Evaluation of the Lattice-Boltzmann Equation Solver PowerFLOW for Aerodynamic Applications

Transcription

1 NASA/CR ICASE Report No. 2-4 Ealation of the Lattice-Boltzmann Eqation Soler for Aerodnamic Applications Daid P. Lockard NASA Langle Research Center, Hampton, Virginia Li-Shi Lo ICASE, Hampton, Virginia Bart A. Singer NASA Langle Research Center, Hampton, Virginia ICASE NASA Langle Research Center Hampton, Virginia Operated b Uniersities Space Research Association National Aeronatics and Space Administration Langle Research Center Hampton, Virginia Prepared for Langle Research Center nder Contract NAS-9746 October 2

2 EVALUATION OF THE LATTICE-BOLTZMAN EQUATION SOLVER POWERFLOW FOR AERODYNAMIC APPLICATIONS DAVID P. LOCKARD, LI-SHI LUO, AND BART A. SINGER z Abstract. A carefl comparison of the performance of a commerciall aailable Lattice-Boltzmann Eqation soler () was made with a conentional, block-strctred comptational id-dnamics code () for the ow oer a two-dimensional NACA-2 airfoil. The reslts sggest that the ersion of sed in the inestigation prodced soltions with large errors in the compted ow eld; these errors are attribted to inadeqate resoltion of the bondar laer for reasons related to grid resoltion and primitie trblence modeling. The reqirement of sqare grid cells in the calclations limited the nmber of points that cold be sed to span the bondar laer on the wing and still keep the comptation size small enogh to t on the aailable compters. Althogh not discssed in detail, disappointing reslts were also obtained with for a cait ow and for the ow arond a generic helicopter congration. Ke words. lattice Boltzmann method,, 2D ow past NACA-2 airfoil, aerodnamic simlations, trblence modeling Sbject classication. Flid Mechanics. Introdction. In recent ears, methods of the lattice-gas atomata (LGA) and the lattice Boltzmann eqation (LBE) hae become an alternatie to conentional comptational id dnamics (CFD) methods for arios sstems (see collections of papers [6], proceedings [7, 6], reiews [4, 4, 3], and monographs [38, 5]). The lattice-gas and lattice Boltzmann methods hae been particlarl sccessfl in dealing with id ow applications inoling complex ids and complicated bondaries, sch as mlti-component ids throgh poros media [8, 3]. Althogh the promising potential of the LGA and LBE methods as iable CFD tools has been demonstrated in a nmber of cases of laminar ow with low Renolds nmber, there is no experience to demonstrate that the method can be applied to trblent ow in the presence of a streamlined bod at high Renolds nmber. In fact, application of the LBE method to trblent ows at high Renolds nmber remains as an area of ftre deelopment [44, 3]. The fndamental philosoph of the lattice gas atomata and the lattice Boltzmann eqation is to constrct simple models based on kinetic theor that presere the conseration laws and necessar smmetries sch that the emerging behaior of these models obes the desired macroscopic eqations. The theoretical fondation for sing simple kinetic models to simlate hdrodnamic sstems (or other complex sstems) is based pon the obseration that in natre macroscopic phenomenon sch as hdrodnamics is rather insensitie to the nderling details in the microscopic, or een mesoscopic, dnamics. In hdrodnamic Comptational Modeling and Simlation Branch, Mail Stop 28, NASA Langle Research Center, Hampton, VA ( address: d.p.lockard@larc.nasa.go) Institte for Compter Applications in Science and Engineering, Mail Stop 32C, NASA Langle Research Center, 3 West Reid Street, Bilding 52, Hampton, VA ( address: lo@icase.ed). This research was spported b the National Aeronatics and Space Administration nder NASA Contract No. NAS-9746 while the athor was in residence at the Institte for Compter Applications in Science and Engineering (ICASE), NASA Langle Research Center, Hampton, VA z Comptational Modeling and Simlation Branch, Mail Stop 28, NASA Langle Research Center, Hampton, VA ( address: b.a.singer@larc.nasa.go)

3 sstems, details of microscopic dnamics can aect the nmerical ales of the transport coecients, bt not the form of the Naier-Stokes eqations. Therefore, as long as these models presere conseration laws and necessar smmetries (e.g., Galilean inariance) satisfactoril, the can be sed to simlate hdrodnamic sstems. The primar design criteria for the deelopment of LGA and LBE models are ecienc and the preseration of the phsical properties of the sstem. The LGA and LBE models resemble moleclar dnamics methods or the Boltzmann kinetic eqation in concept, bt do not hae their comptational complexit. Howeer, some articial phsical limitations are tpicall placed on the sstem to improe the ecienc of the algorithm. The kinetic natre of the LGA and LBE methods leads to the following featres that distingish the LGA and LBE methods from an other conentional CFD method. First, the conection operator of the LGA or LBE models is linear in phase space, similar to that of the Boltzmann kinetic eqation bt dierent than the Eler or the Naier-Stokes eqations. Second, the pressre is obtained throgh an eqation of state, as opposed to soling a Poisson eqation in the incompressible Naier-Stokes eqations. Third, with the Naier-Stokes eqations, the constittie relations are inpt from empirical data, while with the LGA and LBE methods, the constittie relations emerge as a reslt of proper modeling of interparticle potentials. Forth, nlike the Naier-Stokes or Eler eqations, in which macroscopic conseration laws are discretized, the LGA and LBE methods tilize a set of discrete particle elocities sch that the consered qantities are presered p to machine accrac in the calclations. In addition, the LGA and LBE methods hae the following comptational characteristics:. Intrinsic parallelism de to nearest neighbor data commnications of the streaming (conection) process and prel local calclation of the collision process; 2. Abilit to handle complex bondaries withot compromising comptational speed. The intrinsic parallelism of the LGA and LBE methods is an appealing featre in light of the clear trend to se massiel parallel compters. The abilit to handle complex bondaries is important to practical applications sch asow throgh poros media. The capabilit to inclde model interactions is crcial to simlations of some phsical sstems sch asmlti-phase or mlti-component ows. Frthermore, the LGA method has the following featres, in addition to the aboe, de to its Boolean natre:. Unconditional stabilit; 2. Memor ecienc; 3. Sitabilit for special-prpose compters. The theoretical adantages of the LGA and LBE methods are still largel ntested in the realm of CFD. Moreoer, precisel what adantages the LGA and LBE methods exhibit oer conentional methods of soling the Naier-Stokes eqations are expected to be problem dependent. The areas where the LGA and LBE methods are more sitable depend on the phsical natre of the problems. Unfortnatel, de to the lack ofresearch eort in this area, a scient amont of nmerical eidence to erif the theoretical adantages of the LGA and LBE methods does not exist, especiall in the areas where conentional CFD methods hae proen to be sefl, sch as in the eld of aerodnamics. Recentl, a commercial CFD software prodct,, based on the lattice-gas and the lattice Boltzmann method became aailable [7]. has been applied to simlate high Renolds nmber trblent external ow oer an atomobile bod in grond proximit [3]. Howeer, has not been sbjected to a set of thorogh alidation tests. The focs of this std is to explore the possibilit of sing intpical aerodnamic applications of CFD simlations of arios congrations. In this report, we present test reslts for a 2-D ow past a streamlined airfoil (NACA-2) [] with Renolds nm- 2

