Thermal fluctuations and boundary conditions in the lattice Boltzmann method

Transcription

1 Thermal fluctuatons and boundary condtons n the lattce Boltzmann method Dssertaton zur Erlangung des Grades "Doktor der Naturwssenschaften" am Fachberech Physk, Mathematk und Informatk der Johannes Gutenberg-Unverstät n Manz Ulf Danel Schller geboren n Georgsmarenhütte Manz November 2008

2 Datum der mündlchen Prüfung: 11. Dezember 2008

3 To my famly

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5 I have not faled. I ve just found 10,000 ways that won t work. THOMAS ALVA EDISON

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7 Zusammenfassung De Lattce-Boltzmann-Methode st en verbretetes Verfahren zur Smulaton hydrodynamscher Wechselwrkungen n wecher Matere und komplexen Flüssgketen. Dabe wrd das Lösungsmttel durch en räumlches Gtter repräsentert, auf dem Telchenpopulatonen entlang der dskreten Kanten zwschen den Gtterpunkten propageren und lokal mtenander kollderen. Dese Mkrodynamk führt auf großen Skalen zu enem hydrodynamschen Strömungsfeld, we es de Naver-Stokes-Glechung beschrebt. In der vorlegenden Arbet werden verschedene methodsche Erweterungen der Lattce-Boltzmann-Methode entwckelt. In komplexen Flüssgketen, z. B. Suspensonen, st de Brownsche Molekularbewegung von zentraler Bedeutung. Se kann jedoch mt der klassschen Lattce-Boltzmann-Methode ncht smulert werden, da de Dynamk vollständg determnstsch st. Es st jedoch möglch, zusätzlche thermsche Schwankungen enzuführen, mt denen fluktuerende Hydrodynamk reproduzert werden kann. In der Arbet wrd mt Hlfe enes verallgemenerten Gtter-Gas-Modells ene systematsche Herletung der Glechgewchtsvertelung aus Prnzpen der Statstschen Mechank präsentert. Der stochastsche Antel der Dynamk wrd als Monte-Carlo-Prozess betrachtet, der dem Prnzp des detallerten Glechgewchts genügen muss. Heraus lässt sch ene Bedngung für de thermschen Fluktuatonen ableten, de nsbesondere besagt, dass alle Frehetsgrade des Systems enschleßlch der knetschen Moden thermalsert werden müssen. Der entwckelte Formalsmus stellt scher, dass de verbesserte fluktuerende Lattce-Boltzmann-Methode sowohl de fluktuerende Hydrodynamk reproduzert als auch konsstent auf der Statstschen Mechank aufbaut. Des könnte de Grundlage für zukünftge Erweterungen der Methode sen, z. B. m Hnblck auf Mult- Phasen-Systeme oder Thermo-Hydrodynamk. En wchtges Anwendungsgebet der Lattce-Boltzmann-Methode st de Mkrofludk. Smulatonen lesten her neben Theore und Experment enen wchtgen Betrag auf dem Weg zum Labor auf dem Chp. Mkrofludk-Systeme zechnen sch durch en hohes Verhältns von Oberflächen zu Volumen aus. Besonderes Augenmerk muss daher auf de Randbedngungen gelegt werden, wobe m Mkroberech de n der Hydrodynamk üblche Haftbedngung an der Oberfläche durch ene Gletbedngung zu ersetzen st. In deser Arbet wrd ene Randbedngung für de Lattce-Boltzmann-Methode konstruert, de de Enstellung der Gletlänge über enen entsprechenden Modellparameter ermöglcht. Es wrd weterhn en neuer Ansatz zur Konstrukton von Randbedngungen untersucht. Ausgangspunkt st dabe de explzte Berückschtgung der gebrochenen Symmetre an ener Oberfläche nnerhalb des Gttermodells. De Lattce-Boltzmann-Methode wrd systematsch auf de gebrochene Symmetre verallgemenert. Am Bespel ener Poseulle- Strömung wrd gezegt, dass ene spezelle Wahl des Kollsonsoperators an der Wand erforderlch st, damt das Strömungsprofl korrekt reproduzert wrd. De systematsche Vorgehenswese führt dabe zu enem erweterten Verständns von Randbedngungen n der Lattce-Boltzmann-Methode, das ncht nur be der Interpretaton von Smulatonsergebnssen hlfrech st, sondern auch zu zukünftgen Verbesserungen der Methode führen könnte.

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9 Abstract The lattce Boltzmann method s a popular approach for smulatng hydrodynamc nteractons n soft matter and complex fluds. The solvent s represented on a dscrete lattce whose nodes are populated by partcle dstrbutons that propagate on the dscrete lnks between the nodes and undergo local collsons. On large length and tme scales, the mcrodynamcs leads to a hydrodynamc flow feld that satsfes the Naver-Stokes equaton. In ths thess, several extensons to the lattce Boltzmann method are developed. In complex fluds, for example suspensons, Brownan moton of the solutes s of paramount mportance. However, t can not be smulated wth the orgnal lattce Boltzmann method because the dynamcs s completely determnstc. It s possble, though, to ntroduce thermal fluctuatons n order to reproduce the equatons of fluctuatng hydrodynamcs. In ths work, a generalzed lattce gas model s used to systematcally derve the fluctuatng lattce Boltzmann equaton from statstcal mechancs prncples. The stochastc part of the dynamcs s nterpreted as a Monte Carlo process, whch s then requred to satsfy the condton of detaled balance. Ths leads to an expresson for the thermal fluctuatons whch mples that t s essental to thermalze all degrees of freedom of the system, ncludng the knetc modes. The new formalsm guarantees that the fluctuatng lattce Boltzmann equaton s smultaneously consstent wth both fluctuatng hydrodynamcs and statstcal mechancs. Ths establshes a foundaton for future extensons, such as the treatment of mult-phase and thermal flows. An mportant range of applcatons for the lattce Boltzmann method s formed by mcrofludcs. Fostered by the lab-on-a-chp paradgm, there s an ncreasng need for computer smulatons whch are able to complement the achevements of theory and experment. Mcrofludc systems are characterzed by a large surface-to-volume rato and, therefore, boundary condtons are of specal relevance. On the mcroscale, the standard no-slp boundary condton used n hydrodynamcs has to be replaced by a slp boundary condton. In ths work, a boundary condton for lattce Boltzmann s constructed that allows the slp length to be tuned by a sngle model parameter. Furthermore, a conceptually new approach for constructng boundary condtons s explored, where the reduced symmetry at the boundary s explctly ncorporated nto the lattce model. The lattce Boltzmann method s systematcally extended to the reduced symmetry model. In the case of a Poseulle flow n a plane channel, t s shown that a specal choce of the collson operator s requred to reproduce the correct flow profle. Ths systematc approach sheds lght on the consequences of the reduced symmetry at the boundary and leads to a deeper understandng of boundary condtons n the lattce Boltzmann method. Ths can help to develop mproved boundary condtons that lead to more accurate smulaton results.

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11 Contents Lst of symbols and notatons xv 1 Introducton 1 2 The lattce Boltzmann method: a modern overvew Hstorcal remarks Knetc theory and contnuum flud mechancs The Boltzmann equaton The Maxwell-Boltzmann equlbrum dstrbuton The lnearzed Boltzmann equaton Hydrodynamc felds and macroscopc equatons Dmensonless formulaton Hermte-Expanson Dscretzaton of the Boltzmann equaton Dscretzaton of velocty space Dscretzaton of confguraton space and tme Choce of truncaton and quadrature The lattce Boltzmann equaton The D3Q19 model Equlbrum dstrbuton Collson operator Statstcal mechancs of the lattce Boltzmann equaton Fluctuatng hydrodynamcs The fluctuatng lattce Boltzmann equaton The generalzed lattce gas model (GLG) Statstcs of the generalzed lattce gas Equlbrum dstrbuton Fluctuatons around equlbrum Stochastc collson operator and detaled balance Asymptotc analyss and the Chapman-Enskog expanson Asymptotc analyss and scalng Chapman-Enskog expanson Zeroth order Frst order Second order x

12 Contents Mergng orders Closng the Chapman-Enskog expanson Explct expressons for f (1) and f (2) Fluctuatons External forces Boundary condtons for lattce Boltzmann models Hydrodynamc boundary condtons Boundary condtons on the Naver-Stokes level Boundary condtons n knetc theory Boundary condtons for lattce Boltzmann models Bounce-back Specular reflectons Dffuse reflectons Advanced closure schemes Interpolaton and extrapolaton schemes Equlbrum nterpolaton Crtcal dscusson of the exstng boundary condtons Partal slp boundary condtons Modelng wall frcton Analytcal soluton of the wall frcton model for Poseulle flow Implementaton of wall frcton: canoncal method Smulaton results for the canoncal mplementaton Theoretcal analyss Force mplementaton revsted: prmtve method Smulaton results for prmtve mplementaton Comparson wth slp-reflecton models Dscusson of the wall frcton model Reduced symmetres n lattce Boltzmann models Boundares and reduced symmetry Lattce sums and nvarant tensors n the reduced symmetry Lattce sums for a locally plane boundary Approaches for constructng the equlbrum dstrbuton Method 0: Drect ansatz for the equlbrum dstrbuton Method I: Dervaton from quadratc functonal Method II: Statstcal mechancs based dervaton Weghts w for the reduced D3Q19 model Boundary equlbrum for reduced D3Q19 model MRT model for the reduced symmetry Collson operator at the boundary Results for the reduced symmetry model The Stokes equaton and the reduced symmetry lattce Boltzmann model x

13 Contents A closer look on the collsons n the reduced mode space Revsed boundary model Attempts for a Chapman-Enskog expanson at the boundary Second and thrd moment n the reduced symmetry Ansotropc Chapman-Enskog expanson: a potental way out? Conclusons, dscusson and outlook 127 Appendx A Implementaton of the lattce Boltzmann method 133 A.1 Usage n ESPResSo A.2 Internal unt conversons A.3 The lattce Boltzmann kernel A.3.1 Nave mplementaton A.3.2 Combned collsons and streamng A.3.3 Data layout optmzaton A.4 Parallelzaton A.5 Thermal fluctuatons A.6 Force couplng A.7 Boundary condtons A.7.1 Bounce-back A.7.2 Specular reflectons A.7.3 Slp reflectons A.7.4 Local boundary collsons B Techncal materal 151 B.1 Hermte tensor polynomals and Gauss-Hermte quadratures B.1.1 Hermte tensor polynomals B.1.2 Gauss-Hermte quadrature B.2 Lattce sums and sotropc lattce models B.2.1 Lattce sums for dscrete velocty sets B.2.2 Lattce sums and Gauss-Hermte quadrature B.3 Theoretcal analyss of the slp boundary condton B.4 Functonal dervaton of the bulk equlbrum dstrbuton C Sourcecode 163 Bblography 165 x

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15 Lst of symbols and notatons A varety of symbols and notatons s used n ths thess. The followng table shall help the reader to fnd the way through. Many formulas n ths work can be wrtten n more than one notaton, the most frequent opton probably beng the choce of vector or ndex notaton. The choces made n the text are an attempt to present the formulas n a way that s at the same tme smple and ntutve, wth the am to gude the reader along the lnes of thought. Therefore, we sometmes delberately swtch between dfferent notatons on the fly. The table below lsts the elementary symbols, some of whch can be ndexed n several ways and combnatons of ndexes can be used. We have not lsted all possble combnatons here. The meanng of combnatons of symbols and notatons should always become clear n the respectve context. Symbol/Notaton Meanng Mscellanea a b scalar product of two vectors a and b a b, ab tensor product of two vectors a and b (cf. footnote 6 on page 15) a : b full contracton of rank-two tensors a and b f g scalar product of two functons f and g gradent or dvergence (nabla operator) r δ δf functonal dervatve wth respect to f ( f) coll general collson operator n knetc theory 1 unt matrx Dmensonless numbers Bo Boltzmann number Kn Knudsen number Ma Mach number Latn letters A, B, C,... coeffcents n the dscrete equlbrum dstrbuton Ã, B, C,... coeffcents n the dscrete boundary equlbrum dstrbuton A, B 1, B 2, C 1,... coeffcents of the lattce tensors n the reduced symmetry a lattce spacng Contnued on next page xv

16 Symbols & Notatons Symbol/Notaton Meanng Contnued from prevous page a (n), a (n) α 1...α n tensor coeffcents n the Hermte expanson b, b k lengths of the bass vectors e k of mode space c s speed of sound c, c α dscrete velocty vectors ĉ, ĉ α dmensonless form of velocty vector, c = ĉ a/τ ED,d n quadrature of degree d n D dmensons usng n nodes E (2), E (3) tensors occurrng n the Chapman-Enskog expanson e k bass vectors of mode space F external force n the Boltzmann equaton f(r, v, t) one-partcle dstrbuton functon n knetc theory f N (r, v, t) truncated Hermte expanson of f(r, v, t) f eq (ρ, u) local equlbrum dstrbuton functon, Maxwell-Boltzmann dstrbuton f (r, t) dscrete velocty populaton n lattce models f eq (ρ, u) local equlbrum populaton of dscrete veloctes f, f populaton numbers n general f post-collsonal value of populaton number f eq, f neq equlbrum and non-equlbrum part of dscrete velocty populaton f (0), f (1), f (2) dfferent orders of the Chapman-Enskog expanson of the populatons f B,eq (ρ, u) local equlbrum populaton at a boundary f ext external force G, G αβ tensor n the forcng term of the lattce Boltzmann equaton g, g α volumetrc force j, j α hydrodynamc momentum densty k B Boltzmann s constant l mfp molecular mean free path L hydrodynamc length scale, typcally the wdth of a channel m mass m p (fcttous) mass of a lattce Boltzmann partcle M transformaton matrx from populatons to modes M B transformaton matrx from populatons to modes at a boundary m, m k hydrodynamc modes or moments m eq k equlbrum value of the moment m k m k post-collsonal value of the moment m k n, n α normal vector of a boundary surface P projector on the boundary normal P = nn T p hydrostatc pressure Contnued on next page xv

17 Symbols & Notatons Symbol/Notaton Meanng Contnued from prevous page p(r, v, t) polynomal n v Q αβγδ correlatons of the fluctuatons of the stress tensor s q fracton of a cut lnk that les outsde a boundary r, r α poston vector r 1, r 1α coarse-graned poston vector r 1 = ɛr S ({ν }) assocated entropy n the generalzed lattce gas model S ({ν }, χ, λ) entropy functonal wth Lagrange multplers s, s αβ fluctuatng stress n Landau-Lfshtz fluctuatng hydrodynamcs T temperature T (n), T α (n) 1...α n lattce tensor or lattce sum (ncludes the weghts w ) t tme varable t 1 coarse-graned tme varable, convectve scale t 1 = ɛt t 2 coarse-graned tme varable, dffusve scale t 2 = ɛ 2 t u, u α hydrodynamc flow velocty u slp slp velocty at a boundary v molecular velocty n knetc theory w, w q weghts n the lattce Boltzmann model, ndexes ndvdual c whle q ndexes a subshell wth the same absolute velocty Greek letters α β γ γ k δ αβ δ αβγδ δ (n), δ (n) α 1...α n g (0), (1), (2) δ B ɛ ζ η αβγδ Maxwell s accommodaton coeffcent slp coeffcent frcton parameter n the hydrodynamc slp boundary condton relaxaton parameter of the k-th moment m k Kronecker delta rank-4 cubc ansotropy, δ αβγδ = δ αβ δ βγ δ γδ (no summaton over double ndces) n-th rank sotropc tensor, sum of all (2n 1)!! products of Kronecker deltas dscrete collson term n the lattce Boltzmann equaton stochastc collson term forcng term dfferent orders of Chapman-Enskog expanson of the collson term slp length at a boundary expanson parameter n the Chapman-Enskog expanson frcton parameter n the wall frcton model vscosty tensor Contnued on next page xv

18 Symbols & Notatons Symbol/Notaton Meanng Contnued from prevous page η b bulk vscosty η s shear vscosty θ reduced (dmensonless) temperature κ 4 coeffcent of the ansotropc term of the rank-4 lattce sum λ egenvalue of the BGK collson operator (sngle-relaxaton-tme approxmaton) λ b egenvalue of the bulk stress modes, related to bulk vscosty λ g egenvalue of the knetc (ghost) modes λ k egenvalues of the k-th mode m k of the collson operator λ s egenvalue of the shear stress modes, related to shear vscosty λ ρ, λ j, λ Π Lagrange multplers λ, λ α Lagrange multpler λ (1) α, λ (2) α low velocty expanson of the Lagrange multpler λ B Lagrange multpler at a boundary λ B,(1) α, λ B,(2) α low velocty expanson of the Lagrange multpler at a boundary µ mass densty parameter ν, ν nteger populaton numbers n the generalzed lattce gas model ν post-collsonal nteger populaton number Π, Π αβ pressure or stress tensor Π, Π αβ post-collsonal stress tensor Π eq, Π eq αβ equlbrum (Euler) stress tensor Π neq, Π neq αβ non-equlbrum (Newtonan) stress tensor Π (0), Π (1) dfferent orders of the Chapman-Enskog expanson of the stress tensor ρ densty σ, σ αβ, σ αβ Naver-Stokes vscous stress tensor (overlne denotes traceless part) σ r, σαβ r, σr αβ random fluctuatng stress n the lattce Boltzmann model (overlne denotes traceless part) σ 2 coeffcent of the rank-2 lattce sum σ 4 coeffcent of the sotropc term of the rank-4 lattce sum τ lattce Boltzmann tme step Φ eq, Φ eq αβγ equlbrum value of the thrd-moment Φ (0), Φ (0) αβγ lowest order Chapman-Enskog expanson of the thrd moment ϕ k ampltude of the random nose for the k-th moment m k χ ndcator varable χ Lagrange multpler χ (1), χ (2) low velocty expanson of the Lagrange multpler χ B Lagrange multpler at a boundary Contnued on next page xv

19 Symbols & Notatons Symbol/Notaton Meanng Contnued from prevous page χ B,(1), χ B,(2) low velocty expanson of the Lagrange multpler at a boundary ω(v) weght functon n knetc theory ω(m m ) transton probablty from m to m for stochastc collsons Callgraphc letters B(v v) boundary scatterng kernel n knetc theory B j dscrete boundary scatterng kernel C lnear collson operator n knetc theory C BGK BGK collson operator n knetc theory F ({f }) functonal of the populaton numbers F B ({f }) functonal at a boundary H (n), H α (n) 1...α n n-th order Hermte tensor polynomal L, L j dscrete lnear collson operator L M collson operator n mode space N normalzaton factor xx

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21 1 Introducton Soft matter systems and n partcular complex fluds are recevng ongong research nterest from both theoretcal and expermental perspectves. Typcal examples are collodal dspersons and polymer solutons. More recently, evolved systems such as membranes, vescles and cells have attracted growng nterest. A decsve property of such systems s the presence of a herarchy of tme and length scales, rangng from the atomstc scale of the molecular nteractons to the macroscopc scale of contnuum hydrodynamcs. The length scales n between are called mesoscopc and are pvotal for the rch dversty of phenomena that can be observed n soft matter. The addtonal mesoscopc length scale provded by partcles dspersed n a solvent can lead to a qualtatve change of the behavor n complex fluds compared to the pure solvent. For example, the addton of a small amount of polymer to water can lead to a sgnfcant reducton of drag when the soluton s pumped through tubes or channels. Such phenomena are nduced by the physcs that takes place on the mesoscopc scale, whch therefore can not be neglected n a theoretcal descrpton of complex fluds. The behavor of soft matter s usually governed by a varety of dfferent physcal mechansms, e.g., thermodynamcs and phase transtons, electroknetc and rheologcal effects, nonlnear phenomena and nstabltes. A complete analytcal treatment ncludng all these mechansms s certanly out of reach, and avalable theoretcal predctons are generally based on more or less crude approxmatons. For ths reason, soft matter and complex fluds belong to the realm of computer smulatons. Smulatons can serve as a powerful tool to nvestgate the prncples underlyng the expermental observatons. They allow one to study smple model systems extensvely and under well-defned condtons, where the nfluence of dfferent nteractons can be systematcally dentfed and solated. In ths way, computer smulatons form another pllar besdes analytcal theory and experment, that s ndspensable for ganng a better understandng of soft matter. The latter s clearly of paramount mportance for practcal applcatons, rangng from engneerng processes to medcal dagnoss and therapy. The development of successful and effcent computer smulaton methods s therefore an mportant branch of contemporary physcal research. Effcent smulaton models for complex fluds are mostly based on the concept of coarsegranng. The number of degrees of freedom n a complex flud s so huge that an explct treatment of all of them s mpractcable. Moreover, the dynamcs of the dspersed partcles takes place on much longer tme scales than the solvent dynamcs, so that followng every solvent molecule s trajectory s unnecessary and wastes a lot of computng tme. Ths latter observaton, whch s the so-called separaton of scales, s at the heart of the mprovements acheved through coarse-granng. Snce the molecular detals of the solvent are rrelevant on the scales of the dspersed partcles, t can be treated as a contnuous hydrodynamc medum that s governed by the Naver-Stokes equaton. Complex fluds can typcally be descrbed 1

22 1 Introducton n the ncompressble and creepng flow lmt,.e., the Mach number and the Reynolds number are small. In ths lmt, the effect of the solvent on the dspersed partcles can be modeled n terms of a Stokes frcton. One of the smplest approaches to nclude solvent effects s Langevn dynamcs, where each dspersed partcle s subject to a frcton force proportonal to ts velocty. The dffuson of the partcles n Langevn dynamcs s consstent wth the Ensten relaton. However, the smple frcton force can not capture the momentum transport through the solvent. If a dspersed partcle moves, the flow feld of the solvent s changed. Ths perturbaton propagates through the solvent and affects the moton of other partcles. The correlatons medated by ths effect are called hydrodynamc nteractons. They are of a long-ranged nature and ther effcent mplementaton n computer smulatons s a topc stll undergong actve development. Among the technques for smulatng hydrodynamc nteractons, the so-called mesoscopc methods have proven partcularly useful for soft matter applcatons. They are based on representng the solvent degrees explctly but va smplstc models on a mesoscopc scale. Partcle based methods, such as dsspatve partcle dynamcs and mult-partcle collson dynamcs, use explct solvent partcles that represent a collectve lump of flud. Besdes partcle methods, the lattce Boltzmann method has become a popular approach to smulate complex fluds. Lattce Boltzmann s bult on a specal space-tme dscretzaton of knetc theory, where the solvent s modeled n terms of partcle dstrbutons on a regular lattce. Although t was orgnally devsed as an alternatve tool for computatonal flud dynamcs, t has successfully been appled to a varety of soft-matter systems. The reason for the success of the lattce Boltzmann method s ts versatlty orgnatng from the possblty to pad the plan solvent dynamcs wth specfc detals descrbng the structure of complex fluds. Numerous examples show that lattce Boltzmann s a very flexble smulaton method for many dfferent knds of systems, rangng from suspensons over reactve and mult-phase flows to turbulence. Besdes hydrodynamc nteractons, thermal fluctuatons play an mportant role n many soft matter systems. They arse from the underlyng mcroscopc dynamcs of the solvent molecules and are responsble for the observed Brownan moton of dspersed partcles. On the hydrodynamc level, thermal fluctuatons n the hydrodynamc varables can be descrbed wthn the framework of fluctuatng hydrodynamcs. A smulaton method for complex fluds should therefore be capable of reproducng the fluctuatng hydrodynamc equatons and provde a means for adjustng the strength of the thermal fluctuatons at wll. In the lattce Boltzmann method, the evoluton of the hydrodynamc varables s completely determnstc, hence Brownan moton s not automatcally reproduced. It s possble though, to equp the lattce Boltzmann method wth thermal nose n a way that s consstent wth fluctuatng hydrodynamcs. However, ths approach lacks a rgorous justfcaton n terms of the underlyng statstcal mechancs. In fact, some crtcsm of the orgnal fluctuatng lattce Boltzmann was put forward and a modfed approach was suggested by Adhkar et al. [1]. The ultmate clarfcaton of ths debate was only acheved recently durng the course of ths thess [2]. 2

23 1 Introducton Another more recent development n soft matter research s an ncreasng trend of turnng towards mcrofludcs. Modern expermental methods make t possble to fabrcate mcrofludc devces wth dmensons on the mcro- or even the nanoscale. The development of the frst MEMS (mcro-electromechancal systems) and µtas (mcro total analyss systems) has led to an ever-growng nterest n the so-called lab-on-a-chp paradgm. Mcrochps are desgned to dsplace, transport, manpulate and separate very small flud volumes. The am s to be able to conduct all typcal elements of physco-chemcal processng mxng, reactng, analyzng and so on on a mcroscopc scale. Potental applcatons emerge from the rapd developments n molecular botechnology and nclude, for example, boanalytc devces that can be used for medcal dagnoss. Wth the reduced spatal dmensons of mcrofludc devces the surface-to-volume rato s sgnfcantly ncreased. Therefore the flow through mcrochps s largely domnated by surface and nterface effects and flud-sold nteractons play an mportant role. Wth regard to smulaton methods ths means that specal emphass has to be put on the boundary condtons. There s ample expermental evdence that the classcal no-slp boundary condton becomes napproprate n mcrochannels when the Knudsen number exceeds a value of roughly 0.1. The flud does not stck to the boundary any more, but an effectve slp velocty s observed that s determned by surface roughness and the physco-chemcal propertes of the boundary. Whle the no-slp boundary condton s the standard boundary condton n lattce Boltzmann smulatons, apparent slppage has been rarely addressed. In general, boundary condtons n lattce Boltzmann smulatons are often based on heurstc arguments, and a systematc and unfed framework s stll lackng. The am of ths thess s to advance the development of the lattce Boltzmann method n order to make t applcable to the flow of complex fluds n mcrofludc devces. For ths purpose, two major ponts shall be addressed. Frst, the fluctuatng lattce Boltzmann equaton shall be revsted n order to restore the statstcal mechancs orgn of the thermal fluctuatons. Ths wll make the method consstent wth both statstcal mechancs and contnuum hydrodynamcs at the same tme. The second major topc shall be boundary condtons for the lattce Boltzmann model. A novel way of mplementng a tunable slp boundary condton based on a mesoscopc wall frcton model wll be devsed. Furthermore, a conceptually new approach to boundary condtons shall be explored that systematcally treats the reduced symmetry at the wall. The purpose of the latter pont s twofold: On one hand, we am at a general local scheme for boundary condtons that can effcently be mplemented n a parallel computng envronment. On the other hand we seek a better understandng of boundary condtons wth respect to the symmetry breakng nduced by the boundary. For both thermal fluctuatons and boundary condtons, another goal s to develop an effcent and versatle mplementaton of the methods for use n the ESPResSo software package [3, 4]. The remander of ths work s structured as follows. In chapter 2, the lattce Boltzmann method s revewed from a modern perspectve. Instead of followng the hstorcal route startng out wth lattce gas automata, we base the presentaton on the knetc nature of the lattce Boltzmann equaton. In chapter 3, the statstcal mechancs of the lattce Boltzmann equaton wll be developed. The connecton between the lattce Boltzmann equaton and 3

24 1 Introducton macroscopc hydrodynamcs s establshed by asymptotc analyss n terms of the Chapman- Enskog expanson, whch wll be the subject of chapter 4. The followng chapters are devoted to boundary condtons. In chapter 5, standard boundary condtons are revewed, and a novel boundary condton for a tunable slp s devsed. The conceptually new treatment of reduced symmetres at the boundary wll be ntroduced and dscussed n detal n chapter 6. Fnally, chapter 7 closes wth conclusons and dscusson. The appendx contans detals on the lattce Boltzmann mplementaton n the ESPResSo package. It also contans some of the more techncal dervatons whch would have dsturbed the flow of the man text. 4

25 2 The lattce Boltzmann method: a modern overvew Ths secton ntroduces the theoretcal background of the lattce Boltzmann method for flud mechancs. In vew of ts development over the past twenty years and ts popularty, t s not surprsng that there s a vast corpus of exstng lterature on LB (see for example the book by Succ [5] or the revews [6 8]). Nevertheless, some detals n the understandng of the lattce Boltzmann method have only been worked out recently, e.g., the fluctuatng lattce Boltzmann equaton [2, 8], and some open questons stll reman for example wth respect to multphase LB models. In the course of ts evoluton the theoretcal nsghts nto LB have led to a formal framework that s rather dfferent from the orgnal descrpton related to lattce-gas automata. Therefore, t s useful to gve an overvew of the lattce Boltzmann theory from ths modern perspectve, whch forms the bass for the further developments n ths work. The materal n ths secton s manly a revew of exstng work. Nevertheless, some effort was put on a comprehensve and unfed presentaton of the lattce Boltzmann framework. The remander of ths secton s organzed as follows: After some short hstorcal remarks, the foundatons of knetc theory and ts relaton to contnuum flud mechancs wll be ntroduced. Then t wll be dscussed how the Boltzmann equaton can be systematcally dscretzed, leadng to an a-pror dervaton of the lattce Boltzmann equaton. Fnally, the constructon of lattce Boltzmann models wll be outlned, and as a specfc example, the D3Q19 model wll be explaned n detal. 2.1 Hstorcal remarks The hstorcal roots of the lattce Boltzmann method le n the lattce gas automata (LGA), a specal class of cellular automata amng at smulatng flud dynamcs n terms of dscrete mcroscopc models [9, 10]. They were nspred by the observaton that the macroscopc flow behavor s smlar for many fluds even when the mcroscopc structure s qute dfferent. Whle the mcroscopc detals may nfluence the dmensonal values of transport coeffcents, the form of the macroscopc hydrodynamc equatons depends solely on symmetres and conservaton laws. An early precursor of the LGA was ntroduced by Kadanoff and Swft [11] already n The frst lattce gas model for fluds was proposed by Hardy, Pomeau and de Pazzs [12] and became known as the HPP model. It conserves mass and momentum and leads to sound waves, but t can not reproduce the Naver-Stokes equaton because the underlyng square lattce lacks suffcent rotatonal nvarance. Ths ssue was resolved n the FHP model by Frsch, Hasslacher and Pomeau [13] by usng a trangular lattce wth hexagonal symmetry. The FHP model was the frst lattce gas automaton flud 5

26 2 The lattce Boltzmann method: a modern overvew that successfully reproduces the Naver-Stokes equaton, yet only n two dmensons. The development of three-dmensonal lattce gas automaton fluds was frst hndered by the fact that there s no lattce wth suffcent symmetry whch s at the same tme space-fllng [9]. Fortunately, the way out was found n four dmensons where the face-centered hypercubc (FCHC) lattce has suffcent symmetry [14]. The FCHC can be projected nto threedmensonal space yeldng a lattce gas automaton for three-dmensonal Naver-Stokes hydrodynamcs. The theoretcal foundatons of the LGA were presented, for example, by Frsch et al. [10] and Wolfram [15]. These works contan already essental nsghts that are also at the heart of the lattce Boltzmann method, namely, that besdes conservaton of mass and momentum, the sotropy of tensors up to fourth rank s requred for Naver- Stokes behavor, whch has to be guaranteed by a suffcent symmetry of the underlyng lattce. Naver-Stokes behavor s obtaned n the double lmt of small Mach number and small Knudsen number [10, 16]. Concernng the symmetry propertes, Hasslacher ponted out that nstead of usng hgh symmetry, one can also use multple speeds correspondng to dfferent neghbor shells of the lattce [see 9]. The sotropy of the lattce tensors n these model s acheved by ntroducng speed dependent weghts for the dfferent subshells. A frst mult-speed model has been proposed by d Humères, Lallemand and Frsch [14] as early as n 1986 and turns out to be the LGA analogon of what s today used under the name D3Q19 model. Lattce gas automata became very popular because, due to ther Boolean nature, they provded an easy to mplement and round-off free method to smulate flud flows. They were, however, plagued by several dseases [9], the most severe of whch beng the lack of Gallean nvarance and statstcal nose [5]. To overcome the statstcal nose nherent to the Boolean occupaton numbers, one can ntroduce ensemble-averaged populatons. Ths had already been done by Frsch et al. [10] to calculate the vscosty through lnear response theory, whch could be vewed as the frst occurrence of a lattce Boltzmann equaton. The frst real lattce Boltzmann (LB) model, where the populatons are used as dynamc varables, was put forward by McNamara and Zanett [17]. In ther model, the collson rules were derved from the mcrodynamcs of the underlyng LGA. A substantal smplfcaton can be made by lnearzng the collson operator around the equlbrum, as ntroduced by Hguera and Jmenez [18]. Not only dd ths reduce the complexty of the collson operator sgnfcantly, but t was also an mportant step n realzng the drect relaton of lattce Boltzmann and contnuum knetc theory. An mmedate follow-up s the sngle relaxaton tme approxmaton tantamount to the Bhatnagar-Gross-Krook (BGK) collson operator [19 21]. Wth the advent of the lattce BGK model, the lattce Boltzmann method had bascally fledged nto a self-standng form, the man ngredents of whch are the local equlbrum dstrbuton and the lnear collson operator. In a 1992 semnal paper, Qan et al. [22] have presented a whole famly of LB models and coned the now common nomenclature DnQm for n-dmensonal models wth m veloctes. Subsequently, many studes were conducted to gan a better understandng of the lattce Boltzmann method and to devse further refnements. One of them s the (re-)ntroducton of a mult-relaxaton-tme (MRT) collson operator to overcome the lmtatons of fxed Prandtl number and fxed rato of bulk and shear vscosty n the lattce BGK model [23 6

27 2.1 Hstorcal remarks 25]. The MRT model provdes more flexblty to tune the macroscopc behavor of the lattce flud and, on top of that, t s much more stable than lattce BGK [26]. In the vew of ths author, the MRT model s ndeed the generc case of the lattce Boltzmann method, and lattce BGK s merely a specal choce of relaxaton parameters. Another mportant seres of studes was concerned wth the H-theorem n the lattce Boltzmann method [27 32]. It was shown by Wagner [27] that the usual polynomal form of the dscrete local equlbrum dstrbuton does not admt a H-theorem. Karln et al. [28 30] have developed a framework to derve local equlbra from entropy functons n such a way that a dscrete H-theorem can be proven. The approach has become known as entropc lattce Boltzmann and can be used to stablze numercal algorthms [31]. More mportantly, t has brought the conceptual advance that nstead of prescrbng an ansatz for the equlbrum dstrbuton, t can be supported by an approprate entropy functon. Along a dfferent lne, Shan and Luo and ther coworkers have establshed a systematc a-pror dervaton of the lattce Boltzmann equaton from contnuum knetc theory [33 35]. In ths sense, the lattce Boltzmann equaton can be seen as a fully dscretzed verson of the contnuum Boltzmann equaton. Ths was a major fndng that shows that the LB model s ndeed more than just a plan Naver- Stokes solver and potentally led to a conceptual shft n devsng models of complex flud behavor [5]. The mentoned shft was also supported by many practcal developments. A major push n modelng complex fluds were the smulatons of partcle suspensons poneered by Ladd [36 38]. In order to account for Brownan moton, Ladd ntroduced fluctuatons n the lattce Boltzmann equaton. Ths was an mportant step towards lnkng the hydrodynamc varables to the statstcal fluctuatons of the lattce Boltzmann populatons. It consttutes the startng pont for applcatons of LB to varous soft matter systems, where t s meanwhle one of the standard methods for hydrodynamc nteractons [8]. A lot of nterest s attracted to smulatons of multphase flows. The frst multcomponent LB method was developed by Gunstensen et al. based on color-component lattce gas models [9]. In the Shan-Chen model, explct nteracton potentals are ntroduced to model nterface forces between the phases [39, 40]. An mprovement over these phenomenologcal approaches are the free energy models ntroduced by Swft et al. [41, 42]. They try to devse a local equlbrum dstrbuton that s consstent wth the thermodynamcs of the nterface. A fully consstent multphase approach n terms of statstcal mechancs s to date not avalable. Another actve topc are thermal fluds where heat transport has to be modeled. It s well known that a quadratc equlbrum can not reproduce the heat transport equaton. To ths end, thrd-order terms were ncluded by Alexander et al. [43], and Qan and Chen have used larger velocty sets to devse thermal LB models [44, 45]. However, these models suffer from the lack of a rgorous justfcaton n terms of statstcal mechancs. Nowadays, t s well perceved that lattce Boltzmann s more than a Naver-Stokes solver. As a dscrete knetc scheme, t s capable of smulatng behavor n the non-hydrodynamc regme up to Knudsen numbers as hgh as Kn O(1) [46]. It s realzed that the systematc dervaton from the Boltzmann equaton n terms of quadratures gves rse to a whole herarchy of lattce Boltzmann models. Shan et al. have recently presented a systematc account of hgher approxmatons to the Boltzmann equaton beyond the Naver-Stokes level [47]. 7

28 2 The lattce Boltzmann method: a modern overvew The framework can be used to devse LB models for thermohydrodynamcs, whch requres to employ hgher level expansons and leads to larger velocty sets. Numercal results obtaned by Ansumal et al. [48] wthn the entropc lattce Boltzmann method gve strong support that the LB herarchy s ndeed capable of representng knetc theory beyond the hydrodynamc regme. In the followng, we wll pck up the sprt of these latest developments n the followng and outlne the lattce Boltzmann method as a fully dscretzed approxmaton to knetc theory. 2.2 Knetc theory and contnuum flud mechancs Hydrodynamc nteractons n a complex flud are medated by the flow feld of the solvent, the tme evoluton of whch can be descrbed by the contnuty equaton and the Naver- Stokes equaton 1 [49] t ρ + (ρu) = 0, r t (ρu) + r (ρu u) = r p + (2.1) r σ + g, where ρ s the mass densty, ρu = j the momentum densty, p the scalar pressure, σ the devatorc (vscous) stress tensor, and g an external volumetrc force. The devatorc stress has the form σ αβ = η αβγδ u γ r δ, (2.2) where the vscosty tensor s gven by η αβγδ = η s (δ αγ δ βδ + δ αδ δ βγ ) + ( η b 2 ) 3 η s δ αβ δ γδ (2.3) wth η s and η b beng the shear and bulk vscosty of the flud, respectvely. These equatons descrbe the flud as a contnuum n the hydrodynamc lmt, that s, on large length and tme scales. Therefore we wll refer to equatons (2.1) as the macroscopc descrpton of the flud dynamcs. The mass densty ρ and the momentum densty j descrbe the state of the flud and are termed the macroscopc or hydrodynamc felds. An mportant feature of the Naver-Stokes equaton s ts unversalty,.e., t apples to a whole class of fluds and ts structure s ndependent of the mcroscopc nteractons that can be qute dfferent, for example, n a lqud compared to a gas. The mcroscopc detals of the dynamcs are subsumed n the transport coeffcents, n other words, the rrelevant degrees of freedom have been projected out and do not show up n the structure of the 1 The full Naver-Stokes-Fourer descrpton of contnuum flud mechancs ncludes also the heat transport equaton, whch we delberately omt here because we wll concentrate on an sothermal flud. 8

29 2.2 Knetc theory and contnuum flud mechancs macroscopc equatons any more. Ths s an essental feature of contnuum flud mechancs because t means that, n order to smulate a solvent, we can use a wde range of mcromodels 2, as long as they gve rse to the Naver-Stokes equaton on the macroscopc level and reproduce the correct values for the transport coeffcents. However, such models have to be devsed wth care snce t s not a-pror clear whch are the basc physcal propertes they have to retan. The development of models that reproduce the relevant physcs whle gnorng rrelevant detals s called coarse-granng and forms a feld of vvd research n computatonal physcs and chemstry. A common approach n coarse-granng s to follow a bottom-up strategy: startng wth the mcroscopc equatons, smplfed mesoscopc models are derved that reproduce the macroscopc dynamcs. The mcroscopc dynamcs of the partcles (atoms or molecules) of a flud s descrbed by Newton s equaton of moton. In prncple, these equatons can be solved for gven ntal condtons, for example by a molecular dynamcs (MD) algorthm. However, ths s usually not feasble for a flud snce the number of partcles whose trajectores one has to compute s on the order of the Avogadro number Av Another problem s that such a system falls nto the regme of Lyapunov nstablty, such that t s mpossble to compute a determnstc trajectory due to the omnpresent round-off errors. 3 On the other hand, the observables used to descrbe the state of the flud also depend on a large number of partcles. The macroscopc propertes are qute stable and nsenstve to devatons n the ntal condtons. Ths suggests that the global observables can be descrbed as statstcal averages over a large number of partcle trajectores. The exact knowledge of the ndvdual trajectores s not requred any more, hence we can use a probablstc pcture to descrbe the moton of the partcles tself. These assumptons are the bass of knetc theory [50] The Boltzmann equaton A central quantty n knetc theory s the (sngle-partcle) dstrbuton functon f(r, v, t) depctng the probablty to fnd a partcle wth velocty v around the pont r at tme t. That s, the quantty f(r, v, t) dr dv (2.4) represents the mean number of molecules n the phase-space volume drdv. The dstrbuton functon f s lnked to the macroscopc observables by ts moments, for example the mass, momentum and energy denstes are gven by m f dv = ρ(r, t), m vf dv = ρu(r, t), (2.5) v 2 m f dv = ρe(r, t). 2 2 From ths vewpont, the mcro-model s n prncple arbtrary, but t wll turn out that there are a number of prerequstes that have to be fulflled n order to obtan Naver-Stokes behavor. 3 In fact, t s mpossble to specfy the ntal condtons precsely, and the computed trajectores always dverge exponentally from the exact soluton. 9

30 2 The lattce Boltzmann method: a modern overvew The tme evoluton of the dstrbuton functon f s governed by the equaton [50, 51] ( t + v r + F ) m f(r, v, t) = ( f) v coll, (2.6) where m s the mass of the partcles and F an external force actng on them. The left hand sde descrbes the streamng of the partcles along ther trajectores. The term ( f) coll on the rght hand sde descrbes the change of the dstrbuton due to collsons between the partcles. It s a shorthand notaton and contans the nformaton about the mcroscopc nteractons between the partcles. In prncple, t contans the two-partcle dstrbuton functon whch tself s governed by a dynamcal equaton nvolvng the three-partcle dstrbuton functon and so on. Ths nfnte herarchy of equatons s called the BBGKY herarchy and stems from the Louvlle equaton for the full phase space probablty densty [50]. A closed form of equaton (2.6) can be obtaned when pont partcles are consdered and the collsons between them are assumed to be bnary and uncorrelated. Ths s the so-called molecular chaos assumpton (or Stosszahlansatz) that leads to the celebrated Boltzmann equaton ( t + v r + F m v ) f(r,v, t) = dv 1 dω σ( v rel, Ω) v rel [f(r, v, t)f(r, v 1, t)) f(r, v, t)f(r, v 1, t)], (2.7) where v rel = v 1 v s the relatve velocty before the bnary collson, σ(v rel, Ω) s the scatterng cross secton, and v and v 1 are the post-collsonal veloctes characterzed by the scatterng angle Ω [see Ref. 50 for detals]. In spte of the closed form, the ntegrodfferental equaton (2.7) s n general complcated to solve The Maxwell-Boltzmann equlbrum dstrbuton A pvotal role n knetc theory s played by the local equlbrum dstrbuton, that s, a soluton of the Boltzmann equaton whch, n the absence of external forces, s ndependent of r and t, or equvalently, s a collsonal nvarant satsfyng ( f eq ) coll = 0. From the collson term n the Boltzmann equaton (2.7) we can deduce the condton of detaled balance f(r, v 1, t)f(r, v 2, t) = f(r, v 1, t)f(r, v 2, t). (2.8) Detaled balance mples that the logarthm of f s an addtve nvarant, hence n thermodynamc equlbrum, ln f must be a lnear combnaton of the collsonal nvarants ln f = γ 0 + γ v + γ 4 v 2. (2.9) The parameters γ can be expressed n terms of the hydrodynamc felds whch leads to the Maxwell-Boltzmann equlbrum dstrbuton [50 52] ( ) 3 ] m f eq 2 ρ (v) = [ 2πk B T m exp m(v u)2, (2.10) 2k B T where k B s the Boltzmann constant. 10

31 2.2 Knetc theory and contnuum flud mechancs The lnearzed Boltzmann equaton In practce, t s usually sutable to assume that the actual dstrbuton functon devates only slghtly from the local equlbrum dstrbuton. In ths case, we have f = f eq + f neq, f neq 1, (2.11) f eq and the collson term can be lnearzed around the equlbrum. Thus we get the lnearzed Boltzmann equaton ( t + v r + F ) m f(r, v, t) = Cf neq, (2.12) v where C s a lnear operator. It can be shown that the operator C s self-adjont wth respect to the scalar product 1 g h = f eq (v) g (v)h(v) dv, (2.13) and ts egenvalues are negatve or equal to zero. It s clear from the defnton that C has the degenerate egenvalue zero correspondng to the collsonal nvarants. The smplest form of the lnear collson operator s the BGK approxmaton (after Bhatnagar, Gross and Krook [19]), whch assumes a collson frequency λ such that durng the tme nterval dt a fracton λ dt of partcles s relaxed to equlbrum. The collson operator then becomes C BGK f neq = λf neq. (2.14) Ths expresson s much easer to treat n analytcal calculatons and many lattce Boltzmann models are based on the BGK approxmaton. We wll use the BGK approxmaton n some of the dervatons that follow n order to keep the formal presentaton smple. In most cases, a general lnear operator does not ntroduce addtonal complcatons and can be treated along the same lnes Hydrodynamc felds and macroscopc equatons We have already seen that the connecton between the knetc level and the hydrodynamc level s obtaned by calculatng moments of the dstrbuton functon f. In general, a moment s gven by m ψ (r, t) = ψ(v)f(r, v, t) dv, (2.15) where ψ(v) s a polynomal n the components of v. The ntegraton over velocty space s essentally an averagng process whch reflects the statstcal nature underlyng the knetc theory pcture. The equatons of moton for these averages,.e., the macroscopc dynamcs, 11

