Lattice Boltzmann simulation of flow around bluffbodies

Transcription

1 Louisiaa State Uiversity LSU Digital Commos LSU Master's Theses Graduate School 2004 Lattice Boltzma simulatio of flow aroud bluffbodies Kevi Tubbs Louisiaa State Uiversity ad Agricultural ad Mechaical College, Follow this ad additioal works at https//digitalcommos.lsu.edu/gradschool_theses Part of the Physical Scieces ad Mathematics Commos Recommeded Citatio Tubbs, Kevi, "Lattice Boltzma simulatio of flow aroud bluff-bodies" (2004). LSU Master's Theses https//digitalcommos.lsu.edu/gradschool_theses/3890 This Thesis is brought to you for free ad ope access by the Graduate School at LSU Digital Commos. It has bee accepted for iclusio i LSU Master's Theses by a authorized graduate school editor of LSU Digital Commos. For more iformatio, please cotact gradetd@lsu.edu.

2 LATTICE BOLTZMANN SIMULATION OF FLOW AROUND BLUFF-BODIES A Thesis Submitted to the Graduate Faculty of the Louisiaa State Uiversity ad Agricultural ad Mechaical College i partial fulfillmet of the requiremets for the degree of Master of Sciece I The Departmet of Physics ad Astroomy by Kevi Tubbs B.S., Souther Uiversity, Bato Rouge, 2001 May 2004

3 TABLE OF CONTENTS ABSTRACT... iii CHAPTER 1 INTRODUCTION Motivatio Backgroud Origiality of Work NUMERICAL REVIEW Grid Based Methods Discrete Vortex Methods Lattice Boltzma Methods GOVERING EQUATIONS Kietic Descriptio Legths Scales Navier Stokes Equatios The Fudametal Boltzma Equatio Discrete Velocity Boltzma Equatio Lattice Boltzma Equatio Viscosity Grid Geeratio Boudary Coditios NUMERICAL MODEL Computatioal Domai Lattice Boltzma Solver Force Evaluatio Results Sigle Bluff Body Multiple Bluff Bodies CONCLUSIONS...36 REFERENCES...37 APPENDIX A CHAMPMAN ENSKOG EXPANSION...40 APPENDIX B LB SAMPLE CODE...43 VITA...54 ii

4 ABSTRACT I this work, results from a 2-D Lattice Boltzma (LB) solver are preseted simulatig flow past rectagular square cyliders at low Reyolds umbers (< 250). The LBGK equatio is a hyperbolic equatio that approximates the Navier Stokes equatios i the early icompressible limit. It is a system of 9 oe dimesioal partial differetial Hamiltoia-Jacobia equatios, cosistig of a advectio ad diffusive portio. The LB method is a alterative computatioal fluid dyamics (CFD) method used to umerically predict icompressible viscous flow. The curret LB method uses a statistical mechaics formulatio to solve the Boltzma equatio. The LB model captures the oliear Navier Stokes advectio terms usig liear streamig operators. I this thesis, the LB model is classified as a explicit, Lagragia, fiite-hyperbolicity ad weakly compressible approximatio of the Navier Stokes equatios. The mometum flux tesor is captured locally as opposed to a pressure field elimiatig the eed to solve the Poisso equatio. This allows the fluid structure iteractios (FSI) behavior to be calculated elegatly at the iterface through the mesoscopic mometum trasfer betwee the fluid ad structure. At this level, the forces are simultaeously calculated. The LB equatios are discretized both i time ad phase space usig a stadard D2Q9 lattice model. Validatio tests for flow aroud sigle square cyliders at differet aspect ratio at low Reyolds umbers are preseted. Good agreemet with other ivestigators is achieved. Flow past multiple bluff bodies (represetig buildig i a city) is also preseted. The vortex sheddig simulatios preseted provide prelimiary idicatios i terms of St that the LB method ca be used to simulate high Re flow. iii

5 CHAPTER 1 INTRODUCTION 1.1 Motivatio Fluid flow aroud bluff bodies is wide area of study with importat applicatios to differet areas of sciece ad egieerig. These flows rage from lamiar to turbulet regimes. A measure of lamiar or turbulet flow is foud through the Reyolds umber (defied as Re=U d/v, where U is the costat iflow velocity, d is the bluff body diameter ad v is the kiematic viscosity). Fluid bluff-body ivestigatios ca is experimetal, umerical or theoretical. The majority of ivestigatios are experimetal. Experimetal ivestigatios are mostly limited to lower Re (< 5000) due to model scale restrictios. Recetly, umerical ivestigatios of fluid dyamics ofte described as Computatioal Fluid Dyamics (CFD) are becomig more popular. Numerical ivestigatios are curretly capable of modelig fluids at higher Re (< 10 6 ); however, much more research is eeded to exted curret to higher Re. Numerical ivestigatios are limited oly by the computatio cost of solvig goverig equatios. The higher the Re the more importat smaller legth scales become, which icreases the computatioal cost. Kietic theory presets the goverig equatios at a smaller legth scale which may allow smaller legth scales to be models with less computatioal cost. The aim of this work is to apply a alterative formulatio based o kietic theory ad statistical mechaics to umerically ivestigate fluid dyamics aroud sharp edge bluff bodies ad evaluate its usefuless i CFD. I this work, a fluid bluff body model based o the Lattice Boltzma (LB) equatios is preseted. Fluid flow is approximated through a 2-D 1

6 viscous early icompressible fluid flow solver past sigle ad multiple square cyliders. The solutios preseted i this thesis are limited to low Re (<250). 1.2 Backgroud A bluff body is oe i which the flow uder ormal circumstaces separates from a large sectio of body surface thus creatig a massive wake regio dowstream. We cosider bluff bodies of various horizotal legth to width ratios with sharp edges, i.e. a square cylider, exposed to a cross flow with a costat free stream velocity, U. Flow separatio, boudary layers ad shear layers characterize the flow flied bluff body disturbace. The iermost portio of the free shear layers moves more slowly tha the outermost portio of the layers which are i cotact with free stream, the free shear layers ted to roll up ito discrete, swirlig vortices. At a Re of approximately 50, the vorticial regio dowstream of the body emaates ito the pheomeo of vortex sheddig characterized by a usteady periodic flow situatio i which the separated vortices are shed alterately from the upper ad lower side of the body. The sheddig of vortices is described with referece to the Strouhal umber, defied as St=f s d/u, where f s is the sheddig frequecy. The ear wake flow usteadiess gives rise to fluctuatig drag ad lift forces which ca result i body vibratio. Cylider vibratio ca, (i) icrease the vortex stregth, (ii) icrease the spa wise wake correlatio ad (iii) force the sheddig frequecy to match the atural frequecy of the body (lock-i or sychroizatio). I additio, we ote that above a critical Re of aroud 250, the flow problem may become three dimesioal. Ivestigatios of vorticial istabilities i wakes represet a widely researched area i fluid dyamics. Bluff body vortex dyamics have bee studied for more tha a cetury. 2

