AlAA 91-0770 Stabilization of the Burnett Equations and Application to High-Altitude Hypersonic Flows X. Zhong R. W. MacCormack D. R. Chapman Stanford Univ., Stanford, CA

29th Aerospace Sciences Meeting January 7-10, 1991/Reno,Nevada For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics 370 L’Enfant Promenade, S.W., Washington, D.C. 20024

STABILIZATION OF THE BURNETT EQUATIONS AND APPLICATION TO HIGH-ALTITUDE HYPERSONIC FLOWS

AIM-91.0770

Xiaolin Zhong * Robert W. MacCormack Dean R. Chapman t Department of Aeronautics and Astronautics Stanford University, Stanford, CA 94305

and the AFE operate mainly at altitudes in the continuum transitional regime. Anticipated applications here would be to their aerodynamic stability and heating parameters as well as to flow-field radiation. A similar situation exists for some Mars return vehicles that would use high-altitude aerobtaking to change orbit. Still other applications involve certain aerothermodynamic computations for the upper portion of the ascent trajectory of vehicles such as the NASP which have cowl lips and leading edges with relative!^ small radius of curvature. These are subjected to very severe heating rates in the continuum transitional regime where the numerical computations using the Yavier-Stokes equations are inaccurate. Additional relevant applications are to hypersonic flow-field radiation at high altitudes, which can be important both to the heating rate on vehicles such as the AOTV, as well as to the hard-body radiation signature of a missile traveling through the upper atmosphere.

Abstract It is shown from both analytical investigation and numerical computations that the 1D and plane-2D Burnett equations are unstable t o disturbances of small wavelengths. This fundamental instability arises in numerical computations when the grid spacing is less than the order of a mean free path, and precludes Burnett flow-field computations above a certain maximum altitude for any given vehicle . A new set of equations termed the "augmented Burnett equations" has been developed, and shown t o be stable both by a linearized stability analysis and by direct numerical computations for 1D and 2D flows. The latter represent the first known Burnett solutions for 2D hypersonic flow over w' a blunt leading edge. Comparison of these solutions with the conventional Navier-Stokes solutions reveals that the difference to be small at low altitudes, but significant at high altitudes. Burnett CFD appears to be especially important for predicting aerodynamic parameters sensitive to flow-field details, such as radiation, at high attitudes.

It is noted that a completely different approach to that investigated herein for circumventing the inaccuracy of Navier-Stokes C F D is to use particulate-flow computations such as the DSMC method of Bird['] and the particle simulation method of Baganad'' This type of flow simulation, however, can require relatively large amount of computer time, especially at lower altitudes. Hence the development of an advanced set of continuum equations having a reasonable accuracy should be much more comvutationallv efficient.

1. I n t r o d u c t i o n

A number of advanced hypersonic vehicles are anticipated to operate in the continuum transitional regime a t high altitudes where the thickness of the bow shock waves is a sizable or dominant part of the shock detachment distance. Under these conditions, CFD codes for the flow past the vehicles must compute through the structure of hypersonic shock waves. However, it has long been known that the conventional Navier-Stokes equations are inaccurate for this purpose, and hence we need to develop some other set of constitutive equations, more advanced than Navier-Stokes, to provide realistic continuum-flow computations for hypersonic flows at these high altitudes. The development of an advanced set of continuum equations of motion is necessary for certain practical applications. Aeroassisted vehicles such as the AOTV V'

In order to develop the advanced set of constitutive equations, Fiscko and Chapmani4. 1' reinvestigated and proposed the Burnett equations[6], which are higher order approximations to the Boltzmann equation than the Navier-Stokes equations. They found that the Burnett equations provide much greater accuracy than the Navier-Stokes equations for one-dimensional shock wave structure in monatomic gases. Aowever, both an analytical analysisrq and the past computational experience[*] showed that the Burnett equations are unstable to very small wavelength disturbances encountered in fine-mesh numerical solutions. This unstability makes it impossible to apply the Burnett equa-

*GraduateStudent 'Professor, Member AlAA 'Professor, FeUow A N A Copyright @American Institute of Aeronautics and Astronautics, Inc. 1981. All right. reserved.

