Analysis of Lattice Boltzmann Boundary Conditions

Transcription

1 Analyss of Lattce Boltzmann Boundary Condtons Dssertaton zur Erlangung des akademschen Grades des Doktors der Naturwssenschaften (Dr. rer. nat.) an der Unverstät Konstanz Mathematsch-Naturwssenschaftlche Sekton Fachberech Mathematk und Statstk vorgelegt von M.Sc. Zhaoxa Yang Tag der mündlchen Prüfung : 25. Jul 27 Referent: Prof. Dr. Mchael Junk Referent: Prof. Dr. Vncent Heuvelne

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3 to my parents

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5 Preface Ths dssertaton s the summaton of the work at the Unversty of Kaserslautern, the Unversty of Saarland and the Unversty of Constance startng from the end of 21, whch s fnancally supported by the German Research Foundaton (Deutsche Forschungsgemenschaft). I would lke to sncerely express my deepest thanks and grattude to my supervsor Prof. Dr. Mchael Junk for provdng me the opportunty to acheve ths work, and for the gudance and the numerous dscussons and suggestons, and for the careful proof readng of ths dssertaton. Ths work would not be possble wthout hs constant support and wsdom. I would also lke to express my grateful thanks to Prof. Dr. Vncent Heuvelne for refereeng my PhD thess. I sncerely thank Prof. Dr. L-Sh Luo and Prof. Dr. Wen-an Yong for useful dscussons and encouragement and sharng useful references. My thanks also go to those people, from whom valuable comments are suppled n ICMMES and dfferent workshops. I gratefully thank all my frends n Bejng, Shenzhen, Qngdao, Hongkong, Kaserslautern and Konstanz for ther care and help and support. I keep each happy moment wth them n my memory forever. Specal thanks to my parents for gvng me lfe and endless love wherever I am. Thanks to my sster and brothers for ther care, understandng, support, and uncondtonal acceptance to me. I am grateful for that from the deepest of my heart.

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7 Abstract In ths dssertaton, we nvestgate a class of standard lnear and nonlnear lattce Boltzmann methods from the pont of vew of mathematcal analyss. Frst we study the consstency of the lattce Boltzmann method on a bounded doman by means of asymptotc analyss. From the analyss of the lattce Boltzmann update rule, we fnd a representaton of the lattce Boltzmann solutons n form of truncated regular expansons, whch clearly exhbt the relaton to solutons of the Naver-Stokes equaton. Through the analyss of the ntal condtons and the well-known bounce back boundary rule, we demonstrate the general procedure to ntegrate the boundary analyss process n the whole analyss, and fnd that our approach can relably predct the accuracy of the lattce Boltzmann solutons as approxmatons to Naver-Stokes solutons. Next, a rgorous convergence proof s acheved for the class of standard lnear and nonlnear lattce Boltzmann methods consdered n ths thess. Concentratng on realzatons of Drchlet velocty boundary condtons, we then nvestgate the consstency of several exstng mplementatons, predct ther accuracy, and ther advantages and shortcomngs. In order to overcome a general drawback of the methods, we construct a class of purely local boundary treatments. All of these methods lead to a second order accurate velocty and a frst order accurate pressure. A careful numercal comparson of ther propertes such as stablty, mass conservaton and error behavor s presented, as well as a gude for choosng a boundary mplementaton among the varous possbltes. Regardng Naver-Stokes outflow condtons whch are hardly studed n the lattce Boltzmann lterature, we deal wth three knd of Neumann-type condtons. We have proposed ther mplementatons n the lattce Bolzmann framework, and brefly carry out ther consstency analyss. Several numercal results demonstrate the capablty of these outflow treatments. For the unsteady benchmark problem lke flows around fxed cylnders n an nfntely long channel, the proposed do-nothng and zero normal stress condtons perform very well. For the steady flow, all of the methods produce convncng results.

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9 Zussamenfassung In deser Arbet werden Konsstenz, Konvergenz und Randbedngungen für ene Klasse von Gtter-Boltzmann Verfahren behandelt, welche m wesentlchen zur numerschen Lösung strömungsdynamscher Glechungen we der Naver- Stokes Glechung engesetzt werden. Insbesondere stehen Randbedngungen vom Drchlet bzw. Neumann Typ m Mttelpunkt. De Konsstenzanalyse basert auf ener asymptotschen Entwcklung der numerschen Gtter-Boltzmann Lösung. Dabe wrd de numersche Lösung durch ene abgeschnttene reguläre Entwcklung approxmert, anhand derer sch de Verbndung zur Naver-Stokes Glechung herstellen läßt. Zunächst wrd de Konsstenzanalyse zur Untersuchung von Anfangsbedngungen und der bekannten Bounce-back Randbedngung herangezogen und bespelhaft erläutert. Es stellt sch heraus, daß der analytsche Zugang ene zuverlässge Vorhersage der Genaugket (Konvergenzordnung) des Gtter-Boltzmann Verfahrens ermöglcht. De theoretschen Untersuchungen werden durch enen Konvergenzbewes für das Gtter-Boltzmannverfahren abgerundet. Dabe werden neben perodschen Randbedngungen auch de klassschen Bounce-back Randbedngungen betrachtet. Im anwendungsorenterten Tel werden verschedene, berets exsterende Umsetzungen der Drchlet Randbedngung n Bezug auf Konsstenz und sonstge Vor- und Nachtele verglchen. Des führt zur Konstrukton ener neuen, lokalen (En-Knoten) Randbedngung, welche de Schwächen anderer Randbedngungen überkommt, ohne wesentlche Vortele enzubüßen. Konsstenzanalysen und numersche Test zegen, daß alle Randbedngungen von zweter Approxmatonsgüte m Geschwndgketsfeld snd, während der Druck nur mt erster Genaugketsordnung berechnet werden kann. Ergänzend werden Tests zur Stabltät und Massenerhaltung der Randbedngungen durchgeführt. Trotz hrer praktschen Bedeutung snd Ausflußbedngungen für de Naver- Stokes Glechung bsher kaum n der Gtter-Boltzmann Lteratur dskutert worden. Zur Umsetzung deser Neumann-artgen Randbedngungen werden her dre verschedene Ansätze verfolgt: de gewöhnlche Neumannbedngung, de Bedngung verschwndender Normalspannung sowe de do-nothng Bedngung. Anhand numerscher Smulatonen statonärer und transenter Kanalströmungen (mt feststehendem Hnderns) werden de Bedngungen erprobt. Während m statonären Falle alle Randbedngungen zufredenstellende Resultate lefern, überzeugen allen de beden letzteren be zetabhänggen Strömungen.

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11 Contents 1 Introducton Bref hstory Scope of ths work Consstency Convergence Boundary condtons Publcatons Synopss of ths work Lattce Boltzmann method Prelmnares and Notatons Descrpton of the lattce Boltzmann method Intal condton Boundary condton Perodc boundary Bounce back boundary Consstency of LBM on a general bounded doman Formal asymptotc expanson Asymptotc expanson for nonlnear LBM Asymptotc expanson for lnear LBM Model flows wth analytc solutons Lnear flow Poseulle flow Decayng Taylor vortex Crcular flow Analyss of ntal condton

12 CONTENTS 3.4 Numercal tests of ntal condtons Analyss of bounce back rule Numercal test of bounce back rule Defnton of consstency order Convergence of the lattce Boltzmann method Stablty of the lnear BM Functon space Reformulaton of lattce Boltzmann algorthms on perodc domans Reformulaton of lattce Boltzmann algorthms wth the bounce back rule Norms and stablty The truncated asymptotc expanson and moment relatonshp Convergence of the lnear LBM Convergence of the nonlnear LBM Recurson nequalty Convergence results Consstency of the truncated expanson Perodc cases Cases wth the bounce back rule Comments on convergence and consstency Perodc case Drchlet case Drchlet boundary condtons and consstency analyss Improvements of bounce back rule FD The boundary-fttng method (FH) and ts mprovement (MLS) Bouzd rule (BFL) Mult-reflecton method (MR) One pont boundary scheme (POP 1 ) Numercal comparson Comparson of accuracy Stablty behavor Investgaton of total mass Summary

13 CONTENTS 6 Neumann type outflow boundary condtons Asymptotc analyss of two known outflow schemes Neumann boundary (NBC) Zero normal shear stress boundary (ZNS) Do nothng condton (DNT) Experments and dscusson Flow around a crcular cylnder Flow around a square cylnder The nfluence of ntal values The nfluence of Drchlet boundary Summary Conclusons 146 A Fgures for the smoothly started flows around a crcular cylnder 147 A.1 Zero Neumann condtons (NBC) A.2 Zero normal stress condton (ZNS) A.3 Do nothng condton (DNT) B Fgures for the mpulsvely started flow around a crcular cylnder 154 C Fgures for the unsteady flows around a square cylnder 156 C.1 Zero Neumann condtons (NBC) C.2 Zero normal stress condton (ZNS) C.3 Do-nothng condton (DNT) D Defnton of V and f and c s 161 D.1 D2Q D.2 D3Q D.3 D3Q D.4 D3Q Bblography 166

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15 Chapter 1 Introducton 1.1 Bref hstory The lattce Boltzmann method (LBM) orgnated from the lattce gas automata (LGA) [8, 28, 23, 77, 22, 57, 39, 36, 6, 66, 67, 11, 12] by takng ensemble averages on the related evoluton equatons. The motvaton for the transton from LGA to LBM s to get rd of the nose due to the fluctuaton of partcle numbers. The lattce Boltzmann method as an ndependent numercal method s frst ntroduced by McNamara and Zannet [57] n 1988 to smulate hydrodynamc problems. Later n order to mprove the computatonal effcency whch s manly pulled down by the cumbersome collson process, the lnearzed collson operator [36] s preferred. For smplcty, some varatons are developed ncludng the well-known sngle relaxaton tme BGK model [6] and the MRT (mult-relaxaton tme) model [17]. Snce then the applcaton of the lattce Boltzmann method s smplfed and greatly ncreased. Wthout tracng back to ts predecessor, the lattce Boltzmann method can also be vewed as a specal fnte dfference scheme for a contnuous Boltzmann equaton wth fnte number of veloctes by usng a small Mach number expanson and by dscretzng the phase space and tme n a coherent manner. The detaled dervaton can be found n [33, 3]. From ths pont of vew, many numercal technques for solvng PDEs can be used to mprove the lattce Boltzmann method. Although the lattce Boltzmann method s formulated on the level of partcles, ts prncpal applcaton focuses on the macroscopc behavor. Hstorcally the Chapmann-Enskog analyss wth a formal multscale expanson [22] shows that the Euler equatons are recovered on the fast convectve tme scale, and the Naver-Stokes equaton from the slower dffusve tme scale n the near ncompressble lmt (due to low Mach number). Because of the knetc nature, the lattce Boltzmann method has certan features that dstngush t from conventonal CFD methods to solve the Naver-Stokes equaton drectly. A great deal of practce has demonstrated the success of the lattce Boltzmann method as well as ts varatons to meet dfferent applcatons, partcularly to smulate complex fluds such as porous meda flow, multphase flow, multcomponent flow, granular flow, etc.. 1

16 2 CHAPTER 1. INTRODUCTION As for the boundary condton, the bounce back rule [77, 51] s the classcal approach n lattce Boltzmann smulatons, whch s used to smulate the nonslp sold wall and later proved not to be accurate enough [16, 8, 24, 34]. Thereafter much effort has been made to construct more accurate boundary schemes than the bounce back rule. The early boundary treatments appeared as varatons of the bounce back rule and were restrcted to very smple regular boundary geometres ([71, 62, 63, 4, 54, 14, 82, 81]), for example flat walls. The velocty and/or densty must be known at the node where the boundary scheme s formed. Besdes, these methods are more or less dependent on some flow propertes too. The so-called hydrodynamc boundary condton n [63] based on the fxed nternal energy beng equal to the square of sound speed for the steady-state hydrodynamc feld, s such an nstance. Later, more endeavor s dedcated to seekng for methods whch deal wth curved boundares. In 1998, Fllpova suggested a purely local method FH ([21, 2]). However the algorthm nvolves the factor 1/(τ 1) and s unstable when τ s close to 1, where τ s the tme relaxaton parameter n the collson operator. In [58, 59], Me et al. suggested an mprovement (called MLS) to ths method by nvolvng one more neghbor node and thus enlargng the stablty regon, but wthout overcomng the nherent drawback of FH. In 21, Bouzd et al. [7] proposed a dfferent lnk-based method (BFL) by means of lnear nterpolaton wth one or two neghbor nodes. Later Gnzburg and d Huméres submtted a more general soluton n [25]. Besde the exceptons lke MR n [25], the other methods mentoned here can be seen as drect extensons of the bounce back rule whch wll be demonstrated clearly later. From the pont of vew of mathematcal analyss, the lattce Boltzmann method s a knetc relaxaton method [1] for the macroscopc equatons. The ssues of convergence, consstency and stablty are as mportant as for other fnte dfference methods. A von Neumann stablty analyss s carred out n [72, 49] for lnearzed lattce Boltzmann methods. In varous publcatons, accuracy and stablty are checked numercally [68, 64, 78, 53, 76, 38, 4, 56, 1]. However, rgorous analyss results are very lmted untl now. For example, Elton [19, 18] studed the convergence, stablty and consstency of lattce Boltzmann methods for the vscous Burgerś equaton and advecton-dffuson systems. For a class of lattce Boltzmann methods solvng for Naver-Stokes equaton on a perodc doman, the consstency [42, 45] and stablty have also been acheved. However, the analyss of the lattce Boltzmann method on general bounded domans wth general boundary condtons s stll under consderaton. Besdes, most of already matured boundary condtons reflect only the Drchlet case for velocty at the macroscopc level. Other boundary treatments nvolvng dervatves of the flud velocty are much less studed. The am of ths thess s to fll some of these gaps.

17 1.2. SCOPE OF THIS WORK Scope of ths work Consstency The classcal Chapman Enskog analyss wth two tme scales, whch s usually taken as bass for the analyss of lattce Boltzmann schemes (see for example [22, 5, 12, 13, 31, 33]), leads to a compressble Naver-Stokes equaton. The ncompressble Naver-Stokes equaton s obtaned consecutvely by another lmt process. In [43], we have successfully appled the regular expanson and multscale expanson n asymptotc analyss to fnte dfference methods and meanwhle compared wth other classcal consstency analyss methods such as the truncaton error analyss and the modfed equaton approach. The merts of the approach are: () The asymptotc analyss predcts and analyzes the behavor of fnte dfference solutons order by order very precsely. Usually the lower order coeffcents produce the exact soluton of the partal dfference equatons, the hgher order coeffcents show the numercal error. () Asymptotc analyss also makes t possble to nvestgate the behavor of ntal layers, boundary layers and oscllaton, by changng the scale approprate to the layer thckness. () Boundary condtons can be drectly ntroduced nto the analyss process and the overall consstency accuracy can be predcted exactly. Compared to other fnte dfference methods for the ncompressble Naver- Stokes equaton, the lattce Boltzmann method s very partcular, snce the hydrodynamc macroquanttes are connected to the mcroquanttes by means of averages accordng to knetc theory. More precsely, from a mathematcal pont of vew, the soluton of the Naver-Stokes equaton results from an asymptotc sngular lmt of the lattce Boltzmann soluton. We show that the chosen approach can smply and easly provde a consstency analyss of boundary algorthms and smultaneously of the whole scheme. The success and advantages of the asymptotc analyss approach ndcates that t s a very approprate tool for the lattce Boltzmann method on a bounded doman. In [42], the authors also advocate the regular expanson by llustratng the consstency of a wde class of lattce Boltzmann equaton on the whole space or on a perodc doman. A comparson wth the Chapman-Enskog expanson s also gven at last. Comparatvely, the asymptotc analyss relates the numercal soluton to the exact soluton and the error n a more straghtforward manner. In ths work, we contnue the asymptotc analyss as n [42] and apply t to the consstency analyss of lattce Boltzmann methods on general bounded domans. We wll see that t plays an mportant role n the rgorous convergence proof too Convergence Very few results about convergence have been presented so far. For nstance, Elton [18, 19] establshed the convergence theory for nonlnear convectve-dffusve

18 4 CHAPTER 1. INTRODUCTION lattce Boltzmann methods, but hs results apply to schemes wth non-lnear collson operators satsfyng an H-theorem. Moreover, hs work s focused on the ntal value problem,.e. the spatal doman s ether the whole space or perodc. The use of truncated regular asymptotc expansons n the proof of convergence can be dated back to [73], and has been successfully appled to the lattce Boltzmann case n [18, 19]. We contnue the dea n ths work to prove the convergence for a class of standard lnear and nonlnear lattce Boltzmann methods based on a rgorous stablty result [44] Boundary condtons Implementng boundary condtons n the lattce Boltzmann settng s dffcult because there s no one-to-one mappng between the varables of the algorthm, the so-called partcle dstrbutons, and the predetermned hydrodynamc macroquanttes gven at the boundary. For example, f Drchlet boundary condtons are to be mplemented, one cannot drectly prescrbe the velocty at boundary nodes but one has to set the partcle dstrbutons n such a way that the average velocty satsfes the requred condtons. Typcally, the requred number of condtons on the knetc level exceeds the avalable condtons from the Naver-Stokes problems. Ths ndcates that the knetc condtons have to be chosen carefully n order to avod the appearance of extra condtons on the Naver-Stokes level whch would render the problem ll posed (leadng to an unwanted behavor on the grd scale lke boundary layers, oscllatons etc.). For other cases lke outflow condtons or condtons at free surfaces, the stuaton s addtonally complcated by the fact that the relaton between the partcle dstrbutons and the quanttes specfed n the boundary condtons s less evdent than n the Drchlet case. In ths work the Drchlet and outflow boundary condtons are studed n detal. Our ams are: () to analyze the consstency of exstng lattce Boltzmann boundary condtons, () to construct new accurate boundary condtons and carry out the related consstency analyss, and () to offer a practcal gude for choosng a boundary mplementaton among all the possbltes Publcatons Ths work s supported by the Deutsche Forschungsgemenschaft (DFG). Several results acheved n ths thess have already been publshed n collaboraton wth Prof. Junk. However, the presentaton here s generally more detaled wth more numercal examples and a unform use of MRT models n the dervaton. 1. Analyss of lattce Boltzmann boundary condtons, Proc. Appl. Math. Mech. 3(23) Asymptotc analyss of fnte dfference methods, Appl. Math. Comput. 158(24)26731.

19 1.3. SYNOPSIS OF THIS WORK 5 3. Asymptotc Analyss of lattce Boltzmann boundary condtons, J. Stat. Phys. 121(25), One-pont boundary condton for the lattce Boltzmann method, Phys. Rev. E, 72,1(25). 5. Convergence of lattce Boltzmann methods for Stokes flows n perodc and bounded domans, Int. J. of Comp. Flud Dynamcs, Vol. 2 No. 6(26). 6. Convergence of lattce Boltzmann methods for Naver-Stokes flows n perodc and bounded domans, submtted to Numer. Math. 7. Outflow condtons for the lattce Boltzmann method, Progress n Computatonal Flud Dynamcs, to appear 1.3 Synopss of ths work Snce the analyss of lattce Boltzmann methods together wth the varous boundary treatments s the man task, the layout of ths work s ordered accordng to the macroscopc boundary condtons. In chapter 2, we descrbe the lattce Boltzmann methods studed n ths work. In partcular, the requred propertes of the collson operators, dscrete velocty sets and the related weght functons are dscussed. It s stressed that the consdered class of lattce Boltzmann methods contans the generally used BGK [6] and MRT [17] models as well as D2Q9, D3Q15, D3Q19 and D3Q27 dscrete velocty sets and weght functons [67, 58]. In secton 2.3, a feasble ntal condton s proposed followed by a lst of boundary condtons for well-posed Naver-Stokes problems. To some of them, the correspondng treatments n the lattce Boltzmann framework are also ntroduced. The Drchlet condton and Neumann-type outflow condtons are carefully studed n chapters 5 and 6. In chapter 3, the man task s to demonstrate the procedure of consstency analyss of lattce Boltzmann methods on general bounded domans by means of asymptotc analyss. The frst step s to obtan the explct expressons for the coeffcents f (k) n the regular asymptotc expanson of the lattce Boltzmann soluton f (n,j) = f () (t n,x j ) + hf (1) (t n,x j ) + h 2 f (2) (t n,x j ) +... at least to ffth order n h. Ths s done n secton 3.1 for both lnear and nonlnear lattce Boltzmann methods by revewng the results of [42]. For lnear flows and the polynomal Poseulle flow, the coeffcents f (k) are computed explctly. Based on these known coeffcents we obtan a truncated expanson, whch s also a representaton of the lattce Boltzmann solutons to some extent.

20 6 CHAPTER 1. INTRODUCTION Ths truncated expanson clearly shows a relaton between the Naver-Stokes solutons and the lattce Boltzmann solutons, as well as the numercal error terms governed by several Oseen-type equatons. The frst order error terms can dsappear f the correspondng ntal and boundary values are zero. The second order error terms, however, usually can not be zero. Next, the analyss of ntal condtons [42, 44] s recalled n secton 3.3. Three ntal treatments are explaned and numercally tested. The most mportant nformaton s that the frst order and second order accurate treatments can be generally mplemented. On the contrary, the thrd order accurate approach s generally not feasble because the requred values depend on the ntal tme dervatve of flud pressure. It s also remarked that the second order accurate ntal condton generally leads to a frst order accurate pressure and second order accurate velocty. Last we come to the analyss of boundary condtons whch s the core part of ths chapter. In secton 3.5 we take the bounce back rule as an example to demonstrate the boundary scheme analyss n detal. We nsert the regular expanson n the bounce back rule, and do Taylor expanson at a sutable pont on the boundary, then collect terms accordng to the order of h. By mposng the resdue of the boundary scheme to be of hgh order n h, we acheve the accuracy of the boundary scheme as well as of the entre algorthm. In the case of the bounce back rule we encounter condtons at second order whch have a structure that s very typcal and common n the resdue for other lattce Boltzmann boundary schemes and whch cannot be satsfed by smooth functons. Ths ndcates rregular behavor whch s actually observed n the error of the velocty (see the followng fgure for the cases of lattce Boltzmann solutons of the crcular flow (Left) n a unt dsk and the Taylor vortex flow n a unt square) x Fgure 1.1: Left: Plot of velocty error (horzontal component ( ) and vertcal component ( )) at t =.6 for the crcular flow. Rght: Plot of horzontal component of velocty error along cuts x =.1 ( ), x =.3 ( ) for the decayng Taylor vortex flow.

21 1.3. SYNOPSIS OF THIS WORK 7 In chapter 4, a rgorous convergence proof for lnear and nonlnear lattce Boltzmann methods s gven n the case of perodc boundares as well as the Drchlet case wth bounce back rule at half lnks. Summarzng the consstency and convergence results, we have found that, n perodc cases, the regular expanson representaton s generally consstent and convergent to the lattce Boltzmann soluton n the same order, whch s the order of the resdue concerned wth the ntal treatments. Ths concluson concdes wth the analytcal results n secton 3.3 and s verfed by the numercal tests n secton 3.4. It showed that the convergence order of moments to the Stokes or Naver-Stokes solutons on a perodc doman s generally determned only by the ntal error, and s up to two for both the velocty and the pressure of the flud, provded that the solutons are suffcently regular. In Drchlet cases, due to the coarse estmate, we do not acheve as good consstency and convergence results as n the perodc cases. Nevertheless, from the convergence proof of the nonlnear lattce Boltzmann method, we see that, the grd sze s requred to be sutably small f the Naver-Stokes soluton has strong gradent and the Reynolds number s hgh. In chapter 5, the consstency and accuracy of boundary schemes smulatng velocty Drchlet condtons s studed by usng the asymptotc analyss developed n chapter 3. Frst, we nvestgate several exstng boundary schemes and dsplay ther advantages and drawbacks. It turns out that the fnte dfference technques, FH, MLS, and BFL can be consdered as mprovements of the bounce back rule, n the sense that the leadng order error terms of the bounce back method are removed, and generally lead to 2nd order accurate velocty and 1st order accurate pressure. Mult-reflecton methods (MR) try to use the nformaton at three nodes to get hgher accuracy up to 3rd order. However, the fnte dfference technque, MLS, BFL and MR are not local and employ two or three nodes. For nodes where the requred neghbors are not avalable, these methods are no more applcable. Usually the bounce back rule s suggested as a remedy at these ponts. However we fnd that the low order error of bounce back rule s transported gradually everywhere nto the doman. the FH method s local, but leads to unwanted velocty condton at certan nodes. the fnte dfference technque and the MR method show very bad stablty n our numercal tests. FH and MLS methods depend on 1/(τ 1) and 1/(τ 2) respectvely whch eventually leads to nstablty when τ 1 or 2. Only BFL shows satsfyng stablty. In order to overcome the shortcomngs of the above-mentoned methods whle remanng the advantages, a class of boundary treatments POP θ (θ [,1]) s developed n secton 5.3 whch has the followng characterstcs:

22 8 CHAPTER 1. INTRODUCTION POP θ are local. POP s explct and POP 1 s purely mplct. the numercal stablty of POP θ s smlar to the one of BFL. POP θ leads to 2nd order accurate velocty and 1st order accurate pressure. In addton, n all of our numercal tests POP θ leads to smaller error and less total mass varatons compared to BFL. The followng fgures are examples to llustrate ther behavors e e Fgure 1.2: Error contour lnes of x-component velocty for the statonary lnear flow (3.82) wth ν =.1 and boundary condtons: BFL (left) and POP 1 (rght). x 1 7 x log ρ(t) 2 1 log ρ(t) log 1 (h) log 1 (h) Fgure 1.3: Devaton of the average densty from ts ntal value ( ρ(t)) versus tme on a grd of sze h = 1/1( ), 1/2( ), 1/3( ), 1/4(+) and 1/5( ) for a crcular flow. Rght: POP 1 ; Left: BFL. In chapter 6, lattce Boltzmann algorthms are constructed to deal wth Neumanntype outflow boundary condtons ncludng the Neumann boundary condton (NBC), the Do-nothng condton (DNT) and the zero normal stress condton (ZNS). The related consstency s also presented whch shows that all these

23 1.3. SYNOPSIS OF THIS WORK 9 treatments lead to 1st order accuracy wth respect to the correspondng outflow condtons. As a benchmark problem, we use the flow around a fxed cylnder n an nfnte long channel to evaluate these outflow boundary condtons. One aspect s to check the drag coeffcent C d, the lft coeffcent C l and the pressure dfference P between the front and end pont of the cylnder, whch can reflect how the outflow boundary condton nfluences the nner flow relatvely far away from the outflow boundary. Another aspect s to check the behavor of the flow at the outflow boundary tself. For the steady flow around the cylnder, all these three outflow condtons work very well and produce results for C d, C l and P n the reference nterval f the grd s suffcently fne. Besdes, the length of channel does not have a strong effect on these quanttes. However, for the unsteady flow around the cylnder, we fnd that the length of the channel has obvous effects on the values of C d, C l and P. The NBC outflow condton has a strong mpact on the nner flow. On the contrary, ZNS and DNT lead to less varatons of the velocty and pressure. Only a slght phase dfference occurs when the channel length s vared. Fgure 1.4 shows the comparson of the pressure along the center lne of the channel. Fgures show plots of the flud streamlnes. When the channel s too short, the streamlnes are deformed apparently f NBC s employed. On the contrary, ZNS and DNT show a very smlar flow structure even n short channels x x x Fgure 1.4: The pressure along the center lne of the channel for the unsteady flow wth ZNS (left), DNT (mddle) and NBC (rght) outflow condtons respectvely. The symbols +, and stand for the length/wdth rato of the channel of 2, 3 and 5. The comparson s restrcted to the common x-regon of the three channels.

24 replacemen1 CHAPTER 1. INTRODUCTION Fgure 1.5: The truncated streamlnes wth NBC n channels of dfferent length/wdth rato 2 (left), 3 (mddle) and 5 (rght). A clear channel length dependence s vsble Fgure 1.6: The truncated streamlnes wth ZNS (left) and DNT (rght) n channels wth length/wdth rato 2 (upper row), 3 (mddle row) and 5 (lower row). The channel length dependence s much weaker than that for NBC condton.

25 Chapter 2 Lattce Boltzmann method The lattce Boltzmann method has many varatonal models due to dfferent collson operators, varous dscrete velocty sets and the nvolved numercal technques after realzng that the lattce Boltzmann method s a fnte dfference approxmaton of the contnuous dscrete Boltzmann equaton [31, 33, 41], for example lattce Boltzmann model on non-unform lattce. There are also a lot of generaton models of the lattce Boltzmann method to meet varous applcatons such as multphase flows, subgrd scale modelng, flows nvolvng energy, etc. Naturally not all knds of lattce Boltzmann models could be studed n one thess. Ths chapter s dedcated to hghlght a standard class of lattce Boltzmann models used n ths text, whch s supposed to be a numercal solver for the nondmensonal ncompressble Naver-Stokes equaton: u =, t u + p + (u )u = ν 2 u + G, (2.1) wth an ntally dvergence free velocty feld and a compatble boundary condton satsfyng u(,x) = ψ(x), x Ω, (2.2) u(t,x) = φ(t,x), x Ω (2.3) Ω φ n =. Here Ω R d s the flow doman, u(t,x) s the velocty feld of the flud, p(t,x) represents the flow pressure, ν s the flud vscosty. G(t, x) stands for the body force. Before the detaled descrpton some basc notatons and defnton are gven. 2.1 Prelmnares and Notatons In ths secton we ntroduce the notaton whch s used n the later parts of ths work. 11

26 12 CHAPTER 2. LATTICE BOLTZMANN METHOD Throughout ths text, the doman Ω denotes a bounded, non-empty, open set n space R d (d = 2,3). Ω s ts boundary and Ω ts closure. The lattce Boltzmann method s formulated at the dscrete nodes n the doman Ω. At each node there s a fnte number of possble velocty drectons. For example, the followng fgure depcts the node dstrbuton n a two dmensonal doman wth a velocty set of D2Q9 type (see appendx D.1 for detals) Ω Ω Fgure 2.1: Node dstrbuton plot n a 2D case wth velocty drectons of the D2Q9 model. Flud nodes n Ω are marked by flled crcles, the non-flud nodes are marked by hollow crcles. The connecton between two nodes along some velocty drecton s called lnk. The nodes are classfed nto two types, flud nodes and non-flud nodes (see the above fgure). A flud node s called an ordnary node f all of ts closest neghbors along the lnks are flud nodes. Correspondngly, a flud node called boundary node has at least one non-flud node as neghbor. In fgure 2.1, the black crcles are ordnary nodes. The grey nodes are boundary nodes and the whte crcles are non-flud nodes. In addton, a lnk s called boundary lnk (see dashed lne n fgure 2.1) f t connects a boundary node and a non-flud node. For the varables n our work we adopt the followng conventons, The bold face symbols such as f,u,v always represent vectors. Greek subscrpts stand for the spatal coordnates. Roman subscrpts represent the ndces of dscrete veloctes. Let V = {c 1,...,c N } R d denote a fnte dscrete velocty set, and 1 be the vector n R N wth all entres one, we ntroduce d vectors n R N based on V, v α = [c 1α,...,c Nα ] T (2.4) where c jα s the αth component of vector c j, and α s taken from 1 to d. Next we defne d operators V α : R N R N by V α f = dag(v α )f = [c 1α f 1,...,c Nα f N ] T. (2.5)

27 2.1. PRELIMINARIES AND NOTATIONS 13 Among V α, any two of them commute, V α V β = V β V α, α, β {1,2,...,d}. (2.6) Obvously V α s a lnear mappng on the vector space R N and V α 1 = v α. A vector operator s hence generated by settng V = [V 1,...,V d ] T, and V : R N R d N, Vf = [V 1 f,...,v d f] T, f R N. (2.7) Then V 2 = V 2 1 +V V 2 d s an operator from RN to R N. Let denote the nner product n the space R d. Observng that = [ 1,..., d ] T s also a vector operator wth the same dmenson as V, we can thus defne a composton by V = d V α α, (2.8) α=1 so that V s an operator from F = C 1 (R d, R N ) to R N. Snce V α and α commute, V and also commute wth respect to,.e., V = V. If f s an arbtrary functon n F, then ( V)f = [(c 1 )f 1,...,(c N )f N ] T. (2.9) Moreover, for an arbtrary vector π n R d, (π V) s agan an operator from R N to R N. It turns out to be (π V)f = d π α V α f = [(π c 1 )f 1,...,(π c N )f N ] T. (2.1) α=1 Further let, denote the nner product on the vector space R N. If f, g R N are two arbtrary vectors, then the followng products wll occur frequently n the lattce Boltzmann context, 1,f = N f, (2.11) =1 1,V α f = v α,f = V β g,v α f = g,v β V α f = N c α f, (2.12) =1 N c α c β f g. (2.13) Snce V s a vector operator, we can apply the nner product to each of ts component, for example, =1 g,vf = [ g,v 1 f,..., g,v d f ] T. (2.14) It s easy to prove that g,vf = Vg,f and n partcular, 1,Vf = [ v 1,f,..., v d,f ] T = N c f. (2.15) =1

28 14 CHAPTER 2. LATTICE BOLTZMANN METHOD In addton f a tensor product between two vectors a and b s defned by (a b) j = 1 2 (a b j + a j b ), (2.16) we can defne a matrx operator V V and calculate g,v Vf αβ = ( V α g,v β f ) αβ. (2.17) Moreover, a related calculaton 1,V(V )f = 1,V Vf (2.18) s frequently used n ths work. Fnally the : product between matrces A and B, A : B = A j B j,j (2.19) s ntroduced. 2.2 Descrpton of the lattce Boltzmann method To begn wth, we dscretze the space R d by a regular cubc lattce usng a unform grd sze h. The grd ponts are x j = hj wth j Z d. Collectng all the grd ponts n Ω, we get a dscretzaton of the doman Ω. Lkewse, we dscretze the tme doman smlarly by placng a grd on the temporal nterval [, ) wth grd spacng t and grd ponts t n = n t, n N. In the correspondng lattce Boltzmann setup to smulate (2.1), the tme step s t = h 2. Ths rato between tme and space step s related to the dffusve scalng of the knetc equatons. For detals we refer to [42]. On ths lattce, the update rule of the lattce Boltzmann method s descrbed by f (n + 1,j + c ) = f (n,j) + J(f) (n,j) + g (n,j), (2.2) where V = {c 1,...,c N } R d s the fnte dscrete velocty set. The numbers f (n,j) represent the partcle dstrbutons related to velocty c at tme level t n and node x j. The functon g models the body force (n appendx D, the set V, f and c s for several well known models are gven) g (n,j) = c 2 s h 3 f c G(t n,x j ). J s the collson operator, and chosen n ths work to be of general relaxaton type J(f) = A(f eq (f) f), (2.21) where A s a lnear mappng and f eq s a so-called equlbrum functon. The common used models for ths knd of relaxaton type collson nclude the famlar BGK model and MRT. Obvously, the collson s determned by several basc ngredents: the dscrete velocty set V, the equlbrum f eq and the lnear

29 2.2. DESCRIPTION OF THE LATTICE BOLTZMANN METHOD 15 mappng A. Wthout restrcton to some specfc model, we state our assumptons contanng the wdely used models such as BGK and MRT. The velocty set V admts a symmetry property,.e., V = V. (2.22) Those velocty sets for the well known models D2Q9, D3Q15, D3Q19 and D3Q27, whch are dsplayed n appendx D, all possess ths property. We use the equlbrum functon recommended n [32] for the ncompressble lattce Boltzmann model, whch s based on the assumpton that the flud densty slghtly fluctuates around a constant ρ. In ths work we set, wthout loss of generalty, ρ = 1. (2.23) Hence the equlbrum functon s of a polynomal form: [ ] f eq (f) = F eq (ˆρ,û) = f ˆρ + c 2 s û c + c 4 s 2 (û c ) 2 c 2 s 2 û 2 (2.24) wth respect to the total mass densty ˆρ and the average velocty û (or more precsely, the average momentum ρû), ˆρ = N f, û = =1 N c f. (2.25) Related to the velocty set V, the weght functon f = F eq (1,), whch s also called constant equlbrum dstrbuton, obeys the symmetry property and s dentfed by the followng constrants N f = 1, =1 =1 f = f, f c = c, (2.26) N c α c β f = c 2 s δ αβ, (2.27) =1 N c α c β c γ c δ f = c 4 s(δ αβ δ γδ + δ αδ δ βγ + δ αγ δ βδ ). (2.28) =1 For the convenence n the later use, we splt the equlbrum functon nto two parts f eq (f) = f L (f) + f Q (f,f), (2.29) namely a lnear part f L and a quadratc part f Q, here f s the vector wth components of f. Defnng them n a concse way, the lnear part s wrtten as f L (f) = F L (ˆρ,û), F L (ˆρ,û) = (ˆρ + c 2 s û V)f, for any f F wth ˆρ and û defned by (2.25). The quadratc part s defned by f Q (f,s) = F Q (û,ŵ), (û,ŵ) = c 4 [ s (û ŵ) : (c c c 2 2 si)f ], F Q

30 16 CHAPTER 2. LATTICE BOLTZMANN METHOD where s F and 1,Vs = ŵ. Due to the propertes of f n (2.27) (2.28), we can fnd out and 1,F L (ˆρ,û) = ˆρ, (2.3) 1,VF L (ˆρ,û) = û, (2.31) 1,V VF L (ˆρ,û) = c 2 s ˆρI, (2.32) 1,F Q (û,ŵ) =, (2.33) 1,VF Q (û,ŵ) =, (2.34) 1,V VF Q (û,ŵ) = û ŵ. (2.35) After gvng the equlbrum functon f eq, condtons to determne the lnear mappng A are requred n order to completely fx the collson operator. These are: () A s symmetrc; () A s postve sem-defnte; () K = {1,v 1,...,v d } generates the kernel of A. (v) AΛf = c 2 s/µλf wth Λ = V V 1/d V 2 I and µ = ν + c 2 s/2, where condton (v) means that the components of Λf are egenvectors of the collson matrx A wth egenvalue c 2 s/µ. Remark 1 Let Q be the orthogonal projecton onto the kernel of A and P := I Q the projecton on the complement. Then we defne A = (A P R N) 1 P to be the peseudonverse of A, and A has the followng propertes: QA = A Q =, PA = A P = A, AA = A A = P (2.36) Remark 2 The so-called BGK collson operator J(f) = 1 τ (f(eq) f) s a specal case consdered here. A = 1 τ P wth τ = µ/c2 s s a partcular choce whch satsfes all the above condtons () to (v), and A = τp. Moreover P(f (eq) f) = (P + Q)(f (eq) f) = (f (eq) f), (2.37) snce f (eq) f s orthogonal to the kernel of A whch s easly checked by observng (2.3) to (2.35). Snce the equlbrum functons have a lnear and quadratc part and A s a constant matrx, the collson operator also conssts of two parts, wth a lnear collson operator J(f) = J L (f) + J Q (f,f), J L (f) = A(f L (f) f), (2.38)

31 2.2. DESCRIPTION OF THE LATTICE BOLTZMANN METHOD 17 and a quadratc operator J Q (f,f) = Af Q (f,f). (2.39) Besdes, observng that the term concerned wth the quadratc collson operator s the only nonlnear part n the lattce Boltzmann scheme (2.2), hereafter the lattce Boltzmann method wth only a lnear collson operator s called the lnear lattce Boltzmann method, whereas the lattce Boltzmann method wth both lnear and quadratc collson operators s called nonlnear lattce Boltzmann method. In ths work, f not partcularly ponted out, the results hold for the nonlnear case usually and for the lnear case by droppng the quadratc equlbrum functon. Whle equpped wth proper ntal values f (,j), x j Ω and boundary condtons at boundary nodes, the lattce Boltzmann method becomes a complete system. The evoluton conssts of two processes, one s a collson process whch s descrbed by the rght hand sde of (2.2),.e., f c (n,j) = f (n,j) + J(f) (n,j) + g (n,j). (2.4) Ths process models the local nteracton among partcles at a node. Second, the transport process realzes the advecton of partcles n one tme nterval, f (n + 1,j + c ) = f c (n,j). (2.41) At an ordnary node, partcles smply move to one of ther neghbors wth a certan velocty n V. When a node x j s next to the boundary and thus some of ts neghbors are out of the doman (see fgure 2.2), c c h hq j x j x j = x j hq j c Ω Fgure 2.2: Intersecton of lnks and boundary gve rse to x j Ω. for example x j hc, then a partcular treatment for f (n + 1,x j ) must be ntroduced. Ths treatment depends on the geometry of Ω and the predetermned boundary settngs, for nstance, the prescrbed average velocty along the boundary. Therefore the transport process dffers respectvely to the varous boundary treatments. Snce analyss of the boundary condtons s one of our man goals n ths work we descrbe the frequently used ntal condtons, and gve a lst of exstng boundary condtons n computatonal flud dynamcs wth ther possble treatments n the lattce Boltzmann framework.