7 Oe of the first mathematical treatmets of vortex sheddig was give by vo Karma (1911) ad the staggered vortex cofiguratio i the wake of a body is therefore usually referred to as the vo Karma vortex street. The majority of ivestigatios have bee carried out for the flow aroud a circular cylider, e.g., see comprehesive review by Zdravkovich (1997). From a egieerig poit of view, it is also ecessary to study flow aroud other bluff body shapes, such as sharp-edged rectagular cross-sectioal cyliders e.g. Kisley (1990). Structures that typically have rectagular or ear rectagular cross sectios iclude architectural features o buildigs, the buildigs themselves, beams, feces ad occasioally stays ad supports i iteral ad exteral flow geometries. The strogest history is still experimetal ivestigatios e.g. Williamso (2004). Due to icreasig computer power, umerical solutios ad isight ito the physics have become attractive. Traditioally, CFD solvers are developed based o a cotiuum mechaics approach. Alteratively, the goverig fluid equatios ca be derived based o kietic theory ad statistical mechaics. Due to its kietic origi, this approach has some attractive features computatioally ad physically that may provide more isight ad flexibility ito uderstadig fluid bluff-body iteractios Origiality of Work To evaluate whether the umerical model preseted i this thesis is useful i CFD, the solutios must be validated agaist other ivestigators. Ivestigatios of fluid flow aroud sigle bluff bodies have bee coducted by may ivestigators. I this thesis, validatios are first preseted for sigle bluff bodies after which ew cotributios are preseted i terms of fluid multiple bluff body solutios. This has iterest i differet areas of sciece ad egieerig. 3

8 CHAPTER 2 REVIEW OF NUMERICAL METHODS Numerical predictio of fluid flow ivolves solutios of oliear equatios. Viscous icompressible flow of a Newtoia fluid is govered by mass cotiuity ad Navier Stokes mometum equatios. This set of partial differetial equatios ca be formulated mathematically ad solved umerically i a umber of ways. Traditioally, umerical models solve a discretized form of the Navier Stokes from a cotiuum approach. I this chapter three formulatios are discussed. Alteratively, the fluid behavior ca be described through the Lattice Boltzma equatio whose formulatio origiates i kietic theory. 2.1 Grid Based Methods The approximate, traditioal discrete methods used i computatioal fluid dyamics are usually either the structured fiite-differece methods or the discrete weighted-residual methods such as fiite-volume or fiite-elemet methods. These methods have bee heavily researched over the past 30 years. However, the process of grid geeratio is ot at the same level of maturity as the approximate solutio methods, especially for three-dimesioal problems, as discussed recetly by Douglas et al., (2002). Furthermore, grid geeratio eve i two-dimesioal (2D) ca be time cosumig, especially whe the grids ivolve bluff-bodies. 4

9 2.2 Discrete Vortex Methods A umber of umerical models are available that do ot require meshig of the flow domai, for example the Discrete Vortex Method (DVM) e.g. Chori (1973). Chori also refied the discrete vortex approach by proposig a operator-splittig method which treated the advective ad diffusive flow process usig separate umerical schemes. To this ed, the DVM has bee popular for may decades e.g., Leoard, (1980); Stasby & Dixo (1983); ad Lewis (1991). I particular, the method of source paels has bee used for studyig aerodyamic iteractios amog various compoets of a aircraft. 2D simulatios have bee very successful, however, predictio of lock-i i relatio to bluff bodies has ot, e.g., Larse & Walther (1998). This is possibly due to problems with replicatig turbulet boudary layers e.g., Hut (2000). Moreover, three-dimesioal (3D) simulatios are further troublesome. A extesive review of DVM is give by Sarpkaya (1989). The DVM is based o itroducig vorticity as a umber of blobs at the body surface ad tracig their paths through the flow field. The lack of meshig i the DVM is oe its strogest advatages as it esures that arbitrary geometries ca be aalyzed. However, the solutio teds to become chaotic e.g. Sarpkaya (1989) which ca oly be avoided by settig certai parameters to obtai the expected solutio. This requires several aalyses to be coducted to calibrate the model ad leaves questios i describig the true physics of the flow by lettig the model reach a solutio with kowig aythig about that solutio. 2.3 Lattice Boltzma Methods I the past years, there has bee cosiderable research i developig ad expadig the Lattice Boltzma (LB) method for differet fluid dyamics problems. 5

10 Comprehesive reviews of LB methods ad the progress of their developmet have bee give e.g., Che & Doole (1998); Yu et al. (2003). The LB method was developed as a improvemet of the method of lattice gas automata (LGA). It was first itroduced by McNamara & Zaetti (1998). Prior to this theoretical cotributios have bee give e.g., Qia et al. (1992); Che et al. (1992); d Humieres (1992). Although ispired by LGA, LB methods are better uderstood as a self-cotaied method for solvig the Boltzma equatio. The LB method ca be developed from fudametal priciples, as show by He & Lou, The most popular form of lattice Boltzma equatio is the Lattice-BGK (LBGK) icorporatig a sigle time approximatio of the Boltzma equatio first itroduced by Bhatagar et al. (1954). This LB approach is motivated by studies doe i molecular dyamics ad physics. Although the scope of this research is limited to icompressible fluid dyamics, LB methods are ot limited to just that. Research is beig coducted i the applicatio of LB methods to other fields such as FSI ad blood flow istabilities i artificial heartvalue geometries e.g. Krafczyk et. al (2001), compressible flow at high Mach umbers, e.g. Yu ad Zhao (2000) ad soud wave propagatio i 2D urba eviromet us the related TLM techiques, e.g. Luthi et al. (1996). Geeralized LB methods are also beig ivestigated as a discrete umerical solutio method for partial differetial equatios, e. g. umerical solutios to Schrödiger equatio i Quatum Mechaics, e.g. Succi (2001). This study is cocered with the LB method as a alterative treatmet for fluid dyamics. The LBGK scheme is a secod order method for solvig icompressible flows. LB methods differ from Navier Stokes solvers i various aspects icludig theoretical 6

