0
REVIEW ARTICLES

Burnett equations for simulation of transitional flows

[+] Author and Article Information
Ramesh K Agarwal

Aerospace Research and Engineering Center, Washington University, St Louis MO 63130rka@me.wustl.edu

Keon-Young Yun

National Institute for Aviation Research, Wichita State University, Wichita KS 67260-0093 keon-young.yun@wichita.edu

Appl. Mech. Rev 55(3), 219-240 (Jun 10, 2002) (22 pages) doi:10.1115/1.1459080 History: Online June 10, 2002
Copyright © 2002 by ASME
Your Session has timed out. Please sign back in to continue.

References

Ivanov  MS and Gimelshein  SF (1998), Computational hypersonic rarefied flows, Annu. Rev. Fluid Mech. 30, 469–505.
Gad-el-Hak  M (1999), The fluid mechanics of microdevices, ASME J. Fluids Eng. 121, 5–33.
Bird GA (1994), Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Oxford Science Publ, New York NY.
Lee  CJ (1994), Unique determination of solutions to the Burnett equations, AIAA J. 32, 985–990.
Comeaux KA, Chapman DR, and MacCormack RW (1995), An analysis of the Burnett equations based in the second law of thermodynamics, AIAA Paper No. 95-0415, Reno NV.
Grad  H (1949), On the kinetic theory of rarefied gases, Commun. Pure Appl. Math. 2, 325–331.
Holway  LH (1964), Existence of kinetic theory solutions to the shock structure problem, Phys. Fluids 7, 911–913.
Weiss  W (1996), Comments on existence of kinetic theory solutions to the shock structure problem, Phys. Fluids 8, 1689–1690.
Levermore  CD (1996), Moment closure Hierarchies for kinetic theory, J. Stat. Phys. 83, 1021–1065.
Levermore  CD and Morokoff  WJ (1998), The Gaussian moment closure for gas dynamics, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 59, 72–96.
Groth CPT, Roe PL, Gombosi TI, and Brown SL (1995), On the nonstationary wave structure of 35-moment closure for rarefied gas dynamics, AIAA Paper No 95-2312, San Diego CA.
Brown S (1996), Approximate Riemann solvers for moment models of dilute gases, PhD Thesis, Univ of Michigan, Ann Arbor MI.
Myong R (1999), A new hydrodynamic approach to computational hypersonic rarefied gas dynamics, AIAA Paper No 99-3578, Norfolk VA.
Eu BC (1992), Kinetic Theory and Irreversible Thermodynamics, John Wiley & Sons, New York, NY.
Oran  ES, Oh  CK, and Cybyk  BZ (1998), Direct simulation Monte Carlo: Recent advances and application, Annu. Rev. Fluid Mech. 30, 403–441.
Roveda  R, Goldstein  DB, and Varghese  PL (1998), Hybrid Euler/particle approach for continuum/rarefied flows, J. Spacecr. Rockets 35, 258–265.
Boyd  I, Chen  G, and Candler  G (1995), Predicting failure of the continuum fluid equations in transitional hypersonic flows, Phys. Fluids 7, 210–219.
Struchtrup H (2000), Some remarks on the equations of Burnett and Grad, Proc of the Workshop on Mathematical Models for Simulation of High Knudsen Number Flows, Inst for Math and Applications, Univ of Minnesota, Minneapolis MN.
Jin S and Slemrod M (2000), Regularization of the Burnett equations via relaxation, J. Stat. Phys. (to appear).
Burnett  D (1935), The distribution of velocities and mean motion in a slight non-uniform gas, Proc. London Math. Soc. 39, 385–430.
Chapman S and Cowling TG (1970), The Mathematical Theory of Non-Uniform Gases, Cambridge Univ Press, New York NY.
Fiscko KA and Chapman DR (1988), Comparison of Burnett, super-Burnett and Monte-Carlo solutions for hypersonic shock structure, Proc of 16th Int Symp on Rarefied Gas Dynamics, Pasadena CA, 374–395.
Zhong X (1991), Development and computation of continuum higher order constitutive relations for high-altitude hypersonic flow, PhD Thesis, Stanford Univ, Stanford CA.
Welder WT, Chapman DR, and MacCormack RW (1993), Evaluation of various forms of the Burnett equations, AIAA Paper No 93-3094, Orlando FL.
Balakrishnan R and Agarwal RK (1996), Entropy consistent formulation and numerical simulation of the BGK-Burnett equations for hypersonic flows in the continuum-transition regime, Proc of Int Conf on Numerical Methods in Fluid Dynamics, Springer-Verlag, Monterey CA.
Balakrishnan  R and Agarwal  RK (1997), Numerical simulation of the BGK-Burnett for hypersonic flows, J. Thermophys. Heat Transfer 11, 391–399.
