Structure of normal shock waves: Direct numerical analysis of the Boltzmann equation for...

Structure of normal shock waves: Direct numerical analysis of theBoltzmann equation for hardsphere moleculesTaku Ohwada Citation: Phys. Fluids A 5, 217 (1993); doi: 10.1063/1.858777 View online: http://dx.doi.org/10.1063/1.858777 View Table of Contents: http://pof.aip.org/resource/1/PFADEB/v5/i1 Published by the AIP Publishing LLC. Additional information on Phys. Fluids AJournal Homepage: http://pof.aip.org/ Journal Information: http://pof.aip.org/about/about_the_journal Top downloads: http://pof.aip.org/features/most_downloaded Information for Authors: http://pof.aip.org/authors

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Structure of normal shock waves: Direct numerical analysis of the Boltzmann equation for hard-sphere molecules

Taku Ohwada Department of Aeronautical Engineering, Kyoto University, Kyoto 606-01, Japan

(Received 15 May 1992; accepted 15 September 1992)

The structure of normal shock waves is investigated on the basis of the standard Boltzmann equation for hard-sphere molecules. This fundamental nonlinear problem in rarefied gas dynamics is analyzed numerically by a newly developed finite-difference method, where the Boltzmann collision integral is computed directly without using the Monte Carlo method. The velocity distribution function, as well as the macroscopic quantities, is accurately obtained. The numerical results are compared with the Mott-Smith and the direct simulation Monte Carlo results in detail. The analytical solution for a weak shock wave based on the standard Boltzmann equation is also presented up to the second order of the shock strength together with its explicit numerical data for hard-sphere molecules. “

1. INTRODUCTION

The structure of normal shock waves has extensively been studied as one of the most fundamental nonlinear problems in rarefied gas dynamics.‘-” This problem has also been used as a quantitative test case for numerical methods of the Boltzmann equation such as the Monte Carlo method.7-‘3 Nowadays, the accurate numerical solution is available (or can be obtained using a personal com- puter) for the model equations15-i7 such as the Bhatnagar- Gross-Krook (BGK) model. As for the original Boltzmann equation, however, the standard solution has not yet been reported.

Recently a numerical method of the linearized Boltz- mann equation for hard-sphere molecules has been developed” and geometrically simple problems have been analyzed accurately. ‘g-2’ In this method, the collision integral is computed directly without using the Monte Carlo method. Similar to the case of the linear problems, direct numerical approaches seem to be advantageous for the pre- cise analysis of normal shock waves, although the nonlinear collision term is more complex and requires an enor- mous amount of computation.

The direct numerical analyses of this problem based on the full Boltzmann equation have also been tried by Chorin” and Tan et aI. Judging from the form of the distribution function predicted by Mott-Smith solution,” the lattice systems used in their computation (at most five points for each velocity variable) are not sufficient to es- tablish the reliable solution. As for the BGK model equation, it is necessary for the reliable computation to use at least 10-20 lattice points in each velocity component. In their methods, however, it is very difficult (or almost im- possible) to carry out such computation at the present stage.

In this paper, the author proposes a new direct numerical method of the standard Boltzmann equation for hard- sphere molecules, where the collision term is computed accurately as well as efficiently, to obtain the accurate so-

lution of the normal shock wave. The numerical results are compared with the Mott-Smith” and the direct simulation Monte Carlo results. The analytical solution for a weak shock wave based on the standard Boltzmann equation is also derived up to the second order of the shock strength together with explicit numerical data for hard-sphere molecules.

II. FORMULATION OF PROBLEM

A. Problem and notations

Consider a plane shock wave perpendicular to a flow. The flow is in the Xl direction, where Xi are the space rectangular coordinates. The gas is uniform at upstream infinity (X1= -CO) and downstream infinity (Xi = CO) and the whole flow is stationary. We analyze the structure of the shock wave on the basis of the standard Boltzmann equation for hard-sphere molecules.

We summarize the main notations in this paper: po, Ue, and To are, respectively, the upstream density, flow speed, and temperature of the gas; po= kpoTdm, where k is the Boltzmann constant and m is the mass of the molecule; Xi= ( @c/2)x; Ze is the mean-free path of the gas molecules at upstream conditions (for a hard-sphere molecular system, Ze=[v%rd( PO/m)]-‘, where (T is the diam- eter of the molecule); (2kTo/m) 1’2~i is the molecular velocity; po(2z-kTdm) -3’2f(Xi,Si) is the velocity distribution function of the gas molecules; pop, (2kTo/m) 1’2Ui, TOT, pOp, pO(psij+pij)9 and po(2kTo/m) lnQi are the density, flow velocity, temperature, pressure, stress tensor, and heat flow vector of the gas, respectively, where Si/ is the Kronecker delta; and M= (5kTo/3m) -1’2U0 is the upstream Mach number.

5. Basic equation and boundary condition

The standard Boltzmann equation in this steady and spatially one-dimensional case is written as

217 Phys. Fluids A 5 (I), January 1993 0899-8213/93/O? 0217-l 8$06.00 @ 1993 American Institute of Physics 217


!G-1 g-a f,fl -y[ flf, (1)

where G[f,f] and ~[f] are the gain term and collision frequency, respectively. The G[f,g] and ~[f] are usually expressed in the form

G [ f,gl (xl,Ci;.) = f f(xl,G )SCXltgi)

X B( I v,cliI ,v)do(ai)dgl dc2 &3 9 (2)

I’II fl (xlLi)

= s

f(xl,6j)B( 1 vfajJ~V)d~(Orj)d(l dgzd5;) (3)

where

Vi=~i-~j, V= ( V~) 1’2, (4)

fi=[j+aj( Vpj), (Tiil=cj-ai( Kpj),

oi is a unit vector, dO(cr[) is the solid angle element in the direction of ap and B is a non-negative function, the form of which depends on the intermolecular force law. For hard-sphere molecules, B is given by

B(j VPi/,V)=4,,;2 1 VPil- (5)

The domains of integration with respect to gi and CLi in Eqs. (2) and (3) [and in all the following integrals with respect to the molecular velocity (gi or ~i:i, and unit vector (ai or ai)] are the whole velocity space and all directions. We show another expression of G[f,g] for later use:

G [ f*gl (Xl&i)

xB( Ijy(~,vy)\j~aa@ tii) xdCld& 45 2 (6)

G !i f,gl (x,,Ci>

=2 s f(x l~~i+~i)g(xl~~i+ w i) X B(Cp &?T@)C-” ds( JJ’i)&Tld& d63, (7)

where 5,. is a unit vector, c= (SF) 1’2, Wi is a vector in a plane orthogonal to pi, dS( Wi) is its surface element, and

the domain of integration with respect to Wi is the whole plane. The relation between OLi and Si is

zii=[-Vi+2ai( VjXj)]/V,

1 2v dWai) =; vFj+ v

d do(&).

While oi moves over the whole unit sphere pi moves over it twice. Equation (6) is derived from Rqs. (2) and (4) using the above relation [cf. Ref. 24 for the derivation of Eq. (7) from Eqs. (2) and (4)].

