Lattice Boltzmann Method in Theory and in Application to Coupled Problems

Transcription

1 Lattce Boltzmann Method n Theory and n Applcaton to Coupled Problems Master Thess Danel Heubes supervsng Prof. Dr. Mchael Günther Dr. Andreas Bartel Unversty of Wuppertal Faculty of Mathematcs and Natural Scences Char of Appled Mathematcs / Numercal Analyss Prof. Dr. Mchael Günther July 2, 2010

2 Declaraton of Authorshp I declare that I have authored ths thess ndependently, that I have not used other than the declared sources, and that I have explctly marked all materal whch has been quoted ether lterally or by content from the used sources. Wuppertal, July 2, 2010 Danel Heubes)

3 Contents Introducton and Overvew 1 1 Lattce Boltzmann Method for Flud Flows Hstorcal Background and Bascs Cellular Automata Lattce Gas Automata The Boltzmann Equaton Contnuty Equaton Momentum Equaton BGK Approxmaton Dscretzng Boltzmann Equaton Lattce BGK Equaton Influence of Dscretzaton Dscretzaton Models Models of Dmenson Models of Dmenson Models of Dmenson Dscrete Equlbrum Dstrbuton n D3Q Chapman-Enskog Expanson Terms of Frst and Second Order Computaton of P 1) ˆx, ˆt) Results up to Second Order Incompressble Naver Stokes Equaton Summary and Algorthm Enhancements of the Lattce Boltzmann Method Boundary Condtons Perodc Boundary Condtons Bounceback Boundary Condtons Velocty Boundary Condtons Pressure Boundary Condtons Addtonal Force Terms I

4 Contents II 2.3 Incorporaton of a Thermal Component Determnaton of Macroscopc Equaton Numercal Scheme Buoyancy Effects Adjustment of the Macroscopc Equaton Realzaton n LBM Numercal Tests for Enhanced Lattce Boltzmann Channel Flow Poseulle Flow Flow Past an Obstacle Temperature Evoluton - Raylegh-Bénard Convecton Outlook 94 A Outsourced Calculatons 95 A.1 Dot Product of a Matrx Valued Functon A.2 Computaton of a Tme Dervatve A.3 Integraton of PDE for Energy Dstrbuton A.4 Computaton of Π 2) ˆx, ˆt) References 104

5 Lst of Fgures 1.1 Evoluton of a one-dmensonal, two-state, two-neghbor cellular automata. Intal states top lne) are chosen randomly wth 50% probablty to have state 1 black square) and 50% probablty to have state 0 whte square). Subsequent tme levels are shown wth progressng lnes from top to bottom Trangular lattce used n the FHP model Lattce vectors of the FHP model Connecton of cells tny empty crcles) and lattce vectors n the case of FHP model Collson rules used n the FHP model. Empty crcles ndcate a state of 0, whereas sold crcles are occuped cells of state 1. Left column contans two partcle collsons, rght column contans three partcle collsons Lattce veloctes n one dmensonal space Lattce veloctes for a trangular a) and square b) lattce n two dmensons Lattce veloctes for cubc lattce n three dmensons Comparson of nteror and boundary lattce ponts after streamng step. Crcles refer to nteror lattce ponts and squares to boundary lattce ponts. Dashed arrows ndcate unknown values, whereas sold arrows refer to known ones Perodc boundary condtons n D2Q9. Sold arrows depct the prestreamng state for an easter boundary lattce pont, whereas the dashed arrows show the post-streamng state Effectve physcal boundary when bounceback boundary condtons are appled to D2Q9. Empty squares refer to bounceback lattce ponts Prncple of bounceback boundary condtons. Dotted arrows ndcate that values of densty dstrbuton were recalculated n the collson step Illustraton of a boundary lattce pont n D3Q19 after streamng step. Dashed arrows ndcate unknown values, whereas sold arrows are referred to known ones Two- and three-dmensonal channels. Arrows show the drecton of flow and dashed lnes llustrate the open sdes of the channel III

6 Lst of Fgures IV 3.2 Velocty profle n a two-dmensonal Poseulle flow smulaton wth pressure boundary condtons Velocty profle n a two-dmensonal Poseulle flow smulaton wth perodc boundary condtons and a force actng on the flud Densty feld n a two-dmensonal Poseulle flow wth pressure boundary condtons Velocty profle n a three-dmensonal Poseulle flow smulaton wth pressure boundary condtons Velocty profle n a three-dmensonal Poseulle flow smulaton wth pressure boundary condtons. Vew from top of the graph n fgure Contour plot of velocty profle n a three-dmensonal Poseulle flow smulaton wth pressure boundary condtons Cut-out of the velocty profle gven n precedng fgures along the tenth row whch corresponds to the lne wth x 3 = Contour plot of the velocty profle of a three-dmensonal Poseulle flow n a channel wth oblong cross secton Obstacle n a channel Streamlnes of a pressure drven flow past an obstacle. Addtonal streamlnes start behnd the obstacle for better llustraton Modulus of velocty for a pressure drven flow past an obstacle Modulus of velocty for a pressure drven flow past an obstacle n three dmensons. Only two slces are plotted, on whch streamlnes are prnted Convecton cells dsplayed at an mage secton of a two-dmensonal Raylegh- Bénard experment Temperature profle between two boundares mantaned at two dfferent temperatures wthout consderng the nfluence of gravty. Boundary at the bottom s colder than the one at the top Vector feld llustratng the flud s movement n a smulaton of twodmensonal Raylegh-Bénard convecton Resultng temperature profle n a smulaton of two-dmensonal Raylegh- Bénard convecton. Lnes are sotherms connectng ponts that have same temperature Smulaton of three-dmensonal Raylegh-Bénard convecton wth a lattce Boltzmann method

7 Lst of Tables 1.1 Weghts for equlbrum dstrbuton functon for several models Result of the product c α cβ cj for D3Q19 model. Lattce veloctes are gven n 1.48) and empty spaces ndcate a result of zero V

8 Introducton and Overvew Flows are n general descrbed by the Naver-Stokes equatons. When smulatng real physcal processes where flud flows play a major role, one s especally nterested n solvng the Naver-Stokes equatons. Unfortunately, analytcal solutons of the Naver- Stokes equatons are n general not present. Fndng an approxmaton of the soluton numercally s a common procedure n the cases when no analytcal solutons are present. Several possbltes to do ths are known, conventonal methods n computatonal flud dynamcs solve the Naver-Stokes equatons drectly. In recent years, an alternatve approach for fndng approxmatng solutons of the Naver-Stokes equatons appeared. Ths approach s based on solvng the dscrete Boltzmann equaton and s known as the lattce Boltzmann method. The Boltzmann equaton descrbes phenomena of the flud mcroscopcally based on partcle denstes and collsons. Usng the lattce Boltzmann method to smulate flud flows means to not solve the Naver-Stokes equatons drectly, nstead one uses the aspect that from the Boltzmann equaton the Naver-Stokes equatons can be recovered n a hydrodynamc lmt. Snce ths method was mentoned for the frst tme, the lattce Boltzmann method becomes ncreasngly popular especally as an approach for smulatng flows through complex meda. Apart from the Naver-Stokes equatons, the evoluton of temperature n a flud s descrbed by an addtonal equaton. The form of ths equaton depends on the effects that are consdered. In a very basc form, the advecton-dffuson equaton s suffcent to descrbe the temperature evoluton. Ths equaton explctly depends on the flud s velocty whose evoluton s descrbed by the Naver-Stokes equatons. In ths way, the advecton-dffuson equaton s coupled to the Naver-Stokes equatons. A couplng of dfferent manner appears f the flud flow s also nfluenced by the temperature evoluton, and not only the other way round. If one s nterested n numercally smulatng physcal processes descrbed by the full set of coupled equatons, a method s necessary whch consders these couplngs. Hence, one cannot solve each equaton separately. Ths work s concerned wth the topcs descrbed above. We organze t as follows, n chapter 1 we gve a short hstorcal background of lattce Boltzmann methods and derve the lattce Boltzmann method for flud flows n detal. Afterwards, we show n a mult-scale expanson, that the numercal method approxmates the ncompressble Naver-Stokes equatons. Hence, we verfy that the lattce Boltzmann model can successfully be used to smulate flud flows. The complete frst chapter can bee seen as an detaled theoretcal look on the man concept of the lattce 1

9 Introducton and Overvew 2 Boltzmann method. Nevertheless, the man concept alone s nsuffcent to apply the numercal method to real physcal problems. In chapter 2 we enhance the method determned n the foregong chapter. We present dfferent types of boundary condtons. Wthout boundary condtons an mplementaton of the lattce Boltzmann method, and thus the applcaton to physcal problems, s mpossble. Moreover, we present a lttle modfcaton of the man concept whch allows us to smulate flud flows under the nfluence of an actng force. Ths force needs not necessarly to be homogeneous. The focus of chapter 2 however les on solvng the advecton-dffuson equaton for the temperature coupled to the Naver-Stokes equatons. Therefore, we deduce an energy conservaton equaton drectly from the Boltzmann equaton and transform t to the advecton-dffuson equaton. We propose a numercal method to approxmate the soluton of the advecton-dffuson equaton. The proposed method s closely related to the lattce Boltzmann method for flud flows. Agan n a mult-scale expanson we verfy that method. We fnsh ths chapter by descrbng buoyancy effects macroscopcally and showng how these effects can be realzed wth the prevously determned methods. In the last chapter of the man part, chapter 3, we present some numercal tests to emphasze the capablty of the presented methods to smulate problems descrbed by the Naver-Stokes equatons n conjuncton wth the advecton-dffuson equaton. We smulate a two- and a three-dmensonal Poseulle flow and compare our smulaton wth the analytcal soluton of t. Furthermore, we smulate a flow past a rectangular and a cubod obstacle, respectvely. These smulatons show the capablty of the lattce Boltzmann method for flud flows. Afterwards, we take also the second method nto account, thus we can approxmately solve problems satsfyng the Naver-Stokes equatons and the advecton-dffuson equaton. We test the coupled method by smulatng Raylegh- Bénard convecton. We perform the smulaton of Raylegh-Bénard convecton n two as well as three dmenson. The smulaton of Raylegh-Bénard convecton consders especally buoyancy effects. Approxmatng solutons of the heat equaton s also possble wth our model and we gve one example. Fnally, subsequent to the last chapter of the man part we gve an outlook. There we talk about some nce features the determned methods can be used for and we menton some further enhancements whch, n our opnon, are very nterestng.

10 CHAPTER 1 Lattce Boltzmann Method for Flud Flows In ths chapter we develop a numercal method capable of smulatng the moton of a flud flow. Ths method s known as the lattce Boltzmann method LBM) and satsfes the ncompressble Naver-Stokes equatons. The LBM s a relatvely new numercal scheme. It has recently acheved consderable success n smulatng flud flows n porous meda [23] and assocated transport phenomena [10]. Ths method has been shown to be partcularly useful n applcaton nvolvng nterfacal dynamcs. Whle tradtonal methods n Computatonal Flud Dynamcs CFD) solve the Naver-Stokes equatons drectly [19, 36] and related [13, 52]), the lattce Boltzmann method solves the Boltzmann equaton from whch the Naver-Stokes equatons can be recovered. The Boltzmann equaton descrbes phenomena of the flud mcroscopcally based on partcle denstes and collsons. All calculatons wthn ths chapter concernng the Boltzmann equaton and the lattce Boltzmann method are performed for the three dmensonal case and are kept general as long as easly possble. At some ponts the calculatons are done for a specfc three dmensonal model D3Q19), but they can also be transferred for most models analogue. In the subsequent secton we brefly descrbe cellular automata CA) and we ntroduce lattce gas automata LGA) whch are closely related to CA and can be seen as the forebears of the lattce Boltzmann method [42]. Afterwards, we present the Boltzmann equaton and show that the Naver-Stokes equatons are recovered. Then we develop the numercal scheme, the lattce Boltzmann method, based on the Boltzmann equaton where we approxmate the collson term by a sngle tme relaxaton term. The lattce Boltzmann method, or more precsely the lattce Bhatnagar-Gross-Krook LBGK) method due to the approxmaton of the collson term, s acheved by dscretzng the Boltzmann equaton approprately. Fnally, we show by a mult-scale expanson that the numercal scheme satsfes the ncompressble Naver-Stokes equatons Hstorcal Background and Bascs For a better understandng of the lattce Boltzmann method, we prmarly begn ths chapter wth a short ntroducton to cellular automata and lattce gas gas automata. 3

11 1.1. Hstorcal Background and Bascs 4 Hstorcally, the lattce Boltzmann method s based on lattce gas automata. Probably, the latter can be better understood wth at least lttle knowledge of cellular automata Cellular Automata A cellular automaton s a dscrete model whch was ntroduced by John von Neumann [66]. A formal descrpton of a D-dmensonal cellular automata can be made by ntroducng a set C Z D descrbng the poston of the cells C C, = 1,..., m c [34]. The cells are arranged regularly and each cell holds a state. The state of cell C s denoted by s S, wth the fnte set of possble states S Z, whch does not depend on the cell tself. At dscrete tme levels the states are updated smultaneously by prescrbed determnstc rules. These local rules are gven by a mappng r : S mn S, where m n gves the number of elements n the neghborhood N. For nteror cells the elements of the neghborhood N = {N 1,..., N mn } Z D have to be nterpreted as relatve coordnates of neghborng cells, ths means that C + N j C for all Z and j = 1,..., m n. Boundary cells are updated due to some approprate update rules. We assume that 0,..., 0) N,.e. the updated state of cell C depends at least on the current state of cell C. As an llustratng example, we consder a one-dmensonal cellular automaton consstng of 400 cells wth only two possble states at each cell. We assume the two possble states are S = {0, 1} and the updatng rule depends only on the two nearest adjacent cells and the cell tself,.e. N = { 1, 0, 1}. Thus, the cellular automaton s fully descrbed by the updatng rules, more precsely by the values r S, = 0,..., 7 on the rght hand sdes of 1.1). The state of each cell s updated by the followng rule s = rs 1, s, s +1 ), = 2,..., 399, where s s the updated state and s 1, s +1 are the states of the left and rght adjacent cell, respectvely. States of cell C 1 and C 400 are set to 0 and are not updated. Eght combnatons of states determne the rule r0, 0, 0) = r 0, r0, 0, 1) = r 1, r0, 1, 0) = r 2, r0, 1, 1) = r 3, r1, 0, 0) = r 4, r1, 0, 1) = r 5, r1, 1, 0) = r 6, r1, 1, 1) = r ) Snce we allow two states there exst 2 8 = 256 dfferent updatng rules. Fgure 1.1 llustrates the evoluton of the cellular automaton wth specfed values r 0,2,5,7 = 0 and r 1,3,4,6 = 1, also known as rule 90 after the notaton of Wolfram [70].

12 1.1. Hstorcal Background and Bascs 5 Fgure 1.1.: Evoluton of a one-dmensonal, two-state, two-neghbor cellular automata. Intal states top lne) are chosen randomly wth 50% probablty to have state 1 black square) and 50% probablty to have state 0 whte square). Subsequent tme levels are shown wth progressng lnes from top to bottom Lattce Gas Automata Lattce gas automata are derved from classcal cellular automata after some modfcatons [69]. These modfcatons smplfy the constructon and applcaton of automata to gven physcal processes. For nstance, unlke the frst lattce gas automaton, ntroduced by Hardy, de Pazzs and Pomeau HPP model) [25], today there exst lattce gas automata whch lead to the Naver-Stokes equatons n the macroscopc lmt [71]. Hence, they are capable to smulate flud flows successfully. Despte of ther relatvely smple nature, lattce gas automata can be appled n less smple themes such as for nstance n the smulaton of flows through porous meda [9]. As the basc prncple of the lattce Boltzmann method and lattce gas automata s the same, we dscuss ths prncple here n some detal. The frst ntroduced lattce gas automaton the HPP model) was proposed as a new however not workng technque for the numercal study of the Naver-Stokes equatons. Instead of a drect ntegraton of partal dfferental equatons PDEs) LGA are based on the smulaton of a very smple mcroscopc system. Here, partcles are allowed to move on a regular lattce, and local collson rules are ntroduced on the nodes whch

13 1.1. Hstorcal Background and Bascs 6 conserve the number of partcles and momentum. The followng descrpton wll be done generally [69] but wth a supportng accompanyng example. The chosen example uses a trangular lattce see fgure 1.2) and was ntroduced by Frsch, Hasslacher and Pomeau FHP model) n 1986 [20]. The FHP model was the frst LGA whch successfully yelds the Naver-Stokes equatons n the macroscopc lmt. The major dfference between the HPP and FHP model s the usage of a trangular lattce n the latter compared to a square lattce n the former. Ths leads to an specfc sotropc tensor of rank four n FHP but not n HPP whch s necessary to acheve the Naver-Stokes equatons, see [69] for detals. The frst step of a formal descrpton of LGA les n defnng a regular lattce n space. Dependng on the lattce a number of k lattce vectors d, = 1,..., k, are ntroduced whch connect nearest neghbors. For the FHP model they read d = cos π 3 ), sn π 3 )), = 1,..., 6, and are shown n fgure 1.3. Furthermore, all partcles shall have the same mass m. Then, each lattce node conssts of a set of k cells wth bnary state. The state of each cell s gven by { 0, cell s not occuped by a partcle n x, t) =, = 1,..., k. 1, cell s occuped by a partcle Here, t and x ndcate tme and the poston of the correspondng node, respectvely. Each cell may contan at most one partcle. We combne all k state functons of the Fgure 1.2.: Trangular lattce used n the FHP model.

14 1.1. Hstorcal Background and Bascs 7 Fgure 1.3.: Lattce vectors of the FHP model. same node n n 1 x, t) n nx, t) = 2 x, t).. n k x, t) Lke cellular automata, LGA are dscrete tme models, the states of nodes or cells are updated at the dscrete tme levels smultaneously, where the updatng process conssts of two parts. One part s referred to as collson and the other as streamng. Connectng each cell of a node to a correspondng lattce vector see fgure 1.4) helps determnng collson rules as well as understandng both updatng parts. In the streamng part partcles are transfered from node to node, more precsely the state of cell at node wth poston x s transfered to the state of cell at node wth Fgure 1.4.: Connecton of cells tny empty crcles) and lattce vectors n the case of FHP model.

15 1.1. Hstorcal Background and Bascs 8 poston x + d,.e. n x + d, t + t) = n x, t), = 1,..., k. 1.2) The tme step t s set to 1, hence lattce vectors can also be nterpreted as lattce veloctes snce d and d t have the same numercal value. The other part, collson, s best descrbed by local rules whch only depend on and alternate the state of the node,.e. the state of cells at that node. The collson rules should be chosen such that they conserve the number of partcles and momentum. Snce ther choce s nfluenced by the appled lattce, we cannot gve the rules n general, hence we only gve the rules of the orgnal FHP model, see fgure 1.5. In the frst ntroduced FHP model only two and three partcle collsons are consdered. For the two partcle head on collson there are two possble outcomes whch conserve the number of partcles and momentum and one has to take a choce randomly. Confguratons not lsted n fgure 1.5 are not affected due to collson. An mplementaton can be realzed by frst changng the states of cells at each node based on the collson rules, and afterwards applyng the streamng process. The followng equaton contans both parts of ths procedure, thus the complete evoluton of LGA s gven by n x + d, t + t) n x, t) = Ψ nx, t)), = 1,..., k. 1.3) The left hand sde descrbes the streamng part, compare 1.2), and the rght hand sde s a collson functon. When expressng the states after applyng the collson rules by ñx, t) we can defne the collson functon by Ψ nx, t)) = Ψ 1 nx, t)). Ψ k nx, t)) where nx, t) denotes the states before collson. = ñx, t) nx, t),

16 1.2. The Boltzmann Equaton 9 Fgure 1.5.: Collson rules used n the FHP model. Empty crcles ndcate a state of 0, whereas sold crcles are occuped cells of state 1. Left column contans two partcle collsons, rght column contans three partcle collsons The Boltzmann Equaton The classcal Boltzmann equaton reads [61] fx, v, t) + v fx, v, t) = Qf, f), 1.4) defned for v, t > 0 and to avod the treatment of boundares we nvestgate the equaton for x n the current chapter. The symbol expresses the partal dfferental operators wth respect to space x. The Boltzmann equaton s an evoluton equaton n tme t for the sngle partcle dstrbuton fx, v, t). The value fx, v, t) dx dv for all nfntesmal small dx and dv represents the number of partcles whch at tme t have poston x and velocty v, multpled by the constant partcle mass m. In general the rght hand sde of 1.4) represents the collson term whch s here expressed as a collson ntegral [69] Qf, f) = [ σω) v w fx, v, t)fx, w, t) fx, v, t)fx, w, t) ] dω dw. 1.5) S 2

