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Simulation of natural convection in an inclined polar cavity using a finite-difference lattice Boltzmann method

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Abstract

Natural convection heat transfer in an inclined polar cavity was studied using a Finite-difference lattice Boltzmann method (FDLBM) based on a double-population approach for body-fitted coordinates. A D2G9 model coupled with the simplest TD2Q4 lattice model was applied to determine the velocity field and temperature field. For both velocity and temperature fields, the discrete spatial derivatives were obtained by combining the upwind scheme with the central scheme, and the discrete temporal term is obtained using a fourth-order Runge-Kutta scheme. Studies were carried out for different Rayleigh numbers and different inclination angles. The results in terms of streamlines, isotherms, and Nusselt numbers explain the heat transfer mechanism of natural convection in an inclined polar cavity due to the change of Rayleigh number and inclination angle.

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References

  1. S. Y. Chen and G. D. Doolen, Lattice Boltzmann method for fluid flows, Annual Review of Fluid Mechanics, 30 (1998) 329–364.

    Article  MathSciNet  Google Scholar 

  2. S. Succi, The Lattice Boltzmann Equation for Fluid Dynamics and Beyond, Oxford: Clarendon Press (2001).

    MATH  Google Scholar 

  3. Z. D. Qian, J. D. Yang and W. X. Huai, Numerical simulation and analysis of pressure pulsation in Francis hydraulic turbine with air admission, Journal of Hydrodynamics, 19 (4) (2007) 467–472.

    Article  Google Scholar 

  4. Y. L. Wu et al., Simulations of unsteady cavitating turbulent flow in a Francis turbine using the RANS method and the improved mixture model of two-phase flows, Engineering with Computers, 27 (3) (2011) 235–250.

    Article  Google Scholar 

  5. X. L. Tang et al., Numerical investigations on cavitating flows with thermodynamic effects in a diffuser-type centrifugal pump, Journal of Mechanical Science and Technology, 27 (6) (2013) 1655–1664.

    Article  MathSciNet  Google Scholar 

  6. B. Ji et al., Numerical simulation of cavitation surge and vortical flows in a diffuser with swirling flow, Journal of Mechanical Science and Technology, 30 (6) (2016) 2507–2514.

    Article  Google Scholar 

  7. Y. H. Yan et al., DNS study on the formation of lambda rotational core and the role of TS wave in boundary layer transition, Journal of Turbulence, 17 (6) (2016) 572–601.

    Article  MathSciNet  Google Scholar 

  8. F. Yang et al., Numerical study on transverse-axis rotary viscous pump and hydropulser mechanism, International Journal of Nonlinear Sciences and Numerical Simulation, 7 (3) (2006) 263–268.

    Article  Google Scholar 

  9. X. L. Tang, S. Y. Yang and F. J. Wang, Researches on twophase flows around a hydrofoil using Shan-Chen multiphase LBM model, Journal of Mechanical Science and Technology, 30 (2) (2016) 575–584.

    Article  Google Scholar 

  10. H. H. Liu, A. J. Valocchi and Q. J. Kang, Threedimensional lattice Boltzmann model for immiscible twophase flow simulations, Physical Review E, 85 (4) (2012) 046309.

    Article  Google Scholar 

  11. H. H. Liu et al., Phase-field-based lattice Boltzmann finitedifference model for simulating thermocapillary flows, Physical Review E, 87 (1) (2013) 013010.

    Article  Google Scholar 

  12. H. R. Ashorynejad, A. A. Mohamad and M. Sheikholeslami, Magnetic field effects on natural convection flow of a nanofluid in a horizontal cylindrical annulus using Lattice Boltzmann method, International Journal of Thermal Sciences, 64 (2) (2013) 240–250.

    Article  Google Scholar 

  13. M. Nazari, L. Louhghalam and M. H. Kayhani, Lattice Boltzmann simulation of double diffusive natural convection in a square cavity with a hot square obstacle, Chinese Journal of Chemical Engineering, 23 (1) (2015) 22–30.

    Article  Google Scholar 

  14. S. Tajiri, M. Tsutahara and H. Tanaka, Direct simulation of sound and underwater sound generated by a water drop hitting a water surface using the finite difference lattice Boltzmann method, Computers and Mathematics with Applications, 59 (7) (2010) 2411–2420.

    Article  MathSciNet  MATH  Google Scholar 

  15. S. Foroughi, S. Jamshidi and M. Masihi, Lattice Boltzmann method on quadtree grids for simulating fluid flow through porous media: A new automatic algorithm, Physica A Statistical Mechanics & Its Applications, 392 (20) (2013) 4772–4786.

