lattice gas models for nonideal gas fluids

P h y s i c a D 47 (1991) 97-111 Nor th -Ho l l and

L A T T I C E G A S M O D E L S F O R N O N I D E A L G A S F L U I D S

Shiyi CHEN, Hudong CHEN, Gary D. DOOLEN, Y. C. LEE, H. ROSE Center for Nonlinear Studies and T-Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

and

Helmut BRAND Center for Nonlinear Studies, Los Alamos National Laboratory, and Department of Physics, University of Essen, W-~3 Essen 1, Germany

i~eceived 20 March 1990

A la t t ice gas mode l wi th a nonidea l gas e q u a t i o n of s t a t e is p resen ted . T rans i t i ons be tween t he solid and gas phase are descr ibed. C o m p u t e r s i m u l a t i o n s of app l i ca t ions of th i s m o d e l to shock waves are d iscussed . Genera l i za t ion of th i s mode l to l iquid c rys ta l flow is also out l ined.

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

Since Frisch, Hasslacher and Pomeau (FHP) [1] proposed the lattice gas au tomata for solv- ing Navier-Stokes equations, much theoretical and computational progress has been made [2-5]. Lat- tice gas au tomata offer several advantages over most numerical schemes: first, lattice gas methods approximate physical systems using parallel algo- rithms, making the design of a parallel dedicated lattice gas machine possible; second, a lattice gas computer program is much simpler than tradi- tional numerical schemes; third, memory is used with maximum efficiency; fourth, because particle number, momentum and energy are conserved ex- actly during each scattering process, there is no roundoff error in the model and the scheme is un- conditionally stable.

The collision processes for most lattice gas models only happen for some particular configura- tions in a same site, such as two- or three-body collisions. During the advection process, particles move along the lattice directions to their nearest neighbors. Thus, particles in a lattice gas behave like classical particles constrained to move on a lattice with an interaction potential identical to

hard spheres with zero radii. The equation of state of these lattice gas models approximates an ideal gas law.

Many interesting physical phenomena occur for systems which have the properties of a nonideal gas. The simplest model in classical thermody- namics is the Ising model, in which only nearest- neighbor interactions are used. Recently, we in- t roduced nearest-neighbor interaction potentials for particles in an FHP-like model [3]. It is interesting that this lattice gas model approximates the Navier-Stokes equations and its equation of state differs considerably from that of an ideal gas. Computer simulations of this model show a first- order phase transition below the critical point. There is a coexistence region for temperatures below this critical point.

In this paper, we present a detailed investiga- tion of this model and discuss applications, including the shock wave simulations, and an extension of this model to two-dimensional liquid-crystal systems. In section 2, we describe the model. In section 3, we discuss the properties of the model. Section 4 presents some detailed applications. In section 5, we present the model for a nematic liquid crystal and the results of some of our simula-

0 1 6 7 - 2 7 8 9 / 9 1 / $ 0 3 . 5 0 (~) 1991 - Elsevier Science P u b l i s h e r s B.V. (Nor th -Hol l and)

98 S. Chen et aL / Lattice gas mode l s /or nonideal gas fluids

tions. A short s u m m a r y appears in section 6.

2. L a t t i c e gas m o d e l for n o n i d e a l gas f lu ids

To model a sys tem with a nonideal gas equation of state and with a realistic energy equation [6,7], a lattice gas model must have more than one speed. In addition, particle interactions must include a potential energy, which can depend on density and tempera ture .

The original 2D F H P lattice gas model [1] con- sists of identical unit-mass particles on a hexagonal lattice and allows six possible particle states along the six different linear momen tum directions. An exclusion rule is imposed so that no more than one particle at a given site can have the same momen tum state. I f we use Na (a ) (a = 1 , . . . , 6 ) to denote the particle occupation in state a at site a , then Na = 0 or 1. For simplicity, the collision rules between moving particles in this paper only include 2R, 2L, 3S and 4S in the notat ion of Wolfram [8]. In addition to the moving particles, the new lattice gas model allows at each site a another kind of particle (a "bound pair" with zero momentum) , with occupation N0(z) (= 1, 0). The mass of a bound pair is twice as that of a free particle. Note tha t this zero speed particle is not the same as a rest particle in the tradit ional seven-bit lattice gas model, where an additional rest particle is allowed with the same mass as a moving particle, but with no interactions between rest particles at different sites [9]. Here we have introduced a square-well potential energy between nearest-neighbor bound pairs:

, = e

---- 0 otherwise,

where to = const. >_ 0 and c is the distance between lattice sites. The constant can be set equal to unity without loss of generality, if one rescales the tempera ture . A bound pair only has nonzero potential energy with those bound pairs at its six nearest-neighbor sites. Free-particle interactions remain unchanged from the original F H P model. Bound pairs possess a total potential energy, E = 1 - ~ e o E . , a N o ( a ) N o ( a + ~ ) , which varies according to the distribution of the

bound pairs. A transit ion between bound pairs and free pairs (with opposite momentum) is in- cluded. The ratio of the probabilit ies for the system to change from one state to another is propor- tional to e x p ( - f ~ A E ) , where A E is the potential energy difference between the two states and ~ is the reciprocal t empera ture defined for the canonical ensemble. The transit ion process between free particles and bound pairs is a Markov process. It can be shown that the canonical ensemble is the eqflilibrium invariant measure for this lattice gas system with t empera ture 1 /~ [10].

In this simplest model, the following transitions are allowed: (i) a pair of oppositely directed free particles may form a bound pair with zero net momentum, and (ii) a bound pair may become two oppositely directed free particles. The potential energy change associated with a binding transition at a site is A E ( z ) 6 = - t 0 N 0 ( a + With constant t empera tu re everywhere, the up- dating rules are specified as follows: to avoid mul- tiple transitions, the lattice is divided into three independent sublattices, each of them a hexagonal lattice with a lattice constant, v/3c; particles on the same sublattice are separated by more than one lattice unit and hence do not mutual ly interact. At each t ime step, the updat ing of the system associated with the transit ion process is done in parallel in three steps, each step involv- ing only one sublattice. A binding probabil i ty ¢ = A exp ( - f~AE) / [1 + e x p ( - ~ A E ) ] (A < 1) is as- signed at each site of a sublattice. The unbinding probability, ¢, for bound pairs is 1 - ¢ . A transit ion is not allowed if it leads to a state which has more than one particle per microstate. For example, for No = 1, ¢ is sampled and, if successful, one of the three paired momen tum directions is chosen with equal probability. An unbinding is allowed only if there are no free particles occupying the chosen pair of directions. For No = 0, one of the three paired momen tum directions is chosen with equal probability. If the chosen pair of the free particle s tates is occupied, ¢ is sampled and, if successful, a binding occurs such that the pair of free particles form a bound pair and No becomes 1. For fixed/3, A = 1 leads to the shortest t ime for the system to reach equilibrium. Streaming and elastic collision processes also occur at each t ime step. The F H P model is the special case with ~ = - c ~ .

