Two Dimensional Simulation of Fluid Flow and Heat Transfer in the Transition Flow Regime using a Lattice Boltzmann Approach
The significant effects of the interactions between the
system boundaries and the near wall molecules in miniaturized
gaseous devices lead to the formation of the Knudsen layer in which
the Navier-Stokes-Fourier (NSF) equations fail to predict the correct
associated phenomena. In this paper, the well-known lattice
Boltzmann method (LBM) is employed to simulate the fluid flow and
heat transfer processes in rarefied gaseous micro media. Persuaded
by the problematic deficiency of the LBM in capturing the Knudsen
layer phenomena, present study tends to concentrate on the effective
molecular mean free path concept the main essence of which is to
compensate the incapability of this mesoscopic method in dealing
with the momentum and energy transport within the above mentioned
kinetic boundary layer. The results show qualitative and quantitative
accuracy comparable to the solutions of the linearized Boltzmann
equation or the DSMC data for the Knudsen numbers of O (1) .
[1] C. M. Ho and Y. C. Tai, "Micro-electro-mechanical systems (MEMS)
and fluid flows," in Ann. Rev. Fluid Mech. 30, p. 579, 1998.
[2] S. Roy, S. Chakraborty, "Near-wall effects in micro scale Couette flow
and heat transfer in the Maxwell-slip regimes," in Microfluid Nanofluid,
3, 437-449, 2007.
[3] J. Chun and D. L. Koch, "A direct simulation Monte Carlo method for
rarefied gas flows in the limit of small Mach number," Phys. Fluids, 17,
107107, 2005.
[4] G. E. Karniadakis, A. Beskok, and N. Aluru, Micro flows and nano
flows: Fundamentals and simulation, Springer, New York, 2005.
[5] C. Cercignani, Slow rarefied flows: Theory and application to Microelectro-
mechanical systems, Birkhäuser Verlag, Basel, Switzerland,
2006.
[6] N. G. Hadjiconstantinou, A. Garcia, M. Bazant and G. He, "Statistical
error in particle simulations of hydrodynamic phenomena," in J.
Comput. Phys., 187, 274, 2003.
[7] F. Sharipov, L. M. G. Cumin, and D. Kalempa, "Plane Couette flow of
binary gaseous mixture in the whole range of the Knudsen number," in
Eur. J. Mech. B/Fluids, 23, 899, 2004.
[8] S. Chen and G. D. Doolen, "Lattice Boltzmann method for fluid flows,"
in Ann Rev Fluid Mech, 30:329-64, 1998.
[9] X. Nie, G. D. Doolen and S. Chen, "Lattice Boltzmann simulations of
fluid flows in MEMS," in J Stat Phys, 107, (1/2), 279, 2002.
[10] Y. H. Zhang, R. S. Qin, Y. H. Sun, R. W. Barber, and D. R. Emerson,
"Gas flow in microchannels - A lattice Boltzmann method approach," in
J. Stat. Phys. 121, 257, 2005.
[11] W. S. Jiaung and J. R. Ho, "Lattice Boltzmann study on size effect with
geometrical bending on phonon heat conduction in a nanoduct," in J
Appli. Physics, V. 95, N. 3, 2004.
[12] D. Yua, R. Meia, L. S. Luo and W. Shyy, "Viscous flow computations
with the method of lattice Boltzmann equation," in Progress in
Aerospace Sciences, 39, 329-367, 2003.
[13] N. S. Martys and H. D. Chen, "Simulation of multicomponent fluids in
complex three-dimensional geometries by the lattice Boltzmann
method," in Phys. Rev. E, 53, 743, 1996.
[14] P. Zhou and C.W. Wu, "Numerical simulation of electrocapillary driven
flows," in Micro and Nanosystems, 1, 57-62, 2009.
[15] Z. L. Guo and T. S. Zhao, "Discrete velocity and lattice Boltzmann
models for binary mixtures of nonideal fluids," in Phys. Rev. E, 68,
035302(R), 2003.
