Numerical Method of Weakly Compressible Poiseuille Flow Using Lattice Boltzmann Method

Abstract:

The present work focuses on the numerical simulation of isothermal and weakly compressible Poiseuille flow in a planar channel using the Lattice Boltzmann method with multiple times of relaxation (MRT-LBE) coupled to the Finite Difference method (FDM). The active fluid considered is the air under low Mach number assumption. The flow is two-dimensional, laminar and all the physical properties are constants except the density which varies in the sense of the Boussinesq approximation. The effects of the compressibility, the inclination angle and the Reynolds number on the dynamical and thermal fields are studied numerically. The results are presented in terms of streamlines, isotherms and transverse velocity.

• Periodical: Defect and Diffusion Forum (Volume 384)
• Pages: 99-116
• DOI: https://doi.org/10.4028/www.scientific.net/DDF.384.99
• Online since: May 2018
• Authors: Sanae Ouajdi*, Fayçal Moufekkir, Ahmed Mezrhab, Jean Pierre Fontaine
• Keywords: Compressible, FDM, Isothermal, MRT-LBM, Poiseuille Flow

References

[1] G.C. Georgiou, M.J. Crochet, Time-dependent compressible extrudate-swell problem with slip at the wall, J. Rheol. 38 (1994) 1745–1755.

DOI: 10.1122/1.550524

[2] G.C. Georgiou, M.J. Crochet, Compressible viscous flow in slits with slip at the wall, J. Rheol. 38 (1994) 639–654.

DOI: 10.1122/1.550479

[3] Z.Y. Guo, X.B. Wu, Further study on compressibility effects on the gas flow and heat transfer in a microtube, Microscale Thermophys. Eng. 2 (1998) 111–120.

DOI: 10.1080/108939598200024

[4] D.C. Venerus, Laminar capillary flow of compressible viscous fluids, J. Fluid Mech. 555 (2006) 59–80.

DOI: 10.1017/s0022112006008755

[5] E.B. Arkilic, M.A. Schmidt, K.S. Breuer, Gaseous slip flow in long microchannels, J. MEMS.6 (1995) 167–178.

DOI: 10.1109/84.585795

[6] C. Cai, Q. Sun, I.D. Boyd, Gas flows in microchannels and microtubes, J. Fluid Mech. 589 (2007) 305–314.

DOI: 10.1017/s0022112007008178

[7] E. Taliadorou, M. Neophytou, GC. Georgiou, Perturbation solutions of Poiseuille flows of weakly compressible Newtonian liquids, J Non-Newton Fluid Mech, 158 (2009) 162–169.

DOI: 10.1016/j.jnnfm.2009.06.003

[8] DC. Venerus, DJ. Bugajsky, Laminar flow in a channel. Phys Fluids( 2010) 22:046101.

DOI: 10.1063/1.3371719

[9] K. Housiadas, GC. Georgiou, Perturbation solution of Poiseuille flow of a weakly compressible Oldroyd-B fluid. J Non-Newton Fluid Mech 166 (2011) 73–92.

DOI: 10.1016/j.jnnfm.2010.10.007

[10] KD. Housiadas, GC. Georgiou, IG. Mamoutos, Laminar axisymmetric flow of a weakly compressible viscoelastic fluid, Rheol Acta.

DOI: 10.1007/s00397-011-0610-x

[11] S. Poyiadji, KD. Housiadas, K Kaouri, GC. Georgiou, Asymptotic solutions of weakly compressible Newtonian Poiseuille flows with pressure-dependent viscosity, European Journal of Mechanics B/Fluids 49 (2015) 217–225.

DOI: 10.1016/j.euromechflu.2014.09.002

[12] KD. Housiadas, GC. Georgiou, New analytical solutions for weakly compressible Newtonian Poiseuille flows with pressure-dependent viscosity, International Journal of Engineering Science 107 (2016) 13–27.

DOI: 10.1016/j.ijengsci.2016.07.001

[13] P. Lallemand, L.S. Luo, Hybrid finite difference thermal lattice Boltzmann equation, International Journal of Modern Physics B, Vol. 17, Issue 01-02 (2003) 41-47.

DOI: 10.1142/s0217979203017060

[14] P. Lallemand, L.S. luo, Theory of the lattice Boltzmann method: Acoustic and thermal properties in two and three dimension, Physical Review E, Vol.68 (2003) 036706(25).

DOI: 10.1103/physreve.68.036706

[15] A. Mezrhab, M. Bouzidi, P. Lallemand, Hybrid lattice Boltzmann finite difference simulation of Convective flows, Computer and Fluids, Vol. 33 (2004) 623-641.

DOI: 10.1016/j.compfluid.2003.05.001

[16] P. Lallemand, L.-S. Luo, Theory of the lattice Boltzmann method: dispersion, dissipation, isotropy, Galilean invariance, and stability, Phys. Rev. E 61 (2000) 6546–6562.

DOI: 10.1103/physreve.61.6546

[17] J. Hardy, Y. Pomeau, O. de Pazzis,Time evolution of two dimensional model system. I. Invariant states and time correlation functions, J. Math. Phys, Vol. 14(12), (1973) 1746-1759.

DOI: 10.1063/1.1666248

[18] S. Ulam, Random processes and transformations, Proc. International Congress of Mathematicians, Vol. 2 (1952) 264-275.

[19] M. Bouzidi, M. Firdaouss,P. Lallemand, Momentum transfer of a Boltzmann lattice fluid with boundaries, Physics of Fluids, Vol. 13, No. 11 (2001) 3452-3459.

DOI: 10.1063/1.1399290

[20] I. Ginzbourg , PM. Adler, Boundary flow condition analysis for the three- dimensional Lattice Boltzmann model, Phys. II, 4 (1994) 191–214.

DOI: 10.1051/jp2:1994123