1

tions to practical flows in two and three dimensions above a certain altitude for any vehicle. Therefore, we need to overcome the instability of the Burnett equations in order to apply the equations to practical flow problems. Guided by the linearized stability analysis, this paper will develop a new set of equations termed the “ augmented Burnett equations” to stablize the conventional Burnett equations. We will show that the augmented Burnett equations are stable and yield essentially the same results as t h e conventional Burnett equations when stable solutions exist for the later equations. T h e new set of equations has been tested by a theoretical stability analysis as well as 1D and 2D flow computations following the objectives below.

where

U=(

p:.),

v

P=Pm, e = p (c,T

+ -)U22

and u l l and q1 are the viscous stress and thc heat flux terms, of which the relations with the gradients of the flow variables are termed the constitutive equations. Eq. (1) together with t h e two constitutive equations for 611 and y1 given in the next section form a complete set of governing equations for the one-dimensional gas flow.

The Research Objectives 1. To develop augmented Burnett equations which overcome the instabilities encountered when finemesh computations are attempted with the coilventional Burnett equations.

3. The Constitutive Equations

2. T o test the computational stability of both the conventional Burnett equations and the augmented Burnett eqnaiions by solving them numerically with progressively refined meshes for both one-dimensional hypersonic shock structure, and two-dimensional flows past a blunt leading edge.

Tlie gas flow regime can be characterized by tlic Kundsen number K n , which is defined as: Kn

x

=L

(2)

where X is the mean free path and L is the macroscopic characteristic length of the flow. When ICn increases from 0 through the order of 1 to 03, the gas flow changes from the translational equilibrium regime through the transitional regime to the Cree molecule regime. The constitutive equations for a gas flow of small I u2

3R 2

(59) where the rotational heat conductivity tiF and translational heat conductivity K for diatomic gases are approximated by the Eucken's relation[25]:

Like the particle simulation by Feiereisen, we used hard-sphere gas model in the computations. The viscosity for hard-sphere gas is:

T h e following rotational relaxation model was used in the computations:

with TP

=

ax

(63)

where

and T is the translational temperature; T,is the rotational temperature. T h e stress u,; and translational heat transfer qi are given in the preceding sections. T h e rotational heat flux terms q?, are as follows:

rc

aT

R,

e = e,

C",

3 p 4 p~

P

In the computations, complete accommodation surface was assumed, Le., 7 = 1 and i3 = 1, which were the same as in the particle simulation. Case IV. : Double Ellipse Flow ( A4, Anele of Attack = 30")

-

= 25.0.

~~

The flow conditions of the case were:

I

= 25.0 = 0.28 Angle of Attack = 30" A, = 1.05 x 10-3nL T, = 13.5'K T, = 620.OaK Nose Radius = 0.00375 m M,

Kn,

Figure 29 shows the body-fitted computational grid of 68 x 62 grid points for the double ellipse geometry. Stability of the Burnett equations: The computations for the Navier-Stokes and the augmented Burnett equations were stable, while the computations for conventional Burnett equations were unstable. These results again showed t h a t the augmented Burnett e q u a tions stablize the conventional Burnett equations in two dimensions. The Burnett Flow Field and the Navier-Stokes Flow Field Compared with the Particle Simulation: Figures 30, 31 and 32 are density, velocity and translational '4 temperature distributions along the stagnation line. T h e results are compared with the particle simulation

, . ~

..-

5. Computation times with the Burnett equations are

results. In both figures, the results of the augmented Burnett equations agree better with those of the particle simulation than those of the Navier-Stokes equations. The results indicate the Burnett equations to be more accurate than the Navier-Stokes equations. I t should be pointed out t h a t the difference between the Navier Stokes flow field and the Burnett flow field is not large because the Kundsen number of the flow was about 0.28 and the artificial hard-sphere model w a s used in the computations. Still the tw-dimensional augmented Burnett equations result in a thicker bow shock wave and agree very well with the particle simulation results. Figures 33 shows the rotational and translational temperature along the stagnation line. The rotational temperature lags behind the translational temperature across the shock wave. Figurcs 34,35, 3G and 37 are the density and transla tional t,emperature contours for the case. The density contonrs of the augmented Burnett equations and the Fciereisen’s resnlts are plott,cd together for comparison. The figures show the density and temperature contours for t,he Augment,ed Brirnet,t equat,ions agree well with the particle simulation results.

only modestly greater than Navier-Stokes with the same grid system; T h e Burnett solutions required about 4C-percent more CPU time than NavierStokes for both the 1D and 2D flows investigated. 9. Acknowledgements

This research was supported by SDIO/IST managed by the Army Research Office under contracts DAAL03-8GK-0139 and DAAL03-90-G-0031-P00002. We would also like to acknowledge the Aerot,hcrmodynamics Branch of NASA Ames Research Center for providing computer time on the Cray-YMP computer