32 18 CHAPTER 2. LATTICE BOLTZMANN METHOD 2.3 Intal condton A feasble ntalzaton proposed n [71] s f (,j) = f eq (1 + c 2 s h 2 p(,x j ),hψ(x j )) c 2 s h 2 A V (V ψ)(x j ) (2.42) where p(,x) s the pressure correspondng to the ntal velocty feld ψ. It s obtaned by solvng the Posson equaton p = (ψ ψ) + G. 2.4 Boundary condton Here we lst some typcal boundary condtons for the velocty feld u and pressure p. A smlar lst can be found n [75] and [54]. 1. perodc boundary Assume the problem s perodc along the coordnate axes wth a perod π R d, where π s the perod n th drecton. Hence Ω = d =1 [,π ), the solutons are completely characterzed by ther values on the perodcty cell. Ths mples that, for example, the velocty feld satsfes u(t,x 1,...,,...,x d ) = u(t,x 1,...,π,...,x d ), and smlar condtons follow for the pressure and ther dervatves. In ths work we take π = 1 wthout loss of generalty. 2. symmetrc boundary The symmetrc boundary s usually realzed by settng the component of velocty normal to the boundary to be zero, u(t,x) n(x) =, x Ω, Here n(x) denotes the unt outer normal vector at the pont x Ω. 3. Drchlet boundary In many applcatons, the flud moves n a bounded doman. We treat the behavor of flow along the boundary by the Drchlet boundary condton,.e., the velocty components are prescrbed. For most sold surfaces whch are mpermeable to flud, the flud stcks to ther surfaces. Hence, there s no slp and no penetraton, and the flud partcles on the wall move wth the velocty of the wall. Namely, the velocty component normal to the wall s set to be zero, the tangental velocty s dentcal to the velocty of the wall. In the specal case of a statonary wall, the velocty s zero.

33 2.4.1 Perodc boundary outflow boundary Outflow condtons are often used at artfcal boundares to smulate a larger flow doman. One possblty s to use the so-called zero Neumann condton u n (t,x) =, x Ω. There are more outflow condtons mentoned later n chapter pressure boundary Ths knd of boundary condton allows to stmulate the flow experments where a flow s nduced by applyng a pressure dfference. From the mathematcal analyss n [35], t s ponted out that the ncompressble Naver- Stokes equaton (2.1) wth the prescrbed average pressure condton p(t,x)n ν n u(t,x) = P Γ n, x Γ (2.43) s a well-posed problem. Where P Γ s the known average pressure on the boundary segment Γ. 6. free surface Free surfaces occur at the nterface between two fluds, for example water and ol or ar. Such nterfaces requre a knematc condton whch relates the moton of the free nterface to the flud veloctes at the free surface and a dynamc condton whch s concerned wth the force balance at the free surface. In the followng we address the mplementatons for two of the above mentoned boundary condtons n the lattce Boltzmann settng, whch wll be analyzed n the next chapter. Addtonal mplementatons wll be dscussed n chapter 5 and chapter Perodc boundary In case that a flow has a perodc structure n the whole space, the computatonal doman Ω can be restrcted to a regon of the sze of one perod. To be specfc, let us assume that the perodcty cell s gven by Ω = [,1) d and that h = 1/m for some m > 1. Defnng the modulo addton (j + m k) = (j + k ) mod m for vectors j,k Z d, we can formulate the advecton step n the perodc doman unformly by f (n + 1,j + m c ) = f c (n,j), = 1,...,N, x j Ω, (2.44) no matter f x j s an ordnary node or a boundary node Bounce back boundary In order to approxmate the Drchlet boundary condton (2.3) on a general bounded doman, the bounce back rule s appled at every boundary node x j to

34 2 CHAPTER 2. LATTICE BOLTZMANN METHOD the components of f whch belong to ncomng veloctes c. The bounce back rule has the form f (n + 1,j) = f c (n,j) + 2hc 2 s f φ(t n,x j ) c, (2.45) where x j s the ntersecton pont of the lnk along c and the boundary Ω. The remanng components are treated wth the usual update rule (2.41).

35 Chapter 3 Consstency of LBM on a general bounded doman The consstency analyss of lattce Boltzmann methods (ncludng those models n chapter 2) on a perodc doman has been ntensvely studed n the last few years [22, 31, 33, 42]. The advantage of applyng asymptotc analyss to the lattce Boltzmann method s gven n [42] where the perodc case s dscussed. In ths chapter, we ntend to extend the consstency analyss to the lattce Boltzmann methods ntroduced n chapter 2 on a general bounded doman by means of asymptotc analyss. As explaned n chapter 2, a typcal lattce Boltzmann method conssts of two parts. One part s the usual update rule whch holds at the majorty of the nodes n Ω. The other part s the specal treatment at the boundary nodes. Correspondngly, the consstency analyss splts nto two parts as well, the analyss of the nteror algorthm and the analyss of the boundary condton. The asymptotc expanson s taken as the regular expanson suggested n [42]: f (n,j) = f () (t n,x j ) + hf (1) (t n,x j ) + h 2 f (2) (t n,x j ) +... (3.1) wth smooth and h ndependent functons f (k). Accordng to [43], the analyss of the update rule yelds explct expressons of the coeffcents n the asymptotc expanson, whch are determned by a seres of partal dfferental equatons (among them, the one for the coeffcents n the lowest relevant order s usually the target problem). We nsert the expanson (3.1) nto the update rule (2.2), do Taylor expanson at node (t n,x j ), and defne f (s) (s k) n such a way that f () h k f (k) satsfes (2.2) wth hgh accuracy. Then we substtute the asymptotc expanson nto the boundary scheme, do Taylor expanson at sutable boundary ponts, and check whch boundary condton leads to a resdue of hgh order n h. In ths way, consstency nformaton about the full algorthm s acheved. Ths chapter gves an outlne of a typcal analyss: 1. Secton 3.1 derves expressons for the coeffcents n (3.1) from the update rule of the algorthm, the relaton between the coeffcents f (k) and the 21

36 22 CHAPTER 3. CONSISTENCY OF LBM ON A GENERAL BOUNDED DOMAIN soluton of the ncompressble Naver-Stokes equaton, and the partal dfferental equatons for the hgher order error terms. The approach s based on the assumpton that the numercal soluton can be descrbed wth smooth functons up to hgh orders n h. Snce the update rule s the same as n the perodc case, the nvestgaton parallels the one n [42]. 2. In secton 3.3, whle summarzng the consstency analyss of the exstng ntal condtons n [42, 44], a practcally feasble ntal condton wth possble hgher accuracy s proposed so that we can concentrate our attenton partcularly on the boundary condtons n the later parts. 3. In secton 3.5, takng the well known bounce back rule as example, a careful descrpton of the boundary consstency analyss s llustrated. For the lattce Boltzmann D2Q9 model wth BGK [6] type collson operator, a smlar result has already been presented n our artcle [44]. 3.1 Formal asymptotc expanson The am of ths secton s to compute the coeffcents n (3.1). Frst we see that the asymptotc expanson of the moments ˆρ and û are correspondngly defned by ˆρ = ρ + hρ 1 + h 2 ρ , (3.2) where the kth order coeffcent of the moments û = u + hu 1 + h 2 u , (3.3) ρ k = f (k), u k = f (k) c are derved from f (k). From the knowledge of f (k) we expect to derve the relaton between the numercal values ˆρ, û and the soluton of the ncompressble Naver-Stokes equaton. To begn wth we nsert (3.1) nto the left hand sde of the lattce Boltzmann algorthm (2.2) and fnd expressons of the form f (k) (t n+1,x j+c ) = f (k) (t n + h 2,x j + hc ). Snce the functons f (k) are assumed to be smooth, we can perform a Taylor expanson around the pont (t n,x j ). Ths process yelds f (k) (t n + h 2,x j + hc ) = f (k) + h(c )f (k) + h 2 ( t + (c ) 2 /2)f (k) + h 3 (c )( t + (c ) 2 /6)f (k) +... where the rght hand sde s evaluated at (t n,x j ). Generalzng ths expanson to arbtrary orders, we formally obtan an nfnte seres f (k) (t n + h 2,x j + hc ) = f (k) (t n,x j ) + h r D r ( t,c )f (k) (t n,x j ) (3.4) r=1

37 3.1. FORMAL ASYMPTOTIC EXPANSION 23 where D r (θ,σ) are polynomals defned by D r (θ,σ) = 2a+b=r θ a σ b a!b! (note that θ replaces the tme dervatve and σ the drectonal space dervatve c ). Altogether, the expanson of the left hand sde of (2.2) s f (n + 1,j + c ) = ( ) h m f (m) m 1 (t n,x j ) + D m r ( t,c )f (r) (t n,x j ), m r= (3.5) Usng the operator (V ) ntroduced n secton 2.1, the rght hand sde of equaton (3.5) can be wrtten also n a vector form f (n + 1,j + c ) = ] m 1 h [f m (m) + D m r ( t,v )f (r) (t n,x j ). (3.6) m r= Next, we have to expand the rght hand sde of (2.2) wth the substtuton of f by the expanson (3.1). Due to the nonlnear term n the collson operator, a mxng of orders occurs. Insertng the expansons (3.2), (3.3) of ˆρ and û nto f eq, we obtan wth where f eq f eq,(m) f eq,(m) (f(n,j)) = f eq,() (f (),...,f (m) ) = f L(m) (s (),...,s (m) ) = f L(m) (t n,x j ) + hf eq,(1) (t n,x j ) + h 2 f eq,(2) (t n,x j ) +... (f (),...,f (m) ) + f Q(m) (f (),...,f (m) ), (s (),...,s (m) ) + f Q(m) (s (),...,s (m) ), f L(m) (s (),...,s (m) ) = f L (s (m) ) = F L ( 1,s (m), 1,Vs (m) ), f Q(m) (s (),...,s (m) ) = f Q (s(k),s (l) ) = F Q ( 1,Vs (k), 1,Vs (l) ) k+l=m k+l=m are functons wth m varables s (k) F. In partcular, f L(m) (f (),...,f (m) ) = F L (ρ m,u m ), (3.7) f Q(m) (f (),...,f (m) ) = F Q (u k,u l ) (3.8) k+l=m represent the lnear and quadratc part of f eq,(m) respectvely. Note that the arguments f (),...,f (m) are sometmes suppressed for brevty. Thus, the rght hand sde of (2.2) turns out to be n vector form [ h m f (m) + A(f eq,(m) f (m) ) + gδ m3 ](t n,x j ). (3.9) m

38 24 CHAPTER 3. CONSISTENCY OF LBM ON A GENERAL BOUNDED DOMAIN Subtractng (3.9) from (3.6), we conclude that the expanson (3.1) satsfes the lattce Boltzmann update rule (2.2) wth a resdue = m [ m 1 h m r= D m r ( t,v )f (r) A(f eq,(m) f (m) ) gδ m3 ]. (3.1) Defnng ξ () =, ξ (m) (f (),...,f (m 1) ) = m 1 r= we can rewrte (3.1) n the more compact form D m r ( t,v )f (r) gδ m3, (3.11) = [ ] h m ξ (m) A(f eq,(m) f (m) ). (3.12) m We remark that n the resdue (3.12), the coeffcent of order h m depends only on the same and lower order functons f (k) (k m). Hereafter, we ntend to fnd f (m) order by order so that the resdue s removed order by order too. Snce the equlbrum functons f eq,(m) have a lnear and a quadratc part, we consder the lnear and nonlnear lattce Boltzmann update rule separately. Before gong to detals, we collect some propertes of the equlbrum functons f eq,(m). Referrng to the defnton (3.7) and (3.8), we apply the equaltes (2.3)-(2.35) and obtan 1,f eq,(m) (s (),...,s (m) ) = 1,f L(m) (s (),...,s (m) ) = 1,s (m), (3.13) 1,Vf eq,(m) (s (),...,s (m) ) = 1,Vf L(m) (s (),...,s (m) ) = 1,Vs (m), (3.14) 1,f Q(m) (s (),...,s (m) ) =, (3.15) 1,Vf Q(m) (s (),...,s (m) ) =, (3.16) 1,V Vf L(m) (s (),...,s (m) ) = c 2 s 1,s(m) I, (3.17) 1,V Vf Q(m) (s (),...,s (m) ) = 1,Vs (k) 1,Vs (k). (3.18) From (3.13) and (3.14), t follows that k+l=m 1,f (m) f eq,(m) =, 1,V(f (m) f eq,(m) ) =, whch mples that f (m) f eq,(m) s orthogonal to the kernel of the matrx A. Thus P(f (m) f eq,(m) ) = f (m) f eq,(m), Q(f (m) f eq,(m) ) =. (3.19)

39 3.1.1 Asymptotc expanson for nonlnear LBM Asymptotc expanson for nonlnear LBM In order to remove the zeroth order of the resdue (3.12), we have to fnd a functon f () whch satsfes We can easly verfy that equaton (3.2) and A(f eq,() (f () ) f () ) =. (3.2) f () = f eq,() (f () ) (3.21) are equvalent. In fact, any soluton of (3.21) satsfes the problem (3.2). On the other hand, multplyng (3.2) wth A from the left and usng A A = P as well as the equalty (3.19), we obtan equaton (3.21). Notce that f () = F L (ρ,u ) + F Q (u,u ) (3.22) wth any ρ and u s a soluton of equaton (3.21). In vew of ρ = 1,f () = ρ, u = 1,Vf () = u, the functons ρ, u are actually the moments ρ, u of f (). We conclude that defnng f () accordng to (3.22) together wth any arbtrarly prescrbed ρ = ρ and u = u removes the zeroth order resdue n (3.12). Next, to remove the frst order term n resdue (3.12), the equaton ( A f eq,(1) (f (),f (1) ) f (1)) ξ (1) (f () ) = (3.23) must be fulflled by some f (1). We frst prove that a necessary and suffcent condton to solve (3.23) s 1,ξ (1) (f () ) =, 1,Vξ (1) (f () ) =. (3.24) Snce A s symmetrc and has a kernel wth bass {1, v 1,...,v d }, t follows from (3.23) that 1,ξ (1) (f () ) =, v α,ξ (1) (f () ) =, (3.25).e., ξ (1) (f () ) must be orthogonal to the kernel of A. Moreover, (3.25) s equvalent to (3.24) by applyng (2.15), hence (3.24) s a necessary condton to solve (3.23). On the other hand, f (3.24) holds, t follows that so that any soluton of Pξ (1) (f () ) = ξ (1) (f () ), Qξ (1) (f () ) =, (3.26) f (1) = f eq,(1) (f (),f (1) ) A ξ (1) (f () ) (3.27) satsfes equatons (3.23). Ths can be easly verfed by nsertng (3.27) nto (3.23) and applyng (3.26) as well as AA = P. Conversely, solutons of (3.23)

40 26 CHAPTER 3. CONSISTENCY OF LBM ON A GENERAL BOUNDED DOMAIN satsfy (3.27) too, whch s a consequence of (3.19) as well as A A = P and (3.26). Based on (3.27), we can defne f () and f (1). To see ths, we frst notce that (3.24) s equvalent to partal dfferental equatons for u and ρ by takng (3.21) and ξ (1) (f (),f (1) ) = (V )f () nto account. They are 1,ξ (1) (f () ) = u =, 1,Vξ (1) (f () ) = c 2 s ρ + (u )u =. (3.28) Assumng we have moments u and ρ satsfyng (3.28), and arbtrary functons u 1 and ρ 1, then f (1) defned by f (1) = F L (ρ 1,u 1 ) + F Q (u,u 1 ) A ξ (1) (f () ) (3.29) satsfes (3.23). Ths s checked by drect computaton usng (3.7) and (3.8). Hence (3.24) s also a suffcent condton to fnd f (1). We stress that the necessary and suffcent condton to solve (3.23) for f (1) gves a constrant on the lower order moments. More precsely f () s restrcted by equaton (3.22) wth the moments u and ρ governed by (3.28). In the followng, we choose a partcular soluton of (3.28), u =, ρ ρ, ρ s a constant, (3.3) whch turns out to be compatble wth the regme n whch lattce Boltzmann algorthms are usually appled. In fact, a relaton to the ncompressble Naver- Stokes equaton s only obtaned n ths regme. Hence f () = ρ f (3.31) s constant n t and x, so that the dervatve terms of u, ρ and f () dsappear n the resdue (3.12) and the functons ξ (k) are actually ndependent of f (). For f (1) defned n (3.29) whch removes the 1st order resdue n (3.12), t has the consequence f (1) = f eq,(1) = f L(1). (3.32) Next, we remove the second order term n resdue (3.12). f (2) s requred to satsfy ( A f eq,(2) (f (),f (1),f (2) ) f (2)) ξ (2) (f (),f (1) ) =, (3.33) ξ (2) (f (),f (1) ) = (V )f (1). (3.34) Proceedng as above, we fnd that a suffcent and necessary condton to solve (3.33) for f (2) s 1,ξ (2) (f (1) ) =, 1,Vξ (2) (f (1) ) =, (3.35) whch are actually equatons for the moments ρ 1 and u 1, u 1 =, ρ 1 =. (3.36)

41 3.1.1 Asymptotc expanson for nonlnear LBM 27 At ths pont, we restrct our consderatons to expansons wth a constant ρ 1 = ρ 1. It turns out, that these expansons are stll flexble enough to descrbe the behavor of the lattce Boltzmann solutons to hgh orders. (In contrast, settng u 1 to be zero would generally stall the expanson at order 3). In addton, under the condton (3.35), equaton (3.33) s equvalent to f (2) = f eq,(2) (f (),f (1),f (2) ) A ξ (2) (f (),f (1) ). (3.37) Prescrbng arbtrary values of ρ 2 and u 2, and restrctng ρ 1 = ρ 1 and u 1 followng the equatons (3.36), we can defne f (2) = F L (ρ 2,u 2 ) + F Q (u 1,u 1 ) A ξ (2) (f (),f (1) ). (3.38) whch satsfes (3.37). As a result, the second order term n the resdue (3.12) s removed. Contnung to remove the thrd order term n the resdue (3.12), we seek f (3) to satsfy A(f eq,(3) (f (),...,f (3) ) f (3) ) ξ (3) (f (),f (1),f (2) ) =, ξ (3) (f (),f (1),f (2) ) = (V )f (2) (V )2 f (1) + t f (1) g. (3.39) Agan we fnd a suffcent and necessary condton to solve (3.39) We frst note that 1,ξ (3) (f (1),f (2) ) =, 1,Vξ (3) (f (1),f (2) ) =. (3.4) 1,(V ) 2 f (1) = : p (1) = c 2 s 2 ρ 1 =, where p (1) = 1,V Vf (1), and the equaton (3.36) for ρ 1 s appled. Due to the symmetry of V and f 1,Vf =, 1,g = c 2 s G 1,Vf =, so that the actual form of the frst equaton n (3.4) s Let us see the second equaton n (3.4) whch gves u 2 =. (3.41) t u 1 + p (2) ,V(V )2 f (1) = G. (3.42) wth p (2) = 1,V Vf (2). Applyng (2.28) to the second order dervatve term, we obtan 1,V(V ) 2 f (1) = c 2 s 2 u 1. (3.43) Proceedng further we need to know p (2) whch s based on the second order coeffcent f (2) = F L (ρ 2,u 2 ) + F Q (u 1,u 1 ) A (V )f L(1). (3.44)

42 28 CHAPTER 3. CONSISTENCY OF LBM ON A GENERAL BOUNDED DOMAIN Takng ρ 1 = nto account and ntroducng the symmetrc part of the velocty dervatve S[u] = ( u + u T )/2, (3.45) we fnd A (V )f (1) = c 2 s S[u 1 ] : A (V Vf ). (3.46) Snce the trace of S[u 1 ] equals 2 u 1 whch vanshes due to the ncompressblty condton n (3.36), V V can be replaced by ts traceless part Λ = V V V 2 /di. Applyng the property (v) of A, we have A (V )f (1) = µc 4 s S[u 1 ] : Λf, (3.47) whch gves the last term on the rght hand sde of (3.44) a more explct form. In vew of (2.28), (2.32) and (2.35), we can now compute p (2) from (3.44) whch gves rse to p (2) = c 2 s ρ 2I + u 1 u 1 2µS[u 1 ]. (3.48) Wth substtuton of (3.43) and (3.48) nto (3.42), the second equaton n (3.4) s also reformulated to an explct form. In summary, the suffcent and necessary condton (3.4) s equvalent to the equatons u 2 =, t u 1 + u 1 u 1 + c 2 s ρ (3.49) 2 = ν u 1 + G, where we have used the defnton µ = ν c2 s. The functon f (3) has to be a soluton of f (3) = f L(3) + f Q(3) A ξ (3) (f (),f (1),f (2) ). (3.5) To obtan a specfc form of the possble solutons to (3.5), we assume that ρ 1 = ρ 1 s constant, and that u 1 and c 2 s ρ 2 solve our target problem, the ncompressble Naver-Stokes equaton (2.1). Further, we choose a dvergence free feld u 2 n the defnton of f (2). Then ξ (3) and f Q(3) can be computed and we set f (3) = F L (ρ 3,u 3 ) + F Q (u 1,u 2 ) A ξ (3) (f (1),f (2) ) (3.51) wth arbtrary values of ρ 3 as well as u 3, whch s a soluton of (3.5) and hence satsfes (3.39). In ths way, also the thrd order term s removed. As for the fourth order resdue, f (4) s searched so that A(f eq,(4) (f (),...,f (4) ) f (4) ) ξ (4) (f (),...,f (3) ) =. (3.52) The suffcent and necessary condton to solve (3.52) s consequently 1,ξ (4) =, 1,Vξ (4) =, (3.53)

43 3.1.1 Asymptotc expanson for nonlnear LBM 29 whch s equvalent to the equaton governng the moments ρ 3 and u 3, u 3 = c 2 s t ρ 2 1 G, 2 (3.54) t u 2 + (u 1 )u 2 + u 2 u 1 + c 2 s ρ 3 = ν u 2. (3.55) Wth arbtrary values of ρ 4 and u 4, and ρ 3, u 3 as well as u 2 determned by (3.54),(3.55), (3.41) and the other already known lower order moments, the functon f (4) s defned by f (4) ) = F L (ρ 4,u 4 )+F Q (u 1,u 3 )+F Q (u 2,u 2 ) A ξ (4) (f (),...,f (3) ). (3.56) It solves the equaton f (4) = f eq,(4) (f (),...,f (4) ) A ξ (4) (f (),f (2),f (3) ) and removes the 4th order term n (3.12). Gong to the 5th order term n the resdue (3.12), we defne f (5) by f (5) = F L (ρ 5,u 5 ) + F Q (u 1,u 4 ) + F Q (u 2,u 3 ) A ξ (5) (f (),...,f (4) ), (3.57) n whch ρ 5 and u 5 are taken arbtrarly and ρ 4, u 3 as well as u 4 are controlled by the followng equatons wth t u 3 + u 1 u 3 + u 3 u 1 + c 2 s ρ 4 = ν u 3 + R (3.58) u 4 = (3.59) R = ( t ρ G)u 1 + B 1 G + B 2 u 1 + B 3 ρ 2 + B 4 u 1 u 1 + B 5 ρ 3 + B 6 u 2 + B 7 u 2 u 2. Here B 1,B 2,B 3,B 4,B 5,B 6 are hgher order dfferental operators: B 1 and B 5 are lnear combnatons of t and 2, B 2 s a lnear combnaton of a t ( ) b wth 2a+b = 4, and B 3, B 4 and B 6 are lnear combnatons of a t ( ) b wth 2a+b = 3; B 7 s a frst order spatal dervatve operator. The related combnaton parameters depend on the constant equlbrum f and the lnear mappng A. Because they act on the soluton (u,p) = (u 1,c 2 s ρ 2) and the data G of the target Naver-Stokes problem, R plays the role of an explctly gven source term n the equaton for u 3. Successvely we could defne hgher order coeffcents f (m) to remove the hgher order resdues. As a result, smlar equatons for the hgher order moments ρ m and u m of f (m) appear. The emergng source terms n ther equatons depend on the temporal and spatal dervatves of the lower order moments and the force lke (3.54), (3.58) for u 3. It s remarked that the mssng nformaton on ntal and boundary condtons for u 1, ρ 2, u 2, ρ 3, u 3 and ρ 4 as well as the other moments wll follow from correspondng parts of the lattce Boltzmann algorthm. The smoothness of

44 3 CHAPTER 3. CONSISTENCY OF LBM ON A GENERAL BOUNDED DOMAIN these moments s determned by the regularty of the ntal and boundary values and source terms n the correspondng dfferental equatons. In partcular, f (m) are smooth enough f these data are suffcently smooth. It s noted that n vew of u =, ρ = ρ, ρ 1 = ρ 1, the moments of the lattce Boltzmann soluton have the expansons û = hu 1 + h 2 u 2 + h 3 u , (3.6) ˆρ = ρ + h ρ 1 + h 2 ρ 2 + h 3 ρ (3.61) In case u 2, only a frst order accurate Naver-Stokes velocty can be extracted from the numercal soluton u = 1 hû hu 2 h 2 u (3.62) In case that u 2 has rregular ntal or boundary values, f (2) cannot be regular, so that ρ 2 s expected to be not smooth ether. Consequently, we expect the numercal result to be nconsstent to the Naver-Stokes pressure. Further, we note that the frst order error term u 2 n velocty s controlled by a homogeneous generalzed Oseen problem ((3.41), (3.55)). Therefore u 2 = s the soluton f the ntal and boundary values of u 2 are zero, so that the velocty feld of the Naver-Stokes equaton s approxmated wth a second order accuracy. As for the pressure, the value of ρ 3 behaves as the frst order error term. In case that rregular ntal or boundary values of ρ 3 or u 3 can not be avoded, f (3) has no regularty n return. Therefore we expect that the pressure can be obtaned from the numercal soluton by p = c 2 s h 2 (ˆρ ρ h ρ 1 ) hc 2 s ρ , (3.63) wth at most frst order accuracy. On the other hand, when a smooth f (3) s obtaned and f ρ 3 s also accompaned wth zero ntal values, we fnd ρ 3 = so that a second order accurate pressure feld can be extracted from the lattce Boltzmann soluton too. The next hgher order errors u 3 and ρ 4 are determned agan by equatons of Oseen type ((3.54) and (3.58)). Introducng a feld ū 3 = ϕ G/2 where ϕ satsfes the equaton ϕ = t ρ 2, we can rewrte the u 3 -equaton as equaton for the ncompressble feld w = u 3 ū 3 (for detals see [46]). Ths equaton can then be solved wth zero ntal value for w and eventually yelds u 3 and ρ 4. But snce R n generally not zero, the solutons u 3 and ρ 4 can not be zero and thus lead to non-trval error terms. Moreover, the velocty error u 3 s not ncompressble (see equaton (3.54)) whch actually reflects the compressblty error of the lattce Boltzmann scheme. As a result, the accuracy of the Naver- Stokes solutons extracted from the moments of the lattce Boltzmann soluton s restrcted by 2nd order for both velocty and pressure.

