Lattice-Boltzmann simulation of creeping generalized Newtonian flows: theory and guidelines. (English) Zbl 1559.76018

Gsell, Simon; D’Ortona, Umberto; Favier, Julien

J. Comput. Phys. 429, Article ID 109943, 20 p. (2021).

Summary: The accuracy of the lattice-Boltzmann (LB) method is related to the relaxation time controlling the flow viscosity. In particular, it is often recommended to avoid large fluid viscosities in order to satisfy the low-Knudsen-number assumption that is essential to recover hydrodynamic behavior at the macroscopic scale, which may in principle limit the possibility of simulating creeping flows and non-Newtonian flows involving important viscosity variations. Here it is shown, based on the continuous Boltzmann equations, that a two-relaxation-time (TRT) model can however recover the steady Navier-Stokes equations without any restriction on the fluid viscosity, provided that the Knudsen number is redefined as a function of both relaxation times. This effective Knudsen number is closely related to the previously-described parameter controlling numerical errors of the TRT model, providing a consistent theory at both the discrete and continuous levels. To simulate incompressible flows, the viscous incompressibility condition \(Ma^2/Re \ll 1\) also needs to be satisfied, where \(Ma\) and \(Re\) are the Mach and Reynolds numbers. This concept is extended by defining a local incompressibility factor, allowing one to locally control the accuracy of the simulation for flows involving varying viscosities. These theoretical arguments are illustrated based on numerical simulations of the two-dimensional flow past a square cylinder. In the case of a Newtonian flow, the viscosity independence is confirmed for relaxation times up to \(10^4\), and the ratio \(Ma^2/Re=0.1\) is small enough to ensure reliable incompressible simulations. The Herschel-Bulkley model is employed to introduce shear-dependent viscosities in the flow. The proposed numerical strategy allows to achieve major viscosity variations, avoiding the implementation of artificial viscosity cut-off in high-viscosity regions. Highly non-linear flows are simulated over ranges of the Bingham number \(Bn \in [1, 1000]\) and flow index \(n \in [0.2, 1.8]\), and successfully compared to prior numerical works based on Navier-Stokes solvers. This work provides a general framework to simulate complex creeping flows, as encountered in many biological and industrial systems, using the lattice-Boltzmann method.

© Elsevier Inc.

MSC:

76A05	Non-Newtonian fluids
76M28	Particle methods and lattice-gas methods

Keywords:

lattice-Boltzmann; creeping flows; non-Newtonian flows

References:

[1]	Guo, Z.; Shu, C., Lattice Boltzmann Method and Its Applications in Engineering, Advances in Computational Fluid Dynamics, 2013, World Scientific · Zbl 1278.76001
[2]	Krüger, T.; Kusumaatmaja, H.; Kuzmin, A.; Shardt, O.; Silva, G.; Viggen, E. M., The Lattice Boltzmann Method: Principles and Practice, 2017, Springer International Publishing · Zbl 1362.76001
[3]	Frisch, U.; d’Humieres, D.; Hasslacher, B.; Lallemand, P.; Pomeau, Y.; Rivet, J.-P., Lattice gas hydrodynamics in two and three dimensions, Complex Syst., 1, 4, 649-707, 1987 · Zbl 0662.76101
[4]	Higuera, F.; Succi, S.; Benzi, R., Lattice gas dynamics with enhanced collisions, Europhys. Lett., 9, 4, 345, 1989
[5]	Chen, S.; Doolen, G. D., Lattice Boltzmann method for fluid flows, Annu. Rev. Fluid Mech., 30, 1, 329-364, 1998 · Zbl 1398.76180
[6]	Shan, X.; Yuan, X.-F.; Chen, H., Kinetic theory representation of hydrodynamics: a way beyond the Navier-Stokes equation, J. Fluid Mech., 550, 1, 413, 2006 · Zbl 1097.76061
[7]	Chapman, S.; Cowling, T. G., The Mathematical Theory of Non-uniform Gases, 1952, Cambridge University Press: Cambridge University Press Cambridge · Zbl 0049.26102
[8]	Karniadakis, G.; Beskok, A.; Aluru, N., Microflows and Nanoflows: Fundamentals and Simulation, Vol. 29, 2006, Springer Science & Business Media
[9]	Ginzburg, I.; d’Humieres, D., Multireflection boundary conditions for lattice Boltzmann models, Phys. Rev. E, 68, 6, Article 066614 pp., 2003
[10]	Talon, L.; Bauer, D.; Gland, N.; Youssef, S.; Auradou, H.; Ginzburg, I., Assessment of the two relaxation time Lattice-Boltzmann scheme to simulate Stokes flow in porous media, Water Resour. Res., 48, 4, 2012
[11]	Lauga, E., Continuous breakdown of Purcell’s scallop theorem with inertia, Phys. Fluids, 19, 6, Article 061703 pp., 2007 · Zbl 1182.76429
[12]	Ginzburg, I.; Steiner, K., A free-surface lattice Boltzmann method for modelling the filling of expanding cavities by Bingham fluids, Philos. Trans. R. Soc. Lond. A, Math. Phys. Eng. Sci., 360, 1792, 453-466, 2002 · Zbl 1001.76083
[13]	Khirevich, S.; Ginzburg, I.; Tallarek, U., Coarse- and fine-grid numerical behavior of MRT/TRT lattice-Boltzmann schemes in regular and random sphere packings, J. Comput. Phys., 281, 708-742, 2015 · Zbl 1351.76238
[14]	Lallemand, P.; Luo, L.-S., Theory of the lattice Boltzmann method: dispersion, dissipation, isotropy, Galilean invariance, and stability, Phys. Rev. E, 61, 6, 6546-6562, 2000
[15]	Ginzburg, I.; Verhaeghe, F., Two-relaxation-time lattice Boltzmann scheme: about parametrization, velocity, pressure and mixed boundary conditions, Commun. Comput. Phys., 52, 2008
[16]	Gsell, S.; D’Ortona, U.; Favier, J., Explicit and viscosity-independent immersed-boundary scheme for the lattice Boltzmann method, Phys. Rev. E, 100, 3, Article 033306 pp., 2019
[17]	d’Humières, D.; Ginzburg, I., Viscosity independent numerical errors for lattice Boltzmann models: from recurrence equations to “magic” collision numbers, Comput. Math. Appl., 58, 5, 823-840, 2009 · Zbl 1189.76405
[18]	Irgens, F., Rheology and Non-Newtonian Fluids, 2014, Springer International Publishing · Zbl 1270.76001
[19]	Krüger, T.; Varnik, F.; Raabe, D., Shear stress in lattice Boltzmann simulations, Phys. Rev. E, 79, 4, Article 046704 pp., 2009
[20]	Gabbanelli, S.; Drazer, G.; Koplik, J., Lattice Boltzmann method for non-Newtonian (power-law) fluids, Phys. Rev. E, 72, 4, Article 046312 pp., 2005
[21]	Boyd, J.; Buick, J.; Green, S., A second-order accurate lattice Boltzmann non-Newtonian flow model, J. Phys. A, Math. Gen., 39, 46, 14241-14247, 2006 · Zbl 1148.82314