11 ad computatioal iterpretatios. Some of the advatages of the LB method iclude a reductio from secod order to first order partial differetial equatios, simplificatio of o-liear modelig, Poisso freedom, computatioal efficiecy ad accuracy, simple fluid iterface boudary coditios ad a mathematical framework allowig molecular level modelig. The method also has limitatios icludig accurate velocity pressure boudary coditios ad grid resolutio which lead to low Re modelig. Efforts are curretly uderway to exted the LB method to higher Re with local grid refiemet, force evaluatio, umerical stability ad turbulece modelig beig areas of mai cocer. Detailed discussios of these issues have bee give e.g., Yu et al., (2003); Nourgaliev et al. ( 2003). To date, the LB approach has bee successfully used for simulatig icompressible 2-D ad 3-D flows for Re ragig from 200 to 5000 (e.g., Che & Doole, 1998; Schafer & Turek, 1996). 7

12 CHAPTER 3 GOVERNING EQUATIONS 3.1 Kietic Descriptio Kietic theory assumes that the fluid is described by a large umber of molecular costituets whose motios obey Newtoia mechaics. The objective of the theory is ot to kow the motio of every idividual molecule, but the collective behavior for which oe eeds a statistical descriptio of the system. The statistical descriptio of a fluid at or ear equilibrium is cotaied i the sigle-particle distributio fuctio, f(r, e, t), where r represets spatial coordiates, e represets microscopic velocity of molecules ad time t. It is defied such that [f(r, e, t) d 3 r d 3 e] is the umber of particles i a phase space cotrol elemet [d 3 r d 3 e]. The kietic theory describes the trasport equatios describig the time ad spatial evolutio of the distributio fuctio with differet collisio processes dictated by the ature of the iteractios betwee molecules, give by f + e + a f ( r, e, t) = t r e t coll (1) where a is the exteral force actig o the particle. I this case the equatios are equivalet to the Navier Stokes equatios with relevat trasport coefficiets such as shear ad bulk viscosity. The trasport equatio used is the Boltzma equatio, described i sectio Legth Scales The assumptio of a fluid at or ear equilibrium is fudametal to a kietic descriptio. Collisios at a molecular level drive the fluid molecules towards a global equilibrium of speed U ad temperature T. The process happes at three differet dyamical stages correspodig to three legth scales microscopic, mesoscopic ad 8

13 macroscopic. Each legth scale has a correspodig set of goverig equatios ad time scales. Qualitatively, the approach to equilibrium is cotrolled by the time scales t c, t µ ad t f, e.g. Bogoliubov (1962). t c is defied as the duratio of a collisioal evet ad is proportioal to ratio of the effective diameter of the particle, s, to the velocity of the particle, e, (i.e. t c ~ s / e). This is cosidered the atomistic or may-body regime where each particle is govered by Newtoia dyamics. t µ is defied as the mea flight-time betwee two collisios ad is proportioal to ratio of the mea free path, l µ, of the particle to the velocity of the particle, (i.e. t µ ~ l µ / e). This is cosidered the kietic regime where a collectio of particles are govered by the Boltzma equatio. t f is defied as the miimum fluid dyamic (covective, diffusive) time scale ad is proportioal to the ratio of the typical macroscopic scale, l M, to the flow speed,u, ad the ratio of the typical macroscopic scale squared to the kiematic viscosity, v, (i.e. t f ~ mi[ l M / u, l 2 M / v ). This is cosidered the macroscopic regime where a ifiitesimal cotrol volume is govered by the Navier Stokes equatios. O the time iterval, 0 < t < t c, May body iteractios relax may-body distributio fuctios to the sigle particle distributio fuctio. O the time iterval t c < t < t µ, the sigle particle distributio fuctio relaxes to a local equilibrium distributio with smooth space-time depedet flow speed ad temperature. O the time iterval, t µ < t < t f, the local equilibrium distributio drifts slowly to a global equilibrium distributio with costat macroscopic speed ad temperature. These three dyamical stages describe the coectios betwee microscopic ad macroscopic legth scales usig itermediate mesoscopic legth scales. I this view, fluid dyamics ca be see as a mea field approximatio emergig from a perturbative treatmet of the 9

14 kietic equatios. Chart 1 summarizes the hierarchy of these legth scales icreasig from right to left. Chart 1 Diagram of legth scales ad correspodig goverig equatios Legth Scales Atomistic Kietic Fluid dyamics Microscopic Mesoscopic Macroscopic Newtoia Dyamics Boltzma Equatio Navier Stokes 0<t< t c t c <t< t µ t µ <t<t f 3.3 Navier Stokes Equatios I this thesis, icompressible Newtoia fluids ad their iteractio with sharp edged bluff bodies are of iterest. The goverig equatios ca be described by the Navier Stokes equatios. The Navier Stokes equatio icludig body forces ad turbulece via the Reyolds Stress Tesor is give by U i { t Acceleratio U i 1 U U j i U j P + = δ ij + µ + ρuiu j + ρg + F (2) S x j ρ x j x j x i Mea Re yolds Body Forces Covectio Pr essure Stress Stress Stress Mea Viscous Tesor Tesor where U is the velocity of the flow field, u is the fluctuatig turbulet velocity, x is the geeralized coordiate of the system i the directio of the idices i ad j. The left had side describes the acceleratio ad covectio of the fluid flow. The depedet variables are the velocity (U) ad the pressure (P) while the costats are fluid desity (ρ) ad the 10