10. Appendix Tlie Derivative Transformation Equations The equations of the grid transformation are: 2

A new set of equations termed the “augmented Burnet,t equations” has been developed which overcome the instability of the conventional Burnett equations to small wavelength perturbations. We have computed both I D shock wave structures and 2D flows past blunt leading edges using the new equations. The analytical analysis and numerical test cases liave demonstrated the following properties of the new equations:

where

i

1. The augmented Burnett equations are always stable in t h r theoretical analysis as well as in both the

I D and 2 0 numerical computation tests produced datc.

t,O

2. ’rhe new equations maintain the same accuracy as the convcntional nurnctt equations.

d

Y r lJ = -ycjJ = -xr/J = xc/J

J

=

a 6 c

=

X C Y ~ - X ~ Y C

The second-order derivative transformation equations are:

3 . At low altitudes ( l i n 5 0.1 ), the difference between the 2D computatiotial results of the Burnett equations and those of t,lie Navier-Stokes equations is small. 2 O(1) ), the difference bc. twecii the 2D computational results ( especially T ) of the Burnett equations and those of the NavierStokes equations is significant, which makes it preferable to use the Burnett equations instead of the Navier-Stokes equations in this regime. Burnett CFD appears to he especially important for predicring aerodynamic parameters sensitive to flow-fieid details, such as radiation.

4. At high altitudes ( K n

,

Z(t,%T)

The transformation equations for the derivat,ives from curvilinear coordinates (r ) to Cartesian coordinates ( t , y , t ) can be derived by the chain rules. The first-order derivative transformation equations are:

8. Conchisions

“

=

Y = YK>%T) t = r

13

References

where the coefficients such as a2 and dv can he computed by Eq. ( 6 7 ) and (68). T h e third-order derivative transformation equations are

[ l ] G . A. Bird. Monte Carlo simulation of gas flows.

A n n . Rev. Fluid Mech., 1O:ll-13, 1978.

a3

a3

-=a

-+3a2b-

ax3

a ~ 3

W

[2] D. Baganoff. Vectorization of a particle code used in the simulation of rarefied hypersonic flow. Symposium o n Computational Technology f o r Flight Vehicles, Washington, D. C., November 1990.

131 D. Baganoff and J. D. McDonald. A collisionselection rule for a particle simulation method suited to vector computers. Physics of Fluids A, July 1990.

a3 --

-

aXvy

[4] Kurt A. Fiscko. S t u d y of Continuum Ifigher Closure Models Evaluated by a Statistical Theory of Shock Structure. PhD thesis, Stanford University, 1988.

a3 + a2d)am7+ b2d -+ (b2c + 2abd) -+

a3 a2c + (2abc

[5] Kurt A . Fiscko and Dean R. Chapman. Comparison of Burnett, Super-Burnett, and Monte Carlo solutions for hypersonic shock structure. In 16th International S y m p o s i u m of Rarefied Gas Dynomics, 1988.

at3 a3

a3

atw

a03

[6] S. Chapman and T. G. Cowling. The Mathematzcs Theory of Non-Uniform Gases. Cambridgc University Press, 1960.

(73)

[7] A. V. Bobylev. T h e Chapman-Enskog and Grad methods for solving the Boltzmann equation. Sou. Phys. Dokl., 2 7 ( 1 ) , January 1982.

a x w - c2a -+ (2acd + bc’) amq+ a3

a3 --

a3

at3 a3

.

+

bd2 - ( a d 2 803

.

‘v

[SI J . D. Foch. On higher order hydrodynamic theories of shock structure. Acta physical Austriaca, suppl. X., 123-140, 1973.

+ 2cdb) -+ a3

ata02

[9] C . S . Wang Chang and G. E. Uhlenbeck. On (.he transport phenomena in rarefied gases. Studies in Statistical Machanics, V:1-17, 1948.

a + dz, a

[ l o ] F. E. Lumpkin 111. Development and Evaluation of Continuum Models f o r Trarislational-Rotational Nonequilibrium. PhD thesis, Stanford University, March 1990.

(74)

80

a3 a3 = c3 + 3c2d a3 +

at3

ay3

d3

am7

-+ 3cdZ ataOZ a1?3

3cc

a2

+

-+ 3 d d

y at2

3(cdy

+dc

-+ a2

aq2

1121 L. C . Woods. On the thermodynamics of nonlinear constitutive relations in gasdynamics. J. of Fluid Mechanics, 101(2):225-242, 1980.

