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