15 kiematic viscosity (µ). The right-had side represets the mea pressure, viscous ad turbulece effects ad the body forces are represeted by gravitatioal force (ρg) ad forces due to FSI (F S ) where g is gravity. I this thesis, we cosider oly lamiar flow aroud fixed bluff bodies. The Reyolds Stress Tesor ad body forces are eglected ad (1) reduces to U i { t Acceleratio U i 1 + U j = x j ρ x j Covectio U U i j Pδ ij + µ x j xi Mea Pr essure Mea Viscous Stress Stress Tesor (3) Equatio (2) ca be separated ito o-dissipative ad dissipative terms through evaluatig the cotributios of the mometum flux tesor. The o-dissipative cotributio is give as Π ij x j 123 No Dissipative = x j ( Covectio Pr U { { ) iu j + Pδij Mea essure Stress (4) ad the dissipative cotributio is give as where tesor. U U i j π ij = µ + (5) x j x j x j xi Dissipatio MeaViscous StressTesor Π lm is the o-dissipative mometum flux tesor ad π lm is the dissipative flux A statistical mechaics approach based o kietic theory is also capable of describig a fluid motio govered by the Navier Stokes equatios. The preset umerical model is based o kietic theory which attempts to describe the macroscopic 11

16 fluid behavior usig the laws of mechaics ad probability theory. Provided that the fluid is ear a state of equilibrium ad the costitutive relatios betwee stress ad strai are obeyed, kietic theory ca represet the Navier Stokes equatios. I the statistical mechaics approach, the o-dissipative cotributios are related to the equilibrium distributio while the dissipative cotributios are related to the small departures from equilibrium or o-equilibrium distributios. This coectio betwee the two approaches is give i geeral as Π ij = m f ( eq) e e de i j (6) ( eq) π = m f e e de. (7) ij i j The expressios for (6) ad (7) are derived i Appedix A. 3.4 The Fudametal Boltzma Equatio The Boltzma equatio was first itroduced by Boltzma (1872). Followig the derivatio outlied by Nourgaliev et al. (2003), the Boltzma equatio is first preseted i dimesioal format. The equatio is derived by explicitly defiig the collisio term i equatio (1). The Boltzma equatio relates the time evolutio ad spatial variatio of a collectio of molecules to a collisio operator that describes the iteractio of the molecules. Two major assumptios were made i developig the collisio operator (i) oly biary collisios are take ito accout, ad (ii) the velocity of a molecule is ucorrelated with its positio. The first assumptio is valid if the gas is sufficietly dilute i.e. ideal gas. The secod assumptio relates to the assumptio of molecular chaos i which the collisio operator is expressed i terms of the sigle particle distributio fuctio, f. Without this assumptio the collisio operator would ivolve a two particle probability distributio fuctio ad i geeral equatio (1) would be replaced by a set of 12

17 N coupled equatios to accout for multi-particle iteractios. This set of coupled equatio is kow as the BBGKY (Bogolyubov, Bor, Gree, Kirkwood ad Yvo) equatios. Uder these assumptios, Boltzma expressed the collisio term of (1) as f t = 3 (0) (0) (0) (0 dθ d e σ ( Θ) e e ( f f f ) ) coll (8) where Θ is the scatterig agle of the biary collisio {e, e (0) } {e, e (0) } with fixed velocities e, e (0) ; where uprimed quatities f e, e (0) ad primed quatities e, e (0) f deote the velocity ad sigle particle distributio fuctio before ad after collisio; ad σ (Θ) is the differetial cross sectio of the collisio, e.g. Huag (1963). The collisio itegral of (8) ca be greatly simplified for ear equilibrium states by implemetig a sigle time relaxatio approximatio, (BGK) collisio model. The sigle time relaxatio approximatio states that durig a time iterval t c a fractio t c /τ = 1/τ* of the particles i a ifiitesimal volume udergoes collisios, drive the sigle particle probability distributio fuctio to the equilibrium value give by f eq ρ = (2πRT ) D / 2 ( e u) exp 2RT 2 (9) where D, R, T, ρ ad u are the dimesio of space i.e. 2D D=2, gas costat, temperature, macroscopic desity ad velocity, respectively. The BGK collisio operator is the give as f t coll = f eq f f f = τ tcτ * eq (10) where τ is a relaxatio time. The Boltzma equatio with the BGK collisio operator is give as 13

18 + e f ( r, e, t) = t r f f τ eq (11) with exteral body forces, i.e. gravity, electro-magetic, movig body, etc., are eglected i this model. The fluid flow i this model is completely drive by pressure or velocity boudary coditios. The mesoscopic quatities of the Boltzma equatio are liked to the macroscopic quatities of fluid dyamics by itegratio of particle distributio fuctio over mometum space. The macroscopic variables desity, velocity ad kietic eergy are calculated as the first, secod ad third momets of the sigle particle distributio fuctio respectively. ρ = ρ = [ f ] de; ρu = [ f e] de; ρe = 1 2 eq eq [ f ] de; ρu = [ f e] de; ρe = [ f ( e u) ] [ f eq 2 ( e u) ] (12) with kietic eergy, E, give by E = N F 2 where N F is the umber of degrees of freedom of a particle ad k B is the Boltzma costat (k B = J/K). 3.5 Discrete Velocity Boltzma Equatio The Boltzma Equatio with BGK collisio model is discretized i velocity space by itroductio a fiite set of velocities, v i, ad associated distributio fuctios, f i (r, v i, t). The equatio ca be o-dimesioalized by the characteristic legth scale, L, the referece speed, U, the referece desity, ρ r, ad the time betwee particle collisios, t c, givig k B T i t + c i i = τε ( ) ( ) 1 eq i i (13) 14

19 Equatio (13) ad all of subsequet equatios are preseted i o-dimesioal variables defied as eˆ c i i = = L ˆ U U t = tˆ L ˆ τ = τ t c i = fˆ ρ i r U ε = t c L where i is the geeral umber of discrete velocities used to approximate the cotiuous distributio fuctio, e.g. i ca assume values from 0 to ifiity. The expasio parameter, ε, or Kudse Number ca be iterpreted as the ratio of collisio time to characteristic time. The LB correspods to a specific discretizatio of the Boltzma equatio. Equatio (13) ca be discretized by expasio of particle distributio fuctio i terms of Kudse Number, choosig t= tˆ U/L. The selectig the lattice spacig divided by the time step to equal the lattice velocity c = x*/ t*. The subscript deotes the specific discrete velocities of the LB discretizatio. The set of subscript alphas are a subset of the geeral set of subscripts i. This results i a Lagragia formulatio of discretized phase space. Choosig t= t c, oe has the o-dimesioal lattice Boltzma BGK equatio give i o-dimesioal lattice uits (lu) 1 (eq) ( x + c t, t + t) ( x, t) = ( ) (14) τ where the represets discrete velocities of the LB discretizatio of the Boltzma equatio. = 0, 1,, 8 for the 2-D ie speed lattice (D2Q9) ad = 0, 1,, 18 for the 3-D fiftee speed lattice (D3Q19). = 0 represets the rest particle ad oe zero values correspod to lattice vectors i the directio of earest eighbors. 15