4 bers (Re = U L=, where U is the free-stream elocit, L is the chord length, and is the kinematic iscosit) ranging from 5 to 5 6 sing and compare the reslts with those from a conentional CFD soler [48, 5, 47, 4, 39, 29]. Or test reslts regarding are rather negatie: in its present state,, release 3. is inadeqate for aerodnamic applications. Neertheless, with frther deelopment, LGA and LBE methods ma proe to be sefl tools for aerodnamics. This report is organized as follows. In the next section, the basic notion of lattice-gas atomata and the lattice Boltzmann eqation is brie reiewed. The lattice Boltzmann eqation is presented as a special nite-dierence form of the continos Boltzmann eqation, and the bondar conditions and trblence models in the lattice Boltzmann method are discssed. Sections 3 and 4 brie describe and, respectiel. Section 5 presents the simlation reslts of ow past a NACA-2 airfoil for Renolds nmbers between 5 and 5 6. Section 6 discsses the reslts and concldes this report. 2. Lattice-Gas Atomata and Lattice Boltzmann Eqation. 2.. Lattice Gas Atomata. In a series of articles pblished in the 98's [53], Wolfram showed that celllar atomata, despite their simple constrction, hae scient complexit to accomplish niersal compting: that is, beginning with a particlar initial state, the eoltion of some atomaton cold realize an chosen nite algorithm. Based pon kinetic theor and the preios experience of the Hard-Pomeade Pazzis (HPP) model [2, 22]thatatwo-dimensional sqare lattice does not possess scient smmetr for hdrodnamics, Frisch, Hasslacher, and Pomea [9] and Wolfram [54] independentl discoered that a simple celllar atomaton on a two-dimensional trianglar lattice can simlate the Naier-Stokes eqations. The two-dimensional Frisch-Hasslacher-Pomea (FHP) model was immediatel generalized to a three-dimensional LGA model [2]. The LGA model proposed b Frisch, Hasslacher, and Pomea, and Wolfram eoles on a two-dimensional trianglar lattice space. The particles hae momenta that allow them to moe from one site on the lattice to another in discrete time steps. A particlar lattice site is occpied b either no particle or one particle with a particlar momentm pointing to a nearest neighbor site. Therefore, at most a lattice site can be simltaneosl occpied b six particles, hence this model is called the 6-elocit model or FHP-I model. The eoltion of the LGA model consists of two steps: collision and adection. The collision process is partiall described in Fig.. For example, two particles colliding with opposite momenta will rotate their momenta 6 clockwise or conter-clockwise with eqal probabilit. In Fig., we do not list those congrations which can be easil obtained b rotational transformation, and which are inariant nder the collision process. The particle nmber, the momentm, and the energ are consered in the collision process locall and exactl. (Becase the FHP model has onl one speed, the energ is no longer an independent ariable, it is eqialent to the particle nmber. Howeer, for mlti-speed models, the energ is an independent ariable.) The eoltion eqation of the LGA can be written as: (2.) n (x + e t ;t+ t )=n (x; t)+c ; where n is the Boolean particle nmber with the elocit e, C is the collision operator, x isaector in the lattice space with lattice constant x, t denotes discrete time with step size t. We sall set both x and t to nit. The sbscript is an index for elocit; as illstrated in Fig., for the FHP model, rns from to 6. In the gre, collisions that can be obtained b rotations of the inpt and otpt states are not shown. Also note that momentm mst be consered dring collisions, therefore man otpt states mst be eqal to their inpt states. These triial collisions are not shown in the gre. After colliding, particles adect to the next site according to their elocities. Fig. 2 illstrates the eoltion of the sstem in one time 3

5 Inpt State Otpt State Inpt State Otpt State Fig.. Collisions of FHP LGA model. Note that the gre does not inclded those collisions that can be obtained b appling rotations of mltiple =3 to inpt and otpt states simltaneosl. Fig. 2. Eoltion of FHP LGA model. Solid and hollow arrows represent particles with elocities corresponding to times t and t +, respectiel. That is, the hollow arrows are the nal congrations of the initial congrations of solid arrows after one ccle of collision and adection. step from t to t + t. In this gre, solid and hollow arrows represent particles with corresponding elocit at time t and t + t, respectiel. The sstem eoles b iteration of the collision and adection processes. (2.2) According to the collision rles prescribed in Fig., the collision operator, C, can be written as follows: C (fn (x; t)g) = X s; s (s, s ) ss Y n s (, n ) (,s) ; where s fs ;s 2 ; :::; s 6 g and s fs ;s 2; :::; s 6g are possible incoming and otgoing congrations at a gien site x and time t, respectiel; ss is a Boolean random nmber in space and time that determines the transition between state s and s satisfing the following normalization condition: (2.3) X s ss =; 8 s : The Boolean random nmber ss mst also hae rotational smmetr, i.e., for an states s and s, ss is inariant if states s and s are both sbjected to simltaneos clockwise or conterclockwise rotations. It is obios that for Boolean nmbers n and s, the following eqation holds: (2.4) where ns (2.5) n s (, n ) (,s) = ns ; is the Kronecker delta smbol with two indices. Therefore, Eq. (2.2) can be written as C (fn (x; t)g) = X s; s (s, s ) ss ns ; 4

6 Table 2. Collision table for 6-Bit FHP model. Inpt State Otpt State where ns ns n2s 2 nb s b. Eq. (2.2), or (2.5), is rather abstract. A specic example of the collision operator for the two-bod collision is (2.6) C (2) = (2) R n +n +4 n n +2 n +3 n +5 + (2) L n +2n +5 n n + n +3 n +4, ( (2) R + (2) L ) n n +3 n + n +2 n +4 n +5 ; where n, n is the complement of n, (2) R and (2) L are Boolean random nmbers which determine the otcome of head-on two-bod collisions. The Boolean random nmbers reect the randomness of the otcomes of the two-bod collision. smmetr grop statisticall (on aerage), the mst satisf Obiosl, for the collision operator to satisf the complete lattice (2.7) h (2) R i = h(2) L i ; where hi denotes the ensemble aerage. The conseration laws of the particle nmber, momentm, and energ of the LGA micro-dnamics can be written as follows: (2.8a) (2.8b) (2.8c) X X X (s, s )=; (s, s )e =; (s, s ) 2 e2 =: In practice, the collision can be implemented with arios algorithms. One can either se logical operations [as indicated b Eq. (2.6)], or b table-lookp. The collision rles shown in Fig. can also be represented b a collision table, as shown b Table 2.. In Table 2., each bit in a binar nmber represents a particle nmber n, =; 2; :::; 6; from right to left. The limitation of table lookp is the size of the table, which is2 b, where b is the nmber of bits of the model. Both logic operations and table lookp can be extremel fast on digital compters, and especiall so on dedicated compters [49, 2] Hdrodnamics of Lattice Gas Atomata. The ensemble aerage of Eq. (2.) leads to a lattice Boltzmann eqation: (2.9) f (x + e t ;t+ t )=f (x; t)+ ; 5