The nondimensional macroscopic variables are given by the moments off:

p=$n j-f 4142 d5; 9 (84

1 ui=3/2

s Ci.f 4’142 dC3 2 (8b)

-i-i- P

T=& J (Cj-uj)2.f d<, dC2 dC3 9 (8~)

P=pT, (8d)

Pij= s

(ci-ui) (Cj-uj)f dC1 dC2 dC3-J’aij 9 (gel

Q~=A J (ci-ui) (cj-uj12f dS1 db dC3 * (80

The boundary condition at upstream and downstream is

f-exp[-(Cl- J5/6M)2-&~~l (XI-+-CO),

f -+pJT3’2 ewC- [K1-~d)2+~~+&/~d}

(x 1-m 1, (9)

where pd, ud, and T, are given by the Rankine-Hugoniot relation:

(loa)

ud= ,,%%(M2+3)/i%& (lob)

T,= (5M2- 1) (M2+3)/16M2. (1Oc)

218 Phys. Fluids A, Vol. 5, No. 1, January 1993 Taku Ohwada 218


C. Similarity solution and collision term

Consider the function in the form Wj= Ymj+ZlZi,

where

f=T(xbS,,cr), L= (~;+wz- (11)

The term c, 87/&t is obviously a function of x1, c,, and 5; From the invariance of the collision operators G and Y under the rotation of pi, i.e.,

G[ f(xl,R6i>,f(xl,RCi) 1 (xl,Ci;i)

=G[ f(Xl,ifi)d(X1,Ei;.) 1 (x,3(i), (12)

ylI f(xlsRCii) 1 (xlsLZi) ‘Y[ f(xl,{j) I (XIJt;,),

where R is any rotation matrix of 3 x3, it is seen that G[f,T] and ~[f] are functions of x1, cl, and [,.. Thus the form of 7 in Eq. (11) is compatible with the Boltzmann equation ( 1). It is easily seen that 7 is also compatible with the boundary condition (9).

Corresponding to the form of fI in Eq. ( 11)) the macroscopic quantities are expressed in the form:

(134 F&d&~, ~0s 6 sin G YZ)

P,,= -2P22’ -2P33

(m721,m2,m3) =(O,-sin e, cos E),

(nl,y12,n3) = (sin I$-cos 8 case,-cos 8 sin e),

and t;i, mi, and ni are orthogonal to each other. Then, noting the relation dS( Wi) =dY dZ and dg, dc2 dc3 = c2 sin 8 de dt3 d,.$, we have

G[ L,M I= JOT J:T Jorn J:, s_mm ~&d-&6

co s s co Cry 4-r d<, 9 --m 0

4 m T=----m- s s 3%- P -*

m [(~,-ul)2+~~ll~sdS,d~l, 0

(13c)

Glb~2Ld = (Sdrcos *,L sin $1,

(cl,c2,{3) = (5 cos f3,{ sin 8 cos E& sin 8 sin e),

= jov sf, Jam Jim s_mm FG(~&-JJ% cos c?, sin .?, Y,Z)dY dZ d[ dFdde, (144

where

4 m =;,m s s

m (<I - ~1 I”&.? 4, dC, -03 0

-pT (other Pii=O), (13d)

QI=--$ srn SD (51-~1)[(51-~,)~+~~lfrf --m 0

As a preparation for the numerical method described in the next section, we show the explicit form of G~~(~&W’(~&,.,)1 and that of 4UC&,)l, where s”,= (g:+w2 and L and M are arbitrary functions of 5, and <,.. In Eq. (7)) we express ci and f;i in terms of cylindrical and polar coordinates, respectively, i.e.,

=2Lg1 +g cos f3,J)M(g, +Z sin em

Xsin 8 B(& Jm),

(14b) J= (cf + c2 sin2 8 + 2& sin 8 cos E3 1’2,

K=(gf+ Y2+z2 60s~ 8--Y&sin s-2Zi$.cos 8 cos 5)1’2.

In Eq. (3), we express {i and ci in terms of cylindrical coordinates, i.e., (gd2;,{3) = (Ed, ~0s 1c1L sin $1 and ( g1,t2,g3) = ($-Jr cos +,LJ, sin 4), and express (Ti in terms of the polar variables 8, and E,, where 8, is the polar angle between Vi and ai and E, is the azimuth angle in a plane orthogonal to VP Then, noting the relation dfl (a,) = sin Bcx dea de, and d{, d(-z d$, = cr d$ dc, dl,, we have

= J:, s,” j-1, s,” s_:, FJ!z-1&C1& cos a%) x 6 de, d? 4, dt, , (Isa)

and write ?Vi in the form where



FvK&~l& COS 6%) = wd-,I B( VI ~~~e,[,~~g.sine,,

~=[(~~-j-,)2+~~+~~-2~~~cos~1"2. (15b)

III. NUMERICAL METHOD

A. Finite-difference scheme of the Boltzmann equation

In the actual computation, we replace the region - CO <xl< CO by a finite interval - Xx,< D. The constant D is chosen large enough so that the deviation of the distribution function from the upstream (downstream) Max- wellian around x1 = - D (x, = D) is negligibly small or

smaller than the error of the numerical computation. At x1 = f D we impose the boundary condition

and solve the two-point boundary-value problem (1) and ( 16) by numerical iteration.

Let xji’ (i= --ND ,..., 0 ,..., N,, (--ND) = -D

(ND) xi

3 = D) and (&‘,~!k’) (j=-N, ,..., 0 ,..., JV,,,’ k= l,...,H) be the space lattice points and_those in the clcr @ane (&SO), respectively. We denote f(~{~),~r,~,.) and S(X~‘),&),C!~)) at the nth iterative step by fl(ci,c,) and fljk, respectively, and use the following iterative formulas:

where

$jkEyI: .fy(iZl,&r) I (C1=Cfi)tSr=!$k))t (19)

and the upper and lower signs go together in Eq. (17a). The finite-difference expression of the boundary condition (16) is just the values at the lattice points.

In Eqs. (18) and (19), we first approximate fl(cl,cr) with respect to j, in a similar way to the finite element method:

f;(~*L,> = j=$N f3SYmqK1L (20) m

where Yj(~l) are basis functions. The basis functions are chosen in such a way that a function is approximated by a sectionary quadratic function that takes the exact values at the lattice points. The explicit definition of Y -(cl) will be

(3 given in subsection B. Next, we expand fF(c,’ ,c,) by the sequence of Laguerre polynomials:

H-l

f;(Q),&) = mso a&&,,(<3)exp -$ Y ( 1

(21)

where L,(y) is the Laguerre polynomial of degree m. The expansion coefficient a$,, is given by

aTjm = 2 s

* fr(51”,~,)Lm(Sf)Srexp 0

(22)

where the set L,(y)exp(--y/2) (m=0,1,2,...) is complete on L,(O, 03 ) and satisfies the orthogonality conditions

s OE L;(v) Lj(~)exp(-Y)&=Sij - 0 We introduce a nonuniform c,. lattice system defined by

c;“‘= & (k= l,...,H), (234

where

L&k) =o (=vz, for I<m), (23b)

and approximate fF([f”,[,) in Eq. (22) by the Lagrange interpolation polynomial with the weight function exp(-&2):

f;(p < 9 r 1

=exp (-$) ~,.fYjkeXP(~) JI(g). (24) Ifk

Using the Christoffel-Darboux formula,25 we have

n aijm = ki, wmkfljk,

(25)

hAzk)expW2) wmk= f

CH- I,H- ,LH- I (zk) n (zk-zh ld<H

l#k

where ci/ is the coefficient of xi in Lj(x). Combining Eqs. (20), (21), and (25) and arranging the result in the order of power of cP we obtain