32 2 The lattce Boltzmann method: a modern overvew are the local conservaton laws obtaned by multplyng the Boltzmann equaton (2.7) wth the collsonal nvarants. Ths yelds ρ t + j r = 0, j t + Π r = ρ (2.16) m F, where j = ρu and the pressure tensor Π(r, t) = (v v)f(r, v, t) dv (2.17) was ntroduced. Ths tensor descrbes the flow of macroscopc momentum due to mcroscopc moton of the partcles. Its equlbrum value can be computed from the Maxwell- Boltzmann dstrbuton (2.10) Π eq = (v v)f eq dv = p1 + ρu u, (2.18) where p = ρ k BT s the scalar thermodynamc pressure. Snce v v s not a collsonal nvarant, the pressure tensor has a non-equlbrum contrbuton whch has to be determned m from the dstrbuton functon f. Ths means that wthout explct knowledge of the dstrbuton functon we can not obtan a closed form of the equaton system (2.16) n terms of ρ and u. Comparson wth the Naver-Stokes equaton (2.1) shows that we have to requre Π = Π eq + Π neq = p1 + ρu u σ. (2.19) The mssng lnk s an explct expresson for the non-equlbrum pressure tensor Π neq whch has to match the vscous stress tensor σ. What remans to be done to come full crcle s to fnd a closure for the equaton system (2.16). Ths s usually done wth certan approxmatons. In the Chapman-Enskog method, a closure s obtaned by expressng the dstrbuton functon f and the hgher moments n terms of ρ and u and ther gradents. It s based on the assumpton that these macroscopc varables vary on scales much larger then the characterstc mcroscopc scales (lmt of small Knudsen number). The Chapman-Enskog method wll be explaned n detal n chapter 4. An alternatve approach s to expand the dstrbuton functon n Hermte polynomals. Such an expanson was used by Grad [53] to obtan partal dfferental equatons for the 13 hydrodynamcally sgnfcant moments. Snce there s a close connecton between the lattce Boltzmann method and the Hermte expanson, we sketch the procedure n the followng paragraphs. A systematc non-perturbatve procedure was presented by Levermore [54] whch leads to a whole herarchy of closed systems. However, snce we are only nterested n the Naver-Stokes behavor, we wll not dscuss ths approach further Dmensonless formulaton For what follows, t wll be useful to non-dmensonalze the Boltzmann equaton. The absence of physcal unts s also needed for the mplementaton of a computer algorthm. 12

33 2.2 Knetc theory and contnuum flud mechancs In order to remove the unts, we ntroduce a length scale l 0 and a tme scale t 0. We choose them n such a way that l 0 /t 0 = c 0 s a characterstc velocty. The form of the Maxwell- Boltzmann dstrbuton suggests to use c 0 = k B T 0 /m whch s the speed of sound n the flud at a characterstc temperature T 0. The dmensonless rato of the actual temperature T and T 0 wll be denoted by θ. Wth these defntons, the Maxwell-Boltzmann dstrbuton comes as f eq (v) = ρ exp [ (2πθ) 3/2 ] (v u)2 2θ (2.20) where v and u have been made dmensonless by scalng wth c 0, and ρ by scalng wth m/l 3 0, respectvely. Wth the above choce of the characterstc scales the Boltzmann-BGK equaton keeps the same form when λ s understood as a dmensonless relaxaton frequency. For the rest of ths secton, we wll stck to the dmensonless formulaton whch has the advantage that the moments have the same unts Hermte-Expanson Accordng to Grad [53], the dstrbuton functon f can be expanded n the bass of the Hermte polynomals as f(r, v, t) = ω(v) where the weght functon ω(v) s gven by n=0 1 n! a(n) (r, t)h (n) (v), (2.21) [ ] ω(v) = (2π) 3 2 exp v2. (2.22) 2 Here, both the Hermte polynomals H (n) (cf. appendx B.1) and the coeffcents a (n) are tensors of order n. The latter are gven by a (n) (r, t) = H (n) (v)f(r, v, t) dv. (2.23) The motvaton behnd ths expanson s that the coeffcents a (n) (r, t) are lnear combnatons of the moments,.e., the lower order expanson coeffcents are drectly related to the hydrodynamc varables by the followng denttes: a (0) = f dv = ρ, a (1) = vf dv = ρu, (2.24) a (2) = (v v 1) f dv = Π ρ1. 13

34 2 The lattce Boltzmann method: a modern overvew The hydrodynamc varables of the Naver-Stokes level are completely determned by the frst coeffcents. Consequently, the macroscopc equatons can be represented by partal dfferental equatons for the coeffcents a (n). From equaton (2.21) and (2.23) t follows that a truncaton of the Hermte expanson at a certan order does not change the expanson coeffcents up to that order, because the Hermte polynomals are mutually orthogonal. Ths means that we can approxmate the dstrbuton functon by the frst N Hermte polynomals f N (r, v, t) = ω(v) N n=0 1 n! a(n) (r, t)h (n) (v) (2.25) wthout changng the moments up to order N,.e., the conservaton equatons for the lower order moments are not affected by the truncaton. 4 The gan of the truncaton s that the system of partal dfferental equatons for the a (n) s now determned and can be used to obtan a closed set of hydrodynamc equatons. In ths way, Grad used the thrd order approxmaton to obtan hs 13-moment system. 5 Because of ts specal propertes, the Hermte expanson s also partcularly well suted for further dscretzatons. Ths wll be used n the next secton to derve the lattce Boltzmann equaton from the lnear Boltzmann equaton. 2.3 Dscretzaton of the Boltzmann equaton Havng acheved a closure of the moment equaton system, we now turn to the problem of dscretzng the Boltzmann equaton n order to make t accessble for a computer smulaton. For the sake of smplcty, we focus on the Boltzmann-BGK equaton wthout a forcng term n ths secton t f + v r f = λ (f f eq ). (2.26) We seek a dscrete representaton of phase space and tme (r, v, t) such that (2.26) can be turned nto a fnte-dfference scheme where all quanttes are evaluated at dscrete ponts. We wll do ths stepwse by frst dscretzng velocty space by means of a Gauss-Hermte quadrature. Ths yelds a so-called dscrete velocty model (DVM). If chosen approprately, the abscssae of the quadrature naturally lead to a dscretzaton of confguraton space n form of a regular lattce. Fnally, the dervatves n the Boltzmann equaton are replaced by fnte dfferences. Ths route of dscretzng the Boltzmann equaton follows the work of Shan and He [35], Shan et al. [47]. The Gauss-Hermte quadrature was already used earler by He and Luo [33, 34] to derve the lattce Boltzmann equaton n a slghtly dfferent way than the one presented here. 4 The truncated terms may, however, affect the dynamcs of the hydrodynamc varables. Therefore the truncaton s really an approxmaton whose valdty wll be justfed later. 5 In prncple, there are 20 moments for the thrd order approxmaton, but only thrteen are consdered hydrodynamcally sgnfcant n Grad s moment system. 14

35 2.3 Dscretzaton of the Boltzmann equaton Dscretzaton of velocty space In the followng, we assume that the dstrbuton functon can be approxmated by ts truncated Hermte expanson f N. We wll see later that ths approxmaton corresponds to the lmt where the typcal hydrodynamc flow velocty u s small compared to the speed of sound c s, that s, the Mach number Ma = u/c s s small. The expanson coeffcents are then gven by a (n) = H (n) (v)f N (r, v, t) dv = ω(v) p(r, v, t) dv. (2.27) Snce p = H(n) f N s a polynomal of degree less than or equal to 2N, we can use the Gaussω Hermte quadrature explaned n appendx B.1 to calculate the ntegral n (2.27) usng the values of p at a set of dscrete veloctes c a (n) = w p(r, c, t) = H (n) (c )f N (r, c, t) w. (2.28) ω(c ) The nodes c and the weghts w are gven by the quadrature and depend on the chosen degree of the quadrature. We now ntroduce f (r, t) = w f N (r, c, t) ω(c ) (2.29) whch are functons of space and tme only. It s mportant to note that the truncated dstrbuton functon f N s completely determned by the f. Therefore, wthout approxmaton the hydrodynamc varables can be wrtten as ρ = a (0) = ρu = a (1) = f, f c, (2.30) Π = a (2) + a (0) 1 = f c c Ths s already the form of the hydrodynamc felds that wll be used n the lattce Boltzmann method. 6 The Maxwell-Boltzmann equlbrum dstrbuton can be expanded n Hermte polynomals n the same way. Ths s necessary because f eq has non-zero Hermte coeffcents at all orders, such that the conservaton laws for the collsonal nvarants hold exactly only when f eq s truncated smlarly to f [35]. Ths corresponds to a projecton nto the subspace spanned by the Hermte polynomals up to the respectve order. Replacng f N by f eq n 6 From now on, we wll skp the symbol n the tensor product. In the opnon of the author, ths makes the structure of the formulas more vsble to the reader s eye. For the more mathematcally nclned readers, please accept ths apology for the lack of notatonal rgor. 15

36 2 The lattce Boltzmann method: a modern overvew equaton (2.27) and nsertng nto (2.21) we get the Hermte expanson of the Maxwellan. Up to second order t reads [ f eq,(2) (v) ω(v)ρ 1 + u v uu : (vv 1) + 1 ] 2 (θ 1)(v2 D). (2.31) For an sothermal system θ = 1 and the last term n the brackets vanshes. Then ths expresson s equal to the Taylor expanson of f eq up to terms of order u 2. The latter has been used by He and Luo [33, 34] to derve the lattce Boltzmann equaton on a slghtly dfferent route. In analogy to (2.29), we ntroduce the equlbrum dstrbuton for the dscrete veloctes f eq = w f eq,(2) (c ) ω(c ) [ = w ρ 1 + u c + 1 ] 2 uu : (c c 1). (2.32) Fnally, we formulate the Boltzmann-BGK equaton n terms of the dscrete velocty dstrbutons f. Takng equaton (2.26) at c and multplyng agan wth w /ω(c ) we arrve at t f + c r f = λ(f f eq ). (2.33) Ths set of dfferental equatons n space and tme consttutes a dscrete velocty model (DVM) and represents an approxmaton to the contnuous Boltzmann-BGK equaton [55, 56]. The moments ρ and u are preserved, an mportant feature f one ams at the hydrodynamc lmt of the knetc equatons. It s also possble to preserve hgher moments by gong to hgher order Hermte approxmatons. The relaton of the DVM and the lattce Boltzmann equaton has been emphaszed by Luo [57], n partcular wth respect to some rgorous results concernng thermodynamcs and the H-theorem. In ths context t should be noted that the postvty of the Maxwell-Boltzmann dstrbuton s sacrfced n the fnte Hermte expanson. Ths s of relevance when the stablty of the lattce Boltzmann method s concerned. Even more mportant, t has the consequence that no H-theorem exsts for the truncated equlbrum dstrbuton,.e., t s not guaranteed that any ntal dstrbuton wll converge to the equlbrum dstrbuton. For ths reason, the latter s sometmes also termed pseudo-equlbrum. The lack of an H-theorem has motvated alternatve approaches whch have led to the development of the entropc lattce Boltzmann models [28 30, 32]. The concept of entropy wll be used later n ths work when we dscuss the statstcal mechancs of the lattce Boltzmann equaton, cf. chapter 3. We close ths secton here by quotng the unscaled dscrete equlbrum dstrbuton f eq [ = w ρ 1 + u c + uu : (c ] c 1), (2.34) c 2 s 2c 4 s where c s = k B T/m s the sothermal speed of sound. Note that we have delberately made the transton to a mass densty here, whereas before the dstrbuton functons where number denstes. 16

37 2.3 Dscretzaton of the Boltzmann equaton Dscretzaton of confguraton space and tme In order to dscretze space and tme, we rewrte the dscrete velocty equaton (2.33) as an ordnary dfferental equaton df dt + λf = λf eq. (2.35) Ths can be formally ntegrated over a tme τ to gve f (r + τc, t + τ) = e λτ f (r, t) + λe λτ τ For small τ, we can to a frst approxmaton wrte f eq 0 e λt f eq (r + t c, t + t ) dt. (2.36) (r + t c, t + t ) = f eq (r, t) + t f eq (r + τc, t + τ) f eq (r, t) + O(τ 2 ). (2.37) τ Expandng the exponental as well and neglectng all terms of order O(τ 2 ), we arrve at f (r + τc, t + τ) = f (r, t) λ [f (r, t) f eq (r, t)]. (2.38) Ths equaton s now fully dscrete n phase space and tme. It s to be noted that the use of the BGK approxmaton does not mean a loss of generalty here. The result of the full dscretzaton for the Boltzmann equaton wth a general lnear operator reads f (r + τc, t + τ) = f (r, t) + L j [ fj (r, t) f eq j (r, t)]. (2.39) Equaton (2.39) s nothng but the famous lattce Boltzmann equaton (LBE). The dervaton shows that t s merely a fnte dfference approxmaton to the contnuous Boltzmann equaton. In partcular, the expresson for the equlbrum dstrbuton s a result of the projecton onto the lower order Hermte polynomals and the weghts w are a pror known through the choce of the quadrature. Ths s n contrast to the alternatve approach where the weghts are determned wthn the Chapman-Enskog expanson such that the correct hydrodynamc equatons are obtaned. What remans to be done at ths stage n order to complete the development of the lattce Boltzmann method s to choose an approprate quadrature to obtan the dscrete veloctes c and the correspondng weghts w Choce of truncaton and quadrature As ponted out n the prevous secton, the truncated Hermte expanson f N preserves the moments up to order N. In what follows, we wll use the second order truncaton whch preserves ρ, j and Π. We wll only be concerned wth sothermal models, hence the heat flux contaned n the thrd moment s not of prmary nterest. However, n the Chapman- Enskog analyss t turns out that the thrd moment enters the dynamcs of the pressure tensor, 17

38 2 The lattce Boltzmann method: a modern overvew Quadrature LB model q b q w q c q E1,5 3 2 D1Q ± 3 E2,5 9 4 D2Q (0, 0) (± 3, 0), (0, ± 3) (± 3, ± 3) E3, D3Q (0, 0, 0) (± 3, 0, 0), (0, ± 3, 0), (0, 0, 3) (± 3, ± 3, ± 3) E3, D3Q (0, 0, 0) (± 3, 0, 0), (0, ± 3, 0), (0, 0, 3) (± 3, ± 3, 0), (± 3, 0, ± 3), (0, ± 3, ± 3) E3, D3Q (0, 0, 0) (± 3, 0, 0), (0, ± 3, 0), (0, 0, 3) (± 3, ± 3, 0), (± 3, 0, ± 3), (0, ± 3, ± 3) (± 3, ± 3, ± 3) Table 2.1: Gauss-Hermte quadratures of degree 5 n dfferent dmensons and the correspondng lattce Boltzmann models. Followng Shan et al. [47], the namng conventon ED,d n denotes a degree-d quadrature n D dmensons wth n abscssae. The vectors c q wth the same value of q = c 2 /3 form a symmetry class wthn whch the weght w q does not vary. By scalng the c q wth 3 sublattces of the standard cubc lattce are obtaned. such that n prncple the second order truncaton s not suffcent to reproduce the Naver- Stokes equaton [47]. The error s, however, of order O(Ma 3 ) and can be neglected. 7 The calculaton of the Hermte coeffcents of the Nth order expanson nvolves polynomals up to degree 2N. The quadrature therefore must have a degree n 2N,.e., for the second order approxmaton we need a quadrature of degree n 4. In addton to the accurateness, the quadrature s requred to nduce a regular lattce,.e., the nodes c should leave the spatal grd nvarant under the transformaton r r + τc. For detals on the producton rules for three dmensonal quadratures, we refer to appendx B.1. It turns out that some of the commonly used lattce Boltzmann models, e.g., the D2Q9, D3Q19 and D3Q27 models after the namng conventon of Qan et al. [22], stem from a degree-5 quadrature. The correspondng quadratures are lsted n table 2.1, where we follow the nomenclature of Shan et al. [47]. An mportant property of the quadratures s that they automatcally mply sotropy of the lattce tensors of rank up to the degree of the quadrature [59] T (n) = { 0 n odd w c... c =, n m. (2.40) δ (n) n even Ths s an mportant requrement to obtan hydrodynamc behavor. If nstead of a quadrature an ad-hoc ansatz for the lattce model s used, the weghts w have to be determned such 7 The second order truncaton leads to errors n the vscosty of order O(u 2 ) whch are related to ncomplete Gallean nvarance of the hgher moments [58, 59]. 18

39 2.4 The lattce Boltzmann equaton that sotropy of the lattce tensors up to a certan rank s satsfed. Ths approach s worked out n more detal n secton 2.5 for the D3Q19 model. It should be remarked that a lattce Boltzmann model does not necessarly need to correspond to an exact quadrature. For example, the D3Q13 model does not stem from a quadrature and stll obeys sotropy and Gallean nvarance [24]. However, the models related to quadratures usually yeld better accuracy and stablty [60]. All the quadratures lsted n table 2.1 yeld regular cubc lattces because ther veloctes can be expressed by nteger multples of a common constant,.e., 3. In general, the rato of the veloctes of a quadrature can be rratonal such that they do not connect the nodes of a smple lattce. Ths holds n partcular for most hgher degree quadratures, e.g., E3,7, 27 whch have to be consdered for thermo-hydrodynamcs, or the Burnett level momentum flow. In ths case, the smple lattce Boltzmann method can not be used and one has to resort to more complex algorthms, for example the nterpolaton scheme by He et al. [61] or the volumetrc scheme by Chen [62]. Havng obtaned an explct quadrature, the lattce Boltzmann equaton s fully specfed and an algorthm can n prncple be mplemented. The dervaton presented here s systematc and shows the underlyng knetc nature of the lattce Boltzmann method. However, when workng on algorthms for computer smulatons, t s useful to also have a more vvd understandng of the physcs behnd the equatons. In the next secton, we wll therefore depct the lattce Boltzmann equaton from a more algorthmc vew n order to demonstrate how t works. 2.4 The lattce Boltzmann equaton As we have seen, the lattce Boltzmann equaton s n essence a systematcally dscretzed form of the contnuous Boltzmann equaton. However, to be algorthmcally useful, the dscretzatons of phase space and tme should be chosen coherently n such a manner that the dscrete veloctes form the lnks of a regular lattce. Then we can nterpret the dscrete dstrbuton functons f (r, t) as quanttes assgned to a lattce ste r at tme t. We wll refer to the f as populatons of the lattce ste r n ths context. The lattce Boltzmann equaton descrbes the dynamc evoluton of these populatons, whch can be wrtten n two parts as f (r + τc, t + τ) = f (r, t) = f (r, t) + (f(r, t)) (2.41) These two parts can be llustrated as follows: the transton from f to f s an nstantaneous local process where the populatons are redstrbuted among the dfferent velocty drectons accordng to the operator. Ths process s due to the collsons at the mcroscopc level and s therefore called the collson step. The second part of the lattce Boltzmann equaton s the streamng step whch assgns the post-collsonal populaton f at r and tme t to the new populaton at r + τc and tme t + τ. One can thnk of the streamng as a whole populaton movng along a lnk c of the lattce, but ths should not be confused wth the 19

40 2 The lattce Boltzmann method: a modern overvew real moton of sngle partcles. The nformaton about the latter s not ncluded n the knetc pcture and can not be recovered at ths level. The lattce Boltzmann equaton s thus a truly mesoscopc approach to flud dynamcs, as t contans more nformaton than just the macroscopc Naver-Stokes descrpton whle at the same tme not representng the full mcroscopc degrees of freedom. The combnaton of collsons and streamng leads to an update scheme for the whole lattce. Because of the smple structure and the localty of most of the operatons, the scheme s very well suted for hghly effcent mplementatons on parallel computng platforms. We wll deal wth mplementng an effcent lattce Boltzmann kernel n some detal n appendx A. The equlbrum dstrbuton appearng n equaton (2.39) was derved n the prevous secton from the standard Maxwell-Boltzmann equlbrum, based on the systematc dscretzaton of velocty space. As ponted out by d Humères [23], t s also admssble to consder the set of dscrete veloctes as a free choce and use the form of the equlbrum dstrbuton as an ansatz wth certan parameters that allow the model to be tuned for specfc propertes. Bascally, ths means that the type of lattce can be chosen wth respect to certan constrants. The ntroduced free parameters n the equlbrum dstrbuton correspond to choosng the equlbrum values of the conserved and non-conserved moments. 8 Smlarly, the collson operator provdes some freedom through ts egenvalues whch are unspecfed parameters so far. Snce we am at constructng a mesoscopc model for flud mechancs, the choce of these parameters s dctated by the ntrnsc propertes of the Naver-Stokes equaton, that s, the symmetres and conservaton laws of macroscopc flud mechancs. Whle the explct lnk wll be establshed wthn the Chapman-Enskog expanson, we antcpate the results of chapter 4 at ths pont and lst the requrements that the constructon of a lattce Boltzmann model has to meet: 1. The lower moments correspondng to hydrodynamc felds have to satsfy mass conservaton: f = momentum conservaton: f c = f eq = ρ, (2.42) f eq c = ρu, (2.43) Euler form of equlbrum stress: f eq c c = p1 + ρuu. (2.44) 8 It can be shown that the form of the hydrodynamc equatons s completely determned by the lower moments of the equlbrum dstrbuton [59]. The number of free parameters depends on the order at whch the Hermte expanson s truncated. Ths determnes also the order to whch the equlbrum moments agree wth those of the Maxwell-Boltzmann dstrbuton. 20

41 2.5 The D3Q19 model 2. The collson operator must obey mass conservaton: momentum conservaton: Naver-Stokes form of the devatorc momentum flux: (f neq = 0, (2.45) c = 0, (2.46) + f neq )c c = 2σ. (2.47) Before we proceed to the systematc dervaton of these requrements, we llustrate them n terms of the D3Q19 model, whch s one of the most popular lattce Boltzmann models for smulatng complex fluds. 2.5 The D3Q19 model Equlbrum dstrbuton The D3Q19 model s based on a three-dmensonal regular cubc lattce and the set of 19 dscrete veloctes conssts of the zero velocty vector, the sx nearest neghbors and twelve next-nearest neghbors on the cubc lattce, see fgure 2.1. The correspondng dmensonless lattce vectors ĉ are the columns of the matrx C = and the velocty vectors are c = ĉ a/τ. The ansatz for the equlbrum dstrbuton s the low Mach number expanson f eq (ρ, u) = w ρ ( 1 + Au c + B(u c ) 2 + Cu 2). (2.48) We have to retan at least terms up to O(u 2 ) n order to obtan the quadratc term n the Euler stress. Sometmes, the non-lnear terms are neglected whch corresponds to the creepngflow lmt [63, 64]. However, the uu-terms ensure Gallean nvarance of the Euler stress [65, 66] and wll not be neglected here. For symmetry reasons, the w must be ndependent of the drecton of the c and must only depend on the length c. The other coeffcents A, B and C are ndependent of c. Snce mass conservaton s to be satsfed ndependently of the value of u, we can use u = 0 n (2.48) to obtan the normalzaton condton for the weghts w = 1. (2.49) 21

42 2 The lattce Boltzmann method: a modern overvew Fgure 2.1: Illustraton of the D3Q19 model. There are 19 populatons on every lattce ste: sx that move to the nearest neghbors durng streamng (red), twelve that move to to the next nearest neghbors (blue), and one populaton that stays on the same lattce ste (yellow). The remanng condtons on the moments on f eq ( = ρ ( f eq f eq c α = ρ f eq c α c β = ρ yeld ) 1 + AT α (1) u α + BT (2) αβ u αu β + Cu α u α, ) T α (1) + AT (2) αβ u β + BT (3) αβγ u βu γ + T α (1) u β u β, ( ) T (2) (3) αβ + AT αβγ u γ + BT (4) αβγδ u γu δ + T (2) αβ u γu γ, (2.50) where the lattce sums T (n) α 1...α n = w c α1... c αn. (2.51) are nvolved. The lattce sums are treated n appendx B.2. Usng (B.22), (B.24) and (B.26) n (2.50) we get f eq = ρ + Bσ 2 ρu α u α + Cρu α u α, f eq c α = Aσ 2 ρu α, (2.52) f eq c α c β = ρσ 2 δ αβ + Bκ 4 δ αβγδ ρu γ u δ + Bσ 4 (δ αβ δ γδ + δ αγ δ βδ + δ αδ δ βγ ) ρu γ u δ + Cσ 2 δ αβ ρu γ u γ. By comparson wth (2.42) to (2.44), we can fnd a unque soluton for the coeffcents of the equlbrum dstrbuton and the lattce tensors: A = 1, B = 1, C = 1, c 2 s 2c 4 s 2c 2 s σ 2 = c 2 s, σ 4 = σ2, 2 κ 4 = 0. (2.53) 22

43 2.5 The D3Q19 model The weghts are then gven by w = 1 3 for ĉ 2 = 0 (zero velocty), w = 1 18 for ĉ 2 = 1 (nearest neghbors), (2.54) w = 1 36 for ĉ 2 = 2 (next-nearest neghbors), and the speed of sound has the fxed value c 2 s = 1 3 ( a τ ) 2. (2.55) Ths soluton corresponds to the case where the mass densty s equally dstrbuted among the dfferent subshells. It was noted earler that ths s benefcal for the stablty of the D3Q19 model, and the maxmum entropy formalsm provdes a deeper explanaton for ths mproved stablty. Ths was already recognzed by Karln et al. [30], albet the formalsm was slghtly dfferent there. We can now wrte down the equlbrum dstrbuton for the D3Q19 model: ( f eq (ρ, u) = w ρ 1 + u c + (u c ) ) 2 u2, (2.56) c 2 s 2c 4 s 2c 2 s whch s the same form as obtaned by the truncated Hermte expanson n equaton (2.34) Collson operator Havng constructed the equlbrum dstrbuton, we now turn to the lnear collson operator. It has to satsfy mass and momentum conservaton and must be compatble wth the symmetres of the D3Q19 lattce. Furthermore, t has to yeld the Naver-Stokes form of the devatorc momentum flux, whch ndcates that the form of the collson operator s related to the vscosty of the flud. The precse lnk wll be establshed wthn the Chapman-Enskog expanson. In ths subsecton, we develop a representaton for the general lnear collson operator. ( (f(r, t)) = L j fj f eq ) j = Lj f neq j. (2.57) The smple BGK collson operator corresponds to the choce L j = λδ j, whch has some drawbacks such as fxed rato of bulk and shear vscosty. A more general collson operator s provded by the mult-relaxaton tme model (MRT) of d Humères et al. [25], whch uses a dagonal representaton of L j n the so-called mode space. The bass vectors e k of mode space are constructed by orthogonalzng polynomals of the dmensonless velocty vectors ĉ. The correspondng orthogonalty relaton s w e k e l = b k δ kl, (2.58) 23

44 2 The lattce Boltzmann method: a modern overvew where the weghts from the equlbrum dstrbuton enter. Ths s not necessary n general, but t has the advantage that the knetc moments have no projecton on the equlbrum dstrbuton. The backward relaton s then b 1 k e ke kj = w 1 δ j. (2.59) The b k are normalzaton factors wth respect to the scalar product k b k = w e 2 k. (2.60) Wth ths, we can apply Gram-Schmdt orthogonalzaton to polynomals of the ĉ n a carefully chosen order. The frst bass vectors are gven by e 0 = 1, e 1 = ĉ x, e 2 = ĉ y, e 3 = ĉ z. (2.61) They correspond to the mass and the momentum, respectvely. The next sx bass vectors are obtaned from quadratc polynomals e 4 = ĉ 2 1, e 5 = 3ĉ 2 x ĉ 2, e 6 = ĉ 2 y ĉ 2 z, e 7 = ĉ x ĉ y, e 8 = ĉ x ĉ z, e 9 = ĉ y ĉ z. (2.62) They correspond to bulk (k = 4) and shear (k = ) modes. Up to here, the polynomals are complete. The hgher order polynomals are not complete due to degeneraces n the D3Q19 model,.e., ĉ α = ĉ 3 α. Nevertheless, by sortng out the degeneraces we can construct a complete bass. Orthogonalzaton of the non-degenerate hgher order polynomals yelds the remanng bass vectors e 10 = (3ĉ 2 5)ĉ x, e 11 = (3ĉ 2 5)ĉ y, e 12 = (3ĉ 2 5)ĉ z, e 13 = (ĉ 2 y ĉ 2 z)ĉ x, e 14 = (ĉ 2 z ĉ 2 x)ĉ y, e 15 = (ĉ 2 x ĉ 2 y)ĉ z, e 16 = 3ĉ 4 6ĉ 2 + 1, e 17 = (2ĉ 2 3)(3ĉ 2 x ĉ 2 ), e 18 = (2ĉ 2 3)(ĉ 2 y ĉ 2 z). (2.63) 24

45 2.5 The D3Q19 model The normalzaton factors of these bass vectors are b = ( 1, 1 3, 1 3, 1 3, 2 3, 4 3, 4 9, 1 9, 1 9, 1 9, 2 3, 2 3, 2 3, 2 9, 2 9, 2 9, 2, 4 9, 4 3) T. (2.64) The bass vectors are used to calculate a complete set of moments, the so-called modes m k = e k f. (2.65) The back transformaton from mode space to the populatons can be obtaned usng (2.59) and s gven by f = w b 1 k m ke k. (2.66) k By constructon, the moments orgnatng from polynomals up to quadratc order yeld the hydrodynamc varables ρ = m 0, j x = m 1 a/τ, j y = m 2 a/τ, j z = m 3 a/τ, Π αα = (m 0 + m 4 ) (a/τ) 2, Π xx = 1 3 m 5(a/τ) 2, Π yy = m 5 3m 6 (a/τ) 2, 6 Π zz = m 5 + 3m 6 (a/τ) 2, 6 Π xy = m 7 (a/τ) 2, Π xz = m 8 (a/τ) 2, Π yz = m 9 (a/τ) 2, (2.67) where we have decomposed the pressure tensor nto ts trace and the traceless part Π αβ = Π αβ Π γγδ αβ. (2.68) The frst moments m 0 to m 3 are the conserved hydrodynamc modes,.e., mass densty and momentum densty. The equlbra of the non-conserved hydrodynamc modes can be 25

46 2 The lattce Boltzmann method: a modern overvew expressed as functons of the mass densty and the momentum densty: m eq 4 = j2 x + jy 2 + jz 2, ρ m eq 5 = j2 x jy 2, ρ m eq 6 = 2j2 x jy 2 jz 2, ρ m eq 7 = j xj y ρ, m eq 8 = j xj z ρ, m eq 9 = j yj z ρ. (2.69) The hgher moments m 10 to m 18 are related to the addtonal degrees of freedom due to the knetc representaton. They wll therefore be referred to as knetc modes. 9 By constructon, they have no equlbrum part. Although the knetc modes do not nfluence the dynamcs n the hydrodynamc lmt, they can have an effect through the underlyng knetc model. Ths becomes mportant when addng fluctuatons to the model and n the case of boundary condtons. It should be remarked that n prncple one can choose the equlbrum values of all the non-conserved moments at wll, as long as they are compatble wth the symmetry of the lattce. The chosen values then enter the MRT algorthm (cf. equaton (2.71) below) nstead of those obtaned from the explct expresson for the equlbrum dstrbuton. However, t was shown n Lallemand and Luo [26] that the addtonal parameters, f chosen to satsfy Gallean nvarance and sotropy, reduce to the ones that come out of the equlbrum dstrbuton. We wll therefore use the expressons (2.69) wthout any addtonal parameters. The transformaton to mode space can be wrtten n matrx-vector form m = M f (2.70) where the entres of the transformaton matrx M are the components of the bass vectors e j. For the D3Q19 model, the matrx s gven n fgure 2.2. Snce M represents a bass transformaton, the collson operator can be represented n mode space as L f neq = M 1 ( MLM 1) M f neq = M 1 L M m neq (2.71) where L M = M L M 1. In mode space, we can choose the collson operator to be dagonal such that the collsons descrbe a lnear relaxaton of the non-equlbrum moments m neq k = (1 + λ k ) m neq k. (2.72) 9 The knetc modes are sometmes also called ghost modes, because they have no relevance on the hydrodynamc level. 26

47 2.5 The D3Q19 model M = Fgure 2.2: The transformaton matrx M for the D3Q19 model for the bass vectors gven n equatons (2.61), (2.62) and (2.63). It s to be noted that ths representaton depends on the order of the c. 27

48 2 The lattce Boltzmann method: a modern overvew The choce of the egenvalues λ k must reflect the symmetres of the lattce, that s, the modes that are related by a symmetry must have the same egenvalue. For the conserved modes m 0 to m 3, the egenvalue s rrelevant because m neq k = 0. The remanng modes can be grouped nto sx symmetry classes usng the explct expresson for the bass vectors; the bulk mode m 4, the shear modes m 5 to m 9, and four groups of knetc modes: the trplet m 10 to m 12, the trplet m 13 to m 15, the snglet m 16 and the doublet m 17 and m 18. From (2.67) we get that the egenvalues λ b and λ s for the bulk and shear modes correspond to the relaxaton of the trace and the traceless part of the pressure tensor Π neq αα Π neq αβ = (1 + λ b )Π neq αα, = (1 + λ s )Π neq αβ. (2.73) Wthn the Chapman-Enskog expanson t wll be shown that ths relaxaton process leads to the correct Newtonan vscous stress, where the vscostes are gven n terms of the egenvalues η s = ρc2 sτ 2 + λ s, 2 λ s η b = ρc2 sτ 2 + λ b. 3 λ b (2.74) The egenvalues for the knetc modes are not related to any macroscopc transport coeffcents and ther value s rrelevant on the Naver-Stokes level. They do, however, nfluence the dynamcs on the knetc level. Ths s mportant n the case of boundary condtons where the knetc egenvalues can be tuned to mprove the accuracy of the boundary condton. Moreover, the knetc egenvalues are related to the nose strength n the fluctuatng lattce Boltzmann model, whch wll be dscussed n secton

49 3 Statstcal mechancs of the lattce Boltzmann equaton The lattce Boltzmann model presented n the prevous secton s completely determnstc and does not nclude fluctuatons. Ths s a result of the coarse-granng procedure, where the averagng process for the sngle-partcle dstrbuton leads to a mean populaton number whle the nformaton about the varance s dropped. The connecton to statstcal mechancs of the mcroscopc degrees of freedom s thus lost. However, fluctuatons can be essental at the mesoscopc level. Brownan moton, for example, s an mmedate consequence of the fluctuatons of the solvent molecules beyond the hydrodynamc scale. Besdes Brownan moton, fluctuatons are also relevant for non-lnear phenomena and they have great mpact on many crtcal phenomena. Whle other smulaton methods for hydrodynamcs, e.g., dsspatve partcle dynamcs (DPD) [67 71] and mult-partcle collson dynamcs (MPCD) [7, 72 80], nclude those fluctuatons automatcally va the underlyng partcle representaton of the solvent, the lattce Boltzmann method must be extended by sutable ncluson of thermal fluctuatons. On the contnuum level, ths extenson corresponds to descrbng the mesoscale dynamcs of a flud by fluctuatng hydrodynamcs [81]. In ths secton, we wll ntroduce fluctuatng hydrodynamcs and the fluctuatng lattce Boltzmann model. The orgnal verson of the fluctuatng lattce Boltzmann equaton was developed by Ladd [36 38] and connects the fluctuatons of the populatons to the fluctuatng stress by solvng a dscrete Langevn equaton. Although the dervaton gves rse to the correct fluctuatng hydrodynamcs, t does not provde a drect lnk to the statstcal mechancs of the model. Adhkar et al. [1] observed n practcal smulatons that n fact the varance of the fluctuatng hydrodynamc quanttes s not fully captured wthn Ladd s approach. They demonstrated that ths can be mproved by addng fluctuatons to the non-hydrodynamc moments as well, but wthout a detaled theoretcal clarfcaton of the statstcal mechancs. In the course of ths work, we succeeded n fndng a new dervaton of the fluctuatng lattce Boltzmann equaton that provdes a consstent lnk to statstcal mechancs [2]. The presentaton here wll be based on the latter dervaton, whch also gves a clear quanttatve meanng to the fluctuatons of the populatons. 3.1 Fluctuatng hydrodynamcs We frst present the equatons of fluctuatng hydrodynamcs as put forward n [81]. The basc dea s that the structure of the hydrodynamc equatons (2.1) remans unchanged n the presence of fluctuatons because they are conservaton equatons for mass and momentum 29

50 3 Statstcal mechancs of the lattce Boltzmann equaton that hold n general. Fluctuatons n the flud can only lead to local momentum fluxes 1 that vansh globally. Hence they must enter the equatons as a dvergence, whch can be acheved by addng a fluctuatng part s to the stress tensor σ σ σ + s. (3.1) The propertes of the fluctuatng stress s have to be consstent wth the thermodynamcs of the flud. They can be derved wthn the theory of Gauss-Markov processes [82] or by usng methods from Langevn analyss [83]. The fluctuatng stress must have zero mean and must be delta-correlated n space and tme s αβ = 0, (3.2) s αβ (r 1, t 1 ) s γδ (r 2, t 2 ) = 2Q αβγδ δ(r 1 r 2 )δ(t 1 t 2 ). (3.3) Another way of lookng at the fluctuatons s to see them as the balancng forces as opposed to the vscous dsspaton n the flud. From ths pont of vew, the propertes of the nose follow from the approprate fluctuaton dsspaton relaton. The non-dsspatve part of hydrodynamcs,.e., an Euler flud s descrbed by the Hamltonan H EF = ( ρ ) dr 2 u2 + ɛ(ρ), (3.4) where ɛ(ρ) s the nternal energy densty of the flud, whch s related to the pressure p by ρ 2 ɛ ρ 2 = p ρ. (3.5) The latter equaton can be used to construct ɛ(ρ). The Louvlle operator for the Hamltonan system s [ ] δ L EF = dr α j α δρ + βπ E δ αβ, (3.6) δj α where δ and δ δρ δj α denote functonal dervatves wth respect to mass densty and momentum densty, and Π E αβ = pδ αβ + ρu α u β s the Euler stress. L EF can be used to wrte the Fokker- Planck equaton for the Euler flud t P ({ρ}, {j}) = L EF P ({ρ}, {j}). (3.7) Ths formulaton can now easly be extended to nclude dsspatve (vscous) and fluctuatng (stochastc) parts by addng the approprate terms to the Louvlle operator for the Euler flud,.e., we replace L EF L EF + L v + L s. (3.8) 1 We restrct the dscusson to an sothermal flud and the momentum equaton. The heat transport equaton can n prncple be treated n the same way by ntroducng local fluctuatng heat fluxes. 30

51 3.2 The fluctuatng lattce Boltzmann equaton The vscous and fluctuatng operators can be obtaned from the Naver-Stokes equaton by a Kramers-Moyal expanson [see Ref. 84] and are gven by δ L v = η αβγδ dr β γ u δ, δj α L s = Q αβγδ δ δ dr β γ. δj α δj δ (3.9) The fluctuaton dsspaton relaton requres that the Boltzmann factor exp( H EF /k B T ) s a statonary soluton of the Fokker-Planck equaton. Therefore we have to requre ( 0 = (L v + L s ) exp H ) EF k B T ] ( δ δ δ = [η αβγδ dr β γ u δ + Q αβγδ dr β γ exp H ) (3.10) EF. δj α δj α δj δ k B T Ths condton s satsfed f [ ] δ 0 = η αβγδ β γ u δ + Q αβγδ β γ δj [ δ = η αβγδ β γ u δ 1 k B T Q αβγδ β γ u δ whch yelds ( exp ] exp H ) EF k B T ( H EF k B T ), (3.11) Q αβγδ = k B T η αβγδ. (3.12) Usng ths result n (3.3) we get the correlatons for the traceless and trace parts of the fluctuatng stress tensor s αβ (r 1, t 1 ) s γδ (r 2, t 2 ) = 2η s k B T (δ αγ δ βδ + δ αδ δ βγ 23 ) δ αβδ γδ δ(r 1 r 2 )δ(t 1 t 2 ), s αα (r 1, t 1 ) s ββ (r 2, t 2 ) = 18η b k B T δ(r 1 r 2 )δ(t 1 t 2 ), s αβ (r 1, t 1 ) s γγ (r 2, t 2 ) = 0. (3.13) These are the same expressons as orgnally put forward by Landau and Lfshtz [81]. They ntroduce fluctuatons nto the hydrodynamc equatons n a thermodynamcally consstent way. The dervaton presented here ponts out that the stochastc momentum flux can be seen as the counterbalance to the vscous frcton. The fluctuaton dsspaton relaton between the dsspatve and the fluctuatng part assures that the correct sothermal ensemble s obtaned. We wll see later that such a balance s mportant for every degree of freedom n the system, even for those that are rrelevant on the hydrodynamc level. 3.2 The fluctuatng lattce Boltzmann equaton Fluctuatons can be ncorporated nto the lattce Boltzmann equaton by addng a stochastc contrbuton to the collson term = L j f neq j +. (3.14) 31

52 3 Statstcal mechancs of the lattce Boltzmann equaton Mass and momentum conservaton mply that the zeroth and frst moment of the stochastc contrbuton must vansh = 0, c = 0. (3.15) It s assumed that the fluctuatons are uncorrelated n space and tme such that the stochastc collson operator s stll local. The mean of the fluctuatons vanshes = 0, and the covarance matrx j has to generate the correct fluctuatons on the hydrodynamc level, cf. equaton (3.13). The stochastc collson operator gves rse to a random contrbuton to the non-equlbrum stress σ r αβ = c α c β. (3.16) The modfcaton of equatons (2.73) for the stochastc collson operator reads Π neq αα Π neq αβ = (1 + λ b )Π neq αα + σ r αα, = (1 + λ s )Π neq αβ + σ r αβ. (3.17) It s to be noted that σαβ r corresponds to fluctuatons on the lattce tme scale τ, as opposed to the fluctuatons s αβ of the hydrodynamc momentum flux on the knetc tme scale. The connecton between the two scales wll become explct wthn the Chapman-Enskog expanson. The relaton between the hydrodynamc fluctuatons and the random stresses on the lattce level s σ r αα = λ b s αα, σ r αβ = λ s s αβ, (3.18) whch leads to the followng correlatons for the random stresses σ r αβ σγδ r 2η s k B T λ 2 = s (δ a 3 αγ δ βδ + δ αδ δ βγ 23 ) τ δ αβδ γδ, σ r αα σββ r 18η b k B T λ 2 = b (3.19), a 3 τ σ r αβ σγγ r = 0. The delta dstrbutons of equaton (3.13) have been replaced here by the lattce unts a 3 and τ 1 to reflect the localty of the stochastc collsons on the dscrete lattce. These are the expressons for the random stresses as orgnally derved by Ladd [36, 37], whch guarantee correct fluctuatng hydrodynamcs at the macroscopc level. They can be drectly mplemented n the MRT lattce Boltzmann model by addng the random contrbutons n mode space durng the collson phase. In the orgnal mplementaton of Ladd and varous followups by other authors [38, 85 88], the random contrbuton was only mposed on the stress modes, whle the knetc modes where not thermalzed but projected out entrely durng the collson phase. Although ths s perfectly consstent wth fluctuatng hydrodynamcs, t was 32

