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Novel wavelet-homotopy Galerkin technique for analysis of lid-driven cavity flow and heat transfer with non-uniform boundary conditions

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Abstract

In this paper, a brand-new wavelet-homotopy Galerkin technique is developed to solve nonlinear ordinary or partial differential equations. Before this investigation, few studies have been done for handling nonlinear problems with non-uniform boundary conditions by means of the wavelet Galerkin technique, especially in the field of fluid mechanics and heat transfer. The lid-driven cavity flow and heat transfer are illustrated as a typical example to verify the validity and correctness of this proposed technique. The cavity is subject to the upper and lower walls’ motions in the same or opposite directions. The inclined angle of the square cavity is from 0 to π/2. Four different modes including uniform, linear, exponential, and sinusoidal heating are considered on the top and bottom walls, respectively, while the left and right walls are thermally isolated and stationary. A parametric analysis of heating distribution between upper and lower walls including the amplitude ratio from 0 to 1 and the phase deviation from 0 to 2π is conducted. The governing equations are non-dimensionalized in terms of the stream function-vorticity formulation and the temperature distribution function and then solved analytically subject to various boundary conditions. Comparisons with previous publications are given, showing high efficiency and great feasibility of the proposed technique.

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References

  1. IDERIAH, F. J. K. Prediction of turbulent flow driven by buoyancy and shear. ARCHIVE Journal of Mechanical Engineering Science 1959–1982 (vols 1–23), 22(6), 287–295 (1980)

    Google Scholar 

  2. TALUKDAR, P. Mixed convection and non-gray radiation in a horizontal rectangular duct. Numerical Heat Transfer Part A Applications, 59(3), 185–208 (2011)

    Article  Google Scholar 

  3. PREMACHANDRAN, B. and BALAJI, C. Conjugate mixed convection with surface radiation from a horizontal channel with protruding heat sources. International Journal of Heat and Mass Transfer, 49(19/20), 3568–3582 (2006)

    Article  MATH  Google Scholar 

  4. CHAMKHA, A. J., HUSSAIN, S. H., and ABD-AMER, Q. R. Mixed convection heat transfer of air inside a square vented cavity with a heated horizontal square cylinder. Numerical Heat Transfer Part A Applications, 59(1), 58–79 (2011)

    Article  Google Scholar 

  5. IMBERGER, J. and HAMBLIN, P. F. Dynamics of lakes, reservoirs, and cooling ponds. Annual Review of Fluid Mechanics, 14, 153–187 (1982)

    Article  MATH  Google Scholar 

  6. OOSTHUIZEN, D. P. H. Unsteady natural convection in a partially heated rectangular cavity. Journal of Heat Transfer, 109(3), 798–801 (1987)

    Article  Google Scholar 

  7. CHA, C. K. and JALURIA, Y. Recirculating mixed convection flow for energy extraction. International Journal of Heat and Mass Transfer, 27(10), 1801–1812 (1984)

    Article  Google Scholar 

  8. BHUVANESWARI, M., SIVASANKARAN, S., and KIM, Y. J. Numerical study on double diffusive mixed convection with a soret effect in a two-sided lid-driven cavity. Numerical Heat Transfer Part A Applications, 59(7), 543–560 (2011)

    Article  Google Scholar 

  9. IWATSU, R., HYUN, J. M., and KUWAHARA, K. Mixed convection in a driven cavity with a stable vertical temperature gradient. International Journal of Heat and Mass Transfer, 36(6), 1601–1608 (1993)

    Article  Google Scholar 

  10. TORRANCE, K. E., DAVIS, R., EIKE, K., GILL, P., GUTMAN D., HSUI, A., LYONS, S., and ZEIN, H. Cavity flows driven by buoyancy and shear. Journal of Fluid Mechanics, 51(2), 221–231 (2006)

