NASA Technical Memorandum 105693

Numerical Solution of a Three-DimensionalCubic Cavity Flow by Using theBoltzmann Equation

Danny P. HwangLewis Research CenterCleveland, Ohio

Prepared for the18th International Symposium on Rarefield Gas Dynamicssponsored by the University of British ColumbiaVancouver, Canada, July 26-31, 1992

NASA

https://ntrs.nasa.gov/search.jsp?R=19920021054 2020-01-18T01:08:52+00:00Z

NUMERICAL SOLUTION OF A THREE-DIMENSIONAL CUBIC

CAVITY FLOW BY USING THE BOLTZMANN EQUATION

Danny P. HwangNational Aeronautics and Space Administration

Lewis Research CenterCleveland, Ohio 44135

SummaryA three-dimensional cubic cavity flow has been analyzed for diatomic

gases by using the Boltzmann equation with the Bhat nagar- Gross- Krook(B—G—K) model. The method of discrete ordinate was applied, and thediffuse reflection boundary condition was assumed. The results, which showa consistent trend toward the Navier-Stokes solution as the Knudsonnumber is reduced, give us confidence to apply the method to a three-dimensional geometry for practical predictions of rarefied-flowcharacteristics. The CPU time and the main memory required for a three-dimensional geometry using this method seem reasonable.

Symbolsa defined in equations (5) and (6)

Eor energy level (equal to the roots of the Laguerre polynomialof degree n)

F Maxwellian distribution functionf distribution functionimax maximum i indexjmax maximum j indexKn Knudson numberk Boltzmann constantkmax maximum k index

Aerospace Engineer.

k a , kp, k weighting coefficient of the odd equally spaced quadratureL lengthin of a moleculen number particle densityR gas constant

Rol Gauss-Laguerre weighting coefficientT temperatureu macroscopic velocity (flow velocityv molecular velocityw velocity of moving surfaceY vertical distanceZR rotational relaxational parameterA mean free pathIL viscosityV collision frequency" nondimensional quantitySubscriptsel elasticeq equilibriumi, j, k index of physical spaceinel inelasticT totalt translationalx x-componenty y-componentz z-componenta, 0 ry index of molecular velocity spaceU discrete energy level1 reference condition

IntroductionWith space vehicles orbiting the world at hypersonic speed, numerical

methods to cover all flow regimes become more important. Advances incomputer hardware in recent years allow methods based on the kinetictheory to be used for practical applications. Many two-dimensionalproblems have been solved by the kinetic approach. 1-7 A limited numberof three-dimensional problems have been solved recently by the direct-simulation Monte Carlo (DSMC) method. 8"10 However, there are no three-dimensional numerical solutions based on the discrete ordinate method"applied to a Boltzmann equation.

A cubic cavity flow problem has been solved for diatomic gases usingthe Boltzmann equation with the Bhatnagar-Gross-Krook (B—G—K)model. 12,13 Because neither theoretical nor experimental data are availablefor cubic cavity flow in free molecule flow, slip flow, and transition flowregimes, efforts have been made to obtain a continuum flow solution as

2

closed as possible so that the results from this study can be compared withthe results of the Navier-Stokes solution. 14

Governing EquationThe geometry of the problem considered is shown in figure 1. A cubic

cavity with the same dimension on all sides is filled with diatomic gas (suchas air), and the wall temperature is assumed to be the same on all walls.The top surface (j = jmax surface) is moving at a constant speed, andsteady-state conditions are reached and analyzed numerically.

