Varmazyar, M., habibi, M., Mohammadi, A. (2018). Investigation on Instability of Rayleigh-Benard Convection Using Lattice Boltzmann Method with a Modified Boundary Condition. Journal of Computational Applied Mechanics, 49(2), 231-239. doi: 10.22059/jcamech.2017.243410.197

Mostafa Varmazyar; Mohammadreza habibi; Arash Mohammadi. "Investigation on Instability of Rayleigh-Benard Convection Using Lattice Boltzmann Method with a Modified Boundary Condition". Journal of Computational Applied Mechanics, 49, 2, 2018, 231-239. doi: 10.22059/jcamech.2017.243410.197

Varmazyar, M., habibi, M., Mohammadi, A. (2018). 'Investigation on Instability of Rayleigh-Benard Convection Using Lattice Boltzmann Method with a Modified Boundary Condition', Journal of Computational Applied Mechanics, 49(2), pp. 231-239. doi: 10.22059/jcamech.2017.243410.197

Varmazyar, M., habibi, M., Mohammadi, A. Investigation on Instability of Rayleigh-Benard Convection Using Lattice Boltzmann Method with a Modified Boundary Condition. Journal of Computational Applied Mechanics, 2018; 49(2): 231-239. doi: 10.22059/jcamech.2017.243410.197

Investigation on Instability of Rayleigh-Benard Convection Using Lattice Boltzmann Method with a Modified Boundary Condition

^{1}Department of Mechanical Engineering, Shahid Rajaee Teacher Training University, Tehran, Iran

^{2}Research Institute of Petroleum Industry, Tehran, Iran

Receive Date: 18 October 2017,
Revise Date: 01 November 2017,
Accept Date: 20 November 2017

Abstract

In this study, the effects of Prandtl number on the primary and secondary instability of the Rayleigh-Benard convection problem has been investigated using the lattice Boltzmann method. Two different cases as Pr=5.8 and 0.7 representing the fluid in liquid and gas conditions are examined. A body forces scheme of the lattice Boltzmann method was presented. Two types of boundary conditions in the presence of body forces are analyzed by the moment method and applied to a Poiseuille flow. Characteristic velocity was set in such a way that the compressibility effects are negligible. The calculations show that the increment of Prandtl number from 0.7 to 5.8 causes to create a secondary instability and onset of the oscillation in the flow field. Results show that at Pr=5.8, when the Rayleigh number is increased, a periodic solution appeared at Ra=48,000. It is observed that the dimensionless frequency ratio for Ra= 105 with Pr=5.8 is around 0.0065. The maximum Nusselt number for Ra = 105 with Pr=5.8 are estimated to be 5.4942.

[1] A. Soloviev, B. Klinger, Open ocean convection, Elements of Physical Oceanography: A derivative of the Encyclopedia of Ocean Sciences, pp. 414, 2009.

[2] M. M. Holland, C. M. Bitz, M. Eby, A. J. Weaver, The role of ice-ocean interactions in the variability of the North Atlantic thermohaline circulation, Journal of Climate, Vol. 14, No. 5, pp. 656-675, 2001.

[3] A. Wirth, B. Barnier, Tilted plumes in numerical convection experiments, Ocean Model, Vol. 12, pp. 101-111, 2006.

[4] D.-J. Yao, J.-R. Chen, W.-T. Ju, Micro–Rayleigh-Bénard convection polymerase chain reaction system, Journal of Micro/Nanolithography, MEMS, and MOEMS, Vol. 6, No. 4, pp. 043007-043007-9, 2007.

[5] N. Agrawal, V. M. Ugaz, A buoyancy-driven compact thermocycler for rapid PCR, Clinics in laboratory medicine, Vol. 27, No. 1, pp. 215-223, 2007.

[6] M. Hennig, D. Braun, Convective polymerase chain reaction around micro immersion heater, Applied Physics Letters, Vol. 87, No. 18, pp. 183901, 2005.

[7] S. Banerjee, A. Mukhopadhyay, S. Sen, R. Ganguly, Natural convection in a bi-heater configuration of passive electronic cooling, International Journal of Thermal Sciences, Vol. 47, No. 11, pp. 1516-1527, 2008.

[8] A. Chaehoi, F. Mailly, L. Latorre, P. Nouet, Experimental and finite-element study of convective accelerometer on CMOS, Sensors and Actuators A: Physical, Vol. 132, No. 1, pp. 78-84, 2006.

