boundary methods

7/27/2019 Boundary Methods

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Journal of Statistical Physics. Vo l , 84. Nos. 5/6. 1996

Local Second-Order Boundary Methods forLatt ice Boitzmann ModelsI. G i n z b o u r g I a n d D . d ' H u m i ~ r e s 2

Received August 2. 1995A new way to implement solid obstacles in lattice Boltzmann models is presen-ted. The unknown populations at the boundary nodes are derived from thelocally known populations with the help of a second-order Chapman-Enskogexpansion and Dirichlet boundary conditions with a given momentum. Steadyflows near a flat wall, arbitrarily inclined with respect to the lattice links, arethen obtained with a third-order error. In particular, Couette and Poiseuilleflows are exactly recovered without the Knudsen layers produced for inclinedwalls by the bounce back condition.K EY W O R D S : Lattice Boltzmann equation; boundary conditions; Cha pman-Enskog expansion: Knudsen layer.

1 . I N T R O D U C T I O NL a t t i c e - g a s ~ a n d l a t t ic e B o l t z m a n n m o d e l s ~2' 31 h a v e b e e n i n t r o d u c e d d u r -i n g t h e p a s t 1 0 y e a r s a s a l t e r n a t i v e w a y s t o s i m u l a t e f lu i d fl o w s . T h e y a r er e s p e c ti v e ly s im p l i fi ed m o l e c u l a r - d y n a m i c s o r B o l t z m a n n e q u a t i o n s , r es tr ic -t e d t o r e g u l a r l a t ti c e s a n d l i m i t e d s e t s o f v e l o c i t ie s . F u r t h e r i n f o r m a t i o na b o u t t h e o r y a n d a p p l i c a t io n s o f t h e se m o d e l s c a n b e f o u n d i n tw o e x t e n -s i v e r e v i ew s : r ef. 4 f o r l a t t i c e - g a s m o d e l s a n d r ef. 5 f o r l a tt i c e B o l t z m a n nm o d e l s .

A s in m o l e c u l a r -d y n a m i c s o r B o l t z m a n n e q u a t i o n s , t h e m a c r o s c o p icf i e l d s ( p r e s s u r e , v e l o c i t y , . . . ) a r e n o t h a n d l e d e x p l i c i t l y i n l a t t i c e - g a s o rt CNRS, Applications Scientifiques du Calcul lntensif , Universit6 Paris Sud, 91405, OrsayCedex, France; e-mail: [email protected].

2 CNRS and Universit6 Pierre et Marie Curie, Laboratoire de Physique Statistique de I'ENS,75231 Paris Cedex 05, France: e-mail: [email protected].

9270022.4715/96/0900.0927509.50/0 (t~ 1996 Plenum PublishingCorporation


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9 2 8 G i n z b o u r g a n d d ' H u m i ~ r e s

lattice Boltzmann models, but are computed a p o s t e r i o r i from the localparticle populations. Consequently, boundary conditions are implementedimplicitly by supplying the unknown populations coming from outside thefluid; i .e., each boundary condition, such as solid walls, inlet, outlets,...,requires the knowledge of an appropriate closure relation for theseunknown populations.

The most commonly used condition to implement solid walls is the so-called b o u n c e b a c k r u l e , where the populations leaving the fluid return tothe node of departure with the opposite velocity. This rule is not only verysimple, but also enforces mass conservation in the computational domain.Indeed, this implicit annihilation of the velocity raises the question of theexact location of the no-slip walls. This problem becomes especially impor-tant for effective simulations of flows when the distance between walls canbe of the order of a few lattice units and hence accurate location of solidwalls with respect to the lattice is crucial. Examples of such flows can befound in porous media (6-9) or due to the motion of Solid particles inSuspension.(~~

When the bounceback rule is applied, linear analysis"a) of stationaryflows invariant by translation along the wall locates no-slip walls exactlyhalfway between the boundary fluid nodes and the next ones inside thesolid. Since this analysis is based on a first-order Chapman-Enskog expan-sion, it gives an exact location of no-slip walls for linear flows only.A second-order analysis of this problem "5-~7) has shown that the actual posi-tion of the no-slip walls depends also on the size of computational domain,the kinematic viscosity, and one of the eigenvalues of the collision matrixunrelated to any physical parameter of the simulated flow. For Poiseuilleflows parallel to one of the population velocities, this second-order analysisspecifies exactly the distance between no-slip walls and boundary nodes,either with the bounceback rule for some eigenvalues of the collisionmatrix (~5. 16) or for some mixing of bounceback and specular reflections. (17)

Unlike the above methods in which the unknown incoming popula-tions are computed from the known outgoing ones, some authors haveproposed alternative methods in which these unknown populations aredirectly computed from Dirichlet boundary conditions (known velocitieson the solid wall). In the approaches proposed by Ziegler(~s) and Noble e ta l . ( 1 9 ) for walls parallel to a link of the FHP lattice, the boundary condi-tions enforce prescribed velocities on the boundary nodes. It turns out thatZiegler's method is a special case of a mixture of bounceback and specularreflections, for which the first-order Chapman-Enskog expansion locatesthe wall on the boundary nodes. However, this method has the samelimitation as the bounceback rule: the position of the solid wall is exact forCouette flows only. The second method provides second-order accuracy


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L a t ti c e B o l tz m a n n M o d e l s 929

but seems difficult to extend when the wall is no longer located on nodes(inclined walls) or when the number of unknown populations is larger thanthe number of velocity components.Skordos (z~ and Ginzbourg c~7) have proposed more general methods inwhich the unknown populations are computed from first-order ~2~ or second-order (17) Chapman-Enskog expansions, the derivatives being approximatedby a finite-difference scheme. In Skordos' method the pressure (or the den-sity) is assumed to be known at the solid. In Ginzbourg's method the densityis kept unknown and the unknown populations are obtained as solutions ofa linear system derived from the second-order Chapman-Enskog expansion.Since finite-difference approximations of derivatives are used in thesemethods, they require access to the populations of the neighboring latticenodes, and thus they are in general nonlocal. The introduct ion of finite dif-ferences into lattice Boltzmann models also makes these methods noticeablyless stable.Moreover, Knudsen-type layers appear near the walls when they areinkclined with respect to lattice links and bounceback or a combination ofbounceback and specular reflections are used, as shown in this paper. Thisproblem seems difficult to solve on regular lattices when finite differenceapproximations are used, since they assume that Dirichlet conditions areimposed on some set of points obtained from the discretization of theinclined solid wall rather than on the real wall itself.In this paper we introduce a new boundary approach, referred as thelocal second-order boundary (LSOB) method, which does not use finite dif-ferences, but computes locally at each boundary node all the derivativesnecessary for the second-order expansion of the unknown populations fromthe known ones. The main idea of this method is to split the hydrodynamicfields-momentum, its first- and second-order derivatives, and density-intotwo sets: the known fields given by the boundary conditions and theunknown fields. The linear system relating these unknown fields to theknown populations is then written at each boundary node using two kindsof known populations: those arriving from neighboring liquid nodes andthose which should leave it and propagate into the solid. It follows that thenumber of known populations is equal to the number of velocities definingthe lattice Boltzmann model. We will show that this number is alwayslarger than the number of unknown hydrodynamic fields, at least forDirichlet conditions. At this stage the difficulty is to show tha t the problemis numerically well posed, i.e., that a subset of linear equations can beextracted in order to obtain an invertible linear system. Assuming this criti-cal step successfully solved, the unknown boundary populations arrivingfrom the solid are then obtained as linear combinations of a subset of theknown populations at this node. The coefficients of these combinations

822/84/5-6-3


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depend upon the inclination of the solid wall, its velocity, the distancebetween the wall and the boundary node, and the eigenvalues of the colli-sion matrix. For a fixed wall with a general Dirichlet condition, these coef-ficients are computed only once at each boundary node during theinitialization procedure and they are then used during the usual iterationsteps.With the LSOB method Dirichlet conditions on arbitrary smoothwalls, such as planes, spheres, cylinders .... can be introduced into steady3D flows with a third-order error. In particular, Couette and Poiseuilleflows can be simulated exactly for any inclined channels of an arbitrarysmall width. Thus, the method solves with a third-order error bothproblems: location of no-slip walls and appearance of nonhydrodynamicsolutions. In principle, this method can be extended to walls moving witha velocity very small compared to the particle velocities. In this case theclosure systems at the boundary nodes have to be recomputed each timethe wall is moved, with potential difficulties when the wall crosses thelattice nodes.

Thus, the paper is organized as follows. Section 2 is devoted to ageneral exposition of the method within the following framework: the 3Dflow is assumed to be stationary and the fluid domain is bounded by aninfinite flat solid wall, arbitrarily inclined with respect to the lattice axes,with given Dirichlet conditions. Although the method applies to latticeBoltzmann models on triangular FHP and facecentered hypercubic(FCHC) lattices, the symbolic derivation will be given for the FCHCmodel only. For simplicity the technical details are worked out only for afiat wall kept parallel to one axis of the lattice and are given in section 3.The extension of the results to more general cases, such as smooth surfaces,is almost straightforwardy but requires cumbersome formulas which can-not be easily given within the scope of this paper.These results are applied to some particular flows in section 3 and inAppendix A. The numerical results are compared to those obtained by thebounceback rule. They show that the new method gives the exact solutionsand removes the nonhydrodynamic modes produced by the bouncebackcondition when the velocity field can be written as a second-order polyno-mial in space coordinates.

Two difficult questions raised by this new method are briefly studiedin Appendices A and B. In Appendix A we show that the total mass isautomatically conserved by LSOB methods when stationary Couette orPoiseuille flows are simulated in inclined channels. The derivation is onlygiven for a simplified discretization of the wall. The linear stability analysisof the lattice Boltzmann method with the new boundary conditions is givenin Appendix B for a very simple geometry.


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Lattice Boltzmann Mo dels 9312 . G E N E R A L F R A M E W O R K O F T H E L O C A L S E C O N D - O R D E RB O U N D A R Y M E T H O D

2 .1 . L a t t i c e B o l t z m a n n E q u a t i o n a n d S e c o n d - O r d e rC h a p m a n - E n s k o g E x p a n s i o n

In this paper, we will consider the standard lattice Boltzmann equa-tion 12"31 in which the nonlinear terms in the equilibrium distribution areneglected [see relation (6)]. For the lattice nodes r surrounded by theliquid fraction of the medium, the lattice Boltzmann equation with anexternal force F is given by

bmNAr + C,, t + 1) = N~(r, t) + ~ ~:~0Nj(r, ) + d ' p ( r , t ) F . C , ,j ~ l

i E { 1 , . . . , b , , , }(1 )

where N~ is the population moving with velocity C~, ~ is the collisionmatrix, and d' is a model-dependent parameter given in relation (10).Density p and momentum j are defined in the standard way

bmp(r, t)= ~" N i ( r , t ) (2)i= lbmj(r, t ) = Y ' N i ( r , t ) C i (3)i= l

Their conservation is imposed on the collision matrix d byd " 1 =. q' . C~=0, V ~ { 1 ..... D} (4)

where D is the dimension of the physical space, 1 is the b,,,-vector { 1 .... 1},and the vector C~ is built from the components of the b , , populationvelocities in direction cc The collision matrix is fully defined by the choiceof its nonzero eigenvalues and the corresponding eigenvectors. These eigen-vectors are given for FHP models in ref. 14 (see also Appendix B) and forFCHC models in refs. 17 and 21. The relation between the eigenvalues andthe coefficients of the usual collision matrix can be found in ref. 22. Notethat the term N j - N~ q is replaced by Nj in the collision operator Z ~jNjsince the nonlinear terms in the equilibrium distribution N~q are neglected.

When the ratio e between the lattice unit and a characteristic length ofthe medium is smaller than one, the second-order Chapman-Enskog


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9 3 2 G i n z b o u r g a n d d ' H u m i( ~ re s

expan s i on g i ves t he fo l low i ng app ro x i m a t i on fo r any so l u t i on o f t he l a tt ic eBo l t zm ann equa t i on (1 ) :

N ~ ( r , t ) = N ~ e q ~ ( r , t ) + N ~ ( r , t ) + O ( e 3 ) , i e { I,. .. , b, ,} (5)wh ere N I eq~ is a func t ion of the loca l con serv ed quan t i ti es , N I tl is O(e) , an dthus a func t ion of the f ir st der iva t ives of these quan t i ti es , and NI -~ is O(e") ,and thus a func t ion o f the i r seco nd der iva t ives . Th e f i rs t tw o t e rm s in ther ig h t h a n d s id e o f ( 5 ) ar e i so t r o p i c f o r t h e F H P a n d F C H C l at ti ce s a n dhave t he sam e exp res s i on fo r any Ca r t e s i an coo rd i na t e sys t em . Th i s i s nol onge r t rue fo r t he second-o rde r t e rm N ~2~ w hich i s anis ot ro pic . In theseque l, the Ca r t e s i an coo rd i na t e sys t em {x , y} o r {x , y , z} i s chosen a l ong" la t ti ce axes , " i. e ., such th a t on e ve loc i ty is w r i t t en (1 , 0 ) for F H P m od el sand three veloci t ies are wri t ten (1, 0 , 0) , (0 , 1 , 0) , and (0, 0 , 1) for the 3Dp r o je c ti o n o f F C H C m o d e ls .