20 y x z y x a) b) Fig. 1 Lattice geometry ad velocity for a) 2-D ie speed D2Q9 model ad b) 3-D ietee speed D3Q19 model The lattice geometry ad discrete velocities are show i Fig. 1. The light ceter circle is the locatio of the ode ad rest particles. Solid ad dashed vectors ad correspodig dark circles represet particle speeds ad locatios after oe time step. It is oted that the two lattices preseted are ot the oly choices. Lattice geometries are selected based o ability to recover mass, mometum ad eergy coservatio. I this thesis, the stadard LB method preseted above is used. The LBGK equatio is a hyperbolic equatio that approximates the Navier Stokes equatios i the early icompressible limit. It is a system of 9 oe dimesioal partial differetial Hamiltoia-Jacobia equatios, cosistig of a advectio ad diffusive portio. The LB method ca be viewed as a special fiite differece approximatio to solvig the discrete velocity Boltzma equatio. Other approaches like FVM have bee used, e.g. Peg et al. (1999); Xi et al. (1999). 3.6 Formal Lattice Boltzma Equatio The formulatio preseted above is a discretizatio of the discrete Boltzma equatio which is viewed as a extesio of the LGA method. To uderstad ad 16

21 improve the LB method for fluid dyamics ivestigatios, a soud theoretical foudatio ad coectio to the cotiuous Boltzma equatio must be established. He ad Lou (1997) demostrated that the LB equatio ca be viewed as a specific fiite differece (FD) approximatio of the cotiuous Boltzma equatio. This FD approach ivolves a time discretizatio coupled to discretizatio of a 4-D phase space i two dimesios or 6- D phase space i three dimesios. The phase space discretizatio ca be geeralized as a) c) eq b) c c eq = A + B c u i i + C u 2 + D c c u u i ε j i j (15) where is directio of the discrete velocities ad i ad j=1, 2, 3 are the Cartesia directio of the coordiate system. Macroscopic cotiuum fluid equatios are derived usig the multi-scale Chapma Eskog perturbative expasio procedure as derived by Chapma (1970). The discrete equilibrium distributio fuctio i c) is called the Chapma-Eskog expasio. The coefficiets A, B, C ad D are such that mass coservatio, mometum coservatio ad viscous stress tesor are recovered durig the Chapma Eskog expasio procedure. Through this procedure the equilibrium distributio fuctio is determied to be a costat temperature ad small velocity (low Mach umber) approximatio of the Maxwellia equatio (9), as follows eq 2 [ c 2RT ] 2 2 ρ exp ( c u) ( c u) u Ο( u ) D / 2 2 (2πRT ) RT 2( RT ) 2RT (16) The phase space discretizatio establishes the structure of the lattice ad the form of the equilibrium distributio fuctio. The discretizatio must be cosistet with the macroscopic variables defied through itegratio i mometum space, equatio (12). I 17

22 the derivatio by He ad Lou (1997), these itegral equatios have a geeral from ad are approximated by Gaussia quadrature eq eq ψ ( c ) ( x, c, t) dc wψ ( c ) ( x, c, t) (17) Where ψ(c) = [1;c i ;(c i c j ); (c i c j c k ); ] ad ω are polyomials of microscopic velocity, c, ad weights of Gaussia quadrature, respectively. Usig (17) the LB method ca be liked to macroscopic hydrodyamic variables as follows ρ = ρ = ; ρu = = c ; ρe 1 2 eq eq ; ρu = c ; ρe = 1 2 eq ( c ( c u) 2 u) 2 (18) where eq eq ( x, t) w ( x, c, t); ( x, t) w ( x, c, t) (19) The selectio of the abscissas of the quadrature equatio (17) determies the structure or symmetry of the lattice. The details of the procedure to fid the required abscissas of the quadrature ad correspodig approximatios of the Maxwellia are give by He ad Lou (1997) for 6-, 7-, ad 9-speed lattice models i 2-D ad 15- ad 27- speed lattice models i 3-D. This is aalogous to selectig discrete velocities of the discrete Boltzma equatio. Costraits are imposed o the structure of the lattice based o the Chapma Eskog procedure likig the Boltzma equatio to the Navier Stokes equatios. The procedure ivolves the followig momets of the equilibrium distributio fuctio Mass coservatio Mometum coservatio Eergy coservatio ψ ( c) = 1; c ; c c i ψ ( c) = 1; c ; c c ; c c c i i i i i j j ψ ( c) = 1; c ; c c ; c c c ; c c c c j i i j j k k i j k l (20) 18

23 The importace of this formulatio is that with the chose abscissas of the Gaussia quadrature equatio (17), the momets of the equilibrium distributio fuctio, equatio (20) ca be treated exactly. The Boltzma equatio i the limits ad costraits of the Chapma Eskog procedure are a exact solutio to Navier Stokes equatios. I this view, the validity of the LB method ca rest o the rigorous results of the Boltzma equatio Viscosity The LB method is a approximatio of the Navier-Stokes equatios. By simulatig the dyamics of a fluid at the molecular level, the viscosity is modeled from a molecular poit of view. The viscosity of a fluid is a property of the material ad i geeral depeds o local desity ad temperature. I this LB model, temperature is take to be costat ad the viscosity is a fuctio of the desity of the fluid. The desity of a fluid at a molecular level depeds o the mea free path. The mea free path of the fluid ca be related to the relaxatio time or average time betwee particle collisios. The viscosity is the a fuctio of the relaxatio time whose exact form is determied by derivig the Navier Stokes equatios from the LB equatio. I order to derive the Navier Stokes equatios from the LB equatio, the Chapma Eskog expasio is used (e.g. see Appedix A for full derivatio.) The Chapma Eskog expasio correspods to a multi-scale expasio to first order i space ad secod order i time. The time ad space coordiates are rescaled as 2 = ε t t = εt, x = ε, (21) 1, 2 1 x t ad the correspodig Taylor expasio of the derivative is t = t 2 ε ε, = ε. (22) 1 + t 2 x x 1 19