7 where f mhn i and hc i, where m is the particle mass. In addition, correlations among colliding particles are assmed to be negligible, i.e., (2.) hn n n i = hn ihn ihn i : The aboe approximation is eqialent to the celebrated moleclar chaos assmption of Boltzmann (Stosszahlansatz). With the moleclar chaos approximation, the lattice Boltzmann collision operator is gien b (2.) (ff (x; t)g) = X s; s (s, s ) A ss Y f s (, f ) (,s) ; where A ss h ss i is the transition probabilit from state s and s. The hdrodnamic moments are gien b: (2.2) = X f ; = X ef ; " = X 2 (e, ) 2 f ; where,, and " are the mass densit, the elocit, and the internal energ densit, respectiel. Eq. (2.9) can be expanded in a Talor series of t p to the second order: (2.3) (@ t + er)f +(@ t + er) 2 f = : The eqilibrim distribtion, f (), which is the soltion of (ff g) =, mst be a Fermi-Dirac distribtion becase the sstem is a binar one [54], that is, (2.4) f () = + exp(a + b e) ; where coecients a and b are fnctions of and 2 in general. Becase the coecients a and b in f () cannot be determined exactl [54], f () mst be expanded in a Talor series of the small elocit (low Mach nmber) expansion. With the small elocit expansion of the eqilibrim f () and throgh Chapman-Enskog analsis, one can obtain the following hdrodnamic eqations from the FHP LGA model [9, 54]: (2.5a) (2.5b) where g is a fnction of, (2.6a) (2.6b) t + r t + r(g) =,rp + r 2 ; P = c 2 s, g 2 ; c 2 c s = p c; 2 =, c x ; c s is the sond speed, c = x = t, and is an eigenale of the linearized collision operator [38]: f =f () The defects of the LGA hdrodnamics are obios from the aboe eqations [Eqs. (2.5a) and (2.6a)]:. Simlations are intrinsicall nois becase of the large ctation in n ; 6

8 2. The factor g() is not nit, ths Galilean inariance is destroed; 3. Increasing the Renolds nmber Re is diclt; 4. The eqation of state depends on 2 (nphsical); 5. Sprios (nphsical) consered qantities exist de to the simple smmetr of the lattice-gas atomata. Howeer, all these defects can be xed b sing more sophisticated LGA models [2], or the other alternatie the lattice Boltzmann eqation Lattice Boltzmann Eqation. Historicall, models of the lattice Boltzmann eqation [32, ] eoled from their Boolean conterparts: the lattice-gas atomata [9, 54]. Recentl, the lattice Boltzmann eqation has been shown to be a special discretized form of the continos Boltzmann eqation [23, 24]. For the sake of simplicit bt withot lose of generalit, the Boltzmann eqation with the Bhatnagar- Gross-Krook (BGK) approximation [5] is sed in the following analsis. The BGK Boltzmann eqation can be written in the form of an ordinar dierential eqation: (2.7) D t f + f = f () ; where D t +r is the Lagrangian deriatie along the microscopic elocit, f f(x; ; t) is the single particle distribtion fnction, is the relaxation time de to collision, and f () is the Maxwell-Boltzmann distribtion fnction: (2.8) f () (2RT) exp (, )2, ; D=2 2 in which D is the dimension of the space; = k B T=m is the normalized temperatre; T, k B and m are temperatre, the Boltzmann constant, and particle mass, respectiel. The macroscopic ariables are the moments of the distribtion fnction f with respect to elocit : (2.9) = Z fd ; = Z fd ; = 2 Eqation (2.7) can be formall integrated oer a time interal t : Z (, ) 2 fd : (2.2) Z f(x + t ; ; t+ t )=e,t= f(x; ; t)+ t e,t= e t = f () (x + t ; ; t+ t ) dt : Assming that t is small enogh and f () is smooth enogh locall, and neglecting the terms of the order O( 2 t ) or smaller in the Talor expansion of the right hand side of Eq. (2.2), we obtain (2.2) f(x + t ; ; t+ t ), f(x; ; t)=, [f(x; ; t), f () (x; ; t)] ; where = t is the dimensionless relaxation time. The eqilibrim distribtion fnction f () can be expanded asatalor series in. B retaining the Talor expansion p to 2,we obtain: (2.22) f (eq) = (2) D=2 exp, 2 2 ( ) ( ) , 2 For the prpose of deriing the Naier-Stokes eqations, the aboe second-order expansion is scient. (2.23) To derie the Naier-Stokes eqations, the following moment integral mst be ealated exactl: Z m f (eq) d ; 2 : 7

9 Fig. 3. Discrete elocities of the 9-elocit model on a sqare lattice. where m 3 for isothermal models. The aboe integral contains the following integral which can be ealated b Gassian-tpe qadratre: (2.24) Z X I = exp(, 2 =2) () d = W exp(, 2 =2) ( ) ; where () is a polnomial in, and W and are the weights and the abscissas (or discrete elocities) of the qadratre, respectiel. Accordingl, the hdrodnamic moments of Eqs. (2.9) can be compted b qadratre as well: (2.25) X = f ; X = f ; = 2 X (, ) 2 f ; where f f (x; t) W f(x; ;t): We shall se the 9-elocit (9-bit) isothermal LBE model on sqare lattice space as an example to illstrate the deriation of LBE models: the eoltion eqation (2.2) on a discretized phase space and time with a proper eqilibrim distribtion fnction leads to the Naier-Stokes eqations. To derie the 9-elocit LBE model, a Cartesian coordinate sstem is sed, and accordingl, we set () =x mn. The integral of Eq. (2.24) becomes: (2.26) where (2.27) I =( p 2) (m+n+2) I m I n ; I m = Z +, e,2 m d ; and = x = p 2 or = p 2. Natrall, the third-order Hermite formla is the optimal choice to ealate I m for the prpose of deriing the 9-elocit LBE model, i.e., I m = P 3 j=! j m j : The three abscissas ( j ) and the corresponding weights (! j ) of the qadratre are: (2.28) =, p 3=2 ; 2 =; 3 = p 3=2 ;! = p =6 ;! 2 =2 p =3 ;! 3 = p =6 : Then, the integral of Eq. (2.26) becomes: (2.29) I =2 [! 2 2 ()+ 4X!! 2 ( )+ 8X = =5! 2 ( )] ; where is the zero elocit ector for =, the ectors of p 3 (; ) and p 3 (; ) for = {4, and the ectors of p 3 (; ) for = 5{8. Note that the aboe qadratre is exact for (m + n) 5. Now the momentm space is discretized with nine discrete elocities f j =; ; ; 8g, as shown in Fig. 3. To obtain the 9-elocit model, the congration space is discretized accordingl, i.e., it is discretized 8

10 into a sqare lattice with lattice constant x = p 3 t. It shold be stressed that the temperatre has no phsical signicance here becase we are onl dealing with an isothermal model. We can therefore choose x to be a fndamental qantit instead, ths p 3 = c x = t,or = c 2 s = c 2 =3, where c s is the sond speed of the model. B comparing Eqs. (2.24) and (2.29), we can identif the weights dened in Eq. (2.24): (2.3) where (2.3) w = W =2 exp( 2 =2) w ; 8 >< >: 4=9; =; =9; =; 2; 3; 4; =36; =5; 6; 7; 8: Then, the eqilibrim distribtion fnction of the 9-elocit model is: (2.32) f (eq) = W f (eq) (x; ;t)=w + 3(e ) + 9(e ) 2 c 2 2c 4, 3 2 ; 2c 2 where (2.33) e = 8 >< >: (; ); =; (cos ; sin ) c; =; 2; 3; 4; (cos ; sin ) p 2c; =5; 6; 7; 8; and =(, )=2 for = {4, and (, 5)=2+=4 for = 5{8, as shown in Fig. 3. The Naier-Stokes eqation deried from the aboe LBE model is: t + r =,rp + r 2 ; where the eqation of state is the ideal gas one, p = c 2 s, the sond speed c s = c= p 3, and the iscosit (2.35) = 3 (, =2)c x =(, =2)c 2 s t for the 9-elocit model. Similarl, we can also derie two-dimensional 6-elocit, 7-elocit, and three-dimensional 27-elocit LBE models [24]. In the aboe deriation, the discretization of phase space is accomplished b discretizing the momentm space in sch awa that a lattice strctre in congration space is simltaneosl obtained. That is, the discretization of congration space is determined b that of momentm space. Of corse, the discretization of momentm space and congration space can be done independentl. This consideration has two immediate conseqences: arbitrar mesh grids and signicant enhancement of the Renolds nmber in LBE hdrodnamic simlations. To implement arbitrar mesh grids with the LBE method, one rst discretizes the congration space b generating a mesh adapted to the phsics of the particlar problem. Then at each grid point, one can discretize momentm space as before. Now, a local LBE is bilt on each mesh grid point. The eoltion of this discretized Boltzmann eqation (DBE) consists of the following three steps. The rst two steps are the sal collision and adection process as in the preios LBE models. After collision and adection, interpolation follows. The interpolation process is what distingishes the DBE from the LBE method. Becase the mesh 9