S u c h s e c o n d - o r d e r C h a p m a n - E n s k o g e x p a n s i o n s c a n b e f o u n d i nre f. 23 fo r t he F H P m ode l and s t eady f low s , and i n re f. 16 fo r F C H Cm ode l s w i t h r es t pop u l a t i ons and t i m e-depe nden t f low s. Fo r s t eady f low sa n d f o r b o t h F H P a n d F C H C m o d e l s w i t h o u t r e st p o p u l a t i o n s , th e d if-fe ren t o rders a re g iven by

N l . e q ' ( r , t ) = d ( r , t ) + d ' j . C , + O ( l l J ' - I I ) ( 6 )t )= d' V [ , . 0 , . ] {2~ -jp 3, F H PN ; ( r , t ) = d ' v Z 0 7 ~ ' ' { - ~ - ~ 2 , F C H C,_ ~ } - - ~ + T ~ p p , t = ( 8 )

w h ere ~ and f l a r e t aken i n {x , y} o r {x , y , z} , c 2 is t he sq ua re o f pop u l a -t ion veloci t ies , the densi ty per cel l d i s given by

d ( r , t ) = p ( r , t ) ( 9 )b ,.and t he m ode l ' s pa ram e t e r d ' i s g i ven by

d ' - D {2 , F H Pc ' - b , , ' D = 4, F C H C ( 10)Th e f i r s t -order e igenvec tors of the co l li s ion ma t r ix , a ssoc ia t ed wi th theeigenvalue 2~, , are

rQ ~ p = C i~ C i p - - ~ d ~ ,p , V o w , l ( 1 1 )


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La t t i c e B o l t z m a n n M o d e l s 9 33

and t he seco nd-o rde r e i genvec t o r s o f t he co l li s ion m a t r i x , a s soc i a t ed w i t hthe e igenvalue 22 , a re

D + 2T i ~ # # = C i ~ , C2 C;~ C~p, V ~ : / : f l (12)

H erea f t e r , t he F C H C l a tt ice is a s sum ed ; t he i m p l em en t a t i on o f t he newb o u n d a r y t e c h n iq u e f o r th e F H P l at ti ce c a n b e d e r iv e d in a s im i la r w a y ( a ne x a m p l e c a n b e f o u n d i n A p p e n d i x B ) .

2 .2 . G e n e r a l iz e d B o u n d a r y C o n d i t io n sW hen t he re a re no so l i d w a l l s , a l l popu l a t i ons a re de f i ned by t he l a t -t i c e Bo l t zm ann equa t i on (1 ) . W hen so l i d w a l l s a re i n t roduced , boundarynodes Zo a p p e a r , f o r w h i c h t h e p o p u l a t i o n s c a n b e d i v i d e d in t w o n o n e m p t y

subse t s : know n local p o p u l a t i o n s N ' i~ w h i ch a r r i ve f rom t he ne i ghbor i ngfluid nodes, and unknown i ncom i ng p op u l a t i ons N i i " w h i ch sho u l d a r r ivef rom t he " so l id" nod e Z o - C i ( see F i g . 1). Le t loc,. (z0) and / i i n ( z 0 ) be t he

I - I I I - I - - - - - - I - - - - -I I I I I II I I I I II - - - 1 - - - " 1" - - - - I - - - - - I - - - - I - - - -I I I I 1% J m " I t

/I i I I I ~ I I- - ~ " + - - ~ - I - - I - i ~ - I - - - -I I Z o l i N " i

Z '

F i g . 1 . x - z s e c t i o n o f t h e F C H C l a t t i c e f o r a s o l i d w a l l p a r a l l e l t o t h e y a x i s .


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s ubs e t s o f i nd i ce s i f o r t he pop u l a t i on s It;~ 162 an d in~ (Z0) , r espec t ive ly ,w i t h/Ioc+=t z 0 ) t . . ) / i n ( z 0 ) = { 1 . .. . b,,,} (13)I'i~ u I in(Zo) = { ~ }

L e t L l~ b e t h e n u m b e r o f lo c a l p o p u l a t i o n s NI+~ a n d L iu t h e n u m b e r o finu n k n o w n p o p u l a t i o n s N i ;n i n t h e n o d e z o ( L l ~ = b . , ) .I n a d d i t i o n , to e a c h u n k n o w n p o p u l a t i o n N i n (z 0), th e r e c o r r e s p o n d s a

k n o w n o n e N O U t ( z o ) l e a v i n g t h e l a t t i c e w i t h a n o p p o s i t e v e l o c i t y :C j = - C i , i E / i n ( z 0 ) , j E I ~ (14)

N o t e t h a t N ~ d e f i n e d f o r i ~ I~ is t h e p o s t c o l l i si o n p o p u l a t i o ng i v e n b y t h e r i g h t - h a n d t e r m o f ( 1) , w h i l e N ~~ is t h e p o s t - p r o p a g a t i o np o p u l a t i o n . L e t L ~ b e t h e n u m b e r o f l e a v in g p o p u l a t i o n s N ~ i n t h e n o d ez0 ( L ~ = L in an d L I~ + L ~ = b m ) .

T h e g o a l o f a g e n e r al iz e d b o u n d a r y c o n d i t i o n is t o c o m p u t e t h e L inu n k n o w n p o p u l at io n s N ii~ , i n t h e n o d e Zo i n s u c h a w a y t h a t s o m e g i v e nc o n d i t io n s , f o r in s t a n c e , a g i v e n v a l u e js o ~ o f t h e m o m e n t u m , w o u l d b ei m p o s e d o n a g i v e n b o u n d a r y s u r f a c e w i t h a t h i r d - o r d e r e r r o r O ( e 3 ) .A s y s t e m a t i c w a y t o a c h i e v e th i s g o a l is t o c o m p u t e a l l t h e u n k n o w nq u a n t i t ie s a p p e a r i n g in t h e d if f e re n t t e r m s ( 6 ) - ( 8 ) o f t h e s e c o n d - o r d e rC h a p m a n - E n s k o g e x p a n s i o n ( 5 ) as li n e ar fu n c t io n s o f t h e b , , , k n o w np o p u l a t i o n s N I ~162n d N ~ f u lf il li n g a ll th e b o u n d a r y c o n s t r a i n t s w i t h t h ep r e s c r i bed accu r acy . I f t h i s l i nea r p r o b l em can be s ucces s f u ll y s o l ved , t heu n k n o w n p o p u l a t i o n s a r e o b t a i n e d a s l i n ea r fu n c t i o n s o f t h e k n o w n o n e s :N i i n ( z o ) = d(zo) + d ' j s~ C i

+ Y . P o < Z o ) { N ~ ~ 1 7 6j E lloc+ E e , j ( Z o ) { N ~ U t ( z o ) - d ( z o ) - d ' ( J ~ ~

j E f o u l i ~ l i " ( z o)(15)

w h e r e t h e l o c a l d e n s i t y i s k e p t u n k n o w n a n d i t s v a l u e m u s t b e a u t o m a t i -c a l l y s u p p l i e d b y t h e a l g o r i t h m a l o n g w i t h t h e m a t r i x P ( z o ) - - ( P 0 . ( Z o ) ) .

T h e e x i st e n ce o f s u c h a n a l g o r i t h m is n o t o b v i o u s a p r i o r i f o r g e n e r a lb o u n d a r y c o n d i t i o n s a n d s u r f a c e s . I n t h i s p a p e r w e e x h i b i t a n a l g o r i t h mf o r a D i r ic h l e t b o u n d a r y c o n d i t i o n o n a n i n fi n it e f la t w a l l p a ra l l e l to af o u r f o l d s y m m e t r y ax i s o f t h e 3 D F C H C l a tt ic e . S e v e r a l p a r t s o f t h e d e r iv a -t i o n a r e k e p t i n t h e i r m o s t g e n e r a l f o r m , a l l o w i n g t h e r e su l ts t o b e e x t e n d e d


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Lattice Boltzmann M od els 935

i n a r a t h e r s t r a ig h t f o r w a r d m a n n e r t o m o r e g e n e r a l o r i e n t a ti o n s o f f ia tw a ll s, to s m o o t h s u rf a c es , o r t o d i f fe r e n t b o u n d a r y c o n d i t i o n s , a t t h ee x p e n s e o f a s l i g h tl y m o r e c o m p l i c a t e d a l g e b r a . T h e e x t e n s i o n o f t h em e t h o d t o t h e d i ff ic u l t p r o b l e m o f s u rf a c e s w i t h s h a r p e d g e s, s u c h a s aco r ne r be t w een t w o p l anes , i s l e f t f o r f u t u r e w or k .

2 .3 . U n k n o w n D e r i v a t i v e s o n a F l a t W a l l

I n s p e c t i n g t h e t e r m s ( 6 ) - ( 8 ) e n t e r i n g t h e s e c o n d - o r d e r C h a p m a n -E n s k o g e x p a n s i o n , o n e s ee s t h a t t h e y m a y i n v o lv e u p t o t w e n t y fiv eu n k n o w n q u a n t i ti e s th e 3 c o m p o n e n t s o f t h e m o m e n t u m , t h e i r 9 fi rs t- o rd e rde r i va t i ves , 12 o f t he i r 18 s ec on d- o r de r de r i va t i ves , and t he dens i t y . T h i sn u m b e r o f u n k n o w n s i s l ar g er t h a n t h e n u m b e r o f k n o w n p o p u l a t io n s , 18f o r t h e 3 D F C H C l at ti c e, a n d t h e y c a n n o t b e e x t r a c t e d f r o m N li~ a n dN ~ w i t h o u t f u r th e r i n f o r m a t i o n . I n d e e d t h is i n f o r m a t i o n m u s t b es o u g h t f r o m t h e b o u n d a r y c o n d i t i o n s ; f o r i n s t a n c e , D i r i c h l e t b o u n d a r yc o n d i t i o n s o n a s m o o t h w a l l p r o v i d e , e v e r y w h e r e o n t h i s w a l l , n o t o n l y t h ev e l o c i ty c o m p o n e n t s , b u t a l so a ll th e i r d e r iv a t i v e s a l o n g a n y d i r e c t io n st angen t t o t he w a l l . I n o r de r t o keep t he a l geb r a a s s i mp l e a s pos s i b l e , i nt h is p a p e r t h e m e t h o d w i ll b e g iv e n e x p li c it ly f o r t h e D i r i c h l e t b o u n d a r yc o n d i t io n w i t h a u n i f o r m m o m e n t u m jso ~ o n a f i a t w a ll.

