A Multiple-Grid Lattice Boltzmann Method for Natural Convection under Low and High Prandtl Numbers
Abstract
:1. Introduction
2. Numerical Method
2.1. Multiple-Grid Lattice Boltzmann (MGLB)
3. Results
3.1. Natural Convection Inside a Heated Square Cavity: Benchmark Test
3.2. Natural Convection with SRT Method for Prandtl Numbers 0.01 and 100
3.3. Natural Convection with MGLB Method
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
References
- Aidun, C.K.; Clausen, J.R. Lattice-Boltzmann Method for Complex Flows. Annu. Rev. Fluid Mech. 2010, 42, 439–472. [Google Scholar] [CrossRef]
- D’Orazio, A.; Corcione, M.; Celata, G.P. Application to Natural Convection Enclosed Flows of a Lattice Boltzmann BGK Model Coupled with a General Purpose Thermal Boundary Condition. Int. J. Therm. Sci. 2004, 43, 575–586. [Google Scholar] [CrossRef]
- Connington, K.; Lee, T. A Review of Spurious Currents in the Lattice Boltzmann Method for Multiphase Flows. J. Mech. Sci. Technol. 2012, 26, 3857–3863. [Google Scholar] [CrossRef]
- He, X.; Luo, L.S. A Priori Derivation of the Lattice Boltzmann Equation. Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. 1997, 55. [Google Scholar] [CrossRef] [Green Version]
- Shih, H.C.; Huang, C.L. Image Analysis and Interpretation for Semantics Categorization in Baseball Video. Proc. ITCC 2003 Int. Conf. Inf. Technol. Comput. Commun. 2003, 94, 379–383. [Google Scholar] [CrossRef]
- Li, Z.; Yang, M.; Zhang, Y. Double MRT Thermal Lattice Boltzmann Method for Simulating Natural Convection of Low Prandtl Number Fluids. Int. J. Numer. Methods Head Fluid 2016, 26. [Google Scholar] [CrossRef]
- Perumal, D.A.; Dass, A.K. A Review on the Development of Lattice Boltzmann Computation of Macro Fluid Flows and Heat Transfer. Alex. Eng. J. 2015, 54, 955–971. [Google Scholar] [CrossRef] [Green Version]
- McNamara, G.; Alder, B. Analysis of the Lattice Boltzmann Treatment of Hydrodynamics. Phys. A Stat. Mech. Appl. 1993, 194, 218–228. [Google Scholar] [CrossRef]
- Alexander, F.J.; Chen, S.; Sterling, J.D. Lattice Boltzmann Thermohydrodynamics. Phys. Rev. E 1993, 47, R2249–R2252. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Qian, Y.H. Simulating Thermohydrodynamics with Lattice BGK Models. J. Sci. Comput. 1993, 8, 231–242. [Google Scholar] [CrossRef]
- Nie, X.; Qian, Y.H.; Doolen, G.D.; Chen, S. Lattice Boltzmann Simulation of the Two-Dimensional Rayleigh-Taylor Instability. Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. 1998, 58, 6861–6864. [Google Scholar] [CrossRef]
- He, X.; Chen, S.; Zhang, R. A Lattice Boltzmann Scheme for Incompressible Multiphase Flow and Its Application in Simulation of Rayleigh-Taylor Instability. J. Comput. Phys. 1999, 152, 642–663. [Google Scholar] [CrossRef]
- McNamara, G.R.; Garcia, A.L.; Alder, B.J. Stabilization of Thermal Lattice Boltzmann Models. J. Stat. Phys. 1995, 81, 395–408. [Google Scholar] [CrossRef]
- Li, Q.; Luo, K.H.; He, Y.L.; Gao, Y.J.; Tao, W.Q. Coupling Lattice Boltzmann Model for Simulation of Thermal Flows on Standard Lattices. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 2012, 85, 16710. [Google Scholar] [CrossRef] [Green Version]
- Nabavizadeh, S.A.; Talebi, S.; Sefid, M.; Nourmohammadzadeh, M. Natural Convection in a Square Cavity Containing a Sinusoidal Cylinder. Int. J. Therm. Sci. 2012, 51, 112–120. [Google Scholar] [CrossRef]
