The lattice Boltzmann method: Fundamentals and acoustics

Transcription

1 Erlend Magnus Vggen The lattce Boltzmann method: Fundamentals and acoustcs Thess for the degree of Phlosophae Doctor Trondhem, February 2014 Norwegan Unversty of Scence and Technology Faculty of Informaton Technology, Mathematcs and Electrcal Engneerng Department of Electroncs and Telecommuncatons

2 NTNU Norwegan Unversty of Scence and Technology Thess for the degree of Phlosophae Doctor Faculty of Informaton Technology, Mathematcs and Electrcal Engneerng Department of Electroncs and Telecommuncatons Erlend Magnus Vggen ISBN (prnted ver.) ISBN (electronc ver.) ISSN Doctoral theses at NTNU, 2014:55 Prnted by NTNU-trykk

3 Abstract The lattce Boltzmann method has been wdely used as a solver for ncompressble flow, though t s not restrcted to ths applcaton. More generally, t can be used as a compressble Naver-Stokes solver, albet wth a restrcton that the Mach number s low. Whle that restrcton may seem strct, t does not hnder the applcaton of the method to the smulaton of sound waves, for whch the Mach numbers are generally very low. Even sound waves wth strong nonlnear effects can be captured well. Despte ths, the method has not been as wdely used for problems where acoustc phenomena are nvolved as t has been for ncompressble problems. The research presented ths thess goes nto three dfferent aspects of lattce Boltzmann acoustcs. Frstly, lnearsaton analyses are used to derve and compare the sound propagaton propertes of the lattce Boltzmann equaton and comparable flud models for both free and forced waves. The propagaton propertes of the fully dscrete lattce Boltzmann equaton are shown to converge at second order towards those of the dscrete-velocty Boltzmann equaton, whch tself predcts the same lowest-order absorpton but dfferent dsperson to the other flud models. Secondly, t s shown how multpole sound sources can be created mesoscopcally by addng a partcle source term to the Boltzmann equaton. Ths method s straghtforwardly extended to the lattce Boltzmann method by dscretsaton. The results of lattce Boltzmann smulatons of monopole, dpole, and quadrupole pont sources are shown to agree very well wth the combned predctons of ths multpole method and the lnearsaton analyss. The excepton to ths agreement s the mmedate vcnty of the pont source, where the sngularty n the analytcal soluton cannot be reproduced numercally. Thrdly, an extended lattce Boltzmann model s descrbed. Ths model alters the equlbrum dstrbuton to reproduce varable equatons of state whle remanng smple to mplement and effcent to run. To compensate for an unphyscal bulk vscosty, the extended model contans a bulk vscosty correcton term. It s shown that all equlbrum dstrbutons that allow varable equatons of state must be dentcal for the one-dmensonal D1Q3 velocty set. Usng such a D1Q3 velocty set and an sentropc equaton of state, both mechansms of nonlnear acoustcs are captured successfully n a smulaton, mprovng on prevous sothermal smulatons where only one mechansm could be captured. In addton, the effect of molecular relaxaton on sound propagaton s smulated usng a model equaton of state. Though the partcular mplementaton used s not completely stable, the results agree well wth theory.

4 Preface There are a few thngs that I would lke to tell you about ths thess before you dve n. Ths thess s splt nto two parts. Part I covers the underlyng theory: Flud mechancs, acoustcs, the knetc theory of gases, and fnally the lattce Boltzmann method tself. Part II bulds drectly on ths background, and covers the research that was done n the course of my ph.d. project. Throughout ths thess you wll occasonally see small notes n the margn. Whenever new and mportant terms are ntroduced, these margn notes gve a short defnton. Occasonally, these notes may be repeated for varous reasons: Readers may have skpped past a prevous defnton n an earler chapter, a dfferent formulaton mght make more sense n lght of the surroundng text, or I may smply have consdered a concept crtcal enough to repeat. At conferences I have surprsngly often met other students who tell me that they have used my Master s thess to learn the lattce Boltzmann method. Ths has been tremendously nsprng, and has led me to take extra care to make Part I of the thess thorough (though hopefully not off-puttngly thorough) and readable. As ntroductons to the lattce Boltzmann method that are easly readable, thorough, and freely avalable are thn on the ground, one of my goals has been to make Part I just such an ntroducton. I hope that I have succeeded n ths goal, though ths s of course up to you to decde. Whle my Master s thess wll stll offer a qucker and smpler ntroducton to the lattce Boltzmann method, Part I of ths thess puts more emphass on the physcal background of the method, whch s n my opnon essental to truly understand t. Ths thess s submtted to the Norwegan Unversty of Scence and Technology (NTNU) n partal fulflment of the requrements for the degree of Phlosophae Doctor (ph.d.). The ph.d. project ran for four years, wth one year spent on teachng dutes. The work was carred out at the Acoustcs Research Center at the Department of Electroncs and Telecommuncatons, wth Professor Ulf Krstansen as supervsor. I hope you fnd t nterestng. v

5 Acknowledgements Throughout these four years as a ph.d. student, there have been many people who have aded me n varous ways. For ths I am deeply grateful. I would have had a much harder tme early on f Jors Verschaeve had not spent many an hour helpng me to understand the lattce Boltzmann method. Smlarly, my later work would have been much more dffcult f not for the occasonal comment from Paul Dellar. Hs deep nsght has been nvaluable to me and my work. There are many other researchers wth whom I have had nterestng and useful dscussons on a wde varety of scentfc topcs. I would especally lke to thank Tmm Krüger, Jonas Lätt, Tm Res, Tor Ytrehus, Davd Packwood, Manuel Hasert, Martn Schlaffer, and Gullaume Dutlleux. Also, I would lke to thank all the other frendly people whose company I have enjoyed at conferences. Thanks to them, the non-scentfc aspects of these conferences have never been dull. I am grateful for the company and frendshp of some of my fellow travellers through the ph.d. program at NTNU. Whle they are too many for me to start lstng names, I would especally lke to thank Anders Løvstad for beng excellent company n the offce throughout much of our tme as ph.d. students, though I do not mss the roar of hs computer. Fnally, I am very grateful to Ulf Krstansen for beng as avalable and affable an advsor as any ph.d. student could hope to have. To all of you: Thank you. Erlend Magnus Vggen Trondhem, September 2013 v

6 Contents I Background 1 1 Introducton Mcroscopc, mesoscopc, and macroscopc scales Connecton A smple mcroscopc model: The FHP lattce gas Ths thess Goals Thess structure Related publshed artcles Mathematcal notaton and lst of symbols Fundamental theory Index notaton Examples Flud mechancs The Euler model The Naver-Stokes-Fourer model Acoustcs Ideal wave equaton Vscous and thermovscous wave equaton Molecular relaxaton processes Acoustc multpoles and aeroacoustcs Nonlnear acoustcs The knetc theory of gases The dstrbuton functon and ts moments Pressure and heat Equlbrum The Maxwell-Boltzmann dstrbuton Pecular velocty moments at equlbrum The Boltzmann equaton The collson operator Macroscopc conservaton equatons Mass conservaton Momentum conservaton v

7 Contents v Energy conservaton Equlbrum: The Euler model The Chapman-Enskog expanson Fndng the dstrbuton functon perturbaton Fndng the moment perturbatons The Naver-Stokes-Fourer model Hgher-order Boltzmann equaton approxmatons Boltzmann s H-theorem The lattce Boltzmann method The dscrete-velocty Boltzmann equaton Moments and constrants Moment-based Chapman-Enskog expanson Velocty sets Dgresson: Lnearsed DVBE The lattce Boltzmann equaton Frst order dscretsaton Second order dscretsaton Summary: The lattce Boltzmann method Lattce Boltzmann unts Alternatve collson operators Multple relaxaton tme Regularsed Entropc Smple boundary condtons II Research Acoustc lnearsaton analyss Isothermal Naver-Stokes-Fourer model Absorpton and dsperson Magntude ratos and phase dfferences Dscrete-velocty Boltzmann equaton Lnearsaton process Propertes of forced and free waves Comparson wth relaxaton processes Comparson to other models Ansotropy n two dmensons Lattce Boltzmann equaton Lnearsaton process Results Example: Exact wave ntalsaton Summary and dscusson

8 v Contents 6 Mesoscopc acoustc sources Source terms for the Boltzmann equaton Macroscopc conservaton equatons Lnear wave equaton Source terms for the lattce Boltzmann equaton Frst order dscretsaton Second order dscretsaton Multpole bass Numercal experments Plane waves Multpoles n two dmensons Summary and dscusson Varable equaton of state The extended model Moments and constrants Macroscopc equatons Bulk vscosty correcton General equlbrum requrements Lnearsaton analyss Isentropc equaton of state and nonlnear acoustcs The sentropc lattce Boltzmann model D2Q9 stablty: Comparson to another model Physcal nonlnear acoustcs case Nonlnear acoustcs smulaton Molecular relaxaton Verfcaton by smulaton Summary and dscusson Dscusson and concluson 209 Bblography 213

9 Part I Background 1

10 1 Introducton Many scentfc artcles on the lattce Boltzmann method begn wth a farly dense paragraph on the method and ts capabltes, whch typcally goes somethng lke ths: The lattce Boltzmann (LB) method s a recent advance n computatonal flud dynamcs (CFD). Whle tradtonal CFD methods drectly dscretse and solve the macroscopc equatons of flud mechancs, the LB method solves a dscrete knetc equaton whch reproduces the flud mechancs equatons n the macroscopc lmt. It s straghtforward to mplement and parallelse effcently, whle beng versatle enough to smulate multphase flows, multcomponent flows, flows of complex fluds, flows n complex geometres such as porous meda, thermal flows, and turbulent flows. Cellular automaton A dscrete model wth very smple rules that can typcally result n very complex behavour Lattce gas A cellular automaton for smulatng gases, based on partcles movng around on a lattce, ther collsons conservng mass and momentum A paragraph ths succnct can of course not gve a full pcture of the method. However, t does manage to pant much of ths pcture n broad strokes. Let us now pant some of the fner strokes by expandng on the three sentences of ths paragraph. As the frst sentence states, the LB method has not been around for as long as most other CFD methods. Hstorcally, t grew out of the feld of cellular automata, and specfcally lattce gases, whch we wll look at brefly n secton The frst lattce gas was descrbed n 1973 [1], though t was not untl 1986 that a lattce gas that could be used to correctly smulate flud flow was proposed [2]. An artcle was publshed soon after n 1988 on a modfcaton to lattce gases n order to avod some of ther problems when smulatng flud flow [3]. Ths artcle can be consdered the frst artcle on the lattce Boltzmann method. The second sentence of the paragraph mples that the LB method solves the equatons of flud mechancs ndrectly by solvng somethng else, somethng smpler. Whle ths may seem too good to be true, there s ndeed a good physcal reason why t works. The lattce Boltzmann method s a dscretsaton of the Boltzmann equaton, an equaton whch descrbes gases at a more detaled level than the equatons of flud mechancs, whle stll havng a smpler form. If we smooth away these detals n the rght way, we end up wth the equatons famlar from flud mech- 2

11 1.1 Mcroscopc, mesoscopc, and macroscopc scales 3 ancs. In secton 1.1 we wll look at the relaton between the dfferent descrptons of a gas. As for the thrd sentence, t s a truth wth modfcatons. It s ndeed true that the basc LB method s both smple and farly powerful. However, t s almost a trusm that f you take somethng smple and bolt on somethng complex, the end product becomes complcated. LB models that are more accurate or that can capture more complex physcs are ndeed more dffcult to understand and to mplement. Even so, capturng complex physcs usng LB may stll be smpler than when usng more tradtonal CFD methods. Fnally, whle parallelsng LB for smple cases s not dffcult, such smple parallelsatons can become qute neffcent for more complex cases; general and effcent parallelsaton of LB s dffcult [4]. The rest of ths ntroductory chapter s splt nto two sectons. The frst secton ntroduces the dfferent scales, orlevels of detal, at whch we can descrbe matter, and the connecton between these scales for gases. A very smple lattce gas model for trackng the molecules of a gas s descrbed, and the connecton to the less detaled but more useful lattce Boltzmann method s hnted at. The dscusson n ths secton s kept at as smple a level as possble. The second secton descrbes ths thess, gong nto ts ams, ts structure, other publcatons by the author durng the same research project, and ts mathematcal notaton. Parallelsng Separatng a computer program nto peces that can run n parallel, whch may ncrease ts speed as the peces can be run smultaneously on several dfferent processors 1.1 Mcroscopc, mesoscopc, and macroscopc scales Consder a glass of water. To the human senses, the water seems contnuous and unform. If we were able to look at the flud at sze scales on the order of nanometers, though, we would see that the flud s nether contnuous nor unform; t conssts of nvddual molecules whch are constantly shftng around. We would be able to see that some areas of ths cloud of molecules are denser than others, as sketched n Fgure 1.1. However, snce our eyesght s far from good enough to perceve ths level of detal, ths non-unformty s evened out due to the law of large numbers, and we perceve the water as unform. We usually descrbe the propertes and movement of the water usng tangble terms lke densty and, f the glass of water s strred, flud velocty. Mass densty, ρ Ths coarse level of detal whch we can perceve drectly, we call the macroscopc scale. The equatons of flud mechancs, whch descrbe how these macroscopc varables evolve, are correspondngly equatons for the macroscopc scale. In the mcroscopc scale, we look at a much more complete (and probably overwhelmng!) level of detal. Instead of regardng the flud as a smooth contnuum, we look at all the dfferent molecules of whch t conssts. Each molecule, ndexed by, has a mass m, a poston x =(x, y, z ), Mass per physcal volume n kg/m 3 Flud velocty, u The local velocty of the flud n m/s Macroscopc scale Where we use tangble terms to descrbe physcal systems Mcroscopc scale Where we descrbe a physcal system through all ts ndvdual components

12 4 Chapter 1 Introducton Fgure 1.1: Molecules n a glass of water. When zoomng n far enough, we see that the flud s not qute unform. a velocty ξ, n addton to addtonal varables to descrbe ts nternal state: typcally ts rotaton and ts vbraton. The moton of each of these molecules can be descrbed by Newton s laws of moton, though some quantum mechancs must also be appled f the nternal confguraton s consdered Connecton Expectaton value, E(...) For a random varable, the expectaton value s what we would fnd f we could measure t an nfnte amount of tmes and average the results. In that sense, t s an dealsed average. Momentum densty, ρu Momentum per physcal volume n kg/s m 2 Energy densty, ρe Knetc energy per physcal volume n J/m 3 Snce the mcroscopc and macroscopc pctures descrbe the same physcal system, t must be possble to lnk them somehow. Somehow, the wldly fluctuatng mcroscopc world must tell us somethng about the smooth and contnuous macroscopc world. To lnk them, we must consder the expectaton values of the mcroscopc system. Thus, the macroscopc mass densty ρ(x, t) at poston x and tme t s found by addng up the mass m of the partcles that we would expect to fnd n a tny volume V at x, such as the rghtmost volume n Fgure 1.1. Mathematcally, ρ(x, t) = lm V 0 E ( ) 1 V m. (1.1a) x V Smlarly, the momentum densty can be found from addng the momentum m ξ of each partcle n the volume, ( ) 1 ρu(x, t) = lm E V 0 V m ξ. (1.1b) x V Fnally, the energy densty can be found by addng the knetc energy 1 2 m ξ 2 of each partcle, ( ) 1 ρe(x, t) = lm E V 0 2V m ξ 2. (1.1c) x V If we know the densty and the momentum densty, then we can obvously drectly fnd the flud velocty as u = ρu/ρ.

13 1.1 Mcroscopc, mesoscopc, and macroscopc scales 5 If the flud s at rest wth u = 0, the partcles are stll n moton at the mcroscale. However, ther drectons of velocty are equally dstrbuted, so that the vector sum n (1.1b) s zero. The energy densty (1.1c) s not zero; t measures the ntensty of the nternal partcle moton n the gas. We shall later see n Chapter 3 that t s related to the temperature. There s also the mesoscopc scale, where the level of detal s somewhere between the extremely detaled mcroscopc scale and the tangble macroscopc scale. Instead of trackng every partcle, we track the dstrbuton of partcles. Whle ths s a farly abstract concept, we wll gve an example of ths at the end of secton and go on to descrbe the theory of gases at the mesoscopc scale, known as the knetc theory of gases, n Chapter 3. On the mcroscopc and mesoscopc scales, deal gases are by far the smplest type of flud to deal wth. In a gas whch s not too dense, the molecules are far enough apart that ther nteracton s approxmately always through one-on-one collsons. For a very dense gas, the molecules are much closer together and the assumpton that collsons are always one-on-one s no longer a suffcent approxmaton. Thngs are even more dffcult n lquds, where the molecules are held close to each other by ntermolecular attractng forces, meanng that there s a contnuous nteracton between the molecules. The feld of knetc theory s therefore most well-developed for dlute gases; the knetc theory of lquds s a much more dffcult topc [5]. On the macroscopc scale, though, the dfference between gases and lquds s much less clear, as the same equatons hold for both. The dfference s manfested manly by a dfference n materal parameters. For nstance, lquds tend to have a sgnfcantly hgher speed of sound than gases. For a gas whch s not too dense, then, the dfferent levels of descrpton are connected as shown n Fgure 1.2. The most detaled descrpton s on the mcroscopc scale, and ths can be approxmated va mesoscopc knetc theory to fnd the equatons of macroscopc flud mechancs.* We can see the general equatons of flud mechancs as splnterng nto two dfferent, ncompatble descrptons, dependng on whch approxmatons are taken. For engneerng flud mechancs where flud flows are the topc, the flud s almost always assumed to be ncompressble. Incompressblty means that the densty s constant, whch s ncompatble wth the compressble phenomenon of sound. For acoustcs, the flud s almost always assumed to be at rest or nearly so, wth sound waves dsturbng the flud only very slghtly. In ths thess, we wll only barely look nto the mcroscopc descrpton, and ncompressble flud mechancs wll be mentoned only n passng. We wll be lookng prmarly at knetc theory, general flud mechancs, *We wll look at the connecton between knetc theory and flud mechancs n detal n secton 3.8. Mesoscopc scale A statstcal descrpton n between the mcroscopc and macroscopc scale

14 6 Chapter 1 Introducton Newtonan mechancs (+ quantum mech.) Mcroscopc Level of approxmaton Knetc theory Flud mechancs Mesoscopc Macroscopc Engneerng flud mechancs Acoustcs Fgure 1.2: The herarchy of descrptons of a gas. Ths thess wll focus on the three hghlghted descrptons. Fgure 1.3: The sx possble partcle veloctes n the FHP lattce gas and acoustcs. Ths wll dfferentate ths thess somewhat from most of the lattce Boltzmann lterature, whch s focused on applcatons n ncompressble flud mechancs A smple mcroscopc model: The FHP lattce gas A lattce gas s a mcroscopc model whch attempts to smulate the behavour and nteracton of the ndvdual partcles n a gas n as smple a fashon as possble. In the followng, we wll go brefly through the FHP model from 1986 [2], named after ts nventors Frsch, Hasslacher, and Pomeau. Ths s the smplest possble lattce gas whch can reproduce the behavour of a flud. Rules In a lattce gas, a large number of partcles exst on a lattce, a regular grd of nodes. In the FHP model, ths lattce s hexagonal; lattce gases

15 1.1 Mcroscopc, mesoscopc, and macroscopc scales 7 on square lattces turned out to be unable to reproduce correct flud behavour. Each partcle has one out of sx possble veloctes shown n Fgure 1.3, whch pont from the partcle s current node to a neghbourng node. In each node, zero to sx partcles may be present smultaneously as long as they all have unque veloctes. Every tme that the clock tcks forwards by one tme step Δt, two rules are appled: Streamng. All partcles move from ther current node to the neghbourng node n the drecton of ther velocty vector. Collson. If two or three partcles meet head on n a node, ther outgong veloctes are changed from ther ncomng veloctes. The two-partcle case has two possble resolutons whch are chosen at random wth equal probablty. These rules are llustrated n Fgure 1.4. After the streamng step, partcles n each node may collde, and after the collson step, partcles are prepared to stream to neghbourng nodes. Later versons of the FHP model ntroduced partcles at rest (where ξ = 0) and addtonal collson types, but we wll not go nto these here. Both rules are n accordance wth fundamental mechancal laws. The streamng rule embodes Newton s frst law; the partcles veloctes are constant untl forces are appled to them n collsons. Wthn each node, these collsons obey the conservaton of mass m, momentum m ξ, and energy m ξ 2 /2. Macroscopc varables and statstcal nose Snce a lattce gas s a mcroscopc model, the mass and momentum n the nodes of the system wll always be fluctuatng, even when the system s at an equlbrum.* Whle ths statstcal nose s a desred property f such fluctuatons are the topc of study, they are hghly undesred when tryng to smulate nce and smooth flud flows. To use ths model to predct anythng on the macroscopc scale, we would need to approxmate the expectaton values n (1.1) somehow. Several methods may be used n order to do ths. One such method utlses the law of large numbers by averagng over several nearby nodes, whch can be compared wth expandng the small volume V n (1.1). Another s averagng the results over several tme steps. These two methods wll ndeed reduce the statstcal nose, at the cost of smoothng out the macroscopc soluton slghtly. In order to reduce ths smoothng, the smulaton resoluton could be ncreased, puttng more nodes nsde the *There are two trval exceptons to ths. One, f the lattce contans no partcles. Two, f the lattce s completely stuffed wth partcles, wth no empty spaces.

16 8 Chapter 1 Introducton (a) Streamng rule (b) Double collson rule; the resolutons are chosen randomly wth equal probablty (c) Trple collson rule Fgure 1.4: Rules of the FHP model. Incomng and outgong partcles,.e. partcles before and after the collson step, are represented wth arrows pontng nto and out of nodes, respectvely.

17 1.1 Mcroscopc, mesoscopc, and macroscopc scales 9 same physcal space. A thrd, qute demandng, method s to smultaneously smulate several smlar systems such that each system s ntalsed wth the same macroscopc state but dfferent mcroscopc states. The macroscopc varables can be estmated by averagng the results over ths ensemble of systems. Stll, the problem of statstcal nose can only ever be reduced n these ways, never entrely removed. Lattce gas smulatons of flud flow typcally end up wth a sgnfcant amount of nose n ther results [6, 7], lmtng ther usefulness. Towards the mesoscopc scale A clever soluton to the problem of statstcal nose was found ndependently by dfferent groups [3, 8]. Instead of trackng partcles that are ether there or not there, t s possble to track partcle dstrbutons; essentally the expectaton value of the number of partcles. If we hypothetcally had an ensemble of nfntely many smlar systems, the partcle dstrbuton can be seen as the average over the systems of the partcle number n each state. In another sense, t s the probablty of fndng a partcle n that state. For example, f the partcle dstrbuton s 0 for a certan poston, tme, and partcle velocty, there s never a partcle n that partcular state. If the partcle dstrbuton s 1, there s always a partcle n that state. Ths partcle dstrbuton functon s usually denoted as f (x, t). now ndexes the dfferent possble veloctes ξ (sx for the FHP lattce gas) nstead of dfferent partcles. Thus, f (x, t) can be seen as the probablty of fndng a partcle wth velocty ξ at poston x and tme t. Wehave now gone from the mcroscale lattce gas to a correspondng mesoscale descrpton. The lattce Boltzmann method was hstorcally derved n ths way, as a modfcaton to lattce gases usng the dstrbuton functon f.we could now proceed along these lnes to fully derve the lattce Boltzmann method,* but there s also another, more physcal, way to derve t from the knetc theory of gases. Ths thess wll follow the latter type of dervaton, as ths gves a very valuable nsght nto the underlyng physcal aspects and ther connecton to the numercal method. We wll therefore stop here for now, and leave ths dscusson as an example of a mesoscale descrpton of a gas. We wll pck up ths thread agan n Chapter 3, where we wll look at the knetc theory of gases. In Chapter 4, the lattce Boltzmann method wll be derved as a dscretsaton of the Boltzmann equaton, a cornerstone equaton of ths knetc theory. *Indeed, that was what ths author dd n hs Master s thess [9].

18 10 Chapter 1 Introducton 1.2 Ths thess Let us now take a break from the theory to consder the purpose and the structure of ths thess Goals There are two relevant sets of goals here. The frst set s the goals of the ph.d. research project that underles ths thess, and the second set s the goals of the thess tself. Project goals The ttle of the research project was Acoustc propertes and methods of lattce Boltzmann. The project had two goals. Frst, to mprove the current understandng of the propagaton of sound waves n LB smulatons. Second, to fnd new and develop exstng methods for LB acoustcs. Thess goals The prmary goal of ths thess s of course to present the research that was done n the course of the research project. By explotng the addtonal space afforded by a thess, the dea s to present the research n a more comprehensve and comprehensble fashon than scentfc artcles allow for. The hope s to make ths research as smple as possble to understand and to develop further. The secondary goal s to attempt to gve a clearly wrtten, reasonably thorough, and freely avalable ntroducton to the lattce Boltzmann method. A far amount of ntroducton would anyhow be necessary n ths thess as background for the research. Extendng that ntroducton makes t more valuable for other people who are tryng to learn the fundamentals of the lattce Boltzmann method Thess structure Ths thess s dvded nto two man parts. Part I contans background materal for Part II, ncludng an ntroducton to the lattce Boltzmann method and ts underlyng physcs. Part II descrbes the results of the research that was carred out n the course of the project. The structure of the thess and the man nformaton flow between the dfferent chapters and parts s shown n Fgure 1.5. The thess contans the followng eght chapters: Ch. 1 Part I: Background Introducton: Ths chapter.

19 1.2 Ths thess 11 Part I Background Part II Research Chapter 1 Introducton Chapter 5 Acoustc lnearsaton analyss Chapter 2 Fundamental theory Chapter 6 Mesoscopc acoustc sources Chapter 3 The knetc theory of gases Chapter 7 Varable equaton of state Chapter 4 The lattce Boltzmann method Chapter 8 Dscusson and concluson Fgure 1.5: The structure of the thess. The arrows represent the man flow of nformaton. Ch. 2 Ch. 3 Ch. 4 Ch. 5 Ch. 6 Ch. 7 Ch. 8 Fundamental theory: Introducton to ndex notaton, flud mechancs, and acoustcs The knetc theory of gases: The mesoscopc descrpton of a gas and ts connecton to flud mechancs The lattce Boltzmann method: Dervaton of LB from knetc theory and descrpton of ts bascs Part II: Research Acoustc lnearsaton analyss: Dervaton and comparson of the sound propagaton propertes of varous flud models, ncludng LB Mesoscopc acoustc sources: Dervaton of mesoscopc sound sources that can be mplemented n LB Varable equaton of state: An extended LB model that allows changng the flud s equaton of state and bulk vscosty s appled to smulate nonlnear acoustcs and molecular relaxaton Dscusson and concluson: A summary of the research and some deas on how t may be contnued

20 12 Chapter 1 Introducton Related publshed artcles In the course of ths ph.d. project, fve scentfc artcles have been publshed. All were wrtten solely by ths author, and all are freely and legally avalable onlne at ths tme of wrtng. In chronologcal order, they are: The lattce Boltzmann method n acoustcs (2010) [10] Proceedngs of the 33rd Scandnavan symposum on physcal acoustcs Abstract: The lattce Boltzmann method, a method based n knetc theory and used for smulatng flud behavour, s presented wth partcular regard to usage n acoustcs. A pont source method of generatng acoustc waves n the computatonal doman s presented, and smple smulaton results wth ths method are analysed. The smulated waves transent wavefronts n one dmenson are shown to agree wth analytcal solutons from acoustc theory. The phase velocty and absorpton coeffcents of the waves and ther devatons from theory are analysed. Fnally, the physcal tme and space steps relatng smulaton unts wth physcal unts are dscussed and shown to lmt acoustc usage of the method to small scales n tme and space. Comments: Ths artcle uses the LB pont source presented n [9] to smulate sound waves propagatng n 1D, 2D, and 3D. Ths pont source s nferor to the mesoscopc source method presented n Chapter 6, for reasons that wll be explaned n that chapter. Nevertheless, the artcle was a frst step to some results that are gven n Chapters 5 and 6. The lmt descrbed n the abstract s fnal sentence has turned out to be less bad than assumed; n Chapter 6 we wll see that usng another partcle collson model softens the constrants causng ths lmt. Vscously damped acoustc waves wth the lattce Boltzmann [11] method (2011) Phlosophcal Transactons of the Royal Socety A 369, pp Abstract: Acoustc wave propagaton n lattce Boltzmann Bhatnagar- Gross-Krook smulatons may be analysed usng a lnearzaton method. Ths method has been used n the past to study the propagaton of waves that are vscously damped n tme, and s here extended to also study waves that are vscously damped n space. Its valdty s verfed aganst smulatons, and the results are compared wth theoretcal expressons. It s found n the nfnte resoluton lmt k 0 that the absorpton coeffcents and phase dfferences between densty and velocty waves match theoretcal expressons for small values of ωτ ν, the characterstc number for vscous acoustc dampng. However, the phase veloctes and ampltude ratos between the waves ncrease ncorrectly wth (ωτ ν ) 2, and agree wth theory only n the nvscd lmt k 0, ωτ ν 0. The

21 1.2 Ths thess 13 actual behavour of smulated plane waves n the nfnte resoluton lmt s quantfed. Comments: Ths artcle partally forms the bass of Chapter 5. The numercal results n ths artcle are mproved and gven analytcally n that chapter. Sound propagaton propertes of the dscrete-velocty [12] Boltzmann equaton (2013) Communcatons n Computatonal Physcs 13, pp Abstract: As the numercal resoluton s ncreased and the dscretsaton error decreases, the lattce Boltzmann method tends towards the dscretevelocty Boltzmann equaton (DVBE). An expresson for the propagaton propertes of plane sound waves s found for ths equaton. Ths expresson s compared to smlar ones from the Naver-Stokes and Burnett models, and s found to be closest to the latter. The ansotropy of sound propagaton wth the DVBE s examned usng a two-dmensonal velocty set. It s found that both the ansotropy and the devaton between the models s neglgble f the Knudsen number s less than 1 by at least an order of magntude. Comments: Ths artcle also partally forms the bass of Chapter 5. Acoustc multpole sources for the lattce Boltzmann [13] method (2013) Physcal Revew E 87, p Abstract: By ncludng an oscllatng partcle source term, acoustc multpole sources can be mplemented n the lattce Boltzmann method. The effect of ths source term on the macroscopc conservaton equatons s found usng a Chapman-Enskog expanson. In a lattce wth q partcle veloctes, the source term can be decomposed nto q orthogonal multpoles. More complex sources may be formed by superposng these basc multpoles. Analytcal solutons found from the macroscopc equatons and an analytcal lattce Boltzmann wavenumber are compared wth nvscd multpole smulatons, fndng very good agreement except close to sngulartes n the analytcal solutons. Unlke the BGK operator, the regularzed collson operator s proven capable of accurately smulatng two-dmensonal acoustc generaton and propagaton at zero vscosty. Comments: Ths artcle partally forms the bass of Chapter 6. Acoustc multpole sources from the Boltzmann equaton (2013) [14] Proceedngs of the 36th Scandnavan symposum on physcal acoustcs Abstract: By addng a partcle source term n the Boltzmann equaton

22 14 Chapter 1 Introducton of knetc theory, t s possble to represent partcles appearng and dsappearng throughout the flud wth a specfed dstrbuton of partcle veloctes. By dervng the wave equaton from ths modfed Boltzmann equaton va the conservaton equatons of flud mechancs, multpole source terms n the wave equaton are found. These multpole source terms are gven by the partcle source term n the Boltzmann equaton. To the Euler level n the momentum equaton, a monopole and a dpole source term appear n the wave equaton. To the Naver-Stokes level, a quadrupole term wth neglgble magntude also appears. Comments: Ths s a companon artcle to the prevous artcle, also partally formng the bass of Chapter 6. A preprnt was publshed at arxv.org [15], wth dentcal content but dfferent formattng. Acoustc equatons of state for smple lattce Boltzmann velocty sets Submtted to Communcatons n Computatonal Physcs Abstract: The most wdely used lattce Boltzmann (LB) method uses an sothermal equaton of state. Ths s not suffcent to smulate a number of acoustc phenomena where the equaton of state cannot be approxmated as lnear and constant. It s possble to mplement varable equatons of state by alterng the LB equlbrum dstrbuton. For smple velocty sets wth velocty components ξ α { 1, 0, 1} for all, these equlbra necessarly cause error terms n the momentum equaton. These error terms are shown to be ether correctable or neglgble at the cost of weakenng the compressblty further. For the D1Q3 velocty set the equlbrum dstrbuton s shown to be unque. Its sound propagaton propertes are found for both forced and free waves wth applcablty beyond D1Q3. Fnally, the equlbrum dstrbuton s appled to a nonlnear acoustcs smulaton where both mechansms of nonlnearty are smulated wth good results, provng that the compressblty of the method s stll suffcently strong even for nonlnear acoustcs. Comments: Ths artcle covers several of the topcs of Chapter 7, though n a more concse manner that allows for a clearer exposton Mathematcal notaton and lst of symbols In ths thess, some accents, subscrpts and superscrpts of symbols used n the matematcal notaton carry a certan meanng. These are shown n Table 1.1. Note that no notatonal dfference s used between vectors, matrces, or hgher-order tensors, as all are tensors. Ths thess makes extensve use of ndex notaton, a style of notaton commonly used n flud mechancs as an alternatve to vector and tensor notaton. Ths style of notaton wll be explaned n secton 2.1.

23 1.2 Ths thess 15 Table 1.1: Accents, subscrpts and superscpts, appled to the example symbol λ Mark Descrpton λ Vector or tensor λ α Arbtrary element of a spatal frst-order tensor (vector) λ λ αβ Arbtrary element of a spatal second-order tensor (matrx) λ λ αβγ Arbtrary element of a spatal thrd-order tensor λ λ 0 Rest state value or deal value of λ λ Devaton from rest state; λ = λ 0 + λ λ Ampltude of the devaton λ λ (0) Equlbrum value λ (n) nth order perturbaton around equlbrum value λ neq Nonequlbrum value λ λ (0) λ ph Value of λ n physcal unts λ la Value of λ n smplfed lattce unts λ Nondmensonalsed value ˆλ Complex phasor or related quantty Throughout a mathematcal text of ths sze, a large number of mathematcal symbols must necessarly be defned. In order to stay as consstent as possble wth the lterature, some symbols have receved several dfferent meanngs, though an effort s made n the text to avod confuson. For future reference, the symbols that are used repeatedly throughout ths thess are shown n Table 1.2.

24 16 Chapter 1 Introducton Table 1.2: Regularly used symbols throughout ths thess Symbol Descrpton A Multpole transformaton matrx b Bulk vscosty correcton constant B Partcle source bass B Bulk vscosty correcton term c Speed of sound c 0 Ideal, small-sgnal speed of sound c ξ Velocty set constant (equals c 0 outsde Chapter 7) c Isothermal speed of sound (equals c 0 outsde Chapter 7) c V Heat capacty at constant volume c p Heat capacty at constant pressure d Number of spatal dmensons d Number of nner degrees of freedom d tot Total number of degrees of freedom e Internal energy E Total energy E(...) Expectaton value f Dstrbuton functon f Dscrete-velocty dstrbuton functon f Modfed dscrete-velocty dstrbuton functon F (0) Rest state dstrbuton functon F Body force G Green s functon h Enthalpy H Exctaton fracton of an nner degree of freedom H Boltzmann entropy functon j Partcle source term j Dscrete-velocty partcle source term J Moment of partcle source term k B Boltzmann constant Kn Knudsen number m Molecular mass m Moment vector M Moment transformaton matrx Ma Mach number p Pressure Pr Prandtl number q Heat flux Q Mass source term R Specfc gas constant k B /m s Entropy S Heavsde step functon S Stran rate tensor t Coordnate n tme t mfp Mean free tme between collsons t r Retarded tme t x/c 0 t shock Lossless plane-wave shock formaton tme T Temperature T MRT relaxaton matrx T Acoustc multpole tensor u Flud velocty v Pecular velocty ξ u V Volume

25 1.2 Ths thess 17 w w W x x mfp x shock X X m α x α t β γ Γ δ αβ δ(x) Δx Δt ɛ ε κ λ μ μ B ν ν B ξ ξ Π Π ρ σ σ τ τ ν τ κ τ m Ω Ω Weghtng coeffcent Complex error functon, aka. Faddeeva functon Wndow functon Coordnate n physcal space Mean free path between collsons Lossless plane-wave shock formaton dstance Acoustc vscosty number ω 0 τ ν Acoustc relaxaton number ω 0 τ m Spatal absorpton coeffcent Temporal absorpton coeffcent Coeffcent of nonlnearty Heat capacty rato c p /c V, aka. adabatc ndex Bulk vscosty correcton tensor Kronecker delta Drac delta functon Spatal resoluton Temporal resoluton Chapman-Enskog smallness parameter Macroscopc smallness parameter Thermal conductvty Acoustc wavelength Dynamc shear vscosty Dynamc bulk vscosty Knematc shear vscosty Knematc bulk vscosty Partcle velocty or coordnate n velocty space Dscrete partcle velocty Moment of f Moment of f, f nequal to the correspondng Π Mass densty Stress tensor Devatorc stress tensor Knetc relaxaton tme Vscous relaxaton tme Thermal relaxaton tme Molecular relaxaton tme Collson operator Collson matrx

26 2 Fundamental theory To ensure that ths thess s approachable for people from varous scentfc backgrounds, we wll frst go through some of the fundamental theory of flud mechancs and acoustcs before delvng nto the specfcs of knetc theory and the lattce Boltzmann method. Some of the topcs n ths chapter wll be covered n more detal n later chapters. 2.1 Index notaton Index notaton, lke vector notaton, s a notaton style for sets of quanttes assocated wth dfferent spatal drectons. It s commonly used n flud mechancs and less commonly n acoustcs, and wll be used throughout ths thess. Throughout physcs, many equatons deal wth behavour whch s smlar n multple spatal drectons. As a smple example, Newton s second law can be wrtten as three equatons, F x = ma x, F y = ma y, F z = ma z, (2.1a) Tensor For our purposes we can see tensors as mathematcal objects generalsng vectors and matrces. Scalars are tensors of zeroth order, vectors are of frst order, matrces are of second order, and hgher orders are also possble. The order corresponds to the number of ndces requred to pont to a component of the tensor. wth one equaton for each spatal drecton. In fact, many classc works on flud mechancs and acoustcs, wrtten before other knds of notaton became common, used ths expansve type of notaton [16, 17]. Three ndvdual but very smlar equatons would be gven together; one for each spatal drecton. Usng vector notaton, whch can more generally be called tensor notaton, these three equatons can be wrtten as one sngle equaton, F = ma. Here, the x, y, and z components of F and a are mplctly related. Another alternatve style of notaton s ndex notaton, F α = ma α. (2.1b) (2.1c) The equaton s wrtten as n (2.1a), except that a generc ndex α s used. Ths allows expressng the system of equatons as explctly as n (2.1a), but wth a sngle equaton nstead of three. 18

27 2.1 Index notaton 19 The sngle ndex used n (2.1c) ndcates that F α and a α are vector components. Index notaton can also be used to pont to generc components of hgher order tensors. A generc component of the second-order tensor (or matrx) A would be A αβ, and smlarly a component of the thrd-order tensor R s R αβγ. Ths llumnates a strength of ndex notaton: The order of the tensor s mmedately clear from the number of unque ndces. Another mportant strength comes from the summaton conventon ntroduced by Ensten [18]: Repeatng an ndex twce n a sngle term mples summaton over all possble values of that ndex. Thus, a α b α = a α b α = a x b x + a y b y + a z b z = a b. (2.2) α The dot product has about the same economy of notaton n tensor and ndex notaton. In ths thess, Greek ndces and the summaton conventon are used for spatal components. For components of general, non-spatal tensors, ndces, j, k,... are used, and the summaton conventon s not used. Ths practce s common n the feld of lattce Boltzmann research. Other vector operatons can also be easly expressed usng ndex notaton. The outer product n tensor and ndex notaton s A = a b A αβ = a α b β. Yet another strength of ndex notaton becomes apparent; n vector/tensor notaton, we must ntroduce a new symbol for ths partcular operaton. In ndex notaton, however, the meanng s mmedately clear: The α, β component of the matrx A s gven by the product of the α component of a and the β component of b. The correspondng downsde s that a large number of ndces can result n a notaton that looks somewhat messy. Usng ndex notaton, we can also generalse coordnate notaton, wrtng a general component of the spatal coordnate vector x =(x, y, z) = (x 1, x 2, x 3 ) as x α. In ths way, we can express e.g. gradents n ndex notaton, λ(x) λ(x) x α. Most common vector and tensor operatons can be convenently expressed n ndex notaton, as shown n Table 2.1. The cross product requres use of the Lev-Cvta symbol, whch s the thrd order tensor +1 f (α, β, γ) s (1, 2, 3), (3, 1, 2) or (2, 3, 1), ε αβγ = 1 f (α, β, γ) s (3, 2, 1), (1, 3, 2) or (2, 1, 3), 0 f α = β, β = γ, orγ = α. (2.3) The cross product wll not be used at any pont later n ths thess, however.