L e t { x ', y ' , : ' } d e n o t e t h e o r t h o g o n a l c o o r d i n a t e s y s t e m a s s o c ia t e dwi th the f l a t wal l : the _- ' - ax i s i s perpendicu lar to the wal l , i t s pos i t ive un i tv e c t o r n b e i n g d i r e c t e d o u t w a r d f r o m t h e f lu i d ; t h e x ' a n d y ' a x e s a repa r a l l e l to t he w a l l ( see F i g . 1 ). Fo r a ny b o un da r y l a t t i c e no de Zo app ea r -s o , be i t s o r t hogona l p r o -ng f r om t he d i s c r e t i za t i on o f the s o l i d w a l l , l e t z oj e c t i o n o n t h e w a l l a n d ~ ( Z o) i ts a l g e b r a ic d i s t a n c e f r o m Z o :

, ~ ( Z o ) = ( Z o s o , ,- z o ) - n (16)W i t h th e se n o t a t i o n s a n d a u n i f o r m m o m e n t u m o n t h e f ia t w a ll, t h e

firs t- a n d s e c o n d - o r d e r d e r i v at iv e s o f t h e m o m e n t u m a l o n g x ' a n d y ' a r ee x a c t l y e q u a l t o z e r o e v e r y w h e r e o n t h e w a l l , a n d c o n s e q u e n t l y a t t h epo in ts z os~

~ j ~ ' ~ j ~ '( Z o ~ = 7 y , ( z ~ ~O x '0 2 J ~ ' ~o, a 2 J ~ ' 02j,. . . . ( Z o )Zo)z o ) ~ol s o lO x '2 - O x ' y ' - O y 2 = 0 , o E = { x ' , y ' , z ' }

( 1 7 )


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A f u r t h e r r e d u c t i o n o f th e n u m b e r o f u n k n o w n d e r iv a t iv e s c o m e s f r o mt h e c o n t i n u i t y e q u a t i o n f o r s t e a d y f lo w s

o j : , / o j ~ . , a & ' ,Oz--W=- ~ , + ~ y , ) (18)a n d i t s d e r i v a t i v e s a l o n g t h e t h r e e a x e s

" ) . ? .02j_-, = _ ( O-'jx, + O 'jy, "~Oo~'O z' \Oo~' O x' Oo~'o y J ' o d e { x ' , y ' , z ' } ( 1 9 )

T h u s o n l y t w o f i r s t- o r d e r a n d s ix s e c o n d - o r d e r d e r i v a t i v e s o f t h em o m e n t u m a re n o n z e r o a n d u n k n o w n o n a f la t w a ll w i t h a u n i f o rmm o m e n t u m . F o r t h e s u b s e q u e n t d e v e l o p m l e n t , i t i s c o n v e n i e n t t o w r i t et h es e d e ri v a ti v es t im e s d ' a s t h e c o m p o n e n t s o f a g e n e r a l u n k n o w n v e c t o rX(zo):

{ o j . , . , o j , , , o ' - j ~ . , o g x , 0 2 + o ' - j , , , o : j . , . , o : Z , , , " ~ so ,X ( z o ) = d ' \ O z " O z" Oz 2 ' Ox' O z" Oz 2 ' Oy' "Oz" Oy' Oz" Ox' Oz ' } ( z ~

(20)A s s u m i ng 6 o f the o r d e r o f a f ew l a t t ic e u n i t s , i .e ., o f o r d e r e , t hem o m e n t u m a n d i ts fir st- a n d s e c o n d - o r d e r d e r i v a ti v e s o n Z o c a n n o w b e

o b t a i n e d f r o m t h e f o l lo w i n g T a y l o r e x p a n s i o n s:L , ( Z o ) - . ~ o , - o S O '- - J~ , t O ( Z ~ ~ "k-2 ~2 0"J='z , z( Z o ) + O ( e 3 ) , o ( = { x ' , y ' , z ' } (21)

o j ~ . o o ' - L . , s o , ,( z 0 ) : Oz--; z ~ ~ + o ~ t z o ~ + O ( e 3 ) ,~ ( Z o ) OzJ='6 ~ ( z ~ ~ + O ( ~ 3 )

O z.(z~ = ~ ~ ( z~~ + 0 (~3) ,Oj:,0 p ' ( Z o ) = o ( ~ 3 ) , { ~ ' , ~ , } = { x ' , y , }

a n d9 -

= , = { x , , y , }

1 ~ , , ~ ' } = i x , , y ' }

- - - - - ( Z o ) + o ( ~ ' ) , ~ ' = { x ' , y ' }& ' O z' (Zo) & ' O z't h e o t h e r s e c o n d - o r d e r d e r i v a t i v e s b e i n g o f O ( e 3 ) .

(22)

f l ' = { x ', y ' , z '} ( 2 3 )


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L a t t ic e B o l t z m a n n M o d e l s 9 3 7

Then the momentum at the boundary node Zo and its first and secondderivatives can be expressed with a third-order error as linear combinationsof eight momentum derivatives on the wall. A first strategy, worked outexplicitly in the next subsections, is to consider the vector X and the localcell density d(zo) as the only unknowns of the problem. In Appendix A,using Couette and Poiseuille flows as examples, a second strategy is givenin which the Taylor expansion is used for the momentum only and startsfrom the boundary node Zo; the number of unknown second-orderderivatives on Zo is kept equal to six using (23) and it can be shown, withthe help of a symmetry argument, the continuity equation, and relations(22), that there are only five unknown quantities entering the first-orderterm of the Chapman-Enskog expansion:

0 j ; 0 j ; 0 j ; , 0 j ; , 0 j ; o j ; ,Oz--' o y " O z" oy ' .4 '3 IZo) (24)This second form of the problem requires more known populations to finda solution and may be difficult to use in some complex geometry. However,in some situations, it gives simpler formulas for which the analyticalcalculations can be pushed farther than with the first approach.

2 .4 . S e c o n d - O r d e r A p p r o x i m a t i o n n e a r a F l a t S o l i d W a l lLet the C'i be the population velocities expressed in the inclined coor-

dinate system: C'i= { C~x,, C;),,, C;.,}. Using (19) and (21), we can relate theequilibrium solution (6) to X by

N~eq~=d(zo)+d'jS~ C'i+ 89 X] 9C'i (25)where the [3 x8] matrices K ~ and K 12) relate respectively the first- andsecond-order derivatives coming into the Taylor approximation (21) of themomentum to the vector X:(000000

K I t ~ = 1 0 0 0 0 00 0 0 0 0 0

(26)

and (010000K12) = 0 0 0 1 0 00 0 - 1 0 - 1 0

(27)


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I n o r d e r t o r e l a t e t h e f i r s t - o r d e r c o r r e c t i o n N l i 1 ) t o X, a l l t hed e r i v a ti v e s a p p e a r i n g i n ( 7 ) c a n b e e x p r e s s e d i n t e r m s o f t h e d e r i v a ti v e s~ o l u s i n g t h e T a y l o r a p p r o x i m a t i o n s ( 2 2 ) . T h i ss t i m a t e d o n t h e w a l l a t z 0 ,f o r m a l p r o c e d u r e is s im p l i fi e d b e c a u s e t h e f i r s t -o r d e r t e r m N ~ ~ i s i n v a r i a n tu n d e r a n y s p a c e i s o m e t r y a n d h a s t h e s a m e f o r m i n a n y o r t h o g o n a l c o o r -d i n a t e s y s t e m . T h u s N I ~ b e c o m e s i n t h e i n c li n e d c o o r d i n a t e s y s t e m{ x ' , y ' ; z ' }

w h e r e [c s ( 1 1 ) ]

a t ~ J c t ,N t ' ) ( r)i ~ ~ , . , ~ - 7 ( r ) a i a , # ,

9

Qs~,p, = C ~, C;p, - D 8~ '/r,

( 2 8 )

U s i n g t h e n t h e c o n t i n u i t y e q u a t i o n ( 1 8 ) , w e c a n w r i t e N l.t)(r) i n t h e f o l lo w -i n g v e c t o r i a l f o r m :

w h e r e

lN l ~ )( r) = ~ - ~ E ' " ( r ) . Q 'i

E I I ) = d ' (E , . , : , , E,- 'x ' , E: , ,=, , Ey,y, , E, ., ;, ,) (r ) (31 )Q', = ( Q,_,. '_- ' ,Q,.,-'.,-'- Q w :' , Q ;y':,, Qs.,,,y' Q ,_-': ', Q , . , - ' y ' ) ( 3 2 )

a n d t h e c o e f f i c ie n t s E ~,a , a r e g i v e n b y

E ~ ,p ,( r) = ~ - 7 + ~7s / ( r ) , s 4= fl', { s f l '} ~ {x ' , y ' , z ' } ( 33)0J~t ' rE~,~,(r ) = ~7~, ( ) , o ~ ' e { x ' , y ' , z ' } ( 3 4 )

A p p l y i n g t h e r e l a t io n s ( 2 2 ) a n d ( 2 3 ) t o t h e c o e ff i c ie n t s ( 3 3 ) a n d ( 3 4 ) a t t h eb o u n d a r y n o d e s , w e r e l a t e t h e f i r s t - o r d e r t e r m N l ~ ) ( Z o ) t o t h e v e c t o r X b yN C J ) = z , li q - [ G l ' ) ' X ] ' Q i + [G I2 )" X ] " Q i (3 5)

(30)

{ ~ ' , f l ' } e { x ' , y ' , z ' } ( 2 9 )


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L a t t i c e B o l t z m a n n M o d e l s 9 3 9

wh ere the [5 x 8] m at r i c es GI t l an d G ~21 are g iven b y( i 0 0 0 0 0 0 ! )0 0 0 0 0G r 1 0 0 0 0 0 (36)0 0 0 0 0 00 0 0 0 0 0a n d ( i 0 1 0 0 0 0 1 )0 - - 1 0 0 0G I ~ I = 0 0 0 1 0 00 0 0 0 - 1 00 0 0 0 0 1 (37)

B y a n a l o g y w i t h t h e f i rs t -o r d e r c o r r e c t i o n , t h e s e c o n d - o r d e r t e r m N ~ 2~i n ( 8 ) can be b r i e f l y w r i t t en a s

Nii2~(r)_ v E, ,_)~ ( r ) . T ~ ( 3 8)t h e s i x - c o m p o n e n t v e c t o r s E ~2~ (r ) a n d T i b e i n g

E I'-) (r ) = d ' ( E ...... E ........ E , .. . E = , ,, ,, E y x . , - , Ex,.y)(r) (39)T i = ( T i . , . = = , ri=.,.,., T /., .. . T i - , ~ r , r i y . , - . , - , T i .,- .,,_ ~ .) (40)

T h e t h i r d - o r d e r e i g e n v e c t o r s T i= pp a r e d e f i n e d i n ( 1 2) ; f o r F C H C m o d e l ,t he coe f f i c ien t s E ~ /jp a r e r e l a t ed t o t he s eco nd - o r de r d e r i va t i ves by [ c f. ( 8 ) ]

[0-'J= ' 2 {x , z} (41 )2 j [ 1 ]T h e t e r m N ~2~ i n ( 38 ) is no t i nv a r i a n t b y an a r b i t r a r y s pace i s om e t r y . C o n-s e q u e n t l y , i t s f o r m c a n c h a n g e w h e n t h e i n c l i n e d c o o r d i n a t e s y s t e m i s u s e dand w e can / ao t s i mp l y r ep l ace { x , y , z } w i t h { x ' , y ' , z '} i n t h i s t e r m asdo ne f o r t he f i r s t - o r de r t e r m N I ~) i n ( 28 ). O n t he o t he r ha nd , u s i ng ( 19 )a n d ( 23 ), w e c a n e x p r e ss t h e c o e f f ic i e n ts ( 41 ) i n t e r m s o f t h e u n k n o w ns e c o n d - o r d e r d e r i v a t i v e s a t t h e s o l i d w a l l

vE{21(Zo) = ~ IN " X ] " T i (42)


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H e r e , t h e [ 6 8 ] m a t r i x ~ d e p e n d s o n l y u p o n t h e i n c l in a t i o n o f t h e s o l i dw a l l; it s t w o f ir st c o l u m n s , c o r r e s p o n d i n g t o t h e f i r s t- o r d e r d e r i v a t i v e s i n X ,a r e i d e n t i c a l l y z e r o [ c f. (2 7 ) a n d ( 3 7 ) ] . I n S e c t i o n 3 , t h e m a t r i x ,~ i s w r i t t e nf o r a s o l i d w a l l p a r a l l e l t o a l a t t i c e a x i s .

S u b s t i t u ti n g r e l a ti o n s ( 2 5 ), (3 5 ), a n d ( 4 2 ) i n to t h e C h a p m a n - E n s k o ge x p a n s i o n ( 5 ), t h e s e c o n d - o r d e r a p p r o x i m a t i o n o f t h e p o p u l a t i o n s a t t h eb o u n d a r y n o d e s

N i ( z o ) = d ( z o ) + d ' j s ~ Ci + X ( z 0 ) " e i ( z 0 ) ( 4 3 )w h e r e t h e v e c t o r e i ( Z o ) i s g i v e n b y

1 . ( . ) , ,e i = 6 K ~ I I ' . C ' ; + ~ o - K - " C i

1 G I , ) , . Q , i + ~ G ( 2 ) , . Q ' i

+ v ~ t~ 9 T i , i ~ { 1 . . . . b . , } ( 4 4 )T h e s u p e r s c r ip t t d e n o t e s t h e tr a n s p o s e o p e r a t o r . T h u s , e i g h t c o m p o n e n t so f t h e v e c t o r X (z o ) a r e to b e f o u n d a t e ac h b o u n d a r y n o d e i n o r d e r t o c o n -s t r u c t t h e p o p u l a t i o n s o l u t io n . I t is i m p o r t a n t t o r e p e a t h e r e t h a t l o c a l d e n -s i t y d ( z o ) i s a l s o c o n s i d e r e d t o b e u n k n o w n .