- Prasianakis, N.I.; Karlin, I.V. Lattice Boltzmann Method for Thermal Flow Simulation on Standard Lattices. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 2007, 76. [Google Scholar] [CrossRef] [Green Version]
- Fattahi, E.; Farhadi, M.; Sedighi, K. Lattice Boltzmann Simulation of Natural Convection Heat Transfer in Eccentric Annulus. Int. J. Therm. Sci. 2010, 49, 2353–2362. [Google Scholar] [CrossRef]
- Souayeh, B.; Ben-Cheikh, N.; Ben-Beya, B. Numerical Simulation of Three-Dimensional Natural Convection in a Cubic Enclosure Induced by an Isothermally-Heated Circular Cylinder at Different Inclinations. Int. J. Therm. Sci. 2016, 110, 325–339. [Google Scholar] [CrossRef]
- Parmigiani, A.; Huber, C.; Chopard, B.; Latt, J.; Bachmann, O. Application of the Multi Distribution Function Lattice Boltzmann Approach to Thermal Flows. Eur. Phys. J. Spec. Top. 2009, 171, 37–43. [Google Scholar] [CrossRef]
- Dixit, H.N.; Babu, V. Simulation of High Rayleigh Number Natural Convection in a Square Cavity Using the Lattice Boltzmann Method. Int. J. Heat Mass Transf. 2006, 49, 727–739. [Google Scholar] [CrossRef]
- Li, Z.; Yang, M.; Zhang, Y. Lattice Boltzmann Method Simulation of 3-D Natural Convection with Double MRT Model. Int. J. Heat Mass Transf. 2016, 94, 222–238. [Google Scholar] [CrossRef] [Green Version]
- D’Humières, D.; Ginzburg, I.; Krafczyk, M.; Lallemand, P.; Luo, L.S. Multiple-Relaxation-Time Lattice Boltzmann Models in Three Dimensions. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 2002, 360, 437–451. [Google Scholar] [CrossRef] [PubMed]
- Wang, J.; Wang, D.; Lallemand, P.; Luo, L.S. Lattice Boltzmann Simulations of Thermal Convective Flows in Two Dimensions. Comp. Math. Appl. 2013, 65, 262–286. [Google Scholar] [CrossRef]
- Contrino, D.; Lallemand, P.; Asinari, P.; Luo, L.S. Lattice-Boltzmann Simulations of the Thermally Driven 2D Square Cavity at High Rayleigh Numbers. J. Comput. Phys. 2014, 275, 257–272. [Google Scholar] [CrossRef] [Green Version]
- Mezrhab, A.; Amine Moussaoui, M.; Jami, M.; Naji, H.; Bouzidi, M. Double MRT Thermal Lattice Boltzmann Method for Simulating Convective Flows. Phys. Lett. Sect. A Gen. At. Solid State Phys. 2010, 374, 3499–3507. [Google Scholar] [CrossRef]
- Xu, A.; Shi, L.; Xi, H.D. Lattice Boltzmann Simulations of Three-Dimensional Thermal Convective Flows at High Rayleigh Number. Int. J. Heat Mass Transf. 2019, 140, 359–370. [Google Scholar] [CrossRef] [Green Version]
- Karlin, I.V.; Ferrante, A.; Öttinger, H.C. Perfect Entropy Functions of the Lattice Boltzmann Method. Europhys. Lett. 1999, 47, 182–188. [Google Scholar] [CrossRef] [Green Version]
- Pareschi, G.; Frapolli, N.; Chikatamarla, S.S.; Karlin, I.V. Conjugate Heat Transfer with the Entropic Lattice Boltzmann Method. Phys. Rev. E 2016, 94. [Google Scholar] [CrossRef] [PubMed]
- Hajabdollahi, F.; Premnath, K.N. Central Moments-Based Cascaded Lattice Boltzmann Method for Thermal Convective Flows in Three-Dimensions. Int. J. Heat Mass Transf. 2018, 120, 838–850. [Google Scholar] [CrossRef] [Green Version]
- Sharma, K.V.; Straka, R.; Tavares, F.W. New Cascaded Thermal Lattice Boltzmann Method for Simulations of Advection-Diffusion and Convective Heat Transfer. Int. J. Therm. Sci. 2017, 118, 259–277. [Google Scholar] [CrossRef]