8. Numerical kernels dk pqab and “$a

We introduce a uniform lattice system for c,, i.e.,

@‘=jh (-N,<j<NJ, (33) where

A$a= 5 Dmkfnijk, k=l

(27)

H--I

Dmk= 2 ~mlwlk, Mnd=

0 (m>O,

C,I (O<m<O. (28)

ISO

Substituting Eq. (26) into Eqs. (18) and (19), we have

Gjkz- (29) p=-N, q=-N,, a=0 b=O

where h is a constant, and introduce the following sequence of localized functions:

~~I(I;,)=~EY(C,--~~~)

~Vzr+l(~l)=Yoo[~l-(2z+l)hl (Z=O,* 1,*2,. . . 1,

Wa)

where

A$=v[Yp(f;,)g’“] (g,=p g =gCk)) r ,r * 9 (32)

(30) (34b)

=CH-i,H-,LH-l(Zk)~kl , g, @k--d . c ff#k

where

i

G-h)G-2W2h2 (O<c,<2h),

Yd5,) = K,+h) G+2hV2h2 (--2h<<, co),

0 (otherwise),

with

E,=exp( -&2).

The complexity of the collision term appears only in the numerical kernels $&, and A$ both of which can be computed and stored in advance. Using the numerical kernels, Ghk and %fjk can be computed efficiently in each iteration. The numerical kernel methods have also been pro- posed in Refs. 18 (for the linearized Boltzmann equation) and 23 (for the nonlinear one).

It can easily be verified that Laguerre polynomial expansion (21) with Eq. (25) is equivalent to Eq. (24) using the following equality derived from the Christoffel- Darboux formula:

H-l

2 ‘%&kk) ‘$,h) m=O

Therefore, in Eqs. (26)-(28) fl(<&) is approximated by a function that takes the exact values at the lattice points (~~fi,[~k’). The macroscopic quantities are also computed using Eqs. (26)-(28).

-(&h)(9,+Wh2 (--h<C,<h), ~ODGl) =

0 (otherwise).

In the computation of fi$,b and A$, we use the following two sets of basis functions (the reason why two sets are necessary will be seen later):

Yj(~*)=\u,(~*) (i=O,il,~l~2; * I, for j=21), (35)

and

yi(Cl)=Gi-1(51-h) (i=O,*l,&2; . ., for j=21+1). (36)

From the uniformity of the t;, lattice system and prop- erties of the operators G and Y, some relations among fi,&,b and those among A$ are derived. They are summarized as follows.

(i) From Eqs. (33) and (34), $;,(c,) satisfies

(37)

The operators G and Y are invariant under the translation of the velocity variables:

G[ f((i),g((i)] (Ci)=G[ f(Ci+ai)d6i+ai)I (LYi-ai)t (381

VI f(&i) 1 (Ci) ‘Y[ f(6i+ai) 1 ([imail (39)



where ai is an arbitrary constant vector. In Eqs. (3 1) and (32), we consider the two cases j=21 and j=21+1 and make use of Eqs. (35)-( 39) with ai= (c{“,O,O). Then, we have

fl gab= fiOlk p-iq--i& (40)

Ajk,AW PO P--/SC (41)

It should be noted that only one set of basis functions is necessary to derive Eqs. (40) and (41 j when they are in the same shape (e.g., hat function).

(ii) From the mirror image relations

Gp(-5,) =Kp(5d (42)

and

G[ L(&&;,),~(61&) 1 (o&j

=G[LI-~,,~,),M(-~,,~,)l(O,~,), (43)

we have

fi’J,k pwb =Ry.&-q,ob. (44)

In a similar way,

AO.k = AW Pa -PG. (45)

(iii) For hard-sphere molecules, the following symme- try relations are derived from Eqs. (5) and (6) :

‘2 ftgl =GkJ-I, (46)

=G[f,(~l)g2(~,)lf2(~1)g1(~r)l. (47)

Then, we have

@A m&k pqab - qpba, (48)

aO,k -@A pqab - pqba’ (49)

Owing to the above relations, we can greatly save the memory size required for the data storage of the numerical kernels. Let the number of the lattice points in each velocity component be of the order of N. Then the number of independent fig&, is reduced from O(N6) to O(3N5/8) and that of A$ is reduced from O(p) to O(N3). In Tan’s method,23 the distribution function is expanded by a set of basis functions in fi space, each of which is constant ( = 1) in a cube and vanishes outside it. That is, the distribution function is approximated by a piecewise constant function that takes the exact values at the lattice points. The num-

ber of the independent elements of the numerical kernel for the gain term is reduced from O(M) to O(64N6) using the uniformity of the pi lattice and Eq. (38). This method can be applied to 2-D and 3-D problems in principle, since the similarity solution ( 11) is not used.

From Eqs. (14), (31), and (35) and ci’)=O, @$, is expressed in the form:

qgb==2 Jo= j:, jam s_mm jmm GpG 03s eG?

7

( )

-2

x exp -2 G,(Z sin f3)E~b exp -2 ( 1

(504

Jk= [ (g:k))2+c2 sin2 0+2~~~k’ sin 8 cos Cl 1’2,

(Sob) Kk= [ (<Lk)j2+ Yz+z2 ~0s~ e-2Yg!k) sin E

-22~~;~) cos 8 cos 51 l12.

For hard-sphere molecules, B( 6, ,/m) depends only on the first argument Ecf. Eq. (5)]. In this case the inner threefold integral in Eq. (50a) can be written as the product of an integral with respect to 6 and a twofold one with respect to Y and 2. Furthermore, from Eq. (34) the domain of integration with respect to 8 and E can be reduced to one-fourth. Thus, we have

@.k 1 d2 r

wb=m o s s o r;a(e,c-@,(e,adFde (p-o),

(51aj

(51b)

where

$(e,a =sin 6 s ?)- 0

\v,(~cos ej2kaexp

\V,(Zsin O)izb exp

(53)

[cf. Eqs. (44), (48), and (49) for the other cases of (p,q)].

The integration in Eqs. (52) and (53) can be carried out analytically (cf. Appendix A) and the twofold integrals in Eqs. ( 5 1 j are computed by the Gauss-Legendre formula.‘5 Next., we express A?;?” using Eqs. (5), (15), (32), and (35)

222 Phys. FIGids A, Vol. 5, No. 1, January 1993 Taku Ohwada 222


and S\“’ =0 and carry out threefold integration with re- ture of a weak shock wave analytically up to O(2) on the spect to eat 8,, and 5. Then we have basis of the standard Boltzmann equation.

vz m AOk=-

pa Tr s s m rg+ (&+gk’j21 1’2 --m 0

xexp

where E(x) is the complete elliptic integral of the second kind:

s

d2 E(x) = (l-xsin2y) 1’2 dy (O<x<l).

0

The twofold integral in Eq. (54) is computed by the Gauss-Legendre formula limiting the domain with respect to & suitably.

