Lattice Boltzmann Simulations in the Slip and Transition Flow Regime with the Peano Framework

ABSTRACT

We present simulation results of flows in the finite Knudsen range, that is in the slip and transition flow regime. Our implementations are based on the Lattice Boltzmann method and are accomplished within the Peano framework. We validate our code by solving two- and three-dimensional channel flow problems and compare our results with respective experiments from other research groups. We further apply our Lattice Boltzmann solver to the geometrical setup of a microreactor consisting of differently sized channels and a reactor chamber. Here, we apply static adaptive grids to further reduce computational costs. We further investigate the influence of using a simple BGK collision kernel in coarse grid regions which are further away from the slip boundaries. Our results are in good agreement with theory and non-adaptive simulations, demonstrating the validity and the capabilities of our adaptive simulation software for flow problems at finite Knudsen numbers.

We present simulation results of flows in the finite Knudsen range, that is in the slip and transition flow regime. Our implementations are based on the Lattice Boltzmann method and are accomplished within the Peano framework. We validate our code by solving two- and three-dimensional channel flow problems and compare our results with respective experiments from other research groups. We further apply our Lattice Boltzmann solver to the geometrical setup of a microreactor consisting of differently sized channels and a reactor chamber. Here, we apply static adaptive grids to further reduce computational costs. We further investigate the influence of using a simple BGK collision kernel in coarse grid regions which are further away from the slip boundaries. Our results are in good agreement with theory and non-adaptive simulations, demonstrating the validity and the capabilities of our adaptive simulation software for flow problems at finite Knudsen numbers.

Cite this paper

P. Neumann and T. Rohrmann, "Lattice Boltzmann Simulations in the Slip and Transition Flow Regime with the Peano Framework,"*Open Journal of Fluid Dynamics*, Vol. 2 No. 3, 2012, pp. 101-110. doi: 10.4236/ojfd.2012.23010.

P. Neumann and T. Rohrmann, "Lattice Boltzmann Simulations in the Slip and Transition Flow Regime with the Peano Framework,"

References

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[2] T. Q. Li, Y. He, G. Tang and W. Tao, “Lattice Boltzmann Modeling of Microchannel Flows in the Transition Flow Regime,” Microfluidics and Nanofluidics, Vol. 10, No. 3, 2011, pp. 607-618. doi:10.1007/s10404-010-0693-1

[3] G. H. Tang, W. Q. Tao and Y. L. He, “Lattice Boltzmann Method for Gaseous Microflows Using Kinetic Theory Boundary Conditions,” Physics of Fluids, Vol. 17, No. 5, 2005, Article ID: 058101. doi:10.1063/1.1897010

[4] F. Toschi and S. Succi, “Lattice Boltzmann Method at Finite Knudsen Numbers,” Europhysics Letters, Vol. 69, No. 4, 2005, pp. 549-555. doi:10.1209/epl/i2004-10393-0

[5] F. J. Uribe and A. L. Garcia, “Burnett Description for Plane Poiseuille Flow,” Physical Review E, Vol. 60, No. 4, 1999, pp. 4063-4078. doi:10.1103/PhysRevE.60.4063

[6] F. Verhaeghe, L.-S. Luo and B. Blanpain, “Lattice Boltzmann Modeling of Microchannel Flow in the Slip Flow Regime,” Journal of Computational Physics, Vol. 228, No. 1, 2009, pp. 147-157. doi:10.1016/j.jcp.2008.09.004

[7] K. Xu, “Super-Burnett Solutions for Poiseuille Flow,” Physics of Fluids, Vol. 15, No. 7, 2003, pp. 2077-2080. doi:10.1063/1.1577564

[8] Y.-H. Zhang, X.-J. Gu, R. W. Barber and D. R. Emerson, “Capturing Knudsen Layer Phenomena Using a Lattice Boltzmann Model”, Physical Review E, Vol. 74, 2006, Article ID: 046704.