S. Chen et hi. / Lattice gas models for nonideal gas fluids 99

The mic rodynamica l evolution of this s imple sys tem is described by the following set of local kinetic equations:

N a ( ~ + ea, t + 1) = N , ( ~ , t ) + Aa + IIa,

a = 1 , . . . , 6 ,

N0(a~, t + 1) = N0(~, t) + II0, (1)

where Aa represents the usual F H P cont r ibut ion from pure elastic collisions for the free part icles [2,8]. Ha (a = 0 , . . . , 6) is the addi t ional contr ibu- t ion f rom the t rans i t ion processes:

II~ = / 3 + [ 1 - N~(a~, t)] - B~N~(~e,t) ,

a : 0 , . . . , 6, (2)

where /3 + and Ba ( = 0, or 1) are the creat ion and annihi la t ion opera tors for Na due to the transi- t ion processes, which depend on the part icle occupa t ions at site ~ and the configurat ion of the bound pairs at the six neares t -neighbor sites. The detailed form of Ha guarantees tha t the part i- cle occupa t ion for each state is either 0 or 1. From the explicit expressions for / ~ and 13 + [3], one can show tha t the conservat ion of mass and

6 m o m e n t u m satisfies: ~ , ,=1 Ha + 2II0 = 0 and 6

~ a = l ea l - Ia = O.

To derive an approx ima te equat ion of state, we use the mean field approx imat ion which assumes: (i) there are no correlat ions between part icles at the same site and same time, (Na(ae, t) Nb(~,, t)) = ( N a ( x , t ) ) ( N b ( ~ z , t ) l , a ~ b, where ( ) represents the ensemble average; and (ii) homogeneous particle d is t r ibut ions (Na(~) ) = (Na(a~')). I t can be shown using an H theorem [2] t ha t the equil ibrium dis t r ibut ion will have the form

1 a = 1 , . . . , 6 , (3)

1 + exp(a + ~ea " u ) ' a =

and

fo= 1

(4) 1 + exp(2a - ~E) '

where the Sa = (Na/ (a = 1 , . . . , 6 ) represent the free-particle equi l ibr ium dis t r ibut ions and S0 = (No) represents the bound-pa i r dis t r ibut ion, u is the velocity and e is the average potent ia l energy per bound pair. a and /3 are Lagrange multipli- ers, which are funct ions of u , p and E, and can

be de te rmined from the conservat ion of mass and m o m e n t u m . Densi ty and velocity are defined by

p= f +2f0, a

pu = Z oSo. (5) a

Using the C h a p m a n - E n s k o g expansion, the hy- d r o d y n a m i c equat ions for this model are

Op/Ot + V . pu = 0,

O , ( p u ) + v . (pg(p),,,,) = - V ( p + p h ( p ) u 2) + VV2(pu) , (6)

where g(p) = p(pf - 3)/pf(pf - 6), pf is the free part icle densi ty and p is the kinetic pressure de termined f rom the equat ion of state, which is equal to one-half of the free-particle density,

6 Pf = ~ a = l fa- The veloci ty-dependent term van- ishes as the macroscopic velocity approaches zero. The explicit form of h(p) can be derived by substi- tu t ing the first-order velocity expansion of fa and S0 into the stress tensor 1-Iij = ~ fa(ea)i(ea)j.

In fig. 1, we show the mean field approx imat ion pressure indicated by the solid lines for four reciprocal t empera tu res (0.2, 0.6, 1.0 and 1.2). V is

1 . 5

. 5 -

i i i i

D D D D O ~ ~

- 0 .5 i .5 v

Fig. 1. Pressure versus specific volume for four different reciprocal temperatures (/3 = 0.2, 0.6, 1.0 and 1.2). The solid line represents the mean field approximation. The computer results are shown by ×, o, + and [] respectively.

100 S. Chert et a L / Lattice gas models for nonideal gas fluids

64

48

>-~ a s

16

- 0 - 0

t - - 0

" " ' " ~" ": " ' " " * ~ ~ " " r~'~ ~'" , -t~. , ¢ . d t . - " r.

,.,X: r .g~., .~ • . . . . . W;eXt .';.'A~ ," " . ~ t F,¢:. c :. - ' :_ '~ #'A - ' ' a " . ' ~ ' ' . X ~ " • L--J x. , : - .,:..._. ":~.*~ , " : , ~ " ~ , , 4 " " " . ; . . . . - ' '

~,• . , ~ - * ,-, . ~ . - . : - t l , . . - " , i . . . r . . . . . : . ~ " ~ " ; IH : ~ . i , . : . " ' . . : ~ ' ~ . : ' : " . ~ - . - - i - t

.._¢o . . . . .o°tto

16 32 4B 64

X t = 8 .000

64

48

>-~ a2

16

64

48

32

16

- 0 - 0

64

t -- 4 , 0 0 0

18 32 48

X

t : 1 2 . 0 0 0

64

4B

>-~ a s

16

- 0 - 0 - 0 16 32 48 64 - 0 16 32 48 64

X X Fig. 2. Instantaneous distribution of bound pairs at time t = 0, 4000, 8000 and 12000 in the two-phase coexistence region with V = 0.5 and ~3 = 1.2.