17 1.2. The Boltzmann Equaton 10 Ths collson ntegral s based on several assumptons [54]. One condton s that the partcles nteract only n two-partcle collsons, ths means we assume that nteractons nvolvng more then two partcles can be neglected. For all two-partcle collsons we assume that they appear locally n the sense that they take place at a sngle pont x. A smlar condton holds for tme t, t s assumed that the duraton of a collson s neglgble. Partcles nvolved n a collson are assumed to be uncorrelated, and the collson tself s modeled as an elastc collson, meanng that knetc energy and especally momentum are conserved. The veloctes before collson are denoted by v and w, and the veloctes after collson by v and w. Snce the collson s elastc the followng two equatons have to be fulflled note that partcles have same mass m): v + w = v + w and v 2 + w 2 = v 2 + w 2. The post-collson veloctes v and w can be computed n dependence of the pre-collson veloctes v, w and the mpact angle. In 1.5) σω) denotes the dfferental collson cross secton and the nner ntegraton s done over all possble sold angles Ω. We defne macroscopc quanttes over ntegrals of the partcle dstrbuton fx, v, t). In detal, the mass densty ρx, t) s gven by ρx, t) = and the macroscopc velocty ux, t) s defned as ρx, t)ux, t) = fx, v, t) dv 1.6) vfx, v, t) dv. 1.7) Wth these defntons we are able to deduce the contnuty equaton and the conservaton equaton for momentum from the Boltzmann equaton. Furthermore, we defne the temperature T x, t) by ρx, t) 3 k B T x, t) = 2 m v ux, t) 2 fx, v, t) dv, 1.8) 2 wth Boltzmann constant k B and partcle mass m. Ths equaton wll be used later but not n the dervaton of the contnuty equaton and momentum equaton Contnuty Equaton We derve the contnuty equaton by ntegratng the Boltzmann equaton over velocty space. The left hand sde of the Boltzmann equaton s denoted as Bx, v, t) n the

18 1.2. The Boltzmann Equaton 11 followng. Bx, v, t) dv := fx, v, t) dv + v fx, v, t) dv = fx, v, t) dv + vfx, v, t) dv = ρx, t) + ρx, t)ux, t)). For the rght hand sde t follows [5] Qf, f) dv = 0, 1.9) whch leads by consderng both sdes to the contnuty equaton Momentum Equaton ρx, t) + ρx, t)ux, t)) = 0. In order to deduce the momentum equaton we ntroduce a general form of the dotproduct. Notaton. For a matrx valued functon a 1,1 x) a 1,2 x)... a 1,n x) a 1 x) T a Ax) = 2,1 x) a 2,2 x)... a 2,n x)... = a 2 x) T. a m,1 x) a m,2 x)... a m,n x) a m x) T and the nabla operator = x 1 x 2. x n 1.10)

19 1.2. The Boltzmann Equaton 12 we compute the dot-product of the nabla operator and a matrx valued functon as follows: a 1 x) Ax) = a 2 x).. a m x) Wth help of ths notaton, we derve the momentum equaton by frst multplyng the Boltzmann equaton wth v and then ntegratng over velocty space. The arsng ntegrals read [5] vqf, f) dv = ) and vbx, v, t) dv = vfx, v, t) dv + v v fx, v, t)) dv. 1.12) The frst ntegral on the rght hand sde of the latter equaton s easy to compute vfx, v, t) dv = vfx, v, t) dv = ρx, t)ux, t)), 1.13) where one has just to use the defnton of the macroscopc velocty u. For the second ntegral on the rght hand sde of 1.12) we use the followng calculaton [18]: v [v fx, v, t)] dv = = = vv T fx, v, t) dv v ux, t) + ux, t)) v ux, t) + ux, t)) T fx, v, t) dv v ux, t)) v ux, t)) T fx, v, t) dv 1.14) + ux, t)ux, t) T fx, v, t) dv R 3 + v ux, t)) ux, t) T fx, v, t) dv R 3 + ux, t) v ux, t)) T fx, v, t) dv,

20 1.2. The Boltzmann Equaton 13 where the frst step can be seen by a calculaton n appendx A.1. The mxed ntegrals n 1.14) vansh due to v ux, t)) ux, t) T fx, v, t) dv = vux, t) T fx, v, t) dv ux, t)ux, t) T fx, v, t) dv R 3 T = vfx, v, t) dv ux, t) ux, t)ux, t) T fx, v, t) dv and an analogue computaton for = ρx, t)ux, t)ux, t) T ux, t)ux, t) T ρx, t) = 0 ux, t) v ux, t)) T fx, v, t) dv. Thus, the equaton 1.14) smplfes to v [v fx, v, t)] dv = v ux, t)) v ux, t)) T fx, v, t) dv + ux, t)ux, t) T fx, v, t) dv = v ux, t)) v ux, t)) T fx, v, t) dv + ρx, t)ux, t)ux, t) T ) and combnng the latter equaton wth 1.11), 1.13) yelds ρx, t)ux, t)) + ρx, t)ux, t)ux, t) T ) = v ux, t)) v ux, t)) T fx, v, t) dv. The ntegral represents the momentum flux tensor, whose entry at poston, j) satsfes [30] v u x, t) ) v j u j x, t) ) fx, v, t) dv = px, t)δ j σ,j x, t),

21 1.3. BGK Approxmaton 14 wth px, t) denotng the pressure. On the rght hand sde, px, t)δ j s the porton of the equlbrum and therefore σ,j x, t) s referred to a correcton term whch expresses the dynamc porton n the momentum flux tensor. It holds for the correcton tensor [ ] σx, t) = ρx, t)ν ux, t) + ux, t)) T, wth knematc vscosty ν. In vector notaton, the momentum flux tensor exhbts the form v ux, t)) v ux, t)) T fx, v, t) dv = px, t)i ρx, t)ν [ ux, t) + ux, t)) T ], wth dentty matrx I. Eventually, t follows the momentum conservaton equaton ρx, t)ux, t)) + ρx, t)ux, t)ux, t) T ) = px, t) + [ ]) ρx, t)ν ux, t) + ux, t)) T. 1.15) 1.3. BGK Approxmaton The collson ntegral 1.5) possesses a rather complcated form. To acheve an effcent numercal scheme ntegratng the Boltzmann equaton 1.4) a less dffcult form of the collson term would gratefully be taken. In the followng, we motvate the approxmaton ntroduced by Bhatnagar, Gross and Krook [3] whch today s frequently used when dealng wth lattce Boltzmann methods, these schemes are often called LBGK-methods. In these schemes the complcated collson ntegral s replaced by a sngle tme relaxaton term. Ths term s chosen n such a way that t emulates specfc propertes of the orgnal collson ntegral. In the prevous secton we have already used cf. 1.9) and 1.11)) some propertes of the collson ntegral. We have seen that those propertes were necessary to derve the conservaton laws. It can be shown [5], that there exst fve elementary nvarants whch read ψ 1 = 1, ψ 2, ψ 3, ψ 4 ) = v and ψ 5 = v ) and lead to vanshng ntegrals Qf, f) dv = 0, vqf, f) dv = 0 and v 2 Qf, f) dv = 0.

22 1.3. BGK Approxmaton 15 It even holds the followng equvalence Qf, f)ϕv) dv = 0 ϕv) = 5 s ψ = α + β v + γ v 2, 1.17) =1 wth real scalars α, γ, s R = 1,..., 5) and a real vector β whch can be chosen arbtrary for the backward drecton of the equvalence. Furthermore, there exst postve functons fx, v, t) whch lead to vanshng collson ntegrals and t holds the mplcaton Qf, f) = 0 fx, v, t) = exp a + δ v + b v 2), wth a R, 0 > b R and δ. For our further nvestgaton, an mportant functon satsfyng the latter feature s the Maxwell dstrbuton ) f M m 3/2 ) m x, v, t) := ρx, t) exp v ux, t) 2, 1.18) 2πk B T x, t) 2k B T x, t) wth Boltzmann constant k B, partcle mass m and temperature T x, t) [61]. Therefore, especally t holds [5] Qf M, f M ) = ) The Maxwell dstrbuton s also relevant n the famous H-Theorem whch we wll state subsequent to the followng lemma. Ths lemma s ntended as a preparaton for the proof of the H-Theorem. Lemma 1. Let ϕ be an arbtrary functon of the velocty v. If ϕ satsfes the condton ϕv) + ϕw) ϕv ) ϕw ) = 0, for all pre-collson veloctes v and w, and post-collson veloctes v and w, then t holds wth α R, β and γ R. ϕv) = α + β v + γ v 2, Proof. We only have to show that ϕ satsfes the followng equaton Qf, f)ϕv) dv = 0, because then 1.17) mples the clamed shape for the functon ϕ.

23 1.3. BGK Approxmaton 16 For smplcty we use the shortcuts f v = fx, v, t), f w = fx, w, t), f v = fx, v, t), f w = fx, w, t). In the followng computaton we use that the collson ntegral s nvarant under exchanges of veloctes. An analogue procedure s descrbed more detaled also n the proof of the subsequent theorem below. Qf, f)ϕv) dv = σω) v w f v f w f v f w ) ϕv) dω dw dv S 2 = 1 σω) v w f v f w f v f w ) ϕv) + ϕw)) dω dw dv 2 = 1 4 S 2 σω) v w f v f w f v f w ) S 2 ϕv) + ϕw) ϕv ) ϕw ) ) dω dw dv. By usng the assumed condton t follows Qf, f)ϕv) dv = 0. Theorem 1 H-Theorem). For any postve functon fx, v, t) wth fnte zeroth to second moments satsfyng the Boltzmann equaton 1.4) the functon s Hx, t) := a) bounded from below for all x and t, b) non-ncreasng n tme,.e. fx, v, t) ln fx, v, t)) dv Hx, t) 0, for all x, and moreover t holds that the equalty s only attaned f fx, v, t) s a Maxwellan dstrbuton 1.18) ndependent of x. The functon Hx, t) s related to the local entropy by Sx, t) = k B Hx, t), wth the Boltzmann constant k B. The H-theorem states that the entropy s maxmzed by Maxwellan dstrbutons whch are the equlbrum dstrbutons n the contnuous case.

24 1.3. BGK Approxmaton 17 Outlne of a proof. For part a) a hnt from [6] was used. Another argumentaton for the boundedness can be found n [15]. Part b) s close to the proof n [69]. a) As one can easly show the functon gs) = lns)+s 1) s mnmzed for s = 1. Snce t holds g1) = 0 we get the general nequalty lns) + s 1) 0, vald for postve s R +. By substtutng s = y z t follows z lnz) z lny) + y z 0, 1.20) whch s vald for all postve real numbers y, z R +. For fxed x and t R the functon fx, v, t) mples a densty, velocty and temperature by 1.6), 1.7) and 1.8), respectvely. Hence, for all x and t R we can defne a Maxwellan dstrbuton Mv) by ) m 3/2 Mv) = ρ exp m ) v u 2, 2πk B T 2k B T where ρ, u and T are gven by fx, v, t). From nequalty 1.20) we derve the ntegral nequalty fx, v, t) ln fx, v, t)) dv fx, v, t) ln Mv)) dv + fx, v, t) dv Mv) dv. 1.21) The ntegrals n the second lne of 1.21) cancel out, snce they are both equal to ρ, due to constructon of Mv). The frst ntegral on the rght hand sde of 1.21) can be smplfed as follows fx, v, t) ln Mv)) dv = ln [ ρ m ) ] 3/2 2πk B T m k B T fx, v, t) dv v u 2 fx, v, t) dv. 2 Especally, t follows that the rght hand sde n the latter equaton s bounded from below, hence we have shown the proposton Hx, t) = fx, v, t) ln fx, v, t)) dv >. b) Let x be arbtrary but fxed. Dfferentatng the ntroduced functon Hx, t)

25 1.3. BGK Approxmaton 18 yelds Hx, t) = fx, v, t) [1 + ln fx, v, t))] dv. Snce fx, v, t) s satsfyng the Boltzmann equaton 1.4) the latter equaton can be wrtten as Hx, t) = Qf, f) [1 + ln fx, v, t))] dv v fx, v, t)) [1 + ln fx, v, t))] dv and wthout dscusson we assume that the second term on the rght hand sde vanshes [69]. For smplcty we use agan the shortcuts f v = fx, v, t), f w = fx, w, t), f v = fx, v, t), f w = fx, w, t), and thus we can wrte now Hx, t) = σω) v w f v f w f v f w ) [1 + ln f v )] dω dw dv. 1.22) S 2 Ths ntegral s nvarant under exchange of v and w because σω) s nvarant under such exchange: Hx, t) = S 2 σω) w v f v f w f w f v ) [1 + ln f w )] dω dw dv. 1.23) Addng 1.22) and 1.23) by takng half and half each we get Hx, t) = 1 2 S 2 σω) v w f v f w f v f w ) [2 + ln f v f w )] dω dw dv, 1.24) whch s now nvarant under exchange of {v, w} and {v, w } because for each collson there exsts an nverse collson wth the same cross secton. Hence, we obtan Hx, t) = 1 2 S 2 σ Ω) v w f v f w f v f w ) [2 + ln f v f w )] dω dw dv,

26 1.3. BGK Approxmaton 19 whch s equvalent to Hx, t) = 1 2 S 2 σω) v w f v f w f v f w ) [2 + ln f v f w )] dω dw dv, 1.25) snce dv dw = dv dw, v w = v w and σω) = σ Ω). Now addng 1.24) and 1.25) agan by takng half and half each we get Hx, t) = 1 4 S 2 From the nequalty σω) v w f v f w f v f w ) [lnf v f w ) lnf v f w )] dω dw dv. b a)lna) lnb)) < 0, a > 0, b > 0, a b, t follows that the ntegrand s never postve and we can conclude 1.26) Hx, t) 0, where the equalty s attaned f and only f f v f w f v f w = ) for all v, w that result from v, w by collson. Moreover, equaton 1.27) mples fx, v, t) = 0, for all v whch can be seen by Hx, t) = = fx, v, t) [1 + ln f v )] dv σω) v w f v f w f v f w ) dω dw [1 + ln f v )] dv. S 2 Note that 1.27) also mples lnf v ) + lnf w ) lnf v ) lnf w ) = 0, and from Lemma 1 one can then conclude that f the ntegral 1.26) vanshes, then

27 1.3. BGK Approxmaton 20 lnf v ) s an ntegraton nvarant and therefore of the form 1.17), hence lnf v ) = lnfx, v, t)) = αx, t) + βx, t) v + γx, t) v 2. Ths can be transformed to fx, v, t) = exp αx, t) + βx, t) v + γx, t) v 2) and wth the substtuton γx, t) 1 2 ˆγx, t) we can wrte t as fx, v, t) = δx, t) exp 1 2 ˆγx, t) βx, t) 2) v, ˆγx, t) ) wth δx, t) = exp αx, t) + βx,t) 2 2ˆγx,t). The unknowns δx, t), ˆγx, t) and βx, t) can be determned wth use of the macroscopc quanttes. Snce fx, v, t) satsfes the Boltzmann equaton the followng ntegral equatons have to be fulflled: δx, t) exp 1 2 ˆγx, t) βx, t) 2) v dv = ρx, t), ˆγx, t) vδx, t) exp 1 2 ˆγx, t) βx, t) 2) v dv = ρx, t)ux, t), ˆγx, t) v ux, t) 2 δx, t) exp ˆγx, t) βx, t) 2) v dv = ρx, t) 3 ˆγx, t) 2 It follows δx, t) = ρx, t)2π) 3/2 ˆγx, t)) 3/2, m ˆγx, t) = k B T x, t), βx, t) = ˆγx, t)ux, t), whch leads to a Maxwellan dstrbuton k B T x, t) m. ) m 3/2 ) m fx, v, t) = ρx, t) exp v ux, t) 2. 2πk B T x, t) 2k B T x, t) A further restrcton for the unknowns comes from the above mplcaton that the

28 1.3. BGK Approxmaton 21 dervatve wth respect to tme should equal zero. For all v : [ fx, v, t) = v δx, t) exp 1 2 ˆγx, t) βx, t) 2)] v = 0, ˆγx, t) note the valdty of 1.19). Ths restrcton leads to the condton that the functon fx, v, t) may not depend on x, hence also the unknowns are ndependent of x. One can even further argue that n a closed system all unknowns also have to be ndependent of tme t, snce the macroscopc quanttes are conserved. Ths shows vanshes only f the functon fx, v, t) s a Maxwellan 1.18) wth each densty, velocty and temperature spatally constant. that Hx,t) Recaptulatng the already known propertes of the collson ntegral state fve ntegraton nvarants cf. 1.17)) and a vanshng collson term cf. 1.19)) for a Maxwellan dstrbuton. From the prevous theorem we derve the last desred property. For all space ponts x the H-theorem shows a tendency of the dstrbuton functon fx, v, t), more precsely of the collson ntegral Qf, f), towards a Maxwellan dstrbuton. Emulatng the mentoned propertes of the collson ntegral motvates the choce of the so called BGK approxmaton Ωf) for the collson term whch was already ntroduced n It s a sngle tme relaxaton term, expressed by Ωf) := 1 ] [fx, v, t) f eq) x, v, t), 1.28) τ c where τ c s the relaxaton tme and f eq) x, v, t) denotes the equlbrum state. The local equlbrum state used n the BGK approxmaton depends on spatal poston x and tme t, the reason for ths becomes evdent when consderng the property of ntegraton nvarants: 1 ψ k Ωf) dv = ψ k fx, v, t) dv ψ k f eq) x, v, t) dv = 0, τ c for k = 1,..., 5 and the ψ k gven by 1.16) [69]. Ths means at any poston x and tme t the equlbrum dstrbuton functon f eq) x, v, t) has to have the same densty, velocty and temperature as mpled by fx, v, t). The latter values are not unform n space and tme, hence a unform and steady equlbrum dstrbuton cannot be chosen. Ths means the local equlbrum dstrbuton f eq) x, v, t) s a Maxwellan dstrbuton where the densty, velocty and temperature are mpled by fx, v, t) va 1.6), 1.7) and 1.8), respectvely.

29 1.4. Dscretzng Boltzmann Equaton Dscretzng Boltzmann Equaton Startng wth the contnuous Boltzmann equaton wth BGK approxmaton fx, v, t) + v fx, v, t) = 1 ] [fx, v, t) f eq) x, v, t) τ c 1.29) the BGK lattce Boltzmann equaton, also called the lattce BGK LBGK) equaton, can be derved, see also [28]. The dervaton s done by a sutable dscretzaton of space, velocty space and tme, then a fnte dfference scheme s appled [69]. We am to acheve a numercal method whch s correct up to second order n macroscopc velocty u Lattce BGK Equaton In a frst step a velocty) dscrete Boltzmann equaton s derved, where the contnuous velocty space s dscretzed. Instead of v a fnte set of possble veloctes v s ntroduced V = v = v 1 v 2 v 3 : = 0,..., n v. 1.30) How some sutable V can be constructed s dscussed n the subsequent secton. Then the dscrete Boltzmann equaton reads f x, t) + v f x, t) = 1 [ ] f x, t) f eq) x, t), = 0,..., n v, 1.31) τ c wth a new dstrbuton functon fx, v, t) f x, t) and besdes a dscrete equlbrum dstrbuton functon s ntroduced f eq) x, v, t) f eq) x, t). We wll handle ths dscrete equlbrum dstrbuton later on n ths secton. For fxed x and t we call each f x, t) a populaton of the dstrbuton. A nondmensonalzaton of 1.31) s acheved by ntroducng scalng quanttes, more precsely a reference velocty U, a reference densty n r, the characterstc length scale L and the tme between partcle collsons t c. In the substtuted varables c = c 1 c 2 c 3 wth accordng operator and rght hand sde = 1 U v, ˆx = 1 L x, ˆt = t U L, ˆτ c = τ c, 1.32) t c ˆ = L, F ˆx, ˆt) = f x, t), F eq) ˆx, ˆt) = f eq) x, t), n r n r

30 1.4. Dscretzng Boltzmann Equaton 23 the nondmensonalzed dscrete Boltzmann equaton s gven by F ˆx, ˆt) ˆt usng the scalng parameter + c ˆ F ˆx, ˆt) = 1 [ F ˆx, ˆt) F eq) εˆτ c ] ˆx, ˆt), = 0,..., n v, 1.33) ε = t c U L. 1.34) In order to use a fnte dfference scheme for 1.33), a spatal dscretzaton and a dscretzaton n tme s necessary. We denote the step sze n tme as t and also scale t to acheve a nondmensonalzed quantty. In the further dervaton we see that t = t c should be used, snce ths choce smplfes the lattce BGK equaton. Thus t holds ˆt = t U L and t = t c. 1.35) A characterstc of the dscretzaton used n lattce Boltzmann methods s a connecton of the underlyng dscretzatons of tme, space and velocty space. For the nondmensonalzed) spatal dscretzaton we demand that for all veloctes c there exst n the dscretzed space a ˆx = ˆx 1, ˆx 2, ˆx 3 ) T such that ˆx = c ˆt 1.36) s fulflled. Ths means, f ˆx s a pont of the spatal dscretzaton, then also ˆx + c ˆt s one. In the subsequent secton we deal wth some popular dscretzatons for lattce Boltzmann methods. The dervatve wth respect to tme n 1.33) s approxmated wth an Euler scheme F ˆx, ˆt) ˆt F ˆx, ˆt + ˆt) F ˆx, ˆt), 1.37) ˆt smlarly each component of the gradent s approxmated. It s necessary for our purpose to evaluate the approxmaton of that gradent at the new tme pont ˆt + ˆt, snce the unknown n the fnally resultng scheme comes from ths part. Wth the abbrevatons ˆx ˆx = 0, 2 ˆx = ˆx 2, 3 ˆx = ) 0 0 ˆx 3

31 1.4. Dscretzng Boltzmann Equaton 24 these approxmatons read ˆ F ˆx, ˆt + ˆt) F ˆx+ 1 ˆx,ˆt+ ˆt) F ˆx,ˆt+ ˆt) ˆx 1 F ˆx+ 2 ˆx,ˆt+ ˆt) F ˆx,ˆt+ ˆt) ˆx 2 F ˆx+ 3 ˆx,ˆt+ ˆt) F ˆx,ˆt+ ˆt) ˆx ) Hence, provded the functon F ˆx, ˆt + ˆt) s total dfferentable wth respect to ˆx, and consderng the second term of the left hand sde n 1.33) one obtans wth use of 1.36), c ˆ F ˆx, ˆt + ˆt) = c 1 F ˆx + 1 ˆx, ˆt + ˆt) F ˆx, ˆt + ˆt) ˆx 1 + c 2 F ˆx + 2 ˆx, ˆt + ˆt) F ˆx, ˆt + ˆt) ˆx 2 + c 3 F ˆx + 3 ˆx, ˆt + ˆt) F ˆx, ˆt + ˆt) + O ˆx ) ˆx 3 = F ˆx + c 1 ˆt, ˆt + ˆt) F ˆx, ˆt + ˆt) ˆt + F ˆx + c 2 ˆt, ˆt + ˆt) F ˆx, ˆt + ˆt) ˆt + F ˆx + c 3 ˆt, ˆt + ˆt) F ˆx, ˆt + ˆt) + O ˆt) ˆt F ˆx + c ˆt, ˆt + ˆt) F ˆx, ˆt + ˆt) + O ˆt) 1.40) ˆt wth c 1, c 2 and c 3 defned analogue to 1.38) and usng O ˆx ) = O ˆt). Thus, the error made n the last step, the frst approxmaton, s of the same order as the approxmatons 1.39). Alternatvely, one can obtan the approxmaton 1.40) for c ˆ F ˆx, ˆt + ˆt) more ntutvely by approxmatng drectly the drectonal dervatve. The drectonal dervatve a F ˆx, ˆt + ˆt) along a normalzed vector a can be wrtten as a F ˆx, ˆt + ˆt) = a F ˆx, ˆt + ˆt).

