Ab Initio Derivation of the Cascaded Lattice Boltzmann Automaton

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AB INITIO DERIVATION OF THE CASCADED LATTICE BOLTZMANN AUTOMATON

Martin Christian Geier ii iv AB INITIO DERIVATION OF THE CASCADED LATTICE BOLTZMANN AUTOMATON vi AB INITIO DERIVATION OF THE CASCADED LATTICE BOLTZMANN AUTOMATON

Vorgelegt von Martin Christian Geier University of Freiburg – IMTEK

Dissertation zur Erlangung des Doktorgrads der Fakultät für Angewandte Wissenschaften der Albert-Ludwigs-Universität Freiburg im Breisgau viii Department of Microsystems Technology IMTEK

University of Freiburg Freiburg im Breisgau, Germany

Author Martin Christian Geier

Thesis period August 2003 to September 2006

Referees Jan G. Korvink and Oliver Paul

Supervisors Jan G. Korvink and Andreas Greiner

Disput a t ion Decemb er 21th 2006 D ec lar at ion I ch er k lär e h ier m it, d as s ich d ie vor liegen d e Ar - beit ohne unzulässige Hilfe Dritter und ohne Be- nutzung anderer als der angegebenen Hilfsmittel ange- fertigt habe. Die aus anderen Quellen direkt oder indirekt übernommenen Daten und Konzepte sind unter Angabe der Quelle gekennnzeichnet. Insbeson- dere habe ich hierfür nicht die entgeltliche Hilfe von Vermittlungs- oder Beratungsdiensten (Promotions- beraterinnen oder Promotionsberater oder anderer Personen) in Anspruch genommen. Niemand hat von mir unmittelbar oder mittelbar geldwerte Leistungen für Arbeiten erhalten, die im Zusammenhang mit dem Inhalt der vorgelegten Dissertation stehen. Die Arbeit wurde bisher weder im In- noch im Ausland in gleicher oder ähnlicher Form einer anderen Prüfungsbehörde vorgelegt.

Freiburg, 5th September 2006

Martin Christian Geier In memory of the 100th anniversary of the dead of Ludwig Boltzmann xii I am conscious of being only an individual struggling weakly against the stream of time. But it still remains in my power to contribute in such a way that, when the theory of gases is again revived, not too much will have to be rediscovered. –Ludwig Boltzmann ( 20.4.1844 5.9.1906) ∗ †

xiii xiv Contents

1 Introduction 7

2 Lattice Boltzmann - state of the art 13 2.1 Lattice gas and lattice Boltzmann cellular automata ...... 13 2.1.1 Lattices ...... 14 2.1.2 Streaming and scattering ...... 16 2.1.3 Particle representations ...... 16 2.2 Lattice Boltzmann scattering operators ...... 17 2.2.1 Single relaxation time operators ...... 17 2.2.2 Multiple relaxation time operators ...... 18 2.2.3 Entropic operators ...... 18 2.3 Theoretical background of the lattice Boltzmann automaton . . 20 2.4 SRT lattice Boltzmann automaton in 2D ...... 21 2.5 Boundary conditions ...... 23 2.6 Limitations ...... 24 2.6.1 Turbulence and the lattice Boltzmann automaton . . . . 25 2.6.2 Scale separation ...... 26 2.6.3 Navier Stokes and higher order schemes ...... 29 2.6.4 Closing under-determined equations with entropy . . . . 31 2.6.5 Entropy and closure ...... 32 2.7 Endnotes ...... 34

3 Minimal kinetic model 37 3.1 Basic principles ...... 37 3.2 1D prototype ...... 38 3.2.1 Equilibrium ...... 39 3.2.2 Scattering rate ...... 41 3.3 Generalization to more than one dimension ...... 43 3.4 Admissible operations ...... 44 3.4.1 The generic collision ...... 44

xv Contents

3.4.2 The generic commutative operator ...... 45 3.4.3 Invariance under translation ...... 45 3.4.4 Invariance under rotation ...... 46 3.4.5 Invariance under collisions ...... 47 3.5 Elementary collisions in two dimensions ...... 48 3.5.1 Moments ...... 48 3.5.2 Collision vectors ...... 51 3.5.3 Equilibria ...... 53 3.6 The collision cascade ...... 56 3.7 Endnotes ...... 59

4 Properties of the cascaded lattice Boltzmann automaton 63 4.1 Basic differences to other lattice Boltzmann models ...... 63 4.1.1 Transport coefficients ...... 64 4.1.2 Algorithmic differences ...... 65 4.2 Effects of high order corrections on stability ...... 66 4.2.1 A thought experiment ...... 66 4.2.2 Aliasing ...... 74 4.2.3 Over-relaxation of third order moments ...... 78 4.3 Simple tests ...... 79 4.3.1 Frozen-in velocity peak at rest ...... 79 4.3.2 Frozen-in velocity peak in motion ...... 84 4.3.3 Step advection with finite viscosity ...... 84 4.4 Measurement of the viscosity error ...... 92 4.4.1 Shear wave decay experiment ...... 97 4.4.2 Determining the numerical viscosity ...... 97 4.5 Speed bounds ...... 100 4.6 Cascaded digital lattice Boltzmann ...... 102 4.7 Endnotes ...... 105