[9] X. Shan, “Lattice Boltzmann in Microand Nano-Flow Simulations,” IMA Journal of Applied Mathematics, Vol. 76, No. 5, 2011, pp. 650-660. doi:10.1093/imamat/hxr009

[10] C. E. Colosqui, D. M. Karabacak, K. L. Ekinci and V. Yakhot, “Lattice Boltzmann Simulation of Electromechanical Resonators in Gaseous Media,” The Journal of Fluid Mechanics, Vol. 652, 2010, pp. 241-257. doi:10.1017/S0022112010000042

[11] P. Neumann and T. Neckel, “A Dynamic Mesh Refinement Technique for Lattice Boltzmann Simulations on Octree-Like Grids,” Computational Mechanics, 2012. doi:10.1007/s00466-012-0721-y

[12] G. Karniadakis, A. Beskok and N. Aluru, “Microflows and Nanoflows: Fundamentals and Simulation,” Springer, New York, 2005.

[13] W.-M. Zhang, G. Meng and X. Wei, “A Review on Slip Models for Gas Microflows,” Microfluidics and Nanofluidics, 2012. doi:10.1007/s10404-012-1012-9

[14] S. Succi, “The Lattice Boltzmann Equation for Fluid Dynamics and Beyond,” Oxford University Press, Oxford, 2001.

[15] P. L. Bhatnagar, E. P. Gross and M. Krook, “A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems,” Physical Review, Vol. 94, 1954, pp. 511-525. doi:10.1103/PhysRev.94.511

[16] D. D’Humières, “Generalized lattice Boltzmann Equations, Rarefied Gas Dynamics: Theory and Simulations,” Progress in Astronautics and Aeronautics, Vol. 159, 1992, pp. 450-458.

[17] D. D’Humières, I. Ginzburg, M. Krafczyk, P. Lallemand and L.-S. Luo, “Multiple-Relaxation-Time Lattice Boltzmann Models in Three Dimensions,” Philosophical Transactions of the Royal Society A, Vol. 360, No. 1792, 2002, pp. 437-451. doi:10.1098/rsta.2001.0955

[18] P. Lallemand and L.-S. Luo, “Theory of the Lattice Boltzmann Method: Dispersion, Dissipation, Isotropy, Galilean Invariance, and Stability,” Physical Review E, Vol. 61, No. 6, 2000, pp. 6546-6562. doi:10.1103/PhysRevE.61.6546

[19] X. D. Niu, C. Shu and Y. T. Chew, “A Lattice Boltzmann BGK Model for Simulation of Micro Flows,” Europhysics Letters, Vol. 67, No. 4, 2004, pp. 600-606. doi:10.1209/epl/i2003-10307-8

[20] M. Sbragaglia and S. Succi, “Analytical Calculation of Slip Flow in Lattice Boltzmann Models with Kinetic Boundary Conditions,” Physics of Fluids, Vol. 17, No. 9, 2005, Article ID: 093602. doi:10.1063/1.2044829

[21] Y. Zhang, R. Qin and D. R. Emerson, “Lattice Boltzmann Simulation of Rarefied Gas Flows in Microchannels,” Physical Review E, Vol. 71, 2005, Article ID: 047702.

[22] L. D. Izarra, J.-L. Rouet and B. Izrar, “High-Order Lattice Boltzmann Models for Gas Flow for a Wide Range of Knudsen Numbers,” Physical Review E, Vol. 84, 2011, Article ID: 066705.

[23] J. Meng and Y. Zhang, “Gauss-Hermite Quadratures and Accuracy of Lattice Boltzmann Models for Nonequilibrium Gas Flows,” Physical Review E, Vol. 83, 2011, Article ID: 036704.

[24] G. H. Tang, W. Q. Tao and Y. L. He, “Gas Slippage Effect on Microscale Porous Flow Using the Lattice Boltzmann Method,” Physical Review E, Vol. 72, 2005, Article ID: 056301.

[25] G. H. Tang, W. Q. Tao and Y. L. He, “Three-Dimensional Lattice Boltzmann Model for Gaseous Flow in Rectangular Microducts and Microscale Porous Media,” Journal of Applied Physics, Vol. 97, 2005, Article ID: 104918.