    Article  Google Scholar 

  11. KHANAFER, K. M. and CHAMKHA, A. J. Mixed convection flow in a lid-driven enclosure filled with a fluid-saturated porous medium. International Journal of Heat and Mass Transfer, 42(13), 2465–2481 (1999)

    Article  MATH  Google Scholar 

  12. AL-AMIRI, A. M., KHANAFER, K. M., and Pop, I. Numerical simulation of combined thermal and mass transport in a square lid-driven cavity. International Journal of Thermal Sciences, 46(7), 662–671 (2007)

    Article  Google Scholar 

  13. KHANAFER, K. M., AL-AMIRI, A. M., and Pop, I. Numerical simulation of unsteady mixed convection in a driven cavity using an externally excited sliding lid. European Journal of Mechanics-B/Fluids, 26(5), 669–687 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. ABDELKHALEK, M. M. Mixed convection in a square cavity by a perturbation technique. Computational Materials Science, 42(2), 212–219 (2008)

    Article  Google Scholar 

  15. GANGAWANE, K. M. Computational analysis of mixed convection heat transfer characteristics in lid-driven cavity containing triangular block with constant heat flux: effect of Prandtl and Grashof numbers. International Journal of Heat and Mass Transfer, 105, 34–57 (2017)

    Article  Google Scholar 

  16. TIWARI, R. K. and DAS, M. K. Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids. International Journal of Heat and Mass Transfer, 50(9), 2002–2018 (2002)

    MATH  Google Scholar 

  17. SHEREMET, M. A. and Pop, I. Thermo-bioconvection in a square porous cavity filled by oxytactic microorganisms. Transport in Porous Media, 103(2), 191–205 (2014)

    Article  MathSciNet  Google Scholar 

  18. TALEBI, F., MAHMOUDI, A. H., and SHAHI, M. Numerical study of mixed convection flows in a square lid-driven cavity utilizing nanofluid. International Communications in Heat and Mass Transfer, 37(1), 79–90 (2010)

    Article  Google Scholar 

  19. CHAMKHA, A. J. Hydromagnetic combined convection flow in a vertical lid-driven cavity with internal heat generation or absorption. Numerical Heat Transfer Part A: Applications, 41(5), 529–546 (2002)

    Article  Google Scholar 

  20. PEKMEN, B. and TEZER-SEZGIN, M. MHD flow and heat transfer in a lid-driven porous enclosure. Computers and Fluids, 89(89), 191–199 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. AHMED, S., OZTOP, H., MANSOUR, M. A., and ABU-HAMDEH, N. MHD mixed thermobioconvection in porous cavity filled by oxytactic microorganisms. Thermal Science, 2016, 319 (2016) https://doi.org/10.2298/TSCI16/005319A

    Google Scholar 

  22. OZTOP, H. F. and DAGTEKIN, I. Mixed convection in two-sided lid-driven differentially heated square cavity. International Journal of Heat and Mass Transfer, 47(8), 1761–1769 (2004)

    Article  MATH  Google Scholar 

  23. CHAMKHA A. J. and ABU-NADA E. Mixed convection flow in single- and double-lid driven square cavities filled with water-Al2O3 nanofluid: effect of viscosity models. European Journal of Mechanics-B/Fluids, 36(10), 82–96 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. ISMAEL, M. A., POP, I., and CHAMKHA, A. J. Mixed convection in a lid-driven square cavity with partial slip. International Journal of Thermal Sciences, 82, 47–61 (2014)

    Article  Google Scholar 

  25. SHEREMET, M. A. and POP, I. Mixed convection in a lid-driven square cavity filled by a nanofluid: Buongiorno’s mathematical model. Applied Mathematics and Computation, 266, 792–808 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. CHATTOPADHYAY, A., PANDIT, S. K., SARMA, S. S., and POP, I. Mixed convection in a double lid-driven sinusoidally heated porous cavity. International Journal of Heat and Mass Transfer, 93, 361–378 (2016)

