AIAA Paper 2008-3865.Google Scholar
Niu, X.D., Shu, C., Chew, Y.T. & Wang, T.G. 2004 Investigation of stability and hydrodynamics of different lattice Boltzmann models. J. Stat. Phys. 117, 665–680.10.1007/s10955-004-2264-xCrossRefGoogle Scholar
Pérez, J.M., Aguilar, A. & Theofilis, V. 2017 Lattice Boltzmann methods for global linear instability analysis. Theor. Comput. Fluid Dyn. 31, 643–664.10.1007/s00162-016-0416-7CrossRefGoogle Scholar
Pérez, J.M., Rodriguez, D. & Theofilis, V. 2012 Linear global instability of non-orthogonal incompressible swept attachment-line boundary-layer flow. J. Fluid Mech. 710, 131–153.10.1017/jfm.2012.354CrossRefGoogle Scholar
Pier, B. 2002 On the frequency selection of finite-amplitude vortex shedding in the cylinder wake. J. Fluid Mech. 458, 407–417.10.1017/S0022112002008054CrossRefGoogle Scholar
Pierrehumbert, R.T. & Widnall, S.E. 1982 The two-and three-dimensional instabilities of a spatially periodic shear layer. J. Fluid Mech. 114, 59–82.10.1017/S0022112082000044CrossRefGoogle Scholar
Saad, Y. 1980 Variations on Arnoldi's method for computing eigenelements of large unsymmetric matrices. Linear Algebr. Applics. 34, 269–295.10.1016/0024-3795(80)90169-XCrossRefGoogle Scholar
Sipp, D. & Lebedev, A. 2007 Global stability of base and mean flows: a general approach and its applications to cylinder and open cavity flows. J. Fluid Mech. 593, 333–358.10.1017/S0022112007008907CrossRefGoogle Scholar
Tatsumi, T. & Yoshimura, T. 1990 Stability of the laminar flow in a rectangular duct. J. Fluid Mech. 212, 437–449.10.1017/S002211209000204XCrossRefGoogle Scholar
Theofilis, V. 2000 a Globally unstable basic flows in open cavities. AIAA Paper 1965-2000.10.2514/6.2000-1965CrossRefGoogle Scholar
Theofilis, V. 2000 b On steady-state flow solutions and their nonparallel global linear instability. Adv. Turbul. VIII, 35–38.Google Scholar
Theofilis, V. 2003 Advances in global linear instability analysis of nonparallel and three-dimensional flows. Prog. Aerosp. Sci. 39, 249–315.10.1016/S0376-0421(02)00030-1CrossRefGoogle Scholar
Theofilis, V. 2011 Global linear instability. Annu. Rev. Fluid Mech. 43, 319–352.10.1146/annurev-fluid-122109-160705CrossRefGoogle Scholar
Theofilis, V., Fedorov, A., Obrist, D. & Dallmann, U.C. 2003 The extended Görtler–Hämmerlin model for linear instability of three-dimensional incompressible swept attachment-line boundary layer flow. J. Fluid Mech. 487, 271–313.10.1017/S0022112003004762CrossRefGoogle Scholar
Touihri, R., Ben Hadid, H. & Henry, D. 1999 On the onset of convective instabilities in cylindrical cavities heated from below. Part 1. Pure thermal case. Phys. Fluids 11, 2078–2088.10.1063/1.870070CrossRefGoogle Scholar
Wang, B.-F., Ma, D.-J., Chen, C. & Sun, D.-J. 2012 Linear stability analysis of cylindrical Rayleigh–Bénard convection. J. Fluid Mech. 711, 27–39.10.1017/jfm.2012.360CrossRefGoogle Scholar
Wanschura, M., Kuhlmann, H.C. & Rath, H.J. 1996 Three-dimensional instability of axisymmetric buoyant convection in cylinders heated from below. J. Fluid Mech. 326, 399–415.10.1017/S0022112096008373CrossRefGoogle Scholar
Wettlaufer, J.S., Worster, M.G. & Huppert, H.E. 1997 Natural convection during solidification of an alloy from above with application to the evolution of sea ice. J. Fluid Mech. 344, 291–316.10.1017/S0022112097006022CrossRefGoogle Scholar
Xin, S. & Quéré, P.L. 2001 Linear stability analyses of natural convection flows in a differentially heated square cavity with conducting horizontal walls. Phys. Fluids 13, 2529–2542.10.1063/1.1388054CrossRefGoogle Scholar
Zebib, A. 1987 Stability of viscous flow past a circular cylinder. J. Engng Maths 21, 155–165.10.1007/BF00127673CrossRefGoogle Scholar
Zhang, M., Martinelli, F., Wu, J., Schmid, P. & Quadrio, M. 2015 Modal and non-modal stability analysis of electrohydrodynamic flow with and without cross-flow. J. Fluid Mech. 770, 319–349.10.1017/jfm.2015.134CrossRefGoogle Scholar
Zhao, C.Y., Dai, L.N., Tang, G.H., Qu, Z.G. & Li, Z.Y. 2010 Numerical study of natural convection in porous media (metals) using Lattice Boltzmann Method (LBM). Intl J. Heat Fluid Flow 31, 925–934.10.1016/j.ijheatfluidflow.2010.06.001CrossRefGoogle Scholar