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

Published online by Cambridge University Press:  25 September 2009

XIAO-JUN GU  and
DAVID R. EMERSON
XIAO-JUN GU*
Affiliation:
Computational Science and Engineering Department, STFC Daresbury Laboratory, Warrington WA4 4AD, UK
DAVID R. EMERSON
Affiliation:
Computational Science and Engineering Department, STFC Daresbury Laboratory, Warrington WA4 4AD, UK
*
†Email address for correspondence: xiaojun.gu@stfc.ac.uk
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Abstract

The method of moments is employed to extend the validity of continuum-hydrodynamic models into the transition-flow regime. An evaluation of the regularized 13 moment equations for two confined flow problems, planar Couette and Poiseuille flows, indicates some important limitations. For planar Couette flow at a Knudsen number of 0.25, they fail to reproduce the Knudsen-layer velocity profile observed using a direct simulation Monte Carlo approach, and the higher-order moments are not captured particularly well. Moreover, for Poiseuille flow, this system of equations creates a large slip velocity leading to significant overprediction of the mass flow rate for Knudsen numbers above 0.4. To overcome some of these difficulties, the theory of regularized moment equations is extended to 26 moment equations. This new set of equations highlights the importance of both gradient and non-gradient transport mechanisms and is shown to overcome many of the limitations observed in the regularized 13 moment equations. In particular, for planar Couette flow, they can successfully capture the observed Knudsen-layer velocity profile well into the transition regime. Moreover, this new set of equations can correctly predict the Knudsen layer, the velocity profile and the mass flow rate of pressure-driven Poiseuille flow for Knudsen numbers up to 1.0 and captures the bimodal temperature profile in force-driven Poiseuille flow. Above this value, the 26 moment equations are not able to accurately capture the velocity profile in the centre of the channel. However, they are able to capture the basic trends and successfully predict a Knudsen minimum at the correct value of the Knudsen number.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 636 , 10 October 2009 , pp. 177 - 216
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

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.Google Scholar
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.CrossRefGoogle ScholarPubMed
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.Google Scholar
Baker, L. L. & Hadjiconstantinou, N. G. 2005 Variance reduction for Monte Carlo solutions of the Boltzmann equation. Phys. Fluids 17, 051703.Google Scholar
Bird, G. 1994 Molecular Gasdynamics and the Direct Simulation of Gas Flows. Clarendon.CrossRefGoogle Scholar
Bobylev, A. V. 1982 The Chapman–Enskog and Grad method for solving the Boltzmann equation. Sov. Phys.-Dokl. 27, 29–31.Google Scholar
Cercignani, C. 1988 The Boltzmann Equation and Its Applications. Springer.Google Scholar
Cercignani, C. 2000 Rarefied Gasdynamics: From Basic Concepts to Actual Calculations. Cambridge University Press.Google Scholar
Chapman, S. & Cowling, T. G. 1970 The Mathematical Theory of Non-uniform Gases. Cambridge University Press.Google Scholar
Chen, S. & Doolen, G. D. 1998 Lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech. 30, 329–364.Google Scholar
Chikatamarla, S. S., Ansumali, S. & Karlin, I. V. 2006 Grad's approximation for missing data in lattice Boltzmann simulations. Europhys. Lett. 74, 215–221.CrossRefGoogle Scholar
Debnath, L. 1997 Nonlinear Partial Differential Equations for Scientists and Engineers. Birkhauser.CrossRefGoogle Scholar
Ferziger, J. H. & Peric, M. 1999 Computational Methods for Fluid Dynamics, 2nd edn. Springer.CrossRefGoogle Scholar