The distribution function at the energy level E00 fff , is governed by theBoltzmann equation with the Bh atnag ar- Gross- Krook-Morse model for thecollision terms:

vx ax + vy a_ + vz — vel( F to — U + vinel(F ia — fo) (1)

where

nm 13/2e 2M ((Vx-Ux)2+(VY—UY)2+(Va—Ua)2)

2c ( )

tv a 21rkTt

f/2— M ( (V

. Ux)2+(VY—UY)2+(Va—Ua)2)

Fiv q— nae m e 2kTT

(3)2^rkTT

The elastic collision frequency vel 15,16 is given by

nkTtvel —

(4)(1 + a)µ

whereas the inelastic collision frequency vine, is related to vel in theformula of

vine, = a vel (5)

ZR _ 5(1 + a)(g)

3a

where the rotational relaxational parameter Z R must be obtained byexperiment. The viscosity-temperature relation of the form

3

µ = Tt(7)

µi T1

is assumed and the value of s can be found in reference 17; for the currentstudy, a and s are equal to 0.4 and 0.758, respectively, for air. Thesubscript 1 indicates the reference condition. The reference condition usedin this study is the equilibrium flow in the cubic cavity when the top surfaceis stationary.

The reference viscosity is related to the reference mean free path aiby the relation

µi = 1 mn,A,(2,rRT 1 ) 1/2(8)

Combining equations (4) to (8) gives

_ 18 nkTt T _egel

5(1 + a) Ajmnj(21rRTj1/2 ^Tjl(9)

All macroscopic quantities, which will be listed in the nondimensionalform, can be calculated in terms of f,'.

Nondimensionaliz ationA characteristic velocity used for non dimensionalization is defined as

V 1 = (2RT 1 ) 1/2(10)

The definitions of nondimensional variables are defined as follows:

n = n T = T (11)n i T1

uX uY uZllX = ; ilY = ; ll Z =

V 1V1 V 1

vX VY v vX = _;vY = _;vz=

V1V1 V1

4

EoEa kT

1

x yzX = ^ Y = _ ^ Z = _

L 1L1 L1

ve1 L 1_vel

V 1

33 3

f _ foV1

FFtov l F FioVl

o to

_io

n l nl nl

Equation (1) becomes

°x + °y + °z — vel((Fto - fa) + a(Fio _ i,)) (12)

where

+(Vy-uy)2+2

1

13/2

ei t

Fto = na — ^rTt

fix) 2+(Vy-uy)2+(w,-u,)Z

1Fiv = nOeq /2 e TT

-I TT

8n 1 -evel = Tt

5(1 + a) FKn

A lKn = L

5

na _ f+ - f+ - +

J ` wfadvXdvYdvE

0

I_Eoexp1

nQeq = fi Il TT

(_ Esexp

s l TT

n RQ exp Ea 1 -(from ref. 1)

TT,, TT

uXl

JJ f+ f+ -

J + wfQVXdvXdvydvzn

0

u _ 1J

f+ °°J

f+ w f

Q* wfovydvXdvydvZ

n

uZ _ ] r+ f+

-f+ ^fQVVdvXdvYdvE

nJ +

0

lit — , J ±3n J t f ± u.) 2 + (^Y - uY)2

+ (vZ - u")2]f,dv.dvydvz

naEan

TT = 3T t + ?Ti5 5

6

Discretization of Distribution Function and Numerical SchemeThe distribution function at energy level v, fQ, was discretized with

equal spacing in the physical space, the arguments of the odd equally spacedquadrature (see appendix and ref. 18) was the spacing for the velocity space,and the roots of the Laguerre polynomial of degree 4 was used for thespacing for the internal energy space. The value of the distribution functionat these discretized points was calculated by solving the Boltzmann equationusing the finite difference method in the physical space.

The difference scheme is the first-order upwind difference based on themolecular velocity; for example,

afP _ f —

fv,i-1 when Vx >- 0N AX

and

Cif _ fQ , i+ l l, k , a ,Q,7 - fo,il,k,a,^,7 when Vx 0

The distribution functions were then integrated for all macroscopicproperties by using the odd equally spaced quadrature, e.g., the particledensity at energy level Q and at the physical point (1, j, k) is

( ( ( n' n, n'

J ± - .1 ± w .1 + „ i od3xdvydv E = kakpk7f v(1^],k,a,p,7}

where k a , kp, and k,y are the weighting coefficients of the odd equallyspaced quadrature for the velocity components 8, , va, and - 7V respec-tively. These properties were summed over the energy space, and thensaved and used for the next iteration.