- Chen, Z.; Shu, C.; Tan, D. A Simplified Thermal Lattice Boltzmann Method without Evolution of Distribution Functions. Int. J. Heat Mass Transf. 2017, 105, 741–757. [Google Scholar] [CrossRef]
- Chen, Z.; Shu, C.; Tan, D.; Wu, C. On Improvements of Simplified and Highly Stable Lattice Boltzmann Method: Formulations, Boundary Treatment, and Stability Analysis. Int. J. Numer. Methods Fluids 2018, 87, 161–179. [Google Scholar] [CrossRef]
- Chen, Z.; Shu, C.; Wang, Y.; Yang, L.M.; Tan, D. A Simplified Lattice Boltzmann Method without Evolution of Distribution Function. Adv. Appl. Math. Mech. 2017, 9, 1–22. [Google Scholar] [CrossRef]
- Chen, Z.; Shu, C.; Tan, D. High-Order Simplified Thermal Lattice Boltzmann Method for Incompressible Thermal Flows. Int. J. Heat Mass Transf. 2018, 127, 1–16. [Google Scholar] [CrossRef]
- Dorari, E.; Eshraghi, M.; Felicelli, S.D. A Multiple-Grid-Time-Step Lattice Boltzmann Method for Transport Phenomena with Dissimilar Time Scales: Application in Dendritic Solidification. Appl. Math. Model. 2018, 62, 580–594. [Google Scholar] [CrossRef]
- Sakane, S.; Takaki, T.; Ohno, M.; Shibuta, Y. Simulation Method Based on Phase-Field Lattice Boltzmann Model for Long-Distance Sedimentation of Single Equiaxed Dendrite. Comput. Mater. Sci. 2019, 164, 39–45. [Google Scholar] [CrossRef]
- López, J.; Gómez, P.; Hernández, J.; Faura, F. A Two-Grid Adaptive Volume of Fluid Approach for Dendritic Solidification. Comput. Fluids 2013, 86, 326–342. [Google Scholar] [CrossRef] [Green Version]
- Sakane, S.; Takaki, T.; Ohno, M.; Shibuta, Y.; Aoki, T. Acceleration of Phase-Field Lattice Boltzmann Simulation of Dendrite Growth with Thermosolutal Convection by the Multi-GPUs Parallel Computation with Multiple Mesh and Time Step Method. Model. Simul. Mater. Sci. Eng. 2019, 27. [Google Scholar] [CrossRef]
- Nabavizadeh, S.A.; Eshraghi, M.; Felicelli, S.D. A Comparative Study of Multiphase Lattice Boltzmann Methods for Bubble-Dendrite Interaction during Solidification of Alloys. Appl. Sci. 2018, 9, 57. [Google Scholar] [CrossRef] [Green Version]
- Hortmann, M.; Perić, M.; Scheuerer, G. Finite Volume Multigrid Prediction of Laminar Natural Convection: Benchmark Solutions. Int. J. Numer. Methods Fluids 1990, 11, 189–207. [Google Scholar] [CrossRef]
- Mohamad, A.A.; Viskanta, R. Transient Natural Convection of Low-Prandtl-number Fluids in a Differentially Heated Cavity. Int. J. Numer. Methods Fluids 1991, 13, 61–81. [Google Scholar] [CrossRef]
- Guo, Z.; Zheng, C.; Shi, B.; Zhao, T.S. Thermal Lattice Boltzmann Equation for Low Mach Number Flows: Decoupling Model. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 2007, 75. [Google Scholar] [CrossRef]
- Pesso, T.; Piva, S. Laminar Natural Convection in a Square Cavity: Low Prandtl Numbers and Large Density Differences. Int. J. Heat Mass Transf. 2009, 52, 1036–1043. [Google Scholar] [CrossRef]
- Ahlers, G.; Xu, X. Prandtl-Number Dependence of Heat Transport in Turbulent Rayleigh-Bénard Convection. Phys. Rev. Lett. 2001, 86, 3320–3323. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Kao, P.H.; Yang, R.J. Simulating Oscillatory Flows in Rayleigh-Bénard Convection Using the Lattice Boltzmann Method. Int. J. Heat Mass Transf. 2007, 50, 3315–3328. [Google Scholar] [CrossRef]