Gad-El-Hak, M. 1999 The fluid mechanics of microdevices – the Freeman scholar lecture. J. Fluids Engng 121, 5–33.CrossRefGoogle Scholar
García-Colín, L. S., Velaso, R. M. & Uribe, F. J. 2008 Beyond the Navier–Stokes equations: Burnett hydrodynamics. Phys. Rep. 465, 149–189.Google Scholar
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.Google Scholar
Grad, H. 1949 a Note on N-dimensional Hermite polynomials. Comm. Pure Appl. Math. 2, 325–330.Google Scholar
Grad, H. 1949 b On the kinetic theory of rarefied gases. Comm. Pure Appl. Math. 2, 331–407.Google Scholar
Grad, H. 1952 The profile of steady plane shock wave. Comm. Pure Appl. Math. 5, 257–300.CrossRefGoogle Scholar
Grad, H. 1963 Asymptotic theory of the Boltzmann equation. Phys. Fluids 6, 147–181.CrossRefGoogle Scholar
Gorban, A. N., Karlin, I. V. & Zinovyev, A. Y. 2004 Constructive methods of invariant manifolds for kinetic problems. Phys. Rep. 396, 197–403.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Guénette, R. & Fortin, M. 1995 A new mixed finite element method for computing viscoelastic flows. J. Non-Newton. Fluid Mech. 60, 27–52.CrossRefGoogle Scholar
Hadjiconstantinou, N. G. 2005 Oscillatory shear-driven gas flows in the transition and free-molecular-flow regimes, Phys. Fluids 17, 100611.CrossRefGoogle Scholar
Hadjiconstantinou, N. G. 2006 The limits of Navier–Stokes theory and kinetic extensions for describing small-scale gaseous hydrodynamics. Phys. Fluids 18, 111301.CrossRefGoogle Scholar
Harley, J. C., Huang, Y., Bau, H. B. & Zemel, J. N. 1995 Gas flow in micro-channels. J. Fluid Mech. 284, 257–274.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Jin, S. & Slemrod, M. 2001 Regularization of the Burnett equations via relaxation. J. Stat. Phys. 103, 1009–1033.Google Scholar
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.CrossRefGoogle Scholar
Karlin, I. V., Gorban, A. N., Dukek, G. & Nonnenmacher, T. F. 1998 Dynamic correction to moment approximations. Phys. Rev. E 57, 1668–1672.Google Scholar
Kogan, M. N. 1969 Rarefied Gas Dynamics. Plenum.CrossRefGoogle Scholar
Leonard, B. P. 1979 A stable and accurate convection modelling procedure based on quadratic upstream interpolation. Comput. Mech. Appl. Mech. Engng 19, 59–98.CrossRefGoogle Scholar
Levermore, C. D. 1996 Moment closure hierarchies for kinetic theories. J. Stat. Phys. 83, 1021–1065.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.Google Scholar
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.CrossRefGoogle Scholar
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.Google Scholar
Mansour, M. M., Baras, F. & Garcia, A. L. 1997 On the validity of hydrodynamics in plane Poiseuille flow. Physica A 240, 255–267.Google Scholar
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.CrossRefGoogle Scholar
Maurer, J., Tabeling, P., Joseph, P. & Willaime, H. 2003 Second-order slip laws in microchannels for helium and nitrogen. Phys. Fluids 15, 2613–2621.Google Scholar
Maxwell, J. C. 1879 On stresses in rarified gases arising from inequalities of temperature. Phil. Trans. R. Soc. Lond. 170, 231–256.Google Scholar
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.Google Scholar
Muller, I. & Ruggeri, T. 1993 Extended Thermodynamics. Springer.CrossRefGoogle Scholar
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.Google Scholar
Patankar, S. V. 1980 Numerical Heat Transfer and Fluid Flow. McGraw-Hill.Google Scholar
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.Google Scholar
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.Google Scholar
Reitebuch, D. & Weiss, W. 1999 Application of high moment theory to the plane Couette flow. Contin. Mech. Thermodyn. 11, 217–225.Google Scholar
Renardy, M. 2000 Mathematical Analysis of Viscoelastic Flows. SIAM.CrossRefGoogle Scholar
