A high-order moment approach for capturing non-equilibrium phenomena in the transition regime

Journal of Fluid Mechanics

Volumn 636, Issue , 2009, Pages 177-216

  Gu, Xiao Jun ^a   Emerson, David R ^a  

^a STFC Daresbury Laboratory ^*  (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

MASS TRANSFER; METHOD OF MOMENTS; MONTE CARLO METHODS; VELOCITY;

CONTINUUM HYDRODYNAMICS; COUETTE AND POISEUILLE FLOWS; DIRECT SIMULATION MONTE CARLO; HIGHER ORDER MOMENTS; NON-EQUILIBRIUM PHENOMENA; REGULARIZED 13 MOMENT EQUATIONS; TEMPERATURE PROFILES; TRANSITION FLOW REGIME;

PIPE FLOW;

CHANNEL; COUETTE FLOW; HYDRODYNAMICS; MONTE CARLO ANALYSIS; SIMULATION; TEMPERATURE PROFILE; VELOCITY PROFILE;

EID: 72549115893     PISSN: 00221120     EISSN: 14697645     Source Type: Journal    
DOI: 10.1017/S002211200900768X     Document Type: Article

References (81)

1. Alves, M. A., Oliveira, P. J. & Pinho, F. T. 2003 A convergent and universally bounded interpolation scheme for the treatment of advection. Intl J. Numer. Meth. Fluids 41, 47-75.
2. Aoki, K., Takata, S. & Nakanishi, T. 2002 Poiseuille-type flow of a rarefied gas between two parallel plates driven by a uniform external force. Phys. Rev. E 65, 026315.
3. Aoki, K., Yoshida, H., Nakanishi, T. & Garcia, A. L. 2003 Inverted velocity profile in the cylindrical Couette flow of a rarefied gas. Phys. Rev. E 68, 016302.
4. Baker, L. L. & Hadjiconstantinou, N. G. 2005 Variance reduction for Monte Carlo solutions of the Boltzmann equation. Phys. Fluids 17, 051703.
5. Bird, G. 1994 Molecular Gasdynamics and the Direct Simulation of Gas Flows. Clarendon.
6. Bobylev, A. V. 1982 The Chapman-Enskog and Grad method for solving the Boltzmann equation. Sov. Phys.-Dokl. 27, 29-31.
7. Cercignani, C. 1988 The Boltzmann Equation and Its Applications. Springer.
8. Cercignani, C. 2000 Rarefied Gasdynamics: From Basic Concepts to Actual Calculations. Cambridge University Press.
9. Chapman, S. & Cowling, T. G. 1970 The Mathematical Theory of Non-uniform Gases. Cambridge University Press.
10. Chen, S. & Doolen, G. D. 1998 Lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech. 30, 329-364.
11. Chikatamarla, S. S., Ansumali, S. & Karlin, I. V. 2006 Grad's approximation for missing data in lattice Boltzmann simulations. Europhys. Lett. 74, 215-221.
12. Debnath, L. 1997 Nonlinear Partial Differential Equations for Scientists and Engineers. Birkhauser.
13. Ferziger, J. H. & Peric, M. 1999 Computational Methods for Fluid Dynamics, 2nd edn. Springer.
14. Gad-El-Hak, M. 1999 The fluid mechanics of microdevices-the Freeman scholar lecture. J. Fluids Engng 121, 5-33.
15. Garćia-Coĺin, L. S., Velaso, R. M. & Uribe, F. J. 2008 Beyond the Navier-Stokes equations: Burnett hydrodynamics. Phys. Rep. 465, 149-189.
16. Gaskell, P. H. & Lau, A. K. C. 1988 Curvature-compensated convective transport: SMART, a new boundedness-preserving transport algorithm. Intl J. Numer. Meth. Fluids 8, 617-641.
17. Grad, H. 1949a Note on N-dimensional Hermite polynomials. Comm. Pure Appl. Math. 2, 325-330.
18. Grad, H. 1949b On the kinetic theory of rarefied gases. Comm. Pure Appl. Math. 2, 331-407.
19. Grad, H. 1952 The profile of steady plane shock wave. Comm. Pure Appl. Math. 5, 257-300.
20. Grad, H. 1963 Asymptotic theory of the Boltzmann equation. Phys. Fluids 6, 147-181.
21. Gorban, A. N., Karlin, I. V. & Zinovyev, A. Y. 2004 Constructive methods of invariant manifolds for kinetic problems. Phys. Rep. 396, 197-403.
22. Gu, X. J., Barber, R. W. & Emerson, D. R. 2007 How far can 13 moments go in modelling microscale gas phenomena? Nanoscale Microscale Thermophys. Engng 11, 85-97.
23. Gu, X. J. & Emerson, D. R. 2007 A computational strategy for the regularized 13 moment equations with enhanced wall-boundary conditions. J. Comput. Phys. 225, 263-283.
24. Guénette, R. & Fortin, M. 1995 A new mixed finite element method for computing viscoelastic flows. J. Non-Newton. Fluid Mech. 60, 27-52.