5 Simulation of turbulent flow 107 5.1 Under-resolved turbulence ...... 107 5.2 Turbulent wake ...... 108 5.2.1 Energy spectrum ...... 109 5.2.2 Convergence study ...... 109 5.3 Turbulence induced by Kelvin Helmholtz instability ...... 112 5.3.1 Resolution of turbulent flow ...... 118 5.4 Under-resolved laminar flow ...... 121 5.5 Endnotes ...... 121

xvi Contents

6 Conclusions 127 6.1 Eﬃciency ...... 127 6.2 Turbulent ﬂow ...... 128 6.3 Complexity ...... 130 6.4 Future challenges ...... 132 6.5 Fields of applications ...... 135 6.6 Endnotes ...... 137

Appendices 139

A Minimal cascaded lattice Boltzmann automaton in 3D 141 A.1 Scattering matrix ...... 142 A.2 Length of collision vectors ...... 144

B Mapping lattice units to the real world 157

Bibliography 159

xvii xviii List of Figures

2.1 D2Q7 and D2Q9 lattices ...... 14 2.2 D3Q13, D3Q15, D3Q19, and D3Q27 lattices ...... 15

3.1 Elementary collisions D2Q9 ...... 54

4.1 Transport of singular velocity peak (SRT) ...... 70 4.2 Transport of singular velocity peak (cascaded LB) ...... 71 4.3 Comparison of the velocity dependence of viscosity on the SRT lattice Boltzmann automaton and the cascaded lattice Boltz- mann automaton ...... 73 4.4 Flipping-over of velocity curvature due to κxxy ...... 80 4.5 Diffusion of peak (SRT model) ...... 81 4.6 Diffusion of peak (none equilibrated) ...... 82 4.7 Diffusion of peak (equilibrated) ...... 83 4.8 Diffusion of peak with velocity (SRT model) ...... 85 4.9 Diffusion of peak with velocity (non equilibrated) ...... 86 4.10 Diffusion of peak with velocity (equilibrated) ...... 87 4.11 Step profile transport with SRT model ...... 89 4.12 Step profile transport with entropic lattice Boltzmann model . 90 4.13 Step profile transport with cascaded lattice Boltzmann model . 91 4.14 Step profile transport with entropic lattice Boltzmann model (gradient) ...... 93 4.15 Step profile transport with cascaded lattice Boltzmann model (gardient) ...... 94 4.16 Step profile transport with cascaded lattice Boltzmann model (gardient, ν = 10−3) ...... 95 4.17 Step profile at rest with cascaded lattice Boltzmann model . . . 96 4.18 Measurement of numerical viscosity...... 99

5.1 Turbulent wake behind plate ...... 109

xix List of Figures

5.2 Energy spectrum of turbulent wake ...... 110 5.3 Turbulent wake behind plate (coarse) ...... 111 5.4 Turbulent wake behind plate (averaged) ...... 113 5.5 Turbulent wake behind plate (variance) ...... 114 5.6 Onset of Kelvin Helmholtz instability (coarse) ...... 115 5.7 Free decay of turbulence ...... 116 5.8 Onset of Kelvin Helmholtz instability (fine) ...... 117 5.9 Energy spectrum of the decay of Kelvin-Helmoltz instability . 119 5.10 Compensated energy spectra ...... 120 5.11 Minimal resolution vortex shedding (isosurface) ...... 122 5.12 Minimal resolution vortex shedding (flow field) ...... 123 5.13 Minimal resolution stationary vortex ...... 124

xx Abstract

In this thesis, an executable cellular automaton model for athermal fluids is derived from first principles. The time evolution of a fluid is described by deterministic streaming of particles and their collisions. It is shown that the assumption of a symmetric equilibrium and the invariance under shift, rotation, and the commutation of the collision operators are sufficient for the unique determination of all model parameters. No other physical or mathematical justification is necessary for the derived model, which will be called the cascaded lattice Boltzmann automaton. The method of central moments is introduced for the derivation of the cascaded lattice Boltzmann automaton. The method of central moments is a mathematical tool applied to gain independence of the frame of reference for systems with an intrinsic frame of reference. The first step in the procedure is to determine the intrinsic frame of reference. This is done by the determination of all characteristic points of the the system. The second step is to apply a coordinate transform to the system to transfer it into its characteristic frame of reference. Operations are constraint to the characteristic frame of reference. The last step is the transformation back into the original frame of reference. For simple fluid dynamics the only characteristic point is the mean of the velocity distribution. The cascaded lattice Boltzmann automaton is compared to other lattice Boltzmann automata and is found to have superior accuracy and stability characteristics. The role of higher order velocity moments which are typically neglected in other lattice Boltzmann automata is investigated in detail. The instabilities which plague other lattice Boltzmann automata are traced back to the wrong behavior of third order velocity moments. Without the independence from the frame of reference gained by the method of central moments, third order velocity moments are wrong and cause an error in viscosity which is of second order in Mach number. The application of the method of central moments reduces this error to fourth order in Mach number. More important is the qualitative differences of the errors. The error in the original lattice Boltzmann automaton is shown to be always negative while the error in the cascaded lattice Boltzmann automaton is always positive. Negative errors in