Rhie, C. M. & Chow, W. L. 1983 Numerical study of turbulent flow past an airfoil with trailing edge separation. AIAA J. 21, 1525–1532.Google Scholar
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.Google Scholar
Sharipov, F. 1999 Non-isothermal gas flow through rectangular microchannels. J. Micromech. Microengng 9, 394–401.Google Scholar
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.Google Scholar
Shen, C., Fan, J. & Xie, C. 2003 Statistical simulation of rarefied gas flows in micro-channels. J. Comput. Phys. 189, 512–526.Google Scholar
Siewert, C. E. 2003 The linearized Boltzmann equation: concise and accurate solutions to basic flow problems. Zeitsch. Angew. Math. Phys. 54, 273–303.CrossRefGoogle Scholar
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.Google Scholar
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.Google Scholar
Struchtrup, H. 2004 Stable transport equations for rarefied gases at higher orders in the Knudsen number. Phys. Fluids 16, 3921–3934.CrossRefGoogle Scholar
Struchtrup, H. 2005 Macroscopic Transport Equations for Rarefied Gas Flows. Springer.Google Scholar
Struchtrup, H. 2008 Linear kinetic transfer: moment equations, boundary conditions, and Knudsen layer. Physica A 387, 1750–1766.Google Scholar
Struchtrup, H. & Torrihon, M. 2003 Regularization of Grad's 13 moment equations: derivation and linear analysis. Phys. Fluids 15, 2668–2680.Google Scholar
Struchtrup, H. & Torrihon, M. 2007 H Theorem, regularization and boundary conditions for linearized 13 moment equations. Phys. Rev. Lett. 99, 014502.Google Scholar
Struchtrup, H. & Torrihon, M. 2008 Higher order effects in rarefied channel flows. Phys. Rev. E 78, 046301.Google Scholar
Succi, S. 2001 The Lattice Boltzmann Equation for Fluid Dynamics and Beyond. Clarendon.Google Scholar
Sun, Q. & Boyd, I. D. 2004 Drag on a flat plate in low-Reynolds-number gas flows. AIAA J. 42, 1066–1072.Google Scholar
Tij, M., Sabbane, M. & Santos, A. 1998 Nonlinear Poiseuille flow in a gas. Phys. Fluids 10, 1021–1027.Google Scholar
Tij, M. & Santos, A. 1994 Perturbation analysis of a stationary nonequilibrium flow generated by an external force. J. Stat. Phys. 76, 1399–1414.Google Scholar
Toro, E. F. 1999 Riemann Solvers and Numerical Methods for Fluids Dynamics: A Practical Introduction, 2nd edn. Springer.Google Scholar
Torrilhon, M. 2006 Two-dimensional bulk microflow simulations based on regularised Grad's 13 moment equations. Multiscale Model. Simul. 5, 695–728.CrossRefGoogle Scholar
Torrilhon, M. & Struchtrup, H. 2004 Regularized 13 moment equations: shock structure calculations and comparison to Burnett models. J. Fluid Mech. 513, 171–198.CrossRefGoogle Scholar
Torrilhon, M. & Struchtrup, H. 2008 Boundary conditions for regularized 13-moment-equations for micro-channel-flows. J. Comput. Phys. 227, 1982–2011.Google Scholar
Truesdell, C. & Muncaster, R. G. 1980 Fundamentals of Maxwell's Kinetic Theory of a Simple Monotomic Gas. Academic.Google Scholar
Uribe, F. J. & Garcia, A. L. 1999 Burnett description for plane Poiseuille flow. Phys. Rev. E 60, 4063–4078.CrossRefGoogle ScholarPubMed
Uribe, F. J., Velasco, R. M. & García-Colín, L. S. 2000 Bobylev's instability, Phys. Rev. E 62, 5835–5838.Google Scholar
White, F. M. 1991 Viscous Fluid Flow. McGraw-Hill.Google Scholar
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.CrossRefGoogle Scholar
Xu, K. & Li, Z. 2004 Microchannel flow in the slip regime: gas-kinetic BGK-Burnett solutions. J. Fluid Mech. 513, 87–110.Google Scholar
Xu, K., Liu, H. & Jiang, J. 2007 Multiple-temperature kinetic model for continuum and near continuum flows. Phys. Fluids 19, 016101.Google Scholar
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.Google Scholar
Zheng, Y., Garcia, A. L. & Alder, B. J. 2002 Comparison of kinetic theory and hydrodynamics for Poiseuille flow. J. Stat. Phys. 109, 495–505.Google Scholar