t i le specific volume, 1/p. The s imula t ion was done for zero veloci ty in a sys tenl wi th 64 × 64 l a t t i ce sites. W'e also used sys t em sizes of 128 × 128 si tes and 256 × 256 si tes to test the sy s t em size dependence. The s imula t ions i nd i ca t ed t h a t there is less t han a 1% change for a sys t em size b igger t han 64 x 64 sites. The ×, +, + and [] in fig. 1 represent s in lu la t ion resul t s for different rec iproca l t empe r - a tu res , /3 0.2, 0.6, 1.0 and 1.2. These s inmla t ions c lear ly show a cr i t ica l po in t for 1 ~ /3 1.2 where Op/cgV = 0. In fig. 2, we present the b o u n d - p a i r d i s t r i b u t i o n s for /3 ~ 1.2 and dens i ty p 2 at t imes t 0, 4000, 8000, and 12000. The in i t ia l cond i t ion for the b o u n d - p a i r dens i ty is 0.2 wi th a r a n d o m spa t i a l d i s t r i bu t i on . The sanle bound- pa i r dens i ty and pressure exis ts be tween t - 8000 and ~ 12000, bu t there is a difference ill the i r s pa t i a l d i s t r i bu t ions . Tile spa t i a l l y homogeneous p ressure and s p a t i a l l y i nhomogeneous bound pa i r

d i s t r i b u t i o n s t rong ly suggests the coexis tence of two phases . Also we no ted t ha t the b o u n d pa i r s t end to coalesce at low t e m p e r a t u r e .

In fig. 3, we presen t 9(P) versus p at equil ib- r ium for four values of t i le inverse t en lpe ra tu r e : - o c ( F H P model ) , 0.2, 0.6 and 1.0. I t is inter- • s t ing t ha t for rec iproca l t e m p e r a t u r e near 0.6, there is a large range of densi t ies , all wi th approx- ima t e ly the same value for 9(P). This is an impor - t an t p roper ty , which can be used for s imu la t ing compress ib le flows wi th large dens i ty var ia t ions .

3. P r o p e r t i e s o f t h e m o d e l

3.1. A mapping between our model and the Ising model for static properties

Using a s imple m a p p i n g , it, can be shown tha t t i le invar ian t measu re of t i le b o u n d pa i rs is equiv-

S. Chen et al. / Lattice gas models for nonideal gas fluids 101

1.0

0.S

0.6

0.4

0.2

0.0 0.0

+

+ + +

+

1.0 2.0 3.0 4.0 5.0

and A(S(.) ~ S'(.)) is the transit ion probabil i ty from configuration S(.) to S'( .) , which satisfies the the total mass and momen tum conservation:

~-~(N" - Ni )OiA(g --~ Y ' i x ) = O, i

E ( N [ - N i )~ iA(N -* N ' Ix ) = O, VN, N', x, i

O i = O , 1 , i = l , . . . , b . (10)

At equilibrium, the system is translat ionally and rotat ionaily invariant, so we can write A(S(.)

S'( . ) ) in a formal factorized form: A(S(.) S'(.)) = 1-I=~LA(N ~ N']x) and from the transition rules discussed in section 2, we can write the following generalized detailed-balance relation:

P

Fig. 3. The convec t ion coefficient g(p) versus average den- s i ty per cell. The sol id l ines represen t the m e a n field re-

su l t s and x , o, and + represen t the s i m u l a t i o n resu l t s for = 0.24, 0.6 and 1.0. The lowest sol id l ine gives the ana-

ly t i ca l resu l t s for fl --- - o ¢ ( F H P la t t i ce gas model ) .

alent to that of the Ising model on a t r iangular lattice with a prescribed external magnetic field, which depends on density and temperature .

In order to simplify the problem, let us define the phase space, F, as the set of all possible as- signments of the configuration of the lattice L: S(.) = (Ni(x) = 0,1; i = O, 1 , . . . , b ; x • L), where i = 0 indicates a bound pair and b is the number of distinguishable moving particle s tates per site. The probabil i ty at t ime step t for a given configuration P( t, S(. ) ) satisfies

P(t, S(.)) >_ O, ~ P(t, S ( - ) ) = 1. (7) s( .)cr

Since our model is a Markov process, the Liou- ville equation for P(t , S(.)) can be writ ten as

P(t + I, TS'( .))

= ~ A(S(.) -~ S'(.)) P(,, S(.)), (s) s(.)~r

where T is the advection operator ,

Tgi (~ f i ) ~ Ni(~[~ + e i ) , i : 1 , . . . , b , (O)

A(S(.) --* S'( .)) = exp{[E(S(-)) - E(S'(.))]/3}, A(S'(.) -~ S(.))

(11)

where E(S(.)) is the total potential energy for a given configuration S(-):

E = - ~eol E No(x) No(x + el). (12) ze,i

The following form is obtained for the time- independent solution of the Liouville equation (8) using eqs. (9), (10) and (11) for systems with total mass and momen tum conservation:

P(S(.))

b

zC L i=O

b

" z E L i=0

where si(x) is the occupation number for a momen tum state at x and where ~ is the part i t ion function:

b

S( . )EF eEL i=0

b

(1,) zEL i=0

102 S. Chen et al. / Latt ice gas models for nonideal gas f luids

In the t h e r m o d y n a m i c l imi t , we rep lace the 6- func t ions by e x p ( - # N - 3, - 7r), where # and ~/ a re chosen to sa t i s fy mass and m o m e n t u m con- s t r a in t s : (N} = NT and (Tr / = O, where NT is the t o t a l pa r t i c l e n u m b e r in the sy s t em and N and r¢ are the t o t a l mass and t o t a l m o m e n t u m for a given conf igura t ion . We o b t a i n a p r o b a b i l i t y d i s t r i b u t i o n which fac tor izes the b o u n d - p a i r and f ree -par t i c le dependence :

P(S(.))

e x p [ - E ( S ( . ) ) ~ 3 - g(s( .))# - r e ( s ( . ) ) . 3,] g~

(15)

The r educed b o u n d - p a i r p r o b a b i l i t y is

Po(S(')) = e x p [ - E ° ( S ( ' ) ) / 3 - gT(So(.))p] f~o

- P 0 ( S 0 ( ' ) ) , ( 1 6 )

since Eo(S(')) = E0(S0( ' ) ) , where

.4

.2 A o

'7 X/

t

o C'Q

II

- . 2

' ' ' ' I ' ' ' ' I ' ' ' ' [ ' ' ' '

o : f l = l . 0

× : f l = 1.1

- . 4 x.___.__.-->e--

t , , , I , , , , I , , , , I , , , r

- .1 -0 .05 0 0.05 .1 Bo

Fig. 4. Magnetization versus magnetic field for the Ising model (from lattice gas simulation without FHP-type collision) for the hexagonal lattice. The simulation confirms that the critical reciprocal temperature is near 1.09.