[9] Z.-Y. Guo, Z.-X. Li, Size effect on microscale single-phase flow and heat transfer, International Journal of Heat and Mass Transfer, Vol. 46, No. 1, pp. 149-159, 2003.

[10] X. J. Hu, A. Jain, K. E. Goodson, Investigation of the natural convection boundary condition in microfabricated structures, International Journal of Thermal Sciences, Vol. 47, No. 7, pp. 820-824, 2008.

[11] M.-S. Nazari, M. Kayhani, A Comparative Solution of Natural Convection in an Open Cavity using Different Boundary Conditions via Lattice Boltzmann Method, Journal of Heat and Mass Transfer Research(JHMTR), Vol. 3, No. 2, pp. 115-129, 2016.

[12] G. Sobamowo, Analysis of Heat transfer in Porous Fin with Temperature-dependent Thermal Conductivity and Internal Heat Generation using Chebychev Spectral Collocation Method, Journal of Computational Applied Mechanics, pp. -, 2017.

[13] G. C. Rana, R. Chand, V. Sharma, A. Sharda, On the onset of triple-diffusive convection in a layer of nanofluid, Journal of Computational Applied Mechanics, Vol. 47, No. 1, pp. 67-77, 2016.

[14] G. Ahlers, S. Grossmann, D. Lohse, Heat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection, Reviews of modern physics, Vol. 81, No. 2, pp. 503, 2009.

[15] A. Puhl, M. M. Mansour, M. Mareschal, Quantitative comparison of molecular dynamics with hydrodynamics in Rayleigh-Benard convection, Physical Review A, Vol. 40, No. 4, pp. 1999, 1989.

[16] E. D. Siggia, High Rayleigh number convection, Annual review of fluid mechanics, Vol. 26, No. 1, pp. 137-168, 1994.

[17] G. Freund, W. Pesch, W. Zimmermann, Rayleigh–Bénard convection in the presence of spatial temperature modulations, Journal of Fluid Mechanics, Vol. 673, pp. 318-348, 2011.

[18] S. Weiß, G. Seiden, E. Bodenschatz, Pattern formation in spatially forced thermal convection, New Journal of Physics, Vol. 14, No. 5, pp. 053010, 2012.

[19] G. Seiden, S. Weiss, J. H. McCoy, W. Pesch, E. Bodenschatz, Pattern forming system in the presence of different symmetry-breaking mechanisms, Physical review letters, Vol. 101, No. 21, pp. 214503, 2008.

[20] M. Hossain, J. Floryan, Heat transfer due to natural convection in a periodically heated slot, Journal of Heat Transfer, Vol. 135, No. 2, pp. 022503, 2013.

[21] M. Hossain, J. M. Floryan, Instabilities of natural convection in a periodically heated layer, Journal of Fluid Mechanics, Vol. 733, pp. 33-67, 2013.

[22] P. Ripesi, L. Biferale, M. Sbragaglia, A. Wirth, Natural convection with mixed insulating and conducting boundary conditions: low-and high-Rayleigh-number regimes, Journal of Fluid Mechanics, Vol. 742, pp. 636-663, 2014.

[23] S. Chen, G. D. Doolen, Lattice Boltzmann method for fluid flows, Annual review of fluid mechanics, Vol. 30, No. 1, pp. 329-364, 1998.

[24] X. He, L.-S. Luo, A priori derivation of the lattice Boltzmann equation, Physical Review E, Vol. 55, No. 6, pp. R6333, 1997.

[25] S. Succi, 2001, The lattice Boltzmann equation: for fluid dynamics and beyond, Oxford university press,

[26] D. Yu, R. Mei, L.-S. Luo, W. Shyy, Viscous flow computations with the method of lattice Boltzmann equation, Progress in Aerospace Sciences, Vol. 39, No. 5, pp. 329-367, 2003.

[27] A. J. Moghadam, TWO-FLUID ELECTROKINETIC FLOW IN A CIRCULAR MICROCHANNEL (RESEARCH NOTE), International Journal of Engineering-Transactions A: Basics, Vol. 29, No. 10, pp. 1469, 2016.

[28] P. Pashaie, M. Jafari, H. Baseri, M. Farhadi, Nusselt number estimation along a wavy wall in an inclined lid-driven cavity using adaptive neuro-fuzzy inference system (ANFIS), International Journal of Engineering-Transactions A: Basics, Vol. 26, No. 4, pp. 383, 2012.