    Article  Google Scholar 

  27. SHARIF, M. A. R. Laminar mixed convection in shallow inclined driven cavities with hot moving lid on top and cooled from bottom. Applied Thermal Engineering, 27(5), 1036–1042 (2007)

    Article  Google Scholar 

  28. CHENG, T. S. and LIU, W. H. Effects of cavity inclination on mixed convection heat transfer in lid-driven cavity flows. Computers and Fluids, 100, 108–122 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. ESFE, M., AKBARI, M., KARIMIPOUR, A., AFRAND, M., HAHIAN, O., and WONGWISES, S. Mixed-convection flow and heat transfer in an inclined cavity equipped to a hot obstacle using nanofluids considering temperature-dependent properties. International Journal of Heat and Mass Transfer, 85, 656–666 (2015)

    Article  Google Scholar 

  30. SIVASANKARAN, S., MALLESWARAN, A., LEE, J., and SUNDAR, P. Hydro-magnetic combined convection in a lid-driven cavity with sinusoidal boundary conditions on both sidewalls. International Journal of Heat and Mass Transfer, 54(1/2/3), 512–525 (2011)

    Article  MATH  Google Scholar 

  31. AHMED, S. E., MANSOUR, M. A., and MAHDY, A. MHD mixed convection in an inclined liddriven cavity with opposing thermal buoyancy force: effect of non-uniform heating on both side walls. Nuclear Engineering and Design, 265(1/2/3), 938–948 (2013)

    Article  Google Scholar 

  32. BASAK, T., ROY, S., SINGH, S. K., and POP, I. Analysis of mixed convection in a lid-driven porous square cavity with linearly heated side wall(s). International Journal of Heat and Mass Transfer, 53(9), 1819–1840 (2010)

    Article  MATH  Google Scholar 

  33. BASAK, T., ROY, S., SINGH, S. K., SHARMA, P. K., and POP, I. Analysis of mixed convection flows within a square cavity with uniform and non-uniform heating of bottom wall. International Journal of Thermal Sciences, 48(5), 891–912 (2009)

    Article  Google Scholar 

  34. BASAK, T., ROY, S., and CHAMKHA, A. J. A Peclet number based analysis of mixed convection for lid-driven porous square cavities with various heating of bottom wall. International Communications in Heat and Mass Transfer, 39(5), 657–664 (2012)

    Article  Google Scholar 

  35. DENG, Q. and CHANG, J. Natural convection in a rectangular enclosure with sinusoidal temperature distributions on both side walls. Numerical Heat Transfer Part A: Applications, 54(5), 507–524 (2008)

    Article  Google Scholar 

  36. SIVASANKARAN, S., SIVAKUMAR, V., and PRAKASH, P. Numerical study on mixed convection in a lid-driven cavity with non-uniform heating on both sidewalls. International Journal of Heat and Mass Transfer, 53(19), 4304–4315 (2010)

    Article  MATH  Google Scholar 

  37. SIVASANKARAN, S., SIVAKUMAR, V., and HUSSEIN, A. K. Numerical study on mixed convection in an inclined lid-driven cavity with discrete heating. International Communications in Heat and Mass Transfer, 46, 112–125 (2013)

    Article  Google Scholar 

  38. HSU, T. H. and ANG, S. G. W. Mixed convection in a rectangular enclosure with discrete heat sources. Numerical Heat Transfer Part A Applications, 38(6), 627–652 (2000)

    Article  Google Scholar 

  39. AL-AMIRI, A. and KHANAFER, K. Fluid-structure interaction analysis of mixed convection heat transfer in a lid-driven cavity with a flexible bottom wall. International Journal of Heat and Mass Transfer, 54(17/18), 3826–3836 (2011)

    Article  MATH  Google Scholar 

  40. ABU-NADA, E. and CHAMKHA, A. J. Mixed convection flow of a nanofluid in a lid-driven cavity with a wavy wall. International Communications in Heat and Mass Transfer, 57, 36–47 (2014)