1 viscosity are destabilizing since they might cause the viscosity to become negative which is not allowed by the second law of thermodynamics. Positive errors in viscosity cannot cause numerical instabilities. In addition, the magnitude of the errors in the cascaded lattice Boltzmann automaton is significantly lower. Numerical viscosity is in general very low in the new method. It is further shown that the time evolution of third order moments have to be decoupled from the evolution of second order moments. The evolution of second order velocity moments is modeled by an over-relaxation process in order to obtain small viscosities. It is shown that the application of the same over-relaxation process to third order moments causes intolerable aliasing effects. Third order moments describe the curvature of the velocity field and their over-relaxation causes a flipping-over of the local curvature. Low wavelength components of the flow field get unstable if over-relaxation of third order moments is allowed. Therefore, the cascaded lattice Boltzmann automaton applies over-relaxation only to second order moments and equilibrates all higher order moments. Computational evidence for the superiority of the cascaded lattice Boltz- mann automaton over other methods of computational fluid dynamics is shown by some computational experiments. The flow around an rectangular obstacle in 3D at a Reynolds number of 1400000 is presented. Even though the simulation is under-resolved by a factor of more than 40000, it is shown that the fully developed turbulent wake complies with the Kolmogorov theory of turbulent flow. Simulations of the free decay of turbulence show that the flow field contains turbulent features complying with the Kolmogorov theory down to the length of a few grid spacings irrespective of the lack of resolution in the simulation. Finally it is shown that the cascaded lattice Boltzmann automaton captures the correct transition from laminar steady to laminar unsteady flow behind a cylinder with a diameter of a single grid spacing. The vortices are smaller than one grid spacings in this case and yet they are modeled with sufficient accuracy to capture the relevant physical effects of the problem.

2 Zusammenfassung

In dieser Arbeit wird ein Zellularautomatenmodel für athermische Fluide ab intio abgeleitet. Die zeitliche Entwicklung eines Fluids wird durch determisti- schen Flug und Stoss von Partikeln beschrieben. Es wird gezeigt, dass die Annahme einer symmetrischen Gleichgewichtsverteilung und der Invarianz unter Translation und Rotation sowie die Kommutation der Stossoperato- ren ausreichen um alle Modellparameter eindeutig zu bestimmen. Die Her- leitung des neuen Modells, das als der kaskadierte Gitterboltzmannautomat bezeichnet werden soll, benötigt keine darüber hinausgehende physikalische oder mathematische Rechtfertigung. Für die Ableitung des kaskadierten Git- terboltzmannautomaten wird die Methode der zentralen Momente eingeführt. Die Methode der zentralen Momente ist ein mathematisches Hilfsmittel mit dem sich für ein System das ein intrinsiches Bezugssystem aufweist Unab- hängigkeit von dem Bezugssystem erlangen lässt. Im ersten Schritt wird das intrinsische Bezugssystem ermittelt indem alle charakteristischen Punkte des Systems bestimmt werden. Der zweite Schritt besteht in einer Koordinaten- transformation in das charakteristische Bezugssystem. Alle Operationen werden in dem charakteristischen Bezugssystem ausgeführt. Im letzten Schritt wird das System wieder in sein Ursprungsbezugssystem zurücktransformiert. Der einzige charakteristische Punkt für einfache Fluide ist der Schwerpunkt der Geschwindigkeitsverteilungsfunktion. Der kaskadierte Gitterboltzmannautomat wird mit anderen Gitterboltz- mannautomaten verglichen und es wird festgestellt, dass er bessere Stabilitäts und Genauigkeitseigenschaften aufweist. Die Rolle der höheren Geschwindig- keitsmomente, die gewöhnlich in anderen Gitterboltzmannautomaten vernach- lässigt werden, wird genauer untersucht. Die Insatbilitäten die andere Gitter- boltzmannautomaten plagen, können auf ein falsches Verhalten der dritten Ge- schwindigkeitsmomente zurückgefürt werden. Ohne die Unabhängigkeit vom Bezugsrahmen, die durch die Methode der zentralen Momente erreicht wird, sind die dritten Geschwindigkeitsmomente falsch und verursachen einen Feh- ler in der Viskosität der zweiter Ordung in der Machzahl ist. Die Verwendung der Methode der zentralen Momente reduziert diesen Fehler auf Ordnung vier