25. Hadjiconstantinou, N. G. 2005 Oscillatory shear-driven gas flows in the transition and free-molecular-flow regimes, Phys. Fluids 17, 100611.
26. Hadjiconstantinou, N. G. 2006 The limits of Navier-Stokes theory and kinetic extensions for describing small-scale gaseous hydrodynamics. Phys. Fluids 18, 111301.
27. Harley, J. C., Huang, Y., Bau, H. B. & Zemel, J. N. 1995 Gas flow in micro-channels. J. Fluid Mech. 284, 257-274.
28. Jin, S., Pareshi, L. & Slemrod, M. 2002 A relaxation scheme for solving the Boltzmann equation based on the Chapman-Enskog expansion. Acta Math. Appl. Sinica 18, 37-62.
29. Jin, S. & Slemrod, M. 2001 Regularization of the Burnett equations via relaxation. J. Stat. Phys. 103, 1009-1033. (Pubitemid 33385634)
30. Karlin, I. V. & Gorban, A. N. 2002 Hydrodynamics from Grad's equations: what can we learn from the exact solutions? Ann. Phys. 11, 783-833.
31. Karlin, I. V., Gorban, A. N., Dukek, G. & Nonnenmacher, T. F. 1998 Dynamic correction to moment approximations. Phys. Rev. E 57, 1668-1672.
32. Kogan, M. N. 1969 Rarefied Gas Dynamics. Plenum.
33. Leonard, B. P. 1979 A stable and accurate convection modelling procedure based on quadratic upstream interpolation. Comput. Mech. Appl. Mech. Engng 19, 59-98.
34. Levermore, C. D. 1996 Moment closure hierarchies for kinetic theories. J. Stat. Phys. 83, 1021-1065.
35. Levermore, C. D., Morokoff, W. J. & Nadiga, B. T. 1998 Moment realizability and the validity of the Navier-Stokes equations for rarefied gasdynamics. Phys. Fluids 10, 3214-3226.
36. Lilly, C. R. & Sader J. E. 2007 Velocity gradient singularity and structure of the velocity profile in the Knudsen layer according to the Boltzmann equation. Phys. Rev. E 76, 026315.
37. Lilly, C. R. & Sader J. E. 2008 Velocity profile in the Knudsen layer according to the Boltzmann equation. Proc. R. Soc. A 464, 2015-2035.
38. Liu, I. S. & Rincon M. A. 2004 A boundary value problem in extended thermodynamics-one-dimensional steady flows with heat conduction. Contin. Mech. Thermodyn. 16, 109-124.
39. Mansour, M. M., Baras, F. & Garcia, A. L. 1997 On the validity of hydrodynamics in plane Poiseuille flow. Physica A 240, 255-267.
40. Marques, W. & Kremer, G. M. 2001 Couette flow from a thirteen field theory with slip and jump boundary conditions. Contin. Mech. Thermodyn. 13, 207-217.
41. Maurer, J., Tabeling, P., Joseph, P. & Willaime, H. 2003 Second-order slip laws in microchannels for helium and nitrogen. Phys. Fluids 15, 2613-2621.
42. Maxwell, J. C. 1879 On stresses in rarified gases arising from inequalities of temperature. Phil. Trans. R. Soc. Lond. 170, 231-256.
43. Mizzi, S., Gu, X. J., Emerson, D. R., Barber, R. W. & Reese, J. M. 2008 Computational framework for the regularised 20-moment equations for non-equilibrium gas flows. Intl J. Numer. Meth. Fluids 56, 1433-1439.
44. Muller, I. & Ruggeri, T. 1993 Extended Thermodynamics. Springer.
45. Ohwada, T., Sone, Y. & Aoki, K. 1989 Numerical analysis of the Poiseuille and thermal transpiration flows between two parallel plates on the basis of the Boltzmann equation for hard-sphere molecules. Phys. Fluids A 1, 2042-2049.
46. Patankar, S. V. 1980 Numerical Heat Transfer and Fluid Flow. McGraw-Hill.
47. Rajagopalan, D., Armstrong, R. C. & Brown, R. A. 1990 Finite element methods for calculation of steady, viscoelastic flow using constitution equations with a Newtonian viscosity. J. Non-Newton. Fluid Mech. 36, 159-192.
48. Reese, J. M., Gallis, M. A. & Lockerby, D. A. 2003 New directions in fluid dynamics: non-equilibrium aerodynamic and microsystem flows, Phil. Trans. R. Soc. Lond. A 361, 2967-2988.
49. Reitebuch, D. & Weiss, W. 1999 Application of high moment theory to the plane Couette flow. Contin. Mech. Thermodyn. 11, 217-225.
50. Renardy, M. 2000 Mathematical Analysis of Viscoelastic Flows. SIAM.
51. Rhie, C. M. & Chow, W. L. 1983 Numerical study of turbulent flow past an airfoil with trailing edge separation. AIAA J. 21, 1525-1532.
52. Shan, X., Yuan, X.-F. & Chen, H. 2006 Kinetic theory representation of hydrodynamics: a way beyond the Navier-Stokes equation. J. Fluid Mech. 550, 413-441.