: 2 N o ( Z ) , z E L

a o = ~ e x p [ - E 0 i S 0 ( . ) ) / 3 - N0(S0( . ) )#] s ( . ) c r

and # = #(¢3, p). In o rde r to c o m p a r e wi th the Is ing model , we

i n t r o d u c e the fol lowing t r a n s f o r m a t i o n ,

1 1] N 0 ( Z ) = ~ [ s ( z ) + , (17)

where s ( z ) = 1 , - 1 for No = 1,0, respect ively . The p o t e n t i a l energy for the t o t a l sy s t em can be w r i t t e n as

a n d the p r o b a b i l i t y d i s t r i b u t i o n has the form

, ( 1 ) p0(s0(.)) = exp - e,0 s(z) s(z + e,) ~, i

x exp(-/3Bo(/3, p) ~ s(z)) , (19) ag

which is the exac t Is ing p r o b a b i l i t y d i s t r i b u t i o n

in an ex te rna l magne t i c field wi th J = ' ~e0 and B 0 1 i 1 = ~eob+B o. At the cr i t ica l po in t , B~ = - ~eob.

For the two-d imens iona l t r i a n g u l a r Is ing mode l , the cr i t ica l rec iproca l t e m p e r a t u r e , 13¢, is

log 3 ~ c - 2 J ( 2 0 )

1 for our model , we o b t a i n a cr i t ica l Se t t ing J = rec ip roca l t e m p e r a t u r e 13c = log 3 ~ 1.0986. In fig. 1, we see t h a t the cr i t ica l r ec ip roca l t e m p e r a t u r e is nea r 1.1. In fig. 4, we presen t the Ising mode l

1 resul t s for m a g n e t i z a t i o n M = ~ - (No) for a la t - t ice gas sy s t em wi th an ex te rna l magne t i c field B~, s imp ly by removing the coll ision process from the l a t t i ce gas code. A sha rp j u m p in m a g n e t i z a t i o n is observed at the cr i t ica l t e m p e r a t u r e .

3.2. Sound speed and measurements

T h e sound speed is defined by Op/Op for l a t t i ce gas sys tems . Because we place the l a t t i ce gas sys- t em in con tac t w i th a u n i f o r m - t e m p e r a t u r e hea t b a t h , the sound speed is the i so the rma l sound speed. I t is poss ib le to measure th is sound speed d i r ec t ly as the p r o p a g a t i o n speed of a dens i ty fluc- t u a t i o n in the long-wave leng th and low-frequency


limit. In order to do that , we define the following Laplace-Four ie r t ransformat ion :

~(k, s) = at e x p ( - s t )

/ × exp(ik • r) p(r, t) dr, (21)

where s = e + ico. We assume tha t the densi ty and velocity in eq.

(6) can be separa ted into a constant par t and fluc- tua t ing part , p = P0 + ~P and u = ~u, where we have also assumed tha t the constant par t of the velocity is zero. We can obta in a dispersion rela- t ion using (21) and el iminat ing higher-order fluc- tua t ions in (6),

s ~ ( k , s) + i k . ~(k, s) = 6~(k, 0),

( s + v k 2 ) ~ ( k , s ) ÷ c 2 i k E p = j ( k , O ) . (22)

Here, j = po~u and c 2 = Op/Op. After simple manipula t ion , one obta ins the following relation:

<~p(k,s)~p*(k,O)) s + L,k 2 = (23)

0) 0)> , ( s + .k2) + c. k2 "

Defining the densi ty correlat ion funct ion as

/ S(k, w) = at e x p ( - w t )

× exp(ik • r ) F ( r , t) dr, (24)

where F(lr - r ' l , t - t') = (p(~' , t)p(r ' , t ' )) . We have

S(k , w) (~p(k, s) ~P* (k, 0)) S(k, 0) - 2Re lim ~ o (6p(k, O) 6p*(k, 0))

which has peaks at w = csk, -c~k, for small wave- numbers .

The sound speed of the latt ice gas systems has been de termined by measur ing the densi ty correlat ion funct ion for k = 2~r/L, (k = 0 represents the constant densi ty mode) in a sys tem with 128 × 128 lattice sites. The y-direct ion space average has been used to replace the ensemble average. Ini t ial ly zero velocity and cons tant densi ty are imposed on the system. After the sys tem approaches equil ibrium, we run an addi t ional T = 16384 t ime

1500.0

/k 3

U~ k/ =in t

0,0 50.0 100.0 150.0 200.0

6)

F ig . 5. T h e p o w e r s p e c t r u m fo r w a v e n u m b e r k = 27r /L : t h e

dash-dotted line represents the FHP result and the solid line represents the present model, both at density p = 2.42. The frequency of the peak is related to the sound speed of the system by ca = csk.

steps to obta in a t ime series for the Fourier t ransformat ion and use this to obtain the densi ty correlat ion power spec t rum. In fig. 5, we show the un- normal ized power densi ty spec t rum, (S(w) S*(¢v)), for the F H P - I model ( da sh -do t t ed line), which has a peak at w = 91.8 × 2felT, which corresponds to a sound speed of 0.707, accura te within 2%. The same me thod has been used for the new model (solid line in fig. 1). Except near the critical point , the densi ty spec t rum always has a peak at csk, where cs is the sound speed. Because of the long-range correlat ion, the measurement of sound speed is not accura te near the critical point. We present typical sound speed measurements for the densi ty p = 2.42. In fig. 6, we give the sound speed as a funct ion of the reciprocal t empera ture . We see tha t the sound speed decreases toward zero as /3 approaches the critical reciprocal t empera tu re of 1.1. The existence of small sound speeds is impor t an t for supersonic flows, as we will discuss in section 4.