32 1.4. Dscretzng Boltzmann Equaton 25 Thus, ) c ˆ F c ˆx, ˆt + ˆt) = c c ˆ F ˆx, ˆt + ˆt) = c c 1 c F ˆx, ˆt + ˆt) c F ˆx + c = c lm h, ˆt + ˆt) F ˆx, ˆt + ˆt) h 0 h c F ˆx + c c ˆx, ˆt + ˆt) F ˆx, ˆt + ˆt) ˆx = F ˆx + c ˆt, ˆt + ˆt) F ˆx, ˆt + ˆt), ˆt agan 1.36) was used. Pluggng n the fnte dfferences 1.37) and 1.40) n 1.33) yelds for all = 0,..., n v reverse order) 1 [ ] F ˆx, ˆt) F eq) ˆx, ˆt) εˆτ c = F ˆx, ˆt + ˆt) F ˆx, ˆt) ˆt + F ˆx + c ˆt, ˆt + ˆt) F ˆx, ˆt + ˆt) ˆt = F ˆx + c ˆt, ˆt + ˆt) F ˆx, ˆt). ˆt 1.41) Notcng that due to 1.34) and 1.35) we have ˆt ˆτ c ε = t U L ˆτ c t c U L = 1ˆτ c. We multply the outcome of 1.33),.e. 1.41), by ˆt to eventually obtan the desred lattce BGK equaton, t reads F ˆx + c ˆt, ˆt + ˆt) F ˆx, ˆt) = 1ˆτ [ ] F ˆx, ˆt) F eq) ˆx, ˆt). 1.42) c Comparng 1.3) and 1.42) drectly shows the close relaton between lattce gas automata and the lattce Boltzmann method. The specal dscretzaton n lattce Boltzmann methods,.e. the relaton 1.36), causes the dstrbutons to move n one tme step exactly from a lattce pont tll another lattce pont Influence of Dscretzaton The dscretzaton of the velocty space has an mmedate effect on the calculaton of the macroscopc densty 1.6) and velocty 1.7). An ntegraton s no longer requred,

33 1.4. Dscretzng Boltzmann Equaton 26 nstead a summaton over all possble veloctes s suffcent. We acheve an adapted densty and velocty n v ˆρˆx, ˆt) = F ˆx, ˆt) 1.43) =0 n v ˆρˆx, ˆt)ûˆx, ˆt) = c F ˆx, ˆt). 1.44) =0 Another nfluence from the dscretzaton of the velocty space s gven for the equlbrum dstrbuton n the collson term. We have ntroduced a dscrete equlbrum dstrbuton n 1.31). We choose ths dscrete dstrbuton such that the desred propertes cf. the prevous secton 1.3) are kept vald. Especally, ths means we demand n v ρx, t) = f eq) x, t), ρx, t)ux, t) = =0 n v =0 vf eq) x, t). 1.45) At least two popular methods exst how dscrete dstrbutons can be computed [33, 45, 69]. The dscrete) equlbrum dstrbutons are not unque, what we wll also see n the subsequent secton. The method we choose uses an ansatz functon whose shape s derved from a Taylor expanson of the contnuous equlbrum dstrbuton [45]. Omttng all arguments, then a Taylor expanson of the Maxwellan dstrbuton n u around zero up to second order reads: m f eq) = ρ 2πk B T m = ρ 2πk B T m ρ 2πk B T [ ) 3/2 exp m ) v u 2 2k B T ) 3/2 exp m ) 2k B T v 2 ) 3/2 exp m ) 2k B T v m k B T v u) + 1 m 2 k B T m exp v u) m k B T ) 2 v u) 2 m 2k B T u 2 ] ) 2k B T u 2 For the dscrete equlbrum dstrbuton, for all = 0,..., n v, functons of the shape. 1.46) f eq) x, t) = Ãx, t) + B x, t)v ux, t)) + C x, t)v ux, t)) 2 + D x, t) ux, t) 2

34 1.5. Dscretzaton Models 27 are taken. And assocated nondmensonalzed dstrbuton functons are then gven by F eq) ˆx, ˆt) = ˆx, ˆt) + ˆB ˆx, ˆt)c ûˆx, ˆt)) + Ĉˆx, ˆt)c ûˆx, ˆt)) 2 + ˆD ˆx, ˆt) ûˆx, ˆt) ) The coeffcent functons Ãx, t), B x, t), C x, t), D x, t) and those of 1.47) ˆx, ˆt), ˆB ˆx, ˆt), Ĉˆx, ˆt), ˆD ˆx, ˆt) are functons dependng also on the densty ρx, t) and ˆρˆx, ˆt), respectvely. Ther explct magntude depends strongly on the appled dscretzaton. In the subsequent secton we wll calculate them for a specfc dscretzaton. Another possblty to compute the dscrete equlbrum dstrbuton s based on the H-theorem Theorem 1). They can also be computed by maxmzng the local entropy under gven constrants [33] Dscretzaton Models In ths secton we present some common choces for the dscretzaton used n lattce Boltzmann methods. For a specfc choce, known as D3Q19, we calculate the dscrete equlbrum dstrbuton functon. For the other presented dscretzatons we only gve the magntude of the correspondng coeffcent functons. The dscretzatons are descrbed by nondmensonalzed lattce veloctes whch s suffcent, snce there s a close connecton n the dscretzaton, see secton 1.4 for detals. Classcally, the nondmensonalzaton s done such a way that the smallest velocty s normalzed. There s a wde-spread notaton whch goes back to Qan et al. [51], t reads DxQy, where x denotes the spatal) dmenson and y the number of lattce veloctes. The latter corresponds to n v + 1 n 1.30). For all presented dscretzaton models we set c 0 = 0, ths velocty s related to a rest state Models of Dmenson 1 When consderng a one dmensonal space, the smplest dscretzaton s the dvson n ntervals of same length where then the set of lattce ponts conssts of the nterval boundares. The D1Q3 model as well as the D1Q5 model are based on such a dscretzaton. In addton to the zero velocty c 0 we have n the D1Q3 model the veloctes c 1 = 1 and c 2 = 1. For each drecton there appear only sngle velocty vectors. By contrast, the D1Q5 model s a so called mult-speed model, where velocty vectors of dfferent Fgure 1.6.: Lattce veloctes n one dmensonal space.

35 1.5. Dscretzaton Models 28 length appear for same drectons. The full set of veloctes read here c 0 = 0, c 1 = 1, c 2 = 1, c 3 = 2, c 4 = 2. Fgure 1.6 llustrates these both one dmensonal models Models of Dmenson 2 At the begnnng of ths secton we gave a short ntroducton to lattce gas automata. There we had as an example the FHP model, and we have mentoned the HPP model whch are both two dmensonal. The FHP model uses a trangular lattce see also fgure 1.2) and the HPP a square lattce. The former can also be successfully used when dealng wth lattce Boltzmann methods for the smulaton of flud flows. The assocated dscretzaton of FHP s referred to as D2Q7 wth correspondng veloctes ) )) 2π 2π c = cos, sn, = 1,... 6, 6 6 fgure 1.7 a) depcts ths model. The HPP model uses a four velocty dscretzaton, addng a rest state we can state the D2Q5 model wth lattce veloctes c 1 = 1, 0), c 2 = 0, 1), c 3 = 1, 0), c 4 = 0, 0), but note that the HPP model faled to yeld the Naver-Stokes equatons, because of nsuffcent symmetry of the lattce. A very common dscretzaton n two dmensons s the D2Q9 model, used e.g. n [37]. That model conssts of each four velocty vectors of a) b) Fgure 1.7.: Lattce veloctes for a trangular a) and square b) lattce n two dmensons.

36 1.5. Dscretzaton Models 29 length 1 and 2, they read c 1 = 1, 0), c 2 = 0, 1), c 3 = 1, 0), c 4 = 0, 1), c 5 = 1, 1), c 6 = 1, 1), c 7 = 1, 1), c 8 = 1, 1), see fgure 1.7 b) for a vsualzaton. Several extensons of the D2Q9 model exst. As only one example the mult-speed D2Q13 model exhbts the addtonal velocty vectors c 9 = 2, 0), c 10 = 0, 2), c 11 = 2, 0), c 12 = 0, 2) Models of Dmenson 3 In three dmensons the analogue to the square lattce n two dmensons s a cubc lattce. Connectng nearest neghbors we acheve sx lattce veloctes of length 1, twelve velocty vectors of length 2 and eght of length 3, see fgure 1.8 a), b) and c), respectvely. Dfferent combnatons of these lattce veloctes yeld to well-known three dmensonal models at a tme [68]. We get the D3Q15 model by takng the velocty vectors of length 1 and 3, hence c 1 = 1, 0, 0), c 2 = 0, 1, 0), c 3 = 1, 0, 0), c 4 = 0, 1, 0), c 5 = 0, 0, 1), c 6 = 0, 0, 1), c 7 = 1, 1, 1), c 8 = 1, 1, 1), c 9 = 1, 1, 1), c 10 = 1, 1, 1), c 11 = 1, 1, 1), c 12 = 1, 1, 1), c 13 = 1, 1, 1), c 14 = 1, 1, 1). Extendng ths model by the twelve 2-velocty vectors we obtan the D3Q27 model, that s c 15 = 1, 1, 0), c 16 = 1, 1, 0), c 17 = 1, 1, 0), c 18 = 1, 1, 0), c 19 = 1, 0, 1), c 20 = 1, 0, 1), c 21 = 1, 0, 1), c 22 = 1, 0, 1), c 23 = 0, 1, 1), c 24 = 0, 1, 1), c 25 = 0, 1, 1), c 26 = 0, 1, 1). The already mentoned D3Q19 model s constructed wth lattce veloctes of length 1 and 2. For ths model we compute the dscrete equlbrum dstrbuton functon n the followng. The trangular and square lattce n two dmensons, as well as the cubc lattce n the three dmensonal space are space fllng dscretzatons. But also advantageous nonspace fllng dscretzaton models were proposed, see for more nformaton [47].

37 1.5. Dscretzaton Models a) 3 11 b) c) 5 Fgure 1.8.: Lattce veloctes for cubc lattce n three dmensons Dscrete Equlbrum Dstrbuton n D3Q19 In ths subsecton we compute the dscrete equlbrum dstrbuton functon n the D3Q19 model, analogue to [69]. Ths means we compute the coeffcent functons n 1.47) for all = 0,..., 18. A reasonable choce for the coeffcents ensures the nonnegatvty for all F eq), later on n the calculaton the demand of non-negatvty wll be

38 1.5. Dscretzaton Models 31 used. The full set of lattce veloctes n D3Q19 read as follows: c 0 = 0, 0, 0), c 1 = 1, 0, 0), c 2 = 0, 1, 0), c 3 = 1, 0, 0), c 4 = 0, 1, 0), c 5 = 0, 0, 1), c 6 = 0, 0, 1), 1.48) c 7 = 1, 1, 0), c 8 = 1, 1, 0), c 9 = 1, 1, 0), c 10 = 1, 1, 0), c 11 = 1, 0, 1), c 12 = 1, 0, 1), c 13 = 1, 0, 1), c 14 = 1, 0, 1), c 15 = 0, 1, 1), c 16 = 0, 1, 1), c 17 = 0, 1, 1), c 18 = 0, 1, 1). The coeffcent functons should depend only on the mass densty. We assume that for fxed ˆx and ˆt, ˆx, ˆt) and Âjˆx, ˆt) gven n 1.47) dffer only f c and c j have dfferent values, for, j = 0,..., 18. Same for the ˆB ˆx, ˆt), Ĉ ˆx, ˆt) and ˆD ˆx, ˆt), = 0,..., 18. Therefore we can rewrte the coeffcents n 1.47) by A 0 A 0 ˆx, ˆt) := Â0ˆx, ˆt), C 0 C 0 ˆx, ˆt) := Ĉ0ˆx, ˆt), B 0 B 0 ˆx, ˆt) := ˆB 0 ˆx, ˆt), D 0 D 0 ˆx, ˆt) := ˆD 0 ˆx, ˆt), A 1 A 1 ˆx, ˆt) := ˆx, ˆt), B 1 B 1 ˆx, ˆt) := ˆB ˆx, ˆt), = 1,... 6, C 1 C 1 ˆx, ˆt) := Ĉˆx, ˆt), D 1 D 1 ˆx, ˆt) := ˆD ˆx, ˆt), = 1,... 6, A 2 A 2 ˆx, ˆt) := ˆx, ˆt), B 2 B 2 ˆx, ˆt) := ˆB ˆx, ˆt), = 7,... 18, C 2 C 2 ˆx, ˆt) := Ĉˆx, ˆt), D 2 D 2 ˆx, ˆt) := ˆD ˆx, ˆt), = 7,... 18, whch yelds a reduced system of equatons wth 10 free parameters F eq) ˆx, ˆt) = A 0 + D 0 ûˆx, ˆt) 2, = 0 A 1 + B 1 c ûˆx, ˆt)) + C 1 c ûˆx, ˆt)) 2 + D 1 ûˆx, ˆt) 2, = 1,..., 6 A 2 + B 2 c ûˆx, ˆt)) + C 2 c ûˆx, ˆt)) 2 + D 2 ûˆx, ˆt) 2, = 7,..., ) where B 0 and C 0 do not appear, because c 0 ûˆx, ˆt) = 0 and hence they do not have a contrbuton. The parameters are actually stll coeffcent functons, dependng on spatal poston, tme and densty. In the computaton we fx ˆx and ˆt, that s why we omt the functon arguments here and n the followng computaton. We obtan from a nondmensonalzed verson of 1.45) the followng general constrants n v ˆρ = F eq), =0 n v ˆρû = =0 c F eq), 1.50)

39 1.5. Dscretzaton Models 32 whch mply, when usng the lattce veloctes 1.48) explctly, two equatons for the free parameters n v = 18): ˆρ = F eq) F eq) 18 + =1 =7 = [ A 0 + D 0 û 2] + [ F eq) 6A 1 + B 1 [ A 2 + B 2 c u) =7 } {{ } =0 ] 6 6 c u) +C 1 c u) 2 +6D 1 û 2 =1 } {{ } =0 +C 2 18 c u) 2 =7 } {{ } =8 û 2 =1 } {{ } =2 û 2 ] +12D 2 û 2 = A 0 + 6A A 2 + 2C 1 + 8C 2 + D 0 + 6D D 2 ) û 2, ˆρû = 2B 1 + 8B 2 )û Ths can be decomposed nto three constrants ˆρ = A 0 + 6A A 2, 0 = 2C 1 + 8C 2 + D 0 + 6D D 2, 1.51) ˆρ = 2B 1 + 8B 2, and we have only three equatons for ten unknowns. Clearly, the soluton s not unque. Addtonal constrants can be mposed, thus we demand constrants [69] whch transform the zeroth order) momentum flux tensor P α,β ) α,β=1,2,3 = 18 =0 c α c β F eq) nto the form P = ˆρûû T + pi, 1.52) whch reads here ˆρû 1 û 1 + p ˆρû 1 û 2 ˆρû 1 û 3 P = ˆρû 1 û 2 ˆρû 2 û 2 + p ˆρû 2 û 3, ˆρû 1 û 3 ˆρû 2 û 3 ˆρû 3 û 3 + p wth pressure p, dentty matrx I and c j, ûj denotng the jth component of c and û, respectvely. Takng agan the lattce veloctes 1.48) nto account the entres n the

40 1.5. Dscretzaton Models 33 tensor can be computed, we obtan P k,k = 2A 1 + 8A 2 ) + 2C 1 + 4C 2 ) û k) 2 + 4C2 + 2D 1 + 8D 2 ) û 2, k = 1, 2, 3, P j,k = 8C 2 û j û k, j, k = 1, 2, 3, j k. Here, we can deduce the constrants to acheve the momentum flux tensor n the desred form, we get ˆρ = 2C 1 + 4C 2, 0 = 4C 2 + 2D 1 + 8D 2, 1.53) ˆρ = 8C 2, and the pressure p s then gven by the porton whch s ndependent of the velocty û,.e. n ths model by p = 2A 1 + 8A 2. From the frst and thrd equaton of 1.53) we can conclude C 1 = 1 4 ˆρ and C 2 = 1 8 ˆρ, and hence the second equaton of 1.51) and 1.53) each can be rewrtten. The not yet determned unknowns A 0, A 1, A 2, B 1, B 2, D 0, D 1 and D 2 are restrcted by only four equatons ˆρ = A 0 + 6A A 2, 3 2 ˆρ = D 0 + 6D D 2, ˆρ = 2B 1 + 8B 2, 1.54) 1 2 ˆρ = 2D 1 + 8D 2. The constrants we mposed to transform the momentum flux tensor do not lead to a unque equlbrum dstrbuton. Some further arbtrary constrants can be mposed, we adapt the dea of lnear relatons from [69] to the gven dscretzaton model here: A 0 = 3rA 1, A 1 = ra 2, B 1 = rb 2, D 0 = 3rD 1. for an addtonal free parameter r R. The thrd equaton of 1.54) smplfes ˆρ = B 2 2r + 8) B 2 = ˆρ 2r + 4),

41 1.5. Dscretzaton Models 34 weghts for c = 0 c = 1 c = 2 c = 3 c = 2 D1Q3 D1Q5 D2Q5 D2Q7 D2Q9 D3Q15 D3Q19 D3Q Table 1.1.: Weghts for equlbrum dstrbuton functon for several models. and from the frst equaton of 1.54) we get ˆρ = 3A 2 r 2 + 2r + 4) A 2 = ˆρ 3r 2 + 2r + 4). When we equate the rght hand sde of the second equaton of 1.54) wth three tmes the rght hand sde of the forth equaton we obtan D 1 3r + 6) + 12D 2 = 6D D 2 rd 1 4D 2 = 0. Wth the addtonal condton D 1 = rd 2 we can resolve the latter equaton, ths yelds r 2 4)D 2 = 0 and we can conclude r = ±2. The soluton r = 2 yelds negatve equlbrum dstrbutons, hence we take r = 2. Ths mples A 2 = 1 36 ˆρ A 1 = 2 36 ˆρ A 0 = ˆρ, B 2 = 1 12 ˆρ B 1 = 2 12 ˆρ. And fnally the magntudes of D 0, D 1 and D 2 can be obtaned by the forth equaton of 1.54) 1 2 ˆρ = 2D 1 + 8D 2 = 12D 2 D 2 = 1 24 ˆρ,

42 1.6. Chapman-Enskog Expanson 35 whch mples D 1 = 2 24 ˆρ and D 0 = ˆρ. Snce all free parameters n 1.49) are now computed, we can state the dscrete equlbrum dstrbuton for the D3Q19 model: [ 1 0 ˆx, ˆt) = ˆρˆx, ˆt) [ 1 F eq) F eq) ˆx, ˆt) = ˆρˆx, ˆt) F eq) j ˆx, ˆt) = ˆρˆx, ˆt) ], ûˆx, ˆt) c ûˆx, ˆt)) c ûˆx, ˆt)) 2 1 ] 12 ûˆx, ˆt) 2, [ c j ûˆx, ˆt)) c j ûˆx, ˆt)) 2 1 ] 24 ûˆx, ˆt) 2, for = 1,..., 6 and j = 7,..., 18. Wrtng these equaton as F eq) 0 ˆx, ˆt) = ω 0 ˆρˆx, ˆt) [1 32 ] ûˆx, ˆt) 2 F eq) ˆx, ˆt) = ω ˆρˆx, ˆt) [1 + 3c ûˆx, ˆt)) + 92 c ûˆx, ˆt)) 2 32 ] 1.55) ûˆx, ˆt) 2 for = 1,..., 18, wth ω 0 = 1 3, ω j = 1 18, ω k = 1, j = 1,..., 6, k = 7,..., 18, 1.56) 36 s more sutable for an mplementaton. Ths result concdes wth the Taylor expanson of the contnuous equlbrum dstrbuton 1.46) n the sense that the coeffcent of the u 2 term wthn the square bracket s 0.5 tmes the coeffcent of the v u term. Same for the coeffcent of the v u) 2 term whch s 0.5 tmes the coeffcent of the v u term squared. The pressure n ths D3Q19 model s gven by pˆx, ˆt) = 2A 1 + 8A 2 = 1 3 ˆρˆx, ˆt). 1.57) The dervaton of dscrete equlbrum dstrbutons for other models s analogue to our calculaton presented here. For the other models stated n the current secton possble equlbrum dstrbutons are gven n table 1.1 whch correspond to equlbrum dstrbuton functons n the shape of 1.55) [44, 46, 59, 61]. In table 1.1 only the weghts ω are gven Chapman-Enskog Expanson In secton 1.2 we have shown that the contnuous Boltzmann equaton 1.4) covers macroscopc equatons. The computatonal smulaton s based on the lattce BGK equaton 1.42), a dscretzed verson of the Boltzmann equaton 1.4) wth an approxmatng term 1.28) for the collson ntegral. In order to show that the lattce BGK equaton also satsfes some meanngful macroscopc equatons, we use a Chapman-Enskog expanson. For more detals and the background of the Chapman-Enskog expanson see for nstance