    Article  MathSciNet  Google Scholar 

  16. Z. L. Guo, B. C. Shi and C. G. Zheng, A coupled lattice BGK model for the Boussinesq equations, International Journal for Numerical Methods in Fluids, 39 (4) (2002) 325–342.

    Article  MathSciNet  MATH  Google Scholar 

  17. D. A. Wolf-Gladrow, Lattice Gas Cellular Automata and Lattice Boltzmann Models, Lecture notes in mathematics, Berlin: Springer Press (2000).

    Book  MATH  Google Scholar 

  18. M. B. Reider and J. D. Sterling, Accuracy of discretevelocity BGK models for the simulation of the incompressible Navier-Stokes equations, Computers & Fluids, 24 (4) (1995) 459–467.

    Article  MathSciNet  MATH  Google Scholar 

  19. R. Mei and W. Shyy, On the finite difference-based lattice Boltzmann method in curvilinear coordinates, Journal of Computational Physics, 143 (2) (1998) 426–448.

    Article  MathSciNet  MATH  Google Scholar 

  20. Z. L. Guo and T. S. Zhao, Explicit finite-difference lattice Boltzmann method for curvilinear coordinates, Physical Review E, 67 (6) (2003) 066709.

    Article  Google Scholar 

  21. V. Sofonea and R. F. Sekerka, Viscosity of finite difference lattice Boltzmann models, Journal of Computational Physics, 184 (2) (2003) 422–434.

    Article  MathSciNet  MATH  Google Scholar 

  22. L. Wu, M. Tsutahara and S. Tajiri, Finite difference lattice Boltzmann method for incompressible Navier-Stokes equation using acceleration modification, Journal of Fluid Science and Technology, 2 (1) (2007) 35–44.

    Article  Google Scholar 

  23. S. C. Fu, R. M. C. So and W. W. F. Leung, Stochastic finite difference lattice Boltzmann method for steady incompressible viscous flows, Journal of Computational Physics, 229 (17) (2010) 6084–6103.

    Article  MATH  Google Scholar 

  24. H. Chen, Volumetric formulation of the lattice Boltzmann method for fluid dynamics: Basic concept, Physical Review E, 58 (3) (1998) 3955.

    Article  Google Scholar 

  25. M. Stiebler, J. Tolke and M. Krafczyk, An upwind discretization scheme for the finite volume lattice Boltzmann method, Computers & Fluids, 35 (8) (2006) 814–819.

    Article  MathSciNet  MATH  Google Scholar 

  26. F. Dubois and P. Lallemand, On lattice Boltzmann scheme, finite volumes and boundary conditions, Progress in Computational Fluid Dynamics, 8 (1-2) (2008) 11–24.

    Article  MathSciNet  MATH  Google Scholar 

  27. V. Patil and K. N. Lakshmisha, Finite volume TVD formulation of lattice Boltzmann simulation on unstructured mesh, Journal of Computational Physics, 228 (14) (2009) 5262–5279.

    Article  MathSciNet  MATH  Google Scholar 

  28. T. Lee and C.-L. Lin, A characteristic Galerkin method for discrete Boltzmann equation, Journal of Computational Physics, 171 (1) (2001) 336–356.

    Article  MATH  Google Scholar 

  29. Y. Li, E. J. LeBoeuf and P. K. Basu, Least-squares finiteelement scheme for the lattice Boltzmann method on an unstructured mesh, Physical Review E, 72 (4) (2005) 046711.

    Article  Google Scholar 

  30. N. Cao et al., Physical symmetry and lattice symmetry in the lattice Boltzmann method, Physical Review E, 55 (1) (1997) R21.

    Article  Google Scholar 

  31. G. D. V. Davis, Natural convection of air in a square cavity: A benchmark numerical solution, International Journal for Numerical Methods in Fluids, 3 (3) (1983) 249–264.

    Article  MATH  Google Scholar 

  32. M. Hortmann, M. Peric and G. Scheuerer, Finite volume multigrid prediction of laminar natural convection: Benchmark solutions, International Journal for Numerical Methods in Fluids, 11 (2) (1990) 189–207.

    Article  MATH  Google Scholar 

  33. G. Barakos, E. Mitsoulis and D. Assimacopoulos, Natural convection flow in a square cavity revisited: Laminar and turbulent models with wall functions, International Journal for Numerical Methods in Fluids, 18 (7) (1994) 695–719.