T h e p o p u l a t i o n s N ~ t ) l e a v in g t h e l a tt ic e a f t e r c o l l is i o n a t t i m et - 1 c a n b e d e ri v e d f r o m ( 4 3 ) w i t h t h e h el p o f t h e la t t ic e B o l t z m a n n e q u a -t i on ( 1 ):N~ t ) = d(z o , t - 1 ) + d' j s~ Ci

+ X (z o, t -1 ) 'e ~ t - l ) F . C , , i e{ l , .. ., b ., } ( 4 5 )w i t h

bme ~ ~ , .~ j "e j i 6 { 1 . .. .. b , ,,} (4 6)j = l

S i n c e t h e v e c t o r s C ~ ,, Q ~ ,p ,, a n d T ~ p p a r e t h e e i g e n v e c t o r s o f th e c o l l i s i o nm a t ri x , t h e d e r i v a t i o n o f e ~ i s s t r a i g h t f o r w a r d . F o r s t a t i o n a r y s o l u t i o n s ,w e c a n r e p l a c e d ( z o , t - 1 ) b y d ( z o , t ) a n d X ( z o , t - 1 ) b y X ( z 0 , t ) in t h er i g h t - h a n d s i d e o f (4 5 ). H o w e v e r , t h e f o r c in g t e r m i n ( 4 5 ) is a c t u a l l yc a l c u l a t e d a t t im e t - 1 [ s e e ( 1 ) ] a n d i s h e n c e k n o w n .


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Latt ice Boltzman n Mo dels 9412 .5 . S e c o n d - O r d e r S o l u t i o n f o r t h e U n k n o w n P o p u l a t io n s

2 .5 .1 . L i n e a r E q u a t i o n s f o r t h e U n k n o w n C o e f f ic i e n t s . W em u s t n o w c o m p u t e X ( z o , t ) f r o m ( 4 3 ) - ( 4 6 ) . T h i s s te p r e q u i r e s a na p p r o p r i a t e s u b s et o f k n o w n p o p u l a t i o n s f r o m t h e s e ts { ,,~ oc,vi ~z 0, t )} an d( o r ) {I ~ o , t ) } . L e t IU ~r be a ne w s e t o f i nd i c e s f o r t he s e c ho s e n po p -u l a ti o n s . E a c h , j e P ~ " ( z0 ) h a s a c o r r e s p o n d i n g i n d e x i e i t h e r i n P ~ o ri n I ~ L e t u s d e n o t e N US~(Zo, t ) t h e c o r r e s p o n d i n g k n o w n c o n t r i b u t i o nt o ( 4 3 ) o r ( 4 5 ) a n d e ~ ~ t h e c o r r e s p o n d i n g v e c t o r e i o r e ~ I n t h e fi rs t c a se ,

N ~ ( Z o , t ) = "" i^rl~ t ) - d ' j s~ C i ( 47 )a n d i n t h e s e c o n d o n e

Nj~'~(Zo, t'j - . . . .v, t,tZo, t ) - { d ' j S ~ t - 1 ) F ' C , } ( 48 )U s i n g t h e s e n o t a t i o n s , w e c a n e x p r e s s E q s . ( 4 3 ) a n d ( 4 5 ) i n t h e s a m e

w a y :e ~ * ( Z o ) . X ( z o ) = N ~ * ( Z o ) - d ( z o ) , j e l U ~ (Z o ) ( 49 )

I t fo l lo w s t h a t a v e c t o r X c a n b e o b t a i n e d f r o m ( 4 9 ) f o r a n y s u b s e t o f e i g h tpo pu l a t i o ns N ~ s~ s uc h t ha t t he c o r r e s p on d i n g [ 8 x 8 ] m a t r i x e us~ is i nv e r -t i b l e . I n S e c t i on 3 , w e s how t ha t t he r e e x i s t s a t l e a s t one s ubs e t w he n t hew a l l i s pa r a l l e l t o t he y a x i s . I n ge ne r a l s e ve r a l s ubs e t s g i v i ng a n i nve r t i b l em a t r i x e us~ c a n be f ou nd , i.e ., t h e r e e x i s ts a f a m i l y o f L S O B c l o s u r e r e l a -t io n s . T h e s t u d y o f t h is f a m i l y a n d t h e e x i s t e n c e o f a p o s s i b l e " b e s t " c h o i c ea m o n g i ts e l e m e n t s i s l ef t f o r f u t u r e w o r k . O f c o u r s e a ll th e a c c e p t a b l eL S O B c l o s u r e r e l a t i o n s y i e l d t h e s a m e N i in p o p u l a t i o n s f o r f lo w s s u c h t h a tt h e O ( e 3) t e rm s v a n i s h i n th e C h a p m a n - E n s k o g e x p a n s i o n . F o r g e n e r a lf low s , a t h i r d - o r d e r d i f f e r e nc e i s e xp e c t e d be t w e e n t he N i ln p o p u l a t i o n sd e r i v e d f r o m d i f fe r e n t s u b se ts .

F o r g e n e ra l D i r ic h le t c o n d i t i o n s ( n o n u n i f o r m m o m e n t u m ) , s o m ed e r i v a t i v e s a p p e a r i n g i n ( 1 7 ) a r e n o n z e r o ( b u t k n o w n ) a n d t h e u n k n o w nd e r i v a t i v e s a r e t h o s e e n t e r i n g t h e v e c t o r X d e f i n e d i n ( 2 0 ) . T h e n t h e l i n e a rs y st em ( 4 9 )" is u n c h a n g e d , b u t t he c o n t r i b u t io n s o f t h e k n o w n n o n z e r od e ri v at iv e s t o th e s e c o n d - o r d e r C h a p m a n - E n s k o g e x p a n s i o n m u s t b ei n c l u d e d i n (4 7 ) a n d ( 4 8 ) , a s d o n e f o r t h e n o n z e r o m o m e n t u m j~o~.

S i n c e t h e l i n e a r s y s t e m ( 4 9 ) i s l o c a l , e a c h b o u n d a r y n o d e c a n b e c o n -s id e r ed i n d e p e n d e n t l y . T h e m e t h o d b e c o m e s e s p ec ia ll y t r a n s p a r e n t f o r t h er a t h e r a r t if i ci a l c a s e o f a k n o w n d e n s i t y d i s t ri b u t i o n . T h u s , l et u s c o n s i d e rthi s case f i rs t .


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942 Ginzbourg and d'Hum i~res2 .5 .2 . L S O B M e t h o d f o r F lo w s w i t h a K n o w n D e n s i ty

( A r t i f i c i a l Case ) . W h e n t h e d e n s i t y i n ( 4 9 ) i s k n o w n a n d p r o v i d e dt ha t d e t [ e ~s~ ] is d if f e r en t f r om ze r o , t he s o l u t i on f o r X can be i m m ed i a t e l yd e r i v e d f r o m ( 4 9) . S u b s t i t u t i o n o f t h is s o l u t i o n i n ( 4 3) g i v es th e u n k n o w np o p u l a t i o n s N ip i n t h e f o l l o w i n g f o r m [ c f. (1 5 ) ] :

N i i n ( zo ) = d { l - ~ , P o } + d ' j S ~ ~ . . P o N ~ , i E / i n ( z o ) ( 5 0 )j e 1 " se . i~ I use

w h e r e t h e e l e m e n t s o f t h e m a t r i x P ( z o ) a r e s i m p l yP / j = e / ' [ e U ~ ] j - l , i~Iin(zo), j~ I~S ~(z o) (51)

H e r e [ e U ~ ] f I d e n o t e s t he j t h c o l u m n o f t he m a t r ix [ e ~ ] - I . N o t e t h a t th es o l u t i o n ( 5 0 ) b e c o m e s d e n s i t y i n d e p e n d e n t i f a n d o n l y i f P is a t r a n s i t i o nm a t r i x

P , j = 1 (52)j e / u s e

N o t e a l s o t h a t t h e c l a s s i c a l r e f l e c t i o n s - - b o u n c e b a c k , s p e c u l a r , o r a l o c a lc o m b i n a t i o n o f t h e t w o - - r e p r e s e n t v e r y p a r ti c u l a r c a se s o f ( 50 ) w h e n ( 52 )is explicit ly i n t r o d u c e d .

2 .5 .3 . L S O B M e t h o d f o r F l o w s w i t h U n k n o w n D e n s i ty( G e n e r a l Case) . L e t u s n o w r e p l a c e t h e u n k n o w n d e n s i t y i n ( 4 3 ) , ( 4 5 )and i n t he l i nea r s y s t em ( 49 ) by i t s de f i n i t i on [ s ee ( 2 ) and ( 9 ) ]

d = b , 7 , 1 { y ' N i + y"i i ~ } ( 5 3 )i e l lC ' C (z o ) i e l i n i z o )

T h e s u b s t i t u t i o n o f ( 5 0) w i t h ( 5 3) i n t h e s u m o f r e l a t io n s ( 4 3 ) f o r p o p u l a -t i ons N i p [ i ~ p n ( z o ) ] gives the sohaion for loca l densiO, w i t h a t h i r d - o r d e rer ror O(z3) :

d = l I ~ N li~ + d ' j ~~ " ~ C i + ~ ~ Pi/. NySe t~C -~r i ~ / l~ Z o ) i e l i n ( z 0 ) i~ m~ ,ol y~ l"~~ (5 4 )

=L'~ Z Z s .j ~ l U S e~ r~ ) i e I i n ( z o )w he re L I~ is the total n u m b e r o f p o p u l a t i o n s N ~;~ a t t he no de z 0 a s de f i nedi n s e c t i o n 2 . 2 . T h u s , t h e l o c a l d e n s i t y i s a u t o m a t i c a l l y o b t a i n e d a s a l i n e a rc o m b i n a t i o n o f t h e p o p u l a t i o n s N ~;~ an d N y S+ p r o v i de d t ha t c Jz i s nonze r o .


17/45


When (54 ) i s subs t i t u t ed i n (50 ) , t he so l u t i on fo r t he unknow n popu l a -t ions b ecom es the fo l l ow i ng li nea r com bi na t i on o f t he p opu l a t i ons A ~ 162and N~ se: c , }

i e l in ( z o )

+p~o~ Z NJPc+ Z P}Sr ~, i E / in (z0 ) (55)j e l l o c j e I u se

w h e r e

j s o , }i - @ zt he e l ement s o f the m at r ix p ,s~ a re

P~ 'r ~ ~ P k j , i e li " (Z o ) , j e I " ~ ( Z o ) ( 57 )k 9 ZO

and the mat r ix P i s i n t roduced in (51) .The p r i nc ipa l r e su l t o f t he r e l a ti ons (55 ) - (5 7 ) is t ha t t he un kn ow npopu l a t i ons a re de t e rm i ned co r rec t l y , w i t h a t h i rd -o rde r e r ro r O (e3 ) , a s a

l inea r com bi na t i on o f t he know n p opu l a t i ons supp l i ed by the l a tt ic eBo l t zm an n equ a t i on ( 1 ). The coe f fi ci en ts o f t h is com bi na t i on a re func t ionsof the inc l ina t ion of the so l id wall , i ts ve loc i ty , t he d i s t ance be tw een thew a l l and t he bo un da ry node , and t he e i genva lues o f t he co l li s ion m a t r i x[ s e e ( 4 4 ) ] . T h e y d e p e n d a l so u p o n t h e lo c a l d i s tr i b u ti o n o f k n o w n a n du n k n o w n p o p u l a t i o n s t h r o u g h t h e t e r m @ z [ s e e ( 5 4 ) ] . F r o m c o n s i d e r a -t ions of l inear s t ab i l it y in som e s impl i f ied geo m et ry (see A ppe ndix B) , thel a t t i ce Bo l t zm ann equa t i on (1 ) w i t h t he bounda ry cond i t i ons (55 ) - (57 ) i sexpec ted to be s t ab le a t l eas t when the so l id wal l i s l oca ted outside thelattic e (5 ~< 0).

F o r e a c h b o u n d a r y n o d e Z o, w e c a n s u m m a r iz e t h e L S O B m e t h o d a sfollow s. Th e d i s t ance t o t he a s sum ed so l id w a l l is com pu t ed and t he vec t o r se ; ( z o ) a r e con s t ruc t ed fo r a ll t he pop u l a t i ons a t t he bo un da ry nod e Zo ,us ing the d es i red inc l ina t ion of the so l id wal l and the chose n e igenv alues )w,and 22 [c f . (44) ] . Th en a s ubs e t o f po pu la t ion s {N~S~(Zo), e lUS~(Zo)} isc h o s e n a n d th e m a t r ix o f b o u n d a r y c o n d i t io n s P "S ~ is c o m p u t e d fr o mthe cor resp on ding m at r ix eUS'(Zo) [c f. (51) , (54) , and (57) ] . Al l t hese s t epsa re done on l y once fo r each bounda ry node du r i ng t he i n i t i a l i za t i on p roce -d u r e . T h e n t h e u n k n o w n p o p u l a t i o n s a r e c o m p u t e d f r o m t h e r e l a t i o n s(55 ) - (5 7 ) fo r each ti m e s t ep o f t he i te r a t i on p rocedure .