28 20 Chapter 2 Fundamental theory Table 2.1: Examples of common operatons n tensor and ndex notaton Operaton Tensor notaton Index notaton Vector dot product λ = a b λ = a α b α Vector cross product c = a b c α = ε αβγ a β b γ Vector outer product A = a b A αβ = a α b β Tensor contracton λ = A : B λ = A αβ B αβ Gradent a = λ a α = λ/ x α Laplacan Λ = 2 λ Λ = 2 λ/ x α x α 1st order tensor dvergence λ = a λ = a α / x α 2nd order tensor dvergence a = A a α = A αβ / x β 3rd order tensor dvergence A = R A αβ = R αβγ / x γ Another, more wdely useful, symbol s the Kronecker delta symbol, whch s the second order tensor { 1 f α = β, δ αβ = (2.4) 0 f α = β. We can see by nspecton that δ αβ s a generc element of the dentty matrx [ 100 ] I = Ths thess wll alternate between tensor notaton and ndex notaton to some degree, generally usng the former for smpler equatons and the latter for more complcated equatons that beneft from the explctness of ndex notaton Examples Some propertes and methods of ndex notaton are best shown by example. These wll be mplctly used throughout the rest of the thess. Arbtrarness of summaton ndces. When the summaton conventon s used, the summaton ndex s arbtrary. Thus, a α b α = a γ b γ, A αβ x β = A αδ x δ. (2.5) The Kronecker delta and the summaton conventon. delta can change the ndex of other quanttes, The Kronecker δ αβ a β = δ αx a x + δ αy a y + δ αz a z = a α. (2.6)

29 2.2 Flud mechancs 21 The Kronecker delta and trace. The trace of a second order tensor can be taken by multplyng wth the Kronecker delta, δ αβ A αβ = A xx + A yy + A zz = A γγ = Tr(A). (2.7) Trace of the Kronecker delta. From Table 2.1, ths s equvalent to contractng the dentty matrx wth tself. Wth three spatal dmensons, ths becomes [ 100 ] [ 100 ] δ αβ δ αβ = I : I = 010 : 010 = 3. (2.8) Eucldean dstance. The square of the Eucldean dstance s a b 2 =(a α b α )(a α b α )=a α a α 2a α b α + b α b α = a a 2a b + b b. (2.9) Weghted ntegral of a sphercally symmetrc ntegrand. If a functon f (x) s sphercally symmetrc around x = 0, then the ntegral over the entre volume, weghted wth x α x β,s xγ x γ x α x β f (x) dx = δ αβ f (x) dx. (2.10) 3 Ths follows from the ntegrand havng odd symmetry, so that the ntegral becomes zero unless α and β are equal. Also, due to the sphercal symmetry of f (x), weghtng the ntegral by x 2, y 2,orz 2 s equal to weghtng t by (x 2 + y 2 + z 2 )/3 = x γ x γ /3. Multdmensonal ntegraton by parts. If there s a volume Ω bounded by a surface Γ, and two functons u and v that are smooth nsde Ω and on Γ, then u v dω = uv dγ α u v dω, (2.11a) x α x α Ω Γ Ω where Γ α s the surface normal of Γ. In one dmenson ths reduces to normal ntegraton by parts. The multdmensonal ntegraton by parts also holds f one or both of the functons are tensor components, e.g. Ω u α x α v dω = Γ u α v dγ α Ω u α v x α dω. (2.11b) 2.2 Flud mechancs The fundamental equatons of flud mechancs are conservaton equatons. Basc physcs tells us that mass,* momentum, and total energy are always conserved n a closed system. These flud mechancs equatons express these conservaton laws n the case of a macroscopc contnuum. *Barrng nuclear reactons and relatvstc effects, of course. Contnuum Modellng a physcal body as a contnuum, we dsregard that the fact t s made out of atoms and empty space and assume that the body s contnuous, preservng ts macroscopc quanttes even on the mcroscopc scale.

30 22 Chapter 2 Fundamental theory λ(x u dt, t dt) u dt λ(x, t) Fgure 2.1: As a flud partcle moves through the flud, ts propertes may change. There are two wdely used models of such conservaton equatons: the Euler model and the Naver-Stokes-Fourer model. The former model s older, whle the latter model s more detaled. Whle these equatons are usually derved from a contnuum mechancs perspectve, we wll derve them from more physcally fundamental knetc theory n Chapter 3. They wll only be ntroduced brefly n ths secton. Common to both sets s the equaton for conservaton of mass, also known as the contnuty equaton, ρ t + (ρu) = 0. (2.12) Mass densty, ρ Mass per physcal volume n kg/m 3 Flud velocty, u The local velocty of the flud n m/s Materal dervatve, D/Dt The tme dervatve for a movng flud partcle Here, ρ = ρ(x, t) s the mass densty (or just densty) and u = u(x, t) s the flud velocty. Ths equaton connects the rate of change of densty at a pont and the outward or nward mass flux at that pont. One notatonal convenence whch s common throughout flud mechancs s the use of the materal dervatve. Let s say we have a generc quantty λ(x, t). From the total dfferental of ths, we have dλ dt = ( ) ( ) λ dt λ t dt + dxα x α dt. Here, dx α /dt s the flud velocty u. Ths gves us the materal dervatve, usually wrtten wth a captal D, Dλ Dt = λ + u λ. (2.13) t The materal dervatve can be nterpreted as a tme dervatve for a flud partcle movng throughout the flud wth a velocty u(x, t), as shown n Fgure 2.1. Ths flud partcle represents a tny pece of the flud contnuum, small enough that all relevant quanttes are nearly constant nsde of t. The frst term on the rght-hand sde of (2.13) s the rate of change that would occur f the partcle were statonary. The second term s the rate of change caused by the partcle movng nto a dfferent part of the flud where the condtons may be dfferent. The conservaton equatons can ether be wrtten n materal dervatve form, emphassng changes to a flud partcle, or n conservaton form,

31 2.2 Flud mechancs 23 emphassng changes n a statc control volume. For the generc quantty λ(x, t), the two are related as ρλ + ρu ( αλ ρ = λ t x α t + ρu ) α + ρ λ x α t + ρu λ α = ρ Dλ x α Dt, (2.14) wth conservaton form on the left and materal dervatve form on the rght. The parenthess n the mddle s zero due to (2.12). The state prncple of equlbrum thermodynamcs s mportant for flud mechancs. It states that any state varable, such as the densty ρ, State varable the pressure p, the temperature T, the nternal energy e, or the entropy s can be found from any other two state varables through an equaton of state [19, Ch. 2].* The most classc example of ths s the classc deal gas law, p = p(ρ, T) =ρrt, (2.15) where R s the specfc gas constant. Alternatvely, the pressure could be expressed as a functon of densty ρ and entropy s. For deal gases, ths equaton of state s [19, Ch. 2] ( ) p ρ γ = e (s s 0)/c V. (2.16) p 0 ρ 0 The quanttes wth subscrpted zeroes are reference state values. γ = c p /c V s the heat capacty rato or adabatc ndex, whch relate the heat capactes at constant pressure c p and at constant volume c V The Euler model The Euler model s named after Leonhard Euler, who frst derved the mass and momentum conservaton equatons [20]. The mass, momentum, and energy conservaton equatons are Quanttes that descrbe the local thermodynamc state of the system. From the state prncple, knowng two state varables s suffcent to fnd the rest. Pressure, p Force per area exerted by the flud on a real or magnary surface n N/m 2 Equaton of state An equaton relatng any three state varables Specfc gas constant, R The proportonalty constant n the deal gas law, n J/kg K. It s dependent on the mass of the gas molecules. ρ t + (ρu) = 0, ρ Du = p + F, Dt ρ De Dt = p u. (2.17a) (2.17b) (2.17c) F s the external body force densty, whch s typcally gravtatonal. The momentum equaton (2.17b) descrbes how the velocty of a partcle s changed by external body forces and pressure dfferences; hgher pressures push the partcle towards lower pressures. *We wll not go nto detal wth these state varables at ths pont. They wll be descrbed n more detal n Chapter 3. Body force densty, F Force densty of long-range forces, e.g. gravty, n N/m 3

32 24 Chapter 2 Fundamental theory The energy equaton (2.17c) descrbes how the nternal energy s ether decreased by expanson (the partcle pushng on ts surroundngs), or ncreased by compresson (the surroundngs pushng on the partcle). The Euler model s less accurate than the Naver-Stokes-Fourer model whch wll be descrbed next, as t lacks the effects of nternal frcton and heat conducton n the flud. Even so, t s suffcent for use n many dfferent cases n acoustcs and aerodynamcs The Naver-Stokes-Fourer model Cauchy momentum equaton A general equaton for the evoluton of momentum, vald for both solds and fluds Cauchy stress tensor, σ A second-order tensor whch at any pont n the flud specfes the normal and shear stresses n the x, y, and z drectons The equatons of the Naver-Stokes-Fourer model are very smlar to the correspondng equatons of the Euler model. The dfference s addtonal terms that take nto account the effect of nternal frcton and heat conducton n the flud. The momentum equaton n ths model s of the form of the Cauchy momentum equaton, ρ Du α Dt = σ αβ x β + F α, (2.18) a general equaton whch can descrbe momentum conservaton n any contnuum, even a sold. The Cauchy stress tensor σ descrbes the stresses due to nternal forces. The three equatons of the Naver-Stokes-Fourer model are ρ t + (ρu) = 0, ρ Du α Dt = p x α + σ αβ x β + F α, ρ De ) ( δ Dt = αβ p + σ αβ uβ q α. x α x α (2.19a) (2.19b) (2.19c) These equatons nclude the heat flux q and splt the Cauchy stress tensor nto two terms as σ = pi + σ. The pressure term was already present n the Euler momentum equaton (2.17b), but the devatorc stress σ s new. The devatorc stress tensor for a smple flud, whch was frst determned by Stokes [21], s ( ) σ αβ = uβ μ + u α 2 x α x β 3 δ u γ u γ αβ + μ x B δ αβ, (2.19d) γ x γ and the heat flux s assumed to be gven by Fourer s law, q α = κ T x α. (2.19e)

33 2.3 Acoustcs 25 These equatons nclude the materal coeffcents of dynamc shear vscosty μ, dynamc bulk vscosty μ B,* and thermal conductvty κ. Often the knematc shear vscosty ν = μ/ρ and the knematc bulk vscosty ν B = μ B /ρ are used nstead of ther dynamc counterparts. Wth ths form of the devatorc stress tensor, the momentum equaton (2.19b) s also known as the Naver-Stokes equaton. In the momentum equaton (2.19b), the addtonal σ term represents the frcton between adjacent parts of the flud movng at dfferent speeds. σ also occurs n the energy equaton (2.19c), representng the energy ncrease due to frctonal heatng. The fnal q term represents the heat conducton between adjacent parts of the flud wth dfferent temperatures. In many subfelds of flud mechancs, the flud s often consdered to be ncompressble, meanng that the densty ρ s constant. Ths assumpton smplfes the mass conservaton equaton to u = 0. Consequently, the bulk vscosty n (2.19d) becomes rrelevant, as ts term s zero. For ths reason, bulk vscosty s often neglected n flud mechancs. Stll, t s relevant n acoustcs and hgh-velocty compressble flow. Many, f not most, problems n mathematcal flud mechancs reduce to solvng these equatons wth approprate boundary condtons, equatons of state, and approxmatons. The most common boundary condton s the no-slp condton, where the flud at a wall s restrcted to have the same velocty as the wall due to frcton. The physcal orgns of ths no-slp condton are not fully clear even today [23, Ch. 4]. 2.3 Acoustcs The wave equaton s the mathematcal bass for most of acoustcs. Ths equaton can be derved drectly from the conservaton equatons of mass and momentum. Typcally the lnearsed Euler equatons wth a smplfed equaton of state are used, but more detaled equatons can also be employed to derve a more detaled wave equaton. In ths secton, we wll frst show how the smple deal wave equaton s derved and then fnd a more complex wave equaton that takes nto account the effects of vscosty and heat conducton. In both cases, we assume that the sound wave s weak enough that the equatons can be lnearsed. Later we look at the effect of molecular rotaton and vbraton on sound propagaton, and the mathematcal modelng of multpole sound sources. Towards the end of ths secton we wll consder what happens when the sound wave s strong and nonlnear effects occur. All Lnearsaton An approxmaton technque where terms of hgher than frst order n small quanttes are neglected, resultng n lnear equatons *Stokes orgnally assumed the bulk vscosty to be zero. Ths assumpton was later found to be vald only for dlute monatomc gases, as wll be shown n Chapter 3. However, the orgns of bulk vscosty are somewhat controversal even today [22].

34 26 Chapter 2 Fundamental theory of these topcs are relevant to the research work presented n Part II of ths thess. The feld quanttes of acoustcs can be dvded nto two parts, ρ(x, t) =ρ 0 + ρ (x, t), p(x, t) =p 0 + p (x, t), u(x, t) =0 + u (x, t). (2.20a) (2.20b) (2.20c) The subscrpted zeroes denote a constant rest state, and the prmed quanttes are small fluctuatons. Lnearsaton entals neglectng any term where more than one prmed term occurs smulaneously, due to that term s smallness. Lnearsaton s wdely employed n acoustcs due to ts wde range of applcablty. The human ear s threshold of pan s at about 140 db, whch corresponds to a relatve RMS pressure p rms/p n ar, wth p 0 beng the standard atmospherc pressure [24, Chs. 5 & 11]. Thus, lnearsaton s a hghly vald approxmaton for the sound waves we encounter n daly lfe.* Ideal wave equaton The deal wave equaton neglects as many nondeal effects, such as vscosty and heat conducton, as possble. Even so, t s suffcent to descrbe most cases n acoustcs wth very good accuracy. The wave equaton s derved from the lnearsed form of the Euler mass and momentum equatons, (2.17a) and (2.17b). Except for extremely low frequences [24, Ch. 5] or long-range atmospherc or underwater propagaton [25, Ch. 2], the effects of gravtatonal force are also neglgble. The lnearsed and forceless Euler mass and momentum equatons are ρ t + ρ u α 0 = 0, x α ρ 0 u α t + p x α = 0. (2.21a) (2.21b) The sum (2.21a)/ t (2.21b)/ x α gves 2 ρ t 2 2 p = 0. (2.22) Ths s one step away from the wave equaton. Note that the dervatves of ρ and p are equal to the dervatves of ρ and p, snce the rest state values ρ 0 and p 0 are constant. *Ths does not necessarly mean that nonlnear effects can always be completely neglected f the ampltude s not extremely hgh. Nonlnear effects accumulate as a sound wave propagates; they can stll be relevant f a sound wave propagates far enough [24, Ch. 16].

35 2.3 Acoustcs 27 To take the fnal step to the wave equaton, we need to relate p and ρ through an equaton of state. Typcally, the sentropc relaton ( ) p ρ γ = (2.23) p 0 s used [19, 24, 25]. Ths relaton follows from (2.16) and the assumpton of near-constant entropy, s s 0. From ths equaton, p and ρ can be related as p ρ ρ 0 ( ) p = γp 0 = c 2 0 ρ s,0 ρ, (2.24) 0 where the dervatve has been evaluated at the rest state. We wll soon see that c 0 s the deal speed of sound. Usng ths, we can re-express the tme dervatve term n (2.22) as 2 ρ/ t 2 = 2 ρ / t 2 =(1/c 2 0 ) 2 p / t 2, and at last we fnd the deal wave equaton, 1 c p t 2 2 p = 0. (2.25) Ths wave equaton s lnear. For solutons of a sngle frequency ω (whch s the angular frequency, lnked to the natural frequency f as ω = 2π f ) we can therefore use complex phasor notaton, whch s mathematcally smpler to deal wth. The smplest soluton of the wave equaton s a one-dmensonal plane wave, whch n phasor notaton s ˆp (x, t) = ˆp e (ωt kx). (2.26) Here, k = ω/c 0 = 2π/λ s the wavenumber, λ s the wavelength, hats ndcate complex quanttes, and ˆp s the complex ampltude, whch has both a magntude ˆp and a phase ϕ p = arg( ˆp ), Phasor notaton A method of notaton for varables n lnear equatons. It uses complex exponentals for mathematcal smplcty. The real, physcal soluton s found as the real part of the phasor soluton. ˆp = ˆp e ϕ p. Ths ampltude s determned by boundary or ntal condtons. The real, physcal soluton s found by takng the real part of the complex phasor soluton, p (x, t) =Re [ ˆp (x, t) ] = ˆp cos(ωt kx + ϕ p ). (2.27) Thus, the magntude ˆp determnes the physcal wave ampltude and the phase ϕ p determnes the phase shft. It s generally mathematcally easer to deal wth expressons on the phasor form of (2.26) rather than those on the physcal form of (2.27). From any of these solutons we fnd that the soluton value s constant f the argument ωt kx s constant. Equatng the argument at two dfferent tmes and postons, as shown n Fgure 2.2, we fnd that x 2 x 1 = ±c 0 (t 2 t 1 ).

36 28 Chapter 2 Fundamental theory p 0 + ˆp x 2 x 1 t = t 2 t = t 1 p(x, t) p 0 p 0 ˆp x 1 x 2 x Fgure 2.2: Part of a plane pressure wave descrbed by (2.27). From tme t 1 to t 2,the leftmost peak moves from poston x 1 to x 2. Retarded tme, t r A transformed coordnate whch, when held constant, follows a wave n tme and space. Ths proves that c 0 s the speed of sound, and that the choce of n the soluton corresponds to propagaton n the ±x drecton. If the argument ωt kx s constant, then the retarded tme t r = t x/c 0 (2.28) must also be constant. Ths quantty can be seen as a transformed coordnate that follows the wave as t propagates. If the soluton s assumed to be sngle-frequency and complex, all fluctuatng feld quanttes vary as e ωt. Wth ths assumpton, the wave equaton (2.25) becomes the Helmholtz equaton, 2 ˆp + k 2 ˆp = 0. (2.29) Thus, the tme-dependent, hyperbolc wave equaton s transformed nto a tme-ndependent ellptc equaton. The soluton ˆp can stll be a functon of tme, however; the harmonc soluton ˆp (x, t) n (2.26) s also a vald soluton of (2.29) Vscous and thermovscous wave equaton In any flud, vscosty and heat conducton cause some absorpton of sound waves. These effects become relevant at hgh frequences and long propagaton dstances. The effect of vscosty on sound wave propagaton was frst examned by Stokes [21], n the same artcle where he derved the stress tensor for a flud. In ths thess, the case of purely vscous absorpton wll be the most relevant. We wll look at ths n detal, and then take a quck look at the case where both vscous and thermal effects are present.

37 2.3 Acoustcs 29 Purely vscous case In the prevous dervaton of the deal wave equaton (2.25), the sentropc equaton of state (2.23) was used. Vscous effects cause entropy to ncrease, but ths varaton s of second order n the small, prmed terms [25, Ch. 9]. Therefore, ths entropy ncrease s neglected by lnearsaton, and (2.23) can stll be assumed to hold. Ths s not the case f thermal absorpton s present, however, and that case requres use of another equaton of state. Instead of the Euler momentum equaton, the Naver-Stokes momentum equaton s used to derve the wave equaton. The extra stress tensor term must now be consdered n the dervaton. When the spatal dervatve of the momentum equaton s taken, ths term becomes 2 σ αβ ( ) = 43 μ + μ x α x B 2 u γ = τ ν 2 p β x γ t. (2.30) In the last equalty, the mass conservaton equaton (2.19a) and the speed of sound (2.24) have been used, and the vscous relaxaton tme τ ν = 1 ( ) 4 c 2 3 ν + ν B (2.31) 0 has been ntroduced. For most physcal fluds, τ ν s s for gases and s for lquds [24, Ch. 8]. Otherwse the dervaton proceeds as n the deal case, and we fnd the vscous wave equaton, 1 c ( ) p t τ ν 2 p = 0. (2.32) t The only dfference from the deal wave equaton s the term where τ ν appears n front of a tme dervatve. We wll now see how ths causes an absorbed soluton. Returnng to complex phasor notaton, we can n ths case get complex wavenumbers ˆk and frequences ˆω. We denote the correspondng wavenumbers and frequences from the deal case as k 0 and ω 0. Lookng at a wave radated from a source oscllatng at a sngle frequency ω 0 and a constant ampltude, we can assume that the soluton vares n tme as e ω 0t, resultng n a Helmholtz equaton 2 ˆp + ˆk 2 ˆp = 0 (2.33) wth a complex wavenumber ˆk. Assumng a soluton propagatng n +x-drecton and normalsng by k 0 = ω 0 /c 0,wefnd ˆk k 0 = ω0 τ ν (ω 0τ ν ) 3 8 (ω 0τ ν ) 2 + O([ω 0 τ ν ] 3 ). (2.34)

38 30 Chapter 2 Fundamental theory ˆp Ideal Absorbed ˆp p (x, t) 0 ˆp x/λ Fgure 2.3: Comparson of plane wave solutons of two wave equatons: the deal (2.25) and the vscous (2.32), the latter wth an exaggerated absorpton coeffcent α x. Acoustc vscosty number, ω 0 τ ν = X A dmensonless number ndcatng the effect of vscosty on sound propagaton ω 0 τ ν can be seen as a dmensonless parameter determnng the effect of vscosty on sound wave propagaton. Its name and exact form vares throughout the lterature [26, 27]; we wll here call t the acoustc vscosty number. It wll be heavly used later n the thess, where t wll be denoted as X for brevty. Followng the seres expanson around ω 0 τ ν = 0 even further, we fnd a pattern of every even term n ω 0 τ ν beng real and every odd term beng magnary. It s convenent to splt the complex wavenumber nto a real part and an magnary part, ˆk = k α x. (2.35) α x s known as the spatal absorpton coeffcent, for reasons that wll become clear presently. As the basc Helmholtz equaton (2.29) s on the same mathematcal form as ts complex counterpart (2.33), ther solutons are nearly dentcal, wth the dfference that the wavenumber s complex n the latter case. Thus, for a plane wave propagatng n the x-drecton, the soluton s ˆp (x, t) = ˆp e α xx e (ω 0t kx), (2.36) where the complex wavenumber ˆk has been separated nto ts real part k and ts magnary part α x. The former causes the speed of sound c = ω 0 /k to be dfferent from ts deal value c 0 = ω 0 /k 0. The latter causes the wave to be absorbed exponentally wth dstance as ts energy s converted nto heat. The solutons to the deal and vscous wave equatons are compared n Fgure 2.3. Assumng small values of ω 0 τ ν, whch s vald n most gases for

39 2.3 Acoustcs 31 frequences up to 10 8 Hz, (2.34) gves an absorpton coeffcent α x k ( ) 0 2 (ω 0τ ν )= ω c 3 3 ν + ν B, (2.37a) 0 and a real part of the wavenumber k k 0. (2.37b) Thus, for small ω 0 τ ν, the absorpton coeffcent α x scales wth the vscosty and the square of the frequency, and there s neglgble change n the speed of sound. We wll see n Chapter 5 that the predctons made by the Naver-Stokes-Fourer model are ncorrect at hgher than frst order n ω 0 τ ν. As such, (2.37) s the best we can get from ths model. We assumed above that the wave was radated outwards from a source. Ths s sometmes called a forced wave, snce the wave s generated, or forced, by a source, and s absorbed wth the dstance to that source. Another case seen throughout the lterature s based on the assumpton that the wave at t = 0 has nfnte extent and the same ampltude everywhere, ˆp (x,0)= ˆp e k 0x. Ths ntal-value problem causes the angular frequency to be complex nstead of the wavenumber, gvng the soluton ˆω = ω + α t, (2.38) ˆp (x, t) = ˆp e α tt e (ωt k 0x). (2.39) Thus, the wave s absorbed exponentally n tme nstead of n space. Such waves are sometmes called free waves. Whle free waves are not really physcally realsable, unlke forced waves, free waves can be used to benchmark numercal methods by performng smulatons wth perodc boundary condtons that smulate a wave of nfnte extent. We wll look at ths case n more detal n Chapter 5. Thermovscous case In physcal gases, the effect of thermal conducton on sound wave absorpton s of comparable relevancy to the effect of vscosty. However, n sothermal lattce Boltzmann smulatons, whch wll be the focus later n ths thess, there s no heat conducton and thermal absorpton s therefore rrelevant. We wll therefore treat ths case more cursorly than the prevous case of pure vscous absorpton.

40 32 Chapter 2 Fundamental theory If thermal conducton s relevant, the nearly sentropc equaton of state (2.23) no longer approxmately holds, and another equaton of state must be used when dervng the wave equaton. From the energy equaton (2.19c), an alternatve equaton of state can be derved under the assumptons of lnearty and the gas beng deal. Adaptng the method of [25, Ch. 9] to multple spatal dmensons, we fnd (p c 2 0 ρ ) t ( ) = κ 2 p c2 0 ρ 0 c V γ ρ. (2.40) Usng ths equaton of state when dervng the nondeal wave equaton, the approxmate thermovscous wave equaton can be found to be 1 c ( p t 2 1 +[τ ν + τ κ ] ) 2 p = 0, (2.41) t where the thermal relaxaton tme τ κ = 1 c 2 0 κ(γ 1) ρ 0 c p (2.42) has been ntroduced. For ar, τ κ = s, of the same order as τ ν, and for freshwater, τ κ = s, whch s neglgble compared to τ ν [24, App. 10]. Note that (2.41) s not the exact thermovscous wave equaton [25, Ch. 9], but an approxmaton where very small terms of order O(τ ν τ κ ) have been neglected. Comparng the thermovscous wave equaton (2.41) wth the purely vscous wave equaton (2.32), t s clear that ther solutons are very smlar. The only dfference les n an addtonal term n the spatal absorpton coeffcent, ( α x ω c 3 3 ν + ν B + γ 1 ) κ. (2.43) ρ 0 0 c p Wth smulaton methods, such as the sothermal lattce Boltzmann method, that smulate the effect of vscosty on sound waves but not the effect of thermal conducton, the latter effect can be emulated by artfcally ncreasng the bulk vscosty by κ(γ 1)/ρ 0 c p Molecular relaxaton processes In most polyatomc gases, the domnant mechansm of absorpton at low frequences s molecular relaxaton due to relatvely slow transfer of energy between the molecules translatonal degrees of freedom (.e. the energy due to the molecules velocty) and ther nner degrees of freedom (.e. the energy due to rotaton and vbraton of the molecules).

41 2.3 Acoustcs 33 Fgure 2.4: A lnear molecule has two possble orthogonal rotatons, and therefore classcally has two rotatonal degrees of freedom. Nonlnear molecules have three. In lquds, a dfferent type of relaxaton may be present, related to changes n chemcal equlbra between dfferent solutes. Whle the relaxaton mechansms of polyatomc gases and lquds are very dfferent, they can be modelled smlarly. We wll however focus on deal polyatomc gases here, as they are most relevant to ths thess. We wll not delve very deeply nto ths topc, as a full treatment requres some amount of quantum and statstcal mechancs and quckly becomes very complcated. More n-depth treatment can be found elsewhere [28, 29]. The followng treatment s based on smpler descrptons [24, Ch. 8][25, Ch. 9]. In a gas at rest, the energes stored n the translatonal and nner degrees of freedom are at equlbrum. If the gas s not at rest, external dsturbances such as a passng sound wave may cause the translatonal energy to change.* Ths wll push the dfferent degrees of freedom out of equlbrum, and a gradual readjustment to equlbrum wll occur. Ths readjustment happens through molecular collsons. Classcally, collsons wll tend to slowly equlbrate the dfferent degrees of freedom. Quantum mechancally, each collson between molecules may knock them nto a hgher or lower rotatonal or vbratonal energy state wth a certan probablty. Classcally or quantum mechancally, the overall result s the same: An equlbrum s eventually reached between the energes stored n the dfferent degrees of freedom. From the classcal equpartton theorem [19, Ch. 2], energy s dstrbuted evenly among all degrees of freedom when the system s at equlbrum. However, from quantum statstcal mechancs, nner degrees of freedom are not fully excted and cannot store ther full amount of energy unless the temperature T s hgh enough. From the equpartton *We wll see n secton 3.2 that pressure and translatonal energy are proportonal. Thus, a pressure change drectly corresponds to a change of the translatonal energy.

42 34 Chapter 2 Fundamental theory Transl. degs. of freedom Inner deg. of freedom Energy 0 τ m t Fgure 2.5: Sketch of the re-equlbraton of energy after the translatonal energy s suddenly ncreased at t = 0. theorem, the total heat capacty at constant volume s ( ) e c V = = d tot R, (2.44) T 2 where d tot = d tot (T) =3 + d = 3 + H (T). (2.45) s the total number of degrees of freedom at temperature T. Ths contans three translatonal degrees of freedom and d nner degrees of freedom. H (T) s the exctaton fracton for the nner degree of freedom, whch asymptotcally vares from 0 at low temperatures, where the degree of freedom s not actve, to 1 at hgh temperatures, where the degree of freedom s fully actve. Dfferent nner degrees of freedom wll be actve n dfferent amounts at a certan temperature T. At room temperatures, H s nearly 1 for rotatonal states n most molecules, sgnfcantly less than 1 for lower vbratonal states, and nearly 0 for hgher vbratonal states [24, Ch. 8]. For deal gases at equlbrum, the heat capacty at constant pressure c p and the heat capacty rato γ are V c p = c V + R = 2 + d tot R, 2 (2.46) γ = c p = c V d tot (2.47) Monatomc gases, whch have no nner degrees of freedom, have γ = 1 + 2/3 = 5/3. For other gases, γ wll tend to ncrease wth temperature, as the exctaton fractons H (T) ncrease. Hgh-temperature gases consstng of very complcated molecules wth many possble vbratonal states would have a very large number of degrees of freedom, so that γ 1. Let us now look at the smplfed case of a gas of molecules wth only one relevant nner degree of freedom. If the translatonal energy of the

43 2.3 Acoustcs 35 gas s suddenly ncreased, for nstance due to a sudden compresson, the translatonal and nner energes wll re-equlbrate exponentally [24, Ch. 8], as sketched n Fgure 2.5. The characterstc tme for ths process s the molecular relaxaton tme τ m. If the translatonal energy s changed perodcally by a sound wave wth frequency ω 0, the dmensonless product ω 0 τ m determnes the character of the relaxaton process: If ω 0 τ m 1, the equlbraton occurs quckly compared to the dsturbance. The translatonal and nner energes are always nearly n equlbrum. If ω 0 τ m 1, the equlbraton occurs relatvely slowly. The nner energy cannot keep up wth the translatonal energy, and remans nearly constant, or frozen. In ths case, the nner energy s degree of freedom does not play a part n determnng heat capacty. Thus, the adabatc ndex γ wll go asymptotcally wth frequency towards a value γ where the nner degree of freedom s not ncluded n d tot. Wth only a sngle nner degree of freedom, the adabatc ndex s the same as for a monatomc gas, γ = 5/3. Wth the speed of sound determned as n (2.24), we fnd an asymptotc hgh-frequency speed of sound c determned from c 2 = γ p 0 ρ 0. (2.48) Wth γ = 5/3 and γ 0 gven by (2.47), the rato between the asymptotc hgh-frequency and low-frequency speeds of sound s determned solely from the exctaton fracton of the nner degree of freedom, Molecular relaxaton tme, τ m The characterstc tme for re-equlbraton of translatonal energy and an nner energy mode c 2 c 2 0 = 1 + d /3 1 + d /5 d = 5(c2 /c 2 0 1) 5/3 c 2 /c 2, (2.49) 0 where we have from (2.45) that d = H(T), the exctaton fracton of the nner degree of freedom. The effect of molecular relaxaton on acoustc propagaton can be modelled usng a relaxaton equaton of state [25, Ch. 9][22, 30], ( τ m p c 2 t ρ ) ( + p c 2 0 ρ ) = 0. (2.50) Ths reduces to p = c 2 0 ρ at frequences where ω 0 τ m 1, and at frequences where ω 0 τ m 1 t reduces to p = c 2 ρ. Dfferentatng ths equaton of state twce wth respect to tme and usng (2.22) to elmnate ρ (thus neglectng thermovscous effects), we

44 36 Chapter 2 Fundamental theory 2(αx/k0)/(c 2 /c 2 0 1) ω 0 τ m c c c ω 0 τ m Fgure 2.6: Normalsed absorpton coeffcent and speed of sound as functons of ω 0 τ m. fnd the wave equaton for a sngle relaxaton process, ( 2 p ) ( τ m t t 2 c2 2 p + 2 p ) t 2 c2 0 2 p = 0. (2.51) Ths wave equaton can also be found by a more fundamental approach that takes quantum statstcal mechancal effects fully nto account [22]. The effect of relaxaton on sound wave propagaton can now be found by nsertng a plane forced wave tral soluton ˆp = ˆp e (ω0t ˆkx) nto (2.51). The complex wavenumber s then found to be ˆk k 0 = 1 + ω 0 τ m 1 + ω 0 τ m (c /c 0 ) 2. (2.52) The exact expressons for the real and magnary parts of the wavenumber, whch respectvely lead to the true speed of sound c and spatal absorpton coeffcent α x, are qute complcated. However, n most real fluds, c 0 and c are very close [25, Ch. 9]. Assumng the quantty c 2 /c 2 0 1to be very small, the complcated expressons can be smplfed [25, Ch. 9], gvng α x k 0 = 1 2 c 2 c 2 0 ( ) c 2 c ω 0 τ m 1 +(ω 0 τ m ) 2, = 1 +(ω 0τ m ) 2 (c 2 /c 2 0 )4 1 +(ω 0 τ m ) 2 (c 2 /c 2 0 )2. (2.53a) (2.53b) These expressons are plotted n Fgure 2.6. The absorpton over a wavelength s maxmal at ω 0 τ m = 1, and the speed of sound c changes smoothly from c 0 to c.

45 2.3 Acoustcs 37 When several relaxaton processes and thermovscous absorpton occur smultaneously, t s often assumed that the absorpton coeffcents of the ndvdual processes can be summed to fnd a total absorpton coeffcent [25, 31, 32]. In ar, the relevant relaxaton processes are rotatonal and vbratonal relaxaton n ntrogen and oxygen. The rotatonal relaxaton tme s typcally of the order of 10 9 s, comparable to the vscous relaxaton tme τ ν,* and the vbratonal relaxaton tmes for ntrogen and oxygen are typcally of the order of 10 3 s and 10 5 s, respectvely, although the latter two are strongly dependent on humdty [31]. Humdty s relevant because water molecules act as catalysts for the relaxaton process: Collsons between water molecules and ntrogen or oxygen molecules has a hgher probablty of knockng the latter molecules nto hgher or lower vbratonal energy states [29], so that the re-equlbraton process occurs more quckly. Therefore, the relaxaton tmes τ m decrease wth humdty. Due to the very short rotatonal relaxaton tme, the approxmaton ω 0 τ m 1 s fully vald for all but extremely hgh frequences. Usng ths approxmaton n (2.53), wefndc c 0 and α x ω0 2. A comparson of the latter wth the ω 0 τ ν 1 approxmaton of the vscous absorpton coeffcent (2.37a) shows that they have the same form. In fact, several sources [28, 33, 34] suggest that bulk vscosty n dlute polyatomc gases s merely the low-frequency behavour of rotatonal relaxaton. A closer analyss of measurements of rotatonal relaxaton from ths perspectve gves a bulk vscosty n ar of ν B = 0.60ν [33, 34]. It s mportant to emphasse that the modellng of rotatonal relaxaton as a bulk vscosty s only vald at low frequences where ω 0 τ m 1. At hgher frequences, ths wll mspredct the sound wave dsperson from rotatonal relaxaton [22]. However, at the extremely hgh frequences where ths s a problem, the entre Naver-Stokes-Fourer model tself s cast nto doubt, as we wll see n Chapters 3 and 5. We end ths secton on relaxaton wth Fgure 2.7, whch shows the composton of the absorpton coeffcent for ar at representatve condtons. The calculaton was performed usng formulas from the relevant ISO standard [32], whch can also be found elsewhere [25, App. B]. From the fgure we see that the effects of the vbratonal relaxatons clearly domnate at audble frequences Acoustc multpoles and aeroacoustcs Acoustc multpoles are oscllatng sound sources that generate acoustc felds of varous drectonal radaton patterns. These sources can be ether *Note however that vscous absorpton s not a relaxaton process lke the ones descrbed here [26]. Multpole Any of several possble forms of oscllatng sources, for example monopoles, dpoles, and quadrupoles

46 38 Chapter 2 Fundamental theory αx [m 1 ] Frequency [Hz] Total Thermovscous Ntrogen vbr. relax. Oxygen vbr. relax. Fgure 2.7: Absorpton coeffcent and ts composton for ar at K, atmospherc pressure and 70 % relatve humdty. Rotatonal relaxaton for ntrogen and oxygen s represented as bulk vscosty. Green s functon The soluton of a dfferental equaton wth a delta functon nhomogenenty pont sources, localsed at sngle ponts n space, or they can be source denstes dstrbuted throughout the medum. The the frst three orders of multpoles are the most well-known: Monopoles, dpoles, and quadrupoles at zeroth, frst, and second order, respectvely. Before gong nto the causes of multpole sources, we wll look at how they are presented and dealt wth mathematcally. In general, multpole sources can be expressed by source terms n the wave equaton. For the deal wave equaton, we have ( 1 c t 2 2 ) p (x, t) =T 0 (x, t)+ T α(x, t) x α + 2 T αβ (x, t) x α x β (2.54) We shall see that T 0, T α, and T αβ represents the local source strength of monopoles, dpoles, and quadrupoles, respectvely. The soluton of ths equaton s based on Green s functons. In ths dervaton we wll consder tme-harmonc Green s functons, wth a tme dependence of e ωt. Ths Green s functon Ĝ s defned by ( 1 c t 2 2 ) Ĝ(x, t) =δ(x) e ωt, (2.55)

47 2.3 Acoustcs 39 where δ(x) s the Drac delta functon. Thus, the Green s functon s the pressure response to a tme-harmonc pont nhomogenty at x = 0. Alternatvely, the mpulsve Green s functon could be used [35]. Ths s defned smlarly to (2.55), but wth a dfferent source term, δ(x)δ(t), on the rght-hand sde. Thus, the mpulsve Green s functon s the response to an nhomogenty representng a sngle Drac pulse at x = 0 and t = 0. However, snce t s mpossble to fnd analytcal solutons on ths form for two dmensonal cylndrcal waves except as far-feld approxmatons or Fourer ntegrals [25, Ch. 1], we wl not consder the mpulsve Green s functon further. It s possble to use the Fourer transform to transform tme-harmonc solutons to more general solutons. In one, two and three dmensons, the tme-harmonc Green s functons are [36] 1D: Ĝ(x, t) = 1 2k e(ωt k x ), 2D: Ĝ(x, t) = 1 4 Ĥ(2) 0 (k x ) e ωt, 3D: Ĝ(x, t) = 1 4π x e(ωt k x ), (2.56a) (2.56b) (2.56c) where Ĥ n (2) s the nth order Hankel functon of the second knd. The Hankel functon s found from the correspondng Bessel functons of the frst knd J n and second knd Y n, as [24] Ĥ (2) n (x) =J n (x) Y n (x). (2.57) x = x 2 + y 2 + z 2 s the dstance from the orgn, often denoted as r. Now, let us consder a smplfed, tme-harmonc verson of (2.54), ( 1 c 2 0 ) 2 t 2 2 p(x, t) = ˆT 0 (x) e ωt. The source term can be seen as a dstrbuton of nfntely many ndvdual weak pont source terms ˆT 0 (y)δ(x y) dy, snce ˆT 0 (x) = ˆT 0 (y)δ(x y) dy. From the defnton of the Green s functon, each component source radates a pressure feld ˆT 0 (y)ĝ(x y, t) dy. Thus, we can fnd a general soluton by addng the contrbutons of all these ndvdual sources, p (x, t) = ˆT 0 (y)ĝ(x y, t) dy.

48 40 Chapter 2 Fundamental theory The pressure at the lstener pont x s thus found by ntegratng over the contrbutons of all source ponts y. To fnd the soluton of the source term wth a sngle spatal dervatve, we can smply let ˆT 0 (x) ˆT α (x)/ x α n the soluton, p (x, t) = ˆT α (y) Ĝ(x y, t) dy. y α Usng multdmensonal ntegraton by parts (2.11) and assumng that ˆT = 0 nfntely far away, ths becomes p (x, t) = Ĝ(x y, t) ˆT α (y) dy = y α Ĝ(x y, t) ˆT α (y) dy. x α A smlar soluton can be found found for the source term wth a double spatal dervatve. Thus, the tme-harmonc analogue of (2.54), ( ) [ ] 1 2 c 2 t p (x, t) = e ωt ˆT 0 (x)+ ˆT α (x) + 2 ˆT αβ (x) +..., x α x α x β (2.58) s solved by [ p (x, t) = Ĝ(x y, t) ˆT 0 (y)ĝ(x y, t)+ ˆT α (y) x α + ˆT αβ (y) 2 Ĝ(x y, t) x α x β +... ] dy. (2.59) Each term nsde the ntegral represents one type of multpole source, and the dfferent ˆT s ndcate the local source strengths of these. The dfferent dervatves of Ĝ have ther own angular dependence. The far-feld angular dependence patterns for some representatve dervatves are gven n Fgure 2.8. Ĝ tself s omndrectonal, radatng equally n all drectons; a monopole source. Applyng a spatal dervatve n x α drecton to Ĝ results n a radaton pattern known as an x α -dpole, wth a lobe n the x α drecton and a lobe of opposte polarty n the x α drecton. Applyng two spatal dervatves n x α and x β drectons, we get x α x β - quadrupoles. If the two dervatves are the same, we have a longtudnal quadrupole wth two lobes of the same polarty n opposng drectons. If the dervatves are dfferent, we have a lateral quadrupole wth four lobes of varyng polartes n the x α x β plane. Now that we know how to deal wth source terms n the wave equaton, we wll now look at where these source terms may orgnate from n the frst place. We wll later return to ths topc n Chapter 6, where the multpole source orgn wll be dfferent.