Bhatnagar  PL, Gross  EP, and Krook  M (1954), A model for collision process in gas, Phys. Rev. 94, 511–525.
Balakrishnan R, Agarwal RK, and Yun K-Y (1997), Higher-order distribution functions, BGK-Burnett equations and Boltzmann’s H-theorem, AIAA Paper No 97-2552, Atlanta GA.
Yun  K-Y and Agarwal  RK (2001), Numerical simulation of three-dimensional augmented Burnett equations for hypersonic flow, J. Spacecr. Rockets 38, 520–533.
Yun  K-Y, Agarwal  RK, and Balakrishnan  R (1998), Augmented Burnett and Bhatnagar-Gross-Krook-Burnett for hypersonic flow, J. Thermophys. Heat Transfer 12, 328–335.
Reinecke  S and Kremer  GM (1996), Burnett’s equations from a (13+9N)-field theory, Continuum Mech. Thermodyn. 8, 121.
Ferziger JH and Kaper HG (1972), Mathematical Theory of Transport Processes in Gases, Amsterdam, Holland.
Truesdell C and Muncaster RG (1980), Fundamentals of Maxwell’s Kinetic Theory of a Simple Monatomic Gas, Academic Press, New York NY.
Foch  TD (1973), On the higher order hydrodynamic theories of shock structure, Acta Phys. Austriaca, Suppl. X, 123–140.
Agarwal RK, Yun K-Y, and Balakrishnan R (1999), Beyond Navier-Stokes: Burnett equations for flow simulations in continuum-transition regime, AIAA Paper No 99-3580, Norfolk VA.
Bobylev  AV (1982), The Chapman-Enskog and Grad methods for solving the Boltzmann equation, Sov. Phys. Dokl. 27, 29–31.
Luk’shin  AV (1983), On the method of derivation of closed systems for macroparameters of distribution function for Small Knudsen number, Sov. Phys. Dokl. 28, 454–456.
Renardy  M (1984), On the domain space for constitutive laws in linear viscoelasticity, Arch. Ration. Mech. Anal. 85, 21–26.
Biscari P, Cercignani C, and Slemrod M (2000), Time derivatives and frame indifference beyond Newtonian fluids, C. R. Acad. Sci. Paris, to appear.
Joseph DD (1990), Fluid Dynamics of Viscoelastic Liquids, Springer-Verlag, New York NY.
Chen  G-Q, Levermore  CD, and Liu  TP (1994), Hyperbolic conservation laws with stiff relaxation terms and entropy, Commun. Pure Appl. Math. 47, 787–830.
Jin  S and Xin  ZP (1995), The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Commun. Pure Appl. Math. 48, 235–276.
Coquel  F and Perthame  B (1998), Relaxation of energy and approximate Riemann solvers for general pressure laws in fluid dynamics, SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 35, 2223–2249.
Grad  H (1963), Asymptotic theory of the Boltzmann equation, Phys. Fluids 6, 147–181.
Smoluchowski  M (1998), Veder Warmeleitung in verdumteu Gasen, Annalen der Physik and Chemie 64, 101–130.
Beskok  A, Karniadakis  G, and Trimmer  W (1996), Rarefaction and compressibility effects in gas microflows, J. Fluids Eng. 118, 448–456.
Yun K-Y (1999), Numerical simulation of 3-D augmented Burnett equations for hypersonic flow in continuum-transition regime, PhD thesis, Wichita State Univ, Wichita KS.
Steger  JL and Warming  RF (1981), Flux vector splitting of the inviscid gas dynamics equations with application to finite-difference methods, J. Comput. Phys. 40, 263–293.
Reese  JM, Woods  LC, Thivet  FJP, and Candel  SM (1995), A second-order description of shock structure, J. Comput. Phys. 117, 240–250.
Alsmeyer  H (1976), Density profiles in argon and nitrogen shock waves measured by the absorption of an electron beam, J. Fluid Mech. 74, 497–513.
Zhong  X and Furumoto  G (1995), Augmented Burnett equation solutions over axisymmetric blunt bodies in hypersonic flow, J. Spacecr. Rockets 32, 588–595.
Vogenitz  FW and Takara  GY (1971), Monte Carlo study of blunt body hypersonic viscous shock layers, Rarefied Gas Dynamics 2, Pisa, Italy, 911–918.
Moss JN and Bird GA (1984), Direct simulation of transitional flow for hypersonic reentry conditions, AIAA Paper No 84-0223, Reno NV.
Sun  Q, Boyd  ID, and Fan  J (2000), Development of particle methods for computing MEMS gas flows, MEMS 2, 563–569.
Beskok  A and Karniadakis  G (1999), A model for flows in channels, pipes, and ducts at micro and nano scales, Microscale Thermophys. Eng. 8, 43–77.
Oh  CK, Oran  ES, and Sinkovits  RS (1997), Computations of high-speed, high Knudsen number microchannel flows, J. Thermophys. Heat Transfer 11, 497–505.