11 grids can be arbitrar, the distribtion fnction f at one mesh grid point, sa xi, cannot go to another grid point in general throgh the adection process as it can in preios LBE models. Therefore, the interpolation step becomes necessar to constrct f (xi; t)oneach and eer mesh grid point from f (xi + e t ;t) after the adection process. Of corse, interpolation brings in additional nmerical error, bt it can be jstied so long as the error indced b interpolation does not aect the DBE algorithm as a whole [25]. In addition, the separate discretization of momentm and congration space allows one to increase the Renolds nmber signicantl in nmerical simlations withot enlarging mesh sizes or decreasing the iscosit b adjsting [25]. In other words, the limitation posed b the lattice Renolds nmber is completel oercome [26] and the stabilit of the LBE method is greatl improed [25, 26] Bondar Conditions and Trblence Modeling. The bondar conditions for the lattice gas atomata are relatiel simple. The simplest bondar conditions for the elocit eld are bonce back and reectie bondar conditions. The former mimics the no-slip bondar condition and the latter simlates the slip bondar condition. In the bonce back bondar condition, a particle hitting a wall simpl reerses its momentm, whereas in the reectie bondar condition the particle hitting a solid bondar onl reerses its momentm normal to the bondar and maintains its momentm tangential to the bondar. In addition, fll deeloped bondar conditions for the elocit eld can be obtained b eqating the distribtion fnctions in two adjacent lattice lines. The pressre bondar condition can be implemented ia the ideal-gas eqation of state b setting appropriate mass densities in the desired locations. Bod forces can also be implemented b probabilisticall making particles gain momentm along the forcing direction. The Boolean natre of the lattice gas atomata sometimes makes the bondar conditions diclt to alter. Frthermore, the accrac of the aforementioned bondar conditions cold be of rst order (in space), instead of second order. The lattice Boltzmann method has greater freedom than lattice gas atomata methods to implement arios bondar conditions accratel [33]. Varios conentional techniqes sch as interpolation or extrapolation can be applied to improe the accrac of the LBE bondar conditions [33]. Frthermore, arbitrar nonniform meshes [25] and mesh renement [8] can be implemented in the LBE method. There are two tpes of trblence models implemented in the lattice Boltzmann method: sbgrid modeling [27] and wall fnction modeling [3]. The sbgrid modeling is illstrated as follows b sing the example of the Smagorinsk sbgrid model. In a sbgrid model, the total iscosit can be diided in two parts, (2.36) = + s ; where and s are the phsical iscosit (for the resoled scales) and edd iscosit (for the nresoled sbscales), respectiel. The edd iscosit is obtained as follows: (2.37a) (2.37b) (2.37c) s = C 2 2 x j Sj ; j Sj = q 2 P i; j S ij ; S ij = 2 (@ j i i j ) ; where is the resoled elocit eld, Sij is the corresponding strain-rate tensor, and C> is the Smagorinsk constant. The relaxation time is ealated at each grid as follows: (2.38) =6( + s )+ 2 ; where is gien a priori, as determined b the Renolds nmber of the simlated ow. The relaxation time now aries locall to model the sbgrid-scale dnamics, depending on the ow eld.

12 An alternatie to the aboe sbgrid modeling of trblence is throgh a wall fnction [3]. Instead of resoling the bondar laer adjacent to the wall, the comptation starts at the rst cell aboe the srface, where a \slip" elocit U s is tpicall nonzero [45]. The trblent law of the wall is assmed to hold at the location s of the rst cell o the wall; hence, the shear stress at s is the same as the wall shear stress w. A local skin-friction coecient C f is dened throgh the relation (2.39) The local friction elocit U is dened as w C f 2 U s 2 : (2.4) U r w Application of the law of the wall at the distance s from the wall ields the logarithmic prole of the bondar laer (2.4) U s = U ln U s + B ; where is the on Karman constant ( :4) and B is a constant ofintegration (B 5:). From Eqs. (2.39) and (2.4), (2.42) s U s 2 = U C f and the friction coecient can be expressed as (2.43) C f = h ln U s B i 2 : In a simple mixing-length model a trblent, or edd iscosit t is proportional to the mean elocit gradient mltiplied b the sqare of a length scale. If s is in the log laer, s is an appropriate length scale and U =( s ) is the mean elocit gradient, hence (2.44) t = s U : Anagnost et al. [3] assme that the lattice iscosit is eqal to the edd iscosit t at the height s of the rst grid cell. Wh sch an assmption is necessaril alid is not clear; howeer, this assmption leads to the basic form of the relationship for the local skin-friction coecient (2.45) C f = ln, B 2 : The eect of local aderse pressre gradients can be incorporated into the model b modifing the relationship between the lattice iscosit and the edd iscosit. The modied eqation for C f is (2.46) C f = ln, 2 2 ( + (@x p)) + B 2 ; where the fnction (@ x p) is negatie inanaderse pressre gradient and zero otherwise. The C f eectiel determines a \slip" at the bondar. This \slip" bondar condition can be accomplished b a proper ratio between bonce-back and reection of the distribtion fnction colliding with the wall.

13 3.. is a software package based on lattice gas celllar atomata [7]. ses 3 speeds in three-dimensional space [45]. The eqilibrim distribtion fnction in the software is of Maxwellian tpe: (3.) f (eq) = w (T ) exp 2T (2e, 2 ) ; where w (T ) can be determined b the conseration constraints [2]. In particlar, for a3speedmodel[45]: (3.2) w (T )= 8 >< >: d (3T 2, 3T +); jej =; 2T (2, 3T ); 24 je j =; T (3T, ); 24 je j = p 2; where d is an adjstable positie constant and is sall set to 6. The distribtion fnction in is represented b an eight-bit integer [45]. Some special care mst be taken to ensre exact preseration of the conseration laws. ses a strctred, cbic Cartesian mesh (oxels), and a grid renement scheme which renes the grid size in each spatial dimension eqall and sccessiel b a factor of 2. The bondar srface is tessellated into trianglar planes (srfels), of which each side has a length that is less than = the linear dimension of a oxel in the region. Varios bondar conditions are implemented in b specifing a elocit prole, static pressre, and mass x at the bondar [7]. At the bondar, momentm xes are compted and redistribted to enforce desirable bondar conditions [3]. Becase is a commercial software package, and sorce code was naailable to the athors, specic implementation details discssed in Sections 2 and 3 ma not be completel releant to the release of the software sed in this work. 4.. A standard compter code, [48, 5, 47, 4, 39, 29], was sed to compte ow elds for comparison with reslts. The compter code was deeloped at NASA Langle Research Center to sole the three-dimensional, time-dependent, thin-laer (in each coordinate direction) Renolds-aeraged Naier-Stokes (RANS) eqations sing a nite-olme formlation. The code ses pwind-biased spatial dierencing for the iniscid terms and x-limiting to obtain smooth soltions in the icinit of shock waes. The iscos deriaties are compted b second-order central dierencing. Flxes at the cell faces are calclated b Roe's x-dierence-splitting method [36]. An implicit three-factor approximate factorization method is sed to adance the soltion in time. Here, onl stead soltions hae been compted and conergence is accelerated with local time-stepping, grid seqencing, and mltigridding. 4.. Trblence Modeling. The ow soler proides a wide range of trblence models from which tochoose. In this work, most of the cases emploed the model of Spalart and Allmaras [42]. One case sed Wilcox's k-! model [52]. In the model of Spalart and Allmaras [42] the trblent edd iscosit is obtained directl from the soltion of a transport eqation. The model is tned for aerodnamic ows, is robst, and is applicable to ows with complex geometr. Seeral stdies for both stead and nstead ows (Rogers et al. [37] and Rmse et al. [4]) show the model to hae better predictie capabilit than algebraic and other oneeqation models for a wide range of aerodnamic ows. The implementation ses Version Ia of the model [43]. The Wilcox k-! model [52] is a two-eqation model, one for the trblent kinetic energ k, the other for!, which is proportional to the trblent energ dissipation rate per nit kinetic energ in the implementation [29]. The resltant trblent edd iscosit is the ratio k=!. 2