49 2.3 Acoustcs (a) Monopole (b) x-dpole (c) xx-quadrupole (d) xy-quadrupole Fgure 2.8: Far-feld (.e. x ) dependence on the angle to the x axs for (a) Ĝ, (b) Ĝ/ x, (c) 2 Ĝ/ x 2, (d) 2 Ĝ/ x y. Lobes wth negatve polarty are drawn wth dashed lnes. These angular dependence patterns hold for both the two- and threedmensonal Green s functons Ĝ, and ther common names are gven n the subcaptons. We begn wth the mass and momentum conservaton equatons (2.17) of the Euler model. On conservaton form, these are ρu α t ρ t + ρu α x α = Q, (2.60a) + ρu αu β x β = p + F. (2.60b) A mass source term Q wth unts kg/s m 3 has been added to the mass equaton. Ths can model the pulsatons of small bodes throughout the flud [24, Ch. 5][35, Ch. 1]. Dervng the wave equaton from these equatons as before, we get ( 1 c 2 0 ) 2 t 2 2 p = Q t F α + 2 ρu α u β. (2.61) x α x α x β By comparson wth equaton (2.54), we explctly fnd the multpole

50 42 Chapter 2 Fundamental theory source strengths T 0 = Q t, T α = F α, T αβ = ρu α u β. (2.62) We could also perform a more general dervaton by usng the Naver- Stokes-Fourer model and not usng the lnear approxmaton p = c 2 0 ρ. Ths would result n an exact quadrupole strength of [35, 37] T αβ = ρu α u β +(p c 2 0 ρ )δ αβ σ αβ. In most cases, however, the two last terms are neglgble, and the frst term can even be approxmated well as ρ 0 u α u β [35, 37]. Thus, monopoles are typcally caused ether by njecton of mass, or by pulsatons of small bodes whch push on the flud; dentcal waves are radated n the two cases f the same amount of mass s njected and pushed [24, Chs. 5 & 7]. Dpoles are typcally caused by body forces actng on the flud. Quadrupoles are typcally caused by areas wth rapd spatal varatons n ρu α u β, such as an area of turbulent flow. The fact that such flow generates sound s famlar from daly lfe; consder for nstance the nose from the turbulent flow of ar blown out of a mouth. The study of aerodynamc sound generaton s called aeroacoustcs. Aeroacoustc studes are especally mportant to fndng out how to reduce the nose generated by vehcles, n partcular arcraft. The study of aeroacoustc computer smulatons s called computatonal aeroacoustcs. Such studes started n the 1980s, and the two man types of methods used are: Hybrd methods, where the flow and acoustc felds are found separately. The ncompressble flow feld s analysed to fnd the acoustc source strength of the flow, whch s then used to compute the acoustc feld n the surroundng doman. Drect methods, where the acoustc feld comes out as a natural part of a numercal soluton to the compressble Naver Stokes or Euler equatons. Whle hybrd methods are usually far more effcent than drect methods, they cannot smulate the feedback of the acoustc feld on the flow feld. Therefore, hybrd methods cannot be used n cases where ths feedback s crtcal to the process of sound generaton, such as the sngng rsers problem currently studed n the natural gas ndustry [38 42]. It s also dffcult to capture complex geometres wth such methods. Drect methods have no such problems, but performng tradtonal numercal smulatons of the compressble Naver Stokes equaton s far more complex and computatonally demandng than usng hybrd methods. We can therefore hope that the lattce Boltzmann method wll

51 2.3 Acoustcs 43 prove to be an useful drect method for computatonal aeroacoustcs. In fact, several publcatons have been made on ths topc recently [42 50] Nonlnear acoustcs At hgh sound wave ampltudes, the assumptons behnd the lnearsaton performed prevously break down, and the equatons of sound propagaton become nonlnear.* The prmary consequence of ths nonlnearty s that the sound wave propagaton speed can no longer be assumed to be constant; peaks propagate more quckly and troughs more slowly. There are two separate mechansms behnd ths effect. The frst mechansm s the dependence of the sound speed c on local state varables. Smply put, a sound wave peak s compressed and therefore has a slghtly hgher temperature, whch leads to a slghtly hgher local sound speed. Conversely, a trough s rarefed and has a lower temperature and a lower sound speed. Evaluatng the speed of sound from the sentropc relaton (2.23) wthout assumng that the state varables are close to the rest state varables p 0 and ρ 0,wefnd ( ) p c 2 = = γp ( ) ρ γ 1 ( ) p (γ 1)/γ ρ ρ = c2 0 = c 2 0. (2.63) ρ 0 p 0 Wth some effort, the speed of sound can also be expressed exactly through the acoustc flud velocty [51][25, Ch. 2]. For a plane wave, c = c 0 + γ 1 u, (2.64) 2 the scalar u beng the flud velocty n the propagaton drecton of the sound wave. The second mechansm s self-advecton. The partcle velocty u of a sound wave, whch ponts towards and aganst the drecton of sound propagaton n peaks and troughs respectvely, also contrbutes to transport of the sound wave. Consequently, the total local wave propagaton speed s c + u, or where β s the coeffcent of nonlnearty, c + u = c 0 + βu, (2.65) β = γ In other words, peaks propagate more quckly and troughs propagate more slowly than the small-sgnal sound speed c 0. As a consequence, the *Ths nonlnearty also means that complex phasor notaton can no longer be used; t only works for lnear equatons.

52 44 Chapter 2 Fundamental theory Harmoncs Wave components wth frequences that are nteger multples of a fundamental frequency peaks may eventually catch up to the troughs, steepenng the sound wave and dstortng t nto a shock wave as shown n Fgure 2.9. Ths dstorton also results n the ntroducton of harmoncs. These harmonc components have hgher frequences than the fundamental and are consequently absorbed to a larger degree. In a way, the propagaton of hgh-ampltude sound waves may be seen as a competton between the effects of nonlnearty, whch cause dstorton and harmoncs, and the effects of absorpton, whch absorb the harmoncs and thus dampen the dstorton. Nonlnear acoustcs can be modeled by a number of dfferent equatons [51]. To derve these, some approxmatons must be made. Most mportantly, the effects of nonlnearty must be assumed to be suffcently small. Let us ntroduce a smallness order parameter ε, so that the fluctuatng feld quanttes of pressure p, densty ρ, velocty u, and so forth are all O(ε). Another parameter whch s small up to very hgh ultrasonc frequences s the acoustc vscosty number X = ω 0 τ ν. To derve reasonably smple model equatons, an approxmaton scheme that neglects all terms of hgher order of smallness than O(ε 2 ) and O(εX) must be used [51]. By comparson, n the lnearsaton process used n the prevous sectons we neglected all terms above the orders O(ε) and O(εX). Perhaps the smplest model equaton s the Burgers equaton, whch descrbes the nonlnear steepenng of a plane wave [51].* In dmensonless form, the Burgers equaton for the propagaton of a forced wave wth frequency ω 0 and ntal ampltude p s [52] p x p = p + 2 p α t x r t 2. r (2.66a) The tlde ndcates parameters whch have been reduced to nondmensonal form as p = p /p, x = x/x shock, t r = ω 0 t r, α x = α x x shock, (2.66b) where x shock s the shock formaton dstance,.e. the dstance t takes for the peak to catch up to the trough for a plane wave n a lossless medum, gven by x shock = ρ 0c 3 0 p βω 0. (2.66c) By rewrtng the equaton nto ths dmensonless form, we see that t reles only on the ntal condton p(0, t r ) and the nondmensonalsed absorpton coeffcent α x. *The Burgers equaton s not lmted to acoustcs; t s used as a model equaton for shock waves n other felds as well.

53 2.3 Acoustcs 45 p = p /p x = 0 x = 0.25 x = 0.5 x = 0.75 x = 1 x = 1.25 x = t r = ω 0 t k 0 x Fgure 2.9: Nonlnear dstorton of a sound wave; the soluton to the dmensonless Burgers equaton (2.66a) for α x = 0.01 and M = 300. One method to solve the Burgers equaton nvolves a transformaton to the frequency doman. The soluton varable p can be wrtten as a superposton of frequency components, [ ] p( x, t r )= Re p n ( x) e n t r, (2.67) n=1 p n ( x) beng the complex ampltudes of these frequency components. Truncatng the number of components to M, the Burgers equaton can be transformed nto a set of coupled ordnary dfferental equatons for the components p n [52], d p n d x = n2 α x p n + n 4 ( n 1 m=1 p m p n m + 2 M m=n+1 ) p m p m n. (2.68) Ths set of ODEs can be solved usng a number of standard methods. Here, Matlab s fourth-order Runge-Kutta method wth adaptve tme steps (the ode45 functon) was used. The results for α x = 0.01 and an ntal condton p 1 (0) =1, p n =1 (0) =0 are shown n Fgure 2.9.

54 3 The knetc theory of gases A gas at rest s not merely a statc contnuum. Instead, t s teemng wth actvty at the molecular level: Most of the molecules fly around at speeds hgher than the speed of sound, and a sngle molecule typcally colldes wth bllons of other molecules wthn the space of a sngle second. In prncple, ths behavour can be descrbed by trackng every sngle molecule n the gas and calculatng how they collde wth each other. Ths s naturally mpossble n practce for many reasons, e.g. the enormous number of partcles and the lack of knowledge of ther ntal states. Knowng the poston (x, y, z) =x and velocty (ξ x, ξ y, ξ z )=ξ = dx/dt of a sngle molecule means knowng 6 dfferent varables. Consequently, knowng the poston and velocty of N molecules means knowng 6N varables. In addton, f the molecules have nternal structure as descrbed n secton 2.3.3, varables descrbng ther rotatonal and vbratonal state are also necessary. Consderng that a sngle gram of oxygen conssts of over molecules, t becomes clear that t s mpossble to descrbe a tangble amount of gas at the mcroscopc level of detal. Instead of attemptng a full descrpton of the gas, we can attempt a statstcal one, where we consder dstrbutons of partcles; essentally expectaton values of the densty of partcles that have a partcular poston and velocty.* We go from a mcroscopc descrpton to a mesoscopc descrpton of the gas, as descrbed n Chapter 1. In ths chapter, a short ntroducton wll be gven to ths feld, whch s known as the knetc theory of gases. We wll start by lookng at the statstcal descrpton of the gas and how any dstrbuton of partcles may be connected to the macroscopc varables famlar from flud mechancs. Then the Boltzmann equaton, whch descrbes the evoluton of these partcle dstrbutons, wll be derved. Fnally, we wll see how the conservaton equatons of flud mechancs may be derved from ths equaton. It s necessary to make certan assumptons about the gas to keep the followng dscusson at a suffcently smple and nstructve level. Frst, we wll assume that the molecules are all dentcal and consst of only one atom. If the molecules have no detaled nternal structure, *For a clear yet bref explanaton of the path from the full descrpton to the statstcal descrpton, see e.g. [53]. 46

55 3.1 The dstrbuton functon and ts moments 47 ξ z dξ z dx ξ ξ y x y ξ x x Fgure 3.1: Vectors and nfntesmal volumes n velocty space (left) and physcal space (rght). all ther knetc energy s translatonal, or coupled to ther velocty; they cannot rotate or vbrate as descrbed n secton The knetc theory of polyatomc gases s sgnfcantly more complcated, and may be found elsewhere [54 58]. Second, Bohr s correspondence prncple states that the quantum behavour of a system reduces to classcal behavour when the system becomes large enough. We can therefore largely dsregard quantum mechancal effects and assume classcal behavour at the statstcal level of descrpton. The prmary goal of ths chapter s to gve an ntroducton to the aspects of knetc theory that are the most relevant for the lattce Boltzmann method. We wll cover the basc prncples, the smplest model for collsons between partcles, and the lnk to flud mechancs. Dscusson of less relevant aspects wll be kept to a mnmum. 3.1 The dstrbuton functon and ts moments The dstrbuton functon f (x, ξ, t) can be seen as a generalsaton of densty; Dstrbuton functon, f t represents the densty of partcles n both physcal space and velocty space smultaneously. In other words, f (x, ξ, t) ndcates the densty of partcles wth poston x and velocty ξ at tme t. The dstrbuton functon lets us fnd other, more famlar quanttes. f (x, ξ, t) dξ s the spatal densty of partcles whch have veloctes whch le nsde an nfntesmal velocty space volume dξ at ξ. Furthermore, f (x, ξ, t) dx dξ s the mass of partcles wth such veloctes and postons whch le nsde an nfntesmal physcal space volume dx at x. Ths dstrbuton functon s suffcent to fnd macroscopc propertes such as flud densty, flud velocty, and nternal energy n the flud. These propertes can be found as moments, where the dstrbuton functon f (x, ξ, t) s weghted wth some functon of ξ and ntegrated over the Indcates the densty of partcles wth poston x and velocty ξ at tme t Moment An ntegral over all veloctes wth f weghted wth a functon of velocty as the ntegrand

56 48 Chapter 3 The knetc theory of gases entre velocty space. These moments, whch lnk the mesoscopc and macroscopc scales, are somewhat smlar to the equatons (1.1) whch lnk the mcroscopc and macroscopc scales. For the smplest moment we do not weght wth anythng. As mentoned prevously, f (x, ξ, t) dξ s the spatal densty of partcles that have veloctes wthn a certan nfntesmal velocty range. Thus, f we ntegrate over all the veloctes,.e. the entre velocty space, we get the Mass densty, ρ physcal mass densty, most commonly smply called densty, Mass per physcal volume n kg/m 3 ρ(x, t) = f (x, ξ, t) dξ. (3.1) Now, f we weght wth ξ, we get ξ f (x, ξ, t) dξ, whch s the momentum densty of the partcles n the gven velocty range. Integratng over all the Momentum densty, ρu veloctes gves us the total momentum densty, Momentum per physcal volume n kg/s m 2 ρu(x, t) = ξ f (x, ξ, t) dξ. (3.2) Flud velocty, u The average velocty of the partcles n m/s Here, u s the average velocty of the partcles, whch corresponds to the flud velocty n flud mechancs. Smlarly, weghtng wth 1 2 ξ 2 and ntegratng gves the energy densty, Energy densty, ρe Knetc energy per physcal volume n J/m 3 ρe(x, t) = 1 2 ξ 2 f (x, ξ, t) dξ. (3.3) Specfc energy, E Knetc energy per mass n J/kg Internal energy densty, ρe The energy densty component due to random thermal movement of partcles n the gas Here, E s the specfc energy. We have assumed here that the knetc energy s gven only by the translatonal movement of the partcles; ths s only correct f the gas s monatomc, as we earler assumed. For a polyatomc gas, there would also be contrbutons from rotatonal and vbratonal energy, whch cannot be as easly represented through f (x, ξ, t). Ths total knetc energy may be splt up nto two parts: The knetc energy densty due to the bulk movement of the flud, 1 2 ρ u 2, and the nternal energy densty ρe, whch s due to random thermal movement of partcles, and s ndependent of the flud velocty u. Thus, we can wrte ( ρe = ρ e u 2). (3.4) Here, e s the specfc nternal energy. As an example of nternal energy, let us look at a somewhat unrealstc case where all partcles move n the same drecton, so that the dstrbuton functon s a Drac delta functon, f (x, ξ, t) =ρδ(ξ u). Usng ths to fnd the energy densty, we fnd ρe = 1 2 ρ u 2. Comparng wth (3.4), we see

57 3.1 The dstrbuton functon and ts moments 49 ξ z v ξ u ξ y ξ x Fgure 3.2: The velocty ξ s splt nto the flud velocty u and the pecular velocty v. that ρe = 0 n ths case where all partcles have the same velocty. Thus, we see that the nternal energy comes from the partcle veloctes devaton from the mean. We wll see later n secton 3.3 that ths devaton and the nternal energy grows larger wth temperature. It s useful to splt the partcle velocty ξ nto two components, as shown n Fgure 3.2: The flud velocty u and the pecular velocty v, ξ = v + u. (3.5) Pecular velocty, v The devaton of the partcle velocty ξ from the flud velocty u Because t descrbes the devaton from the mean velocty, the pecular velocty cannot contrbute to the momentum. Ths may also be shown mathematcally, v f dξ = ξ f dξ u f dξ = ρu ρu = 0. (3.6) Here the flud velocty has been moved outsde the ntegral, as u(x, t) does not depend on ξ. In (3.3), we can substtute wth (3.5). Usng (3.6) and (3.1), we fnd ρe = 1 2 (v + u) (v + u) f (x, ξ, t) dξ = 1 2 v 2 f dξ + u v f dξ u 2 f dξ v 2 f dξ ρ u 2. = 1 2 By comparson wth (3.4) we fnd the moment equaton for the nternal energy densty, ρe(x, t) = 1 2 v 2 f (x, ξ, t) dξ. (3.7)

58 50 Chapter 3 The knetc theory of gases v dz dy Fgure 3.3: A sngle partcle bounces elastcally off a surface da = dy dz. In the tme t takes for the partcle to reach the surface, all the partcles wth the same velocty nsde the upper collson prsm around the partcle s path wll also have ht the surface, and wll have bounced nto the lower collson prsm. 3.2 Pressure and heat Pressure, p Force per area exerted by the flud on a real or magnary surface n N/m 2 Fluds exert a force on adjacent surfaces. The area densty of ths force s gven by the pressure p(x, t) at the surface. In general, the pressure s an nternal thermodynamc quantty of the flud. It ndcates the force densty whch would be exerted on a surface placed at that pont, f that surface s at rest relatve to the flud. The actual, mcroscopc reason why fluds exert a pressure on surfaces s that the partcles n the flud keep bouncng off the surface. Snce momentum s always conserved, the momentum change mparted to the surface s equal and opposte to the momentum change of a partcle bouncng off t. In the followng, we wll make the assumpton that the dstrbuton functon s approxmately sphercally symmetrc around v = 0, so that all velocty drectons are equally probable, and the pecular velocty does not tend to any partcular drectons. Ths assumpton wll be dscussed further n secton 3.3. Let s say that we have placed a surface n the y z-plane at rest relatve to the flud, and that a partcle bounces elastcally off t, as shown n Fgure 3.3. Before and after the collson, the partcle s momentum relatve to the wall n the x-drecton s mv x and mv x respectvely, havng changed by 2mv x n the collson. The same momentum change s mparted to the wall. We can use the dstrbuton functon to fnd the total momentum

59 3.2 Pressure and heat 51 change mparted to the wall over an nfntesmal tme perod dt. If the aforementoned partcle has a velocty ξ, the total mass of the partcles wth smlar veloctes httng the surface da = dy dz n the perod dt s f (x, ξ, t) dξ dx. Here, dx = dx dy dz = v x dt dy dz s the volume of the collson prsm shown n Fgure 3.3. Thus, the total mass of partcles httng the wall wth such veloctes s v x f (x, ξ, t) dξ dt dy dz. To fnd the total momentum mparted to the surface by these partcles, we multply ther mass by ther velocty change 2v x. Thus, ther total mparted momentum change to the surface s 2v 2 x f (x, ξ, t) dξ dt dy dz. (3.8) Of course, ths s only an nfntesmal part of all possble veloctes. Ths surface may be ht by partcles wth any velocty ξ where v x < 0,.e. the veloctes that pont toward the surface. To fnd the momentum change mparted by all partcle collsons, we must ntegrate (3.8) over the velocty half-space where v x < 0. Alternatvely, snce we have assumed that f s sphercally symmetrc around v = 0, we can smply ntegrate over the entre velocty space and dvde by 2. Thus, the momentum change mparted to the wall by all partcle collsons durng the perod dt s dt dy dz v 2 x f (x, ξ, t) dξ. To get the force from the momentum change, we smply need to dvde by dt, as per Newton s second law. To get the pressure from the force, we dvde by the area dy dz. Thus, the pressure felt by the surface n Fgure 3.3 s p = v 2 x f (x, ξ, t) dξ. Snce we have assumed that the partcles are dstrbuted symmetrcally about v = 0, we could nstead have placed the wall n Fgure 3.3 n the x z-plane or the x y-plane and found equvalent defntons for the pressure wth v 2 y and v 2 z nsde the ntegral, respectvely, nstead of v 2 x. In fact, we can take the average of these three equvalent defntons, replacng v 2 x wth 1 3 (v2 x + v 2 y + v 2 z)= 1 3 v 2. Havng done ths, the pressure ntegral s on a famlar form, p = 1 3 v 2 f (x, ξ, t) dξ = 2 3 ρe. (3.9) The last equalty follows from comparson wth (3.7). We have thus found that the pressure of a gas s proportonal to ts nternal energy densty!

60 52 Chapter 3 The knetc theory of gases Specfc gas constant, R The proportonalty constant n the deal gas law, n J/kg K Temperature, T A thermodynamc quantty related to e, n K Boltzmann s constant, k B A physcal constant relatng e and T, approx J/K Ths mght also have been expected beforehand; wth a hgher nternal energy densty, partcles tend to move faster. Faster-movng partcles bounce off a surface harder and more often. Equaton (3.9) s an equaton of state for the gas, relatng the pressure p, the densty ρ and the specfc nternal energy e. We may compare ths to the deal gas law [19, Ch. 2.5], p = ρrt = ρ k BT m, (3.10) where R = k B /m s the specfc gas constant, T s the temperature, and k B s Boltzmann s constant. Combnng equatons (3.9) and (3.10), we fnd e = 3 2 p ρ = 3 2 RT = 3 k B T 2 m, (3.11) whch relates the specfc energy wth the other thermodynamc quanttes. Ths equalty could also be predcted by the equpartton theorem of statstcal mechancs prevously descrbed n secton 2.3.3, whch predcts an average molecular energy me of 1 2 k BT for each degree of freedom [19, Ch. 2]. Our monatomc gas has three degrees of freedom, one for each coordnate n three-dmensonal space. More complcated molecules have addtonal degrees of freedom, as they may also rotate and vbrate. We may use (3.11) to calculate the heat capactes for ths gas [19, Ch. 2]. The specfc heat capacty at constant volume s Specfc heat capacty at constant volume, c V The rate of ncrease of e ( ) aganst T f the volume (or ρ) e s kept constant, n J/kg K c V = = 3 T V 2 R = 3 k B 2 m. (3.12) Specfc heat capacty at constant pressure, c p The rate of ncrease of the enthalpy h = e + p/ρ aganst T f the pressure s kept constant, n J/kg K Heat capacty rato, γ The dmensonless rato c p /c V The specfc heat capacty at constant pressure can be found through a property of deal gases [19, Ch. 2], c p c V = R = k ( ) B (e + p/ρ) m c p = = 5 T p 2 R = 5 k B 2 m. (3.13) The rato between the two heat capactes s often more useful than the heat capactes themselves. For our monatomc deal gas, t s found from (3.12) and (3.13) to be γ = c p c V = 5 3, (3.14) whch fts well wth tabulated values for noble gases (see e.g. [19, App. F]). As we saw n secton 2.3.1, the speed of sound n a gas depends on ths heat capacty rato.

61 3.3 Equlbrum 53 ξ z v u ξ y ξ x Fgure 3.4: A sphercal sosurface of a dstrbuton functon whch s sphercally symmetrc around ξ = u. At any pont on ths sosurface, f has the same value. 3.3 Equlbrum When two partcles collde, ther veloctes are changed. Ther new veloctes depend on ther pre-collson postons and veloctes and the ntermolecular forces durng the collson. However, we can assume that collsons tend to dstrbute the partcles veloctes evenly n all drectons around ther mean velocty u. Ths means that f we take a gas of partcles wth any ntal dstrbuton and leave t for long enough, t wll eventually reach an equlbrum state where all drectons of the pecular velocty v are equally probable.* The Maxwell-Boltzmann dstrbuton Ths even dstrbuton of veloctes means that the dstrbuton functon s only dependent on the pecular velocty. The dstrbuton functon has the same value for pecular veloctes where v 2 = v 2 x + v 2 y + v 2 z s the same, as shown n Fgure 3.4. We can therefore smplfy the notaton for the dstrbuton functon at equlbrum to f (0) ( v ). We also assume that the dstrbuton functon s separable n the dfferent v coordnates, so that f (0) ( v ) = f (0) x (v x ) f (0) y (v y ) f (0) z (v z ). For a constant v 2, f (0) ( v ) s constant, and ln f (0) x (v x )+ln f (0) y (v y )+ln f (0) z (v z )=const.. *There s an nterestng paradox hdden here. At the mcroscale, where we look at the nteracton of ndvdual partcles, the Newtonan dynamcs of the partcles are tme reversble: Flppng all the partcle veloctes would cause the system to retrace ts steps back n tme. Why, then, does f always go to equlbrum? The short answer s that f s part of a statstcal descrpton; the mcroscopc system tends towards dsorder, and the equlbrum state s the most dsordered state. A fuller explanaton s found n [58, III.9]. Equlbrum dstrbuton functon, f (0) The dstrbuton functon of a gas that has been left undsturbed for long enough

62 54 Chapter 3 The knetc theory of gases Ths s solved only f the equlbrum dstrbutons for the dfferent drectons are of the form ln f x (0) (v x )=a bv 2 x f x (0) (v x )= e a e bv2 x, where a and b are two constants that are ndependent of v. Thus, the equlbrum dstrbuton s fn the form f (0) ( v ) = e 3a e b(v2 x+v 2 y+v 2 z) = e 3a e b v 2. (3.15) The constants a and b can be determned from the moments of f (0). Frst, we fnd the moment of densty, ρ = f (0) ( v ) dξ = e 3a e bv2 x dv x e bv2 y dv y e bv2 z dv z ( = e 3a π ) 3 2. b We see that e 3a = ρ(b/π) 3/2, and the equlbrum dstrbuton becomes ( ) 3 f (0) b 2 ( v ) =ρ e b v 2. (3.16) π Fnally we can determne b usng the moment of energy. Snce f (0) ( v ) s sphercally symmetrc around v = 0, we can perform the substtuton dξ = 4π v 2 d v. The ntegral s ρe = ( ) 3 v 2 f (0) b ( v )4π v 2 2 d v = 2ρπ v 4 e b v 2 d v = 3ρ π 4b, 0 whch lets us dentfy b usng several dfferent thermodynamc quanttes, b = 3 4e = ρ 2p = m 2k B T. (3.17) The two last equaltes follow from (3.11). Thus, the equlbrum dstrbuton has the possble forms ( f (0) 3 ( v ) =ρ 4πe ( m = ρ 2πk B T ) 3 ( 2 e 3 v 2 ρ /4e = ρ ) 3 2 e m v 2 /2k B T, 2πp ) 3 2 e ρ v 2 /2p (3.18) dependng on whch thermodynamc varables we use to descrbe t. Ths equlbrum dstrbuton s called the Maxwell-Boltzmann dstrbuton. It was frst found by James Clerk Maxwell usng a dervaton smlar to the one we have used here, and was later found by Ludwg Boltzmann usng more rgorous statstcal mechancs.

63 3.3 Equlbrum 55 4π v 2 f (0) /ρ T = 75 K T = 150 K T = 300 K T = 600 K T = 1200 K v Fgure 3.5: The normalsed dstrbuton of radal veloctes n a neon gas at equlbrum Pecular velocty moments at equlbrum It wll later be necessary to know several dfferent v-weghted moments of the equlbrum dstrbuton. We wll now fnd the frst seven general moments of ths knd, wth f (0) on the general form gven by (3.16). The parameter b can be related to the macroscopc state varables through (3.17). The dfferent moments of f (0) are tensors of dfferent orders. These are most convently descrbed usng ndex notaton, whch has been prevously ntroduced n secton 2.1. The zeroth-order moment tensor s already famlar. It s smply the densty, f (0) dξ = ρ. (3.19a) The frst-order moment has already found for a general dstrbuton functon n (3.6). For the equlbrum specal case f = f (0) t s stll the same, v α f (0) dξ = 0. (3.19b) Ths could also have been seen from symmetry consderatons: Snce f (0) s symmetrc around v = 0 and v α s antsymmetrc around v α = 0, the ntegrand s antsymmetrc, and the ntegral s zero. Ths s the case for all of the odd moments, as the product of an odd number of antsymmetrc functons s always also antsymmetrc. Snce f (0) s rotatonally nvarant around v = 0, the moment ntegrals are ndependent of the coordnate axes n velocty space, and the resultng tensors must be sotropc, whch means that t s ndependent of the choce of coordnate system. For example, the tensor δ αβ s one f both ndces are the same, whch s true for any coordnate system. Isotropc tensor A tensor whch has dentcal components for any choce of orthogonal coordnate system

64 56 Chapter 3 The knetc theory of gases The general second-order sotropc tensor s C 1 δ αβ [59], where C 1 s a constant. The constant can be found by calculatng the ntegral vx v x f (0) dξ, and we fnd that the second-order moment tensor s v α v β f (0) dξ = ρ 2b δ αβ. (3.19c) The thrd-order moment tensor s v α v β v γ f (0) dξ = 0, (3.19d) due to antsymmetry. The general form of the fourth-order sotropc tensor s C 1 δ αβ δ γδ + C 2 δ αγ δ βδ + C 3 δ αδ δ βγ [59]. Snce the moment tensors are nvarant wth respect to ndex order (e.g. changng β and γ makes no dfference, as vx v x v y v y f (0) dξ = v x v y v x v y f (0) dξ), these constants must be equal, and can be found by takng the ntegral v x v x v y v y f (0) dξ. The fourthorder moment tensor s therefore v α v β v γ v δ f (0) dξ = ρ (2b) 2 (δ αβδ γδ + δ αγ δ βδ + δ αδ δ βγ ). (3.19e) The ffth-order moment tensor s also antsymmetrc, whch means that v α v β v γ v δ v ɛ f (0) dξ = 0. (3.19f) Smlar to the approach taken to fnd the second and fourth moment tensors, the somewhat unweldy sxth-order moment tensor can be found to be v α v β v γ v δ v ɛ v ζ f (0) dξ (3.19g) = ρ (2b) 3 [ δ αβ δ γδ δ ɛζ + δ αβ δ γɛ δ δζ + δ αβ δ γζ δ δɛ + δ αγ δ βδ δ ɛζ + δ αγ δ βɛ δ δζ + δ αγ δ βζ δ δɛ + δ αδ δ βγ δ ɛζ + δ αδ δ βɛ δ γζ + δ αδ δ βζ δ γɛ + δ αɛ δ βγ δ δζ + δ αɛ δ βδ δ γζ + δ αɛ δ βζ δ γδ + δ αζ δ βγ δ δɛ + δ αζ δ βδ δ γɛ + δ αζ δ βɛ δ γδ ]. The seven moment tensors whch we have seen show a clear pattern: The odd-order moment tensors are zero, whle the moment tensors wth even order 2n are ρ/(2b) n multpled wth the sum of all Kronecker delta product permutatons of the 2n dfferent ndces. 3.4 The Boltzmann equaton So far we have talked about the dstrbuton functon, ts moments, and ts value at equlbrum, but we stll know nothng about how t actually evolves wth tme. We wll therefore now derve the equaton that descrbes the evoluton of the dstrbuton functon.

65 3.5 The collson operator 57 Fgure 3.6: Collson between two molecules modeled as hard spheres. The dstrbuton functon s a functon of x, ξ, and t. Therefore, ts total dfferental wth respect to t s ( ) ( ) ( ) d f f dt = dxα f dξα f dt + + x α dt ξ α dt t dt. dx α /dt s the partcles velocty ξ α.dξ α /dt s ther acceleraton, whch by Newton s second law s gven by the body force densty as dξ α /dt = F α /ρ. Body force densty, F Thus, the equaton can be rewrtten as ( ) f f t + ξ α + F ( ) α f = d f x α ρ ξ α dt. If the rght hand sde of ths equaton s zero, the equaton becomes a sort of advecton equaton, descrbng collsonless propagaton of the partcle dstrbuton f. Ths propagates wth the velocty of ts partcles, ξ, whch s tself affected by the force F. In general, the rght hand sde d f /dt s a source term that ndcates the rate of change of f, due to collsons causng partcles to change ther drectons. Snce collsons can only happen between partcles whch are at the same place at the same tme, d f (x, ξ, t)/dt for a partcular choce of x, ξ, and t depends on the dstrbuton functon f for all ξ at the same x and t. Wrtng ths equaton on vector form wth d f /dt rewrtten as the collson operator Ω( f ), wefnd f t + ξ f + F ρ ξ f = Ω( f ). (3.20) Here, ξ f s the gradent of f n velocty space. Ths equaton s called the Boltzmann equaton after Ludwg Boltzmann, who devsed t n the late 19th century. Force densty of long-range forces, e.g. gravty, n N/m 3 Collson operator, Ω An operator whch can be appled to f to gve ts rate of change 3.5 The collson operator The collson operator may have many dfferent forms, as long as t fulfls certan condtons. Three quanttes are always conserved n a collson:

66 58 Chapter 3 The knetc theory of gases Mass, momentum, and, f the collsons are elastc,* translatonal energy. These conservaton condtons are expressed mathematcally as Mass conservaton: Momentum conservaton: Ω( f ) dξ = 0, ξ Ω( f ) dξ = 0, Energy conservaton: { ξ 2 Ω( f ) dξ = 0, v 2 Ω( f ) dξ = 0. (3.21a) (3.21b) (3.21c) The two energy conservaton condtons are equvalent, whch can be shown usng the dentty (2.9), v 2 = ξ u 2 = ξ 2 2ξ u + u 2, and the mass and momentum conservaton condtons. Another crteron for collson operators s that they must ensure that the dstrbuton functon always evolves towards equlbrum. We wll not dscuss ths further before secton 3.9, except statng that ths crteron s fulflled by all collson operators dscussed n ths secton. Boltzmann s orgnal collson operator s of the form of a complcated double ntegral over velocty space. It essentally consders the outcome of all possble two-partcle collsons, for any choce of ntermolecular forces between the partcles. Ths collson operator fulfls condtons (3.21), but s very cumbersome. Alternatve collson models were later proposed. The goal was to fnd a collson model that was smpler than Boltzmann s orgnal one, but whch stll gave a largely correct macroscopc behavour. In the scentfc feld of lattce Boltzmann methods, varants of the BGK collson operator, Ω( f )= 1 τ ( f f (0)), (3.22) Relaxaton tme, τ A tme constant ndcatng how quckly a gas relaxes towards an equlbrum are generally used. Here, τ s called the relaxaton tme. Ths collson operator proposed by Bhatnagar, Gross, and Krook n 1954 as a very smple model for partcle collsons [60]. As dscussed n secton 3.3, the partcle dstrbuton wll tend to relax to an equlbrum. The BGK operator captures ths behavour by drectly modelng the relaxaton process nstead of attemptng to follow the detals of the collsons. *For the monatomc gas we are consderng, collsons are elastc. However, for polyatomc gases as consdered n secton 2.3.3, collsons may be nelastc or superelastc due to energy beng transferred between translatonal and nner (.e. rotatonal or vbratonal) degrees of freedom. In that case, the conservaton condton must apply to the total energy,.e. the sum of translatonal and nner energy, as n e.g. [55].

67 3.6 Macroscopc conservaton equatons 59 The BGK operator can easly be shown to satsfy conservaton of mass, momentum, and energy. As the equlbrum dstrbuton f (0) has the same moments of densty, momentum, and energy as the dstrbuton functon f, the BGK operator (3.22) satsfes (3.21). As an example, say that we have a dstrbuton functon whch s spatally homogeneous ( f (ξ, t) =0), but n a non-equlbrum state at t = 0. Neglectng external forces, the Boltzmann equaton for ths case s f (ξ, t) t = 1 τ ( ) f (ξ, t) f (0) (ξ), whch s solved by ( ) f (ξ, t) = f (0) (ξ)+ f (ξ,0) f (0) (ξ) e t/τ. Ths soluton descrbes a dstrbuton functon that relaxes exponentally to the equlbrum dstrbuton wth a tme constant τ. From the results that wll be found n secton 3.8.3, t can be shown that τ for ordnary gases lke ar at room temperature. However, beng only a smplfed model of the result of partcle collsons n a gas, the BGK collson operator s not as exact as Boltzmann s orgnal collson operator. For nstance, the Boltzmann equaton wth the BGK operator predcts a Prandtl number Pr = 1, whereas wth Boltzmann s collson operator we get a value Pr 2/3, whch agrees wth experments wth monatomc gases [58, Ch.II]. Ths wll be dscussed further n secton Prandtl number, Pr A dmensonless number relatng the strengths of vscosty and thermal conductvty 3.6 Macroscopc conservaton equatons By takng the approprate moments of the Boltzmann equaton (3.20) we can fnd general conservaton equatons for the three collson nvarants of mass, momentum, and energy. It wll be useful to defne a common notaton for all the moments of f ; Π 0 = Π αβ = f dξ = ρ, Π α = ξ α ξ β f dξ, Π αβγ = ξ α f dξ = ρu α, ξ α ξ β ξ γ f dξ, (3.23) and so forth. From ther defntons, the moment tensors are clearly nvarant wth a swtch of ther ndces, e.g. Π xy = Π yx. For the force term t wll be useful to know some moments of f / ξ α. Usng multdmensonal ntegraton by parts (2.11), these can be found to

68 60 Chapter 3 The knetc theory of gases be f ξ α dξ = 0, ξ α f ξ β dξ = ξ α ξ α f ξ β dξ = ξα ξ β f dξ = ρδ αβ, (3.24a) (3.24b) ξα ξ α ξ β f dξ = 2ρu β. (3.24c) The surface ntegrals vansh usng the assumptons that f, ξ α f, and ξ α ξ α f vanshes as ξ. These assumptons are qute safe, as they are necessary for the conserved quanttes to be fnte. Also, t s reasonable to assume that f wll not be very far off n any case from the equlbrum f (0), whch goes exponentally to zero Mass conservaton We frst take the zeroth moment of all terms n the Boltzmann equaton (3.20), t f dξ + x α ξ α f dξ + F α ρ f dξ = ξ α Ω( f ) dξ. Snce t and x are not functons of ξ, ther dervatves have been moved outsde the ntegrals. Also, ξ does not depend on x as t s merely a coordnate n velocty space, and we have used that ξ α f / x α = (ξ α f )/ x α. The ntegrals on the left hand sde are the densty moment (3.1), the momentum moment (3.2), and zero (by 3.24a), respectvely. The rght sde s also zero, by conservaton of mass (3.21a). Thus, the zeroth moment of the Boltzmann equaton becomes ρ t + ρu α x α = 0, (3.25) whch s exactly the contnuty equaton. Ths equaton does not depend on the specfc form of the dstrbuton functon f, only ts conserved moments Momentum conservaton Takng the frst moment of the Boltzmann equaton and usng the same assumptons as for the zeroth moment, we get t ξ α f dξ + x β ξ α ξ β f dξ + F β ρ f ξ α dξ = ξ β ξ α Ω( f ) dξ.

69 3.6 Macroscopc conservaton equatons 61 By equatons (3.2), (3.23), (3.24b), and (3.21b), ths reduces to ρu α t + Π αβ x β = F α. (3.26) The moment Π αβ = ξ α ξ β f dξ can be nterpreted as the flow n the α drecton of the momentum component n the β drecton, whch s ξ β f dξ. The drectons can also be vce versa, as Π αβ s symmetrc. It can be resolved nto two parts by usng ξ α ξ β =(u α + v α )(u β + v β ) and the fact that moments of only the pecular velocty are zero: Π αβ = (u α u β + u α v β + v α u β + v α v β ) f dξ = ρu α u β σ αβ. (3.27) Here, the frst term, ρu α u β, represents the macroscopc flow of momentum, and the second term, σ αβ = v α v β f dξ, (3.28) represents a dffuson of momentum. Thus, the frst moment of the Boltzmann equaton results n the conservaton form of the Cauchy momentum equaton (2.18), ρu α t + ρu αu β x β = σ αβ x β + F α. (3.29) σ αβ can be dentfed as the Cauchy stress tensor, whch completely defnes the state of stress at any pont n the flud. We see from ts defnton (3.28) that t s symmetrc,.e. σ αβ = σ βα. The Cauchy stress tensor (3.28) s determned by the form of f, and we can therefore not fnd a fully macroscopc momentum conservaton equaton before we know more about f. Cauchy stress tensor, σ A second-order tensor whch at any pont n the flud specfes the normal and shear stresses n the x, y, and z drectons Energy conservaton To fnd an energy conservaton equaton, we take the ξ β ξ β moment of the Boltzmann equaton, ξ t β ξ β f dξ + ξ α ξ x β ξ β f dξ + F α f ξ α ρ β ξ β dξ = ξ ξ β ξ β Ω( f ) dξ. α By dvdng by 2 and usng equatons (3.3), (3.23), (3.24b), ths reduces to ρe t Π αββ x α = F α u α. (3.30) As 1 2 ξ βξ β f dξ = 1 2 ξ 2 f dξ s related to the translatonal energy of partcles, the moment Π αββ represents the flow of energy n the α drecton.