3.3. Viscosity measurements

The measurement of k inemat ic viscosity has been implemented using forced channel flow be-

104 S. Chen et aL / /Lat t ice gas models for nonideal gas fluids

, v , , ~ , , , , ] . . . . ] , , ,

0

. 4

fL 5~

Z

.6

0 12 .4 .6 .8

fl

Fig. 6. The sound speed versus the reciprocal temperature f~ for p = 2.42. The sound speed decreases with decreasing temperature.

tween two parallel plates (Poisseuille flow). The forcing scheme is similar to that used by Kadanoff et al. [11]. To mainta in energy conservation, no transit ions are allowed between moving particles and bound pairs during the forcing process. A system size of 256 × 64 sites was used. The boundary condition in the x-direction is periodic and the y- direction is non-slip (bounce back). The viscosity of the system is given by,

3 - L 2 (25)

u 32jm~. f y,

where jma. is the averaged x-direction momen tum along the centerline of the channel; f is the forcing for each t ime step per unit area and Ly is the lattice size in the y-direction. After equilibration, a t ime average is used to obtain a parabolic x- direction momen tum distribution. In fig. 7, measurements of viscosity versus density are given for several reciprocal temperatures , ~. The solid line describes/3 -- - o o (FHP). The ×, % + and [] represent ~ -- - c o , 0.6, 1.0 and 1.2, respectively. The viscosity increases with increasing /3 (i.e. with a decrease of tempera ture) for a given density in the coexistence region (1 < p < 3). Moreover, a higher density is associated with a lower viscosity for a fixed temperature . This occurs because the mean free pa th decreases for the system with a reduced density in the regime between zero and one-half. In the very low density region, two moving particles

1 . 5

?> o 1

' ' ' I . . . . I . . . . I ' ' ' - ' I . . . .

4- +

_ x ~ x £ ~

0 1 , , , , I , , , , I , , ~ , 1 , , , , I , ~ , ,

0 1 2 3 4 ! P

Fig. 7. Simulation results for kinematic viscosity v e t

density for fl = -oo,0 .6 ,1 .0 , and 1.2 ( × , o , + and O). T solid line represents the analytical result for the FHt lattice gas.

having a head-on configuration will have a sm probability. This causes the moving particles a bound pairs to have almost the same tempera t t distribution, as can be seen in the P - V diagr~ in fig. 1.

The total density as a function of distance fr( one wall for different fl are presented in fig. 8 : the case p = 2. This density is in the domain coexistence (fig. 1). We find that the density d tr ibution across the channel is neither const~ nor symmetr ic for low temperature . For/3 higl than the critical value, the system still beha, like the F H P lattice gas model. If the velocity is ways less than 0.1, then the nonphysical veloci dependent te rm in the equation of state shol not seriously affect the distribution of particles space. Hence, the nonuniform density distributi across the channel comes from a variation of te perature. When we reduce the temperature , 1 bound pairs concentrate at the center of the ch~ nel because of the increased chance of a head- collision. This phenomenon is observed in exp imental solid-gas or solid-liquid flows [12]. ~I density variation across the channel does not aft the momen tum profile in the simulation. The n malized momentum profiles for different /3 (e~ for coexistence region) are still parabolic. But


3.0

0..

2.5

2.0 i

1.5

1,0

0.5 0.0

+÷~÷+÷+++-~'+++++++ + +

++ + + ÷

++ +

' 000000 +++ +iO0 ¢ ++ O

"00 + ÷ ++ +

O ++

++

~÷÷÷+*÷

] I I

16.0 32.0 48.0 64.0

Channel Width

Fig. 8. The total densi ty dis t r ibut ion as a function of channel width for forced channel flow for the case p = 2. The solid circle, square, d iamond and plus signs represent /3 = - o o , 0.6, 1.0 and 1.2, respectively.

velocity has changed due to the definition in eq. (5). In fig. 9, we show a typical velocity distribution from our simulation ([3) along channel direction y for a density p = 2, and ,3 -- 1.0 compared

1

.8

N []

Z

.2

0 0 20 40 60 80

Y

Fig. 9. The normalized channel flow velocity d is t r ibut ion along the y-direction for ~ = 1.0. Squares represent the present lattice gas s imulat ion results and the solid line is a parabolic profile.

with a parabolic profile (solid line). Nonsymmet- ric velocity is observed. The phenomenon of the change of velocity distribution with a change of void ratio has also been observed in other two- phase flow experiments [13].

4. O n e - d i m e n s i o n a l s h o c k w a v e s

The model discussed in section 3 can be con- sistently applied to supersonic flows because the sound speed is very small when the system is near the critical point, as shown in subsection 3.2. The hydrodynamic equation (6) is obtained assuming a small macroscopic velocity expansion. The sound speed in the FHP- I lattice gas is l /v /2 . Hence, it is not consistent to use the original lattice gas model to simulate a flow with Mach number bigger than 1. Also, because the convection coefficient, g(p), in Navier-Stokes equation is density dependent for previous models, it will not consistent to rescale g(p) at different density. The present model pro- vides the possibility of keeping g(p) almost constant over a large density region as shown in section 2.

An impor tant phenomenon in supersonic flows is shock wave formation: the discontinuous macroscopic solution for the Navier-Stokes equation and the microscopic transit ion from a uniform up- s t ream flow to a uniform downstream flow. The main difference between our model and the usual hydrodynamic discontinuous solution is that the shock wave phenomena usually depend on temperature or entropy. Our model has a constant temperature. Thus only pressure, density and velocity fields can be measured. In this paper, we are inter- ested in one-dimensional shock waves, part icularly the piston problem [14]. We initialize the flow with uniform velocity along the x-direction and place a fixed wall on the right. The fluid is compressed when it strikes the wall and forms a wave moving ups t ream (in the negative x-direction). To obtain one-dimensional results, we use periodic boundary conditions in the y-direction and average over the y-direction. The fluid velocity in the region between the wave interface and the wall is zero. We define the Mach number of the system to be the ratio of the ups t ream (incoming) velocity to the sound speed of the fluid. The Mach number can be

106 S. Chen et al. / Lattice gas models for nonideal gas fluids

var ied by changing the incoming velocity. Usua l ly shocks are s t ud i ed in the reference f rame of the shock. Then all quan t i t i e s are t i m e - i n d e p e n d e n t . Because we are i n t e r e s t ed in the shock wave s t ruc- tu re and the speed of the shock, we use a l abora - to ry f rame of reference.

In the F H P la t t i ce gas, the sound speed does not d e p e n d on dens i ty and t e m p e r a t u r e . The p ropa - ga t ion speed of a dens i ty d i scon t inu i ty will move at the speed of sound. T h e numer i ca l s imula t ions for a s ix -b i t l a t t i ce gas mode l have d e m o n s t r a t e d the ex is tence of a macroscop ic d i scon t inuous solu- t ion for the F H P l a t t i ce gas and conf i rm tha t the in terface p r o p a g a t i o n speed is the sound speed.