53 3.3 The generalzed lattce gas model (GLG) shown by Adhkar et al. [1] that the procedure leads to poor accuracy on shorter length scales. The reason s that the procedure neglects the knetc nature of the lattce Boltzmann method, whch ncludes more degrees of freedom than just the hydrodynamc ones, namely the hgher order knetc modes. These are non-conserved modes that take part n the dsspatve processes n the flud, whch suggests that they should be thermalzed as well. It was shown numercally that addtonal nose on the knetc modes sgnfcantly mproves the accuracy on short length-scales [1]. In the course of ths thess, ths observaton could be clarfed theoretcally from a statstcal mechancs vewpont, makng use of a generalzed lattce gas model [2]. The generalzed lattce gas model and the dervaton of the statstcal mechancs of the fluctuatng lattce Boltzmann model wll be the subject of the followng secton. 3.3 The generalzed lattce gas model (GLG) One of the motves that drove the development of the lattce Boltzmann method was the am to cope wth the large statstcal nose nherent to the precedng lattce gas automaton models. Snce the dynamc quanttes n the lattce gas automata are boolean varables, a consderable amount of samplng s needed to obtan good data statstcs for the hydrodynamc felds. To crcumvent ths neffcency, the lattce Boltzmann method replaces the boolean varables wth ther ensemble-averaged populatons. Ths yelds smooth dynamc varables that are not subject to nose any longer. On the other hand, the complete absence of fluctuatons means that the connecton to the underlyng statstcal propertes of the populatons s lost, whch makes t necessary to rentroduce the fluctuatons a posteror. In the prevous secton we have shown how ths can be done for fluctuatng hydrodynamcs. However, that approach fals to restore the connecton to the statstcs of the underlyng mcro-model. For smulatons of soft matter systems t s of pvotal mportance to sample a well defned statstcal ensemble. In order to put the lattce Boltzmann method back onto the fundament of statstcal mechancs, we have developed the conceptual model of a generalzed lattce gas (GLG). In the GLG model, the equlbrum dstrbuton of the populaton numbers can be derved from fundamental statstcal consderatons. For ths purpose, we ntroduce an ensemble of populaton numbers on the local lattce ste where each velocty drecton c can be occuped by an nteger number ν of partcles. The evoluton equaton for a sngle realzaton of the occupances has the same form as n the lattce gas and lattce Boltzmann models ν (r + c h, t + h) = ν (r, t) = ν (r, t) + (ν(r, t)). (3.20) The collson operator of ths model redstrbutes partcles among the dfferent velocty drectons.the dfference to the LG and LB models les n the nature of the ν, for n a lattce gas the occupaton varables are boolean whereas n lattce Boltzmann real-valued varables are used. The ensemble pcture behnd the occupances allows to quantfy the dfference by 33

54 3 Statstcal mechancs of the lattce Boltzmann equaton lookng at the fluctuatons of the varables ν. The dmensonless Boltzmann number (Bo) s defned by ν 2 Bo = ν 2, (3.21) ν where the angle brackets denote ensemble averages. In the LG models, Bo 1,.e., the fluctuatons are on the same order as the mean whch corresponds to a fully mcroscopc model. Conversely, erasng any fluctuatons as n the determnstc LB models leads to Bo = 0. Our am s to ntroduce thermal fluctuatons n such a way that fluctuatng hydrodynamcs s obtaned and at the same tme statstcal consstency s retaned at the mcroscopc level. The connecton between the nteger varables ν and the hydrodynamc varables can be establshed by ntroducng the mass densty µ = m p a 3, (3.22) where m p s the mass of a partcle and a s the lattce spacng. The relaton between the GLG occupances ν and the LB mass denstes f s then f = µν, (3.23) and the hydrodynamc felds can be obtaned as usual n lattce Boltzmann Statstcs of the generalzed lattce gas The statstcs on the mcroscopc level s governed by the probablty dstrbuton of the occupaton numbers. In a homogeneous equlbrum state, we can consder the ndvdual occupances of the dfferent velocty drectons as ndependently sampled from a large reservor. The probablty dstrbuton for an ndvdual ν s then Possonan p(ν ) = νν e ν, (3.24) ν! whch s n accordance wth the phase-space occupances n an deal gas [89]. ν = ν denotes the mean number of partcles occupyng c. Ths mean occupaton number ν can be wrtten n terms of the total number of partcles ν on the lattce ste ν = w ν = w ν, (3.25) where, for symmetry reasons, the weghts w depend on the absolute speed of c only and not on the drecton. The total occupaton s related to the mass densty by µ ν = ρ. From Posson statstcs t follows that ν 2 ν 2 = ν and the varance of the LB mass densty f 2 f 2 = µ f = m p /a 3 f s controlled by the mass m p of a partcle. The latter can be related to the temperature through the deal gas equaton of state m p = k BT c 2 s. (3.26) 34

55 3.3 The generalzed lattce gas model (GLG) Ths means that f 2 f 2 T and the thermal fluctuatons can thus be controlled by the temperature as desred. The jont probablty dstrbuton P of the occupaton numbers s the product of the ndvdual Possonans, but subject to the constrants that t has to correspond to a gven total mass and momentum ( ) ( ν ν P ({ν }) e ν δ µ ) ( ν ρ δ µ ) ν c j. (3.27) ν! Usng Strlng s approxmaton for the factoral n (3.27) we can ntroduce an assocated entropy for the occupaton numbers S ({ν }) = (ν ln ν ν ν ln ν + ν ), (3.28) and the probablty P can be rewrtten as 2 ( P ({ν }) exp [S ({ν })] δ µ ) ( ν ρ δ µ ν c j ). (3.29) Equlbrum dstrbuton We now take as the equlbrum dstrbuton ν eq of the GLG the most probable set of occupaton numbers for gven values of mass and momentum. It can be obtaned by maxmzng P, or equvalently, by maxmzng the entropy S subject to the constrants. We take the constrants nto account va Lagrange multplers and maxmze the functonal ( ) ( ) S({ν }, χ, λ) = S({ν }) + χ ν ρ + λ ν c j. (3.30) µ µ Dfferentaton wth respect to ν, χ and λ results n the followng equaton system S ν + χ + λ c = 0, µ ν ρ = 0, (3.31a) (3.31b) µ ν c j = 0. (3.31c) 2 The assocated entropy and equaton (3.29) can also be derved by consderng a Bernoull experment where partcles are selected wth probablty p 0 such that ν = Np 0 partcles are drawn n total: p(ν) = N! ( ν ) ν ( 1 ν ) N ν. ν!(n ν)! N N The entropy (3.28) then follows from ln p(ν) n the lmt N at fxed ν [see also Ref. 8]. 35

56 3 Statstcal mechancs of the lattce Boltzmann equaton The formal soluton of (3.31a) s ν eq = ν exp (χ + λ c ), (3.32) where the Lagrange multplers χ and λ are functons of mass and momentum and have to be determned from the constrants (3.31b) and (3.31c). Due to the nonlnearty of the equaton system, the explct soluton s dffcult to obtan. However, n analogy to the low Mach number approxmaton n the conventonal lattce Boltzmann models, we seek a polynomal approxmaton for ν eq. We expand the equlbrum dstrbuton around the reference state where the flud s at rest. For ths case, we can wrte the equlbrum dstrbuton explctly ν (0) = ν eq (j = 0) = w ν. (3.33) The soluton for non-vanshng momentum s obtaned by perturbng around j = 0, that s, we expand the Lagrange multplers as ν eq χ = ɛ n χ (n), λ = n=1 ɛ n λ (n), j = ɛj (1) (3.34) where ɛ s a formal parameter that wll be set to one at the end. Expandng the equlbrum dstrbuton up to second order n the Lagrange multplers yelds [ = w ν 1 + ɛχ (1) + ɛλ (1) c n= ɛ2 (χ (1) + λ (1) c ) 2 + ɛ 2 χ (2) + ɛ 2 λ (2) c +... ]. (3.35) The constrants for mass and momentum should be satsfed by ths expanson at all orders. The zeroth order does ths by constructon. The hgher orders can be solved recursvely. On the frst order, we have µ ν ( ) w χ (1) + λ (1) c = 0, µ ν [ ] (3.36) w (1 + χ (1) )c + λ (1) c c = j (1). For the evaluaton, we use the symmetry propertes of the lattce sums of a cubc lattce derved n secton B.2. The result for the Lagrange multplers on the frst order s χ (1) = 0, λ (1) = j(1) µ νσ 2. (3.37) Insertng ths result nto the next order we obtan the equatons for the second order µ ν ( ) w χ (2) j (1) c = 0, 2 µ νσ 2 µ ν ( ) w λ (2) c c = 0, (3.38) 36

57 3.3 The generalzed lattce gas model (GLG) and usng the lattce sums agan we get the second order Lagrange multplers ( ) χ (2) = 1 2 j (1), λ (2) = 0. (3.39) 2σ 2 µ ν The procedure can be systematcally carred out to hgher orders. Results up to 8th order are for example gven by Ansumal [90]. Here, we shall be satsfed wth the second order approxmaton. Combnng (3.37) and (3.39) nto (3.35) we get the explct soluton for the equlbrum dstrbuton up to quadratc terms n j ν eq = w ν [ 1 + j c µ νσ σ 2 2 ( ) 2 j c 1 µ ν 2σ 2 Convertng the GLG occupaton numbers nto LB mass denstes, f eq ( ) ] 2 j. (3.40) µ ν = µν eq, we have exactly the same form of the equlbrum dstrbuton as used n the standard lattce Boltzmann models, cf. equatons (2.34) and (2.56). It should be remarked that the procedure descrbed here s very smlar to the entropc lattce Boltzmann approaches of Karln and Succ [29], Karln et al. [30], Ansumal et al. [91]. In that framework, however, the am s to obtan an H-theorem for lattce Boltzmann models, and the dervaton starts from a convex H-functonal. The resultng equlbrum has, n contrast to the dervaton presented here, no connecton to an underlyng probablty dstrbuton and consequently the entropc lattce Boltzmann s focused on determnstc models wthout fluctuatons Fluctuatons around equlbrum The equlbrum dstrbuton s the most probable set of populatons of a lattce ste for gven mass and momentum. The actual populatons ν fluctuate around those mean values accordng to the probablty dstrbuton P. Usng the Fourer representaton of the δ-dstrbutons we can rewrte P as [ ( P ({ν }) dq dk exp S({ν }) + q µ ) ( ν ρ + k µ )] ν c j. (3.41) The expresson n the square brackets s dentcal to the functonal S({ν }, q, k) where the Lagrange multplers χ and λ have been replaced by q and k, respectvely. The soluton ({ν eq }, q 0, k 0 ) obtaned from the equaton system (3.31a) above s a saddle pont, around whch we can expand to second order S ({ν }, q, k) = S ({ν eq }, q 0, k 0 ) + µ (q q 0 ) (ν ν eq ) 2 2ν eq (ν ν eq ) + µ (k k 0 ) c (ν ν eq ), (3.42) 37

58 3 Statstcal mechancs of the lattce Boltzmann equaton where we have used the explct form (3.28) of the entropy. Ths yelds 2 S ν ν j = 1 eq {ν } ν eq δ j, 2 S ν q = µ, 2 S ν k = µc. (3.43) Insertng the expanson nto (3.41) we can wrte the probablty dstrbuton for ν neq ν eq P ({ν neq }) exp d(q q 0 ) d(k k 0 ) [ (ν neq ) 2 2ν eq +µ (q q 0 ) ν neq + µ (k k 0 ) ] ν neq c. = ν (3.44) where we have transformed the varables of the Fourer ntegrals and absorbed all constant factors n the normalzaton. Fnally, we rentroduce the δ-dstrbutons and obtan [ P ({ν neq }) exp ] ( (ν neq ) 2 2ν eq δ µ ) ( ν neq δ µ ) ν neq c. (3.45) Ths expresson shows that the fluctuatons around the equlbrum have a Gaussan dstrbuton subject to constrants. The varance s ν eq and depends on drecton, whch s a consequence of the broken Gallean nvarance. However, snce the non-equlbrum populatons are small compared to the equlbrum value, we can approxmate the latter by the lmt of vanshng flud velocty u = 0. In ths case, Gallean nvarance s restored and the varance lm u 0 νeq = w ν (3.46) becomes ndependent of drecton. 3 The fnal result for the fluctuatons, wrtten n terms of the lattce Boltzmann populatons f, s [ P ({f neq }) exp ] ( ) ( ) (f neq ) 2 δ f neq δ f neq c, (3.47) 2µw ρ whch shows agan that the fluctuatons are controlled by µ. In order to look at the fluctuatons of the hydrodynamc varables, we transform to modes accordng to equaton (2.65). The probablty dstrbuton for the non-equlbrum parts n 3 Consequently, the weghts w do not depend on drecton. The approxmaton can be justfed wthn the Chapman-Enskog expanson. It turns out that up to second order, the macroscopc dynamcs s not changed by the approxmaton. 38

59 3.3 The generalzed lattce gas model (GLG) mode space s P ({m neq k }) exp [ k ] (m neq k )2 δ (m neq k 2b k µρ ) [ ] exp (mneq k )2 2µb k ρ k 3 k>3. (3.48) Here, the constrants have been elmnated because the conserved moments m k, k = do not fluctuate and hence do not contrbute to P. The fluctuatons n mode space are ndependent and Gaussan wth varance µb k ρ Stochastc collson operator and detaled balance Havng obtaned the probablty dstrbuton for the statstcal fluctuatons of the non-equlbrum moments explctly, we turn to the mplementaton n the lattce Boltzmann equaton. Smlar to secton 3.2, we ntroduce a stochastc collson operator that adds nose to the determnstc dynamcs.we extend the update rule for the moments by an addtonal random nose n the followng way m neq k = γ k m neq k + ϕ k r k, (3.49) where γ k = 1 + λ k and r k s a Gaussan random number wth zero mean and unt varance. The ampltude ϕ k of the random nose for the k-th mode remans to be determned. Ths can be acheved by nterpretng the update rule (3.49) as a Monte-Carlo process: the fluctuatons of the modes are sampled by random moves. Such a process has to satsfy detaled balance to generate the correct dstrbuton. The condton of detaled balance s ω (m neq k m neq k ) exp [ (mneq k )2 2µb k ρ ] = ω (m neq k m neq k ) exp [ (m neq k ) 2 2µb k ρ ]. (3.50) The probablty for a move from the pre-collson moment m neq k to the post-collson moment m neq k s equal to the probablty of generatng the Gaussan random nose ϕ k r k = m neq k γ k m neq k ω (m neq k m neq k ) = [ 1 exp 2πϕ 2 k (m neq k ] γ k m neq k )2. (3.51) 2ϕ 2 k The probablty of the reverse transton s obtaned analogously. Combnng (3.50) and (3.51) we get [ ] [ ] ω (m neq k m neq k ) exp (m neq ω (m neq k m neq k ) = k ) 2 /(2µb k ρ) exp (m neq k γ k m neq k [ ] )2 /(2ϕ 2 k = [ ) ]. exp (m neq k )2 /(2µb k ρ) exp (m neq k γ k m neq k ) 2 /(2ϕ 2 k ) (3.52) Takng the logarthm yelds [ (m neq k )2 (m neq k ) 2 (1 γk 2) (m neq k )2 (m neq k ) 2] =, (3.53) 2µb k ρ 2ϕ 2 k 39

60 3 Statstcal mechancs of the lattce Boltzmann equaton whch s satsfed f and only f ϕ 2 k = µb k ρ ( 1 γ 2 k ) ρk B T = c 2 sa b ( ) 3 k 1 γ 2 k. (3.54) Ths relaton holds for all modes n the system, where γ k = 1 apples to conserved modes wthout fluctuatons. The value γ k = 0 projects out the determnstc part and makes the mode entrely random. It s mportant to note that the relaton (3.54) ensures consstent samplng of the fluctuatons on the mcroscopc level. Therefore, t guarantees that detaled balance s satsfed on all scales. Ths s n contrast to the prevous fluctuatng lattce Boltzmann, where only the stress modes are thermalzed whle the knetc modes are entrely projected out,.e., γ k = 0 and ϕ k = 0 at the same tme. Our dervaton shows that such a procedure volates detaled balance, because ω (m neq k 0) = 1 and ω (0 m neq k ) = 0, and explans why t leads to the observed poor thermalzaton. The fluctuatons of the knetc modes are needed for detaled balance and proper thermalzaton beyond fluctuatng hydrodynamcs. The relaton (3.54) has a general nterpretaton: every degree of freedom n the system that s subject to dsspaton,.e., γ k 1, needs random fluctuatons to counterbalance dsspaton. That s, the number of random varables needed to thermalze the system must be equal to the number of non-conserved degrees of freedom. Ths s necessary to ensure that for every trajectory the reverse trajectory can be generated as well, whch s another formulaton of detaled balance. Otherwse the system wll not reach thermal equlbrum. Although ths general argument seems rather trval, the consequences for the lattce Boltzmann equaton have long been overlooked. It s among the benefts of the development of the generalzed lattce gas model that t makes a rgorous statstcal mechancs dervaton of these concepts possble n the framework of the fluctuatng lattce Boltzmann equaton. It should also be noted that the expressons obtaned n secton 3.2 for the fluctuatng stress reman vald. But they have now a bottom-up justfcaton n terms of the statstcal fluctuatons on the mcroscopc level, whereas they were prevously derved top-down by comparson wth the macroscopc equatons. For the connecton between the mcroscopc level and the macroscopc hydrodynamcs we once agan refer to the Chapman-Enskog expanson descrbed n chapter 4. 40

61 4 Asymptotc analyss and the Chapman-Enskog expanson The lattce Boltzmann method s based on the dea that the mesoscale knetc descrpton of the system gves rse to hydrodynamc behavor on the macroscale. The formal connecton between the dfferent levels of descrpton can be acheved wthn an asymptotc analyss of the lattce Boltzmann equaton. In partcular, such an analyss makes the relaton between the parameters of the lattce Boltzmann model and the macroscopc transport coeffcents explct. There exst dfferent approaches to lnk the mcroscopc dynamcs of a system to a reduced descrpton n terms of macroscopc varables. Most of them are multscale methods that are based on the separaton of scales,.e., dfferent physcal mechansms can be dstngushed accordng to the tme and length scales they are governed by. The mechansms do not nterfere dynamcally and can thus be treated separately. Formally, ths separaton can be treated by multscale expanson technques [10, 65, 92]. The commonly used method for asymptotc analyss of the lattce Boltzmann equaton s the Chapman-Enskog expanson, whch wll be descrbed n the followng sectons. 4.1 Asymptotc analyss and scalng The separaton of scales for the knetc descrpton manfests tself n dfferent transport phenomena. The herarchy of tme scales n a flud ranges from the tme between partcle collsons over the tme needed for a flud element to travel a typcal dstance up to the dffuson tme scale. Smlarly, the length scales range from the molecular mean free path l mfp to the typcal macroscopc length scale L. The rato of these two length scales defnes the Knudsen number Kn = l mfp /L. In the lattce Boltzmann method, the mean free path corresponds to the lattce spacng a. To dstngush between dfferent phenomena, the relaton between the tme and length scales characterzng the transport processes s mportant, the so-called scalng. In a flud, we can expect two types of transport processes to be relevant. Frst wave-lke phenomena, that obey convectve scalng t x,.e., the tme scale t s lnearly related to the length scale x. And second dffusve phenomena wth dffusve scalng t ( x) 2 where the tme scale s quadratcally related to the length scale. Ths suggests to consder three dfferent tme scales wthn the analyss of the lattce Boltzmann method: the lattce tme scale τ, the convectve tme scale t 1 and the dffusve tme scale t 2. The separaton of length scales s guaranteed by a small Knudsen number. Ths s an mportant premse whch, n addton to the low Mach number assumpton, sets the lmts wthn whch the lattce Boltzmann method can reproduce hydrodynamc behavor. 41

62 4 Asymptotc analyss and the Chapman-Enskog expanson 4.2 Chapman-Enskog expanson The lattce Boltzmann equaton descrbes the system on mcroscopc lattce scales. In order to analyze the dynamcs on hydrodynamc scales, we have to coarse-gran tme and space. We ntroduce a small dmensonless scalng parameter ɛ, where the above suggests that ɛ = Kn s a natural choce. 1 A coarse-graned length scale s ntroduced by wrtng r 1 = ɛ r, (4.1) whch corresponds to measurng postons wth a coarse-graned ruler, e.g., nstead of nanometers we can only resolve the poston up to mcrometers. Further, we ntroduce the convectve tme scale t 1 and the dffusve tme scale t 2 by t 1 = ɛt, t 2 = ɛ 2 t. (4.2) The tme-scales can be nterpreted as the dfferent hands of a clock: The lattce tme t s the sweep hand countng every clock-tck, t 1 s the mnute hand, whereas t 2 s the hour hand. In the course of the LB algorthm, the hands advance accordng to the scalng (4.2). One tme step corresponds to t t + τ, whle t 1 t 1 + ɛτ and t 2 t 2 + ɛτ 2. Measurng a coarse-graned tme corresponds to readng-off the mnute and hour hand (t 1, t 2 ). To analyze the dynamcs on the coarse-graned scales, we wrte the lattce Boltzmann varables f as functons of r 1, t 1 and t 2. The determnstc lattce Boltzmann equaton s then f (r 1 + ɛτc, t 1 + ɛτ, t 2 + ɛ 2 τ) = f (r 1, t 1, t 2 ) + (f(r 1, t 1, t 2 )). (4.3) Whle on the lattce scale r and t are dscrete varables, the coarse-graned varables can be consdered as contnuous because ɛ s assumed to be very small. The lattce Boltzmann equaton wrtten n terms of the coarse-graned varables can therefore be Taylor-expanded. Up to order O(ɛ 2 ), we get f (r 1 + ɛτc, t 1 + ɛτ, t 2 + ɛτ 2 ) = f (r 1, t 1, t 2 ) + ɛτ [ + ɛ 2 τ + τ t 2 2 ( t 1 + c ( t 1 + c ) f (r 1, t 1, t 2 ) r 1 ) ] 2 f (r 1, t 1, t 2 ). r 1 (4.4) Smlarly to the space-tme varables, also the LB populatons and the collson operator are expanded n powers of the scalng parameter ɛ f = f (0) + ɛf (1) + ɛ 2 f (2) + O(ɛ 3 ), (4.5a) = (0) + ɛ (1) + ɛ 2 (2) + O(ɛ 3 ). (4.5b) 1 In other words, the Chapman-Enskog expanson s a perturbaton expanson n the Knudsen number. 42

63 4.2 Chapman-Enskog expanson Because the collson operator s a functon of the LB populatons, we can also wrte [5, 93] (f) = (f (0) ) + ɛ f j j f (1) j f (0) ( + ɛ 2 f j j f (2) j + ) (4.6) 2 f (0) f j f k f (1) j f (1) k + O(ɛ 3 ). j,k f (0) Snce the conservaton laws hold on all scales,.e., ndependently of ɛ, the collson operator must satsfy mass and momentum conservaton at all orders, for all k. (k) = 0, (k) c = 0, (4.7) Insertng the expansons (4.4), (4.5a) and (4.5b) nto (4.3) we get the (quas-)contnuous and scale separated verson of the lattce Boltzmann equaton ( ɛ + c t 1 ) [ f (0) + ɛ 2 + τ ( ) ] 2 + c r 1 t 2 2 t 1 r 1 ( ) + ɛ 2 + c f (1) = 1 t 1 τ r 1 f (0) ( ) (0) + ɛ (1) + ɛ 2 (2), (4.8) where we have neglected all terms of order O(ɛ 3 ). The dfferent orders n (4.8) can be treated separately and we get a herarchy of equatons at dfferent powers of ɛ: O(ɛ 0 ) : (0) = 0, ( O(ɛ 1 ) : + c t 1 r [ 1 O(ɛ 2 ) : + τ t 2 2 (4.9a) ) f (0) = 1 τ (1), (4.9b) ) ] 2 ( t 1 + c r 1 ( f (0) + + c t 1 r 1 ) f (1) = 1 τ (2). (4.9c) In the followng, we wll nvestgate these equatons by constructng the moments on the dfferent scales Zeroth order On the zeroth order, the collson operator (0) vanshes and from (4.9a) t follows that f (0) s a collsonal nvarant. The latter can hence be dentfed wth the equlbrum dstrbuton f eq. The local conserved varables must be moments of the equlbrum dstrbuton 43

64 4 Asymptotc analyss and the Chapman-Enskog expanson only, that s, mass densty ρ and momentum densty j can be wrtten as f eq = ρ, f eq c = j. (4.10) The vanshng zeroth order allows to smplfy the collson operator. Neglectng all terms of order O(ɛ 2 ) we obtan = ɛ (1) = ɛ j f j f (0) f (1) j = j ( L j fj f eq ) j, (4.11) where L j = f f j. Ths justfes the use of a lnear collson operator n the lattce (0) Boltzmann method [18, 93] Frst order The zeroth and frst moment of the ɛ-order equaton (4.9b) are ρ + j = 0, t 1 r 1 j + Π (0) = 0, t 1 r 1 (4.12a) (4.12b) where Π (0) = f (0) c c s the equlbrum momentum flux. These are exactly the nvscd flud equatons when the equlbrum momentum flux s equal to the Euler stress Π (0) = p1 + ρuu, (4.13) where p s the scalar flud pressure. For further reference, we calculate the second moment equaton on ths order whch yelds Π (0) + Φ (0) = 1 t 1 r 1 τ (1) c c = 1 τ (Π (1) Π (1) ). (4.14) Φ (0) = f (0) c c c s the equlbrum thrd moment and Π (1) s the ɛ-order of the postcollsonal momentum flux. 44

65 4.2 Chapman-Enskog expanson Second order Before we construct the moments on the ɛ 2 -order, we rewrte equaton (4.9c) by nsertng the ɛ-order whch elmnates the second dervatves f (0) + 1 ( ) ( ) + c f (1) + f (1) = 1 t 2 2 t 1 r 1 τ (2). (4.15) We have wrtten f (1) = f (1) + (1) for the O(ɛ) post-collsonal populaton. The equatons for the zeroth and frst moment then come as ρ = 0, (4.16a) t 2 j + 1 ) (Π (1) + Π (1) = 0. (4.16b) t 2 2 r 1 Here we have used that the zeroth and frst moment of f (1) vansh. In the followng, we wll merge the moment equatons of the dfferent orders to obtan a sngle equaton n the varables r and t Mergng orders The macroscopc felds depend on the coarse-graned varables r 1, t 1 and t 2, and thus ndrectly on the lattce length and tme r and t. The dervatves wth respect to the lattce varables come as r = ɛ, r 1 t = ɛ + ɛ 2. t 1 t 2 (4.17) Up to the order O(ɛ 2 ) of the Chapman-Enskog expanson, the second-order of the populatons f (2) and the collson operator (2) do not show up n the zeroth and frst moment equatons, therefore we can set f eq = f (0), f neq = ɛf (1), = ɛ (1). (4.18) We merge (4.12a) and (4.12b) wth (4.16a) and (4.16b) and obtan the combned equatons for mass and momentum t ρ + r j = 0, t j + r Πeq r (Π neq + Π neq ) = 0, (4.19a) (4.19b) 45

66 4 Asymptotc analyss and the Chapman-Enskog expanson where Π eq = Π (0) and Π neq = ɛπ (1). The mass equaton s the contnuty equaton whch s automatcally satsfed by the lattce Boltzmann equaton. The momentum equaton resembles the the Naver-Stokes equaton f Π eq equals the Euler stress and the pre- and postcollsonal non-equlbrum stresses are related to the Newtonan vscous stress σ, cf. equaton (2.19) Π neq + Π neq = 2σ (4.20) In order to evaluate ths expresson explctly, we need to use the specfc equlbrum dstrbuton and the collson operator of the LB model. Below we wll use the D3Q19 model ntroduced n secton 2.5 to fnalze the Chapman-Enskog expanson Closng the Chapman-Enskog expanson Usng the O(u 2 ) polynomal expanson of the equlbrum dstrbuton, we can calculate the equlbrum thrd moment Φ (0) αβγ = f eq c α c β c γ = ρu δ c 2 s w c α c β c γ c δ = ρc 2 s (u α δ βγ + u β δ αγ + u γ δ αβ ), (4.21) where we have exploted that only the even-rank lattce sums contrbute. Equaton (4.14) then becomes Π (1) αβ Π(1) αβ = τ ( ) ρc 2 t s δ αβ + ρu α u β + τc 2 s 1 (ρu α δ βγ + ρu β δ αγ + ρu γ δ αβ ). (4.22) r 1γ The tme dervatve of the Euler stress can be expressed wth the help of ɛ-order moment equatons (4.12a) and (4.12b) ( ) ρc 2 t s δ αβ + ρu α u β 1 = ρc 2 sδ αβ t 1 ρ + u β = ρc 2 sδ αβ ( r 1γ ρu γ ( ) ( ) ρu α + u α ρu β u α u β ρ t 1 t 1 t ) ( ) ( 1 c 2 su β δ αγ ρ c 2 r su α δ βγ ρ 1γ r 1γ ) + O(u 3 ), (4.23) where we have neglected terms of O(u 3 ). Insertng ths nto (4.22) yelds Π (1) αβ Usng (4.17) we can wrte ths n unscaled form Π neq αβ ( Π(1) αβ = ρc2 sτ u β + ) u α. (4.24) r 1α r 1β ( Πneq αβ = ρc2 sτ u β + ) u α. (4.25) r α r β 46

67 4.2 Chapman-Enskog expanson Wth the lnear collson operator L j, alternatve expressons for the non-equlbrum stresses were obtaned n (2.73) Π neq αβ Π neq αα = (1 + λ s ) Π neq αβ, = (1 + λ b ) Π neq αα. (4.26) From ths equaton system we obtan a soluton for the non-equlbrum stresses n terms of velocty gradents ( Π neq αβ = ρc2 sτ u β + u α 2 ) u γ δ αβ, λ s r α r β 3 r γ (4.27) Π neq αα = 2ρc2 sτ u α. λ b r α Ths yelds the vscous stresses σ αβ = 1 2 ( Π neq αβ σ αα = 1 2 (Π neq αα ) + Π neq αβ = ρc2 sτ 2 + λ s 2 λ s + Π neq αα) = ρc 2 sτ 2 + λ b u α. λ b r α ( u β + u α 2 u γ δ αβ r α r β 3 r γ ), (4.28) By comparng wth the Newtonan form we fnd the relaton between the egenvalues λ s and λ b and the shear and bulk vscostes ( η s = ρc2 sτ 2 + λ s 1 = ρc 2 2 λ sτ + 1 ), (4.29) s λ s 2 η b = ρc2 sτ 2 + λ b = 2 ( 1 3 λ b 3 ρc2 sτ + 1 ). (4.30) λ b 2 Ths result closes the Chapman-Enskog expanson of the lattce Boltzmann equaton. The addtonal 1 n the brackets s a lattce correcton orgnatng at the 2 ɛ2 -scale. Ths correcton s the reason that, despte the underlyng lattce structure, Gallean nvarance s restored at the macroscopc level. Strctly speakng, there comes another correcton term due to the O(u 3 ) terms n (4.23). Ths would be of the form r β r γ ρu α u β u γ, and hence there are O(u 2 ) correctons to the vscosty [58]. The resultng naccuracy n the momentum equaton can be compensated by ncludng hgher orders n the Hermte expanson correspondng to thrd order velocty terms n the equlbrum dstrbuton [47]. However, for nearly ncompressble flows at low Mach number the correctons are very small and can be neglected completely. The hgher order terms of the populatons f (2) and (2) do not contrbute to the momentum equaton up to O(ɛ 2 ). Therefore the egenvalues of the collson operator correspondng to the knetc modes are rrelevant on the Naver-Stokes level. The reason s that the Naver- Stokes equaton contans only gradents of the velocty feld. The second dervatves of the flow feld only contrbute to the knetc modes, whch do not appear on the macroscopc level. However, the second dervatves are mportant for accurate boundary condtons, as 47

68 4 Asymptotc analyss and the Chapman-Enskog expanson shown by Gnzburg and d Humères [64]. In that case, the knetc egenvalues have to be tuned to specfc values, sometmes called magc values, that yeld second-order accurate boundary condtons. More down-to-earth, they can be obtaned from the second-order Chapman-Enskog soluton. We wll therefore derve explct expressons for f (1) and f (2) n the followng Explct expressons for f (1) and f (2) Instead of usng the thrd equlbrum moment Π (0), we can also evaluate the ɛ 1 -scale equaton (4.9b) drectly. Usng the polynomal expanson for the equlbrum dstrbuton and nsertng the mass and momentum equaton we get ( + c α t 1 r 1α ) f (0) = ( + c α t { 1 = w c 2 s = w c 2 s r 1α r 1α ) w c 2 s [ ρc 2 s + ρu β c β + ρu βu γ 2c 2 s [ ρc 2 s + ρu β c β + ρu βu γ 2c 2 s [ c 2 r s ρu α + ( ) ] ρc 2 sδ αβ + ρu α u β cβ 1α u β 2c 2 s u γ 2c 2 s ( cβ c γ c 2 sδ βγ ) ] ( cβ c γ c 2 sδ βγ ) ] c α ( ) cβ c γ c 2 ( ) sδ βγ ρc 2 r s δ αγ + ρu α u γ 1α ( ) cβ c γ c 2 ( ) sδ βγ ρc 2 r s δ αβ + ρu α u β 1α } ( ) +u β u γ cβ c γ c 2 sδ βγ (ρu α ) r { 1α ( ) ρu β cα c β c 2 r sδ αβ 1α + ρu β u γ ( ) cα c r 1α 2c 2 β c γ c 2 sc α δ βγ c 2 sc β δ αγ c 2 sc γ δ αβ s u β ρ ( } ) c α c β c 2 r sδ αβ + O(u 3 ) 1α = ρ u β E (2) αβ r + ρu β u γ E (3) αβγ 1α r 1α 2c + O(u3 ), s (4.31) where we have ntroduced tensors E (2) and E (3) as shorthand notatons E (2) αβ = w ( ) cα c c 2 β c 2 sδ αβ, s E (3) αβγ = w ( ) (4.32) cα c c 3 β c γ c 2 sc α δ βγ c 2 sc β δ αγ c 2 sc γ δ αβ. s Note that these tensors are dmensonless and have no projecton onto the conserved modes. The tme dervatves n (4.31) have been expressed n terms of spatal dervatves wth the 48

69 4.2 Chapman-Enskog expanson help of the moment equatons, and terms of O(u 3 ) have been neglected accordng to the low Mach number approxmaton. Wth the lnear collson operator, the ɛ 1 -scale equaton now comes as 1 τ j L j f (1) j = ρ u β E (2) αβ r + ρ 1α 3 u β E (2) γγ r δ αβ + 1α r 1α ρu β u γ 2c s E (3) αβγ (4.33) where we have decomposed the tensor E (2) αβ nto ts trace and traceless part. Now we use the specfc form of the collson operator n mode space. By projectng equaton (4.33) onto the modes we can assgn the terms to the dfferent symmetry classes: E (2) αβ has a projecton onto the shear stress modes only, E (2) γγ δ αβ has a projecton onto the bulk stress modes only, whle E (3) αβγ has projectons onto the knetc modes only. We can therefore use the egenvalues of the collson operator to wrte f (1) = ρτ λ s r 1α u β E (2) αβ + ρτ 3λ b r 1α u β E (2) γγ δ αβ + τ λ g r 1α ρu β u γ 2c s E (3) αβγ, (4.34) where we have assumed that all knetc (ghost) modes have the same egenvalue λ g. From ths, we can determne the non-equlbrum stresses Π (1) αβ = = ρc2 sτ λ s Π (1) αα = = 2ρc2 sτ λ b f (1) c α c β = ρτ r 1γ u δ u δ E (2) γδ λ s r c αc β 1γ (δ αγ δ βδ + δ αδ δ βγ 23 ) δ αβδ γδ, f (1) c α c α = ρτ 3λ b r 1β u β, u β r 1β E (2) γγ c αc α (4.35) n accordance wth (4.27). We know already, that there are no contrbutons from f (2) and to the Naver-Stokes equaton. However, here we are nterested n the full Chapman- (2) Enskog soluton of the populatons f up to second order. We therefore contnue wth the ɛ 2 -scale equaton (4.9c), [ 1 L j f (2) j = + τ ( ) ] 2 ( ) + c α f (0) + + c α f (1) τ t j 2 2 t 1 r 1α t 1 r 1α = ( ) ( ) (4.36) f (0) c α L j f (1) j + f (1). t 2 t 1 2 r 1α We are nterested n the soluton f (2) up to second dervatves n the velocty feld and gnore all hgher order terms. That s, n the followng we wll neglect all thrd and hgher dervatves of the velocty, and all second and hgher dervatves of the mass and the nonlnear terms. j 49

70 4 Asymptotc analyss and the Chapman-Enskog expanson Frst we evaluate the t 2 tme dervatve of the equlbrum dstrbuton f (0) = [ w ρc 2 t 2 t 2 c 2 s + ρu β c β + ρu βu γ ( ) ] cβ c s 2c 2 γ c 2 sδ βγ. (4.37) s On the t 2 -scale, the flud s ncompressble ρ t 2 = 0. The tme dervatve of the momentum can be replaced by spatal dervatves usng the O(ɛ 2 ) momentum equaton ρu α = 1 ( t 2 2 r 1β Π (1) αβ = η s 2 r 1β r 1γ u δ + Π(1) αβ ) (δ αγ δ βδ + δ αδ δ βγ 23 δ αβδ γδ ) + η b 2 r 1β r 1γ u δ δ αβ δ γδ. (4.38) The t 2 dervatve of the nonlnear term produces terms of the form u α of order ɛ 2 u 2 and can also be neglected. r 1β r 1γ u δ whch are Next we turn to the t 1 tme dervatve of the frst-order soluton f (1). From equaton (4.34) we see that t produces only terms that are at least of order ɛ 2 u 2, hence t 1 f (1) can be neglected. Droppng also the second dervatve of the nonlnear term, equaton (4.36) comes as 1 τ j L j f (2) j = w c 2 s = w η s c 2 s = w η s c 2 s ρu β c β + c α t 2 2 r 1α r 1γ u δ 2 r 1α ( 1 2 j L j f (1) j + f (1) (c δ δ αγ + c γ δ αδ 23 c αδ γδ ) + w η b u c 2 δ c α δ γδ s r 1α r 1γ [( 1 + c α r 1α ) ρτ u β E (2) βγ λ s r 1γ ( ) ρτ u β E (2) λ b 3 r 1γ 2 r 1α r 1γ u δ + w η b c 2 s c α r 1α 2 γγ δ βγ (c δ δ αγ + c γ δ αδ 23 c αδ γδ ) u δ c α δ γδ r 1α r 1γ [ w η s u β (c β c γ 13 ) r c δc δ δ βγ 1γ c 4 s + w η b 2c 4 s ) ] ( ) u β cδ c δ 3c 2 s δβγ r 1γ ] (4.39a) 50

71 4.2 Chapman-Enskog expanson ( 2 w 1 = η s u β c r 1α r 1γ c 2 s c 2 α c β c γ c β δ αγ c γ δ αβ s c αδ βγ 1 ) c δ c δ c α δ βγ 3η b 2 3c 2 s ( 2 w 1 u β c r 1α r 1γ c 2 s 3c 2 δ c δ c α δ βγ 5 ) s 3 c αδ βγ (4.39b) = η s c s 2 u β r 1α r 1γ E (3) αβγ η b 2c s 2 u β r 1α r 1γ E (3) αδδ δ βγ, where the traceless part of the thrd-rank tensor s to be understood n the followng way: E (3) αβγ = E(3) αβγ 1 3 E(3) αδδ δ βγ. (4.40) Snce E (3) (2) αβγ projects only on the knetc modes, the fnal result for f s f (2) = η sτ 2 u β E (3) αβγ c s λ g r 1α r 1γ η bτ 2c s λ g 2 u β r 1α r 1γ E (3) αδδ δ βγ. (4.41) Puttng all parts together and usng unscaled varables, we get the second order Chapman- Enskog soluton for the lattce Boltzmann populatons f f = w ρ + w c α + ρτ λ s c 2 s ρu α + ρu αu β E (2) 2c 2 αβ s r α u β E (2) αβ + ρτ 3λ b η sτ 2 u β E (3) αβγ c s λ g r α r γ r α u α E (2) γγ + η bτ 2c s λ g 2 u β r α r β E (3) αδδ. τ ρu β u γ c s λ g r α 2 E (3) αβγ (4.42) A smlar expresson has been derved by Gnzburg and d Humères [64]. There, however, the two-relaxaton tme model was used whch makes the expressons slghtly smpler. In contrast, we have here derved the more general soluton for the case of ndependent relaxaton rates for shear and bulk modes. The knetc modes are all relaxed wth the same egenvalue λ g. In prncple, we could also use ndependent egenvalues for the knetc modes by splttng the tensor E (3) nto symmetry-related parts. Snce ths makes the calculatons unnecessarly tedous, we have not done ths here. The second-order soluton (4.42) shows that the Chapman-Enskog procedure yelds an expanson of the LB populatons n terms of the conserved hydrodynamc felds and the gradents of the velocty feld, whle gradents of the mass densty do not contrbute. The equlbrum dstrbuton depends on the mass and momentum denstes exclusvely, and the non-equlbrum contrbutons are obtaned as dervatves of the flow velocty wth ncreasng order. On the Naver-Stokes level, only the gradents of the velocty play a role whch enter on the ɛ-scale. As a consequence of the lattce dscretzaton, the ɛ 2 -scale yelds an addtonal correcton to the vscosty whch s not present n the Chapman-Enskog expanson of the contnuous Boltzmann equaton [50, 52]. The varous truncatons we have made do 51

72 4 Asymptotc analyss and the Chapman-Enskog expanson not effect the dynamcs on the Naver-Stokes level as long as the Mach number s small. In partcular, the error terms stemmng from the ɛ 3 -scale are neglgble [65], and the truncaton error n the vscosty s of order O(u 2 ) [58]. In the contnuous case, the Chapman-Enskog expanson can be contnued at hgher orders to yeld the Burnett and super-burnett equatons. For the lattce Boltzmann equaton, ths requres to take nto account hgher-order Hermte approxmatons of the equlbrum dstrbuton, but n prncple t s also possble. It should also be remarked that we have only looked at the mass and momentum equatons, but not the heat transport equaton (cf. [94] for a treatment of thermal transport). Another type of asymptotc analyss was carred out by Junk et al. [92], whch dffers from the Chapman-Enskog expanson n some aspects. Frst, a purely dffusve scalng t x 2 s used, whle acoustc effects are consdered as numercal artfacts. Second, all quanttes ncludng the hydrodynamc varables are cast n a regular expanson, whereas n the Chapman-Enskog procedure the hydrodynamc felds are usually not expanded (for reasons explaned n Ref. [8]). The Chapman-Enskog expanson makes the requrements for a successful lattce Boltzmann model explct,.e., symmetry requrements and conservaton laws. Ths makes t possble to construct LB models wthout referrng to the Hermte-expanson of the contnuous Boltzmann equaton. From another pont of vew, asymptotc expansons lke Chapman-Enskog can also be vewed as a means of valdatng a gven LB model. Ths s specfcally mportant when one pursues extensons of the conventonal LB models lke mult-phase lattce Boltzmann. Furthermore, ntal and boundary condtons have to be valdated by asymptotc expanson to show that they are well-behaved n terms of Naver-Stokes hydrodynamcs. For ntal condtons, ths has been carred out by Me et al. [95], Caazzo [96]. Asymptotc expansons have been appled to varous boundary condtons by Junk and Yang [97] and have led to the development of more accurate reflecton rules by Gnzburg and d Humères [64]. Ths wll be dscussed n more detal n chapter Fluctuatons In the analyss of the precedng subsectons, we have not taken nto account fluctuatons. It s straghtforward to ncorporate the fluctuatng part by settng and usng the stochastc collson operator = j L jf neq j Π neq + Π neq = 2 (σ + s) (4.43) Π neq αβ Π neq αα = (1 + λ s )Π neq αβ + σ r αβ, = (1 + λ b )Π neq αα + σ r αα. + such that Usng (4.25) we can elmnate the post-collsonal stress Π neq and get ( Π neq αβ = ρc2 sτ u β + u α 2 ) u γ δ αβ 1 σ r λ s r α r β 3 r γ λ αβ, s Π neq αα = 2ρc2 sτ u α 1 σ λ b r α λ αα. r b (4.44) (4.45) 52

73 4.2 Chapman-Enskog expanson Insertng nto (4.43) leads to σ αβ + s αβ = ρc2 sτ λ s λ s σ αα + s αα = ρc 2 sτ 2 + λ b λ b ( u β + u α 2 ) u γ δ αβ + 1 σ r r α r β 3 r γ λ αβ, s u α + 1 σ r α λ αα, r b (4.46) from whch we can read off s αβ = 1 λ s σ r αβ, s αα = 1 λ b σ r αα. (4.47) Therefore the fluctuatons s αβ on the hydrodynamc level are related to the random stresses σαβ r n the stochastc collson operator va s αβ s γδ = 1 λ 2 s σ r αβ σ r γδ, s αα s ββ = 1 σ r λ 2 αα σββ r, (4.48) b s αβ s γγ = 1 σ r λ s λ αβ σγγ r. b The fluctuatons on the hydrodynamc level are thus dfferent from the fluctuatons on the lattce Boltzmann level because the former are present on the convectve tme scale t 1, whle the latter enter on the lattce tme scale τ. Comparng wth the expressons (3.13) for fluctuatng hydrodynamcs, s αβ s γδ = 2k BT a 3 τ η αβγδ = 2k BT η s (δ a 3 αγ δ βδ + δ αδ δ βγ 23 ) τ δ αβδ γδ + 2k BT η b δ a 3 αβ δ γδ, τ (4.49) and takng nto account the results (4.29) and (4.30) for the shear and bulk vscostes we fnally arrve at σ r αβ σγδ r ρc 2 = sk B T λ a 3 s (2 + λ s ) (δ αγ δ βδ + δ αδ δ βγ 23 ) δ αβδ γδ, σ r αα σββ r 6ρc 2 = sk B T (4.50) λ a 3 b (2 + λ b ), σ r αβ σγγ r = 0. In secton 3.3.4, we have shown that for consstency wth statstcal mechancs, the varance of the k-th mode must satsfy ϕ 2 k = µρb k(1 γ 2 k ). Usng the formulas (2.67), the b k-values 53