70 62 Chapter 3 The knetc theory of gases It may be resolved to another form, 1 2 Π αββ = 1 2 (u α u β u β + u α v β v β + 2v α v β u β + v α v β v β ) f dξ = 1 2 ρu α u 2 + ρu α e u β σ αβ + q α = ρu α E u β σ αβ + q α, where the frst term represents macroscopc advecton of energy, the second term can be dentfed as related to the work done by the Cauchy stresses, and the last term, q = 1 2 v v 2 f dξ, (3.31) represents the dffuson of energy. Thus, ths moment of the Boltzmann equaton results n a conservaton equaton for the energy, ρe t + ρu αe x α = σ αβu β x α + F α u α q α x α. (3.32) By subtractng u α (3.29) from ths (carefully swtchng some of the repeated ndces), we can fnd a smpler conservaton equaton for the nternal energy, ρe t + ρu αe x α = σ αβ u β x α q α x α. (3.33) Heat flux, q Macroscopcally; the flow rate of heat. Mcroscopcally; the rate of dffuson of energy. In J/s m 2. Ths s a general form of the energy equatons gven n secton 2.2. The frst term on the rght hand sde descrbes the ncrease n energy due to work done by the stresses, whereas the last term, whch descrbes dffuson of energy, can be dentfed wth the heat flux vector from the macroscopc energy equaton. In ths equaton, both the Cauchy stress σ and the heat flux q depend of the form of the dstrbuton functon. Lke the momentum conservaton equaton, the energy equaton s not fully determned untl we know more about the form of f. 3.7 Equlbrum: The Euler model The smplest assumpton we can make about the form of the dstrbuton functon s that t s always at equlbrum,.e. f f (0). (In the next secton we wll see that ths corresponds to assumng that the dstance between gas partcles s very small compared to any relevant macroscopc length.) It s then smple to fnd an equlbrum expresson for the stress tensor, σ (0), and for the heat flux, q (0).

71 3.8 The Chapman-Enskog expanson 63 Referrng to the moments of the equlbrum dstrbuton found n secton 3.3.2, we fnd that the stress tensor and heat flux at equlbrum are = v α v β f (0) dξ = pδ αβ (3.34) σ (0) αβ and q (0) α = 1 2 v α v β v β f (0) dξ = 0, (3.35) respectvely. Wth the assumptons that σ σ (0) and q q (0), the momentum and energy conservaton equatons become ρu α t ρe t + ρu αu β x β + ρu αe x α = p x α + F α, = p u α x α. (3.36a) (3.36b) These and the equaton of contnuty correspond to Euler s equatons of flud dynamcs, prevously gven n materal dervatve form n (2.17). However, these equatons lack the vscous stresses and heat conducton of the Naver-Stokes-Fourer model. The assumpton that the dstrbuton functon s at an equlbrum s clearly not suffcent to reproduce all the phenomena of contnuum flud mechancs. Ths shows that vscosty and heat conducton are connected wth the relaxaton of the dstrbuton functon to equlbrum. 3.8 The Chapman-Enskog expanson In the prevous secton we saw that the Euler equatons can be derved from the Boltzmann equaton under the assumpton that the gas s always at equlbrum. Ths mples that more detaled flud models, such as the Naver-Stokes-Fourer model, are connected wth the devaton from equlbrum. We can see ths more clearly f we nondmensonalse the Boltzmann equaton, by replacng each varable wth the product of a dmensonless counterpart (denoted wth a tlde) and an approprate characterstc number. Usng the characterstc length x 0, characterstc velocty ξ 0, characterstc tme t 0 = x 0 /ξ 0, mean free path x mfp, and mean free tme t mfp = x mfp /ξ 0,weget t = t x 0 /ξ 0, x = x x 0, ξ = ξ ξ 0, τ = τ x mfp /ξ 0, F = F ρξ 2 0 /x 0, f = f ρ/c 3 0. (3.37) Here we have assumed that the relaxaton tme n the BGK collson operator s related to the tme between partcle collsons. Mean free path, x mfp The average dstance travelled between collsons by a partcle Mean free tme, t mfp The average tme between collsons for a partcle

72 64 Chapter 3 The knetc theory of gases Replacng all the varables and dervatves n the Boltzmann equaton (3.20) wth ther dmensonless counterparts, we get ( x mfp f x 0 t + ξ f α + F f ) α = 1 τ ( f f ) (0). (3.38) x α ξ α Knudsen number, Kn The rato between the mean free path and a characterstc length. If Kn 1, the gas can be seen as a contnuum. On the left hand sde we have the Knudsen number, Kn = x mfp x 0 = t mfp t 0. (3.39) If Kn 0, then both sdes of (3.38) must also go to zero. The rght sde gong to zero mples that f f (0),.e. that the dstrbuton functon s very close to equlbrum, whch agan mples that the Euler equatons s an approxmate macroscopc descrpton of the flud flow for ths case. As we ncrease Kn, the dstrbuton functon s devaton from equlbrum becomes more sgnfcant, and we need a better macroscopc model, such as the Naver-Stokes-Fourer model, to properly descrbe the flud flow. As the dstrbuton functon s devaton from equlbrum becomes more mportant wth larger Kn, t would be natural to approxmate f by f (0) wth a small perturbaton whch ncreases wth Kn. We ntroduce a smallness parameter ɛ whch serves to label a term s order n the Knudsen number,.e. we let Kn ɛkn. Gong back to dmensonal quanttes agan, we expand the dstrbuton functon around equlbrum wth terms n ncreasng order of Kn, f = f (0) + ɛ f (1) + ɛ 2 f (2) +..., (3.40) where the ɛs ndcate that f (1) / f (0) = O(Kn), f (2) / f (0) = O(Kn 2 ), and so forth. Snce the rato between the left and rght hand sdes of (3.38) s O(Kn), the Boltzmann equaton wth f expanded now becomes [ ] t + ξ f ( α + F α f (0) + ɛ f (1) + ɛ 2 f (2) +...) x α ξ α = 1 ( ) (3.41) ɛ f (1) + ɛ 2 f (2) ɛτ The reason for ntroducng ɛ s to order the terms accordng to ther Knudsen number order. We assume that terms of dfferent order n Kn are sem-ndependent. Thus, (3.41) may be seen as a herarchy of equatons; one equaton at O(Kn 0 ), one at O(Kn 1 ), and so forth.* Ths *The dfferent equatons are generally connected through the tme dervatve, whch s expanded nto components at each order n Kn. In the followng we wll not need to consder ths, as we wll only need the O(Kn) equaton. However, we wll need to consder ths n the later Chapman-Enskog dervaton n secton

73 3.8 The Chapman-Enskog expanson 65 expanson technque for attackng the Boltzmann equaton s named after Sydney Chapman and Davd Enskog, who dscovered t ndependently of each other n the 1910s. At the O(Kn 0 ) level n (3.41), wehave f (0) on the left hand sde and f (1) on the rght. Thus, we can wll be able to fnd f (1) as a functon of f (0). Smlarly, we can n prncple fnd f (2) from f (1), f (3) from f (2),and so forth. We have seen prevously that f and f (0) have the same moments of densty, momentum, and energy. Therefore, we can assume that the contrbutons of the hgher order terms f (1), f (2),... to these moments are zero, f (n) dξ = ξ f (n) dξ = ξ 2 f (n) dξ = v 2 f (n) dξ = 0 (3.42) for n 1. Fndng the stress tensor (3.28) and heat flux vector (3.31) from the expanded dstrbuton functon (3.40), we fnd that the stress tensor and heat flux vector are necessarly also expanded n Kn, σ = σ (0) + ɛσ (1) + ɛ 2 σ (2) +..., q = q (0) + ɛq (1) + ɛ 2 q (2) +..., (3.43a) (3.43b) where σ (n) αβ = v α v β f (n) dξ, q (n) α = 1 2 v α v 2 f (n) dξ. (3.43c) As our goal s to fnd macroscopc conservaton equatons beyond the Euler equatons from the Boltzmann equaton, we ultmately want to fnd at least the frst-order moment perturbatons σ (1) and q (1). There are several paths to these moments. One s to fnd an expresson for f (1), and calculate the moments from that. Another s to take several dfferent moments at dfferent Kn orders n the expanded Boltzmann equaton and fnd the stress tensor and heat flux moments through other unknown moments, whch must be traced back to the known, conserved moments. In the followng, we wll take the former approach, usng a dervaton smlar to [61, Ch. 7]. The latter approach wll be taken later n secton Fndng the dstrbuton functon perturbaton At O(Kn 0 ), the expanded Boltzmann equaton (3.41) s f (0) t + ξ f (0) + F ρ ξ f (0) = f (1) τ. (3.44)

74 66 Chapter 3 The knetc theory of gases Takng the moments of mass, momentum or energy of ths wll gve the Euler equatons as descrbed n secton 3.7; the rght sde dsappears by (3.42). Instead, we dvde both sdes by f (0), rearrange, and use the chan rule n reverse, f (1) f (0) = τ = τ f (0) [ f (0) t [ ln f (0) t + ξ α f (0) x α + F α ρ ] f (0) ξ α ln f (0) +(u α + v α ) + F α x α ρ ] ln f (0). ξ α (3.45) The rest of ths secton wll consst of resolvng the dervatves of ln f (0) nto dervatves of the macroscopc varables, n order to fnd an entrely macroscopc relatonshp (3.52) between f (1) and f (0). The logarthm of the equlbrum dstrbuton functon (3.18) has a reasonably smple form, ln f (0) = 3 2 ln ( 3 4π ) ( ) 3 + ln ρ 3 2 ln e ξ u 2, (3.46) 4e but the dervatves n (3.45) are not straghtforward, wth the excepton of ln f (0) = 3 ( ) 3v α ξβ ξ ξ α 4e ξ β 2ξ β u β + u β u β = α 2e. (3.47a) The equlbrum dstrbuton functon f (0) s unquely determned by the conserved quanttes of densty, momentum, and energy,.e. f (0) = f (0) (ρ(x, t), u(x, t), e(x, t), ξ). The dependence n tme and space s only through the conserved quanttes, and we may therefore use the chan rule for the tme and space dervatves, ln f (0) t ln f (0) ρ = ρ t ln f (0) ln f (0) = x α ρ + ln f (0) u β u β t ln f (0) + e ρ ln f (0) u β ln f (0) + + x α u β x α e e t, e x α. (3.47b) (3.47c) These dervatves of ln f (0) can be easly resolved, ln f (0) = ρ ρ ln ρ = 1 ρ, (3.47d) ln f (0) = 3 (ξ α ξ α 2ξ α u α + u α u α ) = 3v β u β 4e u β 2e, (3.47e) ln f (0) = ( 3 v ) e e 4e 2 ln e = 1 ( 3 v 2 3 ). e 4e 2 (3.47f)

75 3.8 The Chapman-Enskog expanson 67 Insertng all of (3.47) nto (3.45), we fnd f (1) [ ( 1 ρ = τ f (0) ρ t +(u α + v α ) ρ x α + 1 ( 3 v 2 3 e 4e 2 ) + 3v ( β uβ 2e t )( e t +(u α + v α ) e x α +(u α + v α ) u ) β x α ) 3 2ρe F αv α ]. (3.48) The tme dervatves may be replaced usng the conservaton equatons. As we are workng at the O(Kn 0 ) level n the Boltzmann equaton, we must use the conservaton equatons that apply at ths level,.e. the Euler equatons found n secton 3.7. On materal dervatve form, the full set of Euler equatons are ( ) t + u α ρ = ρ u α, x α x α ( ) ( ) ( ) t + u α u x β = 1 p + F α ρ x β = 1 2ρ e 2e ρ + F β ρ 3 x β 3 x β, β ( ) t + u α e = p u α = 2e u α. x α ρ x α 3 x α Insertng ths nto (3.48), we get f (1) [ = τ u α + v α ρ + 3v ( β 2 e 2e ρ + F ) β f (0) x α ρ x α 2e 3 x β 3ρ x β ρ + v u β α x α ( 3 v )( 2 u α + v ) α e 3 ] 4e 2 3 x α e x α 2ρe F αv α. Some of the terms n ths equaton cancel, and we are left wth f (1) [( 3 v 2 = τ 5 ) vα e + 3 ( )] u β v f (0) α v 4e 2 e x α 2e β v 2 u α. (3.50) x α 3 x α The last parenthess s a famlar tensor n a dfferent form. Swtchng the ndces n one half of the frst term and expandng the second, we get ( ) [ ( ) ] u β v α v β v 2 u α 1 uβ = v α v x α 3 x β + u α 1 α 2 x α x β 3 δ u γ αβ x γ = v α v β S αβ, where S αβ = 1 2 ( ) uβ + u α x α x β 1 3 δ u γ αβ (3.51) x γ s the symmetrc stran rate tensor. The trace of ths tensor, whch may be found by multplyng wth δ αβ, s zero.

76 68 Chapter 3 The knetc theory of gases Fnally, we have the frst-order perturbaton of the dstrbuton functon n a reasonably smple form, [ ( f (1) = τ f (0) 1 e 3 v 2 5 ) v α + 3S ] αβ e x α 4e 2 2e v αv β. (3.52) As predcted, f (1) / f (0) s O(Kn). Ths can be shown by performng a nondmensonalsaton such as the one at the start of secton Fndng the moment perturbatons Now that we know f (1), we can fnd the frst-order moment perturbatons σ (1) and q (1) drectly usng (3.43c). Frst we fnd the frst-order perturbaton of the stress tensor, = v α v β f (1) dξ σ (1) αβ = τ [ 1 e ( e 3 v 2 5 x α 4e 2 ) v α v β v γ f (0) dξ + 3S γδ 2e ] v α v β v γ v δ f (0) dξ. By (3.19d) and (3.19f), the frst ntegral dsappears. By (3.19e), the second ntegral s v α v β v γ v δ f (0) dξ = 4 9 ρe2 ( δ αβ δ γδ + δ αγ δ βδ + δ αδ δ βγ ). Snce the stran rate tensor S γδ s traceless, the frst term n the parenthess becomes zero. The two others, appled to S γδ, together gve 2S αβ. Thus, the frst-order perturbaton to the stress tensor s σ (1) αβ = 4 3 ρeτs αβ = pτ ( uβ x α + u α ) 2 x β 3 δ u γ αβ. (3.53) x γ Ths corresponds exactly to the devatorc stress tensor σ αβ n (2.19d) wth a shear vscosty μ = 2 3 ρeτ = pτ = ρrtτ (3.54) and a bulk vscosty μ B = 0. Next, we fnd the the frst-order heat flux perturbaton, q (1) α = 1 2 v α v 2 f (1) dξ = τ [ ( 1 e 3 v 2 5 ) v 2 v α v 2 e x β 4e 2 β f (0) dξ + 3S ] βγ v 2 v α v 2e β v γ f (0) dξ. By (3.19f), the second ntegral s zero. The frst part of the frst ntegral s 3 v 4 v α v 4e β f (0) dξ = 3 1 4e 3 δ αβ v 6 f (0) dξ = 70 9 δ αβρe 2

77 3.8 The Chapman-Enskog expanson 69 The last equalty can most easly be shown by nsertng for f (0) and performng the ntegral usng the sphercal symmetry of the ntegrand. Fnally we have the second part of the frst ntegral, whch can smlarly be shown to be v 2 v α v β f (0) dξ = 5 6 δ αβ v 4 f (0) dξ = 50 9 δ αβρe Thus, the frst-order heat flux perturbaton s q (1) α = The Naver-Stokes-Fourer model ρeτ e x α. (3.55) Now that we know the frst-order perturbatons σ (1) and q (1),wehavethe flud model one order hgher n Kn than the Euler model. We reabsorb the smallness parameter ɛ, lettng ɛkn Kn. Then we approxmate σ σ (0) + σ (1) and q q (0) + q (1). Insertng ths approxmate stress tensor nto the Cauchy momentum equaton (3.29), we fnd exactly the Naver-Stokes mass conservaton equaton ρu α t + ρu αu β = ) ( δ x β x αβ p + σ αβ + F α, (3.56) β wth σ αβ gven by (3.53). Ths corresponds exactly to (2.19b) n conservaton form, but wth known values of the shear vscosty μ = pτ and bulk vscosty μ B = 0. Ths confrms that there s zero bulk vscosty for a monatomc dlute gas. Insertng the stress tensor and heat flux vector nto the general energy conservaton equaton (3.33), usng (3.11) to get a temperature dervatve n the heat flux, we fnd the energy equaton ρe t + ρu ( ) αe = δ x αβ p + σ αβ uβ + κ T (3.57) α x α x α x α wth a thermal conductvty κ = 5 3 ρerτ = 5 2 prτ = 5 2 ρr2 Tτ. (3.58) These two equatons, together wth the contnuty equaton (3.25), form the Naver-Stokes-Fourer model prevously n (2.19). Ths model may be derved ether usng knetc theory as done here, or from contnuum mechancs. In the contnuum dervaton, the transport coeffcents μ and

78 70 Chapter 3 The knetc theory of gases κ are emprcal materal parameters, whereas wth the knetc dervaton the transport coeffcents are gven only through the relaxaton tme τ. The hghly smplfed BGK model of collsons gves the same form of the macroscopc equatons as the full Boltzmann collson operator; n fact, ths can be seen as a requrement for any collson model. However, from any dervaton based n knetc theory, the transport coeffcents are determned by the choce of collson operator. In our case the relaxaton tme τ comes from the BGK operator; Boltzmann s orgnal collson operator results n dfferent transport coeffcents, gven by other parameters. In fact, assumng dfferent ntermolecular forces n Boltzmann s operator would change the resultng transport coeffcents slghtly [62]. Also, an underlyng assumpton of ths entre chapter s that the gas s monatomc. It s also possble to descrbe polyatomc gases n knetc theory [54 58]. Ths requres a dfferent collson model and results n dfferent transport coeffcents. However, the knetc theory of polyatomc gases s sgnfcantly more complcated. From the vscosty (3.54), the thermal conductvty (3.58), and the heat capacty at constant pressure (3.13), we can fnd the Prandtl number Pr = c pμ κ = 1. (3.59) Ths s a weakness of the BGK collson operator; as mentoned prevously, Boltzmann s orgnal collson operator results n transport coeffcents that gve a Prandtl number of Pr 2/3, a value whch corresponds well wth measurements on monatomc gases [58]. However, ths s not a problem for sothermal lattce Boltzmann models, where the thermal conductvty s rrelevant Hgher-order Boltzmann equaton approxmatons Burnett model The flud model found by takng Chapman-Enskog one step further than the Naver-Stokes-Fourer model In the prevous sectons, we found the frst-order perturbaton f (1) to the dstrbuton functon, and ts correspondng frst-order moment perturbatons σ (1) and q (1). Of course, t s possble to go further, fndng f (2), σ (2), and q (2). Whle we get the Euler model by assumng f f (0) and the Naver- Stokes-Fourer model by assumng f f (0) + f (1), the assumpton of f f (0) + f (1) + f (2) gves us an even more detaled pcture, called the Burnett model. A far tougher dervaton leads to moment perturbatons σ (2) and q (2) whch contan several new terms and several new transport coeffcents that cannot be predcted by contnuum theory [57, Ch. 15]. Snce f (2) / f (0) s O(Kn 2 ), the extra terms of the Burnett model are usually neglgble, as Kn 1 n most flows of nterest. However, the dfferences between the Naver-Stokes-Fourer model and the Burnett model are sgnfcant for the propagaton of sound waves at very hgh fre-

79 3.8 The Chapman-Enskog expanson 71 quences, where the acoustc Knudsen number gven from the wavelength λ as Kn = x mfp /λ goes towards one. In fact, measurements on plane sound wave propagaton n rarefed noble gases have ndcated that the Burnett model gves a better descrpton than the Naver-Stokes-Fourer at such hgh frequences [27, 63]. The two models depart sgnfcantly n ther predctons of sound speed and sound wave absorpton at Kn 0.1, and the Burnett model agrees very well wth measurements of sound propagaton n noble gases up to Kn 1. As t s mathematcally very tough to get to even the Burnett level, approxmatons of even hgher order can nstead be found through partcular assumptons on the form of f. Poneerng work for plane sound waves, where f was assumed to be of the form of an nfntesmal forced plane wave around an equlbrum state, was done by Wang Chang and Uhlenbeck [64] (later collected n [65]). It was later found that even hgherorder approxmatons to the Boltzmann equaton paradoxcally gves a poorer agreement wth experments than the Burnett model [27]. A very hgh-order approxmaton to the Boltzmann equaton was later found for the propagaton of plane sound waves, of the form of a power seres n Kn up to O(Kn 32 ) [66]. However, the seres coeffcents ncrease so rapdly that the seres dverges unless Kn s small. Ths s an example of an asymptotc seres, whch s non-convergent unless a parameter tends to a certan lmt (n ths case Kn 0), and whch usually s most useful and accurate when truncated to a small number of terms. Stll, such seres can often be approxmated beyond ther range of convergence, and n ths case the Shanks transformaton gave promsng results [66]. Smlarly, t has been suggested that the Chapman-Enskog expanson f = f (0) + f (1) + f (2) +... tself may be asymptotc [57, Ch. 15]. If so, truncatng the expanson earler could gve a better soluton than truncatng t later. For sound wave propagaton, t seems that the Burnett model gves the best agreement, although t s only sgnfcantly better than the Naver-Stokes-Fourer model for about an order of magntude n Kn. The agreement between the Burnett model and measurements n ths sngle case mght also be only a fortunate concdence and cannot be taken as absolute proof that the Burnett model s generally superor. Snce the Naver-Stokes-Fourer model may be derved ndependently ether from contnuum theory, the results of the Chapman-Enskog expanson to frst order can be trusted. However, the Burnett model, or any hgher-order models for that matter, cannot be found from any other dervaton, and have therefore hstorcally been vewed wth some suspcon [67]. Snce the Burnett model dffers very lttle from the Naver- Stokes-Fourer model at low Kn, the dfference s neglgble n most practcal cases. Other approaches than the Chapman-Enskog expanson can also be

80 72 Chapter 3 The knetc theory of gases used to fnd macroscopc equatons from the Boltzmann equaton. Some of the resultng models have been seen to at least agree well wth measurements of the speed of sound over the entre range of Kn [68]. 3.9 Boltzmann s H-theorem One thermodynamc quantty whch has not been dscussed yet n ths chapter s entropy. It was shown by Boltzmann hmself that a quantty H can be found from the dstrbuton functon f whch has many of the same propertes as thermodynamc entropy. H can only evolve n one drecton, and t reaches an extremum when the system s at an equlbrum. It was later shown for deal gases that H s proportonal to the entropy. The frst step n fndng H s to see from the chan rule that f f ln f =(1 + ln f ) t t. Ths s vald for any dervatve n the Boltzmann equaton, not just / t. Thus, multplyng the Boltzmann equaton wth (1 + ln f ), t thus becomes ( t + ξ α + F ) α f ln f =(1 + ln f ) Ω( f ). x α ρ ξ α Integratng ths over velocty space, we fnd f ln f dξ + ξ α f ln f dξ = t x α ln f Ω( f ) dξ. (3.60) The force term dsappears smlarly to (3.24), as f ln f f when f 0 as ξ. Also, one term on the rght sde dsappears by mass conservaton, (3.21a). Equaton (3.60) s lke a conservaton equaton for the quantty f ln f, but wth a source term on the rght sde. Usng the BGK collson operator, the rght sde can be shown to be ln f Ω( f ) dξ = ( f ln f (0) = 1 ( f ln τ = 1 f (0) ln τ ) f (0) Ω( f ) dξ + ln f (0) Ω( f ) dξ ) ( ) f (0) f dξ ( f f (0) )( 1 f ) f (0) dξ 0. (3.61) Here, the ntegral ln f (0) Ω( f ) dξ can be shown to dsappear by nsertng for f (0) and usng the mass and energy conservaton propertes of Ω( f ). The last nequalty follows from the dentty (1 x) ln x 0 for x > 0. For x = 1 (.e. f = f (0) ), t s zero.

81 3.9 Boltzmann s H-theorem 73 Thus, (3.60) s equvalent to H t + H α x α 0, (3.62) where H = f ln f dξ, H α = ξ α f ln f dξ. (3.63) Here, H α s the flux of H, smlarly to how ρu s the flux of ρ. From the nequalty, we see that H s not necessarly conserved lke mass n the contnuty equaton, but wll decrease f the system s not at equlbrum. However, as mentoned prevously, the nequalty n (3.62) becomes an equalty at equlbrum. Ths means that H wll decrease untl the system reaches an equlbrum, where H reaches ts lowest value. Ths s very smlar to how thermodynamc entropy ncreases untl the system reaches an equlbrum state. In fact, for an deal gas H s proportonal to the entropy densty ρs [58, Entropy densty, ρs 69], ρs = k B H. (3.64) m However, for a non-deal gas, where the equaton of state s affected by ntermolecular forces, ths equalty does not hold [69]. The nequalty (3.62) can also be shown from Boltzmann s orgnal collson operator. In fact, t s an mportant crteron for any collson operator, n addton to the conservaton crtera dscussed n secton 3.5, as t states that molecular collsons wll nvarably drve the dstrbuton of partcles towards an equlbrum. Entropy ( mxedupness ) per physcal volume n J/K m 3

82 4 The lattce Boltzmann method Isothermal flud A flud wth constant temperature In the last chapter we derved the Boltzmann equaton and saw that the famlar equatons of flud mechancs follow. In practce, s s extremely hard to fnd analytcal solutons for the Boltzmann equaton, except n trval cases lke the spatally homogeneous example n secton 3.5 and other smplfed cases [70]. In fact, t s also extremely hard to fnd solutons for the general equatons of flud mechancs, so they are smplfed n almost every case. In engneerng flud mechancs, the flud s often consdered ncompressble (.e. the densty ρ s consdered constant). In acoustcs, vscosty s usually neglected and the equatons are lnearsed so that the flow feld s consdered as a small perturbaton around a rest state. However, f we somehow could fnd a soluton for the Boltzmann equaton, we would smultaneously be fndng a soluton to the less general but more famlar equatons that follow from t. Snce t s usually too dffcult to attack the Boltzmann equaton analytcally, we must try to solve t numercally nstead. When dscretsng most transport equatons, t s suffcent to dscretse n only physcal space and tme. Wth the Boltzmann equaton, however, the man varable f s a functon of coordnates n physcal space, velocty space, and tme. We must therefore dscretse t n two separate steps. Frst, we restrct the contnuous space of veloctes ξ to a fnte dscrete set ξ,a set whch should deally be as small as possble. Then, we smultaneously dscretse n space and tme. The result of ths dscretsaton process can be qute convenently mplemented on a computer as the lattce Boltzmann method. There are many approaches to ths dscretsaton, but n ths chapter we wll emphasse clarty and brevty over generalty. Throughout, as necessary, we wll refer to artcles wth other, more general dervatons. In ths chapter, we wll derve the smplest and most common varety of the LB method: The sothermal, deal gas, forceless varety. As the flud s sothermal wth a constant temperature T 0, the deal gas equaton of state s p = ρrt 0. From ths follows a constant speed of sound and a smplfed equaton of state whch does not nvolve the temperature, ( ) p c 2 0 = = k BT 0 ρ m p = c 2 0ρ. (4.1) 74

83 4.1 The dscrete-velocty Boltzmann equaton 75 Comparng ths wth the physcal sentropc equaton of state (2.23) and speed of sound (2.24), we fnd that ths sothermal equaton of state corresponds to the physcal assumpton of γ = 1. From (2.47), ths tself mples an nfnte number of nner degrees of freedom n the molecules that make up the gas. 4.1 The dscrete-velocty Boltzmann equaton The frst step n dscretsng the Boltzmann equaton s to dscretse velocty space. One very general method for ths s based on approxmatng f (0) usng a truncated bass of Hermte polynomals and a Gauss-Hermte quadrature [72, 73]. The order of the quadrature determnes the number of veloctes ξ requred. Wth suffcently hgh orders, ths method can preserve the behavour of the Boltzmann equaton to arbtrary level n the Chapman-Enskog expanson [72]. However, hgher levels requre larger numbers of veloctes, whch makes the lattce Boltzmann method more dffcult to mplement and more resource demandng. Instead of ths general method, we wll use a mathematcally smpler method n ths dervaton. The frst step s to approxmate the Maxwell- Boltzmann dstrbuton (3.18) by expandng t up to O(u 2 ), f (0) (x, ξ, t) = ρ (2πc 2 0 )3/2 e (ξ αξ α 2ξ α u α +u α u α )/2c 2 0 ρ (2πc 2 0 )3/2 e ξ αξ α /2c 2 0 ( 1 + ξ αu α c (ξ αu α ) 2 2c 4 0 u αu α 2c 2 0 Here, (4.1) has been used, and terms of O(u 3 ) have been neglected. Whle stoppng at O(u 2 ) may seem somewhat arbtrary, the followng subsectons wll motvate ths choce. Next, we dscretse velocty space, restrctng ξ to a fnte set of veloctes ξ. Thus, the dstrbuton functon f (x, ξ, t) becomes f (x, t), representng the densty at (x, t) of partcles wth velocty ξ. We also replace the coeffcent e ξ αξ α /2c 2 0/(2πc 2 0 )3/2 n front of the expanson above wth a sngle weghtng coeffcent w, endng up wth the classc [74] dscrete equlbrum dstrbuton f (0) ( = ρw 1 + ξ αu α c ξ αu α ξ β u β 2c 4 0 u αu α 2c 2 0 ). ). (4.2) Hermte polynomals An orthogonal polynomal sequence useful n knetc theory [53, 71, 72] Gauss-Hermte quadrature An approxmaton method for certan ntegrals: e x 2 f (x) dx w f (x ) Ths s arguably the optmally stable sothermal polynomal dscrete equlbrum dstrbuton [75]. We wll see n the followng subsecton how the velocty sets defned Velocty set A dscrete set of velocty vectors ξ and accompanyng weghtng coeffcents w

84 76 Chapter 4 The lattce Boltzmann method by ξ and w must be constraned n order to reproduce hydrodynamcs correctly. Havng dscretsed velocty space and usng the the dscrete analogue of the BGK operator (3.22), the Boltzmann equaton becomes the dscretevelocty Boltzmann equaton (DVBE), f t + ξ f α = 1 ( f x α τ f (0) ). (4.3) Moments and constrants Of course, the velocty set cannot be chosen randomly. For the DVBE to gve the same mass and momentum conservaton equatons as found from the contnuous Boltzmann equaton n secton 3.8, we wll show n the next subsecton that the zeroth to thrd moments of f (0) must be equal to those of f (0),.e. f (0) (x, t) =ρ(x, t), (4.4a) ξ α f (0) (x, t) =ρu(x, t), (4.4b) ξ α ξ β f (0) (x, t) =Π (0) αβ (x, t), (4.4c) ξ α ξ β ξ γ f (0) (x, t) =Π (0) αβγ (x, t). (4.4d) We wll now show that these equaltes gve us a set of constrants (4.11) on the velocty set. As we soon shall see, the last of these equaltes can only be approxmately fulflled wth f (0) gven as n (4.2). The result of ths dscrepancy wll be a largely nsgnfcant error term n the momentum equaton. For (4.4a) to hold, ρ = [ = ρ f (0) w + u α c 2 0 w ξ α + u αu β 2c 2 0 ( 1 c 2 0 )] w ξ α ξ β δ αβ w,.e. the contents of the square brackets must equal 1. Gven that w are

85 4.1 The dscrete-velocty Boltzmann equaton 77 constants, ths holds only f w = 1, (4.5) w ξ α = 0, (4.6) w ξ α ξ β = c 2 0 δ αβ. (4.7) These are three of the constrants on the velocty vectors ξ and the weghtng coeffcents w. The other moments n (4.4) wll supply us wth addtonal such constrants. Smlarly, for (4.4b) to hold, ρu α = ξ α f (0) [ = ρ u β c 2 0 w ξ α ξ β } {{ } =u β δ αβ =u α ( ) + 1 u βu β 2c 2 0 w ξ α } {{ } =0 + u βu γ 2c 4 0 w ξ α ξ β ξ γ ], whch only holds f w ξ α ξ β ξ γ = 0. (4.8) From (3.27), (3.34), and (4.1) we fnd Π (0) αβ = ξ α ξ β f (0) dξ = ρu α u β + ρc 2 0 δ αβ, and thus we requre ρu α u β + ρc 2 0 δ αβ = [ ( ) = ρ 1 u γu γ 2c 2 0 For ths to hold, u γ u δ 2c 4 0 ξ α ξ β f (0) w ξ α ξ β } {{ } =c 2 0 δ αβ + u γ c 2 0 whch gves us our ffth constrant, w ξ α ξ β ξ γ } {{ } =0 + u γu δ 2c 4 0 w ξ α ξ β ξ γ ξ δ u γu γ 2 δ αβ = ρu α u β, w ξ α ξ β ξ γ ξ δ ]. w ξ α ξ β ξ γ ξ δ = c 4 ( ) 0 δαβ δ γδ + δ αγ δ βδ + δ αδ δ βγ. (4.9)

86 78 Chapter 4 The lattce Boltzmann method Usng (3.19c), the thrd moment of f (0) s Π (0) αβγ = ξ α ξ β ξ γ f (0) dξ (uα ) = u β u γ + u α v β v γ + u β v α v γ + u γ v α v β f (0) dξ = ρu α u β u γ + ρc 2 ( ) 0 uα δ βγ + u β δ αγ + u γ δ αβ. However, as the polynomal (4.2) for f (0) lacks any O(u 3 ) term, t s mpossble to exactly reproduce ths moment n the dscrete case. The correspondng dscrete moment* Π (0) αβγ must therefore lack the ρu αu β u γ term, so that Π (0) αβγ = ξ α ξ β ξ γ f (0) dξ = ρc 2 ( ) 0 uα δ βγ + u β δ αγ + u γ δ αβ = ρ w ξ α ξ β ξ γ (1 u δu δ 2c 2 0 Ths gves us our sxth and fnal constrant, + u δ c 2 0 ) ξ δ + u δu ɛ ξ 2c 4 δ ξ ɛ. 0 w ξ α ξ β ξ γ ξ δ ξ ɛ = 0. (4.10) In summary, the constrants on the velocty set are w = 1, w ξ α = 0, w ξ α ξ β = c 2 0 δ αβ, w ξ α ξ β ξ γ = 0, w ξ α ξ β ξ γ ξ δ = c 4 0 w ξ α ξ β ξ γ ξ δ ξ ɛ = 0. ( δαβ δ γδ + δ αγ δ βδ + δ αδ δ βγ ), (4.11a) (4.11b) (4.11c) (4.11d) (4.11e) (4.11f) These condtons can be seen as symmetry propertes of ξ and w, smlar to the symmetry propertes of f (0) n secton The odd moments *We separate the notaton for the moments of f from the notaton of the moments of f by a breve accent. Whle the frst three equlbrum moments agree n both cases (.e. Π (0) 0 = Π (0) 0, Π α (0) = Π α (0), Π (0) αβ = Π (0) αβ ) so that no dfference n notaton s requred, the hgher moments do not agree exactly due to the truncaton of f (0) n (4.2) (.e. Π (0) αβγ = Π (0) αβγ, etc.). We wll only use the accent to denote dscrete moments f there s such a dscrepancy.

87 4.1 The dscrete-velocty Boltzmann equaton 79 dsappear due to even symmetry, and the even moments are sotropc tensors. These symmetry propertes must be fulflled for any DVBE to solve the conservaton equatons for mass and momentum. The number of condtons motvates truncatng the polynomal n f (0) to O(u 2 ):Ifwe nclude hgher-order terms, lke some extended (e.g. thermal) LB methods requre, we get more of these constrants (ncludng one analogous to the somewhat bulky (3.19g)), whch n turn requres a larger set of veloctes [76, Ch. 2] Moment-based Chapman-Enskog expanson In the last subsecton we saw that the zeroth to thrd moments of f (0) and f (0) are equal (to O(u 2 )) f the symmetry condtons (4.11) on ξ and w are upheld. We wll now prove that ths s suffcent for the DVBE (4.3) to behave essentally dentcally to the Boltzmann equaton (3.20) to the Naver-Stokes level. The Chapman-Enskog expanson, prevously seen n secton 3.8, s used for ths proof. However, we wll use a dfferent method here than prevously. In secton 3.8, the frst perturbaton of the dstrbuton functon, f (1), was found from f (0), and subsequently used to fnd the stress tensor and heat flux vector perturbatons σ (1) and q (1). In the method to be through the known moments of f (0), usng a dervaton smlar to [77]. In ths moment-based expanson, a new mathematcal trck must be used n addton to the perturbaton expanson f = f (0) + ɛ f (1) + ɛ 2 f (2) + used here, we fnd relatons for the unknown moments of f (1)... In order to usefully close the system of equatons for the moments, we need to perform a multple-scale expanson of tme n orders of Kn, lettng t t 1 + ɛ 1 t [78, Ch. 1]. Ths technque s used n general perturbaton theory to deal wth expansons that result n unbounded terms at each order [79, Ch. 11]. A common explanaton s that t 1 as a tme scale dealng wth fast phenomena lke advecton and t 2 as a tme scale dealng wth slower phenomena lke dffuson. Instead of expandng as τ ɛτ as n secton 3.8, we expand the dervatves as t ɛ t 1 + ɛ 2 t , ɛ, (4.12) x α x α whch s comparable to multplyng wth ɛ n (3.41). Ths wll not make a sgnfcant dfference here, but wll be convenent later when expandng the fully dscrete lattce Boltzmann equaton. Whle the tme dervatve expanson s necessary for the Chapman- Enskog expanson to work, the multple-scale expanson of tme makes

88 80 Chapter 4 The lattce Boltzmann method lttle physcal sense. Another, more physcal, motvaton for the dervatve expanson s that dfferent phenomena affect the tme dervatve at dfferent orders n the Knudsen number. The tme dervatve must therefore be splt nto components for each order n ɛ. When the expanson s separated nto multple equatons accordng to the order of ɛ, all of these phenomena wll then be able to affect the tme dervatve. Wth these expansons, the DVBE becomes ( ɛ + ɛ 2 ) ( +... f (0) t 1 t + ɛ f (1) ) +... (4.13) 2 +ξ α ɛ ( f (0) x + ɛ f (1) ) +... = 1 ( ɛ f (1) α τ + ɛ 2 f (2) ) Separatng terms nto frst and second order n ɛ, wefnd ( ) O(ɛ): + ξ t α f (0) 1 x = 1 α τ f (1), (4.14a) O(ɛ 2 f (0) ( ) ): + + ξ t 2 t α f (1) 1 x = 1 α τ f (2). (4.14b) As n the fully contnuous Boltzmann equaton case (3.42), the conservaton of mass and momentum mply that f (n) dξ = ξ f (n) dξ = 0 for n 1. (4.15) Thus, the zeroth to second moments of (4.14a) are ρu α t 1 Π (0) αβ t 1 ρ t 1 + ρu α x α = 0, (4.16a) + Π (0) αβ = 0, (4.16b) x β + (0) Π αβγ = 1 x γ τ Π (1) αβ, (4.16c) where Π (1) αβ = ξ α ξ β f (1). Smlarly, the zeroth and frst moments of (4.14b) are ρu α t 2 ρ t 2 = 0, (4.17a) + (1) Π αβ = 0. (4.17b) x β We now proceed by recombnng the moment equatons at dfferent orders n ɛ to produce conservaton equatons. For the mass conservaton

89 4.1 The dscrete-velocty Boltzmann equaton 81 equaton, we smply reverse the expanson (4.12) n the sum ɛ(4.16a) + ɛ 2 (4.17a), whch results n the normal contnuty equaton ρ t + ρu α = 0. (4.18) x α Smlarly recombnng (4.16b) and (4.17b), we fnd a momentum conservaton equaton ρu α + ( ρu α u t x β + ρc 2 0 δ αβ + ɛ Π (1) ) αβ = 0. (4.19) β However, ths equaton nvolves the as of yet unknown moment Π (1) αβ. From a closer look at the equaton, we can already now expect ths moment to play the role of the stress tensor. The necessty of expandng the tme dervatve nto several components can now be seen. If t had not been expanded, (4.17b) would be statng that Π (1) αβ / x β = 0, snce no component of the tme dervatve would exst at O(ɛ 2 ). Consequently, the stress tensor would not have appeared n the momentum conservaton equaton. Whle the Chapman- Enskog analyss n secton 3.8 was performed wthout ntroducng any tme dervatve expanson, the unused and unnecessary terms n that dervaton at hgher orders n ɛ smlarly do not stand up to close scrutny wthout ths expanson. Usng (4.16c), we may fnd Π (1) αβ and Π (0) αβγ / x γ. Usng the product rule corollares ρu α u β t 1 ρu α u β u γ = u α ρu β t 1 by resolvng the dervatves Π(0) αβ / t 1 + u β ρu α t 1 u α u β ρ t 1, ρu β u γ ρu α u γ ρu γ = u α + u x γ x β u α u γ x β, γ x γ n addton to (4.16a) and (4.16b), we can resolve the former dervatve, Π (0) αβ t 1 = ρu αu β t 1 = u α ρu β t 1 + ρc2 0 δ αβ t 1 + u β ρu α t 1 = u α x γ ( ρu β u γ + ρc 2 0 δ βγ + u α u β ρu γ x γ = ρu αu β u γ x γ c 2 0 ρ u α u β + c 2 0 t δ ρ αβ 1 t 1 ) ( ) u β ρu α u γ + ρc 2 0 x δ αγ (4.20) γ c 2 0 δ ρu γ αβ x γ ( u α ρ x β + u β ρ x α ) c 2 0 δ ρu γ αβ. x γ

90 82 Chapter 4 The lattce Boltzmann method The latter dervatve s more easly resolved, Π (0) αβγ x γ = = c 2 0 ρc 2 ( ) 0 uα δ x βγ + u β δ αγ + u γ δ αβ γ ) ( ρu α + ρu β x β x α + c 2 0 δ ρu γ αβ. x γ (4.21) Wth these two dervatves resolved, (4.16c) becomes ( Π (1) αβ = ρc2 0 τ u α + u ) β + τ ρu αu β u γ. (4.22) x β x α x γ The last term s an error term due to the ρu α u β u γ term whch s mssng n Π (0) αβγ due to the truncaton to O(u2 ) n the equaton (4.2) for f (0).Ina correspondng dervaton for an sothermal, fully contnuous Boltzmann equaton, we would have found ( Π (1) αβ = ρc2 0 τ u α + u ) β. x β x α A closer look at the terms n (4.22) reveals that the error term s neglgble f u 2 c 2 0,.e. f Ma2 1. Whle ths condton excludes usng ths basc LB method for smulatons of transsonc and supersonc flow, the condton s otherwse not very strct. We wll see n secton 7.2 that the error term s neglgble even for smulatons of nonlnear acoustcs. Neglectng the error term, we can nsert (4.22) nto (4.19) to fnd the momentum conservaton equaton ρu α t + ρu αu β x β = p x α + σ αβ x β. (4.23) Ths s the forceless Naver-Stokes momentum equaton (2.19b) wth pressure p = ρc 2 0, shear vscosty ν = ρc2 0τ = pτ, and bulk vscosty ν B =(2/3)ν. Ths bulk vscosty appears solely due to the sothermal equaton of state [77]. Thus, we have shown that the dscrete-velocty Boltzmann equaton (4.3), gven the velocty set condtons (4.11), correctly reproduces the mass and momentum conservaton equatons of flud mechancs, wth the excepton of a O(u 3 ) error term whch s neglgble f Ma Velocty sets So far, we have found the condtons (4.11) on the velocty set gven by ξ and w and proven that these are suffcent for the dscrete-velocty

91 4.1 The dscrete-velocty Boltzmann equaton 83 ξ 3 ξ 2 ξ 6 ξ 2 ξ 5 ξ 2 ξ 0 ξ 1 ξ 4 ξ 0 ξ 1 ξ 3 ξ 0 ξ 1 (a) D1Q3 ξ 5 ξ 6 (b) D2Q7 ξ 7 ξ 8 ξ 4 (c) D2Q9 Fgure 4.1: Velocty vectors of smple one- and two-dmensonal velocty sets Boltzmann equaton (4.3) to reproduce the mass and momentum equatons of the Naver-Stokes-Fourer model. It s now tme to look at specfc velocty sets that fulfl these condtons. In the early years of lattce Boltzmann, velocty sets were developed by fndng weghtng coeffcents w that fulfl condtons such as (4.11), gven a certan choce of velocty vectors [74]. These velocty vectors were chosen for ther ablty to tle physcal space; each velocty brngs partcles from one pont on a regular perodc grd (or lattce) to another neghbourng pont (unless the velocty s zero, n whch case the partcles are mmoble). Later t was dscovered how to develop both velocty vectors and weghtng coeffcents smultaneously through the Gauss-Hermte quadrature method mentoned prevously [72, 73]. In the followng, we wll consder the former method, whch s mathematcally smpler, although more ad-hoc and less general. In LB termnology, t s common to denote dfferent velocty sets and the lattces that they form as DdQq, d beng the number of spatal dmensons, and q the number of veloctes. The velocty vectors of the most smple velocty sets n one and and two dmensons, namely D1Q3, D2Q7, and D2Q9, are shown n Fgure 4.1. Each of these velocty sets has a zero velocty vector ξ 0 = 0, whch corresponds to partcles beng at rest. The one-dmensonal D1Q3 lattce s a lne of regularly spaced ponts. The two-dmensonal D2Q7 and D2Q9 lattces are hexagonal and square, respectvely. Whle D2Q7 has the advantage of havng fewer veloctes than D2Q9 and thus beng less computatonally expensve, the square D2Q9 lattce s much easer to deal wth n computer mplementatons. Thus, the latter lattce s more commonly chosen for two-dmensonal smulatons. In the D1Q3 and D2Q7 velocty sets, the nonzero velocty vectors ξ =0 all have the same magntude, whch we defne as Δx/Δt.* In the D2Q9 *The notatonal choce of Δx and Δt foreshadows the dscretsaton of space and tme. Lattce A regular perodc grd of nodes n physcal space. In LB, lattces are defned by the velocty vectors ξ.