In the present numerical method (hard-sphere molecules), each element of the numerical kernel for the gain term is reduced to a twofold integral, which is advantageous for the accuracy. If the basis function expansion is used with respect to c, instead of the Laguerre polynomial expansion, each of them is computed by fivefold numerical integration and its domain is very complicated.

IV. ANALYSIS OF WEAK SHOCK

A. Preliminaries

It is well known that the Navier-Stokes equation gives a hyperbolic tangent profile for a weak shock wave (Tay- lor’s solutionZ6). According to it, the ratio of the shock thickness to the mean-free path is of the order of the re- ciprocal of the shock strength, i.e., S/Z,=O( I/E), where S is a suitably defined shock thickness parameter and E is a shock strength parameter, e.g., e=iti- 1. For a very weak shock ( E( 1 ), more accurate computation is required in the collision term, since the error induced by the large computation of it accumulates over the wide range of x1 and it affects the small variation greatly. In this case, however, analytical approaches become advantageous similar to the case of small Knudsen numbers [for the other extreme limit (EN 1)) an analytical approach has been suggested by Grad27].

The weak shock profile is also derived from the Boltz- mann equation by a power series expansion of E under the assumption S/1,=0( l/e), which has been shown by Hu’s and Cercignani.“‘29 According to their results, the lowest order of the expansion is a constant sonic state and the hyperbolic tangent profile appears in the next order [O(E)]. A similar systematic analysis using the BGK model was carried out by Miyake3’ and the result up to 0( e’) has been obtained [analytically up to O(E’)]. Before proceed- ing to the numerical computation, we investigate the struc-

B. Analysis

The analysis is carried out in a similar way to the Hilbert expansion.2*29 We introduce a new space coordinate q =exI, where e=M- 1. For the convenience of analysis, we use the following independent variables and func- tjon: n ($,,${3) = (51 - $m,g2,m, $c= (W2,

f(17,51,52,53)=f(XllS1,~2,2,5‘3). Then the Boltzmann equation ( 1 j for j is written as

i $1 f,+ &f al(T,~ij 1 371 =E Q[ .?Jl (rl,ti)t (55)

where

Q[ &I =#I &?I +W,~l-4 &?-Y[&.? I.

We expand ?into a power series of E around the upstream Maxwellian f. = exp ( - p) :

~(rl,~i)=~o?bl:1+~*(77,~i)E+~2(17,~i)E2+ * . * 1. (56)

Substituting Eq. (56) into Eq. (55) and assuming a~i/arl=O( l), i.e., S/1,=0( l/e), and arranging the result in the order of power of E, we have the following sequence of linear integral equations:

n A

x?r fOLfOrbl1 =(A (57a)

. . . ,

- l+;=, Q[ ~04~~~ohnl (03).

ISI lr;m

(57c)

The boundary condition (16) is expanded in a similar way to Eq. (56). The integral equations are, in principle, solved from the lowest order. The homogeneous integral equation [(57a)] has nontrivial solutions { l,~,,~} and the coefficient functions of { 1,c1,t2} in +,, functions of v, are deter- mined by the solvability conditions of (n + 1) th and (n +2)th integral equations and the boundary condition for 4,. The position of solution is arbitrary in this problem



and we determine it in such a way that the density corresponding to f&,, p”(v), satisfies p”(O) =[p”( - co j +p"(+co)1/2.

C. Analytical result

We show the results up to O(g) below. The distribution function is

#,=a(q)(- JEl*+Pj, (584

~,=b,(rl)+b2(r])~l+b3(rl)P+a(rl)2[- mt,

~(&5/2)+(15/4)(~~-~~/3j+(1/8)

x (4&20p+15)] + ,/%A~’ exp( - &//2,)

XaW2[ Jrs/s(%:-{2/3j B(f) -&4(t) It

where

(58b)

a(v)=[l+exp(- $&//z~)l-‘,

~,(~)=(l5/2)~(r1)[-~~+(~~-1/2)~(~)1,

b2(77) = J15/2-w3b7) +a(rl) [5&-l

-(%--5/2)dv)l>, (59)

b3(rlj =dqj2(5/2+exp( - I./G/~)

XUirlf~3 log[l+exp( 1/3071/A,)] +&}),

a,= J15/2C--8(~+&‘J/A:+ l/A,], (60)

h= - WW2+83/J/A~

&= y&,, j15=5/24-A3 log 2.

A

The A (5) and B(c) are the solutions of the following integral equations:

ZQ[ ~ohuh~oi = -.fo&&5m7 (61)

I om pA(l)exp( --$)dl==O,

2Q[ ?o(&&, B(&,j-d= -2 j-o&&3), (62)

and ‘yi are constants given by

224 Phys. Fluids A, Vol. 5, No. 1, January 1993

y1=T6(Bj, y2=21&4), y3=fdABA

Y4=-(5/2)y1+Is(B)+(1/2)16(BC), (63)

y5= --6y,+.u&4)+214(AG),

where

I,#) = (8/15)rr-,‘2 I

m &Fb%xp( -&& 0

and C(t) and G(l) are defined by

2Q[ ~005',~0(~-~~3~B(~~1=~0(~-~/3)[~(~) -31, (64)

The macroscopic quantities corresponding to the above distribution function are

p=li- w)drlkS [hw i- w)w7> k% - . .,

(664

u1= J5/6+ [ $I= JmQr]j]E+ [b2(77)/2

+ ~iEmz(~j2]~+ * * *, (66bj

T=l+a(r])E+tb3(rl)-(11/4)a(r1j21EZ+. . *, (66c)

= lOy,R,’ exp( - ,/%~/A,)L~(~)~E~+ . . . ,

(66dj

Q,=-(5/4) $&~~A.;‘exp(-- \I?;Tjg/A,j

Xa(q)22+ * . . . (66e)

The accurate numerical data of A(t) and B(t) for hard- sphere molecules were first obtained by Pekeris and Alterman3’ and the recomputed data are tabulated in Ref. 32. The quantities of pi for hard-sphere molecules are

y,= 1.270 042 427, y2= 1.922 284 066,

y3= 1.947 906 335, y,=O.635 021 214, (67)

ys=O.961 142 033, y,=4.893 662 449.

The above results up to O(E) are also derived from the second-order (Navier-Stokes j solution in Chapman- Enskog theory33 when a similar perturbation expansion is

Taku Ohwada 224


applied.‘7129 Similarly the above results up to O(z) are derived from the third-order (Burnett) solution. The functions A(l3, B(t), CC&, and G(c) and the constants y) except y= appear in the asymptotic theory of slightly rarefied gas at finite Reynolds numbers derived by Sone and Aoki.3k36 The values of y4 and y5 for hard-sphere molecules in Refs. 35 and 36, which have been obtained by Ohwada, are not accurate and contain errors at the fourth figure. They are corrected in Eq. (67) with more accuracy.

V. NUMERICAL RESULTS

A. Numerical computation

We prepared three lattice systems for the molecular velocity space. In each system, f, lattice points and I& lattice points are defined by Eq. (33) and Eq. (23), respectively, with the following values of N,, NP, h, and H:

(Ml) N,n=44, N,=56, h=0.15, H=lO, (M2) N,=26, N,=34, h=0.25, H=14, (M3) N,,=44, N,=56, h=0.15, H=14.

