In hypersonic flows about space vehicles in low earth orbits or flows in microchannels of microelectromechanical devices, the local Knudsen number lies in the continuum–transition regime. Navier–Stokes equations are not adequate to model these flows since they are based on small deviation from local thermodynamic equilibrium. To model these flows, a number of extended hydrodynamics or generalized hydrodynamics models have been proposed over the past fifty years, along with the direct simulation Monte Carlo (DSMC) approach. One of these models is the Burnett equations which are obtained from the Chapman–Enskog expansion of the Boltzmann equation [with Knudsen number (Kn) as a small parameter] to O(Kn2). With the currently available computing power, it has been possible in recent years to numerically solve the Burnett equations. However, attempts at solving the Burnett equations have uncovered many physical and numerical difficulties with the Burnett model. As a result, several improvements to the conventional Burnett equations have been proposed in recent years to address both the physical and numerical issues; two of the most well known are the “augmented Burnett equations” and the “BGK–Burnett equations.” This paper traces the history of the Burnett model and describes some of the recent developments. The relationship between the Burnett equations and the Grad’s 13 moment equations is elucidated by employing the Maxwell–Truesdell–Green iteration. Numerical solutions are provided to assess the accuracy and applicability of Burnett equations for modeling flows in the continuum–transition regime. The important issue of surface boundary conditions is addressed. Computations are compared with the available experimental data, Navier–Stokes calculations, Burnett solutions of other investigators, and DSMC solutions as much as possible.

1.
M. S.
Ivanov
and
S. F.
Gimelshein
, “
Computational hypersonic rarefied flows
,”
Annu. Rev. Fluid Mech.
30
,
469
(
1998
).
2.
M.
Gad-el-Hak
, “
The fluid mechanics of microdevices
,”
J. Fluids Eng.
121
,
5
(
1999
).
3.
G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows (Oxford Science, New York, 1994).
4.
G.
Koppenwallner
, “
Low Reynolds number influence on the aerodynamic performance of hypersonic lifting vehicles
,” Aerodynamics of Hypersonic Lifting Vehicles,
AGARD, CP-428
11
,
1
(
1987
).
5.
R. P.
Nance
,
D. B.
Hash
, and
H. A.
Hassan
, “
Role of boundary conditions in Monte Carlo simulation of microelectromechanical systems
,”
J. Spacecr. Rockets
12
,
447
(
1998
).
6.
C. J.
Lee
, “
Unique determination of solutions to the Burnett equations
,”
AIAA J.
32
,
985
(
1994
).
7.
K. A. Comeaux, D. R. Chapman, and R. W. MacCormack, “An analysis of the Burnett equations based in the second law of thermodynamics,” AIAA Paper No. 95-0415, 1995.
8.
H.
Grad
, “
On the kinetic theory of rarefied gases
,”
Commun. Pure Appl. Math.
2
,
325
(
1949
).
9.
L. H.
Holway
, “
Existence of kinetic theory solutions to the shock structure problem
,”
Phys. Fluids
7
,
911
(
1964
).
10.
W.
Weiss
, “
Comments on existence of kinetic theory solutions to the shock structure problem
,”
Phys. Fluids
8
,
1689
(
1996
).
11.
C. D.
Levermore
, “
Moment closure hierarchies for kinetic theory
,”
J. Stat. Phys.
83
,
1021
(
1996
).
12.
C. D.
Levermore
and
W. J.
Morokoff
, “
The Gaussian moment closure for gas dynamics
,”
SIAM (Soc. Ind. Appl. Math.) J. Appl. Math.
59
,
72
(
1998
).
13.
C. P. T. Groth, P. L. Roe, T. I. Gombosi, and S. L. Brown, “On the nonstationary wave structure of 35-moment closure for rarefied gas dynamics,” AIAA Paper No. 95-2312, 1995.
14.
S. Brown, “Approximate Riemann solvers for moment models of dilute gases,” Ph.D. thesis, University of Michigan, Ann Arbor, MI, 1996.
15.
R. Myong, “A new hydrodynamic approach to computational hypersonic rarefied gas dynamics,” AIAA Paper No. 99-3578, 1999.
16.
B. C. Eu, Kinetic Theory and Irreversible Thermodynamics (Wiley, New York, 1992).
17.
E. S.
Oran
,
C. K.
Oh
, and
B. Z.
Cybyk
, “
Direct simulation Monte Carlo: Recent advances and application
,”
Annu. Rev. Fluid Mech.
30
,
403
(
1998
).
18.
R.
Roveda
,
D. B.
Goldstein
, and
P. L.
Varghese
, “
Hybrid Euler/particle approach for continuum/rarefied flows
,”
J. Spacecr. Rockets
35
,
258
(
1998
).
19.
I.
Boyd
,
G.
Chen
, and
G.
Candler
, “
Predicting failure of the continuum fluid equations in transitional hypersonic flows
,”
Phys. Fluids
7
,
210
(
1995
).
20.
D.
Burnett
, “
The distribution of velocities and mean motion in a slight nonuniform gas
,”
Proc. London Math. Soc.
39
,
385
(
1935
).
21.
S. Chapman and T. G. Cowling, The Mathematical Theory of Nonuniform Gases (Cambridge University Press, New York, 1970).
22.
K. A. Fiscko and D. R. Chapman, “Comparison of Burnett, super-Burnett, and Monte Carlo solutions for hypersonic shock structure,” in Proceedings of the 16th International Symposium on Rarefied Gas Dynamics, 1988, p. 374.
