Investigation Accuracy and Stability of Modified Inter-Particle Interaction Forcing Scheme in Shan and Chen Method for Simulating Stationary Bubbles in Multiphase Flow

Document Type : Research Article


Department of Mechanical Engineering, Shahid Rajaee Teacher Training University, Tehran, I.R. IRAN


The current study investigated the accuracy and stability of an improved inter-particle interaction force term by incorporating force methods and different equations of state. In order to evaluate, eight schemes for incorporating the force term and five equations of state, Shan and Chen (S-C), Carnahan-Starling, Peng-Robinson (P-R), Van der Waals (VdW), and Soave-Redlich-Kwong (SRK), are utilized. Furthermore, affections of relaxation time and reduced temperature on accuracy, stability, and amount of spurious velocities are considered. Among several methods of incorporating the force term, the exact difference method with an improved model and Carnahan-Starling equation of state is shown to have better accuracy and stability and can give more accuracy with spurious velocities which have been greatly reduced. Compared with existing models, the improved model for the inter-particle interaction force term reduced the error of simulation, The minimum error rate accrued for the improved model with the exact difference method of incorporating force term and Carnahan-Starling equation of state and its value is equal to 0.4% while in the identical condition this error rate for equation (9) inter-particle interaction force term is 3.6%. Subsequently, in different relaxation times, a maximum range of stability and accuracy occurred for the exact different method which has better operation in a range of 0.8 to 1 relaxation time. The minimum error of this method with the Carnahan-Starling equation of state in relaxation time 0.8 and reduced temperature 0.54 is 1.8%. By comparing the variation of spurious velocities in the range of 0.57 to 2 relaxation time is demonstrated spurious velocities are increased when temperature decreased.