24 The Chapma Eskog expasio assumes that the diffusio time scale, t 2, is much slower tha the covectio time scale t 1. The sigle particle distributio fuctio,, is expaded about equilibrium as = ( 0) + ε (1) + ε 2 (2) + O( ε 3 ). (23) Applyig the Chapma Eskog expasio ad takig the icompressible flow limits as described by Che ad Doole (1998), the traditioal cotiuum mass ad mometum are recovered. Comparig the coefficiets of the dissipative flux tesor i the cotiuum mechaics approach, equatio (5), ad the statistical mechaics approach, equatio (7), the viscosity coefficiet is obtaied. The expressio for viscosity is the give a fuctio of relaxatio parameter v = (τ ½) c s. I this expressio, the relaxatio parameter, τ, presets the physical viscosity of the fluid while the factor of ½ is a product of the umerical discretizatio. This umerical viscosity is a artifact of the lattice ad ca be cosidered a propagatio viscosity Grid Geeratio The LB method is based o grid boud particles movig alog a set of discrete velocities. The computatioal grid i the LB formulatio is a set of spatial poits coupled to directioal liks coectig them formig a lattice. Every positio i the lattice ca be reached by liear combiatio traslatios alog the discrete velocities which lik them. This allows oe to solve equatios 2N-dimesioal phase space o a N-dimesioal computatioal space. The LB equatio (14) is solved o a 2-D computatioal lattice. The computatioal grid cosists of a 2-D spatial grid of dimesios x y ad a v - dimesioal stecil placed at each spatial ode. Where x is the umber of odes i the x 20

25 directio ad y is the umber of odes i the y directio ad v is the umber of discrete velocities of the model. The v =9 stecil correspods to the discrete velocities of the D2Q9 velocity model ad all possible liks betwee eighborig odes. Figure 2 shows the regular spatial grid with discrete velocity stecil imposed. I the streamig step a distributio of particles will propagate from each spatial ode via oe of the eight liks or stays at rest o the ode. The grid is represeted computatioally as a v x y array Boudary Coditios Fig. 2 Diagram of spatial grid ad discrete velocity stecil combied to form lattice. Two classes of boudary coditios are frequetly ecoutered i CFD ope boudaries ad solid walls. Ope boudary coditios iclude lies or plaes of symmetry, periodic cross sectios, ifiity, ad ilet ad outlet. O these boudaries, velocity or pressure is usually specified i the macroscopic descriptio of fluid flows. Velocity or pressure coditios ca be prescribed usig odal (Dirichlet) or elemet (Neuma) boudary coditios. A odal boudary coditio prescribes a specific scalar value whereas a elemet boudary coditio prescribes derivatives. 21

26 Oe challege of the LB method is that the boudary coditios for the distributio fuctios are ot kow. Oe must costruct suitable boudary coditios based o macroscopic flow variables. At symmetric ad periodic ope boudaries, coditios o distributio fuctios are trivial. Solid boudary coditios ca be satisfied approximately by solvig for the ukow distributio fuctios ( ). After the collisio step, (x f ) at the fluid ode x f i the fluid regio is kow for all, but (x b ), the distributio fuctio streamig from the solid ode x b to a fluid ode x f is ot kow. To complete the streamig step, (x b,t) is eeded because it exactly gives (x b,t+δt) after streamig. For the o-slip boudary coditio, these distributio fuctios must be chose such that the macroscopic velocity is set to zero o the boudary. A popular boudary coditio is to employ a bouce back scheme, as described by Ziegler (1993). I this scheme, the mometum from post collisio particle is bouced back i the opposite directio after the particle hits the wall. Reversig the mometum attaches the particle to the solid surface ad sets the particle s velocity equal to the velocity of the solid wall. For the free slip boudary coditio, the distributio fuctios are chose such that the tagetial motio of the fluid flow is free ad o mometum is to be exchaged with the wall alog the tagetial directio. This is achieved i a similar maer to the bouce back scheme; however, the oly the ormal compoets are reversed while the tagetial compoets are allowed to stream freely. 22

27 CHAPTER 4 NUMERICAL MODEL 4.1 Computatioal Domai The boudary coditios prescribed o the computatioal domai show i Figure 3. At the ilet boudary, Γ 1, a costat velocity profile is prescribed usig equilibrium distributio fuctios applied to the first lattice colum. At the top ad bottom boudary coditios, Γ 2 ad Γ 3 respectively, free-slip boudary coditios are applied with free stream velocity. The top ad bottom boudaries are located a distace (Z-d)/2 above ad below the bluff body. At the outlet of the domai, Γ 4, a costat pressure ad zero velocity gradiet i the x directio is applied. The outlet is located a distace Y away from the bluff body. No-slip boudary coditios are prescribed at the walls of the bluff-body, Γ 5. The bluff body is located a distace X from the ilet. The oslip boudary coditio implemeted i this thesis, is based o secod order accurate boudary coditio (half-bouce back scheme) as described by Che & Doole, (1998). Γ 2 Γ 1 y Γ 5 Γ 4 Z x Γ 3 X Y Fig. 3 Computatioal domai ad boudary coditios. 23

28 4.2 Lattice Boltzma Solver The majority of grid based methods preseted i sectio 2.1 are solved via primitive variable solutios which have advatages ad disadvatages. A advatage of primitive variable solutio procedures is that they ca be applied to 3-D flow problems i a straight forward maer. A commo feature i these methods is the solutio of the Poisso equatio for pressure or pressure correctio. Solvig this equatio is the majority of the computatioal cost ad represets the mai disadvatage. Icompressible viscous fluid flow is described mathematically by the Navier Stokes equatios cosistig of mometum equatios coupled to the cotiuity equatio. These equatios have four ukows i 3-D ad three ukows i 2-D, correspodig to the pressure ad compoets of velocity. Solvig these equatios as oe large oliear system of equatios is very expesive to solve. For this reaso, most solvers solve pressure ad velocity weakly coupled. These solutio methods ca be categorized ito two mai types artificial compressibility ad pressure correctio methods. The artificial compressibility method was origially proposed by Chori (1967). It is based o solvig a modified form of the cotiuity equatio for compressible flow. At each time step, a Poisso equatio is usually solved for pressure to drive the flow solutio to a steady state. There are several algorithms that ca be cosidered pressure correctio methods such as the Marker ad Cell, (MAC) method proposed by Harlow ad Welch (1965) ad the Semi-Implicit Method for Pressure Liked Equatios (SIMPLE) method first itroduced by Patakar ad Spaldig (1972). I these methods, at each time step, a itermediate velocity is estimated. The estimated velocity is used to solve the Poisso equatio to obtai a ew correspodig pressure field. The steps are take to advace 24