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9 4 4 G i n z b o u r g a n d d ' H u m i 6 r e s

2 . 6 . C o n s e r v a t i o n o f M a s sI t is n o t e w o r t h y t h a t t h e r e is n o r e a s o n f o r th e s u m o f t h e u n k n o w n

p o p u l a t i o n s c o m p u t e d w i t h th e L S O B m e t h o d t o b e e q u a l to th e s u m o ft h e p o p u l a t i o n s l e a v i n g t h e l a t t i c e . T h u s , i n g e n e r a l , o n e s h o u l d e x p e c t t od e t e c t s o m e c h a n g e i n t h e t o t a l m a s s AM(t),

AM (t) Z ~ N ~ t ) - Z Z i ,= N i ( Z o , t ) ( 5 8 )z0 iel~ 1-O i~l in( z0 )i . e . , a nonz e r o m a s s f l ux a c r o s s t he bounda r i e s .

T h i s p o s s i b i l i t y t h a t t h e t o t a l m a s s i s n o t c o n s e r v e d i n g e n e r a l f l o w sr e p r e s e n t s t h e p r i n c i p a l d i f fe r e n c e b e t w e e n e x p l ic i t m e t h o d s a n d c l as s ic a lr e f l e c t i o n s . A l t h o u g h i t i s t a n t a l i z i n g t o e n f o r c e t h e m a s s c o n s e r v a t i o n i nt h e L S O B m e t h o d u s in g l o ca l ly t h e a d d i ti o n a l c o n s t r a i n t A M ( t ) = 0 a t e a c ht i m e s t e p t, w e s h o w i n th e n e x t s e c ti o n t h a t l o c a l c o n s e r v a t i o n o f m a s sl ea d s to n o n h y d r o d y n a m i c b o u n d a r y l ay e rs fo r C o u e t te a n d P o is eu i l lef l ow s i n i nc l i ne d c ha nne l s .

S i n c e t h e l o c a l m a s s is e x a c t ly c o n s e r v e d b y t h e c o ll is i o n o p e r a t o r , t h et o t a l m a s s i n a n y b o u n d e d d o m a i n is a l so e x a c t l y c o n s e r v e d f o r a n y s t e a d yf lo w . I t f o ll o w s t h a t z J M - 0 f o r a n y s t e a d y fl o w i f t h e s u m m a t i o n s in ( 5 8 )a r e t a k e n f o r a ll t h e b o u n d a r y n o d e s o f t h e c o m p u t a t i o n a l d o m a i n .H o w e v e r e v e n i f t h e l a c k o f m a s s d i s a p p e a r s f o r s t e a d y f lo w s , it c a n e x i s tw h e n t h e s t a t i o n a r y r e g i m e i s n o t y e t r e a c h e d . I n a d d i t i o n , t h e g l o b a l c o n -s e r v a t i o n o f m a s s d o e s n o t i m p l y t h a t t h e t o t a l m a s s f lu x a c r o s s a n y s id eo f t h e d o m a i n s h o u l d b e e x a c t l y z e r o . F o r p a r t i c u l a r f lo w s , e x a c t l y g i v e n b yt he s e c o n d - o r d e r C h a p m a n - E n s k o g e x p a n s i o n a n d h a v i n g a z e ro m a ss f lu xa c r o ss s o m e s id e s o f t h e c o m p u t a t i o n a l d o m a i n , t h e L S O B m e t h o d g iv e s az e r o m a s s f l ux a c r o s s t he s e s i de s . T h i s i s t he c a s e f o r t he t w o w a l l s pa r a l l e lt o C o u e t t e a n d P o i s e u i l le f lo w s , a s s h o w n a n a l y t i c a l ly i n A p p e n d i x A . F o rm o r e g e n e r a l s t e a d y flo w s , t h e b e s t w h i ch c a n b e e x p e c t e d f ro m L S O Bm e t h o d i s A M = O ( 3 ) , s ince t he t e rm s O(e 3 ) h a v e b e e n n e g l e c t e d .

L e t u s fi rs t c o m p u t e A M f o r a g e n e r a l s t e a d y f l o w n e a r a s o l id b o u n d -a r y p a r a l l e l to t h e x y p l a n e o f t h e F CH C l a t t i c e , w i t h o u t f o r c i n g t e r m s(F_, = 0 ) . S u b s t i t u ti n g in ( 5 8 ) t h e s e c o n d - o r d e r a p p r o x i m a t i o n s o f Nip a n dN ~ g i v e n b y ( 4 3 ) - ( 4 6 ) y i e ld s t h e f o l lo w i n g m a s s f l u x a c r o s s t h e b o u n d a r y :

o 2 j = o ,M - - 1 2 + z 2

H e nc e , w he n t he no r m a l ve l oc i t y is z e r o a t t he w a l l (j ~o t = 0 ) , t he m a s s f l uxi s p r o p o r t i o n a l t o 02j~~ 2. E x c e p t f o r t h e p a r t i c u l a r w a l l l o c a t i o n s6 - - - 2 / 3 a n d ~ = 0 ( w a ll lo c a t e d o n t h e b o u n d a r y l a tt ic e n o d e s ), t h e m a s s


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L a t t ic e B o l t z m a n n M o d e l s 9 45

f lu x is o f o r d e r O ( 2 ) w h e n t h e d e r i v a t i v e 0 2 j ~ l /O z 2 i s O ( 1 ) , e . g . , s t a g n a t i o nf l o w n e a r a n # f i n i t e s o l i d p l a n e , t25~ A s i m i l a r c a l c u l a t i o n f o r a s o l i d w a l lp a r a l l e l t o t h e y a x i s a n d i n c l i n e d a t 4 5 ~ w i t h t h e x a x i s ( s t il l p a r a l l e l t os o m e o f t h e F C H C v e lo c it ie s ) sh o w s a d e p e n d e n c e o f A M w i t h 82j~~ '2w h i c h d o e s n o t v a n i s h f o r A = 0 a n d ~ = - 2 /3 , b u t f o r o t h e r v a l u e s o f t h em i n i m u m d i s t a n c e b e t w e e n t h e l a t t i c e n o d e s a n d t h e w a l l . P r e l i m i n a r yc a l c u l a t i o n s f o r m o r e g e n e r a l o r i e n t a t i o n s o f t h e w a l l h a v e s h o w n a s im i l a rd e p e n d e n c e o f t h e m a s s f lu x w i t h t h e s e c o n d - o r d e r d e r i v a t i v e s o f t h em o m e n t u m . U n f o r t u n a t e l y t h e c o e ff ic ie n ts d e p e n d u p o n t h e d e ta i ls o f t h ed i sc r e ti z a ti o n o f t h e b o u n d a r y a n d w e d o n o t e v e n k n o w i f t h e y c a n v a n i s hf or s o m e v a l u e s o f t h e m i n i m u m d i s ta n c e b e t w e e n t h e la t t ic e n o d e s a n d t h ew a l l , a s o b s e r v e d f o r t h e p r e v i o u s o r i e n t a t i o n s .F r o m t h e s e r e s u l t s , i t a p p e a r s t h a t t h e u s e r i s f a c e d w i t h t w o c h o i c e s :t o k e e p th e t h i r d - o r d e r p r e c is i o n o f t h e L S O B m e t h o d a t th e p o s s ib l ee x p e n se o f a s e c o n d - o r d e r m a s s f lu x a c r o s s s o m e s id es o f t h e b o u n d a r y , o rt o p r e s c r i b e a z e r o m a s s f l ux a c r o s s e a c h s i de o f t h e b o u n d a r y ( f o r i n s ta n c e ,b y a n o n l o c a l d is t r ib u t i o n o f A M b e t w e e n t h e u n k n o w n p o p u l a t i o n s a t th eb o u n d a r y n o d e s ) a t th e e x p e n s e o f a s e c o n d - o r d e r p r e c is i o n w h e n82j~?J/Sz '2 is O ( 1 ). T h e a c t u a l c h o i c e w i ll p r o b a b l y d e p e n d o n t h e p r o p e r t i e so f t h e s i m u l a t e d f lo w .

3 . F L A T S O L I D W A L L P A R A L L E L T O T H E y A X I S3 . 1 . 3 D F l o w

I n t h e p r e v i o u s s e c t i o n s w e h a v e n o t m a d e a n y a s s u m p t i o n a b o u t t h ei n c l in a t i o n o f t h e s o li d b o u n d a r y w i t h t h e l a t ti c e a x es . H o w e v e r , in o r d e rt o d e m o n s t r a t e t h e m e t h o d i n s o m e d e t a i l a n d , i n p a r t i c u l a r , i n o r d e r t od e t e r m i n e t h e s u b s e t s o f th e p o p u l a t i o n s N ~ se, o n e m u s t f in d a l l th ep o s s ib l e l o ca l n e i g h b o r h o o d o f t h e b o u n d a r y n o d e s a p p e a r i n g f r o m t h e d is -c r e t i z a t io n o f a n i n c li n e d f l at w a l l o n t h e F C H C l at ti ce . T h i s r a t h e r c o m -p l ic a te d p r o b l e m b e c o m e s m u c h s i m p l e r if t he s o l id w a l l is a s s u m e d t o b ep a r a l l e l t o o n e a x i s , s a y y , b u t a r b i t r a r i l y i n c l i n e d w i t h t h e t w o o t h e r o n e s ,x a n d z . T h i s s i m p l i f i e d g e o m e t r y s t i l l r e m a i n s q u i t e g e n e r a l c o m p a r e d t ot h e s o l i d w a l l s p a r a l l e l t o a l a t t i c e p l a n e t h a t a r e u s u a l l y c o n s i d e r e d .

L e t 0 b e t h e a n g l e o f t h e w a l l w i t h t h e x a x is ; th e a s s o c i a t e do r t h o g o n a l c o o r d i n a t e s y s t e m is

x ' - -- x co s 0 + z s in 0z ' = - x s i n 0 + z co s 0 ( 6 0 )y t . .~ . y

822/84/5-6-4


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a nd t he [ 6 8 ] m a t r i x ~ - i n ( 42 ) is g i ve n by r e l a t i ons

E x : : = c os 0 ( C O S 2 0 - - 2 s in 2 0) O z , 2

+ sin 0 (7 cos '- 0 -- 2 sin-" 0) +O 2 j x ,E_..,.,. = s i n 0 (s in 2 0 - - 2 cos-" 0) O z , 2

+ co s 0 (2 cos-" 0 - 7 s in 2 0) 0 2 j ' 'O x ' O z ', o j , , , . . + o 9 . < ?Eyz z = C O S - g ~ n '- s m 20 . Ox ' Oz ' J

a ' - j . .. .E._;,>,= 2 c o s 0 0 y ' O z '"9. "9., , 0 2 jr . ~ O -'j. ,. , + 0- '~), , "~E ...... = s i n ' - v ~ - - s i n 2 0' { O y ' O z ' O x ' O z ' J

o ' s ; .E~, . y = - 2 s in 0 0 y ' O z '

3 s in 0 c os 2 0 0 2 j y 'O y ' O z '

3 s in 2 0 cos O'- j>, ,0 0 3 , ' Oz ' ( 61 )

A n a l y z i n g t h e r e l a t i o n s ( 2 7 ) , ( 3 7 ) , a n d ( 6 1 ) , o n e s e e s t h a t t h e d e r i v a t i v e s9 . t02j . , . , /Oy ' Oz ' a n d O- j > , , / Ox Oz ' c o m e i n t o ( 4 3 ) w i t h e q u a l c o e f f i c i e n t s e ,v a n d

e;8 fo r wa l l s pa ra l l e l t o t h e y ax i s :ve , v = - - Q i x . r ) , e , s = e n , i ~ { I . . . . . b , , } ( 62 )2 q , " a 2 - -

H e n c e , i n o r d e r t o o b t a i n d e t [ e u se] ~ 0 , t h e s e c o n d - o r d e r a p p r o x i m a t i o n( 4 3 ) m u s t b e w r i t t e n

N i(zo ) = d(z o) + d ' j s~ Ci + X U(zo) 9elJ(Zo) (63)w h e r e t h e s e v e n t h c o m p o n e n t o f X " ( zo ) is t h e s u m o f t h e l as t tw o c o m -p o n e n t s o f t h e v e c t o r X ( z o)

x , . t Z o ) = d , ( a j x , a j ~ , a g x , a k , . , a g ~ , a g , , , a g x ,\ O z " O z " O z 2 ' O x ' O z " 0 2 ' 2 , O y ' O z " O y ' O z ' 0 " - j; , ~ s o l- - + ~ / ( z 0 )( 6 4 )


21/45

Lattice Bol tzman n Mo dels 947a n d t h e s e v e n c o m p o n e n t s o f th e v e c t o r e l I a r e t h e f ir st s e ve n c o m p o n e n t so f t he v e c t o r e,. d e f i ne d i n ( 44 ) . N o t e t ha t f o r a s o l i d w a l l pa r a l l e l t o t h e( x , y ) l a tt ic e p l a n e ( 0 = 0 ~ a n d l o c a t e d e x a c t l y o n th e l a st l a tt ic e n o d e s( 6 = 0 ), t h e l a s t c o m p o n e n t e ll o f ell b e c o m e s e q u a l t o z e r o [ c f. ( 6 2 ) ] . I no r d e r t o a v o i d d e t [ e ~ r = 0 , o n l y t h e f i rs t s ix c o m p o n e n t s o f X l I a n d e l im u s t b e u s e d in ( 63 ) f o r t h e p a r ti c u l a r c as e 0 = 0 ~ 6 = 0 .