92 84 Chapter 4 The lattce Boltzmann method velocty set, not all nonzero velocty vectors have the same magntude; the dagonal velocty vectors ξ 5 ξ 8 have the hgher magntude of 2Δx/Δt. Wth the D1Q3 velocty vectors defned, we can derve the weghtng coeffcents w that satsfy the symmetry condtons (4.11). Frst, for (4.11b) to hold, w 1 = w 2 = w s (the letter s beng an abbrevaton of short). Then, the other nonzero symmetry condtons form the system of equatons w = w 0 + 2w s = 1, w ξx 2 = w ξx 4 = (Δx Δt )2 (2w s )=c 2 0, (Δx Δt )4 (2w s )=3c 4 0, (4.24a) the soluton of whch s c 0 =( Δx Δt )/ 3, w 0 = 2/3, w s = 1/6. (4.24b) For the D2Q9 velocty vectors, we set w 1 = w 2 = w 3 = w 4 = w s and w 5 = w 6 = w 7 = w 8 = w l for smlar reasons (the letter l beng an abbrevaton of long). Agan, we get a system of equatons from the nonzero symmetry condtons, w ξx 2 = w ξy 2 = w = w 0 + 4w s + 4w l = 1, w ξx 2 ξ2 y = w ξx 4 = w ξy 4 = (Δx Δt )2 (2w s + 4w l )=c 2 0, (Δx Δt )4 (4w l )=c 4 0, (Δx Δt )4 (2w s + 4w l )=3c 4 0. (4.25a) Ths s solved by c 0 =( Δx Δt )/ 3, w 0 = 4/9, w s = 1/9, w l = 1/36. (4.25b) A smlar dervaton can be performed for the D2Q7 lattce. In summary, the velocty vectors and weghtng coeffcents for the smple oneand two-dmensonal lattces are gven n Table 4.1. In three dmensons, the smplest velocty sets are D3Q15, D3Q19, and D3Q27 [74]. These three velocty sets are shown graphcally n Fgure 4.2, and are fully descrbed n Table 4.2. Note that D1Q3 s a one-dmensonal projecton of D2Q9, whch tself s a two-dmensonal projecton of D3Q15, D3Q19, and D3Q27. Ths means, for example, that a D3Q27 smulaton of a case where all varables are nvarant n the y and z drecton could be performed usng D1Q3 wth the same results.

93 4.1 The dscrete-velocty Boltzmann equaton 85 (a) D3Q15 (b) D3Q19 (c) D3Q27 Fgure 4.2: Velocty vectors of the smple three-dmensonal velocty sets. Velocty vectors to the sx nearest neghbours are n black, those to the twelve second nearest neghbours are n dark grey, and those to the eght thrd nearest neghbours are n grey.

94 86 Chapter 4 The lattce Boltzmann method Table 4.1: Smple one- and two-dmensonal velocty sets (a) D1Q3, c 0 =( Δx Δt )/ 3 ξ /( Δx Δt ) w 2 0 (0) (1) ( 1) 6 (b) D2Q7, c 0 =( Δx Δt )/2 ξ /( Δx Δt ) w 1 0 (0, 0) (1, 0) 12 2 ( 1 2, 3 2 ) ( 1 2, 3 2 ) ( 1, 0) ( 1 2, 3 2 ) ( 1 2, 3 2 ) 1 12 (c) D2Q9, c 0 =( Δx Δt )/ 3 ξ /( Δx Δt ) w 4 0 (0, 0) (1, 0) (0, 1) ( 1, 0) (0, 1) 9 5 (1, 1) ( 1, 1) ( 1, 1) (1, 1) 1 36 Table 4.2: Smple three-dmensonal velocty sets. (...) cycle ndcates all possble spatal and sgn permutatons of the gven vector coordnates. (a) D3Q15, c 0 =( Δx Δt )/ 3 ξ /( Δx Δt ) w 2 0 (0, 0, 0) (±1, 0, 0) cycle (±1, ±1, ±1) cycle 72 (b) D3Q19, c 0 =( Δx Δt )/ 3 ξ /( Δx Δt ) w 1 0 (0, 0, 0) (±1, 0, 0) cycle (±1, ±1, 0) cycle 1 36 (c) D3Q27, c 0 =( Δx Δt )/ 3 ξ /( Δx Δt ) w 8 0 (0, 0, 0) (±1, 0, 0) cycle (±1, ±1, 0) cycle (±1, ±1, ±1) cycle All of the LB velocty sets descrbed n ths secton contan a zero velocty vector ξ 0 wth a correspondng weghtng coeffcent w 0. Ths s n contrast to the smple lattce gas descrbed n secton 1.1.2, n whch no zero velocty partcles exst. However, these rest veloctes are necessary for the lattce Boltzmann method to behave correctly; otherwse, the velocty set condtons (4.11) cannot all be smultaneously fulflled. Ths can be demonstrated by tryng to remove the rest partcles from the D1Q3 and D2Q9 velocty sets by settng w 0 = 0: The systems of equatons (4.24a) and (4.25a) would both then become underdetermned and unsolvable.

95 4.2 The lattce Boltzmann equaton Dgresson: Lnearsed DVBE Instead of keepng terms to O(u 2 ) n the Taylor expanson of f (0),we could lnearse the expanson. Consequently, the DVBE wll be fully lnear, whch means that the resultng macroscopc equatons wll also necessarly be lnear. Ths lnearsed DVBE wll not be used untl later chapters, but showng t here serves to emphasse how mportant the equlbrum dstrbuton s to the resultng physcs of the model. As n secton 2.3, we assume that there s only a very small perturbaton n densty and velocty from the rest state. Thus, we can lnearse the equlbrum dstrbuton (4.2), ( ) f (0) = w ρ + ρ 0u α ξ α c 2 0. (4.26) Ths s, n fact, the only alteraton we make n ths secton to the prevously descrbed DVBE. Usng the symmetry propertes n (4.11), we fnd wthout approxmaton the dscrete equlbrum moments Π (0) 0 = ρ, Π α (0) = ρ 0 u α, Π (0) αβ = ρc2 0 δ αβ, Π (0) αβγ = ρ 0c 2 ( ) (4.27) 0 uα δ βγ + u β δ αγ + u γ δ αβ. Followng the Chapman-Enskog dervaton n secton wth these moments, we exactly fnd Π (1) αβ = τ Π (0) αβ + (0) ( Π αβγ u α = p 0 τ + u ) β, (4.28) t 1 x γ x β x α and the macroscopc mass and momentum conservaton equatons ρ t + ρ u α 0 = 0, x α ( ) (4.29a) u α ρ 0 = p u α + p 0 τ + u β t x α x β x α. (4.29b) These equatons are lnearsed versons of the correspondng equatons of the Naver-Stokes-Fourer model (2.19), wth vscostes μ = p 0 τ and μ B = 2μ/3. No error terms had to be neglected n ths dervaton, unlke the prevous dervaton of the DVBE where the momentum equaton ganed a O(u 3 ) error term. 4.2 The lattce Boltzmann equaton Whle the dscrete-velocty Boltzmann equaton (4.3) s dscrete n velocty space, t s stll contnuous n physcal space and tme. To fnd

96 88 Chapter 4 The lattce Boltzmann method Characterstcs Curves n the functon varables (e.g. space and tme) along whch a hyperbolc partal dfferental equaton becomes an ordnary dfferental equaton Lattce unts A smplfed nonphyscal choce of unts for smulatons, where Δx = Δt = 1 the fully dscrete lattce Boltzmann equaton, we must perform further dscretsaton. The Boltzmann equaton and the DVBE are both hyperbolc equatons, and a possble way of dscretsng these s to ntegrate along characterstcs. We assume that we can wrte the dstrbuton functon as f = f (x(a), t(a)), where a denotes the poston along the characterstc. The total dfferental of f n a, assumng no external forces, s ( ) ( ) d f da = f dt t da + f dxα x α da = 1 τ ( f f (0) ). (4.30) The rght equalty holds f the total dfferental s the left-hand sde of the DVBE (4.3), whch s true f dt da = 1, dx α da = ξ α. Thus, allowng a mnor abuse of notaton, f can be wrtten as f (x + ξ a, t + a).* Integratng (4.30) from a = 0toa = Δt (.e. from one tme step to the next) and usng the fundamental theorem of calculus on the left sde, we fnd f (x + ξ Δt, t + Δt) f (x, t) = 1 τ Δt 0 [ f (x + ξ a, t + a) f (0) ] (x + ξ a, t + a) da. (4.31) Whle the left sde s exact, the ntegral on the rght-hand sde must be approxmated. There are two man ways to do ths, whch we shall descrbe n the followng subsectons. When mplementng the lattce Boltzmann equaton on a computer, t s convenent to scale tme and space so that the lattce resoluton Δx and tme resoluton Δt both equal one. Ths choce of unts s called lattce unts. We wll use ths choce henceforth, and go nto how to convert between lattce and physcal unts n secton Frst order dscretsaton Wth the most common frst order dscretsaton, the ntegral n (4.31) s approxmated wth the rectangle method, gvng the frst order lattce Boltzmann equaton, f (x + ξ, t + 1) f (x, t) = 1 τ [ f (x, t) f (0) ] (x, t). (4.32) *Strctly speakng, ths should be f (x 0 + ξ a, t 0 + a), where x 0 and t 0 are constants defnng the startng pont, a = 0, of the characterstc. Thence, the abuse of notaton. It s correct but slghtly msleadng to call ths dscretsaton frst order, as the resultng scheme wll turn out to have second order accuracy n space and tme.

97 4.2 The lattce Boltzmann equaton 89 The resultng numercal scheme s fully explct, as all populatons f of the next tme step s determned by the populatons f of the current tme step. However, the rectangle method s generally only a frst order accurate approxmaton. Therefore, we should determne ts actual numercal accuracy usng the Chapman-Enskog expanson as n secton However, the left-hand sde of (4.32) must frst be de-dscretsed. Truncatng ts Taylor expanson to second order, we fnd f (x + ξ, t + 1) f (x, t) ( ) ( ) t + ξ α f x α 2 t 2 + 2ξ α + ξ t x α ξ β f α x α x. β Due to the perturbaton of the dervatves n (4.12), thrd order dervatve terms n ths expanson wll be O(ɛ 3 ) and therefore not contrbute at the Naver-Stokes level. Usng ths expresson for the left-hand sde, and performng the Chapman-Enskog expanson as n secton 4.1.2, we get O(ɛ) : ( ) + ξ t α 1 x α f (0) = 1 τ f (1) (4.33a) to frst order n ɛ, whch s dentcally to the correspondng DVBE equaton (4.14a). To second order, we fnd an ntally bulky expresson whch can be smplfed by subtractng 1 2 ( / t 1 ξ β / x β )(4.33a), gvng O(ɛ 2 f (0) ( )( ) : + + ξ t 2 t α 1 1 ) f (1) 1 x α 2τ = 1 τ f (2). (4.33b) There s one dfference here from (4.14b): the parenthess (1 1/2τ). By comparson wth the correspondng DVBE expressons, we can see that the followng dervaton would result n the same equaton for Π (1) αβ, but that the (1 1/2τ) factor changes the shear vscosty to ( ) ν = c 2 0 τ 1 2. (4.34) Thus, whle the rectangle method s only frst order accurate, the effect of the second order error terms s only to alter the vscosty by c 2 0 / Second order dscretsaton Wth the less common, second order dscretsaton, the ntegral n (4.31) s approxmated wth the trapezum rule, gvng f (x + ξ, t + 1) f (x, t) = 1 [ f 2τ (x + ξ, t + 1) f (0) (x + ξ, t + 1) (4.35) + f (x, t) f (0) ] (x, t),

98 90 Chapter 4 The lattce Boltzmann method whch can be rewrtten nto the cleaner form f (x + ξ, t + 1) f (x, t) = 1 [ ] f neq 2τ (x + ξ, t + 1)+ f neq (x, t) (4.36) usng f neq = f f (0) for the nonequlbrum part of f. Ths s not a fully explct system of equatons, as there s a codependence between f (x + ξ, t + 1) and f (0) (x + ξ, t + 1). We shall later see that a substtuton can be done to make t explct, but we wll frst examne at the macroscopc behavour of ths dscretsaton. Usng the same methods as n the prevous secton, we fnd the Taylor expanson of (4.36) to second order, ( ) ( ) t + ξ α f x α 2 t 2 + 2ξ α + ξ t x α ξ β f α x α x β = 1 [ ( ) ] (4.37) 2 f neq 2τ + t + ξ α f x, α and subsequently fnd the frst and second order terms n ɛ, ( ) O(ɛ) : + ξ t α f (0) 1 x = 1 α τ f (1), (4.38a) ( ) O(ɛ 2 ) : f (0) t + + ξ 2 t α f (1) 1 x = 1 α τ f (2). (4.38b) These equatons correspond exactly to those of the DVBE, (4.14). Thus we know that ths dscretsaton s a fully consstent approxmaton to the DVBE, wth vscosty ν = c 2 0 τ. The mplct dscretsaton (4.36) can be made fully explct through the substtuton f (x, t) = f (x, t)+ 1 2τ f neq (x, t), (4.39) after whch (4.36) becomes the fully explct second order lattce Boltzmann equaton, f (x + ξ, t + 1) f (x, t) = 1 [ f (x, t) f (0) ] τ + 1/2 (x, t). (4.40) By (4.39) and (4.15), the two frst moments of f are equal to the correspondng moments of f, f = ρ = f, ξ f = ρu = ξ f. (4.41a) (4.41b)

99 4.2 The lattce Boltzmann equaton 91 Therefore, f (0) can be constructed from f as f t were f. However, the second moment of f dffers, ( ξ α ξ β f = ) Π 2τ αβ 1 2τ Π(0) αβ. (4.41c) Comparng the lattce Boltzmann equatons of frst and second order, (4.32) and (4.40) respectvely, we fnd that they are nearly dentcal except for the the second moment and the denomnator of the collson operator. The latter may be seen as a redefnton of τ that compensates for the vscosty dfference. Thus, for smple LB models, the dfference between the two dscretsatons s not relevant and we wll henceforth use the smpler frst order verson. However, more complex extended LB models (e.g. [80]) may requre usng the second order dscretsaton. Also, f an alternate DVBE model s devsed, performng a second-order dscretsaton wll always result n a numercal scheme whch s consstent wth the DVBE model to the Naver-Stokes-Fourer level, unlke the frst-order dscretsaton whch may only be consstent to the Euler level. As we have seen for ths basc model, ths nconsstency n the frst-order dscretsaton was manfested as a change n the numercal vscosty Summary: The lattce Boltzmann method The precedng sectons have been qute mathematcal, focusng on the dervaton of the lattce Boltzmann equaton. Therefore, t s prudent to now gve a short, more practcal summary of the lattce Boltzmann method. In a regular spatal grd of nodes (a lattce), each node contans a number of dstrbuton functons f (x, t). These represent the densty of partcles wth velocty ξ n the node at x, at tme t. These velocty vectors are chosen so that partcles are brought from one node to ts neghbours (or reman statonary n the case ξ 0 = 0) durng one tme step. Some smple velocty sets n one, two, and three dmensons are shown n Fgures 4.1 and 4.2, and are specfed n Tables 4.1 and 4.2. The dstrbuton functons f can be used to fnd the more famlar macroscopc quanttes of mass densty and momentum densty, ρ(x, t) = f (x, t), ρu(x, t) = ξ f (x, t). The flud velocty s found as u(x, t) =( ξ f )/( f ). The pressure s determned through the sothermal relaton p = c 2 0ρ, where the deal

100 92 Chapter 4 The lattce Boltzmann method speed of sound c 0 s determned by the choce of velocty set (though n most cases, c 0 = 1/ 3). Once n each tme step, the partcles n each node collde, whch s modelled as a relaxaton of the dstrbuton functon f towards the equlbrum dstrbuton f (0) [ = ρw 1 + ξ u c (ξ u) 2 2c 4 0 u ] u 2c 2. 0 Ths s constructed from the macroscopc moments of densty ρ and flud velocty u, found usng the above relatons. The weghtng coeffcents w are determned by the choce of velocty set. What happens n each tme step s ths: In each node, collsons between ncomng partcles f are represented by a relaxaton wth characterstc tme τ towards the equlbrum dstrbuton f (0), resultng n a new dstrbuton of partcles whch s streamed on to the neghbourng nodes. All ths s represented by the lattce Boltzmann equaton, f (x + ξ, t + 1) = f (x, t) 1 τ [ f (x, t) f (0) ] (x, t). (4.42) The rght-hand sde represents the dstrbuton of partcles after collsons have taken place, and the left-hand sde represents these partcles appearng n neghbourng nodes n the next tme step. Ths very smple numercal scheme s suffcent to evolve the macroscopc densty ρ and flud velocty u n accordance wth the mass and momentum conservaton equatons of flud mechancs, often known as the contnuty equaton and the compressble Naver-Stokes equaton. In the latter, we get a knematc shear vscosty ν = c 2 0 (τ 1/2) and a knematc bulk vscosty ν B =(2/3)ν. (Note that there are some problems n the low-vscosty lmt τ 1/2, whch wll be dscussed n secton 4.3.) There s also an error term n the momentum conservaton whch s neglgble only f Ma 2 =(u/c 0 ) 2 1. In summary, the lattce Boltzmann algorthm conssts of the followng steps n each node: Macroscopc quanttes: From the dstrbuton of partcles f, calculate the node s flud densty ρ and flud velocty u. Equlbrum dstrbuton: From the flud densty ρ and flud velocty u, calculate the node s equlbrum dstrbuton f (0). Collson: The post-collson dstrbuton functons are calculated accordng to the rght-hand sde of (4.42). Streamng: The post-collson dstrbuton functons are streamed to neghbourng nodes accordng to ther veloctes, completng (4.42).

101 4.2 The lattce Boltzmann equaton 93 Streamng Fgure 4.3: Partcle dstrbutons f (black) streamng from the central node to ts neghbours, from whch new dstrbutons (black) are streamed back. Apart from the streamng step, each step s completely local wthn each node, meanng no quanttes outsde that node are used. In the streamng step, partcles only stream to neghbourng nodes. Ths localty s a very mportant property of the lattce Boltzmann method whch allows for massve parallelsaton of the algorthm. However, the steps above requre some startng condton. The most smple (but not necessarly the most accurate!) way to ntalse the smulaton s to calculate f (0) throughout the system based on specfed values of ρ and u, set f = f (0) everywhere, and start drectly on the streamng step. Ths summarses how the lattce Boltzmann method works n normal flud nodes. So far we have sad nothng about boundary condtons, whch are handled dfferently to the steps above. We wll touch on boundary condtons n secton Lattce Boltzmann unts Relatng lattce unts to physcal unts s a surprsngly trcky topc, especally as t actually depends on what s beng smulated. If LB s beng used to smulate ncompressble flow, sound propagaton n the smulaton s consdered physcally rrelevant as sound propagaton cannot occur n an ncompressble flud. Sound-lke propagaton mght especally occur at the start of a smulaton run, but s consdered to be a transent error due to the ntal condton of the flow feld beng dfferent from the steady-state condton. Other sound-lke propagaton mght occur due to the non-nfnte nformaton propagaton speed of the numercal model.* *Dsturbng a theoretcal ncompressble flud at a pont, the dsturbance s felt throughout the entre flud mmedately. In a compressble flud and LB smulatons (even ncom-

102 94 Chapter 4 The lattce Boltzmann method As the speed of sound s not a physcally relevant quantty n an ncompressble flud, t does not put any constrants on the LB unts n ncompressble cases. Ths topc s well explaned elsewhere [81], and we shall not go nto t here as ncompressblty s ncompatble wth acoustcs. Instead, we shall look at the case of compressble flow, where the unts are qute constraned. The tme and space resolutons Δx and Δt relate quanttes n lattce unts and physcal unts. For nstance, the physcal and lattce flud speeds are connected as u ph = u la Δx Δt. Snce u la can be freely scaled throughout the system, ths does not form a constrant on the unts. However, the speed of sound s a constant and does form a constrant, c 0,ph = c 0,la Δx Δt. (4.43) Smlarly, the knematc vscosty, wth physcal unts m 2 /s, forms another constrant, ν ph = ν la Δx 2 Δt. (4.44) As c 0,ph and ν ph are determned by the smulated case, we now have two equatons for the two unknowns Δx and Δt. Solvng them, we fnd Δx = ν ph ν la c 0,la c 0,ph, (4.45a) Δt = ν ph ν la ( c0,la c 0,ph ) 2. (4.45b) The only tunable parameter we are left wth s the relaxaton tme τ, whch determnes the lattce vscosty ν la. To avod Δx and Δt becomng prohbtvely small so that an extreme number of nodes and tme steps are requred to resolve the physcal case to be smulated, the relaxaton tme τ should generally be made as small as possble. However, the BGK collson operator can be qute unstable and naccurate for small τ. In secton 4.3 we wll look at some alternatve collson operators whch may perform better than BGK as ν la becomes very small. Densty can also be represented n lattce unts, for nstance wth the rest state densty normalsed to ρ 0,la = 1. Δx and Δt are clearly nsuffcent to convert between lattce and physcal densty as ther unts are metres and seconds, respectvely. Convertng densty to physcal unts requres pressble ones), the dsturbance propagates wth the speed of sound. From the defnton of the speed of sound (2.24), we see that t actually goes to nfnty as the flud becomes ncompressble.

103 4.2 The lattce Boltzmann equaton 95 unts of klograms. We therefore defne the densty converson factor C ρ so that ρ ph = C ρ ρ la, (4.46) wth unts kg/m 3. Δx, Δt, and C ρ are suffcent to convert the pressure to physcal unts. From the sothermal equaton of state 4.1, the pressure n lattce and physcal unts s From these, we fnd p ph = c 2 0,ph ρ ph, p la = c 2 0,la ρ la. (4.47) p ph = C ρ ( c0,ph c 0,la ) 2 ( ) Δx 2 p la = C ρ p Δt la. (4.48) Thus, convertng from lattce to physcal pressure requres the use of both Δx, Δt, and C ρ. It may seem from ths that the LB method s restrcted to smulatng unphyscal sothermal fluds, snce the equaton of state (4.47) gves a fully lnear relatonshp between pressure and densty. However, as only the pressure gradent p occurs n the momentum equaton, the absolute pressure does not matter.* In cases where the state varables vary lttle from constant rest values, e.g. p = p 0 + p where p p 0, (4.49) the pressure devaton p can be expressed as a lnearsed state functon of densty ρ and entropy s, ( ) ( ) p p = ρ p + s. (4.50) ρ s s ρ In nearly sentropc cases lke acoustcs, s s neglgble as dscussed n secton Usng the defnton of the speed of sound (2.24), (4.49) becomes p = p 0 + c 2 0 ρ = p 0 + c 2 0 (ρ ρ 0). (4.51) Therefore, n the momentum equaton, p c 2 0 ρ. Ths s a vald approxmaton, no matter the equaton of state, as long as the processes nvolved are nearly sentropc and ρ ( p/ ρ) s (ρ 2 /2)( 2 p/ ρ 2 ) s. Snce the absolute pressure does not matter n the mass and momentum conservaton equatons smulated by the LB method, the rest pressure p 0 s rrelevant and we can n prncple choose t arbtrarly, gnorng ts true but rrelevant value of p 0 = c 2 0 ρ 0. *The absolute pressure only plays a role n the energy equaton, whch s not relevant unless an extended energy-conservng LB method s used.

104 96 Chapter 4 The lattce Boltzmann method 1.1 τ = 2 τ = 1 τ = 0.6 τ = 0.51 f (t)/ f (0) t Fgure 4.4: BGK relaxaton wth dfferent relaxaton tmes n the spatally homogeneous case descrbed by (4.52) wth f (0)/ f (0) = Alternatve collson operators The BGK operator s the smplest known collson operator for lattce Boltzmann, that s stll suffcent to reproduce most hydrodynamc behavour. However, t has ts problems, most mportantly that t s prone to nstablty and naccuracy at low vscostes. In a spatally homogenous case analogous to the one n secton 3.5, the frst order scheme (4.32) gves ( f (t + 1) = 1 1 ) f τ (t)+ 1 τ f (0). (4.52) If τ = 1, f s fully relaxed to f (0) n one tme step. If τ > 1, t s underrelaxed; gradually relaxed towards f (0).If1/2 < τ < 1, t s overrelaxed, oscllatng around f (0) wth a decayng ampltude. Havng τ < 1/2 would be an unstable overrelaxaton wth the ampltude ncreasng nstead of decayng. Thus, τ = 1/2 s the lnear stablty lmt. The dfferent cases of relaxaton are llustrated n Fgure 4.4. When a very low vscosty s requred and τ 1/2, the BGK overrelaxaton s close to the lnear stablty lmt, causng f to tend to decay very slowly towards f (0) n an oscllatory manner. The combned effect of such oscllatons and partcle streamng can cause values of f to go far away from equlbrum and even become negatve. If a number of such far-off-equlbrum dstrbutons f are streamed nto the same node, a very small or even negatve densty ρ = f may result and destablsng effects may occur [82]. For nstance, f the densty ρ goes towards zero, the flud velocty u =( ξ f )/ρ may explode. If ths happens, f wll then be

105 4.3 Alternatve collson operators 97 overrelaxng past an entrely unrealstc dstrbuton functon f (0), lkely gong even further away from the correct local equlbrum dstrbuton. Note that overrelaxaton can occur only due to the dscretsaton of tme. Wth the contnuous BGK operator, the dstrbuton functon f s always relaxed towards f (0), but never past t. Therefore, the contnuous BGK operator s not potentally unstable n the same way as the dscrete BGK operator. Before the dscrete BGK operator became common, the collson operator was often wrtten n a more general way usng a collson matrx Ω j [8], ( f (x + ξ, t + 1) f (x, t) = Ω j j f j f (0) ) j. (4.53) The collson matrx acts on the nonequlbrum dstrbuton vector f j f (0) j, resultng n a vector of changes to the dstrbuton functon. Other lnear dscrete collson operators, ncludng BGK, can be found as a specal case of ths general lnear dscrete collson operator. In the BGK case, Ω j = δ j /τ. A number of alternatve, more complex collson operators have been proposed to mprove on the occasonally problematc BGK operator, though they generally buld on the same prncple of relaxng towards equlbrum. We wll brefly go through the man ones as they all gve a valuable addtonal understandng of the LB method Multple relaxaton tme Multple relaxaton tme (MRT) operators evolved out of early experments wth general collson operators as descrbed by (4.53) [8]. The dstrbuton functons f can be represented as a q-dmensonal vector f, whch can be transformed to another bass. Wth MRT models, ths bass s q hydrodynamc and nonhydrodynamc moments of f. The hydrodynamc moments are typcally Π 0 = ρ, Π α = ρu, and Π αβ, the moments whch are relevant for the lnk to hydrodynamcs. The nonhydrodynamc moments are not drectly relevant for the hydrodynamcs behavour of the model, but must usually be present to fll out the moment bass. The relaxaton to equlbrum s performed n the moment bass nstead of the f bass lke the BGK operator. After relaxaton, the moments are then transformed back to the f bass for streamng. The advantage of relaxng n the moment bass s that dfferent moments can be relaxed at dfferent rates.

106 98 Chapter 4 The lattce Boltzmann method The transformaton to moment bass s done usng the moment transformaton matrx M j, where Mf = m, Mf (0) = m (0). (4.54) m and m (0) are the resultng moment vectors of the transformaton. The smplest possble practcal example s the D1Q3 velocty set descrbed n secton Slghtly abusng the notaton of the ndces of f by usng f + = f 1 for the partcles movng n the +x-drecton and f = f 2 for those movng n the x-drecton, (4.54) becomes f f 0 f + f (0) f (0) 0 f (0) + = = ρ ρu Π xx ρ ρu Π (0) xx,. (4.55a) (4.55b) D1Q3 can be handly transformed nto a fully hydrodynamc bass. However, other velocty sets also need some nonhydrodynamc moments to fll out the moment bass, as the number of hydrodynamc quanttes s smaller than the number of veloctes. For nstance, D2Q9 has sx ndependent hydrodynamc varables: Π 0, Π x, Π y, Π xx, Π xy, and Π yy. The other three possble moments needed to fll out the bass cannot be entrely hydrodynamc, as they have to be lnearly ndependent from the fully hydrodynamc moments whch have been accounted for already. Choosng Π xxy, Π xyy, and Π xxyy for the other moments as n [83, Ch. 4], we get f f f f f 4 = f f f f 8 Π 0 Π x Π y Π xx Π xy Π yy Π xxy Π xyy Π xxyy. (4.56) Ghost modes Nonhydrodynamc behavour that coexsts wth the hydrodynamc behavour n LB smulatons The nonhydrodynamc moments are often called ghost modes. These moments are not relevant to the Chapman-Enskog expanson, and therefore affect the flud behavour n LB smulatons only ndrectly. Usng the dscrete BGK collson operator, these moments decay to equlbrum wth the relaxaton tme τ, lke the other moments.

107 4.3 Alternatve collson operators 99 The q q collson matrx Ω s assumed to be dagonalsable, or expressable as Ω = M 1 TM (4.57) where the relaxaton matrx T s usually dagonal. The generalsed LBE (4.53), left-multpled wth M, results n a relaxaton equaton n moment space, ( m out = m + T m m (0)). (4.58) Relaxaton matrx The matrx n the MRT collson operator contanng ndvdual relaxaton tmes for each moment m out can be seen as the post-collson moment vector, or the moment vector of the partcles beng streamed out of the node. We see that each element n the dagonal matrx T s a relaxaton tme for one of the moments n m. In the BGK specal case, all relaxaton tmes are equal, and T = τ 1 I so that Ω = 1 τ M 1 IM = 1 τ I; the case mentoned prevously. More generally, the dfferent moments can have dfferent relaxaton tmes. For the conserved moments, the relaxaton tmes do not matter, as ρ = ρ (0) and u = u (0). For symmetry reasons, the non-conserved hydrodynamc moments Π αβ should all have the same relaxaton tme;* from secton we have that the relaxaton of ths second order moment determnes the vscosty of the model. The post-collson moments m out cannot be propagated drectly, and must be converted back to the dstrbuton functon bass lke f = M 1 m before streamng. Thus, n prncple, the MRT algorthm works by streamng, converson to moment bass, relaxaton of the moments, converson back to the dstrbuton functon bass, and streamng agan. In practce, t s more effcent to compute Ω drectly and perform the relaxaton as n (4.53). The usefulness of MRT les n settng dfferent relaxaton tmes for the dfferent nonhydrodynamc moments. For a gven lattce and a gven moment bass, an analyss can be carred out to fnd the optmal nonhydrodynamc relaxaton tmes to optmse certan aspects of the behavour of the LB method [83 86]. A downsde s that such analyses do not gve unversal results. The results are specfc to each velocty set, each choce of moment bass, and each desred optmal behavour. These analyses can also be dffcult both to perform and to comprehend. However, one smple general-purpose choce that can vastly mprove the accuracy and stablty of the LBM s to choose a relaxaton tme of 1 *Note however that t s possble to use some specal technques when relaxng Π αβ to allow settng the shear and bulk vscosty ndependently [83, Ch. 4].

108 100 Chapter 4 The lattce Boltzmann method for the nonhydrodynamc moments [8, 78]. Thus, the nonhydrodynamc moments are always fully relaxed n each tme step nstead of oscllatng analogously to the underrelaxaton shown n Fgure 4.4. In secton 6.3 we shall look at a case where ths choce allows accurate smulatons wth τ = 1/2,.e. wth no vscosty at all. One argument leveled aganst MRT s that t s a numercal technque wth no correspondng physcal bass n knetc theory [87]. However, that does not n tself make the method less valuable Regularsed The regularsed collson operator [76, 88, 89]* s based on a dfferent dea than the MRT operator. The dea s to fully reconstruct the dstrbuton functon n the collson step, f (x + ξ, t + 1) = ( f (0) + f [1] ) 1 τ f [1]. (4.59) By comparng ths scheme wth (4.32), we see that f [1] has been substtuted for f neq = f f (0). f [1] s an approxmaton for f (1), constructed from the ncomng dstrbuton functons f. f (1) tself can be found from the Chapman-Enskog expanson of the lattce Boltzmann equaton, smlarly to how f (1) was found n secton The constructon s desgned so that f [1] only contans the Π αβ terms from f (1) that are necessary to reproduce the correct momentum equaton. The detals of how f [1] s constructed are somewhat complcated and can be found elsewhere [76], but the result s f [1] = w ( ) ( 2c 4 ξ α ξ β c 2 0 δ αβ ξ jα ξ jβ f j f (0) ) j, (4.60) 0 j resultng n a regularsed lattce Boltzmann equaton f (x + ξ, t + 1) = f (0) + (1 1/τ)w 2c 4 0 ( ) ξ α ξ β c 2 0 δ αβ j ξ jα ξ jβ ( f j f (0) ) j. (4.61) Whle the prncple of the regularsed collson operator s dfferent from that of the MRT operator, the regularsed operator can also be expressed as a general lnear LB collson operator as n (4.53) and be analysed usng MRT prncples. The result s that the regularsed scheme *The prncples behnd the regularsed collson operator have also been presented n dfferent contexts n other artcles [90, 91].

109 4.3 Alternatve collson operators 101 s equvalent to the aforementoned general-purpose MRT scheme that relaxes the second order moments Π αβ wth a relaxaton tme τ, and the other moments wth a relaxaton tme of 1 [76, 89]. Ths tells us that f [1] can be seen as an mproved approxmaton to f (1). Both have the same moment Π αβ, but f [1] s constructed wth the ghost modes set drectly to equlbrum. The regularsed collson operator offers an nterestng nsght to the LB method. However, snce t s n practce equvalent to the aforementoned smple general-purpose MRT method, whch one to use of the two s to some degree a matter of taste and famlarty Entropc There are several dfferent approaches to entropc LB methods (e.g. [75, 82]). Only ther common general concepts wll be presented here as a bref overvew. For detals on the mplementaton of entropc methods t wll be necessary to refer to the lterature. The orgnal motvaton for ntroducng entropc LB methods was to mprove the stablty of LB methods. In the process of dscretsng the Boltzmann equaton n velocty and physcal space, the H-theorem s lost, and the drect dscrete analogue of (3.62), f ln f s not a vald H-functon [82]. Instead other approprate functons must be found that can play the role of entropy. Such functons must be convex,.e. monotoncally decreasng towards ther mnmum, whch corresponds to the hghest entropy. The value of f at ths mnmum thus defnes the equlbrum dstrbuton functon f (0). An entropc LB method s therefore based on specfyng a vald alternatve H functon, fndng f (0) as ts mnmum (ether mplctly [82] or explctly [75]), and ensurng that H s never ncreased n the collson step. Wth the BGK collson operator, t s possble for the H functon to ncrease n collsons when overrelaxng, especally when τ s close to the stablty lmt of τ = 1/2.* Therefore, there can be a boundary on the lowest possble value of τ; the one where the post- and pre-collson values of H are the same. *Even f H s monotoncally decreasng towards f (0), t s not necessarly symmetrc around f (0). Snce overrelaxaton moves f past f (0), strong overrelaxaton could move f to a pont where H s hgher than before.