[16] X. He, S. Chen and G. D. Doolen, "A novel thermal model for the lattice
Boltzmann method in incompressible limit," in J. Comput. Phys., 146,
282, 1998.
[17] Y. Shi, T. S. Zhao and Z. L. Guo, "Thermal lattice Bhatnagar-Gross-
Krook model for flows with viscous heat dissipation in the
incompressible limit," in Phys. Rev. E, 70, 066310, 2004.
[18] G. H. Tang, X. J. Gu, R. W. Barber, and D. R. Emerson, "Lattice
Boltzmann simulation of nonequilibrium effects in oscillatory gas flow,"
in Phys. Rev. E, 78, 026706, 2008.
[19] Ansumali S. and Karlin I. V., "Kinetic boundary conditions in the lattice
Boltzmann method," in Phys. Rev. E, 66, 026311, 2002.
[20] Y. H. Zhang, X. J. Gu, R. W. Barber and D. R. Emerson, "A thermal
lattice Boltzmann model for low speed rarefied gas flow," Daresbury
Laboratory Technical Report, TR- 2006-002, 2006.
[21] X.D. Niu, C. Shu and Y.T. Chew, "A thermal lattice Boltzmann model
with diffuse scattering boundary condition for micro thermal flows," in
Computers & Fluids, 36, 273-281, 2007.
[22] L. Zheng, B. C. Shi and Z. H. Chai, "Lattice Boltzmann method for
simulating the temperature jump and velocity slip in microchannels," in
Commun. Comput. Phys., Vol. 2, No. 6, 1125-1138, 2007.
[23] T. Lee and C. L. Lin, "Rarefaction and compressibility effects of the
lattice-Boltzmann-equation method in a gas microchannel," in Phys. Rev.
E, 71, 046706, 2005.
[24] Y. H. Zhang, R. S. Qin and D. R. Emerson, "Lattice Boltzmann
simulation of rarefied gas flows in microchannels," in Phys. Rev. E, 71,
047702, 2005.
[25] H. P. Kavehpour, M. Faghri and Y. Asako, "Effects of compressibility
and rarefaction on gaseous flows in microchannels," in Numer Heat
Transfer, Part A: Applications, 32, 677 - 696, 1997.
[26] G. H. Tang, Y. H. Zhang and D. R. Emerson, "Lattice Boltzmann
models for nonequilibrium gas flows," in Phys. Rev. E, 77, 046701,
2008.
[27] Y. H. Zhang, X. J. Gu, R. W. Barber and D. R. Emerson, "Capturing
Knudsen layer phenomena using a lattice Boltzmann model," in Phys.
Rev. E, 74, 046704, 2006.
[28] M. Ashrafizaadeh, and M. Shamshiri, "Two dimensional simulation of
fluid flow in the Knudsen layer using a lattice Boltzmann approach," in
Proc. 13 Annual and 2nd international Fluid Dynamics Conference,
University of Shiraz, Shiraz, 2010, submitted for publication.
[29] G. H. Tang, Y. H. Zhang, X. J. Gu, R. W. Barber, and D. R. Emerson,
"Lattice Boltzmann model for thermal transpiration," in Phys. Rev. E,
79, 027701, 2009.
[30] Y. Sone, S. Takata and T. Ohwada, "Numerical analysis of the plane
Couette flow of a rarefied gas on the basis of the linearized Boltzmann
equation for hard-sphere molecules," in Eur. J. Mech. B-Fluids, 9, 273,
1990.
[31] M. A. Gallis, D. J. Rader and J. R. Torczynski, "Calculations of the nearwall
thermophoretic force in rarefied gas flow," in Phys. Fluids, 14,
4290, 2002.
[1] C. M. Ho and Y. C. Tai, "Micro-electro-mechanical systems (MEMS)
and fluid flows," in Ann. Rev. Fluid Mech. 30, p. 579, 1998.