M e a s u r e m e n t s of the shock speed have been car- r ied out for our phase t r an s i t i on l a t t i ce gas mode l wi th a dens i ty of 2.42 a n d / 3 = 0.7. The m e a s u r e d sound speed for th is m a t e r i a l is 0.16. Th is s imula- t ion was done in a sy s t em wi th 1024× 1024 l a t t i ce sites. A cons t an t ve loc i ty b o u n d a r y at x = 0 was s a m p l e d each t ime s tep from the equ i l ib r ium dis- t r i b u t i o n s (3) and (4). The in i t ia l cond i t ion is the un i fo rm veloc i ty field, s amp led aga in f rom eqs. (3) and (4). The p r o p a g a t i o n speed of the in terface in the p resen t mode l is lower t han the F H P model . The g(p) var i a t ion is large for a sy s t em wi th high Mach number . Hence, the present m o d e l has much b e t t e r Ga l i l ean invar iance for weak shock flows. For example , for the sys t em wi th Mach n u m b e r 1.2, we have gz(p = 2.49) = 0.473; the down- s t r e a m dens i ty is p = 3.8, which gives gl = 0.475, where subsc r ip t 2 refers to the incoming dens i ty and subsc r ip t 1 refers to the high dens i ty region. I f the Mach n m n b e r is 1.875, g2(P - 2.42) - 0.442 and gl(P = 4.25) = 0.313. We see t h a t the first case has a p p r o x i m a t e l y the same g(p) before and af ter the shock wave, while there is a big change in g(p) for the second case. The p r o p a g a t i o n speed of the shock in ter face also d e p e n d s s t rong ly on Mach number . The first examp le has a p ropaga - t ion speed of 0.375 and the second example has a value of 0.401. Using the conserva t ion of mass and the m e a s u r e d densi ty , we have a p r o p a g a t i o n speed of 0.38 in the first case. In fig. 10, we presen t the shock interface for t ime s teps from t = 0 to t = 1000 for every 100 t ime s teps for a sy s t em wi th P2 = 2.42 and P2 z 1.62. Even though the re is some noise because of the finite space average in the y -d i rec t ion , we see t ha t the p r o p a g a t i o n speed is s t i l l a cons tan t .

2.4 ~ ' , . ~ . i l , d i : . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . -

512 761 1024

512 768 1024

--.i . . . . . . . . . . . . . ' . . . . -

512 768 1024 X

Fig. 10. Evolution of the discontinuous interfaces for (a) pressure; (b) density and (c) velocity from t - 0 to t = 1000. Time interval is 100.

The s t ruc tu r e of the shock wave is an in teres t - ing p rob l e m for s t a t i s t i ca l mechanics reasons. A c o n t i n u u m descr ip t ion , such as the N a v i e r - S t o k e s equa t ion , is l im i t ed to the macroscop ic in forma- t ion and cannot descr ibe the in ter face s t ruc tu r e in de ta i l . In fig. 11, we present typ ica l p ressure and dens i ty profi les for a no rma l shock wave at

2.5

c~ 2

1.5 100 110 120 130 140 150

X/2

' 1 ' ' 1

4

3

2r,,,,l,,,,l .... i, ,i,,

i00 110 120 130 140 150 x / 2

Fig. 11. Instantaneous simulation pressure (a) and density (b) profiles for a normal shock wave with Mach number 1.875.

S. Chen et al. / Lattice gas models for nonideal gas fluids 1 0 7

t ime t = 2000 for a Mach number of 1.875. The detailed s tructure is very similar to other simulations [15]. A two-cell x-direction average has been used to smooth the noise in the simulation. The mean free path is defined by the ratio of the mean microscopic velocity to the collision frequency. For our model and the density in this calculation, the mean free path is about 3 lattice lengths [5]. The transit ion thickness is about 10 mean free paths in our simulation. Also from our simulations, we find that the shock thickness depends on Mach number. A strong shock produces a short shock thickness.

The relation between pressure, density and velocity before and after a shock interface is given by the Rankine-Hugoniot relation [14], which is derived assuming conservation of mass, momentum and energy. Energy conservation for our case is trivial because our system has a constant tem-

.perature. The viscosity term in the momen tum equation is nonzero only in the transit ion region. Including the dependence on g(p) and h(p) in (6), we have the following relations:

pu -- const.

p + p(g(p) + h(p))u 2 ~- const. (26)

valid throughout the shock in the frame of reference of the shock.

Because of the complicated form of g(p) and h(p), it is difficult to get a simple expression for the Rankine-Hugoniot relation. In fig. 12, we show the simulation results for the relation between P2/pl and P2/pl for the incoming velocities of 0.2, 0.25, 0.3, 0.35 and 0.37. There is a 3% error in the measured da ta due to noise. The crosses in the plot represent for the analytical solution of a perfect gas from the formula

p2 _ ( ~ + l p 2 1) / ( ~ + 1 P 2 ) .

PI - - ]- Pi - - ]- Pl

The diamonds represent the lattice gas simulation. Because these two systems are different, one should not expect exact agreement, but there is qualitative agreement. It will be interesting to work out a scaling relation between different ma- terials for the Rankine-Hugoniot relation. If one can find it, then all simulations for realistic systems can be carried out using lattice gas simulations.

:Z

c ~

2.5

1.5

, , , i [ , , , , i t , , , i , , i , I . . . .

O O

×

×

O

O

I i l J , I j , i r r , i l ~ I I I I I I I I I I

1.5 1.6 1.7 1.8 1.9 2 p2//pl

Fig. 12. The Rankine-Hugoniot relation for perfect gases (×) and a similar relation for the present model from computer simulation (o).