Main Subjects

[1] Cheng P., Wu H.Y., Mesoscale and Microscale Phase-Change Heat Transfer, Advances in heat Transfer, 39: 461-563 (2006).
[2] Gong S., Cheng P., Lattice Boltzmann Simulations for Surface Wettability Effects in Saturated Pool Boiling Heat Transfer, International Journal of Heat and Mass Transfer, 85: 635-646 (2015).
[3] Sukop M.C., Huang H., Lin C.L., Deo M.D., Oh K., Miller J.D., Distribution of Multiphase Fluids in Porous Media: Comparison between Lattice Boltzmann Modeling and Micro-X-Ray Tomography, Physical Review E., 77(2): 026710 (2008).
[4] Ishii M., Hibiki T., "Thermo-Fluid Dynamics of Two-Phase Flow", Springer Science & Business Media, (2010).
[5] جابرزاده س.، حقیقی اصل ع.، صفدری س.ج.، رشیدی ع.، بررسی تأثیر افزودن الکل و فعال کننده‌های سطحی بر روی هیدرودینامیک راکتور هوا بالابر در شرایط سه فازی، نشریه شیمی و مهندسی شیمی ایران، (4)33: 69 تا 77 (1393).
[۶] رحیمی م.ر.، مطالعه اختلاط فازها در سینی غربالی با استفاده از دینامیک سیالات محاسباتی، نشریه شیمی و مهندسی شیمی ایران، (1)31: 101 تا 113 (1391).
[7]  Brennen CE, Brennen CE. "Fundamentals of Multiphase Flow", Cambridge University Press (2005).
[8] Izadpanah A.A., Vafaei S.M., Varaminian F., Multi-Component-Multiphase Flash Calculations for Systems Containing Gas Hydrates by Direct Minimization of Gibbs Free Energy, Iranian Journal of Chemistry and Chemical Engineering (IJCCE), 25(3): 27-34 (2006).
[9] Bahramian A.R., Kalbasi M., CFD Modeling of TiO2 Nano-Agglomerates Hydrodynamics in a Conical Fluidized Bed Unit with Experimental Validation, Iranian Journal of Chemistry and Chemical Engineering (IJCCE), 29(2): 105-120 (2010).
[10] Gorji M., BOZORG M.B., KAZEMINI M., CFD Modeling of Gas-Liquid Hydrodynamics in a Stirred Tank Reactor, Iranian Journal of Chemistry and Chemical Engineering (IJCCE) 26(2): 85-96 (2007).
[11]  باباپور ع.، پیشکارآذری ر.، گلستانه س.ا.، قاضی طباطبائی ز.، شبیه‌سازی مدیریت گرمایی مواد نانوکامپوزیت تغییرفازدهنده توسط فناوری CFD، نشریه شیمی و مهندسی شیمی ایران، (4)37: 195 تا 210 (1397).
[۱۲]  بابایی گورچین لو ف.، علی حسینی ا.، مدل سازی فرایند تبخیر سطحی آب در سد های آبی با استفاده از CFD (مطالعه موردی سد امیرکبیر کرج)، نشریه شیمی و مهندسی شیمی ایران، (4)35: 125 تا 136 (1395).
[13] Huang H., Sukop M., Lu X., "Multiphase Lattice Boltzmann Methods: Theory and Application", John Wiley & Sons, (2015).
[15] Gunstensen A.K., Rothman D.H., Zaleski S., Zanetti G., Lattice Boltzmann Model of Immiscible Fluids, Physical Review A., 43(8): 4320 (1991).
[16] Shan X., Chen H., Lattice Boltzmann Model for Simulating Flows with Multiple Phases and Components, Physical Review E., 47(3): 1815 (1993).
[17]  Shan X., Chen H., Simulation of Nonideal Gases and Liquid-Gas Phase Transitions by the Lattice Boltzmann Equation, Physical Review E., 49(4): 2941 (1994).
[18] Swift M.R., Osborn W.R., Yeomans J.M., Lattice Boltzmann Simulation of Nonideal Fluids, Physical Review Letters., 75(5): 830 (1995).
[19] Swift M.R., Orlandini E., Osborn W.R., Yeomans J.M., Lattice Boltzmann Simulations of Liquid-Gas and Binary Fluid Systems, Physical Review E., 54(5): 5041 (1996).
[20] He X., Chen S., Zhang R., A Lattice Boltzmann Scheme for Incompressible Multiphase Flow and its Application in Simulation of Rayleigh–Taylor Instability, Journal of Computational Physics., 152(2): 642-663 (1999).
[21] Kupershtokh A.L., Medvedev D.A., Karpov D.I., On Equations of State in a Lattice Boltzmann Method, Computers & Mathematics with Applications., 58(5): 965-974 (2009).
[22] Hyväluoma J., Harting J., Slip Flow over Structured Surfaces with Entrapped Microbubbles, Physical Review Letters., 100(24): 246001 (2008).
[23] Sbragaglia M., Benzi R., Biferale L., Chen H., Shan X., Succi S., Lattice Boltzmann Method with Self-Consistent Thermo-Hydrodynamic Equilibria, Journal of Fluid Mechanics., 628: 299-309 (2009).
[25] Bao J., Schaefer L., Lattice Boltzmann Equation Model for Multi-Component Multi-Phase Flow with High Density Ratios, Applied Mathematical Modelling., 37(4): 1860-1871 (2013).
[26] Yu Z., Fan L.S., An Interaction Potential based Lattice Boltzmann Method with Adaptive Mesh Refinement (AMR) for Two-Phase Flow Simulation, Journal of Computational Physics., 228(17): 6456-6578 (2009).