[2] S. Roy, S. Chakraborty, "Near-wall effects in micro scale Couette flow
and heat transfer in the Maxwell-slip regimes," in Microfluid Nanofluid,
3, 437-449, 2007.
[3] J. Chun and D. L. Koch, "A direct simulation Monte Carlo method for
rarefied gas flows in the limit of small Mach number," Phys. Fluids, 17,
107107, 2005.
[4] G. E. Karniadakis, A. Beskok, and N. Aluru, Micro flows and nano
flows: Fundamentals and simulation, Springer, New York, 2005.
[5] C. Cercignani, Slow rarefied flows: Theory and application to Microelectro-
mechanical systems, Birkhäuser Verlag, Basel, Switzerland,
2006.
[6] N. G. Hadjiconstantinou, A. Garcia, M. Bazant and G. He, "Statistical
error in particle simulations of hydrodynamic phenomena," in J.
Comput. Phys., 187, 274, 2003.
[7] F. Sharipov, L. M. G. Cumin, and D. Kalempa, "Plane Couette flow of
binary gaseous mixture in the whole range of the Knudsen number," in
Eur. J. Mech. B/Fluids, 23, 899, 2004.
[8] S. Chen and G. D. Doolen, "Lattice Boltzmann method for fluid flows,"
in Ann Rev Fluid Mech, 30:329-64, 1998.
[9] X. Nie, G. D. Doolen and S. Chen, "Lattice Boltzmann simulations of
fluid flows in MEMS," in J Stat Phys, 107, (1/2), 279, 2002.
[10] Y. H. Zhang, R. S. Qin, Y. H. Sun, R. W. Barber, and D. R. Emerson,
"Gas flow in microchannels - A lattice Boltzmann method approach," in
J. Stat. Phys. 121, 257, 2005.
[11] W. S. Jiaung and J. R. Ho, "Lattice Boltzmann study on size effect with
geometrical bending on phonon heat conduction in a nanoduct," in J
Appli. Physics, V. 95, N. 3, 2004.
[12] D. Yua, R. Meia, L. S. Luo and W. Shyy, "Viscous flow computations
with the method of lattice Boltzmann equation," in Progress in
Aerospace Sciences, 39, 329-367, 2003.
[13] N. S. Martys and H. D. Chen, "Simulation of multicomponent fluids in
complex three-dimensional geometries by the lattice Boltzmann
method," in Phys. Rev. E, 53, 743, 1996.
[14] P. Zhou and C.W. Wu, "Numerical simulation of electrocapillary driven
flows," in Micro and Nanosystems, 1, 57-62, 2009.
[15] Z. L. Guo and T. S. Zhao, "Discrete velocity and lattice Boltzmann
models for binary mixtures of nonideal fluids," in Phys. Rev. E, 68,
035302(R), 2003.
[16] X. He, S. Chen and G. D. Doolen, "A novel thermal model for the lattice
Boltzmann method in incompressible limit," in J. Comput. Phys., 146,
282, 1998.
[17] Y. Shi, T. S. Zhao and Z. L. Guo, "Thermal lattice Bhatnagar-Gross-
Krook model for flows with viscous heat dissipation in the
incompressible limit," in Phys. Rev. E, 70, 066310, 2004.
[18] G. H. Tang, X. J. Gu, R. W. Barber, and D. R. Emerson, "Lattice
Boltzmann simulation of nonequilibrium effects in oscillatory gas flow,"
in Phys. Rev. E, 78, 026706, 2008.
[19] Ansumali S. and Karlin I. V., "Kinetic boundary conditions in the lattice
Boltzmann method," in Phys. Rev. E, 66, 026311, 2002.
[20] Y. H. Zhang, X. J. Gu, R. W. Barber and D. R. Emerson, "A thermal
lattice Boltzmann model for low speed rarefied gas flow," Daresbury
Laboratory Technical Report, TR- 2006-002, 2006.