45 3.1.1 Asymptotc expanson for nonlnear LBM 31 In summary, we have the followng lst of possble accuracy constellatons Table 3.1: Possble accuracy of Naver-Stokes solutons extracted from the moments of lattce Boltzmann soluton. expected accuracy of (u, p) cases 1, u 2 and ρ 2 rregular 2, 1 u 2 = and ρ 3 rregular 2, 2 u 2 = and ρ 3 = For a more careful dscusson of possble accuracy orders we refer to secton 3.3 and secton 3.5. From the above nformaton, the frst several f (k) (k 5) n (3.1) can be defned under the assumpton that the nvolved partal dfferental equatons possess suffcently smooth solutons. In partcular, the coeffcents wth trval lower order moments ρ = ρ, u = and ρ 1 = ρ 1 are of our specal nterest, whch render (u 1 = u, c 2 sρ 2 = p) to be the soluton of the mpressble Naver-Stokes problem (2.1), (u 2, ρ 3 ) to be the soluton of equaton (3.41), (3.55), and (u 3, c 2 sρ 4 ) to satsfy the equaton (3.54), (3.58). We set u 5 =, ρ 5 = and u 4 = as smple, specfc choce. Usng these moments n the formula of f (m) = F L (ρ m,u m ) + F Q (u k,u l ) A ξ (m), m = 1,...,5, k+l=m we arrve at the defnton of the frst fve coeffcents n the expanson (3.1) for the nonlnear lattce Boltzmann solutons: f () = ρ f ; f (1) = F L ( ρ 1,u 1 ); f (2) = F L (ρ 2,u 2 ) + F Q (u 1,u 1 ) A V f (1) ; f (3) = F L (ρ 3,u 3 ) A ( t f (1) + (V )f (2) + 1 ) 2 (V )2 f (1) G ; f (4) = F L (ρ 4,) + F Q (u 1,u 3 ) + F Q (u 2,u 2 ) A ( t f (2) + (V )f (3) (V )2 f (2) + D 3 ( t,v )f (1) ) ; f (5) = F Q (u 2,u 3 ) A ( t f (3) + (V )f (4) (V )2 f (3) ) A ( D 4 ( t,v )f (1) D 3 ( t,v )f (2)). (3.64) Combnng them, we thus obtan a truncated asymptotc expanson of the lattce Boltzmann solutons ˆf(n,j) = f () (t n,x j ) + hf (1) (t n,x j ) + + h 5 f (5) (t n,x j ). (3.65)

46 32 CHAPTER 3. CONSISTENCY OF LBM ON A GENERAL BOUNDED DOMAIN Due to the constructon, the truncated expanson ˆf removes the resdue terms n (3.1) up to order h 5 and thus satsfes the lattce Boltzmann update rule up to terms of order h 6. We summarze ths result n the followng theorem. Theorem 1 Assume that solutons u, p of the Naver-Stokes equaton (2.1), solutons u 2,ρ 3 of (3.41), (3.55) and solutons u 3,ρ 4 of (3.54), (3.58) have the followng regularty u C 5 ([,T], Ω), p C 4 ([,T], Ω), u 2 C 4 ([,T], Ω), ρ 3 C 3 ([,T], Ω), u 3 C 3 ([,T], Ω), ρ 4 C 2 ([,T], Ω), then the predcton functon ˆf defned by (3.65) and (3.64) together wth arbtrary constants ρ and ρ 1 satsfes ˆf (n + 1,j + c ) = ˆf (n,j) + J ( ˆf)(n,j) + g (n,j) + ˆr (n,j) (3.66) at any node x j Ω wth x j + hc Ω and there s a h ndependent constant κ 6 such that ˆr (n,j) = O(h κ ), nh 2 T. (3.67) The proof s very straghtforward. Insertng the predcton ˆf nto the scheme (2.2) and usng Taylor s theorem at (t n,x j ) for each f (k), k = 1,...,5, wth the Lagrange form of the truncaton error at order 6 k, we fnd the unformly bounded expresson ˆr (n,j) = h 6 5 k=1 D 6 k f (k) (t ξk, (n),x ηk, (j)), (3.68) wth t ξk, (n) = t n + ξ k, h 2,ξ k, [,1] and x ηk, (j) = x j + η k, hc,η k, [,1] Asymptotc expanson for lnear LBM In ths case, the quadratc terms n the collson operator vansh so that f eq = f L as well as f eq,(m) = f L(m). The terms resultng from f Q(m) n the defnton of f (m) and n the moment equatons also dsappear. Exactly as n the nonlnear case, we can defne f () = ρ f, f (1) = F L (,u 1 ) f (2) = F L (ρ 2,u 2 ) A (V )f L(1) by takng arbtrary u 2, ρ 2, ρ = ρ, u =, ρ 1 = ρ 1, together wth u 1 satsfyng u 1 = to remove the th, 1st and 2nd order terms n the resdue (3.12). In order to remove the 3rd order term, f (3) s defned by f (3) = F L (ρ 3,u 3 ) A (V )ξ (3) f (),(f (1),f (2) )

47 3.1.2 Asymptotc expanson for lnear LBM 33 wth arbtrary functons u 3 and ρ 3, as well as u 1, u 2 and ρ 2 controlled by the equatons u 2 =, t u 1 + c 2 s ρ 2 = ν u 1 + G. Next, to make the 4th order resdue vansh, we defne f (4) by f (4) = F L (ρ 4,u 4 ) A (V )ξ (4) (f (),...,f (3) ) wth arbtrary u 4 and ρ 4. However, the moments u 2, u 3, ρ 2 and ρ 3, whch appear n ξ (4), are governed by the equaton t u 2 + c 2 s ρ 3 = ν u 2, u 3 = c 2 s tρ 2 1 G. 2 Proceedng further, we arrve at f (5) to remove the 5th order resdue n (3.1), whch s defned as f (5) = F L (ρ 5,u 5 ) A (V )ξ (5) (f (),...,f (4) ). The moments u 3, u 4 and ρ 4, whch are needed n the calculaton of ξ (5), are determned by the equaton wth source term t u 3 + c 2 s ρ 4 = ν u 3 + B, u 4 = B = A 1 p + A 2 u + A 3 G, where A 1,A 2,A 3 are dfferental operators wth coeffcents dependng only on the dscrete veloctes, the equlbrum weghts f and the matrx A. Agan u 5 and ρ 5 can be random values. In summary, we collect the results about the moments. In partcular, let (c 2 s ρ 2, u 1 ) satsfy the Stokes equaton u 1 =, (3.69) t u 1 + c 2 s ρ 2 = ν u 1 + G, (3.7) assume that (u 2, ρ 3 ) also follow a Stokes equaton, t u 2 + c 2 s ρ 3 = ν u 2, u 2 = (3.71) and that (u 3, ρ 4 ) are governed by u 3 = c 2 s t ρ G t u 3 + c 2 s ρ 4 = ν 2 u 3 + B. (3.72)

48 34 CHAPTER 3. CONSISTENCY OF LBM ON A GENERAL BOUNDED DOMAIN Then we defne the leadng order coeffcents up to 5th order for the lnear lattce Boltzmann solutons: f () = ρ f, f (1) = F L ( ρ 1,u 1 ), f (2) = F L (ρ 2,u 2 ) A (V )f (1), f (3) = F L (ρ 3,u 3 ) A ( t f (1) + (V )f (2) (V )2 f (1) g), f (4) = F L (ρ 4,) A ( t f (2) + (V )f (3) (V )2 f (2) + D 3 f (1) ), f (5) = A ( t f (3) + (V )f (4) (V )2 f (3) + D 4 f (1) + D 3 f (2) ). Wth a smlar proof as n the nonlnear case, we obtan (3.73) Theorem 2 Assume that solutons u, p of the Stokes equaton (3.69), solutons u 2,ρ 3 of (3.71) and solutons u 3,ρ 4 of (3.72) have the followng regularty u C 5 ([,T], Ω), p C 4 ([,T], Ω), u 2 C 4 ([,T], Ω), ρ 3 C 3 ([,T], Ω), u 3 C 3 ([,T], Ω), ρ 4 C 2 ([,T], Ω), then the predcton functon ˆf defned by (3.65) and (3.73) wth random constants ρ and ρ 1 satsfes ˆf (n + 1,j + c ) = ˆf (n,j) + J L ( ˆf)(n,j) + g (n,j) + ˆr (n,j) (3.74) at any node x j Ω wth x j + hc Ω and there s h ndependent constant κ 6 ˆr (n,j) = O(h κ ), nh 2 T. (3.75) Agan the hgher order terms u k and ρ k (k 3) behave as the error when the exact soluton of the Stokes equaton s approxmated by the lattce Boltzmann solutons. The frst order error terms u 2, ρ 3 are determned by a Stokes problem (3.71) whch can be satsfed wth u 2 = and ρ 3 =. Note that u 3 s, n general, not dvergence free whch agan reflects the weak compressblty observed n the lattce Boltzmann soluton. As before we can relate (3.72) to a standard problem. Introducng auxlary varables ω, π and ϕ we obtan a soluton to (3.72) va u 3 = ω + ϕ 1 2 G, c2 s ρ 4 = π t ϕ + νc 2 s tρ 2 (3.76) provded ϕ satsfes the Posson equaton and ω,π are solutons of the standard Stokes problem ϕ = c 2 s t ρ 2 (3.77) ω =, t ω + π = ν ω + B ( t ν )G.

49 3.2. MODEL FLOWS WITH ANALYTIC SOLUTIONS 35 Referrng to [42], we see that all the moments u k and ρ k+1 wth even k 2 satsfy Stokes problems lke (3.71) whch can be fulflled by u k = and ρ k+1 = n case of zero ntal and boundary values. Whereas the moments u k and ρ k+1 wth odd k 3 are governed by Stokes problems smlar to (3.72) and hence are generally not trval. In summary, we can say that the expanson coeffcents are defned n terms of solutons to Stokes type problems. Remark 3 Snce u 2 and ρ 3 are governed by the homogeneous Oseen type equaton (3.41), (3.55) n the nonlnear case or Stokes equaton (3.71) n the lnear case, u 2 = and ρ 3 = may result f the correspondng ntal and boundary values are all zero. Obvously, Theorem 1 and 2 hold n case u 2 = and ρ 3 =, too. Remark 4 In general u 3 and ρ 4 for both the lnear and the nonlnear case due to the dvergence condton n (3.54), (3.58) or (3.72), whenever the soluton of Stokes or Naver-Stokes equaton s tme dependent. Hence the hghest order of accuracy s expected to be h 2 for both velocty and pressure f the soluton of the Stokes (3.69) or Naver-Stokes equaton (2.1) s extracted from the lattce Boltzmann soluton by (3.62) and (3.63). Remark 5 Observng (3.73) and (3.64) closely, we see that formally the only dfference s the emergng quadratc equlbrum n f (m) wth even m. Therefore n the later sectons and chapters, we wll consder the nonlnear lattce Boltzmann method unless partcularly ponted out. Results for the lnear case can always be deduced by droppng the nfluence of the quadratc terms. 3.2 Model flows wth analytc solutons Ths secton descrbes several flows whch are governed by the Stokes equaton (3.69) or ncompressble Naver-Stokes equaton (2.1). All these flows have analytcal velocty and pressure felds whch we can obtan explctly. For the smple lnear flow and the polynomal Poseulle flow, we can even compute the complete asymptotc expanson dscussed n the prevous secton Lnear flow The so-called lnear flows refer to velocty felds whch are lnear functons wth respect to the spatal varables. The general form of the velocty and pressure s descrbed by u(t,x) = α(t)bx, p(t,x) = 1 2 α(t)xt B 2 x, x R d, (3.78) where B s a constant matrx whch has a zero trace. Actually tr(b) = guarantees the ncompressblty, snce u = α(t)tr(b). In addton, let B 2 be symmetrc n order to get p = α(t)b 2 x.

50 36 CHAPTER 3. CONSISTENCY OF LBM ON A GENERAL BOUNDED DOMAIN The functon α(t) s supposed to be havng at least a frst order dervatve. Wth the calculaton t u = α (t)bx, (u )u = α 2 (t)b 2 x, u =, t s easy to valdate that the felds u and p satsfy the Naver-Stokes equaton (2.1) n the entre space wth a body force defned by G(t,x) = α (t)bx + α(t)(α(t) 1)B 2 x. (3.79) When α(t) s a constant, for example α(t) = 1, the flow s statonary and the correspondng body force vanshes. In 3D space, the matrx B must be symmetrc to guarantee that B 2 s symmetrc. We have used B = (3.8) wth α(t) = 1 and α(t) = t 2 as two test problems n secton 5.3. In 2D space, t s not necessary to requre symmetry of the matrx B, snce B 2 s always a dagonal matrx wth the dentcal dagonal entres B B 12B 21 for any B wth tr(b) =. Three choces of B have been used as statonary lnear flows n ths context, u(x) = B x, p(x) = 1 2 xt B 2 x, B = ( 1 1 ), S[u] =. (3.81) u(x) = B 1 x, p(x) = 1 2 xt B 2 1x, B 1 = ( ), S[u] = B 1. (3.82) u(x) = B 2 x, p(x) = 1 2 xt B 2 2 x, B 2 = ( 1 1 ), S[u] = B 2. (3.83) The lnear flows move wth dfferent velocty profles correspondng to dfferent matrces B, however the pressure always shows the same behavor only wth dfferent magntude. For example n the 2D case, the sobars of pressure are concentrc crcles around the orgn, snce B 2 s a multple of the dentty. The sobars of exact pressure and the stream lnes of flows n the unt square [,1] 2 are plotted n the followng fgures.

51 3.2.1 Lnear flow Fgure 3.1: Left: sobars of pressure for the lnear flows; Rght: streamlnes of the velocty feld for the lnear flow (3.81) Fgure 3.2: Streamlnes of the velocty feld for the lnear flows (3.82) (left) and (3.83) (rght). Expanson coeffcents for statonary lnear flow For the statonary lnear flow (α(t) = 1 n (3.78)) there s no body force. Moreover, all the terms about tme dervatves dsappear, and the spatal dervatves hgher than second order are zeros. Hence the source term appearng n (3.58) for u 3 vanshes. In fact, we can successvely fnd that all the source terms for the hgher order moments dsappear. Hence the partal dfferental equatons for hgher( 2) order moments are homogeneous Oseen type equaton as (3.41), (3.55) and may have zero solutons. Therefore n ths case, the asymptotc expanson (3.1) may have only a fnte number of terms up to h 4, f = f + hf (1) + h 2 f (2) + h 3 f (3) + h 4 f (4)

52 38 CHAPTER 3. CONSISTENCY OF LBM ON A GENERAL BOUNDED DOMAIN provded ths s compatble wth ntal and boundary condtons. In ths case, the coeffcents are explctly (for nonlnear equlbrum) f (1) = c 2 s u Vf, f (2) = (c 2 s p + c 4 s 2 (u u) : (V V c2 si))f c 2 s A (V V) : Bf, ( ) f (3) = c 2 s A V T B 2 x + c 2 s (u V)(V V) : B u BV f, f (4) = A (A 1 2 I)c 2 s ( (V V) : B2 (BV) T BV + c 2 s ((V V) : B)2 )f. For the lnear equlbrum, the expanson coeffcents become smpler because the statonary velocty feld u = Bx along wth a zero pressure feld p = and a zero body force G = s a soluton of the Stokes equaton f tr(b) =. We fnd the nonzero coeffcents, Poseulle flow f (1) = c 2 s u Vf, f (2) = c 2 s A (V V) : Bf. The Poseulle flow s consdered n a 2D nfntely long channel between two boundares y = λx + µ 1 and y = λx + µ 2 wth µ 1 µ 2. The flow s ntally at rest and drven by a constant body force F parallel to the boundary. Therefore the flow moves parallel to the boundary, of course, to the centerlne y = λx + 1/2(µ 1 + µ 2 ) too. Let ω denote the wdth of the channel, ν the shear vscosty, and z the dstance to the centerlne. Then the flow velocty parallel to the boundary has a profle U(z) = F 2µ (z ω 2 )(z + ω ), (3.84) 2 whch satsfes the Naver-Stokes equaton (2.1) combned wth a zero velocty component vertcal to the boundary and a zero pressure. U U λ = λ Fgure 3.3: Sketch of the velocty profle for Poseulle flow. Usually, the channel s chosen parallel to the coordnate axs,.e., λ =. For more general λ, an easy mplementaton of perodc boundary condtons at the two ends of the channel s stll possble f the slope λ s chosen to be ratonal so that partcles leavng at the rght end can reenter at the left end wth a sutable shft.

53 3.2.2 Poseulle flow 39 Expanson coeffcents for Poseulle flow Here the asymptotc expanson coeffcents are gven for a partcular Poseulle flow. Ths Poseulle flow s parallel to the coordnate axs and wthout loss of generalty, we assume that the channel s between two planes y = and y = H. The statonary velocty feld has components u(x,y) = F y(h y), v(x,y) =. 2ν The body force s a constant G = (F,) T and the pressure s zero. Snce the x component velocty s a 2nd order polynomal functon of y and the y component velocty s zero, almost all the dervatves of ths flow are zero except u y = F (2y H), 2ν 2 u y 2 = F ν. Due to the body force beng nonzero, we obtan a constant u 3 = (F/2,) T. Altogether, for ths flow the asymptotc expanson (3.1) has only three terms f = f + hf (1) + h 2 f (2) + h 3 f (3) wth f (1) = c 2 s uv 1 f, f (2) = c 4 s 2 u2 (V1 2 c2 s 1)f cs 2 u y A V 1 V 2 f, f (3) = c 2 s 2 FV 1f c 2 s u u y A V 2 (3V1 2 1)f + c 2 2 u s y 2 A (A 1 2 I)V 1V2 2 f. f the nonlnear lattce Boltzmann method s appled. Snce ths flow actually satsfes the followng smple equaton ν u + F =, t s a soluton of the Stokes equaton too. Smulated by the lnear lattce Boltzmann method, the coeffcents are smpler: f (1) = c 2 s uv 1f, f (2) = c 2 u s y A V 1 V 2 f, f (3) = c 2 s 2 FV 1f + c 2 2 u s y 2 A (A 1 2 I)V 1V2 2 f.

54 4 CHAPTER 3. CONSISTENCY OF LBM ON A GENERAL BOUNDED DOMAIN Decayng Taylor vortex The velocty feld and pressure of ths flow are defned by u(t,x,y) = 1 cos(ax)sn(by)exp( 2ανt), (3.85) a v(t,x,y) = 1 sn(ax)cos(by)exp( 2ανt), (3.86) b p(t,x,y) = 1 ( cos(2ax) 4 a 2 + cos(2by) ) b 2 exp( 4ανt), (3.87) where the constant α s equal to (a 2 + b 2 )/2, ν s the knematc vscosty, and u and v are the x-component and y-component velocty respectvely. Ths flow s perodc wth a perod 2π/a n x drecton and 2π/b n y drecton. A test example s a = 2π, b = 2π and Ω = [,1] 2, ths s exactly the Taylor vortex n one perod Fgure 3.4: Streamlnes of velocty feld (left) and contour lnes of pressure (rght) for the decayng Taylor vortex flow n one perod Crcular flow The so-called mpulsvely started crcular flow has been descrbed, for example, n [7]. The radus of the nfntely long crcular cylnder s taken as R = 1. Usng dmensonal reducton, we are led to a 2D Naver-Stokes problem posed on the unt dsk Ω. Intally the flow s at rest wth a unform densty and s drven by an angular velocty ω = U θ /R, where U θ s the constant tangental velocty of the boundary. The nteror angular velocty u θ (t,r) s: wth w θ (t,r) = k u θ (t,r) = U θr R + w θ(t,r), (3.88) a k J 1 (µ k r R )exp( νµ2 k R 2 t).

55 3.3. ANALYSIS OF INITIAL CONDITION 41 Here ν s the knematc vscosty and µ k s the kth root of the Bessel functon of order 1,.e., J 1 (µ k ) =. The coeffcent a k s defned by a k = 2U θ µ k J (µ k ). In addton, the pressure p(t, r) s centrally symmetrc (the orgn s the center) and appears as an ntegraton of the angular velocty, p(t,r) = r 1 s u2 θ (t,s)ds. Obtanng the exact velocty and pressure for ths model, obvously nvolves numercal ntegraton and an approxmaton of u θ. For the former we take a second order accurate ntegraton method, the latter s calculated by takng the sum of the frst 5 terms n the seres only u θ (r) p(r) r r Fgure 3.5: Streamlnes of velocty feld (left) and p and u θ (rght) as functons of the radus r at tme t =.6. The test problems descrbed here are used to verfy our theoretcal predctons about the accuracy of ntal and boundary condtons presented n the followng sectons. 3.3 Analyss of ntal condton A frequently used ntal settng s of the form f(,j) = f eq (1,hψ(x j )), (3.89) wth a prescrbed vector feld ψ whch s dvergence free. Usng the structure of the equlbrum dstrbuton, we can rewrte t as ( ) f(,j) = f + hc 2 s ψ Vf + h 2c 2 s s (ψ V)2 ψ 2 )f. xj 2 (c 2

56 42 CHAPTER 3. CONSISTENCY OF LBM ON A GENERAL BOUNDED DOMAIN Insertng the expanson (3.1) nto the left hand sde and usng the formulas for the coeffcents (3.64) leads to the resdue ( ρ 1)f + h [ ρ 1 f + c 2 s (u 1 (,x j ) ψ(x j )) Vf ] + h 2 ( ρ 2 (,x j )f + c 2 s u 2 (,x j ) Vf A (V )f (1) (,x j ) + R 2 (x j ) ) + k=3 h k f (k) (,x j ) (3.9) where R 2 (x j ) = c 2 s 2 (c 2 s (u 1 (,x j ) V) 2 u 1 (,x j ) 2 )f Obvously only f c 2 s 2 (c 2 s (ψ(x j ) V) 2 ψ(x j ) 2 )f. ρ = 1 (3.91) the zeroth order term s removed. In order to remove the frst order term, we have to fnd ρ 1 and u 1 satsfyng the relaton ρ 1 f + c 2 s (u 1 (,x j ) ψ(x j )) Vf =, c V. Snce the second term depends on c and ρ 1 s a constant, only ρ 1 =, u 1 (,x) = ψ(x), (3.92) can remove the frst order terms and R 2 (x j ) n (3.9) s consequently removed, too. In other words, among all truncated expansons provded by Theorem 1, those whch satsfy addtonally ρ = 1, ρ 1 = and u 1 (,x) = ψ(x) are accurate representatons of the numercal solutons. In order to fnd even better representatons, we try to locate truncated expansons whch remove also the second order contrbuton ρ 2 (,x j )f + c 2 s u 2 (,x j ) Vf A (V )f (1) (,x j ) (3.93) n the resdue of the ntal condtons (3.89). It s noted that ψ(x) s the prescrbed ntal velocty feld (2.2) of the ncompressble Naver-Stokes problem (2.1), the related ntal pressure p(, x) can be obtaned by solvng the Posson equaton p = (ψ ψ) + G. (3.94) As a result, the moment ρ 2 (,x) = c 2 s p(, x) generally does not vansh and does not cancel the term A (V )f (1) ether. Hence n order to remove (3.93), u 2 can not be zero. However, n the general case S[ψ], the second order term (3.93) can not be removed, because no smooth u 2 exsts so that (3.93) dsappears.

57 3.3. ANALYSIS OF INITIAL CONDITION 43 To see ths, let us apply property (v) of the matrx A and rewrte (3.93) as c 2 s p(,x j)f + c 2 s u 2(,x j ) Vf µc 4 s (V V)f : S[ψ](x j ). (3.95) Assume there s a smooth functon u 2 (,x) so that (3.95) s equal to zero,.e., the followng equaton u 2 (,x j ) c = p(,x j ) + µc 2 s (c c ) : S[ψ](x j ) (3.96) holds for all c V. Snce V s symmetrc, we have for all c V and c = c, u 2 (,x j ) c = p(,x j ) + µc 2 s (c c ) : S[ψ](x j ), u 2 (,x j ) c = p(,x j ) + µc 2 s (c c ) : S[ψ](x j). (3.97) Observng that the left hand sde of (3.97) s lnear n c and the rght hand sde quadratc, we obtan that u 2 (,x j ) c = u 2 (,x j ) c = u 2 (,x j ) c, (3.98) so that only u 2 (,x j ) = satsfes (3.98). Ths contradcts wth the prevous dscusson. Only n specfc stuatons (for example ψ(x) =, G t= = ), the condton u 2 (,x j ) = can be concluded from (3.93). Thus, n the general case, we can not defne f (2) such that the second order term n the ntal resdue s removed. Only f () and f (1) are properly determned so that the lattce Boltzmann algorthm wth ntalzaton (3.89) s only expected to yeld a frst order accurate velocty and an nconsstent pressure (see the numercal examples n secton 3.3). The dffculty s already ndcated by takng averages of (3.89), whch yelds ˆρ(,j) = 1, û(,j) = hψ(x j ) and thus mples ρ = 1, ρ k (,x j ) = k 1 (3.99) clearly, the value of ρ 2 (,x j ) s nconsstent to the ntal pressure n case p(,x). To obtan an algorthm whch yelds second order accurate pressure and velocty for general Naver-Stokes ntal values, we have to make sure that the low orders up to h 3 appearng n (3.9) can be removed. It s easy to see that the contrbutons of order h 2 and h 3 n (3.9) dsappear wth u 2 (,x j ) =, ρ 3 (,x j ) =, f we use the modfed ntalzaton f(,j) = f eq (1 + c 2 s h2 p(,x j ),hψ(x j )) h 2 c 2 s A (V )(V ψ)(x j )f + h 3 f (3) (,x j ) (3.1)

58 44 CHAPTER 3. CONSISTENCY OF LBM ON A GENERAL BOUNDED DOMAIN The moment u 3 t= contaned n f (3) t= = c 2 s V u 3 t= f ( A (V ) 3ν ψ 3ψ ψ + 3 ) 2 (3(V ψ)2 ψ 2 ) f + c 2 s (A ) 2 (V ) 2 (V ψ)f + A g t= s defned as u 3 = φ wth φ beng the soluton of the Posson problem φ = c 2 s tp G/2. In summary, two Posson equatons have to be solved and several dervatves of the ntal velocty are requred n order to obtan a second order accurate pressure. Apart from the smple condton (3.89) and the mprovement (3.1) we also test the ntermedate choce (2.42) whch s actually the ntalzaton (3.1) wth f (3) = and thus removes the h 2 contrbuton of (3.9) n the general case but not the part of order h 3 (ths ntalzaton has also been proposed n [71]). It generally leads to a frst order accurate pressure and a second order accurate velocty. Remark 6 The ntalzatons (3.89) and (2.42) are practcally feasble for many flows. On the contrary (3.1) s generally not achevable due to the unknown feld u 3 dependng on the tme dervatve of flud pressure. Only n specal cases, u 3 can be calculated explctly. For example, the Poseulle flow yelds u 3 = G/2. Remark 7 The smple ntal condton (3.89) s dentcal to the ntermedate choce (2.42) n case ψ(x) =, and even dentcal to the mprovement (3.1) f G(,x) = and t p(,x) =. The later equaton s fullflled f t G(,x) vanshes. Let I (α ), α = 1,2,3 denote the followng ntalzaton routnes I (1) (x) = f eq (1,hψ(x)), I (2) (x) = f eq (1 + c 2 s h2 p(,x), hψ(x)) h 2 c 2 s A (V )(V ψ(x))f, I (3) (x) = I (2) + h 3 f (3) (,x), and ˆr (α ) be the correspondng resdue functon, (3.11) 5 ˆr (1) = h 2 [ρ 2 (,x j )f A (V )f (1) ] + h k f (k) (,x j ), 5 ˆr (2) = h 3 f (3) (,x j ) + h k f (k) (,x j ), k=4 ˆr (3) = h 4 f (4) (,x j ) + h 5 f (5) (,x j ). k=3 (3.12) Summarzng the above results, we have the followng theorem,

59 3.4. NUMERICAL TESTS OF INITIAL CONDITIONS 45 Theorem 3 Assume that solutons u, p of the Naver-Stokes equaton (2.1) and (2.2), u 3,ρ 4 of (3.54), (3.58) have the followng regularty u C 5 ([,T], Ω), p C 4 ([,T], Ω), u 3 C 3 ([,T], Ω), ρ 4 C 2 ([,T], Ω). Then the predcton functon ˆf defned by (3.65), (3.64) and u 2 =, ρ 3 = as well as ρ = 1, ρ 1 = satsfes ˆf (,j) = I (α ) (x j ) + ˆr (α ) (,j),. (3.13) where p(, x) s calculated from (3.94). Moreover, there exsts an h-ndependent constants κ α + 1 such that ˆr (α ) (,j) = O(h κ ), x j Ω. (3.14) Proof: The regularty of u,p,u 3 and ρ 4 guarantees that the coeffcents n are contnuous n the closed bounded doman Ω, and hence bounded. ˆr (α ) 3.4 Numercal tests of ntal condtons For the purpose of numercally testng the ntal condtons, we consder three perodc problems n two dmensons. The frst problem s the decayng Taylor vortex wth analytc solutons (3.85) (3.87). The other two problems are constructed by multplyng tme dependent polynomals to the soluton of the frst one,.e., u (n) = t n u and p (n) = t n p, n =,1,2. Obvously n = corresponds to the decayng Taylor vortex flow. In general, the felds u (n),p (n) are solutons of (2.1) f we ntroduce a force term G (n) = nt n 1 u + t n (t n 1)(u )u, n =,1,2. The lattce Boltzmann method s based on a D2Q9 velocty set V and the BGK collson operator wth one tme relaxaton parameter. All the results below are calculated by settng the parameters a = b = π and ν =.5. The 2-perodc flow s computed n the regon Ω gven by x < 2 and y < 2. Numercal tests are mplemented for h { 1 1, 1 2, 1 3, 1 4, 1 5 }. The error between exact and numercal values s measured n terms of M e = max t n T max x j Ω ˆM(n,j) M(t n,x j ), (3.15) where, ˆM and M represent the numercal and exact values respectvely. Here M s ether u (n) or p (n) wth correspondng ˆM gven by ( ) 1 hû(n,j) = f (n,j)c, or h c 2 s h2(ˆρ 1) = c 2 f s h 2 (n,j) 1.

60 46 CHAPTER 3. CONSISTENCY OF LBM ON A GENERAL BOUNDED DOMAIN Let us frst consder the smple ntal condton (3.89) appled to all three cases. In the case n =, we have ρ 2 (,x) = 3p () (,x). Accordng to our analyss, we therefore expect the lattce Boltzmann pressure to be of order h (nconsstent) and the velocty to be only frst order accurate. Ths s supported by the numercal results n fgure 3.6 and tables 3.2 and Fgure 3.6: The logarthmc error of pressure (left) and velocty (rght) versus log 1 h for the flows u (n),p (n) (n = ( ), n = 1( ), n = 2(+)). The errors are normalzed to stress the dfferent slopes (see tables 3.2, 3.3 for error values). For n = 1, ρ 2 (,x) = c 2 s p (1) (,x) = and u (1) (,x) = so that the second order contrbuton of (3.9) vanshes. However, the ntal value of u 3 s now dfferent from zero because t satsfes Specfcally, we have u 3 t= = t ρ 2 t= G(1) t= = 3p () t=. u 3 (,x) = 3 8 ( sn(2ax) A 3, sn(2by) ) B 3, (3.16) thus a frst order accurate pressure and a second order accurate velocty are expected. Ths predcton s supported by the numercal experment (fgure 3.6 and tables 3.2, 3.3). Table 3.2: Error of pressure. The order s obtaned wth least squares. h n = n = 1 n = 2 1/ / / / / order

61 3.4. NUMERICAL TESTS OF INITIAL CONDITIONS 47 Table 3.3: Error of velocty. The order s obtaned wth least squares. h n = n = 1 n = 2 1/ / / / / order In the last case n = 2, also the ntal tme dervatve of the pressure vanshes so that u 3 t= = apart from ρ 2 (,x) = and u 1 (,x) =. Hence, both pressure and velocty are second order accurate (fgure 3.6, and tables 3.2, 3.3)). In the case n =,1 where the smple condton (3.89) fals to gve second order velocty and pressure, we now show that the mproved condton (3.1) works. The requred feld u 3 t= s gven by (3.16) n the case n = 1 and by u 3 (,x) = 3 ( sn(2ax) 2 να a 3, sn(2by) ) b 3, n the case n =. The result of the numercal experment s shown n fgure Fgure 3.7: Error of pressure (left) and velocty (rght) versus log 1 h usng the mproved ntalzaton routne for the flow u (n),p (n) (n = ( ), n = 1( )). The accuracy s second order n both cases. The effect of the mproved ntal condton on the reducton of the spatal error s demonstrated n fgure 3.8. We stress that even though the order of velocty does not ncrease when usng (3.1) nstead of (2.42), the velocty error s smoother f u 3 s ntalzed correctly usng condton (3.1) as shown n fgure 3.8

62 48 CHAPTER 3. CONSISTENCY OF LBM ON A GENERAL BOUNDED DOMAIN 1 x x x Fgure 3.8: Tme evoluton of the maxmal spatal error of pressure (left) and velocty (rght) for the frst 1 tme steps (h = 1/1). Results for the ntal condton (3.89) are gven by a sold lne, condtons (2.42) and (3.1) are represented by dashed and dotted lnes. The lower plots are magnfcatons of the upper ones. 3.5 Analyss of bounce back rule The purpose of ths secton s to llustrate the boundary analyss procedure usng the well-known bounce back rule. The accuracy of the bounce back rule has already been studed n [16, 8, 24]. The phenomena, whch we encounter when analyzng the bounce back rule, are very typcal and common for other lattce Boltzmann boundary schemes. We start by nsertng the regular asymptotc expanson (3.1) nto the bounce back rule (2.45) and use the specfc form (3.64) of the coeffcents. We calculate the resdue at ponts x j on the boundary (see fgure 2.2). The computatons are very smlar to the analyss of the update rule and we can use the same operators D k. Takng nto account that x j = x j + hq j c (see fgure 2.2), the drectonal dervatve s now q j c nstead of c. In accordance wth (3.5) we have (defnng D (θ,σ) = 1) f (n + 1,j) = m h m m k= D m k ( t,q j c )f (k) (t n,x j ). The rght hand sde of (2.45), whch s denoted as f b (n,j) hereafter, has a structure smlar to the collson product. Expandng also around x j, we arrve at f b (n,j) = h2c 2 + m h m m k= s f c φ(t n,x j ) D m k (,q j c ) [ f (k) + (A(f eq,(k) f (k) )) + g δ m3 ](t n,x j ).

63 3.5. ANALYSIS OF BOUNCE BACK RULE 49 Fnally, collectng terms of equal order n the bounce back rule, we obtan the resdue = h2c 2 s f c (u 1 (t n,x j ) φ(t n,x j )) ( + h 2 f (2) f (2) + 2q j(c )f (1) (A(f eq,(2) f (2) )) ) (t n,x j ) + O(h 3 ). (3.17) Apparently, when u 1 (t,x) = φ(t,x), the frst order resdue s removed. Hence the boundary value of f (1) s acheved and u 1 turns out to be the velocty feld of the full boundary value problem (2.1), (2.3). Proceedng to order h 2, we frst calculate A(f eq,(2) f (2) ) = AA (V )f (1) = P(V )f (1). Snce 1,(V )f (1) = u 1 =, 1,V(V )f (1) = c 2 s ρ 1 =, (V )f (1) s orthogonal to the kernel of the matrx A. Therefore Q(V )f (1) = so that A(f eq,(2) f (2) ) = P(V )f (1) = (P + Q)(V )f (1) = (V )f (1). Next, we compute f (2) f (2) = 2c 2 s (u 2 c )f + (A (V )f (1) ) (A (V )f (1) ). Recallng equalty (3.47), we know A (V )f (1) s quadratc wth respect to c V so that the last two terms n the above equaton are equal and thus cancel out. Altogether we fnd the second order resdue 2c 2 s f [ c u 2 (t n,x j ) (q j 1 2 )(c c ) : S[u 1 ](t n,x j ) ] (3.18) after reformulatng (c )(c u 1 ) = c c : S[u 1 ]. We wll see, n the followng, that t s generally mpossble to extract boundary values for u 2 from (3.18). We need to nvestgate ths relaton carefully. General behavor. It s mportant to notce that for general geometres the values q j cannot be wrtten as smooth functons of the ponts x j because the dfference between the values q j correspondng to neghborng nodes x j s generally of order one whle the dstance between the nodes s of order h. Typcally x j oscllates wth a frequency proportonal to 1/h along nclned boundares. Consequently, (3.18) cannot be removed by any smooth functon u 2 for general flows, wth S[u 1 ]. Ths tells us that f (2) cannot be determned as a smooth functon. Hence we can expect that the lattce Boltzmann soluton f generally exhbts rregular behavor at order h 2. As a consequence, the Naver-Stokes velocty u 1 can only be expected to be frst order accurate when extracted from û and that the pressure s no longer recoverable from ˆρ (nconsstent pressure).

64 5 CHAPTER 3. CONSISTENCY OF LBM ON A GENERAL BOUNDED DOMAIN Specfc case of vanshng stress S[u 1 ] =. Settng u 2 = removes the resdue term (3.18). Snce the homogeneous Oseen type equaton (3.41), (3.55) wth a zero boundary value renders the soluton u 2 =, the lattce Boltzmann soluton gves rse to a second order accurate Naver-Stokes velocty. An example s the lnear flow (3.81) n secton 3.2 whch possesses S[u] =. Specfc geometres q j 1 2. The stuaton may be better n the case of specfc geometres where q j s constant (and thus smooth) along connected components of the boundary Ω. For example, n the partcular case q j = 1/2 where the boundary s located half a lnk dstance away from the boundary nodes, the second term n the resdue (3.18) s zero, and u 2 (t,x) = removes the resdue. Ths may lead to u 2 = n the whole doman and the expanson of f shows rregular behavors only at thrd order (whch can be checked analytcally by carryng out the expanson to order three). Thus the pressure can be expected wth frst order and velocty wth second order accuracy n the case q j = 1/2. Specfc geometres q j Constant 1 2. However, for other constant values of q j, the stuaton turns out to be qute dfferent. Despte the fact that the second term of (3.18) s now smooth, (3.18) can typcally not be removed by any smooth u 2. Ths s due to the fact that c u 2 s lnear n c whle (c )c u 1 s quadratc whch, n general, leads to conflcts f equalty s requred for a lnearly dependent set of vectors c. To gve a specfc example, we consder the statonary lnear flow (3.82) n a half space Ω = (, ) R. Usng the D2Q9 model, the ncomng drectons at the boundary Ω are ( 1,) T, ( 1,1) T, and ( 1, 1) T. If we assume, for example, q j = then the followng three condtons on u 2 = (u x 2,uy 2 )T follow from (3.18) u x 2 = 2, u x 2 + u y 2 = 1, ux 2 u y 2 = 1. (3.19) However, by addng the second and the thrd condton, we fnd u x 2 = whch obvously contradcts the frst condton. Consequently, we cannot construct a regular coeffcent f (2) to remove the resdue (3.18). The pressure s predcted to be nconsstent and a frst order accurate velocty s expected. A more extreme example s the statonary lnear flow (3.83) n a unt square Ω = [,1] 2 wth q j =. Only at four corners, the relaton (3.18) cannot be removes. For example at the rght top corner (1, 1), there are fve ncomng drectons when mplementng the D2Q9 model, c 3 = ( 1,) T, c 4 = (, 1) T, c 6 = ( 1,1) T, c 7 = ( 1, 1) T, c 8 = (1, 1) T. The correspondng condtons followng from (3.18) are u x 2 =, uy 2 =, ux 2 + uy 2 = 2, ux 2 uy 2 = 2, ux 2 uy 2 = 2, and the contradcton s obvous. As a consequence, the nconsstency of pressure stll arses (see the fgure 3.1 n secton 3.6).