5. A n e x t e n s i o n o f t h e m o d e l : o r d e r p h a s e t r a n s i t i o n

The difference between the present lattice gas model and previous FHP- type lattice gas models is, so far, only the existence of the interaction potentials between bound-pair particles at nearest-neighbor sites. Another interaction can be added which depends on the orientation of particles. For this extension, a directional degree of freedom must be introduced. Suppose that , in addition to having a spatial coordinate, x, and linear momentum, ~ , we let all particles have an orientational degree of freedom. For simplicity, we limit ourselves to the simplest model which allows bound-pair particles to have an orientation which is independent of the linear momen tum and interaction potentials. Hence, the bound-pair particles have anisotropic properties. This requires an additional degree of freedom, the molecular orientation t?, which can vary continually from 0 ° to 360 ° in two-dimensional space. We restrict t? to be a mul- tiple of 60 ° because of the hexagonal lattice. The general interaction Hamil tonian between two particles at different sites and orientations can be rep- resented by H = H(Oij, r). Here t?ij is the relative angle between particles i and j and r is the dis-

108 S. Chen et al. / Lattice gas models for nonideal gas fluids

tance between them. In order to extend the model to liquid crystals, we restrict the Hamil tonian to be an even function of 0. Physically this restriction is related to the indistinguishability of head and tail of the macroscopic properties of liquid crystals. We have designed lattice rules which allow a transit ion between an orientationally ordered state and a disordered state. This corresponds to the nemat ic- isotropic transition. There are two types of potentials for liquid crystals [16]: separable and nonseparable potentials. The separable potential requires that the Hamil tonian with a dependence on 0 and r can be writ ten as a summa- tion of products of a function of angle times a function of the distance: H(O, r) = ~,~ H,~(O)H~(r). It has been proved [17,18] rigorously that no long- range order can exist in systems with separable potentials. In contrast, for nonseparable interactions, the existence of the true long-range order is not excluded. In this paper, we only s tudy the separable case and take Ht(r) to be the same as in the model discussed earlier. There are many possibilities for H(0) [16,19]. For D-dimensional problems, it is interesting to s tudy the following interaction potential:

D cos2(0) - 1 U(0) = n - 1 (27)

For this Hamil tonian, a randomly distr ibuted ori- entat ion will give zero average for (H(0)). A com- pletely parallel orientation will have (H(0)) = 1. Here ( ) is the ensemble average. When the interactions between orientations are isotropic, it is possible to see two-phase transit ions of the Koster l i tz-Thouless type [20]. The low-temperature ordered phase is characterized by quasi-long- range order, with a power-law decay for the correlation function, g2(r), defined as follows:

g 2 ( r ) = ( c o s { 2 [ 0 ( r ) - 0 ( 0 ) ] } 5 , (28 )

where 8 ( r ) - 0(0) is the angular difference between an orientation at a given position and the orien- tat ion at distance r. The ( ) can be replaced by a summat ion over all space. In order to test the effect of the position interaction potential H(r) on H(0) , we introduce the paramete r X, the ratio of H(r) to H(0). Then H(r, O)in this paper (D = 2) is

H(r, 8) = xH(r ) + 2 cos2(0) - 1, (29

where H(r) o = e0N0(x) ~ i = 1 N0(~ + ~i) and 0 i the relative orientation angle between the nearest neighbor bound pairs.

We define the order paramete r for the syste~ to be the ensemble average of H(0) . All the opei ations in section 2 (collision, s t reaming and tran sition) will apply. The transit ion will have to ae count for binding energy. Because of the indistin guishability of head and tail, we only need to ca] culate AEI (i = 1, 2, 3) for three possible oriente tions. Mean field theory results can be obtaine¢ but in this paper we only present simulation re sults. The lattice gas simulation has been don for a system with a 64 × 64 lattice sites. A nine bit lattice gas code was developed containing si: moving states and three bound orientation state, The equilibration process from an arbi t rary initi~ condition depends strongly on density and tern perature. Usually, when the system is not highl oriented, the t ime to approach equilibrium is re] atively short. If the system is near H(O) -- 1, the the equilibrium t ime is very long.

In fig. 13, we present a plot of (H(8)) versus j for density, p = 2.4 (fig. 13a) and p = 4.8 (fi~ 13b) for X -- 0,0.5 and 1.0 denoted by [], × an o, respectively. We see that the transit ion is n¢ the first-order where there would be a jump i the order parameter . Rather, a continuous beha~ ior is found, where one has a big change of th value of the order parameter over a small inte~ nal of fl values. The continuum transit ion is mol likely for a higher density system for the same and more likely for a high tempera ture (small/~ for the same density system. This is the same the first-order phase transit ion as we have seen f( bound pairs in sections 2 and 3.

In plate I, we show snapshots of the orientl tionat equilibrium distribution for a system at di ferent temperatures , but with the same densit p = 4.8, at t ime t = 10000 after a random initi orientation. The color coding represents the oriel tation. Yellow represents the x-direction orient~ tion, red represents the orientation having an a~ gle 60 ° with x-direction; and black represents tl orientation having an angle 120 ° with x-directio: The pat tern formation depends on the initial co~ dition. As is clear from the plots, there is a increa:


i i I I

o

.6 A

v

V .4

.2

.8

.6 A

V .4

.2

X

X 0

X

0

0

X

O

0 X 0 E]

D

(3

E]

13

[ ]

[ ]

, , i , ' ' ' ' 1 ' ' '

a :~^~ ~ I , ; ~ I ~ ,

o 1 2 fl

' ' ' x x 6 ~ l ' ' ' rh ' '

0 E] [ ]

X

X o [ ]

D

X

X o

X O 0 b

~ , , , I , I,, ,

0 i 2

Fig. 13. The nematic order parameter (H(0)) as a function of/3 for X = 0 (D),0.5 (o) and 1.0 (×): for density p = 2.4 (a) and p = 4.8 (b). Note that orientational order increases as X is increased for fixed density, and as the density is increased for fixed X.

of the a l i gnmen t of the o r i en t a t i on wi th increasing/3 .

A s imu la t i on for d e t e r m i n i n g the spa t i a l dependence of the angu l a r co r re la t ion was run which used p -- 4.8, X = 0.5 a n d / 3 = 1.2. A typ ica l cor- r e l a t ion func t ion g2(r) versus spa t i a l s e p a r a t i o n r is p re sen ted in fig. 14 in log - log form at t -- 20000 af te r a r a n d o m in i t i a l condi t ion . Th i s t ime is long enough to al low the s y s t e m to re lax to equ i l ib r ium. We found t h a t the spa t i a l decay r a t e is not expo- nent ia l . Also we observe t ha t for r sma l l e r t h a n 7,

.8

.5

.2

I I I I I I I I I I i i . 1

1 2 5 10 20 50 r

Fig. 14. Log-log plot of the correlation function (cos{2[O(r)- 0(0)]}) (circles) versus spatial separation. We can see that t h e r e i s a n algebraic decay over the range from r = 20 to 50. Here p = 4.8, X ---- 0.5 and t = 20000.