53. Sharipov, F. 1999 Non-isothermal gas flow through rectangular microchannels. J. Micromech. Microengng 9, 394-401. (Pubitemid 30519338)
54. Shavaliyev, M. S. 1993 Super-Burnett corrections to the stress tensor and the heat flux in a gas of Maxwellian molecules. J. Appl. Math. Mech. 57, 573-576.
55. Shen, C., Fan, J. & Xie, C. 2003 Statistical simulation of rarefied gas flows in micro-channels. J. Comput. Phys. 189, 512-526.
56. Siewert, C. E. 2003 The linearized Boltzmann equation: concise and accurate solutions to basic flow problems. Zeitsch. Angew. Math. Phys. 54, 273-303.
57. Sone, Y., Takata, S. & Ohwada, T. 1990 Numerical-analysis of the plane Couette-flow of a rarefied-gas on the basis of the linearized Boltzmann-equation for hard-sphere molecules. Eur. J. Mech. B 9, 273-288.
58. Struchtrup, H. 2003 Grad's moment equations for microscale flows. In Symposium on Rarefied Gasdynamics 23 (ed. A. D. Kestdever & E. P. Muntz), AIP Conference Proceedings, vol.663, pp. 792-799.
59. Struchtrup, H. 2004 Stable transport equations for rarefied gases at higher orders in the Knudsen number. Phys. Fluids 16, 3921-3934.
60. Struchtrup, H. 2005 Macroscopic Transport Equations for Rarefied Gas Flows. Springer.
61. Struchtrup, H. 2008 Linear kinetic transfer: moment equations, boundary conditions, and Knudsen layer. Physica A 387, 1750-1766.
62. Struchtrup, H. & Torrihon, M. 2003 Regularization of Grad's 13 moment equations: derivation and linear analysis. Phys. Fluids 15, 2668-2680.
63. Struchtrup, H. & Torrihon, M. 2007 H Theorem, regularization and boundary conditions for linearized 13 moment equations. Phys. Rev. Lett. 99, 014502.
64. Struchtrup, H. & Torrihon, M. 2008 Higher order effects in rarefied channel flows. Phys. Rev. E 78, 046301.
65. Succi, S. 2001 The Lattice Boltzmann Equation for Fluid Dynamics and Beyond. Clarendon.
66. Sun, Q. & Boyd, I. D. 2004 Drag on a flat plate in low-Reynolds-number gas flows. AIAA J. 42, 1066-1072.
67. Tij, M., Sabbane, M. & Santos, A. 1998 Nonlinear Poiseuille flow in a gas. Phys. Fluids 10, 1021-1027.
68. Tij, M. & Santos, A. 1994 Perturbation analysis of a stationary nonequilibrium flow generated by an external force. J. Stat. Phys. 76, 1399-1414.
69. Toro, E. F. 1999 Riemann Solvers and Numerical Methods for Fluids Dynamics: A Practical Introduction, 2nd edn. Springer.
70. Torrilhon, M. 2006 Two-dimensional bulk microflow simulations based on regularised Grad's 13 moment equations. Multiscale Model. Simul. 5, 695-728.
71. Torrilhon, M. & Struchtrup, H. 2004 Regularized 13 moment equations: shock structure calculations and comparison to Burnett models. J. Fluid Mech. 513, 171-198.
72. Torrilhon, M. & Struchtrup, H. 2008 Boundary conditions for regularized 13-moment-equations for micro-channel-flows. J. Comput. Phys. 227, 1982-2011.
73. Truesdell, C. & Muncaster, R. G. 1980 Fundamentals of Maxwell's Kinetic Theory of a Simple Monotomic Gas. Academic.
74. Uribe, F. J. & Garcia, A. L. 1999 Burnett description for plane Poiseuille flow. Phys. Rev. E 60, 4063-4078.
75. Uribe, F. J., Velasco, R. M. & Garćia-Coĺin, L. S. 2000 Bobylev's instability, Phys. Rev. E 62, 5835-5838.
76. White, F. M. 1991 Viscous Fluid Flow. McGraw-Hill.
77. Xu, K. 2001 A gas-kinetic BGK scheme for the Navier-Stokes equations and its connection with artificial dissipation and Godunov method. J. Comput. Phys. 171, 289-335.
78. Xu, K. & Li, Z. 2004 Microchannel flow in the slip regime: gas-kinetic BGK-Burnett solutions. J. Fluid Mech. 513, 87-110.
79. Xu, K., Liu, H. & Jiang, J. 2007 Multiple-temperature kinetic model for continuum and near continuum flows. Phys. Fluids 19, 016101.
80. Zhang, Y.-H., Gu, X. J., Barber, R. W. & Emerson, D. R. 2007 Modelling thermal flow in the transition regime using a lattice Boltzmann approach. Europhys. Lett. 77, 30003.
81. Zheng, Y., Garcia, A. L. & Alder, B. J. 2002 Comparison of kinetic theory and hydrodynamics for Poiseuille flow. J. Stat. Phys. 109, 495-505.