Boundary ConditionsFor a constantly moving surface, perfectly diffuse reflection is assumed to

specify the interaction of the molecules with the surface of the moving plate.Molecules which strike the moving surface (j = jmax surface} are emittedwith a Maxwellian velocity distribution characterized by the plate temper-ature TN, and the plate velocity w; that is,

_ y a

is = nW 1 13/2

e Tw

irTN,

7

where the density of molecules diffusing from the plate n W is not knowna priori and must be found by applying the condition of zero mass fluxnormal to the plate at the surface, i.e.,

0 -

f f f vy fadvXdvydv Q + f f f vy fadvXdr dv " = 0

where f+ is the distribution function for - Y > 0 and f or

< 0. In the discreteis the distri-

bution function for v ordinate form, this gives therelation at (1, j, k)

2(7r)1 /2 n n' n., n"'

nw i1 12 E E L E k.k0k7vpfv,a,Q,1a=1 a=-n' 0=1 7=-n"'

W

Similarly, a zero-mass-flux boundary condition was applied to all othersurfaces, and a Maxwellian velocity distribution function characterized bythe plate temperature was used for the molecules emitted from the surface.

Because of the symmetry of the flow inside the cubic cavity, half of thecubic cavity was used for the computation. At the symmetry plane (i.e.,k = kmax plane, the symmetry boundary condition was applied; that is,the distribution function of the outgoing molecule is equal to the distri-bution function of the incoming molecule at the symmetry plane.

Computational ProceduresFor the present study, the initial condition assumed was that the flow

everywhere was characterized by n = 1 and T = 1. As mentionedpreviously, the i = jmax surface was selected to be the moving surface.The computation began from the j = 2 surface with a known value forfa at the j = 1 surface. On each j = constant surface, the computationwas performed along the i direction first and then marching toward thesymmetry plane (k = kmax) starting with fo ++ The symbol fo ++ isdefined as the subdistribution function at energy level v for v X > 0,vY > 0, and vz > 0. At the symmetr7+plane, the symmetry boundary con-dition was applied (i.e., f ++ = f + ^. Knowing fa } at the sym-metry plane, the computation was marched backward from k = kmax - 1toward k = 1. Similarly, the computation continued for f a ++ and fa + .

After the computation reached the j = jmax surface, the particledensity was found at this surface by applying the no-flux boundarycondition and then the computation marched downward from i = jmax - 1to j = 1 surface. The order of computation was f 0

+-+

fo , fa -+ and f^ The particle density on the surfaces could thenbe found by using the no-flux boundary condition and the iteration

continued. The fourth-order Gauss-Laguerre quadrature was used for theenergy space, and four sets of the third-order odd equally spaced quadraturewere used for the molecular velocity space with the equal spacing of 0.15.(See the appendix.) The convergence was assumed when the maximumparticle density increment was less than 10-4.

ResultsAs mentioned previously, efforts have been made to calculate the flow

with as small a Knudson number as possible so that the results could becompared with the solution of the Navier-Stokes equation. The results ofthis study are all based on w = 0.1. Comparisons between the Boltzmannsolution and the Navier-Stokes solution 14 are given in figures 2 to 4.

Figure 2 shows the velocity profiles along the centerline for freemolecule flow (Knudson nu;ftbcr (Kn) = 100), slip flow (Kn = 0.1), andnearly continuum flow (Kn = 0.03). It clearly shows that the flow slips onthe moving surface for Kn = 100 and Kn = 0.1; that is, the flow velocityon the surface is less than the speed of the moving wall. It also shows thatthe no-slip boundary condition of the Navier-Stokes equation was almostrecovered for Kn = 0.03.

The comparison of the velocity vector plot on the symmetry planebetween the Navier-Stokes solution and the Boltzmann solution is given infigure 3. The general shapes of the primary vortex are similar except thatthe center of the vortex for the Navier-Stokes solution is slightly fartherdownstream than that for the Boltzmann solution.