[29] M. Varmazyar, M. Bazargan, Development of a thermal lattice Boltzmann method to simulate heat transfer problems with variable thermal conductivity, International Journal of Heat and Mass Transfer, Vol. 59, pp. 363-371, 2013.

[30] M. Jafari, M. Farhadi, K. Sedighi, E. Fattahi, Effect of wavy wall on convection heat transfer of water-al2o3 nanofluid in a lid-driven cavity using lattice boltzmann method, International Journal of Engineering-Transactions A: Basics, Vol. 25, No. 2, pp. 165, 2012.

[31] H. Ashorynejad, M. Sheikholeslami, E. Fattahi, Lattice boltzmann simulation of nanofluids natural convection heat transfer in concentric annulus, International Journal of Engineering-Transactions B: Applications, Vol. 26, No. 8, pp. 895, 2013.

[32] M. Varmazyar, M. Bazargan, A. Moahmmadi, A. Rahbari, Error Analysis of Thermal Lattice Boltzmann Method in Natural Convection Problems with Varying Fluid Thermal Diffusion Coefficient, Modares Mechanical Engineering, Vol. 16, No. 12, pp. 335-344, 2016.

[33] S. Chen, H. Chen, D. Martnez, W. Matthaeus, Lattice Boltzmann model for simulation of magnetohydrodynamics, Physical Review Letters, Vol. 67, No. 27, pp. 3776, 1991.

[34] B. M. Riley, J. C. Richard, S. S. Girimaji, Assessment of magnetohydrodynamic lattice Boltzmann schemes in turbulence and rectangular jets, International Journal of Modern Physics C, Vol. 19, No. 08, pp. 1211-1220, 2008.

[35] X. He, S. Chen, R. Zhang, A lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of Rayleigh–Taylor instability, Journal of Computational Physics, Vol. 152, No. 2, pp. 642-663, 1999.

[36] T. Inamuro, T. Ogata, S. Tajima, N. Konishi, A lattice Boltzmann method for incompressible two-phase flows with large density differences, Journal of Computational Physics, Vol. 198, No. 2, pp. 628-644, 2004.

[37] S. Tilehboni, K. Sedighi, M. Farhadi, E. Fattahi, Lattice Boltzmann simulation of deformation and breakup of a droplet under gravity force using interparticle potential model, International Journal of Engineering-Transactions A: Basics, Vol. 26, No. 7, pp. 781, 2013.

[38] M. H. Rahimian, R. Haghani, Four different types of a single drop dripping down a hole under gravity by lattice Boltzmann method, Journal of Computational Applied Mechanics, Vol. 47, No. 1, pp. 89-98, 2016.

[39] M. Varmazyar, M. Bazargan, Modeling of Free Convection Heat Transfer to a Supercritical Fluid in a Square Enclosure by the Lattice Boltzmann Method, Journal of Heat Transfer, Vol. 133, No. 2, pp. 022501, 2011.

[40] L.-S. Luo, Unified theory of lattice Boltzmann models for nonideal gases, Physical review letters, Vol. 81, No. 8, pp. 1618, 1998.

[41] M. Varmazyar, M. Bazargan, Numerical Investigation of the Piston Effect of Supercritical Fluid under Microgravity Conditions Using Lattice Boltzmann Method, Modares Mechanical Engineering, Vol. 17, No. 5, pp. 138-146, 2017.

[42] X. Shan, Simulation of Rayleigh-Bénard convection using a lattice Boltzmann method, Physical Review E, Vol. 55, No. 3, pp. 2780, 1997.

[43] F. Busse, H. Frick, Square-pattern convection in fluids with strongly temperature-dependent viscosity, Journal of fluid mechanics, Vol. 150, pp. 451-465, 1985.

[44] X. He, S. Chen, G. D. Doolen, A novel thermal model for the lattice Boltzmann method in incompressible limit, Journal of Computational Physics, Vol. 146, No. 1, pp. 282-300, 1998.

[45] R. Clever, F. Busse, Transition to time-dependent convection, Journal of Fluid Mechanics, Vol. 65, No. 04, pp. 625-645, 1974.

[46] P.-H. Kao, R.-J. Yang, Simulating oscillatory flows in Rayleigh–Benard convection using the lattice Boltzmann method, International Journal of Heat and Mass Transfer, Vol. 50, No. 17, pp. 3315-3328, 2007.