In the (Ml) system, ~~ivm’ = -6.6, $N” = 8.4, <F” =0.3712..., and 65” = 5.4699... . There are 1010 lattice points in the <i& plane. In the (M2) system, ciNm, = - 6.5, {is’ = 8.5, <:“=0.3158 . . . . and ~$~‘=6.6607... . There are 854 lattice points in the gig,. plane. In the (M3) system, 5, lattice points are the same as those in (Ml > and

(a) --4 -2 0 2 4 G

1.0 1 ,A 2, = 0

f(%G

TABLE I. The present numerical computation.

M Lattice system

Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 Case 9

1.2 @3,M3) 1.2 (S3,M3) for 7 1.59 WW3) 2.0 WW’) 2.5 W,M3) 3.0 WLM3) 3.0 (glN3) 3.0 (SU42) 3.0 6041)

5; lattice points are those in (M2). There are 1414 lattice points in the c1cr plane.

The space interval - D&x, < D is considered for two cases, i.e., 0=20 and 30. We prepared three lattice system for x1. In each system, the space lattice points are given by Eq. (Bl ) (cf. Appendix B) with the following values of D, ND, and d:

(Sl) D=20, ND=25, d=O.l, (S2) D=20, ND=50, d=O.l, (S3) D=30, ND=50, d=0.2.

In the (Sl) system, there are 51 lattice points with the minimum width 0.200 around xl =0 and the maximum one

FIG. 1. The distribution function ~(x,.<&) at three typical points in the gas for M=1.2 (case 2): (a) x,=-14.444, (b) x,=0, (c) x,=14.444. The surface is shown by &=O, I;,=t$!“)/v”L, and [, =g12j)/vZ iines [(M3) lattice, k=l,..., 14, j= -22 ,..., 281. The c,=O lke is drawn using Eq. (26).



2.554 around xi = j=20. The (S2) system is finer than (Sl); there are 101 lattice points with the minimum width 0.100 around x,=0 and the maximum one 1.283 around x1 = =!=20. The (S3) system is coarser than (S2) but the space interval is extended (0=30). In this system there are 101 lattice points with the minimum width 0.200 around x,=0 and the maximum one 1.777 around x1= *30.

The numerical computation has been carried out for M= 1.2, 1.59, 2, 2.5, and 3. We started the iteration from a suitable smooth state, e.g., the corresponding numerical solution of the BGK model. As the accuracy test, we have carried out the computation for different lattice systems in the case of M=3. We-have also carried out the computation for ~(~i,~i,~~) =f(~,,2-*‘*~~,2-~‘*5~) in the case of M= 1.2, because the computation for 7 is advantageous for Laguerre polynomial expansion (21), since the distribution function is nearly Maxwellian, i.e., f-exp( --cF) and therefore T-exp( - &2), and the leading term of the expansion is proportional to exp( -&2). The present numerical computation is summarized in Table I. We have also confirmed that in each case shown in Table I the accurate numerical results are obtained for the BGK model.

For the comparison, we have also carried out the direct simulation Monte Carlo (DSMC) (hard-sphere molecules; M=3) using Bird’s no-time-counter (NTC) method and the piston boundary condition at downstream boundary” under the following condition: the space interval is -2O<x,<20 and is divided into 100 uniform cells (cell size AX=0.4); the number of the simulation particles P,, is about 32 000; the initial state is the upstream Maxwellian for x1 < 0 and the downstream one for x1 > 0; the time step is At=O.Ol, where (7r~/8k7’,)“*Zot is the time; the start- ing time of sampling is t=800; the sampling interval is At,=O.S; the number of the flow-field samples is N,=8000.

The numerical computation was carried out on Fujitsu FACOM VP-2600 at the Data Processing Center, Kyoto University. The CPU time during one iteration step is about 100 set for the (M3 S2) lattice system.

5. Distribution function and macroscopic quantities . The distribution function 7 at typical points in the gas

for M= 1.2 (case 2), 2 (case 4), and 3 (case 6) are shown in Figs. l-3. The density, flow velocity, temperature of the gas versus x1 for M=1.2, 1.59, 2, 2.5, and 3 (cases 2-6) are tabulated in Tables II-V. For the convenience of comparison, we define the position of a shock wave, x1 =S, as that of the point with the average density, i.e., p(S) = ( 1 + pd)/2, and introduce a new space coordinate 5, =x1 -S. The profiles of normalized density p = ( p - 1 )/ (Pd-l),flow~elocity~i = (u, - Q)/( J5/6M- ud),and temperature T=(T-l)/(Td-1) vs Yi for M=1.2, 2, and 3 (cases 2, 4, and 6) are shownin Figs. 4-6 together with the Mott-Smith4 corresponding to the moment equation of sc:f dc, dcZ dc3 (#=u*; there are some typo- graphical errors in Ref. 4), the DSMC results of Erwin et al. ‘* [hard-sphere molecules (s=O.5); M= 1.2

(P,-20 000, N,-35 000) and M=2 (P,- 10 000, N,- 13 000)], and the present DSMC result (M=3). The agreement of the Mott-Smith with the present numerical results is good for M=2 and 3 and that of the DSMC result for M=2 and 3 is excellent [the DSMC result’* for M=2.5 (P,- 10 000, N,-26 000) is also in good agreement with the present numerical one]. The profiles of the stress tensor P,, and the heat flow vector Q1 vs 2, are shown in Figs. 7 and 8 together with the Mott-Smith and the analytical result (66d) and (66e) [M= 1.2 (~=0.2)]. The distribution function of the present numerical result for M=3 (case 6) is compared with those of the Mott- Smith and the’present DSMC in Fig. 9. A distinct difference can be seen between the Mott-Smith and the present numerical result. The DSMC result agrees well with the present numerical one at the microscopic level as well.

C. Accuracy of computation

We summarize the main accuracy data of the present computation.

(i) The absolute value of the numerical collision term, [ ~j~-~j~~’ I, for the upstream Maxwellian and that for the downstream one, which should theoretically be zero, are less than

[3.3x IO-’ (Ml)],

[2.5x 1O-4 CM2113

[3.0x 1O-5 (M3)],

for all the cases of the Mach number, where O(fljk) = 1. (ii) From the conservation laws, the fluxes of mass,

moment, and energy are constant. The differences of them from the exact values at all the space lattice points are less than

[O.OS% for M=1.2, (S3,M3) (case l)],

[0.009% for M=1.2, (S3,M3) (case 2)],

[O.OS% for M=1.59, (S2,M3) (case 3)],

10.03% for M=2, (S2,M3) (case 4)],

[0.02% for M=2.5, (S2,M3) (case 5)],

[0.02% for M=3, (S2,M3). (case 6)],

[0.02% for M_=3, (Sl,M3) (case 7)],

[0.09% for M=3, (S2,M2) (case S)],

[0.4% for M=3, (S2,Ml) (case 9)].

For the present DSMC result (M=3), they are less than 0.5%.