110 102 Chapter 4 The lattce Boltzmann method Streamng Fgure 4.5: A fully perodc D2Q9 lattce of 4 4 nodes. The upper left node s connected to neghbourng nodes on the opposte edges of the lattce. Fgure 4.6: A system wth perodc boundares emulates an nfntely large perodc system. 4.4 Smple boundary condtons Perodc BC A boundary condton where the edge of the system s connected to the opposte edge Obvously, the lattce Boltzmann method cannot smulate nfntely large systems. The system sze must be fnte, and the edges of the system must be handled by applyng some boundary condton (BC). Many of these boundary condtons may also be appled wthn the flud, for nstance to create walls wth no-slp boundary condtons where u = 0. The lterature on varous LB boundary condtons s extensve, and we wll only go nto the bare mnmum here, referrng to other boundary condtons at the end of ths secton. In every tme step, unknown partcles dstrbutons f enter the fnte system from magnary nodes on the outsde. A LB boundary condton must at least specfy the value of these unknown ncomng partcle dstrbutons, and the entre boundary of the system must be covered by such boundary condtons. The smplest boundary condton s the perodc one, where the edge of the system s connected to the opposte edge. Thus, a node on the edge of the system streams some of ts partcles to nodes at the other sde of the system, as shown n Fgure 4.5. Havng a perodc boundary, where nformaton that leaves on one sde re-enters on the other, s equvalent to havng an dentcal copy of the system on both sdes of tself. In the case shown n Fgure 4.6,

111 4.4 Smple boundary condtons 103 Streamng Fgure 4.7: Md-grd bounceback. The dstrbutons streamed towards the wall rebound and reappear wth opposte velocty n ther node of orgn after streamng. wth hard walls on two sdes and perodc boundares on the other two, the smulaton emulates an nfntely long perodc duct, wth the actual smulated system as a secton of ths. Smple no-slp boundary condtons can be mplemented n LB smulatons through the prncple of bounceback. Whle there are many varatons on bounceback BCs wth varyng capabltes and accuracy [92], they all share the dea of bouncng partcles back to ther node of orgn. One bounceback method whch s smple to mplement and retans the second-order accuracy of the LB scheme s the md-grd bounceback BC [78]. Wth ths method, walls can be placed at the centrelnes between nodes. Whenever partcle dstrbutons are streamed towards such walls, they reappear after streamng n ther node of orgn wth ther veloctes reversed, as shown n Fgure 4.7. We can look at ths reflecton of partcles from several perspectves. One s that the partcles stream towards the wall, ht t at tme t + 1/2, have ther velocty reversed, and reappear at tme t + 1 n the node where they began. Thus, the wall should be exactly n between the nodes. Another perspectve s that the partcles propagate through the wall, but that ghost nodes on the other sde (marked n grey n Fgure 4.7) stream dentcal partcle dstrbutons back. Thus, at t + 1/2, the partcle dstrbutons that meet at the wall cancel each other s velocty out, resultng n a net flud velocty u = 0 at the wall. We can conceptually separate the partcle bounceback nto normal bounceback and tangental bounceback. The bounceback n the normal drecton ensures that the normal component of the flud velocty s zero. It therefore also ensures that the walls actually block the flud, so that no net mass actually enters the wall, and that the partcle dstrbutons that enter the flud from the wall are known. The tangental bounceback ensures that the tangental component of the flud velocty s zero, resultng n a no-slp condton. Bounceback BCs Boundary condtons where no-slp walls are acheved by reflectng partcles back the way they came

112 104 Chapter 4 The lattce Boltzmann method Non-reflectng BCs Boundary condtons that let waves propagate smoothly out of the system wthout beng reflected back n It s also possble to not nclude the tangental bounceback. Havng partcles bounce specularly off the wall nstead results n a free-slp condton [78], where the normal velocty on the wall s stll zero but there s no constrant placed on the tangental velocty. In the feld of acoustcs, walls are generally assumed to be free-slp for mathematcal smplcty, as the acoustc effects of no-slp condtons are usually only partcularly relevant for enclosed geometres such as small ppes [24, 25]. A number of other, more complex, boundary condtons may also be used to mplement no-slp boundary condtons [93]. However, as ther relatve accuracy and stablty vary between dfferent applcatons, there s no sngle method that s superor n every case. Some smulatons requre nlets and outlets, where flud can enter or ext the system wth a specfed flud velocty u or densty ρ. The Zou-He boundary condton [94] s often used for ths purpose. However, ths boundary condton has stablty problems at hgh Reynolds numbers [93]. In cases where there s generaton of sound waves or smlar unsteady flow patterns, such nlet and outlet boundary condtons can cause errors, as they reflect pressure waves back nto the smulaton doman [95]. To avod such reflectons, we can nstead use non-reflectng boundary condtons (NRBCs). One wdely used type of NRBC throughout computatonal physcs s the perfectly matched layer. The concept s to put an absorbng layer around the computatonal doman. Ths layer s n prncple perfectly matched to the medum nsde the doman n such a way that an ncomng wave should not be reflected due to the dfference n materal parameters. As the wave propagates nto the absorbng layer, t s exponentally damped. Multple artcles have proposed LB mplementatons of ths concept [96, 97]. Other artcles have used smlar but smpler sponge layer areas where the relaxaton tme τ ncreases smoothly towards the edge of the doman [98, 99]. The ncreased τ causes sound waves to be absorbed more quckly, and the gradual ncrease means that there s lttle reflecton from the layer. However, one dsadvantage common to all such methods s that they requre a thck layer of many nodes around the computatonal doman. Snce each of these nodes must also be updated n each tme step, the requred smulaton workload and tme can ncrease sgnfcantly. Another type of NRBC s the characterstc boundary condton. These take advantage of the fact that the hyperbolc system of conservaton equatons can be separated nto a number of characterstcs.* On the boundary, one characterstc belongs to waves enterng the system, and the ampltude of such waves can be set to zero. Agan, multple artcles have been publshed on ths topc [100, 101]. In addton to these methods, there also exst varous other NRBC methods for lattce Boltzmann [102]. It s dffcult to know what to *Such characterstcs were prevously descrbed n secton 4.2.

113 4.4 Smple boundary condtons 105 choose among ths smorgasbord of dfferent methods, and there would certanly be room for a revew artcle that compares these dfferent lattce Boltzmann NRBC methods.

114

115 Part II Research 107

116 5 Acoustc lnearsaton analyss Acoustc lnearsaton analyss s a mathematcal method for studyng the absorpton and dsperson of propagatng sound waves. The goal of ths chapter s to apply ths method to both the dscrete-velocty Boltzmann equaton and the lattce Boltzmann equaton. In the latter case, we wll end up wth an equaton whch very accurately descrbes how sound propagates n lattce Boltzmann smulatons. Lnearsaton analyss s based on nsertng tral solutons wth weakly fluctuatng feld varables nto the governng equatons of the medum of propagaton. The frst artcle wth such an analyss for fluds was Stokes 1845 artcle on the flud stress tensor [21]. From that pont on, varous theores were advanced for the detals of the governng equatons of dfferent meda. The predctons of sound wave propagaton from these governng equatons could then be tested by experments. An account of the research up to 1953, much of whch s stll farly current, was gven by Truesdell [26]. For gases, the pcture of separate, smultaneous effects of vscosty, thermal conducton, and molecular relaxaton has been found to successfully descrbe the absorpton and dsperson of sound for frequences suffcently low that Kn = x mfp /λ 1[28, 29, 31, 32]. However, for hgher frequences wth Kn 1 and above, thermovscous behavour domnates but the assumptons behnd the Naver-Stokes-Fourer model are no longer vald. There s even today a msmatch between current thermovscous models and measurements at these extreme frequences [68]. Usually, the feld varables are assumed to be on steady-state plane wave form wth complex angular frequency ˆω and wavenumber ˆk,* ˆρ(x, t) ρ 0 ˆρ (x, t) ρ 0 ˆρ û(x, t) = 0 + û (x, t) = 0 + û e ( ˆωt ˆkx). (5.1) ˆp(x, t) p 0 ˆp (x, t) p 0 ˆp The prmed varables, such as ˆp (x, t) = ˆp e ( ˆωt ˆkx), are assumed to be nfntesmally small, so that any term where they appear more than once can be neglected. By puttng such harmonc solutons nto the *As shown n secton the plane wave form s not necessary for forced waves snce a complex Helmholtz equaton (2.33) can be found for these. Therefore, the wavenumbers found for forced waves are vald for any type of wave, and not merely plane waves. 108

117 109 governng equatons, the equatons are smplfed as / t ˆω and ˆk. These smplfed equatons can then be solved for ˆω or ˆk to fnd all possble wavelke modes allowed by the governng equatons. As dscussed n secton 2.3.2, the real and magnary parts of the complex frequency and wavenumber are responsble for dfferent effects. Splttng them as ˆω = ω + α t, ˆk = k αx, (5.2) we fnd e ( ˆωt ˆkx) = e (ωt kx) e α tt e α xx. (5.3) Thus, the magnary parts of ˆω and ˆk are clearly absorpton coeffcents n tme and space, respectvely. The real parts ω and k determne the wave s phase speed, c = ω/k. (5.4) In the low-frequency lmt ω 0, k 0, or n an deal medum as descrbed by the Euler equatons (2.17) and the deal wave equaton (2.25), ˆω = ω 0 and ˆk = k 0 are real, and the deal speed of sound s c 0 = ω 0 /k 0. (5.5) In general, ˆω and ˆk are partly determned by boundary or ntal condtons. As descrbed n secton there are two man cases. In the frst, ˆω = ω 0 s real. In the second, ˆk = k 0 s real. The frst case of forced or spatally absorbed waves s a boundary value problem. In smplfed form, there may be a source at x = 0 whch fluctuates at a sngle frequency ω 0. The flud surroundng the source necessarly also fluctuates at ths frequency, causng a wave proportonal to e (ω0t ˆkx) = e (ω0t kx) e α xx (5.6) to be propagated away from the source. The second case of free or temporally absorbed waves s an ntal value problem. At t = 0, there exsts a plane wave of nfnte extent and wavenumber k 0.Att > 0, the wave s damped as t propagates,.e. e ( ˆωt k 0x) = e (ωt k 0x) e α tt. (5.7) Forced wave A wave generated by a source, absorbed exponentally wth the dstance to ths source. Free wave A spatally perodc wave wth a spatally constant ampltude whch decreases exponentally wth tme. However, t s very dffcult to magne any physcal cause of ths knd of sound wave.* Even a standng wave needs to be generated by a source, makng t a forced wave. Stll, the free wave case s useful n benchmarks of numercal methods, as the spatally perodc nature of the wave allows usng a small but perodc smulaton doman. *Kelvn-Helmholtz nstabltes, caused e.g. by wnd blowng over a water surface, are free waves [103, Ch. 5]. However, these are not sound waves.

118 110 Chapter 5 Acoustc lnearsaton analyss p (x,0) x p (x,0) x p (0, t) t p (0, t) t (a) Forced wave (b) Free wave Fgure 5.1: Sketch of forced and free waves. Forced waves are absorbed n space, whereas free waves are absorbed wth tme. Acoustc vscosty number, X A dmensonless number ndcatng the effect of vscosty on sound propagaton These two cases of forced waves and free waves are sketched n Fgure 5.1. Note that these two types of waves are so physcally dfferent that the results for one cannot be used to derve results for the other [26]. Wth an sothermal equaton of state as assumed n Chapter 4 there s no heat conducton. Only vscosty affects the absorpton and dsperson of waves, through the vscous relaxaton tme τ ν = 1 ( ) 4 c 2 3 ν + ν B (5.8a) 0 prevously ntroduced n secton Ths usually appears together wth the frequency ω 0. Together they form the dmensonless acoustc vscosty number. Usng the notaton of [26], ths s X = ω 0 τ ν. (5.8b) For audble sound, ths number s very small: A 20 khz sound wave n normal ar gves X Only for extremely hgh ultrasonc frequences of 1 GHz do we get X 1. As ˆk and ˆω turn out to be functons of X only, t s useful to compare ther expressons for dfferent models of the form of seres expansons around X = 0, ˆk = 1 + a k 1 X + a 2 X 2 + a 3 X 3 + a 4 X 4 + O(X 5 ), 0 (5.9) ˆω = 1 + b ω 1 X + b 2 X 2 + b 3 X 3 + b 4 X 4 + O(X 5 ). 0 The exact expressons can be both extremely cumbersome and hard to compare. Relatng the vscosty to the mean free path x mfp [62], a closer look at X reveals that ω 0 c 0 /λ ν ν B c 0 x mfp } X x mfp λ = Kn. (5.10)

119 5.1 Isothermal Naver-Stokes-Fourer model 111 Therefore, the acoustc vscosty number X represents an acoustc Knudsen number, whch relates the mean molecular collson dstance x mfp wth the acoustc wavelength λ. For the extremely hgh frequences where X 1, the wavelength s comparable to the fne-granedness of the gas n whch t s propagatng. In secton 3.8 on the Chapman-Enskog expanson we showed that the Euler model s only vald to O(Kn 0 )=O(X 0 ). Wth the deal acoustcs that follow from ths model, we fnd ˆk/k 0 = 1 and ˆω/ω 0 = 1. Above O(X 0 ), where the model cannot be trusted, t makes an ncorrect predcton. Smlarly, the Naver-Stokes-Fourer model was shown vald only to O(Kn 1 )=O(X 1 ); ts predctons above ths order n X are not to be trusted [27]. A smlar technque to ths lnearsaton analyss, called Von Neumann analyss, s used n numercal analyss [104, Chs. 9 & 10]. There, a wavelke tral soluton s nserted nto a dscrete numercal scheme to examne how the soluton would evolve n tme. The man purpose of ths analyss s usually to determne the stablty lmt of the numercal scheme; beyond ths lmt the unstable soluton may ncrease exponentally n tme nstead of remanng constant or decreasng as above. The analyss can also determne the artfcal dsperson and absorpton that may be caused by dscretsaton error n the numercal scheme. Wth the lattce Boltzmann equaton, the lnearsaton analyss of sound propagaton wll let us separate the effects of the physcal model from such dscretsaton errors stemmng from the numercal scheme. For the reader who does not wsh to read through everythng, the end products of the LBE analyss come from the egenvalue problem (5.59). Two equatons on the seres expanson form of (5.9) are found: (5.63) for forced waves, and (5.64) for free waves. These seres expansons can be used to predct the wavenumber or frequency n lattce Boltzmann smulatons wth good accuracy. For the forced wave case, a slghtly cumbersome but exact soluton (5.62) s also avalable. Many of the calculatons later on n ths chapter are far too complcated to carry out by hand, and therefore the computer algebra system Maple was used for these. In some cases, the results were so cumbersome that they would requre an unfeasble amount of space to dsplay here. When necessary, these results wll be presented here n seres expanson form nstead. Von Neumann analyss A technque n numercal analyss to determne the numercal stablty, numercal absorpton, and numercal dsperson of a numercal scheme. 5.1 Isothermal Naver-Stokes-Fourer model From the Naver-Stokes-Fourer model (2.19), a vscous (2.32) and a thermovscous (2.41) wave equaton can be found. In the sothermal case, the energy equaton and ts correspondng thermal effects are not relevant,

120 112 Chapter 5 Acoustc lnearsaton analyss and the vscous wave equaton 1 c p t 2 ( ) 1 + τ ν 2 p = 0 (5.11) t descrbes sound propagaton wth no approxmatons other than lnearsaton. Whle the acoustc behavour of ths model s well-known [24 26], ts analyss wll be nstructve Absorpton and dsperson Dsperson relaton An equaton connectng the angular frequency ˆω and the wavenumber ˆk Insertng the tral soluton (5.1) nto the vscous wave equaton, we mmedately fnd a dsperson relaton ˆω2 c (1 + ˆωτ ν ) ˆk 2 = 0. (5.12) Ths relaton can be solved easly for ether forced waves or free waves. Forced waves If the frequency s assumed real, we fnd the complex wavenumber ˆk 1 = ±, ˆω = ω 0. (5.13) k X Here, the ± sgn corresponds drectly to propagaton drecton. Ths relaton holds n general, not only for the plane wave tral soluton: In secton t was derved through the complex Helmholtz equaton (2.33), whch s ndependent of the spatal form of the soluton. The postve soluton above, whch corresponds to a wave propagatng n the +x-drecton, predcts an absorpton and dsperson of α x k 0 = 1 + X X 2, c = k 0 c 0 k = 2 + 2X X (5.14) At very hgh frequences where X 1, ths predcts that the phase speed ncreases as X. Ths mples that the phase speed grows wth the frequency wthout bound, whch s clearly unphyscal. Stll, ncorrect predctons at very hgh frequences could be expected, snce we know that the Naver-Stokes-Fourer model s only vald for X 1. In the seres expanded form of (5.9), (5.13) becomes ˆk k 0 = X 3 8 X X X4 + O(X 5 ). (5.15)

121 5.1 Isothermal Naver-Stokes-Fourer model 113 Free waves If the wavenumber s assumed real, we fnd the complex frequency ˆω = X ω 0 2 ± 1 ( ) X 2, ˆk = k0. (5.16) 2 Agan, the ± sgn corresponds to propagaton drecton. The magnary part s always postve, so that the wave ampltude always decreases wth tme. One surprsng feature of ths soluton s that t predcts that free waves cannot propagate beyond X = 2. At ths pont, the real part of ˆω vanshes, resultng n a fully magnary soluton. For 0 < X < 2, the absorpton coeffcent and dsperson are α t ω 0 = X 2, c c 0 = ω ω 0 = 1 ( ) X 2. (5.17) 2 Whle the phase speed was predcted to ncrease wth X for forced waves, we see that the phase speed for free waves s predcted to decrease wth X. In seres expanded form, (5.16) s ˆω ω 0 = X 1 8 X X4 + O(X 6 ). (5.18) In the seres expansons for both forced and free waves, there s a clear pattern of even terms n X beng real and odd terms beng magnary. Ths pattern also apples beyond O(X 5 ), and seems to contnue ndefntely. Thus, even terms affect dsperson and odd terms affect absorpton. We wll see that ths pattern also holds for the other models consdered n ths chapter Magntude ratos and phase dfferences Not only does an ncrease n X cause absorpton and dsperson, t also affects the complex ampltudes of the wave components: momentum ρ 0 û, densty ˆρ, and pressure ˆp. As explaned n secton and shown n Fgure 5.2, the magntude of these complex ampltudes, e.g. ˆρ, determnes the physcal ampltude, whle ther complex phase angles or arguments, e.g. arg( ˆρ ), determnes the physcal phase shft. Therefore, ρ 0 û / ˆρ = ρ 0 û / ˆρ represents the rato of the physcal ampltudes of the momentum and densty components of the wave, and arg(ρ 0 û / ˆρ )=arg(ρ 0 û ) arg( ˆρ ) represents the dfference n phase of the components. If ths phase dfference s nonzero, the peaks of the wave components are stuated at dfferent postons. Argument A mathematcal functon that returns the complex angle of a complex number, e.g. arg(ẑ) =ϕ for ẑ = r e ϕ

122 114 Chapter 5 Acoustc lnearsaton analyss ρ 0 û c 0 ˆρ ρ 0 û (x, t) c 0 ˆρ (x, t) ( ) 1 ρ0 û k arg ˆρ c 0 ˆρ ρ 0 û x Fgure 5.2: Exaggerated sketch of dfferences n ampltude and phase of the momentum and densty components of a wave. For an acoustc vscosty number X = 0, correspondng to the Euler model and the deal wave equaton, these dfferences are zero and the two wave components would overlap. In the Naver-Stokes-Fourer case, these relatve ampltudes can be determned usng the mass and momentum conservaton equatons, whch n lnearsed one-dmensonal form are ρ t + ρ u 0 x = 0, (5.19a) u ρ 0 t + p ( ) 4 x = 3 μ + 2 μ u B x 2. (5.19b) Insertng the tral solutons (5.1) and rearrangng, we fnd ρ 0 û ˆρ = ˆωˆk, (5.20a) ˆp ρ 0 û = ˆωˆk ( ) 4 ˆk 3 ν + ν B. (5.20b) By nsertng the forced or free wave solutons of ˆk and ˆω from the prevous secton, these ampltude ratos become explct functons of X. These equatons can also be used to relate ˆp and ˆρ, ˆp ˆρ = ρ 0û ˆρ ˆp ρ 0 û = ( ˆωˆk ) 2 c 2 0 ˆωτ ν. (5.20c) Ths could seem to predct a phase dfference between pressure and densty, contrary to the sentropc assumpton of p = c 2 0 ρ whch was used to derve the vscous wave equaton. However, by nsertng any of the two solutons (5.13) or (5.16) and smplfyng, we fnd ˆp ˆρ = c2 0. (5.20d)

123 5.2 Dscrete-velocty Boltzmann equaton 115 Thus, ths s stll consstent wth the orgnal sentropc assumpton.* In fact, the dsperson relaton (5.12) can be found drectly from (5.20c) and (5.20d). Such an approach wll be followed n secton 5.2 on the lnearsaton of the dscrete-velocty Boltzmann equaton. Forced waves Substtutng the complex wavenumber (5.13) of the forced wave case nto (5.20a), we fnd the magntude rato 1 ρ 0 û c 0 ˆρ = k 0 ˆk = X 2 (5.21a) = X X4 + O(X 6 ), and the phase dfference ( ) ( ) ρ0 û arg ˆρ = arg k0 ˆk = 1 2 arctan(x) = 1 2 X 1 6 X3 + O(X 5 ). (5.21b) Ths shows that the ampltude of the momentum component wll ncrease relatve to the ampltude of the densty component as X ncreases. Also, the momentum component of the wave wll propagate ahead of the densty component for nonzero X. Free waves Substtutng the complex frequency of the free wave case, we fnd the magntude rato and phase dfference 1 ρ 0 û c 0 ˆρ = ˆω ω0 = 1 for X 2, (5.22a) ( ) ( ) ( ) ρ0 û arg ˆρ = arg ˆωω0 = arctan X for X 2 (5.22b) 4 X 2 = 1 2 X X3 + O(X 5 ). As n the forced wave case, the momentum component s ahead of the densty component, but the rato between the momentum and densty ampltudes are predcted to be constant. 5.2 Dscrete-velocty Boltzmann equaton As seen n secton 4.1, the dscrete-velocty Boltzmann equaton (4.3) (DVBE) s a sem-dscretsed form of the Boltzmann equaton where the contnuous velocty space of the Boltzmann equaton s restrcted to a *Ths dsproves a statement n one of ths author s prevous publcatons [11].

124 116 Chapter 5 Acoustc lnearsaton analyss dscrete set of veloctes. In addton, the equlbrum dstrbuton s approxmated as the seres (4.2), whch s truncated to O(u 2 ). The DVBE can be seen as an ntermedary step n the dscretsaton of the Boltzmann equaton. The fnal step s the lattce Boltzmann equaton (LBE), where space and tme have also been dscretsed. Consequently, as the numercal resoluton of the LBE s mproved, the numercal errors n space and tme are reduced and the LBE converges towards the DVBE. Snce the DVBE and sothermal Naver-Stokes-Fourer models do not exactly agree, sound propagaton wth the LBE wll not exactly agree wth the Naver-Stokes-Fourer model ether, even wth an nfntely fne numercal resoluton. The fneness of the dscretsaton of velocty space and the approxmaton of the equlbrum dstrbuton determnes how well the DVBE can capture the Boltzmann equaton. We saw n secton 4.1 that the DVBE presented there cannot do any better than the sothermal Naver-Stokes- Fourer model wth a O(u 3 ) error term. However, wth a dscrete velocty space contanng more veloctes and a less truncated equlbrum dstrbuton, the behavour of the Boltzmann equaton could be captured correctly to the Burnett level and beyond [72] Lnearsaton process Analogously to the wavelke tral soluton (5.1), let us assume that the soluton to the DVBE s of the form ˆf (x, t) =F (0) + ˆf e ( ˆωt ˆkx). (5.23) Here, ˆf e ( ˆωt ˆkx) s an nfntesmal fluctuaton around the equlbrum rest state F (0).* For ths one-dmensonal problem t s suffcent to use the D1Q3 velocty set ntroduced n secton (4.1.3), wth veloctes (ξ, ξ 0, ξ + )= ( 1, 0, 1), weghtng coeffcents (w, w 0, w + ) = (1/6, 2/3, 1/6), and speed of sound c s = 1/ 3. Snce hgher-dmensonal velocty sets such as D2Q9, D3Q15, D3Q19, and D3Q27 have D1Q3 as ther one-dmensonal projecton, usng D1Q3 here s suffcent to predct the behavour of waves propagatng along a man (.e. x, y, orz) axs n any of these other sets. In secton on the sotropy of the D2Q9 velocty set, ths wll be demonstrated. *A note on notaton: Whle rest states have been denoted wth a subscrpted zero elsewhere, we use a captal F here snce denotng t as f 0 would lead to confuson wth the zero-velocty partcle dstrbuton f 0. F (0) should not be confused wth the body force densty F, whch wll not appear n ths chapter.

125 5.2 Dscrete-velocty Boltzmann equaton 117 Snce the fluctuaton s nfntesmal, we can lnearse the equlbrum dstrbuton functon lke n secton 4.1.4, ( ) ˆf (0) = w ˆρ + ρ 0û c 2 ξ. (5.24a) 0 Ths can then be splt nto two parts assocated wth the rest state and the fluctuaton, F (0) = ρ 0 w, (5.24b) ( ) ˆf (0) = w ˆρ + ρ 0û c 2 ξ. (5.24c) 0 The moments of ths soluton are entrely analogous to the prevously used tral soluton (5.1), [ ] [ ] [ ] [ ] ˆf (x, t) ˆρ(x, t) ρ0 ˆρ = = + ξ ˆf (x, t) ρ 0 û(x, t) 0 ρ 0 û e ( ˆωt ˆkx). (5.25) The pressure can be found drectly through the sothermal equaton of state p = c 2 0ρ, or the pressure fluctuaton can equvalently be found from the fluctuaton equlbrum dstrbuton (5.24c) usng the velocty set constrant (4.11c) as (0) ξ ξ ˆf = ˆΠ xx (0) = c 2 0 ˆρ = ˆp. (5.26) Insertng the tral soluton (5.23) nto the DVBE, applyng the dervatves and rearrangng, we fnd the harmonc lnearsed dscrete-velocty Boltzmann equaton, [ ( 1 + τ ˆω ˆkξ )] ˆf = ˆf (0). (5.27) Ths relates the dstrbuton functon ampltude ˆf to ts equlbrum (0) counterpart ˆf. Due to the presence of ξ on the left hand sde, any moment of ths equaton wll relate that moment of ˆf wth the moment of one order hgher. Thus, t would seem that ths equaton leads to an nfnte herarchy of moments. However, we wll soon see that ths s not so, because of the fnte number of veloctes. Due to the conservaton of mass and momentum, the zeroth and frst moments of the fluctuaton dstrbuton functon and ts equlbrum counterpart are dentcal, ˆf = ˆf (0) = ˆρ, ξ ˆf (0) = ξ ˆf = ρ 0 û.

126 118 Chapter 5 Acoustc lnearsaton analyss Consequently, the zeroth, frst, and second moments of (5.27) gve ρ 0 û ˆρ = ˆωˆk, (5.28a) ˆΠ xx ρ 0 û = ˆωˆk, (5.28b) ˆΠ xx = c2 0 ˆρ + ˆkτ ˆΠ xxx. (5.28c) 1 + ˆωτ Due to the lmted number of veloctes, the number of ndependent moments s also lmted. As a consequence, the thrd moment s nonunque and s gven by the frst moment, ˆΠ xxx = ξ ξ ξ ˆf =( Δx Δt )2 ξ ˆf = 3c 2 0 ρ 0û. (5.29) Thus, the system of moments (5.28) s closed, and we can fnd a dsperson relaton ( ) 2 = ˆωˆk ˆΠ ( ) xx 1 c 2 ˆρ = 0 ˆρ + 3c 2 0ˆkτρ 0 û 1 + ˆωτ ˆρ = c ˆωτ ˆωτ. We can relate the BGK relaxaton tme τ to the vscous relaxaton tme τ ν usng the DVBE values of the shear and bulk vscostes, ν = τc 2 } 0 τ ν B = 2ν/3 ν = 2τ. (5.30) Thus, the above dsperson relaton s ( ˆωˆk ) 2 = c ˆωτ ν / ˆωτ ν /2. (5.31) Ths may now be used to fnd expressons for ˆk or ˆω for forced or free waves, respectvely. If ˆk and ˆω are known, the values of ˆf can also be found f necessary. Adaptng (4.55a) and usng (5.28), ˆf ˆρ ˆf 1 0 = ρ 0 û = ˆρ ˆω/ˆk ( ˆω/ˆk) 2 ˆf + Invertng the matrx results n an equaton for the values of ˆf, ˆf ˆf 0 = ˆρ ( ˆω/ˆk) 2 /2 ˆω/2ˆk ˆω/ˆk = ˆρ 1 ( ˆω/ˆk) 2. ˆf ( ˆω/ˆk) 2 ( ˆω/ˆk) 2 /2 + ˆω/2ˆk (5.32) ˆΠ xx

127 5.2 Dscrete-velocty Boltzmann equaton 119 In the deal flud case X 0, these values are ˆf (c ˆf 2 0 = ˆρ 0 c 0)/2 1 c 2 0. (c c 0)/ Propertes of forced and free waves ˆf + From the dsperson relaton (5.31), the absorpton and dsperson propertes of the DVBE model can be found. In addton, after havng found ˆk or ˆω, (5.28a) gves us the complex ampltude rato of the momentum and densty wave components. Forced waves Assumng ˆω real and solvng (5.31) for ˆk, wefnd ˆk 1 + X/2 = ± k X/2, ˆω = ω 0. (5.33a) Agan, we have two possble solutons, one for each propagaton drecton. Ths wavenumber equaton s actually dentcal n form to the wavenumber equaton (2.52) for molecular relaxaton processes, wth τ m = τ and c /c 0 = 3. We wll soon go nto ths connecton. As n the molecular relaxaton case, the expressons for the real and magnary parts of ˆk/k 0 are not smple. We wll make do wth the seres expanson ˆk k 0 = X 5 8 X X X4 + O(X 5 ). (5.33b) A quck comparson wth the equvalent Naver-Stokes-Fourer expanson (5.15) shows an agreement at O(X) but not above. Thus, the same absorpton s predcted to the lowest order, but the predcted dsperson s dfferent. A more detaled comparson s deferred to secton The magntude ratos and phase dfferences found from the soluton are arg 1 ρ 0 û c 0 ˆρ = k 0 ˆk = X2 1 2 X4 + O(X 6 ), ) ( ρ0 û ˆρ ) = arg ( k0 ˆk Free waves, propagatng mode (5.33c) = 1 2 X X3 + O(X 5 ). (5.33d) In the free wave case, ˆk s assumed real and (5.31) s solved for ˆω. An examnaton of the equaton reveals that t s a cubc equaton n ˆω, wth

128 120 Chapter 5 Acoustc lnearsaton analyss three solutons. These solutons become very cumbersome and are most easly found usng computer assstance. The three dfferent solutons correspond to three modes of propagaton. As n the Naver-Stokes-Fourer case, two of these modes correspond to normal plane wave propagaton n the ±x-drecton. However, there s also a thrd, non-propagatng soluton where ω = Re( ˆω) = 0. We frst consder the soluton that propagates n the x-drecton. As the exact soluton s too cumbersome, we go drectly to the seres expanson, ˆω ω 0 = X X X X4 + O(X 5 ), ˆk = k0. (5.34a) Agan we can fnd by comparson wth the Naver-Stokes-Fourer equvalent (5.16) that the expansons agree only to O(X). The magntude ratos and phase dfferences found from ths soluton are 1 ρ 0 û c 0 ˆρ = ˆω ω0 = X X4 + O(X 6 ), (5.34b) ( ) ( ) ρ0 û arg ˆρ = arg ˆωω0 = 1 2 X X3 + O(X 5 ). (5.34c) Free waves, dffusve mode The purely magnary soluton corresponds to a mode where the waves do not propagate, they are merely absorbed. Ths soluton s therefore dffusve, smlarly to solutons of the heat equaton. Even so, ths soluton s not related to dffuson of heat, whch cannot happen n the sothermal DVBE. On seres expanson form, the dffusve soluton s ˆω ω 0 = 2X 1 X 1 4 X3 + O(X 5 ), ˆk = k0. (5.35a) Note that the absorpton coeffcent goes to nfnty as X 0. Thus, for mmedate relaxaton to equlbrum, ths mode s absorbed nstantly. The magntude ratos and phase dfferences are also on a substantally dfferent form than prevously, 1 ρ 0 û c 0 ˆρ = ˆω ω0 = 2X 1 X 1 4 X3 + O(X 5 ), (5.35b) ( ) ( ) ρ0 û arg ˆρ = arg ˆωω0 = π 2. (5.35c) For low X, the momentum component of the wave domnates over the densty component. Also, the two components are out of phase by π/2, so that the peaks and troughs n one concde wth zeroes n the other.

129 5.2 Dscrete-velocty Boltzmann equaton Comparson wth relaxaton processes Secton descrbed how relaxaton processes affect absorpton and dsperson. In gases, the nature of such processes s usually the transfer of energy between translatonal and nner (.e. vbratonal and rotatonal) degrees of freedom of the gas molecules. For a sngle relaxaton process wth no addtonal thermovscous effects, the result s the relaxaton wave equaton (2.51), ( 2 p ) ( τ m t t 2 c2 2 p + 2 p ) t 2 c2 0 2 p = 0. Here, τ m s the relaxaton tme of the process, and c s the frozen speed of sound. The latter occurs at frequences where ω 0 τ m 1, frequences so hgh that the relaxaton process cannot keep up wth the rapd changes n translatonal energy. Consequently, the nner energy s statonary, or frozen. Insertng the plane wave tral soluton (5.1) nto ths wave equaton, applyng the dervatves and rearrangng results n a dsperson relaton for relaxaton, ( ) 2 = c ˆωˆk ˆωτ m (c /c 0 ) 2 0. (5.36) 1 + ˆωτ m A comparson wth the correspondng DVBE dsperson relaton (5.31) shows that the two equatons are dentcal, wth the substtutons c /c 0 = 3 and τm = τ ν /2 = τ. Thus, the wave modes found for the DVBE case correspond exactly to the wave modes of for a sngle relaxaton process wthout any thermovscous nfluence. Whle ths connecton would seem sgnfcant, t s not clear what t means. The DVBE and relaxaton cases are not very smlar. The major smlarty s that both cases nvolve relaxaton to a changng equlbrum state. However, the DVBE gves a macroscopc stress tensor wth vscosty lke the sothermal Naver-Stokes-Fourer model, whle the relaxatonal wave equaton does not. It s based on the more lmted Euler model wth a non-sothermal relaxatonal equaton of state. By some early authors, absorpton and dsperson n fluds was assumed to stem completely from relaxatonal phenomena. Absorpton was assumed to occur only when the densty gets out of phase wth the pressure [105]. Even the effect of vscosty was treated n ths way, though t can be shown not to cause a phase lag between densty and pressure n the lmt of weak waves [25, Ch. 9]. Even though such assumptons may have been wrong n general [26], a relaxaton model could successfully be made to ft ether the absorpton or the dsperson of a purely thermovscous model to lowest order [22]. As we soon shall see, the DVBE fts the absorpton to lowest order when the shear vscosty s matched as μ = pτ.

130 122 Chapter 5 Acoustc lnearsaton analyss Comparson to other models Wth known sound propagaton behavour for the Naver-Stokes-Fourer, DVBE, and sngle relaxaton models, a comparson s now possble. It s not necessary to consder relaxaton separately from the DVBE, as we have shown them to qualtatvely predct the same type of absorpton and dsperson. Forced waves Maxwell molecules Molecules wth a partcular form of the ntermolecular force feld whch smplfes Boltzmann s orgnal collson operator Ths secton s augmented by calculatons done by Greenspan [27] for forced waves wth the Burnett model. He found equatons for ˆk/k 0 for deal gases of Maxwell molecules wth otherwse arbtrary materal propertes. To adapt hs expressons to our sothermal case, we set γ = 1 as n (4.1). From (2.46) and (2.47), ths mples heat capactes whch go to nfnty, whch determnes Greenspan s thermal Reynolds number as 1/s 0. Fnally, Greenspan s assumpton of zero bulk vscosty determnes hs vscous Reynolds number as 1/r = 3γX/4. Wth these assumptons, Greenspan s Naver-Stokes-Fourer model s dentcal to the one descrbed n secton 5.1. Hs dsperson relaton for forced waves wth the Burnett model becomes the bquadratc equaton ( ) 4 ( ) 2 3 ˆk 4 k0 X 2 + ˆk k0 (1 + X) 1 = 0, whch has the four solutons ˆk 2 = ± k 0 3X 2 ± 1 + 2X + 2X 2 (1 + X). (5.37) Two of these solutons are normal propagatng waves n the ±x-drecton, whle the other two are waves that propagate, but wth an absorpton coeffcent where α x as X 0. We wll only consder the normal x-propagatng soluton n the followng. Gatherng the exact solutons of the sothermal Naver-Stokes-Fourer model, the sothermal Burnett model, and the DVBE model, we have N-S-F: Burnett: DVBE: ˆk k 0 = X, ˆk 2 = k 0 3X X + 2X 2 (1 + X), ˆk 1 + X/2 = k X/2. (5.38a) (5.38b) (5.38c)

131 5.2 Dscrete-velocty Boltzmann equaton 123 Whle the analytcal expressons are dffcult to compare by themselves, the seres expanson around X = 0 s more nstructve, N-S-F: Burnett: DVBE: ˆk k 0 = X 3 8 X O(X 5 ), ˆk k 0 = X 6 8 X X X4 + O(X 5 ), ˆk k 0 = X 5 8 X X X4 + O(X 5 ). (5.39a) (5.39b) (5.39c) As ponted out prevously, the terms always alternate between real (dsperson) and magnary (absorpton). All of the models agree up to O(X), meanng that they predct the same absorpton to the lowest order. However, there s dsagreement above ths pont, meanng that they predct dfferent dsperson even to the lowest order. It should not be a surprse that the Naver-Stokes-Fourer model does not match the rest, as ts dervaton was found by truncatng the Chapman- Enskog expanson to O(X); t should not be trusted above ths order. That the Burnett and DVBE models dsagree requres some more explanaton, however. There are at least two reasons: Frstly, the Burnett model ncludes hgher-order transport coeffcents, whch can be seen as second-order vscostes and conductvtes. Ther relaton to the normal vscosty and conductvty s determned by the choce of collson operator and molecular model, although the results seem not to vary greatly between dfferent molecular models [27]. Even so, Greenspan s Burnett expressons were derved va Boltzmann s orgnal collson operator whle the DVBE expressons were derved usng the BGK model. As the BGK model has weaknesses such as an ncorrect Prandtl number,.e. an ncorrect rato between conductvty and vscosty (as seen n secton 2.2.2), t s not unlkely that t also predcts a wrong rato between the transport coeffcents at the Naver-Stokes-Fourer level and the Burnett level. Secondly, when dervng the DVBE, the equlbrum dstrbuton was approxmated as a truncated seres n u and velocty space was dscretsed. The order of the seres approxmaton determnes both the requred sze of the dscrete velocty set and to whch level n the Chapman- Enskog expanson the DVBE wll be accurate [72]. Wth the truncaton to O(u 2 ), we saw that the momentum equaton was not even fully accurate to the Naver-Stokes level, as t contaned a O(u 3 ) error term whch dsappears durng lnearsaton. To get a momentum equaton whch s fully accurate to the Burnett level, the equlbrum dstrbuton would need to be approxmated to a hgher order. Even so, any terms n f (0) above O(u) would dsappear n the lnearsaton. In fact, all terms above frst order n the sothermal Hermte expanson of f (0) are removed by lnearsaton [72]. The order of the

132 124 Chapter 5 Acoustc lnearsaton analyss Bnomal seres The seres expanson of a functon of the form f (x) =(1 + x) a Hermte expanson determnes what level of the Chapman-Enskog expanson to whch a DVBE s vald. Consequently, ths would suggest that the lnearsed sothermal equlbrum dstrbuton (5.24) s vald at every level n the Chapman-Enskog expanson. The Naver-Stokes-Fourer expanson n (5.39a) s a typcal bnomal seres, whch converges for X 1 and dverges for hgher values, even though the orgnal functon (5.38a) s vald for any physcal (.e. real and non-negatve) value of X. If the coeffcents n the correspondng Burnett and DVBE seres expansons are always of larger magntude, then by the comparson test the convergence range of these seres cannot be larger. As explaned n secton 3.8.4, seres lke these are asymptotc; they do not converge unless X s suffcently small. Consequently, these seres expansons are not very useable unless X 1. For X 1, the exact normalsed wavenumbers (5.38) asymptotcally go to N-S-F: lm X 1 ˆk 1 =(1 ) k 0 2X, (5.40a) Burnett: DVBE: lm X 1 lm X 1 ˆk k 0 = ˆk k 0 = , (5.40b) 3X 3X X. (5.40c) CFL condton A stablty condton n numercal mathematcs. For hyperbolc equatons t states that nformaton must not propagate n the smulaton more slowly than the characterstcs of the soluton (e.g. the speed of sound). Thus, we see that the normalsed speed of sound c/c 0 = Re(k 0 /ˆk) unphyscally goes to nfnty for the Naver-Stokes-Fourer and Burnett models, whle t plateaus at 3 for the DVBE model. Snce c 0 = 1/ 3, the maxmum speed of sound s c = 1. Ths equals the speed of partcles n the D1Q3 velocty set, and correspondngly the speed at whch nformaton propagates. In fact, any hgher speeds of sound would mean that the Courant- Fredrchs-Lewy condton would be volated and the DVBE model would necessarly be unstable [104, Ch. 10]. From ths, we can suspect that extended velocty sets would have dfferent behavour for X 1. Interestngly, measurements n real gases at very hgh X ndcate the same qualtatve behavour as the DVBE model [63]. We now know the sound propagaton predcted from the models at X 1 and at X 1. To compare them for X 1, we can plot the exact absorpton and dsperson as n Fgure 5.3. From these plots, we fnd that the three models behave almost dentcally up to X 0.1, where only the lowest-order terms n X are felt. The DVBE and Burnett models behave very smlarly up untl about X 1, n partcular the dsperson. As the thermal Burnett model has been seen to match measurements better than the thermal Naver-Stokes-Fourermodel [27, 63], we can assume that for moderate X, sound propagates

133 5.2 Dscrete-velocty Boltzmann equaton Re(ˆk/k0) =c0/c Naver-Stokes Burnett DVBE X Naver-Stokes Burnett DVBE Im(ˆk/k0) =αx/k Fgure 5.3: Comparson of the normalsed nverse dsperson (upper) and the normalsed absorpton (lower) of forced waves for three dfferent models. X

134 126 Chapter 5 Acoustc lnearsaton analyss 10 1 Naver-Stokes Burnett DVBE k0/ˆk = ρ0û / ˆρ /c X arg(k0/ˆk) =arg(ρ0û / ˆρ ) Naver-Stokes Burnett DVBE Fgure 5.4: Comparson of the normalsed magntude ratos (upper) and the phase dfferences (lower) of the momentum and densty components of forced waves for two dfferent models. X

135 5.2 Dscrete-velocty Boltzmann equaton 127 more correctly n the sothermal DVBE model than n the sothermal Naver- Stokes-Fourer model tself. Nether the predctons of the Naver-Stokes-Fourer model nor those of the Burnett model match well wth measurements of sound propagaton after X 1[27]. The problem of fndng models that can predct sound propagaton well was prevously explaned n secton Up to ths pont, we have only consdered absorpton and dsperson. We should also take a bref look at the relatve ampltudes and phase dfferences of the momentum and densty components of the sound wave. As descrbed prevously, these quanttes are found from the relaton ρ 0 û ˆρ = ˆωˆk. Snce ths relaton s a consequence of the mass conservaton equaton whch holds for all the models, we can also nclude the Burnett model n ths comparson. The magntude ratos and ampltude dfferences of the three models are N-S-F: Burnett: DVBE: 1 ρ 0 û c 0 ˆρ = X X4 + O(X 6 ), ( ) ρ0 û arg ˆρ = 1 2 X 1 6 X3 + O(X 5 ), 1 ρ 0 û c 0 ˆρ = X X4 + O(X 6 ), ( ) ρ0 û arg ˆρ = 1 2 X X3 + O(X 5 ), 1 ρ 0 û c 0 ˆρ = X X4 + O(X 6 ), ( ) ρ0 û arg ˆρ = 1 2 X X3 + O(X 5 ). (5.41a) (5.41b) (5.41c) Agan, we fnd that the three models agree only up to O(X), as could be expected. There s no agreement at O(X 2 ) and above, just as for the absorpton and dsperson. The exact values are plotted n Fgure 5.4. Agan, we see that the Burnett and DVBE solutons are farly close. The phase dfferences for the Naver-Stokes-Fourer model and the Burnett model converge to ( / ) arctan(1) =π/4 and to arctan , respectvely. Both values can be found from the asymptotcs (5.39). Free waves For completeness, we wll also take a look at the case of free waves. We wll not go nto the same depth as for forced waves; the remarks that can be made are mostly smlar, and free waves are physcally less relevant. Greenspan s Burnett soluton s only gven for the more physcally relevant case of forced waves and cannot be used here. We wll gnore

136 128 Chapter 5 Acoustc lnearsaton analyss the dffusve DVBE mode, whch has no analogue n the Naver-Stokes- Fourer model. As the exact DVBE soluton for propagatng free waves s too cumbersome, we jump straght to the seres expansons, N-S-F: DVBE: ˆω ω 0 = X 1 8 X O(X 6 ), ˆω ω 0 = X X X O(X 5 ). (5.42a) (5.42b) The exact values are plotted n Fgure 5.5. The correspondng seres expansons for magntude ratos and phase dfferences are N-S-F: DVBE: 1 ρ 0 û c 0 ˆρ = 1, ( ) ρ0 û arg ˆρ = 1 2 X X3 + O(X 5 ), 1 ρ 0 û c 0 ˆρ = X X4 + O(X 6 ), ( ) ρ0 û arg ˆρ = 1 2 X X3 + O(X 5 ). (5.43a) (5.43b) The exact values are plotted n Fgure 5.6. Three aspects are dentcal n the forced and free wave cases: Frstly, the two models agree up to O(X) but not beyond that order. Secondly, from the fgures we see that the models agree well up to around X 0.1. However, n the free wave case, the Naver-Stokes-Fourer model entrely changes character at X = 2, the pont at whch the wave stops propagatng. Thrdly, the hgh-x lmt of the DVBE phase speed s c = Ansotropy n two dmensons Isotropy/ansotropy Unformty/nonunformty wth respect to drecton The prevous DVBE results n ths secton were found usng the D1Q3 velocty set, and therefore hold only for hgher-dmensonal velocty sets when sound propagates along any of the man (.e. x, y, orz) axes as explaned prevously. Snce hgher-dmensonal velocty sets are not themselves sotropc, and n fact cannot be, we mght expect some amount of ansotropy: The wave propagaton propertes may depend on angle, at least for hgh X. We now use a more general tral soluton; a plane wave propagatng at an angle θ to the x axs, ˆf (x, y, t) =F (0) + ˆf e ( ˆωt ˆk x x ˆk y y), (5.44) where ˆk x = ˆk cos(θ), ˆky = ˆk sn(θ). For θ = 0, ths reduces to the prevously used tral soluton.