43 1.6. Chapman-Enskog Expanson 36 [7, 54, 61, 69] Terms of Frst and Second Order The procedure used n the current and subsequent subsecton was nspred by [10, 27, 32]. Begnnng the calculaton, we employ a Taylor expanson of the left hand sde of 1.42) and obtan for all = 0,..., n v up to order ˆt 2 : ) F ˆx + c ˆt, ˆt + ˆt) = ˆt ˆt + c F ˆx, ˆt) ˆt 2 2 ˆt + c ) F ˆx, ˆt). The Knudsen number s a dmensonless number defned as the rato of the partcle mean free path to the characterstc length scale. Snce t c gves the tme between collsons, and L s the characterstc length scale, the parameter ε defned n 1.32) can be consdered as the Knudsen number. Due to 1.35) the nondmensonalzed tme step ˆt equals the parameter ε. The complete lattce BGK equaton after a Taylor expanson of the left hand sde up to second order then reads ) ε ˆt + c F ˆx, ˆt) + 1 ) 2 2 ε2 ˆt + c F ˆx, ˆt) = 1 [ ] F ˆx, ˆt) F eq) ˆx, ˆt). ˆτ c 1.58) We expand the dstrbutons F ˆx, ˆt) around the equlbrum dstrbuton, we get where the zeroth order F 0) F ˆx, ˆt) = F 0) ˆx, ˆt) + εf 1) ˆx, ˆt) + ε 2 F 2) ˆx, ˆt) + Oε 3 ), 1.59) ˆx, ˆt) represents the equlbrum dstrbuton F eq) ˆx, ˆt). The defntons of the macroscopc densty 1.43) and velocty 1.44) as well as the demanded constrants 1.50) n the constructon of the equlbrum dstrbutons yeld n v n v ˆρˆx, ˆt) = F ˆx, ˆt) = F 0) ˆx, ˆt), =0 n v =0 n v ˆρˆx, ˆt)ûˆx, ˆt) = c F ˆx, ˆt) = =0 =0 c F 0) ˆx, ˆt). 1.60) Hence, we can conclude n v =0 n v =0 F 1) ˆx, ˆt) = c F 1) ˆx, ˆt) = n v =0 n v =0 F 2) ˆx, ˆt) = 0, c F 2) ˆx, ˆt) = )

44 1.6. Chapman-Enskog Expanson 37 In a smlar manner as the dstrbuton F ˆx, ˆt) we expand the dfferental operator wth respect to tme by whereas the spatal dfferental operator obeys ˆt = 1) ˆt + ε 2) ˆt + Oε2 ), 1.62) = 1). 1.63) We substtute 1.59), 1.62) and 1.63) n 1.58) and splt terms wth respect to orders of ε. The terms of order ε read ) 1) ε ˆt + c 1) F 0) ˆx, ˆt) = 1ˆτ εf 1) ˆx, ˆt). 1.64) c Takng a further dervatve ths equaton mples the followng relaton between F 0) ˆx, ˆt) and F 1) ˆx, ˆt) ) 2 ) 1 1) 2 ˆt + c 1) F 0) ˆx, ˆt) = 1 1) 2ˆτ c ˆt + c 1) F 1) ˆx, ˆt). 1.65) The terms of order ε 2 are gven by [ ) ε 2 1) ˆt + c 1) F 1) ˆx, ˆt) + 2) ˆt F 0) ˆx, ˆt) ) ) 2 ˆt + c 1) F 0) ˆx, ˆt) = 1ˆτ ε 2 F 2) ˆx, ˆt), c whch can be smplfed by usng 1.65) to [ ε ) ) ] 1) 2ˆτ c ˆt + c 1) F 1) ˆx, ˆt) + 2) ˆt F 0) ˆx, ˆt) = 1ˆτ ε 2 F 2) ˆx, ˆt). c 1.66) In the contnuous case the macroscopc equatons were obtaned by an ntegraton of the Boltzmann equaton, more precsely the contnuty equaton was acheved n ths manner. For the momentum equaton the frst moment of the Boltzmann equaton was computed,.e. an ntegraton of the Boltzmann equaton multpled wth v led to that equaton. Now consderng the dscretzed equaton we take the summaton over all possble veloctes nstead. The frst order terms are acheved by equaton 1.64),

45 1.6. Chapman-Enskog Expanson 38 meanng n v =0 ε 1) ˆt + c 1) ) F 0) ˆx, ˆt) = n v =0 1ˆτc ) εf 1) ˆx, ˆt). Dvson by ε and usng 1.60) and 1.61) for the left and rght hand sde, respectvely, yelds 1) ˆt ˆρˆx, ˆt) + 1) ˆρˆx, ˆt)ûˆx, ˆt) ) = ) Analogue to the contnuous case, summng 1.64) after multplyng wth c gves for the momentum wth 1) ˆt ˆρˆx, ˆt)ûˆx, ˆt) ) + 1) P 0) ˆx, ˆt) = 0, 1.68) n v P 0) ˆx, ˆt) = c c T F 0) ˆx, ˆt) =0 where the equaton was dvded by ε and the condtons 1.60) and 1.61) were used. The expanson 1.59) mples an expanson for the momentum flux tensor wth P ˆx, ˆt) = P 0) ˆx, ˆt) + εp 1) ˆx, ˆt) + ε 2 P 2) ˆx, ˆt) + Oε 3 ), n v P ˆx, ˆt) = c c T F ˆx, ˆt). =0 In the constructon of the equlbrum dstrbuton see secton 1.5) we prescrbed the form of the zeroth order momentum flux tensor P 0) ˆx, ˆt), cf. 1.52), hence equaton 1.68) changes to 1) ˆt ˆρˆx, ˆt)ûˆx, ˆt) ) + 1) ˆρˆx, ˆt)ûˆx, ˆt)ûˆx, ˆt) T ) = 1) pˆx, ˆt). 1.69) Clearly, ths step s only vald, f the momentum flux tensor s gven n the form 1.52), that condton to the equlbrum dstrbuton s satsfed by the models presented n the prevous secton. For the second order terms 1.66), a smple summaton yelds [ 1 1 ) ) ] 1) 2ˆτ c ˆt + c 1) F 1) ˆx, ˆt) + 2) ˆt F 0) ˆx, ˆt) = ε2 ˆτ c ε 2 n v =0 n v =0 F 2) ˆx, ˆt),

46 1.6. Chapman-Enskog Expanson 39 whch reduces by usng 1.60), 1.61) and dvson by ε 2 to 2) ˆt ˆρˆx, ˆt) = ) Summaton of 1.66) after multplyng wth c gves the terms of the momentum equaton of second order. It follows after dvson by ε ) [ 1) 2ˆτ c ˆt n v =0 c F 1) ˆx, ˆt) + 1) P 1) ˆx, ˆt) wth the frst order momentum flux tensor n v P 1) ˆx, ˆt) = c c T F 1) ˆx, ˆt). =0 ] + 2) ˆt n v =0 c F 0) ˆx, ˆt) = 0 Note that 1.61) was already used for the rght hand sde, applyng 1.60) and 1.61) also for the left hand sde smplfes ths equaton to [ 1) 1 1 ) ] P 1) ˆx, ˆt) + 2) ˆρˆx, ˆt)ûˆx, ˆt) ) = ) 2ˆτ c ˆt Computaton of P 1) ˆx, ˆt) Unlke P 0) ˆx, ˆt) we do not know P 1) ˆx, ˆt) from prevous calculatons, hence we have to compute t. From equaton 1.64) we get the relaton F 1) ˆx, ˆt) = ˆτ c 1) ˆt + c 1) ) F 0) ˆx, ˆt), whch can be used n the computaton of P 1) ˆx, ˆt). It follows P 1) ˆx, ˆt) = ˆτ c [ 1) ˆt [ 1) = ˆτ c n v =0 c c T F 0) n v ˆx, ˆt) + c c T =0 =0 ˆt P n v 0) ˆx, ˆt) + c c T The entres of the matrx R 1) ˆx, ˆt) are gven by R 1) nv α,β = 3 c α c β =0 j=1 c j 1) F 0) ˆx, ˆt) = ˆx j c 1)) F 0) c 1)) F 0) ˆx, ˆt) } {{ } =:R 1) ˆx,ˆt) 3 j=1 1) ˆx j n v =0 ] ˆx, ˆt) ]. 1.72) c α c β cj F 0) ˆx, ˆt), 1.73)

47 1.6. Chapman-Enskog Expanson 40 wth α, β = 1, 2, 3 runnng through the dmenson of space. The matrx R 1) ˆx, ˆt) cannot be stated n general, the equlbrum dstrbuton F 0) as well as the lattce veloctes c have to be taken nto account. Hence we consder the D3Q19 model agan. Usng the explct form 1.55) of the equlbrum dstrbuton n ths model and the lattce veloctes 1.48), t follows when omttng all functon arguments F 0) 1 F 0) 3 = 1 3 ˆρû 1, F 0) 2 F 0) 4 = 1 3 ˆρû 2, F 0) 5 F 0) 6 = 1 3 ˆρû 3, F 0) 7 F 0) 9 = 1 6 ˆρû 1 + û 2 ), F 0) 10 F 0) 8 = 1 6 ˆρû 1 û 2 ), F 0) 11 F 0) 13 = 1 6 ˆρû 1 + û 3 ), F 0) 14 F 0) 12 = 1 6 ˆρû 1 û 3 ), F 0) 15 F 0) 17 = 1 6 ˆρû 2 + û 3 ), F 0) 18 F 0) 16 = 1 6 ˆρû 2 û 3 ). Wth help of the latter equatons and the values gven n table 1.2 we can calculate the nner sums of 1.73). A general expresson for these sums s obtaned by [23] 18 =0 c α c β cj F 0) ˆx, ˆt) = 1 [ ] 3 ˆρˆx, ˆt) δ αβ û j ˆx, ˆt) + δ αj û β ˆx, ˆt) + δ βj û α ˆx, ˆt), 1.74) { 1, f = j wth α, β, j = 1, 2, 3 and Kronecker delta δ j =. Wth ad of the latter 0, f j expresson t follows for the matrx 1.73) n case of the D3Q19 model and all other models whch satsfy 1.74) wth n v nstead of 18) R 1) α,β = 3 j=1 1) 1 [ ] ) ˆx j 3 ˆρˆx, ˆt) δ αβ û j ˆx, ˆt) + δ αj û β ˆx, ˆt) + δ βj û α ˆx, ˆt) = 1 3 1) ˆρˆx, ˆt)û j ) ˆx, ˆt)δ 1) αβ + 3 ˆx j=1 j ˆx α = 1 δ αβ 1) ˆρˆx, ˆt)ûˆx, ˆt) )) + 1) ˆρˆx, ˆt)û β ˆx, ˆt) 3 ˆx α ˆρˆx, ˆt)û β ˆx, ˆt) )+ 1) ˆρˆx, ˆt)û α ˆx, ˆt) ) ˆx β ˆρˆx, ˆt)û α ˆx, ˆt) ) ). ˆx β )+ 1) For the further calculaton of the rght hand sde of 1.72) we use the representaton cf. 1.52)) P 0) α,β ˆx, ˆt) = ˆρˆx, ˆt)û α ˆx, ˆt)û β ˆx, ˆt) + pˆx, ˆt)δ αβ,

48 1.6. Chapman-Enskog Expanson 41 j α β Table 1.2.: Result of the product c α cβ cj for D3Q19 model. Lattce veloctes are gven n 1.48) and empty spaces ndcate a result of zero. and 1.67) for the frst addend n the latter equaton. Thus, we obtan wth the pressure densty relaton 1.57) P 1) α,β ˆx, ˆt) = ˆτ c [ 1) ˆt = ˆτ c [ 1) ˆt ) ˆρˆx, ˆt)û α ˆx, ˆt)û β ˆx, ˆt) + 1) ˆt pˆx, ˆt)δ αβ 1 3 δ 1) αβ ˆt ˆρˆx, ˆt) 1) ) ˆρˆx, ˆt)û β ˆx, ˆt) + 1 1) ˆρˆx, ˆt)û α ˆx, ˆt) ) ] ˆx α 3 ˆx β ) ˆρˆx, ˆt)û α ˆx, ˆt)û β ˆx, ˆt) + 1 1) ) ˆρˆx, ˆt)û β ˆx, ˆt) 3 ˆx α ] + 1 1) ˆρˆx, ˆt)û α ˆx, ˆt) ) 3 ˆx β, 1.75) whch can also be wrtten n vector notaton by [ P 1) ˆx, ˆt) 1) = ˆτ c ˆρˆx, ˆt)ûˆx, ˆt)ûˆx, ˆt) T ) + 1 ˆt 3 1)[ˆρˆx, ˆt)ûˆx, ˆt) T ] + 1 1)[ˆρˆx, ˆt)ûˆx, ˆt) T ]) ] T. 3

49 1.6. Chapman-Enskog Expanson 42 In the latter expresson the tme dervatve exhbts the form 1) ˆt ) ˆρˆx, ˆt)û α ˆx, ˆt)û β ˆx, ˆt) = û α ˆx, ˆt) 1) pˆx, ˆt) û β ˆx, ˆt) 1) pˆx, ˆt) 1.76) ˆx β ˆx α 3 1) ) ˆρˆx, ˆt)û α ˆx, ˆt)û β ˆx, ˆt)û j ˆx, ˆt), ˆx j j=1 1.77) whch s computed n detal n appendx A.2. Substtutng the rght hand sde of the latter equaton n 1.75) and usng product rule for the spatal dervatve the outcome then reads [ P 1) α,β ˆx, ˆt) = ˆτ c û α ˆx, ˆt) 1) pˆx, ˆt) û β ˆx, ˆt) 1) pˆx, ˆt) ˆx β ˆx α + 1 û β ˆx, ˆt) 1) ˆρˆx, ˆt) + ˆρˆx, ˆt) 1) û β ˆx, ˆt) 3 ˆx α ˆx α 3 1) ˆx j j=1 = ˆτ c [ ˆρˆx, ˆt) 3 1) +û α ˆx, ˆt) 1) ˆρˆx, ˆt) + ˆρˆx, ˆt) 1) û α ˆx, ˆt) ˆx β ˆx β ) ] ˆρˆx, ˆt)û α ˆx, ˆt)û β ˆx, ˆt)û j ˆx, ˆt) ) û β ˆx, ˆt) + 1) û α ˆx, ˆt) ˆx α ˆx β 3 j=1 1) ˆρˆx, ˆt)û α ˆx, ˆt)û β ˆx, ˆt)û j ˆx, ˆt)) ], ˆx j where the pressure densty relaton 1.57) was used. Note that the second step was not a general one, snce the pressure s n general not gven by 1.57). When usng a dscrete equlbrum dstrbuton n the type of 1.55), the O û 3 ) term n the latter equaton should be neglected to be consstent wth the approxmaton of the equlbrum dstrbuton whch tself s an approxmaton of order O û 2 ) cf. 1.46)). Thus we acheve n vector notaton P 1) ˆx, ˆt) = ˆτ c 3 ˆρˆx, ˆt) Results up to Second Order [ ) ] T 1) ûˆx, ˆt) + 1) ûˆx, ˆt). From prevous calculatons we already know the frst order approxmatons 1.67) and 1.69) of the contnuty equaton and momentum equaton, respectvely. The second order approxmaton for the contnuty equaton s also known and gven above by 1.70). )

50 1.6. Chapman-Enskog Expanson 43 Eventually, wth the fnal equaton from the prevous subsecton the second order approxmaton 1.71) for the momentum equaton reads 2) ˆt ˆρˆx, ˆt)ûˆx, ˆt) ) = 2ˆτ [ c 1 ) ] T 1) ˆρˆx, ˆt) 1) ûˆx, ˆt) + ˆρˆx, ˆt) 1) ûˆx, ˆt) ) Addng 1.67) and 1.70) leads to an approxmaton up to second order of the contnuty equaton ˆt ˆρˆx, ˆt) + ˆρˆx, ˆt)ûˆx, ˆt) ) = ) Smlarly, by addton of the frst and second order approxmatons 1.69) and 1.78) the resultng term s an approxmaton up to second order of the momentum equaton, t reads ˆρˆx, ˆt)ûˆx, ˆt) ) + ˆρˆx, ˆt)ûˆx, ˆt)ûˆx, ˆt) T ) ˆt [ = pˆx, ˆt) + ν ˆρˆx, ˆt) ûˆx, ˆt) + ˆρˆx, ˆt) ûˆx, ˆt) ) ] T, 1.80) wth ν = 2ˆτc 1 6, beng the knematc vscosty, as we wll see n the subsequent subsecton Incompressble Naver Stokes Equaton Summarzng the computaton above, the man results are 1.79) and 1.80). The queston arsng s how these two equatons can be nterpreted. A further calculaton wll answer ths. We show below that the equatons can be nterpreted as the ncompressble Naver-Stokes equatons. Recallng the followng equatons [65]: For an ncompressble flud the Naver-Stokes equatons read u + u ) u = ˆp + ˆν u, u = 0, 1.81) wth velocty u, knematc vscosty ˆν and ˆp equal to the rato of scalar pressure p and densty ρ. We assume that changes n ˆρˆx, ˆt) are neglgble, what s vald f changes n the densty are small enough. In the followng the densty s treated as beng a constant, thus equaton 1.79) changes to ûˆx, ˆt) = 0, 1.82) condtonng the macroscopc velocty to be dvergence free. The approxmatng momen-

51 1.7. Summary and Algorthm 44 tum equaton 1.80) also changes, we obtan the followng ˆρˆx, ˆt) ˆtûˆx, ˆt) + ûˆx, ˆt)ûˆx, ˆt) T ) ) [ = pˆx, ˆt) + ν ˆρˆx, ˆt) ûˆx, ˆt) + ûˆx, ˆt) ) ] T. Usng 1.82) we can smplfy on the left hand sde as follows ûˆx, ˆt)ûˆx, ˆt) T ) = ûˆx, ˆt) ûˆx, ˆt) + ûˆx, ˆt) ûˆx, ˆt) ) = ûˆx, ˆt) ûˆx, ˆt). Also the rght hand sde of equaton 1.82) can be smplfed [ ûˆx, ˆt) + ûˆx, ˆt) ) ] T = ûˆx, ˆt) + ûˆx, ˆt) ) T = ûˆx, ˆt) ) + ûˆx, ˆt) = ûˆx, ˆt), wth Laplacan whch s appled to each component of ûˆx, ˆt). Usng both smplfcatons and dvson by ˆρˆx, ˆt), we acheve ˆtûˆx, ˆt) + ûˆx, ˆt) ûˆx, ˆt) = pˆx, ˆt) + ν uˆx, ˆt), 1.83) wth pˆx, ˆt) = pˆx,ˆt). Note that the gradent of the pressure was not neglected, although ˆρˆx,ˆt) t s related to the densty. By not neglectng t, we observe that equatons 1.82) and 1.83) are exactly the same as the ncompressble Naver-Stokes equatons 1.81) Summary and Algorthm We fnsh the current chapter by summarzng the chapter so far and recaptulatng the most mportant equatons and results from the foregong sectons. Afterwards, we present an algorthm for the lattce BGK method n pseudo code. Startng from lattce gas automata, the bass of lattce Boltzmann methods, where explaned: Sngle partcles collde and hop on a regular lattce, where the collsons are descrbed by local rules. If the lattce exhbts a suffcent symmetry one can successfully smulate flud flows wth lattce gas automata, for nstance wth the FHP model. A counterexample s the HPP model, whch fals to yeld the Naver-Stokes equatons. Snce, n general, a flud conssts of a huge number of molecules and lattce gas automata consder only sngle partcles an average over many smulatons wth dfferent ntal condtons has to be buld n order to gan physcally meanngful results. The statstcal nose emergng by ths averagng s elmnated n lattce Boltzmann methods. Here, nstead of sngle partcles a stochastc dstrbuton moves and "colldes" on a lattce.