(iii) The macroscopic quantities p, ui, and T vs F, for different lattice systems (cases 6-9) (M=3) are compared in Tables VI-VIII, where the tabulated data are obtained using the interpolation3’ for x,. The macroscopic quantities



x1 = -2.045

(a) -6 ,--4 -2 0 2 4 6 G 8

CT 6

(b) -6 -4 I = ‘? 0 * $ ‘1 ,$ 8 (d) -6 -4 -i 0 2 4 6 iI 8

FIG. 2. The distribution function ~(x,,~,,&) at four typical points in the gas for M=2 (case 4). (a) x, = -2.045, (b) x1= -0.800, (c) x1 = -0.400, (d) x,=1.611. The surface is shown by 5;=0, &=ctk), and c,=c, (*j) lines [(M3) lattice, k= 1 ,..., 14, j= -22 ,..., 281. The &=O line is drawn using Eq. cm.

p, ul, and T vs z1 for case 1 and case 2 (M= 1.2) and the analytical result (66a)-(66c) (M=1.2 or e=O.2) are compared in Table IX in a similar way.37 It is seen that the difference between case 1 and case 2 is very small and the difference between the numerical result and the analytical one is 0(&.

(iv) The maximum difference of the distribution function from the upstream (downstream) Maxwellian in the molecular velocity space, i.e.,

where F,( c&> is the upstream (downstream) Maxwell- ian, tends to decay almost exponentially as i-+ -ND (i+NJ and it ceases to do so on the way due to the numerical error except for case 2 (in this case it continues to decrease). The AfM is less than 3.7 X low4 in 1 x1 1 > 26.5 for (case 2) and 1.8X 10m3, 9.2X 10m4, 4.3X lo-“, and 3.2~ 10d4 for case 3, case 4, case 5, and case 6, respectively, in Ix1 1 > 17.5.

(v) We judged the convergence of iteration from the variation of density profile. Let p,,(x] ) be the density pro-


file at nth iterative step and let xi =S, be its shock position, i.e., p,(S,) = ( 1 + pd)/2. We define the variation of its shape during 20 iterations by

Aps=max{ 1 pmon(x!‘)--AS) I -p 20(n--1)W) I,

i= -ND+5,--ND+ 10 ,..., ND- 10,ND-5},

where

The Aps was measured every 20 iterations. The criterion of the convergence is that Aps is less than 3 x 10e5 for case 9 and 1 X 10m5 for the other cases. It should be noted that AS approached a small but finite value in each case [O(AS) = lo-* for cases 1 and 9 and lOA for the other cases]. This phenomenon, called the shift phenomenon, also occurs for the model equation and it is considered to be a consequence of the invariance of the solution under

Taku Ohwada 227


1.0 T1 1.0 1 x1 = -2.045 i 21 = 0.400

f(zl,rl,sr, :

i

G

6 I-- (a) -6 -4 -2 0 2 4 6 cl 8 (c) -6 -4 -2 0

1.0 1.0 : -0.800 1 x1 = 21 = 1.611

f(Zl, Cl, CT) ; PC% Cl, 4-T) j :

G 6

(b) -6 -4 -2 0 2 4 (d) -6 -4 -2 0 2 4

FIG. 3. The distribution function ~(x,,~,,&) at four typical points in the gas for M=3 (case 6). (a) xt = -2.045, (b) x1= -0.800, (c) x1= -0.400, (d) x1 = 1.611. The surface is shown by 5;=0, &.={i’), and {I =C{‘” lines [(M3) lattice, k= 1,...,14, j= -22,...,28]. The c,=O line is drawn using Eq. (26).

TABLE II. The density p(x,), flow velocity u,(x~), and temperature T(x,) for M=1.2 (case 2). The data in the parentheses are the exact values at x, = * ~0.

Xl P Ul T

--m (1.000) (1.095) f 1.000) -26.461 ‘1.000 1.095 - 19.851 1.002 1.093 - 14.444 1.009 1.086 - 10.514 1.023 1.071

-7.895 1.042 1.052 -6.188 1.060 1.034 -5.001 1.075 1.019 - 4.060 1.089 1.006 -3.215 1.102 0.994 - 2,403 1.116 0.982 - 1.601 1.130 0.970 -0.800 1.144 0.958

0.000 1.158 0.946 0.800 1.172 0.935 1.601 1.186 0.924 2.403 1.198 0.914 3.215 1.211 0.905 4.060 1.222 0.896 5.001 1.234 0.888 6.188 1.246 0.879 7.895 1.261 0.869

10.515 1.276 0.858 14.444 1.288 0.850 19.85 1 1.295 0.846 26.461 1.297 0.845

00 (1.297) (0.844)

‘1.000’ 1.002 1.008 1.022 1.038 1.053 1.066 1.076 1.086 1.096 1.105 1.114 1.123 1.132 1.140 1.147 1.154 1.160 1.166 1.172 1.179 1.186 1.191 1.194 1.194

(1.195)

TABLE III. The density p(x,) for M= 1.59, 2, 2.5, and 3 (case 6). The data in the parentheses are the exact values at x1 = f 00.

Xl M= 1.59 M=2 M=2.5 M=3 (case 6)

- ;7%6 - 12.688

- 8.833 - 6.087 -4.321 -3.241 -2.551 - 2.045 -1.611 - 1.203 -0.800 -0.400

0.000 0.400 0.800 1.203 1.611 2.045 2.551 3.241 4.321 6.087 8.833

i2.688 17.446 c.3

(1.000) (1.000) fl.000) (1.000) 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.003 1.001 1.000 1.000 1.018 1.007 1.004 1.003 1.057 1.034 1.024 1.020 1.109 1.088 1.072 1.063 1.160 1.155 1.142 1.131 1.206 1.230 1.230 1.223 1.252 1.314 1.341 1.345 1.300 1.410 1.480 1.507 1.350 1.518 1.647 1.715 1.402 1.633 1.834 1.957 1.453 1.749 2.024 2.207 1.503 1.857 2.197 2.435 1.551 1.952 2.341 2.618 1.594 2.032 2.453 2.752 1.633 2.098 2.535 2.845 1.670 2.151 2.595 2.907 1.705 2.195 2.639 2.949 1.743 2.234 2.672 2.978 1.781 2.265 2.693 2.994 1.812 2.28 1 2.701 2.999 1.826 2.286 2.703 3.000 1.830 2.287 2.703 3.001 1.831 2.287 2.703 3.001

(1.829) (2.286) (2.703) (3.000)



TABLE IV. The flow velocity U, (x1) for M= 1.59,2,2.5, and 3 (case 6). The data in the parentheses are the exact values at x1 = f CO.

M= 1.59 M=2 M=2.5 M=3 (case 6)

--17%6 -12.688

-8.833 -6.087 -4.321 -3.241 -2.551 L~2.045 -1.611 -1.203 -0.800 -0.400

0.000 0.400 0.800 1.203 1.611 2.045 2.551 3.241 4.321 6.087 8.833

12.688 17.446 co

(1.451) (1.826) (2.282) (2.739) 1.452 1.826 2.282 2.739 1.451 1.826 2.282 2.739 1.448 1.825 2.282 2.738 1.426 1.813 2.273 2.730 1.373 1.765 2.228 2.684 1.308 1.678 2.129 2.575 1.252 1.581 1.998 2.420 1.203 1.485 1.855 2.240 1.159 1.390 1.702 2.037 1.117 1.295 1.542 1.817 1.075 1.203 1.386 1.597 1.036 1.118 1.244 1.400 0.999 1.044 1.128 1.241 0.966 0.984 1.039 1.125 0.936 0.935 0.975 1.046 0.911 0.899 0.930 0.995 0.889 0.871 0.900 0.963 0.870 0.849 0.880 0.942 0.852 0.832 0.865 0.929 0.833 0.817 0.854 0.920 0.815 0.806 0.848 0.915 0.801 0.801 0.845 0.913 0.795 0.799 0.844 0.913 0.793 0.799 0.844 0.913 0.793 0.799 0.844 0.913

(0.793) (0.799) (0.844) (0.913)

TABLE V. The temperature T(x,) for M= 1.59, 2, 2.5, and 3 (case 6). The data in the parentheses are the exact values at x1 = f CO.