137 5.2 Dscrete-velocty Boltzmann equaton Re( ˆω/ω0) =c/c Naver-Stokes DVBE X 10 2 Naver-Stokes DVBE 10 1 Im( ˆω/ω0) =αt/ω Fgure 5.5: Comparson of the normalsed dsperson (upper) and the normalsed absorpton (lower) of free waves for two dfferent models. X

138 130 Chapter 5 Acoustc lnearsaton analyss 10 2 Naver-Stokes DVBE ˆω/ω0 = ρ0û / ˆρ /c X 10 0 Naver-Stokes DVBE arg( ˆω/ω0) =arg(ρ0û / ˆρ ) Fgure 5.6: Comparson of the magntude ratos (upper) and the phase dfferences (lower) of free waves for two dfferent models. X

139 5.2 Dscrete-velocty Boltzmann equaton 131 Π 0 Π x Π y Π xx Π xy Π yy Π xxx Π xxy Π xyy Π yyy Π xxxx Π xxxy Π xxyy Π xyyy Π yyyy Π xxxxx Π xxxxy Π xxxyy Π xxyyy Π xyyyy Π yyyyy Fgure 5.7: The nne ndependent moments of the D2Q9 velocty set and the dependence of hgher-order moments on these. It s natural to capture ths behavour wth the D2Q9 velocty set ntroduced n secton Ths s a two-dmensonal projecton of the D3Q15, D3Q19, and D3Q27 velocty sets, n the same way as the D1Q3 velocty set s a one-dmensonal projecton of all of these. Consequently, the results found here wll also be vald for these three-dmensonal velocty sets for plane waves whch propagate normal to at least one of the man axes. We wll smplfy ths secton by makng full use of lattce unts, settng Δx = Δt = 1. As n the D1Q3 case, the answer wll not actually depend on the value of these. Also, to avod unnecessarly heavy notaton, we suppress the breve accent n the notaton for the moments Π of the dscrete dstrbuton functon f. There should be no chance of msunderstandng, as the moments Π of the contnuous dstrbuton functon f are not relevant here. Lke n (3.23), we agan ntroduce a consstent notaton for the complex lnearsed moments, ˆΠ 0 = ˆΠ αβ = ξ α ξ β ˆf, ˆf = ˆρ, ˆΠ α = ξ α ˆf = ρ 0 û α, ˆΠ αβγ = ξ α ξ β ξ γ ˆf, (5.45) and so forth. In the D1Q3 case, we found that there were only three ndependent moments: ˆΠ 0, ˆΠ x, and ˆΠ xx; the hgher-order moment ˆΠ xxx was found n (5.29) to depend on ˆΠ x. By the same dervaton there are nne ndependent moments n the D2Q9 velocty set [83, Ch. 4], and hgher-order

140 132 Chapter 5 Acoustc lnearsaton analyss moments are gven from these as ˆΠ xxx = ˆΠ x, ˆΠ xxxy = ˆΠ xy, ˆΠ xxxyy = ˆΠ xyy, ˆΠ yyy = ˆΠ y, ˆΠ xyyy = ˆΠ xy, ˆΠ xxyyy = ˆΠ xxy. (5.46) Ths dependence on the nne ndependent moments s also shown graphcally n Fgure 5.7. For ths two-dmensonal plane wave, the fluctuatng equlbrum dstrbuton functon s generalsed to [ ] ˆf (0) = w ˆΠ 0 + ˆΠ ɛ ξ ɛ c 2 0. (5.47) The equlbrum moments of ths can be found usng the symmetry propertes (4.11) to be ˆΠ (0) αβ = c 2 0 ˆΠ 0 δ αβ, (5.48a) ˆΠ (0) ( ) αβγ = c2 0 ˆΠ αδ βγ + ˆΠ β δ αγ + ˆΠ γδ αβ, (5.48b) ˆΠ (0) αβγδ = c4 0 ˆΠ 0 ( ) δαβ δ γδ + δ αγ δ βδ + δ αδ δ βγ. (5.48c) The harmonc lnearsed dscrete-velocty Boltzmann equaton (5.27) n generalsed form s (1 + ˆωτ) ˆf = ˆf (0) + τˆk ɛ ξ ɛ ˆf. (5.49) In the D1Q3 case, we took all the ndependent moments of ths and related them to each other usng the hgher-order moments dependence on the ndependent moments. In the D2Q9 case, we wll do essentally the same thng. However, t s now more complcated due to the hgher number of moments and the two-dmensonal nature of the plane wave. The zeroth-order moment of (5.49) s ˆω ˆΠ 0 (ˆkx = ˆΠ x + ˆk ) y ˆΠ y, (5.50a) and the two frst-order moments are ˆω ˆΠ x = (ˆkx ˆΠ xx + ˆk y ˆΠ xy), ˆω ˆΠ y = (ˆkx ˆΠ xy + ˆk y ˆΠ yy). (5.50b) For the second-order moments, we must start usng the equlbrum moments and the moment dependence relatons, resultng n ), (1 + ˆωτ) ˆΠ xx = c 2 0 ˆΠ 0 (ˆkx + τ ˆΠ x + ˆk y ˆΠ xxy ), (1 + ˆωτ) ˆΠ xy = τ (ˆkx ˆΠ xxy + ˆk y ˆΠ xyy (1 + ˆωτ) ˆΠ yy = c 2 0 ˆΠ 0 + τ (ˆkx ˆΠ xyy + ˆk y ˆΠ y ). (5.50c)

141 5.2 Dscrete-velocty Boltzmann equaton 133 Smlarly, the thrd-order moments are (1 + ˆωτ) ˆΠ xxy = c 2 0 ˆΠ y + τ (ˆkx ˆΠ xy + ˆk ) y ˆΠ xxyy, (1 + ˆωτ) ˆΠ xyy = c 2 0 ˆΠ x + τ (ˆkx ˆΠ xxyy + ˆk ) y ˆΠ xy, (5.50d) and the fourth-order moment s (1 + ˆωτ) ˆΠ xxyy = c 4 0 ˆΠ y + τ (ˆkx ˆΠ xyy + ˆk y ˆΠ xxy ). (5.50e) To deal wth ths complcated system of equatons we put t n matrx form, 1+ ˆωτ ˆk x τ ˆk y τ c 4 0 ˆΠ xxyy ˆk x τ 1+ ˆωτ 0 0 ˆk y τ 0 0 c ˆΠ xyy ˆk y τ 0 1+ ˆωτ 0 ˆk x τ 0 c ˆΠ xxy 0 ˆk x τ 0 1+ ˆωτ 0 0 ˆk y τ 0 c 2 0 ˆΠ yy 0 ˆk y τ ˆk x τ 0 1+ ˆωτ ˆΠ 0 0 ˆk y τ ˆωτ 0 ˆk x τ c 2 xy = 0. 0 ˆΠ xx ˆk y ˆk x 0 ˆω 0 0 ˆΠ y ˆk y ˆk x 0 ˆω ˆk y ˆk x ˆω (5.51) As ths system becomes too dffcult to handle unaded, t s necessary to use a computer algebra system to deal wth t. We must now try to coax a dsperson relaton from ths system. Performng Gaussan elmnaton on the matrx, the last row becomes an equaton g( ˆω, ˆk, θ, X) ˆΠ 0 = 0, where the functon g s too cumbersome to ft here. Snce we can safely assume that ˆΠ 0 = 0, ths reduces to a dsperson relaton g( ˆω, ˆk, θ, X) =0. Ths dsperson relaton can be solved for ˆk for forced waves, or for ˆω for free waves. Whle the exact solutons are extremely complcated and not really comprehensble to look at, t s possble to fnd nterestng nformaton from the seres expansons. Also, the exact solutons can be plotted for dfferent angles θ. Due to the symmetry of the D2Q9 velocty set, we only need to consder the angular nterval 0 θ π/4; the ansotropy s determned by the angle to any man axs, as shown n Fgure 5.8. Forced waves For forced waves where ˆω = ω 0, the seres expanson of the normalsed wavenumber s ˆk k 0 = X 5 8 X [ [ sn2 (θ) cos 2 (θ) ] 13 sn2 (θ) cos 2 (θ) ] X 4 + O(X 5 ). X 3 ˆΠ x ˆΠ 0 (5.52)

142 134 Chapter 5 Acoustc lnearsaton analyss Fgure 5.8: Drectons of wave propagaton, supermposed on the D2Q9 velocty vectors. The lne styles of the dfferent propagaton drectons correspond wth Fgures 5.9 and The wave propagaton propertes have a smlar angular perodcty as the velocty set. Ths ndcates that the angular dependence of DVBE sound propagaton does not start untl O(X 3 ). Consequently, both absorpton and dsperson s sotropc to lowest order n X. For angles θ = 0, π/2, π,3π/2, where the propagaton drecton concdes wth a man axs, ths seres expanson reduces to the correspondng D1Q3 seres expanson (5.39c) as predcted. The exact dsperson and absorpton s plotted for fve dfferent angles n the nterval 0 θ π/4 n Fgure 5.9, and the exact magntude ratos and phase dfferences are plotted smlarly n Fgure Whle these fgures show practcally sotropc behavour up to X 0.1, sgnfcant angular varatons are found beyond that pont. In partcular, the dsperson and magntude ratos have entrely dfferent lmts at X for dfferent angles. For the dsperson, the X lmt of the phase speed c s cos(θ) for 0 θ π/4. Ths asymptotc phase speed s shown for all angles n Fgure At θ = π/4, when sound propagates parallel to the dagonal partcle veloctes, the speed of sound s c = 1/ 2, half the speed ξ {5 8} of the dagonal partcles. The absorpton stll behaves qualtatvely the same for any angle, although the poston of the absorpton peak s a functon of angle, and the peak tself s somewhat uneven for angles not concdng wth a man axs.

143 5.2 Dscrete-velocty Boltzmann equaton ) = c0/c Re (ˆk/k θ = 0 θ = 1 16 π θ = 2 16 π θ = 3 16 π θ = 4 16 π X = αx/k θ = 0 θ = 1 16 π θ = 2 16 π θ = 3 16 π θ = 4 16 π Im (ˆk/k 0 ) Fgure 5.9: Comparson of the normalsed nverse dsperson (upper) and the normalsed absorpton (lower) of forced waves for a D2Q9 velocty set and fve dfferent angles. X

144 136 Chapter 5 Acoustc lnearsaton analyss k0/ˆk = ρ0û / ˆρ /c θ = 0 θ = 1 16 π θ = 2 16 π θ = 3 16 π θ = 4 16 π X arg(k0/ˆk) =arg(ρ0û / ˆρ ) θ = 0 θ = 1 16 π θ = 2 16 π θ = 3 16 π θ = 4 16 π Fgure 5.10: Comparson of the magntude ratos (upper) and the phase dfferences (lower) of forced waves for for fve dfferent angles for a D2Q9 velocty set and fve dfferent angles. X

145 5.3 Lattce Boltzmann equaton 137 Fgure 5.11: Phase speed c for X = 0 ( ) and X ( ) as functon of angle, supermposed on and drawn to the same scale as the D2Q9 velocty vectors. Ths holds both for forced and for free waves. Free waves In the free wave case, the correspondng seres expanson s [ ] ˆω ω 0 = X X sn 2 (θ) cos 2 (θ) [ ] sn2 (θ) cos 2 (θ) X 4 + O(X 5 ) X 3 (5.53) Agan, the expanson reduces to the D1Q3-based expanson (5.42b) for the angles θ = 0, π/2, π,3π/2. In the same way as for the forced wave case, the dsperson and absorpton are plotted n Fgure 5.12 and the magntude ratos and phase dfferences n Fgure The comments for the forced wave case hold for the free wave case as well. The asymptotc phase speed s the same as n the forced wave case; the same as shown n Fgure Lattce Boltzmann equaton In the last secton we examned n detal the sound propagaton propertes of the dscrete-velocty Boltzmann equaton. The lattce Boltzmann equaton s smply ths DVBE, dscretsed n space and tme. For nfntely fne tme and space resoluton Δt and Δx, the sound propagaton of the LBE must therefore behave lke the DVBE. Wth a fnte resoluton, numercal errors wll be felt for the dsperson and absorpton. The dervaton s smlar to the one for the DVBE, wth one major dfference: The DVBE has exact dervatves n space and tme, whereas for the LBE we must evaluate functons at dfferent ponts and tmes. Consequently, the LBE analyss wll be smlar to a von Neumann analyss.

146 138 Chapter 5 Acoustc lnearsaton analyss Re ( ˆω/ω0) = c/c θ = 0 θ = 1 16 π θ = 2 16 π θ = 3 16 π θ = 4 16 π X Im( ˆω/ω0) =αt/ω θ = 0 θ = 1 16 π θ = 2 16 π θ = 3 16 π θ = 4 16 π Fgure 5.12: Comparson of the normalsed dsperson (upper) and the normalsed absorpton (lower) of free waves for a D2Q9 velocty set and fve dfferent angles. X

147 5.3 Lattce Boltzmann equaton 139 ˆω/ω0 = ρ0û / ˆρ /c θ = 0 θ = 1 16 π θ = 2 16 π θ = 3 16 π θ = 4 16 π X arg( ˆω/ω0) =arg(ρ0û / ˆρ ) θ = 0 θ = 1 16 π θ = 2 16 π θ = 3 16 π θ = 4 16 π Fgure 5.13: Comparson of the magntude ratos (upper) and the phase dfferences (lower) of free waves for for fve dfferent angles for a D2Q9 velocty set and fve dfferent angles. X

148 140 Chapter 5 Acoustc lnearsaton analyss Smlar work has been done before for the LBE, though only for the free wave case. Sterlng and Chen performed an early lnearsaton stablty analyss [106]. Lallemand and Luo [84], and later Res and Phllps [107], performed an analyss for wavelke perturbatons the D2Q9 velocty set and the MRT collson operator. Dellar compared the results of a BGK D1Q3 analyss wth measurements of sound absorpton n smulatons, fndng full agreement [77]. Wlde gave numercal results for the D2Q9 and D3Q19 velocty sets [44]. Maré et al. compared the sound propagaton of lattce Boltzmann and hgh-order fnte-dfference- Runge-Kutta schemes for the wave equaton, concludng that the LBE s the faster method for a gven dsperson error [108]. However, the prevous work s lackng n two areas. Frstly, t consders only free waves, where the analyss s the smplest to perform, but where the results are not partcularly physcally relevant and cannot be used to derve results for the more relevant forced waves [26]. Secondly, none of the prevous work separates the effects of the model (DVBE) and the numercal error. Ths current work wll both consder forced waves and the separaton of model and error, though t wll refran from analysng more complex cases than the sothermal D1Q3 model and a medum at rest Lnearsaton process As done when lnearsng the DVBE, we here use the D1Q3 velocty set wth partcle veloctes (ξ, ξ 0, ξ + )=( 1, 0, 1), weghtng coeffcents (w, w 0, w + )=(1/6, 2/3, 1/6), and speed of sound c s = 1/ 3. Ths D1Q3 analyss wll predct how waves propagate along man axes n any lattce that can be projected to D1Q3, and wll therefore not nclude the effects of ansotropy. The tral soluton s on the same form as for the DVBE, ˆf (x, t) =F (0) + ˆf (x, t) =F(0) + ˆf e ( ˆωt ˆkx). (5.54) The dfference s that x and t are dscrete here, whereas they were contnuous n the DVBE case. The equlbrum dstrbutons are the same as n (5.24). For the fluctuatng ampltude, t s ˆf (0) = w ( ) ˆρ + ρ 0û [ c 2 ξ = w ˆf + ˆf 0 + ˆf ( ˆf + ˆf ) ] ξ. 0 (5.55) Snce the frst order LBE (4.32) and the second order LBE (4.40) are on the same form, t does not matter whch one we use for the lnearsaton. We wll use the more common frst order LBE here. Subtractng the background rest state from t, we get a lnearsed LBE, ˆf (x + ξ, t + 1) = ( 1 1 τ ) ˆf + τ 1 (0) ˆf. (5.56)

149 5.3 Lattce Boltzmann equaton 141 Usng the plane wave tral soluton and the equlbrum dstrbuton of the fluctuaton, we fnd ( ) [ ˆf e ( ˆω ˆkξ ) = 1 τ 1 ˆf + w τ ˆf + ˆf 0 + ˆf ( ˆf + ˆf ] ) ξ. (5.57) Evaluatng for the three dfferent dstrbuton functons, we get the system of equatons ( ) ˆf e ˆω e ˆk = 1 3τ 1 ˆf + 6τ 1 ˆf 0 3τ 1 ˆf +, (5.58a) ˆf 0 e ˆω = 3τ 2 ˆf ( ) + 1 3τ 1 ˆf 0 + 3τ 2 ˆf +, (5.58b) ˆf + e ˆω e ˆk = 3τ 1 ˆf + 6τ 1 ˆf ( ) τ 1 ˆf +. (5.58c) In matrx form, ths s 1 3 e ˆk(3 1/τ) e ˆk/2τ e ˆk/τ 2/τ 3 1/τ 2/τ e ˆk/τ e ˆk/2τ e ˆk(3 1/τ) ˆf ˆf 0 ˆf + = e ˆω ˆf ˆf 0 ˆf +. (5.59) The matrx form of the system of equatons s a typcal egenvalue problem  ˆf = e ˆω ˆf, wth e ˆω as the egenvalue. For the free wave case, ths s smple to handle: Wth ˆk = k 0 known, ˆω can be found drectly from the matrx s egenvalues. The forced wave case s more dffcult. Here, the egenvalue e ˆω = e ω 0 s known, but the matrx contans the unknown complex wavenumber ˆk. In a prevous publcaton, ths author used a numercal search to fnd the value of ˆk that gave the correct egenvalue e ω 0 [11]. However, ths approach does not lead to analytcal expressons. An analytcal technque whch works both for free and for forced waves uses the characterstc polynomal of (5.59), det(â I e ˆω )=g( ˆω, ˆk, τ) =0. (5.60) The characterstc polynomal g, whch also doubles as a dsperson relaton, s unweldy and we do not gan much by dsplayng t here. However, t can be solved for ether ˆk or ˆω, leadng to useful results for forced or free waves, respectvely. The egenvector also contans useful nformaton. For any forced or free wave propagaton mode, the complex ampltude rato of the momentum and densty wave components are ρ 0 û ˆρ = ˆf + ˆf ˆf + ˆf 0 + ˆf +. (5.61) A smulaton of free waves can be ntalsed exactly usng (5.54) f the values of ˆf are known. On the other hand, f the ntalsaton s performed usng standard acoustcal expressons based on the Euler model,

150 142 Chapter 5 Acoustc lnearsaton analyss several wave modes wll be present smultaneously. The ntal state wll typcally contan the ntended wave mode wth a small contrbuton of the mode of wave propagaton n the opposte drecton, leadng to a slght standng wave effect n the smulaton. We wll demonstrate ths n secton Results Analytcally fndng the egenvectors ˆf n (5.59) seems to be a problem too dffcult to handle for the computer algebra system used here. We wll therefore restrct ourselves to the absorpton and dsperson propertes that can be found from ˆk and ˆω. However, t was found numercally that the magntude ratos and phase dfferences correspond wth the DVBE lnearsaton results for forced waves (5.41c) and free waves (5.43b) n the lmt of an nfntely fne numercal resoluton [11]. To determne whether solutons of the egenvalue problem (5.59) can be used to predct LB sound propagaton, the predcted LB waves should be compared wth actual waves. Indeed, ths has been done prevously both for free waves [77] and for forced waves [11]. We wll also see n the next chapters that the solutons accurately predct LB wave propagaton even for two-dmensonal smulatons of non-plane waves. Forced waves In the case of forced waves, where ˆω = ω 0, we can solve the characterstc polynomal for ˆk. There are two solutons, one for each drecton of propagaton. For propagaton n the +x-drecton, the exact analytcal soluton s ˆk = ln [ 3τ(ζ 2 ζ + 1 ζ 1 )+ζ 2 + ζ 1 (3 + ] 3Ξ), (5.62) 4 + 6τ(ζ 1) 2ζ where the shorthands ζ = e ω 0 and Ξ =(ζ + 1)(ζ 1) 2 (τζ + 1 τ)(3τζ 2 ζ + 3 3τ) have been used. Whle ths soluton s complcated, t s far less so than the exact solutons that have been judged too cumbersome to dsplay elsewhere n ths chapter. If we seres expand ths exact soluton, we can separate between the numercal error and the physcal behavour by nestng seres expansons n ω 0 nto a seres expanson n X. The angular frequency ω 0 n lattce unts expresses the numercal resoluton as t determnes the number of

151 5.3 Lattce Boltzmann equaton X = 0 X = 0.1 X = 0.5 Re(ˆk/k0) =c0/c ω X = 0 X = 0.1 X = 0.5 Im(ˆk/k0) =αx/k Fgure 5.14: The normalsed nverse dsperson (upper) and the normalsed absorpton (lower) of forced waves as functons of the numercal resoluton ω 0 for three dfferent values of X. ω 0

152 144 Chapter 5 Acoustc lnearsaton analyss tme steps that resolve a perod of the wave. The seres expanson s [ ] ˆk k 0 = ω ω4 0 + O(ω6 0 ) ] 1 2 [1 X ω2 0 + O(ω4 0 ) 5 8 X2 [ ω2 0 + O(ω4 0 ) ] + O(X 3 ). (5.63) Ths seres expanson has at least two nterestng consequences. Frstly, when the numercal resoluton s nfntely refned as ω 0 goes to zero, ths seres expanson goes to the DVBE soluton (5.39c), as expected. Secondly, the numercal error s second-order n ω 0. Ths shows analytcally that LB s a second order scheme, at least n the lnear lmt, even wth the frst order dscretsaton of the DVBE. The exact dsperson and absorpton are shown n Fgure 5.14.* The most strkng result s how the character of the soluton changes at ω Beyond ths pont, even waves n the nvscd lmt X = 0are heavly absorbed. A closer look reveals that ths s the pont where λ = 2πc/ω 0 = 2; the shortest wavelength whch s possble to resolve n dscretsed space. Free waves For free waves, ˆk = k 0. The wavenumber k 0 determnes the number of nodes per wavelength, and expresses the numercal resoluton lke ω 0 dd n the forced wave case. There are three dfferent solutons of the characterstc polynomal, smlar to the three wave modes found for the DVBE case. Two modes propagate n opposte drectons. The thrd mode s dffusve and does not propagate. Unlke the forced wave case, the mathematcal expresson of all the three modes are too cumbersome to wrte out here. For the DVBE model, the dffusve mode s mmedately absorbed when X = 0,.e. when the soluton relaxed mmedately to equlbrum. For the LBE, the dffusve mode s mmedately absorbed when τ = 1. Ths smlarly corresponds to mmedate relaxaton to equlbrum, but t does not correspond to X = 0. For general values of τ, numercal expermentaton ndcates that the dffusve egenvalue s e ˆω = 1 1/τ for nfntely fne numercal resoluton,.e. k 0 0. Ths leads to the ampltude of the dffusve wave mode beng under- or overrelaxed analogously to the BGK relaxaton shown n Fgure 4.4. Consequently, f *Because of a roundoff error due to lmted numercal precson, the soluton orgnally plotted by the computer algebra system dverged for very small ω 0. The soluton from the seres expanson (5.63) had to be substtuted n Fgure 5.14 where the exact soluton was n error. Ths also had to be done n the free wave case.

153 5.3 Lattce Boltzmann equaton X = 0 X = 0.1 X = 0.5 Re( ˆω/ω0) =c/c k 0 Im( ˆω/ω0) =αt/ω X = 0 X = 0.1 X = k 0 Fgure 5.15: The normalsed dsperson (upper) and the normalsed absorpton (lower) of free waves as functons of the numercal resoluton k 0 for three dfferent values of X.

154 146 Chapter 5 Acoustc lnearsaton analyss ths mode s present at ntalsaton n a smulaton at low vscosty, t wll be slow to dsappear. The mode propagatng n the x-drecton can be expressed as the seres expanson [ ] [ ] ˆω ω 0 = k k4 0 + O(k6 0 ) X 1 + O(k 4 0 ) X2 [ k2 0 + O(k4 0 ) ] + O(X 3 ). (5.64) As n the forced wave case, ths reduces to the DVBE soluton (5.42b) as k 0 0. The numercal accuracy s second order except the lowest order absorpton, where the the lack of a k 2 0 term has been shown n an earler numercal study [11]. Indeed, the exact dsperson and absorpton, plotted n Fgure 5.15, show that the absorpton changes slowly wth k 0. In addton, there s always zero absorpton for X = 0, unlke the forced wave case. Accuracy We have seen that the lattce Boltzmann equaton s second-order accurate n space and tme, at least n the lnear lmt. As the numercal resoluton s mproved, the LBE soluton converges to the soluton of the dscretevelocty Boltzmann equaton, on whch the LBE s based. Ths DVBE predcts the same lowest-order absorpton but dfferent dsperson to the sothermal Naver-Stokes-Fourer model. The seres expansons of ˆk/k 0 (5.63) and ˆω/ω 0 (5.64) quantfy the severty of numercal errors n absorpton and dsperson. Let us determne when the numercal dsperson s one percent, meanng that numercal errors cause sound waves to propagate one percent more slowly than they should. For forced and free waves at low X, the seres expansons ndcate to a good approxmaton that ths occurs when ω0 2/12 = 0.01 and k2 0 /36 = 0.01, respectvely. Relatng these to the wavelength, both lead to λ 0 = 2π/ Thus, wth more than 10.5 nodes per wavelength, the numercal dsperson s less than one percent Example: Exact wave ntalsaton The usefulness of the egenvalue problem (5.59) s not lmted to predctng the wavenumber ˆk of forced waves or the frequency ˆω. The egenvector ˆf can also be useful when ntalsng a smulaton. Not so much n the forced wave case where the wave s generated by a source,* but more so *Such acoustc sources wll be descrbed n Chapter 6.

155 5.3 Lattce Boltzmann equaton 147 n the free wave case where the wave must already exst fully formed when the smulaton s started. Let us look at a smple one-dmensonal free wave case. Wth one wavelength of a wave of spatally constant ampltude ntalsed nsde a perodc system, an nfnte wave s smulated. The ntal condtons thus fulfl ρ (x,0)=ρ cos( k 0 x + ϕ ρ ), u (x,0)=u cos( k 0 x + ϕ u ), (5.65a) (5.65b) for 0 x λ 1, wth the wavenumber beng k 0 = 2π/λ. The perodc boundary condton connects the nodes at x = 0 and x = λ 1, makng them neghbours. One oft-used ntalsaton method for ths case s to ntalse at equlbrum wth the ampltudes and phases ntalsed usng the Euler-level relatonshps ρ 0 u = c 0 ρ and ϕ ρ = ϕ u [ ]. These can be found from (5.22) wth X = 0. Ths results n an ntal dstrbuton functon f (x,0)=ρ 0 w + ρ w (1 + ξ /c 0 ) cos( k 0 x + ϕ ρ ). (5.66) In another, smlar ntalsaton method the choce u = 0 s made nstead [77, 112], so that the ntal condton represents the superposton of two plane waves propagatng n opposte drectons.* However, ths type of ntalsaton s nexact for two reasons. Frstly, t neglects the small magntude rato dfference and phase shft caused by vscosty and dscretsaton error, as descrbed prevously n ths chapter. Secondly, the system s ntalsed at equlbrum wth f neq = 0, a state whch ends mmedately when the smulatons start unless τ = 1. Instead, the free wave can be ntalsed usng the egenvalue problem (5.59). Nonlnear effects notwthstandng, ths method should be exact snce s based on an exact soluton of the same problem. As the wavenumber k 0 s known from the system sze λ, we can drectly numercally fnd the egenvector ˆf from the matrx  and use t to ntalse the smulaton as ( ) f (x,0)=ρ 0 w + Re ˆf e k 0x. (5.67) The two methods of ntalsaton can be compared drectly to each other by smulaton. We choose a system length of λ = 11 whch as prevously shown should gve a dsperson error of less than one percent. Ths choce corresponds to ω and k A relaxaton tme of τ = 0.52 s chosen, resultng n an acoustc vscosty number X The ampltude was chosen as ρ /ρ 0 = 10 6 n order to avod nonlnear effects. *The Euler-level ampltude relatonshp s n general ρ 0 u = ±c 0 ρ ; the sgn corresponds to propagaton n the ±x-drecton.

156 148 Chapter 5 Acoustc lnearsaton analyss Normalsed ampltude Euler nt. Egenvec. nt. e α tt t Fgure 5.16: Normalsed ampltude of the fundamental frequency component absorbed n tme, for two dfferent methods of free wave ntalsaton, along wth the predcted Naver-Stokes model absorpton. Smulaton parameters are λ = 11, τ = 0.52 (leadng to X 0.013), and ρ /ρ 0 = The comparson of the absorpton n tme for the two dfferent ntalsaton methods s shown n Fgure 5.16, along wth the absorpton (5.17) predcted from the Naver-Stokes-Fourer model. The egenvector ntalsaton (5.67) and the Naver-Stokes-Fourer model agrees to plottng accuracy. However, wth the Euler ntalsaton (5.66) the wave msbehaves, wth a strong perodc rpple n the ampltude. The errors n the Euler ntalsaton are caused by the wave beng ntalsed n a way whch s not fully consstent wth the desred +xpropagatng wave mode. The ntalsaton s based on Euler-level expressons whch do not take nto account the effects of vscosty and dscretsaton error. Consequently, the actual ntalsed state s a superposton of two waves: A strong one propagatng n the +x-drecton and a weak one propagatng n the x-drecton. The rpple n Fgure 5.16 comes from constructve and destructve nterference of the two waves. They are ntalsed at constructve nterference, and nterfere twce n each wave perod. If ths explanaton of the Euler ntalsaton errors holds, the rpple perod must be half of the wave s perod, or approxmately λ/2c 0 9.5, whch matches the observed rpple perod qute well. Of course, ths example s somewhat exaggerated, wth farly large values of k 0 and X. If these quanttes are smaller, the devatons between the two ntalsaton methods, whch should scale wth k 2 0 and X2, wll also decrease. Even so, the egenvector-based ntalsaton should stll be chosen f possble for the hghest possble accuracy.

157 5.4 Summary and dscusson Summary and dscusson In ths chapter, a lnearsaton analyss was appled to three comparable, but dfferent models of a flud: The sothermal Naver-Stokes-Fourer (N-S-F) model, the Boltzmann equaton wth dscretsed velocty space (DVBE), and the fully dscretsed lattce Boltzmann equaton (LBE). The analyss resulted n a predcton of how sound would propagate for each model; most mportantly, how the speed of sound and the absorpton of sound would be altered by governng parameters. Forced wave results for a fourth model, called the Burnett model, were taken from the lterature [27]. For the N-S-F model, the sngle governng parameter s the dmensonless acoustc vscosty number X = O(Kn), whch descrbes the effect of vscosty on sound propagaton. Ths model and the DVBE and Burnett models do not agree beyond O(X). Ths s reasonable consderng that the N-S-F model can be derved as a O(Kn) approxmaton to the Boltzmann equaton. Above ths order, that model s predctons cannot be trusted. The DVBE model s an ntermedary step between the Boltzmann equaton and the lattce Boltzmann equaton, where velocty space has been dscretsed but not physcal space and tme. The sound propagaton propertes of ths model also depend on the velocty set used when dscretsng the velocty space, and the sound waves angle of propagaton. For the D2Q9 velocty set, sound propagaton s ansotropc at O(X 3 ) and at hgher orders. One partcularly nterestng result of ths ansotropy s the maxmum speed of sound at dfferent angles. When a sound wave propagates along a man axs wth the D2Q9 velocty set, dsperson causes the phase speed to go asymptotcally to c = ξ {1 4} = 1asX. Ths lmt s reasonable, as a hgher c would break the CFL condton. However, as the propagaton angle moves away from a man axs, the asymptotc phase speed decreases to a mnmum of 1/ 2 for dagonal propagaton, as shown n Fgure In Chapter 7, whch s about changng the model s equaton of state and speed of sound, we wll see that ncreasng the D2Q9 sound speed c 0 above 1/ 2 leads to nstablty. Fnally, the lnearsaton analyss on the lattce Boltzmann equaton tself shows that sound n LB smulatons propagates as predcted by the DVBE model, wth the addton of dscretsaton error terms whch start at second order n the numercal resoluton. The sound propagaton propertes of the LBE thus go asymptotcally to those of the DVBE as the resoluton s ncreased. All analyses show major dfferences n the results for forced waves (generated by a source and absorbed wth dstance to the source) and free waves (plane waves of nfnte extent and constant ampltude, absorbed wth tme). As an example taken from the sothermal N-S-F model,

158 150 Chapter 5 Acoustc lnearsaton analyss the phase speed of forced waves ncreases wth frequency, whle for free waves t decreases wth frequency. It s not possble to predct the sound propagaton propertes of one type of wave from those of the other [26]. Whle both types of wave wll be used later n ths thess, only forced waves are physcally realsable. For smulatons of sound n the audble range, the acoustc vscosty number X s generally so small that the O(X 2 ) devatons between the models are entrely neglgble. The dscretsaton error may as always be decreased by ncreasng the numercal resoluton. The sound propagaton expressons for the DVBE model and the LBE are of course not fully general. These results depend on the choce of equlbrum dstrbuton, propagaton angle, and the choce of collson operator. However, for the fully one-dmensonal case of plane wave propagaton along a man axs wth D2Q9, D3Q15, D3Q19, or D3Q27 velocty sets, the problem can be projected down to D1Q3. Then, any orgnal lnear collson operator of the form of (4.53) reduces to BGK, as there s only one remanng nonconserved parameter and consequently only one relevant relaxaton tme. As for dependence on the equlbrum dstrbuton, we wll look at a lnearsaton analyss of a more general equlbrum dstrbuton n Chapter 7. Plane waves have some dfferent propertes to other smple waveforms such as cylndrcal and sphercal waves. For nstance, for X = 0, sphercal waves have a phase dfference between the pressure and velocty component whch goes asymptotcally to π/2 as the dstance to the orgn of the wave goes to 0 [24, Ch. 5]. However, snce forced waves lead to a Helmholtz equaton where ˆk s ndependent of the waveform, as seen n secton 2.3.2, we may expect that the plane wave wavenumber predcton s accurate also for other waveforms. In fact, we wll see n Chapter 6 that the predcted wavenumber (5.62) accurately descrbes the propagaton of a smulated cylndrcal wave. The work n ths secton may be extended by performng thorough analyses on the sound propagaton propertes of forced waves n the LBE for other velocty sets than D1Q3. Ths s necessary to descrbe the ansotropy of the dscretsaton error for each velocty set.* In the lterature, such analyses currently only exst for free waves. *Smlarly to how the D1Q3 results are vald for 1D propagaton n a number of other velocty sets, such D2Q9 results would be vald for 2D propagaton n the D3Q15, D3Q19, and D3Q27 velocty sets.