74 4 Asymptotc analyss and the Chapman-Enskog expanson n (2.64) for the D3Q19 model and c 2 s = 1/3(a/τ) 2, we get σ αα σ ββ = ϕ 2 4 = 6µρc 4 s(1 γ 2 b ) σ r xxσ r xx = 1 9 ϕ2 5 = 4 3 µρc4 s(1 γ 2 s) σ r yy σ r yy = σ r zz σ r zz = σ r yyσzz r 1 = 36 ϕ ϕ2 6 = 4 3 µρc4 s(1 γs) 2 σ r xx σ r yy = σ r xx σ r zz = 1 18 ϕ2 5 = 2 3 µρc4 s(1 γs) 2 (4.51) σ r xy σ r xy = ϕ 2 7 = µρc 2 4(1 γs) 2 σ r xzσ r xz = ϕ 2 8 = µρc 2 4(1 γs) 2 σ r yz σ r yz = ϕ 2 9 = µρc 2 4(1 γs). 2 Pluggng n γ k = 1 + λ k and µc 2 s = k B T/a 3 we see that these expressons are equvalent to (3.19). Ths shows that the result does not only recover fluctuatng hydrodynamcs, but t s at the same tme consstent wth statstcal mechancs. The fluctuatons of the knetc modes do not nfluence the hydrodynamc behavor because the non-equlbrum parts of Φ and other knetc modes do not appear at the Naver-Stokes level. They are only mportant for proper thermalzaton on mcroscopc scales External forces So far we have only looked at the lattce Boltzmann equaton wthout external forces, where momentum s strctly conserved. In many applcatons t s desrable to be able to transfer momentum to the flud by an external force densty g(r, t). In the same sprt as n the case of fluctuatons, we ncorporate the effect of the external force by addng an addtonal term g to the collson operator = j L j f neq j + g. (4.52) Whle the external forces have no effect on the mass densty, they transfer an amount gτ of momentum to the flud n one tme step. Therefore, the zeroth and frst moment of g have to satsfy g = 0, (4.53a) g c = gτ. (4.53b) Snce the momentum before and after the collson dffer, but the collsons are assumed to take place nstantaneously, the hydrodynamc momentum densty s not unquely defned. Any value between the pre- and the post-collsonal value could be used. Consequently, there s an ambguty whch value to use for calculatng the equlbrum dstrbuton f eq. In the 54

75 4.2 Chapman-Enskog expanson lterature, dfferent propostons have been made to defne the momentum densty [8, 64, 65]. Here, we contnue wthout an a-pror defnton and use the Chapman-Enskog expanson to deduce an approprate choce afterwards. For ths purpose, we ntroduce the followng notatons to dstngush between the momentum denstes obtaned from the dfferent orders of the Chapman-Enskog expanson j = f c = j (0) + ɛj (1), (4.54) where j (0) = j (1) = f (0) c, f (1) c. (4.55) Snce momentum s not conserved, j (1) s not necessarly equal to zero. The forcng term must enter the Chapman-Enskog expanson at order O(ɛ), hence g = ɛ g(1) + ɛ 2 g(2) + O(ɛ 3 ). (4.56) As prevously, we can expand the lattce Boltzmann equaton and evaluate the moments at dfferent orders of ɛ. The frst three moments at O(ɛ) are ρ + j (0) = 0, t 1 r 1 j (0) + Π (0) = g (1), t 1 r 1 Π (0) + Φ (0) = 1 ) (Π (1) Π (1). t 1 r 1 τ (4.57) Here we dentfy f (0) wth the equlbrum dstrbuton f eq, where we plug n u (0) = j (0) /ρ for the flow velocty. We can also evaluate Π (0) and Φ (0). Ths yelds a smlar result as n (4.24), but wth addtonal terms due to the forcng contrbuton n the momentum flux Π (1) αβ ( Π(1) αβ = ρc2 sτ u (0) β r + ) ( ) u (0) α + τ u (0) α g (1) β + g α (1) u (0) β + O(u 3 ). (4.58) 1α r 1β A second relaton s agan obtaned from the collson operator Π (1) αβ Π(1) αβ = λ sπ (1) αβ + λ b 3 Π(1) γγ δ αβ + g(1) c α c β. (4.59) 55

76 4 Asymptotc analyss and the Chapman-Enskog expanson Solvng the equaton system as before yelds ( Π (1) αβ + Π (1) αβ = ρc2 sτ(2 + λ s ) u (0) β λ s r + ) u (0) α 1α r 1β + τ(2 + λ ( ) s) u (0) α g (1) β + g α (1) u (0) β 2 λ s λ s Π (1) αα + Π (1) αα = 2ρc2 sτ(2 + λ b ) λ b u (0) α r 1α + 2τ(2 + λ b) u (0) α g α (1) λ b 2 λ b g(1) c α c α. g(1) c α c β, (4.60) The addtonal terms due to the forcng can be compensated f the second moment of the collson operator s made to satsfy g(1) c α c β = (2 + λ s)τ 2 g(1) c α c α = (2 + λ b )τu (0) α g (1) α. ( ) u (0) α g (1) β + g α (1) u (0) β, (4.61) Proceedng to the order O(ɛ 2 ) where the zeroth and frst moment equatons are ρ + (j (1) + 12 ) t 2 r τg(1) = 0, 1 j (0) + (j (1) + 12 ) t 2 t τg(1) + 1 ) (Π (1) + Π (1) = g (2). 1 2 r 1 Insertng the above result for Π (1) n the momentum equaton yelds j α (0) + ( j α (1) + 1 ) t 2 t 1 2 τg(1) α [ 2 + λs + ρc 2 sτ r 1β 2λ s ( u (0) β r + u (0) α 1α r 1β ) λ b 3λ b where we have used (4.60) and (4.61). After mergng orders we arrve at t ρ + ( j α + 1 ) r α 2 τg α = 0, ( j α + 1 ) t 2 τg α r β + ( ) ρc 2 r sδ αβ + ρu (0) α u (0) β β [ η s ( r α u (0) β + r β u (0) α 2 ) u (0) γ δ αβ 3 r γ ] u (0) γ δ αβ = g α (2), r 1γ + η b r γ u (0) γ δ αβ ] = g α. (4.62) (4.63) (4.64) 56

77 4.2 Chapman-Enskog expanson Ths can be cast n the form of the Naver-Stokes equaton by usng the followng defnton for the hydrodynamc momentum densty: j j (0) j τg = f c + 1 τg. (4.65) 2 Note that ths mples f eq c = j, f neq c = 1 2 τg. (4.66) The defnton (4.65) corresponds to the arthmetc mean of the pre- and post-collsonal momentum densty. Ths has been determned as the optmal value prevously, n both numercal and theoretcal studes. However, n those works the redefned value was not plugged n the equlbrum dstrbuton. Whle Ladd and Verberg [65] used j and accordngly f neq c = 0, n Gnzburg and d Humères [64] two dfferent values are used: j for the lnear part n f eq, and j + 1 τg for the nonlnear part. Snce only the non-lnear part 2 enters the equlbrum stress tensor, ths makes no dfference n the usual mplementaton of an MRT model. However, the mxed use of dfferent momentum denstes makes the theoretcal dervaton rather obscure. In contrast, our redefnton s strctly compatble wth the Chapman-Enskog expanson and all spurous terms are canceled for a proper choce of the forcng term. Let us defne G αβ = 2 + λ s (u α g β + g α u β 23 ) 2 u γg γ δ αβ λ b u γ g γ δ αβ. (4.67) 3 The forcng term s determned from the condtons (4.53a), (4.53b) and (4.61), and can be wrtten as g = w [ τ g c c G : ( c s 2c 2 c c 2 s1 )]. (4.68) s Ths expresson leads to the Naver-Stokes equaton wth the same vscostes as n the case wthout forcng. Moreover, no addtonal assumptons about the external force have to be made,.e., the result holds for tme-varyng and nhomogeneous forces as well. It was frst derved by Guo et al. [98], whle n other works only constant or homogeneous forces were treated [65]. 57

78 58

79 5 Boundary condtons for lattce Boltzmann models The lattce Boltzmann method has become a popular approach for smulatng complex fluds and soft matter. Ths s due to ts mesoscopc nature and the underlyng knetc pcture, whch opens up the possblty to augment the model wth more mcroscopc nformaton beyond the Naver-Stokes level. For example, one can smulate partculate suspensons where the solute partcles do not only nteract whch each other drectly, but also va the exchange of momentum that s propagated through the surroundng solvent, so called solventmedated or hydrodynamc nteractons. In ths context, the lattce Boltzmann method plays the role of an effcent solver for the momentum propagaton n the solvent and meanwhle s an establshed alternatve to other mesoscopc solvent models, such as dsspatve partcle dynamcs and mult-partcle collson dynamcs (see for example Yeomans [7] for a recent comparson of lattce Boltzmann and mult-partcle collson dynamcs). The presence of solute partcles makes t necessary to deal wth nteractons between the flud and the sold phase. For geometrcally extended objects lke collods or walls, boundary condtons have to be nvoked at the object s surface to prevent the flud from leakng nto the sold. Another case where boundary condtons become mportant are confned flows, that s, the flud s bounded n a geometrcal doman of relatvely small dmensons. Confned flows can exhbt nterestng behavor even at low Reynolds numbers [16], and the observed phenomena depend strongly on the precse nature of the boundary condton. Even a straght wall can gve rse to complcated effects, such as boundary layer effects and the Kramers problem [5, 6]. Such effects are of paramount mportance n modern mcrofludcs: The so called lab on a chp -paradgm has led to the constructon of mcrofludc devces where fluds can be manpulated on the mcro- or even the nanoscale. Fgure 5.1 shows two examples of such mcrofludc devces. Due to the large surface-to-volume rato of such structures, the flow behavor s to a wde extent domnated by surface and nterface effects. A successful smulaton of the flow through a mcrofludc devce therefore depends crtcally on approprate modelng of the boundary condtons. One of the man objectves of ths thess was the development and mplementaton of boundary condtons for the lattce Boltzmann method. It s sometmes perceved that boundares can be readly mapped to the lattce and hence boundary condtons are smple to mplement, e.g., by the bounce-back rule [5]. On closer nspecton, however, t turns out that numerous dffcultes arse and an effcent and at the same tme accurate treatment of the boundary can be an ntrcate affar [101]. The challenges one s faced wth whle developng boundary condtons shall be dscussed n detal n ths chapter. It s organzed as follows: In secton 5.1 we dscuss boundary condtons n hydrodynamcs and knetc theory. Then we move on to boundary condtons n lattce Boltzmann models n secton 5.2, where we revew the 59

80 5 Boundary condtons for lattce Boltzmann models From [99]. Reprnted wth knd permsson from Prof. Charles Baroud, LadHyX, Ecole Polytechnque. From [100]. Reprnted wth permsson from AAAS. Fgure 5.1: Two examples of mcrofludc devces. In the left devce, a water stream s njected nto an ol phase and passes a gate after whch droplets are formed. The rght devce s a mcrofludc memory. Two fluds are njected nto the devce, a brght one and a dark one. By mposng a pressure peak n the narrow channel on the rght, the flow can be swtched such that the dark flud streams to the top and the brght flud streams to the bottom. In ths way the state of the devce can serve to store a bnary number. commonly used boundary models. In secton 5.3, we turn to the phenomenon of boundary slppage and develop a novel boundary condton for tunable slp. Algorthmc boundary condtons Before we proceed wth the boundary condtons at the surface of sold objects, we shall menton another type of boundary condtons that s omnpresent n computer smulatons. Due to memory lmtatons, any computer smulaton s necessarly restrcted to a fnte smulaton volume. Hence, one has to defne a rule for what should happen at the boundares of the fnte smulaton doman. Such rules are usually also referred to as boundary condtons. Let us call them algorthmc boundary condtons here, n order to dstngush from sold-flud or hydrodynamc boundary condtons. The most commonly used algorthmc boundary condtons are perodc boundares, where any mass porton that leaves the smulaton doman on one sde, smultaneously re-enters the doman at the opposte sde. Ths rule conserves mass and usually momentum by constructon 1, and the smulated system becomes effectvely nfnte whlst fnte-sze correctons are taken nto account. Another possblty are n- and outflow boundary condtons, where the mass and momentum flux at the doman boundary s prescrbed by some rule. For example ths could be known values from an alternatve smulaton of a smlar system. A specal case of n- and outflow boundary condtons 1 It s possble to modfy the momentum of the perodcally re-enterng mass porton, but ths s then usually not referred to as perodc boundary condtons any more. For example, Lees-Edwards boundary condtons can be appled to generate a shear-profle n the smulaton doman. 60

81 5.1 Hydrodynamc boundary condtons are open boundares, where mass just flows out of the smulaton doman. Fnally, t s of course possble to employ a sold boundary condton at the doman boundary. In ths case, the dstncton between algorthmc and hydrodynamc boundary condtons s mmateral. Algorthmc boundary condtons wll be dscussed n conjuncton wth the mplementaton of a lattce Boltzmann kernel n appendx A. In the remander of ths secton, we wll focus exclusvely on hydrodynamc boundary condtons at sold surfaces. 5.1 Hydrodynamc boundary condtons Boundary condtons on the Naver-Stokes level The Naver-Stokes equaton of classcal hydrodynamcs s a partal dfferental equaton. From the vewpont of mathematcs, the partal dfferental equaton tself s not enough to determne a unque soluton, but we have to pose the boundary value problem, that s, the ntal and boundary condtons have to be specfed.the boundary value problem corresponds to flterng from all admssble solutons to the Naver-Stokes equaton exactly the ones that satsfy the specfc ntal and boundary condtons. Typcal boundary condtons for the Naver-Stokes equaton are ether Drchlet or Neumann boundary condtons. The Drchlet boundary condtons prescrbes the value of the flow velocty at the boundary u(r B ) = U B, (5.1) whle the Neumann boundary condton prescrbes a value to the normal dervatves of the velocty and can be wrtten n the form n u r = h B. (5.2) rb In classcal hydrodynamcs, t s usually assumed that the flud at the surface moves along wth the same velocty as the sold object. Ths s the so called stck or no-slp boundary condton. For an object at rest t reduces to u(r B ) = 0, (5.3).e., the flow velocty at the boundary s zero. Conversely, the free-slp or full-slp boundary condton assumes that the flud gldes freely over the surface, whch can be expressed by a vanshng normal velocty gradent at the surface. To ensure mpermeablty of the sold, the normal velocty must equal the normal velocty of the surface. n u r = 0, n u(r B ) = n U B. (5.4) rb The full-slp boundary condton s an example of a mxed Drchlet and Neumann boundary condton, whch n conjuncton wth a curved boundary s sometmes referred to as a 61

82 5 Boundary condtons for lattce Boltzmann models Cauchy boundary condton. Vrtually all applcatons n classcal hydrodynamcs make use of the no-slp boundary condton. It s well justfed on the macroscopc level where the characterstc scales of the flow are much larger then molecular scales,.e., at small Knudsen number. The no-slp boundary condton s also often used as a reference case for the development of boundary condtons n knetc models lke lattce Boltzmann. In mcrofludc devces, the separaton between molecular and hydrodynamc length scales s less pronounced and fnte Knudsen number effects come nto play. Flows can generally be dvded nto dfferent regmes accordng to the value of the Knudsen number: contnuum flows at small Knudsen numbers Kn 0.001, slp flows at ntermedate Knudsen numbers Kn 0.1, the transton regme at hgher Knudsen numbers 0.1 < Kn < 10, and free molecular flows beyond Kn 10 [102, 103]. The contnuum hydrodynamc descrpton of fluds remans vald up to Kn 0.1. Mcroflows are typcally n the slp-flow regme, where the Naver-Stokes equaton remans vald but the no-slp boundary condton breaks down [104, 105]. It has been observed n varous experments that the velocty of the flow does not entrely vansh at the surface [102, ]. The appearance of the apparent slp velocty s a consequence of the mcroscopc structure of the surface and ts nteractons wth the flud, e.g., the wettng propertes. The effects of apparent slp n a mcrofludc devce may nclude a reducton of the surface stresses such that the flow throughput can sgnfcantly be enhanced [110, 111]. The latter observaton suggests to formulate a smple lnear consttutve equaton for the stress at the surface [111] n σ = γ (u(r B ) U B ) γ u slp. (5.5) Combnng ths wth the expresson (2.2) for the Newtonan vscous stress n the bulk we arrve at the followng slp-flow boundary condton n u(r B ) = n U B, n u r = γ u slp. (5.6) rb η s If we choose the coordnate system such that the boundary normal s n z-drecton and the flow s n x-drecton, we can rewrte (5.6) as the Naver slp boundary condton [112] u slp = η s u x u x = δ B γ z z. (5.7) zb zb Here the slp length δ B = η s /γ s ntroduced, whch can be llustrated as the dstance nto the sold at whch the lnearly extrapolated flow profle s equal to zero, cf. fgure 5.2. The lmtng case δ B = 0 corresponds to the no-slp boundary condton, whereas for δ B equaton (5.7) resembles the full-slp boundary condton. A negatve slp length ndcates an apparent change n the flow drecton close to the boundary. Accordng to ts defnton, the slp length depends on the vscosty η s and the coeffcent γ. It s thus the rato of a bulk property and a surface property. In other words, the parameter that truly descrbes the propertes of the surface s not the slp length δ B but the coeffcent γ. Therefore, the latter 62

83 5.1 Hydrodynamc boundary condtons Wall Hydrodynamc boundary u x(z) u slp Slp length δ B z B z Fgure 5.2: Illustraton of the slp-length. The flow profle has a fnte slp velocty u slp at the hydrodynamc boundary z B. The slp length s defned as the dstance at whch the lnearly extrapolated profle s equal to zero. should be vared to model dfferent sold-flud nterfaces. Ths suggests furthermore that t s desrable to be able to mplement the coeffcent γ drectly n a smulaton method. One such method s developed wthn the lattce Boltzmann model n secton Boundary condtons n knetc theory Boundary condtons n knetc theory have the same mathematcal orgn as Naver-Stokes boundary condtons: the Boltzmann equaton s a dfferental equaton and thus the boundary value problem has to be solved. However, the physcal pcture s qute dfferent because knetc theory descrbes the system at much smaller scales, namely at the level of the onepartcle dstrbuton functon. Therefore molecular detals of the nteracton between flud partcles and the sold surface can not be gnored completely, but have to be ncorporated approprately nto the knetc descrpton. Influencng factors are for example surface roughness and chemcal detals of the sold-flud nteracton such as hydrophobcty or chemcal bondng. At the mesoscopc level of descrpton, the nteractons should be characterzed by ther statstcal propertes [110]. The boundary condtons therefore have to be formulated n terms of the dstrbuton functons and transton probabltes, respectvely. One of the frst systematc accounts of boundary condtons n the knetc theory of fluds was presented by Maxwell n the appendx of [113]. A more recent overvew can be found n the book by Cercgnan [51]. A key feature of knetc boundary condtons s that the velocty space has to be splt nto ncomng veloctes and outgong veloctes accordng to the projecton n v onto the wall normal [ ]. The dstncton between ncomng and outgong veloctes s n general related to a dscontnuty n the dstrbuton functon f. The smplest case s a perfectly flat elastc surface, whch reflects the mpngng partcles n such a way that ther normal velocty component s reversed whle the other components reman unchanged. Ths s 63

84 5 Boundary condtons for lattce Boltzmann models called specular reflecton and s descrbed by the relaton f(r B, v, t) = f(r B, v 2n(n v), t), n v > 0. (5.8) However, the dealzed flat elastc surface s an unrealstc assumpton and the specular reflecton rule s n general not applcable n practce. In general, the effect of the surface can be descrbed by a scatterng kernel B(v v) ndcatng the probablty that an mpngng partcle wth velocty v wll be deflected to a new velocty v. The dstrbuton functon for the reflected (outgong) partcles can then be wrtten n the form [51] n v f(r B, v, t) = dv B(v v) n v f(r B, v, t), n v > 0. (5.9) n v <0 The scatterng kernel must be non-negatve. If the surface s mpermeable and non-adsorbng, every mpngng partcle s re-emtted and the scatterng kernel satsfes the normalzaton condton B(v v) dv = 1. (5.10) n v>0 Ths s equvalent to the statement that the normal hydrodynamc momentum densty at the boundary vanshes: n ρu = dv n vf(r B, v, t) = dv n v f(r B, v, t) dv n v f(r B, v, t) n v>0 n v<0 = dv dv B(v v) n v f(r B, v, t) dv n v f(r B, v, t) n v>0 n v <0 n v<0 = dv n v f(r B, v, t) dv n v f(r B, v, t) = 0, n v <0 n v<0 (5.11) where we have nserted (5.9) and used (5.10). Specular reflectons are just the specal case where B(v v) = δ (v v + 2n(n v)). (5.12) If the dynamcs of the system at the molecular level s tme-reversble, the scatterng kernel satsfes the detaled balance condton for thermal equlbrum (cf. fgure 5.3) [51] n v B(v v)f eq (v ) = n v B( v v )f eq (v), (5.13) where f eq denotes the Maxwell-Boltzmann equlbrum dstrbuton. Ths mples that the equlbrum dstrbuton automatcally satsfes the boundary condton n v f eq (v) = dv n v B(v v)f eq (v ). (5.14) n v <0 64

85 5.1 Hydrodynamc boundary condtons v v v v n v da Fgure 5.3: Illustraton of boundary condtons n knetc theory. (Left) Detaled balance: the blue scatterng process v v has the same probablty as the red tme reversed process v v. (Rght) A partcle emergng from a surface element da wth a velocty v propagates nto a skewed volume element dv = da n v dt. Ths s the reason for the factor n v n the knetc boundary condtons. In other words, f the ncomng partcles have a Maxwellan dstrbuton, then the outgong partcles have the same Maxwellan dstrbuton. It should be noted that (5.14) s a weaker requrement than (5.13). The constructon of the scatterng kernel B(v v) s a formdable task because of the complexty of the underlyng mcroscopc flud-sold nteractons. Usually one has to resort to smplfed models that satsfy the basc requrements of the boundary condton such as mass conservaton and detaled balance. In hs semnal work, Maxwell [113] put forward a boundary model n whch a fracton of the mpngng partcles s specularly reflected from the surface, whle the remanng fracton s re-emtted accordng to a boundary equlbrum dstrbuton f eq. The latter s assumed to have a Maxwellan form where the temperature T B of the boundary enters. The scatterng kernel for ths model can be wrtten as [51, 113] B(v v) = (1 α) δ (v v + 2n(n v)) + α N n v f eq (v), (5.15) where the factor n v s llustrated n fgure 5.3, and the normalzaton N s ntroduced to satsfy (5.10). For smplcty, we have assumed that the boundary s at rest. The generalzaton to a movng boundary can be easly acheved by substtutng v by v U B. The fracton α of the partcles s reflected dffusvely,.e., they completely loose memory of ther ncomng velocty. After the collson wth the surface, they have accommodated a velocty as f they were evaporated from the surface. For ths reason, the coeffcent α s called accommodaton coeffcent. The value α = 0 corresponds to specular reflectons (5.8). The other lmtng case α = 1 corresponds to a completely dffusve boundary condton where any memory of the state before the surface collson s lost. The scatterng kernel s then ndependent of the ncomng veloctes, and from equaton (5.14) we get [51, 114, 117] B(v) B(v v) = n v f eq (v) n v <0 dv n v f eq (v ). (5.16) Insertng ths nto (5.9) yelds an explct expresson for the outgong dstrbuton functon [117] f(r B, v, t) = f eq n v (v) <0 dv n v f(r B, v, t), n v > 0. (5.17) n v <0 dv n v f eq (v ) 65

86 5 Boundary condtons for lattce Boltzmann models dffuse reflectons α specular reflectons 1 α l mfp Fgure 5.4: Maxwell s dffuse scatterng boundary condton. A fracton α of partcles s dffusely reflected from the wall and accommodated to the boundary equlbrum dstrbuton. The remanng fracton 1 α s specularly reflected. At a fcttous plane nfntesmally close to the surface, half of the partcles mpnge from a dstance a mean free path away from the surface, and the other half s made up of reflected partcles. Ths s the so-called dffusve boundary condton wthn knetc theory. For general values of α the reflected dstrbuton s gven by f(r B, v, t) = (1 α)f(r B, v 2n(n v), t) + αf eq n v (v) <0 dv n v f(r B, v, t). n v <0 dv n v f eq (v ) (5.18) where n v > 0. Once the boundary condton for the dstrbuton functon s specfed, we can proceed to evaluate the hydrodynamc flow velocty at the boundary. As shown above, the normal velocty at the boundary s zero. The flow velocty at the boundary s u(r B ) = 1 vf(r B, v, t) dv ρ = 1 vf(r B, v, t) dv + 1 vf(r B, v, t) dv ρ n v 0 ρ n v>0 = 1 vf(r B, v, t) dv ρ (5.19) n v α vf(r B, v 2n(n v), t) dv ρ n v>0 + α vf eq n v (v) <0 dv n v f(r B, v, t) dv. ρ n v>0 n v <0 dv n v f eq (v ) The evaluaton of ths expresson requres explct knowledge of the dstrbuton functon f(r B, v, t) and, as stated above, there s an essental dscontnuty at the surface of the boundary. Despte ths complcaton, Maxwell used the bulk soluton to approxmate the dstrbuton functon at the surface [113]. He obtaned the tangental flow velocty u(r B ) = u(r B ) at the surface n terms of the bulk velocty at a dstance of the order of the mean free path away from the surface u(r B ) = 1 2 [u(r B + l mfp n) + (1 α)u(r B + l mfp n) + αu B ], (5.20) 66

87 5.1 Hydrodynamc boundary condtons where U B = U B s the absolute velocty of the boundary. The formula s llustrated n fgure 5.4. Conversely, the bulk velocty close to the boundary can be Taylor expanded around the surface velocty u(r B + l mfp n) = u(r B ) + l mfp n u r u rb 2 l2 mfpnn : r r (5.21) rb Solvng these equatons for u(r B ) yelds the celebrated expresson for the slp velocty as an expanson n powers of the Knudsen number [51, 118] [ u slp = u(r B ) U B = 2 α Kn n u ] + Kn2 α ˆr 2 nn : 2 u +..., (5.22) ˆr ˆr ˆrB ˆrB where the spatal varables ˆr = r/l have been scaled wth the characterstc hydrodynamc length L. To frst-order, ths resembles the Naver slp boundary condton, cf. equaton (5.7), where the slp length s gven by δ B = 2 α α l mfp. (5.23) In the context of knetc theory the slp length was earler referred to as Gletungs coeffcent [113, 119]. Agan, the slp length depends on both a bulk (l mfp ) and a surface (α) property. Equaton (5.22) shows that for fnte Knudsen number, Maxwell s knetc boundary condton always leads to a non-vanshng slp velocty. The slp length δ B s on the order of the mean free path, f the accommodaton coeffcent α s close to unty,.e., a purely dffusve boundary condton. Conversely, the slp length dverges n the lmt α 0 correspondng to the full-slp boundary condton. The hydrodynamc no-slp boundary condton s only vald n the lmt of vanshng Knudsen number. The frst-order approxmaton descrbes the slp flow regme up to Knudsen numbers of the order Kn 0.3 [107]. Although beyond ths lmtng value the valdty of the Naver-Stokes equaton s n general questonable, some results have been reported where the ncluson of the second-order slp coeffcent leads to reasonable mprovements [see Refs. 107, 118]. Maxwell s knetc boundary model s n agreement wth a range of expermental and numercal fndngs [109, 120]. At the same tme, t s a very smple model as t descrbes the surface propertes by only one parameter,.e., the accommodaton coeffcent α. Therefore t s a promsng startng pont to devse mesoscopc boundary models for use n computer smulatons [110]. 67

88 5 Boundary condtons for lattce Boltzmann models Fgure 5.5: Lattce representaton of a rgd object. (Left) Node-based representaton. (Mddle) Md-lnk representaton. (Rght) Lnk representaton wth boundary markers. 5.2 Boundary condtons for lattce Boltzmann models Boundary condtons for computer algorthms dffer conceptually from boundary condtons n analytcal theores. Whle n the latter, they are posed as addtonal equatons that flter from the admssble solutons of the dfferental equaton the unque soluton to the boundary value problem, n computer smulatons the boundary condtons are part of the algorthm that generates the sought-after soluton as a trajectory n tme. That s, the boundary condtons are part of the dynamcal updatng scheme and are appled n every teraton step changng the state of the system. Consequently, they have to be constructed n such a way that the theoretcal boundary condton s satsfed durng the course of the smulaton. It can be expected that ths s only possble to a certan degree of accuracy, n the same sense as the dscretzed dynamcs can only mmc the real system approxmately. As the boundary condton s part of the dynamcs, t can potentally deterorate the whole method f ts accuracy s nferor compared to the bulk dynamcs. Therefore, t s hghly desrable to use boundary condtons that attan at least the same accuracy as the method n general. In order to represent boundares n the lattce Boltzmann method, one has to map the sold objects to the lattce structure. Ths can be done n several ways, three of whch are llustrated n fgure 5.5 for the example of a crcle. A smple way s the node based approach where the lattce stes are dvded nto sold, flud and boundary nodes. Sold nodes are completely covered by the sold object; boundary nodes have at least one velocty lnk to a sold node whereas flud nodes are only lnked to boundary nodes or other flud nodes. An alternatve way s the lnk based approach where boundary markers are put on the velocty lnks that connect sold and flud nodes. If no further nformaton s ncluded, the boundary markers are smply located halfway between the sold and the flud nodes, whch yelds a staggered representaton of the sold object. In a more elaborate varant of the lnk based approach, the boundary markers are assumed to le drectly on the boundary surface. Ths requres some addtonal effort to determne the exact ntersecton of the velocty lnk and the boundary, but t has the beneft that the obtaned representaton of the sold object s somewhat more precse. Asde from the dfferent postonng of the boundary nodes or markers, all the dfferent versons have n common that some of the velocty lnks are cut by the surface. On these lnks, the populaton movng from the nteror to the exteror of the 68

89 5.2 Boundary condtons for lattce Boltzmann models bounce back specular reflecton r s=1 r slp reflecton Fgure 5.6: Illustraton of smple md-lnk reflecton rules. (Left) Bounce-back reverses the velocty of the mpngng populaton. (Mddle) Specular reflectons reverse only the normal momentum durng reflecton of the populatons. (Rght) Slp-reflectons combne bounce-back and specular reflectons. sold object s undefned. These unknown populatons have to be specfed by the boundary condton. The role of a lattce Boltzmann boundary condton s thus to defne a rule for the unknown populatons that s compatble wth the dynamcs of the system and produces the desred hydrodynamc boundary condton. Ths can ether be done by dscretzng the knetc boundary condtons dscussed n the prevous secton, or by mposng explct rules and verfyng the macroscopc behavor. In the followng paragraphs, some of the most commonly used boundary condtons for the lattce Boltzmann model are revewed Bounce-back The oldest but nevertheless stll the most wdely used boundary condton for lattce Boltzmann s the bounce-back rule [10]. It was already ntroduced n the context of lattce gas automata and s appled to obtan a hydrodynamc stck boundary condton. The rule reflects the populatons at the boundary nodes by a bounce-back collson, n whch an ncomng populaton s bounced back whereupon ts velocty s reversed. Dependng on the lattce representaton of the sold object, there are two ways to mplement the bounce-back. In the node-based mplementaton, the reverson of the velocty takes place on the boundary nodes r B f (r B + τc, t + τ) = f (r B, t), (5.24) where the ndex s defned by c = c. It can be shown theoretcally [121, 122], that the node-based bounce-back rule leads to a hydrodynamc boundary that s shfted nto the flud by half a lattce spacng. Hence, the stck boundary s effectvely located on the lnks. It has therefore become common to use a lnk-based formulaton [65, 121, 123] f (r B, t + τ) = f (r B, t), (5.25) where r B s now a flud node lnked to a boundary marker by c. The lnk-based bounceback rule s llustrated n fgure 5.6. The stck boundary s located on the boundary markers n the mddle between the nodes. In the followng, we wll refer to the lnk-based mplementaton. 69

90 5 Boundary condtons for lattce Boltzmann models The bounce-back rule has no drect analogon n knetc theory, as a bounce-back process s rather mprobable to occur n the mcroscopc dynamcs. The boundary condton s constructed to enforce a vanshng flow velocty at the boundary. For smple flows, t s possble to solve the lnear lattce Boltzmann equaton wth bounce-back boundary condtons analytcally [ ]. The result for Poseulle flow shows that the bounce-back rule s only frst order exact,.e., the boundary locaton s subject to O(L 1 ) correctons, where L s the channel wdth. The flow velocty profle devates from the exact no-slp soluton by a constant offset [65, 126] a 2 u(r B ) = βu max L. (5.26) 2 The correcton factor β depends on the collson operator and the defnton of the hydrodynamc momentum densty. Wth the defnton (4.65) the result s β = 1 ( 16 3 λ 16 ) 2 λ + 1, (5.27) for the BGK collson operator and β = 1 3 ( ) 8 λ 7 (5.28) for the MRT collson operator wth equal egenvalues λ = λ s = λ b for the bulk and shear modes [65]. A detaled asymptotc analyss of the bounce-back rule shows further that the pressure s at most frst-order accurate, even for boundares that are algned along a lattce drecton. For nclned boundares and arbtrary shaped objects, the order of accuracy reduces to frst order for velocty and zeroth order for the pressure [97]. These defcences have motvated varous attempts to mprove the bounce-back scheme, some of whch wll be dscussed below Specular reflectons An equally smple scheme as bounce-back s the specular reflecton rule [see e.g. 121], where an ncomng populaton s reflected from the wall such that only the normal velocty component changes sgn whle the tangental component s unchanged. The lnk-based formulaton of specular reflectons s gven by f (r B + τ [c n(n c )], t + τ) = f +(r B, t), (5.29) where n s the boundary normal and the ndex + s defned by c + = c 2n(n c ). The specular reflectons are llustrated n fgure 5.6. They drectly correspond to ther counterpart n knetc theory, cf. equaton (5.8), hence they produce a hydrodynamc full slp boundary condton. As stated above, a full slp boundary wthout momentum transfer at the surface s usually not desrable n realstc applcatons. The specular reflecton rule s therefore not wdely appled and s manly used n conjuncton wth alternatve rules. Nevertheless, t can functon as a startng pont to develop more sophstcated boundary schemes, see secton

91 5.2 Boundary condtons for lattce Boltzmann models Dffuse reflectons The bounce-back and specular reflecton rules defne the unknown populaton by a smple reflecton of one sngle populaton. In general, an outgong populaton can be a functon of all ncomng populatons, and an obvous generalzaton of the lnk based reflecton rules s a lnear combnaton of the known populatons [127] f (r B, t) = B j (r B, r B τpc j, t)fj (r B τpc j, t). (5.30) n c j <0 P = 1 nn s the projecton operator that projects c j onto the tangental subspace of the boundary, and B j s a scatterng matrx that satsfes conservaton of mass and normal momentum by the condton B j (r B, r B τpc j, t) = 1. (5.31) n c >0 In the smplest case, the scatterng matrx mplements a combnaton of bounce-back and specular reflectons resultng n a slp boundary condton [110, ]. In fact, comparson of equatons (5.30) and (5.9) shows that B j s nothng but a dscrete verson of the scatterng kernel n knetc theory. Ansumal and Karln used ths to dscretze the dffusve boundary condton 2 and obtaned the dscrete dffusve boundary condton [117], whch can be straghtforwardly generalzed to a dscrete verson of Maxwell s accommodaton condton, cf. equaton (5.18) and [130], f (r B, t + τ) = (1 α)f +(r B τpc, t) + αf eq (ρ B, u B ) n c j <0 n c j fj (r B τpc j, t) n c j <0 n c j f eq j (ρ B, u B ) (5.32) where n (c u B ) > 0. Ths rule consttutes a drect mplementaton of knetc boundary condtons whch s n lne wth the mesoscopc sprt of the lattce Boltzmann method. It can reproduce the Knudsen number dependent wall slp to very good agreement and yelds the same convergence to the hydrodynamc lmt as the Boltzmann equaton [117]. On the other hand, (5.32) s stll a reflecton rule based on ether a node-based or a lnk-based representaton of the boundary surface. The ramfcatons for arbtrary shaped objects are to date not very well explored. Moreover, the mplementaton of the dffusve boundary condton [117] s more complcated compared to the smpler slp-reflecton models [127, 128]. It s probably for these reasons, that the knetc boundary condton s yet rarely used n practcal applcatons, where the bounce back boundary condton s frequently favored for ts strkng smplcty. Ths holds n partcular for arbtrary geometres and partculate suspensons [8, 131]. 2 In prncple ths can be done along the same lnes as n the bulk. However, due to the occurrence of halfspace ntegrals, the quadrature nodes are dfferent, resultng n a lattce msmatch. Ansumal and Karln therefore resort to usng the bulk quadrature for the boundary nodes as well, whch strctly speakng ntroduces addtonal dscretzaton errors [117]. It s also to be noted, that n ther work the equlbrum dstrbuton of the entropc lattce Boltzmann model was used., 71

92 5 Boundary condtons for lattce Boltzmann models Advanced closure schemes Apart from the dffusve reflectons, many other attempts to mprove the accuracy of the bounce-back boundary condton have been proposed. One class of these approaches tres to fnd a soluton for the unknown populatons n terms of the populatons on adjacent lattce stes. Let us refer to ths class as closure schemes. Ther am s to generate a set of populatons that satsfes the desred boundary condtons at the hydrodynamc level,.e., the desred velocty feld (Drchlet condton) and ts gradents (Neumann condton). Zegler [132] combned the nodal bounce-back wth settng the grazng drectons to the average of the ncomng drectons. Ths scheme ensures the no-slp condton by constructon, but t s not mass conservng on the boundary nodes. Skordos [133] addressed the problem of nversely mappng the hydrodynamc felds to the lattce Boltzmann populatons. A modfed collson operator was ntroduced for the boundary nodes, whch relaxes the populatons towards an equlbrum dstrbuton that ncludes velocty gradents as addtonal correcton terms. Although a modfed equlbrum dstrbuton at the boundary s a reasonable assumpton, the ncluson of gradent terms s questonable and lacks a rgorous justfcaton n terms of the Chapman-Enskog expanson. If the velocty gradents are unknown, they must be evaluated usng fnte-dfferences. Moreover, the densty was assumed to be known at the boundary nodes, whch may not always be approprate. Noble and coworkers [134, 135] developed a two-dmensonal closure scheme where the densty s a computed quantty and only the velocty components at the boundary have to be prescrbed. The scheme s based on dvdng the populatons nto groups that stream n from neghborng flud nodes, boundary nodes or sold nodes, respectvely. The latter of these three are the unknown quanttes n an equaton system whch s obtaned from the conservaton laws for mass and momentum. Noble et al. solved ths equatons system for a seven velocty model, and a generalzaton to the three-dmensonal D3Q15 and D3Q18 models was developed by Maer et al. [136]. A smlar technque was used by Inamuro et al. [137, 138] and combned wth the dea of dffusve scatterng. The unknown populatons are drawn from an equlbrum dstrbuton for the wall, but wth an addtonal counterslp velocty n the tangental drecton whch s adjusted to satsfy mass conservaton on the wall. The Inamuro method yelds an equaton system that can be solved for arbtrary lattce models. On another route, Zou and He [139] proposed a closure based on the concept of bounce-back of non-equlbrum parts. The approach was used to derve pressure and velocty boundary condtons for the D2Q9 and D3Q15 models. However, the nherent msmatch between the number of unknown parameters and the number of constrant equatons was only heurstcally solved, but no systematc procedure to cope wth ths problem was devsed [5]. Lätt and coworkers [101, 140] have recently appled bounce-back of non-equlbrum parts together wth a so-called regularzed LBGK model. Ths approach turns out to be a mxture of the Inamuro and the Zou/He methods, whle the Drchlet condton s ensured by mplctly makng use of the moment representaton famlar n MRT models. A more systematc approach was put forward by Gnzbourg and d Humères [141]. The basc dea s to explot the Chapman-Enskog result for the populatons to compute the necessary dervatves of the velocty feld locally at the boundary node wthout usng fnte 72

93 5.2 Boundary condtons for lattce Boltzmann models dfferences. Wth the help of the expresson (4.42) t s possble to wrte the unknown outgong populaton numbers as lnear combnatons of the known populatons and the prescrbed boundary constrants. Whle t s not obvous that a unque soluton for ths lnear equaton system exsts n general, an algorthm for a flat wall algned to a lattce drecton was presented n [141]. For smple flows, t yelds a local second order boundary condton at the expense, however, of a second-order mass flux across the boundary. More recently, Hallday et al. [142] have revsted the method of Gnzbourg and d Huméres and presented an enhanced unfed framework to tackle the boundary closure problem n a systematc way. Snce most of the above closure schemes can be subsumed under ths framework, t deserves to be outlned n a lttle more detal [see also 101]. For smplcty, we restrct the formal presentaton to lattce models wth c α { 1, 0, +1}. 3 If the wall normal s assumed to pont n the postve z-drecton, the populatons can be dvded nto mpngng, grazng and reflected drectons accordng to the z-component of the velocty vector. Closure for equlbrum parts The mass densty at the boundary node can then be wrtten as ρ = f = f + f + f. (5.33) c z <0 c z =0 c z >0 Smlarly, the normal boundary velocty can be wrtten as ρu B,z = f c z = f c z + f c z = f + f. (5.34) c z <0 c z >0 c z <0 c z >0 Elmnatng the unknown populatons yelds an expresson for the densty n terms of the known ncomng populatons ( 1 ρ = 2 f + ) f. (5.35) 1 u B,z c z <0 A smlar expresson was used n the works [101, 134, 137, 139]. Havng determned the densty, t can be used together wth the prescrbed boundary velocty to compute the equlbrum part f eq of the boundary populatons usng the standard equlbrum dstrbuton (2.56). By ths procedure, any Drchlet condton on the hydrodynamc felds can be satsfed. c z =0 Closure for non-equlbrum parts It remans to determne the non-equlbrum parts of the populatons such that the correct velocty gradents at the boundary are recovered. Usng the Chapman-Enskog result (4.42), 3 Hallday et al. [142] have treated only the D2Q9 explctly. The presentaton here s a straghtforward generalzaton. 73

94 5 Boundary condtons for lattce Boltzmann models we are left wth the followng equaton system 0 = f neq + f neq + f neq, c z >0 0 = ρτc 2 s λ c z >0 c z =0 f neq c α + ( uα r β + u β r α c z =0 ) = c z <0 f neq c α + f neq c α, c z <0 c z >0 f neq c α c β + c z =0 f neq c α c β + f neq c α c β. c z <0 (5.36) Ths system s n prncple under-determned, such that there s no general procedure to determne the soluton f neq. Ths s due to the addtonal degrees of freedom n the lattce Boltzmann method compared to the number of hydrodynamc varables. A soluton of (5.36) can be obtaned by choosng a set of the ncomng populatons to take ther known values after streamng. The choce of ths set should nclude a maxmum number of populatons that stream from the bulk to the boundary n order to facltate the couplng of the bulk and the surface. On the other hand, the explct choce must guarantee the solvablty of the equaton system. In [142], ths was acheved by computng the determnant for all possble combnatons and enumeratng all forbdden combnatons. Another dffculty arses n measurng the velocty gradents at the boundary. Gnzbourg and d Humères [141] consdered the velocty dervatves as addtonal unknowns of the closure system, whch makes t even more complcated to solve because more known populatons are requred. Such an approach s thus nfeasble n complex geometres. Therefore, Hallday et al. [142] chose to use fnte dfferences to determne the velocty dervatves, lke Skordos [133]. In [137] the gradents are not necessary for the equlbrum forcng, whereas n [101, 139] they are fxed mplctly by the bounce-back of non-equlbrum parts. In the works [132, ] the gradents were not accounted for explctly, whch s the reason why the respectve schemes are only frst order for arbtrary geometres. The dfferent closure schemes show that there s a trade-off between accuracy and localty because second-order closures requre the velocty dervatves whose local computaton s only possble n smple geometres. Second-order accuracy s n general dffcult to acheve locally, a fact whch has led to the development of non-local nter- and extrapolaton schemes. A comprehensve comparson of several of the above boundary schemes for straght walls algned to the lattce can be found n Latt et al. [101] Interpolaton and extrapolaton schemes One of the frst extrapolaton schemes was ntroduced by Chen et al. [143]. They ntroduced an addtonal layer nsde the sold and extrapolated the populatons at those nodes from the boundary nodes and the frst flud node. After the extrapolaton, an equlbrum forcng s appled at the surface smlar to the Inamuro method. The locaton of the wall s, however, only frst order n the lattce spacng. Ths s because the smple node or lnk-based schemes lead to a staggered representaton of curved boundares. To treat the boundary wth hgher accuracy, t s necessary to use boundary markers that specfy the exact cuttng ponts wth 74