52 1.7. Summary and Algorthm 45 We derved the lattce Boltzmann method from the Boltzmann equaton 1.4)) fx, v, t) + v fx, v, t) = Qf, f) by a sutable dscretzaton. We showed that solvng the Boltzmann equaton s an approprate approach for the smulaton of flud flows, snce Naver-Stokes equatons are recovered from the Boltzmann equaton. Important for recoverng the Naver-Stokes equatons were the defntons of macroscopc quanttes 1.6) and 1.7)) ρx, t) = fx, v, t) dv, ρx, t)ux, t) = vfx, v, t) dv. The assocated dscretzed equatons 1.43) and 1.44)) were n v ˆρˆx, ˆt) = F ˆx, ˆt), =0 n v ˆρˆx, ˆt)ûˆx, ˆt) = c F ˆx, ˆt). In secton 1.3 we motvated an approxmaton of the complcated collson ntegral Qf, f) by a sngle relaxaton term. In addton wth the dscretzed left hand sde of the Boltzmann equaton ths led to the lattce BGK equaton 1.42)) F ˆx + c ˆt, ˆt + ˆt) F ˆx, ˆt) = 1ˆτ [ ] F ˆx, ˆt) F eq) ˆx, ˆt). c The equlbrum dstrbuton F eq) ˆx, ˆt) used n the lattce BGK equaton was derved for the D3Q19 model n secton 1.5. The presented procedure can be appled to other dscretzaton models as well. The dscrete equlbrum dstrbuton arose from a Taylor expanson of the contnuous equlbrum dstrbuton n the macroscopc velocty up to second order. Hence, the presented lattce BGK methods are not sutable for the smulaton of fluds wth hgh veloctes. The deduced dscrete equlbrum dstrbuton was 1.55)) 0 ˆx, ˆt) = ω 0 ˆρˆx, ˆt) [1 32 ] ûˆx, ˆt) 2 F eq) F eq) ˆx, ˆt) = ω ˆρˆx, ˆt) =0, [1 + 3c ûˆx, ˆt)) + 92 c ûˆx, ˆt)) 2 32 ] ûˆx, ˆt) 2, wth weghts ω gven for several models n table 1.1. In secton 1.6 we attested that the lattce BGK method s capable to approxmate phys-

53 1.7. Summary and Algorthm 46 cally meanngful equatons. Ths verfcaton was done by usng a Chapman-Enskog expanson. We saw that f changes n the macroscopc densty are small then LBGK methods approxmate the ncompressble Naver-Stokes equatons. Hence, lattce Boltzmann methods wth BGK approxmaton can be used to smulate flud flows n the ncompressble lmt. Comparson of the ncompressble Naver-Stokes equatons and the result of the Chapman-Enskog expanson mpled that the knematc vscosty n the numercal scheme 1.80) and 1.83)) s gven by ν = 2ˆτ c 1. 6 Ths means the relaxaton parameter ˆτ c can be chosen to tune the vscosty. An relaxaton parameter ˆτ c 1 2, ) leads to a postve vscosty. Fnally, we state the results above n a pseudo code algorthm. As an representatve for lattce BGK methods we choose the dscretzaton model D3Q19. The algorthm llustrates the prncpal dea, nevertheless t s mpossble to mplement t, because the lnes foreach lattce pont ˆx do refer to an nfnte number of lattce ponts. For an mplementaton t s necessary to restrct the flud to a bounded doman, meanng nvestgatng the Boltzmann equaton for x X nstead of x, where X satsfes the condton y z 2 < M, wth M R and for all y, z X. The followng chapter deals, among others, wth ths and resultng topcs.

54 1.7. Summary and Algorthm 47 Algorthm: Sketch: Smple LBGK D3Q19 Intalzaton: Set tme horzon T end for smulaton Defne lattce veloctes: c 0 = [0; 0; 0], c 1 = [1; 0; 0] c 2 = [0; 1; 0], c 3 = [ 1; 0; 0] c 4 = [0; 1; 0], c 5 = [0; 0; 1] c 6 = [0; 0; 1], c 7 = [1; 1; 0] c 8 = [ 1; 1; 0], c 9 = [ 1; 1; 0] c 10 = [1; 1; 0], c 11 = [1; 0; 1] c 12 = [ 1; 0; 1], c 13 = [ 1; 0; 1] c 14 = [1; 0; 1], c 15 = [0; 1; 1] c 16 = [0; 1; 1], c 17 = [0; 1; 1] c 18 = [0; 1; 1] Set ˆt = 0 Choose relaxaton parameter ˆτ c Determne tme step ˆt Set ntal values: foreach lattce pont ˆx do for =0 to 18 do Set ntal values F ˆx, 0) end end Tme Evoluton: whle ˆt < T end do Collson: foreach lattce pont ˆx do Compute densty: ˆρ = 18 =0 Fˆx, ˆt) Compute velocty: û = 1ˆρ 18 =0 cfˆx, ˆt) Compute equlbrum dstrbuton: F eq) 0 = 1 ˆρ[ û 2 ] for =1 to 6 do F eq) = 1 ˆρ [ 1 + 3c 18 û) c û)2 3 û 2] 2 end for =7 to 18 do F eq) = 1 ˆρ [ 1 + 3c 36 û) c û)2 3 û 2] 2 end for =0 to 18 do [ ] F ˆx, ˆt) = F ˆx, ˆt) 1ˆτ c F ˆx, ˆt) F eq) end end end Streamng: foreach lattce pont ˆx do for =0 to 18 do F ˆx + c ˆt, ˆt + ˆt) = F ˆx, ˆt) end end Update: ˆt = ˆt + ˆt

55 CHAPTER 2 Enhancements of the Lattce Boltzmann Method The prevous chapter ntroduced the lattce Boltzmann method 1, nevertheless the focus of the presented model was exclusvely on streamng and collson. However, the smplest lattce Boltzmann models that can be mplemented are based on the concepts presented n the precedng chapter, we enhance the method n ths chapter and thus enlarge the means of ts applcaton. There s a wde range of problems where lattce Boltzmann methods can be appled, t contans, only as examples, magnetohydrodynamcal problems [8] or mult-component flows n porous meda [39]. Roughly speakng, f one consders a specfc problem one also needs a specfc Boltzmann method whch results after an approprate nvestgaton. Hence, we do not present enhancements of the basc lattce Boltzmann method here, n such a way that we are capable of applyng t to plenty of physcal problems. A true strengths of lattce Boltzmann methods les n ther ablty to smulate both sngle-component SC) and mult-component MC) multphase MP) flud flows. Each component refers to a chemcal consttuent of the flud whch has ts own propertes lke vscosty or densty. The phases refer n a sense to the states of matter, such that for example a sngle-component, say H 2 O, multphase system would nvolve the lqud and vapor phases of water. Wth mult-component models t s possble to smulate both mscble and mmscble fluds, such as n [22, 31, 64]. An overvew and ntroducton to some both SC-MP and MC-MP models s gven n [63]. Despte SC-MP models opens the approach of nterestng problems nvolvng for nstance phase separaton [64], capllary rse [53], adsorpton and capllary condensaton [62] we restrct the nvestgaton n ths chapter to snglephase SP) models. These are sngle flud models that represent the behavor of a sngle gas or lqud phase for example. As already mentoned at the end of the precedng chapter an mplementaton of a method demands the restrcton to a fnte number of lattce ponts, meanng to smulate the behavor of the flud n a bounded doman. Consequently, we get boundares and we have to descrbe how the flud behaves on the boundary. The frst secton deals wth boundary condtons, where we gve an ntroducton to the most common ones. In the then subsequent secton, we demonstrate how one can easly add external forces to 1 More precsely the lattce BGK method was ntroduced n the precedng chapter, nevertheless we call t lattce Boltzmann method n the sequel. 48

56 2.1. Boundary Condtons 49 lattce Boltzmann methods. Wth these both enhancements t s possble to smulate for example gravty drven flows n cavtes wth obstacles. Nevertheless, the focus n the current chapter s on the extenson of a SC-SP) lattce Boltzmann method for flud flows by a thermal component, ths treatment s done n the thrd secton. The fourth secton descrbes how the thermal component can nfluence the evoluton of the flud n the sense of buoyancy Boundary Condtons In ths secton we consder lattce Boltzmann models and the flud on a bounded doman. Here, boundedness does not necessarly refer to a stuaton where the flud s enclosed n an mpenetrably boundary, but we consder the flud on the doman of nterest only. In smulatons wth such models we are not nterested n proceedngs whch happen beyond the boundares, explctly. However, we have the opportunty to consder those proceedngs as nput data n our smulaton n the form of boundary condtons. For nstance we can prescrbe the flud s macroscopc velocty on boundares whch could be gven as the results of those proceedngs. Imagne for example a motor n a car and let us assume we are nterested n the moton of the exhaust gas through the exhaust system. In ths case one boundary could be the connecton to the engne where we could prescrbe the pressure or velocty of the exhaust gas, the other boundares of the doman would be the walls of the nvolved exhaust ppes. On the latter we could prescrbe a dfferent behavor of the exhaust gas. The passage where the exhaust gas passes out the exhaust system could be the fnal part of the boundary, and an specfc prescrpton here s also possble. In the latter example most lkely the occurrng boundares are curved. As ths secton s thought as a bref mpresson of how boundares can be consdered n lattce Boltzmann models we restrct the descrpton of boundary condtons here to straght boundares whch are algned wth the man lattce drecton. For boundares not algned wth lattce drectons, schemes based on extrapolatons lke n [11] can be used [72]. A consderaton of curved boundares can be found e.g. n [44]. The restrcton to a bounded doman makes an mplementaton of the lattce Boltzmann method possble, snce then a fnte number of lattce ponts s suffcent. An nteror lattce pont has the full number of neghbors, 18 n the case of the D3Q19 model. For these ponts the man concept of lattce Boltzmann method descrbed n the prevous chapter s stll applcable. Nevertheless, there are lattce ponts whch do not have the full number of neghbors, we denote these lattce ponts as boundary lattce ponts. Recaptulatng the man concept, a streamng and a collson step are successvely executed, n turns. After the streamng step almost all values of the densty dstrbuton populatons) are determned by pre-streamng dstrbuton values of adjacent ponts 2. Consequently, for boundary lattce ponts there are unknown populatons after the streamng step. Fg- 2 The only excepton s the value correspondng to the rest partcle.

57 2.1. Boundary Condtons 50 ure 2.1 llustrates ths stuaton n case of D2Q9 for a south boundary. Provded all populatons are known the collson step can be executed as usual, especally for boundary lattce ponts. A boundary condton n lattce Boltzmann models has the task to fnd the mssng values of the densty dstrbuton, n a way whch leads to a desred macroscopc behavor on the boundary. However, boundary condtons whch replace not only the unknown populatons can be constructed, for example the boundary condtons presented n [58] are nonlocal n the sense that nformaton of neghborng lattce ponts s also used to recompute the densty dstrbuton ncludng also the already known ones. When only the unknown populatons are determned, the acton of a boundary condton can be vewed as and addtonal streamng step from magnary lattce ponts located outsde of the actual numercal doman. Choosng physcal correct boundary condtons s necessary to acheve meanngful results. A smulaton may gan or lose mass durng tme evoluton [12]. Ths necessarly need not to ft wth the real smulatng process. Another effect s that ncorrect densty or velocty values on the boundary can eventually cause negatve dstrbuton values n the nteror [68]. The lattce Boltzmann method tself s correct up to second order n velocty. Implementng nconsstent boundary condtons may affect the total accuracy of results obtaned by that method [38]. It follows that boundary condtons have an mmedate nfluence of the stablty and accuracy of the entre method. Fgure 2.1.: Comparson of nteror and boundary lattce ponts after streamng step. Crcles refer to nteror lattce ponts and squares to boundary lattce ponts. Dashed arrows ndcate unknown values, whereas sold arrows refer to known ones.

58 2.1. Boundary Condtons Perodc Boundary Condtons Perodc boundary condtons are n our opnon the smplest boundary condtons one can thnk of. For boundary lattce ponts they remove the drawback of not havng the complete amount of adjacent lattce ponts n a straghtforward manner. Perodc boundary condtons are appled to a boundary and a correspondng boundary counterpart. The mssng neghborng lattce ponts are replaced wth boundary lattce ponts of the correspondng boundary counterpart, and vce versa. Consder the doman llustrated n fgure 2.1, a possble applcaton of perodc boundary condton could be to lnk parallel the west boundary wth the east one as well as the north wth the south one. Thus, the effectve numercal doman can n ths example be vewed as a torus. In fgure 2.2 the streamng step n D2Q9 s llustrated, where perodc boundary condtons are assumed for the west and east boundary. The extenson to 3D models s straghtforward. Perodc boundary condtons are usually appled f the doman of nterest and the physcal process have a regular symmetry. They are also typcally appled n order to get rd of actual physcal boundares, e.g. f one s nterested n the behavor of a mult-component mxture for a gven ntal state where surface effects play a neglgble role [61, 63]. Implementaton of perodc boundary condtons s not a dffcult task, the only dfference to nteror lattce ponts s that for boundary lattce ponts n equaton 1.42) the post-stream poston ˆx+c ˆt, f t s nonexstent, has to be replaced by a correspondng lattce pont of the boundary counterpart Bounceback Boundary Condtons The second type of boundary condton we present are bounceback condtons whch n lattce Boltzmann methods are usually mplemented f no-slp boundares are present. They are also commonly used for consderaton of sold walls and obstacles. The basc dea s relatvely smple and states that the unknown populatons of the densty dstrbuton are adopted from those of opposte drectons. When a bounceback condton s appled the collson step s omtted at the lattce ponts wth ncomplete amount of dstrbuton values. Hence they dstngush from the boundary lattce ponts and therefore we call them wth a new term as bounceback lattce ponts. The bounceback boundary condton wth wall placed through the bounceback lattce ponts has an nconsstent accuracy wth the lattce Boltzmann method, f nstead the effectve physcal boundary les half way between the bounceback lattce ponts and the frst nteror lattce ponts, the method stll has a second order accuracy [29]. Fgure 2.3 llustrates the poston of the effectve physcal boundary. Remark that boundary lattce ponts are not part of the flud doman whch s why they are also called dry ponts, whereas boundary lattce ponts are called wet ponts. More detaled the prncple of bounceback boundary condtons s the followng. The streamng step transports the densty dstrbuton from an nteror lattce pont to a bounceback lattce pont. In the followng collson step the collson operator s ordnary

59 2.1. Boundary Condtons 52 Fgure 2.2.: Perodc boundary condtons n D2Q9. Sold arrows depct the prestreamng state for an easter boundary lattce pont, whereas the dashed arrows show the post-streamng state. evaluated for nteror lattce ponts and, f exstent, also for boundary lattce ponts. The latter exst when besdes of bounceback type dfferent boundary condtons are appled for other boundares. For bounceback lattce ponts, n place of the collson step, unknown values of the densty dstrbuton are replaced wth the value of opposte drecton. Snce only the values streamng back nto the flud doman are mportant n the subsequent streamng step, a replacement of all values s also possble. Ths extremely smplfes an mplementaton, because the algorthm s then ndependent of the orentaton of the boundary. Fgure 2.4 llustrates the descrpton above. Boundary lattce ponts have to be ntalzed wth an approprate densty dstrbuton. From a mathematcal pont of vew the prncple of bounceback boundary condtons alters the lattce BGK equaton 1.42) nto F ˆx + c ˆt, ˆt + ˆt) F ˆx, ˆt) = 0, for all = 0,..., n v, 2.1) whenever the poston ˆx refers to a bounceback lattce pont. In the latter equaton denotes the ndex of lattce velocty wth opposte drecton to lattce velocty c,.e. c = c. Ths pont of vew shows that bounceback boundary condtons work n the two-dmensonal case as well as n the three-dmensonal one. In both cases they are often used for the handlng of statonary walls and obstacles.

60 2.1. Boundary Condtons 53 Fgure 2.3.: Effectve physcal boundary when bounceback boundary condtons are appled to D2Q9. Empty squares refer to bounceback lattce ponts Velocty Boundary Condtons In ths subsecton we deal wth more sophstcated boundary condtons whch prescrbe the flud s macroscopc velocty on a boundary. We have to convey that macroscopc condton to the mcroscopc model. In the precedng chapter, we already defned the macroscopc quanttes n dependence of the mcroscopc dstrbuton cf. 1.43) and 1.44)), thus t s natural to use these relatons n the consderaton of velocty boundary condtons. In contrast to the case of bounceback boundary condtons, we assume that the effectve physcal boundary passes through boundary lattce ponts, lke shown n fgure 2.1. For a boundary lattce pont ˆx 0 we prescrbe the macroscopc velocty ûˆx 0, ˆt) for a gven tme pont ˆt. The velocty vector conssts of two and three components n the two- and three-dmensonal case, respectvely. The lnkage of the macroscopc velocty to the mcroscopc densty dstrbuton uses the current macroscopc mass densty whch s not and cannot 3 prescrbed for the boundary. Snce the mass densty s an unknown quantty we frst show n the followng paragraph how t can be calculated from the gven quanttes. Afterwards, we return to our man task of calculatng the mssng populatons. 3 In addton to the velocty the densty cannot be prescrbed for the boundary as long as one calculates only the unknown values of the densty dstrbuton. The reason s that the prescrbed velocty and the known values of the dstrbuton unquely mply a mass densty.

61 2.1. Boundary Condtons 54 1) before streamng 2) after streamng / before collson and bounceback after collson and bounceback / 3) before streamng 4) after streamng Fgure 2.4.: Prncple of bounceback boundary condtons. Dotted arrows ndcate that values of densty dstrbuton were recalculated n the collson step. Computaton of Mass Densty The mass densty s an ndependent varable n lattce Boltzmann methods and can be computed for nteror lattce ponts by equaton 1.43). For a boundary lattce ponts 1.43) s not feasble, snce some necessary values for the calculaton are mssng, nevertheless we can calculate the densty out of the known populatons n addton wth the prescrbed velocty. Ths procedure can be found n [35] and reads as follows. We denote the sum of post-stream unknown populatons by ˆρ ˆx 0, ˆt),.e. ˆρ ˆx 0, ˆt) = I F ˆx 0, ˆt),

62 2.1. Boundary Condtons 55 where I = { = 0,..., n v F ˆx 0, ˆt) unknown after streamng step } s the set of ndces of unknown densty dstrbuton values. Smlarly, ˆρ + ˆx 0, ˆt) denotes the sum over drectons opposte to the unknown ones, meanng ˆρ + ˆx 0, ˆt) = F ˆx 0 {, ˆt) wth I + = = 0,..., nv } I, I + where s explaned n 2.1). The remanng populatons whch are those whose correspondng lattce veloctes are tangental to the boundary or zero are summed up and denoted by ˆρ 0 ˆx 0, ˆt). Wth ths notaton the mass densty decomposes, due to 1.43), nto ˆρˆx 0, ˆt) = ˆρ ˆx 0, ˆt) + ˆρ + ˆx 0, ˆt) + ˆρ 0 ˆx 0, ˆt). 2.2) The prescrbed velocty can be used to determne the velocty normal to the boundary û n ˆx 0, ˆt), where the vector normal to the boundary s assumed to pont outsde of the flud n drecton of the boundary. Alternatvely from 1.44) we get the relaton ˆρˆx 0, ˆt)û n ˆx 0, ˆt) = ˆρ + ˆx 0, ˆt) ˆρ ˆx 0, ˆt). 2.3) Hence, we can compute ˆρˆx 0, ˆt) by combnng 2.2) and 2.3), thus t follows ˆρˆx 0, ˆt) = û n ˆx 0, ˆt) 2 ˆρ+ ˆx 0, ˆt) + ˆρ 0 ˆx 0, ˆt) ), 2.4) where on the rght hand sde only known quanttes appear. Computaton of Unknown Densty Dstrbuton We stll am fndng the unknown values of the densty dstrbuton, such that the macroscopc velocty matches the prescrbed one. The number of components the prescrbed velocty vector conssts of gves, by usng 1.44), an equal number of constrants for the unknown values. Ths means n the two-dmensonal case we have two constrants and n the three-dmensonal one we acheve three constrants. The number of unknown values depends on the dscretzaton model. Usng the D2Q9 model we have three unknowns, compare wth fgure 2.1, and n the D3Q19 model we have fve unknowns lke shown n fgure 2.5. In both cases we have more unknown populatons than constrants, hence addtonal relatons are requred. Zou and He [72] suggest as an addtonal relaton that the non-equlbrum part of the dstrbuton normal to the boundary fulflls a bounceback condton. In the lterature the resultng knd of boundary condton s sometmes called Zou and He velocty boundary condton. Followng ths suggeston, we get F n ˆx 0, ˆt) F eq) n ˆx 0, ˆt) = F n ˆx 0, ˆt) F eq) n ˆx 0, ˆt), 2.5)

63 2.1. Boundary Condtons 56 Fgure 2.5.: Illustraton of a boundary lattce pont n D3Q19 after streamng step. Dashed arrows ndcate unknown values, whereas sold arrows are referred to known ones. where n and n s the ndex of lattce velocty normal to the boundary and opposte to the former, respectvely. Note that there exst a lattce velocty c normal to the boundary, snce we assumed that boundares are algned wth the man lattce drecton. For the two-dmensonal case we have three equatons for three unknown values, hence we can easly determne the unknown populatons. The three-dmensonal case s not covered by the foregong consderaton, snce we stll have more unknowns than equatons. Here we focus exclusvely on the D3Q19 model, hence we need to fnd another approprate relaton to acheve a well constraned system of lnear equatons. Fndng ths relaton can however be avoded by bascally extendng the bounceback condton of the non-equlbrum part of the dstrbuton normal to the boundary to all unknowns. Afterwards, a redstrbuton of the densty lke n [38] adjusts the dstrbuton to match the prescrbed velocty. We outlne ths procedure more detaled now. In the prevous paragraph, we have computed the mass densty whch appears n 1.44) on the left hand sde. Together wth the prescrbed velocty the complete left hand sde s gven, we denote the thus prescrbed momentum densty by jˆx 0, ˆt) = ˆρˆx 0, ˆt)ûˆx 0, ˆt). Wthout loss of generalty we assume the boundary to be algned such that n = 1 holds, meanng lattce velocty c 1 = 1, 0, 0) gves the drecton of the vector normal to the boundary cf. 1.48)). After a streamng step F 3 ˆx 0, ˆt), F 8 ˆx 0, ˆt), F 9 ˆx 0, ˆt), F 12 ˆx 0, ˆt) and F 13 ˆx 0, ˆt) are unknown populatons. The value of F 3 ˆx 0, ˆt) can be determned by