65 3.5. ANALYSIS OF BOUNCE BACK RULE 51 Fnally we remark that wthn the specal case of geometres havng constant q j there are some rare cases n whch the lnear and quadratc c -dependence n (3.18) does not lead to ncompatbltes. One such very specal stuaton s the famous Poseulle flow n an axs parallel channel whch therefore s a rather nadequate test case for the general behavor of boundary algorthms. The typcal structure of such a flow n Ω = R (,1) s At the boundary x 2 =, we have ( ) 1 u(t,x) = (x 2 1)x 2, p(t,x) =. S[u] = 1 2 ( ) 1. 1 Agan wth the D2Q9 model, the ncomng drectons are (,1) T, (1,1) T and ( 1,1) T. The correspondng values c T S[u 1]c are, 1,1 so that u 2 (t,x) = (q x2 = 1/2)(1,) T removes (3.18) at the lower boundary (q x2 = s the common value of q j at the lower boundary). Smlarly, u 2 (t,x) = (1/2 q x2 =1)(1,) T elmnates (3.18) at the upper boundary. Usng these boundary values, we have to solve the Oseen-type equaton (3.55) wth the ncompressblty condton (3.41) to determne u 2,ρ 3 nsde the doman. We fnd ρ 3 = and the smooth feld u 2 (t,x) = [ (q x2 = 1/2) + x 2 (1 q x2 =1 q x2 =) ]( ) 1. In fact, we now have û = hu 1 + h 2 u and (ˆρ 1)/3h 2 = +h +O(h 2 ). (actually, the pressure s exact n ths smple case snce (ˆρ 1)/3h 2 = s the exact pressure). The order of accuracy s not nfluenced, we only get a better nsght nto the structure of the error. In summary, f we denote the resdue for the bounce back rule at node x j for the ncomng drecton c by ˆR (n,j) = 2c 2 s f [ c u 2 (t n,x j ) (q j 1 ] 2 )(c c ) : S[u 1 ](t n,x j ) h 2 + O(h 3 ), (3.11) and for non-ncomng drectons c by ˆR (n,j) =, we obtan the followng theorem. Theorem 4 Assume that solutons u, p of the Naver-Stokes equatons (2.1) (2.3) u 3,ρ 4 of (3.54), (3.58) wth sutable boundary condtons have the followng regularty u C 5 ([,T], Ω), p C 4 ([,T], Ω), u 3 C 3 ([,T], Ω), ρ 4 C 2 ([,T], Ω), then the predcton functon ˆf defned by (3.65), (3.64) and u 2 =, ρ 3 = as well as arbtrary constants ρ and ρ 1 satsfes the bounce back rule (2.45) at

66 52 CHAPTER 3. CONSISTENCY OF LBM ON A GENERAL BOUNDED DOMAIN any boundary node x j wth the ncomng drecton c, ˆf (n + 1,j) = ˆf (n,j) + J( ˆf) (n,j) + g (n,j) such that + 2hc 2 s f φ(t n,x j ) c + ˆR (n,j) (3.111) ˆR (n,j) = O(h κ b ), x j Ω, nh 2 T. (3.112) Here κ b 2 s an h ndependent constant, and κ b = 3 for specal flows wth S[u] = or for geometres whch cut the boundary lnks exactly n the mddle (q j = 1/2). Otherwse κ b = 2. Remark 8 Referrng to the dscusson about solvng for u 3,ρ 4 n sectons , a possble boundary treatment are homogeneous Neumann condtons for ϕ and homogeneous Drchlet condtons for ω. In the next secton, numercal examples are used to llustrate the theoretcal predctons about the lattce Boltzmann algorthm wth bounce back rule n the above mentoned cases. 3.6 Numercal test of bounce back rule We apply the lattce Boltzmann method wth the bounce back rule to several boundary value problems for whch exact solutons are known. Agan the velocty set V s chosen to be of D2Q9 type wth BGK collson model and the relaxaton parameter τ = c 2 s ν + 1/2. The frst and second test problems are the statonary lnear problem (3.82) and (3.83) descrbed n secton 3.2 whch we now restrct to the unt square. The thrd problem s the decayng Taylor vortex flow restrcted to the unt square wth parameters A = π 2 and B = π 2. Its exact soluton s suppled by (3.85) (3.87). The fourth one s the Poseulle flow drven by the body force n an nclned channel, the slope of the channel used here s 3/1. The last one s the crcular flow n a dsk wth radus R = 1. For the crcular flow, we ntalze the velocty to the exact value at tme t =.5. For all other test problems, the ntal velocty s provded at tme t =. The termnaton tme s T = 1. Boundary condtons are specfed by evaluatng the exact velocty on the boundary Ω. The numercal tests are carred out on a sequence of grds wth grd sze h { 1 1, 1 2, 1 3, 1 4, 1 5 }. The flud vscosty parameter s fxed as ν =.1. The error s measured n the same way as descrbed n secton 3.4. For the flows n the unt square the boundary nodes are located exactly on the boundary,.e. q j = for all boundary nodes. By settng the grd n ths way, we avod the specfc stuaton q j = 1 2. For the crcular flow and nclned Poseulle flow the boundary nodes exhbt many dfferent q j [,1) no matter how the grds are lad out. The rght fgure n 3.9 shows the logarthmc error of velocty

67 3.6. NUMERICAL TEST OF BOUNCE BACK RULE 53 aganst the logarthmc grd sze, whch decreases whle the grd becomes fner. The least squares slopes have values around 1, whch demonstrates that the bounce back rule can brng out frst order accurate veloctes n general. A smlar plot for pressure s gven on the left of fgure 3.9 but here the error ncreases wth decreasng grd sze, and the slopes are around zero or even negatve, whch means the pressure s zero order accurate. Hence the numercal results concde wth the theoretcal predcton n the general case for the bounce back rule. The tremendous dfference between exact pressure and lattce Boltzmann approxmaton s presented n fgure error of pressure error of velocty Fgure 3.9: Logarthmc pressure (left) and velocty (rght) error versus log 1 h for varous test problems show that the bounce back rule leads to an nconsstent pressure and frst order accurate velocty. ( ) lnear flow (3.82), ( ) lnear flow (3.83), (+) Taylor vortex, ( ) crcular flow and ( ) Poseulle flow Fgure 3.1: Pressure contour lnes compared wth exact soluton (black) for the lnear flow (3.82) and the decayng Taylor vortex flow. In the exceptonal case that all boundary nodes are located half a grd spacng away from the boundary, t should be noted that the choce q j = 1/2 s only

68 54 CHAPTER 3. CONSISTENCY OF LBM ON A GENERAL BOUNDED DOMAIN possble for a very restrcted class of flow geometres Ω. By smulatng the statonary lnear flow (3.82) n a unt square, fgures 3.11 compare the performance of the bounce back rule n the case q j = (all boundary nodes are on Ω) wth the favorable case q j = 1/2 whch can be used for ths smple set Ω. The numercal accuracy demonstrates an ncrease of one order for both pressure and velocty whch agan agree wth the theoretcal analyss. 2 error of velocty 1.5 error of pressure Fgure 3.11: Logarthmc velocty (left) and pressure (rght) error versus log 1 h n the case of the 2D statonary lnear flow (3.82) wth q j = ( ) and q j =.5 ( ). Alltogether, the analyss shows that the bounce back rule generally can not produce a numercal soluton whch allows a regular expanson up to second order. In other words, one wll generally observe a boundary layer or oscllaton for the error û hu 1 of the velocty. Fg.3.12 dsplays the error behavor of the velocty for a lnear problem and the crcular flow and 3.13 for the Taylor vortex flow. Snce the matrx n lnear flow s symmetrc and the parameters a and b are dentcal for the Taylor vortex flow, both flows have some symmetry property. Therefore the error plots along few cuts already show the whole error behavor for these flows. In the case of the Taylor vortex flow, the error vares smoothly wth varable y and oscllates volently wth varable x. In the cases of lnear and crcular flow the boundary layer occurs close to the boundary.

69 3.7. DEFINITION OF CONSISTENCY ORDER Fgure 3.12: Left: plot of (û hu)/(h 2 ) for the lnear flow (3.83) along the cuts x =.1,.3. Rght: plot of (û hu)/(h 2 ) ( ) and (ˆv hv)/(h 2 ) ( ) at t =.6 for the crcular flow. h =.2 for both cases x x Fgure 3.13: Plot of (û hu)/(h 2 ) along several cuts for the decayng Taylor vortex flow. Left: along the cuts y =.1 ( ),.3 ( ). Rght: along the cuts x =.1 ( ),.3 ( ). 3.7 Defnton of consstency order After analyzng the resdue of the ntal condtons and boundary condtons, we can dscuss the consstency order of the numercal scheme more carefully. We remark that, usually, the consstency order s defned based on the order of the resdue n such a way, that t equals the convergence order of the method. To follow ths conventon we assume a certan type of convergence estmate dependng on the varous resdues correspondng to the dfferent parts of the algorthm. Based on the estmate, we defne upper bounds for the convergence order. The general expectaton s that the least upper bound s a reasonable order of consstency. As we have always stated, the lattce Boltzmann method s consdered to solve a

70 56 CHAPTER 3. CONSISTENCY OF LBM ON A GENERAL BOUNDED DOMAIN problem such as (2.1). Moreover, from the asymptotc analyss, we have defned a truncated expanson (3.65) whch shows a clear relaton to the Naver-Stokes soluton. If we have the consstency order of the expanson (3.65) wth respect to the lattce Boltzmann soluton f, we can conclude how well ths expanson represents the lattce Boltzmann solutons and thus clarfy the relaton to the Naver-Stokes equaton. For the mathematcal settng of our dscusson, we frst ntroduce the dscretzaton N h of the tme nterval [,T] and Ω h of the spatal doman Ω for an arbtrary grd sze h H, where H s the set of all the dscretzaton parameters possbly used n the smulatons, whch contans an accumulaton pont at zero. Next, we ntroduce a lnear and normed grd functon space V h equpped wth norm Vh. The lattce Boltzmann soluton f and the truncated expanson ˆf on Ω h are two functons n V h. Furthermore, we defne a grd functon set A = { s s(h) V h, h H n whch each element s a functon of h and grd nodes n Ω h and contans, for example, f and ˆf. Our dscusson s buld on A θ whch may be some subspace of A or A tself. In addton, we denote a complete lattce Boltzmann algorthm by a set of mappngs, E () h : V h V h, = 1,...,m, (3.113) where f s the soluton of E () h (f) =, = 1,...,m, (3.114) Generally we wrte a scheme n a concse way by E m h or Em, whch contans m mappngs representng the nner algorthm, the ntalzaton algorthm, etc. For any s A, E () h (s) s the resdue wth respect to the th mappng. The structure of an arbtrarly grd functon s A θ s analyzed at any possble order k R of h by checkng the resdue of modfcatons s + h k γ wth γ A. Wth the help of the followng set { } ord E m(s) = κ R m () E h (s) V h = O(h κ ), = 1,...,m., (3.115) whch contans the powers of all possble order estmate of the resdues of s, we have the followng defnton Defnton 1 (Undetermned order of s n A θ ) Gven a lattce Boltzmann algorthm E m h and s A θ. If there exsts γ A whch satsfes c 1 γ Vh c > for sutable constants and all h H such that s + h k γ A θ and ord E m(s) ord E m(s + h k γ) then we say that k s an undetermned order of s n A θ under ths lattce Boltzmann scheme. }

71 3.7. DEFINITION OF CONSISTENCY ORDER 57 A drect result of ths defnton s Lemma 1 Let E m h be a lattce Boltzmann algorthm and s A θ. If s has an undetermned order k n A θ under E m h, then k s also an undetermned order of s n A. Intutvely f k s an undetermned order of s n the space A θ, t s already clear that s wll generally not descrbe the numercal soluton correctly n that order. To make ths more clear, we frst ntroduce a general noton of convergence wth the help of set ord E m(s). Defnton 2 (A standard convergence estmate n A θ ) Let f be the soluton of the lattce Boltzmann algorthm E m h. We say that E m h admts a standard convergence estmate n A θ, f there exsts a monotone ncreasng functon σ : R m R such that for any s A θ and any κ ord E m(s) holds f s Vh = O(h σ(κ) ). We call σ(κ) an estmated convergence order of s. Based on the above defntons, we have the followng concluson. Theorem 5 Let f be the soluton of the lattce Boltzmann scheme E m h, whch possesses a standard convergence estmate n A θ based on the monotone ncreasng functon σ : R m R. Assume that s A θ has an undetermned order k n A θ. Then σ(κ) k for any κ ord E m(s). Proof: (By contradcton). Assume there s κ ord E m(s) such that σ(κ) > k. Due to the standard convergence estmate n A θ, f s Vh = O(h σ(κ) ). Hence there s a constant c 1 and some h 1 > such that f s Vh c 1 h σ(κ) for h < h 1. Snce s s not determned n order k under the scheme E m h, there s γ A such that s + h k γ A θ and ord E m(s) ord E m(s + h k γ) Hence κ ord E m(s + h k γ), and there are constants h 2, c 2 such that f s h k γ Vh c 2 h σ(κ) (3.116) holds for any h < h 2. Nether c 1 nor c 2 depend on h. Snce for any h < mn(h 1,h 2 ), h k γ Vh = (h k γ + s f) + (f s) Vh we have (h k γ + s f) Vh + (f s) Vh (c 1 + c 2 )h σ(κ), < γ Vh (c 1 + c 2 )h σ(κ) k. Takng the lmt h of the rght sde, we obtan lm h γ Vh =. Ths contradcts wth γ Vh > c >. Therefore σ(κ) k.

72 58 CHAPTER 3. CONSISTENCY OF LBM ON A GENERAL BOUNDED DOMAIN Ths lttle observaton gves us a hnt how to defne the consstency order of s whch s qute lkely to be equal to the best possble convergence order, f the scheme s convergent n A. Knowng that the estmated convergence order s not bgger than the smallest undetermned order, we defne M s (E m h ) := {k R s s undetermned by Em h at order k n A.}, M θ s (Em h ) := {k R s s undetermned by Em h at order k n A θ.}, to collect all the undetermned orders of s n A and A θ respectvely. Defnton 3 (Consstency order of s w.r.t. the LB algorthm) The consstency order ÕE m(s) of s s equal to nf M s(e m h ). As a drect concluson of Lemma 1, we have Proposton 1 For a gven lattce Boltzmann algorthm and s A θ, holds M θ s M s, nf M s nf M θ s. Usually, computng the consstency order s not a smple task due to a lack of unqueness and exstence results for the equatons governng γ. However, once we know some k M s (E m h ), we can conclude that the consstency order of s s not larger than k. On the other hand, the estmated convergence order may help to get the correct consstency order by means of Theorem 5. Even n the case that the standard convergence estmate of the scheme s only avalable n a subspace A θ, the consstency order of s may stll be obtaned (ths concluson for ˆf wll be verfed n chapter 4).

73 Chapter 4 Convergence of the lattce Boltzmann method In ths chapter we ntend to study the convergence of the lattce Boltzmann method ntroduced n chapter 2. Our consderatons are a drect contnuaton of the stablty result [45] whch provdes sutable norm estmates for the soluton of the lnear lattce Boltzmann method both n the case of perodc domans and on bounded domans n connecton wth the bounce back rule. The general structure of our convergence proof follows the approach n [73]: stablty combned wth an asymptotc expanson of the numercal soluton yelds convergence. Ths dea has been successfully appled to the convergence theory [18, 19] for nonlnear convectve-dffusve lattce Boltzmann methods. Recallng that the nonlnear term n the presented lattce Boltzmann algorthms appears only n the collson part and the collson operator can be dvded nto a lnear and a quadratc part,.e., J(f) = J L (f) + J Q (f,f), we stress that a careful analyss of the lnear case s an mportant step towards a convergence proof of the lattce Boltzmann method wth non-lnear equlbrum functons. In fact, for a contnuous lattce Boltzmann equaton a rgorous convergence proof n the nonlnear case could be obtaned [46] by controllng only the lnear part J L of the collson operator. Our convergence proof s therefore carred out also n two parts: lnear and nonlnear. Ths chapter begns wth a revew of the stablty results from [45], then n secton 4.2 the truncated asymptotc expanson of lnear and nonlnear lattce Boltzmann solutons s recalled. Wth the help of stablty results [45] and truncated expanson, we then gve a rgorous convergence proof for both the lnear (secton 4.3) and the nonlnear (secton 4.4) lattce Boltzmann methods. 4.1 Stablty of the lnear BM The stablty result presented n [45] s formulated for lattce Boltzmann algorthms wth lnear collson operator J L on perodc doman or on bounded domans wth bounce back rule at the boundares. In order to express the norm estmates n a compact form, the lattce Boltzmann algorthm s formulated n 59

74 6 CHAPTER 4. CONVERGENCE OF THE LATTICE BOLTZMANN METHOD evoluton operator notaton. The relaton to the abbrevaton E (1) whch s ntroduced n chapter 3, s also presented. h, E(2) h, E(3) h, Functon space In ths work, the lattce Boltzmann algorthms are set up by dscretzng the tme nterval [,T] (T > ) nto a set of tme ponts t n = nh 2 and the spatal doman Ω nto a node set of x j = jh. We denote the related ndex sets by N h = h 2 [,T] Z, Ω h = h 1 Ω Z d (4.1) for an arbtrary grd sze h. Based on these two sets, we defne a grd functon space G h = {s : Ω h R N }, I : G h G h s the dentty operator. Moreover, we set V h = { s s : N h Ω h R N}, (4.2) n ths chapter and A s based on ths set accordng to the dscusson n secton 3.7. The set H of grd parameters wll be gven dfferently wth respect to the dfferent algorthms. Obvously A contans all grd functons n V h. For an element s V h, we denote s(n) G h for any n N h. Later a dscrete ntegraton norm s ntroduced on G h, and the norm Vh s related to the temporal maxmum of ths dscrete ntegraton norm. It s emphaszed that for a gven grd functon s A, the argument h s often suppressed wthout causng conflcts lke the lattce Boltzmann soluton f and the prevously obtaned regular expanson ˆf. As for the lattce Boltzmann algorthm E m h, we let the second mappng n a scheme E m h correspond to the ntal condton by settng E (2) h (s)() = E α (s), E (2) h (s)(n) =, n,n N h, (4.3) n whch E α s the ntal condton ntroduced n Theorem 3 assocated wth I (α ), E α (s) = s(,j) I (α ) (x j ). (4.4) The lattce Boltzmann soluton satsfes E (2) h (f) =. Apparently E (2) h s requred for problems on both perodc domans and general bounded domans. The other members of E m h wll be gven explctly for the perodc case and bounded case wth the bounce back rule n the next two sectons.

75 4.1.2 Reformulaton of lattce Boltzmann algorthms on perodc domans Reformulaton of lattce Boltzmann algorthms on perodc domans In secton 2.4.1, we have descrbed the case of perodc domans, and have rewrtten the lattce Boltzmann algorthm wth the help of modulo addton. In partcular, the grd sze s chosen to be n the set H = {1/m m N} (4.5) n order to mplement the perodcty condton exactly on the boundary. Here we take up ths descrpton and further defne the assocated shft operator (Sf) (j) = f (j + m c ), = 1,...,N, x j Ω. (4.6) Then we can formulate the lattce Boltzmann algorthm (2.2) n the compact form Sf(n + 1) = f(n) + J(f(n)) + g(n). Here f(n), g(n) G h are the vectors f(n,j) and g(n,j) (j Ω h ) respectvely. In addton, we let E (1) h be the update rule and complement t wth zero equatons at n =, E (1) h (s)(n) = Ss(n) [s(n 1) + J(s(n 1)) + g(n 1)], n N h \ {}, E (1) h (s)() =. (4.7) Obvously, f satsfes E (1) h (f) =. (4.8) Combned wth the ntal mappng E (2) h, a complete lattce Boltzmann scheme s obtaned to solve a perodc problem Reformulaton of lattce Boltzmann algorthms wth the bounce back rule In secton 2.4.2, the bounce back rule has been descrbed, and s appled at every boundary node x j to the components of f whch belong to ncomng veloctes. On the other hand, for each velocty c k V, a boundary node set can be ntroduced, n whch each node possesses the ncomng drecton c k = c k,.e., k Ω = {x j : x j+ck = x j + hc k / Ω}. Usng the new ndex symbol, the bounce back rule at x j k Ω has the form f k (n + 1,j) = f k (n,j) + J k (f(n,j)) + g k (n,j) + 2hF (eq) k (,φ(t n,x jk )) where x jk s the ntersecton pont of the lnk along c k and the boundary Ω.

76 62 CHAPTER 4. CONVERGENCE OF THE LATTICE BOLTZMANN METHOD Agan, the full lattce Boltzmann scheme can be wrtten n terms of a sutable shft operator S : G h G h n the compact form where Sf(n + 1) = f(n) + J(f(n)) + g(n) + b(n), (4.9) ( Sf) k (j) = { f k (j + c k ) x j Ω\ k Ω f k (j) x j k Ω and b(n) G h s defned by { x j Ω \ k Ω, b k (n,j) = 2hF (eq) k (,φ(t n,x jk )) x j k Ω. We stll use E (1) h to denote the update rule, and let E (3) h correspond to the bounce back rule. However, the update rule s mplemented only for the nonncomng veloctes n V at all nodes, and the bounce back rule only for the ncomng drectons at boundary nodes. We defne trval equatons for the mssng drectons, (E (1) h (s)) k(n,j) =, x j k Ω, n N h \ {}, k = 1,...,N, (E (3) h (s)) k(n,j) =, x j Ω \ k Ω, n N h \ {}, k = 1,...,N, E (1) h (s)(,j) =, j Ω h, E (3) h (s)(,j) =, j Ω h. Hence the scheme (4.9) can be translated nto E (1) h (s)(n + 1) + E(3) h (s)(n + 1) = Ss(n + 1) [s(n) + J(s(n)) + g(n) + b(n)] (4.1) for all n,n + 1 N h and the lattce Boltzmann soluton f satsfes E (1) h In addton, the set of dscretzaton parameters (f) + E(3) h (f) =. (4.11) H (, 1], (4.12) contans an accumulaton pont at zero. Let δω h Ω h be the set of all boundary nodes for whch at least one neghbor s mssng, then δω h = N =k kω. Partcularly, n the convergence proof, we requre a property of the number of boundary nodes δω h h d δω h ηh (4.13) for some h-ndependent constant η > whch expresses the fact that the boundary s one dmenson lower than the volume.

77 4.1.4 Norms and stablty Norms and stablty The key pont n the stablty analyss s the defnton of the stablty structure (P,a,λ) consstng of a matrx P R N N of left egenvectors of J L correspondng to the egenvalues λ 1,..., λ N and sutable weghts a > collected n the vector a R N. As ponted out n [45], the frst row of P can be taken as the vector 1 and the next d rows can be chosen as v 1,...,v d wth assocated egenvalues λ 1 =... = λ d+1 =. Stablty s then guaranteed under the condtons λ 2, PJ L = dag(λ)p, P T P = dag(a). (4.14) The frst condton s the well known necessary stablty condton n the lnear case [67]. The second condton states that the rows of P contan the left egenvectors of J L. The thrd condton s an addtonal requrement on the egenvectors of J L, whch can be reformulated as an orthonormalty condton wth respect to the weghted scalar product (for detals see [45]) f,g = N =1 1 a f g. It s requred for provng that both the shft operator and the operator (I +J L ) on the rght hand sde of the lattce Boltzmann equaton (see (4.8), for example) have norms bounded by one. I s the dentty operator. A sutable norm for the stablty estmate s defned n terms of a weghted nner product based on the vector a f,g a = N a f g, f,g R N, =1 accordng to N f 2 a = f,f a = a f 2, f RN. =1 Summng over all nodes of the grd, we obtan a dscrete ntegral norm on G h for grd functons f 2 a,ω h = j Ω h h d f(j) 2 a, f G h. Based on t, a norm for convergence and consstency study s defned on V h, f Vh = max n N h f(n) a,ωh, f V h. The operator norm assocated to a,ωh s gven by B a,ωh = Bf a,ωh sup. f G h \{} f a,ωh

78 64 CHAPTER 4. CONVERGENCE OF THE LATTICE BOLTZMANN METHOD Smlarly, for matrces C R N N, we defne C a = Cf a sup. f R N \{} f a Wth these notatons, we can state the stablty result presented n [45]. Theorem 6 If the lattce Boltzmann model possesses a stablty structure (P, a, λ) wth a symmetrc weght vector (.e. a = a ) satsfyng (4.14), then S and S are sometres and S a,ωh = 1, S 1 a,ωh = 1, S a,ωh = 1, S 1 a,ωh = 1, (4.15) I + J L a 1, I + J L a,ωh 1. (4.16) It s also shown n [45] that the standard BGK, TRT, and MRT models possess the requred stablty structure. We conclude ths summary wth a remark on how to estmate the conserved moments ρ(f) = f and u(f) = f c. Recallng that the frst row of P s 1 T and the next d rows are v T 1,...,vT d, we can also wrte ρ(f) = (Pf) 1 and u α (f) = (Pf) 1+α. Now, n vew of the thrd condton n (4.14) Pf 2 = Pf,Pf = P T Pf,f = f 2 a, so that ρ(f) = (Pf) 1 Pf = f a, u(f) = (Pf) (Pf)2 d+1 Pf = f a. In partcular, we obtan for the dscrete spatal L 2 norms ρ(f) 2 Ω h = j Ω h h d ρ(f(j)) 2 f 2 a,ω h, u(f) 2 Ω h = and for a dscrete maxmal norm j Ω h h d u(f(j)) 2 f 2 a,ω h. (4.17) u(f),ωh = max n N h u(f(n)) Ωh. (4.18) 4.2 The truncated asymptotc expanson and moment relatonshp In ths chapter, we use the truncated asymptotc expanson ˆf defned by (3.65) wth coeffcents (3.64) n the nonlnear case and (3.73) n the lnear case together wth u 2 =, ρ 3 =, ρ = 1 and ρ 1 = and denote t f. More explctly, we have f(n,j) = f + 5 h k f (k) (t n,x j ), (4.19) k=1

79 4.2. THE TRUNCATED ASYMPTOTIC EXPANSION AND MOMENT RELATIONSHIP 65 wth the coeffcents f (1) = F L (,u); f (2) = F L (c 2 s p,) + F Q (u,u) A V f (1) ; f (3) = F L (,u 3 ) A ( t f (1) + (V )f (2) + 1 ) 2 (V )2 f (1) G ; f (4) = F L (ρ 4,) + F Q (u,u 3 ) A ( t f (2) + (V )f (3) (V )2 f (2) + D 3 ( t,v )f (1) ) ; f (5) = A ( t f (3) + (V )f (4) (V )2 f (3) ) A ( D 4 ( t,v )f (1) D 3 ( t,v )f (2)). (4.2) By droppng out the F Q terms, the coeffcents for the lnear problem are obtaned. Here (u, p) s the soluton of Naver-Stokes problem (2.1) or Stokes problem (3.69) wth ntal condton (2.2) and boundary condton (2.3) f requred. (u 3,ρ 4 ) s the soluton of Oseen type problem (3.54), (3.58) or Stokes problem (3.72). The advantage of f compared to f s that ts relaton to the flud velocty u and pressure p s explct and easy to understand. We can recover p and u n leadng order from f by computng moments c 2 s h 2 1 h ( N ) f (n,j) 1 = p(t n,x j ) + h 2 c 2 s ρ 4(t n,x j ), =1 N f (n,j)c = u(t n,x j ) + h 2 u 3 (t n,x j ). =1 In vew of (4.17), the correspondng moments of the lattce Boltzmann soluton satsfy P(n,j) = c2 s h 2 ( N ) f (n,j) 1, U(n,j) = 1 h =1 N f (n,j)c (4.21) =1 p(t n ) P(n) Ωh c2 s h 2 f(n) f(n) a,ωh + h 2 c 2 sρ 4 (t n ) Ωh, u(t n ) U(n) Ωh 1 h f(n) f(n) a,ωh + h 2 u 3 (t n ) Ωh. (4.22) Here, p(t n ),ρ 4 (t n ),u(t n ),u 3 (t n ) G h are defned by restrcton to the grd, for example p(t n )(j) = p(t n,x j ). Consequently, we obtan nformaton about the convergence of the lattce Boltzmann moments to the Naver-Stokes soluton, f the predcton f s suffcently close to the lattce Boltzmann soluton.

80 66 CHAPTER 4. CONVERGENCE OF THE LATTICE BOLTZMANN METHOD Therefore, the man dea of the convergence analyss n the next secton s to show that f s a good approxmaton of the lattce Boltzmann soluton f. By summarzng the results n chapter 3, we frst estmate how well f satsfes the lattce Boltzmann algorthm on perodc domans or on bounded domans wth the bounce back rule. We conclude the result about the ntal condtons n Theorem 3, Theorem 7 Assume that u,p and u 3,ρ 4 have the followng regularty u C 5 ([,T], Ω), u 3 C 3 ([,T], Ω), p C 4 ([,T], Ω), ρ 4 C 2 ([,T], Ω). Then the predcton functon f defned by (4.19) satsfes and there s a constant κ α + 1 such that E (2) h ( f) = ˆr (α ), (4.23) ˆr (α ) () a,ωh = O(h κ ); ˆr (α ) (n) a,ωh =, n N h \ {}. (4.24) As a result we have κ = α + 1 generally (refer to the dscusson followng equaton (3.93)). In partcular cases, for example the case n Remark 7, κ > α +1 wth respect to E α (α = 1,2). When ψ(x) = and t G(,x) =, we have κ = 4. For the even more partcular case of Poseulle flow n a horzontal channel, we have ˆr (3) = and κ can be. In case of perodc domans, the evoluton scheme s dentcal at both ordnary flud nodes and boundary nodes, hence the results of Theorems 1 and 2 can be extended to the perodc case. Theorem 8 Assume that perodc functons u,p and u 3,ρ 4 have the followng regularty u C 5 ([,T], R d ), p C 4 ([,T], R d ), u 3 C 3 ([,T], R d ), ρ 4 C 2 ([,T], R d ), then the predcton functon f defned by (4.19) satsfes E (1) h ( f) = ˆr (4.25) and there exsts a h-ndependent constant κ 6 such that ˆr() a,ωh = ; ˆr(n) a,ωh = O(h κ ), n N h \ {}. (4.26)

81 4.2. THE TRUNCATED ASYMPTOTIC EXPANSION AND MOMENT RELATIONSHIP 67 Proof: Startng wth ˆr (n,j) = O(h κ ) from ether (3.67) or (3.75), there exsts a constant c such that ˆr(n,j) a c h κ. Summng over the doman, we obtan ˆr(n) a,ωh c h κ h d c h κ. j Ω h In the case of bounded domans wth Drchlet boundary condtons (2.3) for velocty, the dfference to the perodc case s that the evoluton does not only consst of the update rule (2.2) but also nvolves the bounce back algorthm (2.45) at nodes next to the boundary. The resdue has been computed and estmated separately n Theorems 1 and 2 for the update rule and Theorem 4 for the bounce back rule. In addton, we have to notce that the resdue ˆr (n,j) has been defned only for the nodes x j Ω\ Ω and ˆR (n,j) only for the nodes x j Ω. We complete them by ˆr (n,j) =, x j Ω, n N h \ {}, ˆR (n,j) =, x j Ω\ Ω, n N h \ {}, ˆr (,j) =, x j Ω, ˆR (,j) =, x j Ω, so that the norm a and a,ωh can be appled. Usng the full lattce Boltzmann scheme (4.9), we summarze these results Theorem 9 Assume that functons u,p and u 3,ρ 4 have the followng regularty u C 5 ([,T], Ω), u 3 C 3 ([,T], Ω), p C 4 ([,T], Ω), ρ 4 C 2 ([,T], Ω), and let the number of boundary nodes δω h fulfll condton (4.13). Then the predcton functon f defned by (4.19) satsfes E (1) h ( f) = ˆr, E (3) h ( f) = ˆR, (4.27) and there exst h-ndependent constants κ 6 and some κ b 2 such that ˆr(n) a,ωh = O(h κ ), ˆR(n) a,ωh = O(h κ b+ 1 2 ), n Nh \ {}, ˆr() a,ωh =, ˆR() a,ωh =, (4.28) κ b 3 for specal flows wth S[u] = or for geometres that q j = 1/2 for all boundary nodes x j Ω and = 1,...,N.

82 68 CHAPTER 4. CONVERGENCE OF THE LATTICE BOLTZMANN METHOD Proof: Because ˆR s non-zero only at boundary nodes j δω h, the summaton over the doman yelds E (3) h ( f) a,ωh = ˆR(k) a,ωh h d δω h Kh κ b C b h κ b Checkng formula (3.11) of ˆR carefully, except the specal flow wth S[u] = we fnd κ b = 2 when the boundary nodes are not located at the mddle of the boundary lnks, and κ b = 3 for the case q j = 1/2. Remark 9 Constants κ and K n the above theorems depends on T and (u,p), (u 3,ρ 4 ). In the next two sectons, we wll show that the error f(n) f(n) a,ωh drectly related to the ntal error f() f() a,ωh and the resdue ˆr, ˆR. s 4.3 Convergence of the lnear LBM We prove that the lnear lattce Boltzmann method admts a standard stablty estmate n A wth respect to the norm a,ωh f t satsfes the stablty estmate (4.15). The convergence and addtonal consstency results follow drectly from the estmated convergence order. The argument starts by ntroducng the dfference e between a truncated regular expanson s A and the soluton f of the lattce Boltzmann method In the perodc case we have e(n,j) = s(n,j) f(n,j). (4.29) Theorem 1 (Estmated convergence order σ for perodc cases) Let E 2 h be a lnear lattce Boltzmann algorthm whch satsfes the stablty estmate (4.15). Then E 2 h admts a standard convergence estmate n A wth respect to the norm Vh and for any s A f s Vh = O(h σ(κ) ), κ ord E 2(s), wth the estmated convergence order σ(κ) = mn(κ 1 2, κ 2 ). Proof: Frst, we notce that the functon σ s monotone, and there exst constants C and C ndependent of h such that E (1) h (s) V h Ch κ 1 and E (2) h (s) V h C h κ 2 for some κ ord E 2(s). Snce E (1) h (f) = and E(2) h (f) =, we have E (1) h (f) E(1) h (s) V h = E (1) h (s) V h Ch κ 1.