95 5.2 Boundary condtons for lattce Boltzmann models C D A B C A D B C D A B r b c r b r b + c r b c r b r b + c r b c r b r b + c q < 1/2 q > 1/2 q q Fgure 5.7: Illustraton of the nterpolaton rules used n the boundary condton by Bouzd et al. [154]. Dependng on the value of q, ether a pre-collson or a post-collson populaton s nterpolated. The rghtmost pcture shows the case where only one lattce node s present between two surfaces. Whle Bouzd s scheme s not applcable any more, equlbrum nterpolaton s stll possble, cf. secton the lattce vectors,.e., for every cut lnk the fracton whch les outsde the sold needs to be known. The applcaton of ths representaton was poneered by Flppova and Hänel [ ] n ther boundary-fttng scheme. The unknown reflected populatons are computed as a lnear combnaton of the ncomng populaton and a fcttous boundary equlbrum, where the velocty nsde the sold s obtaned by lnear extrapolaton from the last flud node and the prescrbed boundary velocty. The weghtng coeffcents of the lnear combnaton are functons of the respectve lnk s cut rato and can be determned by a Chapman-Enskog expanson [145]. However, f the lnk fracton outsde the sold becomes small, the method shows strong nstabltes for pressure drven channel flows [147, 148]. Me et al. [147] reexamned the boundary fttng scheme and proposed an mproved verson by refnng the extrapolaton for the velocty used n the boundary equlbrum. It was numercally shown that the stablty s mproved consderably [147, 149]. A dfferent scheme was put forward by Verberg and Ladd [150, 151] under the name contnuous bounce back. It s related to the volumetrc formulatons of the lattce Boltzmann method [62, 152, 153] and uses specal nterpolated bounce-back rules for lattce cells that are partally occuped by sold objects. The contnuous bounce back rules for general geometres are however qute complcated. Furthermore, t was found that they lead to mpared stablty below a crtcal shear vscosty [151]. On the other hand, the accuracy of the method s not affected by the shape or poston of the flud-sold nterface wth respect to the lattce snce only the fracton of flud per node s needed n the algorthm. In essence, the method of Verberg and Ladd s a specal nterpolaton scheme. A smpler, physcally ntutve nterpolaton scheme has been ntroduced by Bouzd et al. [154]. An essental feature of the approach s that only populatons along one drecton are used for the nterpolaton. Hence, t s enough to consder the one-dmensonal stuatons depcted n fgure 5.7. We seek an nterpolaton scheme for the reflected populaton at the flud node A next to the surface. If we magne that ths populaton was bounced-back by the surface, t would fcttously orgnate at the locaton D. Let the fracton of the lnk outsde the sold be denoted by q. Then two cases have to be dstngushed: If q < 1/2, the fcttous populaton at D can be obtaned by nterpolaton from A and C. In the other case q 1/2, the prestreamng populaton at D can only be obtaned extrapolaton, whch s nferor n terms of stablty and should be avoded. To ths end, the fcttous post-streamng populaton at D 75

96 5 Boundary condtons for lattce Boltzmann models r b 2 c r b c r b r b + c κ 2 κ 1 κ 1 κ 0? q κ 1 Fgure 5.8: Illustraton of the multreflecton boundary condton of Gnzburg and d Humères [64]. The fve blue populatons are weghted wth coeffcents κ l to calculate the unknown red populaton. can be used to nterpolate the sought reflected populaton at A by nterpolaton from C and D. Ths leads to the followng lnear nterpolaton scheme for the unknown populaton at A f (r B, t + τ) = 2qf (r B, t) + (1 2q)f (r B τc, t), q < 1 2, f (r B, t + τ) = 1 2q f (r B, t) + 2q 1 f 2q (r B, t), q 1 (5.37) 2. These expressons are contnuous n q and for q = 1/2 they reduce to the standard bounceback rule. The scheme can straghtforwardly be extended to quadratc nterpolaton and s applcable to movng boundares as well [154, 155]. Rohde et al. [156] have appled the nterpolaton rules to the volumetrc schemes and carred out a theoretcal analyss for plane Poseulle flow. They fnd that the Bouzd scheme s subject to errors n mass conservaton, whereas the volumetrc nterpolaton schemes are mass conservatve. For nclned boundares, however, the methods are stll frst-order accurate. Furthermore, the effectve locaton of the boundary depends on the vscosty, and for example n Poseulle flows, exact parabolc profles are not obtaned for arbtrary nclned channels. Ths was ponted out by Gnzburg and d Humères [64] n a semnal paper, n whch they present a comprehensve analyss of the accuracy of nterpolaton boundary condtons. For ths purpose, they ntroduce the multreflecton boundary condton whch subsumes bounce-back and the lnear and quadratc nterpolaton schemes. The multreflecton rule uses three flud nodes and fve populatons along a lattce drecton. The set-up s depcted n fgure 5.8. The weghtng factors κ l for the nterpolaton are derved by matchng the second-order Chapman-Enskog result wth a Taylor expanson at the boundary. The multreflecton rule for general flows s [64, 131] f (r B, t + τ) = f 1 2q 2q2 (r B, t) f (1 + q) 2 (r 1 2q 2q2 B, t) + f (1 + q) 2 (r τc, t) q2 (1 + q) f 2 (r τc q 2, t) + (1 + q) f 2 (r 2τc, t). (5.38) It consttutes a thrd order knetc accurate boundary scheme and s therewth the most accurate boundary condton avalable. It was also shown n [64] that the standard bounce-back 76

97 5.2 Boundary condtons for lattce Boltzmann models can be tuned to yeld second order accuracy by settng the collson egenvalues for the knetc (ghost) modes to λ g (λ) = λ 8 + λ, (5.39) where λ = λ s = λ b. Ths choce compensates the vscosty dependent correctons to the locaton of the boundary surface Equlbrum nterpolaton Lke all nterpolaton methods, the multreflecton boundary condton has the drawback that nformaton from several flud nodes s needed to determne the unknown populatons. Besdes renderng the scheme non-local ths s problematc n applcatons where sold objects are close together and the dstance s on the order of the lattce spacng. Then there may occasonally not be enough flud nodes avalable n between the sold objects, such that the nterpolaton schemes break down. To overcome ths drawback, Chun and Ladd [131] have very recently proposed to nterpolate only the equlbrum dstrbuton. Ths dea s justfed by the fact that the non-equlbrum dstrbuton enters the Chapman-Enskog expanson an order later than the equlbrum dstrbuton. Thus a boundary condton whch s second-order for the equlbrum dstrbuton, and only frst order for the non-equlbrum dstrbuton, wll stll be suffcent to guarantee overall second-order accuracy. Chun and Ladd suggest to use the Bouzd method for the equlbrum dstrbuton and smple bounceback for the non-equlbrum part. The equlbrum nterpolaton boundary condton s gven by f eq (r B, t + τ) = 2qf eq (r B, t) + (1 2q)f eq (r B τc, t) q < 1 2, (r B, t + τ) = 1 q f eq (r, t) + 2q 1 f eq (r B + qτc ) q q q 1 2, (r B, t + τ) = f neq (r B, t), f eq f neq (5.40) where f eq (r B + qτc ) s the boundary equlbrum. Equaton (5.40) stll requres two flud nodes n between sold objects. If the objects are very close, the equlbrum dstrbuton on the other surface s used and (5.40) s replaced by f eq q (r B, t + τ) = q + 2q 1 f eq (r B, t) + 1 2q q f eq (r B q τc ). (5.41) where q s the fracton of ĉ outsde the second surface, cf. fgure 5.7. The equlbrum nterpolaton rule s second-order accurate and requres only one flud node n between the boundares. Wth the choce (5.39) for the collson egenvalues of the knetc modes, the locaton of the boundary s ndependent of vscosty. It was shown numercally that the equlbrum nterpolaton boundary condton s more accurate than the lnear and quadratc nterpolaton rules [131]. Equlbrum nterpolaton s probably the best compromse between accuracy and smplcty of mplementaton among all boundary condtons presented so far. 77

98 5 Boundary condtons for lattce Boltzmann models Crtcal dscusson of the exstng boundary condtons The body of works on boundary condtons s consderable and qute some progress has been made n recent years. The man fndngs can be summarzed as follows: Smple reflecton rules lke bounce-back n ether node-based or lnk-based formulatons are only frst-order accurate wth respect to general flows. They are affected by correctons to the flow velocty at the surface and a vscosty dependent locaton of the hydrodynamc boundary wth respect to the underlyng lattce. Advanced closure schemes amng at second-order accuracy can mprove matters, but they tend to be complcated to mplement and localty of the scheme has to be sacrfced when velocty gradents are requred. Furthermore, the mproved accuracy s often annuled by nclned or curved boundares. Arbtrary shaped objects can be handled wth nter- and extrapolaton schemes. Snce extrapolaton s connected to deterorated stablty, nterpolaton s generally preferable. Interpolaton rules are easer to mplement than closure schemes and can systematcally be tuned to the desred accuracy. However, nterpolaton schemes are nherently non-local and rely on a mnmum number of nodes avalable between sold objects. In general, there seems to be a trade-off between accuracy and localty of the boundary condton. In addton, some of the methods are faced wth the problem that the local mass conservaton constrant s volated. Another pont has to be made wth respect to the role of the hgher moments of the dstrbuton functon. The Chapman-Enskog analyss of the multreflecton boundary condton shows that the collsonal egenvalue of the knetc modes affects the locaton of the wall. In other words: Boundary condtons are affected by the dynamcs of the knetc modes of the lattce Boltzmann model. Ths suggests that the more flexble MRT collson operator should be favored over the BGK collson operator, whch does not allow to tune the relaxaton rates of the modes separately. Knetc type boundary condtons such as dffusve reflectons have receved broader attenton only recently. A conceptual problem s that the systematc dscretzaton n terms of quadratures leads to a msmatch of nodes and ncompatble lattce structures at the boundary (cf. footnote 2 on page 71). So far, ths could only be resolved by abandonng the systematc expanson and acceptng addtonal dscretzaton errors. A fully consstent adopton of knetc type boundary condtons to the dscrete Boltzmann lattce s to date not avalable. The latter pont gves evdence that the very nature of lattce Boltzmann boundary condtons s stll not understood completely n regard to the mesoscopc orgn of the method. Ths s of partcular relevance n vew of an upcomng trend to use the lattce Boltzmann method for complex flows beyond the Naver-Stokes equaton [46 48, 157]. Much nterest s attracted to smulatons of mcroflows at non-vanshng Knudsen number, especally n the slp flow regme. Most of the boundary condtons descrbed above focus on realzng a stck-boundary condton for the Naver-Stokes equaton. These methods are clearly not capable of reproducng slp-flows n mcrochannels. The apparent slp velocty accordng to (5.26) s merely a numercal artfact [158]. Knetc type boundary condtons can reproduce the apparent slp-effects n mcroflows, but as ponted out, ther dscrete counterparts are affected by dscretzaton errors such that the accuracy of the results s dffcult to assess. 78

99 5.2 Boundary condtons for lattce Boltzmann models It s therefore far to clam that boundary condtons for smulatons n the slp-flow regme need further nvestgaton. The followng specfc ssues are rased: The hydrodynamc stck boundary condton s replaced by the Naver slp condton (5.7) where the slp length δ B enters. The latter s a functon of the boundary coeffcent γ that models the propertes of the boundary. A lattce Boltzmann boundary condton for slp-flow needs to be tunable by an analogous parameter. There s yet no consensus about the best way of mplementng a partal slp boundary condton. It s desrable to have a theoretcal relaton for the slp length and the model parameter and, optmally, an estmate for the numercal errors. These are basc tools needed to compare results wth analytcal theory, experments and other smulaton methods. As the Knudsen number becomes fnte, the knetc nature of the flud becomes more pronounced and effects beyond Naver-Stokes behavor occur, for example, the appearance of Knudsen layers. The lattce Boltzmann method s n prncple a vald tool to smulate such effects because t consttutes an approxmaton to the Boltzmann equaton. Ths knetc nature has to be reflected by the boundary condton as well. Ths brngs up two ponts: frstly, the nfluence of the hgher order moments to the dynamcs can eventually not be neglected any more; secondly, two knds of collson processes have to be taken nto account at the boundary, namely nterpartcle collsons and collsons wth the boundary. Both of these ponts have to be tackled by an approprate choce of the collson operator at the boundary. Another aspect wth respect to the lattce Boltzmann approxmaton of knetc theory s the mpact of dscretzaton errors due to the underlyng lattce structure. In the bulk, the systematc dscretzaton up to a gven degree naturally leads to a quadrature that mples related symmetres of the emanatng lattce. As a consequence, sotropy of tensors up to the rank of the quadrature s automatcally satsfed. Ths stuaton changes at the boundary where the bulk symmetry s broken. To the best of the author s knowledge, systematc half-range quadratures have not been appled and the effects of the broken symmetry at the boundary have not been treated systematcally n any avalable work on lattce Boltzmann boundary condtons. The aforementoned ssues shall be addressed n the remander of ths work. In the next subsecton, a novel way of mposng a partal slp boundary condton n the lattce Boltzmann model s developed. In chapter 6, an attempt s made to devse a conceptually new method for lattce Boltzmann boundary condtons, whch s completely local and takes the reduced symmetry at the boundary nto account n a systematc and consstent fashon. 79

100 5 Boundary condtons for lattce Boltzmann models 5.3 Partal slp boundary condtons In mcrofludc devces, such as those shown n fgure 5.1, the surface-to-volume rato s hgh. Another way of sayng ths s that the dmensons of these devces are small, and so s the typcal length scale of the flud flow. Wth decreasng sze of the devces the Knudsen number ncreases and the flow n typcal mcrochps reaches the slp-flow regme. Expermental studes show that n ths regme, the boundary condton s affected by an nterplay of a varety of physco-chemcal parameters, such as surface charge, hydrophobcty and wettng, surface roughness etc. The most relevant phenomenon s the appearance of an effectve slppage of the flud on the boundary and reduced hydrodynamc stresses, whch can lead to a sgnfcant enhancement of the flow throughput n mcrochannels [see Ref. 109 for an overvew of slp phenomena n experments]. Clearly, the no-slp boundary condton s napproprate n ths stuaton and a more mcroscopc approach s needed to model detals of the flud-surface nteractons. Whle molecular dynamcs and drect smulaton Monte Carlo can shed some lght on specfc aspects of the flud-surface nteractons, such as rarefacton or dewettng [159], they are computatonally too expensve to smulate complex flows on reasonable tme and length scales. The lattce Boltzmann method s much better suted for smulatng flows and, due to ts knetc orgn, t s practcally well suted to handle the propertes of the slp-flow regme [103, 104]. Sbragagla and Succ [46] have recently argued that the lattce Boltzmann approxmaton ndeed remans vald up to Kn O(1). An essental requrement s yet an approprate boundary condton that can model the appearance of partal slppage on a mesoscopc level. In the followng, we wll develop a method to mplement partal slp n the lattce Boltzmann method. It ams at capturng slp at a coarse-graned level, where the detals of the fludsurface nteracton are modeled by a sngle parameter. Ths s n contrast to other works, where the flud-surface nteracton s modeled as an explct potental wthn a mult-phase LB model [7, ]. In the same sprt, we do not ncorporate the roughness of the surface explctly, as t was for example done by Kunert and Hartng [163, 164]. Furthermore, we seek a general coarse-graned model for partal slp whch s not specfc to the lattce Boltzmann method but can be used n other smulaton methods as well, e.g., dsspatve partcle dynamcs. In fact, a collaboratng group has recently succeeded n mplementng tunable-slp boundares n DPD n an analogous fashon [165]. Ths opens the possblty to compare smulaton results from dfferent methods, whch allows to dfferentate the errors that stem from the model from those that arse as artfacts of the specfc mplementaton Modelng wall frcton Our startng pont s the consttutve equaton (5.5) whch leads to the Naver slp boundary condton. Let us consder the case where the boundary normal s n z-drecton and a flow n the x-drecton. Then the slp boundary condton says that the stress exerted on the boundary surface s proportonal to the flow velocty at the boundary [111] σ xz = γ u slp. (5.42) 80

101 5.3 Partal slp boundary condtons Ths suggests that we can model the flud-sold nteractons as an effectve frcton force, whch ncludes all the mcroscopc detals that lead to dsspaton of energy at the surface and thereby decelerate the flow F (r B ) = ζu x (r B ). (5.43) The dea s then to combne specular reflectons at the boundary wth the frcton force, where the frcton coeffcent ζ s meant to control the amount of slppage over the surface. Such an approach has several advantages: frst, t s very general and can be appled to off-lattce partcle-based methods as well. Second, the frcton force s smlar to Langevn lke forces, whch are well establshed n computer smulatons and provde a framework to ensure correct thermodynamcs by obeyng the fluctuaton dsspaton relaton, that s, a lnk to statstcal mechancs s readly avalable for our boundary model. A thrd pont to menton s that the frcton force s local and hence the valdty of local conservaton laws can be ensured. Fnally, the mplementaton of forces n the lattce Boltzmann model s possble wthout major complcatons Analytcal soluton of the wall frcton model for Poseulle flow Before we turn to the mplementaton of the wall frcton model n the lattce Boltzmann method, we dscuss the analytcal soluton n the case of a plane Poseulle flow. Ths serves as a further justfcaton and s used to compare the smulaton results below. Let us consder a statonary flow of an ncompressble flud drven by a volumetrc force f ext n the x-drecton and confned between two plane walls whose normals pont n the z-drecton. It can be descrbed by the Stokes equaton η s 2 u x z 2 = f ext. (5.44) Snce we have bult the boundary condton on a consttutve equaton for the frcton force, there are addtonal terms that enter the bulk equaton. The contnuum transcrpton of the frcton force on the last layer of flud nodes comes n terms of δ-dstrbutons. Let us assume that the walls are located at z = 0 and z = L, then the frcton force acts on the layers z = z B and z = L z B, and we arrve at the dfferental equaton η s 2 u x z 2 = f ext + ζ a 2 u(z)δ(z z B) + ζ a 2 u(z)δ(z L + z B). (5.45) Because of the addtonal specular reflectons, ths equaton has to be solved wth full-slp boundary condtons: u (0) u x(0) z = 0, u (L) u x(l) z = 0. (5.46) The soluton of the second order dfferental equaton s pecewse parabolc, as qualtatvely shown n fgure 5.9, 81

102 5 Boundary condtons for lattce Boltzmann models Integraton constants A 1 0 f A extla f extzb 2 2ζ 2η s f B extl 1 2η s f B ext La fextz2 B 2ζ 2η s flz B 2η s f C ext L 1 η s f C extla fextz2 B 2ζ 2η s Fgure 5.9: Analytcal soluton of the partal slp boundary condton. The flow profle s pecewse parabolc. The table lsts the ntegraton constants appearng n (5.47). u x (z) = f ext 2η s z 2 + A 1 z + A 2, z < z B, u x (z) = f ext 2η s z 2 + B 1 z + B 2, z B z L z B, u x (z) = f ext 2η s z 2 + C 1 z + C 2, z > L z B. (5.47) From the boundary condtons (5.46) we get A 1 = 0 and C 1 = f extl η s. From symmetry around the centerlne of the channel we get B 1 = fextl 2η s. The δ-dstrbutons mply a jump n the frst dervatves of the velocty. The heght of ths jump can be determned by formally ntegratng over an ɛ-nterval centered around the support of the dstrbuton. For the frst δ-dstrbuton n (5.45), we get lm [u ɛ 0 x(z B + ɛ) u x(z B ɛ)] = ζ η s a 2 u x(z B ). (5.48) The second δ-dstrbuton can be treated analogously. Pluggng n the results obtaned so far we arrve at u x (z B ) = f extla 2 = u x (L z B ). (5.49) 2ζ Ths s just another way of expressng that the center pece of the soluton satsfes a slp boundary condton wth a slp velocty u slp = f extla 2 2ζ (5.50) 82

103 5.3 Partal slp boundary condtons j reflectons Fgure 5.10: Canoncal mplementaton of the wall frcton force. Each populaton has a contrbuton from the momentum transfer j. Ths mplementaton leads to artfacts n the velocty profles measured n smulatons. at the hydrodynamc wall locatons z = z B and z = L z B. In the lmt z B 0 we get the velocty profle u x (z) = f ext 2η s z 2 + f extl = f ext 2η s ( z L 2 z + f extla 2 2η s 2ζ ) 2 + f extl 2 + f extla 2, 8η s 2ζ (5.51) whch s nothng but a Poseulle profle shfted by u slp. The tangental stress at the wall s and the slp length s η s u x (0) z = f extl 2, (5.52) δ B = η s ζ a2. (5.53) The latter relaton shows that the force frcton parameter ζ s related to the coeffcent γ n the hydrodynamc consttutve equaton (5.42) Implementaton of wall frcton: canoncal method Force mplementatons n the lattce Boltzmann model have been dscussed by several authors [47, 85, 86, ]. The most strngent way s to dscretze the forcng term n the contnuous Boltzmann equaton n terms of a truncated Hermte expanson [47, 166]. The obtaned result s to second order equvalent to the forcng terms derved wthn the Chapman-Enskog expanson, cf. chapter 4 and [8, 65]. Here, we accordngly modfy the force couplng method of Ahlrchs and Dünweg [85, 86], whch was orgnally developed to couple polymers to the lattce Boltzmann flud. The modfcaton conssts of addng the second-order correcton term derved wthn the Chapman-Enskog expanson. The forcng term s gven by g = w τ c 2 s [ g c + 1 G : ( c 2c 2 c c 2 s1 )], (5.54) s 83

104 5 Boundary condtons for lattce Boltzmann models where the tensor G s gven by equaton (4.67). The volumetrc force s obtaned by the above frcton force g(r B ) = ζ a 3 j(r B ) ρ(r B ). (5.55) In prncple, the modfed defnton (4.65) for the hydrodynamc momentum densty has to be used. Ths brngs up an addtonal complcaton because the redefned momentum contans the force and at the same tme the force s a functon of the momentum. Strctly speakng, the equatons (4.65) and (5.55) have to be solved self-consstently whch would requre an teratve scheme. However, f the velocty at the boundary s small 4, the correctons are neglgble. The localzed force densty s appled on the last layer of flud nodes n front of the boundary. In addton, the specular reflectons are appled on the boundary lnks. The complete scheme s llustrated n fgure We call ths scheme the canoncal mplementaton of the frcton force because t uses the forcng term wthout any further adjustments Smulaton results for the canoncal mplementaton To valdate the new partal slp scheme, we have smulated a Poseulle flow through a plane channel. The channel s algned wth the lattce such that the wall normals pont n the postve or negatve z-drecton, respectvely. The smulaton box has a wdth of 20 lattce spacngs n the z-drecton, where the walls are located at the frst and the last lattce layer. Takng nto account the shft of a/2 for the hydrodynamc boundares, the channel has an effectve wdth of L eff = 19a. Perodc boundary condtons are appled n the x- and y- drectons. The flud s drven by a volumetrc force f ext n the x-drecton. The densty of the flud s ρ = 1.0 and the knematc vscosty s set to ν = 3.0, both measured n lattce unts. In fgure 5.3.4, the velocty profles are shown for a lattce spacng of a = 1.0 and a drvng volumetrc force of f ext = The value of the frcton coeffcent ζ vared from 1.0 to 5.0. The analytcal solutons, cf. (5.47), for the respectve parameter sets are plotted as dotted lnes. The veloctes on the vertcal axs are scaled by the theoretcal maxmum velocty for ζ = 1.0. The measured velocty profles have a parabolc shape n the bulk, as expected. However, they devate from the analytcal predcton sgnfcantly and close to the boundary the profles are dstorted due to a knk on the next nearest flud node. Nevertheless, the velocty on the last flud node vsbly matches the theoretcal value. Ths ndcates that, whle the frcton nduced deceleraton of the flud at the boundary s captured correctly, there s an undesrable jump n the tangental stress at the next nearest layer. Ths has to be nterpreted as an artfact n the smulaton. As the most lkely source of such artfacts s the dscreteness of the lattce structure, the effect of reducng the lattce spacng a was nvestgated. Fgure 5.12 shows the results for a lattce spacng of a = 0.1 and an effectve channel wdth of 4 In the smulatons that were performed, the flud s ntally at rest. Therefore the slp velocty n the statonary state s approached from below. The frcton force ncreases only up to the value where, together wth the vscous stresses n the flud, the external drvng force s balanced. 84

105 5.3 Partal slp boundary condtons Canoncal mplementaton of wall frcton model Velocty profles for Poseulle flow 1 f ext =0.01 a grd =1.0 ζ=1.0 ζ=2.0 ζ=5.0 analytcal solutons v ζ=1.0 v max z / L Fgure 5.11: Smulaton results for the wall frcton model wth the canoncal mplementaton of the frcton force. The profles devate from the analytcal soluton due to the occurrence of a knk next to the wall. 85

106 5 Boundary condtons for lattce Boltzmann models Canoncal mplementaton of wall frcton model f ext =0.01 a grd =0.1 1 ζ=1.0 ζ=2.0 ζ=5.0 analytcal solutons v ζ=1.0 v max z / L Canoncal mplementaton of wall frcton model f ext =0.01 ζ=0.001 a grd = ζ = analytcal 0.8 v v max ζ= z / L Fgure 5.12: Smulaton results for reduced lattce spacngs a. The knk moves closer to the surface, accordng to the value of a, but t does not dsappear. In the lower plot, the bare frcton constant ζ s reduced to keep the effectve frcton n the same range. 86

107 5.3 Partal slp boundary condtons L eff = 199a. Due to the reduced lattce spacng, the knk moves closer to the boundary, but the offset from the analytcal curve remans unchanged n ts relatve order of magntude. Ths outcome s not altered when the lattce spacng s further reduced. Wth a lattce spacng of a = 0.05 and an effectve channel wdth of L eff = 200a the knk s stll clearly vsble. Note that n ths case the frcton coeffcent was reduced to ζ = to keep the absolute veloctes n the same order of magntude to avod nstabltes. 5 These observatons suggest that the knk and the jump n the tangental stress are not just a dscretzaton artfact, but rather are related to the specfc mplementaton of the wall frcton force. Apparently, the canoncal mplementaton leads to an unphyscal tangental stress at the boundary, whch must be nduced by the way the boundary condton s mplemented at the level of the populatons. We also verfed that ths s a frst order effect by omttng the second-order contrbuton from the forcng term (4.68) and the redefnton of the momentum densty (4.65), wthout any vsble effect. To nvestgate ths n more depth, a theoretcal analyss of the mplementaton of the wall frcton force n the lattce Boltzmann model s necessary Theoretcal analyss The analyss n ths secton s nspred by the work of He et al. [126], n whch the lattce BGK equaton was solved analytcally for a two-dmensonal statonary channel flow. We consder the stuaton where ρ = const. and the flow s nvarant n the x and y drectons. Then we can wrte a one-dmensonal lattce Boltzmann equaton f α+ĉ z (t + τ) = f α (t) + j L j ( f α j (t) f eq j (ρ, uα x) ) + g,α, (5.56) where f α denotes the populaton and u α x the flow velocty at r z = α a, and α = 0... L. The forcng term s gven by c x (f ext ζ ) a 3 uα x(δ 1,α + δ α,n ), (5.57) g,α = w τ c 2 s where the Kronecker deltas mplement the wall frcton and second order terms have been dropped. In the statonary case, equaton (5.56) smplfes to f α+ĉ z = j (δ j + L j ) f α j j L j f eq j (ρ, uα x) + g,α. (5.58) The explct expressons for the D3Q19 model are gven n appendx B.3. These expressons can be used to obtan the fnte-dfference soluton of the lattce Boltzmann equaton. The algebra s straghtforward but rather tedous. Therefore, t s also carred out n the appendx. 5 Equaton (5.50) shows that the velocty shft u slp scales wth the square of the lattce spacng. We could also ntroduce an effectve frcton ζ eff = ζ/a 2 whch takes nto account ths scalng. The reducton of the bare ζ then corresponds to keepng the effectve frcton ζ eff roughly constant. 87

108 5 Boundary condtons for lattce Boltzmann models For smplcty, we use the sngle-relaxaton tme approxmaton where all egenvalues of the collson operator are dentcal. The fnal result s u α 1 x 2u α x + u α+1 x η s a λ + 5λ2 ζ = f ext + 6λ 2 a 3 uα x(δ 1,α + δ α,n ) 5λ + 4 ζ 6λ 2 a 3 (uα 1 x δ 1,α 1 + u α+1 x δ α+1,n ), (5.59) whch s a second-order fnte dfference approxmaton of the Stokes equaton. The crucal pont to note here s the fact that there appear four Kronecker deltas on the rght-hand sde of equaton (5.59), whch stem from the frcton force. Two of them, δ 1,α and δ α,n, act on the boundary layers, whch s the expected effect of the wall frcton. The other two however, δ 1,α 1 and δ α+1,n, are addtonal contrbutons actng on the next nearest layer to the boundary. Ther appearance n the second-order fnte dfference approxmaton corresponds to a peak n the second dervatve of the veloctes, whch mples a jump n the frst dervatve and a knk n the velocty profle accordngly. It s a consequence of an effectve delocalzaton of the frcton force, whch dstorts the profle at the next nearest lattce layer. Ths explans why the canoncal mplementaton of the wall frcton force leads to the knk n the profles. Ths effect can also be understood descrptvely by lookng at fgure 5.10 agan: In the frst stage, the force s appled whch changes all populatons on the lattce ste. In the second stage, the reflectons occur whch take one tme step. However, only some of the populatons to whch the force has been appled do actually collde wth the wall, whle the others propagate to the bulk mmedately (the dashed red ones n the mddle pcture). In other words, the effect of the force s splt n two parts where one of them propagates nto the bulk too early and dstorts the flow profle. In concluson, the knk nduced by the canoncal mplementaton of the frcton force can be explaned by undesrable contrbutons on the next nearest lattce stes. These contrbutons wll be elmnated n the followng Force mplementaton revsted: prmtve method In order to avod the knk n the velocty profles, we have to elmnate the Kronecker deltas δ 1,α 1 and δ α+1,n n (5.59). Ther source can be easly tracked down by lookng at the dervaton n the appendx. The contrbutons from δ 1,α 1 and δ α+1,n enterng n (B.41b) and (B.41c) cannot possbly cancel and must be elmnated. In equaton (B.42) the unwanted terms stem from f 11 to f 14 whle n equaton (B.45) they stem from f 7 to f 10. Consequently, all the Kronecker deltas have to be removed n (B.39) from the bulk populatons n order to restore the desred Stokes form of the second-order fnte dfference equaton. Ths means that the only remanng possblty for the frcton force to be appled are the populatons whch actually collde wth the wall. In the D3Q19 model, only four of the populatons colldng wth the wall have a tangental projecton onto the force. Hence, the frcton force can be appled by addng half of t to the populaton wth the postve projecton, and subtractng the other half from the populaton wth the negatve projecton: g = 1 ζτ 3 2 a (c 5 Pu) n c < 0 (5.60) 88

109 5.3 Partal slp boundary condtons j reflectons Fgure 5.13: Prmtve mplementaton of the wall frcton force. Only the populatons colldng wth the wall get a contrbuton from the momentum transfer j. Ths mplementaton leads to correct velocty profles n the smulatons. The resultng scheme s schematcally depcted n fgure We call t the prmtve mplementaton of the frcton force because t leads back to a very smple force applcaton scheme, whch was already dscussed by Ahlrchs and Dünweg [85] for the polymer couplng. In that context, however, the scheme turned out to be an unfavorable choce and was superseded by the canoncal mplementaton [86] Smulaton results for prmtve mplementaton The results for the wall frcton boundary condton n the prmtve mplementaton are shown n fgure Plotted are the measured velocty profles for a Poseulle flow n a plane channel of wdth L eff = 20 lattce spacngs. The flud has a densty of ρ = 1.0 and a knematc vscosty of ν = 3.0. The flow s drven by a volumetrc force of f = 0.01, whch s mplemented as usual. The value of the frcton coeffcent ζ vared from 1.0 to 5.0. The analytcal solutons are plotted as dotted curves. For comparson also the results of the canoncal mplementaton are plotted agan as grey dashed curves. The measured velocty profles for the prmtve mplementaton show a perfect vsble match to the analytcal predcton. There s no knk n the curves and the amount of slppage s reproduced correctly. Ths shows that the prmtve mplementaton of the wall frcton model yelds the desred behavor and s superor compared to the canoncal mplementaton. Ths s n contrast to the bulk case, where the canoncal mplementaton s more favorable Comparson wth slp-reflecton models The prmtve mplementaton of the frcton force has an nterestng connecton to the famly of heurstc boundary condtons motvated by knetc theory [110, 127, 128]. Ths can be seen as follows. Frst we modfy the frcton force at the wall such that t acts on the momentum densty j B = f c, (5.61) n c <0 89

110 5 Boundary condtons for lattce Boltzmann models Prmtve mplementaton of wall frcton model Velocty profles for Poseulle flow 1 f ext =0.01 a grd =1.0 ζ=1.0 ζ=2.0 ζ=5.0 analytcal solutons v γ=1.0 v max z / L Fgure 5.14: Smulaton results for the prmtve mplementaton of the wall frcton model. The match of the measured profles wth the analytcal soluton s vsbly perfect. For comparson, the results from the canoncal mplementaton are plotted agan as dashed grey curves. 90

111 5.3 Partal slp boundary condtons where only the populatons colldng wth the wall contrbute. Ths corresponds to applyng the frcton force after the streamng on the frst lattce layer nsde the wall, before the populatons are reflected. In the steady state, the effect of ths modfcaton s a renormalzaton of the frcton constant. The forcng term (5.60) can then be wrtten as g = 1 ζτ 3 2 a (c 2 Pj B ) = 1 ζτ 3 c 2 a 2 f j Pc j, n c < 0, (5.62) n c j <0 where ζ now has unts of an nverse tme. Snce g s appled to the populatons that collde wth the wall and are reflected, we can combne the forcng term wth the specular reflectons f (r B, t + τ) = f +(r B τpc, t) 1 ζτ 3 c 2 a 2 fj (r B τpc j, t)pc j n c j <0 = (5.63) B j (r B, r B τpc j, t)fj (r τpc j, t), n c j <0 where we have ntroduced the boundary kernel B j (r B, r B τpc j, t) = δ j, ζτ 3 a 2 c Pc j. (5.64) Ths boundary kernel can be wrtten as a matrx. For the specfc case of the D3Q19 model and the wall normal n pontng n the postve z-drecton, we have 1 f ζτ ζτ 2 2 f 11 1 f 5 f 14 = 0 ζτ ζτ f f ζτ 1 0 ζτ f 6 0 f (5.65) f ζτ ζτ f The same goes for the D2Q9 model and when we set s = ζτ, (5.66) exactly the same boundary kernel as for the slp-reflecton model n [110, 127] s obtaned. The parameter s quantfes the reflectvty of the wall,.e., a specularly reflectng wall has s = 1 whle a bounce-back wall has s = 0. Sbragagla and Succ [127] show that a boundary kernel as n (5.65) leads to a slp velocty whch to frst order s gven by u slp = A B Kn u x ˆn, (5.67) rb where n the contnuum lmt of small tme and space ncrements A B = 1 s a c s 1 s τ. (5.68) 91

112 5 Boundary condtons for lattce Boltzmann models Comparng ths wth the expressons from knetc theory, cf. (5.22) and (5.23), we obtan an accommodaton coeffcent of α = 2 2s and a slp length of δ B = s l mfp a 1 s c s τ = 2 ζτ ζτ η s ρc 2 s a τ. (5.69) Ths s nothng but the expresson (5.53) wth a renormalzed frcton constant. The close connecton of the wall frcton model to the knetc slp-reflecton models gves confdence that the frcton force s ndeed a reasonable mesoscopc model for apparent slppage on boundary surfaces Dscusson of the wall frcton model In summary, the smulaton results and the theoretcal analyss show that the wall frcton model can successfully be used to mplement a boundary condton wth a tunable amount of slppage at the boundary. The basc concept s very general and can be appled to other smulaton methods as well [165]. Some care has to be taken when mplementng the wall frcton force n the lattce Boltzmann method. If the frcton force s mplemented n a canoncal way, the resultng Poseulle profles are dstorted by a knk next to the boundary whch s a consequence of undesrable momentum flux contrbutons. These can be elmnated by an apparently more prmtve mplementaton of the frcton force where only the populatons that actually collde wth the wall are changed. Ths result may seem surprsng snce the prmtve force mplementaton has no systematc justfcaton and s usually consdered unfavorable n the bulk [86]. On the other hand, there s no strngent reason to expect that the formula for the bulk case s applcable at the boundary and n fact our analyss shows that ths s not the case. We have rather shown that the correct dscrete Stokes profle can be reproduced wthout artfacts only f the applcaton of the wall frcton s restrcted to the populatons that actually collde wth the wall. In retrospectve, ths stands to reason because t ntroduces an explct asymmetry among the dfferent velocty drectons. The breakng of the bulk symmetry s an effect of the presence of the boundary, whch s explctly bult nto the prmtve wall frcton force mplementaton. In ths sense, ths mplementaton, though called prmtve, s more approprate for boundary condtons. Furthermore, t can be connected to a famly of knetcally motvated slp-reflecton models. The wall frcton force hence seems an approprate mesoscopc smulaton model that gves rse to apparent slp at boundary surfaces. The relaton between the slp length and the wall frcton parameter s known such that t s possble to tune the slp length systematcally. The fact that ths s n drect correspondence to the Naver-slp condton s thereby partcularly appealng, as t makes the boundary condton compatble wth knetc theory and hydrodynamcs at the same tme. The smulaton results presented n ths secton manly serve as a proof of concept, showng that the wall frcton model leads to tunable slp n the desred way. The applcablty of the wall frcton boundary condton n practcal smulatons has yet to be explored. A concernng study of electro-osmotc flow n plane channels and a comparson wth DPD smulatons s currently underway. 92

113 6 Reduced symmetres n lattce Boltzmann models In ths chapter, an attempt s made to tackle the ssue of lattce Boltzmann boundary condtons n a conceptually novel way. As ponted out n the prevous sectons, most of the exstng boundary condtons can only accomplsh second-order accuracy f the localty of the method s abandoned. The respectve nter- and extrapolaton schemes can be rather complcated and requre a mnmum number of nodes, whch can lead to severe dffcultes when the mplementaton has to be parallelzed for executon on modern hgh-performance computng clusters. Moreover, the conservaton laws are not drectly bult nto these rules and, n partcular, strct local mass conservaton s not guaranteed n a number of schemes. One reason for the defcences s the heurstc nature of these boundary condtons,.e., they are based on reflecton rules for the populatons that are constructed on rather ad-hoc assumptons. Whle ths leads to the desred macroscopc boundary condtons, the systematc connecton to conservaton laws and symmetres key features of the lattce Boltzmann method n the bulk s somewhat weakened. From the vewpont of a sound understandng of the foundatons of the method ths s clearly an unsatsfactory stuaton. For ths reason, we have explored another approach for boundary condtons whch s qute dfferent from the exstng ones. It s based on a lattce model that explctly takes nto account the reduced symmetry and respects the local conservaton laws. The man ams of nvestgatng the model are the followng: We attempt to develop a completely local boundary condton that allows straghtforward and easy mplementaton n a parallel computng envronment. The conservaton laws shall be bult nto the rules locally, and the collsons of flud partcles wth each other and wth the boundary shall be modeled consstently nto a lattce Boltzmann collson operator at the boundary. We ntend to get a better understandng of boundary condtons wth respect to the symmetry propertes of the underlyng lattce structure. By treatng the boundary explctly n terms of a reduced symmetry model, we hope to be able to clarfy the mplcatons of the broken symmetry at the boundary on the lattce Boltzmann dynamcs. Ths s partcularly nterestng n regard to moment systems, where self-consstent boundary condtons are mpossble to provde for the moments [48, 170, 171]. In the next secton, we wll explan the basc deas of the new approach and ntroduce the concept of reduced symmetry. Then we wll treat the lattce sums and nvarant tensors for the reduced symmetry. In secton 6.3, we ntroduce several varants to construct the equlbrum dstrbuton n the reduced symmetry. Ths wll be used n the followng subsectons to devse boundary condtons for the reduced symmetry. Fnally, we sketch some attempts to carry out a Chapman-Enskog expanson at the boundary n secton

114 6 Reduced symmetres n lattce Boltzmann models sold phase boundary flud phase Fgure 6.1: Illustraton of local boundary condtons. The dotted lnks pont nto the sold and must not be populated. They are forbdden lnks that are to be excluded from the model. 6.1 Boundares and reduced symmetry The fundamental objectve of the boundary condton s to allow for a completely local update of the lattce Boltzmann varables, that s, only nformaton that s avalable on the local lattce ste s used n the boundary scheme. For ths purpose, t stands to reason to use a node based representaton of the sold object where the surface s located drectly on the lattce stes. For smplcty and as a frst startng pont, we consder the case of a straght wall boundary. Curved boundares could easly be represented by specfyng the surface normal ndvdually on every boundary node. The representaton of a straght wall s schematcally depcted n fgure 6.1. The central dea for the update scheme at the boundary s that t should have a smlar structure as the bulk update n terms of streamng and collsons. The fgure shows, however, that some of the lnks on the boundary node pont nto the sold. If there were populatons streamng on those lnks, they would gve rse to a flud current nto the sold. Consequently, we have to requre that the lnks pontng nto the sold are not populated at the begnnng of the streamng phase. Vce versa, there are lnks that pont nto the flud and whch are undefned after the streamng phase because ther populatons would come from sold nodes. These populatons have to be computed durng the collson phase of the boundary scheme such that ther value can be used n the next streamng step. In addton, we have the tangental lnks whose populatons stream along the surface. They can be propagated n the usual way, but they can also be modfed n the collson phase of the boundary rule. In order to specfy the stuaton formally, let us consder the pre-streamng stuaton where the lnks pontng nto the sold must not be populated f (r B, t) = 0, f c n < 0. (6.1) These lnks are so-called forbdden lnks, and hence they are excluded from the lattce model. For a plane wall on a D3Q19 lattce, there s a total of fve forbdden lnks. The remanng set of 14 lnks depcted n fgure 6.2 forms a subset of the D3Q19 model and obvously has a reduced symmetry compared to the full set. We wll refer to the model wth 14 lnks as the reduced D3Q19 model. The dea s now to systematcally construct a lattce Boltzmann scheme based on the lnks of the reduced D3Q19 model. The constructon s done on the same footng as n the bulk, that s, we assume that the update of the populatons conssts of a streamng step and a 94

115 6.2 Lattce sums and nvarant tensors n the reduced symmetry n Fgure 6.2: Lattce model wth reduced symmetry. The forbdden lnks are excluded from the model. For the D3Q19 model, there are 14 remanng populatons. The smaller set of velocty drectons has a reduced symmetry that has to be taken nto account n a formally consstent fashon. collson step. As n the bulk, the collson step s local and has to satsfy the conservaton laws for mass and momentum n the drectons parallel to the plane. We assume further that the collsons can be mplemented as a lnear relaxaton towards a local equlbrum dstrbuton, the form of whch has yet to be determned. In analogy to the MRT model, the relaxaton s most generally formulated n mode space where the bass has to be constructed from the reduced symmetry D3Q19 vectors. To sum up, the central questons arsng for the constructon of the reduced D3Q19 model are: What s the equlbrum dstrbuton for the reduced symmetry model? How can the reduced mode space be constructed? What s an approprate collson operator for the reduced symmetry? Before we attempt to answer these questons, we frst have to look at the lattce sums and nvarant tensors n the reduced symmetry. 6.2 Lattce sums and nvarant tensors n the reduced symmetry As n the bulk case, we wll assume that the equlbrum dstrbuton can be wrtten as a polynomal expanson n the veloctes. The expanson coeffcents are determned by requrng that the conservaton laws and symmetry propertes are satsfed. The explct calculaton nvolves the moments of the equlbrum dstrbuton, for whch the lattce sums have to be evaluated. For the bulk case, ths was done n appendx B.2. To prepare the dervaton of the equlbrum dstrbuton n the reduced symmetry, we frst dscuss the lattce sums and ther relaton to nvarant tensors n the reduced symmetry. We focus on the reduced D3Q19 model for a plane wall boundary here. Generalzatons to other models or geometrcally complex boundares are tedous and the exstence of a soluton s not guaranteed for arbtrary lattces. 95