Figure 4 shows the surface static pressure looking from the center of thecubic cavity toward the upper corner for both the Navier-Stokes solutionand the Boltzmann solution. The three-dimensional effect of the staticpressure distribution is clearly shown qualitatively. The Boltzmannsolution, even for the case of Kn = 0.03, is still not close to the continuumsolution obtained by the Navier-Stokes equation; however, the consistenttrend toward the Navier-Stokes solution is encouraging.

The symmetry plane velocity vector plots for two different Knudsonnumbers are shown in figure 5. The shape of the primary vortex is clearlyshown even for Kn = 100. The center of the vortex moves upward towardthe center obtained by the Navier-Stokes solution (fig. 3) when the Knudsonnumber is reduced. It also clearly shows that the magnitude of the velocityvectors increases as expected when the Knudson number is reduced.

Figure 6 is a plot of the number particle density on the surfaces forthree Knudson numbers. The distribution patterns for Kn = 100 andKn = 0.03 are completely different. For a free molecular flow (Kn = 100),there are not enough molecular collisions to ensure the high number particledensity at the upper downstream corner as shown for the case of Kn = 0.03.The high number particle density on the upper corner of the downstreamvertical surface (i.e., near i = imax and j = jmax) and the low number

9

particle density on the upper corner of the upstream vertical surface (i.e.,near i = 1 and j = jmax) is a reasonable solution.

The convergence history is shown in figure 7 for Kn = 0.03. Eachiteration took about 40 min on the Cray 2 supercomputer. Fortunately, itrequired less than 50 iterations to reach a convergent solution. In otherwords, it took less than 30 hr of CPU time for Kn = 0.03. The CPU timefor Kn = 100 was less than 10 hr. To reduce the main memoryrequirement, the Maxwellian distribution functions and collision frequencywere calculated repeatedly, as a consequence, extra CPU time was neededfor each iteration. The grid size used in the study was 31 by 31 by 16 inthe physical space, 12 by 12 by 12 in the molecular velocity space and 4levels of internal energy in the energy space. Considerable CPU time canbe saved by improving the integration scheme in the molecular velocityspace. In any case, the discrete ordinate method used in this study can beapplied to simple three-dimensional geometries without using the parallelprogramming technique.

Concluding RemarksA three-dimensional cubic cavity flow was solved for diatomic gases by

using the Boltzmann equation with the Bh at nag ar- Gross- Krook (B-G-K)model. A comparison was made between the Boltzmann solution and theNavier-Stokes solution for the velocity profiles along the centerline, theprimary vortex on the symmetry plane, and the surface static pressure. Thegeneral trend toward the Navier-Stokes solution as the Knudson number isreduced indicates that the solutions are very reasonable and that thediscrete ordinate method can be used with confidence to a three-dimensionalgeometry for practical predictions of rarefied-flow characteristics. Becauseof the robustness of this numerical scheme, it requires less than 50 iterationsto obtain a converged solution. The present method has a potential to bea practical flow simulation method to cover all flow regimes.

Appendix—Laguerre and Odd Equally Spaced Quadratures

Abscissa and Weight Factors for Laguerre Integration

°° n

f e E f (E) dE _ Rif(Ei)0 i=1

where Ei are the abscissas and are the roots of Laguerre polynomials.

For n = 4,

E 1 = 0.322547689619

10

E2 = 1.745761101158E3 = 4.536620296921E4 = 9.395070912301R 1 = 6.03154104342x10 -1R 2 = 3.57418692438x10 -2

R3 = 3.88879085150x10 -2

R4 = 5.39294705561 x 10 -4

Odd Equally Spaced Quadrature

1 nf f(x)dx kif(ai)0 i=1

where the arguments a i are taken to be 1/(2n), 3/(2n),..., 1 — [1/(2n)] andthe weighting coefficients k i for n = 3 are

k l = k3 = 0.375k 2 = 0.25

References'Huang, A.B. and Hwang, D.P., "Test of Statistical Models for Gases With

and Without Internal Energy States," Physics of Fluids, Vol. 16, No. 4,1973, pp. 466-475.