3 .2 . 2 D F l o w sT h e u s e rs o f l a tt ic e B o l t z m a n n m e t h o d s o f t en c o n s i d e r 2 D f lo w s.

W he n t he f l ow i s pa r a l l e l t o s om e l a t t i c e p l a ne , s a y ( x , z ) , it c a n b es i m u l a te d o n t h e F C H C l a tt ic e w i t h p e r io d i c a l b o u n d a r y c o n d i t i o n s i n t h ey d i r e c t io n . I n s u c h a c a s e , th e s e c o n d - o r d e r s o l u t i o n f o r 2 D f lo w s w il l b ea u t o m a t i c a ll y o b t a i n e d w h e n t h e L S O B m e t h o d i s a p p l i ed i n i ts g e n er a l 3 Df or m . H o w e v e r , sin ce th e y c o m p o n e n t o f t h e m o m e n t u m a n d t h e m o m e n -t u m d e r i v a t i v e s a l o n g t h e y a x i s a r e e q u a l t o z e r o f o r s u c h f l o w s , t h es e c o n d - o r d e r a p p r o x i m a t i o n o f t h e p o p u l a t i o n s c a n b e w r i tt e n in th ef o ll o w in g r e d u c e d 2 D f o r m :

N i ( z o ) = d ( z o ) + d ' j ~ ~ C i + X 2 D ( Z o ) " e ~ D ( Z o )j = {j ., ., (x ' , z ' ) , j : , (x ' , z ' ) } ( 65 )

w h e r e: o : . , , o 2 j . , , o ; . s o , ,X 2D (z 0) = d ' \ O z " Oz 2 ' O x - ~ z J t z ~ ) ( 66 )

a n d t h e t h r e e c o m p o n e n t s o f t h e v e c t o r e~~ a r e , a c c o r d i ng l y , t he f i r s t , t h i r d ,a n d f o u r t h c o m p o n e n t s o f t h e v e c t o r e; d e f in e d in ( 4 4 ). T h u s , t h e L S O Bm e t h o d n e e d s o n l y t h r e e p o p u l a t i o n s N } s~ t o d e t e r m i n e b o u n d a r y c o n d i -t io n s w i t h t h i r d o r d e r e r r o r f o r t w o d i m e n s i o n a l flo w s.

3 .3 . S o m e P a r t i c u l a r S e t s o f K n o w n P o p u l a t i o n sI n o r d e r to a p p l y th e L S O B m e t h o d o n a n y b o u n d a r y n o d e Zo o n e

m u s t c h o o s e t h e p o p u l a t i o n s N y Se (Z o) [ s e e ( 4 7 ) - ( 4 9 ) ] i n s u c h a w a y t h a tt he c o r r e s po nd i ng m a t r i x e use is i nve r t i b l e a nd t he va l ue o f ~ i n ( 54 ) isn o n z e r o f o r t h e i n c l i n a t i o n 0 o f t h e g i v e n s o l id w a l l, t h e d i s t a n c e 6 b e t w e e nz o a n d t h e w a l l, a n d t h e c h o s e n e i g e n v a l u e s o f t h e c o l li s io n m a t r ix .


22/45

9 4 8 G i n z b o u r g a n d d ' H u m i ( ~ r e s

At th is po in t i t becomes necessary to label exp l ic i t ly the 18 d i f feren tvelocit ies o f the 3D FC H C mo del ; le t us use the fo l lowing cho ice:C1 = ( 1 , O, 0), C2 = ( 1 , O, 1), C3 = (0, O, 1)C 4 = ( - - l , O , 1 ), C 5 = ( - 1 , 0 , 0 ) , C 6 = (0 , 1, 1)C 7 = ( 0 , - 1 , 1), C 8 = ( 0 , 1 ,0 ) , C 9 = ( 0 , - 1 , 0 )

C l o = ( l , I, 0), C ,I = ( - 1 , 1 ,0 ), C 1 2 = ( - 1 , - 1 , 0 )C i 3 = ( 1 , - -1 , 0 ), C ,4 = ( 0 , 0, - 1 ) , C l s = ( - - 1 , 0, - - 1 )C i 6 : ( 0 , 1, - 1 ) , Cz7 = ( 0 , - 1 , - 1 ) , C I 8 = ( 1 , O, - 1 )

(7 )

W he n the solid wall is paral lel to the y axis an d 0~ ~ f ivek inds o f bo un da ry nod es , de no te d r~ to r 5 in F igs. 2a and 2b , can app earfo r th e n a tu ra l " s t a ir - ty p e" d i s c re t i za tio n o n th e F C H C la tt ice , d e f in ed su chthat the a lgebraic d is tance ~ f rom these nodes to the wal l i s negat ive and

Fig. 2 .

Z ~ [ O U t" x : , , ,o0.

_

2 hII

- - - - , .L _ - -III

II

, - [ - - - -I

o II

-4II

dI

o I o o

I(a )

T w o - d i m e n s i o n a l s e c t io n o f n o n i n c l i n e d a n d i n c l in e d c h a n n e l s p a r a l l e l t o t h e y a x i s :( a) 0 = 0 ~ ( b ) t a n 0 = 3 / 5 .


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1 = , - ~ m

X V ' l l " I- - - - " 1 " ' " - - I, . A W -7 , ,

z . , , / I h ' ~ ' '\ i ~ - " 1 I - - - I - - - ~ - - 4\ j , , , , ' o n l o 4 i ~ i i i

\ ~ ~ " i i q i , I9 ~ _ ~ - - L . . ~ ~ - - . - - - ~ - - - , - - - '" , ~ - " / v " " ' 1 I I t I\ ~ l o t i I I

X i I I - I I - _ _ ~ - - ~ - i ~ . " l I - - - -\ , - - - , , , , T 72h'~ I i o i / / o ~ i o ~ 1L \ _ _ J _ _ _ 2 x ~ . . ~ - . \ u / ;

_ _ _ ~ _ .( b )

F i g . 2 (Continued)

t h e r e a r e a t m o s t t w o c o n s e c u t i v e b o u n d a r y n o d e s i n t h e d ir e c ti o n . T h e s en o d e s r , t o r 5 c a n b e d e s c r i b e d a s f o l l o w s :

n o d e r l : L + = 10, L - = 8 ,n o d e r 2 : L + = 1 1 , L - = 7 ,n o d e r 3 : L + = 13, L - = 5 ,n o d e r 4 : L + = 1 4 , L - = 4 ,n o d e r s : L + = 17, L - = I ,

I ' ~ 1 6 2 . . .. . 9 , 1 1 , 1 2 }I ' ~I ' ~ ..... 1 3 }I ' ~ ..... 1 3 , 1 5 }1 ' ~ ..... 1 7 }

( 6 8 )


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w h e r e L + ( L - ) is t h e n u m b e r o f k n o w n ( re sp . u n k n o w n ) p o p u l a t i o n s i nt h e n o d e Z o ( L + + L - = 18 f o r t h e 3 D p r o j e c ti o n o f t h e F C H C l at ti ce ;i n t h i s ca s e t he popu l a t i ons 1 , 3 , 5 , 8 , and 14 mus t be coun t ed t w i ce ) .N o t e t h a t f o r 0 = 0 ~ a ll t h e b o u n d a r y n o d e s a r e o f th e r3 ty p e (s eeF i g . 2 a ) a n d t h e p a t t e r n { r2 , rs} r e p e a t s i t se l f f o r 0 = 4 5 ~ W h e n 0 is n o ti n th e r a n g e [ 0 ~ 4 5 ~ t h e d e s c r ip t i o n o f t h e o t h e r p o s s i b le b o u n d a r yn o d e s is d e r iv e d f ro m ( 68 ) t h r o u g h t h e s y m m e t r y g r o u p o f t h e F C H Clat t ice .

O b v i o u s l y , it is m u c h s i m p l e r to h a v e t h e s a m e s u b s e t o f th e p o p u l a -t i o n s N ~ ~ f o r a n y l o c a l g e o m e t r y a p p e a r i n g f r o m t h e d i s c r e t iz a t i o n o f ag i ven w a l l . Fo r a s o l i d w a l l pa r a l l e l t o a l a t t i c e ax i s and f o r t he " s t a i r - t ype"d i s c re t i z a ti o n , w e h a v e v e r if ie d t h a t s u c h s u b s e t s c a n b e e x t r a c t e d f r o m t h epo pu l a t i o ns ~r~ oc i.e ., w i t ho u t u s i ng t he pop u l a t i on s N ~ E xce p t f o r s om ep a r t i c u la r c o m b i n a t i o n s o f t h e e ig e n v a lu e s 2 , , 22 a n d t h e p a r a m e t e r 3,t he s e s o l u t i ons g i ve d e t [ e u~r ~ - 0 in ( 49 ) an d ~ z # 0 i n (54 ) w he n t he w a l li s loca ted outside t h e l a t t ic e a n d t h e e i g e n v a l u e s o f t h e c o l l i si o n m a t r i xsa t i s fy the l inear s t ab i l i ty co nd i t io ns ~2~ i.e ., th ey a re t a ke n in the in te rva l3 - 2 , 0 [ .S i m u l a t i n g 2 D f lo w s , o n e c a n t a k e t h e f o ll o w i n g s u b s e t o f k n o w np o p u l a t i o n s N ~~ f o r any l oca l geome t r y r ~ t o r s [ c f . ( 66 ) ]

IU ~ e= { 2 , 3 , 4 } ( 69 )W h e n t h e f lo w is 3 D , o n e c a n t a k e , f o r in s t a n c e , i n o r d i n a r y n o d e s r 3 f o ra n o n i n c l i n e d w a ll ( 0 = 0 ~ o r i n a n y n o d e r l - r 5 a p p e a r i n g f r o m t h e d is -c r e t i z a t io n o f t h e i n c l in e d w a l l t h e f o l l o w i n g s u b s e ts o f th e k n o w n p o p u l a -

I o c .t i o n s N ; .IUS~ = {3, 4 , 5, 6 , 7, 11, 13},IUS~= {3, 4, 5, 6, 7, 11},I"Sr = {2, 3, 4, 6, 7 , 8, 11 },

0 = 0 ~ 3 # 00 = 0 ~ 3 = 0

0 ~ ~(70)

I t s hou l d be m en t i on ed t ha t t he s ubs e t s I use i n ( 69 ) and ( 70 ) a r e n o t t heo n l y s o l u t io n s ; o n e c a n f i n d o t h e r s u b s e t s o f t h e k n o w n p o p u l a t i o n s N li~o r N ~ o r e v e n u s e li n e a r c o m b i n a t i o n s o f t h e s e p o p u l a t i o n s .

H o w e v e r , t h e g e n e r a li z a ti o n o f t h es e b o u n d a r y c o n d i t i o n s t o m o r ec o m p l i c a t e d w a l l s , s u c h a s w a l l s w i t h a m o r e g e n e r a l i n c l i n a t i o n o r s m o o t hs u r faces , w i ll r equ i r e ch a r ac t e r i z i ng a l l t he pos s i b l e t yp es o f b ou n da r yn o d e s a n d f i n d i n g a s y s t e m a t i c a l g o r i t h m t o c h o o s e t h e p o p u l a t i o n s N ~ se a tt h e s e n o d e s . T h e s e t a s k s r e m a i n t o b e d o n e .


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L a t ti c e B o l t z m a n n M o d e l s 9 51

3 .4 . A p p l i c a t i o n o f L S O B M e t h o d t o P o i s e u il le a n dC o u e t t e F l o w s

Po i s eu i l l e and C oue t t e f l ow s pa r a l l e l t o s ome p r i nc i pa l l a t t i c e l i nkr ep r e s en t t he u s ua l t e s t s o l u t i ons . Fo r Po i s eu i l l e f l ow s pa r a l l e l t o a pop u l a -t i o n v e lo c i ty , t h e o n l y p r o b l e m o f l a tt ic e B o l t z m a n n s i m u l a t i o n s w i t hc l a ss i cal r e f l ec t ions i s t he e f f ec ti ve l oca t i on o f no - s l ip w a l l s w i t h r e s pec t t ot h e c o m p u t a t i o n a l do m ain /~ 5-2 ~ W h e n t h e b o u n c e b a c k c o n d i t i o n is u s e d ,i t is s how n i n re f. 16 t ha t t he e f fec ti ve w i d t h o f t he c han ne l pa r a l l e l t o s o m el a tt ic e l in k d e p e n d s u p o n i ts i n c l in a t i o n a n d s o m e f u n c t i o n o f t h e e i g e n -va l ues 2~ and 22 ( see r ef. 24 f o r f u r t he r d i s cus s i ons a bo u t t h i s f unc t i on ) .