83 4.3. CONVERGENCE OF THE LINEAR LBM 69 Accordng to the defnton of E (1) h, we further obtan Wrttng E (1) h E (1) h (f)(n) E(1) h (s)(n) a,ω h Ch κ 1, n N h \ {}, E (1) h (f)() E(1) h (s)() =. n the explct form of (4.8), we thus get Se(n) (I + J L )e(n 1) a,ωh Ch κ 1. Due to lnearty of collson and shft operator, usng the norm estmates of Theorem 6, we mmedately obtan Se(n) a,ωh = e(n) a,ωh, (I + J L )e(n 1) a,ωh e(n 1) a,ωh. Further, applyng the trangle nequalty, we fnd and recursvely Snce nh 2 T, fnally e(n) a,ωh e(n 1) a,ωh + Ch κ 1 e(n) a,ωh e() a,ωh + nch κ 1. (4.3) e(n) a,ωh e() a,ωh + TCh κ 1 2. From the defnton (4.4) of the ntal condton E α, t follows that Hence e() a,ωh = E (2) h (s)() a,ω h = E (2) h (s) V h C h κ 2. e(n) a,ωh C h κ 2 + TCh κ 1 2 max(c,tc)h mn(κ 2,κ 1 2). Note that C h κ 2 max(c,tc)h mn(κ 2,κ 1 2), so that, n the end, The proof s complete. e Vh max(c,tc)h mn(κ 2,κ 1 2). Applyng ths theorem to the specal truncated expanson f, we thus have Theorem 11 (Convergence on perodc domans) Let E 2 h be a lnear lattce Boltzmann algorthm whch satsfes the stablty estmate (4.15). Assume further that the predcton functon f defned n (4.19) satsfes (4.25), (4.26). Then the estmated convergence order σ of f to the soluton f of E 2 h s σ = mn(κ 2,κ ), and for the moments (4.21) of the lattce Boltzmann soluton, there exsts some h-ndependent constant C > such that for all n N h p(t n ) P(n) Ωh Ch mn(2,κ 2), u(t n ) U(n) Ωh Ch mn(2,κ 1), where p(t n ),u(t n ) G h are the restrctons of the Stokes soluton p,u to the grd.

84 7 CHAPTER 4. CONVERGENCE OF THE LATTICE BOLTZMANN METHOD Proof: From Theorem 7 and 8, t follows that [κ,κ ] T ord E 2( f). Further, accordng to Theorem 1, E 2 h possesses a standard convergence estmate n A for the norm Vh. Snce f A, the estmated convergence order of f s thus σ = σ(κ,κ ) = mn(κ 2,κ ). Moreover, f(n) f(n) a,ωh f f Vh max(c,tc)h mn(4,κ ), n N h. The rest of the argument follows wth (4.3) and (4.22) and the constant C = max(c,tc). Altogether we see that the truncated expanson f converges to the soluton of the lnear scheme E 2 h at least wth order κ. More mportantly, the convergence of the moments to the Stokes solutons on a perodc doman s obtaned f the ntal predcton s suffcently accurate. In the case of bounded domans wth bounce back rule as boundary condton for the lattce Boltzmann equaton, the analyss can be carred out smlarly but the resultng convergence order depends addtonally on the resdue order of the boundary algorthm. Theorem 12 (Estmated convergence order σ for Drchlet condtons) Let f be the soluton of a lnear lattce Boltzmann algorthm E 3 h whch satsfes the stablty estmate (4.15). Then E 3 h admts a standard convergence estmate n A n the norm Vh for any s A, f s Vh = O(h σ(κ) ), κ ord E 3(s), and the estmated convergence order s σ(κ) = mn(κ 1 2, κ 2, κ 3 2). Proof: For any s A and any κ ord E 3(s), there s a constant C b ndependent of h such that E (3) h (s)(n) a,ω h E (3) h (s) V h C b h κ 3, n N h. (4.31) In vew of (4.9), the lnearty of the algorthm renders for n > E (1) h (e)(n) + E(3) h (e)(n) = Se(n) [ e(n 1) + J L e(n 1) ], and notng that E (1) h (f) = and E(3) h (f) =, E (1) h (e)(n)+e(3)(n) h (e) a,ωh = E (1) h (f)(n)+e(3) h (f)(n) E(1) h (s)(n) E(3) h (s)(n) a,ω h = E (1) h (s)(n) + E(3) h (s)(n) a,ω h Ch κ 1 + C b h κ 3.

85 4.3. CONVERGENCE OF THE LINEAR LBM 71 Wth the same recursve argument as n the perodc case and (4.15), we obtan e(n) a,ωh e() a,ωh + n(ch κ 1 + C b h κ 3 ). (4.32) Snce nh 2 T and e() a,ωh = E (2) h (s) V h C h κ 2, t leads to e(n) a,ωh C h κ 2 + T(Ch κ 2 + C b h κ 3 2 ) max(c,tc,tc b )h σ(κ), and e Vh max(c,tc,tc b )h σ(κ). From ths theorem, we see that n the case of the bounce back rule the provable convergence orders are not optmal. Theorem 13 (Moments convergence wth Drchlet boundary condtons) Let f be the soluton of a lnear lattce Boltzmann algorthm E 3 h whch admts the stablty estmate (4.15). Assume further that the predcton functon f defned n (4.19) satsfes (4.27), (4.28). Then the convergence order of f s σ = mn(κ,κ,κ b 3 2 ), and for the moments (4.21) of the lattce Boltzmann soluton, there exsts some h-ndependent constant C > such that for all n N h p(t n ) P(n) Ωh Ch mn(2,κ 2,κ b 7 2 ), u(t n ) U(n) Ωh Ch mn(2,κ 1,κ b 5 2 ), where p(t n ),u(t n ) G h are the restrctons of the soluton p,u of (3.69), (2.3) to the grd. Proof: From (4.28) we conclude as n the proof of the prevous theorem that for the truncated expanson f E (1) h ( f) Vh Ch κ, E (3) h ( f) Vh C b h κ b Besdes from Theorem 7, t follows that E (2) h ( f) Vh C h κ. Thus [κ,κ,κ b + 1/2] T ord E 3( f). Applyng Theorem 12, we obtan the estmated convergence order of f, and σ = σ(κ,κ,κ b ) = mn(κ 2,κ,κ b 3 2 ), f(n) f(n) a,ωh f f Vh max(c,tc,tc b )h mn(κ 2,κ,κ b 3 2 ),

86 72 CHAPTER 4. CONVERGENCE OF THE LATTICE BOLTZMANN METHOD The convergence results for the moments agan follow wth (4.22) and the constant C = max(c,tc,tc b ). Due to the low accuracy of the bounce back rule (κ b = 2 n general), the provable convergence rate s qute poor. Theorem 13 only shows that f(n) f(n) a,ωh = O( h) whch proves that the lnear lattce Boltzmann soluton converges to the constant equlbrum f but ths s not enough to show moment convergence to the Stokes soluton. In case that κ b = 3, the stuaton s better, The convergence of the velocty s obtaned, However, the order s 1.5 lower compared to the numercally observed order. 4.4 Convergence of the nonlnear LBM For the nonlnear lattce Boltzmann method, a standard convergence estmate n A s so far not acheved due to the dffculty to estmate the nonlnear mxed term J Q (s,e) for a general s A. However, under the condton on s that u(s),ωh = O(h), the term J Q (s,e) contans a factor h so that ts estmate can be controlled. In the followng, we denote the set of all s whch have ths property by A = { s A Then we show that a slght modfcaton } u(s),ωh = O(h). (4.33) < λ < 2, = d + 2,...,N (4.34) nstead of λ 2 n the stablty structure (4.14) of the lnear collson operator J L s suffcent to derve a convergence order for all s A. As a straghtforward applcaton to f, the convergence result for the nonlnear lattce Boltzmann method s obtaned, although J L s not the complete lnearzaton of the collson operator J. Let s A be gven and satsfy the lattce Boltzmann scheme n the perodc case up to a resdue η..e., E (1) h (s) = η. Snce the lattce Boltzmann solutons fulfll the equatons E (1) h (f) =, we fnd that the dfference e between the grd functon s and the soluton f n the perodc case follows the equaton for all n, n + 1 N h E (1) h (s)(n + 1) = E(1) h (s)(n + 1) E(1) h (f)(n + 1) = Se(n + 1) [ (I + J L )e(n) + J Q (s(n),s(n)) J Q (f(n),f(n)) ] = η(n), where we notce that Se(n + 1) = (I + J L )e(n) + J Q (s(n),s(n)) J Q (f(n),f(n)) + η(n), (4.35) J Q (s,s) J Q (f,f) = 2J Q (s,e) J Q (e,e). (4.36) To estmate the error, we take norm a,ωh on both sdes of (4.35). When J Q, whch s exactly the lnear case, the norm estmates (4.15) can be

87 4.4. CONVERGENCE OF THE NONLINEAR LBM 73 appled to (4.35) drectly and the convergence s easly acheved (see the prevous secton). In case of J Q, however, the rght hand sde of (4.35) appears more complcated than the lnear case because of the quadratc terms (4.36). The straghtforward estmate as n the lnear case wll be too rough. Therefore a more delcate estmate to the rght hand sde of (4.35) s necessary. We observe that the rght hand sde of (4.35) now contans a lnear part and a quadratc part of e. In partcular, the square norm 2 a of the rght hand sde wll lead to terms lke e 2 a and e 4 a. Our am s to obtan a relatvely strct estmate of e a, therefore the effect of e k a wth hgher powers k > 2 needs to be checked. The followng lemma wll eventually help us to estmate e a. It shows that the hgher powers of e a are not a problem provded that the ntal value of e a and the accumulaton of the source term wth respect to tme are small enough. Lemma 2 Gven that α, β, T, h are postve numbers, we set C = α + β/t. Assume further that z(n) and a(n) are nonnegatve for all n N, and z(n) satsfy the nequalty z(n + 1) a(n) + (1 + h 2 α)z(n) + βz 2 (n), n N. (4.37) Fnally assume that χ(n) = z() + n 1 k= a(k) s unformly bounded by then we obtan the estmate χ(n) h 2 e C T /T, nh 2 T. (4.38) z(n) χ(n)e nc h 2 χ(n)e C T, nh 2 T. (4.39) Proof: We prove (4.39) by nducton. To begn wth, we check the case n =. Apparently (4.39) holds snce z() = χ(). Secondly, assume (4.39) holds for all k n. Let us check the case n + 1 now. For convenence the notaton S n = 1 + αh 2 + βz(n) s ntroduced. Substtutng (4.39) and (4.38) nto S n produces S n 1 + αh 2 + βχ(n)e C T 1 + αh 2 + h 2 β/t = 1 + C h 2 e C h 2. Now let us turn to (4.37) whch can be shortened to z(n + 1) a(n) + z(n) S n. Substtutng (4.39) nto t, produces z(n + 1) a(n) + χ(n)e (n+1)c h 2. Snce e x > 1 for all x > and χ(n + 1) = χ(n) + a(n), we fnally obtan z(n + 1) (a(n) + χ(n))e (n+1)c h 2 = χ(n + 1)e (n+1)c h 2. Hence (4.39) s proved.

88 74 CHAPTER 4. CONVERGENCE OF THE LATTICE BOLTZMANN METHOD Recurson nequalty Now let us turn to calculate the norm of (4.35). When we apply 2 a,ω h on both sdes, there wll arse a relaton about z = e 2 a,ω h between two tme levels n and n + 1. Referrng to Lemma 2, we know that there s a bound for z f the factor χ, whch s determned by the ntal error z() and the error source a n each tme step, s sutably small. In order to show that the rght hand sde has the structure used n Lemma 2, we sutably splt the 2 a norm of the rght hand sde of (4.35), namely 2 = (I + J L )e + 2J Q (s,e) J Q (e,e) + η 2 a = e 2 a + η 2 a + Σ 1 + Σ 2 + Σ 3 + Σ 4, (4.4) where wth the help of the abbrevaton δ Q = 2J Q (s,e) J Q (e,e), Σ 1 = 2 η,(i + J L )e a, (4.41) Σ 2 = 2 η,δ Q a, (4.42) Σ 3 = δ Q 2 a, (4.43) Σ 4 = J L e 2 a + 2 e,j L e a + 2 (I + J L )e,δ Q a. (4.44) We remark that the expresson (4.4) of 2 already contans a term e 2 a. Hence whenever a term Σ k s estmated proportonal to e 2 a, we have to make sure that the proportonalty factor contans at least a factor h 2, otherwse Lemma 2 s not applcable. Ths s easy to acheve for Σ 1 and Σ 2 provded that η s controllable. The contrbuton Σ 3 s related to the nonlnear collson and produces terms proportonal to e 2 a and e 4 a because δq contans a lnear and quadratc term n e. Snce J Q only b-lnearly depends on the average veloctes of ts arguments and snce the average velocty of the grd functon s of order h, the term J Q (s,e) 2 a produces the requred factor h 2 n front of e 2 a. Most effort s therefore requred n the estmate of Σ 4 leadng to e 2 a contrbutons whch seem to have no explct factor h 2 n front. These estmates requre detals of the stablty structure whch are summarzed n Lemma 6 below. We begn wth the smpler estmates of Σ 1,Σ 2 and Σ 3. Lemma 3 For Σ 1 = 2 η,(i + J L )e a and Σ 2 = 2 η,δ Q a we have Σ 1 1 h 2 η 2 a + h2 e 2 a, Σ 2 η 2 a + Σ 3. Proof: For Σ 1 apply 2αβ 1 h 2 α 2 + h 2 β 2 and I + J L 1 from Theorem 6. For Σ 2 apply the nequalty 2αβ α 2 + β 2. Lemma 4 There exsts a constant c 1 > such that AF Q (u,v) a c 1 u v, u,v R d. (4.45)

89 4.4.1 Recurson nequalty 75 Proof: Snce AF Q : R d R d R N s a blnear mappng, we have AF Q (u,v) a u v max AF Q (u,v ) a, (4.46) u,v B1 d() where B d 1 () s the unt ball n Rd. Settng c 1 = max u,v B d 1 () AF Q (u,v ) a whch depends on the velocty set V and f and the matrx A, the proof s complete. Applyng ths lemma to Σ 3 yelds Lemma 5 Expresson Σ 3 = δ Q 2 a can be estmated as Σ 3 8 c 2 1 u(s) 2 e 2 a + 2 c 2 1 e 4 a. (4.47) Proof: Frst apply (α + β) 2 2α 2 + 2β 2 to Σ 3 = 2J Q (s,e) J Q (e,e) 2 a, we have Σ 3 8 J Q (s,e) 2 a + 2 J Q (e,e) 2 a (4.48) Usng the defnton of J Q and u(e) = c e, we can transform and J Q (s,e) a = AF Q (u(s),u(e)) a c 1 u(s) u(e), J Q (e,e) a = AF Q (u(e),u(e)) a c 1 u(e) 2. Substtutng u(e) e a from (4.17), we arrve at and J Q (s,e) a c 1 u(s) e a, (4.49) J Q (e,e) a c 1 e 2 a. (4.5) Insertng (4.49) and (4.5) nto (4.48), we reach the end of the proof. The expresson Σ 4 collects all the possble terms contanng e 2 a wthout explct small factor h 2. Its estmate reles on a detaled nvestgaton of the sgns of the partcpatng terms. Usng the algebrac propertes of J L collected n Lemma 6, we show n Lemma 7 that 2 e,j L e s negatve enough to compensate those contrbutons from the other two terms whch do not provde the requred factor h 2 n ther estmate. Lemma 6 (Defnton and propertes of P k ) For the component P n the stablty structure of the lnear collson operator J L, there exst N orthogonal projectors P k so that and P T P k = δ k P T P, P T P = N Pk T P k, Pk T P kj L = λ k Pk T P k, (4.51) =1 P k J L =, P k J Q =, k = 1,...,d + 1. (4.52)

90 76 CHAPTER 4. CONVERGENCE OF THE LATTICE BOLTZMANN METHOD Proof: Frst we construct P k. Denote r T k,k = 1,...,N to be the row vectors of P, then we know, from (4.14) and the subsequent dscusson, that r k are left egenvectors of J L whch are, orthonormal,.e., r T k JL = λ k r T k, r,r k = N l=1 where W = dag(1/a 1,...,1/a N ). We defne 1 a l r l r kl = r T Wr k = δ k, P k = W 2r 1 k r T k, k = 1,...,N. (4.53) Relaton (4.51) can be drectly derved from the propertes of r k. The frst equalty n (4.52) holds snce P k J L = λ k P k, k = 1,...,N (4.54) and the frst d + 1 egenvalues λ k of J L are zeros. As for the second equalty n (4.52), we know for any g holds P k J Q (g,g) = W 1 2r k r T k AF Q (g,g) = snce r k s n the kernel of A. To estmate Σ 4 we need the stablty structure wth a slghtly stronger assumpton on the egenvalues. Lemma 7 Assumng that J L admts the stablty structure (P,a,λ) wth λ k (,2) for k = d + 2,...,N, we fnd Σ 4 c λ Σ 3. Here c λ s a constant only dependng on the egenvalues λ d+2,...,λ N. Proof: We consder the frst two terms n Σ 4, J L e 2 a = P T PJ L e,j L e, 2 e,j L e a = 2 e,p T PJ L e. Successvely applyng (4.51), we obtan N J L e 2 a = λ 2 k P ke 2 2, k=1 N 2 e,j L e a = 2λ k P k e 2 2. k=1 From (4.51) t s also followed that P T P(I + J L ) = N k=1 (1 λ k)p T k P k. Proceedng to the last term n Σ 4, we obtan smlarly 2 (I + J L )e,δ Q a = 2 N (1 λ k ) Pk T P ke,δ Q. k=1 Then applyng (4.52), the frst d + 1 terms n the summaton dsappear due to the structure of δ Q = 2J Q (s,e) J Q (e,e), so that 2 (I + J L )e,δ Q a = 2 N (1 λ k ) Pk T P ke,δ Q. k=d+2

91 4.4.1 Recurson nequalty 77 Then mposng the nequalty 2αβ α 2 + β 2 on the rght sde renders 2 (I + J L )e,δ Q a N k=d+2 [ 1 λ k s k P k e ] P k (δ Q ) 2 2, s k where s k are some arbtrary postve constants. We choose s k as s k = λ k (2 λ k ) > (4.55) so that λ 2 k 2λ k + 1 λ k s k λ 2 k 2λ k + s k = after takng 1 λ k 1 nto account. Defnng c λ = 1 λ k 1 λ k max = max d+2 k N s k d+2 k N λ k (2 λ k ), the result follows summng up all the contrbutons Σ 4 c λ δ Q 2 a = c λ Σ 3. Summarzng the above results, we get a recurson nequalty for e 2 a,ω h, Theorem 14 (Error nequalty for perodc cases) Let the lnear part J L of the collson operator n the lattce Boltzmann method E 2 h admt the stablty condton (4.14), (4.34) and the stablty estmate (4.15) n Theorem 6. Then for any s A and n, n + 1 N h e(n + 1) 2 a,ω h e(n) 2 a,ω h + h 2 α e(n) 2 a,ω h + β e(n) 4 a,ω h + (2 + 1 h 2) η(n) 2 a,ω h, (4.56) for some postve constants α = (1 + 8(2 + c λ ) c 2 1 c2 ), β = 2(2 + c λ) c 2 1. Proof: Snce s A, there s a h ndependent contant c such that u(s(n)) Ωh u(s),ωh c h, n N h. Applyng Theorem 6 to the left sde of (4.35) yelds Se 2 a,ω h = e 2 a,ω h. For the rght hand sde of (4.35) whch we have wrtten n the form (4.4), we collect all the estmates of Σ k, and obtan 2 e 2 a + h 2 e 2 a + (2 + c λ )Σ 3 + (2 + 1 h 2) η 2 a whch hold at all nodes (n,j). Σ 3 8 c 2 1 u(s) 2 e 2 a + 2 c2 1 e 4 a

92 78 CHAPTER 4. CONVERGENCE OF THE LATTICE BOLTZMANN METHOD Apply nequalty a b ( a )( b ) for postve numbers a and b to the frst term n Σ 3, we obtan j Ω h u(s) 2 e 2 a u(s) 2 Ω h e 2 a,ω h h 2 c 2 e 2 a,ω h. (4.57) Then takng summaton of 2 on j Ω h and nsertng (4.57) produces (4.56). So far we have successfully formulated an nequalty n perodc cases for e 2 a,ω h smlar to that n Lemma 2. In the case of bounded domans wth bounce back rule as boundary condton for the lattce Boltzmann equaton, the analyss can be carred out smlarly but the resultng orders are less optmal. The truncated expanson s satsfes (s) = Υ wth an addtonal error source Υ due to the boundary algorthm. For the dfference e = s f, combnng E (1) h (s)+e(3) h (s) = η +Υ and (4.9) we obtan frst E (3) h Se(n+1) = (I +J L )e(n)+j Q (s(n),s(n)) J Q (f(n),f(n))+η(n)+υ(n), whch s smlar to the equaton (4.35) for e n the perodc case. Agan smlar estmate analyss leads to Theorem 15 (Error nequalty for bounded doman) Let the lnear part J L of the collson operator n the lattce Boltzmann algorthm E 3 h admt the stablty condton (4.14), (4.34) and the stablty estmate (4.15) n Theorem 6. Then for any s A e(n + 1) 2 a,ω h e(n) 2 a,ω h + h 2 α e(n) 2 a,ω h + β e(n) 4 a,ω h + (2 + 1 h 2) η(n) + Υ(n) 2 a,ω h (4.58) for some postve constants α = (1 + 8(2 + c λ ) c 2 1 c2 ), β = 2(2 + c λ) c Convergence results Wth the help of Lemma 2 and the recurson nequaltes, we arrve at the followng convergence theorems. Theorem 16 (σ for nonlnear LBM on perodc domans) Let f be the soluton of the lattce Boltzmann algorthm E 2 h whch fulflls the assumptons of Theorem 14. Then for any s A s f Vh = O(h σ(κ) ), κ ord E 2(s), and the correspondent estmated convergence order s { mn(κ 1 2, κ 2 ) κ 2 > 1 and κ 1 > 3 σ(κ) = κ 2 1 or κ 1 3.

93 4.4.2 Convergence results 79 Proof: We prove the results for the cases of κ 2 > 1 and κ 1 > 3. The other cases hold apparently. Let E (1) h (s) = η, then there exsts a constant c such that η() a,ωh =, η(k) a,ωh c h κ 1, k N h \ {}. Thus n 1 (2 + 1 h 2) η(k) 2 a,ω h nc h 2κ1 2 c Th 2κ1 4. k=1 From (4.4), we have e() = E (2) h (s)(), therefore there s another constant c such that e() a,ωh c h κ 2. Frst, we show that there exsts h > so that for all h < h and for all n wth nh 2 T n 1 e() 2 a,ω h + (2 + 1 h 2) η(k) 2 a,ω h h 2 1 T e C T. (4.59) wth C = α + β/t. k= We fnd that the left sde of (4.59) s of sze O(h 2 h mn(2(κ 2 1),2(κ 1 3)) ), where mn(2(κ 2 1),2(κ 1 3)) > for any κ 2 > 1 and κ 1 > 3. On the other hand, the rght sde of (4.59) s h 2 tmes a constant whch depends on lattce Boltzmann model and the bounds of the velocty felds u(s). Hence, nequalty (4.59) leads to a condton on the grd sze h,.e., we can fnd a postve h to ensure that (4.59) holds for all h < h. Now Lemma 2 can be appled to the nequalty (4.56) and we obtan [ n 1 e(n) 2 a,ω h e() 2 a,ω h + (2 + 1/h 2 ) η(k) 2 a,ω h ]e CT, (4.6) k= Usng the nequalty a 2 + b 2 (a + b) 2 for a > and b > on the rght sde and then takng square root on both sdes, we acheve e(n) a,ωh C [ e() a,ωh + h κ 1 2 ] Ch mn(κ 2,κ 1 2). (4.61) Snce e(n) = s(n) f(n), we fnally prove s f Vh max(c,c )h mn(κ 2,κ 1 2). (4.62) Next, we apply the above theorem to our specfc predcton functon f,

94 8 CHAPTER 4. CONVERGENCE OF THE LATTICE BOLTZMANN METHOD Theorem 17 (Convergence of f on perodc domans) Let f be the soluton of a gven lattce Boltzmann algorthm E 2 h whch fulflls the assumptons of Theorem 14. Assume the predcton functon f satsfes Theorem 8. Then f f Vh = O(h σ ), σ = mn(κ 2,κ ), and there exst h-ndependent constants C > and h > so that for all h < h, ( ) p(t n ) P(n) Ωh C h mn(2,κ 2), u(t n ) U(n) Ωh C (h ) mn(2,κ 1) hold for the moments (4.21) of the lattce Boltzmann soluton, where p(t n ),u(t n ) G h are the restrctons of the Naver-Stokes soluton p,u to the grd. Proof: Snce f satsfes Theorem 8, E (1) h ( f) Vh = ˆr Vh = O(h κ ). Further from Theorem 7 for the ntal condton E (2) h, t follows that E(2) h ( f) Vh = O(h κ ). Obvously f A, applyng Theorem 16, we get the estmated convergence order for f. As a straghtforward consequence of nequalty (4.22), the results for the moments are proved. Smlarly, we fnd n the case wth bounded domans Theorem 18 (σ(s) for nonlnear LBM on bounded domans) Let f be the soluton of a gven lattce Boltzmann algorthm E 3 h whch fulflls the assumptons of Theorem 15. Then for any s A θ, σ(κ) = s f Vh = O(h σ(κ) ), κ ord E 3(s), { mn(κ 1 2, κ 2, κ 3 2) when κ 3 > 3, κ 2 > 1, κ 1 > 3 otherwse. Proof: We proof the results for the cases of κ 2 > 1, κ 1 > 3, κ 3 > 3. Let E (1) h (s) = η, and E(3) h (s) = Υ, then there exsts constants c and c b such that η Vh c h κ 1, Υ Vh c b hκ 3. Ths gves rse to n 1 (2 + 1 h 2) η(k) + Υ(k) 2 a,ω h n(c h 2κ1 2 + C b h 2κ3 2 ) k= T(c h 2κ C b h 2κ 3 4 ) = O(h 2 (h 2κ h 2κ 3 6 )).

95 4.5. CONSISTENCY OF THE TRUNCATED EXPANSION 81 Smlarly to the proof of Theorem 16, we fnd that there exsts h > so that for all h < h and n N h n 1 e() 2 a,ω h + (2 + 1 h 2) η(k) + Υ(k) 2 V h h 2 1 T e C T. (4.63) k= whch allows us to apply Lemma 2 to the nequalty (4.58). Hence e(n) a,ωh C [ e() a,ωh + h κ h κ 1 2 ] C [ h κ 2 + h κ h κ 1 2 ] and the result follows. (4.64) Agan, we apply the general result of the theorem above to our specfc predcton functon f, Theorem 19 (Convergence of f on bounded domans) Let f be the soluton of a gven lattce Boltzmann algorthm E 3 h whch fulflls the assumptons of Theorem 15. Assume the predcton functon f satsfes Theorem 9 wth κ b > 5/2, Then f f Vh = O(h σ ), σ = mn(κ 2,κ,κ b 3 2 ) = κ b 3 2, and there exst h-ndependent constants C > and h > so that for all h < h and n N h, p(t n ) P(n) Ωh Ch mn(2,κ 2,κ b 7 2 ), u(t n ) U(n) Ωh Ch mn(2,κ 1,κ b 5 2 ) for the moments (4.21) of the lattce Boltzmann soluton, where p(t n ),u(t n ) G h are the restrctons of the soluton p,u of (2.1), (2.3) to the grd. Proof: Snce [κ,κ,κ b + 1/2] T ord E 3( f), applyng Theorem 18 to f fnally leads to ths theorem by takng κ 6, κ 2 and 1 < κ b 3 2 < 3 2 nto account Consstency of the truncated expanson To assess the consstency order of f to the lattce Boltzmann soluton, we have to fnd the undetermned orders of f n A, whch are related to the undetermned orders of f n A. Recall that the slght dfference of A and A les on the addtonal condton u(s),ωh = O(h). Therefore there s some relaton between M s and M s, whch s descrbed n the followng two Lemmas and holds for f.

96 82 CHAPTER 4. CONVERGENCE OF THE LATTICE BOLTZMANN METHOD Lemma 8 Let E m h be a lattce Boltzmann algorthm Em h and s A. If k M s (E m h ) and k 1, then also k M s(e m h ). Proof: Snce there s a γ A satsfyng c γ Vh c 1 such that s + h k γ A and E () h (s + hk γ) Vh = O( E () h (s) V h ), 1 m. ths lemma follows f s + h k γ A s proved. Snce s A, there s constant C such that u(s),ωh C h. Usng the trangle rule and u(γ) Ω h γ Vh, we fnd that u(s+h k γ),ωh u(s),ωh +h k u(γ),ωh C h+h k c 1 max(c,c 1 )h. In some partcular cases, both sets are even equvalent. Proposton 2 For a gven lattce Boltzmann algorthm and some s A, we have M s = M s f nf M s 1. Proof: Ths can be easly proved wth the help of Lemma 8. Snce a drect checkng of the element of M f s very dffcult, we adopt a dfferent strategy. To obtan an upper bound of the consstency order, we explctly verfy that a certan order s undetermned gvng rse to a specfc element of M f. To acheve a lower bound, we use the estmated convergence order n connecton wth Theorem 5. We ntroduce three numbers for the regular expanson f. κ = In case that m = 2, κ b sup κ 1, κ = sup κ 2, κ b = sup κ 3. κ ord E m( f) κ ord E m( f) κ ord E m( f) s dropped out Perodc cases Proposton 3 (M f for nonlnear perodc cases) Provded that the condtons for the lattce Boltzmann scheme E 2 h and the predcton functon f n Theorem 17 are satsfed, we fnd M f = M f. Proof: Frst, we prove nf M f 1 by contradcton. Assume there s γ satsfyng γ Vh c > such that for any k < 1(k R) hold E (1) h ( f + h k γ) Vh = O(h κ 1 ), κ 1 κ 6, E (2) h ( f + h k γ) Vh = O(h κ 2 ), κ 2 κ.

97 4.5.1 Perodc cases 83 We denote E (1) h ( f + h k γ) = η and E (2) h ( f + h k γ) = ζ, wrte E (1) h form and obtan for n, n + 1 N h, n the explct γ(,j) = h k (ζ ˆr (α ) ), (4.65) Sγ(n + 1) = (I + J L )γ(n) + J Q (γ(n),γ(n)) + J Q (γ(n),2 f(n)) + h k (η(n) ˆr(n)), (4.66) n whch E (1) h ( f) = ˆr and E (2) h ( f) = ˆr (α) have been nserted. Carryng out a smlar estmate as n secton (by replacng e wth γ), we get a recurson nequalty for γ (n the same form of (4.56)), γ(n + 1) 2 a,ω h γ(n) 2 a,ω h + h 2 α γ(n) 2 a,ω h β γ(n) 4 a,ω h + h 2k (2 + 1 h 2) η(n) ˆr(n) 2 a,ω h. (4.67) Further, we prove that there exsts h > so that for all h < h and for all n N h n 1 γ() 2 a,ω h + h 2k (2 + 1 h 2) η(k) ˆr(k) 2 a,ω h h 2 1 T e C T. (4.68) Note that k= η ˆr Vh = O(h 6 + h κ 1 ), γ() Vh = ζ ˆr (α ) a,ωh = h k O(h κ + h κ 2 ), the rght hand sde of (4.68) turns out to be n 1 γ() 2 a,ω h + h 2k (2 + 1 h 2) η(k) ˆr(k) 2 a,ω h k= = h 2 O(h 2(κ 1 k) + h 2(κ 2 1 k) ) + h 2 O(h 6 2k + h 2κ 1 6 2k ) and s thus of sze h 2 O(h mn(2(κ 1 k),6 2k) ). Snce κ α and k < 1, we can fnd some h > to guarantee the nequalty (4.68). Now Lemma 2 can be appled to recurson nequalty (4.67) of γ, whch yelds γ Vh C h κ k, nh 2 T Takng the lmt h, we have lm h γ Vh =. Hence we acheve (, 1) M f =. Accordng to Proposton 2, M f = M f. We check the possble elements of M ˆf and fnd, Lemma 9 If κ, then κ M f(e 2 h ).

98 84 CHAPTER 4. CONVERGENCE OF THE LATTICE BOLTZMANN METHOD Proof: For the sake of convenence, let k = κ. Further choose γ = f to be a constant functon. Consequently γ Vh s constant too. We nsert ˆf + h k γ nto the update rule (2.2) and calculate the resdue. Snce the equlbrum functon has a lnear and a quadratc part, t turns out that E (1) h ( ˆf + h k γ) = E (1) h ( ˆf) ( + h k A f L (γ) + f Q (γ,2 ˆf ) + h k γ) γ. But the last term on the rght hand sde s zero, because f L (γ) = γ and 1,Vγ = whch cancels the term connected to f Q. Hence E (1) h ( ˆf + h k γ) = E (1) h ( ˆf). Now, t follows that E (2) h ( f + h k γ) = E (2) h ( f) + h k γ. Further snce E (2) h ( f + h k γ) Vh E (2) h ( f) Vh + h k γ Vh, (4.69) thus ord E 2( f) ord E 2( f + h k γ). In summary, we have proved the theorem. At ths pont, t s not clear whether (k < κ ) M ˆf(E 2 h ). However, from ths theorem, we can conclude that ˆf may be consstent up to order κ to the lattce Boltzmann scheme E 2 h. For the lnear lattce Boltzmann scheme E 2 h, we fnd that a standard convergence estmate s admtted n A, and combned wth ths theorem, we get the estmated convergence order of f, whch s a lower bound of the consstency order nf M f. For the nonlnear lattce Boltzmann scheme E 2 h, we obtan a standard convergence estmate only n the subspace A. The estmated convergence order of f s supposed to be the lower bound of nf M f. However, from Proposton 3 t follows that nf M f = nf M f. Summarzng ths dscusson, we acheve a concluson about consstency order n perodc cases. Theorem 2 (Consstency order of f n perodc domans) Let the lnear part J L of the collson operator n the lattce Boltzmann scheme E 2 h admt the stablty condton (4.14) and the stablty estmate (4.15). Further let the addtonal condton (4.34) holds for the nonlnear E 2 h. Assume f satsfes the mplcatons of Theorems 7, 8. Then where κ K, (κ,κ ) T ord E 2( f) (4.7) K = (,α + 1] {α + 2,α + 3,...,5} { }, (4.71) and the consstency order s n the nterval, σ = mn(κ 2,κ ) ÕE 2( f) κ. (4.72)

99 4.5.1 Perodc cases 85 In the partcular case that κ 2 κ, the convergence order s equal to the consstency order, σ = ÕE 2( f) = κ. (4.73) Proof: We recall that ˆr (α ) (n) = for all n N h \ {} and ˆr (1) () = h 2 [c 2 s p(,x j)f A (V )f (1) ] + ˆr (2) () = h 3 f (3) (,x j ) + 5 h k f (k) (,x j ), k=4 ˆr (3) () = h 4 f (4) (,x j ) + h 5 f (5) (,x j ). 5 h k f (k) (,x j ), k=3 (4.74) and accordng to Theorem 7, ˆr (α ) Vh = O(h κ ) and κ can be equal or larger than α +1. On the other hand ˆr (α ) s polynomals of h. Due to the regularty of the moments, the coeffcents of ˆr (α ) are unformly bounded. Ths mples that the possble values of κ can only be n K. Hence κ K and (κ,κ )T ord E 2( f) too. Therefore we have a value of κ = κ. Snce the estmated convergence order of f, whch s mn(κ 2,κ ) = mn(κ 2,κ ) (Theorem 11), cannot be larger than the consstency order accordng to Theorem 5. On the other hand, Theorem 9 tells us ÕE 2( f) = nf M f s less or equal to κ. Thus (4.72) holds. In addton, when κ 2 κ, we have mn(κ 2,κ ) = κ. Hence (4.73) holds too. Let us dscuss several cases of (4.73). In general, κ = α + 1 (refer to the dscusson followng equaton (3.93)). Combned wth κ 2 4 α + 1, the convergence order and the consstency order are dentcal, α + 1. Ths mples that, n general, f precsely represents the lattce Boltzmann solutons up to order α + 1. In partcular cases, for example the case n Remark 7, both κ and κ are bgger than α + 1 when E α (α = 1,2) s employed. Thus nf M f > α + 1. When ψ(x) = and t G(,x) =, ˆr (1) () = ˆr (2) () = ˆr (3) (). Hence κ = 4 for all α, and maybe κ = 4 snce f (4) (,x j ) generally, whch mples agan the convergence order and the consstency order are dentcal, and the regular expanson f correctly stands for the lattce Boltzmann soluton up to order 4. In even more partcular cases lke Poseulle flow n a horzontal channel, ˆr (3) = and κ s f E α (α = 3) s appled. Therefore κ =. Actually, n ths case, the resdue from the update rule ˆr(n,j) vanshes, snce Poseulle flow s a statonary, second order polynomal flow. Therefore κ =, too. Ths mples that f s the exact soluton of the lattce Boltzmann method E 2 h.