[27] Pan C., Hilpert M., Miller C.T., Lattice‐Boltzmann Simulation of Two‐Phase Flow in Porous Media, Water Resources Research., 40(1): (2004).
[28] Benzi R., Biferale L., Sbragaglia M., Succi S., Toschi F., Mesoscopic Modeling of a Two-Phase Flow in the Presence of Boundaries: the Contact Angle, Physical Review E., 74(2): 021509 (2006).
[29] Succi S., "The Lattice Boltzmann Equation: for Fluid Dynamics and Beyond", Oxford University Press, (2001).  
[30] He X., Doolen G.D., Thermodynamic Foundations of Kinetic Theory and Lattice Boltzmann Models for Multiphase Flows, Journal of Statistical Physics., 107(1-2): 309-328 (2002). 
[31] Chen L., Kang Q., Mu Y., He Y.L., Tao W.Q., A Critical Review of the Pseudopotential Multiphase Lattice Boltzmann Model: Methods and Applications, International Journal of Heat and Mass Transfer., 76: 210-236 (2014).
[32] Chen S., Doolen G.D., Lattice Boltzmann Method for Fluid Flows, Annual Review of Fluid Mechanics, 30(1): 329-364 (1998).
[33] Li Q., Luo K.H., Kang Q.J., He Y.L., Chen Q., Liu Q., Lattice Boltzmann Methods for Multiphase Flow and Phase-Change Heat Transfer, Progress in Energy and Combustion Science., 52: 62-105 (2016).
[34] Guo Z., Zheng C., Shi B., Discrete Lattice Effects on the Forcing Term in the Lattice Boltzmann Method, Physical Review E., 65(4): 046308 (2002).
[35] Sun K., Wang T., Jia M., Xiao G., Evaluation of Force Implementation in Pseudopotential-based Multiphase Lattice Boltzmann Models, Physica A: Statistical Mechanics and Its Applications., 391(15): 3895-3907 (2012).
[36] Yuan P., Schaefer L., Equations of State in a Lattice Boltzmann Model, Physics of Fluids, 18(4): 042101 (2006).
[37] شیربانی م.، ورمزیار م.، محمدی آ.، آنالیز دقت و پایداری مدل‌های گوناگون تقابل ذره‌ها در روش شبکه بولتزمن چند فازی، نشریه شیمی و مهندسی شیمی ایران، (3)38: 253 تا 262 (1398).
[38] Qian Y.H., d'Humières D., Lallemand P., Lattice BGK Models for Navier-Stokes Equation, EPL (Europhysics Letters)., 17(6): 479 (1992).
[39] Gong S., Cheng P., Quan X., Lattice Boltzmann Simulation of Droplet Formation in Microchannels under an Electric Field, International Journal of Heat and Mass Transfer., 53(25-26): 5863-5870 (2010).
[40] Kang Q., Zhang D., Chen S., Displacement of a Two-Dimensional Immiscible Droplet in a Channel, Physics of Fluids., 14(9): 3203-3214 (2002).
[41] Kang Q., Zhang D., Chen S., Displacement of a Three-Dimensional Immiscible Droplet in a Duct, Journal of Fluid Mechanics., 545: 41-66 (2005).
[42] Sukop M.C., Or D., Lattice Boltzmann Method for Modeling Liquid‐Vapor Interface Configurations in Porous Media, Water Resources Research., 40(1):  (2004).
[43] Zhang R., Chen H., Lattice Boltzmann Method for Simulations of Liquid-Vapor Thermal Flows, Physical Review E., 67(6): 066711 (2003).
[44] Zeng J., Li L., Liao Q., Cui W., Chen Q., Pan L., Simulation of Phase Transition Process using Lattice Boltzmann Method, Chinese Science Bulletin, 54(24): 4596-4603 (2009).
[45] Kupershtokh A.L., Medvedev D.A., Lattice Boltzmann Equation Method in Electrohydrodynamic Problems, Journal of Electrostatics., 64(7-9): 581-585 (2006).
[46] Martys N.S., Shan X., Chen H., Evaluation of the External Force Term in the Discrete Boltzmann Equation, Physical Review E., 58(5): 6855 (1998).
[47] He X., Shan X., Doolen GD., Discrete Boltzmann Equation Model for Nonideal Gases, Physical Review E., 57(1): (1998).
[48] He X., Zou Q, Luo L.S., Dembo M., Analytic Solutions of Simple Flows and Analysis of Nonslip Boundary Conditions for the Lattice Boltzmann BGK Model, Journal of Statistical Physics., 87(1-2): 115-136 (1997).
[49] Guo Z., Zheng C., Shi B., Discrete Lattice Effects on the Forcing Term in the Lattice Boltzmann Method, Physical Review E., 65(4): 046308 (2002).
[50] Shan X., Doolen G., Multicomponent Lattice-Boltzmann Model with Interparticle Interaction, Journal of Statistical Physics., 81(1-2): 379-393 (1995).
[51] Luo L.S., Unified Theory of Lattice Boltzmann Models for Nonideal Gases, Physical Review Letters., 81(8): 1618 (1998).
[52] Ladd A.J., Verberg R., Lattice-Boltzmann Simulations of Particle-Fluid Suspensions, Journal of Statistical Physics., 104(5-6): 1191-1251 (2001).