3 . 4 .1 . P o i s e u i l l e F l o w s in In c l i n e d C h a n n e l s . T w o k in d s o fs t a t i o n a r y P o i s e u i l l e f l o w s c a n b e c o n s i d e r e d i n a c h a n n e l w i t h s o l i d w a l l sp a r a ll e l to t h e y a x is ( y ' = y ) a n d a r b i t r a r i l y i n c li n e d w i t h r e s p e c t to t h e xax is . In the f i rs t case , the f low i s para l l e l to th e x ' ax i s [ see (60 ) ] ; in thes econd cas e t he f l ow i s pa r a l l e l t o t he y ax i s . B o t h f l ow s depend on l y ont h e c o o r d i n a t e z ' a n d t h e f o r c i n g t e r m c a n b e d e s c r i b e d i n a u n i f o r mm a n n e r

0-'j'~, - h ' ~ z ' < ~ h ' , o~' x ' or c~'- P F ~ ' = V O z , 2 , = = Y , P = P o (71)

F o r s u c h f l o w s , t h e s e c o n d - o r d e r C h a p m a n - E n s k o g e x p a n s i o n g i v e s t h ee x a c t s o l u t i o n s i n c e h i g h e r o r d e r d e r i v a t i v e s v a n i s h ,

d' O j~ , . , Q i , , + V__ E~ 2~ (r) . T iN i( r ) = d + d ' j~ ,( r ) c )~ , + ~ [ r ) (72)

Th e t e r m EI2~(r ) 9T i is w r i t t e n i n ( 38 ) - ( 41 ) . W he n t he e x t e r na l f o r ce ( 1 ) i sa p p l i e d t o s i m u l a t e c o n s t a n t p r e s s u r e g r a d i e n t , t h e d e n s i t y d i s c o n s t a n t(s ee d i s cu s s i on i n re f. 24 f o r m or e de t a i ls ) . M or e ov e r , t h e s ec on d- o r de rt e r m E C 21( r ) . T ; i s i nva r i an t by t he r o t a t i on f o r Po i s eu i l l e f l ow s ( s eeA p p e n d i x A ) .

F i g u r e 3 s h o w s t h e p r o fi le s a c r o ss t h e c h a n n e l o f t h e n o r m a l i z e d d i f-f e r e n c e s b e t w e e n t h e n u m e r i c a l a n d e x p e c t e d s o l u t i o n s f o r t h e d e n s i t y a n dt he n o r m a l a n d t a n g e n ti a l m o m e n t a o b t a i n e d w i t h L S O B a n d b o u n c e b a c kc o n d i t i o n s . F o r a l l t h e p r o f i l e s t h e n o r m a l i z a t i o n f a c t o r i s h ' / j ma x, j m ' ~b e i n g t h e t h e o r e t i c a l p e a k m o m e n t u m i n t h e P o i s e u i l l e f l o w a n d h ' t h ee x p e c t e d h a l f - w i d t h o f th e c h a n n e l : h ' = h c o s 0 a n d t a n 0 = 3 / 5.

A s s h o w n b y t h e z e r o - v a l u e l in e s i n F i g. 3, t h e L S O B m e t h o d p r o v i d e sa n e x a c t a n a l y t i c a l s o l u t i o n f o r a n y i n c l i n a t i o n a n d f o r a n y c h a n n e l w i d t h .


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" x-r-'l

0 . 2

0 . 1

o . oQ,.

- 0 . i

- 0 . 2

0 . 2 5 0

0 . 1 2 5

0 . 0 0 0

- 0 . 1 2 5

.0

- 0 . 2 5 0- 5 . 0

h = 5(a )

I I- 2 . 5 0 . 0 2 . 5 5 . 0

Z '

h = 5(b )

I I I '- 2 . 5 0 . 0 2 . 5

Z '5 . 0

F i g . 3. P r o f i l e s o f t h e n o r m a l i z e d d i ff e r e n c e b e t w e e n n u m e r i c a l a n d t h e o r e t i c a l s o l u t i o n f o rt w o P o i s e u i l l e fl o w s : ( a , d ) d e n s i t y p , ( b , e ) t a n g e n t i a l m o m e n t u m j ., ., a n d , ( c , f) n o r m a lm o m e n t u m j ~ , . T h e n o r m a l i z a t i o n f a c t o r i s h ' / j m a x, m a x b e i n g t h e t h e o re t ic a l p e a k m o m e n t u mi n t h e P o i s e u i l l e f l o w a n d h ' t h e e x p e c t e d h a l f - w i d t h o f t h e c h a n n e l ; h ' = h c o s O a n dt a n O = 3 /5 . T h e s o l i d l i n es a r e o b t a i n e d f o r b o u n c e b a c k c o n d i t i o n a n d t h e d o t t e d z e r o - v a l u eo n e s f or t h e L S O B m e t h o d . T h e s i m u l a t i o n s a r e p e r f o rm e d w i t h t h e s a m e v i s co s it y a n d b o d yf o rc e ; a l l t h e e i g e n v a l u e s o f t h e c o l l is i o n m a t r i x a r e e q u a l t o - 1. T h e r e s u l t s a r e s h o w n f o rt w o s i z e s o f t h e c o m p u t a t i o n a l d o m a i n : h = 5 ( a - c ) a n d h = 2 0 ( d - f ) .


27/45


0 . 2

0 . i

,--ik 0 . 0

N"r-3

- 0 . 1

- 0 . 2.0

h=5

I( c )

I- 2 . 5

I0 . 0Z '

I2 . 5 .0

0 . 2

0 . 1

r- 4 o . o

- 0 . i

- 0 . 2

h = 2 0

(d )

k!,,,i . I L- 2 0

I- 1 0

I0

Z '

' I10 20

Fig. 3 (Continued)


28/45


0 . 2 5 0

0 . 1 2 5

~ , . o . o o oX

- 0 . 1 2 5

- 0 . 2 5 0- 2 0

h = 2 0

( e )

I

- 1 0I I0 i 0Z'

20

0 . 2

0 . i

, - - I~ . o . oN

-i==~

- 0 . 1

- 0 . 2-20

h = 2 0

( f )

I I

- i 0 0Z'

Fig. 3 (Continued)

i 0 2 0


29/45


This is due to the fact that for such flows the second-order expansion (72)represents the exact solution and the total mass is automatically conservedby the LSOB method (as shown in Appendix A for the simplified dis-cretization given in Fig. 4). The width of the channel is fixed by the choiceof the computational domain and the distance 3; the inclination is con-trolled by the construction of the LSOB conditions at each boundary node[cf. (43)-(44)]. All the different implementations of LSOB methodsimulate exactly the Poiseuilie solutions: the density can be assumed eitherknown or unknown, and the method can use either the general 3D solution(43)-(44) or its 2D form (65), or even the shorter 1D form given inAppendix A; the possible subsets of the known populations Nys~ for theupper wall can be taken either in (69) or in (70) (the populations for thelower wall being found by symmetry).

III

m

/

9 Jr31~- - - - - 7 " I~ 1 I I

~ I It - - - - - - F - - - - I - - - - - I - - - -I I I II I I I

- - I ' ~ - ~ h ' r - - ~ - - - I - - -I I I II r 5 I I I Il - - j - - - J ' - - - - ~ . . . . IJ i I I I

_ I !

rIo

r 4 I

- - IIo I

o I

, /I

~ Z '

F i g . 4 . " R o u g h " d i s c r e t i z a t i o n o f a s o l i d w a l l p a r a l l e l t o t h e y a x i s f o r t a n 0 - - - m / n , w i t hm = 3 a n d n = 5 .


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956 Ginzbourg and d'Humi(~resA s s how n by t he s o l i d l i ne s i n F i g . 3 , t he s i t ua t i on i s t o t a l l y d i f f e r e n t

f o r t h e b o u n c e b a c k c o n d i t io n w h e n t h e f lo w is n o l o n g e r p a r a ll e l to a s y m -m e t r y a x is o f t h e l at ti c e (0d = x ' , 0 ~ 0 < 4 5 ~ T h e s e re s u lt s s h o w t h a t t h eb o u n c e b a c k c o n d i t io n l ea d s t o u n p h y s i c a l o s c il la ti o n s o f t h e d e n s i ty , w h i c hs h o u l d b e c o n s t an t , a n d t h e n o r m a l c o m p o n e n t o f t h e m o m e n t u m , w h ic hs h o u l d b e ze r o . N o t o n l y d o s im i l ar o s c il la ti o n s a p p e a r o n t h e t a n g e n t ia lm o m e n t u m , b u t a l s o i ts a v e r a g e v a l u e i s s h if te d . W h e n t h e s e ef fe c ts a r ec o m p a r e d f o r t w o d i f fe r e n t w i d t h s , o n e se e s t h a t t h e y a r e lo c a t e d n e a r t h ew a ll s ( b o u n d a r y o r K n u d s e n l a y e r ) a n d t h a t t h e i r a m p l it u d e s a r e a lm o s tc o n s t a n t f o r th e c h o s e n n o r m a l i z a t i o n . T h i s r e s u lt s h o w s t h a t t h e se e r r o r ssca l e l i ke 1/h'; t h e n t h e s i m u l a t i o n s w i t h t h e b o u n c e b a c k c o n d i t i o n a r eo n l y f i rs t -o r d e r a c c u r a te in t h e s iz e o f t h e d o m a i n . T h u s , b o u n c e b a c k c o n -d i ti o ns c a u se t h e ap p e a r a n c e o f K n u d s e n - t y p e n o n h y d r o d y n a m i c s o lu t io nn e a r a n i n c li n e d w a ll. T h e p r i n c i p a l r e a s o n f o r t h a t is local m a s s c o n s e r v a -t i o n e n f o r c e d b y t h e b o u n c e b a c k c o n d i t i o n . I n f a c t , i t c a n b e a n a l y t i c a l l yp r o v e n t h a t t h e m a s s i s n o t l o c a l ly c o n s e r v e d f o r th e s o l u t i o n ( 7 1 ) w h e n an o - s l ip c o n d i t i o n is c o r r e c t l y i m p o s e d a n d t h e c h a n n e l is i n c li n e d w i t hl a t ti c e l inks ; a s i m p l i fi e d p r o o f i s g i ve n i n t he n e x t s e c t i on f o r C o ue t t ef lows.

3 . 4 . 2 . C o u e t t e F l o w in In c l i n e d C h a n n e l s . W e c o n s i d e rs t a t i o n a r y C o u e t t e f l o w i n a n i n c l i n e d c h a n n e l i n o r d e r t o d e m o n s t r a t e a na p p l ic a t i o n o f t h e L S O B m e t h o d t o flo w s w i t h a n o n z e r o m o m e n t u m o na n o b s t a c le . In f a c t, o n e o f t h e a d v a n t a g e s o f t h e L S O B m e t h o d i n c o m -p a r i s o n w i t h t h e c la s si ca l re f le c ti o n s is th a t n o n z e r o m o m e n t u m c a n b ei n t r o d u c e d i n t o th e f lo w in th e s a m e m a n n e r a s z e r o m o m e n t u m .