100 86 CHAPTER 4. CONVERGENCE OF THE LATTICE BOLTZMANN METHOD Cases wth the bounce back rule Lemma 1 If κ, then κ M f(e 3 h ). Proof: Agan we choose γ = f. Moreover, E (3) h ( f + h k γ) = E (3) h ( f) h k A(f L (γ) γ + f Q (γ,h k γ + 2 f)). (4.75) The remander of the proof s smlar to the case n Theorem 9. Recallng the relatonshp between the estmated convergence order and the consstency order, we can dscuss the consstency of f more precsely for the lnear schemes on bounded domans. Proposton 4 (Consstency order of f on bounded domans) Let the collson operator J L n the lnear lattce Boltzmann scheme E 3 h admt the stablty condton (4.14) and the stablty estmate (4.15). Assume f satsfes the mplcatons of Theorem 7, 9. Then ÕE 3( f) [κ b 3/2, κ ]. Proof: From Theorem 1, we know κ M f, thus nf M f κ. In addton σ = κ b 3/2 nf M f for the lnear problem accordng to Theorem 5. For the nonlnear problem E 3 h, we stll can get a upper bound from Theorem 1, nf M f κ. However, M f = M f s not obtaned. Thus we only acheve κ b 3/2 nf M f by applyng Theorem 5. A lower bound for nf M f s not clear. 4.6 Comments on convergence and consstency Perodc case From Proposton 2, we can conclude that the consstency order of our regular expanson f s generally α + 1, whch mples that f represents the lattce Boltzman soluton up to order α + 1 supporng the results n secton 3.3. From Theorem 11 for the lnear lattce Boltzmann method and Theorem 17 for the nonlnear one, f converges to the lattce Boltzmann soluton f at order α + 1 n general, whch s the order of the resdue concerned wth the ntal condtons, snce α + 1 s less or equal to 4. Ths result s also verfed numercally by several flow smulatons n secton 3.4. More mportantly the convergence of the moments to the Stokes or Naver- Stokes solutons on a perodc doman s obtaned f the ntal predcton s suffcently accurate (see Theorem 11 and Theorem 17).

101 4.6.2 Drchlet case 87 Recallng the results n Theorem 7, f satsfes the classcal lattce Boltzmann ntalzaton plus a resdue of order h 2 (α = 1), therefore, the accuracy of the velocty feld s only frst order and our estmate does not mply convergence of the pressure. In the case of the mproved ntalzaton routne (2.42) proposed n [71] (α = 2), we have f f a,ωh = O(h 3 ) so that velocty converges wth second order and pressure wth frst order n h. The optmal convergence order of the lattce Boltzmann method s h 2 for both pressure and velocty. Ths s acheved through the ntalzaton (3.1) (α = 3) whch s ncludng also the thrd order coeffcent f (3). It may be nterestng to note that the smple ntalzaton (3.89) s equvalent to the better choce (2.42) f ψ =. It s even equvalent to the optmal choce (3.1) f addtonally f (3) (,x j ) vanshes dentcally whch follows from G(,x) = and t p(,x) =. The latter requrement can be satsfed by assumng t G(,x) =. In other words, f the ntal state s a flud at rest and f the strength of the spatally smooth force s ntally proportonal to t 2 (smooth start), then the smple ntalzaton (3.89) gves rse to second order accurate pressure and velocty. The predcton functon f precsely represents the lattce Boltzmann soluton n order,1,2 and 3. We conclude that, n the case of perodc boundary condtons, the convergence order s determned only by the ntal error, provded the Stokes or Naver- Stokes soluton s suffcently regular so that Theorem 8 apples. In any case, the convergence order s restrcted to two for both pressure and velocty Drchlet case We stress that the estmate (4.32) and (4.58) are not optmal because they treat the boundary error exactly as the nteror error. However, the reacton of the scheme on the dfferent types of error may not be unform. Smple examples show that nteror error ˆr can really accumulate n tme so that a reducton of the order of ˆr by t 1 = h 2 s realstc. In contrast to ths, the boundary error generally does not accumulate but s rather transported away from the boundary. Therefore the reducton by h 2 of the order of ˆR may be a very coarse estmate. In the lnear case, we only get an upper bound and lower bound for the consstency order of f. The lower bound mples that f correctly determnes the lattce Boltzmann soluton n orders less than κ b 3/2. The convergence order of the truncated expanson f to the lattce Boltzmann soluton f s equal to the resdue order from the bounce back rule reduced by 3/2 (see Theorem 13). Combned wth the low accuracy of the bounce back rule (κ b = 2 n general), the provable convergence rate s qute poor. In fact, for κ b = 2, Theorem 13 only shows that e Vh = O( h) whch proves that the lnear lattce Boltzmann soluton converges to the equlbrum f = F eq (1,) but ths s not enough to show that the average velocty converges to the Stokes soluton. In the nonlnear case, the stuaton s worse because Theorem 19 requres κ b > 5/2 so that the case of general boundares (where κ b = 2) s not even covered.

102 88 CHAPTER 4. CONVERGENCE OF THE LATTICE BOLTZMANN METHOD However, when all boundary lnks are exactly cut n the mddle (q j = 1/2 at all boundary nodes), the order of the bounce back rule ncreases to κ b = 3 so that u(t n ) U(n) Ωh = O( h) provded the ntalzaton s suffcently accurate, for example of the form (2.42). We remark that numercal experments and the formal asymptotc expanson suggest second order accuracy for velocty and frst order pressure n ths case. The loss of 1.5 orders seems to be due only to our coarse estmate. There s also no clear nformaton about consstency, snce no lower bound for the consstency order s found. A fnal word of cauton s for the convergence results n the non-lnear Naver- Stokes case. We can see from the proofs that the error can only be bounded f the grd sze h s sutably small, h < h. It s not straghtforward to gve an explct constant h because t requres the soluton of nonlnear problems. However, the tendency s clear: f the exact soluton s rough (strong gradents), h s small, and f one egenvalue s close to 2 (hgh Reynolds number), h becomes small as well. Hence, the lattce Boltzmann method converges, n prncple, for every Reynolds number, though the grd requrement may be too demandng for practcal computatons.

103 Chapter 5 Drchlet boundary condtons and consstency analyss In ths chapter the lattce Boltzmann boundary schemes dealng wth the Drchlet boundary condton are carefully studed. The consstency analyss of exstng boundary schemes s carred out only for the more accurate methods ncludng FH [21, 2], BFL [7] and MR [25]. We dsplay ther advantages, expose ther drawbacks and verfy the correspondng results numercally. After that, a new boundary recpe s proposed n order to keep the requred vrtues and get rd of the unwanted shortcomngs. Fnally a comparson among these methods s also gven whch may provde a use gude n practce. 5.1 Improvements of bounce back rule In the prevous chapter, the bounce back rule has been carefully nvestgated. The results show that bounce back rule generally leads to a frst order accurate velocty and nconsstent pressure. The reason s that no smooth functon u 2 can remove the second order resdue 2c 2 s f [c u 2 (t n,x j ) (2q j 1)(c )c u 1 (t n,x j )]. (5.1) Therefore, f we modfy the bounce back rule such that the assumpton u 2 = removes one more order of the resdue, the accuracy wll be rased. A frst dea s to modfy the bounce back rule n the form f (n + 1,j) = f b (n,j) h2 c 2 s f (2q j 1)(c )(c u 1 (t n,x j )) (5.2) wth a correcton term that removes the unwanted terms n (5.1). Note, however, that the rule cannot be mplemented n that form because (c )c u 1 nvolves drectonal dervatves of the Naver-Stokes soluton u 1 and these dervatves cannot explctly be wrtten n terms of the boundary data φ. We frst have to replace the space dervatves by a sutable approxmaton, whch s by far not unque so that addtonal constrants are mportant n the desgn lke stablty and smplcty. Below we present several algorthms whch realze approxmatons of the requred dervatve terms, though some of them were not 89

104 9 CHAPTER 5. DIRICHLET BOUNDARY CONDITIONS AND CONSISTENCY ANALYSIS orgnally constructed from ths dea. If we wrte the condtons as addtve correctons to the bounce back scheme, we can drectly use the results n secton 3.5 and just analyze the addtve correcton FD Fnte dfferences are a natural tool to construct an applcable scheme before a more elegant and sophstcated one s nvented. One choce s [û(n,j + c ) hφ(t n,x j ) ] c (1 + q j )h 2 q j 1 2 (c )c u 1 (t n,x j ) ˆϕ (n,j) = [û(n,j) hφ(tn,x j ) ] c q j h 2 q j > 1 2 where the reason to use two dfferent expressons dependng on q j 1/2 or q j > 1/2 s for the sake of stablty. The correspondng boundary algorthm s f (n + 1,j) = f b (n,j) h2 c 2 s f (2q j 1)ˆϕ (n,j) Note that, n general geometres, extra consderatons are requred f for some boundary node a neghbor s mssng n the ncomng drecton, or n other words, f there exst two opposng ncomng drectons at the same node. Ths happens, for example, at corners n 2D and 3D, at edges n 3D, but also along smooth boundares at nodes where ncomng lnks are almost tangental to the boundary (see the followng fgure). Fgure 5.1: Two opposng ncomng drectons at one boundary node. In such stuatons the algorthm above s not applcable f q j 1/2 so that the correspondng drectonal dervatves of u 1 are mssng. However, the dervatves can be computed from the boundary values n these cases by usng approprate fnte dfferences along the boundary. For example, n the case of the square geometry, we can get the frst dervatves of u 1 at the corners by takng one-sded dervatves of the boundary values along the edges of the square.

105 5.1.2 The boundary-fttng method (FH) and ts mprovement (MLS) 91 To analyze the proposed FD scheme, we nsert the expanson (3.6) of û n the formula of ˆϕ, do Taylor expanson at (t n,x j ) and obtan ˆϕ = κ/h(u 1 φ) c + (c )(c u 1 ) + O(h) wth κ = { 1/(1 + q j ), q j < 1 2 1/q j, q j 1 2 Combnng wth (3.17) whch takes care of the analyss of the bounce back part f b, we fnd the resdue hc 2 s f [2 (2q j 1)/ κ](u 1 φ) c h 2 2c 2 s f u 2 c + O(h 3 ). Obvously u 1 = φ and u 2 = at boundary remove the frst and second order resdue terms. Snce u 2 = s also compatble wth the corner treatment, we expect that FD leads to frst order accurate pressure and second order accurate velocty felds The boundary-fttng method (FH) and ts mprovement (MLS) The so-called boundary-fttng method (FH) proposed n [21, 2], s based on a lnear combnaton of f c and f eq at the boundary node wthout reference to neghborng nodes. To motvate ths boundary condton, a Chapman- Enskog analyss s used. Apart from the structural assumptons related to the Chapman-Enskog approach, t requres the addtonal assumpton that the ntrnsc tme scale of the unsteady flow must be large compared wth the advecton tme on the lattce scale. In our analyss, ths assumpton s bult n from the begnnng and we can show the accuracy order wthout any extra assumpton. Snce t was orgnally presented for the BGK collson model wth one tme relaxaton parameter τ, we also only analyze ths class of lattce Boltzmann methods here (note that, formally A = 1 τ I n ths case). As outlned above, we wrte t as a correcton of the bounce back rule whch gves rse to f (n + 1,j) = f b (n,j) + ˆθ (n,j) wth and ˆθ (n,j) = χ j [ f c (n,j) f eq (n,j) + c 2 s f v (n,j) c ], v (n,j) = { (hφ(t n,x j ) û(n,j))/q j, q j 1/2, q j < 1/2. Ths method has an ntrnsc dsadvantage because the parameter χ j { (2q j 1)/τ, q j 1 χ j = 2 (2q j 1)/(τ 1), q j < 1 2

106 92 CHAPTER 5. DIRICHLET BOUNDARY CONDITIONS AND CONSISTENCY ANALYSIS depends on 1/(τ 1) whch eventually leads to nstablty when τ 1. Obvously ths method degenerates to bounce back rule when q j = 1 2. To analyze the algorthm, we nsert the regular expanson (3.1) wth the coeffcents (3.64) nto the expresson for ˆθ (n,j) and perform a Taylor expanson around the node (t n,x j ). Ths gves rse to ˆθ (n,j) = χ j [(1 1 ] τ )(f feq )(n,j) + c 2 s f v (n,j) c = χ j (1 1 τ ) h m+p D p (,q j c )(f (m) feq,(m) )(t n,x j ) p= m= + κc 2 s f χ j h m+p D p (,q j c )u m (t n,x j ) c, q j p= m= hκc 2 s f χ j φ(t n,x j ) c q j n whch an ndcator κ s ntroduced n such a way that { 1, q j 1 κ = 2,, q j < 1 2. Up to thrd order, we have explctly ˆθ = hκc 2 s χ jf /q j(φ u 1 ) c + h 2 (1 2q j )(c )f (1) + O(h 3 ). Combned wth the expanson (3.17) of the bounce back part n the FH condton we get the resdue ( h 2 κ χ ) j c 2 s f (φ u 1 ) c h 2 2c 2 s f u 2 c + O(h 3 ) q j whch s of order h 3 f u 1 = φ and u 2 = on Ω. However, n the case q j = the method FH ntroduces addtonal condtons at nodes wth two opposng ncomng drectons whch generally cannot be satsfed wth a smooth coeffcent f (2). Thus our regular expanson breaks down at second order whch ndcates that pressure wll be nconsstent and velocty only frst order accurate. To derve the addtonal condton, we consder opposng ncomng drectons c and c = c at node x j n the case q j = (for example, the case n fgure 5.1). The algorthm s thus reduced to f (n+1,j) = f (n,j)+2c 2 s f hφ c, f (n+1,j) = f (n,j)+2c 2 s f hφ c, Addng the equatons, we obtan f (n + 1,j) + f (n + 1,j) = f (n,j) + f (n,j)

107 5.1.3 Bouzd rule (BFL) 93 so that the quantty f (n,j) + f (n,j) = C s constant n tme. Insertng the expanson nto ths relaton and expandng around (t n,x j ), we fnd the second order resdue c 2 s f t [ρ (3(u 1 c ) 2 u 1 2 ) τ(c )(u 1 c )], (5.3) n general, can not be removed by the Naver-Stokes soluton c 2 s ρ 2,u 1. As example, we consder a tme dependent lnear flow n secton 3.2. At the upper rght corner (1,1) there are two ncomng drectons c 6 and c 8 f the D2Q9 velocty set s used. The relaton (5.3) at ths corner yelds t [ρ (3(u 1 c ) 2 u 1 2 ) τ(c )(u 1 c )] = 372sn(t)(1 cos(t))+8τ sn(t) whch s certanly not zero. In [58, 59] an mprovement (MLS) of FH s acheved by usng the next neghbor along the lnk to calculate v (n,j) = û(n,j + c ) û(n,j) for q j < 1/2. Then χ j becomes a functon of 1/(τ 2) whch enlarges the regon of stablty but does not overcome the nherent drawback of FH. Moreover, ths modfcaton s no longer defned for opposng ncomng drectons ether Bouzd rule (BFL) By means of nterpolaton, ths boundary algorthm s constructed on the bass of the bounce back rule and the restrcton that the partcles travel a dstance of sze c h n one tme step along the drecton c. It has been numercally demonstrated to gve second order accurate velocty n [7]. Therefore t s not surprsng that BFL can be vewed as a corrected bounce back rule wth a partcular dscretzaton of the requred dervatve (c )c u 1. To see ths, we wrte BFL n the form wth f (n + 1,j) = f b (n,j) + ± (n,j) (n,j) = (1 2q j)[f c (n,j + c ) f c (n,j)], q j 1 2, + (n,j) = (1 2q j)[f (n + 1,j) f c (n,j)], q j > 1 2. Insertng a general expanson (3.1) nto, we fnd (n,j) = (1 2q j) m h m [ [f (m) + A (f eq,(m) f (m) ) ] (t n,x j + hc ) [ ] ] f (m) + A (f eq,(m) f (m) ) (t n,x j ),

108 94 CHAPTER 5. DIRICHLET BOUNDARY CONDITIONS AND CONSISTENCY ANALYSIS where A s the th row of matrx A. Expandng at the boundary pont x j = x j hq j c, and nsertng the coeffcents (3.64), we get (n,j) = h2 c 2 s f (2q j 1)(c )(c u 1 (t n,x j )) + (1 2q j ) m h m [(1 + q j ) k qj k ](c ) k k! m 3 k= ( f (m k) + A (f eq,(m k) f (m k) ) + g δ m3 )(t n,x j ). The frst term n order h 2 s precsely the requred one. The same s true for the leadng order of + (n,j), + (n,j) = h2 c 2 s f (2q j 1)(c )(c u 1 (t n,x j )) + (1 2q j ) m h m[ D k ( t,q j c )f (m k) m 3 k= ( ) D k (,q j c ) f (m k) ](tn + A (f eq,(m k) f (m k) ) + g δ m3,x j ). We can thus conclude that, n general, the BFL rule gves rse to a second order accurate velocty and a frst order accurate pressure. However note that one requres a neghbor flud node x j + hc n the ncomng drecton c when mplementng (n,j) at a boundary node x j. If ths neghbor node s out of the flud doman Ω, one can not realze ths BFL scheme at x j any more. As a remedy t s often suggested to use bounce back rule at such exceptonal pont. However, ths approach generally lowers the accuracy of the whole scheme. In order to llustrate ths phenomenon, a 2D lnear flow (3.82) s numercally smulated by the lattce Boltzmann method wth a D2Q9 velocty set and the smple BGK collson operator wth one tme relaxaton parameter ν + 1/2. Here ν s fxed to be.1. The computatonal doman s chosen to be a square [,1] 2 or a unt dsc x 2 + y 2 1. Apparently at each corner of the square and at each touchng pont between crcle x 2 +y 2 = 1 and coordnate axes, there s a par of opposng ncomng drectons. τ = c 2 s The numercal calculatons show an nconsstent pressure and a frst order accurate velocty f the bounce back rule s adopted to supplement the BFL only at the opposng ncomng drectons (see fgure 5.2). Ths demonstrates that the boundary condton wth lower accuracy at very few nodes may reduce the accuracy of the whole scheme. In contrast to ths, the expected second order accurate velocty and frst order accurate pressure s acheved when a more accurate scheme POP 1 (developed n the later secton 5.3) replaces the bounce back rule.

109 5.1.3 Bouzd rule (BFL) 95 error of velocty error of pressure Fgure 5.2: Logarthmc velocty (left) and pressure (rght) error versus log 1 h n the case of a 2D statonary lnear flow wth dfferent boundary treatments at the ponts where neghbors n ncomng drectons are mssng. The results for crcular geometry, are denoted by (bounce back rule) and (POP 1 ). For the flow n a square, the bounce back result s and POP 1 yelds. We also want to stress that the lower order error s not stuated locally at the node where the bounce back rule s realzed. Instead, the error transports nwards and eventually appears everywhere n the doman. To see ths behavor, the numercal error s checked more carefully nsde the doman. For example we separate the unt dsc nto several rngs R 5 R 3 R = {(x,y) : r 2 1 < x 2 +y 2 < r 2 } r =,r 1 =.5,r 2 =.7, r 3 =.8,r 4 =.9,r 5 = 1., R 1 and the unt square nto several rbbons C 1 = {(x,y) :.3 < x, y <.7}, C 2 = {(x,y) :.2 < x, y <.8}, S 1 = C 1, S 2 = C 2 \ C 1, S 3 = Ω \ C 2. S 3 S 2 S 1 The error s measured n all the subregons for several grd szes. In each subregon the results based on bounce back rule for the exceptonal ponts shows a lower accuracy than those obtaned wth POP 1. For both cases t s

110 96 CHAPTER 5. DIRICHLET BOUNDARY CONDITIONS AND CONSISTENCY ANALYSIS not surprsng that the further the subregon s away from the boundary curve, the smaller the error. Moreover, the use of bounce back rule at the opposng ncomng drectons produces not only a bgger error (see fgure 5.3, 5.4) but also a lower accuracy than POP 1 (see table 5.1, 5.2) n each subregon. It means that the boundary error does not only affect the neghborhood of the boundary curve, but have nfluences n the complete nner part. Table 5.1: Numercal convergence orders of velocty/pressure for crcular flow. rng bounce back POP 1 R / / R / / R / /1.246 Table 5.2: Numercal convergence orders of velocty/pressure for a statonary lnear flow. rbbon bounce back POP 1 S / / S / / S / / error of velocty error of pressure Fgure 5.3: Logarthmc velocty(left) and pressure(rght) error versus log 1 h n the case of a 2D statonary lnear flow n a dsk wth dfferent boundary treatments at the ponts where neghbors n ncomng drectons are mssng. The dashed lnes denote the results wth bounce back rule, sold lnes stand for the results wth POP 1. The symbols,+, refer to errors n the concentrc subregons R 1,R 3,R 5.

111 5.2. MULTI-REFLECTION METHOD (MR) Fgure 5.4: Logarthmc velocty(left) and pressure(rght) error versus log 1 h n the case of a 2D statonary lnear flow n a square wth dfferent boundary treatments at the ponts where neghbors n ncomng drectons are mssng. The dashed lnes denote the results wth bounce back rule, sold lnes stand for the results wth POP 1. The symbols,,+ refer to errors n the subregons S 1,S 2,S Mult-reflecton method (MR) In contrast to the algorthms above, the mult-reflecton method MR presented n [25] s not a modfcaton of the bounce back rule,.e. t does not reduce to bounce back f q j = 1/2 for all nodes. In general, t uses three nodes (.e. two addtonal neghbors) along the ncomng drectons f (n + 1,j) = κ 1 f c (n,j) + κ f c (n,j + c ) + κ 1 f c (n,j + 2c ) + κ 1 f c (n,j) + κ 2 f c (n,j + c ) + 3w f hφ(t n,x j ) c + 3f F p.c. (n,j). In [25], wth the help of three numbers κ t = 3 + 2q j, κ s = 1 + 6q j + 4q 2 j, κ f = 2(1 + q j ) 2 and an arbtrary chosen parameter κ {max(1,κ s /κ t ), 2κ s /κ f }, parameters κ 1, κ, κ 1, κ 1, κ 2, w and F p.c. are defned by κ 1 = κ κ s κ f 1, κ = 2 κ, κ 1 = 2 κ κ s κ t, κ 1 = κ 1 κ 2 1, F p.c. (n,j) = κ Λ 2 (2) f κ s f ν κ 2 = κ κ s (κ t 2) 1, w = 4 κ κ s,

112 98 CHAPTER 5. DIRICHLET BOUNDARY CONDITIONS AND CONSISTENCY ANALYSIS The functon f (2) p.c. n the defnton of F (n,j) s gven by [ f (2) = A (V )f (2) + 1 ] 2 (V )2 f (1) n whch the requred spatal dervatve terms can be approxmated by N =1 (f f eq )φ (V) and φ (V) are sutable thrd order polynomals n V. Two choces of κ are gven n [25]. The method wth κ = 2κ s /κ f s called MR1, and MR2 wth κ = (15 + ( )q j (4 3 3)qj 2 )/15. At boundary nodes where only one neghbor s avalable (.e. the neghbor j+2c s mssng), a modfcaton s suggested n [25] to replace f c (n,j + 2c ) by f c (n,j +c ). However, n the case of opposng ncomng drectons where both neghbors are not avalable, an algorthm s only gven for the case q j = 1/2 when bounce back can be used. In the case of more general q j we have extended the proposed treatment for one mssng neghbor to the case of two mssng neghbors,.e. we replace f c (n,j + c ) wth f c (n,j). Another possblty s to use the FD or POP 1 approach n the later secton 5.3 whch work well n connecton wth FD, FH, MLS, and BFL. The analyss follows the same procedure as the analyss of the bounce back rule n secton 3.5 by nsertng (3.1) and the coeffcents (3.64) and dong Taylor expanson at the pont (t n,x j ). Accordng to our asymptotc analyss, when each boundary node has two neghbors along the lnk, the method MR yelds the resde c 2 s f [ (2 2( k 1 + k ] 2 ))(u 1 c ) ω φ c h + 2c 2 [ 1 ( k 1 + k 2 ) ] u 2 c h 2 s f + 2 k k s [ 2c 2 s f (u ] 2 G) c + t f (1) h (5.4) Snce ω = (2 2( k 1 + k 2 )), the leadng terms vansh f u 1 = φ and u 2 = at the boundary so that second order velocty and frst order accurate pressure felds are obtaned (f the case of mssng neghbors s properly treated). Expandng to hgher orders, we even fnd that MR leads to ncreased precson f on the boundary u 3 = 1 2 (G + tu 1 ). In prncple, ths guarantees a smooth coeffcent f (3) so that rregular behavor would only appear at fourth order. Ths explans the termnology thrd order knetc accuracy used n [25] from the pont of vew of the asymptotc expanson approach. However, smoothness of f (3) also requres proper ntalzaton up to thrd order terms whch s dffcult for general ntal value problems. In fact, the ntal value for u 3 depends on the ntal tme dervatve of the pressure (see (3.54)) whch s generally not known and has to be determned by solvng addtonal Posson equatons. Secondly, the treatment of the boundary nodes havng less than two neghbors may also destroy the smoothness of f (3) so that the pressure s only frst order accurate.

113 5.2. MULTI-REFLECTION METHOD (MR) 99 Addtonally t s valuable to pont out that the replacement of f c (n,j + 2c ) wth f c (n,j + c ) suggested n [25] when only one neghbor s avalable, results n a behavor smlar to the bounce back rule. It mples that the second order resdue s removed only when u 2 satsfes the relaton 2(1 k 1 k 2 )u 2 c = k 1 (c )(u 1 c ) whch n general turns out to be mpossble (a detaled explanaton s already presented n secton 3.5). Hence n ths case the pressure s only th order accurate. If these problems do not appear, lke n the case of Poseulle flow n the channel parallel to the coordnate axs where always two neghbors are avalable and where the ntalzaton of u 3 = G/2 s easy, the method MR yelds ndeed second order accuracy for both velocty and pressure (or even the exact pressure whch s constant n the case of Poseulle flow). Moreover f u 3 s precsely ntalzed, we can reproduce the coeffcents n the expanson (3.1) wth our code and fnd that they are dentcal to the ones n secton In the followng we smulate the Poseulle flow addressed n secton by usng three knds of ntalzatons (3.89), (2.42), (3.1) proposed n secton 3.3. Recall that these ntalzatons are of 1st, 2nd and 3rd order accuracy respectvely. Numercal results ndeed verfy the theoretcal statements about the method MR. Fg.5.5 demonstrates the velocty accuracy produced by the MR method, n whch the lne marked by + shows that the lattce Boltzmann method wth ntalzaton (3.1) can exactly recover u error of velocty error of velocty Fgure 5.5: Logarthmc velocty error versus log 1 h n the case of Poseulle flow n a channel parallel to the x-axs wth dfferent ntal treatments. Left: The lnes wth symbols, refer to errors wth ntalzaton (3.89), (2.42) and have slopes 1.28, 2. respectvely. Rght: The lne marked by + plots the logarthmc velocty error n the case of the ntalzaton (3.1). Also for the statonary lnear problem (3.82) on the unt square, MR1 yelds the exact soluton f the populatons are ntalzed correctly up to fourth order and f the ncomng populatons at the corners and the adjacent nodes are prescrbed exactly up to order four, snce u k = (k 2) and ρ k = (k 3) hold n ths

114 1 CHAPTER 5. DIRICHLET BOUNDARY CONDITIONS AND CONSISTENCY ANALYSIS case. Ths s also numercally verfed. The errors for both velocty and pressure are zero up to machne accuracy. If the ntal and corner populatons are only exact up to thrd order, we recover both second order pressure and velocty (see the lne marked by n Fg.5.6). However, f we do not use the exact populatons as boundary values (they are only avalable n smple cases) but any of the other corner treatments descrbed above, we fnd unstable solutons (ths may be partly due to the fact that we work wth the BGK collson operator and not the multple relaxaton tme approach). Only the combnaton wth the bounce back rule n the case q j = 1/2 worked satsfactorly (see the lne marked by n Fg.5.6) error of velocty error of pressure Fgure 5.6: Logarthmc velocty(left) and pressure(rght) error versus log 1 h n the case of lnear flow (3.82) wth ntalzaton (3.1). The lne marked by represents the results n the case q j = 1/2, stands for the results when the exact 3rd order accurate boundary condton s used. 5.3 One pont boundary scheme (POP 1 ) We remark that all non-local lnk-based boundary schemes face the same problem as FD at boundary nodes wth opposng ncomng drectons although the problem s rarely addressed n lterature. In the examples BFL and MLS above, an addtonal neghbour flud node n each ncomng velocty drecton s requred. In MR two such nodes are needed. Smply applyng the bounce back rule at the boundary nodes where the nterpolaton s not applcable nevtably reduces the accuracy of the whole method (numercal verfcaton has been presented n the part of BFL and MR). FH s actually a local schemes whch yelds second order accurate velocty and frst order accurate pressure, however, ths method s not unformly applcable for all the relaxaton parameters, and n some specal cases, for example at corners of a square wth q j =, t leads to an unexpected condton and an nconsstent soluton. Therefore t s very useful to develop a local, consstent, and suffcently accurate boundary algorthm. For ths purpose a dscretzaton s presented here, whch

115 5.3. ONE POINT BOUNDARY SCHEME (POP 1 ) 11 also starts from the modfcaton (5.2) of the bounce back rule. More precsely, we propose a local approach to approxmate the dervatve term (c )(u 1 c ). Ths dscretzaton uses the fact that the lattce Boltzmann soluton does not only carry nformaton about the Naver-Stokes felds but also about ther dervatves. To see ths, we check the coeffcent f (2) whch carres nformaton about the requred dervatve, c f (1) = c 2 s f c c u 1. (5.5) From the expanson (3.1) and the coeffcents (3.64), we also fnd that Solvng for V f (1) n (5.6), we obtan f = f eq h 2 A (V )f (1) + O(h 3 ). (5.6) (V )f (1) = 1 h 2A(f feq ) + O(h) or wth the post-collsonal dstrbuton f c, (V )f (1) = 1 h 2(f fc + g) + O(h). (5.7) Snce the terms on the rght hand sde are avalable when runnng the lattce Boltzmann algorthm, we thus obtan a frst order approxmaton of the correcton term n (5.2) (usng the th component n (5.7)). More generally, we can use (5.7) n a sem-mplct form by replacng f (n,j) wth a convex combnaton θf (n + 1,j) + (1 θ)f (n,j), θ [,1]. Ths leads to the approxmaton wth c f (1) (t n,x j ) = 1 h 2θf (n + 1,j) + 1 h 2 ˆσ (n,j) + O(h) (5.8) ˆσ = f c g (1 θ)f. (5.9) Now (5.8) can be used n connecton wth (5.5) to defne the modfcaton term n (5.2). The actual boundary condton follows by solvng for f (n+1,j) whch appears on both sdes of equaton (5.2) gvng rse to the famly of rules (1 + θ(2q j 1))f (n + 1,j) = f b (n,j) + (2q j 1)ˆσ (n,j). (5.1) Note that the fully mplct case θ = 1 only works n connecton wth q j because (5.1) needs to be solved for the ncomng populaton ˆf (n + 1,j). Smlar to the orgnal bounce back rule, ths algorthm (5.1) s completely local (no neghbor node needs to be accessed for the evaluaton) and t s lnkbased,.e. the ncomng populaton n drecton c s computed by usng only nformaton concernng drectons c and c. Unfortunately, the algorthm has poor stablty propertes but t serves as an mportant buldng block for the followng approach whch s also based on (5.8) but uses an addtonal local averagng over all lnks. The basc average relaton s (c )c u 1 = c 2 s 2 k ( (c c k ) 2 c 2 s c 2) c k f (1) k (5.11)

116 12 CHAPTER 5. DIRICHLET BOUNDARY CONDITIONS AND CONSISTENCY ANALYSIS whch holds for all the velocty models satsfyng (2.27) and (2.28). Equvalently, usng the fact that u 1 =, we can wrte ths relaton n the form c 2 s f (1 2q j )(c )c u 1 = k K k c k f (1) k (5.12) wth coeffcents K k = c 4 s 2 (1 2q j)f ( (c c k ) 2 c 2 s c 2 c 2 α( c k 2 dc 2 s) ) where α {1,...,d} s any ndex (we take α = d). In (5.12) we now use (5.8) as approxmaton for c k f (1) k whch leads to the equaton f (n + 1,j) = f b (n,j) + K k (ˆσk (n,j) θf k (n + 1,j) ) k to determne the ncomng functons. Let V j be a set of all the ndces of the ncomng drectons at a boundary node x j. Collectng all unknown coeffcents f k (n + 1,j) on the left, we end up wth a lnear system (δ k +θk k )f k (n+1,j) = f b (n,j) θk k f k (n+1,j) K kˆσ k (n,j) k V j k V j Note that the quanttes f k (n + 1,j), k V j on the rght are outgong populatons whch are avalable after the lattce Boltzmann transport step. Invertblty of the matrx L k = δ k +θk k for,k V j beng the ndces of the ncomng drectons, can always be guaranteed wth a sutable choce of θ [,1]. In fact, the parameter θ controls the locaton of the egenvalues of θk k and up to a fnte number of choces the spectrum wll not contan 1. Consequently, δ k + θk k s nvertble for all but fntely many choces of θ. Before runnng the scheme, the nverse of L k has to be assembled for each boundary node. After that, the ncomng populatons can be determned by a combnaton of purely local nformaton. Summng up the above arguments, we wrte the mplementaton of ths scheme at boundary node x j, compute K k for all pars of velocty ndces select θ [,1] compute L k = δ k + θk k for ndces,k V j of the ncomng drectons determne the nverse of L k evaluate ˆσ k (n,j) = f c k (n,j) g k(n,j) (1 θ)f k (n,j) for all drectons k notng that f k (n + 1,j) s avalable for non-ncomng drectons k V j after the transport step, compute for all V j k r (n,j) = f b (n,j) k V j θk k f k (n + 1,j) k K kˆσ k (n,j)

117 5.3. ONE POINT BOUNDARY SCHEME (POP 1 ) 13 determne the requred ncomng populatons ˆf k (n + 1,j) by solvng the lnear system (here the nverse of the small matrx L k s needed) L k f k (n + 1,j) = r (n,j), V j k V j In contrast to the scheme whch uses (5.8) drectly, the proposed scheme s not lnk based because of the averagng over all veloctes n (5.11). Apparently, ths averagng has a stablzng effect. Altogether, the stablty s comparable to the scheme BFL [7]. The advantage s the local evaluaton whch smplfes the mplementaton. Numercal verfcaton For several test flows the new method s also numercally nvestgated. Fgure 5.7 s the error plot of velocty and pressure for POP 1. The least squares slope s around 2 for velocty and 1 for pressure, whch agan demonstrates the second order accurate velocty and frst order accurate pressure n general. 2 error of pressure 3 error of velocty log 1 error log 1 error log 1 h log 1 h Fgure 5.7: Double logarthmc error plots of pressure (left) and velocty (rght) versus grd sze usng POP 1 : ( ) lnear flow (3.82), ( ) lnear flow (3.83), (+) Taylor vortex, ( ) crcular flow and ( ) Poseulle flow Moreover, one statonary and one non-statonary 3D lnear flow (problems (3.78), (3.8)) are used as model problems to test the new proposed one-pont method. The numercal tests for ths knd of lnear flow wll be restrcted to the unt cube Ω = [,1] 3 as computatonal regon and the known values of u are prescrbed at the boundary Ω. Note that n 3D, more opposng ncomng drectons appear whch s the perfect geometry for the new methods POP θ. Because of the smple geometrcal structure, the parameters q j are determned by three values q, where h(q 1,q 2,q 3 ) are the coordnates of the node closest to the orgn. In ths paper q are chosen equally and the common value s denoted q, wth values {,.4} n our test cases. Table 5.3 dsplays all the slopes of the error lnes n Fgures 5.8, 5.9 for statonary lnear flows. Those of the velocty are around 1.9 and the pressure curves show slopes around.9. These numercal results demonstrate that the new boundary

118 14 CHAPTER 5. DIRICHLET BOUNDARY CONDITIONS AND CONSISTENCY ANALYSIS scheme produces a second order accurate velocty and at least a frst order accurate pressure, whch supports the analytcal statements. Observng the error sze n those fgures, t shows that all varants of the method have a very smlar performance, and none of the models D3Q15, D3Q19 and D3Q27 produces promnently smaller errors than the others. Table 5.3: Numercal convergence orders of velocty/pressure for a statonary lnear flow. θ q D3Q15 D3Q19 D3Q / / / / / / / / / / / / L rregular 1.953/ / / / /.911 Fgures 5.1, 5.11 for the nonstatonary flow show the error plots of velocty and pressure versus the grd sze. Whle the error n velocty s essentally dentcal for all the methods, the D3Q27 model gves slghtly better results than the other two models. Table 5.4 dsplays all the slopes of the error lnes n those fgures. These numercal results agan demonstrate that the new boundary scheme produces a second order accurate velocty and at least a frst order accurate pressure. Table 5.4: Numercal convergence orders for velocty/pressure n the case of a non-statonary lnear 3D flow. θ q D3Q15 D3Q19 D3Q / / / / / / / / / / / / L rregular 1.991/ / / / /1.55

119 5.3. ONE POINT BOUNDARY SCHEME (POP 1 ) log 1 error log 1 error log 1 h log 1 h 4 log 1 error log 1 error log 1 h log 1 h log 1 error log 1 h log 1 error log 1 h Fgure 5.8: Logarthmc velocty error versus log 1 h n the case of a 3D statonary lnear flow (3.8) correspondng to the model D3Q15 (1st row), D3Q19 (2nd row), D3Q27 (3rd row) wth q = (left) and q =.4 (rght)., and + denote the results of explct θ =, mxed θ =.7 and mplct θ = 1 schemes.