14 Althogh not reqired b, in cases in which a trblence model is sed, the ow is considered fll trblent from the onset, i.e., no laminar region is assmed. In addition, both the Spalart-Allmaras model and the Wilcox model are integrated all the wa to the wall. As with all trblence models in, the soltion of either trblence model is decopled from the soltion of the ow eqations. Information from the ow eqations that is reqired in the trblence eqation lags one iteration. This is not a problem for scientl conerged soltions. Additional implementation details are aailable in the manal [29] Bondar Conditions. As shown in Fig. 5, a C-grid was sed for the calclations. This grid reqires bondar conditions to be dened on three distinct geometrical srfaces: () the wing bod, (2) the downstream plane, and (3) the external srface that denes the pstream, and pper and lower portions of the comptational domain. For the iniscid calclations with, slip wall conditions ( bondar-condition tpe 5) are sed on the wing bod (srface ). These conditions reqire that the component of elocit normal to the wall be eqal to zero. In this bondar condition approximates the cell-face bondar ale for the densit as the densit of the nearest cell-center point on the grid. The bondar ales for the pressre are obtained similarl. For the iniscid cases, the external comptational srfaces (srfaces 2 and 3) are modeled with the se of local one-dimensional Riemann inariants ( bondar condition tpe 3). The conditions otside the comptational domain are taken from the prescribed free-stream Mach nmber and ow direction. For the iscos calclations, no-slip, no-penetration, adiabatic wall conditions ( bondar-condition tpe 24 with twtpe = and cq =)were sed on the wing bod (srface ). In the iscos cases, the condition imposed at the downstream plane (srface 2) inoled a zeroth-order extrapolation from the comptational interior ( bondar-condition tpe 2). The far-eld, pstream comptational srface (srface 3) had its bondar condition modeled with the local one-dimensional Riemann inariant, as with the iniscid cases. More details regarding the implementation of the bondar conditions can be fond in the manal [29]. 5. Two-Dimensional Simlation of Flow Past NACA-2 Airfoil. 5.. NACA-2 Airfoil. The NACA-2 airfoil is a poplar wing section that is sed for man prposes. Coordinates and a generation formla are gien in Ref. []. Figre 4 shows a NACA-2 wing section. The NACA-2 airfoil has zero camber and a maximm thickness to chord ratio of 2 percent. A 2 percent thickness ratio reslts in maximm lift for man 4 and 5 digit NACA wing sections []. Experimental lift erss angle-of-attack cres, reprodced in graphical form in Ref. [], sggest that on aerodnamicall smooth wings, the NACA-2 airfoil does not experience massiel separated ow p to an angle of attack of 6 for Renolds nmbers of 3, 6, and 9 million. For the wing with a standard roghness, massie ow separation occrs at an angle of attack of abot 2 at a Renolds nmber of 6 million. Becase the NACA- 2 airfoil is smmetric, it generates no lift at zero angle of attack. At an angle of attack of3, the lift coecient is expected to be abot.35. At an angle of attack of7, the lift coecient is expected to be between abot.74 and.78, depending pon conditions Comptational Domain and Bondar Conditions. A series of 2-D simlations of ow past the NACA-2 airfoil at low speed were performed b sing and the reslts are compared with those from. The Mach nmber M is set to.2 in the simlations. Seeral ales of angle of 3

15 x Fig. 4. NACA-2 airfoil. attack,, are selected in the simlations:,3,7, and 2. Both and are three-dimensional codes. Howeer, the simlations of the ow past the airfoil are two-dimensional. In the case of, this is simpl done b sing the two-dimensional soler option in the code. In the case of, the two-dimensional simlations were accomplished b shrinking the third dimension to two cells and emploing a periodic bondar condition [46]. The simlations were done on an SGI Origin2 with 6 R processors. The CPU times and resoltions sed in the simlations of both and are smmarized in Table 5.. For, a tpical comptational domain is L x L =98 in nits of chord length. ses Cartesian grids. The resoltion at the srface is approximatel 5 points per chord. In the simlations there are seen ariable resoltion regions, i.e., the grid spacing is aried from 2 to 2,6. The resoltion sed in is sall 383,724 oxels (grid points) and,275 srfels (srface elements). The aspect ratio of grid spacings is alwas : in the simlations. The bondar conditions sed in are: a free-slip bondar condition for pper and lower bondaries; a driing bondar condition with a xed elocit at the inlet bondar, and a xed static pressre for exit bondar condition. At the low Renolds nmber (Re = 5), a no-slip bondar condition is sed for the airfoil srface. At high Renolds nmbers (Re :5 6 ), the \trblent wall bondar condition" sing a wall fnction is applied to the airfoil srface. is a time explicit scheme with a small time step: the Corant-Friedrichs-Lew nmber is one with respect to the grid. has signicant ctations in its instantaneos reslts becase of its integer natre. Therefore hdrodnamic otpts of are sall aeraged oeranmber of time steps. The interal for the time-aerage for the NACA-2 airfoil simlations is, steps nless otherwise specied. Figre 5 shows the bod-tted comptational mesh (C-grid) sed in simlations. With the grid nondimensionalized with the chord length, the comptational domain consists of a half disc of radis 2 in front of the airfoil and a rectangle of 2 4, as shown in Fig. 5. The resoltion is tpicall 373 4, or 52,593 cells. The grid spacing normal,, and tangential, x, to the srface near the middle of chord are approximatel :5 and :5 for iniscid cases, respectiel, and :2 and :8 for iscos cases, respectiel. With these ne grids, can resole the bondar laer in the simlations. Althogh all ows possess some nsteadiness, for attached ows the ctations tpicall are on a time scale that is so small that the are sall of no practical interest for aerodnamic qantities like lift and drag. Most experimental force data are actall time-aeraged qantities. Some sensors implicitl aerage becase the are incapable of resoling the small, nstead ctations. To obtain mean data from a time-accrate CFD simlation reqires performing a similar time-aerage on the nmerical reslts. This is the approach sed in. Althogh onl fractions of a second are needed to obtain a signicant 4