XI M=1.59 M=2 M=2.5 M=3 (case 6)

-17146 _ 12.688

-8.833 -6.087 -4.321 -3.241 -2.551 -2.045 -1.611 - 1.203 -0.800 -0.400

O.KJO 0.400 0.800 1.203 1.611 2.045 2.551 3.241 4.321 6.087 8.833

12.688 17.446 co

(l.ow (l.cw (1.000) (1.000) 1.000 1.oQo 1.000 1.000 1.000 I.fNo 1.000 1.000 1.004 I.001 1.001 1.001 1.031 1.021 1.021 1.024 1.094 1.098 1.117 1.146 1.167 1.232 1.319 1.423 1.228 1.371 1.566 1.789 1.276 1.496 1.811 2.174 1.319 1.609 2.043 2.554 1.358 1.712 2.254 2.904 1.394 1.801 2.429 3.189 1.426 1.875 2.562 3.393 1.455 1.933 2.652 3.520 1.480 1.976 2.711 3.592 1.502 2.007 2747 3.631 1.519 2.029 2.769 3.651 1.534 2.045 2.782 3.661 1.546 2.056 2.790 3.666 1.558 2.065 2.795 3.668 1.568 2.071 2.797 3.668 1.579 2.075 2.798 3.667 1.586 2.077 2.798 3.666 1.589 2.077 2.797 3.666 1.589 2.077 2.797 3.666 1.589 2.077 2.797 3.666

(1.591) (2.078) (2.798) (3.667)


i.... I I I I _ -20 -10 0 10 20

-6

FIG. 4. T&e profiles of normalized density i?, flow velocity i7,, and temperature T vs 5, for M=1.2. Here, -: the present numerigal result (case 2); - - -: the Mott-Smith;4 0, Cl, and A are “p. U,, and T, respectively, of the DSMC result.‘2

the translation16’17 for x1. If this is not taken into account, the convergence will be poorer than it is.

VI. DISCUSSION

In the present study, the normal shock waves up to moderate strength are analyzed numerically with good accuracy. Since there is a large temperature difference in a strong shock wave, more molecular velocity lattice points (and therefore more memory size for the numerical kernels) may be necessary for the accurate computation [the

- 1.0 G

G 72 2 6

2 c 0.5

0

FIG. 5. The profiles of normalized density F, flow velocity E,, and temperature T vs ?, for M=2. Here, --: the present numerigJ result (case 4); - --: the Mott-Smith;4 0, 0, and A, are p, U;, and T, respectively, for the DSMC result.‘2

Taku Ohwada 229


- 1.0 G 'i; 2 2 ,; 5 4. 0.5

0

I I !

-- ~-- _ I.--..-

-8 -4 0 4 8 51

FIG. 6. TJe profiles of normalized density p, flow velocity CL, and temperature T vs Y, for M=3. Here, --: the present num_erical result (case 6); - - -: the Mott-Smith;4 0, Cl, and A are & c,, and T, respectively, for the present DSMC result. The length of each vertical line shows the magnitude of the standard deviation of the DSMC samples for the density at the corresponding point.

0.030 r Sl(%)

I -----I-

/q,,,

0

-30 -20 -10 0 10 20 i, 30

2.0 / --I--- ‘--‘I

41(e)

1.0

0 -8 -4 0 4 n

EL

FIG. 7. The profile of stress tensor Pi, vs 2, for M= 1.2, 1.59, 2, 2.5, and 3. Here, -: the present numerical result (cases 2-6); ---: the Mott-Smith: --‘--: the analytical result [Eq. (66d) with M= 1.2 (E =0.2)].

0 --30 -20 -10 0 IO 20 30

5.0 r------~P -i- I I -Q1(%) / MC3 / 2.5 / 2.5 1

It : “- b-i

/ a’.’ , : ’ /‘2

oiL / 1 { ,1.59

-& _- i

:.;‘A-=

,,‘_ /:“\

,,’ ,’ ,’

Y..;. ?* ‘L. A”

.,p;-- _.- - .-. _..__ I - . ..___ -++i s -.I

-8 -4 0 4 21 8

1

FIG. 8. The profile of heat flow vector Q, vs ?, for M= 1.2, 1.59, 2, 2.5, and 3. Here, -: the present numerical result (cases 24); ---: the Mott-Smith;4 - * -: the analytical result [Eq. (66e) with M=l.2 (E =0.2)].

memory size for (M3) is about 90 Mbytes]. Furthermore, it becomes difficult to obtain accurate values of a:$, for large a and b, since the integrand in Eq. ( 5 1) becomes very steep for 8. There is a practical upper bound of the number of terms in Laguerre polynomial expansion (21). There- fore, the present numerical method is not suitable for the accurate computation of strong shock waves. In this case it seems to be preferable to use the basis function expansion for 6, instead of the Laguerre polynomial expansion, although the computation of the numerical kernel is geometrically complicated.

The convergence theorem of the DSMC method has been given by Babovsky and Illner for Nanbu’s method.38 According to it, the DSMC solution converges to that of the Boltzmann equation as P, .+ CO, A,-+O, and At-O. In the actual simulation, the final result is obtained as the average of many flow-field samples each of which does not contain so many particles as to capture the actual flow well. Owing to the averaging procedure, the total particle number, P,N, becomes large enough to yield smooth phys- ical quantities. However, the collision effect between the particles which belong to different samples is not taken into account. There is a question whether the final result con-



, .- ..-- “.--... -.-.--7

,-\ : : 2, = -0.580

-4 -2 0 a 4 6 iI

-4 -1 0 2 4 4-l

6

-.“.._..-“.- -........... r--- / 1

, ’ \. i, = 0.620 c,=o.ls,~ ,.i

‘4. : ,' ‘1 <> ‘/

0.75 -“.., .%_ --....

':';y :

--I. .,> '*: ,

"Xsa -,;,& 0 _..,- pi \':,

-4 -2 0 2 4 it

G

verges as N,+ CO keeping P, finite. If it does not, the final result is only a centerline of prediction band the width of which is of the order of the standard deviation. As can be seen in Fig. 6, the standard deviation for the density is not small (the scatter at the microscopic level is larger than that at the macroscopic level); nevertheless, the agreement of the DSMC result with the direct numerical one is quite satisfactory up to the microscopic level. This shows that

TABLE VI. Comparison of the density ~(2,) for different lattice systems (M=3). The data in the parentheses are the exact values at 2, = f a.