64 2.1. Boundary Condtons ) wth n = 1 and n = 3, F 3 ˆx 0, ˆt) = F eq) 3 ˆx 0, ˆt) + F neq) 1 ˆx 0, ˆt). 2.6) We have wrtten n the latter equaton and wll also wrte n the followng the nonequlbrum part of the populaton correspondng to lattce velocty c by F neq) ˆx, ˆt) := F ˆx, ˆt) F eq) ˆx, ˆt). 2.7) A bounceback condton for the non-equlbrum part s used to compute the remanng four unknowns, too, whch yelds F 8 ˆx 0, ˆt) = F eq) 8 ˆx 0, ˆt) + F neq) 10 ˆx 0, ˆt), F 9 ˆx 0, ˆt) = F eq) 9 ˆx 0, ˆt) + F neq) 7 ˆx 0, ˆt), F 12 ˆx 0, ˆt) = F eq) 12 ˆx0, ˆt) + F neq) 14 ˆx 0, ˆt), F 13 ˆx 0, ˆt) = F eq) 13 ˆx0, ˆt) + F neq) 11 ˆx 0, ˆt). 2.8) Wth ths choce the momentum densty normal to the boundary matches the prescrbed one, as shown next. Due to 1.44) we obtan ρˆx 0, ˆt)û 1 ˆx 0, ˆt) = F 1 ˆx 0, ˆt) + F 7 ˆx 0, ˆt) + F 10 ˆx 0, ˆt) + F 11 ˆx 0, ˆt) + F 14 ˆx 0, ˆt) ) F 3 ˆx 0, ˆt) + F 8 ˆx 0, ˆt) + F 9 ˆx 0, ˆt) + F 12 ˆx 0, ˆt) + F 13 ˆx 0, ˆt) ). After nsertng 2.6) and 2.8) for the subtractng terms, we get wth use of a rearranged verson of 2.7), ρˆx 0, ˆt)û 1 ˆx 0, ˆt) = F eq) 1 ˆx 0, ˆt) F eq) 3 ˆx 0, ˆt) + F eq) 7 ˆx 0, ˆt) F eq) 8 ˆx 0, ˆt) = 18 =0 F eq) 9 ˆx 0, ˆt) + F eq) 10 ˆx0, ˆt) + F eq) 11 ˆx0, ˆt) F eq) 12 ˆx0, ˆt) F eq) 13 ˆx0, ˆt) + F eq) 14 ˆx0, ˆt) c 1 F eq) ˆx 0, ˆt), where the sum exactly computes to the prescrbed momentum densty normal to the boundary,.e. to the frst component of jˆx 0, ˆt). Denotng the dfference between jˆx 0, ˆt) and the momentum densty mpled by the dstrbuton wth jˆx 0 j1 ˆx 0, ˆt), ˆt) = j 2 ˆx 0 18, ˆt) := c F ˆx 0, ˆt) jˆx 0, ˆt), 2.9) j 3 ˆx 0, ˆt) =0

65 2.1. Boundary Condtons 58 t holds j 1 ˆx 0, ˆt) = 0. In the foregong calculaton the upper rght ndex 1 always refers to the frst component of the correspondng vector, note that for the gven orentaton of the boundary ths component corresponds to the normal drecton. The prescrbed momentum densty tangental to the boundary s n general not obtaned by the choce 2.8). The fnal step adjusts the formerly four unknown values, such that the momentum densty gets consstent wth the prescrpton. For ths reason we frst compute j 2 ˆx 0, ˆt) and j 3 ˆx 0, ˆt), we get 18 j 2 ˆx 0, ˆt) = c 2 F ˆx 0, ˆt) j 2 ˆx 0, ˆt) = = c 2 F ˆx 0, ˆt) = =0 18 c 2 =0 18 = =0 18 j 3 ˆx 0, ˆt) = =0 =0 F ˆx 0, ˆt) F eq) c 2 F neq) ˆx 0, ˆt), c 3 F neq) ˆx 0, ˆt). c 2 F eq) ˆx 0, ˆt) ) ˆx 0, ˆt) We try to redstrbute j 2 ˆx 0, ˆt) and j 3 ˆx 0, ˆt) over the formerly four unknown populatons, such that follows jˆx 0, ˆt) = 0. Snce c 2 8 = 1 and c2 9 = 1 holds, ncreasng F neq) 8 ˆx 0, ˆt) and decreasng F neq) 9 ˆx 0, ˆt) leads to an ncrease of j 2 ˆx 0, ˆt). The other way round, decreasng F neq) 8 ˆx 0, ˆt) and ncreasng and F neq) 9 ˆx 0, ˆt) leads to a decrease of j 2 ˆx 0, ˆt). An analogue effect s gven for F neq) 12 ˆx 0, ˆt), F neq) 13 ˆx 0, ˆt) and j 3 ˆx 0, ˆt). Hence, the redstrbuted non-equlbrum values F neq) wth the desred property are obtaned by the general equaton F neq) ˆx 0, ˆt) = F neq) ˆx 0, ˆt) 1 2 c2 j 2 ˆx 0, ˆt) 1 2 c3 j 3 ˆx 0, ˆt), 2.10) wth = 8, 9, 12, 13 for the gven orentaton. j 1 ˆx 0, ˆt) s not nfluenced by ths redstrbuton. Boundares of dfferent orentatons, stll algned wth the man lattce drecton, are treated analogue. The procedure descrbed by equaton 2.10) redstrbutes the excess momentum 2.9) unformly over the unknown populatons. Summarzng, velocty boundary condtons can be realzed by computng the macroscopc densty wth 2.4), replacng the unknown populaton normal to the boundary wth help of 2.5) and the remanng four unknown populatons by approprate equa-

66 2.1. Boundary Condtons 59 tons referrng to 2.10). Implementng velocty boundary condtons therefore demands a dstncton of the orentaton of the boundary Pressure Boundary Condtons The last type of boundary condtons we present n the current secton are those whch prescrbe, smlar to velocty boundary condtons, a dfferent macroscopc quantty on the boundary, namely the pressure. Actually, ths means that a macroscopc mass densty s prescrbed on the boundary, snce the pressure s proportonal to the densty, see especally 1.57) for D3Q19. A pressure boundary condton can be combned wth addtonal constrants for the macroscopc velocty, but only the components tangental to the boundary can be prescrbed. The velocty normal to the boundary û n ˆx 0, ˆt) nstead s already unquely determned by the pressure condton, as one can see n 2.4). Rearrangng ths equaton leads to û n ˆx 0, ˆt) = 2 ˆρ +ˆx 0, ˆt) + ˆρ 0 ˆx 0, ˆt) ˆρˆx 0, ˆt) where ˆρˆx 0, ˆt) denotes the prescrbed densty. Hence, the task reads to determne the unknown values of the dstrbuton. Agan a bounceback condton 2.5) for the nonequlbrum part of the dstrbuton normal to the boundary s used [72]. In D2Q9 t remans to solve for two unknowns and provded addtonally a velocty tangental to the boundary s prescrbed ths task s not a dffcult one, snce t means to solve a well constraned system of lnear equatons. If no velocty tangental to the boundary s prescrbed we have only one condton to be fulflled, but snce two populatons need to be determned ths cannot be solved unquely. The only condton s gven by 1, ˆρˆx 0, ˆt) = 8 F ˆx 0, ˆt), =0 and the problem can be solved for nstance by further assumng the two unknown populatons are equal. The task n the three-dmensonal D3Q19) case s to determne four populatons. When we suppose that the pressure boundary condton s gven n addton wth tangental veloctes, the dea descrbed n the second paragraph of the precedng subsecton can adopted here [35, 38]. However, f no addtonal velocty condtons are mposed one can extend the prncple descrbed above for D2Q9 to the three-dmensonal, analogue.

67 2.2. Addtonal Force Terms Addtonal Force Terms When consderng the moton of a flud, the flud would come to rest after some tme due to the vscosty f there are no forces drvng the flud. For nstance, these forces can result from a pressure gradent or an external force lke gravty actng on the flud. There are also other stuatons where an external or nternal force has to be consdered. Anyway, the demand of extendng the smulaton by effects resultng from forces actng on the flud s gven. On the macroscopc scale the Naver-Stokes equatons descrbe the moton of the flud. We already know from the results n the precedng chapter that lattce Boltzmann methods smulate flud motons descrbed by the ncompressble Naver-Stokes equatons cf. 1.81)) provded the densty varaton s small. An addtonal force can be ncorporated to these equatons by smply addng a term expressng the acceleraton due to that force on the rght hand sde of the momentum equaton [60]. Therefore, the complete set of ncompressble Naver-Stokes equatons whch takes nto account addtonal forces reads u = 0, u + u ) u = ˆp + ˆν u + k, 2.11) where k s the acceleraton due to the addtonal force densty K = ρ k. The lattce Boltzmann method has to be altered n such a way that ther hydrodynamcal behavor s descrbed by equatons 2.11). Then the model consders effects comng from addtonal forces n a physcal correct manner. Guo et al. state that n order to obtan the correct contnuty equaton, the flud velocty must be defned such that the effect of the external force s ncluded, and to obtan the correct momentum equaton, the contrbutons of the force to the momentum flux must be canceled [24]. They also propose an approach fulfllng these both condtons. In the followng, we brefly present ther model n the current secton and use t for the consderaton of buoyancy effects below n secton 2.4. Bascally, we can ncorporate addtonal forces by smply modfyng the collson step and usng a slghtly dfferng formula for the calculaton of the macroscopc velocty. Hence, the consderaton of addtonal forces can be done wthout much effort. Recaptulatng the equlbrum dstrbuton 1.55) of D3Q19 F eq) ˆx, ˆt) = ω ˆρˆx, ˆt) [1 + 3c ûˆx, ˆt)) + 92 c ûˆx, ˆt)) 2 32 ] ûˆx, ˆt) 2, = 0,..., 18. We can rewrte ths equaton to [ F eq) ˆx, ˆt) = ω ˆρˆx, ˆt) 1 + c ûˆx, ˆt) [ûˆx, ˆt)ûˆx, ˆt) T ] : [ c c T c 2 c 2 + si ] ] s 2c 4, 2.12) s

68 2.2. Addtonal Force Terms 61 wth c s = 1 3, I beng the dentty matrx and the twce contracton product as follows 4 : Notaton. Let A and B be two matrces of same dmenson, say A = a j ),j=1,...,n and B = b j ),j=1,...,n. Then the twce contracton product A : B s computed as follows A : B = n n a j b j. =1 j=1 Ths means the twce contracton s the summaton of the component wse multplcaton. Equaton 2.12) s a general expresson of the equlbrum equaton. Assumng a dscrete forcng term s gven n a smlar more generalzed form K ˆx, ˆt) = ω [ l 1 + l 2 c c 2 s + l 3 : [ c c T c 2 si ] ] 2c 4, 2.13) s wth unknown scalar l 1, vector l 2 and matrx l 3 each dependng on the force densty Kˆx, ˆt) whch need to be determned. The weghts ω are equal wth those of the equlbrum dstrbuton. On the one hand the modfcaton of the collson term adds K ˆx, ˆt) to the known collson term from the prevous chapter. Hence, ths modfcaton changes 1.42) nto F ˆx + c ˆt, ˆt + ˆt) F ˆx, ˆt) = 1ˆτ [ ] F ˆx, ˆt) F eq) ˆx, ˆt) + K ˆx, ˆt). 2.14) c On the other hand we use the altered equaton u ˆx, ˆt) = 1 nv ) c F ˆx, ˆt) + Kˆx, ˆt) ρˆx, ˆt) 2 =0 2.15) to compute the macroscopc velocty whch s especally used n the computaton of the equlbrum dstrbuton n the collson step. The unknowns n 2.13) have to be determned such that the modfed model leads to the correct macroscopc equatons n a mult-scale expanson. Ths s done n [24], the authors conclude that a feasble choce s gven by l 1 = 0, l 2 = 1 1 ) Kˆx, ˆt), l 3 = 1 1 ) 2u ˆx, ˆt)Kˆx, ˆt) T. 2ˆτ c 2ˆτ c 4 Remark: The pressure/densty relaton 1.57) can be wrtten as pˆx, ˆt) = c 2 sρˆx, ˆt), vald not only for D3Q19. c s gves the speed of sound n the correspondng model.

69 2.3. Incorporaton of a Thermal Component 62 Insertng c s = 1 3 and the latter values n 2.13), t follows K ˆx, ˆt) = 1 1 ) [ ω 3 c u ˆx, ˆt) ) + 9 c u ˆx, ˆt) ) ] c Kˆx, ˆt). 2ˆτ c Usng ths dscrete force term and the lttle model changes 2.14) and 2.15) the resultng equatons n a Chapman-Enskog approxmaton match the ncompressble Naver- Stokes equatons ncludng addtonal forces. Therefore, the model ncorporates addtonal forces n a physcal correct manner. Note that f no forces are present, meanng Kˆx, ˆt) = 0, the modfed model s equvalent to that presented n the prevous chapter Incorporaton of a Thermal Component The lattce Boltzmann method or more precsely the lattce BGK method presented n the prevous chapter s a numercal approach to smulate ncompressble flud flows. Although the method presented n the prevous chapter satsfes the ncompressble Naver- Stokes equatons, an applcaton of the LBGK method to a varety of physcal problems s often possble not untl one consders correspondng boundary and surface condtons. We touched that topc at the begnnng of the current chapter. Despte the consderaton of boundary and surface condtons enables an applcaton to a varety of problems, the method s stll only capable of smulatng sothermal flows. The goal of ths secton s to develop a numercal method to smulate the temperature evoluton. Therefore we propose a method whch extends the lattce BGK method such that the temperature s advected by the flud s flow and dffuses n t, meanng the temperature obeys an advecton-dffuson equaton. Lke vscosty n the lattce BGK method, a parameter whch can be chosen to tune the thermal dffusvty s avalable. The flud tself behaves lke an sothermal flud, that means n areas of dfferent temperatures the flud s vscosty does not dstngush. Consequently, the latter fact lmts the applcaton of the proposed method to problems n whch only a relatvely small temperature range appears or at least to those n whch the nfluence from the temperature on the vscosty s small. Furthermore, the compresson work done by the pressure and the vscous heat dsspaton are neglected. When smulatng the moton of real ncompressble fluds n applcatons these terms are very often) neglgble. A less smple method whch ncorporates also the compresson work done by the pressure and the vscous heat dsspaton can be found n [26]. In lterature, the exstng approaches to add a thermal component to lattce BGK methods can be grouped n at least two categores. On the one hand there s a multspeed approach whch uses only the densty dstrbuton functon e.g. [1, 41]). In these approaches a hgher order approxmaton of the equlbrum dstrbuton s necessary to obtan the energy evoluton equaton and thus an energy conservatng method [40]. On the other hand there s the category of approaches whch do not use the densty dstrbuton of the flud alone. One possblty s to use so called passve-scalar ap-

70 2.3. Incorporaton of a Thermal Component 63 proaches whch are based on multple component models [56], where the temperature s regarded as an addtonal component wthout mass whch behaves lke an passve dffusng one only advected by the flow [55, 57]. These procedures have the advantage that the thermal dffusvty can be tuned ndependently of the vscosty, whereas n multspeed approaches a fxed Prandtl number of usually P r = 1 2 appears. The Prandtl number s a dmensonless quantty descrbng the rato of knematc vscosty and thermal dffusvty. Another possblty s gven by mult-dstrbuton approaches, these schemes are smlar to the passve-scalar approaches because they also use a separate dstrbuton functon to smulate the temperature evoluton. The numercal method we propose n the followng s classfed to them. In ths procedure, the LBGK method smulatng the densty and velocty evoluton can theoretcally be decoupled from the temperature evoluton, thus for nstance a classcal Naver-Stokes solver can be used to smulate the densty and velocty evoluton nstead. Although we have already mentoned above that the proposed method wll satsfy an advecton-dffuson equaton, we determne the desred temperature evoluton equaton n the subsequent subsecton extensvely Determnaton of Macroscopc Equaton In the prevous chapter we dealt wth the LBGK method, where we could show that the numercal method obeys the well known ncompressble Naver-Stokes equatons. Ths subsecton dscusses the determnaton of the general macroscopc equaton the temperature obeys, and thus clears up the queston how the evoluton equaton for the temperature looks lke. The determnaton uses the Boltzmann equaton drectly and s analogue to [26]. Here, we focus on the three-dmensonal case exclusvely, the generalzaton to R d, d {2, 3}, s straghtforward. Snce we already know a relaton between the temperature and the densty dstrbuton fx, v, t) satsfyng the Boltzmann equaton, t s naturally to start wth ths relaton. Recaptulatng that relaton cf. 1.8)) gves ρx, t)ex, t) = v ux, t) 2 fx, v, t) dv, 2.16) 2 wth the nternal energy Ex, t) = 3 k B T x,t) 2 m n. We defne a new dstrbuton, callng t the energy dstrbuton, by gx, v, t) = v ux, t) 2 fx, v, t). 2 For the new dstrbuton we derve a PDE smlar to the Boltzmann equaton. The desred macroscopc temperature evoluton equaton s then obtaned from ths PDE va a Chapman-Enskog expanson.

71 2.3. Incorporaton of a Thermal Component 64 PDE for energy dstrbuton A dervaton of the energy dstrbuton wth respect to tme results n gx, v, t) = = 1 2 v ux, t) 2 ) fx, v, t) 2 [ 2 fx, v, t) v ux, t) + fx, v, t) ] v ux, t) 2. Smlarly, we obtan for the advecton term ) 1 v gx, v, t) = v 2 v ux, t) 2 fx, v, t) = 1 [ fx, v, t)v v ux, t) 2 ) + v ux, t) 2 v fx, v, t) ]. 2 Combnng both equatons results n gx, v, t) [ v ux, t) 2 fx, v, t) + v gx, v, t) = 2 fx, v, t) ] + v fx, v, t) [ 1 v ux, t) v v ux, t) 2)], where the left hand sde of the Boltzmann equaton 1.4) appears n the square bracket on the frst lne of the rght hand sde. Snce the densty dstrbuton fx, v, t) satsfes the Boltzmann equaton, the square bracket can be substtuted by the correspondng rght hand sde Qf, f) of the Boltzmann equaton. A new collson term s ntroduced whch corresponds to a substtuton of the square bracket by the rght hand sde of the BGK equaton 1.29), where a new relaxaton parameter τ d s used, meanng that wth v ux, t) 2 Qf, f) 2 g eq) x, v, t) := v ux, t) 2 2 = v ux, t) 2 ρx, t) 2 v ux, t) 2 2 1τd [ ] ) fx, v, t) f eq) x, v, t) = 1 ] [gx, v, t) g eq) x, v, t), τ d f eq) x, v, t) 2.17) ) m 3/2 ) m exp v ux, t) 2. 2πk B T x, t) 2k B T x, t)

72 2.3. Incorporaton of a Thermal Component 65 It follows a frst PDE for the energy dstrbuton: gx, v, t) + v gx, v, t) = 1 ] [gx, v, t) g eq) x, v, t) τ d [ fx, v, t) 1 v ux, t) v v ux, t) 2)]. 2.18) However, ths equaton can be smplfed, especally the square bracket on the second lne. For ths reason, we compute the dervatve wth respect to tme as follows 1 v ux, t) 2 2 = 1 v ux, t) = v = v ux, t)) v ux, t)) ux, t) ux, t). 1 ux, t) 2 2 ux, t) The second term n the square bracket of 2.18) s smplfed wth help of and v ux, t)) = [v )ux, t)] 1 [ v ux, t) 2 )] 2 = 1 [ v u 1 x, t) 2 ) + v u 2 x, t) 2 ) + v u 3 x, t) 2 ) ] 2 = u 1 x, t) [ v u 1 x, t) )] + u 2 x, t) [ v u 2 x, t) )] + u 3 x, t) [ v u 3 x, t) )] = u 1 x, t) [ v ) u 1 x, t) ] + u 2 x, t) [ v ) u 2 x, t) ] + u 3 x, t) [ v ) u 3 x, t) ] = ux, t) [v )ux, t)]. Usng the latter two equatons, the mentoned second term can be smplfed as follows 1 2 v v ux, t) 2) = 1 2 v v 2 2v ux, t) + ux, t) 2) = v v ux, t)) 1 [ v ux, t) 2 )] 2 = v [v )ux, t)] ux, t) [v )ux, t)] = v ux, t)) [v )ux, t)].