the co r re l a t ion decays fas ter t han a lgebra ic , whi le 20 < r < 50, we observe a l inear decay (an al- gebra ic decay in t he l inear coo rd ina t e ) , th is indi- ca tes a quas i - long- range order . A t r a n s i t i on occurs be tween these two regions. The resul t has also be observed by o the r s [18]. The dependence of g2(r) on dens i ty (p -- 1.8, 2.4 and 4.8) is shown in fig. 15 (wi th X -- 0.5). A t lower densi ty, on ly sho r t - r ange co r re l a t ions a ppe a r . T h e dependence of g2(r) on t e m p e r a t u r e (/3 = 0 .4 ,0 .6 and 1.0) is shown in fig. 16 for a sy s t em wi th X = 0.5 and den- s i ty p -- 4.8. The s m a l l / 3 (high t e m p e r a t u r e ) be- hav io r is r e la ted to sho r t - r ange cor re la t ions which e x p o n e n t i a l l y decay. Th is agrees wi th Frenke l ' s resu l t s us ing Monte Car lo s imu la t ions [18]. The t r a d i t i o n a l co r re la t ions inc lude g21(r) which is defined as g21(r) = (cos{2/[0(r) - 8 ( 0 ) ] } ) . Because of our def in i t ion of o r i en t a t i on for a hexagona l la t - t ice, these h igh-o rde r co r re l a t ions are equivalent to g2(r) .

6 . C o n c l u d i n g r e m a r k s

In th i s pape r , we have s t ud i e d a l a t t i ce gas m o d e l for fluids w i th an equa t ion of s t a t e cor- r e s p o n d i n g to a non idea l gas. Mean field t heo ry

110 S. Chen et al. / Lattice gas models for nonideal 9as fluids

(a) [b)

(c) (d)

P l a t e I. S n a p s h o t s of c o n f i g u r a t i o n s for a s y s t e m of 64 × 64 l a t t i c e s i t e s a t d e n s i t y p = 4.8 a t r e c i p r o c a l t e m p e r a t u r e s fl 0 (a) , /3 0.4 (b) , /3 = 0.6 (c) , a n d /3 = 0.8 (d) for t = 10000 a f t e r a r a n d o m i n i t i a l d i s t r i b u t i o n . No te t he i n c r e a s e in loca l

o r i e n t a t i o n a l o r d e r as /3 i n c r e a s e s . T h e y e l l o w color r e p r e s e n t s t i le p a r t i c l e w i t h o r i e n t a t i o n p o i n t i n g to t he x - d i r e c t i o n , r ed r e p r e s e n t s t h e p a r t i c l e w i t h an o r i e n t a t i o n h a v i n g a 60 ° a n g l e w i t h t he z - a x i s a n d b l a c k r e p r e s e n t s t he p a r t i c l e w i t h an o r i e n t a t i o n h a v i n g a 120 ° a n g l e w i t h t he z - a x i s .

1.0

,.-&-.

0.8

0.6

\"'.L L

0,4

0.2 ~ _ , ~

0 . 0

0.0 10.0

" - - . ...... . . . ....

J J I ,

20.0 500 40.0 500

r-

Fig . 15. T h e c o r r e l a t i o n f u n c t i o n v e r s u s s p a t i a l s e p a r a t i o n for t h e s y s t e m w i t h /3 -- 1.0: p -- 1.8 ( so l i d l ine) , p -- 2.4 ( d a s h e d l ine ) a n d p = 4.8 ( d o t t e d l ine) . N o t e t h a t a s y s t e m w i t h a s m a l l d e n s i t y h a s o n l y a s h o r t - r a n g e c o r r e l a t i o n .

i C .

1.0

0.8

~'.,..

0.6 '"'"

0.4 \,, " '",

0.2 ....... "

00 0.0 10.0 20.0 30.0 400 500

V

Fig . 16. T h e c o r r e l a t i o n f u n c t i o n ve r sus s p a t i a l s e p a r a t i o n for a s y s t e m w i t h d e n s i t y p 4.8 a n d r e c i p r o c a l t e m p e r -

a t u r e s o f / 3 = 0.4 ( so l id l ine) , /3 0.6 ( d a s h e d l ine) a n d /3 1.0 ( d o t t e d l ine) .


and s imula t ions of this model s t rongly suggest the existence of a phase t rans i t ion . Appl ica t ion of this model to two-phase flow in channels , to one-d imens iona l shock waves and to or iental phase t r ans i t ions have shown promis ing results. There are several in te res t ing points wor th ment ion ing . Firs t , this model allows one to specify an arbi- t rary, spat ia l ly dependen t t empera tu re . This fea- ture can be used for research on the pa t t e rn forma- t ion, such as the fo rmat ion of snowflakes. Second, the convect ion coefficient, g ( n ) , in this model depends on t empera tu re . For a large range of density, this coefficient can be a cons tant . This is a desir- able p roper ty to be used to m a i n t a i n the Gal i lean invar iance of the la t t ice gas. Th i rd , the nonideal gas equa t ion of s ta te and phase t r ans i t i on properties have been produced by neares t -ne ighbor inte rac t ions between b o u n d pairs. This long-range in te rac t ion na tu r a l l y in t roduces correla t ions between b o u n d pairs at low t empera tu re . For simula t ing realistic mater ia ls , a long-range in te rac t ion may be necessary.

It is possible to extend the present studies to model l iquid-crys ta l hyd rodynamics and other complex fluids. Resul t s in this paper have demon- s t ra ted the capabi l i ty of s imula t ing a large n u m b e r of rod-like particles. In order to ob ta in a const i tu- tive equa t ion which couples o r ien ta t ion and velocity, a variable associated with angu la r m o m e n t u m must be in t roduced and collision rules tha t con- serve angu la r m o m e n t u m are required.

A c k n o w l e d g e m e n t s

We thank B. Hasslacher, L. Lam and W.H. Ma t thaeus for helpful discussions. This work was suppor ted by the US D e p a r t m e n t of Energy at Los Alamos Nat iona l Labora tory , by D A R P A grant DPP88-50, and by the NASA Innovat ive Research

P r og r a m under grant N A G W 1648. H.R.B. thanks the Deutsche Forschungsgemeinschaf t for suppor t

0f his work.

R e f e r e n c e s

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lattice gas models for nonideal gas fluids

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