23.
X. Zhong, “Development and computation of continuum higher order constitutive relations for high-altitude hypersonic flow,” Ph.D. thesis, Stanford University, Stanford, CA, 1991.
24.
W. T. Welder, D. R. Chapman, and R. W. MacCormack, “Evaluation of various forms of the Burnett equations,” AIAA Paper No. 93-3094, 1993.
25.
S. Jin and M. Slemrod, “Regularization of the Burnett equations via relaxation,” J. Stat. Phys. (to be published).
26.
R. Balakrishnan and R. K. Agarwal, “Entropy consistent formulation and numerical simulation of the BGK–Burnett equations for hypersonic flows in the continuum–transition regime,” in Proceedings of the International Conference on Numerical Methods in Fluid Dynamics (Springer-Verlag, Monterey, 1996).
27.
R.
Balakrishnan
and
R. K.
Agarwal
, “
Numerical simulation of the BGK–Burnett for hypersonic flows
,”
J. Thermophys. Heat Transfer
11
,
391
(
1997
).
28.
P. L.
Bhatnagar
,
E. P.
Gross
, and
M.
Krook
, “
A model for the collision process in gas
,”
Phys. Rev.
94
,
511
(
1954
).
29.
R. Balakrishnan, R. K. Agarwal, and K. Y. Yun, “Higher-order distribution functions, BGK–Burnett equations and Boltzmann’s H-theorem,” AIAA Paper No. 97-2552, 1997.
30.
K. Y. Yun, R. K. Agarwal, and R. Balakrishnan, “Three-dimensional augmented and BGK–Burnett equations,” unpublished report, Wichita State University, Wichita, KS, 1998.
31.
K. Y.
Yun
,
R. K.
Agarwal
, and
R.
Balakrishnan
, “
Augmented Burnett and Bhatnagar–Gross–Krook–Burnett for hypersonic flow
,”
J. Thermophys. Heat Transfer
12
,
328
(
1998
).
32.
H. Struchtrup, “Some remarks on the equations of Burnett and Grad,” in Proceedings of the Workshop on Mathematical Models for Simulation of High Knudsen Number Flows, Institute for Mathematics and Applications, University of Minnesota, Minneapolis, MN, 2000 (to be published).
33.
T. Gökçen, “Computation of hypersonic low density flows with thermochemical nonequilibrium,” Ph.D. dissertation, Stanford University, Stanford, CA, 1989.
34.
M.
von Smoluchowski
, “
Veder Warmeleitung in Verdumteu Gasen
,”
Ann. Phys. Chem.
64
,
101
(
1998
).
35.
A.
Beskok
,
G.
Karniadakis
, and
W.
Trimmer
, “
Rarefaction and compressibility effects in gas microflows
,”
J. Fluids Eng.
118
,
448
(
1996
).
36.
A. V.
Bobylev
, “
The Chapman–Enskog and Grad methods for solving the Boltzmann equations
,”
Sov. Phys. Dokl.
27
,
29
(
1982
).
37.
R. Balakrishnan and R. K. Agarwal, “A comparative study of several higher-order kinetic formulations beyond Navier–Stokes for computing the shock structure,” AIAA Paper No. 99-0224, 1999.
38.
K. Y. Yun and R. K. Agarwal, “Numerical simulation of 3D augmented Burnett equations for hypersonic flow in the continuum–transition regime,” AIAA Paper No. 2000-0339, 2000.
39.
J. L.
Steger
and
R. F.
Warming
, “
Flux vector splitting of the inviscid gas dynamics equations with application to finite-difference methods
,”
J. Comput. Phys.
40
,
263
(
1981
).
40.
J. M.
Reese
,
L. C.
Woods
,
F. J. P.
Thivet
, and
S. M.
Candel
, “
A second-order description of shock structure
,”
J. Comput. Phys.
117
,
240
(
1995
).
41.
H.
Alsmeyer
, “
Density profiles in argon and nitrogen shock waves measured by the absorption of an electron beam
,”
J. Fluid Mech.
74
,
497
(
1976
).
42.
R. Balakrishnan, “Entropy consistent formulation and numerical simulation of the BGK–Burnett equations for hypersonic flows in the continuum–transition regime,” Ph.D. thesis, Wichita State University, Wichita, KS, 1999.
43.
X.
Zhong
and
G. H.
Furumoto
, “
Augmented Burnett equation solutions over axisymmetric blunt bodies in hypersonic flow
,”
J. Spacecr. Rockets
32
,
588
(
1995
).
44.
F. W.
Vogenitz
and
G. Y.
Takara
, “
Monte Carlo study of blunt body hypersonic viscous shock layers
,”
Rarefied Gas Dynamics
2
,
911
(
1971
).
45.
J. N. Moss and G. A. Bird, “Direct simulation of transitional flow for hypersonic reentry conditions,” AIAA Paper No. 84-0223, 1984.
46.
A.
Beskok
and
G.
Karniadakis
, “
A model for flows in channels, pipes, and ducts at micro and nano scales
,”
Microscale Thermophys. Eng.
8
,
43
(
1999
).
47.
C. K.
Oh
,
E. S.
Oran
, and
R. S.
Sinkovits
, “
Computations of high-speed, high Knudsen number microchannel flows
,”
J. Thermophys. Heat Transfer
11
,
497
(
1997
).
This content is only available via PDF.
You do not currently have access to this content.