[26] A. Beskok and G.E. Karniadakis, “A Model for Flows in Channels, Pipes, and Ducts at Micro and Nano Scales,” Microscale Thermophysical Engineering, Vol. 3, 1999, pp. 43-77.

[27] D. W. Stops, “The Mean Free Path of Gas Molecules in the Transition Regime,” Journal of Physics D: Applied Physics, Vol. 3, No. 5, 1970, pp. 685-696. doi:10.1088/0022-3727/3/5/307

[28] G. H. Tang, Y. H. Zhang, X. J. Gu and D. R. Emerson, “Lattice Boltzmann Modelling Knudsen Layer Effect in Non-Equilibrium Flows,” Europhysics Letters, Vol. 83, 2008.

[29] M. Nourmohammadzadeh, M. Rahnama, S. Jafari and A. R. Akhgar, “Microchannel Flow Simulation in Transition Regime Using Lattice Boltzmann Method,” Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, Vol. 226, No. 2, 2012, pp. 552-562. doi:10.1177/0954406211413959

[30] V. Michalis, A. Kalarakis, E. Skouras and V. Burganos, “Rarefaction Effects on Gas Viscosity in the Knudsen Transition Regime,” Microfluidics and Nanofluidics, Vol. 9, No. 4-5, 2010, pp. 847-853. doi:10.1007/s10404-010-0606-3

[31] A. Beskok, G. E. Karniadakis and W. Trimmer, “Rarefaction and Compressibility Effects in Gas Microflows,” Transactions of the ASME, Vol. 118, 1996, pp. 448-456.

[32] M. Mehl, T. Neckel and P. Neumann, “Navier-Stokes and Lattice-Boltzmann on Octree-Like Grids in the Peano Framework,” International Journal for Numerical Methods in Fluids, Vol. 65, No. 1, 2010, pp. 67-86. doi:10.1002/fld.2469

[33] H.-J. Bungartz, M. Mehl, T. Neckel and T. Weinzierl, “The PDE Framework Peano Applied to Fluid Dynamics: An Efficient Implementation of a Parallel Multiscale Fluid Dynamics Solver on Octree-Like Adaptive Cartesian Grids,” Computational Mechanics, Vol. 46, No. 1, 2010, pp. 103-114. doi:10.1007/s00466-009-0436-x

[34] C. Colin and S. Aubert, “High-Order Boundary Conditions for Gaseous Flows in Rectangular Microducts,” Microscale Thermophysical Engineering, Vol. 5, No. 1, 2001, pp. 41-54. doi:10.1080/108939501300005367

[35] C. Shen, D. B. Tian, C. Xie and J. Fan, “Examination of the LBM in Simulation of Microchannel Flow in Transitional Regime,” Microscale Thermophysical Engineering, Vol. 8, No. 4, 2004, pp. 423-432. doi:10.1080/10893950490516983

[36] L. Lei, N. Wang, X. M. Zhang, Q. Tai, D. P. Tsai and H. L. W. Chan, “Optofluidic Planar Reactors for Photocatalytic Water Treatment Using Solar Energy,” Biomicrofluidics, Vol. 4, No. 4, 2010.

[37] H. Chen, “Volumetric Formulation of the Lattice Boltzmann Method for Fluid Dynamics: Basic Concept,” Physical Review E, Vol. 58, No. 3, 1998, pp. 3955-3963. doi:10.1103/PhysRevE.58.3955

[38] M. Rohde, D. Kandhai, J. J. Derksen and H. E. A. van den Akker, “A Generic, Mass Conservative Local Grid Refinement Technique for Lattice-Boltzmann Schemes,” International Journal for Numerical Methods in Fluids, Vol. 51, No. 4, 2006, pp. 439-468. doi:10.1002/fld.1140

[39] N. G. Hadjiconstantinou, “Comment on Cercignani’s Second-Order Slip Coefficient,” Physics of Fluids, Vol. 15, No. 6, 2003, pp. 2352-2354. doi:10.1063/1.1587155

[40] T. Ohwada, Y. Sone and K. Aoki, “Numerical Analysis of the Shear and Thermal Creep Flows of a Rarefied Gas over a Plane Wall on the Basis of the Linearized Boltzmann Equation for Hard-Sphere Molecules,” Physics of Fluids, Vol. 1, No. 9, 1989, pp. 1588-1599. doi:10.1063/1.857304

[41] J. Meng, Y. Zhang and X. Shan, “Multiscale Lattice Boltzmann Approach to Modeling Gas Flows,” Physical Review E, Vol. 83, 2011, Article ID: 046701.