159 6 Mesoscopc acoustc sources Lattce Boltzmann smulatons may deal wth sound waves, but where can these sound waves come from? In some cases, sound s generated aerodynamcally by the nstablty of the flow feld tself [42 50, ]. In other cases, sound waves are set up through ntal condtons; the ntal dstrbuton of f s such that sound waves start propagatng when the smulaton starts runnng.* For free waves, ths ntal dstrbuton s typcally one wavelength of a sound wave, ft n a perodc system so that t represents a wave of nfnte extent [77, ], as done n secton For forced waves, there s typcally an ntal condton where the densty s ncreased n an area, often wth a Gaussan or smlar dstrbuton n space [9, 98, 101, 102, 108, ]. Ths results n a wave propagatng outwards from ths area. We wll now look at yet another case, where sound s generated contnuously throughout the smulaton by some knd of source. One such method uses a tme-varyng boundary condton to generate a sound wave that propagates nto the doman [ ]. Another s based on pnnng the densty and optonally the velocty n one or more nodes to fxed, oscllatng values,.e. ρ = ρ 0 + ρ sn(ω 0 t). (6.1) The earlest publshed nstance of such an LB source was for a heavly modfed LBM for wave propagaton [124], equvalent to the TLM method [125, 126]. Later versons of such sources were mplemented for more standard LBMs for flud flow [9 11, 86, 98, 99, ]. The latter source method has ts dsadvantages. Frstly, t fully relaxes the source node(s) to equlbrum n each tme step, locally removng the nonequlbrum dstrbuton f neq whch s relevant to the physcal behavour of the model. Secondly, t can generate errors n the source node(s) by overwrtng the densty and flud velocty of the background flow. Ths s most easly seen n the lmt ρ = 0, where the method stll *Ths also happens as an undesred sde effect n cases when a flow feld s ntalsed to an ncorrect state, for example when flow n a channel wth an obstructon s ntalsed as a normal Poseulle flow for a channel wthout any obstructon. However, alternatve mplementatons of ths method that negate ths dsadvantage by replacng only f (0), thus leavng f neq ntact, should be possble. 151

160 152 Chapter 6 Mesoscopc acoustc sources Macroscopc approach Mesoscopc approach Knetc eq. Knetc eq. Src. term Src. terms Cons. eqs. Cons. eqs. + src. terms Wave eq. + src. terms Wave eq. + src. terms Fgure 6.1: Two approaches to multpole source terms n the wave equaton. Wth the classc approach (left), source terms are added at the conservaton equaton level. Wth the current approach (rght), a sngle source term s added at the knetc level. pns the densty to ρ 0. Such errors wll propagate outward, pollutng the rest of the flow feld. Thrdly, no expressons, nether regresson- nor theory-based, have been found to relate the ampltudes and phases of the source node and the radated wave. Therefore, t s hard to predct what comes out of such a source. In ths secton we descrbe and verfy a new type of source, based on addng a source term to the knetc equaton; ether the Boltzmann equaton or the lattce Boltzmann equaton. Ths approach s somewhat analogous to the approach n secton 2.3.4, where terms n the conservaton equatons for mass and momentum lead to multpole source terms n the wave equaton. For example, an artfcal mass flux source term n the mass conservaton equaton results n a monopole source term. Wth the current method, we go one level deeper: Instead of source terms n the conservaton equatons, we put a more general partcle source term n the knetc equaton. We wll see that ths causes source terms to appear n the conservaton equatons derved from the knetc equaton. These source terms lead n turn to source terms n the wave equaton. The two dfferent approaches are llustrated n Fgure 6.1. Other extensons to the lattce Boltzmann method also work by addng source terms to the lattce Boltzmann equaton. For example, body forces [130], adjustable speed of sound [111], adjustable bulk vscosty [76, Ch. 3], and axsymmetrc geometres [131, 132] have all been mplemented n a smlar way.

161 6.1 Source terms for the Boltzmann equaton Source terms for the Boltzmann equaton Frst, we wll go through the general case of a source term n the contnuous Boltzmann equaton. In secton 6.2 we wll look at the hghly related and more drectly applcable case of a source term n the lattce Boltzmann equaton. In ths dervaton, we wll make certan approxmatons: We neglect the effects of heat conducton and molecular relaxaton, and use the sentropc equaton of state (2.23), to approxmate p from ρ as p p 0 = ( ) ρ γ. (6.2) Ths equaton of state lets us nclude the effects of equpartton of energy between translatonal and nner degrees of freedom.* It also leads to the deal speed of sound c 2 0 = p/ ρ = γp 0/ρ 0. Generalsng the reference state, we have = γp/ρ. (6.3) c 2 0 Next, we add a source term to the forceless Boltzmann equaton (3.20) wth the BGK collson operator (3.22), ρ 0 f t + ξ f = j 1 ( f f (0)) (6.4) τ where j(x, ξ, t) s the partcle source term, whch descrbes the rate at whch partcles are added nto or removed from the dstrbuton functon f (x, ξ, t). Consequently, the source term lets us specfy not only how much mass s added, but also the velocty dstrbuton of that mass. In fact, no net mass s necessarly added; for nstance j(x, ξ, t) could be an odd functon n velocty space. We wll fnd the moments of j very useful, and we defne them smlarly to the moments (3.23) of f, J 0 (x, t) = j(x, ξ, t) dξ, (6.5a) J α (x, t) = ξ α j(x, ξ, t) dξ, (6.5b) J αβ (x, t) = ξ α ξ β j(x, ξ, t) dξ, (6.5c) and so forth. J 0 represents the nstantaneous mass flux, J α s assocated wth odd symmetres n velocty space, and J αβ s assocated wth even symmetres. *We now assume that the equlbraton process s nstant, meanng that the translatonal and nner degrees of freedom are always n equlbrum, unlke the slower equlbraton process descrbed n secton Partcle source term, j or j An artfcal source term n the (lattce) Boltzmann equaton whch represents the flux of partcles beng added wth poston x and velocty ξ at tme t

162 154 Chapter 6 Mesoscopc acoustc sources Macroscopc conservaton equatons Our goal now s to fnd the wave equaton that results from the Boltzmann equaton wth an added source term. Ths must be found va the conservaton equatons, whch n turn must be found va a Chapman-Enskog expanson of (6.4). As we only need the mass and momentum equatons to derve the wave equaton, t s easest to perform a moment-based expanson as n secton To close the expanded system of equatons usefully, we must assume that the source term s at frst order of smallness,.e. j ɛj. (Expandng j across several orders gves no extra beneft n ths case.) Expandng (6.4) and separatng the orders, we have ( ) O(ɛ): f (0) = j 1 τ f (1), (6.6a) O(ɛ 2 ): f (0) ( + t 2 + ξ α t 1 x α ) + ξ α f (1) = 1 t 1 x α τ f (2). (6.6b) The zeroth to second order moments of the terms at frst order n smallness are ρu α t 1 Π (0) αβ t 1 ρ t 1 + ρu α x α = J 0, (6.7a) + Π (0) αβ = J α, (6.7b) x β + Π (0) αβγ x γ = J αβ 1 τ Π(1) αβ, (6.7c) and the zeroth and frst order moments at second order n smallness are ρu α t 2 ρ t 2 = 0, (6.8a) + Π (1) αβ = 0. (6.8b) x β As before, to fnd the momentum equaton we must fnd Π (1) αβ through Π (0) αβ t 1 + Π (0) αβγ x γ ( u α = p + u ) ( β p + δ x β x αβ + pu ) γ α t 1 x γ (6.9) + ( ) u α J β + u β J α u α u β J 0. The second parenthess prevously dsappeared n the sothermal case where γ = 1. In ths more general case we wll see that t leads to a

163 6.1 Source terms for the Boltzmann equaton 155 γ-dependent bulk vscosty. The parenthess can be smplfed by to get p = p ρ = c 2 ρ 0, t 1 ρ t 1 t 1 p t 1 + pu γ x γ p = p x α ρ ρ = c 2 ρ 0, x α x α = c 2 0 J 0 +(p c 2 0 ρ) u γ x γ = c 2 0 J 0 + p(1 γ) u γ x γ. Ths gves us the frst order perturbaton to the second order moment of f, [ Π (1) αβ = pτ u α + u β + δ x β x αβ (1 γ) u ] γ α x γ (6.10) ) + τ (J αβ δ αβ c 2 0 J 0 u α J β u β J α + u α u β J 0. ρu α t Now we can fnd the mass and momentum conservaton equatons, ρ t + ρu α x α = J 0, (6.11a) + ρu αu β x β = p + σ αβ x α x β + J α x β τ (6.11b) (J αβ δ αβ c 2 0 J 0 u α J β u β J α + u α u β J 0 ). wth a stress tensor [ σ αβ = pτ u α + u β + δ x β x αβ (1 γ) u ] γ. (6.11c) α x γ The bulk vscosty ν B =(5/3 γ)ν contaned n ths stress tensor has been found n prevous lterature for the same lmt of nstant equlbraton of energy between degrees of freedom [55, 133]. Ths bulk vscosty also has some famlar lmts: For a monatomc gas, γ = 5/3 and ν B = 0, as found n secton 3.8. For an sothermal gas, γ = 1 and ν B /ν = 2/3, as found n secton Comparng wth the classc source terms n (2.60), we fnd that the zeroth moment J 0 s equvalent to the mass flux source term Q, as could be expected, and the frst moment J α s analogous to the body force densty F α. The terms n the momentum equaton whch are proportonal to τ have no such analogue Lnear wave equaton From these conservaton equatons, we can derve the wave equaton. We wll perform ths dervaton under the assumpton that the acoustc

164 156 Chapter 6 Mesoscopc acoustc sources fluctuaton and vscosty are both small, so that we can lnearse by droppng the source moment terms that nclude both the relaxaton tme τ and the flud velocty u. Dervng the wave equaton as done prevously n secton 2.3 but wth the nhomogenous contnuty equaton, we fnd 2 σ αβ ( ) = 43 μ + μ x α x B 2 u γ = τ ν 2 p β x γ t + τ 3 γ 2 δ αβ c 2 0 J 0, (6.12) γ x α x β and consequently [ 1 c ( ) ] t τ ν 2 p = J 0 t t J α x α + τ 2 ( J αβ 3δ αβ c 2 0 J 0/γ ) x α x β. (6.13) Comparng wth the general nhomogeneous wave equaton (2.54), the multpole source strengths are T 0 = J ( ) 0 t, T α = J α, T αβ = τ J αβ 3δ αβ c 2 0 J 0/γ. (6.14) As the relaxaton tme τ = μ/p s typcally very small, the quadrupole strength resultng from j s typcally neglgble. Ths could have been expected by the fact that J αβ appears through Π (1) αβ, whch s of one order deeper n the smallness parameter ɛ than the Euler-level terms. Gong to the Burnett level, another order deeper n ɛ, mght lead to a 2 J αβγ / x β x γ term n the momentum equaton, leadng n turn to an octupole term n the wave equaton. However, snce ths term would be at a further level of smallness, t should be even more neglgble than the quadrupole term. 6.2 Source terms for the lattce Boltzmann equaton Now that we know and understand the effect of addng a source term to the Boltzmann equaton, t s tme to nvestgate what happens when we smlarly add a source term to the lattce Boltzmann equaton. We start by nputtng a source term nto the sothermal dscretevelocty Boltzmann equaton (4.3), f t + ξ f α = j x 1 ( f α τ f (0) ). (6.15) Snce we know from the dervaton n that the DVBE results n the same mass and momentum equatons as the contnuous Boltzmann

165 6.2 Source terms for the lattce Boltzmann equaton 157 equaton (wth the excepton of a neglgble error term), we know that the effect of the source term at ths stage n the dscretsaton process corresponds wth that found n the prevous secton. However, we must consder what happens when we dscretse tme and space to get a lattce Boltzmann equaton wth a source term. We ll assume n ths secton that we choose the frst-order dscretsaton of the f -related terms as n secton Thus, by ntegratng along the characterstc as n secton 4.2, the dscretsed LBE s f (x + ξ, t + 1) f (x, t) = τ j (x + ξ a, t + a) da [ f (x, t) f (0) ] (x, t). (6.16) We must now consder how to approxmate the ntegral over j. In the followng dervatons, we wll assume that the source moments are defned as dscrete analogues to (6.5),.e. as J 0 (x, t) = j (x, ξ, t), J α (x, t) = ξ α j (x, ξ, t), J αβ (x, t) = ξ α ξ β j (x, ξ, t), (6.17a) (6.17b) (6.17c) and so forth Frst order dscretsaton Approxmatng the ntegral usng the rectangle method as n secton 4.2.1, the LBE becomes f (x + ξ, t + 1) f (x, t) =j (x, t) 1 τ [ f (x, t) f (0) ] (x, t). (6.18) Performng subsequent Taylor and Chapman-Enskog expansons and separatng the orders, we have O(ɛ): ( ) + ξ t α 1 x α f (0) = j 1 τ f (1), (6.19a) O(ɛ 2 ): f (0) ( + + ξ t 2 t α 1 = 1 ( 2 x α + ξ t α 1 x α )( 1 1 2τ ) f (1) ) j 1 τ f (2). (6.19b)

166 158 Chapter 6 Mesoscopc acoustc sources Followng the prevous dervaton wth the addtonal assumpton of j / t 2 = 0, we end up wth the macroscopc conservaton equatons ρ t + ρu α = J 0 1 ( J0 x α 2 t + J ) α, (6.20a) x α ρu α t + ρu αu β x β = p x α + σ αβ x β + ( t ) J α τj αβ x β (6.20b) + x β (τ 1 2 ) ( δ αβ c 2 0 J 0 + u α J β + u β J α u α u β J 0 ), wth the stress tensor ( σ αβ = ρc2 0 (τ 1 2 ) u α + u ) β. (6.20c) x β x α The vscostes are the same as before for the frst-order dscretsaton, ν = c 2 0 (τ 1/2) and ν B/ν = 2/3. The lnearsed wave equaton derved from these conservaton equatons s [ 1 c ( ) ] ( t τ ν 2 p = 1 1 ) J0 t 2 t t J α x α + τ 2 J αβ x α x β (τ 1 2 ) 2 3δ αβ c 2 0 J 0 x α x β (6.21) wth multpole strengths ( T 0 = 1 1 ) J0 2 t t, T α = J α, T αβ = τj αβ (τ 1 2 )3δ αβc 2 0 J 0. (6.22) Comparng these wth the multpole strengths (6.14) of the undscretsed Boltzmann equaton, we clearly see two effects of the dscretsaton error. Frstly, there s an extra term n the monopole strength, whch s less relevant at low frequences. Secondly, the quadrupole strength does not dsappear wth vscosty as n the contnuous case; n the nvscd case τ = 1/2wehaveT αβ = J αβ /2. In other words, the errors that occur when performng ths dscretsaton of the source term n space and tme have the benefcal effect of gvng non-vanshng quadrupoles even at low vscostes Second order dscretsaton For comparson, let us now look at the second order dscretsaton. Usng the trapeze method as n secton to approxmate the ntegral, the

167 6.2 Source terms for the lattce Boltzmann equaton 159 LBE becomes f (x + ξ, t + 1) f (x, t) = 1 2 [j (x + ξ, t + 1)+j (x, t)] 1 [ f τ (x, t) f (0) ] (x, t). (6.23) Thus, when calculatng f (x + ξ, t + 1) we add partcles both for the current tme step n the current node, and for the next tme step n the neghbourng node. Performng the Taylor and Chapman-Enskog expansons and separatng the orders, we have O(ɛ): O(ɛ 2 ): ( ) + ξ t α 1 x α f (0) ( )( + + ξ t 2 t α 1 1 ) 1 x α 2τ f (0) = j 1 τ f (1), (6.24a) f (1) = 1 τ f (2). (6.24b) Except for the parenthess (1 1/2τ), whch changes the relaton between τ and ν, ths s equvalent to the contnuous case. The conservaton equatons are the same as (6.11), except wth γ = 1, and we therefore get an equvalent wave equaton, [ 1 c ( ) ] t τ ν 2 p = J 0 t t J α x α ( +(τ 1 2 ) 2 J αβ 3δ αβ c 2 0 J ) 0, x α x β (6.25) and equvalent multpole source strengths T 0 = J ( ) 0 t, T α = J α, T αβ =(τ 1 2 ) J αβ 3δ αβ c 2 0 J 0. (6.26) As n the contnuous case, the quadrupole strength dsappears wth the vscosty. Consequently, whle ths dscretsaton s more true to the contnuous case, the frst order dscretsaton s paradoxcally more useful because ts dscretsaton errors allow the generaton of quadrupoles even at τ = 1/ Multpole bass Regardless of the chosen dscretsaton, how we add partcles through j wll determne whch multpole moments J we generate. The queston s now how to add partcles n unque ways that generate only one multpole moment at a tme.

168 160 Chapter 6 Mesoscopc acoustc sources y y x Fgure 6.2: The x y (sold) and x y (dashed) coordnate systems. x For a velocty set wth q veloctes, j can be seen as a q-dmensonal vector, whch can be found from a q-dmensonal orthogonal partcle source bass vector B j as j = AB. The transformaton matrx A j can be chosen so that each component of B j represents the strength of a partcular multpole. As all LB velocty sets are symmetrc and have an odd number of veloctes, one reasonable choce s to have one monopole n addton to (q 1)/2 pars of oddly symmetrc dpoles and evenly symmetrc longtudnal quadrupoles; one such par for each par of opposng veloctes ξ. As we soon shall see, these longtudnal quadrupoles may also be used to construct lateral quadrupoles. As a concrete example, take the D2Q9 velocty set descrbed n secton Here we get one monopole, four dpoles, and four longtudnal quadrupoles, as shown n Fgure 6.3. For the dagonal dpoles and quadrupoles we have ntroduced a new x y coordnate system, rotated π/4 to the x y system, as shown n Fgure 6.2. The partcle source term j s related to the multpole bases B j through the transformaton matrx A j as j 0 j 1 j 2 j 3 j 4 j 5 j 6 j 7 j 8 = w /2 1/2 w 1 1/2 0 1/ w 2 0 1/2 0 1/ w 3 1/2 0 1/ w 4 0 1/2 0 1/ w / 8 0 1/4 0 w / 8 0 1/4 w / 8 0 1/4 0 w / 8 0 1/4 B 0 B x B y B xx B yy B x B y B x x B y y. (6.27) The dagonal multpoles,.e. the ones orented n the x and y drectons, are scaled so that they radate wth the same strength as the multpoles orented n the x and y drectons. From the values of j generated by each multpole bass B j, we can fnd the correspondng source moments J. Table 6.1 shows how the multpole bass defned by (6.27) maps onto the source moments defned n (6.17). The monopole has a quadrupole component due to ts even symmetry. The non-dagonal dpoles and quadrupoles map exactly onto the correspondng source moments; the dagonal ones would smlarly

169 6.2 Source terms for the lattce Boltzmann equaton 161 j = B 0 w (a) Monopole B y /2 B x /2 B x /2 B y /2 (b) x-dpole (c) y-dpole B x / 8 B y / 8 B x / 8 B y / 8 (d) x -dpole (e) y -dpole B yy /2 B xx /2 B xx B xx /2 B yy B yy /2 (f) xx-quadrupole (g) yy-quadrupole B x x /4 B y y /4 B x x /2 B y y /2 B x x /4 (h) x x -quadrupole () y y -quadrupole B y y /4 Fgure 6.3: Graphcal overvew of the nne multpole bases for the D2Q9 velocty set.

170 162 Chapter 6 Mesoscopc acoustc sources Table 6.1: Nonzero moments of the D2Q9 bass multpoles B j B 0 B x B y B xx B yy B x B y B x x B y y J 0 1 J x 1 1/ 2 1/ 2 J y 1 1/ 2 1/ 2 J xx c /2 1/2 J yy c /2 1/2 J xy 1/2 1/2 J yx 1/2 1/2 B xy /4 B xy /4 B x y /2 B x y /2 B x y /2 B xy /4 B xy /4 (a) xy-quadrupole B x y /2 (b) x y -quadrupole Fgure 6.4: Vrtual bases for lateral quadrupoles, constructed from the longtudnal quadrupole bases. map exactly onto hypothetcal source moments n the x y coordnate system. To represent the dpole and quadrupole strengths, the dpole vector and the symmetrc quadrupole tensor can be ntroduced [36], D = [ Tx T y ], Q = [ Txx T yx T xy T yy ]. (6.28) Wth B 0 = 0, both possble dscretsatons of j result n T αβ J αβ. Wth B xx = B yy = 0, we can ntroduce a vrtual bass B xy = B x x = B y y. Referrng to Table 6.1, the quadrupole tensor becomes [ ] 0 Bxy Q. B xy 0 Thus we can get lateral xy quadrupoles by settng B x x = B y y. Smlarly we could get lateral x y quadrupoles by settng B xx = B yy. These vrtual bases are shown n Fgure 6.4. In a sense, the bass we have chosen n (6.27) s overdetermned. We only need to have a bass that can recreate sx dfferent source moments: J 0, J x, J y, J xx, J yy, and J xy. Consequently, not all nne bass multpoles

171 6.3 Numercal experments 163 n (6.27) are necessary; we could make do wth a bass contanng B 0, B x, B y, B xx, B yy, and B xy. However, whle ths would be a complete multpole bass, t would not be a complete bass for the nne dstrbuton functons f. The chosen bass has the advantage of beng trval to extend to other velocty sets, such as the three-dmensonal ones descrbed n secton Numercal experments To ensure that the lattce Boltzmann multpole sources work as expected, we must verfy ther behavour wth numercal experments. We wll use only the frst order dscretsaton of the source term performed n secton Ths secton wll use the lnearsed LBE descrbed n secton Ths lnearty lets us use complex phasor sources, whch makes analyss smpler: Anywhere n the doman, we can fnd the local wave ampltude as ˆp and the local wave phase as arg( ˆp ). Only the fluctuatng part ˆf of ˆf wll be smulated. Explctly, the numercal scheme used s ˆf (x + ξ, t + 1) ˆf (x, t) =ĵ 1 [ ˆf τ (x, t) ] (0) ˆf (x, t), (6.29a) wth an equlbrum dstrbuton ( ) ˆf (0) (x, t) =w ˆρ + ρ 0û αξ α c 2 0 (6.29b) and moments ˆρ = ˆf, ρ 0û α = ξ α ˆf. (6.29c) In the cases smulated here, the boundary condton at the edge of the smulated system wll not matter; the smulaton wll be stopped before the frst wavefront reaches the boundary. The waves smulated here wll all be forced waves, generated by a source. The exact D1Q3 forced wave numercal wavenumber (5.62) wll be used n the followng to match the numercal and the analytcal solutons Plane waves For the frst, smplest case, we smulate forced plane waves generated at one end of the system and propagatng towards the other. Snce ths s an nherently one-dmensonal problem, we use the onedmensonal D1Q3 velocty set wth veloctes (ξ, ξ 0, ξ + )=( 1, 0, 1), weghtng coeffcents (w, w 0, w + )=(1/6, 2/3, 1/6), and speed of sound c s = 1/ 3. The source s a monopole pont source, representng an

172 164 Chapter 6 Mesoscopc acoustc sources nfnte pulsatng plate n the y-z plane, stuated at the leftmost node at x = 0. The partcle source term consequently becomes ĵ (x, t) = w e ω 0t S(t)δ(x). (6.30) Here δ(x) s a dscrete delta functon whch equals 1 for x = 0 and 0 otherwse, whle S(t) s the Heavsde step functon, 0 for t < 0, S(t) = 1 2 for t = 0, 1 for t > 0. (6.31) Ths functon s present to account for the fact that the source does not radate before the smulaton starts at t = 0. Also, the complex source s phase shfted so that ts real, physcal part j = Re ( ĵ ) smoothly starts to ncrease from 0 at t = 1. A symmetry boundary condton was chosen at x = 0, where the unknown dstrbuton functon ˆf +(0, t), whch represents partcles enterng the x = 0 node from outsde the system, s set equal to the known dstrbuton functon ˆf (0, t), whch represents partcles enterng that node from nsde the system. Ths smulates a source whch s radatng equally nto the smulated doman at x > 0 and a symmetrc doman at x < 0. Smple forced wave The frst smulaton s performed wth relaxaton tme τ = 0.6, and source frequency ω 0 = 0.1, leadng to an acoustc vscosty number X = Due to lnearty, the source strength ˆp s arbtrary. The smulaton s stopped when the frst wavefront has reached the poston α x x = 1. Usng secton and approxmatng* the multpole strengths (6.22) of the dscrete source term, we fnd the steady-state analytcal soluton of ths plane wave case as ˆp (x, t) = ˆp c 0 2 e α xx e (ω 0t kx). (6.32) The smulaton results are compared wth ths analytcal soluton n Fgure 6.5. Both the ampltude and the real part of the wave are compared drectly, wthout any artfcal scalng or phase shftng of the solutons. There s an excellent match between the numercal and analytcal solutons except near the frst wavefront, where the numercal soluton has smply not had the tme to propagate farther before the smulaton was stopped. *The second dervatve term n T 0 and the quadrupole contrbutons of B 0 can be shown to approxmately cancel f ω 0 (τ 1/2) 1. Ths also holds true n the two-dmensonal case, as shown later n secton

173 6.3 Numercal experments 165 ˆp LB p LB ˆp Ana. p Ana. ˆp 0 ˆp x Fgure 6.5: Comparson of numercal and steady-state analytcal soluton of a forced plane wave beng radated nto a vscous flud. Smulaton parameters are τ = 0.6, source frequency ω 0 = 0.1, and acoustc vscosty number X = Interestngly, the numercal soluton s termnated wth a bump and a smooth transton to zero, whch wll be examned next. Comparng the ampltudes of the numercal and the analytcal soluton (6.32), we fnd a relatve error of constant ampltude, ˆp num ˆp ana ˆp ana everywhere except near the frst wavefront. The phase error s smlarly constant, at arg( ˆp ana) arg( ˆp num) π. We defer a proper numercal benchmarkng of the lattce Boltzmann sources to the end of ths chapter. Precursors As noted, there s a bump near the frst wavefront n Fgure 6.5. Such transents, often called precursors, appear when a sound wave s radated nto a vscous flud, and have prevously been descrbed n the lterature [134]. Close to the source,.e. for α x x 1, the precursor s weak. However, t s attenuated at a slower rate than the wave tself, and wll be clearly vsble at α x x 1 and ncreasngly sgnfcant above. If the boundary condton at x = 0su = u sn(ω 0 t)s(t), the radated transent soluton s [134] Precursors In general, transents that occur when a wave s radated nto a dspersve medum u (x, t) =u st(x, t)+u tr(x, t), (6.33a)

174 166 Chapter 6 Mesoscopc acoustc sources where the truncated steady-state part u st and the transent part u tr are gven by u st(x, t) =u e αxx sn(ω 0 t kx)s(ωt kx), [ w u tr(x, t) = u 2 e (ω 0t kx) 2 /4α x x Im ( αx x + ω 0t kx 4αx x (6.33b) )], (6.33c) w(...) beng the complex error functon known as the Faddeeva functon. u st (x, t) s a normal steady-state soluton, truncated at the frst wavefront by the Heavsde functon. The transent soluton u tr (x, t) represents the precursor caused by the vscosty. Whle the source n the smulaton s a monopole and therefore has u = 0atx = 0, the specfed velocty boundary condton stll holds for the x > 0 doman. It s merely cancelled at x = 0 by the velocty component of the wave radated nto the x < 0 doman. The numercal soluton s compared wth (6.33) n Fgure 6.6. At the same resoluton as n Fgure 6.5, the solutons match almost to plottng accuracy. At reduced resoluton wth ω 0 = 0.5 and X = 0.02 held constant, the precursor s smeared out and does not ft the analytcal soluton well. Even so, the analytcal and numercal soluton well away from the frst wavefront and precursor match very well, as the exact D1Q3 numercal wavenumber (5.62) s used n the comparson. Note also that the normalsed wavelength α x λ s longer n the lowresoluton case. Ths s as expected from equaton (5.63), whch predcts α x λ = 2π Im(ˆk) Re(ˆk) πx 1 + 5ω2 0 / ω0 2/12 assumng ω0 4 ω2 0,.e. a normalsed wavelength that ncreases wth a coarser numercal resoluton. Invscd forced wave Prevous authors have successfully smulated LB sound waves at a very low vscosty [45, 108, 115, 120]. We wll now attempt to perform a smulaton at τ = 0.5, wth no vscosty at all. The smulaton s set up lke before. The numercal frequency s set to ω 0 = 0.1, and snce the smulaton s nvscd, we have an acoustc vscosty number X = 0. The smulaton results are compared wth the analytcal truncated steady-state soluton n Fgure 6.7(a). The real part p of the numercal soluton s smooth and matches the analytcal soluton very well, apart from a small amount of smoothng near the frst wavefront. The wave ampltude ˆp shows some rpple, especally around the frst wavefront. Snce ths rpple s not present n the real part p of the full complex soluton ˆp, t must be contaned n the magnary part Im( ˆp ).

175 6.3 Numercal experments 167 LB u Ana. u Ana. u st Ana. u tr α x x (a) The same case as n Fgure 6.5: ω 0 = 0.1, τ = 0.6, X = LB u Ana. u Ana. u st Ana. u tr α x x (b) Decreased numercal resoluton: ω 0 = 0.5, τ = 0.52, X = Fgure 6.6: Numercal and analytcal solutons of a forced plane wave n a vscous flud, near the frst wavefront. Note how the dscretsaton error affects the wavelength. As the magnary part of the source functon (6.30) starts sharply as cos(ω 0 t)s(t) unlke the real part whch starts smoothly as sn(ω 0 t)s(t), t s clear that such rpples occur when startng the source sharply. A lkely cause for ths effect s that the Heavsde functon n the source functon ĵ represents a rectangular half-wndow, gvng a truncated steady-state soluton of ˆp (x, t) = ˆp e (ω 0t kx) S(ω 0 t kx). (6.34) From sgnal theory t s well-known that multplyng a sngle-frequency sgnal wth a wndow functon results n spectral leakage, meanng that unwanted frequency components appear n the resultng sgnal. Wth the rectangular wndow, ths effect s especally promnent [135, Ch. 10]. Wndow functon A functon of tme, only non-zero nsde an nterval. Typcally multpled wth a fnte tme sgnal to mnmse spectral leakage. Spectral leakage A spreadng of a sgnal s frequency components whch occurs when the sgnal s fnte n the tme doman

176 168 Chapter 6 Mesoscopc acoustc sources ˆp LB ˆp LB p Ana. ˆp Ana. p 0 ˆp x (a) Rectangular half-wndow ˆp LB ˆp LB p Ana. ˆp Ana. p 0 ˆp x (b) Hann half-wndow Fgure 6.7: Comparson of two methods of startng an acoustc source n a smulaton wth ω 0 = 0.1 and τ = 0.5 Consequently, wth a rectangular half-wndow as n the smulaton, the sgnal contans addtonal frequency components dfferent from ω 0. Snce there s an element of numercal dsperson n the LBM as shown n secton 5.3, these other frequency components propagate wth a slghtly dfferent phase speed. Thus, numercal dsperson near the frst wavefront s lkely the cause of the rpple. In sgnal analyss, other wndow functons can used that have less spectral leakage than a rectangular wndow. No wndow functons can avod t entrely, but one smple wndow functon wth better propertes s the Hann wndow, also known as the von Hann or Hannng wndow [135, Ch. 10]. Adaptng t to a half-wndow coverng one wavelength, we get { [ ] S(t) 12 W(t) = 1 2 cos(ω 0t/2) for t 2π/ω 0, (6.35) 1 for 2π/ω 0 t.

177 6.3 Numercal experments 169 Usng ths Hann half-wndow nstead of the Heavsde functon, the source functon s gvng an analytcal soluton ĵ (x, t) = w e ω 0t W(t)δ(x), (6.36) ˆp (x, t) = ˆp e (ω 0t kx) W(ω 0 t kx). (6.37) The result of a smulaton performed lke as before but wth a Hann half-wndow s shown n Fgure 6.7(b). We see that usng the Hann halfwndow fully elmnates the rpple near the frst wavefront, wth only a mnor devaton remanng Multpoles n two dmensons Havng verfed that the source term works n one dmenson, we turn our attenton to multpole sources n two dmensons. As the pressure responses of the two-dmensonal multpole sources are expressed through dervatves of the two-dmensonal Green s functon (2.56b), Ĝ(x, t) = 1 4 Ĥ(2) 0 (ˆk x ) e ωt, (6.38a) we should fnd the resultng functons of these dervatves for later use. Usng polar coordnates gven through x and θ, the frst dervatves of the Green s functon result n Ĝ(x, t) = ˆk cos θ Ĥ(2) x 4 1 (ˆk x ) e ωt, (6.38b) Ĝ(x, t) = ˆk sn θ Ĥ(2) y 4 1 (ˆk x ) e ωt. (6.38c) The second dervatves result n 2 ˆk 2 { 1 Ĝ(x, t) = x2 4 2 cos2 θ 2 ˆk 2 { 1 Ĝ(x, t) = y2 4 2 sn2 θ [Ĥ(2) 2 (ˆk x ) Ĥ (2) } 1 ˆk x sn2 θ Ĥ (2) 1 (ˆk x ) ] 0 (ˆk x ) e ωt, [Ĥ(2) 2 (ˆk x ) Ĥ (2) 0 (ˆk x ) } 1 ˆk x cos2 θ Ĥ (2) 1 (ˆk x ) e ωt, ] (6.38d) (6.38e) 2 ˆk 2 Ĝ(x, t) = cos θ sn θ Ĥ(2) x y 4 2 (ˆk x ) e ωt. (6.38f)

178 170 Chapter 6 Mesoscopc acoustc sources Also useful s 2 Ĝ/ x α x α, whch becomes remarkably smple, ( 2 ) x y 2 Ĝ(x, t) = ˆk 2 4 Ĥ(2) 0 (ˆk x ) e ωt = ˆk 2 Ĝ(x, t). (6.38g) Wth the excepton of the longtudnal quadrupoles 2 Ĝ/ x 2 and 2 Ĝ/ y 2, each of these functons can be handly separated nto a θ-dependent part and a x -dependent part. Note how the functons Ĝ and 2 Ĝ/ x α x α are smply proportonal. Usng ths, the nhomogeneous wave equaton (6.21), and the map from B 0 to J αα n Table 6.1 we have that the response of a monofrequency bass source B 0 s ˆp (x, t) = ( ω 0 + ω2 0 ) 2 τc 2 0ˆk 2 B0 4 Ĥ (2) 0 (ˆk x ) e ωt, where the three terms n the parenthess are the contrbutons from the correct monopole term, the monopole dscretsaton error term, and the T αβ quadrupole contrbuton of the monopole bass B 0, respectvely. For τ 1/2 and ω0 2 1, we have from (5.63) that ω2 0 /2 τc2 0ˆk 2, and the two latter terms n the parenthess cancel. In other words: For low vscosty and decent numercal resoluton, the unwanted contrbutons of the monopole source bass cancel. The smulatons n ths secton are set up as follows. Usng the D2Q9 velocty set descrbed n secton 4.1.3, we place a pont source n the centre of the two-dmensonal smulaton doman, wth ĵ (x, t) = w e ω 0t W(t)δ(x). (6.39) Agan, δ(x) s a dscrete delta functon whch equals 1 for x = 0 and 0 elsewhere, and W(t) s the Hann half-wndow (6.35). The smulaton s stopped when the frst wavefront reaches the boundary. The smulaton performed are nvscd wth τ = 1/2. We wll see that the BGK collson operator works poorly n ths lmt. Instead we wll for the most part use the general-purpose MRT operator descrbed n secton (4.3.1), where the relaxaton tme of the nonhydrodynamc moments s set to one. Radal varaton of pressure Frst we wll compare the radal pressure varaton,.e. how ˆp vares wth x, of the numercal multpole sources wth the correspondng analytcal solutons. These smulatons were performed wth ω 0 = 0.25, τ = 1/2 (resultng n X = 0), n a doman of nodes. Although the boundary condtons should not matter snce the wave wll not reach the outer boundary, perodc boundares were used. Each smulaton was performed once wth the BGK operator and once wth the MRT operator.

179 6.3 Numercal experments BGK MRT Analytcal 0.05 p /B x (a) Monopole 0.1 BGK MRT Analytcal 0.05 p /B x x (b) x-dpole 0.1 BGK MRT Analytcal 0.05 p /B x x (c) x -dpole 0.1 BGK MRT Analytcal 0.05 p /B xx x (d) xx-quadrupole 0.1 BGK MRT Analytcal 0.05 p /B x x x (e) x x -quadrupole Fgure 6.8: Radal pressure from representatve multpoles, smulated wth BGK and MRT and compared to analytcal solutons. The left-hand mages come from the MRT smulatons, and ndcate the lnes along whch the rght-hand pressures were measured.

180 172 Chapter 6 Mesoscopc acoustc sources The results of the smulatons and comparsons wth analytcal solutons found from (2.59), (6.22), and (6.38) are shown n Fgure 6.8 for fve representatve multpoles: A monopole, x- and x -dpoles, and xx- and x x -quadrupoles. The remanng four multpoles are not necessary to verfy because of the rotatonal symmetry of the D2Q9 velocty set. Most strkngly, the BGK results dsplay strong spurous oscllatons whch domnate the soluton n the dpole and quadrupole case. On the other hand, the MRT results demonstrate a largely excellent agreement wth the analytcal soluton, wth two exceptons: There are mnor errors near the frst wavefront lke n Fgure 6.7(b), and there are major errors n the mmedate vcnty of the source node. The errors close to the source node are lkely caused by the hgh spatal dervatves compared to the spatal resoluton close to the source. Wth a pont source, t s mpossble to refne the resoluton n such a way that the spatal dervatves near the source become managable; as the resoluton s mproved, the effectve sze of the pont source node shrnks. In addton, the sngularty present at the source pont n the analytcal solutons can never be reproduced numercally, snce the partcle source term must be fnte. However, perhaps the problems near the source could be allevated usng a spatally smoothed source that spans several nodes [136], or by usng a wder dpole and quadrupole bass whch nvolves more partcle dstrbutons than the mnmal bass used here. The errors close to the source are present and of smlar magntude n both the BGK and the MRT solutons. The MRT operator used here dffers from the BGK operator only n that t fully supresses nonhydrodynamc moments n each tmestep. Consequently, these results suggest that errors n the nonhydrodynamc moments are created at the source, and the BGK operator propagates them outwards wth the wave whle the MRT operator suppresses them, leavng an accurate soluton. Angular varaton of pressure We have now verfed that the radal varaton of the pressure of the numercal soluton agrees wth that of the analytcal soluton except n the mmedate vcnty of the source f the MRT collson operator s used. We should subsequently verfy that the angular varaton of the pressure s also correct. In order to compare pressure at a constant dstance from the source and dfferent angles, the numercal soluton must be sampled between the nodes. For ths reason, some nterpolaton of the soluton s requred. In order to mprove the qualty of ths nterpolaton, the numercal resoluton was ncreased by 250 % compared to the prevous smulatons. The smulatons were performed wth parameters ω 0 = 0.1, τ = 1/2 (leadng to X = 0), and a doman of nodes.

181 6.3 Numercal experments (a) Monopole (b) x-dpole (c) x -dpole (d) xx-quadrupole (e) x x -quadrupole (f) xy-quadrupole Fgure 6.9: ˆp as functon of angle θ at k x = 25, compared between MRT-based smulatons ( ) and analytcal solutons ( ) and normalsed by the maxmum of the analytcal soluton. Relatve lobe phase s ndcated by plus and mnus sgns.

182 174 Chapter 6 Mesoscopc acoustc sources (a) Rotated x-dpole (b) Rotated xx-quadrupole Fgure 6.10: Angular varaton of two rotated multpoles, plotted as n Fgure 6.9. In addton to the fve multpoles smulated n the prevous comparson, the xy-quadrupole, generated by superposton of the x x - and y y -quadrupoles as descrbed n secton 6.2.3, was smulated. The numercal solutons were measured for all angles at a dstance of k x = 25, correspondng to a dstance of roughly four wavelengths. The results are compared wth the correspondng analytcal solutons n Fgure 6.9, wth both the numercal and the analytcal solutons normalsed by the maxmum of the analytcal solutons. In all cases, there s an excellent agreement between the numercal and analytcal solutons. Rotated multpoles and composte sources The basc multpoles that have been descrbed and verfed so far may be superposed n order to generate more complex sources. As examples of ths, we wll look at multpole rotaton and a hghly drectve composte source. Dpoles and quadrupoles can be rotated wth an angle θ by applyng the rotaton matrx [ ] cos θ sn θ a = (6.40) sn θ cos θ to the dpole vector or the quadrupole tensor (6.28) as [137, App. 6] D rot α = a αβ D β, Q rot αβ = a αγa βδ Q γδ. (6.41) For example, takng an x-dpole and an xx-quadrupole and rotatng results n [ ] [ D rot cos θ, Q rot cos 2 ] θ sn θ cos θ sn θ sn θ cos θ sn 2. θ

183 6.3 Numercal experments Fgure 6.11: The MRT smulaton result of a rotated supercardod source (left) and ts normalsed angular varaton at x = 25/k, plotted as n Fgure 6.9 (rght) Both longtudnal and lateral quadrupoles are present n the resultng quadrupole tensor. Results of smulatons where the bass strengths were chosen to reproduce the dpole vector and quadrupole tensor for a rotaton of 30 are shown n Fgure By superposng dfferent types of multpoles, we can generate composte sources that have other propertes that mght be useful, such as beng focused n a partcular drecton. For nstance, one type of source s generated by requrng that the x-dpole and the xx-quadrupole have the same ampltude at θ = 0 and some partcular dstance x (whch may be n the far-feld lmt). Mathematcally, ths leads to ˆk [Ĥ(2) 2 (x, t) Ĥ (2) ] 0 (x, t) T x Ĝ(x, t) x = T xx 2 Ĝ(x, t) x 2 B x B xx = 4Ĥ (2) 1 (x, t) Ths type of source s called a supercardod [138]. Ths was smulated wth the dpole and quadrupole ampltudes normalsed at x = 25/k and the resultng dpole vector and quadrupole tensor rotated 30. The smulaton results of ths rotated supercardod are shown n Fgure As could be expected by ths pont, the agreement s excellent. Numercal convergence of radated wave It s nsuffcent to merely compare the numercal and analytcal solutons at a partcular resoluton and verfy that they agree well. The source must also be shown to radate a wave whch corresponds ncreasngly well to the analytcal soluton as the numercal resoluton s ncreased. The LBM tself has a second order accuracy [92], meanng that the numercal error,.e. the dfference between a numercal and a correspondng analytcal soluton, decreases proportonally to the square of the numercal resoluton. In.