116 6 Reduced symmetres n lattce Boltzmann models Lattce sums for a locally plane boundary Consder a plane wall whose normal ponts n the postve z-drecton n = (0, 0, 1) T. The velocty vectors of the correspondng reduced D3Q19 model are the columns of the matrx C B = (6.2) Any of these vectors satsfes n c 0. The lattce sums for the reduced symmetry thus come as T α (n) 1...α n = w c α1... c αn. (6.3) n c 0 The n-th lattce sum s a tensor of rank n whch s nvarant under any symmetry transformaton for the cubc lattce and leaves the wall normal nvarant. The most general form of such tensors can be constructed from proper combnatons of δ αβ and n α plus potental cubc ansotropes of the respectve rank, e.g., δ αβγδ. In essence, the tensors can be obtaned from the respectve expressons n the bulk by replacng all occurrences of δ αβ by δ αβ, δ αβ = δ αβ n α n β, (6.4) and accordngly for the ansotropes n hgher rank tensors. Ths procedure yelds the followng general expressons for the lattce sums up to ffth rank T (0) = K, T (1) α = An α, T (2) αβ = B 1 δ αβ + B 2 n α n β, T (3) αβγ = C 1 (n α δ βγ + n β δ αγ + n γ δ αβ ) + C 2 n α n β n γ, T (4) αβγδ = D 1 (δ αβ δ γδ + δ αγ δ βδ + δ αδ δ βγ ) + D 2 (n α n β δ γδ + n α n γ δ βδ + n α n δ δ βγ + n β n γ δ αδ + n β n δ δ αγ + n γ n δ δ αβ ) + D 3 δ αβγδ + D 4 n α n β n γ n δ, T (5) αβγδɛ = E 1 [n α δ βγδɛ + n β δ αγδɛ + n γ δ αβδɛ + n δ δ αβγɛ + n ɛ δ αβγδ ] + E 2 [n α (δ βγ δ δɛ + δ βδ δ γɛ + δ βɛ δ γδ ) + n β (δ αγ δ δɛ + δ αδ δ γɛ + δ αɛ δ γδ ) + n γ (δ αβ δ δɛ + δ αδ δ βɛ + δ αɛ δ βδ ) + n δ (δ αβ δ γɛ + δ αγ δ βɛ + δ αɛ δ βγ ) + n ɛ (δ αβ δ γδ + δ αγ δ βδ + δ αδ δ βγ )] + E 3 [n α n β n γ δ δɛ + n α n β n δ δ γɛ + n α n γ n δ δ βɛ + n β n γ n δ δ αɛ + n α n β n ɛ δ γδ + n α n γ n ɛ δ βδ + n β n γ n ɛ δ αδ + n α n δ n ɛ δ βγ + n β n δ n ɛ δ αγ + n γ n δ n ɛ δ αβ ] + E 4 n α n β n γ n δ n ɛ. (6.5) 96

117 6.2 Lattce sums and nvarant tensors n the reduced symmetry weght ĉ 2 n ĉ ĉ w (0, 0, 0) T w (1, 0, 0) T, ( 1, 0, 0) T, (0, 1, 0) T, (0, 1, 0) T w (1, 1, 0) T, ( 1, 1, 0) T, (1, 1, 0) T, ( 1, 1, 0) T w (0, 0, 1) T w (1, 0, 1) T, ( 1, 0, 1) T, (0, 1, 1) T, (0, 1, 1) T Table 6.1: Weght factors for the equlbrum dstrbuton of the reduced D3Q19 model. There are fve ndependent weghts, accordng to the length of the lnks and the projecton on the boundary normal. The coeffcents n these expressons are related to the weghts w. Usng the vectors of the reduced D3Q19 model n equaton (6.3), we get the followng set of equatons w 0 + 4w 1 + 4w 2 + w 3 + 4w 4 = K (w 3 + 4w 4 ) a τ = A (2w 1 + 4w 2 + 2w 4 ) a2 τ 2 = B 1 (w 3 + 4w 4 ) a2 τ 2 = B 1 + B 2 (2w 4 ) a3 τ 3 = C 1 (w 3 + 4w 4 ) a3 τ 3 = 3C 1 + C 2 (2w 1 + 4w 2 + 2w 4 ) a4 τ 4 = 3D 1 + D 3 (w 3 + 4w 4 ) a4 τ 4 = 3D 1 + 6D 2 + D 3 + D 4 (4w 2 ) a4 τ 4 = D 1 (2w 4 ) a4 τ 4 = D 1 + D 2 (2w 4 ) a5 τ 5 = E 1 + 3E 2 0 = E 2 (6.6) (2w 4 ) a5 τ 5 = E 2 + E 3 (w 3 + 4w 4 ) a5 τ 5 = 5E E E 3 + E 4, where the weghts w 0 to w 4 are assgned to the dfferent classes of velocty vectors as lsted n table 6.1. The free parameters are the fve weghts w whch allow to tune at most fve of the coeffcents of the lattce sums. The remanng coeffcents are related by varous 97

118 6 Reduced symmetres n lattce Boltzmann models degeneraces, e.g., A(a/τ) 3 = (B 1 +B 2 )(a/τ) 2 = (3C 1 +C 2 )(a/τ) = 3D 1 +6D 2 +D 3 +D 4. In the D3Q19 model, we had already seen n the bulk that ĉ α = ĉ 3 α. In the reduced D3Q19 model we have the addtonal degeneracy ĉ z = ĉ 2 z = ĉ 3 z =... whch leads to subtle dependences of the moments. The choce of a complete set of condtons used to determne the weghts w s therefore a crucal step n the constructon of a reduced symmetry lattce Boltzmann model. 6.3 Approaches for constructng the equlbrum dstrbuton In ths secton we treat the equlbrum dstrbuton for the reduced symmetry model. In the course of ths work, we have worked out several approaches to devse the equlbrum dstrbuton. Although the statstcal mechancs based dervaton s the most consstent approach and supersedes the pror ones, the other approaches shall be outlned here as well because n comparng the dfferent methods the mportant features can be hghlghted. In fact, the earler approaches have nspred and eventually led to the development of the statstcal mechancs of the lattce Boltzmann model Method 0: Drect ansatz for the equlbrum dstrbuton In the bulk, the equlbrum dstrbuton s a polynomal expanson n the veloctes up to second order. It can be systematcally derved from the contnuous Maxwell-Boltzmann dstrbuton n terms of an expanson n Hermte tensor polynomals. However, the polynomal expanson can also just be vewed as an ansatz whch s justfed wthn the Chapman- Enskog expanson. At the boundary, we have to generalze the ansatz to the reduced symmetry approprately. Snce the normal vector n of the boundary s an addtonal nvarant, the equlbrum dstrbuton must ncorporate respectve terms u n up to order O(u 2 ). Ths suggest to modfy the bulk ansatz n the followng way: [ f B,eq (ρ, u) = w ρ 1 + Ãu c + B(u c ) 2 + Cu 2 + Du n + Ẽ(u n)2 + F n c + G(u n)(u c ) + H(u n)(n c ) + Ĩ(u u)(n c ) + J(u c )(n c ) + K(n ] c )(n c ). (6.7) The coeffcents have to be determned such that the moment relatons for the hydrodynamc varables hold, ρ = n c >0 f B,eq, j = n c >0 f B,eq c, Π = n c >0 f B,eq c c. (6.8) 98

119 6.3 Approaches for constructng the equlbrum dstrbuton Wth the help of the lattce sums n the reduced symmetry the mass and momentum equatons yeld the followng set of equatons K + A F + (B 1 + B 2 ) K = 1, K C + AĨ + B B 1 = 0, K D + AÃ + A H + (B 1 + B 2 ) J = 0, KẼ + A G + B 2 B = 0, B 1 Ã + C 1 J = 1, B 1 G + 2C1 B = 0, A + (B 1 + B 2 ) F + (3C 1 + C 2 ) K = 0, A D + (B 1 + B 2 )(Ã + H) + (3C 1 + C 2 ) J = 1, A C + (B 1 + B 2 )Ĩ + C B 1 = 0, AẼ + (B 1 + B 2) G + (2C 1 + C 2 ) B = 0. (6.9) These are ten equatons whch s not enough to specfy the soluton for the coeffcents Ã, B, C,... unquely. In prncple, we can get further condtons by requrng that the pressure tensor has the form of the Euler stress. However, some of the addtonal equatons mpose certan condtons on the coeffcents of the lattce sums at the same tme. Ths renders the whole procedure rather complcated and t s not straghtforward how to solve the equaton system. It wll become clear below that n fact we are tryng to determne a soluton for all orders up to O(u 2 ) smultaneously here. Ths s because n the drect ansatz for the equlbrum dstrbuton we can not say more about the orgn of the terms appearng n (6.7). Therefore, the lne of the calculatons s a bt unsystematc for ths approach and we wll not follow t further here. Instead, we develop a much more consstent formalsm to derve the equlbrum dstrbuton from varatonal prncples that can be systematcally generalzed to the reduced symmetry at the boundary Method I: Dervaton from quadratc functonal Although the a-pror ansatz for the equlbrum dstrbuton can be justfed by asymptotc analyss, t s more desrable to devse the equlbrum dstrbuton systematcally from general prncples. In partcular, we have looked for varatonal prncples that have proven successful n many branches of theoretcal physcs. The dea s that the equlbrum dstrbuton can be nterpreted as a statonary state that can be found by mnmzng or maxmzng an approprate functonal, possbly subject to constrants. Ths reasonng s ndeed smlar to the dea of the entropc lattce Boltzmann method [28 32, 91], where a dscrete lattce analogon to the Boltzmann H-functon s sought n order to comply wth a H-theorem. In that case, the equlbrum s not only a statonary soluton of the functonal but also an attractor of the dynamcs. Here, we do not consder the attractng property. We merely search a functonal where wth a fxed-pont soluton that has the usual form of the bulk equlbrum dstrbuton. The am s then to generalze the functonal to the reduced symmetry. 99

120 6 Reduced symmetres n lattce Boltzmann models The bulk equlbrum dstrbuton s a polynomal expanson n the veloctes up to second order. In terms of the moments, ths corresponds to a lnear combnaton of the zeroth, frst and second moment. Snce the equlbrum form of the moments s known, we can ncorporate them as constrants va Lagrange multplers. A general form of the functonal can then be wrtten down as F({f }) = ( F (f ) + λ ρ ρ ) f ( + λ j,α j α ) ( f c α + λ Π,αβ Π eq αβ ) (6.10) f c α c β, where F (f ) denotes a functonal of a sngle populaton f. It should be remarked that the ncluson of the stress tensor as a constrant s not dctated by the local conservaton laws, and conventonally only the real collsonal nvarants, mass and momentum, are ncluded as constrants [32]. In ths sense, the ncluson of the stress tensor yelds an over-constraned equlbrum whch makes t more complcated to guarantee postve entropy producton [29]. That s, however, not our prmary concern here. On the other hand, the stress constrant has some techncal advantages because t automatcally yelds Gallean nvarance of the equlbrum dstrbuton. Furthermore, t s much easer to obtan the correct form of the equlbrum stress tensor n ths way than by fndng a functonal that mples the correct form. It remans to fnd an approprate form for the F. A set of lnear equatons for the Lagrange multplers s only obtaned, f the functonal s quadratc n the populatons. The smplest choce would be F = f 2, but to allow for a relatve weghtng of the dfferent neghbor shells we choose F = f 2 /(2w ) where the weghts w are model dependent. The complete functonal then s F({f }) = ( f 2 + λ ρ ρ ) f 2w ( + λ j,α j α ) ( f c α + λ Π,αβ Π eq αβ The equlbrum dstrbuton s the statonary dstrbuton of ths functonal, f c α c β ). (6.11) f eq = w (λ ρ + λ j,α c α + λ Π,αβ c α c β ), (6.12) where the Lagrange multplers can be determned from the constrant equatons. The calculaton for the bulk case s carred out n appendx B.4. The result s the famlar expresson for the lattce Boltzmann equlbrum dstrbuton n the bulk, whch justfes the above choce of the functonal. The next step s to generalze the varatonal formalsm to the reduced symmetry. At the boundary, the outgong populatons are enforced to be zero. Formally, ths can be ncluded 100

121 6.3 Approaches for constructng the equlbrum dstrbuton as another set of constrants of the form F B = F λ f n c <0 = F χ λ f, (6.13) where the ndcators χ are ntroduced to formally sum over all { 1 f n c < 0 χ = 0 else. (6.14) The boundary equlbrum then comes as f B,eq = w (λ ρ + λ j,α c α + λ Π,αβ c α c β + χ λ ), (6.15) whch dffers from the bulk expresson only by the last term n the brackets. Ths can be exploted to rearrange terms n the constrant equatons whch can be wrtten n the form ρ j α χ λ w = w (λ ρ + λ j,α c α + λ Π,αβ c α c β ), (6.16a) χ λ w c α = w c α (λ ρ + λ j,β c β + λ Π,βγ c β c γ ) (6.16b) Π eq αβ χ λ w c α c β = w c α c β (λ ρ + λ j,γ c γ + λ Π,γδ c γ c δ ), (6.16c) χ f B,eq = 0. (6.16d) The rght-hand sdes of the frst three equatons have the same structure as n the bulk case, hence the Lagrange multplers for the boundary can be obtaned by replacng the moments ρ, j α and Π αβ by the respectve left-hand sdes n (6.16a) to (6.16c). The resultng expressons yeld a lnear equaton system of the form χ A k λ k = χ f eq. (6.17) k In prncple, ths could be solved numercally n order to obtan the λ k. However, a soluton only exsts f the matrx A k s not sngular, whch not necessarly needs to be the case. Ths s because some of the constrant equatons may be degenerate. For example, n the D3Q19 model we have Π zz = j z a τ, (6.18) because ĉ 2 z = ĉ z for a plane wall n the xy-plane. That s, the zz-momentum flux s lnearly dependent on the momentum n z-drecton and therefore j z and Π zz must not be constraned smultaneously. A possble way to avod the dependences between the moments s to reduce the number of stress components that are constraned. Specfcally, the constrants on Π zz, Π xz and Π yz 101

122 6 Reduced symmetres n lattce Boltzmann models are omtted for the plane wall. Ths corresponds to not fxng any normal stress component and s smlar n sprt to the concept of reacton forces n mechancs, whch are ntally consdered unknown and have to be found wth the soluton. The functonal wth the reduced set of constrants reads F B = ( f 2 + λ ρ ρ ) ( f + λ j,α j α ) f c x λ Π,xy f c x c y 2w + λ Π,xx ( ρc 2 s + j2 x ρ f c x c x ) + λ Π,yy ( ρc 2 s f c y c y ) χ λ f (6.19) and yelds an equaton system that can now be solved for the unknown Lagrange multplers. Before the explct calculatons are carred out, some remarks are n order to clarfy the meanng of the weghts w. For symmetry reasons, not every velocty drecton can have an ndvdual weght. In the bulk, the w have to be constant wthn a neghbor shell. On the boundary, where the symmetry s reduced, there s more freedom to vary the weghts: wthn a neghbor shell, the velocty drectons can have dfferent weghts accordng to ther projecton on the boundary normal n c. In the D3Q19 model, for example, there are three dfferent weghts n the bulk whle fve dfferent weghts are possble at the boundary, cf. table 6.1. One could n prncple attempt to just use the bulk weghts for the boundary as well. Ths leads to some problems, however, as wll be dscussed below. Let us now sketch the calculaton of the statonary dstrbuton of the functonal F B. Mnmzaton of (6.19) leads to f B,eq = w ( λρ + λ j,α c α + λ Π,xx c 2 x + λ Π,yy c 2 y + λ Π,xy c x c y + χ λ ), (6.20) and the constrant equatons are ρ = n c 0 j α = n c 0 Π eq xx = n c 0 Π eq yy = n c 0 Π eq xy = n c 0 f B,eq, f B,eq c α, f B,eq c x c x, f B,eq c y c y, f B,eq c x c y. (6.21) The evaluaton of the rght-hand sdes of the constrants nvolves the lattce sums for the 102

123 6.3 Approaches for constructng the equlbrum dstrbuton reduced symmetry. Wth the formulas devsed n secton 6.2 we get the equaton system ρ = Kλ ρ + An α λ j,α + B 1 (λ Π,xx + λ Π,yy ), j α = An α λ ρ + (B 1 δ αβ + B 2 n α n β ) λ j,β + C 1 n α (λ Π,xx + λ Π,yy ), Π eq xx = B 1 λ ρ + C 1 n α λ j,α + (3D 1 + D 3 )λ Π,xx + D 1 λ Π,yy, Π eq yy = B 1 λ ρ + C 1 n α λ j,α + D 1 λ Π,xx + (3D 1 + D 3 )λ Π,yy, Π eq xy = D 1 λ Π,xy. Solvng ths system nvolves some lengthy algebra whch fnally yelds where λ ρ = (B 1 + B 2 )(4D 1 + D 3 ) 2C1 2 R + AC 1 B 1 (B 1 + B 2 ) λ j,x = B 1 1 j x, λ j,y = B 1 1 j y, R λ j.z = 2B 1C 1 A(4D 1 + D 3 ) R + AB 1 KC 1 (Π eq xx + Π eq R yy), λ Π,xx = AC 1 B 1 (B 1 + B 2 ) R ρ + 2B 1C 1 A(4D 1 + D 3 ) j z R (Π eq xx + Π eq yy), ρ + K(4D 1 + D 3 ) 2B1 2 j z R ρ + AB 1 KC 1 j z R + K(B 1 + B 2 ) A 2 (Π eq xx + Π eq 2R yy) + λ Π,yy = AC 1 B 1 (B 1 + B 2 ) R λ Π,xy = D 1 1 Π xy, ρ + AB 1 KC 1 j z R + K(B 1 + B 2 ) A 2 (Π eq xx + Π eq 2R yy) 1 2(4D 1 + D 3 ) (Πeq xx Π eq yy), 1 2(4D 1 + D 3 ) (Πeq xx Π eq yy), (6.22) (6.23) R = K(4D 1 + D 3 )(B 1 + B 2 A2 K ) 2B2 1(B 1 + B 2 ) + 2C 1 (2AB 1 KC 1 ). (6.24) The coeffcents K, A, B 1,... are related to the weghts w. In prncple, any set of weghts that s compatble wth the reduced symmetry at the boundary can be chosen. The systematc determnaton of an approprate set of weghts shall be postponed to the next subsecton. Nevertheless, we can already make some observatons. The coeffcent K s the sum of all the weghts, whch represents a normalzaton and hence the choce K = 1 suggests tself. Furthermore, the coeffcent D 3 s the prefactor of the cubc ansotropy n the fourth-rank lattce sum. Snce we stll have rotatonal nvarance n the xy-plane, the cubc ansotropy should vansh,.e., D 3 = 0. If the bulk weghts were used as a frst guess for the boundary, we would get K = 5/6 and D 3 = 1/18, n contradcton to what has just been proposed. Therefore, the bulk weghts are napproprate at the boundary. 103

124 6 Reduced symmetres n lattce Boltzmann models At ths pont, a gudelne for choosng further condtons for the coeffcents and the weghts s not obvous. Ths s because there s no clear physcal nterpretaton of the quadratc functonal, whch merely forms a formal startng pont for dervng the equlbrum dstrbuton from a varatonal formalsm. The form of the functonal s only justfed a-posteror by the correct outcome for the equlbrum dstrbuton n the bulk. It s much less straghtforward to construct the equlbrum dstrbuton at the boundary because the connecton to physcal prncples remans somewhat obscure. The stuaton can be mproved by usng the underlyng statstcal mechancs of the lattce Boltzmann method to apply t at the boundary. Ths wll be done n the next subsecton, whch supersedes the prevous results Method II: Statstcal mechancs based dervaton In chapter 3, we have dscussed the statstcal mechancs of the lattce Boltzmann method. It provdes a systematc dervaton of the equlbrum dstrbuton from the entropy of the generalzed lattce gas model. The statstcal consderatons reman vald for the reduced symmetry lattce model, hence the entropy at the boundary keeps the same form as n the bulk but wth the sum runnng only over allowed lnks S({ν }) = (ν ln ν ν ν ln ν + ν ). (6.25) n c >0 The constrants of mass and momentum conservaton are agan taken nto account va Lagrange multplers. The functonal to maxmze at the boundary s then ( ) ( ) S({ν }, χ, λ) = S({ν }) + χ ν ρ + λ ν c j. (6.26) µ µ n c >0 n c >0 The formal soluton for the allowed lnks has stll the same form as n the bulk, f expressed n terms of the Lagrange multplers ν B,eq = ν exp (χ + λc ). (6.27) We can agan expand the equlbrum dstrbuton about the reference state where the flud s at rest. Furthermore, we assume that the mass densty of the boundary node s dstrbuted among the velocty drectons accordng to f B,eq (ρ, j = 0) = w ρ, (6.28) whch fxes the normalzaton of the weghts to w = K = 1. (6.29) n c >0 The reduced symmetry s now taken nto account by a larger set of possble weghts w for the dfferent velocty drectons, cf. secton 6.2 and table 6.1. After expandng the 104

125 6.3 Approaches for constructng the equlbrum dstrbuton equlbrum dstrbuton up to second order, we have f B,eq [ = w ρ 1 + ɛχ B,(1) + ɛλ B,(1) c ɛ2 ( χ B,(1) + λ B,(1) c ) 2 + ɛ 2 χ B,(2) + ɛ 2 λ B,(2) c ]. (6.30) Ths looks stll the same as the bulk expresson, but the reduced symmetry s contaned n the Lagrange multplers. On the frst order, the mass and momentum constrants yeld ρ = ρ ( ) 1 + χ B,(1) + Aλ B,(1) z [ ] j α = ρ An α (1 + χ B,(1) ) + (B 1 δ αβ + B 2 n α n β )λ B,(1) (6.31) β where we have used the lattce sums for the reduced symmetry. The soluton for the frstorder Lagrange multplers s χ B,(1) = A P (u(1) z A), λ B,(1) x λ B,(1) y λ B,(1) z = u(1) x, B 1 = u(1) y, B 1 = u(1) z A, P (6.32) where Proceedng to the second order, we get P = B 1 + B 2 A 2. (6.33) 0 = 1 2 (χb,(1) ) 2 + Aχ B,(1) λ B,(1) z + χ B,(2) + Aλ B,(2) z, 0 = A 2 (χb,(1) ) 2 + (B 1 + B 2 )χ B,(1) λ B,(1) z (B 1δ αβ + B 2 n α n β )λ B,(1) α λ B,(1) β + Aχ B,(2) + (B 1 + B 2 )λ B,(2) z (6.34) + C 1 2 λb,(1) α λ B,(1) 0 = B 1 χ B,(1) λ B,(1) x,y α + 2C 1 + C B 1 λ B,(2) x,y (λ B,(1) z ) 2, + 2C 1 λ B,(1) x,y λ B,(1) z, 105

126 6 Reduced symmetres n lattce Boltzmann models and the second-order Lagrange multplers thus are χ B,(2) = ( AZ P P ) (1) ( (u z A) 2 A(AB1 C 1 ) + B ) (1) 1 (u x ) 2 + (u (1) 2 P 2 2P 2 λ B,(2) x = AB 1 C 1 u (1) x (u (1) z A), B 1 B 1 P λ B,(2) y = AB 1 C 1 u (1) y (u (1) z A), B 1 B 1 P λ B,(2) z = AB 1 C 1 (u (1) x ) 2 + (u (1) y ) 2 Z 2P B1 2 P (u (1) z A) 2 P 2, B 2 1 y ) 2, (6.35) where Z = A A(B 1 + B 2 ) (3C 1 + C 2 ). (6.36) These expressons are generally vald for a locally plane boundary whose normal ponts n the postve z-drecton. In the next step, the coeffcents of the lattce sums have to be determned. Ths wll be done by lookng at the pressure tensor. We have already seen n the prevous secton that t s useful to start wth the tangental components Π xx, Π yy and Π xy, for whch we have ρc 2 s + ρu 2 x = ρ [B 1 (1 + ɛχ B(1) + 12 ) (ɛχb,(1) ) 2 + ɛ 2 χ B,(2) +C 1 ( ɛλ B,(1) z + D 1 2 ɛ2 λ B,(1) γ + ɛ 2 χ B,(1) λ B,(1) z λ B,(1) γ + 2D 1 + D 3 2 = ρ [ B 1 + (C 1 AB 1 )ɛλ B,(1) z ( + AB 1(AB 1 C 1 ) B2 1 2P 2 + ( AB 1(AB 1 C 1 ) B2 1 2P 2 ( B1 A B 1AZ P ) + ɛ 2 λ B,(2) z ( ɛλ B,(1) x ) 2 + D C 1(AB 1 C 1 ) + 3D 1 + D 3 B C 1(AB 1 C 1 ) + D 1 B 1 2 B 1P 2 AC 1 C 1Z P + D 1 + D 2 2 ] ( ) ɛλ B,(1) 2 z ) (ɛλ ) B,(1) 2 y ) (ɛλ ) B,(1) 2 x ) ] (ɛλ ) B,(1) 2 z, (6.37) 106

127 6.3 Approaches for constructng the equlbrum dstrbuton and ρc 2 s + ρu 2 y = ρ [B 1 (1 + ɛχ B(1) + 12 ) (ɛχb,(1) ) 2 + ɛ 2 χ B,(2) +C 1 ( ɛλ B,(1) z + D 1 2 ɛ2 λ B,(1) γ + ɛ 2 χ B,(1) λ B,(1) z λ B,(1) γ + 2D 1 + D 3 2 = ρ [ B 1 + (C 1 AB 1 )ɛλ B,(1) z ( + AB 1(AB 1 C 1 ) B2 1 2P 2 + ( AB 1(AB 1 C 1 ) B2 1 2P 2 ( B1 A AZB 1 P ) + ɛ 2 λ B,(2) z ( ɛλ B,(1) x + C 1(AB 1 C 1 ) + D 1 B 1 2 ) 2 + D 2 2 ] ( ) ɛλ B,(1) 2 z ) (ɛλ ) B,(1) 2 x + C 1(AB 1 C 1 ) + 3D 1 + D 3 B 1 2 B 1P 2 AC 1 ZC 1 P + D 1 + D 2 2 ) (ɛλ ) B,(1) 2 y ) ] (ɛλ ) B,(1) 2 z, (6.38) ρu x u y = ρ [ D 1 ɛ 2 λ B,(1) x ] λ B,(1) y. (6.39) Ths almost drectly leads to B 1 = c 2 s, C 1 = AB 1, D 1 = c 4 s, D 2 = B 1 B 2, D 3 = 0. (6.40) Weghts w for the reduced D3Q19 model Snce n the D3Q19 model we have, cf. equaton (6.6) B 1 a 2 τ 2 = 3D 1 + D 3, (6.41) the speed of sound s fxed to c 2 s = 1 a 2, whch s compatble wth the bulk speed of sound. 3 τ 2 If we plug n the results obtaned so far n the equaton system (6.6), we get the followng 107

128 6 Reduced symmetres n lattce Boltzmann models equatons for the coeffcents K = 1, B 1 = c 2 s, B 2 = A a τ c2 s, C 1 = Ac 2 s, C 2 = A a2 τ 2 3Ac2 s = 0, D 1 = c 4 s, D 2 = Ac 2 a s τ c4 s, D 3 = 0, D 4 = A a3 τ + 2 3c4 s 6Ac 2 a s τ. (6.42) Ths system shows that we are left wth only one free parameter A, for whch we have to fnd an addtonal condton. Up to here, only local consttutve equatons for the moments were consdered. At the boundary, however, t s also mportant to take nto account the fluxes between the bulk and the boundary. In partcular, t has to be assured that the boundary does not lead to an accumulaton of mass on the wall, that s, the mass fluxes between the bulk and the surface have to balance. In equlbrum, the amount of mass that streams to the wall has to be compensated by an equal amount of mass that leaves the wall. We wll refer to ths condton as the bulk balance condton. In a pure bulk system, the balance s guaranteed because the equlbrum dstrbuton s the same on all lattce stes. On the boundary, where the equlbrum s dfferent from the bulk, we have to satsfy bulk balance as an addtonal condton. In a flud at rest, the mass streamng from the bulk to the surface s gven by ρ n = f eq (u = 0). (6.43) n c <0 Vce versa, the mass streamng from the surface to the bulk s gven by ρ out = f B,eq. (6.44) The condton of bulk balance then reads n c >0 ρ n = ρ out. (6.45) The evaluaton of the sum of the bulk equlbrum can be carred out explctly for u = 0 and yelds ρ n = f eq = 1 ρ. (6.46) 6 n c <0 108

129 6.3 Approaches for constructng the equlbrum dstrbuton On the boundary, we consequently have to requre that ρ out = ρ/6. We get ρ out = n c >0 whch delvers the desred condton on the coeffcent A w ρ = Aρ τ a, (6.47) A = 1 a 6 τ. (6.48) Now we have enough equatons, namely fve, to determne the fve free weghts of the reduced D3Q19 model. From the equaton system (6.6) we fnally obtan w 0 = 7 18, w 1 = 1 12, w 2 = 1 36, w 3 = 1 18, w 4 = (6.49) Ths s a nontrval result whch was prevously not known Boundary equlbrum for reduced D3Q19 model The boundary equlbrum dstrbuton for the reduced D3Q19 model s now completely specfed. The most nstructve way to wrte t down s n terms of the frst order Lagrange multplers λ B,(1) α. Collectng all ntermedate results we fnally arrve at { [ 1 f B,eq = ρw 1 + ɛ [c α An α ] λ B,(1) α + ɛ 2 2 c αc β B 1 2 δ αβ An α c β Z P n αn β n γ c γ ( AZ + P P 2 + A2 2 + B ) ] } 1 n α n β λ B,(1) α λ B,(1) β 2 (6.50) where the Lagrange multplers can be generally expressed as ( λ B,(1) α = u(1) (1) β u β δ αβ + A ) u(1) β n α n β. (6.51) B 1 P B 1 In the expresson (6.50) we can convenently dentfy the terms that are already present n the bulk equlbrum dstrbuton, and the new terms that are due to the reduced symmetry at the boundary. The x- and y-components of the Lagrange multplers are stll the same as n the bulk, whereas the z-component has to be modfed at the boundary. We can observe that there s a shft A for the normal component u z of the flud velocty n the Lagrange multpler. Ths means that the reference state around whch we expand actually has a nonvanshng velocty component normal to the boundary of u z = 1/6 (a/τ). The reason for ths s the bulk balance condton, whch requres that there s a mass flux from the surface that balances the ncomng mass flux from the bulk. Ths s mportant for the mplementaton 109

130 6 Reduced symmetres n lattce Boltzmann models { equlbrum dstrbuton f B,eq = ρw 1 + ɛ [c α An α ] λ B,(1) α + ɛ [ 2 1 c 2 αc β B 1 2 δ αβ An α c β Z n P αn β n γ c γ ( ) ] } AZ + P + A2 + B 1 P n α n β λ B,(1) α λ B,(1) β Lagrange multplers χ B,(1) A P (u(1) λx B,(1) u (1) x B 1 λy B,(1) u (1) y B 1 λz B,(1) u (1) z A P χ B,(2) (u(1) x ) 2 +(u (1) y ) 2 2B 1 λx B,(2) 0 λy B,(2) 0 z A) + ( AZ P P 2 λ B,(2) z Z P 3 (u (1) z A) 2 ) (u (1) z A) 2 P 2 coeffcents 1 a A 6 τ 1 a B τ 2 B 2 1 a 2 6 τ 2 1 a C τ 3 C a D τ 4 D 2 1 a 4 18 τ 4 D a D τ 4 5 a P 2 36 τ 2 5 a Z τ 3 weghts w w w w w Table 6.2: Summary of the results for the equlbrum dstrbuton of the reduced D3Q19 model. of the algorthm, because t means that we have to plug n the value of the reflected velocty nto the equlbrum dstrbuton, and not the desred hydrodynamc velocty of the boundary. The hydrodynamc velocty s gven by the arthmetc mean of the pre- and post-reflecton veloctes at the boundary node. Ths s n accordance wth the defnton (4.65) of the hydrodynamc velocty n the presence of external forces. To conclude the dervaton of the equlbrum dstrbuton for the reduced D3Q19 model, we summarze the results for the relevant quanttes agan n table MRT model for the reduced symmetry The equlbrum dstrbuton s already enough to mplement a smple BGK model at the boundary. However, as stated above, the collson processes at the surface are more complex and hence t s desrable to have a more flexble collson operator. In the followng, we therefore attempt to construct a MRT-lke collson operator for the boundary. In analogy to the bulk, ths requres frst to construct the bass for mode space n whch the collson operator s assumed to be dagonal. The constructon of the moment bass follows the same reasonng as n the bulk: Startng wth the conserved moments, we systematcally orthogonalze polynomals of the ĉ by the Gram-Schmdt procedure. The orthogonalty relaton w e k e l = b k δ kl (6.52) 110

131 6.4 MRT model for the reduced symmetry has the same form as n the bulk, but the weghts w are replaced wth the weghts for the boundary equlbrum. The same holds for the weghts used to calculate the normalzaton factors b k = w e 2 k. (6.53) The mass mode stll corresponds to the bass vector Next, we orthogonalze the polynomals ĉ α and get e 0 = 1. (6.54) e 1 = ĉ x, e 2 = ĉ y, e 3 = ĉ z 1 6. (6.55) The last of these bass vectors shows that ĉ z s not automatcally orthogonal to the mass mode any more, whch s a result of the mssng party n the reduced symmetry. The form of the bass vector s another hnt that the bulk balance condton for the mass flux s mportant at the boundary. To proceed to the next bass vectors, we orthogonalze quadratc polynomals. The reduced symmetry mples the degeneracy ĉ z = ĉ 2 z such that ĉ 2 z s not ndependent any more and must not be orthogonalzed. We thus get fve bass vectors from quadratc polynomals e 4 = ĉ 2 x + ĉ 2 y 2 3, e 5 = ĉ 2 x ĉ 2 y, e 6 = ĉ x ĉ y, e 7 = ĉ x (ĉ z 1 6 ), (6.56) e 8 = ĉ y (ĉ z 1 6 ). In analogy to the bulk, we can dentfy e 4 as a bulk-lke mode, and e 5 to e 8 as shear lke modes. Contnung the Gram-Schmdt procedure wth the hgher order polynomals and carefully sortng out any degeneraces, we obtan the remanng fve bass vectors e 9 = (ĉ 2 3 5ĉ2 z 7 5 )ĉ x, e 10 = (ĉ 2 3 5ĉ2 z 7 5 )ĉ y, e 11 = (ĉ 2 x + ĉ 2 y 2 3 )(ĉ z 1 12 ), e 12 = (ĉ 2 x ĉ 2 y)(ĉ z 1 4 ), e 13 = ĉ (ĉ2 x + ĉ 2 y) (ĉ2 x + ĉ 2 y)ĉ z 67 55ĉz (6.57) 111

132 6 Reduced symmetres n lattce Boltzmann models These are the knetc modes of the reduced D3Q19 model. From equaton (6.53) we get the normalzaton factors of the boundary bass vectors b = ( 1, 1 3, 1 3, 5 36, 4 9, 4 9, 1 9, 5 108, 5 108, 1 15, 1 15, , 1 12, 28 ). (6.58) 165 Wth these bass vectors, the populatons can be transformed nto the mode space of the boundary va the transformaton m = M B f, (6.59) where the transformaton matrx s obtaned from the bass vectors e k as M B = The relatons between the moments and the hydrodynamc varables at the boundary are ρ = m 0 j x = m 1 a/τ j y = m 2 a/τ j z = (m 3 + m 0 /6) a/τ Π xx = 1 ( m 4 + m ) 2 3 m 0 (a/τ) 2 Π yy = 1 ( m 4 m ) 2 3 m 0 (a/τ) 2 (6.60) Π xy = m 6 (a/τ) 2 Π xz = (m 7 + m 1 /6) (a/τ) 2 Π yz = (m 8 + m 2 /6) (a/τ) 2. To proceed further, the projectons of the mode bass on the equlbrum dstrbuton have to evaluated. Wth the expresson (6.50) for the equlbrum dstrbuton and the mode bass e k, 112

133 6.4 MRT model for the reduced symmetry we get m eq 0 = ρ m eq 1 = ρu x (a/τ) 1 m eq 2 = ρu y (a/τ) 1 ( m eq 3 = ρ u z 1 ) a (a/τ) 1 6 τ m eq 4 = ρ(u 2 x + u 2 y) (a/τ) 2 m eq 5 = ρ(u 2 x u 2 y) (a/τ) 2 m eq 6 = ρu x u y (a/τ) 2 ( m eq 7 = ρu x u z 1 ) a (a/τ) 2 6 τ ( m eq 8 = ρu y u z 1 ) a (a/τ) 2 6 τ m eq 9 = 0 m eq 10 = 0 m eq 11 = 0 m eq 12 = 0 m eq 13 = 0. (6.61) Lke n the bulk, the knetc modes have no projecton on the equlbrum dstrbuton because of the ncluson of the weghts w n (6.52). We further note the specal role of the z-component of the flow velocty, whch always appears n ts orthogonalzed form u z 1/6 (a/τ) n the above expressons. Ths observaton has a connecton to the bulk balance condton, as can be seen from the relaton ρ out a τ = j z. (6.62) It was already noted above that n order to satsfy bulk balance, the reflected momentum has to be used for j z and that t has to be j z = 1/6 ρ (a/τ) for a flud at rest. Here we observe now, that the correspondng moment m 3 vanshes n ths case, that s, for a flud at rest the momentum-lke moments m 1 to m 3 are consstently all zero. Moreover, there s a relaton between the momentum modes and the frst order Lagrange multplers ρc 2 sλ B,(1) x ρc 2 sλ B,(1) y ρp λ B,(1) z = m 1 (a/τ), = m 2 (a/τ), = m 3 (a/τ). (6.63) These relatons are smlar to the ones obtaned n the bulk, except for the z-component whch encodes the effects of the reduced symmetry. It s a strkng feature of our formalsm, that the reduced symmetry can be systematcally ncorporated and automatcally yelds consstent expressons, f the addtonal bulk balance condton s taken nto account. 113

134 6 Reduced symmetres n lattce Boltzmann models Collson operator at the boundary It remans to construct a lnear collson operator at the boundary. As n the bulk, we assume that the operator s dagonal n mode space such that the collsons descrbe a lnear relaxaton of the non-equlbrum part of moments m neq k = (1 + λ k )m neq k. (6.64) The choce of the egenvalues λ k has to be guded by the symmetry propertes of the model. Lookng at the bass vectors, we can dvde the modes at the boundary nto the conserved modes m 0 to m 3, the bulk-lke mode m 4, the shear-lke modes m 5 to m 8 and the knetc modes m 9 to m 13. We choose to relax the bulk and shear modes wth the same egenvalues as n the bulk m neq 4 = (1 + λ b )m neq 4, m neq k = (1 + λ s )m neq k 5 k 8. (6.65) For the knetc modes, we use λ k = 1 whch corresponds to project them out durng the collson step. It should be emphaszed that ths s only a frst guess. Strctly speakng, the choce of the egenvalues has to be justfed by an asymptotc analyss. Ths turns out to be rather complcated and eventually mpossble, such that we have to stck to the guess at ths pont. It wll turn out, unfortunately, that ths choce of the MRT collson operator leads to undesrable artfacts n smulatons of smple Poseulle flow. 6.5 Results for the reduced symmetry model We have tested the reduced symmetry model for boundary condtons n the case of a Poseulle flow n a plane channel, smlar to the case n secton 5.3. The channel has a wdth of L eff = 20a n the z-drecton. Perodc boundary condtons were appled n the x- and y-drecton. The flud has a densty of ρ = 1.0 and a knematc vscosty of ν = 3.0, both measured n lattce unts. The flow s drven by a volumetrc force f ext = The boundary nodes are located on the frst and last lattce layer wth respect to the z-coordnate. On these nodes, boundary condtons are appled n the followng way. After the streamng step, we transform the ncomng populatons nto moment space accordng to equaton (6.59). Note that some of the entres n the matrx M B change ther sgn dependng on whether we treat the top or bottom wall. After the transformaton, we can apply the necessary operatons n moment space: the z-component of the flow velocty s reversed, whle the x- and y- components are decreased by a frctonal force n order to generate a tunable slp boundary condton, cf. secton 5.3. The shear and bulk-lke modes are then relaxed towards ther equlbrum value whch s evaluated wth the ntermedate flow velocty. The egenvalues λ b = λ s = λ are calculated from the vscosty ν. Havng appled these modfcatons to the moments, they are transformed back to the outgong populatons wth the respectve nverse matrx for the space spanned by the reflected drectons. Ths completes the boundary collson process. It should be noted that all operatons requre local nformaton only, and that 114

135 6.5 Results for the reduced symmetry model Local boundary condtons for reduced symmetry model Velocty profles for Poseulle flow 1 f ext =0.01 a grd =1.0 ζ=1.0 ζ=2.0 ζ=5.0 analytcal solutons v ζ=1.0 v max z / L Fgure 6.3: Smulaton results wth the local boundary condton n the reduced symmetry. The profles devate from the analytcal soluton and show a knk next to the boundary. The latter s caused by an unphyscal momentum transfer due to artfacts stemmng from the reduced symmetry. 115

136 6 Reduced symmetres n lattce Boltzmann models only two parameters are nvolved, namely the vscosty ν (a bulk property) and the frcton coeffcent ζ (a surface property). The results for the velocty profles n the channel are shown n fgure 6.5 for three dfferent frcton coeffcents between 1.0 and 5.0. The plots show a parabolc profle but, once more, a knk s present at the flud node next to the boundary. Although vsbly the curves wth the knk look smlar to those observed n secton 5.3, the knk has a dfferent orgn here, as wll be brought out below. To clarfy that the knk s not an artfact related to the force applcaton, we have tred dfferent varants of applyng the force drectly to the populatons rather than n mode space. The results for the velocty profles reman vsbly unchanged. By expermentng wth the egenvalues of the collson operator, the modes m 7 and m 8 can be dentfed as beng relevant for the knk. Ths stands to reason because these modes are connected to the normal stress components Π xz and Π yz whch must have a jump, f the velocty profles has a knk. However, the quanttatve mpact of those modes can not be systematcally assessed so far. In prncple t s requred to conduct an asymptotc analyss or Chapman-Enskog expanson of the boundary condton for the reduced symmetry model. Some attempts to do ths are presented below, but they eventually remaned unsuccessful. Nevertheless, the orgn of the knk can be sem-systematcally dentfed by nvestgatng how the Stokes equaton emerges from the lattce Boltzmann equaton The Stokes equaton and the reduced symmetry lattce Boltzmann model One of the central deas of the boundary condton n the reduced symmetry s that the correct macroscopc boundary values emerge from the dynamcs on the lattce level, n the same way that the lattce Boltzmann equaton generates Naver-Stokes behavor n the bulk. Instead of rgorously provng ths by a Chapman-Enskog expanson, a slghtly more heurstc argument shall be followed here. The lattce Boltzmann equaton conssts of the collson and the streamng phase. Durng the latter, populatons propagate on the lattce and transport nformaton about the hydrodynamc felds from one lattce ste to another. The nterplay of the collsons and the streamng eventually leads to the macroscopc flow profle. To study ths n more detal, one can look at the x-component of the momentum densty. We frst consder the behavor n the bulk. Usng the defnton (2.30) and the lattce Boltzmann equaton (2.39), we have j x (r, t) = f (r, t)c x = f (r, t)c x + f (r, t)c x + f (r, t)c x n c >0 = n c >0 + n c =0 f (r τc, t τ)c x + n c <0 f (r τc, t τ)c x. n c =0 n c <0 f (r τc, t τ)c x (6.66a) 116

137 6.5 Results for the reduced symmetry model If we assume that the flow s n the statonary state and that the profle s nvarant n the x- and y-drecton, the momentum densty s only dependent on the z-coordnate and we can wrte j x (z) = f (z a)c x + f (z)c x + f (z + a)c x. (6.67) c z =+1 c z =0 c z = 1 The second-order Chapman-Enskog result for ths specfc flow s gven by, cf. (4.42) f ρu x = w ρ + w c x + ρu2 x + τ 2c 2 s c 2 s ( 1 λ g + 1 2c 2 s ) ( τ E (2) xx + ρτ ( 1 λ s + 1 2c 2 s ρu 2 x r z τη s c s 2 u x r 2 z ) ux r z E (2) xz ) E (3) xxz + w τ c c 2 x f ext. s (6.68) If we plug ths expresson nto (6.67) and explot that sums over odd polynomals n c x vansh, we obtan f (z a)c x = 1 6 ρu x(z a) + ρa ( ) 1 ux λ c z =+1 s r z + τ z a 6 f ext, f (z)c x = 2 3 ρu x(z) + 2τ 3 f ext, (6.69) c z =0 f (z + a)c x = 1 6 ρu x(z + a) ρa ( ) 1 ux λ s r z + τ z+a 6 f ext, c z = 1 where the bulk weghts w of the D3Q19 model were explctly used. The hydrodynamc flow velocty u x can be Taylor expanded around z, and the same goes for the gradent. If only second dervatve terms are kept we get u x (z ± a) = u x (z) ± a u x r z + a2 z 2 u x r z = u x z±a r z ± a 2 u x z z r 2 z 2 u x r 2 z ±..., z Puttng all results together yelds a dfferental equaton for the flow velocty profle ρu x (z) = 1 6 ρu x(z a) + ρa ( ) 1 ux λ s r z + τ z a 6 f ext (6.70) ρu x(z) + 2τ 3 f ext ρu x(z + a) ρa ( ) 1 ux λ s r z + τ z+a 6 f ext = ρu x (z) + ρa2 2 u x 6 rz 2 ρa2 1 + λ s 2 u x z 3 λ s rz 2 + τf ext z = ρu x (z) ρc2 sτ λ s 2 u x 2 λ s rz 2 + τf ext. z (6.71) 117