3 in der Machzahl. Wichtiger ist aber der qualitative Unterschied der Fehler. Wärend der Fehler in der Ursprünglichen Gitterboltzmannmethode immer zu einer negativen Viskosität führt ist der Fehler in der neuen Methode immer po- sitiv. Negative Fehler in der Viskosität sind destabilisierend da sie dazu führen können das die Viskosität negativ werden kann was laut dem zweiten Haupt- satz der Thermodynamik verboten ist. Positive Fehler in der Viskosität können keine numerischen Instabilitäten verursachen. Zusätzlich ist auch der Betrag der Fehler in der kaskadierten Methode erheblich kleiner. Es wird gezeigt, dass die zeitliche Entwicklung der dritten Momente von der der zweiten Momen- te entkoppelt werden muss. Die zeitliche Entwicklung der zweiten Momente wird durch einen Überrelaxationsprozess modelliert um kleine Viskositäten zu erlangen. Es wird gezeigt, dass die Anwendung des gleichen Überrelaxati- onsprozesses auf die dritten Momente zu intolerablen Aliasingeffekten führt. Dritte Momente beschreiben die Krümmung des Geschwindigkeitsfeldes. Ihre überrelaxierung führt zu einer Inversion der lokalen Krümmung. Kurzwellige Bestandteile des Strömungsfeldes werden instabil wenn die Überrelaxierung dritter Momente zugelassen wird. Daher überrelaxiert der kaskadierte Gitter- boltzmannautomat nur zweite Momente und setzt alle übrigen Momente auf ihren Gleichgewichtszustand. Die Überlegenheit des kaskadierten Gitterboltzmannautomaten gegenüber anderen fluidischen Simulationsmethoden wird anhand einiger rechnerischer Experimente gezeigt. Der Fluss um ein rechteckiges Hindernis bei Reynolds- zahl 1400000 wird untersucht. Obwohl die Auflösung der Simulation um mehr als einen Faktor 40000 zu niedrig ist kann gezeigt werden, dass die völlig ausgebildete turbulente Strömung der Kolmogorov-Theorie gehorcht. Simula- tionen von frei zerfallender Turbulenz zeigen Details die bis hinunter auf weni- ge Gitterlängen der Kolmogorov-Theorie gehorchen ungeachtet der fehlenden Auflösung in der Simulation. Es wird ferner gezeigt, dass der kaskadierte Git- terboltzmannautomat auch den Übergang von laminar statischen zu laminar dynamischen Stömungen hinter einem Zylinder korrekt abbildet wenn der Zy- linder nur den Durchmesser einer einzigen Gitterlänge aufweist. Die Wirbel sind in diesem Fall zwar kleiner als eine einzige Gitterlänge, die relevanten physikalischen Effekte werden aber dennoch mit hinreichender Genauigkeit von dem Modell abgebildet.

4 Acknowledgments

This thesis was only possible in this form due to the nearly unlimited freedom I enjoyed in the group of professor Korvink. Professor Korvink gave me free choice of my topic, my objectives and my working principles. I exploited this freedom to the limit. I chose a topic which was unrelated to other work in the group, even unrelated to MEMS in general. The way I approached my work was rather chaotic. I did what ever came to my mind without a clear aim what I wanted to archive. Very few professors would have trusted in me for so long. Very few Ph. D. students were given a similar opportunity. Therefore, I want to thank professor Korvink deeply: for the trust, for the freedom, and for accepting my decisions and opinions even when he had other directions in mind. I also have to thank my supervisor Andreas Greiner for his mental and scientific support and, of course, for keeping my salary flowing. Together with Mr. Korvink and Mr. Greiner I have to thank Mr. Zhenyu Liu for all the insight from our unnumbered discussions. For enlightening discussion on specific topics I have to thank the following people: Evgenii Rudnyi for his explications of entropy. Jan Lienmann helped me to understand the closure problem of turbulence. Professor An- dreas Bamberger enlightened me on the subject of vortex pairing and the related generation of sound. Professor Oliver Paul pointed me to Noether’s theorem. For technical support with the hardware in the group I have to thank Chris- tian Moosmann, Oliver Rübenkönig, and Jan Lienemann. Furthermore, I have to thank Jan Linemann for the technical support in designing this thesis and for preparing the Latex stylesheets. Andreas Greiner extracted the data points from Mr. Karlin’s paper which was a great help for me in comparing the cascaded lattice Boltzmann automaton to the entropic lattice Boltzmann automaton. For help with C++ programming, lots of tricks, and for the voice of the ”other point of view in fluidics” I want to thank David Kauzlaric.

5 I also want to thank the rest of the simulation group and our Hiwis for the convenient working atmosphere. For discovering errors in my code and in my papers I have to thank Helmut Kühnelt, Alexander Gaulke, Kerstin Tegeler, and Jonas Tölke. I have to thank all the people who discussed with me on the DSFD 2006 conference in Geneva. Even though, the conference was only two weeks before I had to submit this thesis I tried to include some important points from our discussions. My work was partially founded by the German Ministry of Science and Education BMBF under the project SIMOD and by the Provincia Autonoma di Trento and the Consiglio Nazionale di Ricerca (Italy) within the project Microsystem Modeling in the reasearch framework ricerca e formatzione nei microsistemi. The following software was applied in my work: the GNU Compiler Col- lection, the Simpliﬁed Wrapper and Interface Generator, the Ruby scripting language, Gnuplot, Paraview, Povray, and Mathematica. Finally I have to thank my sister Monika for her remark that if I am not able to explain my theory to her, I have obviously not understood it myself. This made me start over again and I spent the last half year of my work with the search for comprehensible justiﬁcations for my theory. And true enough, that led to much deeper understanding of the cascaded lattice Boltzmann automaton than I had before.