120 16 CHAPTER 5. DIRICHLET BOUNDARY CONDITIONS AND CONSISTENCY ANALYSIS log 1 error log 1 error log 1 h log 1 h log 1 error log 1 h log 1 error log 1 h log 1 error log 1 error log 1 h log 1 h Fgure 5.9: Logarthmc pressure error versus log 1 h n the case of a 3D statonary lnear flow (3.8) correspondng to the model D3Q15 (1st row), D3Q19 (2nd row), D3Q27 (3rd row) wth q = (left) and q =.4 (rght)., and + denote the results of explct θ =, mxed θ =.7 and mplct θ = 1 schemes.

121 5.3. ONE POINT BOUNDARY SCHEME (POP 1 ) 17 error of velocty error of velocty log 1 error log 1 error log 1 h log 1 h error of velocty error of velocty log 1 error log 1 error log 1 h log 1 h error of velocty error of velocty log 1 error log 1 error log 1 h log 1 h Fgure 5.1: Logarthmc velocty error versus log 1 h for non-statonary lnear flow (3.8) correspondng to the model D3Q15 (1st row), D3Q19 (2nd row), D3Q27 (3rd row) wth q = (left) and q =.4 (rght)., and + denote the results of explct θ =, mxed θ =.7 and mplct θ = 1 schemes.

122 18 CHAPTER 5. DIRICHLET BOUNDARY CONDITIONS AND CONSISTENCY ANALYSIS 3 error of pressure 3 error of pressure log 1 error 3.5 log 1 error log 1 h log 1 h 3 error of pressure 3 error of pressure log 1 error 3.5 log 1 error log 1 h log 1 h 3.2 error of pressure 3.2 error of pressure log 1 error log 1 error log 1 h log 1 h Fgure 5.11: Logarthmc pressure error versus log 1 h for non-statonary lnear flow (3.8) correspondng to the model D3Q15 (1st row), D3Q19 (2nd row), D3Q27 (3rd row) wth q = (left) and q =.4 (rght)., and + denote the results of explct θ =, mxed θ =.7 and mplct θ = 1 schemes.

123 5.4. NUMERICAL COMPARISON Numercal comparson In the followng, we use the decayng Taylor vortex soluton wth the same parameters as n secton 3.6, the mpulsve crcular flow and the second lnear flow (3.82) to test and compare the boundary schemes addressed n the prevous sectons of ths chapter. The reason to choose these three flows les n that each s amed at testng varous aspects of these methods. Taylor vortex flow and crcular flow are tme-dependent, the lnear flow s statonary, and the crcular flow s used to test the applcablty to curved boundares. Snce the method MR does not yeld stable solutons wth the avalable corner treatments for general values q j, we splt the comparson. In the case q = 1/2, all these schemes reduce to the bounce back rule and a comparson wth MR s now possble because, for q j = 1/2, MR works n combnaton wth the bounce back rule at the corners and gves rse to a frst order accurate pressure and a second order accurate velocty. Frst we consder those schemes whch are modfcatons of the bounce back rule for q = and q =.1 (where q s the common value of q j at lower and left boundary of the square) Comparson of accuracy The numercal convergence rates for pressure and velocty are summarzed n tables 5.5 to 5.7 and the correspondng error plots can be found n fgure 5.12 to All the schemes demonstrate agan the second order accurate velocty and frst order pressure n accordance wth the theoretcal results. Comparng the error sze for the decayng Taylor vortex flow (Fg.5.12), we see that the mplct POP 1 produces smaller error than the explct POP and the mxed one POP.7, and all POP methods gve smaller error than the BFL rule. Table 5.5: Convergence rates for the Taylor vortex flow from secton 3.6 computed wth dfferent boundary algorthms. FD POP POP 1 POP.7 BFL FH pressure velocty

124 11 CHAPTER 5. DIRICHLET BOUNDARY CONDITIONS AND CONSISTENCY ANALYSIS error of pressure error of velocty log 1 error 2 log 1 error log 1 h log 1 h Fgure 5.12: Double logarthmc error plots of pressure (left) and velocty (rght) versus grd sze for the decayng Taylor vortex flow. The boundary schemes are FD ( ), POP ( ), POP 1 (+), POP.7 ( ), BFL( ), and FH( ) Secondly we consder the so-called mpulsvely started crcular flow. Snce now the boundary curve s a crcle, the dstance between boundary nodes and boundary vares arbtrarly between and h,.e., q j [,1). In order to concentrate on the errors caused by the proposed boundary schemes, we want to avod ntal errors due to the non-smooth abrupt start of the cylnder (abrupt start means u = n the doman but u θ = 1 along the boundary). Ths can be acheved, for example, by startng at tme t =.5 nstead of t =. To check the accuracy, the termnaton tme s set to T =.6. In the explct case θ =, the POP scheme s not stable. Fgure 5.13 shows the error of velocty and pressure versus the grd sze for mplct and mxed boundary condtons and the BFL method. It demonstrates the second order accuracy of velocty and frst order accuracy of pressure. More mportantly, these numercal results confrm that the proposed boundary scheme works very well n connecton wth curved boundares f θ >. Table 5.6: Convergence rates for the crcular flow from secton 3.6 computed wth dfferent boundary algorthms. FD POP POP 1 POP.7 BFL FH pressure.828 N velocty 1.87 N

125 5.4.1 Comparson of accuracy Fgure 5.13: Double logarthmc error plots of pressure (left) and velocty (rght) versus grd sze for the mpulsvely started crcular flow. The boundary schemes are FD ( ), POP ( ), POP 1 (+), POP.7 ( ), BFL( ), and FH( ) Next we consder the lnear flow (3.82). The FD method produces the smallest error n ths case. Ths s verfed by the numercal results n fgure The reason mght be that the frst spatal dervatves can be calculated by means of fnte dfference wth a hgher order accuracy. Table 5.7: Convergence rates for the lnear flow (3.82) computed wth dfferent boundary algorthms. FD POP POP 1 POP.7 BFL FH pressure velocty log 1 error 3.5 log 1 error log 1 h log 1 h Fgure 5.14: Double logarthmc error plots of pressure (left) and velocty (rght) versus grd sze for the statonary lnear flow (3.82). The boundary schemes are FD ( ), POP ( ), POP 1 (+), POP.7 ( ), BFL( ), and FH( )

126 112 CHAPTER 5. DIRICHLET BOUNDARY CONDITIONS AND CONSISTENCY ANALYSIS Fnally, we compare the error sze c 2 sρ 3 + O(h) = ((ˆρ 1)/(c 2 s h 2 ) p)/h for pressure and u 3 + O(h) = (û hu)/h 2 for velocty when a smaller vscosty ν =.1 and grd sze h = 1/5 are deployed. The test problem s agan the lnear flow (3.82). Fgure 5.15 shows the contourlnes of errors n the case q j = 1/2, the correcton terms n (5.2) vansh because of the partcular q j. Fgure 5.16 gves the error contour lnes wth exact correcton n (5.2) whch s obtaned by the analytcal solutons. Fgure 5.17 and 5.19 dsplay the error when BFL and FH methods are used. These three plots show smlar profles of the error. However BFL and FH produce much bgger errors than the case q j = 1/2. Fgure 5.18 shows the error plot for the method POP 1 whch yelds almost the same sze of error as n the case q j = 1/2 but wth a dfferent profle. Moreover all plots llustrate the rregularty of the error. FH method produces a relatvely smooth feld u e e e e e e 5 7.1e e e e e Fgure 5.15: Error contour lnes of pressure (left), x-component of velocty (mddel) and y-component of velocty (rght) for the statonary lnear flow (3.82) n case ν =.1 usng the bounce back rule wth q = 1/ e e e e e 6 1.3e e e e e e e e e e Fgure 5.16: Error contour lnes of pressure (left), x-component of velocty (mddel) and y-component of velocty (rght) for the statonary lnear flow (3.82) n the case ν =.1 and q = usng the exactly modfed bounce back rule.

127 5.4.2 Stablty behavor e e Fgure 5.17: Error contour lnes of pressure (left), x-component of velocty (mddel) and y-component of velocty (rght) for the statonary lnear flow (3.82) n the case ν =.1 and q = usng the BFL method e e e e e e e e e Fgure 5.18: Error contour lnes of pressure (left), x-component of velocty (mddel) and y-component of velocty (rght) for the statonary lnear flow (3.82) n the case ν =.1 and q = usng the mplct POP e e e Fgure 5.19: Error contour lnes of pressure (left), x-component of velocty (mddel) and y-component of velocty (rght) for the statonary lnear flow (3.82) n the case ν =.1 and q = wth FH Stablty behavor Tables 5.8 and 5.9 gve an mpresson on the stablty of the dfferent boundary schemes. Here we compare the maxmal error n velocty for a computaton on

128 114 CHAPTER 5. DIRICHLET BOUNDARY CONDITIONS AND CONSISTENCY ANALYSIS a grd wth h = 1/5 and vscosty s rangng from 1/1 to 1. The letter N s used f no error value could be obtaned because of nstablty. Table 5.8: Maxmal velocty error on a fxed grd (h = 1/5) wth nodes on the boundary (q = ) for varyng vscosty. The letter N ndcates nstablty. vscosty 1/1 1/5 1/1 1/6 1 1 FD N N N FH N POP N POP.7 N POP 1 N BFL Table 5.9: Maxmal velocty error on a fxed grd (h = 1/5) wth nodes not on the boundary (q =.1) for varyng vscosty. The letter N ndcates nstablty. vscosty 1/1 1/5 1/1 1/6 1 1 FD N N FH N POP N N N POP.7 N N POP BFL In the general case q > (table 5.9 s a representatve case), we can see that FH, POP 1 and BFL have the best stablty among the modfcatons of the bounce back rule where FH s very naccurate or suffers from nstablty f τ 1. Only when q =, BFL s slghtly more stable than POP 1 whch n turn has a slghtly smaller error. However, on rectangular domans, the best choce of q wth respect to stablty s certanly q = 1/2 n whch case all the consdered methods turn nto the smple bounce back rule. A comparson wth MR for ths case s gven n table 5.1. We have also compared the new method POP θ wth BFL for the crcular flow problem. In ths case, the geometry coeffcents q j range n the whole nterval [,1). On two grds (h = 1/5,1/1) wth termnaton tme T = 1 we have vared the vscosty. Down to ν =.1 both schemes are stable n the sense that they produce bounded solutons.

129 5.4.3 Investgaton of total mass 115 Table 5.1: Maxmal velocty error on a fxed grd (h = 1/5) wth nodes half way from the boundary (q = 1/2) for varyng vscosty. The letter N ndcates nstablty. vscosty BB M R N N Investgaton of total mass Snce the total mass n Ω s a conserved quantty for the ncompressble flow, t s valuable to check to what extent the above mentoned boundary condtons guarantee mass conservaton. The evaluaton crteron s the average densty over the doman Ω. We calculate the devaton from the ntal average densty, so that the sze of devaton at tme t demonstrates the mass loss or ncrease over the tme nterval [,t], where the ntal tme s always denoted by t = wthout loss of generalty. From the consstency analyss we know most of the mprovements of the bounce back rule are based on dscretzaton of f (n + 1,j) = f c (n,j) + 2hc 2 s f φ(t n,x j ) c + h 2 (2q j 1)c 2 s f (c )(u 1 c )(t n,x j ) + O(h 3 ). (5.13) The bounce back rule wth φ = can conserve the total mass exactly, snce all the partcles leavng the doman wll reenter along the opposte drecton. However when φ, the term 2hc 2 s f φ(t n,x j ) c certanly brngs mass nto or out of the doman at each tme step and results n an amount of total mass devaton M b (n + 1) over the tme nterval [t n,t n+1 ] whch s defned by M b (n + 1) = 2hc 2 s f φ(t n,x j ) c. V j j Ω h The correspondng devaton of the average densty from ρ() over the doman at tme t = t n s ρ b (t n ) = 1 n M b (k), (5.14) Ω h where Ω h = O(h d ) s the total number of nodes n Ω. It s known for ncompressble flows that φ must satsfy the addtonal constrant φ(t,x) nds =. Ω In fact, h d 2 M b (n) can be regarded as an approxmaton to the above ntegral at t = t n 1. Assume the accuracy s frst order, k=1 h d 2 M b (n) = O(h), nh 2 [,T], (5.15)

130 116 CHAPTER 5. DIRICHLET BOUNDARY CONDITIONS AND CONSISTENCY ANALYSIS t follows for the densty ρ b (t n ) = O(nh 3 ), nh 2 T, and ρ b (T) = O(h). (5.16) Fg.5.2 shows ρ b (t), t [,.1] for the lnear flow (3.82) wth q l =.1 as well as q b =, smlarly Fg.5.21 treats the Taylor vortex flow n one-sxteenth of one perod(ω = [,π/2] 2 ) wth q l =.19 as well as q b =.37 and Fg.5.22 shows the plot for the crcular flow n the tme nterval [.5,.6]. Both lnear flow and decayng Taylor vortex dsplay that the total mass decreases wth tme. The slopes of the lnes n the rght plot are.953 for lnear flow and 1.32 for the decayng Taylor vortex, whch numercally demonstrates that the devaton of the average densty s of order O(h) at the endng tme T = 1. When h = 1/3 and 1/5, for the crcular flow the total mass varaton s zero by accdent. However n the case of h = 1/1,1/2,1/4 the total mass s ncreasng (see the left plot n Fg. 5.22) wth tme elapsng. The slopes of the lnes n the rght plot of Fg are 2.125(T =.3), 2.37(T =.5), 2.358(T =.7) and 2.428(T =.1), whch llustrate that the devaton of the average densty s of order h 2. x ρ(t) 2 4 log ρ(t) t log 1 (h) Fgure 5.2: Plots for the lnear flow (3.82) wth the bounce back rule. Left: The behavor of ρ(t) wth tme changng for the cases h = 1/1 ( ), 1/2 (+), 1/3 ( ), 1/4 ( ), 1/5 ( ). Rght: Plot of logarthmc ρ(t) vs. logarthmc grd sze at termnal tme T =.3 ( ), T =.5 ( ), T =.7 ( ), and T =.1 ( ).

131 5.4.3 Investgaton of total mass 117 x ρ(t) 2 4 log ρ(t) t log 1 (h) Fgure 5.21: Plots for the Taylor vortex flow wth the bounce back rule. Left: The behavor of ρ(t) wth tme changng for the cases h = 1/1 ( ), 1/2 (+), 1/3 ( ), 1/4 ( ), 1/5 ( ). Rght: Plot of logarthmc ρ(t) vs. logarthmc grd sze at termnal tme T =.3 ( ), T =.5 ( ), T =.7 ( ), and T =.1 ( ). ρ(t) ρ(t) x x 1 9 t t log ρ(t) log 1 (h) Fgure 5.22: Plots for the crcular flow wth the bounce back rule. Left: The behavor of ρ(t) wth tme changng for the cases h = 1/1 ( ), 1/2 (+), 1/4 ( ). Rght: Plot of logarthmc ρ(t) vs. logarthmc grd sze at termnal tme T =.3 ( ), T =.5 ( ), T =.7 ( ), and T =.1 ( ). Remark 1 On some specal geometres, the ncomng drectons at boundary node x j ft the normal drecton n so well that 2c 2 s f φ(t n,x j ) c = φ(t n,x j ) n (for example n a square). If, n addton, the nodes are located on the boundary lnes, and f φ s lnear, M b and ρ b are zero. Remark 11 For the decayng Taylor vortex flow n one perod,.e. Ω = [,2π] [,2π], and lattce nodes whch are regularly dstrbuted n the same way as descrbed n secton2.2, we fnd M b = and ρ b = no matter what q l and q b are. Ths s a consequence of the perodc property and the partcular

132 118 CHAPTER 5. DIRICHLET BOUNDARY CONDITIONS AND CONSISTENCY ANALYSIS structure of the velocty feld. Next we consder the mprovements of the bounce back rule, where the mass fluctuates not only due to the nonzero φ but also due to the dscretzaton of the thrd term n (5.13) wth factor h 2. We have M b (n + 1) = M b (n + 1) + h 2 (2q j 1)c 2 s f (c )(u 1 c )(t n,x j ) + O(h 3 ). V j j Ω h together wth the correspondng devaton of the average densty from ρ() over the doman at tme t = t n ρ b (t n ) = 1 Ω h n M b (k). (5.17) k=1 We use the Taylor vortex flow and the crcular flow to check the densty varaton on dfferent grds. Fgure 5.23 shows the plots for BFL method, fgure 5.24 for FH method and fgure 5.25 for POP 1. In case of the Taylor vortex flow, all three methods demonstrate a mass ncrease of sze O(h 2 ). However n case of the crcular flow, BFL and FH boundary treatments ncrease a mass of sze O(h 1.5 ) whereas POP 1 of sze O(h 4 ). These test cases ndcate that POP 1 has a better mass conservaton property than BFL and FH. log ρ(t) x log 1 (h) log ρ(t) x log 1 (h) log ρ(t) log 1 (h) Fgure 5.23: Plots for the BFL method. Left (Crcular flow) and mddel (Taylor vortex) plots: The behavor of ρ(t) versus tme on the grd sze of h = 1/1 ( ), 1/2 ( ), 1/3 ( ), 1/4 (+) and 1/5 ( ). Rght: Plot of logarthmc ρ(t) vs. logarthmc grd sze at termnal tme T =.1 ( ) for Taylor vortex flow wth a slope of 2.83 and T =.6( ) for crcular flow wth a slope of

133 5.5. SUMMARY 119 log ρ(t) x log 1 (h) log ρ(t) 3.5 x log 1 (h) log ρ(t) log 1 (h) Fgure 5.24: Plots for the FH method. Left (Crcular flow) and mddel (Taylor vortex) plots: The behavor of ρ(t) versus tme on the grd sze of h = 1/1 ( ), 1/2 ( ), 1/3 ( ), 1/4 (+) and 1/5 ( ). Rght: Plot of logarthmc ρ(t) vs. logarthmc grd sze at termnal tme T =.1 ( ) for Taylor vortex flow wth a slope of and T =.6 ( ) for crcular flow wth a slope of x 1 7 x 1 6 log ρ(t) 3 2 log ρ(t) 3 2 log ρ(t) log 1 (h).5.1 log 1 (h) log 1 (h) Fgure 5.25: Plots for the POP 1 method. Left (Crcular flow) and mddel (Taylor vortex) plots: The behavor of ρ(t) versus tme on the grd sze of h = 1/1 ( ), 1/2 ( ), 1/3 ( ), 1/4 (+) and 1/5 ( ). Rght: Plot of logarthmc ρ(t) vs. logarthmc grd sze at termnal tme T =.1 ( ) for Taylor vortex flow wth a slope of and T =.6 ( ) for crcular flow wth a slope of Summary 1. All the numercal experments concde very well wth the theoretcal results and show that the mprovements of bounce back rule produce 2nd order accurate velocty and 1st order accurate pressure. 2. Numercal results show that the proposed scheme POP θ yelds slghtly smaller errors than the other methods. 3. BFL and POP 1 have comparable numercal stablty. BFL requres two lattce nodes generally whereas POP 1 needs only one. In the 2D case very few nodes have opposng ncomng drectons. In the 3D case the opposng ncomng drectons appear more frequently. Take the cube as an example: the opposng ncomng drectons appear not only at corners but

134 12 CHAPTER 5. DIRICHLET BOUNDARY CONDITIONS AND CONSISTENCY ANALYSIS also along the edges. In ths case, the use of POP 1 has clear advantages. As an alternatve, one can also use BFL at most of the ponts and POP 1 only at the exceptonal ones. POP 1 as a completon of BFL thus s very convncng and feasble. 4. Usng a lower order method at these exceptonal nodes leads to a deteroraton of the order everywhere n the doman. 5. If MR s realzable everywhere and stable, t produces smaller errors than the other methods and the error s also smoother. 6. The bounce back rule for the zero Drchlet boundary preserve the total mass precsely, however for non zero Drchlet boundary condton generally leads to mass varatons of sze O(h). The mprovements of the bounce back rule also yeld an O(h) mass loss or ncrease. In all the numercal test cases, POP 1 produces the smallest mass varaton.

135 Chapter 6 Neumann type outflow boundary condtons Ths chapter s devoted to the constructon and analyss of outflow boundary condtons, whch s one of the mportant dffcultes encountered n numercal smulaton n flud dynamcs. Ths dffculty arses because the computatonal doman s bounded whereas the physcal doman s unbounded, for example the flow n a nfnte long channel. Then a sutable open boundary s requred at outflow, because unwanted effects of the outflow boundary mght propagate to the nternal doman of nterest and lead to unrealstc results. For flows governed by Naver-Stokes system, several outflow condtons are avalable. Some of the most popular choces are lsted here. The smple Neumann boundary condton (NBC) [2], u (t,x) = ϕ(t,x), x Ω, t [,T]; (6.1) n or the Neumann-type do-nothng condton (DNT) [35] pn + ν u =, (6.2) n and the zero normal shear stress boundary condton (ZNS) or so-called nofrcton condton [26, 55] are often suggested. ( pi + 2νS[u])n =, S[u] = 1 2 ( u + ut ) (6.3) Another class of frequently used outflow condtons s the so-called convectve boundary condton [65] φ t + c φ n = where φ can be some functon of velocty or pressure, and c s a typcal velocty. The numercal realzaton of these outflows has been successfully demonstrated n the fnte dfference (FD) and fnte element smulatons (FVM) of Naver- Stokes equatons [65, 61, 2, 1, 47, 74]. 121

136 122 CHAPTER 6. NEUMANN TYPE OUTFLOW BOUNDARY CONDITIONS In lattce Boltzmann smulatons, we frequently encounter an extrapolaton method [79] whch operates drectly on the dstrbuton functon f. Another treatment s proposed n [15] whch uses Grad s approxmaton. However, none of these treatments smulate the above mentoned condtons (see secton 6.1). Therefore, we focus on these Neumann type boundary condtons and try to realze ther lattce Boltzmann mplementatons. In order to reduce the nfluence of the outflow boundary condton onto the nner flud, a usually adopted way s use a very large computatonal doman, for example a very long channel flow. Often, the rato of length and wdth of the channel s at least 5. For example, Flppova [21] uses the rato equal to 5, and Lamura et al. [5] uses a rato equal to 25/4. However ths approach s qute expensve snce more memory and CPU tme are needed due to the large computatonal doman. Wth the proposed outflow boundary condton, we hope to reduce the computatonal effort. Snce the outflow boundary s often not specfed by the problem, we can choose t as a sutable straght lne n the 2D case and as a plane n 3D to smplfy the geometry. Especally n the lattce Boltzmann method we can let the lattce nodes be located exactly on the open boundary whch s parallel to the coordnate axs. In ths work, we take use of ths advantage for all cases..e., we assume that n = se k, s { 1,1}, (6.4) where e k s the k -th unt axs vector parallel to n. Let t be the unt tangental drectons. Then we can rewrte the do-nothng condton and zero shear stress condton n an equvalent form. DNT has the form and ZNS reads p + ν u n n =, u t n = ; (6.5) p + 2νS[u] : (n n) =, S[u] : (n t ) = ; (6.6) 6.1 Asymptotc analyss of two known outflow schemes In chapter 3 and 5, the asymptotc analyss has been appled to the lattce Boltzmann algorthm for Drchlet boundary condtons and demonstrates that ths analyss can provde accuracy nformaton for the whole algorthm. Therefore we also apply t to the outflow schemes. Snce the basc procedure s smlar to the one for the Drchlet boundares. Frst we nvestgate two outflow schemes proposed n [79, 15]. If the outflow boundary s parallel to one axs and c s one ncomng drecton at boundary node x j, the method proposed n [79] lnearly extrapolates the unknown value f from nsde f (n + 1,j) = 2f (n + 1,j n) f (n + 1,j 2n).

137 6.2. NEUMANN BOUNDARY (NBC) 123 It s noted that x j n and x j 2n are lattce nodes n Ω. Insertng the regular expanson (3.1) nto ths equaton, then dong Taylor expanson at (t n,x j ), we fnd that the coeffcents of h k (k =,1,2) are zero and the resdue arses Obvously the condton for u 1 : (n ) 2 (u 1 c )h 3 + O(h 4 ). (6.7) (n ) 2 (u 1 c ) = (6.8) can remove the resdue of order h 3. Snce there are generally two lnearndependent ncomng drectons at every node on the outflow boundary, we get the macroscopc boundary condton, (n ) 2 u 1 =. (6.9) whch contans the second order dervatves of the velocty feld. Second let us consder the method n [15], whch reads f (n + 1,j) = f [ˆρ(n,j) + c 2 s c û(n,j)] [ ] + f c 4 s 2 ( f k (n,j)c c ˆρ(n,j)c 2 si) : (c c c 2 si). (6.1) k Applyng the same procedure, we see that ths outflow scheme yelds a resdue [ ( t f (1) A ( t f (1) + (V )f (2) + 1 ) ] 2 (V )2 f (1) g) h 3 + O(h 4 ). Removng the lowest order resdue leads to a boundary condton of u 1 ( t f (1) A ( t f (1) + (V )f (2) + 1 ) 2 (V )2 f (1) g) =. Usng the explct formula 3.64 to replace the coeffcents f (k) by the moments, we fnd an unusual condton on u 1 nvolvng ts tme and second order spatal dervatves. 6.2 Neumann boundary (NBC) For the Neumann boundary condton (6.1), a practcal, second order accurate realzaton n fnte dfference methods s obtaned by ntroducng a so-called ghost node for each boundary node, such that the par of ghost node and boundary node s mrror symmetrc wth respect to the boundary lne Ω. For example, Fg. 6.1 shows a 2D case where x j1 Ω s a flud node next to the outflow boundary node x j on Γ and x j s ts ghost node. Note x j s the ntersecton pont of the boundary curve and the connectng lne between x j1 and x j.

138 124 CHAPTER 6. NEUMANN TYPE OUTFLOW BOUNDARY CONDITIONS n n x j x j x j x j1 x j1 x j Γ Γ Fgure 6.1: Demonstraton of the ghost node x j to a boundary node x j1. The vector n s the outward normal drecton. Left: the boundary lne stuates at half a grd dstance from the flud node x j1. Rght: The flud node x j s exactly located on the boundary. Then we set the velocty value at ghost node so that u(t,x j ) = u(t,x j1 ) + αhϕ(t,x j ), u n (t,x j) = u(t,x j ) u(t,x j1 ) + O(h 2 ) = ϕ(t,x j ) + O(h 2 ), (6.11) αh wth α = 1 for the left case and α = 2 for the rght case n Fg. 6.1 obtanng a second order accurate approxmate. To realze the Neumann condton (6.1) n the lattce Boltzmann setup, we use the above dea and apply t to the average velocty û. Notng that the soluton 1 of Naver-Stokes equaton s approxmated by hû, we set ˆφ(n,j ) = 1 hû(n,j 1) + αhϕ(t,x j ), (6.12) At the ghost node x j we apply the lattce Boltzmann realzatons n chapter 5 for the velocty Drchlet condton lke BFL or POP θ, and eventually acheve a feasble scheme for the Neumann boundary condton on Ω. Let c be an ncomng drecton at ghost node x j. We formulate ths scheme usng BFL n the followng f (n + 1,j ) = f c (n,j + hc ) + 2c 2 s f hˆφ(n,j ) c. (6.13) Asymptotc analyss. The analyss s started by nsertng the expanson (3.1), and then dong Taylor expanson at the pont (t,x j ) whch s the ntersecton of the boundary Ω along the lnk n the drecton of c. In fact x j = x j +hαc /2 and x j = x j +hα(c +n)/2. After a straght forward calculaton and cancelng out all the possble terms we are left wth a resdue of h 2 αc 2 s ( u 1 n ϕ) c f + O(h 3 ). (6.14)

139 6.3. ZERO NORMAL SHEAR STRESS BOUNDARY (ZNS) 125 Ths demonstrates that the Neumann boundary condton removes the lowest order resdue of h 2. In addton, the resdue of order h 3 results from applyng of BFL. Recallng the analyss results about BFL n chapter 5, the resdue of order h 3 s generally not removable. Hence a frst order accuracy of the Neumann boundary condton s generally expected. Moreover, we note that the dfference error n (6.11) resultng from the central dfference approxmaton fnally enters the term of order h 4 n (6.14). However, from the asymptotc analyss we know the error n order h 3 s nevtable, hence usng second order dfferences n (6.11) does not ncrease the accuracy. Another possblty s to use one-sde dfference n (6.11), u n (t,x j) = u(t,x j) u(t,x j1 ) + O(h) = ϕ(t,x j ) + O(h). (6.15) αh/2 In ths case we set correspondngly ˆφ(n,j) = 1 hû(n,j 1) + α 2 hϕ(t,x j). (6.16) We assemble a Drchlet condton at outflow agan based on the BFL scheme f (n + 1,j) = f c (n,j + hc ) + 2c 2 s f hˆφ(n,j) c, α = 2, (6.17) f (n + 1,j 1 ) = f c (n,j 1) + 2c 2 s f hˆφ(n,j) c, α = 1. (6.18) Obvously the asymptotc analyss wll lead to the same leadng order n (6.14) but wth a dfferent coeffcent of h 3 whch now contans the dfference error from (6.15). 6.3 Zero normal shear stress boundary (ZNS) In the prevous part we have mentoned that the outflow boundary s supposed to be parallel to the coordnate axs, and snce only the cubc lattce s consdered, one of the ncomng drectons at a boundary node x j must be opposte to the outer normal drecton n. Moreover t s requred that V contans veloctes n all the axs drectons,.e., there s a lnear ndependent set C = {c m1,...,c md } V, n whch c mk = c mk e k and e k s the k-th unt axs vector. It s noted that the known velocty sets D2Q9, D3Q15, D3Q19, D3Q27 nclude such a set C. The proposed mplementaton at x j of the zero normal shear stress condton (6.3) treats the ncomng drecton opposte to the outer normal drecton n and the others dfferently. For the ncomng drecton c k = n, we set the correspondng dstrbuton functon accordng to f k (n + 1,j) = F eq (1,û(n,j)) ((2νA I)(f f eq )) k (n,j), (6.19) If c s another ncomng drecton, we can express t n the form c = d k=1 c ke k wth respect to the standard bass {e k }. In vew of (6.4), we can rewrte ths

140 126 CHAPTER 6. NEUMANN TYPE OUTFLOW BOUNDARY CONDITIONS relaton n the form c = sc k n + k k c k e k Snce c s ncomng, we have c n <,.e. c k. Moreover, there must be an addtonal c k because c c k. Dependng on the structure of the coeffcents c k, k k we dstngush two cases (n the specfc examples, we assume n = (1,) T for the 2D and n = (1,,) T for the 3D case): () For ncomng drectons c wth c k for all k k, lke ( 1,1) n the D2Q9 case or ( 1,1,1) n the D3Q15 model, we set f (n + 1,j) = F eq (1,û(n,j)) f f k (2νA(f(n,j) f eq (n,j))) k + c 4 s c kc lf k l k,l k N p=1 c pk c pl (f p (n,j) f eq p (n,j)) (6.2) () The remanng ncomng drectons, e.g. ( 1, 1, ) n the D3Q19 model, are treated accordng to f (n+1,j) = f eq (n,j)+ d (c 2 k 1) f fm k c mk 2(f m k (n,j) f eq m k (n,j)) (6.21) k=1 where c mk = c mk e k are the veloctes pontng n the coordnate drectons. We stress that for the D2Q9 and D3Q15 models, only case () appears. The D3Q19 model nvolves () and D3Q27 requres both () and (). Asymptotc analyss After a standard calculaton followng the routne procedure, (6.19) produces a resdue h 2 f k (ρ 2 c 2 s 2ν(c k )(u 1 c k )) Removng the h 2 term n (6.22) yelds 2h 3 ν( t f (1) k x f (2) k xf (1) k g k ) + O(h 4 ). (6.22) ρ 2 2c 2 s ν(c k )(u 1 c k ) = (6.23) Further, notng that c k = n s a constant coordnate drecton, we obtan c 2 s ρ 2 2ν(n )(u 1 n) =. Ths equaton actually corresponds to the frst equaton n (6.6) f we recall that c 2 sρ 2, u 1 are the solutons of ncompressble Naver-Stokes equaton (2.1). Processng of (6.2) leads to a resdue ( h 2 f (ρ 2 c 2 s 2ν(n )(u 1 n)) c 2 s (A Λ : S[u 1 ]) + c 6 s c kc lf k l k,l k N ) c pk c pl (A Λ : S[u 1 ]) p + O(h 3 ). (6.24) p=1

141 6.3. ZERO NORMAL SHEAR STRESS BOUNDARY (ZNS) 127 Snce the frst term of h 2 coeffcent n (6.24) can be removed by (6.23), we observe the left term = c 2 N s (A Λ : S[u 1 ]) c 4 s c pk c pl (A Λ : S[u 1 ]) p. (6.25) c kc lf k l k,l k p=1 Usng the property (v) of matrx A, (6.25) turns out to be = µc 4 s Λ : S[u 1 ] c 4 s c kc lf k l k,l k N c pk c pl Λ p : S[u 1 ]. Takng nto account u 1 = and c 4 N s p=1 c pkc pl Λ p : S[u 1 ] = e k e l : S[u 1 ], we get = µc 4 s Observng that c c = d p=1 c c : S[u 1 ] c kc le k e l : S[u 1 ] k l k,l k. (6.26) k=1 c 2 k e k e k + k k c k c k e k e k + k l k,l k c kc le k e l, (6.27) as well as c k = 1 for all k and d e k e k : S[u 1 ] = u 1 =, (6.28) k=1 we obtan from (6.26) = µc 4 s c k e k e k : S[u 1 ]. (6.29) k k Hence the second condton n (6.6) can remove. A smlar process appled to (6.21) leads to a resdue µc 4 s [ c c : S[u] ] d (c 2 k 1)(e k e k : S[u]). k=1 Insertng (6.27) whle notng that the last term dsappears for the consdered case (), and usng (6.28) we agan obtan that the resdue s same as (6.29).