    Article  Google Scholar 

  41. NASRIN, R. and PARVIN, S. Hydromagnetic effect on mixed convection in a lid-driven cavity with sinusoidal corrugated bottom surface. International Communications in Heat and Mass Transfer, 38(6), 781–789 (2011)

    Article  Google Scholar 

  42. MOUMNI, H., WELHEZI, H., DJEBALI, R., and SEDIKI, E. Accurate finite volume investigation of nanofluid mixed convection in two-sided lid driven cavity including discrete heat sources. Applied Mathematical Modelling, 39(14), 4164–4179 (2014)

    Article  MathSciNet  Google Scholar 

  43. ABBASBANDY, S. and SHIVANIAN, E. Multiple solutions of mixed convection in a porous medium on semi-infinite interval using pseudo-spectral collocation method. Communications in Nonlinear Science and Numerical Simulation, 16(7), 2745–2752 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  44. WILLIAMS, J. R. and AMARATUNGA, K. Introduction to wavelets in engineering. International Journal for Numerical Methods in Engineering, 37(14), 2365–2388 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  45. LEWALLE, J. Wavelet transforms of the Navier-stokes equations and the generalized dimensions of turbulence. Applied Scientific Research, 51(1/2), 109–113 (1993)

    Article  MATH  Google Scholar 

  46. GLOWINSKI, R., PERIAUX, J., RAVACHOL, M., PAN, T. W., WELLS, R. O., and ZHOU, X. Wavelet Methods in Computational Fluid Dynamics, Springer, New York (2010)

    Google Scholar 

  47. YANG, Z. and LIAO, S. A HAM-based wavelet approach for nonlinear ordinary differential equations. Communications in Nonlinear Science and Numerical Simulation, 48, 439–453 (2017)

    Article  MathSciNet  Google Scholar 

  48. YANG, Z. and LIAO, S. A HAM-based wavelet approach for nonlinear partial differential equations: two dimensional Bratu problem as an application. Communications in Nonlinear Science and Numerical Simulation, 53, 249–262 (2017)

    Article  MathSciNet  Google Scholar 

  49. YU, Q., XU, H., and LIAO, S. Coiflets solutions for FÖppl-von KÁrmÁn equations governing large deflection of a thin flat plate by a novel wavelet-homotopy approach. Numerical Algorithms, 1, 1–28 (2018)

    MATH  Google Scholar 

  50. LIU, X., ZHOU, Y., WANG, X., and WANG, J. A wavelet method for solving a class of nonlinear boundary value problems. Communications in Nonlinear Science and Numerical Simulation, 18(8), 1939–1948 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  51. ZHOU, Y. H. and WANG, J. Z. Generalized gaussian integral method for calculations of scaling function transform of wavelets and its applications (in Chinese). Acta Mathematica Scientia, 19(3), 293–300 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  52. ZHAO, Y., LIN, Z., and LIAO, S. An iterative ham approach for nonlinear boundary value problems in a semi-infinite domain. Computer Physics Communications, 184(9), 2136–2144 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  53. GHIA, U., GHIA, K. N., and SHIN, C. T. High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method. Journal of Computational Physics, 48(3), 387–411 (1982)

    Article  MATH  Google Scholar 

  54. BOTELLA, O. and PEYRET, R. Benchmark spectral results on the lid-driven cavity flow. Computers and Fluids, 27(4), 421–433 (1998)

    Article  MATH  Google Scholar 

  55. BRUNEAU, C. H. and SAAD, M. The 2D lid-driven cavity problem revisited. Computers and Fluids, 35(3), 326–348 (2006)

    Article  MATH  Google Scholar 

  56. MARCHI, C. H., SUERO, R., and ARAKI, L. K. The lid-driven square cavity flow: numerical solution with a 1024X1024 grid. Joarnal of the Brazilian Society of Mechanical Sciences and Engineering, 31(3), 186–198 (2009)