[22]	Chai, Z.; Shi, B.; Guo, Z.; Rong, F., Multiple-relaxation-time lattice Boltzmann model for generalized Newtonian fluid flows, J. Non-Newton. Fluid Mech., 166, 5–6, 332-342, 2011 · Zbl 1281.76041
[23]	Conrad, D.; Schneider, A.; Böhle, M., Accuracy of non-Newtonian lattice Boltzmann simulations, J. Comput. Phys., 301, 218-229, 2015 · Zbl 1349.76670
[24]	Talon, L.; Bauer, D., On the determination of a generalized Darcy equation for yield-stress fluid in porous media using a Lattice-Boltzmann TRT scheme, Eur. Phys. J. E, 36, 12, 139, 2013
[25]	Bhatnagar, P. L.; Gross, E. P.; Krook, M., A model for collision processes in gases, I: small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94, 3, 511-525, 1954 · Zbl 0055.23609
[26]	Ginzburg, I.; d’Humières, D.; Kuzmin, A., Optimal stability of advection-diffusion lattice Boltzmann models with two relaxation times for positive/negative equilibrium, J. Stat. Phys., 139, 6, 1090-1143, 2010 · Zbl 1205.82049
[27]	Kuzmin, A.; Ginzburg, I.; Mohamad, A., The role of the kinetic parameter in the stability of two-relaxation-time advection–diffusion lattice Boltzmann schemes, Comput. Math. Appl., 61, 12, 3417-3442, 2011 · Zbl 1225.76233
[28]	Gsell, S.; D’Ortona, U.; Favier, J., Multigrid dual-time-stepping lattice Boltzmann method, Phys. Rev. E, 101, 2, Article 023309 pp., 2020
[29]	Coreixas, C.; Wissocq, G.; Puigt, G.; Boussuge, J.-F.; Sagaut, P., Recursive regularization step for high-order lattice Boltzmann methods, Phys. Rev. E, 96, 3, Article 033306 pp., 2017
[30]	Giraud, L.; d’Humieres, D.; Lallemand, P., A lattice Boltzmann model for Jeffreys viscoelastic fluid, Europhys. Lett., 42, 6, 625, 1998
[31]	Lashgari, I.; Pralits, J. O.; Giannetti, F.; Brandt, L., First instability of the flow of shear-thinning and shear-thickening fluids past a circular cylinder, J. Fluid Mech., 701, 201-227, 2012 · Zbl 1248.76061
[32]	Mossaz, S.; Jay, P.; Magnin, A., Non-recirculating and recirculating inertial flows of a viscoplastic fluid around a cylinder, J. Non-Newton. Fluid Mech., 177–178, 64-75, 2012
[33]	Farnoush, S.; Manzari, M. T., An investigation on the body force modeling in a lattice Boltzmann BGK simulation of generalized Newtonian fluids, Physica A: Stat. Mech. Appl., 415, 315-332, 2014 · Zbl 1395.76005
[34]	Wu, W.; Huang, X.; Yuan, H.; Xu, F.; Ma, J., A modified lattice Boltzmann method for Herschel-Bulkley fluids, Rheol. Acta, 56, 4, 369-376, 2017
[35]	Papanastasiou, T. C., Flows of materials with yield, J. Rheol., 31, 5, 385-404, 1987 · Zbl 0666.76022
[36]	Mossaz, S.; Jay, P.; Magnin, A., Criteria for the appearance of recirculating and non-stationary regimes behind a cylinder in a viscoplastic fluid, J. Non-Newton. Fluid Mech., 21-22, 2010, pp. 1525–1535 · Zbl 1274.76050
[37]	Zou, Q.; He, X., On pressure and velocity boundary conditions for the lattice Boltzmann BGK model, Phys. Fluids, 9, 6, 1591-1598, 1997 · Zbl 1185.76873
[38]	Sen, S.; Mittal, S.; Biswas, G., Flow past a square cylinder at low Reynolds numbers, Int. J. Numer. Methods Fluids, 67, 9, 1160-1174, 2011 · Zbl 1426.76303
[39]	Pantokratoras, A., Steady flow of a power-law non-Newtonian fluid across an unconfined square cylinder, J. Appl. Mech. Tech. Phys., 57, 2, 264-274, 2016
[40]	Nirmalkar, N.; Chhabra, R.; Poole, R., On creeping flow of a Bingham plastic fluid past a square cylinder, J. Non-Newton. Fluid Mech., 171, 17-30, 2012
[41]	Bharti, R. P.; Chhabra, R.; Eswaran, V., Steady flow of power law fluids across a circular cylinder, Can. J. Chem. Eng., 84, 4, 406-421, 2006