142 128 CHAPTER 6. NEUMANN TYPE OUTFLOW BOUNDARY CONDITIONS Fnally, our assumpton on the structure of the velocty model guarantees that all tangental drectons e k, k k appear. In partcular, for the smple D2Q9 model, (6.29) can be removed by u 1 y + u 2 x = whch s the second set of equatons n (6.6). Moreover the coeffcent of h 3 n (6.22) s generally not zero so that the boundary scheme (6.19), (6.2) and (6.21) renders a frst order accurate zero normal stress outflow condton (6.3) Remark 12 From (6.24), we can observe that the rght hand sde n the frst row of (6.2) renders the frst equaton of the zero normal stress condton (6.6). A smplfed scheme thus can be produced by elmnatng these terms: f (n + 1,j) = f eq (n,j) + c 4 s c kc lf k l k,l k N p=1 c pk c pl (f p (n,j) f eq p (n,j)). (6.3) The related asymptotc analyss suggests that ths smplfed scheme also results n the second equaton of (6.6). 6.4 Do nothng condton (DNT) To realze the do-nothng condton (6.2) n lattce Boltzmann smulatons, we combne formulas (6.19) and an extenton of the Neumann boundary scheme (6.17). Agan we use the assumpton that n s opposte to one of ncomng drecton, say, n = c k. Frst let us consder the equaton n (6.5) contanng the pressure p. Ths equaton dffers only by a factor 2 from the one n (6.6), hence a smlar strategy gves rse to f k (n + 1,j) = F eq k (1,û(n,j)) ((νa I)(f f eq )) k (n,j), c k = n. (6.31) Next we turn to the equatons n (6.5) wthout pressure, whch appear as zero Neumann boundary condtons for the velocty components perpendcular to n. Ths suggests to use the dea n secton 6.2. ũ k (n,j) = û k (n,j), e k = sn (6.32) ũ k (n,j) = û k (n,j n), e k e k (6.33) where e k s the axs drecton parallel to n and û = (û 1,...,û d ) T s the velocty average. Hence we see that the equatons (6.32) and (6.33) set up a Drchlet condton at the outflow boundary. Agan BFL or POP θ method n chapter 5 can be employed. We mplement t to the other ncomng drectons excludng c k and acheve f (n + 1,j) = f c (n,j + hc ) + 2c 2 s f ũ(n,j) c, c n. (6.34)

143 6.5. EXPERIMENTS AND DISCUSSION 129 After carrng out the standard asymptotc analyss process n sectons 6.2 and 6.3, t s also verfed that (6.31) and (6.34) approxmate the do nothng condton (6.2) wth at least frst order accuracy. 6.5 Experments and dscusson In the numercal experments, four test problems n 2D space are employed. All these problems deal wth the flow n an nfntely long channel between two sold walls parallel to the x-axs. The flow s at rest ntally and drven by an nflow velocty at the left end. Near the left end, we place a sold obstacle n the form of a crcular or square cylnder. The artfcal outflow boundary s set at the rght end. The boundng walls are fxed. The followng Fg.6.2 shows the prmary confguraton of two models wth a crcular obstacle. (, H).16m u = outflow nflow.15m.1m y.15m (, ) u = 2.2m x Fgure 6.2: Confguraton of 2D test cases wth a crcular obstacle. The cylnder has a slghtly downward offset to the axs of the channel wth equal dstance to the nlet and to the bottom wall. Fgure 6.3 descrbes the setup for both flows wth a square obstacle. (, H) u = outflow nflow x D y (, ) y u = L x Fgure 6.3: Confguraton of 2D test problems wth a square cylnder. D s the heght of the square cylnder. H and L are the wdth and length of the channel. The blockage rato H/D s fxed to be 4. In the lattce Boltzmann setup, the dscrete velocty set s D2Q9, the collson operater s the BGK model wth a sngle tme relaxaton parameter τ. Snce at all nflow and rgd walls the boundary condton s of Drchlet type, ether BFL or POP 1 s used. The outflow condton s realzed wth the methods descrbed n sectons 6.2, 6.3 and 6.4.

144 13 CHAPTER 6. NEUMANN TYPE OUTFLOW BOUNDARY CONDITIONS A statonary crtera s requred when smulatng a statonary flow wth the lattce Boltzmann algorthm. We evaluate the varaton of average densty ˆρ between two neghborng tme levels. If the varaton s small and satsfes max ˆρ(n + 1,x j) ˆρ(n,x j ) < h 6, (6.35) x j Ω the numercal velocty and pressure felds are consdered as approxmatons for the steady flows. There are two aspects to evaluate these outflow boundary condtons. One s to check how the outflow boundary condtons nfluence the nner flow relatvely far away from the open boundary. The other s to check whether the outflow boundary condton lets the flud pass through the open boundary so naturally as f there was no boundary. In order to quantfy how the outflow condtons nfluence the nner flow, some quanttes are checked ncludng the drag coeffcent C d, lft coeffcent C l and the pressure dfference P between the front and end pont of the cylnder. In case of the flow around a crcular cylnder, P = p(.15,.2) p(.25,.2), lkewse n case of a flow around a square, P = p(x,y +D/2) p(x +D,y + D/2). Here (x,y ) s the coordnate of left-bottom corner of the square. As for the second aspect, we consder the smoothness of the flow at outflow. In order to reach these ams, we desgn the experments n two seres. One s fxng the grd sze and settng dfferent rato between the channel length and wdth. The other s fxng the channel length/wdth rato and changng the grd sze. Fnally we want to stress that the numercal results are not only affected by the outflow condtons. Proper grd sze, ntal values and Drchlet boundary treatments wll help us arrve at a relatvely far evaluaton of the outflow schemes. Quanttes around cylnder The drag and lft coeffcents are defned by C d = 2F D Ū 2 D, C l = 2F l Ū 2 D. (6.36) Here F b = (F D,F l ) s called boundary force whch conssts of the drag and lft force, D s the dameter of the cylnder and Ū s the reference velocty. In the lterature we can fnd two methods to evaluate the boundary force along the boundary curve Γ of the cylnder F b = [ pi + 2νS[u]] ndσ. (6.37) Γ Our choce s to calculate the force by ntegraton along the boundary curve. The components of the stress are ether calculated usng a dscrete velocty

145 6.5.1 Flow around a crcular cylnder 131 gradent([29, 3]) or obtaned drectly from the lattce Boltzmann method ([37, 21]) usng the nonequlbrum parts of the dstrbuton functon (see also [9]). Instead of the ntegraton method, another method s based on the momentum exchange orgnally proposed by Ladd [48] to compute the flud force on a sphere suspended n the flow. Ths method s very easy to mplement. The core of ths method s to ntroduce the momentum transfer ψ (n,j) at the boundary node x j (close to the obstacle) on the lnk c at tme level t n, whch s defned by the dfference between the dstrbutons reachng and leavng ths node, ψ (n,j) = c (f (n + 1,j) + f c (n,j)). (6.38) Then the boundary force actng on the cylnder can be approxmated by F b (n) = x j Γ c h V j ψ (n,j), (6.39) where V j s the collecton of all the ncomng drectons at node j, Γ c h s the set of all boundary nodes n the neghborhood of the cylnder. Many numercal experments[52, 6] have been performed for both methods. Each demonstrates ts advantage and weakness. However for the calculaton of the force on a restng crcular cylnder, both artcles agree that the momentum-exchange method works qute well. The artcle [7] nvestgates the momentum transfer n connecton wth the second order boundary condton BFL. In [9] the accuracy of the above momentum-exchange method s analyzed, and a more accurate correcton s proposed. Here we follow the suggeston to take use of the momentum-exchange method n the followng numercal calculatons. Remark 13 Note that the lattce Boltzmann method (2.2) can approxmate the solutons of Naver-Stokes equaton (2.1) wth 2nd order accurate velocty and 1st order accurate pressure generally. Therefore we can only expect the spatal dervatves of velocty to be 1st order accurate. The force evaluaton (6.37) n the lattce Boltzmann method can only be 1st order accurate too. Hence n order to obtan compettve values of C d, C l and P to the reference provded n [7, 69, 27], the grd has to be fne enough Flow around a crcular cylnder The frst two problems are two benchmark flows descrbed detaledly n [7, 69, 27]. These artcles also provde a number of reference values to verfy the followng results. The nflow condton at nlet s defned by U(,y) = 4U m y(y H)/H 2, V (,y) =, (6.4) where H =.41m s the heght of channel. The dameter of the cylnder s D =.1m. The blockage rato D/H s fxed for all the numercal experments. The flud vscosty s ν = 1 3 m 2 /s. The Reynolds number s defned by

146 132 CHAPTER 6. NEUMANN TYPE OUTFLOW BOUNDARY CONDITIONS Re = ŪD/ν wth a mean velocty Ū = 2U m/3. These two test flows are dstngushed by dfferent values of U m. ()U m =.3m/s produces a Reynolds number Re = 2 and leads to a steady flow wth a closed steady recrculaton regon consstng of two vortces behnd the cylnder. ()U m = 1.5m/s yelds a Reynolds number Re = 1 and renders an unsteady flow wth a perodc vortex street appearng behnd the cylnder. Steady flow around a crcular cylnder Table 6.1 and 6.2 show our calculatons of drag coeffcent(c d ), lft coeffcent(c l ), and the pressure dfference( P) for the steady flow. The top row provdes the reference upper and lower bounds. Frst let us observe the results (Table 6.1) n a short channel wth fxed L/H = 2 on three grds respectvely, t s found that fner grds render more accurate values n a relatve short channel and agree well wth the reference values. On the contrary table 6.2 shows the results on a fxed grd wth larger L/H {3,4}. Snce larger L/H values do not mprove the results, we can conclude that all the outflow schemes have approached the real flud movement to some extent. Table 6.1: Comparson results for steady flow around a crcular cylnder n a short channel. L/H grd C D C l P lower bound upper bound NBC ZNS DNT

147 6.5.1 Flow around a crcular cylnder 133 Table 6.2: Comparson results for steady flow around a crcular cylnder n channels of varyng length. L/H grd C D C l P lower bound upper bound NBC ZNS DNT Further, to compare the values of velocty and pressure carefully, we plot them along several cuts n the computatonal doman. The cut lnes are chosen to meet dfferent purposes (See the followng fgure 6.4). The dashed lnes are three vertcal cuts close to the front part of the cylnder (x =.15m), through the center (x =.2m) and at the back (x =.25m), whch are used to show the velocty and pressure behavor around the cylnder. The dashdot lnes are parallel to the sold walls and located next to the walls (y =.1m,.4m) at a dstance of.1m and through the center (y =.2m) of the cylnder respectvely. They are used to observe the velocty and pressure varyng along the x-coordnate. The sold lnes at x = 2H,2H.1m,2H.2m are used to nvestgate the velocty and pressure varyng versus the y-coordnate at outflow. The test grds have the resoluton (L/H = 2). Fgure 6.4: The poston of cutlnes n the computatonal doman. Frst let us see the results near the cylnder whch are gven n fgure 6.5. All the calculatons demonstrate a very reasonable and qute smlar velocty and

148 134 CHAPTER 6. NEUMANN TYPE OUTFLOW BOUNDARY CONDITIONS pressure. Ths agan verfes that all the outflow schemes can lead to the very close values for C d, C l and P. Next observe fgure 6.6 whch dsplays the results near the vertcal lne x = 2H (the sold lnes n fgure 6.4), where the outflow condtons are appled. From the plots of the horzontal velocty component (the frst row n fgure 6.6), we can see that all methods yeld very close values. Near the symmetry axs of the channel, the results of NBC show a lttle devaton, whereas the results of ZNS and DNT concde wth each other. Besdes, the plots of the vertcal velocty component demonstrate that ZNS and DNT produce a boundary layer near the wall and NBC yelds a smooth functon wth values close to zero. In addton, the plots of pressure dsplay agan that ZNS and DNT behave smlarly. Especally ZNS llustrates obvous boundary layers. Now let us look at the plots along the dashdot cutlnes (see fgure 6.7). All the results agree wth each other very well for the ponts a few grd szes away from the rght end at x = 2H. Moreover, before and behnd the cylnder the curves are smooth. However, boundary layers appear near the outflow and they are stronger at ponts close to the wall. The reason may be that the boundary condtons at the corners are ncompatble. Unsteady flow around a crcular cylnder The Reynolds number for ths unsteady flow s Re = 1. The numercal tests are carred out on three grds (L/H = 2), (L/H = 3), 82 41(L/H = 5). Fgure 6.1 plots the quanttes C d, C l and P for 1 tme steps after the ntal transent behavor. All the results show that the values are gettng closer to the reference value as the channel becomes longer, and ZNS and DNT produces much better results than NBC on all grds. Comparng the sze of these quanttes carefully, we fnd that ZNS and DNT yeld C d and P for the short channel (L/H = 2) closer to the reference value than the ones gven by NBC on a longer channel (L/H = 3). A smlar comparson also holds for the channel L/H = 3 wth ZNS or DNT and L/H = 5 wth NBC. Comparatvely, DNT and ZNS nfluences the nner flow most weakly, so that the short channel(l/h = 3) already produces nearly the same values as the one wth large L/H = 5. Another test of the effect of the outflow condton s to nvestgate the vortex street developed behnd the cylnder. To check how ths structure s nfluenced by the outflow condton, we compare velocty and pressure results along the central lne at the same tme level but for varous choces of the channel length. The results are shown n the nterval reflectng the shortest channel (see Fg. 6.11, Fg and Fg for the output of the varous methods). From Fg we see that all quanttes behave dfferently behnd the cylnder wth respect to dfferent L/H ratos. It demonstrates that the NBC outflow condton (6.13) has a comparably strong mpact on the nner flow. On the contrary, ZNS and DNT yeld more smlar varatons of these quanttes. Only a slght phase dfference occurs.

149 6.5.1 Flow around a crcular cylnder x Fgure 6.5: Results for the steady flow. The rows represent the horzontal velocty component, the vertcal component and the pressure respectvely, whle the columns represent the three cut lnes at x =.15m (frst column), x =.2m (mddle column),.25m (rght column). In each plot the methods are dstngushed by: ZNS ( ), NBC ( ) and DNT ( ).

150 136 CHAPTER 6. NEUMANN TYPE OUTFLOW BOUNDARY CONDITIONS Fgure 6.6: Results for the steady flow along three cut lnes x = 2H (left column), x = 2H.1m (mddle column), x = 2H.2m (rght column). The horzontal velocty (1st row), the vertcal velocty (2nd row) and the pressure (3rd row) are shown for three methods ZNS ( ), NBC ( ) and DNT ( ).

151 6.5.1 Flow around a crcular cylnder x x 1 3 x Fgure 6.7: The plots of velocty and pressure along three cut lnes parallel to the wall for the steady flow: y =.1m (left column), y =.2m (mddle column), y =.4m (rght column). The horzontal velocty (1st row), the vertcal velocty (2nd row) and the pressure (3rd row) are shown for three methods ZNS ( ), NBC ( ) and DNT ( ).

152 138 CHAPTER 6. NEUMANN TYPE OUTFLOW BOUNDARY CONDITIONS C d 2 C l P x x x 1 4 Fgure 6.8: The plots of C d (left), C l (mddle) and P(rght) for the unsteady flow wth NBC outflow condton. Sold lne stands for the result n the channel wth L/H = 5, dashed lne for the result wth L/H = 3 and dotted lne for the result wth L/H = 2. The dash-dotted lne s the upper and lower bound for the ampltude of these quanttes. 3.4 C d 1.5 C l P x x x 1 4 Fgure 6.9: The plots of C d (left), C l (mddle) and P(rght) for the unsteady flow wth ZNS outflow condton. Sold lne stands for the result n the channel wth L/H = 5 grds, dashed lne for the result wth L/H = 3 and dotted lne for the result wth L/H = 2. The dash-dotted lne s the upper and lower bound for the ampltude of these quanttes. 3.4 C d 1.5 C l P x x x 1 4 Fgure 6.1: The plots of C d (left), C l (mddle) and P(rght) for the unsteady flow wth DNT outflow condton. Sold lne stands for the result n the channel wth L/H = 5 grds, dashed lne for the result wth L/H = 3 and dotted lne for the result wth L/H = 2. The dash-dotted lne s the upper and lower bound for the ampltude of these quanttes.

153 6.5.1 Flow around a crcular cylnder x x x Fgure 6.11: The velocty components (left: horzontal, mddle: vertcal) and pressure (rght) along the central lne for the unsteady flow wth NBC outflow condton. +, and stand for the result n the channel wth L/H = 2, 3, 5 respectvely x x x Fgure 6.12: The velocty components (left: horzontal, mddle: vertcal) and pressure (rght) along the central lne for the unsteady flow wth ZNS outflow condton. +, and stand for the result n the channel wth L/H = 2, 3, 5 respectvely x x x Fgure 6.13: The velocty components (left: horzontal, mddle: vertcal) and pressure (rght) along the central lne for the unsteady flow wth DNT outflow condton. +, and stand for the result n the channel wth L/H = 2, 3, 5 respectvely.

154 14 CHAPTER 6. NEUMANN TYPE OUTFLOW BOUNDARY CONDITIONS Fgure 6.14: The pressure solnes for three outflow condtons (NBC (left), ZNS (mddle), DNT (rght)) n channels of dfferent length. Magnfed versons can be found n appendx A. Further, n order to observe the behavor of velocty and pressure n the whole doman, we show ther contour or solne plots n the computatonal doman. From Fg whch s the plots for the pressure, we see that the lattce Boltzmann method wth DNT and ZNS can result n a clear vortex street for the channels of dfferent length. However, the perodcty of the flow s not so clear wth NBC near the outflow, n partcular, ths phenomenon s more obvous n the case wth short channel. The same stuaton occurs also n the case of velocty. See the appendx A n whch all the fgures are collected Flow around a square cylnder Numercal computatons are carred out for both a steady and an unsteady flow around a square cylnder n the channel. The south and north sdes of the channel as well as four cylnder walls are consdered to be rgd wth zero velocty u =. The nflow s located at the west end of the channel. The velocty at nflow s descrbed by 4 U(t,y) = U m H2y(H y), V (t,y) =, y [,H]. (6.41) Outflow s at the east end. In order to use the smple bounce back rule at the cylnder wall and the upper and bottom boundng walls, we arrange the poston of the cylnder n such a way that the lattce nodes are located half a grd dstance from the boundary. U m s the maxmal value of the horzontal velocty at nflow. x and y are the coordnates of the lower left corner of the cylnder wth respect to the lower left corner of the channel. Ther values are dfferent for the steady and unsteady flows. Snce for ths knd of flow no expermental values are avalable, we just check whether the outflow condtons preserve the typcal features of the flud behavor. Frst, we calculate a steady flow wth a Reynolds number equal to 4. The square cylnder s stuated symmetrcally at the central lne of the channel.e.,

155 6.5.3 The nfluence of ntal values Fgure 6.15: Streamlne plot for the statonary flow around a square cylnder wth ZNS condton on a grd wth ponts. 2y +D = H. In addton, U m = 1 and x = y = 3D/2. We choose such a test confguraton that the flow s nsde a relatvely short channel (L/H = 2) to hghlght the effect mposed on the nner flow by outflow boundary condtons. The streamlne plots wth all three outflow schemes clearly show the stagnaton zone near the mdpont on the front face of the cylnder, where the flow equally dvdes nto two streams and leads to a par of counterrotatng vortces behnd the cylnder. The length of the recrculaton bubble s approxmately D from the rear face of the cylnder (see Fg.6.15). Moreover, all the lft coeffcent C l s always zero durng the calculatons, whch reflects the symmetry property of the flow wth respect to the central lne of the channel, and demonstrates agan that the symmetry property s exactly recovered by the lattce Boltzmann method. Secondly we smulate an unsteady flow. The confguraton for ths flow s Re = 1, x = D, y = 1.4D and U m = 1 as well as L/H = 5. The wdth of the channel s dscretzed wth 82 steps. Along the length of the channel there are 41 grd ponts. At a montor pont (x m,y m ) = (x +D+.3,y +D/2.2) n the wake of the obstacle, we observe the varaton of the pressure and velocty wth tme (see fgure 6.16). The results show that there s a domnant frequency for the vortex sheddng after the ntal transent. It mples that the vortex street behnd the cylnder s perodc. Fgure 6.17 llustrates the flow behavor n the whole doman. It s shown that ZNS and DNT produce sharper vortces than NBC. From the contour lnes of velocty and solnes of pressure (see fgures n appendx C), we see the perodcty of the flow from the results of ZNS and DNT, whch s however partly volated near the outflow by NBC The nfluence of ntal values When the flow s ntally at rest, the parabolc nflow profle s ncompatble wth the nteror velocty. Such an mpulsve start nevtably leads to an ntal

156 142 CHAPTER 6. NEUMANN TYPE OUTFLOW BOUNDARY CONDITIONS p t p t p t u u u t t t v v v t t t Fgure 6.16: Plots at a montor pont for the unsteady flow around a square cylnder wth NBC (left column), ZNS (mddle column) and DNT (rght column) condtons on a grd wth ponts. After some ntal phase, pressure (1st row), horzontal velocty (2nd row) and vertcal velocty (3rd row) show a perodc behavor.

157 6.5.4 The nfluence of Drchlet boundary 143 Fgure 6.17: The curl of velocty llustrates the vortex sheddng for the unsteady flow around a square cylnder wth NBC (1st row), ZNS (2nd row) and DNT (3nd row) condtons on a grd wth ponts. layer. Usually one expects that after a short transent perod, the ntal dsturbance becomes weak and neglgble whch s actually observed f we use BFL scheme n our experments. However, whle smulatng the flow around the crcular cylnder wth POP 1 beng used to represent the Drchlet condtons, we fnd a dfferent stuaton. Fgure 6.18 demonstrates the velocty and pressure after 2 tme steps, It shows that the rregular behavor does not dsappear but always exsts everywhere n the doman Ω even after a long run (see more fgures n Appendx B). On the other hand, when the nflow velocty s smoothly growng to ts fnal value at the left sde, the flow s drven smoothly. For example, let Ū(,y) = α(t)u(,y) and α(t) changes twce dfferentally from to 1 n a short tme perod [,t ], say t = 1/2. In ths case POP 1 also gves smooth results(fg.6.19) The nfluence of Drchlet boundary In ths part we brefly dscuss the nfluence of the Drchlet boundary approach around the cylnder. In the chapter 5 we have already observed that n many cases, POP 1 ntroduces a smaller error than BFL although both methods are of same order accuracy. We mght also expect the same phenomenon here. We use do-nothng condton (DNT) at outflow to smulate the flow around

158 144 CHAPTER 6. NEUMANN TYPE OUTFLOW BOUNDARY CONDITIONS 2 u p Fgure 6.18: Horzontal velocty plot (upper) and pressure plot (lower) versus x along the central lne usng a resoluton of ponts wth ZNS outflow condton and POP 1 scheme at 2th tme step after an mpulsve start. 2 u 2 p Fgure 6.19: Horzontal velocty plot (upper) and pressure plot (lower) versus x along the central lne usng a resoluton of ponts wth ZNS outflow condton and POP 1 scheme at 2th tme step wth a gradual start.

159 6.6. SUMMARY 145 P P x x 1 4 Fgure 6.2: Plots of P on two grds wth (left) and (rght) ponts around a crcular cylnder usng DNT outflow condton and BFL (dashed lne) and POP 1 (sold lne). the crcular cylnder whch has a curved boundary nterface. In order to stress the nfluence of the Drchlet condton, we start the nflow smoothly. The numercal results for ths confguraton does not dsplay sgnfcant dfference between BFL and POP 1. Only from the values of P on two grds and (see Fg.6.2), we see BFL s slghtly better than POP Summary 1. Three frequently used Neumann type outflow condtons are mplemented n the lattce Boltzmann context. The proposed treatment for each one s of frst order accuracy. 2. Numercal tests show that all outflow treatments render convncng results for the steady flows, provded the grd s fne enough. In the unsteady cases, ZNS and DNT have a better performance than NBC. It s noted that ZNS and DNT have only a slght effect on the nner flow and produce a clear vortex street behnd the obstacle. On the contrary NBC obvously deforms the nner flow and destroys the vortex close to the outflow. 3. The mxed nfluence from ntal condton, Drchlet boundary condton and outflow condton s numercally nvestgated. It turns out that POP 1 needs a smooth start. In ths case, POP 1 and BFL have a smlar performance.

160 Chapter 7 Conclusons We have shown that the asymptotc analyss can be successfully used for the consstency analyss of lattce Boltzmann schemes ncludng boundary condtons. The analyss gves rse to analytc detals about the behavor of the numercal soluton. In partcular, the lattce Boltzmann moments are dentfed as approxmatons to the Naver-Stokes felds. In addton, the convergence of the lattce Boltzmann method n chapter 2 wth perodc boundary or the bounce back rule at half lnks s rgorously proved. Based on the results of our analyss, we have constructed a new class of local boundary condtons (POP θ ) mplementng the velocty Drchlet condton. The numercal tests ndcate that t has some advantages over exstng methods n some aspects such as localty, stablty and sze of error. Three Neumann type outflow condtons whch are frequently used n fnte dfference and fnte element smulatons are also constructed for the lattce Boltzmann method n ths work. For each one we have proposed a frst order accurate treatment. Usng statonary and tme-dependent channel flows, the features of these outflow schemes are nvestgated. For the statonary flow all three outflow schemes produce convncng results, provded the grd s fne enough. In the nonstatonary case, the zero Neumann scheme destroys the vortex close to the outflow and obvously deforms the nner flow. The zero shear stress scheme and the do-nothng scheme behave better and render only a slght effect on the nner flow. Compared wth the results n [7, 69, 21], our results are defntely compettve. All the results about Drchlet boundary and outflow boundary condtons hold for both the lnear and nonlnear cases. 146

161 Appendx A Fgures for the smoothly started flows around a crcular cylnder The followng sectons dsplay the numercal pressure solnes and velocty contourlnes for an unsteady flow around a crcular cylnder (Re = 1), carred out wth varous outflow condtons. The detaled descrpton s found n secton 6.5 The Drchlet boundary condtons at the other parts of the boundary are realzed wth POP 1. The ntalzaton s done wth a smooth start from rest, and the nflow s contnuously amplfed untl saturaton. The grd sze n all smulatons s dentcal, only the aspect rato of the channel changes. 147

162 148 APPENDIX A. FIGURES FOR THE SMOOTHLY STARTED FLOWS AROUND A CIRCULAR CYLINDER A.1 Zero Neumann condtons (NBC) Fgure A.1: The pressure solnes for NBC outflow condton n channels of varyng length at the same tme pont Fgure A.2: The horzontal velocty contour lnes for NBC outflow condton n channels of varyng length at the same tme pont.

163 A.1. ZERO NEUMANN CONDITIONS (NBC) Fgure A.3: The vertcal velocty contour lnes for NBC outflow condton n channels of varyng length at the same tme pont. Fgure A.4: The vortex sheddng u for NBC outflow condton n channels of varyng length at the same tme pont.

164 15 APPENDIX A. FIGURES FOR THE SMOOTHLY STARTED FLOWS AROUND A CIRCULAR CYLINDER A.2 Zero normal stress condton (ZNS) Fgure A.5: The pressure solnes for ZNS outflow condton n channels of varyng length at the same tme pont Fgure A.6: The horzontal velocty contour lnes for ZNS outflow condton n channels of varyng length at the same tme pont.

165 A.2. ZERO NORMAL STRESS CONDITION (ZNS) Fgure A.7: The vertcal velocty contour lnes for ZNS outflow condton n channels of varyng length at the same tme pont. Fgure A.8: The vortex sheddng u for ZNS outflow condton n channels of varyng length at the same tme pont.

166 152 APPENDIX A. FIGURES FOR THE SMOOTHLY STARTED FLOWS AROUND A CIRCULAR CYLINDER A.3 Do nothng condton (DNT) Fgure A.9: The pressure solnes for DNT outflow condton n channels of varyng length at the same tme pont Fgure A.1: The horzontal velocty contour lnes for DNT outflow condton n channels of varyng length at the same tme pont.

167 A.3. DO NOTHING CONDITION (DNT) Fgure A.11: The vertcal velocty contour lnes for DNT outflow condton n channels of varyng length at the same tme pont. Fgure A.12: The vortex sheddng u for DNT outflow condton n channels of varyng length at the same tme pont.

168 Appendx B Fgures for the mpulsvely started flow around a crcular cylnder We show the fgures for the mpulsvely started flow around a crcular cylnder where POP 1 s employed. It s remarked that the zgzag phenomenon appears for all the outflow condtons addressed n chapter 6. Here only the results obtaned wth the ZNS scheme are exhbted 154

169 Fgure B.1: The contour lnes of the horzontal velocty (1st row) and vertcal velocty (2nd row) and pressure solnes (3rd row) for the flow after an mpulsve start of the nflow wth ZNS outflow condton and POP 1 Drchlet condton.

170 Appendx C Fgures for the unsteady flows around a square cylnder Here we show the fgures for the smoothly started flow around a square cylnder wth Re = 1, whch s descrbed n secton 6.5. The grd nodes are so arranged that the bounce back rule at half lnk can be realzed at rgd boundary walls and at the cylnder surface. All the results refer to smulaton tme t = 2.5 and a channel wth aspect rato L/H = 5. C.1 Zero Neumann condtons (NBC) Fgure C.1: The pressure solnes for NBC outflow condton Fgure C.2: The horzontal velocty contour lnes for NBC outflow condton. 156

171 C.2. ZERO NORMAL STRESS CONDITION (ZNS) Fgure C.3: The vertcal velocty contour lnes for NBC outflow condton. Fgure C.4: The vortex sheddng u for NBC outflow condton Fgure C.5: Streamlnes for NBC outflow condton. C.2 Zero normal stress condton (ZNS) Fgure C.6: The pressure solnes for ZNS outflow condton.

172 158 APPENDIX C. FIGURES FOR THE UNSTEADY FLOWS AROUND A SQUARE CYLINDER Fgure C.7: The horzontal velocty contour lnes for ZNS outflow condton Fgure C.8: The vertcal velocty contour lnes for ZNS outflow condton. Fgure C.9: The vortex sheddng u for ZNS outflow condton Fgure C.1: Streamlnes for ZNS outflow condton.

173 C.3. DO-NOTHING CONDITION (DNT) 159 C.3 Do-nothng condton (DNT) Fgure C.11: The pressure solnes for DNT outflow condton Fgure C.12: The horzontal velocty contour lnes for DNT outflow condton Fgure C.13: The vertcal velocty contour lnes for DNT outflow condton. Fgure C.14: The vortex sheddng u for DNT outflow condton.

174 16 APPENDIX C. FIGURES FOR THE UNSTEADY FLOWS AROUND A SQUARE CYLINDER Fgure C.15: Streamlnes for DNT outflow condton.

175 Appendx D Defnton of V and f and c s In ths part, several dscrete velocty sets V and the constant equlbrum dstrbutons f are gven. In addton, the sound speed c s, whch depends on the dscrete velocty set V, s also specfed. Moreover, for all the followng velocty set, c s = 1 3 D.1 D2Q9 On a two dmensonal square lattce, the lattce Boltzmann model D2Q9 has the dscrete velocty set V wth nne elements [67]: c = c 1 = ( 1 ) c 2 = ( 1 ) c 3 = ( 1 ) c 5 = ( 1 1 ) c 6 = ( 1 1 ) c 7 = ( ) 1 1 c 4 = ( 1 ) c 8 = ( 1 1 ). wth correspondng weghts 4/9, =, f = 1/9, = 1,2,3,4, 1/36, = 5,6,7,8 161

176 162 APPENDIX D. DEFINITION OF V AND F AND C S c 6 c 2 c 5 c 3 c c 1 c 8 c 7 c 4 Fgure D.1: The velocty set of the D2Q9 model. On a three dmensonal cubc lattce, there are three frequently used lattce Boltzmann model: ffteen-velocty (D3Q15), nneteen-velocty (D3Q19) and twenty-seven-velocty (D3Q27) models [67]. D.2 D3Q15 The D3Q15 model has the followng set of dscrete veloctes: c = c 1,2 = c 7,...,14 = ( ±1 ) ( ±1 ±1 ±1 ) ( ±1 ) c 3,4 = c 5,6 = ( ±1 ) wth correspondng weghts 2/9, =, f = 1/9, = 1,2,...,6, 1/72, = 7,...,14

177 D.3. D3Q c 9 c 11 c 5 c 7 c 2 c 3 c 13 c 4 c 14 c c 1 c 8 c 6 c 12 c 1 Fgure D.2: The velocty set of the D3Q15 model. D.3 D3Q19 The D3Q19 model has the followng set of dscrete veloctes [58, 33, 31]: c = c 1,2 = c 7,...,1 = ( ±1 ) ( ±1 ) ±1 ( ±1 ) c 3,4 = c 11,...,14 = ( ±1 ) ±1 ( ) c 5,6 = ±1 ( ±1 ) c 15,...,18 = ±1 wth correspondng weghts 1/3, =, f = 1/18, = 1,2,...,6, 1/36, = 7,...,18

178 164 APPENDIX D. DEFINITION OF V AND F AND C S c 14 c 2 c 9 c 18 c 5 c 15 c 3 c 11 c 7 c c 8 c c 4 12 c 6 c 16 c 1 c 17 c 1 c 13 Fgure D.3: The velocty set of the D3Q19 model. D.4 D3Q27 The D3Q27 model [58, 33, 31] has the followng set of dscrete veloctes: c = c 1,2 = c 7,...,1 = ( ±1 ) c 19,...,26 = ( ±1 ) ±1 ( ±1 ±1 ±1 ) ( ±1 ) c 3,4 = c 11,...,14 = ( ±1 ) ±1 ( ) c 5,6 = ±1 ( ±1 ) c 15,...,18 = ±1 wth correspondng weghts 8/27, =, f 2/27, = 1,2,...,6, = 1/54, = 7,...,18 1/216, = 19,...,26

179 D.4. D3Q c 25 c 22 c 8 c 2 c 14 c 2 c 12 c 1 c 18 c 24 c 4 c 16 c 5 c c 6 c 15 c 3 c 23 c 17 c 9 c 26 c 11 c 1 c 13 c 19 c 7 c 21 Fgure D.4: The velocty set of the D3Q27 model.

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