## Accuracy of higher-order lattice Boltzmann methods for microscale flows with finite Knudsen numbers.(English)Zbl 1147.82027

Summary: Accuracy of the lattice Boltzmann (LB) method for microscale flows with finite Knudsen numbers is investigated. We employ up to the eleventh-order Gauss-Hermite quadrature for the lattice velocities and diffuse-scattering boundary condition for fluid-wall interactions. Detailed comparisons with the direct simulation Monte Carlo (DSMC) method and the linearized Boltzmann equation are made for planar Couette and Poiseuille flows. All higher-order LB methods considered here give improved results as compared with the standard LB method. The accuracy of the LB hierarchy, however, does not monotonically increase with the order of the Gauss-Hermite quadrature at moderate and large Knudsen numbers. The results also show the sensitivity to a quadrature chosen, even when the Gauss-Hermite quadratures have the same order of formal accuracy. Among the schemes investigated here, D2Q16 is the most efficient method and offers a quantitative prediction in the slip and transition regimes. The higher-order LB methods predict the Knudsen layer up to $$Kn=$$O$$(0.1)$$. The Knudsen layer, however, rapidly disappears when the Knudsen number approaches unity due to a finite number of the lattice velocities, while it is still present for $$Kn=$$O$$(1)$$ in the Boltzmann equation. It is also found that the higher-order LB methods adopted here do not capture the asymptotic behavior of the Boltzmann equation at large Knudsen numbers.

### MSC:

 82C40 Kinetic theory of gases in time-dependent statistical mechanics 76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics 76M28 Particle methods and lattice-gas methods 82C80 Numerical methods of time-dependent statistical mechanics (MSC2010)

### Keywords:

lattice Boltzmann method; kinetic theory; microscale flow
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### References:

 [1] Ho, C.-M.; Tai, Y.-C., Micro-electro-mechanical-systems (MEMS) and fluid flows, Annu. rev. fluid mech., 30, 579-612, (1998) [2] Hadjiconstantinou, N.G., The limits of navier – stokes theory and kinetic extensions for describing small-scale gaseous hydrodynamics, Phys. fluids, 18, 111301, (2006) · Zbl 1146.76400 [3] Knudsen, M., Die gesetze der molecular stromung und dieinneren reibungstromung der gase durch rohren, Annu. phys., 28, 75, (1909) · JFM 40.0825.02 [4] Chapman, S.; Cowling, T.G., The mathematical theory of non-uniform gases, (1970), Cambridge University Press Cambridge · Zbl 0098.39702 [5] Cercignani, C., The Boltzmann equation and its applications, (1988), Springer-Verlag New York · Zbl 0646.76001 [6] Bird, G.A., Molecular gas dynamics and the direct simulation of gas flows, (1994), Oxford University Press Oxford · Zbl 0709.76511 [7] Wagner, W., A convergence proof for bird’s direct simulation Monte Carlo method for the Boltzmann equation, J. stat. phys., 66, 1011-1044, (1992) · Zbl 0899.76312 [8] McNamara, G.R.; Zanetti, G., Use of the Boltzmann equation to simulate lattice-gas automata, Phys. rev. lett., 61, 2332, (1988) [9] Qian, Y.H.; d’Humières, D.; Lallemand, P., Lattice BGK models for navier – stokes equation, Europhys. lett., 17, 479-484, (1992) · Zbl 1116.76419 [10] Chen, S.; Doolen, G.D., Lattice Boltzmann method for fluid flows, Annu. rev. fluid mech., 30, 329-364, (1998) · Zbl 1398.76180 [11] Benzi, R.; Succi, S.; Vergassola, M., Thee lattice Boltzmann-equation – theory and applications, Phys. reports, 222, 145-197, (1992) [12] Succi, S., The lattice Boltzmann equation for fluid dynamics and beyond, (2001), Oxford University Press Oxford · Zbl 0990.76001 [13] He, X.; Luo, L.S., A priori derivation of the lattice Boltzmann equation, Phys. rev. E, 55, R6333, (1997) [14] Shan, X.; He, X., Discretization of the velocity space in the solution of the Boltzmann equation, Phys. rev. lett., 80, 65, (1998) [15] Nie, X.B.; Doolen, G.D.; Chen, S.Y., Micro-electro-mechanical-systems (MEMS) and fluid flows, J. stat. phys., 107, 279-289, (2002) [16] Lim, C.Y.; Shu, C.; Niu, X.D.; Chew, Y.T., Application of lattice Boltzmann method to simulate microchannel flows, Phys. fluids, 14, 2299-2308, (2002) · Zbl 1185.76227 [17] Ansumali, S.; Karlin, I.V., Kinetic boundary condition in the lattice Boltzmann method, Phys. rev. E, 66, 026311, (2002) [18] Toschi, F.; Succi, S., Lattice Boltzmann method at finite Knudsen numbers, Europhys. lett., 69, 549-555, (2005) [19] Ansumali, S.; Karlin, I.V.; Frouzakis, C.E.; Boulouchos, K.B., Entropic lattice Boltzmann method for microflows, Physica A, 359, 289-305, (2006) [20] Lee, T.; Lin, C.L., Rarefaction and compressibility effects of the lattice-Boltzmann-equation method in a gas microchannel, Phys. rev. E, 71, 046706, (2005) [21] Sofonea, V.; Sekerka, R.F., Boundary conditions for the upwind finite difference lattice Boltzmann model: evidence of slip velocity in micro-channel flow, J. comp. phys., 207, 639-659, (2005) · Zbl 1213.76150 [22] Zhang, R.; Shan, X.; Chen, H., Efficient kinetic method for fluid simulation beyond navier – stokes equation, Phys. rev. E, 74, 046703, (2006) [23] Li, B.; Kwok, D.Y., Discrete Boltzmann equation for microfluidics, Phys. rev. lett., 90, 124502, (2003) [24] Luo, L.-S., Comment on discrete Boltzmann equation for microfluidics, Phys. rev. lett., 92, 139401, (2004) [25] Guo, Z.; Zhao, T.S.; Shi, Y., Physical symmetry, spatial accuracy, and relaxation time of the lattice Boltzmann equation for microgas flows, J. appl. phys., 99, 074903, (2006) [26] Shan, X.; Yuan, X.-F.; Chen, H., Kinetic theory representation of hydrodynamics: a way beyond navier – stokes equation, J. fluid mech., 550, 413-441, (2006) · Zbl 1097.76061 [27] Ansumali, S.; Karlin, I.V.; Ottinger, H.C., Minimal entropic kinetic models for hydrodynamics, Europhys. lett., 63, 798-804, (2003) [28] Chikatamarla, S.S.; Ansumali, S.; Karlin, I.V., Entropic lattice Boltzmann models for hydrodynamics in three dimensions, Phys. rev. lett., 97, 010201, (2006) · Zbl 1228.82078 [29] Chikatamarla, S.S.; Karlin, I.V., Entropy and Galilean invariance of lattice Boltzmann theories, Phys. rev. lett., 97, 190601, (2006) · Zbl 1228.82079 [30] Philippi, P.C.; Hegele, L.A.; dos Santos, L.O.E.; Surmas, R., From the continuous to the lattice Boltzmann equation: the discretization problem and thermal models, Phys. rev. E, 73, 056702, (2006) [31] Ansumali, S.; Karlin, I.V.; Arcidiacono, S.; Abbas, A.; Prasianakis, N.I., Hydrodynamics beyond navier – stokes: exact solution to the lattice Boltzmann hierarchy, Phys. rev. lett., 98, 124502, (2007) [32] Stroud, A.H., Approximate calculation of multiple integrals, (1971), Prentice-Hall, Inc. · Zbl 0379.65013 [33] He, X.; Shan, X.; Doolen, G., Discrete Boltzmann equation model for nonideal gases, Phys. rev. E, 57, R13-R16, (1998) [34] Mei, R.; Shyy, W., On the finite difference-based lattice Boltzmann method in curvilinear coordinates, J. comp. phys., 143, 426-448, (1998) · Zbl 0934.76074 [35] van Leer, B., Towards the ultimate conservative difference scheme. v. a second-order sequel to godunov’s method, J. comp. phys., 32, 101-136, (1979) · Zbl 1364.65223 [36] Bardow, A.; Karlin, I.V.; Gusev, A.A., General characteristic-based algorithm for off-lattice Boltzmann simulations, Europhys. lett., 75, 3, 434-440, (2006) [37] Bardow, A.; Karlin, I.V.; Gusev, A.A., Multispeed models in off-lattice Boltzmann simulations, Phys. rev. E, 77, 025701, (2008) [38] Zhang, Y.-H.; Gu, X.J.; Barber, R.W.; Emerson, D.R., Capturing Knudsen layer phenomena using a lattice Boltzmann model, Phys. rev. E, 74, 046704, (2006) [39] Sofonea, V.; Sekerka, R.F., Diffuse-reflection boundary conditions for a thermal lattice Boltzmann model in two dimensions: evidence of temperature jump and slip velocity in microchannels, Phys. rev. E, 71, 066709, (2005) [40] C. Cercignani, Slow Rarefied Flows, Birkh$$\ddot{\operatorname{a}}$$user Basel, 2006. [41] Kim, S.H.; Pitsch, H.; Boyd, I.D., Slip velocity and Knudsen layer in the lattice bolzmann model for microscale flows, Phys. rev. E, 77, 026704, (2008) [42] Cercignani, C., Theory and application of the Boltzmann equation, (1975), Academic Press · Zbl 0403.76065 [43] Sbragaglia, M.; Succi, S., Analytic calculation of slip flow in lattice Boltzmann models with kinetic boundary conditions, Phys. fluids, 17, 093602, (2005) · Zbl 1187.76469 [44] Willis, D.R., Comparison of kineric theory analyses of linearized Couette flow, Phys. fluids, 5, 127-135, (1962) · Zbl 0102.41101 [45] Qian, Y.-H.; Zhou, Y., Complete Galilean-invariant lattice BGK models for the navier – stokes equation, Europhys. lett., 42, 4, 359-364, (2006) [46] Cercignani, C.; Lampis, M.; Lorenzani, S., Variational approach to gas flows in microchannles, Phys. fluids, 16, 3426-3437, (2004)
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