Xl (M3S2) (M3Sl) (ML=) W1S7-) --Q) ( 1.000) (1.000) [ 1.000) (l.OoQ)

- lO.OuO 1.000 I.000 1.000 0.999 -3.000 1.058 1.057 1.058 1.056 -2.000 1.165 1.165 1.165 1.164 -1.000 1.451 1.451 1.450 1.449

0.000 2.000 2.000 2.000 2.000 1.000 2.563 2.562 2.563 2.565 2.ooo 2.855 2.855 2.855 2.857 3.ooo 2.956 2.956 2.957 2.959

10.000 3.000 3.000 3.002 3.009 cis (3.000) (3.000) (3.000) (3.000)

FIG. 9. Comparison of the distribution function ~($,<i,<,) for M=3 at ?I = -0.580, 0.620, and 1.420. Here, -: the present n_umerical result (case 6); - - -: the Mott-Smith: 0, Ll, +, and x show f of the present DSMC corresponding to <,=0.15, 0.75, 1.35, and 1.95, respectively.

the averaging procedure works quite well in the spatially inhomogeneous t ime-independent problem.

ACKNOWLEDGMENTS

The author wishes to express his heartfelt thanks to Professor Yoshio Sone of Kyoto University for valuable discussion in connection with the analysis of a weak shock

TABLE VII. Comparison of the flow velocity ~~(2,) for different lattice systems (M=3). The data in the parentheses are the exact values at ,Y,=*CO.

XI CM3,S2) (M3Sl) (MZ32) OfL.52) -00 (2.739) (2.739) (2.739) (2.739)

- 10.000 2.739 2.739 2.739 2.740 - 3.000 2.589 2.591 2.590 2.592 -2.000 2.350 2.350 2.350 2.353 -1.000 1.888 1.887 1.889 1.891

0.000 1.369 1.369 1.370 1.371 1.000 1.069 1.069 1.069 1.070 2.000 0.959 0.959 0.960 0.962 3.000 0.927 0.927 0.927 0.929

10.000 0.913 0.913 0.913 0.913 M (0.913) (0.913) (0.913) (0.913)



TABLE VIII. Comparison of the temperature r(Z,) for different lattice systems (M=3). The data in the parentheses are the exact values at x;=kCO.

XI WC=) (M3Sl) ML=) (ML=) Xexp --m’ ( 1.000) (1.000) (1.000) ( 1.000)

- 10.000 1.000 1.000 1.000 0.994 - 3.000 1.388 1.386 1.387 1.378 -2.000 1.945 1.945 1.943 1.933 - 1.000 2.798 2.799 2.796 2.786

0.000 3.419 3.419 3.418 3.408 1.000 3.621 3.621 3.619 3.608 2.000 3.662 3.662 3.660 3.648 3.000 3.668 3.668 3.666 3.653

10.000 3.666 3.666 3.663 3.647 co (3.667) (3.667) (3.667) (3.667)

wave; and to Professor E. Phillip Muntz of University of Southern California for providing the DSMC data of normal shock waves. Thanks are also due to Professor Kazuo Aoki of Kyoto University for his encouragement during this research and careful reading of the manuscript; and to Professor Andrzej Paiczewski of University of Warsaw for recommending the present study strongly.

This work was supported by Hattori Hoko foundation.

APPENDIX A: DETAIL OF NUMERICAL KERNEL

Since \iJc,) is a sectionary quadratic function that vanishes outside a finite interval, we consider the following integrals instead of Eqs. (52) and (53) :

F~‘(x,y,B,F) =sin 0 s m (5 cos e>‘u(c cos 8;x,y)$

0

ig’(x,y$,S) = Cc s s m (2 sin @‘U(Zsin B;x,y)gib --rr -m

(AZ)

where U( t;x,y) is the rectangular impulse function defined by

I 1 (X<t<Y),

U(t;x,y I= 0 (otherwise).

Then f$(8,F) and @$(8,g) are expressed by the linear combination of ~~‘(x,y,B,~) and that of @‘(x,y,&F), respectively, with suitable replacement of (x,y), e.g.,

r;,(e,a = [ - F$Y0,2h,e,c) +2hF2°(0,2h,e,q ]/h2.

The integration in Eqs. (Al) and (A2) is carried out analytically:

(A31

(ij.:;;;;) Gk( 3;), ~ (A41

TABLE IX. Comparison of the density p(F,), flow velocity u,(X,), and temperature T(?,) (M= 1.2). The data in the parentheses are the exact values at X; = f m. The data in the square brackets are the limiting values as & -+ f 00.

xi (Case 1)

-03 ( 1.000) - 25.000 1.000 - 15.000 1.006 - 10.000 1.022 - 5.000 1.067

0.000 1.149 5.000 1.228

10.000 1.272 15.000 1.290 25.000 1.298

m (1.297)

P

(Case 2)

( 1.000) 1.000 1.007 1.023 1.068 1.149 1.227 1.271 1.288 1.297

(1.297)

Eq. (66a) (Case 1)

[l.ooO] (1.095) 1.000 1.095 1.006 1.089 1.021 1.072 1.066 1.027 1.150 0.954 1.230 0.892 1.274 0.862 1.291 0.850 1.299 0.844

[1.300] (0.844)

u1

(Case 2)

(1.095) 1.095 1.088 1.071 1.026 0.954 0.893 0.862 0.850 cl.845

(0.844)

Eq. (666) (Case 1)

[1.095] (1.000) 1.095 1.000 1.089 1.005 1.072 1.020 1.027 1.058 0.952 1.116 0.892 1.161 0.865 1.182 0.854 1.189 0.850 1.193

[0:849] (1.195)

T

(Case 2)

(1.000) 1.000 1.006 1.022 1.060 1.117 1.162 1.183 1.191 1.194

(1.195)

Es. (66c)

[LOOO] 1 .ooo 1.006 1.021 1.059 1.119 1.163 1.180 1.187 1.190

[1.190]



i&=exp ( _ (%“‘,: sin2 3

~0s~ 8 sin’ 0 0

cos 8 0

sin2 0

\ 0 0

sin’ eh0s3 8 0 0

0 sin eh0s2 8 0

l- o -

cos t

1 sin 6

\/ 1 2gLk’ cos z ( [Jk’ ) 2 cm2 “\

li 0 1

i O 0 <Sk’ cos 5

I

, (A61 1

,f;‘= i a (c;@ sin ~)2~&.,-,.)+/ 0 r=O y (1=0,1,2,3),

(A7)

b .b F;‘= c

0 y &,-,@,,,2r+[ (~=0,1,2), r-0

Fk=ES(y tan e+I;Jk) cos F) --E,(x tan e+g-:k) cos F), (A9)

~&=E,(Y cot e--gk) cos 3 --E,(x cot e-gLk) cos ~1,

(Al01

with

E,(z)= Jfzi.+exp( -G)dw,

K(z) =-2-l exp( -S/2) + (s- l)E,-,(z),

Eo(z)=VZerf(z/v%, E,(z)= -exp( -2/z),

& JIrn Pexp( -$dz,

g&1*3*5* * *(2S-1) &G, E0== JG,

0

0 o =l, erfiz)=

s

z exp( -w2)dw.

0

APPENDIX 6: SPACE LATTICE POINTS

The space lattice poirits xi’) are defined by

erf(3.5) -erf 3 5 [ . (%z)])

x(D-50d) (~=%..,ND-I),

x~“~) = D (i=ND), (Bl)

x1 C-4 = -xf) 2

where the lattice points concentrate around xl =O.


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