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1 1 Introduction cellular automata. A cellular automaton is a state machine which describes the evolution of observable entities by rules acting on the entities. The different mathematical dialects are in principle equivalent. There are long winded methods to translate expressions from one dialect to the other but they are in general not very useful. The main difference between the dialects is to be seen in the fact that operations which are very simple in one dialect are very involved in the other dialect. Differential equations are compact statements on the general properties of systems regardless of its specific state. Cellular automata, on the other hand, are advantageous for the description of specific systems being in specific states. If we can express a theory in terms of state machine rules, we have, at least in principle, solved it. A cellular automaton is an executable model. Knowledge of the rules and the initial configuration implies knowledge of all forthcoming configurations to later times. We only need to apply the rules to the system state which can easily be done by a digital computer. This idea has been applied to solve the Navier Stokes equation by means of a cellular automaton today known as the lattice Boltzmann automaton. A formal link to the Navier Stokes equation was desired in the historically context since the Navier Stokes equation represented the accepted theory. The objective for the development of the lattice Boltzmann automaton was hence not to derive a new theory in another mathematical dialect but merely to translate the Navier Stokes equation from the dialect of differential equations to the dialect of cellular automata. The technique that was applied successfully to this task is the Chapman Enskog expansion, an expansion technique in Knudsen and Mach number (ratios of system size to mean free path and system speed to sound speed, respectively) which can also be used to derive the continuous Navier Stokes equation from the continuous Boltzmann transport equation, the equation describing the evolution of a single particle distribution function in time, space, and momentum. The procedure is reminiscent of the craft of solving differential equations insofar as it is not possible to obtain the lattice Boltzmann automaton from the Chapman Enskog expansion directly. It is only possible to prove for a given (actually a guessed) cellular automaton that it complies with the Navier Stokes equation in a long wavelength limit. Only rudimentary parameters for the model can be obtained from a comparison between the physical Navier Stokes equation and lattice Navier Stokes equation. Everything else is due to scientific inspiration. This turns out to be a problem. Different assumptions made in the process lead to different models with different properties notwithstanding that they are all supposed to solve exactly the same equation.

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1 1 Introduction money, CPU hour, gigabytes of memory and so on. We could deﬁne a model performance like this:

T×R if C < R P= C (1.1) 0 else The model performance of the Schrödinger equation for describing life is zero because the cost we are able to spend is smaller than the cost we had to spend in order to solve it. It is necessary to recognize that this is a hard constraint. A model is infeasible even if the cost for its application is only very slightly larger than the amount we are able to spend. For example, if a finite element model is just a little bit too large for the address space of the largest available computer it is as useless as no model at all. In applied sciences our objective is always to solve concrete problems given a concrete amount of resources. That we could increase the trustworthiness of a model if we had other resources is not of practical interest for us. In this thesis we are not going to solve concrete technical problems. Our objective here is instead to derive a model that can be applied to concrete technical problems. That is to say, we are not taking the physicists point of view and optimize T but we take the applied science point of view and optimize P. The model performance equation favors simple models. The educated guess comes at low cost and has a non-zero trustworthiness. Simple models are in fact very common in engineering sciences. However, sometimes a threshold trustworthiness is indispensable below which the results are worthless. The thesis is structured as follows: in the second chapter the lattice Boltz- mann automaton is explained in the way it is nowadays commonly used. In the third chapter, a novel methodology is introduced to obtain an hydrodynamic cellular automaton without reference to other models. The new methodology allows us to derive an hydrodynamic lattice Boltzmann automaton from first principle. Even though the new model is completely independent from the Navier Stokes equation, it will become evident in chapter four that it solves the Navier Stokes equation to a much higher accuracy than previously proposed lattice Boltzmann automata. The analytical evidence will be supplemented with numerical experiments in chapter five. There we will see that the new lattice Boltzmann automaton solves hitherto intractable problems with ease. In particular, we will see simulations of fully developed turbulent flow which were obtained without any ad hock turbulence modeling. Unlike other hydrodynamic modeling techniques, the cascaded lattice Boltzmann automaton, as we shall call the new model, captures turbulent behavior ”out of the box”.

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2 2 Lattice Boltzmann - state of the art

Figure 2.1: Momentum state distribution of the D2Q7 and the D2Q9 lattices. The volume of the spheres indicates the weight of the states in the equilibrium distribution.

2.1.1 Lattices

The nature of lattice Boltzmann automata allows only for specific momentum sets or lattices. The number of lattices that found their way into applications is very small. The first lattice that was found to comply with the laws of fluid dynamics is the one due to Frisch, Hasslacher, and Pomeau [3]. It applies a hexagonal distribution of nodes. Each node has 6 links to six nearest neighbors, each having a 60 degrees angle to the next link. We call this lattice the D2Q6 lattice (two dimensions, six speeds). By allowing for a resting particle we obtain a D2Q7 lattice. Today, the most popular lattice in two dimensions is the square lattice with links to four next neighbors, four diagonal neighbors and a resting link (see figure 2.1). We call it the D2Q9 lattice. Common lattices in three dimensions are the D3Q27, the D3Q19, the D3Q15, and the D3Q13 latices [4, 5] (see figure 2.2). There are also lattices with links pointing beyond neighbors to fare away nodes [6]. Such lattices are necessary to include energy conservation, a topic we will not touch in this thesis. A condition for the lattice to be valid is the isotropy of second rank velocity tensors [7]. We will later see that the derivation of the cascaded lattice Boltz- mann automaton results in a more stringent constraint on the lattice so that lattices like the D2Q6, D2Q7, D3Q19, D3Q15, D3Q13, and many more are not

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2 2 Lattice Boltzmann - state of the art valid for the cascaded lattice Boltzmann automaton. The isotropy of second rank tensors does not constrain the homogeneity of momentum space. The momentum state of the D2Q6 lattice, for example, is clearly inhomogeneous since the velocity of a particle in x-direction imposes restrictions on its velocity in y-direction. The standard theory of the lattice Boltzmann automata is ignorant of violations of the homogeneity of the momentum space.