29 both the pressure ad velocity i time. Differet variatios of this method have bee used to simulate flow past bluff bodies, (e.g. Davis ad Moore 1982; Okajima et al. 1993). The SIMPLE algorithm ivolves solvig a Poisso equatio for a pressure correctio istead of pressure. This improves covergece compared to the projectio method because the differece betwee estimated ad fial velocities is smaller. Variatios of the SIMPLE method have bee developed to icrease computatioal efficiecy ad stability, e. g. SIMPLER, e.g. Patakar ad Spaldig (1972) ad SIMPLEC, e.g. Pierre (1988). These algorithms are employed to obtai a steady state at each time step. The LB solver i this thesis has the advatage of straight forward 3-D implemetatio but elimiates the disadvatage of solvig the Poisso equatio. To solve for umerically, equatio (14) is solved usig a basic stream ad collide algorithm. The advectio part is performed i the streamig part of the algorithm ad diffusio i the collisio part. Equatio (14) is computed i two steps collisio step ( eq [ ( x, t) ( x, t) ] 1 ( x, t + t) = ( x, t) (24a) τ ~ ) streamig step ~ ( x+ c t, t + t) = ( x, t + t) (24b) Where ~ deotes the post collisio state of the distributio fuctio. Note the left-hadside of (24a) ad the right-had-side of (24b) are at time level t + δt as these equatios are solved explicitly. The equilibrium distributio for is give as 3 = + u + u c c 2c 2c 2 ( ) eq ρω 2 ( c u) u (25) where ω is the weightig factor of a Gaussia quadrature give by 25

30 4 9, = 0, ω = 1 9, = 1,2,3,4 (26) 1 36, = 5,6,7,8 Macroscopic variables desity ad mometum i phase space are calculated as the momets of the distributio fuctios give i equatio (18). The pressure is determied from the pseudo equatio of state for a ideal gas p=ρc 2 s, where c s is the pseudo speed of soud i the lattice model. 4.3 Force Evaluatio Estimatio of fluid forces o bluff bodies ca be hadled either by pressure stress itegratio approach or mometum exchage approach. Comparisos of these methods were performed by Yu et al., (2003) ad the mometum exchage approach was foud to be more accurate because it avoids extrapolatios. For this reaso, the mometum exchage method was chose for force estimatio. The total resultat fluid force, F, o a fixed bluff body is obtaied as F = d allx = 1 b N e β [ ( x, t) + ( x + e δt, t) ] δx δt b b β (27) where N d is the umber of o-zero lattice velocity vectors, the subscript β deotes the lattice directio opposite of the directio, = -β = 1,2, 8 i 2-D ad 1,2,,18 i 3-D. Equatio (4) is evaluated at the midpoit of the fluid lattice odes at x f = (x b + e β δt,t) ad the solid lattice odes at x b = (x f + e δt, t) givig the fluid solid mometum exchage per uit time. Note the distributio fuctios used here are i a post collisio state. The ier summatio describes the mometum exchage betwee a solid ode at x b ad all possible eighborig fluid odes aroud that solid ode. The outer summatio icludes the force cotributed by all boudary odes x b. Equatio (4) is applicable i both 2-D ad 3-D LB 26

31 models. The estimated force is used to calculate characteristic drag, C d, ad lift, C l, coefficiets defied as C Fx = 1 2 ρu d 2 H C Fy = 1 2 ρu l 2 d 4.4 Results I the followig, solutios are preseted for flow past fixed sigle ad multiple bluff bodies o 2-D uiform grids Sigle Bluff Body First, solutios are preseted for flow aroud a sigle square cylider for Re = 100 ad 250. The computatioal domai is 100d 30d where d=0.166 m deotes the cylider diameter. The cylider is located at X=20d from the ilet ad vertically cetered (Z/2) i the domai. The outlet is located at Y=80d. Simulatios were carried out for three grid resolutios, , , ad with 10, 20, ad 40 lattice odes o the diameter respectively. x ad t i lattice uits (lu) are coupled by the uiform lattice spacig of x/ t =1. The lattice uits are related to physical uits through the Re. LB parameters are chose so that the Re i lattice uits equal the Re i physical uits. These LB parameters traslate ito physical spatial ad time steps give i Table 1. Table 1 LB parameters for Re=250. Grid Resolutio x (m) t (s) τ(lu) Grid sesitivity test were carried out. Grids 2 ad 3 coverged to the same Strouhal umber, St. Therefore, the remaiig simulatios relate to grid 2. St of

32 ad was obtaied for Re= 100 ad 250 respectively. These LB results fairly agree with the results obtaied by other ivestigators, as show i Table 2. Table 2 Sigle Strouhal No. Reyolds No Preset LB Model Okajima (1982) Davis et al. (1984) Sohakar et. al (1998) Davis et al. (1982) implemeted a 2D Fiite Volume (FV) solver to simulate flow over rectagular cross sectios i ifiite ad cofied domais. Davis et al. showed good results for Re below Frake et al. (1990) implemeted a 2D FV third orderaccurate scheme to solve usteady flow past a square cylider for Re 300. We ote that Frake et al. reported o umerical problems due to sharp edged geometry. Okajima et al. (1992) implemeted a 2D Fiite Differece (FD) solver for low Re ad a Discrete Vortex (DV) method for high Re o flow past rectagular cyliders of varyig aspect ratio. Low Re results successfully captured chages i flow patter. Okajima simulated high Re ad foud jumps i St. Mahir (2002) implemeted a differet high order FD solver for flow aroud square cylider for Re 250 with results similar to experimetal results. Sohakar et al. (1999) developed a FV SIMPLE code for flow aroud sharp edge bodies i 2D ad 3D. I the followig, the horizotal legth, H, to diameter aspect ratio, (a=h/d), is varied for the cases of a=2,3 ad 4 with same computatioal domai for Re=100 as show i figure 5. St, lift, C l ad drag, C d are predicted. Figure 4 shows the time history of drag ad lift force coefficiets for a=1. The time histories are harmoic with the lift coefficiet fluctuatio about a mea close to zero as expected. Solutios are listed i Table 3 for differet sharp edge bodies. The solutios are compared with Sohakar et al. 28