138 6 Reduced symmetres n lattce Boltzmann models Π (z 1,t 1) xz jx(z,t) Π (z+1,t 1) xz j x j x j + x Fgure 6.4: Schematc llustraton of the emergence of the Stokes equaton on the lattce. The full dfference equaton arses from a combnaton of terms that carry nformaton about the flow profle from both sdes of any gven lattce ste. At the boundary, the nformaton from one sde s mssng, thus the Stokes equaton s not reproduced. Fnally we arrve at η s 2 u x r 2 z = f ext, (6.72) whch s nothng but the Stokes equaton. The mportant pont to note here s that the Stokes equaton emerges from the combnaton of terms streamng to the node at z from both sdes, as llustrated n fgure 6.4. If ether one of the terms wth argument z a or z + a s omtted, the resultng equaton s dfferent from the Stokes equaton. In other words, the Stokes equaton s nherently a bulk equaton. Therefore t s not necessarly fulflled at a boundary node, where nformaton streams n from only one sde. As a consequence, t can not be expected that the correct Stokes profle s generated by the boundary condton n the reduced symmetry. To be more precse, ths s because the nformaton about how the profle extrapolates beyond the boundary s mssng. The local boundary scheme hence has to mpose the desred velocty at the boundary, n the very sense of a Drchlet boundary condton. In ths aspect, local boundary condtons dffer crucally from the smple lnkbased reflecton schemes. Followng ths lne of thnkng further, t becomes clear that n fact all hydrodynamcally relevant moments have to be prescrbed correctly at the boundary. Besdes the mass and momentum, ths ncludes the gradents of velocty feld,.e., the nonequlbrum part of the stress tensor. The need to fx Π neq was already ponted out by Lätt [101, 140] n regard to varous closure relatons. In the reduced symmetry, however, the attempt to fx the normal stresses at the boundary reveals another complcacy A closer look on the collsons n the reduced mode space The tangental stresses Π xz and Π yz are related to the modes m 7 and m 8 n the reduced symmetry, cf. expressons (6.60). Consequently, n order to prescrbe the tangental stresses we have to fnd the correct values to fx m 7 and m 8. However, a second look reveals another subtlety here. In the reduced symmetry mode space, Π xz does not only depend on m 7 but also on m 1 j x. Lookng at ths relaton n another way, ths means that the stress component Π xz has projectons on the populatons n the parallel subspace,.e., c = (±1, 0, 0). 118

139 6.6 Revsed boundary model Ths leads to a problem when relaxng the modes durng the collson process because there s an nherent couplng of degrees of freedom that one would rather relax ndependently. For example, f we were to relax the stress component Π xz, ths would mply a relaxaton of m 7 and m 1 smultaneously accordng to equaton (6.60). Ths s clearly not feasble as t would volate momentum conservaton. Conversely, the relaxaton of the mode m 7 can not be nterpreted n terms of the hydrodynamc varables, snce the reduced symmetry mxes m 7 wth the conserved mode m 1. It remans unclear, how the relaxatons n the reduced mode space can be made consstent wth the hydrodynamc felds n the bulk. We beleve that more lght could be shed on ths ssue by asymptotc analyss. Snce our attempts have remaned unfrutful so far, ths consttutes an unresolved problem. 6.6 Revsed boundary model Although the above analyss may delver some doubts about the feasblty of the reduced symmetry approach, we shall show that the model can stll be tamed to our needs, at least for the smple Poseulle flow. For ths purpose, we wll take the varous ponts of the analyss as hnts for addtonal modfcatons of the boundary condton n the reduced symmetry. The frst essental observaton s that none of the lsted ssues s related to the equlbrum dstrbuton, whch we can therefore keep n ts current form. We have to modfy the boundary scheme to mpose the Drchlet condtons on the hydrodynamc varables. Instead of modfyng the momentum densty by a frcton force, we prescrbe the desred slp velocty at the boundary by settng the momentum modes to the correspondng values. Fxng of the nonequlbrum stresses s yet not straghtforward because of the structure of the mode space. Therefore we resort to an ndrect way usng a bounce-back scheme for the non-equlbrum parts of the populatons f +B,neq = f B,neq, (6.73) where c = c. Ths does not alter the conserved moments and t assures that the nonequlbrum stress before and after the reflecton s the same f B,neq c c = f B,neq c c. (6.74) n c 0 n c 0 Furthermore, the relaxaton at the wall has to be modfed as the MRT-lke model can not be mplemented consstently. In ths context, we resort to the smpler BGK model wth a sngle relaxaton rate λ. Ths has the advantage that t does not rely on the structure of the mode space and t can drectly be formulated n terms of the populatons f B,neq = λf B,neq. (6.75) The dfference compared to the MRT-lke relaxatons used above s that the knetc modes are not projected out nstantaneously, but they are relaxed at the same rate as the hydrodynamc modes. 119

140 6 Reduced symmetres n lattce Boltzmann models In summary, we are now back to a boundary closure scheme that expresses the outgong populatons n terms of the ncomng populatons at the boundary node ( ) f B = f B,eq λ f B f B,eq, (6.76) where the equlbrum dstrbutons are calculated by the formula for the reduced symmetry. The modfed boundary condton wth bounce-back of non-equlbrum parts and BGK-lke relaxaton was once more appled to the case of Poseulle flow n a plane channel. The setup of the system was as descrbed above and the same parameters for the flud were used. Instead of varyng the frcton coeffcent, several values for the slp velocty at the boundary were mposed, rangng from stck boundares to u slp = 0.4. The results are plotted n fgure 6.5. There s no knk vsble n the curves and the match wth the analytcal soluton s vsbly perfect. The amount of slppage can be controlled by the prescrbed value of the slp velocty u slp as desred. Ths shows that the modfcatons to the boundary condton can ndeed avod the artfacts observed for the earler attempts. The reason s that the boundary scheme s now capable of controllng the momentum flux at the boundary node, an essental prerequste to reproduce the Stokes equaton at the hydrodynamc level. However, t should be noted that ths s only possble by avodng the mode space of the reduced symmetry model and thereby the advantages of MRT-lke models are sacrfced. Moreover, a rgorous justfcaton n terms of a Chapman-Enskog expanson s stll mssng. On the pro-sde, t should be ponted out that the results for the Poseulle flow show that reduced symmetry lattce Boltzmann models are a feasble approach to treat rgd boundares. The equlbrum dstrbuton for such models can be derved systematcally n the statstcal mechancs framework and can be readly appled to straght boundares. The resultng boundary scheme satsfes the conservaton laws and s completely local, whch makes t very useful for parallel mplementatons. It has yet to be shown whether the novel boundary condton yelds acceptable results for more complcated flows than plane Poseulle flow. 120

141 6.6 Revsed boundary model Reduced symmetry wth bounce back of nonequlbrum & BGK Velocty profles for Poseulle flow f ext = 0.01 a grd = 1.0 Ma ~ 10-2 Re ~ 3-6 u stck u max u slp = 0.0 (stck) u slp = 0.1 u slp = 0.2 u slp = 0.4 analytcal solutons z / L Fgure 6.5: Smulaton results for the reduced symmetry model wth bounce-back of non-equlbrum parts and a BGK collson operator. No knk s vsble n the profles and an accurate match to the analytcal soluton s acheved. 121

142 6 Reduced symmetres n lattce Boltzmann models 6.7 Attempts for a Chapman-Enskog expanson at the boundary The constructon of the boundary condton n the reduced symmetry s manly based on heurstc arguments so far. A rgorous proof of ts correctness and an assessment of the order of accuracy requres a more systematc analyss. The Chapman-Enskog expanson s the standard technque to derve the hydrodynamc equatons n the bulk case. In ths secton, we attempt to apply the Chapman-Enskog expanson to the reduced symmetry model. Although we have not completely succeeded n fndng a consstent way of dong ths, t s worthwhle to sketch some peces of the puzzle n order to pont out some of the dffcultes nvolved wth the boundary Second and thrd moment n the reduced symmetry Some of the components of the pressure tensor were not explctly evaluated durng the constructon of the equlbrum dstrbuton. Snce they are needed n the Chapman-Enskog expanson, we evaluate them now. From equatons (6.60) and (6.61) we mmedately get Π B,eq xx Π B,eq yy = ρc 2 s + ρu 2 x = ρc 2 s + ρu 2 y Π B,eq xy = ρu x u y, Π B,eq xz = ρu x u z, Π B,eq yz = ρu y u z, Π B,eq zz = ρu z a τ. (6.77) The last expresson makes the degeneracy of the zz-stress and the z-momentum explct. On the lowest order n the Chapman-Enskog expanson, t s also necessary to evaluate the thrd moment Φ of the equlbrum dstrbuton. At the boundary we get Φ B,eq αβγ = f B,eq c α c β c γ n 0 [ ( = ρ 1 + ɛχ B,(1) + 1 ) 2 (ɛχb,(1) ) 2 + ɛ 2 χ B,(2) w c α c β c γ n 0 ( ) + ɛλ B,(1) δ + ɛ 2 χ B,(1) λ B,(1) δ + ɛ 2 λ B,(2) δ ɛ2 λ B,(1) δ λ B,(1) ɛ ] w c α c β c γ c δ c ɛ. n 0 n 0 w c α c β c γ c δ (6.78) 122

143 6.7 Attempts for a Chapman-Enskog expanson at the boundary Usng the lattce tensors for the reduced symmetry and the explct values for the reduced D3Q19 model we can determne the components of the thrd moment as Φ B,eq xxx = ρu x a 2 τ 2, Φ B,eq yyy = ρu y a 2 τ 2, Φ B,eq a 2 zzz = ρu z τ, 2 Φ B,eq xxy = ρc2 s a A τ Φ B,eq xyy = ρc2 s a A τ ( = ρc 2 s u z + A c 2 s Φ B,eq xxz Φ B,eq yyz Φ B,eq xzz Φ B,eq yzz = ρc 2 s Φ B,eq xyz = 0. = ρu x u z a τ, = ρu y u z a τ, ( a ) u y τ u yu z, ( a ) u x τ u xu z, u 2 x A 2c 2 s u 2 y ( u z A u 2 2c 2 x + A u 2 s c 2 y s ), ), (6.79) These expressons are clearly very dfferent from the bulk expresson (4.21). If they were plugged nto the frst order equaton (4.9b), the dervatve Π eq αβ t + Φ eq αβγ (6.80) 1 r γ would yeld addtonal spurous terms n the normal components. However, t s doubtful whether the Chapman-Enskog expanson drectly carres over from the bulk to the boundary. Ths wll be dscussed n the next subsecton Ansotropc Chapman-Enskog expanson: a potental way out? The pvotal pont n the Chapman-Enskog expanson s the scale-separated verson of the lattce Boltzmann equaton (4.8) whch can then be solved stepwse by nsertng the soluton on lower orders nto the equatons on hgher orders. In the dervaton of the scale-separated equatons two expanson steps are nvolved. The frst step s the ntroducton of the scalng parameter ɛ and the expanson of the populatons n powers of ɛ. The second step s the ntroducton of the coarse-graned varables r 1, t 1, t 2 and the Taylor expanson of the populatons wrtten n terms of the coarse-graned varables. In the phlosophy of our boundary condton, we have assumed that the noton of an equlbrum dstrbuton s stll vald and 123

144 6 Reduced symmetres n lattce Boltzmann models that the populatons relax towards that equlbrum durng the collsons. In ths sense, the expanson n powers of ɛ remans vald f B = f B,eq + ɛf B,(1) + ɛ 2 f B,(2) (6.81) However, snce the equlbrum dstrbuton on the boundary s dfferent from the equlbrum dstrbuton n the bulk, there s a spatal dscontnuty n the populatons at the boundary. Therefore the Taylor expanson around a boundary node s not applcable. We have to avod expandng the populatons n the drecton of the dscontnuty. In the case of a straght boundary wth normal n the z-drecton ths means that the Taylor expanson can only be executed n the x- and y-drecton but not n the z-drecton. In ths way, we may try to expand the populatons as f (r 1x + ɛτc x, r 1y + ɛτc y, r z + τc z, t 1 + ɛτ, t 2 + ɛτ 2 ) = f (r 1x, r 1y, r z + τc z, t 1, t 2 ) ( ) + ɛτ + c x + c y f (r 1x, r 1y, r z + τc z, t 1, t 2 ) (6.82) t 1 r 1x r 1y [ + ɛ 2 τ + τ ( ) ] 2 + c x + c y f (r 1x, r 1y, r z + τc z, t 1, t 2 ). t 2 2 t 1 r 1x For drectons wth c z = 0 ths reduces to the same expresson as n the bulk, thus the ansotropy nduced by the boundary does not affect the parallel drectons. In the normal drecton, however, there s an addtonal non-local dfference term nvolvng populatons at r z and r z + τc z whch contrbutes at all orders of the expanson. In partcular the zerothorder equaton becomes f (0) (r 1x, r 1y, r z + τc z, t 1, t 2 ) f (0) (r 1x, r 1y, r z, t 1, t 2 ) = (0) (r 1x, r 1y, r z, t 1, t 2 ). (6.83) That s, the collson operator has non-vanshng contrbutons at the zeroth order due to the dscontnuty at the boundary. The dfference-term on the left hand sde leads to dffcultes when one tres to construct the moment equatons because the sum f (0) (r 1x, r 1y, r z + τc z, t 1, t 2 ) runs through dfferent locatons,.e., t s not a local moment on a lattce ste. It s therefore unclear how macroscopc equatons should be derved wthn the ansotropc Chapman-Enskog expanson. There s actually an even more severe problem, whch has to do wth the reflectons at the wall. Let us look at an outgong lnk c n > 0 at a boundary node. The post-streamng populaton of ths lnk s by constructon zero r 1y f (r B, t) = 0, n c > 0, (6.84) because no populatons stream from the sold nto the boundary node. Consequently we have (r B, t) = f (r B, t) f eq (r B, t) = f eq (r B, t). (6.85) f neq 124

145 6.7 Attempts for a Chapman-Enskog expanson at the boundary Ths leads to a contradcton because accordng to the expanson (6.81), the left-hand sde s of the order O(ɛ 1 ) whle the rght-hand sde s of the order O(ɛ 0 ). Hence, the suggested ansotropc Chapman-Enskog expanson s faced wth an nconsstency. The source are the reflectons at the wall, whch nstantaneously change the populatons by an O(1) term. Another way to look at ths s to notce that the subspace of the full lattce model that s allowed to be populated s changed by the reflectons,.e., the reduced symmetry before and after the reflectons s not the same but s changed by a party transformaton. In ths sense, the equlbrum dstrbuton s not an nvarant under reflectons. We have not succeeded n ncorporatng ths nto a consstent Chapman-Enskog procedure, whch we therefore have to leave as an open ssue. In concluson, t can be sad that the reduced symmetry seems so far naccessble to the Chapman-Enskog expanson. The reasons are the dscontnuty of the populatons at the wall and the reduced symmetry, whch preclude a straghtforward expanson of the lattce Boltzmann equaton. A rgorous asymptotc analyss turns thus out to be hghly complcated. Probably t even requres to develop more sophstcated mathematcal technques whch are beyond the scope of ths work [cf. Refs. 92, 97 and the references theren]. 125

146 126

147 7 Conclusons, dscusson and outlook Ths thess presents aspects of method development for smulatons of complex fluds and mcroflows wthn the lattce Boltzmann model. The lattce Boltzmann model was chosen because t has proven to be successful n smulatng hydrodynamc nteractons n soft matter systems. It has an establshed foundaton n terms of knetc theory and can be systematcally lnked to the hydrodynamc level by methods from asymptotc analyss. The propertes of the solvent, such as densty and vscosty, can be drectly adjusted by the respectve smulaton parameters. In addton, lattce Boltzmann s very flexble and extensble, for example, t s possble to couple coarse-graned representatons of collods and polymers to the hydrodynamc flow feld. The bascs of the lattce Boltzmann method and ts asymptotc analyss were ntroduced n chapters 2 and 4. One major topc of ths work was the treatment of thermal fluctuatons n the lattce Boltzmann model, whch was dscussed n chapter 3. Prevously, there was a lack of clarty about the correct way of addng thermal fluctuatons to the lattce Boltzmann varables. In partcular, t was debated whether the knetc modes have to be thermalzed even though they yeld no contrbuton to the hydrodynamc equatons. To tackle ths problem, the generalzed lattce gas model was developed and subsequently used to derve the equlbrum dstrbuton of the lattce Boltzmann populatons wthn a maxmum entropy formalsm. In dong so, the connecton to the underlyng statstcal mechancs was restored. If the collson operator ncludes thermal fluctuatons, t can be vewed as a Monte-Carlo process. The statstcal perspectve then mples that detaled-balance has to be satsfed, whch leads to thermal fluctuatons that are smultaneously consstent wth statstcal mechancs and hydrodynamcs. The crucal result s that detaled-balance s only satsfed f the knetc modes are thermalzed accordngly. Ths ultmately clarfes the role of fluctuatons n the lattce Boltzmann model, a queston whch, untl now, was not answered satsfactorly. The statstcal mechancs framework of the lattce Boltzmann equaton bears potental for future extensons. One partcular case where the generalzed lattce gas model wll be useful s the development of advanced multphase models. The dffculty n ths context s to couple the thermodynamcs of the nterface between the phases to the hydrodynamcs of the fluds n a consstent way. The exstng multphase models are constructed n a rather heurstc fashon. We beleve that the statstcal mechancs approach helps to mprove these algorthms systematcally. In chapters 5 and 6, boundary condtons for the lattce Boltzmann model were dscussed. Boundary condtons become partcularly mportant n mcrofludc devces where the surface to volume rato s large. In ths case, the flud flow can be strongly affected by fludsold nteractons at the boundary. The classcal hydrodynamc stck boundary condton s 127

148 7 Conclusons, dscusson and outlook only vald up to Knudsen numbers of the order of 0.1. Beyond ths value, apparent slp s observed at the boundary and the no-slp condton s no longer approprate. Most work on boundary condtons n the lattce Boltzmann model has focused on the no-slp boundary condton. In ths work, a boundary condton for tunable slp was developed that s based on the dea of a mesoscopc frcton force at the boundary. The model has the advantage that t only ntroduces a sngle addtonal parameter for the boundary condton, whch can be used to tune the amount of slppage. Moreover, the concept of a frcton force at the boundary can be drectly appled n other mesoscopc smulaton methods, such as dsspatve partcle dynamcs or mult-partcle collson dynamcs. In the lattce Boltzmann method, the detals of the mcroscopc mplementaton of the frcton force are hghly relevant. Ths was demonstrated by the comparson of the canoncal and the prmtve mplementaton n secton 5.3. Whle the canoncal force mplementaton was known to be superor n the bulk, t has to be dscarded at the boundary n favor of the prmtve force mplementaton. The reason was dentfed by an analytcal analyss of the flow profle revealng that the canoncal mplementaton s affected by spurous momentum terms that lead to a knk n the measured velocty profle. In contrast, the prmtve mplementaton produces smooth velocty profles that match the analytcal predcton. Ths shows that the wall frcton model can be used to model a slp velocty n mcroflows. The frcton parameter s related to the slp length, whch s a quantty that can be ether measured n experments or ftted to avalable data. The method can thus be used to study the mpact of the slp length on the behavor of mcroflows. So far, only Poseulle flow n a plane channel geometry has been smulated, whch served as a basc test case. Snce ths s a very specal stuaton, t does not say too much about the general behavor of the boundary condton. For example, the accuracy of the method n curved geometres remans to be nvestgated. Also patterned surfaces wth locally varyng boundary frcton are nterestng and partcularly relevant n regard to generalzatons of the boundary condton to the recently ntroduced concept of tensoral slp [172]. Furthermore, t would be very nterestng to compare the lattce Boltzmann mplementaton of the wall frcton model to ts counterpart n dsspatve partcle dynamcs. Wth regard to the Naver slp boundary condton n hydrodynamcs, several other open questons reman to be nvestgated. Recent expermental results suggest that the lnear consttutve equaton for the slp velocty has to be replaced by a second-order boundary condton above a crtcal Knudsen number of 0.3. Such a nonlnear dependence can probably not be modeled accurately wthn the wall frcton model. It needs to be nvestgated how far the valdty of the wall frcton model extends. Another pont concerns thermal effects n mcroflows. It s known that, besdes velocty slp, flows n mcrochannels show a temperature jump at the wall. Moreover, the vscous dsspaton at low Reynolds number can lead to consderable heat generaton n the flud. It s therefore questonable whether the sothermal assumpton s stll vald n mcroflows. On the hydrodynamc level, the Naver- Stokes-Fourer descrpton mght be more approprate n ths case. As for the boundary condton, t s unclear whether the slp length s enough to capture the temperature jump or whether another parameter s needed. These questons could be tackled by usng a thermal lattce Boltzmann model whch s able to reproduce the heat transport equaton. Ths was beyond the scope of ths thess, but t forms an nterestng topc for future research. 128

149 7 Conclusons, dscusson and outlook In chapter 6, a new conceptual approach to mplementng lattce Boltzmann boundary condtons was developed. It s based on the noton of reduced symmetry, whch s a consequence of the fact that at the boundary some of the lnks of the lattce pont nto the sold and must have zero flud populatons. The remanng set of allowed lnks has a reduced symmetry compared to the bulk symmetry of the lattce. The dea n chapter 6 was to use ths set of lnks as the bass for a reduced symmetry lattce Boltzmann model, whch can be systematcally constructed wthn the framework developed earler n ths thess. Namely, the generalzed lattce gas model can be used to derve the equlbrum dstrbuton n the reduced symmetry. The boundary condton should then be ncorporated n the collson process n the reduced symmetry. It turned out, however, that t s not clear how to set up an approprate MRT model at the boundary because the mode space n the reduced symmetry s affected by degeneraces of the model that obscure the physcal nterpretaton of the moments. A back door was found by resortng to BGK-lke collsons and usng bounce-back of the non-equlbrum parts to fx the normal stresses at the boundary. The resultng boundary condton requres only local nformaton and can therefore be easly mplemented n a parallel smulaton envronment. However, a systematc analyss n terms of a Chapman- Enskog expanson proves to be hghly complcated, such that a rgorous assessment of the accuracy remans an open ssue. Nevertheless, the exploraton of reduced symmetry models sheds some lght on the effects of the small velocty set at the boundary. An nterestng queston for future work would be, for example, whether the degeneraces n mode space can be removed by usng a hgher number of dscrete veloctes that nclude more neghbor shells. In the course of ths work, t became clear that the techncal detals n the reduced symmetry are qute nvolved and requre careful treatment. Applcatons of the newly devsed methods therefore had to take a back seat and manly served as proof (and even more frequently as dsproof) of the concepts. There s no doubt that many more practcal smulatons are needed to further verfy the applcablty and accuracy of the methods. To round up ths dscusson, some routes that could be followed n the future shall be outlned. A feld that attracts non-ceasng nterest n the lattce Boltzmann communty s turbulence. Wth regard to soft matter the phenomenon of turbulent drag reducton has receved growng attenton. It s known that dlute polymer solutons show a substantally reduced drag when pumped through channels. One hypothess to explan ths effect s that there s an nterplay between polymer dynamcs and turbulence that damps vortex structures near the wall. The lattce Boltzmann method s an deal canddate to study ths effect n smulatons, because both the flud-polymer couplng and the boundary effects are readly ncluded. The queston s whether suffcent tme and length scales can be smulated wthn acceptable computng tme. More sophstcated multscale technques that try to cope wth the problem are currently under actve development. Another potentally very nterestng applcaton s to study electrophoress and electroosmoss n mcrochannels. The presence of charged partcles adds another level of complexty to a soft matter system as electroknetc effects come nto play. In a channel flled wth a polyelectrolyte, the walls usually become charged through the release of counterons 129

150 7 Conclusons, dscusson and outlook nto the soluton. The counterons form a charged Debye layer close to the wall, whch can be acted on by an external electrc feld thereby generatng a plug flow n the channel. The electro-osmotc moblty s determned by the chemco-physcal propertes of the surface, and a fnte slp length on the wall ncreases the moblty. Ths effect could be exploted to mprove the effcency of electro-osmotc pumps. So far, there are only a few smulatons of electro-osmotc flow whch mostly focus on the free-flow case. The nterplay of boundary condtons and electro-knetc effects has not been studed extensvely and forms an nterestng applcaton for the tunable slp boundary condton developed. Gong one step further n the method development, one could thnk about how polarzaton effects of the solvent can be ncorporated n the smulaton methods. A dfferent applcaton feld whch s more related to the fluctuatng lattce Boltzmann s formed by multphase flows. For example, the flow behavor of emulsons s hghly relevant for commercal applcatons n the cosmetc or food ndustres. A number of nterestng questons arses concernng the phase behavor under flow, e.g., mxng, demxng and varous knds of pattern formaton. Of partcular nterest s the dynamcs of lqud droplets n mcrodevces. Multphase flows are governed by the nterplay of nterface and surface effects. Concernng boundary condtons, the wettng propertes of the dfferent phases have to be taken nto account. The dffculty n a computer smulaton s that both thermodynamcs and hydrodynamcs have to be accurately reproduced. As to the lattce Boltzmann method, s has to be taken nto account that the speed of sound and the equaton of state n the dfferent phases may vary. Hence, a standard D3Q19 model s clearly not approprate for multphase flows. A varable speed of sound requres a larger set of dscrete veloctes. The generalzed lattce gas model s a promsng startng-pont for the development of new methods that treat the thermodynamcs of nterfaces n a consstent way. Apart from droplets, ths could be combned wth electroknetc methods to study the behavor of jets and sprays. Fnally, a future drecton that encompasses several of the above mentoned aspects s the smulaton of deformable partcle suspensons. Deformable partcles are omnpresent n bologcal systems, the most promnent examples beng vescles and cells. In contrast to rgd collods, vescles and cells make t necessary to treat ther ntrnsc deformaton mechancs. The couplng of the elastc energy of the deformable partcles and the vscous dsspaton n the flud can lead to vscoelastc behavor of the suspenson, whch n turn may gve rse to non-newtonan rheologcal effects. An mportant example where these effects are hghly relevant s blood, the flow behavor of whch s strongly determned by the deformablty of the red blood cells. The flow of blood through mcrovessels s subject to mgraton and aggregaton mechansms of the cells whch cause a shear-thnnng behavor of blood. The mcroscopc mechansms underlyng the blood flow are under vvd nvestgaton, and an effcent smulaton method could help to enhance the scentfc progress. For the treatment of deformable partcles wth the lattce Boltzmann method, a combnaton of force-couplng schemes and boundary condtons seems a promsng approach whch has so far not been explored much. 130

151 7 Conclusons, dscusson and outlook These examples are, of course, far from beng exhaustve. Nevertheless, they gve an mpresson of what mght be done wth lattce Boltzmann smulatons and why method development for computer smulatons contnues to be an nterestng and challengng research topc. Altogether, t can be sad that the road s paved wth exctng applcatons for the lattce Boltzmann method. 131

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153 A Implementaton of the lattce Boltzmann method In ths appendx, the lattce Boltzmann mplementaton n the ESPResSo software package [3, 4] s descrbed. The mplementaton features a D3Q19 model wth a MRT collson operator and thermal fluctuatons. Molecular dynamcs partcles can be coupled to the LB flud by the method of Ahlrchs and Dünweg [85]. Varous boundary condtons are mplemented as well. The algorthms are parallelzed usng the message passng nterface MPI [173]. A.1 Usage n ESPResSo In the ESPResSo software package, every smulaton s a sequence of TCL commands. The lattce Boltzmann method can be used wth the command lbflud whch has the followng syntax: lbflud (<lbvarable> <value>)+ The possble choces for <lbvarable> are lsted n table A.1. The parameter values <value> have to be gven n the MD unt system used throughout the TCL scrpt, whch typcally means Lennard-Jones unts. Although any varable can be set ndvdually, t s recommended to use the lbflud command n the form lbflud grd <grd> tau <tau> dens <dens> vsc <vscosty> where abbrevated forms of the varable names can be used. The couplng of MD partcles to the LB flud can be swtched on separately by LB varable grd tau densty vscosty frcton ext_force Descrpton lattce spacng a lattce Boltzmann tme step τ average densty ρ of the flud knematc vscosty ν of the flud frcton coeffcent ζ bare for the flud-partcle couplng external volumetrc force g drvng the flud flow Table A.1: ESPResSo varables for the varous parameters of the lattce Boltzmann model. They can be set wth the lbflud command. 133

154 A Implementaton of the lattce Boltzmann method lbflud frcton <frcton> where <frcton> s the bare frcton coeffcent for the Stokes drag force. The fluctuatng lattce Boltzmann model s mplemented as a thermostat. Thermal fluctuatons can be swtched on wth the command thermostat lb <temp> where <temp> s the desred temperature. The thermostat command sets the temperature for both the fluctuatons of the LB populatons and the random nose balancng the frcton force. Boundares are mplemented as constrants and can be defned wth the constrant command (see the ESPResSo documentaton for detals). So far, only wall constrants are supported by the lattce Boltzmann mplementaton. The type of lattce Boltzmann boundary condtons can be chosen wth the command lbboundares <type> <parameters>* The possble <type>s that have been mplemented so far comprse bounce_back, specular_reflecton, slp_reflecton and partal_slp. A.2 Internal unt conversons Snce ESPResSo s orgnally a Molecular Dynamcs package, the smulaton parameters set on the TCL level are typcally measured n Lennard-Jones unts. We have decded to keep ths conventon for the lattce Boltzmann parameters. Internally, however, lattce unts are used where a = 1, τ = 1 and m p = 1. The converson of the unt systems s done n the ntalzaton routnes. Ths has the advantage that all conversons are collected at a central place and need to be executed only when parameters change. The only pont where a unt converson has to be done on-the-fly s the calculaton of the flud-partcle couplng. The recpe for unt conversons s straghtforward: A quantty that has dmensons (mass) k (length) l (tme) m s transformed from MD to LB unts through dvson by m p k a l τ m where the respectve values of m p, a and τ n the MD unt system are used. The quanttes n lattce unts can then be used n the equatons where all occurrences of m p, a and τ have been dropped. The reverse converson goes along the same lnes. 134

155 A.3 The lattce Boltzmann kernel A.3 The lattce Boltzmann kernel In the basc lattce Boltzmann algorthm, the followng steps have to be performed n each tme-step update t t + 1: 1. Calculaton of the local moments m k from the actual populatons f accordng to m k = f ê k. 2. Calculaton of the equlbrum moments m eq k from the mass densty ρ = m 0 and the momentum denstes j x = m 1, j y = m 2 and j z = m Relaxaton of the moments towards ther equlbrum value m k = m eq k + (1 + λ k)(m k m eq k ). 4. Back-transformaton from the moments to the populatons accordng to f = w b 1 k m kê k. 5. Propagaton of the post-collsonal populatons f (r, t) along the lattce lnks ĉ to the new populatons f (r + ĉ, t + 1). The steps one to four consttute the collson phase, whle the ffth step s the streamng step. By countng the number of arthmetc operatons, t s found that the collson phase s the computatonally more ntensve part. The streamng step just shfts data n memory. On modern computng hardware, however, the last step turns out to be the most tme consumng because floatng pont operatons can be executed very fast whle memory access s lmted by bandwdth and latency. k A.3.1 Nave mplementaton The above algorthm s straghtforward to mplement. A crucal observaton s that the collson phase n steps one to four s completely local and can be executed ndependently for all cells whereas the propagaton step s non-local and replaces the old populatons wth the new ones. The smplest possblty for the mplementaton s the followng: One uses two separate sweeps for collsons and streamng, each consstng of three nested loops for the three spatal drectons. In the frst sweep, the collsons are executed and n the second sweep, the post-collsonal populatons are loaded and wrtten back to the shfted poston. The streamng step has to be done n a carefully chosen order to not overwrte any relevant data. At the boundares of the doman, an addtonal halo layer s needed to mplement perodc boundary condtons and for the parallelzaton, respectvely. Ths halo layer has to be flled wth the perodc mages of the populatons at the begnnng of the streamng step. The nave mplementaton could look lke n the followng code snppet. 135

156 A Implementaton of the lattce Boltzmann method for (x = 1; x <= n_grd[0]; x++) { for (y = 1; y <= n_grd[1]; y++) { for (z = 1; z <= n_grd[2]; z++ { /* collsons */ double moments[19], m_eq[19]; lb_calc_moments(x,y,z,f,moments); lb_calc_equlbrum(x,y,z,moments,m_eq); lb_relaxaton(x,y,z,moments,m_eq); lb_calc_f(x,y,z,f,moments); } } } lb_halo_update(); for (x = n_grd[0]; x >=0 ; x--) { for (y = n_grd[1]; y >= 0; y--) { for (z = m_grd[2]; z >= 0; z--) { /* streamng upwards n memory */ lb_stream_up(x,y,z,f); } } } for (x=1; x <= n_grd[0]+1; x++) { for (y=1; y <= n_grd[1]+1; y++) { for (z=1; z <= n_grd[2]+1; z++ { /* streamng downwards n memory */ lb_stream_down(x,y,z,f); } } } A.3.2 Combned collsons and streamng Closer nspecton of the memory access pattern n the nave mplementaton reveals that every populaton s read and wrtten twce, once n the collson phase and once n the streamng phase. On modern computng hardware, where memory access s a bottleneck, ths lmts the performance of the mplementaton. The stuaton can be mproved by reducng the number of data transfers from and to memory, whch s acheved by combnng collsons and streamng n one loop [174, 175]. Ths leads to ether the pull scheme or the push scheme, both of whch are llustrated n fgure A.1. In the pull scheme, the propagaton s realzed frst whle n the push scheme, the propagaton s the last step after the collsons. In order not to overwrte any relevant data, the new populatons are stored n a separate array and after each tme step the roles of the two arrays are nterchanged. An mplementaton wth combned collsons and streamng could look lke n the followng code snppet. lb_halo_update(); for (x = 1; x <= n_grd[0]; x++) { for (y = 1; y <= n_grd[1]; y++) { for (z = 1; z <= n_grd[2]; z++ { 136

157 A.3 The lattce Boltzmann kernel pull collsons collsons push Fgure A.1: Illustraton of the pull scheme (stream-collde) and the push scheme (collde-stream). In the pull scheme, the populatons are read from the neghbor stes for the collson process, whle n the push scheme, they are wrtten to the neghbor stes after the collsons. #fdef PUSH #endf #fdef PULL #endf double moments[19], m_eq[19]; lb_calc_moments(x,y,z,f[0],moments); lb_calc_equlbrum(x,y,z,moments,m_eq); lb_relaxaton(x,y,z,moments,m_eq); lb_calc_f_and_push(x,y,z,moments,f[1]); lb_pull_f_and_calc_moments(x,y,z,f[0],moments); lb_calc_equlbrum(x,y,z,moments,m_eq); lb_relaxaton(x,y,z,moments,m_eq); lb_calc_f(x,y,z,moments,f[[1]); } } } swap_ponters(f[0],f[1]); A.3.3 Data layout optmzaton In the combned mplementaton every populaton s read and wrtten only once n each tme step. Although ths seems already optmal, the performance can yet depend crtcally on the actual data layout n memory. Frst, t has to be noted that every array s a sequental data structure n memory, no matter how many logcal dmensons there are. In other words, any array s physcally onedmensonal n memory. Consequently, the three nested loops over the three spatal drectons bol down to a loop over one-dmensonal memory locatons. The crucal pont s how the logcal dmensons are mapped to the physcal memory locatons and n whch order the sequental data n memory s accessed. Ths s especally mportant on cache-based arch- 137

158 A Implementaton of the lattce Boltzmann method tectures where non-consecutve data access s connected to severe performance mparment. The reason s the way n that data s transferred from man memory to the caches. Cache memores are ntermedate memores that are located on the processor chp and provde hgh bandwdth and low latency at the expense of beng much smaller than the man memory. They are organzed n cache lnes that are always fetched from or stored to man memory as a whole contguous block of data. Snce not the whole man memory fts nto cache, cache lnes have to be replaced frequently durng the course of the program. The am s to use the cache lnes as effcent as possble, that s, accessng data should lead to a mnmum number of fetch and store operatons for the same cache lne. Optmally, all entres of a cache lne are used once t resdes n the cache memory. Ths can be acheved by organzng the data layout such that consecutvely used data s contguous n memory and can be grouped nto the same cache lne. The lattce Boltzmann populatons can be stored n a fve dmensonal array wth three ndces x, y and z for the spatal dmensons, one ndex for the velocty drecton, and one ndex t {0, 1} for the double bufferng. The mappng between the fve-dmensonal array and the lnear memory layout s defned by the order of the ndces. The ndex t s typcally the slowest ndex. 1 Intutvely, one would group together the 19 veloctes of each lattce ste whch s acheved when the ndex s the fastest. Ths yelds an array of structures, or the so-called collson optmzed data layout [174] whch s schematcally depcted n fgure A.2. Durng the collson phase, the populatons f 0:18 (x, y, z) have to be fetched from memory whch typcally nvolves two cache lnes of 16 doubles. The populatons f 0:12 (x, y, z+1) are then automatcally resdent n the cache and can be used n the next teraton. However, n the propagaton phase the populatons are stored to non-contguous locatons and, n the worst case, 19 cache lnes are stored of whch only one entry was modfed. Ths s a rather neffcent access pattern and, n addton, t s lkely to generate many cache msses, especally f the cache sze s small. The collson optmzed data layout s therefore sub-optmal on many hardware archtectures [174]. A better performance s acheved wth structures of arrays, or the so-called propagaton optmzed data layout (see fgure A.1). It groups the populatons for one velocty drecton together,.e., the ndex s the next slowest followng the ndex t. The ndex z s now the fastest and should be terated n the nnermost loop. As the populatons of a local lattce ste are not contguous n memory, the collson phase now requres 19 cache lnes to be loaded. The same holds for the propagaton phase, where also 19 cache lnes are accessed. Gven that the cache can keep at least 38 lnes resdent at the same tme, the cached data can be reused n successve teratons over z. In ths way, all cache entres are used once they have been loaded from man memory, or vce versa, they are modfed before they are 1 Snce the order of ndexng has dfferent semantcs n dfferent programmng languages, the terms slow and fast are used to descrbe the poston of the ndces. Slowest means that all other ndces run through ther whole range before the slowest ndex s ncreased by one. Thus, the slowest ndex corresponds to the frst array ndex n C s row major order, whle t corresponds to the last ndex n Fortran s column major order. In ths termnology, the fastest ndex always addresses consecutve memory locatons. 138

159 A.3 The lattce Boltzmann kernel Collson optmzed layout (0,0,0) =0 (0,0,0) (0,0,0)... (0,0,0) =1 =2 =18 (0,0,1) =0 (0,0,1) (0,0,1) (0,0,1)... =1 =2 =18 (0,0,2) =0... (0,1,0) (0,1,0) (0,1,0) (0,1,0) (0,1,1) (0,1,1) (0,1,1) (0,1,1) (0,1,2) =0 =1 =2 =18 =0 =1 =2 =18 = (1,0,0) (1,0,0) (1,0,0) (1,0,0) (1,0,1) (1,0,1) (1,0,1) (1,0,1) (1,0,2) =0 =1 =2 =18 =0 =1 =2 =18 =0... Propagaton optmzed layout =0 =0 =0 =0 =0 = =0 =0 =0... (0,0,0) (0,0,1) (0,0,2) (0,1,0) (0,1,1) (0,1,2) (1,0,0) (1,0,1) (1,0,2) =1 =1 =1 =1 =1 = =1 =1 =1... (0,0,0) (0,0,1) (0,0,2) (0,1,0) (0,1,1) (0,1,2) (1,0,0) (1,0,1) (1,0,2) =2 =2 =2 =2 =2 = =2 =2 =2... (0,0,0) (0,0,1) (0,0,2) (0,1,0) (0,1,1) (0,1,2) (1,0,0) (1,0,1) (1,0,2)... Fgure A.2: Illustraton of the collson optmzed (top) and the propagaton optmzed data layout (bottom). Colors are used to dstngush dfferent lattce stes. The black box ndcates the populatons that have to be fetched for the collson step. In the propagaton optmzed data layout, more cache lnes have to be loaded n the collson phase, but they can be exploted durng the propagaton phase. In practce, the propagaton optmzed data layout s superor on many computer archtectures. 139

160 A Implementaton of the lattce Boltzmann method stored back. The propagaton optmzed data layout s hence much more effcent and shows superor performance on many common hardware archtectures [174]. A subtle problem wth the propagaton optmzed data layout s that cache thrashng can occur f the fastest array dmenson z s a power of two. Ths s because assocatve caches map physcal memory locatons to specfc cache locatons. Wth a power of two n the fastest array dmenson, subsequent z-planes are mapped to the same cache address whch causes many cache msses. The effect s a severe performance breakdown. A power of two n the z dmenson should therefore be avoded, for example by array paddng. Further optmzatons wth respect to the memory access pattern are possble. For example one could use blockng technques or loop splttng. However, these technques are hghly specfc and tend to make the program code less extensble. They shall therefore not be dscussed further here. The lattce Boltzmann kernel of ESPResSo uses the propagaton optmzed data layout. The memory for the lattce Boltzmann populatons s allocated durng the ntalzaton. statc vod lb_realloc_flud() { nt ; } lbflud[0] = realloc(*lbflud,2*lbmodel.n_veloc*szeof(double *)); lbflud[0][0] = realloc(**lbflud, 2*lblattce.halo_grd_volume*lbmodel.n_veloc*szeof(double)); lbflud[1] = (double **)lbflud[0] + lbmodel.n_veloc; lbflud[1][0] = (double *)lbflud[0][0] + lblattce.halo_grd_volume*lbmodel.n_veloc; for (=0; <lbmodel.n_veloc; ++) { lbflud[0][] = lbflud[0][0] + *lblattce.halo_grd_volume; lbflud[1][] = lbflud[1][0] + *lblattce.halo_grd_volume; } The frst ndex of the array lbflud s the ndex t for the double bufferng. The second ndex s the ndex for the veloctes. lbflud[t][] s the spatal array for the velocty drecton. It s addressed wth a sngle ndex whch can be calculated from the spatal coordnates x, y and z va ndex = get_lnear_ndex(x,y,z,lblattce.halo_grd); After a sweep through the whole lattce, the ponters for the source and destnaton arrays are swapped wth double **tmp; tmp = lbflud[0]; lbflud[0] = lbflud[1]; lbflud[1] = tmp; 140

161 A.4 Parallelzaton doman decomposton 3 4 Fgure A.3: Schematc llustraton of the doman decomposton scheme used for parallelzng the lattce Boltzmann kernel. On each processor, the physcal doman s surrounded by a halo regon that serves to communcate the populatons that cross the doman boundares to the neghbor processors. Accessng the populatons s straghtforward: the populaton f (x, y, z) s read n the collson phase wth lbflud[0][][ndex], and n the streamng phase the populaton f (x + c x, y + c y, z + c z ) s wrtten wth lbflud[1][][ndex+shft], where shft s calculated from c x, c y and c z. The algorthm s bascally ndependent of the data layout. The crucal pont s the approprate ntalzaton of the array lbflud, whch has to be done n a way that leaves access to the populatons transparent. A.4 Parallelzaton Parallelzaton can be a trcky ssue and s certanly one of the most error-prone parts of any smulaton software. The lattce Boltzmann method s comparatvely easy to parallelze due to the specfc structure of the update rule. The regular lattce suggests to use a doman decomposton scheme where the whole smulaton box s dvded nto smaller rectangular subdomans. Each subdoman s taken care of by an ndvdual processor. Whle the collson phase can be executed ndvdually on every processor, the streamng phase requres the exchange of data between dfferent processors. In ESPResSo, the message passng nterface MPI s used for communcaton between the dfferent processes. In order to facltate the nter-processor data-exchange, an artfcal halo regon s ntroduced around the physcal doman of each process, cf. fgure A.3. The halo regon contans the populatons that leave or enter the local physcal doman to or from another processor s doman. It has to be communcated n every tme step, ether before the collson loop over the nternal lattce stes (pull scheme), or after the collson loop (push scheme). The halo s dvded nto planes that 141

162 A Implementaton of the lattce Boltzmann method drecton send populatons receve populatons postve x f 1, f 7, f 9, f 11, f 13 f 2, f 8, f 10, f 12, f 14 negatve x f 2, f 8, f 10, f 12, f 14 f 1, f 7, f 9, f 11, f 13 postve y f 3, f 7, f 10, f 15, f 17 f 4, f 8, f 9, f 16, f 18 negatve y f 4, f 8, f 9, f 16, f 18 f 3, f 7, f 10, f 15, f 17 postve z f 5, f 11, f 14, f 15, f 18 f 6, f 12, f 13, f 16, f 17 negatve z f 6, f 12, f 13, f 16, f 17 f 5, f 11, f 14, f 15, f 18 Table A.2: The populatons that have to be communcated between processes n the respectve drectons. In each of the sx necessary communcatons, fve populatons are sent and fve populatons are receved. have to be communcated to the respectve neghbor processes n the three spatal drectons. In the doman decomposton scheme, the structure of the memory areas and the processes between whch they are exchanged reman statc. Ths nformaton can hence be stored n a C-struct, the HaloCommuncator, whch s set up durng the ntalzaton. It contans the nformaton about the sx dfferent communcatons (two per spatal drecton) and the respectve halo planes. The structure of the data s represented usng MPI datatypes. Whle an x-plane s contguous n memory, the y- and z-planes are strded. By defnng approprate MPI datatypes, the exact layout s hdden n an abstracton layer. The communcaton procedure can then be mplemented genercally by explotng the nformaton stored n the HaloCommuncator. The actual data transfer s handled by the call MPI_Sendrecv(s_buffer, 1, datatype, r_node, REQ_HALO_SPREAD, r_buffer, 1, datatype, s_node, REQ_HALO_SPREAD, MPI_COMM_WORLD, &status); n the functon vod halo_communcaton(halocommuncator *hc, vod *base). The beneft of ths abstract parallelzaton scheme s that the detals of the mplementaton can be hdden n separate functons. Hence, n the lattce Boltzmann kernel, a sngle call to halo_communcaton at the rght place s suffcent. In ths way, the algorthm and the parallelzaton are clearly separated, whch reduces the rsk of programmng errors consderably and makes the code much more readable. Nonetheless, the parallelzaton scheme can be easly extended f necessary. In practce, there s stll potental for mprovement. One pont s the observaton that only a subset of all velocty drectons can cross the doman boundary n a gven drecton, namely those fve wth a postve projecton on that drecton. Hence, there s no need to communcate all 19 populatons of the halo plane but t s suffcent to send the fve populatons that leave the doman and receve the fve that enter. Table A.2 lsts the populatons that have to be communcated n the dfferent drectons. The set of populatons that s sent to the rght neghbor and the set receved from the left neghbor s always the same. It s therefore convenent to combne the sent-to-rght and receve-from-left nto a sngle MPI_Sendrecv call. The optmzed communcaton routne for the x-drecton, for example, could look lke the followng: 142