[42] A. Dupuis, E. M. Kotsalis and P. Koumoutsakos, “Coupling Lattice Boltzmann and Molecular Dynamics Models for Dense Fluids,” Physical Review E, Vol. 75, 2007, Article ID: 046704.

[43] P. Neumann and N. Tchipev, “A Coupling Tool for Parallel Molecular Dynamics-Continuum Simulations,” Proceedings of the International Symposium on Parallel and Distributed Computing, 2012 Accepted.

[1] Y. Zheng, A. L. Garcia and B. J. Alder, “Comparison of Kinetic Theory and Hydrodynamics for Poiseuille Flow,” Journal of Statistical Physics, Vol. 109, No. 314, 2002, pp. 495-505. doi:10.1023/A:1020498111819

[2] T. Q. Li, Y. He, G. Tang and W. Tao, “Lattice Boltzmann Modeling of Microchannel Flows in the Transition Flow Regime,” Microfluidics and Nanofluidics, Vol. 10, No. 3, 2011, pp. 607-618. doi:10.1007/s10404-010-0693-1

[3] G. H. Tang, W. Q. Tao and Y. L. He, “Lattice Boltzmann Method for Gaseous Microflows Using Kinetic Theory Boundary Conditions,” Physics of Fluids, Vol. 17, No. 5, 2005, Article ID: 058101. doi:10.1063/1.1897010

[4] F. Toschi and S. Succi, “Lattice Boltzmann Method at Finite Knudsen Numbers,” Europhysics Letters, Vol. 69, No. 4, 2005, pp. 549-555. doi:10.1209/epl/i2004-10393-0

[5] F. J. Uribe and A. L. Garcia, “Burnett Description for Plane Poiseuille Flow,” Physical Review E, Vol. 60, No. 4, 1999, pp. 4063-4078. doi:10.1103/PhysRevE.60.4063

[6] F. Verhaeghe, L.-S. Luo and B. Blanpain, “Lattice Boltzmann Modeling of Microchannel Flow in the Slip Flow Regime,” Journal of Computational Physics, Vol. 228, No. 1, 2009, pp. 147-157. doi:10.1016/j.jcp.2008.09.004

[7] K. Xu, “Super-Burnett Solutions for Poiseuille Flow,” Physics of Fluids, Vol. 15, No. 7, 2003, pp. 2077-2080. doi:10.1063/1.1577564

[8] Y.-H. Zhang, X.-J. Gu, R. W. Barber and D. R. Emerson, “Capturing Knudsen Layer Phenomena Using a Lattice Boltzmann Model”, Physical Review E, Vol. 74, 2006, Article ID: 046704.

[9] X. Shan, “Lattice Boltzmann in Microand Nano-Flow Simulations,” IMA Journal of Applied Mathematics, Vol. 76, No. 5, 2011, pp. 650-660. doi:10.1093/imamat/hxr009

[10] C. E. Colosqui, D. M. Karabacak, K. L. Ekinci and V. Yakhot, “Lattice Boltzmann Simulation of Electromechanical Resonators in Gaseous Media,” The Journal of Fluid Mechanics, Vol. 652, 2010, pp. 241-257. doi:10.1017/S0022112010000042

[11] P. Neumann and T. Neckel, “A Dynamic Mesh Refinement Technique for Lattice Boltzmann Simulations on Octree-Like Grids,” Computational Mechanics, 2012. doi:10.1007/s00466-012-0721-y

[12] G. Karniadakis, A. Beskok and N. Aluru, “Microflows and Nanoflows: Fundamentals and Simulation,” Springer, New York, 2005.