[21] X.D. Niu, C. Shu and Y.T. Chew, "A thermal lattice Boltzmann model
with diffuse scattering boundary condition for micro thermal flows," in
Computers & Fluids, 36, 273-281, 2007.
[22] L. Zheng, B. C. Shi and Z. H. Chai, "Lattice Boltzmann method for
simulating the temperature jump and velocity slip in microchannels," in
Commun. Comput. Phys., Vol. 2, No. 6, 1125-1138, 2007.
[23] T. Lee and C. L. Lin, "Rarefaction and compressibility effects of the
lattice-Boltzmann-equation method in a gas microchannel," in Phys. Rev.
E, 71, 046706, 2005.
[24] Y. H. Zhang, R. S. Qin and D. R. Emerson, "Lattice Boltzmann
simulation of rarefied gas flows in microchannels," in Phys. Rev. E, 71,
047702, 2005.
[25] H. P. Kavehpour, M. Faghri and Y. Asako, "Effects of compressibility
and rarefaction on gaseous flows in microchannels," in Numer Heat
Transfer, Part A: Applications, 32, 677 - 696, 1997.
[26] G. H. Tang, Y. H. Zhang and D. R. Emerson, "Lattice Boltzmann
models for nonequilibrium gas flows," in Phys. Rev. E, 77, 046701,
2008.
[27] Y. H. Zhang, X. J. Gu, R. W. Barber and D. R. Emerson, "Capturing
Knudsen layer phenomena using a lattice Boltzmann model," in Phys.
Rev. E, 74, 046704, 2006.
[28] M. Ashrafizaadeh, and M. Shamshiri, "Two dimensional simulation of
fluid flow in the Knudsen layer using a lattice Boltzmann approach," in
Proc. 13 Annual and 2nd international Fluid Dynamics Conference,
University of Shiraz, Shiraz, 2010, submitted for publication.
[29] G. H. Tang, Y. H. Zhang, X. J. Gu, R. W. Barber, and D. R. Emerson,
"Lattice Boltzmann model for thermal transpiration," in Phys. Rev. E,
79, 027701, 2009.
[30] Y. Sone, S. Takata and T. Ohwada, "Numerical analysis of the plane
Couette flow of a rarefied gas on the basis of the linearized Boltzmann
equation for hard-sphere molecules," in Eur. J. Mech. B-Fluids, 9, 273,
1990.
[31] M. A. Gallis, D. J. Rader and J. R. Torczynski, "Calculations of the nearwall
thermophoretic force in rarefied gas flow," in Phys. Fluids, 14,
4290, 2002.
@article{"International Journal of Mechanical, Industrial and Aerospace Sciences:62787", author = "Mehdi Shamshiri and Mahmud Ashrafizaadeh", title = "Two Dimensional Simulation of Fluid Flow and Heat Transfer in the Transition Flow Regime using a Lattice Boltzmann Approach", abstract = "The significant effects of the interactions between the
system boundaries and the near wall molecules in miniaturized
gaseous devices lead to the formation of the Knudsen layer in which
the Navier-Stokes-Fourier (NSF) equations fail to predict the correct
associated phenomena. In this paper, the well-known lattice
Boltzmann method (LBM) is employed to simulate the fluid flow and
heat transfer processes in rarefied gaseous micro media. Persuaded
by the problematic deficiency of the LBM in capturing the Knudsen
layer phenomena, present study tends to concentrate on the effective
molecular mean free path concept the main essence of which is to
compensate the incapability of this mesoscopic method in dealing
with the momentum and energy transport within the above mentioned
kinetic boundary layer. The results show qualitative and quantitative
accuracy comparable to the solutions of the linearized Boltzmann
equation or the DSMC data for the Knudsen numbers of O (1) .", keywords = "Fluid flow and Heat transfer, Knudsen layer, Lattice
Boltzmann method (LBM), Micro-scale numerical simulation,
Transition regime.", volume = "5", number = "1", pages = "244-8", }