16 Fig. 5. A tpical bod-tted mesh sed in. (left) the entire comptational domain with grid resoltion 373 4, and (right) the mesh arond the airfoil, which is the center part of the domain. sample in an experiment, considerable compter resorces are reqired to generate all of the reqired samples from a nmerical simlation. To simplif and accelerate the soltion procedre, CFD schemes commonl sole eqations that hae been time-aeraged so that all of the dependent ariables are actall time-mean qantities. In the case of the Naier-Stokes eqations applied to trblent ows, this process is called Renolds aeraging and prodces seeral additional terms which mst be modeled to close the sstem of eqations. This modeling of the eects of the small scale, nstead trblent ow on the mean qantities is nontriial, bt man models hae been deeloped and alidated for man important ows. Becase the eqations are nonlinear, the tpical soltion procedre is to iterate ntil the soltion becomes stead. Man acceleration techniqes hae been deeloped that take adantage of the fact that the intermediate soltion need not be time accrate. This is the approach sed in Simlation Reslts. The local reslts discssed below inclde and, the x and components of the elocit eld normalized b the inlet elocit U ; and the pressre coecient (5.) = p, p ; 2 U 2 where p is the local pressre, p is the far-eld pressre, and is the far-eld densit. The elocit and pressre are measred at e stations: x=l =, =4, =2, 3=4, and, where L is the chord length. In addition, the lift coecient (5.2) C l = F l=a 2 U 2 ; and the drag coecient (5.3) C d = F d=a 2 U 2 ; are obtained b integrating the lift force F l and the drag force F d per nit span oer the srface of the airfoil. The resltant lift coecients C l and drag coecients C d are smmarized in Table 5.. The Renolds nmbers Re in these simlations range from 5 to :5 9. For Re = 5 (Case a and b in Table 5.), the reslts were obtained b directl simlating the laminar ow. For Re 5 3, trblence modeling was sed in both and (Cases 2{6inTable 5.). ses wall fnctions, described in Sec. 2.4, to model the bondar laer. incorporates arios trblence models in the 5

17 Table 5. Smmar of reslts for simlations for ow past a NACA-2 airfoil. (Top) Drag coecient C d and lift coecient C l obtained from and. The nmbers in parentheses are bonds estimated from the nstead reslts of the simlations. (Bottom) Resoltions and CPU times of simlations sing and. Case Re C d C l C d C l a , ,3 b 5.74,.538,5.876,.2,3 2a : :769,5.2245,:2,3 2b : :924,9.236,:358,3 2c : ,:66,7 3 3 : : a 3 iniscid b 3 :5 9, c 3 : : (,.7,.9) (.72,.2) 6 2 : (,.54,,:2) (.7,.56) Case Re N N r CPU Time s (hrs) VoxelSrfel CPU Time s (hrs) a (.) (.2) b (6.4) (9.7) 2a : (.5) (6.4) 2b : (4.9) (7.5) 2c : (4.9) 3 3 : (8.3) (6.9) 4a 3 iniscid (.5) 4b 3 : (8.6) 4c 3 : (3.3) 5 7 : (8.3) (7.) 6 2 : (6.3) (8.7) code. Two trblence models were sed in the simlations of ow pastanaca-2 airfoil: the Spalart- Allmaras model and the k-! model. The k-! model was onl sed in Case 2c of Table 5. to demonstrate the leel of ariabilit in the reslt with dierent trblence models. The reslts show little dependence on the trblence model for this ow. The remainder of simlations sed the Spalart-Allmaras model. Case : Re = 5, =. The simlations sed two resoltions: and 373 4, and sed 59; 6 oxels and 828 srfels (Case a), and 48; 8 oxels and 275 srfels (Case b). The ale of drag coecient C d obtained from is abot 3.8% larger than the ale obtained from. Althogh a smmetric airfoil at zero angle of attack shold generate zero lift, the lift coecient aried between :2. The soltions maintained zero lift to an accrac almost two orders of magnitde better. 6

18 C d.8.82 Re=5 M=.2 α= Drag Lift C l Time Step Fig. 6. Time histor of C d and C l in the simlation. Re = 5, M=:2, and =. At a Renolds nmber Re = 5, the ow is laminar and the reslts were obtained b direct nmerical simlation no trblence modeling was sed. Along the srface of airfoil, the no-slip bondar condition is sed in. After 25, time steps, a stead-state soltion was attained. Figre 6 shows the time histor of C d and C l. The ariations of C d and C l are on the order of,4. Figres 7, 8, and 9 show (x; ), (x; ), and (x; ) at e ertical sections of x=l =,=4, =2, 3=4, and. These gres show the reslts with the higher resoltion case (Case b). The dierences between the reslts of and are alread isible at Re = 5. The dierence in the x- component ofelocit is relatiel small. Howeer, the relatie dierence in the -component ofelocit is signicantl greater, and the dierence is amplied downstream. Similar discrepancies exist in the ertical pressre distribtion and the srface pressre distribtion. Althogh the no-slip condition is satised in the simlations, the gres do not show! near the airfoil srface. This is becase the data were extracted at niforml spaced points in the direction, and a point on the airfoil srface is sall not inclded. This is also tre for the data. Case 2: Re=:5 6, =. When Re = :5 6, trblence modeling was sed. sed the Spalart-Allmaras model (Case 2a and 2b in Table 5.) and k-! model (Case 2c). sed the wall fnction discssed in Section 2.4 (Case 2a and 2b). The ale of drag C d obtained from is abot 66% larger than the ale obtained from. The ale of C l from the simlation is 2 to 6 orders of magnitde larger than that of. As before, for =, the ale of C l shold be. Figres,, and 2 show (x; ), (x; ), and (x; ) at the e ertical sections, for the reslts with the higher resoltion (Case 2b). The dierences between the reslts and reslts are qite signicant on the srface of the airfoil, and relatiel negligible in the ow region awa from the srface. Obiosl, the dierences in the ow elds near the bondar reslt in the signicant dierence in the ale of drag C d. This raises the qestion of adeqac of trblence modeling b awall fnction in, een for cases in which the ow is attached. Case 3: Re = 5 3, =3. The reslts of the drag coecient C d obtained from the two methods are er close in this case: C d from is onl abot 2.% larger than that from. With = 3, the lift coecient is non-zero. The lift coecient C l obtained from simlations is 7