2Huang, A.B., Hwang, D.P., Giddens, D.P., and Srinivasan, R., "High-Speed Leading Edge Problem," Physics of Fluids, Vol. 16, No 6, 1973,pp. 814-824.

3Dogra, V.K. and Moss, J.N., "Hypersonic Rarefied Flow About Plates atIncidence," AIAA Paper 89-1712, 1989.

¢Chung, C., DeWitt, K., Jeng, D., and Keith, T., "Rarefied Gas FlowThrough Two-Dimensional Nozzles," AIAA Paper 89-2893, 1989.

5Chung, C.H., Keith, T.G., Jeng, D.R., and DeWitt, K.J., "NumericalSimulation of Rarefied Gas Flow Through a Slit," Journal ofThermophysics and Heat Transfer, Vol. 6, No. 1, 1992, pp. 27-34.

6Koura, K., "Direct Simulation of Vortex Shedding in Dilute Gas FlowsPast an Inclined Flat Plate," Physics of Fluids, A, Vol. 2, No. 2, 1990,pp. 209-213.

7Chung, C.H., DeWitt, K.J, and Jeng, D.R., "New Approach in Direct-Simulation of Gas Mixtures," AIAA Paper 91-1343, 1991.

8Lin, T.C., McGregor, R.D., Wong, J.L., and Grabowsky, W.R.,"Numerical Simulation of 3D Rarefied Hypersonic Flows," AIAAPaper 89-1715, 1989.

9Sreekanth, A.K. and Davis, A., "Rarefied Gas Flow Through RectangularTubes: Experimental and Numerical Investigation," Rarefied GasDynamics: Space-related Studies, International Symposium,

11

Y(1)

t Upper surface moves atspeed of w

X(i)

E.P. Muntz, D.P. Weaver, and D.M. Campbell, eds., AIAA,Washington, DC, 1989, pp. 257-272.

loCelenligil, M. and Moss, J., "Direct Simulation of Hypersonic RarefiedFlow About a Delta Wing," AIAA Paper 90-0143, 1990.

"Huang, A.B., Report No. 4, "A General Discrete Ordinate Method for theDynamics of Rarefied Gases," Georgia Institute of Technology, Schoolof Aerospace Engineering Rarefied Gas Dynamics and PlasmaLaboratory, 1967.

12Bhatnagar, P.L., Gross, E.P., and Krook, M., "A Model for CollisionProcesses in Gases," I. Small Amplitude Processes in Charged andNeutral One-Component Systems," Physical Review, Vol. 94, No. 3,1954, pp. 511-525.

13 Huang, A.B., "Nonlinear Rarefied Couette Flow with Heat Transfer,"Physics of Fluids, Vol. 11, No. 6, 1968, pp. 1321-1326.

14 Hwang, D.P. and Huynh, T.H., "A Finite Difference Scheme for Three-Dimensional Steady Laminar Incompressible Flow," NASA TM-89851,1987.

15 Morse, T.F., "Kinetic Model for Gases with Internal Degrees of Freedom,"Physics of Fluids, Vol. 7, No. 2, 1964, pp. 159-169.

16Holway, L.H., Jr., "New Statistical Models for Kinetic Theory: Methodsof Construction," Physics of Fluids, Vol. 9, No. 9, 1966, pp. 1658-1673.

17Chapman, S. and Cowling, T.G., The Mathematical Theory of Non-Uniform Gases: An Account of the Kinetic Theory of Viscosity,Thermal Condition, and Diffusion in Gasses, Cambridge UniversityPress, London, UK, 1958, p. 223.

18Hwang, D.P., "A Kinetic Theory Description of Rarefied Gas Flows withthe Effect of Rotational Relaxation," Ph.D. Thesis, Georgia Instituteof Technology, Atlanta, GA, 1971.

Use half thecube forcomputation

Symmetry plane J

z(k)

Figure 1.--Geometry.

12

Symmetryplane

1.0

.8

m EDQs ID

q0.6 — qO

qOq0

r

EID

.4

IMCD

.2

0-.04 0

00 Do 0 o 0 A

O WA3A

O Kn=100q Kn = 0.1

Q Kn = 0.03A Re = 100, incompressible

Navier-Stokes(from ref. 14)

.04 .08 .10

ux

w = 0.1

/

/

Centerline

Figure 2.—Velocity profiles along centerline.