[13] W.-M. Zhang, G. Meng and X. Wei, “A Review on Slip Models for Gas Microflows,” Microfluidics and Nanofluidics, 2012. doi:10.1007/s10404-012-1012-9

[14] S. Succi, “The Lattice Boltzmann Equation for Fluid Dynamics and Beyond,” Oxford University Press, Oxford, 2001.

[15] P. L. Bhatnagar, E. P. Gross and M. Krook, “A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems,” Physical Review, Vol. 94, 1954, pp. 511-525. doi:10.1103/PhysRev.94.511

[16] D. D’Humières, “Generalized lattice Boltzmann Equations, Rarefied Gas Dynamics: Theory and Simulations,” Progress in Astronautics and Aeronautics, Vol. 159, 1992, pp. 450-458.

[17] D. D’Humières, I. Ginzburg, M. Krafczyk, P. Lallemand and L.-S. Luo, “Multiple-Relaxation-Time Lattice Boltzmann Models in Three Dimensions,” Philosophical Transactions of the Royal Society A, Vol. 360, No. 1792, 2002, pp. 437-451. doi:10.1098/rsta.2001.0955

[18] P. Lallemand and L.-S. Luo, “Theory of the Lattice Boltzmann Method: Dispersion, Dissipation, Isotropy, Galilean Invariance, and Stability,” Physical Review E, Vol. 61, No. 6, 2000, pp. 6546-6562. doi:10.1103/PhysRevE.61.6546

[19] X. D. Niu, C. Shu and Y. T. Chew, “A Lattice Boltzmann BGK Model for Simulation of Micro Flows,” Europhysics Letters, Vol. 67, No. 4, 2004, pp. 600-606. doi:10.1209/epl/i2003-10307-8

[20] M. Sbragaglia and S. Succi, “Analytical Calculation of Slip Flow in Lattice Boltzmann Models with Kinetic Boundary Conditions,” Physics of Fluids, Vol. 17, No. 9, 2005, Article ID: 093602. doi:10.1063/1.2044829

[21] Y. Zhang, R. Qin and D. R. Emerson, “Lattice Boltzmann Simulation of Rarefied Gas Flows in Microchannels,” Physical Review E, Vol. 71, 2005, Article ID: 047702.

[22] L. D. Izarra, J.-L. Rouet and B. Izrar, “High-Order Lattice Boltzmann Models for Gas Flow for a Wide Range of Knudsen Numbers,” Physical Review E, Vol. 84, 2011, Article ID: 066705.

[23] J. Meng and Y. Zhang, “Gauss-Hermite Quadratures and Accuracy of Lattice Boltzmann Models for Nonequilibrium Gas Flows,” Physical Review E, Vol. 83, 2011, Article ID: 036704.

[24] G. H. Tang, W. Q. Tao and Y. L. He, “Gas Slippage Effect on Microscale Porous Flow Using the Lattice Boltzmann Method,” Physical Review E, Vol. 72, 2005, Article ID: 056301.

[25] G. H. Tang, W. Q. Tao and Y. L. He, “Three-Dimensional Lattice Boltzmann Model for Gaseous Flow in Rectangular Microducts and Microscale Porous Media,” Journal of Applied Physics, Vol. 97, 2005, Article ID: 104918.

[26] A. Beskok and G.E. Karniadakis, “A Model for Flows in Channels, Pipes, and Ducts at Micro and Nano Scales,” Microscale Thermophysical Engineering, Vol. 3, 1999, pp. 43-77.

[27] D. W. Stops, “The Mean Free Path of Gas Molecules in the Transition Regime,” Journal of Physics D: Applied Physics, Vol. 3, No. 5, 1970, pp. 685-696. doi:10.1088/0022-3727/3/5/307

[28] G. H. Tang, Y. H. Zhang, X. J. Gu and D. R. Emerson, “Lattice Boltzmann Modelling Knudsen Layer Effect in Non-Equilibrium Flows,” Europhysics Letters, Vol. 83, 2008.