19 abot 58% smaller than the reslt from. Becase drag incldes both a frictional component and a component related to lift, the close agreement of the drag coecient is likel to be fortitos. Figres 3, 4, and 5 show (x; ), (x; ), and (x; ) at the eertical sections. The elocit elds obtained from and simlations signicantl disagree near the airfoil srface, which in trn leads to the disagreement in the region awa from the bondar. Frthermore, qalitatie disagreement in pressre (x; ) starts to appear (e.g., (x; ) at x = 3=4 and in Figs. 5). Figre 6 shows the streamlines of and simlations in the icinit of the airfoil. The streamlines do not sta completel attached to the airfoil srface in the simlations. Note that the calclation shows separated ow near the trailing edge while the soltion remains completel attached. The separation shown in Fig. 6 cold also be srmised from the srface pressre distribtion near the trailing edge in Figre 5. Case 4: Iniscid ow, Re =, =3. In the simlation of iniscid ow, an Eler scheme is sed b neglecting the iscos x terms. In the simlation, the Renolds nmberis:5 9, and the free slip bondar condition is applied at the srface of the airfoil [7] to mimic the iniscid eect. In this case, the ale of C d from one simlation is negatie, as indicated in Table 5. (Case 4b). Figres 7, 8, and 9 show (x; ), (x; ), and (x; ) at the e ertical sections, for Case 4a () and 4b (). The reslts of (x; ) from and the simlations are qalitatiel dierent near the airfoil srface. The reslts were compared against those from other CFD codes, and the agreement was excellent. Figre 2 shows the time histor of C d and C l in simlations with two dierent resoltions and comptational domain sizes. In the rst case (Case 4b), the resoltion is 383,724 oxels and,275 srfels, the resoltion on the airfoil srface is 5 points along the chord, the comptational domain is 9 8, and there are seen ariable resoltion regions (2 to 2,6 ). The drag coecient C d conerges to a negatie ale after abot 2, iterations, as shown in the left part of Fig. 2. In the second case (Case 4c), the resoltion is 63,62 oxels and 64 srfels, the resoltion on the airfoil srface is 2 points along the chord, the comptational domain is 2 2, and there are seen ariable resoltion regions (2 to 2,7 ). This case was rn to inestigate whether the original comptational domain was too small. With a coarser resoltion bt a larger comptational domain, the reslts of C d and C l do not seem to conerge after 6, iterations, as shown in the right part of Fig. 2. The ales of C d and C l gien in Table 5. (Case 4c) are the nal ales of the simlations. The are apparentl far awa from the correct ales. It is also noted that the CPU time for the simlation is almost dobled in this case, despite the fact that the resoltion is mch coarser. Case 5: Re = 5 3, =7. With =7, the mean ow shold be stead and attached. Howeer, the simlation displas massie ow separation, and ow eld becomes highl nstead. Figre 2 shows a snapshot of streamlines in both the and the simlation. The two reslts are completel dierent from each other. Becase failed to conerge to a stead mean ow after 3; time steps, the ales of C l and C d can onl be gien in their estimated bonds, as shown in Fig. 22. Figres 23, 24, and 25 show (x; ), (x; ), and (x; ) at the e ertical sections. In this case the ow elds obtained from the two methods do not hae an resemblance; while the soltion from remains completel attached, the soltion from ndergoes a massie separation, as also shown in Fig. 2. 8

20 Case 6: Re = 5 3, =2. At =2, the obseration is similar to the preios case of =7 ; in the simlation the ow separates massiel while it shold not. The mean ow fails to conerge to a stead state, as it shold. Therefore, the ow elds obtained in simlation are qalitatiel incorrect, as shown in Figres 26, 27, and Discssion and Conclsion. In this work we present the reslts of a thorogh inestigation sing to simlate two-dimensional ow past a NACA-2 airfoil with a Renolds nmber ranging from 5 to :5 9. The reslts were compared with the reslts obtained from. In all cases we obsered deciencies with the calclations. We hpothesize that the principal clprit for the obsered deciencies is the the lack of wall-normal grid resoltion near the srface of the airfoil. Becase reqires grid cells to be sqare, properl sizing the grid cells in the wall-normal direction reqires grid cells in the streamwise direction to be so small as to make een a two-dimensional calclation infeasible. The lack of wall-normal grid resoltion is probabl the case for the near-wall elocit and pressre errors that were shown in Figs. 7, 8, and 9. Errors in the near-wall region degrade the accrac of the oerall aerodnamic qantities, sch as the lift and drag coecients. Althogh the se of wall fnctions for the high-renolds nmber trblent cases reliees some of the resoltion reqirements for those cases, the data sggest that the resoltion sed was still inscient. Figre 29 shows a elocit prole normal to the wing srface as compted b. The ariables plotted are in wall coordinates (i.e., + = = and + = U= where = p wall = and wall is the wall shear stress), which emphasize the prole in the near-wall region. The ertical lines indicate the locations of the rst throgh fth grid points from the srface in the grid. Althogh the rst grid point is located at an appropriate position for implementing a wall model, the grid onl has abot 5 points in the entire bondar laer. That leel of grid resoltion is inadeqate for properl compting the bondar laer. In this ow, incorrectl compting the bondar laer prodces catastrophic problems. Flow separation occrs when the downstream momentm of the id in the bondar laer is inscient to maintain the downstream moement of the id against an aderse pressre gradient. The momentm of the id in the bondar laer is decreased b the shear stress at the wall, bt is increased b the (sall trblent) dision of momentm from the free stream into the bondar laer. Incorrectl compting the bondar laer can dramaticall alter where the ow rst separates and the degree of separation. After massie ow separation occrs, the compted ow has little resemblance to its attached-ow conterpart. In the test problems stdied here, minor separation was obsered in the reslts for a case with a three-degree angle of attack. Massie separation was obsered in the reslts for a seendegree angle of attack case. After separation occrred, een the gross aerodnamic qantities cold not be compted well. Two additional tests were carried ot at NASA Langle Research Center: () simlation of threedimensional cait ow [28], and (2) three-dimensional ow past the ROBIN congration (ROtor Bod INteraction), a generic helicopter congration consisting of a bod and an engine-transmission plon [34]. The geometr of the cait ow is er simple. The Mach nmber and the nit Renolds nmber of the problem are.3 and 6 6 =ft., respectiel. The resoltion reqired to obtain reasonable reslts is 9,24, oxels and 286,5 srfels. A rn of 2, time steps reqired approximatel 6 hors on an SGI Origin2 with 6 25MHz processors. Althogh qantitatie comparisons with experimental reslts of mean wall pressre distribtions, the sond pressre leel, and length scales of the ow were not satisfactor 9

21 [28], the simlation did captre the acostic interaction with the shear laer in the ow. Constraints on time and memor size preclded frther simlations with ner resoltions. The ROBIN is a generic three-dimensional helicopter congration. The Renolds nmber of the ow is 9:2 6 based on the bod length (2 meters). Fie leels of grid resoltion were sed in the simlations (2 to 2,4 ). The reslting grid had 4:3 6 oxels. The simlations were compared with reslts obtained b sing Naier-Stokes solers on a strctred grid () and an nstrctred grid (USM3D), and with experimental obserations. The main concerns with the simlations of the ow past ROBIN are: () too large a ale for the maximm pressre coecient ; (2) an excessie amont of oscillation in the pressre coecient distribtions and skin friction coecient distribtions; (3) signicant dierences in srface ow patterns and nondimensional stagnation pressre distribtions from those predicted b the Naier-Stokes solers; and (4) drag and lift coecients that dier signicantl from those predicted b the Naier-Stokes solers [34]. Based on or inestigation of the two-dimensional ow past the NACA-2 airfoil, it is or conclsion that at its present state is inadeqate for aerodnamic applications in two aspects: performance and accrac. is generall mch less accrate than and mch slower as well. Some sccessfl examples sing inclde simlating the ow past a ehicle [3]. The reason for this sccess, we sspect, is that in the case of the ow past a ehicle, the bl bod has geometric discontinities that dictate the location of ow separation. Therefore, the ow separation, which is the most important ow featre in this case, is not aected b other factors as mch. In this case, seems to be able to delier better reslts [3]. Howeer, when dealing with a smooth or streamlined bod sch as an airfoil, does not proide reliable reslts. We also obsered the fact that simlations sing to obtain a stead mean state generall reqire mch larger memor and longer CPU time than the calclations. Althogh the reslts of the aerodnamic simlations reealed a less than adeqate performance of compared with, frther adancements in LGA and LBE methodolog ma enable this approach to become a iable alternatie to traditional CFD. LBE methods hae alread been sed sccessfll for applications inoling poros media and bl bodies, and frther research shold enable sccessfl applications to streamlined aerodnamic bodies. The crrent reslts are presented to illstrate the crrent state of these methods, not to predict their ftre. Some researchers hae alread obtained good soltions for airfoil ows with Renolds nmber Re between 5 and 5 b sing interpolation schemes at bondar and mlti-block grid to handle srface cratre [5]. Contined renements b the LBE commnit and Exa corporation will ndobtedl hae occrred b the time this paper is disseminated. Acknowledgments. The athors wold like to thank Mr. Stanle Willner of the Hdrodnamic/Hdroacostic Technolog Center of the Naal Sea Sstems Command for proiding resorces for this inestigation. The athors are gratefl to Dr. G. Jones and Dr. R. Mineck for sharing and discssing their test reslts sing [28, 34] with the athors, and to Dr. Michael Barton, Prof. Dominiqe d'hmieres, Dr. Manfred Krafczk, Prof. Pierre Lallemand, Prof. Renwei Mei, Dr. Cord Rossow, Dr. Robert Rbinstein, and Prof. Wei Sh for helpfl discssions. We wold also like to thank Dr. James L. Thomas and Mr. William Sellers of NASA Langle Research Center, and Mr. M. Salas of ICASE for their spport of this work. Technical assistance from Exa Corporation dring the test is also acknowledged. 2