    Google Scholar 

  57. RUBIN, S. G. and KHOSLA, P. K. Polynomial interpolation methods for viscous flow calculations. Journal of Computational Physics, 24(3), 217–244 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  58. SCHIREIBER, R. and KELLER, H. B. Driven cavity flows by efficient numerical techniques. Journal of Computational Physics, 49(2), 310–333 (1983)

    Article  MATH  Google Scholar 

  59. VANKA, S. P. Block-implicit multigrid solution of Navier-Stokes equations in primitive variables. Journal of Computational Physics, 65(1), 138–158 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  60. HOU, S., ZOU Q., CHEN S., DOOLEN, G., and COGLEY, A. C. Simulation of cavity flow by the lattice Boltzmann method. Journal of Computational Physics, 118(2), 329–347 (1995)

    Article  MATH  Google Scholar 

  61. GOYON, O. High-Reynolds number solutions of Navier-Stokes equations using incremental unknowns. Computer Methods in Applied Mechanics and Engineering, 130(3/4), 319–335 (1996)

    Article  MATH  Google Scholar 

  62. BARRAGY, E. and CAREY, G. F. Stream function-vorticity driven cavity solution using p finite elements. Computers and Fluids, 26(5), 453–468 (1997)

    Article  MATH  Google Scholar 

  63. ZHANG, J. Numerical simulation of 2D square driven cavity using fourth-order compact finite difference schemes. Computers and Mathematics with Applications, 45(1), 43–52 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  64. GUPTA, M. M. and KALITA, J. C. A new paradigm for solving Navier-Stokes equations: streamfunction-velocity formulation. Journal of Computational Physics, 207(1), 52–68 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  65. ERTURK, E., CORKE, C. T., and GÖkcÖl, C. Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers. International Journal for Numerical Methods in Fluids, 48(7), 747–774 (2005)

    Article  MATH  Google Scholar 

  66. GANZAROLLI, M. M. and MILANEZ, L. F. Natural convection in rectangular enclosures heated from below and symmetrically cooled from the sides. International Journal of Heat and Mass Transfer, 38(6), 1063–1073 (1995)

    Article  MATH  Google Scholar 

  67. WAHEED, M. A. Mixed convective heat transfer in rectangular enclosures driven by a continuously moving horizontal plate. International Journal of Heat and Mass Transfer, 52(21/22), 5055–5063 (2009)

    Article  MATH  Google Scholar 

  68. AHMED, S. E. Mixed convection in thermally anisotropic non-Darcy porous medium in double lid-driven cavity using Bejan’s heatlines. Alexandria Engineering Journal, 55(1), 299–309 (2016)

    Article  Google Scholar 

  69. PATANKAR, S. V. Numerical Heat Transfer and Fluid Flow, CRC Press, Boca Raton (1980)

    MATH  Google Scholar 

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Acknowledgements

Thanks to the anonymous reviewer for their constructive comments and suggestions. The authors also wish to express their thanks to the very competent reviewers for the valuable comments and suggestions.

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Authors and Affiliations

  1. Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration (CISSE), State Key Laboratory of Ocean Engineering, School of Naval Architecture Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China

    Qiang Yu & Hang Xu

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  1. Qiang Yu
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  2. Hang Xu
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Correspondence to Hang Xu.

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Project supported by the National Natural Science Foundation of China (Nos. 11272209, 11432009, and 11872241)

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Yu, Q., Xu, H. Novel wavelet-homotopy Galerkin technique for analysis of lid-driven cavity flow and heat transfer with non-uniform boundary conditions. Appl. Math. Mech.-Engl. Ed. 39, 1691–1718 (2018). https://doi.org/10.1007/s10483-018-2397-9

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  • DOI: https://doi.org/10.1007/s10483-018-2397-9

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