33 (1998) who used a FV SIMPLE Navier Stokes Solver o a o-uiform grid. The ouiform grids iclude high grid resolutio ear the boudary of the bluff body, as steep velocity gradiets ca occur at high Re. The preset LB model is based o uiform grids but captures the similar resposes for various aspect ratios. Compared to Sohakar et al. good agreemet is foud for the St; however, drag coefficiets are higher i the preset LB model. This over predictio is most likely due to uiform grid resolutio, i. e. isufficiet grid poits i boudary layer. 2 Cd Cl x 10 4 Fig. 4 Drag ad lift coefficiets for a=1 Table 3 Drag ad St of sigle sharp edged bodies (Re=100). a Preset LB Model Sohakar (1999) St Cd St Cd St Cd St Cd

34 a) b) c) d) Fig 5 Near wake vorticity cotours for aspect ratios (a) a=1; (b) a= 2; (c) a=3; ad (d) a=4 Figure 5 illustrates the vorticity cotours i the ear wake for each aspect ratio. The dark ad light cotour lies correspodig to top ad bottom shear layers respectively represet opposite sigs of vorticity. The shear layer istabilities i the flow 30

35 cause vortices to shed off of opposite corers as expected. The sheddig period is icreased as the aspect ratio icreases. This agrees with the treds i the data give i Table 3. Figure 6 shows a close up look of pressure cotours for aspect ratio 1, 2, 3 ad 4. The separatio ad reattachmet characteristics are visualized with o reattachmet for a=1 as opposed to the separated flow reattachig for a=2, 3, 4 respectively, affectig the St. a) b) c) Fig 6 Close up look of pressure cotours for aspect ratio a) a=1; b) a=2; (c) a=4; (d) a=4 d) 31

36 4.4.2 Multiple Bluff Bodies The secod series of test preseted test the LB model i the applicatio of multiple bluff body cases. The first model reproduces qualitatively compare flow patter visualizatios aroud buildig complex of Murakami (1990). The preset simulatios were carried out o a computatioal grid. A free stream velocity correspodig to a wid origiatig from the south south-west directio is prescribed o the left ad bottom boudaries. A costat pressure, zero-velocity gradiet is applied to the top ad right boudaries. Figure 7 illustrates the flow field past the array of buildigs. Figure 7a shows the istataeous streamlies of the flow idicatig low velocities i the ear wake of the buildigs as expected. These velocity patters are similar to those of Murakami. Figure 7b shows the correspodig vorticity cotours lies, showig separatio poits occurrig at the sharp corers with opposite sigs of vorticity o opposite corers, as expected. 600 a) b) 0 0 SSW Fig. 7 2-D flow past multiple bluff bodies a) streamlies ad b) vorticity cotour lies

37 At this stage we are cofidet i developig ew multiple bluff body studies at Re 250 with the purpose of reducig bluff body forces. The sigle square bluff body preseted above is ameded by placig two smaller bluff bodies i frot ad i the ear wake respectively. Figure 8 compares the flow patters to the sigle bluff-body studies. For this multiple bluff body study the same computatioal domai ad similar boudary coditios are prescribed. Flow patters were compared to the results of first case study. Figure 8 shows the istataeous streamlies for the (a) sigle square cylider ad multiple square cyliders with (b) iflow disturbace ad (c) ear wake disturbace. Figure 9 shows the istataeous vorticity cotours at two differet time steps for each of the cases i Figure 8. Chages i the wake patter ca be observed for both iflow ad ear wake disturbace. A larger icrease i period ca be observed for iflow disturbace with a smaller icrease for ear wake disturbace. a) b) c) Fig. 8 Streamlies for (a) udisturbed wake flow ad disturbace of wake flow. (b) Iflow ad (c) ear wake. 33

38 a) b) Fig. 9 Vorticity cotours (a) udisturbed wake flow ad disturbace of wake flow. (b) Iflow ad (c) ear wake. c) 34

39 Figure 10 illustrates the vertical velocity compoets at the same poit i the wake. It further illustrates the chages i ear wake velocities due to flow disturbaces caused by the smaller bluff bodies relative to o flow obstructios. Oe ca observe chages i cross-flow forces ad sheddig frequecies relative to the sigle bluff body case (St=0.14). Iflow disturbaces cause lower sheddig frequecy (St=0.08) i agreemet with the larger period i figure 9b. The magitude of velocities is slightly smaller which results i a smaller reductio i cross-flow forces. Wake disturbaces cause smaller chages i sheddig frequecy (St=0.13) which agree with the smaller chage i period i figure 9c. The magitude of velocities is also smaller which results i a larger reductio i cross flow forces, as expected. 0.1 Uy t x 10 4 Fig. 10 Near wake velocity time histories -, fig. 9a; -.-, fig. 9b; --, fig.9c 35

40 CHAPTER 5 CONCLUSIONS I this thesis, a 2-D Lattice Boltzma (LB) solver was developed ad results are validated for flow past rectagular square cyliders at low Reyolds umbers (< 250). The solver was the applied to multiple bluff body cases ad results preseted. The LB method is a alterative computatioal fluid dyamics (CFD) method used to umerically predict icompressible viscous flow. The curret LB method uses a statistical mechaics formulatio to solve the Boltzma equatio. The preseted LB simulatio bee show to capture some key features i vortex dyamics. Results were foud to agree well for flow aroud a sigle square cylider for Re 250. The simulatios captures the qualitative effects of icreasig aspect ratio such as a decrease i both C d ad St which is to be expected, however some predicted values such as lift ad drag coefficiets were slightly higher tha other ivestigators. This over predictio is believed to be cotributed low grid resolutio i the boudary layer as a result of the uiform grid. The model was the applied to multiple bluff bodies to study the flow effects of the sigle bluff body. A reductio i cross-flow forces whether the smaller bluff bodies are located i frot or i the ear wake was observed. The chages i wake patters ad characteristics due disturbaces were cosistet ad agreed with expected theories. The vortex sheddig simulatios preseted provided reasoable results o coarse uiform grids compared to o-uiform FD methods. 36