[29] M. Nourmohammadzadeh, M. Rahnama, S. Jafari and A. R. Akhgar, “Microchannel Flow Simulation in Transition Regime Using Lattice Boltzmann Method,” Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, Vol. 226, No. 2, 2012, pp. 552-562. doi:10.1177/0954406211413959

[30] V. Michalis, A. Kalarakis, E. Skouras and V. Burganos, “Rarefaction Effects on Gas Viscosity in the Knudsen Transition Regime,” Microfluidics and Nanofluidics, Vol. 9, No. 4-5, 2010, pp. 847-853. doi:10.1007/s10404-010-0606-3

[31] A. Beskok, G. E. Karniadakis and W. Trimmer, “Rarefaction and Compressibility Effects in Gas Microflows,” Transactions of the ASME, Vol. 118, 1996, pp. 448-456.

[32] M. Mehl, T. Neckel and P. Neumann, “Navier-Stokes and Lattice-Boltzmann on Octree-Like Grids in the Peano Framework,” International Journal for Numerical Methods in Fluids, Vol. 65, No. 1, 2010, pp. 67-86. doi:10.1002/fld.2469

[33] H.-J. Bungartz, M. Mehl, T. Neckel and T. Weinzierl, “The PDE Framework Peano Applied to Fluid Dynamics: An Efficient Implementation of a Parallel Multiscale Fluid Dynamics Solver on Octree-Like Adaptive Cartesian Grids,” Computational Mechanics, Vol. 46, No. 1, 2010, pp. 103-114. doi:10.1007/s00466-009-0436-x

[34] C. Colin and S. Aubert, “High-Order Boundary Conditions for Gaseous Flows in Rectangular Microducts,” Microscale Thermophysical Engineering, Vol. 5, No. 1, 2001, pp. 41-54. doi:10.1080/108939501300005367

[35] C. Shen, D. B. Tian, C. Xie and J. Fan, “Examination of the LBM in Simulation of Microchannel Flow in Transitional Regime,” Microscale Thermophysical Engineering, Vol. 8, No. 4, 2004, pp. 423-432. doi:10.1080/10893950490516983

[36] L. Lei, N. Wang, X. M. Zhang, Q. Tai, D. P. Tsai and H. L. W. Chan, “Optofluidic Planar Reactors for Photocatalytic Water Treatment Using Solar Energy,” Biomicrofluidics, Vol. 4, No. 4, 2010.

[37] H. Chen, “Volumetric Formulation of the Lattice Boltzmann Method for Fluid Dynamics: Basic Concept,” Physical Review E, Vol. 58, No. 3, 1998, pp. 3955-3963. doi:10.1103/PhysRevE.58.3955

[38] M. Rohde, D. Kandhai, J. J. Derksen and H. E. A. van den Akker, “A Generic, Mass Conservative Local Grid Refinement Technique for Lattice-Boltzmann Schemes,” International Journal for Numerical Methods in Fluids, Vol. 51, No. 4, 2006, pp. 439-468. doi:10.1002/fld.1140

[39] N. G. Hadjiconstantinou, “Comment on Cercignani’s Second-Order Slip Coefficient,” Physics of Fluids, Vol. 15, No. 6, 2003, pp. 2352-2354. doi:10.1063/1.1587155

[40] T. Ohwada, Y. Sone and K. Aoki, “Numerical Analysis of the Shear and Thermal Creep Flows of a Rarefied Gas over a Plane Wall on the Basis of the Linearized Boltzmann Equation for Hard-Sphere Molecules,” Physics of Fluids, Vol. 1, No. 9, 1989, pp. 1588-1599. doi:10.1063/1.857304

[41] J. Meng, Y. Zhang and X. Shan, “Multiscale Lattice Boltzmann Approach to Modeling Gas Flows,” Physical Review E, Vol. 83, 2011, Article ID: 046701.

[42] A. Dupuis, E. M. Kotsalis and P. Koumoutsakos, “Coupling Lattice Boltzmann and Molecular Dynamics Models for Dense Fluids,” Physical Review E, Vol. 75, 2007, Article ID: 046704.

[43] P. Neumann and N. Tchipev, “A Coupling Tool for Parallel Molecular Dynamics-Continuum Simulations,” Proceedings of the International Symposium on Parallel and Distributed Computing, 2012 Accepted.