22 .5.5 x = Re = 5 α = x =.25 Re = 5 α =.5.5 x =.5 Re = 5 α = x =.75 Re = 5 α =.5 x = Re = 5 α = Fig. 7. NACA-2 airfoil with angle of attack =,Re = 5, and M =:2. The x-component of elocit (x; ) at x=l = :, :25, :5, :75, and :, where L is the chord length. The resoltion is 48; 8 oxels and ; 275 srfels for and for. 2

23 x = Re = 5 α = -.5 x =.25 Re = 5 α = x =.5 Re = 5 α = -.4 Re = 5 α = x = Re = 5 α = x = Fig. 8. NACA-2 airfoil with angle of attack =,Re = 5, and M =:2. The -component of elocit (x; ) at x=l = :, :25, :5, :75, and :, where L is the chord length. The resoltion is 48; 8 oxels and ; 275 srfels for and for. 22

24 x = Re = 5 α = -. x =.25 Re = 5 α = x =.5 Re = 5 α = -.5 x =.75 Re = 5 α = x = Re = 5 α = Srface Pressre Re = 5 α = x Fig. 9. NACA-2 airfoil with angle of attack =,Re = 5, and M =:2. The pressre coecient (x; ) at x=l =:, :25, :5, :75, and :, where L is the chord length, and along the srface. The resoltion is 48; 8 oxels and ; 275 srfels for and for. 23

25 x = Re = 5 5 α = x =.25 Re = 5 5 α =.6.5 x =.5 Re = 5 5 α =.5 x =.75 Re = 5 5 α =.5 x = Re = 5 5 α = Fig.. NACA-2 airfoil with angle of attack =,Re =:5 6, and M =:2. The x-component of elocit (x; ) at x=l =:, :25, :5, :75, and :, where L is the chord length. The resoltion is 383; 724 oxels and ; 275 srfels for and for. 24

26 .5.5 Re = 5 5 α = Re = 5 5 α = x = x = Re = 5 5 α = Re = 5 5 α = x =.5 -. x =.75.5 Re = 5 5 α = -.5 x = Fig.. NACA-2 airfoil with angle of attack =,Re =:5 6, and M =:2. The -component of elocit (x; ) at x=l =:, :25, :5, :75, and :, where L is the chord length. The resoltion is 383; 724 oxels and ; 275 srfels for and for. 25

27 Re = 5 5 α = x = -.3 x = Re = 5 5 α = x = x = Re = 5 5 α = -.8 Re = 5 5 α =.2 Re = 5 5 α = Srface Pressre Re = 5 5 α =.5. x = x Fig. 2. NACA-2 airfoil with angle of attack =,Re =:5 6, and M =:2. The pressre coecient (x; ) at x=l =:, :25, :5, :75, and :, where L is the chord length, and along the srface. The resoltion is 383; 724 oxels and ; 275 srfels for and for. 26

28 .5 x = x =.25 Re = Re = x =.5 x =.75 Re = 5 5 Re = x = Re = 5 5 Fig. 3. NACA-2 airfoil with angle of attack =3,Re =:5 6, and M =:2. The x-component of elocit (x; ) at x=l =:, :25, :5, :75, and :, where L is the chord length. The resoltion is 383; 724 oxels and ; 275 srfels for and for. 27

29 x = Re = x =.25 Re = Re = 5 5 x =.5 -. Re = 5 5 x = Re = 5 5 x = -.5 Fig. 4. NACA-2 airfoil with angle of attack =3,Re =:5 6, and M =:2. The -component of elocit (x; ) at x=l =:, :25, :5, :75, and :, where L is the chord length. The resoltion is 383; 724 oxels and ; 275 srfels for and for. 28

30 Re = 5 5 x = x = Re = x =.5 x = Re = 5 5 Re = Re = 5 5 x =.5 Srface Pressre Re = x Fig. 5. NACA-2 airfoil with angle of attack =3,Re =:5 6, and M =:2. The pressre coecient (x; ) at x=l =:, :25, :5, :75, and :, where L is the chord length, and along the srface. The resoltion is 383; 724 oxels and ; 275 srfels for and for. 29

31 α = 3 Re = 5 5 Fig. 6. NACA-2 airfoil with angle of attack =3, Re = 5 3, and M=:2. Streamlines of simlation (left) s. simlation (right). 3

32 x = Re = 5 8 x =.25 Re = x =.5 Re = 5 8 x =.75 Re = x = Re = 5 8 Fig. 7. NACA-2 airfoil with angle of attack =3,Re =:5 9, and M =:2. The x-component of elocit (x; ) at x=l =:, :25, :5, :75, and :, where L is the chord length. The resoltion is 383; 724 oxels and ; 275 srfels for and for. 3

33 x = Re = x =.25 Re = Re = 5 8 x = Re = 5 8 x = Re = 5 8 x = -.8 Fig. 8. NACA-2 airfoil with angle of attack =3,Re =:5 9, and M =:2. The -component of elocit (x; ) at x=l =:, :25, :5, :75, and :, where L is the chord length. The resoltion is 383; 724 oxels and ; 275 srfels for and for. 32

34 Re = 5 8 x = -.6 Re = 5 8 x = Re = 5 8 x =.5 Re = 5 8 x = Srface Pressre Re = Re = 5 8 x = -.5 x Fig. 9. NACA-2 airfoil with angle of attack =3,Re =:5 9, and M =:2. The pressre coecient (x; ) at x=l =:, :25, :5, :75, and :, where L is the chord length, and along the srface. The resoltion is 383; 724 oxels and ; 275 srfels for and for. 33

35 C d Drag Lift Re = 5 8 M= C l C d..2.3 Re = 5 8 M=.2 Drag Lift.5..5 C l Time Step Time Step Fig. 2. Time histor of C d and C l in the simlation. Re = :5 9, M=:2, and =3. (left) Comptational domain L x L =98. The total resoltion is 383; 724 oxels and ; 275 srfels. The resoltion on the airfoil srface is 5 points along the chord. Note that C d conerges to a negatie ale. (right) Comptational domain L x L =22. The total resoltion is 63; 62 oxels and 64 srfels. The resoltion on the airfoil srface is2 points along the chord. Neither C d nor C l conerge after 6; steps of iteration. α = 7 Re = 5 5 Fig. 2. NACA-2 airfoil with angle of attack =7, Re = :5 6, and M=:2. Streamlines of simlation (left) s. simlation (right). C d Drag Lift Re=5 5 α=7.7 C l Time Step Fig. 22. Time histor of C d and C l in the simlation. Re=:5 6, M=:2, and =7. Note that the ale of C d is negatie most of time, and both C d and C l do not conerge after 3; steps of iteration. 34