Figures

Grahic Jump Location
Characteristic trajectories of the 1D Navier-Stokes equations
Grahic Jump Location
Characteristic trajectories of the 1D conventional Burnett equations; Euler equations are used to express the material derivatives D()/Dt in terms of spatial derivatives.
Grahic Jump Location
Characteristic trajectories of the 1D augmented Burnett equations; Euler equations are used to express the material derivatives D()/Dt in terms of spatial derivatives.
Grahic Jump Location
Characteristic trajectories of the 1D super-Burnett equations; Euler equations are used to express the material derivatives D()/Dt in terms of spatial derivatives.
Grahic Jump Location
Plot showing the variation of reciprocal density thickness with Mach number, obtained with the Navier-Stokes, Woods, Simplified Woods 49, and Burnett equations for a hard sphere gas 22. Experimental data was obtained from Alsemeyer 50.
Grahic Jump Location
2D computational grid (50×82 mesh) around a blunt body, rn=0.02 m
Grahic Jump Location
Density distributions along the stagnation streamline for blunt body flow: Air, M=10, and Kn=0.1
Grahic Jump Location
Velocity distributions along the stagnation streamline for blunt body flow: Air, M=10, and Kn=0.1
Grahic Jump Location
Temperature distributions along the stagnation streamline for blunt body flow: Air, M=10, and Kn=0.1
Grahic Jump Location
Comparison of temperature contours for blunt body flow: Air, M=10, and Kn=0.1
Grahic Jump Location
Comparison of Mach number contours for blunt body flow: Air, M=10, and Kn=0.1
Grahic Jump Location
Density distributions along the stagnation streamline for a hemispherical nose: Argon, M=10.95, and Kn=0.2
Grahic Jump Location
Temperature distributions along the stagnation streamline for a hemispherical nose: Argon, M=10.95, and Kn=0.2
Grahic Jump Location
Density distributions along the stagnation streamline for a hemispherical nose: Hard-sphere gas, M=10, and Kn=0.1
Grahic Jump Location
Temperature distribution along the stagnation streamline for a hemispherical nose: Hard-sphere gas, M=10, and Kn=0.1
Grahic Jump Location
Density distributions along the stagnation streamline for a hemispherical nose at an angle of attack: Air, M=10, and Kn=0.1768
Grahic Jump Location
Temperature distributions along the stagnation streamline for a hemispherical nose at an angle of attack: Air, M=10, and Kn=0.1768
Grahic Jump Location
Side view of the grid (61×100 mesh) around a hyperboloid nose of radius rn=1.362 m
Grahic Jump Location
Density distributions along the stagnation streamline for a hyperboloid nose: Air, M=25.3, and Kn=0.227
Grahic Jump Location
Temperature distributions along the stagnation streamline for a hyperboloid nose: Air, M=25.3, and Kn=0.227
Grahic Jump Location
Density distributions along the stagnation streamline for a hyperboloid at an angle of attack: Air, M=25.3, and Kn=0.227
Grahic Jump Location
Temperature distributions along the stagnation streamline for a hyperboloid at an angle of attack: Air, M=25.3, and Kn=0.227
Grahic Jump Location
Comparison of temperature contours for a hyperboloid at an angle of attack: Air, M=25.3,Kn=0.227, and α=15° (front view of exit plane)
Grahic Jump Location
Grid (101×91 mesh) around a NACA 0012 airfoil, c=0.04 m
Grahic Jump Location
Density contours for NACA 0012 airfoil: Air, M=0.8, and Kn=0.014
Grahic Jump Location
Pressure distributions along NACA 0012 airfoil surface: Air, M=0.8, and Kn=0.014
Grahic Jump Location
Slip velocity distributions along NACA 0012 airfoil surface: Air, M=0.8, and Kn=0.014
Grahic Jump Location
Comparison of velocity profiles at various streamwise locations: Knin=0.088,Knout=0.2, and Pin/Pout=2.28    
Grahic Jump Location
Comparison of mass flow rates along the microchannel: Knin=0.088,Knout=0.2, and Pin/Pout=2.28
Grahic Jump Location
Comparison of pressure distribution along the centerline: Knin=0.088,Knout=0.2, and Pin/Pout=2.28
Grahic Jump Location
Comparison of streamwise velocity distributions along the centerline: Knin=0.088,Knout=0.2, and Pin/Pout=2.28
Grahic Jump Location
Comparison of slip velocity distributions along the wall: Knin=0.088,Knout=0.2, and Pin/Pout=2.28
Grahic Jump Location
Microchannel geometry and grid
Grahic Jump Location
Comparisons of contours between Navier-Stokes and augmented Burnett equations: Helium, M=5, and Kn=0.7
Grahic Jump Location
Comparisons of density, temperature, pressure, and Mach number profiles along the centerline of the channel: Helium, M=5, and Kn=0.07
Grahic Jump Location
Comparisons of density, temperature, pressure, and Mach number profiles along the wall of the channel: Helium, M=5, and Kn=0.07
Grahic Jump Location
Comparisons of temperature profiles across the channel at various locations: Helium, M=5, and Kn=0.07
Grahic Jump Location
Comparisons of velocity profiles across the channel at various locations: Helium, M=5, and Kn=0.07

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In