- Silano, G.; Sreenivasan, K.R.; Verzicco, R. Numerical Simulations of Rayleigh-Bénard Convection for Prandtl Numbers between 101 and 104 and Rayleigh Numbers between 105 and 109. J. Fluid Mech. 2010, 662, 409–446. [Google Scholar] [CrossRef]
- Fei, L.; Luo, K.H. Cascaded Lattice Boltzmann Method for Thermal Flows on Standard Lattices. Int. J. Therm. Sci. 2018, 132, 368–377. [Google Scholar] [CrossRef]
- Pandey, A.; Scheel, J.D.; Schumacher, J. Turbulent Superstructures in Rayleigh-Bénard Convection. Nat. Commun. 2018, 9. [Google Scholar] [CrossRef] [Green Version]
- De Vahl Davis, G. Natural Convection of Air in a Square Cavity: A Bench Mark Numerical Solution. Int. J. Numer. Methods Fluids 1983, 3, 249–264. [Google Scholar] [CrossRef]
- Bessonov, O.A.; Brailovskaya, V.A.; Nikitin, S.A.; Polezhaev, V.I. Three-Dimensional Natural Convection in a Cubical Enclosure: A Bench Mark Numerical Solution. In Proceedings of the International Symposium on Advances in Computational Heat Transfer, Çesme, Turkey, 26–30 May 1997. [Google Scholar] [CrossRef]
- Guo, Z.; Shi, B.; Zheng, C. A Coupled Lattice BGK Model for the Boussinesq Equations. Int. J. Numer. Methods Fluids 2002, 39, 325–342. [Google Scholar] [CrossRef]
- Qian, Y.H.; D’Humières, D.; Lallemand, P. Lattice Bgk Models for Navier-Stokes Equation. EPL 1992, 17, 479–484. [Google Scholar] [CrossRef]
- Benzi, R.; Succi, S.; Vergassola, M. The Lattice Boltzmann Equation: Theory and Applications. Phys. Rep. 1992, 145–197. [Google Scholar] [CrossRef]
- Luo, L.S.; Liao, W.; Chen, X.; Peng, Y.; Zhang, W. Numerics of the Lattice Boltzmann Method: Effects of Collision Models on the Lattice Boltzmann Simulations. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 2011, 83. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Pan, C.; Luo, L.S.; Miller, C.T. An Evaluation of Lattice Boltzmann Schemes for Porous Medium Flow Simulation. Comput. Fluids 2006, 35, 898–909. [Google Scholar] [CrossRef]
- Zhao, F. Optimal Relaxation Collisions for Lattice Boltzmann Methods. Comput. Math. Appl. 2013, 65, 172–185. [Google Scholar] [CrossRef]
- Li, Z.; Yang, M.; Zhang, Y. Numerical Simulation of Melting Problems Using the Lattice Boltzmann Method with the Interfacial Tracking Method. Numer. Heat Transf. Part. A Appl. 2015, 68, 1175–1197. [Google Scholar] [CrossRef]
Natural Convection Inside a Heated Square Cavity | ||
---|---|---|
Ref. | Physical Parameter | Numerical Method/Remarks |
Hortmann et al., 1990 [40] | Pr = 0.71 | Finite volume method with uniform and non-uniform grid |
Mohamad and Viskanta, 1991 [41] | Control volume-based finite difference method with non-uniform grid | |
Guo et al., 2007 [42] | Pr = 0.71 | LB double distribution function with a uniform grid |
Pesso and Piva, 2009 [43] | Finite volume method with non-uniform grid | |
Li et al., 2016 [6] | LB double distribution function with MRT collision | |
Chen et al., 2017 [31] | Pr = 0.71 | Simplified thermal LB |
Hajabdollahi and Premnath, 2018 [29] | Pr = 0.71 | Cascaded LB method |
Xi et al., 2019 [26] | Pr = 0.71 | LB double distribution function with MRT collision model |
Rayleigh–Bernard Convection | ||
Ahlers and Xu, 2001 [44] | Experimental | |
Kao and Yang, 2007 [45] | LB double distribution function with a uniform grid | |