Sll7fli.---- »____--.^YYsslllS i S I S l 7 7 r r i__,, i l 1^ J 1 1s111tssY „ l^!lr+JlJlltlsl\tY^•,__rrlllJJJfI

. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .

(a) Navier -Stokes solution (Re = 100). (b) Boltzmann solution (Kn = 0.03).

Figure 3.—Velocity vectors on symmetry plane for Navier-Stokes solution and Boltzmann solution.

13

Page intentionally left blank

MovingI

surface ^

YY

Downstreamce

Movingsurface

^– Downstream/ vertical surface

Symmetryplane

(a) Navier-Stokes solution (Re =100). (b) Boltzmann solution (Kn = 0.03).

Figure 4.--Surface static pressure for Navier-Stokes solution and Boltzmann solution.

.. r - - - _ - _ - - - - - - - - - - .. .

Ir t , . . .- - - - - - - - - - - - - - -

1 1 I r I , . . . . . . _ . . . . . . . . . . . 1 1 1 1

1 I, I. I. .., I I/ 1 1 1 1

1 1 1 1 1 1 1 1 1... 1. 1 1 1 1 1 1

iftitSlr+.r__,Ilfllll1111tt1,1,1 ,rffllll^11t\\t\,...........!l/lJJJ1 ^

t l t \ \ \ \ ._ - - - _ _ - / / /

(a) Kn = 100. (b) Kn = 0.03.

Figure 5.—Velocity vectors on symmetry plane for two Knudson numbers.

15

Page intentionally left blank

Downstream

Moving surface —\ _.Adbk vertical surface

(a) Kn = 0.1.

4

(c) Kn = 0.03.(b) Kn = 100.

C N

I

LEUELS

PIPS

Figure 6.—Number particle density on surfaces for three Knudson numbers.

17

Page intentionally left blank

.008

.007

.006

.005

ro

.004d

003

.002

.001

10 20 30 40 50 60

Iteration

Figure 7.--Convergence history for Kn = 0.03.

19

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1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVEREDJuly 1992 Technical Memorandum

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Numerical Solution of a Three-Dimensional Cubic Cavity Flow by Usingthe Boltzmann Equation

WU-505-62-206. AUTHOR(S)

Danny P. Hwang

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATIONREPORT NUMBER

National Aeronautics and Space AdministrationLewis Research Center E-7080Cleveland, Ohio 44135-3191

9. SPONSORING/MONITORING AGENCY NAMES(S) AND ADDRESS(ES) 10. SPONSORING/MONITORINGAGENCY REPORT NUMBER

National Aeronautics and Space AdministrationWashington, D.C. 20546-0001 NASA TM— 105693

11. SUPPLEMENTARY NOTESPrepared for the 18th International Symposium on Rarefield Gas Dynamics, sponsored by the University of BritishColumbia, Vancouver, Canada, July 26-31, 1992. Responsible person, Danny P. Hwang, (216) 433-2187.

12a. DISTRIBUTION/AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE

Unclassified - UnlimitedSubject Category 34

13. ABSTRACT (Maximum 200 words)

A three-dimensional cubic cavity flow has been analyzed for diatomic gases by using the Boltzmann equation with theBhatnagar-Gross -Krook (B-G-K) model. The method of discrete ordinate was applied, and the diffuse reflection bound-ary condition was assumed. The results, which show a consistent trend toward the Navier-Stokes solution as the Knudsonnumber is reduced, give us confidence to apply the method to a three-dimensional geometry for practical predictions ofrarefied-flow characteristics. The CPU time and the main memory required for a three-dimensional geometry using thismethod seem reasonable.

14. SUBJECT TERMS 15. NUMBER OF PAGES

Boltzmann equation; Cubic cavity flow; Kinetic theory; B- G -K model, Discrete 2016. PRICE CODEordinate method

A0317. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACT

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