2.1.2 Streaming and scattering

The evolution of the lattice gas and lattice Boltzmann automaton is split into two subsequent steps which are repeated over the duration of the simulation: the streaming step and the scattering step. In the streaming step, all links move in their respective direction carrying their particles with them. The set of momentum states is deﬁned so as to match the spatial distribution of nodes. After streaming, each particle has reached another node and the number and type of momentum states on each node is identical before and after streaming. This imposes stringent restrictions on the regularity of both the set of links and the arrangement of nodes. In the scattering step, update rules are applied to the links of each node. The rules depend only on the state of the links on the same node and are ignorant of other variables in the system. Scattering is supposed to comply with given conservation laws, typically for mass and momentum. State variables which are not considered to be conserved should be changed by scattering. That is to say, they should not be conserved by accident.

2.1.3 Particle representations

The basic ingredient of lattice gas and lattice Boltzmann automata are the field of nodes, the set of momentum states, the representation of the particles, and the update rules. The main difference between lattice gas automata and lattice Boltzmann automata is to be found in the representation of the particles and the update rules while the arrangement of nodes and the set of momentum states is potentially identical in both methods. Lattice gas represents particles in a binary fashion. The momentum state is allowed to be either occupied or not occupied. The lattice gas automaton comply with the classic definition of a cellular automaton as it allows only for a finite set of states. Lattice Boltzmann automata represent particles in a fuzzy fashion, dealing only with the likelihood of the state to be occupied which is a real number between zero and one. For actual implementations of the lattice Boltzmann

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2 2 Lattice Boltzmann - state of the art

distribution. Each link occupation si is relaxed to its equilibrium population eq si with the same relaxation parameter ω.

spost = ωseq + (1 ω)s (2.1) i i − i

2.2.2 Multiple relaxation time operators

The MRT operators [5, 10, 11] apply a multitude of different relaxation operators. This is only possible by first transforming the state of the node as defined by its link occupations into an equivalent moment space. Each moment is associated with one physical quantity like velocity in certain directions, energy, and so on. Some moments are conserved quantities, some moments are associated with the local stress in the fluid and some moments are supposed to be irrelevant as long as they are not conserved by accident. The MRT operators leave the conserved moments as they are. The relevant moments are relaxed with the relaxation parameter chosen to match the physical parameters of the fluid and the irrelevant moments relax with a relaxation parameter chosen solely from stability considerations. The equilibrium is the same as in the SRT method, however it has to be transfromed into moment space. The scattering of moment mi reads:

mpost = ω meq + (1 ω )m (2.2) i i i − i i After the scattering, the moments are transformed into occupation space. The MRT lattice Boltzmann automata are considerable more stable than the SRT automata [11]. Sound waves which contaminate the results of SRT simulations can be eﬀectively removed with multiple relaxation times by choosing the bulk viscosity of the ﬂuid much higher than its shear viscosity. However, the stability properties of the MRT lattice Boltzmann models are still not satisfactory.

2.2.3 Entropic operators

Entropic lattice Boltzmann automata are systems designed to be unconditionally stable [12–16]. An additional constrained is imposed on the scattering operation. The method computes the entropy of the particle distribution on the node. Scattering is not allowed to decrease entropy. The equilibrium is the maximizer of entropy and is in general not given in polynomial form. The

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2 2 Lattice Boltzmann - state of the art the lake, where will this line hit the waterline? The nice thing with this idea is that it reduces the problem to an one-dimensional one irrespective of the original number of dimensions of the landscape. Note that the ”waterline” is an object with d 1 dimensions for a d-dimensional search space. The solution − is hence only unique if we reduce the problem to one dimension. Exploiting all degrees of freedom of a D3Q27 lattice would lead to a 25-dimensional search space in which the root of the entropy functional had 24 dimensions.

2.3 Theoretical background of the lattice Boltzmann automaton

The theory of the lattice Boltzmann automaton is tightly bound to an expansion technique, called the Chapman Enskog procedure which was applied to derive the worlds first working lattice gas automaton2 [20]. The Chapman- Enskog expansion can be used to derive the fluid dynamic Navier Stokes equation from the microscopic Boltzmann transport equation by means of a double expansion in Knudsen and Mach number. The Boltzmann transport equation describes the evolution of the continuous probability of finding a classical particle in a state in phase space subject to free streaming, collision, and forcing. The lattice Boltzmann automaton is hence a digital variant of the continuous Boltzmann transport equation and can be interpreted as a discritization thereof. However, the purpose of the lattice Boltzmann automaton is to solve the Navier Stokes equation. The Chapman Enskog procedure can be applied to determine how the macroscopic equation for a lattice Boltzmann automaton would look like. Free parameters can then be fixed by comparing the results to the Navier Stokes equation. The Chapman Enskog procedure is not a straightforward application of algebra. In fact, some assumptions made in the process are not easily justified. The expansion in terms of Knudsen number Kn uses the smallness parameter ǫ = Kn. The Knudsen number is the ratio of the mean free path of a particle and the macroscopic dimension of the considered system. Only a small

2 There is an ongoing argument on who invented the lattice gas algorithm. Stephan Wolfram obtained an US patent [17] on lattice gas even though the algorithm was demonstrated earlier by Frisch, Hasslacher, and Pommeau [3]. Note that [3] was received on October 22nd 1985 while [17] was received on December 27th 1985. It appears unlikely that the involved persons were not aware of each other since Stephan Wolfram was in close contact with Brosl Hasslacher at this time [18, 19].