)a2 i

am a + dYY 5 at

a

CYY

[ l l ] L. C. Woods. Transport processes in dilut gases over the whole range of Knudsen number. part 1 general theory. J . of Flvid Mechonics, 93(3):585607, 1979.

a3

a3

[I31 C. E. Simon. Theory of Shock Structure i n a Maxwell Gas Based o n the Chapman-Enskog Development Through Super-Burneit Order. PhD thesis, University of Colorado, 1976.

(75)

where the coefficients such as arz, cCy and d,, can be computed by Eq. (69), ( 7 0 ) and ( 7 1 ) .

14

‘d

I.

[14] M. Sh. Shavaliev. T h e Burnett approximation of the distribution function and the supper-Burnett contributions to the stress tensor and the heat flux. Journal of Applied Mathematics and Mechanics, 42(4):656-702, 1978.

9

[15] C. S. Wang Chaug. On the theory of the thickness of weak shock waves. Studies in Statistical Machanics, V:27-42, 1948. [16] J . Stager and R. F. Warming. Flur VectorSplitting of the Inuiscid Gasdynamics Equations with Application t o Finite Difference Methods. TM 78650, NASA, 1979. 1171 11. LV. MacCormack. Currenf Status of iVumerical Solution of the iVauier-Stokes Equations. AIAA Paper 85-0032, AIAA, January 1985.

E? -15

-10

.5

5

0

15

10

ATTENUATION COEFFICIENT (

20

0 )

Figure 1: Xaxwellian gas characteristic trajectories for flows in both one and two dimensions. The arrows show the direction in decreasing wavelength.

-

[IS] E. H . Kennard. Kinetic Theory of Gases McGraw-Hill Book Co., Inc., New York, 1938.

E?

91

I

C Y I

z

Y Y Y

[I91 Samuel A Schaaf and P. L. Chambre. Flow of rarefied gases. P n n c e l o n Aemnaultcal Paperbacks, 8 , 1961. *r

w 0 v

[20] .J. C. Tannehill and R. G. Eisler. Numerical computation of the hypersonic leading edge problem using the Burnett equations. The Physics of Fluzds, 19(1):%15, 1976. [21]

0

-15

-10

-5

0

5

15

10

20

ATTENUATION COEFFICIENT [ e )

Figure 2: Maxwellian gas characteristic trajectories for the super Burnett equations. T h e arrows show the direction in decreasing wavelength.

R. Schamberg.

The Fundamental Dzfferenttal Equations and the Boundary Condttions f o r Htgh Speed Slip Flow. PhD thesis, California Institute of Technology, Pasadena, California, 1947

!

[22] Chul-So0 Kim. Experimental studies of supersonic flow past a circular cylinder. J. of Physical Society of Japan. 11(4):439445,April 1956. [23] W. D. Hayes and R. F. Probstein. Hypersonic Flow Theory. Volume I, Academic Press, New York and London, second edition, 1966. [24] W. J . Feiereisen. The hypersonic double ellipse in rarefied flow (problem 6.4). INRIA Workshop on Hypersonic FIoius for Reentry Problems, Antibes, France 22-26, January, 1990.

-15

.5

0

5

15

10

ATTENUATION COEFFICIENT (

0

20

I

Figure 3: Maxwellian gas characteristic trajectories for the augmented Burnett equations in both one and two dimensions. T h e arrows show the direction in decreasing wavelength.

[25] W. G . Vincenti and W. G. Kruger J r . Introduction Y

-10

t o Physical Gas Dynamics. Krieg, 1965.

15

>-ma %#=-

01.

4m

aa

@4

MI

rhl

.WL

Figure 7: The bow shock shape for Case I: Mw = 4.0 K ~ =, 0.67 10-4.

Figure 4: Shock temperature profile in Fiscko’s computations of the Burnett equations when the instability started. ( Maxwellian gas M = 8 )

0

7

I I

4

0

10

20

30

0.m

50

40

0.-

Qool

o m

00

X W

MACH NUMBER(M,)

Figure 8: Density along stagnation streamline for Case I: M , = 4.0, Kn, = 0.67 x

Figure 5 : Argon shock wave inverse density thickness.

. . )

4u

MI

om

Mo

0.W

o m

00

ii $4

XI.)

Figure 9: Pressure along stagnation streamline for Case W I: M , = 4.0, ICn, = 0.67x

Figure 6: Computational mesh for Case I: M , = 4.0, K n , = 0.67 x

16