[47] J. Wang, D. Wang, P. Lallemand, L.-S. Luo, Lattice Boltzmann simulations of thermal convective flows in two dimensions, Computers & Mathematics with Applications, Vol. 65, No. 2, pp. 262-286, 2013.

[48] Z. Guo, C. Zheng, B. Shi, Discrete lattice effects on the forcing term in the lattice Boltzmann method, Physical Review E, Vol. 65, No. 4, pp. 046308, 2002.

[49] S. Chen, D. Martinez, R. Mei, On boundary conditions in lattice Boltzmann methods, Physics of Fluids (1994-present), Vol. 8, No. 9, pp. 2527-2536, 1996.

[50] J. Latt, Hydrodynamic limit of lattice Boltzmann equations, Thesis, University of Geneva, 2007.

[51] J. Latt, B. Chopard, O. Malaspinas, M. Deville, A. Michler, Straight velocity boundaries in the lattice Boltzmann method, Physical Review E, Vol. 77, No. 5, pp. 056703, 2008.

[52] N. S. Martys, H. Chen, Simulation of multicomponent fluids in complex three-dimensional geometries by the lattice Boltzmann method, Physical review E, Vol. 53, No. 1, pp. 743, 1996.

[53] D. R. Noble, S. Chen, J. G. Georgiadis, R. O. Buckius, A consistent hydrodynamic boundary condition for the lattice Boltzmann method, Physics of Fluids (1994-present), Vol. 7, No. 1, pp. 203-209, 1995.

[54] P. Skordos, Initial and boundary conditions for the lattice Boltzmann method, Physical Review E, Vol. 48, No. 6, pp. 4823, 1993.

[55] S.-M. Li, D. K. Tafti, Near-critical CO 2 liquid–vapor flow in a sub-microchannel. Part I: Mean-field free-energy D2Q9 lattice Boltzmann method, International Journal of Multiphase Flow, Vol. 35, No. 8, pp. 725-737, 2009.

[56] R. Allen, T. Reis, A lattice Boltzmann model for natural convection in cavities, International Journal of Heat and Fluid Flow, 2013.

[57] S. Bennett, A lattice Boltzmann model for diffusion of binary gas mixtures, Thesis, University of Cambridge, 2010.

[58] P. L. Bhatnagar, E. P. Gross, M. Krook, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Physical review, Vol. 94, No. 3, pp. 511, 1954.

[59] A. Ladd, R. Verberg, Lattice-Boltzmann simulations of particle-fluid suspensions, Journal of Statistical Physics, Vol. 104, No. 5-6, pp. 1191-1251, 2001.

[60] N. S. Martys, J. F. Douglas, Critical properties and phase separation in lattice Boltzmann fluid mixtures, Physical Review E, Vol. 63, No. 3, pp. 031205, 2001.

[61] T. Inamuro, M. Yoshino, H. Inoue, R. Mizuno, F. Ogino, A lattice Boltzmann method for a binary miscible fluid mixture and its application to a heat-transfer problem, Journal of Computational Physics, Vol. 179, No. 1, pp. 201-215, 2002.

[62] Y. Zhou, R. Zhang, I. Staroselsky, H. Chen, Numerical simulation of laminar and turbulent buoyancy-driven flows using a lattice Boltzmann based algorithm, International Journal of Heat and Mass Transfer, Vol. 47, No. 22, pp. 4869-4879, 2004.

[63] Y. Peng, C. Shu, Y. Chew, Simplified thermal lattice Boltzmann model for incompressible thermal flows, Physical Review E, Vol. 68, No. 2, pp. 026701, 2003.

[64] X. Shan, H. Chen, Simulation of nonideal gases and gas-liquid phase transitions by the lattice Boltzmann Equation, Phys. Rev. E. v49 i4, pp. 2941-2948.

[65] L.-S. Luo, Lattice-gas automata and lattice Boltzmann equations for two-dimensional hydrodynamics, 1993.

[66] W. Reid, D. Harris, Some further results on the Bénard problem, Physics of Fluids (1958-1988), Vol. 1, No. 2, pp. 102-110, 1958.

[67] K. Xu, S. H. Lui, Rayleigh-Bénard simulation using the gas-kinetic Bhatnagar-Gross-Krook scheme in the incompressible limit, Physical Review E, Vol. 60, No. 1, pp. 464, 1999.