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− : se T o ute information. further for ] 9 2 nw + se ( and + c 8 · · · ) ) i .Tesaeo oeis node a of state The ]. ρ e ρ · − − + v c 1 1 ) i ne 2 stevlct vector velocity the is −

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2 2 Lattice Boltzmann - state of the art

Each link is associated with a constant weight wi which is derived from the Maxwell-Boltzmann distribution. Note that only ﬁrst and second order terms in v appear. This is not an accident. It results from the derivation with the Chapman Enskog expansion which must be truncated at second order to obtain the Navier Stokes equation.

The derivation of weighting factors wi is of some interest since we will see the same result from a completely different argument in the next chapter. First we recognize that there are three types of links: the resting link r, the links to the nearest neighbors w, s, e, and n, and the links to the next nearest neighbors nw, sw, se, and ne. So we have obviously only three different weights. The first few moments of the Maxwell-Boltzmann distribution are:

µ0 = ρ (2.11) k T µ = ρ B (2.12) xx m k T 2 µ = 3ρ B (2.13) xxxx m k T 2 µ = ρ B (2.14) xxyy m

Odd moments vanish for v = 0. We calculate moments of the node distribution function at velocity zero and match them to the values from the Maxwell-Boltzmann distribution:

w0 + 4w1 + 4w2 = ρ (2.15) k T 2c2w + 4c2w = ρ B (2.16) 1 2 m k T 2 2c4w + 4c4w = 3ρ B (2.17) 1 2 m k T 2 4c4w = ρ B (2.18) 2 m

with c = 1 being the unit speed. This solves to:

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2 2 Lattice Boltzmann - state of the art from. Since this is an inversion of the velocity prior to collision, the particle must pass velocity zero somewhere between the nodes. The so-called bounce back rule gives a no-slip boundary condition. The location of the boundary is only known within the uncertainty of the lattice itself. No orientation information of the boundary is retained.

2.6 Limitations

The lattice Boltzmann automaton is subject to some limitations. The sim- plifying incompressibility assumption commonly used in computational fluid dynamics is not valid for the lattice Boltzmann automaton because incompressibility implies instantaneous information transfer and an infinite speed of sound. However, the speed of sound is bound to be below one lattice spacing per time step by the nature of the cellular automaton. Still we did not include energy conservation and the lattice Boltzmann automaton should therefore be regarded as a weakly compressible fluid solver. The nature of the cellular automaton imposes a strict limitation on the speed allowed at a node known as the CFL (Courant-Friedrichs-Lewy) condition [21]. The CFL condition is the statement that the propagation of some simulated physical quantity must be slower than the propagation of computational information due to the numerical algorithm. The critical velocity for the lattice Boltzmann automaton is the velocity at which sound waves can propagate in a Galilean invariant fashion:

v = c c (2.25) CFL − s Simulation results are physically meaningless if the CFL condition is violated (if v > vCFL). The violation of the CFL condition typically results in numerical instabilities (by this we understand that some state variables grow until they cannot be represented on a digital computer anymore). The CFL condition might be violated with a realizable distribution of particles on the node. (A realizable distribution in the statistical sense is one in which the occupation number of each link is in the range 0...1.) Another limitation of the lattice Boltzmann automaton is the restriction of the relaxation parmeter ω to be smaller than 2. The viscosity ν in 2.24 becomes negative for ω > 2. Negative viscosity results in instability. This is physically consistent since viscosity is the diffusion coefficient of momentum. However, the lattice Boltzmann automaton is, in general, unstable even if ω is significantly smaller than 2. The reason for this instabilities is subject to some

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2 2 Lattice Boltzmann - state of the art

The term ”relevant scales” is by itself not completely objective. Coming from the continuum theory we could argue that there is a length scale below which dissipation dominates inertial forces so that the velocity and pressure gradients are nearly constant and no eddies are formed below this scale. Ve- locity fluctuations below this scale can be accurately described on a statistical basis as thermal energy. The length η is called the Kolmogorov length and defines the separating length scale below which no vortices exist in the continuum theory [22–24]. It is the scale at which we need to resolve our simulation in order for the continuum theory to be valid. Fluid simulations, lattice Boltzmann or other, do not tend to be stable if the discretization length is larger than the Kolmogorov length. The Kolmogorov length is the length scale for which the Reynolds number is of order unity: ηv Re = 1 (2.26) η ν ≈ Where v is the flow velocity and ν is the kinematic viscosity. Flow at Reynolds numbers smaller than one are not assumed to be influenced by convection. From the microscopic point of view the separation appears to be a little bit arbitrary. The smallest vortex should be the rotation of a single molecule (and possibly it is). Let us investigate the problem of scale separation in fluidic simulation a bit further.