73 2.3. Incorporaton of a Thermal Component 66 By usng all smplfcatons together, the smplfed form of 2.18) becomes our fnal PDE for the energy dstrbuton. It reads gx, v, t) + v gx, v, t) = 1 ] [gx, v, t) g eq) x, v, t) fx, v, t)qx, t), τ d 2.19) wth [ ] ux, t) qx, v, t) = v ux, t)) + v )ux, t). 2.20) Chapman-Enskog Approxmaton Followng [26], the last equaton n the prevous paragraph, 2.19), s used n a Chapman- Enskog approxmaton to determne the macroscopc temperature evoluton equaton. Therefore, we denote the equlbra of the densty and energy dstrbuton by f 0) x, v, t) and g 0) x, v, t), respectvely. The dstrbutons are expanded around ther correspondng equlbrum, t follows fx, v, t) = f 0) x, v, t) + δf 1) x, v, t) + δ 2 f 2) x, v, t) +..., gx, v, t) = g 0) x, v, t) + δg 1) x, v, t) + δ 2 g 2) x, v, t) +..., 2.21) wth δ beng the expanson parameter. We also expand or scale the occurng operators by = δ 1) 2) + δ2 +..., = δ 1). 2.22) Consderng the defnton 2.20) and the operators 2.22), we observe that qx, v, t) has the followng expresson qx, v, t) = δq 1) x, v, t) + δ 2 q 2) x, v, t) +..., 2.23) wth [ ] q 1) 1) ux, t) x, v, t) = v ux, t)) + v 1) )ux, t), [ ] q k) k) ux, t) x, v, t) = v ux, t)), 2 k N.

74 2.3. Incorporaton of a Thermal Component 67 Substtuton of 2.21), 2.22) and 2.23) nto 2.19) yelds the Chapman-Enskog expanson. The frst order terms gve the followng equaton 1) g 0) x, v, t) + v 1) g 0) x, v, t) = g1) x, v, t) τ d f 0) x, v, t)q 1) x, v, t), 2.24) whereas those of order two read [ ] 2) g 0) x, v, t) 1) + +v 1) g 1) x, v, t) = g2) x, v, t) f 0) x, v, t)q 2) x, v, t) τ d f 1) x, v, t)q 1) x, v, t). 2.25) Hgher order terms are dsregarded. Integraton of 2.24) and 2.25) over velocty space leads to the temperature evoluton equaton. Before we ntegrate, we splt q 1) x, v, t) n 2.23) nto two terms q 1) x, v, t) = q 1) 1 x, v, t) + q1) 2 x, v, t), 2.26) where only q 1) 1 x, v, t) contrbutes to the thermal conducton term n the evoluton equaton resultng below, whereas q 1) 2 x, v, t) only affects the terms of the compresson work done by pressure and the vscous heat dsspaton. We set [ ] q 1) 1 x, v, t) = v ux, t)) 1) ux, t) + ux, t) 1) )ux, t), thus the term n the square bracket corresponds to the frst order left hand sde of the ncompressble Naver-Stokes equatons, therefore we can substtute t by the correspondng rght hand sde. We obtan q 1) 1 x, v, t) = v ux, t)) [ 1) px, t) ρx, t) The remanng unknown term q 1) 2 x, v, t) s then gven by the dfference of q1) x, v, t) and q 1) 1 x, v, t). The notaton explaned on page 61 s used n the expresson of ths ].

75 2.3. Incorporaton of a Thermal Component 68 dfference. q 1) 2 x, v, t) = q1) x, v, t) q 1) 1 x, v, t) [ ] 1) ux, t) = v ux, t)) + v 1) )ux, t) [ = v ux, t)) v ux, t)) 1)] ux, t) = [ v ux, t)) v ux, t)) T ] : [ 1) ux, t) ]. Now we ntegrate 2.24) and 2.25), for the sake of clarty we only gve the results here. More nformaton to the calculaton can be found n appendx A.3. Integratng the frst order equaton 2.24) over velocty space yelds 1) ρx, t)ex, t)) + 1) ρx, t)ux, t)ex, t)) = px, t) 1) ux, t). 2.27) When ntegratng the second order terms 2.25), one obtans [26] 2) ) [ ] ρx, t)ex, t)) = 1) ρx, t)κ 1) Ex, t) + Λ 1) x, t) : 1) ux, t), 2.28) where the coeffcent κ s the thermal conductvty and [ ) )] T Λ 1) x, t) = ρx, t)ν 1) ux, t) + 1) ux, t). Fnally, combnng 2.27) and 2.28) leads to the complete evoluton equaton up to second order ρx, t)ex, t)) + ρx, t)ux, t)ex, t)) = ρx, t)κ Ex, t)) + Λ 1) x, t) : [ ux, t) T ] px, t) ux, t). 2.29) The both terms n the second lne refer to the vscous heat dsspaton and the compresson work done by the pressure, respectvely. As explaned above, we assume these terms are neglgble, thus 2.29) reduces to ρx, t)ex, t)) + ρx, t)ux, t)ex, t)) = χ ρx, t)ex, t)), 2.30) wth Laplacan and thermal dffusvty χ. We have treated ρx, t) as a constant,.e. assumed agan that changes n mass densty are neglgble. Equaton 2.30) s the desred advecton-dffuson equaton we have already spoken about at the begnnng of the current secton.

76 2.3. Incorporaton of a Thermal Component Numercal Scheme In ths subsecton, we propose a numercal scheme to smulate the temperature evoluton, lke n [48, 49]. Lke n secton we focus agan on the three-dmensonal case takng the D3Q19 dscretzaton model explctly. The method orgnates n solvng gx, v, t) + v gx, v, t) = 1 ] [gx, v, t) g eq) x, v, t), 2.31) τ d where the equlbrum dstrbuton g eq) x, t) depends on the macroscopc densty and velocty. One possblty to smulate the densty and velocty evoluton s the lattce BGK method, presented n chapter 1. Hence, we understand the numercal scheme for the temperature evoluton as an extenson to the lattce BGK method cf. especally 1.42)) whch should be solved smultaneously. For our purpose, approxmatng the soluton of 2.31) numercally demands a dscretzaton of space. Usng the same dscretzaton as for the lattce BGK method s a natural choce. We propose to use the method gven by [49] g x + c t, t + t) g x, t) = 1 [ ] g x, t) g eq) x, t), = 0,..., 18, 2.32) τ d where the veloctes c are those of the lattce BGK equaton and the equlbrum dstrbuton g eq) x, t) s a dscretzed approxmaton of the contnuous equlbrum dstrbuton 2.17). The quanttes g x, t) descrbe the assocated dscrete energy dstrbuton whch s characterzed by 18 =0 g x, t) = ρx, t)ex, t). 2.33) In the followng two paragraphs, we deduce the dscrete equlbrum dstrbuton and use a Chapman-Enskog expanson to verfy that the method 2.32) recovers the macroscopc equaton, 2.30), determned n the prevous subsecton. Dscrete Energy Equlbrum Dstrbuton The dscrete energy equlbrum dstrbuton s a dscretzed approxmaton of the contnuous equlbrum dstrbuton 2.17). Observng the calculatons n secton 2.3.1, we see that only ntegratons up to frst order of the equlbrum dstrbuton were necessary to derve the macroscopc energy equaton,.e. the ntegrals g eq) x, v, t) dv and vg eq) x, v, t) dv. 2.34)

77 2.3. Incorporaton of a Thermal Component 70 The dscrete equlbrum dstrbuton orgnates n a Taylor expanson of the contnuous dstrbuton n u around zero up to second order, ths expanson reads cf. 1.46)) g eq) = 1 2 v u 2 f eq) = Em 3k B T v u 2 f eq) ) m 3/2 ) [ m m v u 2 = ρe exp 2πk B T 2k B T v 2 3k B T ) m 3/2 ) m = ρe exp 2πk B T 2k B T v 2 [ m v 2 m v 2 3k B T + 3k B T 2 ) mv u) m v k B T 3k B T 4 3 m v 2 3k B T 2 ) ] m 3 2k B T u 2 + O u 3 ), m exp v u) m k B T ) m ) 2 v u) 2 k B T 2 )] 2k B T u 2 where functon arguments were omtted for the sake of clarty. Splttng the terms of the square bracket yelds m g eq) = ρe 2πk B T [ m v 2 ) 3/2 m exp 2k B T v 2 m v 2 3k B T + 3k B T 2 ) mv u) 3 k B T ) m 3/2 ) m + ρe exp 2πk B T 2k B T v 2 [ m v ) 2 2 v u) 2 3k B T 7 3 ) m k B T 2 ) ) m 2 v u) 2 + k B T 2 ] m 2k B T u 2 m v 2 3k B T 5 ) ] m 3 2k B T u 2 +O u 3 ), where then the second addend can be neglected, snce the ntegrals 2.34) are not affected by ths term, meanng that ) m 3/2 ) [ m m v 2 ρe exp 2πk B T 2k B T v 2 3k B T 7 ) ) m 2 v u) 2 3 k B T 2 m v 2 3k B T 5 ) ] m 3 2k B T u 2 dv = 0

78 2.3. Incorporaton of a Thermal Component 71 and ) m 3/2 ) [ m m v 2 vρe exp 2πk B T 2k B T v 2 3k B T 7 ) ) m 2 v u) 2 3 k B T 2 m v 2 3k B T 5 ) ] m 3 2k B T u 2 dv = 0. We are lookng for a dscrete equlbrum dstrbuton, and snce the remanng term s of smlar shape as the densty equlbrum dstrbuton 1.46), we use the same ansatz functon for the dscrete energy equlbrum dstrbuton. Ths means, we try to determne coeffcents A, B, C and D n g eq) x, t) = ω ρx, t)ex, t) [ A + B c ux, t)) + C c ux, t)) 2 + D ux, t) 2], 2.35) for = 0,..., 18, such that the dscrete energy equlbrum satsfes 18 =0 g eq) x, t) = ρx, t)ex, t). 2.36) In 2.35), the appearng weghts ω are gven by 1.56) and lattce veloctes are gven by 1.48). We can avod a drect computaton of the coeffcents A, B, C and D whch would be smlar to the computaton n subsecton Namely, by comparson of m 1.46) and 1.55) we conclude that t holds k B T = 3, hence we try to use [ g eq) x, t) = ω ρx, t)ex, t) c 2 + c 2 2 ) 3c ux, t)) c ux, t)) ux, t) 2 ], whch leads by consderng the lattce veloctes 1.48) for D3Q19 to g eq) 0 x, t) = 1 3 ρx, t)ex, t) [ 3 2 ux, t) 2 ] = 1 2 ρx, t)ex, t) ux, t) 2, g eq) j x, t) = 1 [ 18 ρx, t)ex, t) 1 + c j ux, t)) c j ux, t)) 2 3 ] 2 ux, t) 2, g eq) k x, t) = 1 [ 36 ρx, t)ex, t) 2 + 4c k ux, t)) c k ux, t)) 2 3 ] 2 ux, t) 2, 2.37)

79 2.3. Incorporaton of a Thermal Component 72 for j = 1,..., 6 and k = 7,..., 18. moreover t also holds Summng up verfes that 2.36) s fulflled and 18 =0 c g eq) x, t) = ρx, t)ux, t)ex, t). 2.38) Hence, the numercal scheme 2.32) s now well defned, n the subsequent paragraph we show that t s also meanngful. Verfcaton of Macroscopc Equaton Lke n secton 1.6, a Chapman-Enskog expanson can be used here to recover the advecton-dffuson equaton, and therefore to prove that the proposed method s meanngful from a physcal pont of vew. Most of the calculatons wthn ths verfcaton are analogue to those of secton 1.6. In order to understand the procedure n the current paragraph well, we recommend the reader to recaptulate the calculatons of secton 1.6. We begn the procedure n the same way as we dd when recoverng the Naver-Stokes equatons n the prevous chapter. Ths means we employ a Taylor expanson of 2.32), leadng to cf. 1.58)) ) ˆε + c g x, t) + 1 ) 2 2 ˆε2 + c g x, t) = 1 [ ] g x, t) g eq) x, t), τ d 2.39) where t = ˆε was replaced by a parameter ˆε, smlar to the computaton leadng to 1.58). The energy dstrbuton s expanded as g x, t) = g 0) x, t) + ˆεg 1) x, t) + ˆε 2 g 2) x, t) + Oˆε 3 ), 2.40) where g 0) x, t) denotes the equlbrum dstrbuton g eq) x, t). Furthermore, we use the expanded tme dfferental operator cf. 1.62)) ˆt = 1) ˆt + ˆε 2) ˆt + Oˆε2 ) and the scaled spatal dfferental operator 1.63) = 1). By substtutng these three expansons nto 2.39), we can deduce the frst and second

80 2.3. Incorporaton of a Thermal Component 73 order expanson, respectvely. They are gven analogue to 1.64) and 1.66) by and [ 1 1 2τ d ) 1) 1) + c 1) + c 1) ) ) g 0) x, t) = 1 τ d g 1) x, t), 2.41) ] g 1) x, t) + 2) g0) x, t) = 1 g 2) x, t), 2.42) τ d respectvely. Due to 2.36) and 2.38), summaton of 2.41) over all = 0,..., 18 yelds the frst order approxmaton where we assume cf. 1.61)) 1) ρx, t)ex, t)) + 1) ρx, t)ux, t)ex, t)) = 0, 2.43) 18 =0 g 1) x, t) = 18 =0 g 2) x, t) = ) The correspondng second order approxmaton s obtaned by takng summaton of 2.42), thus we obtan 2) ρx, t)ex, t)) ) Π 2) x, t) = 0, 2.45) 2τ d wth Π 2) x, t) = 18 =0 1) + c 1) ) g 1) x, t). A further nvestgaton of Π 2) x, t) s necessary for our purpose. The authors of [49] state, that one can obtan Π 2) x, t) = 5 9 τ d 2) ρx, t)ex, t)), 2.46) after neglectng some approprate terms, where 2) = s a scaled Laplace operator. Usng ths equaton we rewrte 2.45) as 2) ρx, t)ex, t)) = 5 τ d 1 ) 2) ρx, t)ex, t)). 9 2 The latter equaton combned wth 2.43) gves the Chapman-Enskog approxmaton up

81 2.4. Buoyancy Effects 74 to second order whch reads ρx, t)ex, t)) + ρx, t)ux, t)ex, t)) = χ ρx, t)ex, t)), wth χ = 5 9 τd 1 ) 2 beng the coeffcent of thermal dffusvty. The latter equaton matches wth the desred advecton-dffuson equaton 2.30). Note that an dea of the dervaton of equaton 2.46) s gven n appendx A.4. In the precedng calculatons we have shown that the numercal method gven by 2.32) can be used to smulate the evoluton of the energy Ex, t) whch dffuses and s advected by the flud flow. The rate of dffuson can be tuned by the relaxaton parameter τ d. The energy s related to the temperature T x, t) by Ex, t) = 3 2 all, we are capable to smulate the evoluton of the temperature. k B T x,t) m n. All n 2.4. Buoyancy Effects It s well known that warmer ar rses whch for nstance enables a hot-ar balloon to fly. Ths phenomenon results from the fact that warmer ar has a lower mass densty than colder one and n addton wth gravty ths leads to an effectve force drected upwards. In the last secton, we have developed a model smulatng the temperature evoluton. The temperature n ths model s advected by the flud and dffuses n t. Therefore, the temperature evoluton s drectly nfluenced by the flud flow, but the other way round, we do not consder the fact that temperature, or more precsely temperature dfference, affects the moton of the flud, yet. In the current secton, we descrbe flud flows nvolvng buoyancy effects due to temperature dfferences on the macroscopc scale and demonstrate how one can smulate them easly wth lattce Boltzmann methods Adjustment of the Macroscopc Equaton In the current subsecton we descrbe flud flows nvolvng buoyancy effects due to temperature dfferences on the macroscopc scale. We focus on the second equaton of 2.11), the momentum equaton, n the followng [16, 50]. If the occurrng force term results from gravty, then the momentum equaton can be wrtten as follows u + u ) u = ˆp + ˆν u + g, 2.47) where g s the vector of acceleraton due to gravty. Multplyng 2.47) wth mass densty ρ yelds ρ u + ρ u ) u = p + µ u + ρ g, 2.48)

82 2.4. Buoyancy Effects 75 where µ s the dynamc vscosty. Inspectng ths equaton f the flud s n a statonary equlbrum wthout any moton or at least wth a homogeneous velocty, t reduces to p e = ρ e g 0 = p e + ρ e g, 2.49) where p e and ρ e are the pressure and densty n that equlbrum state, respectvely. We can wrte the pressure as p = p e + p m, 2.50) wth an approprate p m. Actually, the ncompressble Naver-Stokes equatons 2.11), especally 2.48), are satsfed only by fluds wth a constant densty, nevertheless f present densty varatons are small and therefore neglgble we also can take these equatons. Hence, we may assume that the densty can be wrtten n the form wth an approprate ρ m ), one obtans ρ = ρ e + ρ m, 2.51) By nsertng 2.50) and 2.51) nto 2.48) and addng ρ e + ρ m ) u + ρ e + ρ m ) u ) u = p m + µ u + ρ m g. 2.52) The approach presented n the current secton uses the Boussnesq approxmaton whch s based on the assumpton that n the governng macroscopc equatons varatons n mass densty and temperature are neglgble everywhere except n the term descrbng the gravtatonal force [14]. Ths means ρ m = 0, and thus ρ e = ρ s used except n the latter term, where the densty s assumed to depend exclusvely on the temperature but only n the sense of a frst order truncated Taylor seres expanson. Ths lnear relaton s expressed by ρ = ρ e 1 βt T e )), 2.53) wth β the coeffcent of thermal expanson and T e the temperature measured n statonary equlbrum. The Boussnesq approxmaton smplfes 2.52) nto ρ u + ρ u ) u = p m + µ u + ρ m g. 2.54) The assumpton of lnear relaton 2.53) mples ρ ρ e }{{} =ρ m = ρ e βt T e ) ρβt T e ),

83 2.4. Buoyancy Effects 76 snce ρ e ρ. ρ m n 2.54) can be substtuted by the approxmaton ρβt T e ). Dong so and after dvdng by ρ, we acheve u + u ) u = ˆp m + ˆν u βt T e ) g, 2.55) wth ˆp m = pm ρ and ˆν = µ ρ as n 1.81). Eventually, on the macroscopc scale the buoyancy term s fully descrbed as an addtonal force dependng on the temperature dfference. We remark that the pressure gradent was altered and dstngushes from the one n 2.11) Realzaton n LBM In ths subsecton, we show how the buoyancy term descrbed n the precedng subsecton can be consdered n lattce Boltzmann methods. On the one hand, we have a descrpton of the buoyancy term on the macroscopc scale. On the other hand, we have seen n secton 2.2 how force terms can be ncorporated n lattce Boltzmann methods. Both facts can be combned to realze an ncorporaton to the numercal method. As remarked at the end of the prevous subsecton, equatons 2.11) and 2.55) dffer also n the pressure gradent, hence the task s to brng both sdes n lne. We begn ths task wth an nvestgaton of the pressure gradent n 2.55) multpled by ρ. Wth use of 2.49) and 2.50), t follows Hence, 2.55) s equvalent to ρˆp m ) = p m = p p e ) = p + ρ e g. u + u ) ) u = ˆp ρe + ˆν u + ρ βt T e) g, 2.56) and a reasonable approxmaton reads u + u ) u = ˆp + ˆν u + 1 βt T e )) g, 2.57) snce ρe ρ s close to 1. Equaton 2.56) can also be acheved drectly by substtutng 2.53) nto the last term n 2.48). Then, the approxmaton 2.57) s obtaned out of of 2.56) by the same step whch besdes s consstent wth the calculaton n the dervaton of the ncompressble Naver-Stokes equatons n the precedng chapter. There, we also assumed that varatons of mass densty are neglgble. Fnally, the governng equaton respectng the buoyancy effect 2.57) s n lne wth 2.11). Choosng k = 1 βt T e )) g and usng the modfed model presented n secton 2.2, we can ncorporate buoyancy effects due to temperature dfferences n lattce Boltzmann methods.