163 A.4 Parallelzaton /*************** * X drecton * ***************/ count = 5*lblattce.halo_grd[1]*lblattce.halo_grd[2]; sbuf = malloc(count*szeof(double)); rbuf = malloc(count*szeof(double)); /* send to rght, recv from left = 1, 7, 9, 11, 13 */ snode = node_neghbors[0]; rnode = node_neghbors[1]; buffer = sbuf; ndex = get_lnear_ndex(lblattce.grd[0]+1,0,0,lblattce.halo_grd); for (z=0; z<lblattce.halo_grd[2]; z++) { for (y=0; y<lblattce.halo_grd[1]; y++) { buffer[0] = lbflud[1][1][ndex]; buffer[1] = lbflud[1][7][ndex]; buffer[2] = lbflud[1][9][ndex]; buffer[3] = lbflud[1][11][ndex]; buffer[4] = lbflud[1][13][ndex]; buffer += 5; } } ndex += yperod; f (node_grd[0] > 1) { MPI_Sendrecv(sbuf, count, MPI_DOUBLE, snode, REQ_HALO_SPREAD, rbuf, count, MPI_DOUBLE, rnode, REQ_HALO_SPREAD, MPI_COMM_WORLD, &status); } else { memcpy(rbuf,sbuf,count*szeof(double)); } buffer = rbuf; ndex = get_lnear_ndex(1,0,0,lblattce.halo_grd); for (z=0; z<lblattce.halo_grd[2]; z++) { for (y=0; y<lblattce.halo_grd[1]; y++) { lbflud[1][1][ndex] = buffer[0]; lbflud[1][7][ndex] = buffer[1]; lbflud[1][9][ndex] = buffer[2]; lbflud[1][11][ndex] = buffer[3]; lbflud[1][13][ndex] = buffer[4]; buffer += 5; } } ndex += yperod; /* send to left, recv from rght = 2, 8, 10, 12, 14 */ snode = node_neghbors[1]; rnode = node_neghbors[0]; buffer = sbuf; ndex = get_lnear_ndex(0,0,0,lblattce.halo_grd); for (z=0; z<lblattce.halo_grd[2]; z++) { for (y=0; y<lblattce.halo_grd[1]; y++) { buffer[0] = lbflud[1][2][ndex]; buffer[1] = lbflud[1][8][ndex]; buffer[2] = lbflud[1][10][ndex]; buffer[3] = lbflud[1][12][ndex]; buffer[4] = lbflud[1][14][ndex]; buffer += 5; 143

164 A Implementaton of the lattce Boltzmann method } } ndex += yperod; f (node_grd[0] > 1) { MPI_Sendrecv(sbuf, count, MPI_DOUBLE, snode, REQ_HALO_SPREAD, rbuf, count, MPI_DOUBLE, rnode, REQ_HALO_SPREAD, MPI_COMM_WORLD, &status); } else { memcpy(rbuf,sbuf,count*szeof(double)); } buffer = rbuf; ndex = get_lnear_ndex(lblattce.grd[0],0,0,lblattce.halo_grd); for (z=0; z<lblattce.halo_grd[2]; z++) { for (y=0; y<lblattce.halo_grd[1]; y++) { lbflud[1][2][ndex] = buffer[0]; lbflud[1][8][ndex] = buffer[1]; lbflud[1][10][ndex] = buffer[2]; lbflud[1][12][ndex] = buffer[3]; lbflud[1][14][ndex] = buffer[4]; buffer += 5; } } ndex += yperod; free(rbuf); free(sbuf); The above verson s for the push scheme. It can easly be adopted to the pull scheme wth mnmal modfcatons. In ths example, we have not used the MPI datatypes and nstead, the data s packed nto buffers manually. The scheme uses the mnmal number of communcatons between processes (at least n case the neghbors n all drectons are dfferent processes) and no overhead data s transferred. It s n ths sense an optmal scheme. Further optmzatons mght be possble wth respect to the bufferng strategy and for specal process topologes. The latter cases are so far not addressed n the ESPResSo mplementaton. A.5 Thermal fluctuatons The mplementaton of thermal fluctuatons s straghtforward. The ampltudes of the random fluctuatons are calculated from the relaxaton parameters durng the ntalzaton: for (=0; <3; ++) lb_ph[] = 0.0; lb_ph[4] = sqrt(mu*e[19][4]*(1.-sqr(gamma_bulk))); for (=5; <10; ++) lb_ph[] = sqrt(mu*e[19][]*(1.-sqr(gamma_shear))); for (=10; <n_veloc; ++) lb_ph[] = sqrt(mu*e[19][]); The random fluctuatons are added to the modes after the relaxaton. 144

165 A.6 Force couplng double rootrho = sqrt(mode[0]+lbpar.rho); /* stress modes */ mode[4] += rootrho*lb_ph[4]*gaussan_random(); mode[5] += rootrho*lb_ph[5]*gaussan_random(); mode[6] += rootrho*lb_ph[6]*gaussan_random(); mode[7] += rootrho*lb_ph[7]*gaussan_random(); mode[8] += rootrho*lb_ph[8]*gaussan_random(); mode[9] += rootrho*lb_ph[9]*gaussan_random(); /* ghost modes */ mode[10] += rootrho*lb_ph[10]*gaussan_random(); mode[11] += rootrho*lb_ph[11]*gaussan_random(); mode[12] += rootrho*lb_ph[12]*gaussan_random(); mode[13] += rootrho*lb_ph[13]*gaussan_random(); mode[14] += rootrho*lb_ph[14]*gaussan_random(); mode[15] += rootrho*lb_ph[15]*gaussan_random(); mode[16] += rootrho*lb_ph[16]*gaussan_random(); mode[17] += rootrho*lb_ph[17]*gaussan_random(); mode[18] += rootrho*lb_ph[18]*gaussan_random(); Note that both the stress modes and the knetc modes have random fluctuatons. For each fluctuatng mode, a Gaussan random number has to be drawn. Ths s done n the functon double gaussan_random() whch mplements a smple Box-Muller transformaton [176]. A.6 Force couplng The lattce Boltzmann mplementaton n ESPResSo ncludes the couplng of MD partcles (polymers, collods, etc.) to the LB flud. For a theoretcal descrpton of the method, the reader s referred to the orgnal publcaton of Ahlrchs and Dünweg [86]. The calculaton of the couplng force s wrapped n the functon vod calc_partcle_lattce_a(). Frst, the random numbers for the fluctuatng part of the force are pre-drawn for all partcles and communcated. /* draw random numbers for local partcles */ for (c=0;c<local_cells.n;c++) { cell = local_cells.cell[c] ; p = cell->part ; np = cell->n ; for (=0;<np;++) { p[].lc.f_random[0] = lb_coupl_pref*gaussan_random(); p[].lc.f_random[1] = lb_coupl_pref*gaussan_random(); p[].lc.f_random[2] = lb_coupl_pref*gaussan_random(); } } /* communcate the random numbers */ ghost_communcator(&cell_structure.ghost_lbcouplng_comm) ; Ths s necessary to make sure that ghost partcles use the same random number as ther real counterparts. The synchronzaton of random numbers can lead to severe complcatons when strct postvty of the populatons s requred. The reason s that the order of ghost 145

166 A Implementaton of the lattce Boltzmann method and real partcles s nterchanged between neghborng processes. Consequently, a random force does not necessarly lead to negatve populatons on both processes. The revocaton of a bad random number can therefore be trggered on dfferent processes, whch makes t very complcated to synchronze the redrawng of random numbers. In the worst case, the program can get stuck n a loop just redrawng random numbers. Therefore, the strct postvty of the populatons s waved and small negatve values are allowed. Ths should be a rare event f the smulaton parameters have physcally reasonable values. The vscous drag force exerted by the flud on the partcle s calculated n the functon vod lb_vscous_couplng(partcle *p, double force[3]). Frst, the nterpolated flud velocty at the partcle s poston s determned. /* calculate flud velocty at partcle s poston ths s done by lnear nterpolaton (Eq. (11) Ahlrchs and Duenweg, JCP 111(17):8225 (1999)) */ nterpolated_u[0] = nterpolated_u[1] = nterpolated_u[2] = 0.0 ; for (z=0;z<2;z++) { for (y=0;y<2;y++) { for (x=0;x<2;x++) { local_node = &lbfelds[node_ndex[(z*2+y)*2+x]]; f (local_node->recalc_felds) { lb_calc_local_felds(node_ndex[(z*2+y)*2+x],local_node->rho,local_node->j,null); local_node->recalc_felds = 0; local_node->has_force = 1; } local_rho[0] = local_node->rho[0]; local_j[0] = local_node->j[0]; local_j[1] = local_node->j[1]; local_j[2] = local_node->j[2]; nterpolated_u[0] += delta[3*x+0]*delta[3*y+1]*delta[3*z+2]*local_j[0]/(*local_rho); nterpolated_u[1] += delta[3*x+0]*delta[3*y+1]*delta[3*z+2]*local_j[1]/(*local_rho); nterpolated_u[2] += delta[3*x+0]*delta[3*y+1]*delta[3*z+2]*local_j[2]/(*local_rho); } } } Then the vscous drag s calculated and added to the random force. /* calculate vscous force * take care to rescale veloctes wth tme_step and transform to MD unts * (Eq. (9) Ahlrchs and Duenweg, JCP 111(17):8225 (1999)) */ force[0] = - lbpar.frcton * (p->m.v[0]/tme_step - nterpolated_u[0]*agrd/tau); force[1] = - lbpar.frcton * (p->m.v[1]/tme_step - nterpolated_u[1]*agrd/tau); force[2] = - lbpar.frcton * (p->m.v[2]/tme_step - nterpolated_u[2]*agrd/tau); force[0] = force[0] + p->lc.f_random[0]; force[1] = force[1] + p->lc.f_random[1]; force[2] = force[2] + p->lc.f_random[2]; In the next step, the force s transformed to the momentum transfer n lattce unts, cf. Equaton (12) of [86]. 146

167 A.7 Boundary condtons /* transform momentum transfer to lattce unts (Eq. (12) Ahlrchs and Duenweg, JCP 111(17):8225 (1999)) */ delta_j[0] = - force[0]*ntegrate_pref2/tme_step*tau/agrd; delta_j[1] = - force[1]*ntegrate_pref2/tme_step*tau/agrd; delta_j[2] = - force[2]*ntegrate_pref2/tme_step*tau/agrd; Fnally, the momentum transfer s extrapolated to the lattce stes and added to the volumetrc force actng on the flud at that ste. for (z=0;z<2;z++) { for (y=0;y<2;y++) { for (x=0;x<2;x++) { local_f = lbfelds[node_ndex[(z*2+y)*2+x]].force; local_f[0] += delta[3*x+0]*delta[3*y+1]*delta[3*z+2]*delta_j[0]; local_f[1] += delta[3*x+0]*delta[3*y+1]*delta[3*z+2]*delta_j[1]; local_f[2] += delta[3*x+0]*delta[3*y+1]*delta[3*z+2]*delta_j[2]; } } } The force s appled n the functon vod lb_apply_forces(ndex_t ndex, double* mode) whch s called from wthn the man collde-stream loop. It has to be mentoned that ths mplementaton s not fully self-consstent, because the force enters the redefned hydrodynamc momentum densty whch tself determnes the force. To deal wth ths problem one could use an teratve scheme. However, snce the lnear nterpolaton of the flud velocty s only a frst approxmaton, an teratve scheme s probably not very benefcal n vew of the computatonal overhead. A.7 Boundary condtons The mplementaton of boundary condtons depends heavly on the specfc type of boundary condtons,.e., node based or lnk based boundary condtons. In ESPResSo, both types have been mplemented. The lnk based mplementaton comprses bounce-back, specular reflectons and slp-reflecton. Lnk based boundary condtons affect the streamng of the populatons at the boundary. Instead of followng ther velocty lnk, they are deflected to a dfferent drecton. Ths could n prncple be mplemented wthn the streamng step by redrectng the copy operaton of the populaton to the correct target node. However, addtonal f statements would have to be ntroduced n the man loop whch would deterorate the effcency. Furthermore, there s a danger to spol the cache optmzed data layout. Therefore, the streamng step s left unchanged such that the populatons end up on nodes outsde the flud doman. They are then moved to the correct target node n a separate boundary loop. Ths works for any lnk based scheme. The specfc steps for the dfferent boundary condtons are outlned n the followng. 147

168 A Implementaton of the lattce Boltzmann method A.7.1 Bounce-back In the bounce-back rule, the populatons that have moved out of the flud doman have to be coped back to the reversed velocty drecton on the node they came from. The ndex shft to the bounce-back node s pre-calculated and stored n the array next. nt yperod = lblattce.halo_grd[0]; nt zperod = lblattce.halo_grd[0]*lblattce.halo_grd[1]; nt next[19]; next[0] = 0; // ( 0, 0, 0) = next[1] = 1; // ( 1, 0, 0) + next[2] = - 1; // (-1, 0, 0) next[3] = yperod; // ( 0, 1, 0) + next[4] = - yperod; // ( 0,-1, 0) next[5] = zperod; // ( 0, 0, 1) + next[6] = - zperod; // ( 0, 0,-1) next[7] = (1+yperod); // ( 1, 1, 0) + next[8] = - (1+yperod); // (-1,-1, 0) next[9] = (1-yperod); // ( 1,-1, 0) next[10] = - (1-yperod); // (-1, 1, 0) + next[11] = (1+zperod); // ( 1, 0, 1) + next[12] = - (1+zperod); // (-1, 0,-1) next[13] = (1-zperod); // ( 1, 0,-1) next[14] = - (1-zperod); // (-1, 0, 1) + next[15] = (yperod+zperod); // ( 0, 1, 1) + next[16] = - (yperod+zperod); // ( 0,-1,-1) next[17] = (yperod-zperod); // ( 0, 1,-1) next[18] = - (yperod-zperod); // ( 0,-1, 1) + The reverson of the velocty s acheved by usng the followng ndex map nt reverse[] = { 0, 2, 1, 4, 3, 6, 5, 8, 7, 10, 9, 12, 11, 14, 13, 16, 15, 18, 17 }; The bounce-back operaton at the top wall then looks lke /* bottom-up sweep */ for (k=lblattce.halo_offset;k<lblattce.halo_grd_volume;k++) { } f (lbfelds[k].boundary) { } /* bounce back to lower ndces */ lbflud[1][reverse[5]][k-next[5]] = lbflud[1][5][k]; lbflud[1][reverse[11]][k-next[11]] = lbflud[1][11][k]; lbflud[1][reverse[14]][k-next[14]] = lbflud[1][14][k]; lbflud[1][reverse[15]][k-next[15]] = lbflud[1][15][k]; lbflud[1][reverse[18]][k-next[18]] = lbflud[1][18][k]; The mplementaton for the bottom wall works analogously. 148

169 A.7 Boundary condtons A.7.2 Specular reflectons Specular reflectons dffer from the bounce-back rule n the target nodes and the velocty mappng. The shft to the target nodes s agan stored n an array next nt zperod = lblattce.halo_grd[0]*lblattce.halo_grd[1]; nt next[19]; next[0] = 0; // ( 0, 0, 0) next[1] = 0; // ( 1, 0, 0) next[2] = - 0; // (-1, 0, 0) next[3] = 0; // ( 0, 1, 0) next[4] = - 0; // ( 0,-1, 0) next[5] = zperod; // ( 0, 0, 1) next[6] = - zperod; // ( 0, 0,-1) next[7] = 0; // ( 1, 1, 0) next[8] = - 0; // (-1,-1, 0) next[9] = 0; // ( 1,-1, 0) next[10] = - 0; // (-1, 1, 0) next[11] = zperod; // ( 1, 0, 1) next[12] = - zperod; // (-1, 0,-1) next[13] = - zperod; // ( 1, 0,-1) next[14] = zperod; // (-1, 0, 1) next[15] = zperod; // ( 0, 1, 1) next[16] = - zperod; // ( 0,-1,-1) next[17] = - zperod; // ( 0, 1,-1) next[18] = zperod; // ( 0,-1, 1) and the ndex mappng for specular reflectons s gven by nt reflect[] = { 0, 1, 2, 3, 4, 6, 5, 7, 8, 9, 10, 13, 14, 11, 12, 17, 18, 15, 16 }; The sweep through the nodes of the top wall s mplemented as follows: /* bottom-up sweep */ for (k=lblattce.halo_offset;k<lblattce.halo_grd_volume;k++) { } f (lbfelds[k].boundary) { } /* reflect to lower ndces */ lbflud[1][reflect[5]][k-next[5]] = lbflud[1][5][k]; lbflud[1][reflect[11]][k-next[11]] = lbflud[1][11][k]; lbflud[1][reflect[14]][k-next[14]] = lbflud[1][14][k]; lbflud[1][reflect[15]][k-next[15]] = lbflud[1][15][k]; lbflud[1][reflect[18]][k-next[18]] = lbflud[1][18][k]; 149

170 A Implementaton of the lattce Boltzmann method A.7.3 Slp reflectons Slp reflectons are a mxture of bounce-back and specular reflectons. The mplementaton s therefore very smlar to the above. We just quote the example of the top wall agan: double s = lb_boundary_par.slp_pref; double r = s; double **n = lbflud[1]; /* bottom-up sweep */ for (k=lblattce.halo_offset; k<lblattce.halo_grd_volume; k++) { } f (lbfelds[k].boundary) { } /* slp reflect to lower ndces */ n[6][k-zperod] = n[5][k]; n[12][k-zperod] = s*n[14][k] + r*n[11][k-zperod+next[11]]; n[13][k-zperod] = s*n[11][k] + r*n[14][k-zperod+next[14]]; n[16][k-zperod] = s*n[18][k] + r*n[15][k-zperod+next[15]]; n[17][k-zperod] = s*n[15][k] + r*n[18][k-zperod+next[18]]; A.7.4 Local boundary collsons In contrast to the lnk-based boundary condtons, local boundary condtons can be drectly ntegrated nto the man loop. Instead of the normal collsons and streamng, the correspondng sequence of boundary functons has to be called: vod lb_boundary_collsons(nt ndex, double *modes) { } double p[6]; lb_boundary_calc_modes(ndex, modes, p); lb_boundary_relax_modes(ndex, modes, p); lb_boundary_apply_forces(ndex, modes); lb_boundary_calc_n_push(ndex, modes); For the specal case of the bounce-back of non-equlbrum parts combned wth BGKrelaxaton, the whole processng of the boundary collsons s combned n the functon lb_boundary_bb_neq_bgk(ndex, modes). It mmcs the same operatons as n the bulk, but wth modfed values for the reduced symmetry at the boundary. For detals, the reader s referred to chapter 6 of ths thess. 150

171 B Techncal materal Ths appendx collects varous techncal detals whch have been skpped n the man text. B.1 Hermte tensor polynomals and Gauss-Hermte quadratures B.1.1 Hermte tensor polynomals The Hermte tensor polynomals as used by Grad [177] are a complete bass set of Hlbert space wth respect to the scalar product f g = ω(v)f(v)g(v) dv. (B.1) The orthonormalty relaton reads ω(v)h (n) α (v)h (m) β (v) dv = δ mnδ (n) αβ, (B.2) where δ (n) αβ = 1 f α = (α 1,..., α n ) s a permutaton of β = (β 1,..., β m=n ) and zero otherwse. The weght functon assocated wth the Hermte polynomals s gven by [ ] ω(v) = (2π) D 2 exp v2. (B.3) 2 The latter can be used to defne the Hermte polynomals explctly H (n) α 1...α n = ( 1)n ω(v) v α1... v αn ω(v). (B.4) The frst few polynomals are H (0) (v) = 1, H (1) α (v) = v α, H (2) αβ (v) = v αv β δ αβ, H (3) αβγ (v) = v αv β v γ v α δ βγ v β δ αγ v γ δ αβ. (B.5) 151

172 B Techncal materal The Hermte tensor polynomals satsfy the recurrence relaton n v α H α (n) 1...α n = H αα (n+1) 1...α n + δ ααk H α (n 1) 1...α k 1 α k+1...α n. k=1 (B.6) Any square ntegrable functon n the Hlbert space can be expanded n the bass as 1 f(v) = ω(v) n! a(n) α H α (n) (v), n=0 (B.7) where a contracton over the n-fold ndex α = (α 1,, α n ) of the tensors a (n) and H (n) s to be understood. Snce H (n) s symmetrc n α, we wll assume that a (n) s symmetrc n α as well. To obtan the expanson coeffcents, we multply by H (m) (v) and ntegrate f(v)h (m) β (v)dv = = ω(v) 1 n=0 n! a(n) α n=0 = a (m) β, 1 n! a(n) α H (n) δ mn δ αβ α (v)h (m) β (v)dv (B.8) where we have exploted that there appear n! permutatons of α n the contracton a (n) α δ αβ. Thus we obtan the Hermte coeffcents as a (n) = f(v)h (n) (v)dv. (B.9) B.1.2 Gauss-Hermte quadrature The Gaussan quadrature s a means to approxmate the ntegrals ω(v)f(v) dv for a gven functon f(v) by n ω(v)f(v) dv w f(c ), (B.10) where w s a set of weghts and c are called the nodes or abscssae of the quadrature. The am s to fnd a choce of n nodes that maxmzes the degree of precson m of the approxmaton, that s, the degree of a polynomal up to whch (B.10) holds exactly. For the weght functon (B.3) and the ntegraton nterval (, ) n D = 1 dmenson, the Gauss-Hermte quadrature can be appled [178, 179]. The optmal nodes for an n-pont quadrature are the roots of the one-dmensonal Hermte polynomals H n (v) and the weghts are gven by w = = 1 H n 1 (c ) d H dv n(c ) n! [nh n 1 (c )] 2, =1 ω(v)h n 1 (v)h n 1 (v) dv (B.11) 152

173 B.1 Hermte tensor polynomals and Gauss-Hermte quadratures Quadrature c w E1, E1,3 2 ±1 1/2 E1, /3 ± 3 1/6 E 4 1,7 ± 3 6 (3 + 6)/12 ± (3 6)/12 Table B.1: Nodes and weghts of some one-dmensonal Gauss-Hermte quadratures. where the relaton d H dv n = vh n H n+1 = nh n 1 was used and δ (n) αα = n! n one dmenson. The degree of precson of the n-pont quadrature n one dmenson s m = 2n 1. The nodes and the weghts for some one-dmensonal Gauss-Hermte quadratures are lsted n table B.1. The nomenclature ED,m n s adopted from Shan et al. [47], where D s the dmenson of space, n s number of nodes and m s the degree of precson. In hgher dmensons D > 1, there s no unque quadrature procedure avalable. Nevertheless, one can construct quadratures n hgher dmensons from the one-dmensonal quadrature by wrtng ω(v) D α=1 v n α α dv = D ( α=1 = k 1 ω(v α )v n α α dv α ) = k D ( D α=1 w k1 w kd c n1 k 1 c n D kd, k α ) w kα c nα k α (B.12) where n n D n. Ths means that a D-dmensonal quadrature emerges from a combnaton of D one-dmensonal quadratures: ω(v)p(v) dv = w k1...k D p(c k1...k D ), (B.13) where w k1...k D = w k1 w kd and c k1...k D = (c k1,..., c kd ), and p(v) s a polynomal of degree n. For example, usng the one-dmensonal quadrature E1,5, 3 we obtan the quadratures E2,5 9 and E3,5 27 usng the full set of abscssae n two and three dmensons, respectvely (see table 2.1 on page 18). In three dmensons, the number of nodes can be reduced wthout affectng the overall degree of the quadrature. The weghts of the quadrature E3,5 27 can be grouped nto four symmetry classes w q accordng to q = c k1...k 3 2 /3. For D = 3 and n = 5 we have n 1 + n 2 + n 3 5. We can assume n 1 n 2 n 3 wthout loss of generalty. The n 0 for = 1, 2, 3 mples n 1 1. If n 1 = 1, the ntegral (B.12) vanshes for party reasons. The sum on the rght hand sde also vanshes as the weghts are symmetrc wth respect to c 1 = 0. Conversely, f n 1 = 0, p(v) reduces to a two-dmensonal polynomal and the quadrature retans ts degree 153

174 B Techncal materal f the weghts of the three-dmensonal quadrature match the weghts of the two-dmensonal quadrature E 9 2,5: w q=0 + 2w q=1 = = 4 9, w q=1 + 2w q=2 = = 1 9, w q=2 + 2w q=3 = = Ths s an under-determned system that can be rewrtten n the parametrc form 1 w q=0 = 1 (2 + t), 9 (B.14) w q=1 = 1 (2 t), 18 w q=2 = 1 36 t, (B.15) w q=3 = 1 (1 t). 72 The orgnal quadrature E3,5 27 s recovered for t = 2/3. By choosng t = 0 or t = 1 we can omt ether the q = 2 or the q = 3 symmetry class and thus effectvely reduce the number of abscssae. Note that ths does not affect the accuracy of the quadrature. The result are the quadratures E3,5 15 and E3,5 19 that correspond to the D3Q15 and D3Q19 models, respectvely. Besdes the producton formulas we shall demonstrate that quadratures can also be constructed from a predefned set of abscssae c. We note that the orthonormal relaton mples ω(v)h (n) dv = δ 0,n. (B.16) The quadrature (B.13) s of degree m, f and only f n w H (n) (c ) = δ 0,n. =1 (B.17) For a set of abscssae that obeys party symmetry, ths relaton s automatcally satsfed for the odd tensor polynomals. The even tensor polynomals up to n = 4 yeld the followng condtons on the weghts w : w = 1, w c α c β = δ αβ, w c α c β c γ c δ = δ αβ δ γδ + δ αγ δ βδ + δ αδ δ βγ. (B.18) 1 Ths was prevously derved by Shan et al. [47], but equaton (A 20) n ther paper contans typographc errors. 154

175 B.2 Lattce sums and sotropc lattce models We note that ths s closely related to the condtons on the weghts that follow from the physcal requrements wthn the Chapman-Enskog equaton. Ths shows the ntmate relatonshp between the Hermte expanson and the symmetry propertes of the velocty set and the lattce sums. Fnally, we remark that quadratures can also be obtaned on dfferent routes [179]. The smallest number of nodes among the known quadratures s used by the 13-pont quadrature E3,5, 13 whch s of degree 5. 2 In general, t s yet an open problem, how the mnmal quadrature for a gven geometry, weght functon and degree of accuracy can be found. B.2 Lattce sums and sotropc lattce models The lattce sums T (n) = w c... c (B.19) play a pvotal role n the dervaton and analyss of the lattce Boltzmann method. In ths secton, we treat some mportant propertes of the lattce sums wth respect to the sotropy of the lattce model. B.2.1 Lattce sums for dscrete velocty sets Lattce Boltzmann models are often constructed from a gven dscrete velocty set. Ths set then has to satsfy certan symmetry propertes n order to guarantee Gallean nvarance and sotropy of the macroscopc equatons. The requrements can be convenently formulated n terms of the lattce sums, whch have to take the form of certan nvarant tensors. Here we derve the general form of the lattce sums for smple cubc lattces. The n-th rank lattce sum for a gven set of dscrete veloctes s T (n) α 1...α n = w c α1... c αn. (B.20) The w are the weghts from the equlbrum dstrbuton whch we consder as free parameters here. For symmetry reasons, w depends on the length of c but not ts drecton. The lattce sum can therefore be splt nto contrbutons from the dfferent subshells ndexed by q = c 2 T (n) α 1...α n = q w q b q =1 c qα1... c qαn, (B.21) 2 The abscssae of E 13 3,5 correspond to the vertces of an cosahedron. It s therefore not related to the D3Q13 model [24] whch uses the next-nearest neghbors on a cubc grd. 155

176 B Techncal materal where b q s the coordnaton number of the q-subshell. On a smple cubc lattce, the velocty set s nvarant under party transformatons. Hence the odd lattce sums vansh T (n) α 1...α n = 0 f n s odd. (B.22) We proceed to evaluate the sums for n = 0, n = 2 and n = 4 explctly. For n = 0, we get T (0) = q w q b q =1 1 = q b q w q = 1. (B.23) The second-rank lattce sum on the cubc lattce s an sotropc tensor. We obtan T (2) αβ = q w q b q =1 c qα c qβ = q qb q D w qδ αβ = σ 2 δ αβ, (B.24) where σ 2 = q qb q D w q. (B.25) To evaluate the fourth-order lattce sum, we note that a fourth rank tensor that s nvarant under transformatons n the symmetry group of the cubc lattce can be wrtten as a lnear combnaton of a fourth-rank sotropc tensor and a cubc ansotropy δ αβγδ, whch s one f all four ndces are equal and zero otherwse: T (4) αβγδ = κ 4δ αβγδ + σ 4 (δ αβ δ γδ + δ αγ δ βδ + δ αδ δ βγ ). (B.26) For a lattce wth c qα 1, we can evaluate T (4) xxxx = q w q b q =1 c qx c qx c qx c qx = q w q qb q D = κ 4 + 3σ 4, (B.27) and T (4) ααββ = q w q b q =1 c qα c qα c qβ c qβ = q w q b q q 2 = Dκ 4 + D(D + 2)σ 4. (B.28) From ths t follows κ 4 = q σ 4 = q qb q (D + 2 3q) w q, D(D 1) w q qb q (q 1) D(D 1). (B.29) 156

177 B.2 Lattce sums and sotropc lattce models For the D3Q19 model wth q = 0, 1, 2, we get the followng equaton system for the weghts 1 = w 0 + 6w w 2, σ 2 = 2w 1 + 8w 2, (B.30) σ 4 = 4w 2, κ 4 = 2w 1 4w 2. Ths admts a soluton only f σ 2 = 3σ 4 + κ 4. (B.31) Then the soluton s w 0 = 1 2σ 2 κ 4 = 1 6σ 4 3κ 4, w 1 = 1 6 (σ 2 + 2κ 4 ) = 1 2 (σ 4 + κ 4 ), (B.32) w 2 = 1 4 σ 4 = 1 12 (σ 2 κ 4 ), whch s used n the man text to obtan the values (2.54). B.2.2 Lattce sums and Gauss-Hermte quadrature Next we prove equaton (2.40) of the man text. It states that, for a gven lattce wth lnks c and correspondng weghts w, sotropy of the lattce sums T (n) = { 0 n odd w c... c = δ (n) (B.33) n even, s equvalent to w and c beng the weghts and nodes of a Gauss-Hermte quadrature of degree m n. Let us defne p n (v) v }.{{.. v}, then t holds n tmes { 0 n odd ω(v)p n (v)dv = (B.34) δ (n) n even. Snce p n (v) s a tensor polynomal of degree n, t s exactly evaluated by a Gauss-Hermte quadrature of degree m n,.e., w c... c = ω(v)p n (v)dv, (B.35) whch proves one drecton of the equvalence. For the other drecton we note that any polynomal p(v) of degree m s a lnear combnaton of p n (v), n m, hence m ω(v)p(v) dv = ω(v) a k p k (v) dv = m a k k=0 k=0 m 2 ω(v)p k (v) dv = a 2k δ (2k). k=0 (B.36) 157

178 B Techncal materal Then, f the lattce sums are sotropc, we can nsert (B.33) for the δ s n (B.36) and use the defnton of p n (c ) to obtan ω(v)p(v) dv = m a k w c... c = k=0 w m k=0 a k p k (c ) = w p(c ). (B.37) The equalty of the leftmost wth the rghtmost sde mples that w and c are the weghts and nodes of a quadrature of degree m. Ths completes the proof of the equvalence. B.3 Theoretcal analyss of the slp boundary condton The statonary soluton of the lattce Boltzmann equaton for the a one-dmensonal flow profle can be obtaned as follows. For smplcty, we use the sngle relaxaton tme approxmaton for whch the statonary soluton of the lattce Boltzmann equaton reads f α+ĉ z = (1 + λ)f α λf eq j (ρ, uα x) + g,α. (B.38) In the specfc case of the D3Q19 model, the statonary soluton for the 19 populatons s explctly gven by f0 α = ρ ) (1 (uα x) 2, 3 2c 2 s f1 α = ρ ( ) 1 + uα xa 18 c 2 sτ + (uα x) 2 c 2 s f2 α = ρ ( ) 1 uα xa 18 c 2 sτ + (uα x) 2 c 2 s f α 3 = ρ 18, f α 4 = ρ 18, af ext 18λc 2 s + af ext 18λc 2 s ζ + 18λc 2 sa 2 uα x(δ 1,α + δ α,n ), ζ 18λc 2 sa 2 uα x(δ 1,α + δ α,n ), f α 5 = ρ 18, f6 α = ρ 18, f7 α = ρ ( ) 1 + uα xa 36 c 2 sτ + (uα x) 2 c 2 s f8 α = ρ ( ) 1 uα xa 36 c 2 sτ + (uα x) 2 c 2 s f9 α = ρ ( ) 1 + uα xa 36 c 2 sτ + (uα x) 2 c 2 s f10 α = ρ ( ) 1 uα xa 36 c 2 sτ + (uα x) 2 c 2 s af ext 36λc 2 s + af ext 36λc 2 s af ext 36λc 2 s + af ext 36λc 2 s ζ + 36λc 2 sa 2 uα x(δ 1,α + δ α,n ), ζ 36λc 2 sa 2 uα x(δ 1,α + δ α,n ), ζ + 36λc 2 sa 2 uα x(δ 1,α + δ α,n ), ζ 36λc 2 sa 2 uα x(δ 1,α + δ α,n ), (B.39) 158

179 f α 11 = λ ρ 36 f α 12 = λ ρ 36 f α 13 = λ ρ 36 f α 14 = λ ρ 36 f α 15 = λ ρ 36 f α 16 = λ ρ 36 f α 17 = λ ρ 36 (1 + uα 1 x a c 2 sτ (1 uα+1 x a c 2 sτ a c 2 sτ a c 2 sτ (1 + uα+1 x (1 uα 1 x + (1 + λ)f α 1 j, + (1 + λ)f α+1 j, + (1 + λ)f α+1 j, + (uα 1 x ) 2 c 2 s + (uα+1 x ) 2 c 2 s + (uα+1 x ) 2 c 2 s + (uα 1 x ) 2 c 2 s B.3 Theoretcal analyss of the slp boundary condton ) ) ) ) + (1 + λ)f α 1 + (1 + λ)f α+1 + (1 + λ)f α+1 + (1 + λ)f α 1 + af ext 36c 2 s af ext 36c 2 s + af ext 36c 2 s af ext 36c 2 s ζ 36c 2 sa 2 uα 1 x δ 1,α 1, + ζ 36c 2 sa 2 uα+1 x δ α+1,n, ζ 36c 2 sa 2 uα+1 x δ α+1,n, + ζ 36c 2 sa 2 uα 1 x δ 1,α 1, f18 α = λ ρ α 1 + (1 + λ)fj. 36 Let us assume that the drecton of the flow s the x-drecton. The flow velocty n ths drecton s obtaned as ρu α x = f α c x = (f α 1 f α 2 + f α 7 f α 8 + f α 9 f α 10) a τ + (f α 11 f α 14) a τ + (f α 13 f α 12) a τ, (B.40) where we have grouped the populatons n a practcally convenent way. The dfferent subexpressons can be evaluated by pluggng n the explct solutons for the populatons (f1 α f2 α + f7 α f8 α + f9 α f10) α = 2a 9c 2 sτ ρuα x 2f exta + 2ζ 9λc 2 s 9λc 2 sa 2 uα x(δ 1,α + δ α,n ), (B.41a) (f11 α f14) α = (1 + λ)(f11 α 1 f14 α 1 ) λa 18c 2 sτ ρuα 1 x + f exta ζ 18c 2 s 18c 2 sa 2 uα 1 x δ 1,α 1, (B.41b) (f13 α f12) α = (1 + λ)(f13 α+1 f12 α+1 ) λa 18c 2 sτ ρuα+1 x + f exta ζ 18c 2 s 18c 2 sa 2 uα+1 x δ α+1,n, (B.41c) whch leads to ρu α x = (f1 α f2 α + f7 α f8 α + f9 α f10) α a τ + (f 11 α f14) α a τ + (f 13 α f12) α a τ = 2 3 ρuα x λ 6 ρ ( ) u α 1 x + u α+1 λ 2 x + 3λ f extτ ( ) f α 1 11 f14 α 1 + f13 α+1 f12 α+1 + (1 + λ) a τ + 2ζτ 3λa 3 uα x(δ 1,α + δ α,n ) ζτ 6a 3 ( u α 1 x δ 1,α 1 + u α+1 x δ α+1,n ). (B.42) 159

180 B Techncal materal Furthermore, we can wrte (f α 11 f α 14 + f α 13 f α 12) a τ = 1 3 ρuα x + 2f extτ 3λ 2ζτ 3λa 3 uα x(δ 1,α + δ α,n ). (B.43) Usng (B.42) and pluggng n (B.41a) to (B.41c) we get the followng recurrence relatons (f α 11 f α 14) a τ = 1 3 ρuα x + 2f extτ 3λ = 1 3 ρuα x + λ 6 ρuα+1 x (f α 13 f α 12) a τ = 1 3 ρuα x + 2f extτ 3λ 2ζτ 3λa 3 uα x(δ 1,α + δ α,n ) (f α 13 f α 12) a τ + 4 λ 6λ f extτ 2ζτ 3λa 3 uα x(δ 1,α + δ α,n ) + ζτ 6a 3 uα+1 x δ α+1,n (1 + λ) a α+1 (f13 f12 α+1 ), τ = 1 3 ρuα x + λ 6 ρuα 1 x + ζτ 6a 3 uα 1 x 2ζτ 3λa 3 uα x(δ 1,α + δ α,n ) (f α 11 f α 14) a τ + 4 λ 6λ f extτ 2ζτ 3λa 3 uα x(δ 1,α + δ α,n ) δ 1,α 1 (1 + λ) a τ (f α 1 11 f α 1 14 ). (B.44) Addng these two relatons leads to ( ) f α 1 11 f14 α 1 +f13 α+1 f12 α+1 a τ = 1 3 ρuα 1 x 2ζτ + λ 3 ρuα x ρuα+1 x 3λa 3 uα 1 x + 4 λ 3λ f extτ δ 1,α 1 + ζτ 6a 3 uα x(δ 1,α + δ α,n ) 2ζτ (1 + λ) a τ (f α 11 f α 14 + f α 13 f α 12) = 1 3 ρuα 1 x + λ 3 ρuα x ρuα+1 x + 4 λ 3λ f extτ 3λa 3 uα+1 x δ α+1,n 2ζτ 3λa 3 uα 1 x δ 1,α 1 + ζτ 6a 3 uα x(δ 1,α + δ α,n ) 2ζτ ( 1 (1 + λ) 3 ρuα x + 2f extτ 3λ 2ζτ ) 3λa 3 uα x(δ 1,α + δ α,n ) = 1 3 ρ ( ) u α 1 x u α x + u α+1 2 3λ x + 3λ f extτ 2ζτ 3λa 3 uα 1 x + 5λ + 4 ζτ 6λ a 3 uα x(δ 1α + δ α,n ) 2ζτ 3λa 3 uα+1 x δ α+1,n, 3λa 3 uα+1 x δ α+1,n δ 1,α 1 (B.45) 160

181 B.4 Functonal dervaton of the bulk equlbrum dstrbuton whch we can fnally use n (B.42) to obtan ρu α x = 2 3 ρuα x λ 6 ρ ( ) u α 1 x + u α+1 λ 2 x + 3λ f extτ + (1 + λ) a ( ) f α 1 11 f14 α 1 + f13 α+1 f12 α+1 τ + 2ζτ 3λa 3 uα x(δ 1,α + δ α,n ) ζτ ( ) u α 1 6a 3 x δ 1,α 1 + u α+1 x δ α+1,n = 1 λ ρu α x λ (B.46) ρ(u α 1 x + u α+1 x ) λf ext τ 3 6 5λ + 4 ζτ 6λ a 3 (uα 1 x δ 1,α 1 + u α+1 x δ α+1,n ) 8 + 9λ + 5λ2 ζτ + 6λ a 3 uα x(δ 1,α + δ α,n ). Reorderng terms and pluggng n the expresson (4.29) for the vscosty η s, we fnally arrve at the fnte dfference equaton u α 1 x 2u α x + u α+1 x 8 + 9λ + 5λ2 ζ η s = f a 2 ext + 6λ 2 a 3 uα x(δ 1,α + δ α,n ) 5λ + 4 ζ 6λ 2 a 3 (uα 1 x δ 1,α 1 + u α+1 x δ α+1,n ). Ths s equaton (5.59) of the man text. (B.47) B.4 Functonal dervaton of the bulk equlbrum dstrbuton Here, we rederve the bulk equlbrum dstrbuton as the mnmzer of the quadratc functonal (6.11). The statonary dstrbuton of the functonal s f eq = w (λ ρ + λ j,α c α + λ Π,αβ c α c β ). (B.48) The equlbrum dstrbuton has to satsfy the constrants ρ = f eq, j α = Π eq αβ = f eq c α, f eq c α c β. (B.49) Pluggng n (B.48) and usng the lattce sums from secton B.2 n ths appendx, we get ρ = λ ρ + σ 2 λ Π,αα, ρu α = σ 2 λ j,α, 3ρc 2 s + ρu α u α = 3σ 2 λ ρ + 5σ 4 λ Π,αα, (B.50) ρu α u β 1 3 ρu γu γ δ αβ = 2σ 4 λ Π,αβ, 161

182 B Techncal materal where have decomposed the stress tensor and the correspondng Lagrange multpler nto ther trace and traceless part. Moreover, we have assumed κ 4 = 0 to ensure sotropy of fourth-rank tensors. The soluton of the above equaton system s λ ρ = 1 5σ 4 3σ 2 2 λ j,α = 1 σ 2 ρu α, λ Π,αα = λ Π,αβ = 1 2σ 4 1 5σ 4 3σ 2 2 ( 5σ4 ρ 3σ 2 ρc 2 s σ 2 ρu α u α ), ( 3ρc 2 s 3σ 2 ρ + ρu α u α ), (ρu α u β 13 ρu γu γ δ αβ ). (B.51) For convenence, we contnue wth the explct values for the D3Q19 model, where σ 4 = σ2 2 = c 4 s and c 2 s = 1/3. Then the Lagrange multplers are ( λ ρ = ρ 1 1 ) u 2c 2 α u α, s λ j,α = 1 c 2 s ρu α, (B.52) λ Π,αβ = 1 ρu 2c 4 α u β, s and we fnally arrve at the equlbrum dstrbuton [ = w ρ 1 1 f eq = w ρ 2c 2 s u α u α + u αc α c 2 s + u ] αu β c α c β 2c 4 s [ 1 + u αc α + u αu β (c α c β c 2 sδ αβ ) c 2 s 2c 4 s Ths s the famlar expresson for the bulk equlbrum dstrbuton. ]. (B.53) 162

183 C Sourcecode The lattce Boltzmann mplementaton that was developed durng ths work has been ntegrated n the ESPResSo software package [3, 4]. The sourcecode can be found on the accompanyng CD-ROM. It contans the full ESPResSo package whch s dstrbuted under the GNU General Publc Lcense (GPL) [180]. The parts whose development has been ntated by ths author comprse, nter ala, the followng fles: lattce.h lattce.c halo.h halo.c lb-d3q18.h lb-d3q19.h lb.h lb.c statstcs_flud.h statstcs_flud.h lb-boundares.h lb-boundares.c Data structures for lattces and mappng functons Data structures for halo regons and parallelzaton routnes Data structures for D3Q18 model Data structures for D3Q19 model The lattce Boltzmann kernel, ncludng fluctuatons and force couplng Data structures for flud observables and analyss routnes Lattce Boltzmann boundary condtons The author has further contrbuted varous bug-fxes and several extensons to ESPResSo, for example a second order accurate Langevn ntegrator. A full lst s set asde n order to put emphass on those parts that are relevant for ths work. 163

184 164

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