Silano et al., 2010 [46] | Second-order cylindrical coordinate finite-difference scheme | |
Fei and Luo, 2018 [47] | Pr = 0.71 | Cascaded LB method |
Pandey et al., 2018 [48] | Direct numerical simulations (DNSs) |
Umax(y) | Vmax(x) | ||
---|---|---|---|
Current study | 8.85 | 65.03(0.845) | 215.8(0.0391) |
Ref. [42] | 8.7746 | 64.91(0.8516) | 218.90(0.0391) |
Ref. [40] | 8.8251 | 64.84(0.8505) | 220.46(0.0390) |
Pr = 0.005 Ra = 15,000 | Pr = 0.007 Ra = 105 | Pr = 0.01 Ra = 105 | Pr = 0.071 Ra = 105 | Pr = 0.71 Ra = 105 | Pr = 7.1 Ra = 105 | |
---|---|---|---|---|---|---|
Current | 2.08 | 2.65 | 3.18 | 3.76 | 4.45 | 4.68 |
Ref. [41] | 2.10 | N/A | 3.23 | N/A | N/A | N/A |
Ref. [43] | N/A | 2.58 | N/A | 3.80 | 4.48 | 4.72 |
Case No. | n | GR | Time | |||||
---|---|---|---|---|---|---|---|---|
1(SRT) | 100 | 100 | 1 | 1 | 0.5015 | 0.6500 | N/A | N/A |
2(SRT) | 200 | 200 | 1 | 1 | 0.5030 | 0.8000 | 5.345 | 1.000 |
3(MGLB) | 100 | 100 | 2 | 1 | 0.5015 | 0.5750 | 5.105 | 0.193 |
4(MGLB) | 150 | 100 | 1 | 1.5 | 0.5023 | 0.6000 | 5.350 | 0.456 |
5(MGLB) | 150 | 100 | 2 | 1.5 | 0.5023 | 0.5500 | 5.343 | 0.491 |
6(MGLB) | 150 | 100 | 3 | 1.5 | 0.5023 | 0.5333 | 5.350 | 0.544 |
7(MGLB) | 200 | 100 | 1 | 2 | 0.5030 | 0.5750 | 5.467 | 0.719 |
8(MGLB) | 200 | 100 | 2 | 2 | 0.5030 | 0.5375 | 5.461 | 0.737 |
9(MGLB) | 200 | 100 | 3 | 2 | 0.5030 | 0.5250 | 5.455 | 0.789 |
10(MGLB) | 300 | 100 | 1 | 3 | 0.5045 | 0.5500 | 5.538 | 1.965 |
11(MGLB) | 300 | 100 | 2 | 3 | 0.5045 | 0.5250 | 5.539 | 2.000 |
12(MGLB) | 300 | 100 | 3 | 3 | 0.5045 | 0.5167 | 5.535 | 2.053 |
Case No. | n | GR | Time | |||||
---|---|---|---|---|---|---|---|---|
1(SRT) | 100 | 100 | 1 | 1 | 0.6500 | 0.5015 | N/A | N/A |
2(SRT) | 300 | 300 | 1 | 1 | 0.950 | 0.5045 | N/A | N/A |
3(SRT) | 350 | 350 | 1 | 1 | 1.0250 | 0.5052 | 9.2980 | 1.000 |
4(MGLB) | 100 | 100 | 2 | 1 | 0.6500 | 0.5030 | N/A | N/A |
5(MGLB) | 100 | 100 | 3 | 1 | 0.6500 | 0.5045 | N/A | N/A |
6(MGLB) | 100 | 150 | 1 | 1.5 | 0.6500 | 0.5034 | N/A | N/A |
7(MGLB) | 100 | 150 | 2 | 1.5 | 0.6500 | 0.5067 | 8.523 | 0.089 |
8(MGLB) | 100 | 150 | 3 | 1.5 | 0.6500 | 0.5101 | 8.521 | 0.117 |
9(MGLB) | 100 | 200 | 1 | 2 | 0.6500 | 0.5060 | 8.453 | 0.108 |
10(MGLB) | 100 | 200 | 2 | 2 | 0.6500 | 0.5120 | 8.451 | 0.141 |
11(MGLB) | 100 | 200 | 3 | 2 | 0.6500 | 0.5180 | 8.450 | 0.160 |
12(MGLB) | 100 | 300 | 1 | 3 | 0.6500 | 0.5135 | 8.489 | 0.178 |
13(MGLB) | 100 | 300 | 2 | 3 | 0.6500 | 0.5270 | 8.487 | 0.239 |
14(MGLB) | 100 | 300 | 3 | 3 | 0.6500 | 0.5405 | 8.486 | 0.296 |
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Nabavizadeh, S.A.; Barua, H.; Eshraghi, M.; Felicelli, S.D. A Multiple-Grid Lattice Boltzmann Method for Natural Convection under Low and High Prandtl Numbers. Fluids 2021, 6, 148. https://doi.org/10.3390/fluids6040148
Nabavizadeh SA, Barua H, Eshraghi M, Felicelli SD. A Multiple-Grid Lattice Boltzmann Method for Natural Convection under Low and High Prandtl Numbers. Fluids. 2021; 6(4):148. https://doi.org/10.3390/fluids6040148
Chicago/Turabian StyleNabavizadeh, Seyed Amin, Himel Barua, Mohsen Eshraghi, and Sergio D. Felicelli. 2021. "A Multiple-Grid Lattice Boltzmann Method for Natural Convection under Low and High Prandtl Numbers" Fluids 6, no. 4: 148. https://doi.org/10.3390/fluids6040148