L e t t he f l ow be pa r a l l e l t o t he x a x i s ; a s a bove , t he z a x i s i s pe r -pe nd i c u l a r t o t he s o l i d w a l l s [ -s ee ( 60 ) ] a n d h ' i s t he w i d t h o f t he c ha n ne l .T h e v e l o c i t y o f t h e l o w e r w a l l is e q u a l t o u 0 a n d t h e u p p e r o n e i s i m m o b i l e .T h u s , t h e s t a t i o n a r y C o u e t t e s o l u t i o n i s

u , - ,= u o I - - , O ~ z ' < ~ h ' , P = P o ( 7 3 )T h e e x a c t s o l u t i o n o f t h e l a tt ic e B o l t z m a n n e q u a t i o n ( I ) f o r th i s l i n e a r f l o wb e c o m e s [ s e e ( 7 2 ) ]

[ ], ( r ) = d + d ' j,,.,(r) C; _ ,. ,+ ~ -- ~ ', Q,.,.,~, , p = p o ( 7 4 )W h e n a c o m b i n a t i o n o f b o u n c e b a c k a n d s p e c u l a r r ef le c ti o n s is a p p l ie d , af i r s t- o r d e r a n a l y si s g iv e s a n e x a c t l o c a t i o n o f t h e i m m o b i l e w a l l in t h el i n e a r f l o w s , p r o v i d e d t h a t t h e c h a n n e l i s n o t i n c l i n e d ( s e e r e f . 1 4 f o r F H P ,


31/45


re fl 17 f o r F C H C ) . I n p a r t i c u l a r , t he no - s l i p c o nd i t i on is s a ti s fi e d e xa c t l y i nt h e m i d d l e b e t w e e n t h e l a s t n o d e s i n t h e f lu i d a n d t h e n e x t s o li d o n e s , w h e n" p u r e " b o u n c e b a c k r e fl e ct io n is i m p l e m e n t e d . H o w e v e r , o n e o b s e r v e s a t th eb o u n d a r y n o d e s n e a r t h e i m m o b i l e w a ll o f a n i n c li n ed c h a n n e l t h a t t h ev e l o ci ty p e r p e n d i c u l a r t o i t d o e s n o t v a n i s h w h e n b o u n c e b a c k r e fl e ct io n isa pp l i e d , i n a w a y s i m i l a r t o t he P o i s e u i l l e f l ow s how n F i g . 3 . H e nc e , t hee x a c t s o l u t i o n ( 7 3 ) c a n n o t b e o b t a i n e d i n i n c l i n e d c h a n n e l s b y t h e l a t t i c eB o l t z m a n n m e t h o d w i t h b o u n c e b a c k c o n d i t i o n . A g a i n t h e p r o b l e m i sl in k e d t o t h e l o c a l c o n s e r v a t i o n o f m a s s. T h e d i f fe r en c e b e t w e e n t h e L S O Ba n d b o u n c e b a c k m e t h o d s is e a si ly c h e c k e d f o r C o u e t t e f lo w s i n th e x 'd i r e c ti o n a n d a n o d e r s . F o r t h e se n o d e s t h e r e is o n l y o n e i n c o m i n g a n do n e o u t g o i n g p o p u l a t i o n r e s p e c ti v e ly , w i t h v e l o c i ti e s C 18 a n d C 4 [ s e e( 6 7 ) ] . U s i n g ( 7 4 ) a n d t h e f a c t t h a t Oj., . , /Oz' i s c o n s t a n t f o r C o u e t t e f l o w s ,t he m a s s f l ux on t he node s r 5 i s g i ve n f o r t he f i xe d w a l l by

o ut N i n A ' r t 't - ' a Jx ' ( 7 5 )N 4 - - 18 . . . . 4.,:'*,"'~4:' "-[-2 O(Z0)) OZ'w ith C4.,-, = s in 0 - co s 0 an d C4:, = s in 0 + co s O. O bv io us ly thi s m ass f lux isin g e n e r a l n o n z e r o f o r 0 ~ < 0 < 4 5 ~ a n d c a n n o t s a ti sf y th e b o u n c e b a c k c o n -dition N'l" = N4OUt.T o s i m u l a te e x a c t ly C o u e t t e f lo w s , o n e c a n . a p p l y th e L S O B m e t h o din it s g e n e r a l 3 D o r 2 D f o r m w i t h j.~~ (Zo) equa l t o poUo ( r e s p . z e r o ) f o r t heb o u n d a r y n o d e s n e a r t h e m o b i l e ( r e s p . i m m o b i l e ) w a l l . H o w e v e r , s i n c e t h ef l o w i s l i n e a r , o n l y o n e d e r i v a t i v e a z l i s r e q u i r e d t o c o m p u t e e x a c t l yt h e u n k n o w n p o p u l a t i o n s . T h i s i s d i s c u s s e d i n s o m e m o r e d e t a i l i nA p p e n d i x A .

4 . C O N C L U S I O NA g e n e r a l a p p r o a c h t o b o u n d a r y c o n d i t i o n s i n l a t t i c e B o l t z m a n n

m o d e l s h a s b e e n d e s c r i b e d . T h i s a p p r o a c h i s b a s e d o n a s e c o n d - o r d e rC h a p m a n - E n s k o g e x p a n s i o n o f t h e p o p u l a ti o n s . A n e xp l i ci t b o u n d a r ym e t h o d w a s t h e n d e v e l o p e d i n o r d e r t o i n t r o d u c e a r b i t r a r i l y i n c l i n e d , f l a ts o l i d w a l l s w i t h a t h i r d - o r d e r e r r o r . T h e d i s t a n c e ~ b e t w e e n l a t t i c e n o d e sa n d t h e m o d e l e d s o l i d w a l l i s a d j u s t a b l e , p r o v i d e d t h a t s t a b i l i t y c o n d i t i o n sa r e sa tis fie d . T h e r e s u l ti n g s o l u t io n f o r t h e u n k n o w n p o p u l a t i o n s a t b o u n d -a r y n o d e s is e x p r e s s ed i n t h e f o r m o f a l i n ea r c o m b i n a t i o n o f t h e k n o w np o p u l a t i o n s , s u p p l i e d l o c a ll y b y t h e l a tt ic e B o l t z m a n n e q u a t i o n . F r o m t h ep o i n t o f v i e w o f n u m e r i c a l e f f ic ie n c y a n d a d a p t a t i o n t o p a r a l l e l c al c u l a-t io n s , t h e m e t h o d is n o t e s s e n t ia l ly d i ff e r en t f r o m s t a n d a r d , i m p l i c it b o u n d -a r y c o n d i t i o n s .


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958 Ginzbourg and d'Hum i~resT h i s t e c h n i q u e r e d u c es t h e e ff ec ts o f t h e n o n h y d r o d y n a m i c m o d e s n e a r

s o l i d b o u n d a r i e s c a u s e d b y l o c a l o r a l m o s t l o c a l m a s s c o n s e r v a t i o ni nhe r en t t o c l a s s i ca l r e f lec t ions . I t a l l ow s exac t s i m u l a t i o ns o f s t a t i o na r yf l o w s , w i t h a c o n s t a n t - c u r v a t u r e p r o f i l e , i n a r b i t r a r i l y i n c l i n e d c h a n n e l sw i t h m o v i n g o r i m m o b i l e w a l ls (e .g ., P o i s e u il le a n d C o u e t t e f lo w s b e t w e e nt w o p l a t e s ) . T h e s e c o n d - o r d e r b o u n d a r y a l g o r i t h m s d e v e l o p e d e a r l i e r f o rs o l i d w a l l s pa r a l l e l w i t h s ome l a t t i c e l i nks r ep r e s en t ve r y pa r t i cu l a r ca s e s o ft h e L S O B m e t h o d .

T h e g e n e r a l i z a ti o n o f t h is b o u n d a r y t e c h n i q u e t o s m o o t h l y c u rv e dw a l ls is c u r r e n t l y u n d e r s t u d y . S o m e p r e l i m i n a r y r e s u lt s h a v e b e e n o b t a i n e df o r a g e n e r a l i z a t i o n o f t h e m e t h o d t o c i r c u l a r s o l id w a l ls a n d h a v e s h o w nt ha t f l ow s i n a r b i t r a r i l y s ma l l e l l i p t i c p i pes can be s i mu l a t ed e x a c t l y t he s er e s u l t s w i l l be r epo r t ed e l s ew her e .

B e s id e s o t h e r s t r a i g h t f o r w a r d e x t e n s i o n s o f t h is m e t h o d , f o r i n s t a n c et o m u l t i v e l o c i ty la t ti c e B o l t z m a n n m o d e l s o r s o l id w a l l s a rb i t r a r i ly i n c l i n e dw i t h r e s pec t t o t he l a t ti c e , s eve r a l d i f fi cu l t p r o b l em s r em a i n . T he f ir s t oneis r e l a te d t o t h e d i f f e re n t t e c h n i q u e s t o i m p l e m e n t t h e m e t h o d ; a l t h o u g ht hey g i ve t he s ame s o l u t i on f o r C oue t t e o r Po i s eu i l l e f l ow s , t h i s w i l l nol onge r be t r ue f o r gene r a l f l ow s w i t h h i ghe r o r de r de r i va t i ves . T he e f f ec t s o ft h e d i ff e re n t a l g o r i t h m s o n t h e q u a l i t y a n d t h e s t a b i l i ty o f t h e n u m e r i c a ls c h e m e r e q u i r e f u r t h e r i n v e s ti g a ti o n s . F i n a l l y , g e o m e t r ie s w i t h s h a r p e d g e sa r e q u i t e c o m m o n i n e n g i n e e r i n g a p p l i c a t i o n s a n d i t w o u l d b e i m p o r t a n tt o e x t e n d t h e p r e s e n t a p p r o a c h t o s u c h b o u n d a r i e s . F o r t h i s l a s t p r o b l e m ,t w o d i ff ic u lt ie s r e m a i n . T h e f ir st o n e i s t o f i n d a n a p p r o p r i a t e m e t h o d t om a t c h t he con s t r a i n t s o n t he t w o w a l ls . T h e s ec ond d i f f i cu l t y is to s pec i f yt h e e f fe c ti ve a c c u r a c y o r d e r w h e n s o m e d e r i v a ti v e s o f t h e v e l o c i t y f ie ld a r eno t de f i ned on t h e s ha r p edges . D e s p i t e t he s e d i f fi cu lt ie s , p r e l i m i na r ys i m u l a t i o n s o f f lo w s in s q u a r e p i pe s h a v e s h o w n t h a t t h e L S O B m e t h o d ism o r e a c c u r a t e t h a n t h e b o u n c e b a c k r u l e .A P P E N D I X A . P O I S E U I L L E A N D C O U E T T E F L O W SA . 1 . O t h e r F o r m o f L S O B M e t h o d

I n t h i s a p p e n d i x , w e f i r s t d e s c r i b e a n a l t e r n a t i v e w a y t o c o n s t r u c ts e c o n d - o r d e r b o u n d a r y c o n d i t i o n s i n l a t t i c e B o l t z m a n n m o d e l s . I n t h i sc a s e , t h e D i r i c h l e t c o n d i t i o n i s a p p r o x i m a t e d a s

9 ~ O j ~ , 1 ~ , 0 2 j ~ , , , o L ' z ' }J ~ e ( z ~ 1 7 6 + ~ ( z ~ ~176 = { x ' , y ' , (A .1)T h e n ex t s t ep r e l a te s t he d e r i va t i ves i n ( A .1 ) t o t he coe f f i c ien t s ( 31 ) an d( 39 ) o f t h e C h a p m a n - E n s k o g e x p a n s io n . U s i n g ( 2 2), o n e c a n e a si ly s h o w ,


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a t l e a s t f o r f l ow s ne a r a f l a t w a l l , t h a t t he de r i va t i ve s i n ( A . 1 ) c a n ber e p r e s e n t e d a s l i n e a r c o m b i n a t i o n s o f t h e c o e f fi c ie n t s ( 3 1 ) a n d ( 3 9 ). T h i sbe c om e s e s pe c i a l l y s i m p l e f o r P o i s e u i l l e f l ow s , f o r w h i c h ( 72 ) c a n bew r i t t e n

w h e r e

N ,( r ) = d (r ) + d 'j~ ,(r ) C,~, + E~ ,: ,(r) Q,,,: , + E~,__,__,(r)T,~,,._,__,~ ' = x ' o r c ( ' = y ( A . 2 )

d ' a L , ,E~,.,(r) 2q, O z ' ( r ) ( A . 3 )

- ) .v 07~ ,E , , . . . . . d ' - - - - ( A .4 )- - 2 z 8 z '2

T h e v e c t o r T~,:,__, is ob t a i n e d by r o t a t i on o f T ~ : : a s d on e p r e v i ou s l y f o r t hevec tor s Q~ , / j , , and Q~p in Sec t i on 2 :

D + 2T i c e : , : , = C i ~ , - c 2 C ia , C ~ z , ( A . 5 )

T h e n , s u b s t i t u ti n g t h e T a y l o r a p p r o x i m a t i o n ( A .1 ) in ( A .2 ), w e h a v e

w i t h

a n d

N i(z o ) , .sold + d j ~ , C i~ , + X ( z o ) 9e i ( z o )

X(zo)={G,: , ,G, : , : ,}(Zo)e i ( Z o ) = { e i ~ , : , , e/s,:,:,}

( A . 6 )

( A . 7 )( A . 8 )

el=,_-, = Q i ~ , : , + p i C i ~ , ( A . 9 )e i~ ,z , z , = T i ~ , z , z , - - p 2 C i ~ ,

T h e l o c a l p a r a m e t e r s PL a n d P 2 a re r e l a t e d t o t h e d i s t a n c e ~ b e t w e e n Zo a n dt he w a l l by

p ~ = 2 q , ~( A . 10 )22 32P 2 = G


34/45

960 Ginzbourg and d'Hum i6resN o t e t h a t i f

e / = O ( A . 1 1 )i E: / in( 7,0

f o r i n s ta n c e w h e n e ' = x ( O = O ~ o r 0 ( = y , t h e n w e h a v e [ cf . ( 5 1 ) ]Po=0 (A.12)

i e l i n I z 0 )

a n d

~ , P , j = 0 (A .1 3)j e t u s e ( z o ) i e l i n ( z o )H ence , w hen t he cond i t i on ( A . 11 ) i s s a t i s

boundary methods

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