84 CHAPTE Numercal Tests for Enhanced Lattce Boltzmann In the precedng two chapters we have developed the lattce Boltzmann method and some useful enhancements to t. The current chapter deals wth the practcal ssue of the developed numercal method, meanng the applcaton of t. The consderatons n the last two chapters were kept general n such a way that the lattce Boltzmann method was not constructed for a gven physcal problem. Certanly, ths lmts the applcaton to less sophstcated problems. If we take only the man concept of the lattce Boltzmann method nto account together wth approprate boundary condtons, we are able to smulate Poseulle flow e.g. [4, 60]) through a channel. Despte we dedcated our calculatons more to the three-dmensonal case, we smulate Poseulle flow n each two and three dmensons below. Ths shows the models capablty to handle free flud flows. Insertng obstacles nto the channel, we can also demonstrate the capablty to smulate flow past an obstacle. Both smulatons are done n secton 3.1. Takng not only the man concept and boundary condtons nto account but also the second numercal scheme, we can smulate a temperature evoluton n a gven doman. In the absence of a temperature nfluenced flow,.e. wthout consderng buoyancy effects, we can ntalze the flow n an equlbrum state wthout any moton, then the temperature only dffuses n the flud. The resultng process s mathematcally descrbed by the heat equaton. Ths smulaton and ts results are gven n secton 3.2. Moreover, we smulate Raylegh-Bénard convecton e.g. [17, 21]) n the same secton Channel Flow In the current subsecton, we consder flows through a channel n two dmensons as well as three dmensons. Fgure 3.1 llustrates the channels and the arrows show the drecton of flow. On the open sdes of the channel we mpose boundary condtons whch drve the flud through the channel. In detal, we use Zou and He pressure boundary condtons [72] for both ends of the channel. Prescrbng two dfferent pressures yelds a pressure gradent drvng the flud. Thus, we obtan an nflow at the open sde wth hgher pressure and an outflow at the other sde. The pressure gradent s equvalent to an external body force actng on the flud. Hence, the smulaton s equvalent to a smulaton 77

85 3.1. Channel Flow 78 Fgure 3.1.: Two- and three-dmensonal channels. Arrows show the drecton of flow and dashed lnes llustrate the open sdes of the channel. wth perodc boundary condtons and an approprate force term. Although, n our opnon, the mplementaton wth pressure boundary condtons s more sophstcated, we mplement both possbltes. Thus, we emphasze the capablty to smulate external forces correctly and demonstrate that also pressure boundary condtons work fne. For the channel walls a bounceback boundary condton s used. The two dmensonal smulaton s based on the D2Q9 model, where the equlbrum dstrbuton reads F eq) 0 ˆx, ˆt) = ω 0 ˆρˆx, ˆt) [1 32 ] ûˆx, ˆt) 2 F eq) ˆx, ˆt) = ω ˆρˆx, ˆt) [1 + 3c ûˆx, ˆt)) + 92 c ûˆx, ˆt)) 2 32 ] ûˆx, ˆt) 2, wth 4 9, = 0 ω = 1 9, = 1,..., , = 5,..., 8 and lattce veloctes c 0 = 0, 0), c 1 = 1, 0), c 2 = 0, 1), c 3 = 1, 0), c 4 = 0, 1), c 5 = 1, 1), c 6 = 1, 1), c 7 = 1, 1), c 8 = 1, 1). For the three dmensonal smulaton, we use the D3Q19 model. The lattce veloctes are gven n 1.48) and the correspondng equlbrum dstrbuton reads as above but

86 3.1. Channel Flow 79 wth dfferent weghts 1 3, = 0 ω = 1 18, = 1,..., 6., = 7,..., The knematc vscosty s computed by ν = 2ˆτc 1 6 n both models Poseulle Flow The Poseulle flow, also known as Hagen-Poseulle flow, s a steady state flow n a long, straght channel, where the flud s pressure-drven. In our smulatons the channel s parallel to the x 1 -axs and has an unchanged cross secton along the x 1 -axs. The cross secton s a lne segment n two dmensons and a rectangle n three dmensons, compare also wth fgure 3.1. Analytcal Soluton The Poseulle flow s one of only few stuatons where an analytcal soluton of the Naver-Stokes equatons s known [61]. We consder a two-dmensonal channel of wdth 2L and flow n x 1 -drecton. The velocty component n drecton tangental to the man flow drecton s zero and the velocty component n man flow drecton depends only on the poston, meanng the dstance to the walls,.e. ) ) u 1 x, t) u 1 x 2 ) u x, t) = u 2 =. 3.1) x, t) 0 The complete Naver-Stokes equatons 1.81) reduce to 0 = ν u x, t) p x, t) and the dvergence equaton of 1.81) vanshes, snce t s automatcally fulflled by 3.1). In general a pressure gradent s equvalent to a force densty, n the present case we obtan p x, t) = k, 3.2) wth k beng the acceleraton correspondng to the force. Note, p s the pressure dvded by the mass densty ρ 0 whch s assumed constant n the ncompressble Naver-Stokes equatons. The soluton of the Poseulle flow then reads [4, 61] u x, t) = L2 x 2 ) 2 k, 3.3) 2ν

87 3.1. Channel Flow 80 where ν denotes the knematc vscosty and the second space component x 2 = 0 s assumed to be n center of the channel s cross secton. The walls are located at x 2 = L and x 2 = L, respectvely, thus the velocty profle n flow drecton s a parabola wth zero velocty on the boundares. Note that the second component of k, tangental to man flow drecton, s zero, thus same for the velocty component n ths drecton. 2D Smulaton The two-dmensonal smulaton of Poseulle flow s done on a lattce wth n x 1 = 151 lattce ponts n x 1 -drecton and n x 2 = 32 n x 2 -drecton. The frst, x 2 = 1, and last row, x 2 = 32, are bounceback lattce ponts and the effectve boundares le half way between the bounceback lattce ponts and the frst nteror lattce ponts. Hence, the walls are located at x 2 = and x2 = It follows a channel wdth of L = 15. On the left openng, the nlet at x 1 = 1, we mpose a pressure boundary condton prescrbng a constant densty of ρ n = 1.03 where the accordng pressuredensty relaton reads p = 1 3ρ. The outflow boundary condton prescrbes a densty of ρ out = 1 at lattce ponts wth x 1 = n x 1. All other lattce ponts are ntalzed wth zero velocty and constant densty ρ = 1. The pressure gradent reads ) ) 1 ρ out ρ n 3 n px, t) = x = 15000, 0 0 whch mples, by 3.2), an acceleraton of ) k = 1 px, t) = ρ = ) , 3.4) 0 where the average densty was used for ρ 0. Furthermore, n the smulaton a relaxaton parameter ˆτ c = 0.9 s used whch mples a knematc vscosty of ν = Both, the analytcal soluton and the result of the lattce Boltzmann smulaton are shown n fgure 3.2. The numercally computed veloctes match very well wth the analytcal soluton. The sole excepton s near the walls, but note that the effectve channel walls le half way between the bounceback lattce ponts and the frst row of nteror lattce ponts. The frst and last marker n the fgures corresponds to bounceback lattce ponts whch le outsde of the effectve doman, and therefore a velocty gven n these ponts s actually meanngless. Nevertheless, usng the velocty n the bounceback lattce pont to compute the velocty on the boundary by lnear nterpolaton, the zero velocty at the walls s not obtaned by the lattce Boltzmann smulaton. In fgure 3.3 the results of the lattce Boltzmann smulaton wth perodc boundary condtons and a body force accordng to 3.4) s shown. Not strkng, the two velocty profles n fgures 3.2 and 3.3 are equal. The dfference between the two smulatons becomes vsble when nspectng the mass densty feld. When pressure boundary condtons are appled, the more one approaches to the outflow the lower gets the densty, as

88 3.1. Channel Flow Analytcal Soluton Numercal Soluton 0.05 Velocty n Man Flow Drecton x 2 axs Fgure 3.2.: Velocty profle n a two-dmensonal Poseulle flow smulaton wth pressure boundary condtons Analytcal Soluton Numercal Soluton 0.05 Velocty n Man Flow Drecton x 2 axs Fgure 3.3.: Velocty profle n a two-dmensonal Poseulle flow smulaton wth perodc boundary condtons and a force actng on the flud.

89 3.1. Channel Flow 82 shown n fgure 3.4. Ths s an mmedate consequence from the fact that the pressure n lattce Boltzmann methods s proportonal to the densty. In the smulaton wth perodc boundary condtons and an addtonal force, the densty s equally constant ρ = 1. The assumpton of a constant densty whch, as shown n chapter 1, led to the Naver-Stokes equatons s therefore only satsfed n the smulaton wth perodc boundary condtons. For ths reason, we prefer the realzaton wth perodc boundary condtons and an actng body force drvng the flud. 3D Smulaton For the three-dmensonal Poseulle flow n a channel wth square cross secton, as shown n fgure 3.1, there s no analytcal soluton known [4]. Testng the D3Q19 lattce Boltzmann method, we expand the smulaton of the prevous paragraph to three dmensons. Hence, we have 151 lattce ponts n x 1 -drecton and each 32 lattce ponts n x 2 - and x 3 -drecton. The relaxaton parameter s kept at ˆτ c = 0.9, thus the knematc vscosty s ν = At the channel nflow a Zou and He pressure boundary condton s appled. Lke n the two-dmensonal smulaton, we prescrbe a densty of ρ n = The outflow boundary condton s of same knd as the nflow, but t prescrbes a densty of ρ out = 1. The pressure-densty relaton n D3Q19 s p = 1 3ρ, see also 1.57). In fgures 3.5, 3.6 and 3.7 the velocty profle of the smulaton s presented. From the smulaton we get the velocty profle n dscrete ponts, the surface gven n fgure 3.5 s obtaned by lnear nterpolaton. We observe the hghest velocty n the center of the channel, however ths was expected from the results of the two-dmensonal nvestga- Fgure 3.4.: Densty feld n a two-dmensonal Poseulle flow wth pressure boundary condtons.

90 3.1. Channel Flow 83 Fgure 3.5.: Velocty profle n a three-dmensonal Poseulle flow smulaton wth pressure boundary condtons. Fgure 3.6.: Velocty profle n a three-dmensonal Poseulle flow smulaton wth pressure boundary condtons. Vew from top of the graph n fgure 3.5. ton and from physcal reasonng. The velocty profle reflects the result of the twodmensonal case, cuttng the profle along lnes parallel to the x 2 - or x 3 -axs the cut-out s a parabola wth the propertes of the two-dmensonal velocty profle, see fgure 3.8 and compare the structure wth fgure 3.2. Due to frcton at walls the velocty on the wall should be equal to zero, lke n two dmensons. The smulaton above had a square cross secton. Addtonally, we smulate a Poseulle flow n a channel wth an oblong cross secton. The parameters are 60 lattce ponts n x 1 -drecton, 16 n x 2 -drecton and 26 n x 3 -drecton. The chosen relaxaton parameter s ˆτ c = 1.1 and boundary condtons as gven above. Fgure 3.9 shows a contour plot of the resultng velocty profle.

91 3.1. Channel Flow x 3 Drecton x 2 Drecton Fgure 3.7.: Contour plot of velocty profle n a three-dmensonal Poseulle flow smulaton wth pressure boundary condtons Velocty n Man Flow Drecton x 2 axs Fgure 3.8.: Cut-out of the velocty profle gven n precedng fgures along the tenth row whch corresponds to the lne wth x 3 = 6.5.

92 3.1. Channel Flow x 3 Drecton x 2 Drecton Fgure 3.9.: Contour plot of the velocty profle of a three-dmensonal Poseulle flow n a channel wth oblong cross secton Flow Past an Obstacle Consderng channels n the shape of those llustrated n fgure 3.1, we have seen n the prevous subsecton how fluds flow through them. In the current subsecton, we add sold obstacles n the center of the channel. In detal we add a rectangular obstacle n the two-dmensonal channel, see also fgure Analogue, we add a cubod n the center of the three-dmensonal channel. In the smulaton, the obstacles conssts of bounceback lattce ponts, lke the channel Fgure 3.10.: Obstacle n a channel.

93 3.1. Channel Flow 86 walls. In order to acheve a flow past the obstacle we agan use a pressure drven flow realzed by Zou and He pressure boundary condtons. A flow through a two-dmensonal channel wth 151 lattce ponts n x 1 -drecton and 32 n x 2 -drecton s smulated where an obstacle of 7 n x 1 -drecton) tmes 4 n x 2 -drecton) lattce ponts s nserted n the mddle of the channel. Pressure boundary condtons lke n the two-dmensonal Poseulle flow above are appled. The flow s llustrated wth help of streamlnes n fgure 3.11 and the modulus of the velocty s presented n fgure In the latter we can see that the velocty ncreases when passng the obstacle. We also use ths knd of vsualzng the velocty to llustrate the three-dmensonal flow past an obstacle. The flow we smulated conssts of a channel wth 101x32x32 lattce ponts n x 1 -, x 2 and x 3 -drecton, respectvely. The measurements of the centrally nserted cubod are 7x4x4, and the relaxaton parameter s ˆτ c = 1. We use same pressure boundary condtons as for the Poseulle flow above. The ntutve behavor of the flud, passng the obstacle analogue to the two-dmensonal smulaton, can be observed n the lattce Boltzmann smulaton. For sake of smplcty, we only llustrate the smulated flow wth help of a velocty vsualzaton. The graphc s gven n fgure Comparng t wth the two-dmensonal flow of fgure 3.13 underlnes a generalzed behavor whch at least s ntutvely correct. Further study of flows past an obstacle can be found, for nstance, n artcles [2, 67] dealng wth that topc n-depth. Fgure 3.11.: Streamlnes of a pressure drven flow past an obstacle. Addtonal streamlnes start behnd the obstacle for better llustraton.

94 3.1. Channel Flow 87 Fgure 3.12.: Modulus of velocty for a pressure drven flow past an obstacle. Fgure 3.13.: Modulus of velocty for a pressure drven flow past an obstacle n three dmensons. Only two slces are plotted, on whch streamlnes are prnted.

95 3.2. Temperature Evoluton - Raylegh-Bénard Convecton Temperature Evoluton - Raylegh-Bénard Convecton The prevous secton already addressed lattce Boltzmann smulatons nvolvng some of the enhancements we have presented n chapter 2. However, the man enhancement we have focused on n the precedng chapter, ncorporaton of a thermal component, has not been tested, yet. In the current secton, we smulate flows advectng temperature besdes ther dffuson. From a dfferent pont of vew, we can say that we smulate the evoluton of temperature. Furthermore n smulatons below we also make use of the last enhancement we have presented n the prevous chapter, the buoyancy effect. But the frst smulaton we begn wth s a less sophstcated problem. All smulatons are performed wth the temperature model derved n the prevous chapter, ths means we have mplemented equaton 2.32) wth energy equlbrum dstrbuton 2.37) for the three-dmensonal D3Q19 model. The energy equlbrum dstrbuton n the two-dmensonal smulatons D2Q9) s [26, 48] g eq) 0 x, t) = 2 ρx, t)ex, t) ux, t) 2 3 g eq) j x, t) = 1 [ 3 9 ρx, t)ex, t) c j ux, t)) c j ux, t)) 2 3 ] 2 ux, t) 2, g eq) k x, t) = 1 [ 36 ρx, t)ex, t) 3 + 6c k ux, t)) c k ux, t)) 2 3 ] 2 ux, t) 2, wth j = 1,..., 4, k = 5,..., 8 and lattce veloctes c gven for D2Q9 n secton 1.5. The relaxaton parameter τ d n 2.32) tunes the thermal dffusvty χ, where the relatons read and for D2Q9 and D3Q19, respectvely. χ = 2 3 τ d 1 2 ) χ = 5 9 τ d 1 2 ), In the followng, we smulate Raylegh-Bénard convecton RBC), see, e.g., [21] for a detaled dscusson of ths problem. In our smulaton we have two parallel rgd boundares of nfnte length algned horzontal, as noton thnk of two plates. The temperature of both boundares s mantaned at two dfferent temperatures, where the bottom boundary s assumed to be the colder one. In between the boundares a vscous flud s njected. The problem s stated under consderaton of gravty whch acts n drecton from the upper boundary to the bottom one, compare also wth fgure At the begnnng the flud s ntalzed wth a homogeneous densty and zero velocty. Except the boundares, the temperature s ntalzed constant, n our smulaton equal

96 3.2. Temperature Evoluton - Raylegh-Bénard Convecton 89 to the colder boundary temperature. In absence of a flud s moton and consderaton of gravty, the problem s descrbed by the heat equaton wth correspondng boundary condtons whch prescrbe a constant temperature at the boundares. Temperature dffuson would yeld a lnear ncrease of the temperature from top to bottom, see also fgure Roughly speakng, ths concdes to a smulaton of RBC where no nfluence of gravty and especally no buoyancy effects are consdered. However, f buoyancy effects are consdered besdes dffuson also advecton has to be taken nto account. Snce the flud s ntalzed wth zero velocty the complete flud movement results from temperature dfferences. Thus, the flud heats at the lower boundary and gravty drves t to the upper boundary, where t s cooled down. Snce we consder ncompressble fluds, the flud cannot completely accumulate near the upper boundary, hence, not only an upward movement can be observed but also a downward one. Convecton cells as shown n fgure 3.14 appear. In our smulaton we use bounceback condtons for the nteracton of the flud wth the boundares. Due to the gven symmetry of the problem we use perodc boundary condtons n horzontal drecton, both for the densty and the temperature dstrbuton. The densty dstrbuton s ntalzed, such that for all lattce ponts the mass densty ρ s equal to one, and the velocty u equal to zero. The temperature s supposed to mantan on the boundares, correspondng boundary condtons for the energy dstrbuton satsfyng ths property can be realzed as follows. In D2Q9 at the boundares three values of the energy dstrbuton have to be computed, such that cf. 2.33)) 8 g x b, t) = ρx, t)e b x b, t) =0 s fulflled for a gven boundary pont x b and prescrbed energy E b x b, t). See also secton 2.3 for more detaled nvestgaton of the thermal model. Note the energy s proportonal to the temperature. For smplcty we consder only a south boundary gravty Cold Fgure 3.14.: Convecton cells dsplayed at an mage secton of a two-dmensonal Raylegh-Bénard experment. Hot

97 3.2. Temperature Evoluton - Raylegh-Bénard Convecton 90 where g 2, g 5 and g 6 have to be determned. Other boundares can be treated analogue. By brngng the unknown values to the left hand sde and all other terms to the rght hand sde, t follows for the above case Gx b, t) := g 2 x b, t) + g 5 x b, t) + g 6 x b, t) = ρx, t)e b x b, t) g x b, t), {1,3,4,7,8,9} where Gx b, t) can be computed explctly. Wth Gx b, t) known, the unknown values can be calculated accordng to ther relatve weghtng n the equlbrum,.e. g 2 x b, t) = g 5 x b, t) = g 8 x b, t) = ω 2 Gx b, t), ω 2 + ω 5 + ω 8 ω 5 Gx b, t), ω 2 + ω 5 + ω 8 ω 8 Gx b, t). ω 2 + ω 5 + ω 8 Ths knd of boundary condtons can be extended to three dmensons straghtforward, and we use them n the three-dmensonal smulaton. On the upper boundary we prescrbe a temperature accordng to an energy of value E cold = 0, whereas on the lower boundary we set E hot = 1. The energy dstrbuton s ntalzed n equlbrum state, such that E = 0 holds everywhere, except for the lower boundary. On the lower boundary ntalzaton of the energy dstrbuton s done by an equlbrum state wth E = E hot. Frstly, we smulate the temperature evoluton between the lower hot boundary and the upper cold one wthout consderng buoyancy effects or gravty actng on the flud at all. Ths means we approxmate the soluton of the heat equaton wth approprate boundary condtons. The smulaton s done n two dmensons on a 102 tmes 51 lattce, choosng relaxaton parameters ˆτ c = τ d = Snce the flud s ntalzed n a rest state and no forces act on the flud, the flud stays at rest and need not to be smulated. Lke expected, we obtan a statonary state for the temperature profle wth an lnear decrease from the warmer to the colder boundary. Fgure 3.15 llustrates ths result by a color plot of the correspondng energy. Now, we add gravty to our smulaton and we consder buoyancy effects. Usng the modfed model gven n secton 2.2, we add a force accordng to the acceleraton kx, t) = 1 βt x, t) T e ) ) g, wth β the coeffcent of thermal expanson, T e a reference temperature and g the acceleraton due to gravty. Instead of drectly smulatng the evoluton of the temperature, 1 At ths pont we apprecatvely lke to menton the web project on whch offers a good opportunty to swap deas on topcs around lattce Boltzmann methods. Durng the work on ths thess, we got several helpful hnts by readng the dscussons n the forum of ths project. The choce of the lattce dmenson was also nspred by readng on ths web page.

98 3.2. Temperature Evoluton - Raylegh-Bénard Convecton 91 Fgure 3.15.: Temperature profle between two boundares mantaned at two dfferent temperatures wthout consderng the nfluence of gravty. Boundary at the bottom s colder than the one at the top. the algorthm smulates the evoluton of the energy E proportonal to the temperature, therefore we wrte the acceleraton n terms of the energy as kx, t) = 1 ˆβEx, ) t) E e ) g, 3.5) wth an approprate ˆβ. We smulate Raylegh-Bénard convecton n two dmensons by takng ) 0 g = and ˆβ = Further, we set the reference energy E e equal to the average energy correspondng to the temperature profle obtaned n the smulaton above when no gravty s consdered, hence we choose E e = 0.5. Ths smulaton yelds a result n the manner of the descrpton above llustrated n fgure In fgures 3.16 and 3.17 the smulaton results are shown. Fgure 3.16 shows the moton of the flud between the two boundares. We observe a crcular movement. The resultng temperature profle s drawn n fgure 3.17, here we also can see convecton cells. For the smulaton of Raylegh-Bénard convecton n three dmensons a slghtly modfed force term s used. The force term accordng to the acceleraton 3.5), used n the twodmensonal smulaton, contans a constant term whch yelds a statc pressure n the flud. Ths s omtted n our three-dmensonal smulaton, nstead of 3.5) a force accordng to the acceleraton kx, t) = ˆβEx, t) E e )g,

99 3.2. Temperature Evoluton - Raylegh-Bénard Convecton 92 Fgure 3.16.: Vector feld llustratng the flud s movement n a smulaton of twodmensonal Raylegh-Bénard convecton. Fgure 3.17.: Resultng temperature profle n a smulaton of two-dmensonal Raylegh-Bénard convecton. Lnes are sotherms connectng ponts that have same temperature.

100 3.2. Temperature Evoluton - Raylegh-Bénard Convecton 93 s used. Therefore, we smulate RBC only under the nfluence of changes n the actng force due to temperature varatons. Compare ths procedure also wth the consderatons n secton where the constant part was combned wth the pressure gradent. We have chosen the followng parameters 0 ˆβ = 1, ˆτ c = 0.75, τ d = 0.75, g = and we used a 60x60x50 lattce, thus the boundares mantaned at constant temperatures are squares of sze 60x60. All other parameters are adopted from the twodmensonal RBC smulaton above. Fgure 3.18 llustrates the smulaton, we agan observe the presence of convecton cells. Fgure 3.18.: Smulaton of three-dmensonal Raylegh-Bénard convecton wth a lattce Boltzmann method.