    Article  MATH  Google Scholar 

  34. E. Fattahi, M. Farhadi and K. Sedighi, Lattice Boltzmann simulation of mixed convection heat transfer in eccentric annulus, International Communications in Heat & Mass Transfer, 38 (8) (2011) 1135–1141.

    Article  Google Scholar 

  35. S. S. Patel et al., A spectral-element discontinuous Galerkin lattice Boltzmann method for simulating natural convection heat transfer in a horizontal concentric annulus, Computers & Fluids, 95 (2) (2014) 197–209.

    Article  MathSciNet  Google Scholar 

  36. S. S. Lv et al., Numerical calculation of natural convection in a horizontal annular sector channel, Journal of South China University of Technology: Natural Science Edition, 24 (2) (1996) 73–101 (in Chinese).

    Google Scholar 

  37. R. M. Clever and F. H. Busse, Transition to time-dependent convection, Journal of Fluid Mechanics, 65 (4) (1974) 625–645.

    Article  MATH  Google Scholar 

  38. J. E. Fromm, Numerical solutions of the nonlinear equations for a heated fluid layer, Physics of Fluids, 8 (10) (1965) 1757–1769.

    Article  Google Scholar 

  39. P. L. Bhatnagar, E. P. Gross and M. Krook, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Physical Review, 94 (3) (1954) 511–525.

    Article  MATH  Google Scholar 

  40. Z. L. Guo, B. C. Shi and N. C. Wang, Lattice BGK model for incompressible Navier-Stokes equation, Journal of Computational Physics, 165 (1) (2000) 288–306.

    Article  MathSciNet  MATH  Google Scholar 

  41. Z. L. Guo, C. G. Zheng and B. C. Shi, Discrete lattice effects on the forcing term in the lattice Boltzmann method, Physical Review E, 65 (4) (2002) 046308.

    Article  MATH  Google Scholar 

  42. K. Hejranfar and E. Ezzatneshan, Implementation of a high-order compact finite-difference lattice Boltzmann method in generalized curvilinear coordinates, Journal of Computational Physics, 267 (5) (2014) 28–49.

    Article  MathSciNet  MATH  Google Scholar 

  43. K. Hejranfar and E. Ezzatneshan, A high-order compact finite-difference lattice Boltzmann method for simulation of steady and unsteady incompressible flows, International Journal for Numerical Methods in Fluids, 75 (10) (2014) 713–746.

    Article  MathSciNet  MATH  Google Scholar 

  44. Z. L. Guo, C. G. Zheng and B. C. Shi, Non-equilibrium extrapolation method for velocity and pressure boundary conditions in the lattice Boltzmann method, Chinese Physics, 11 (4) (2002) 366–374.

    Article  Google Scholar 

  45. X. Y. He, S. Y. Chen and G. D. Doolen, A novel thermal model for the lattice Boltzmann method in incompressible limit, Journal of Computational Physics, 146 (1) (1998) 282–300.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. School of Energy and Power Engineering, University of Shanghai for Science and Technology, Shanghai, 200093, China

    Fan Yang, Haicheng Yang, Xueyan Guo & Ren Dai

  2. Shanghai Key Laboratory of Multiphase Flow and Heat Transfer in Power Engineering, Shanghai, 200093, China

    Fan Yang, Yonghua Yan, Xueyan Guo & Ren Dai

  3. Department of Mathematics and Computer Science, Alcorn State University, Lorman, MS, 39096, USA

    Yonghua Yan

  4. Department of Mathematics, University of Texas at Arlington, Arlington, TX, 76019-0408, USA

    Chaoqun Liu

Authors
  1. Fan Yang
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  2. Haicheng Yang
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  3. Yonghua Yan
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  4. Xueyan Guo
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Correspondence to Fan Yang.

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Recommended by Associate Editor Seong Hyuk Lee

Yang Fan received the Bachelor’s degree from Northeastern University in 1997, the Master’s degree from Lanzhou University of Technology in 2000 and the Ph.D. degree from Tsinghua University in 2006, in China. Currently, he is an Associate Professor in School of Energy and Power Engineering, University of Shanghai for Science and Technology, China. His researches interests include CFD simulations by lattice Boltzmann methods.

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Yang, F., Yang, H., Yan, Y. et al. Simulation of natural convection in an inclined polar cavity using a finite-difference lattice Boltzmann method. J Mech Sci Technol 31, 3053–3065 (2017). https://doi.org/10.1007/s12206-017-0549-7

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  • DOI: https://doi.org/10.1007/s12206-017-0549-7

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