2.6.2 Scale separation Let us take a look at the equation of motion of a continuous ﬂuid, the Navier Stokes equation:

∂v ρ + v v = p + νρ v (2.27) ∂t ∇ −∇ △ This equation imposes severe difficulties if we try to solve it with common numerical methods like finite differences or finite elements. The convection term v v could be blamed as the source of the difficulties because the non- ∇ linearity in velocity results in a coupling of all length scales (the superposition principle does not hold). However, there is another source of significant difficulties. The velocity vector v appears directly and together with three different differential operators. These differential operators must somehow be implemented numerically. Differentiation requires knowledge of the considered quantity in a ”small” neighborhood. Numerical schemes are typically

26 ore pc ieeta prtr oee,a eghsca have: length we at However, spacing operator. differential space Fourier u ftoDladsrbtosadhsteFuirtransform Fourier the has and distributions Delta two of sum prtrb ipecnrldffrnestencil: difference approximatio central the simple Consider a averagi by precision. stat of operator of In loss kind a some means points. requires always differentiation neighboring that of say knowledge might requires later the ieetainrslsi oso eouin o n give any For resolution. of not met of loss the (spectral edge a With functions in ansatz steps. results dire global time differentiation the apply is and that neighborhood fi spacings methods the use grid of all of the they of smallness practice sizes The In the zero. spacings. to grid go th steps and if time equation and differential nodes exact grid the to converge to designed ∂f Since Equation hr h seikdntstecnouinoperator: convolution the denotes asterisk the where h r-atrbcmsngiil o ml rdsaig a spacings grid small for negligible becomes pre-factor the ∂x F ( x n ) δ ( ≈ v x ∆ + f ilb oecranta htfor that than certain more be will ( 2.28 x ∆ + x ) 2∆ − sasadr eta ieecn tni.I steweighte the is It stencil. differencing central standard a is x x δ ) 2∆ ( − x w x − f ( ( k ∆ x ) x − ∗ ∆ ω ) lim → x lim v f ∆ → ( ( ∆ k 2 x x 0 π x = ) ) ) i o i sin( = sin( = = Z ∆ 2∆ ∆ ω e e x 1 ω w x ∆ − − x ∆ ( i i x x ωx ω Z x ) ) ( ) x v i ( = +∆ ( ∆ sin( 0 = δ k ∇ ( x i x x − ω v 2∆ ) ω ∆ + − x ttesm on because point same the at ∆ ) x e dx x − x ) i ) ω F ∗ − ( x e ls otegrid the to close les − on u knowl- our point n pcnsbetween spacings e δ n ∆ ( gadaveraging and ng fadifferential a of n dw banthe obtain we nd f os,numerical hods), x x sia em we terms istical : . Limitations 2.6 ( ) − ietm steps time nite tyrltdto related ctly x beexception able F ∗ ) ∆ o x n )) f f ( x ( (2.31) (2.29) (2.28) (2.30) (2.32) (2.33) x ) o ) dx 27 d

2 2 Lattice Boltzmann - state of the art

We hence see that this stencil has a low pass characteristic which is true for almost all useful differentiation schemes since they are supposed to improve with higher resolution. This low pass characteristic means that the derivatives are valid only for length scales larger than the grid spacing. The problem is that the discretization can represent the field quantity with the accuracy of the grid spacing while the gradient of the field is known only with lesser accuracy. There are hence three length scales: The large scale on which the discretization is good, the length scale which is much shorter than the grid spacing and is invisible to the numerics, and the intermediate length scale that fits on the grid but is so short that the numerical differentiation filters it out. Now let us assume that the numerical differential operators introduce errors that can be modeled as multiplicative low pass filters in the discretization frequencies. In general, the filters have magnitude and phase but for simplicity the phase change is omitted here and we assume that the errors can be expressed as multiplicative scalar values:

∂ ∂discrete = A (ω ) (2.34) t t t ∂t discrete = A (ω ) (2.35) ∇ ∇ x ∇ discrete = A (ω ) (2.36) △ △ x △ If we try to solve the Navier Stokes equation numerically, we are, in fact, solving the equation:

ρ(A (ω )∂ v + vA (ω ) v)= A (ω ) p + A (ω )νρ v (2.37) t t t ∇ x ∇ − ∇ x ∇ △ x △ The most severe problem here is that the low pass characteristics are diﬀer- ent. In particular we have reason to assume for large ωx:

A (ωBIG) A (ωBIG) (2.38) △ x ≪ ∇ x due to the fact that a second derivative is even more uncertain than a ﬁrst derivative. We may cast A△(ωx)ν in equation 2.37 into an eﬀective viscosity:

νef (ωx)= A△(ωx)ν (2.39) The numerical differentiation has the effect of an altered viscosity at shorter scales. The low pass characteristic of the differential operator implies a reduc- tion of the effective viscosity at high frequencies. This has profound implications on the stability of the numerics. The point is that if the grid spacing

28 ρ offiinsin coefficients sarsl h ytmi o lsd h qainin equation The closed. not is system the result a As hudg oifiiyadw a oslea nnt e fequat of set infinite an ca we solve So to approximations. had that finite we apply always and power schemes infinity merical their to to go according should time in equations ential in tms nt ubro icniute nafiiedmi.T domain. finite a in discontinuities of number finite write a most at ol oeta hnslo ieeti eue o xml,fi example, for Stoke use, Navier we one-dimensional example: if simplified the different sophi consider very look us not Let things are that above hope used we could stencils difference schemes finite order The higher and Stokes Navier 2.6.3 th corrupt grad scale its this and on frequenc errors field numerical possible velocity the all and the consistent frequencies length, high Kolmogorov for the However, than larger is

n t derivatives: its and easm htteslto for solution the that assume We fw ute develop further we If x ∂u n ∂t n u n ecnsltteprildffrnileuto into equation differential partial the split can we and n a ebgbtfiie hshstecneuneta h number the that consequence the has This finite. but big be may naTio expansion: Tailor a in + X i =0 n u ∂u/∂x i u n u − i = +1 and ∂x ∂ ρ u ( ∂u ∂x 2 ∂p/∂x n ( u 2 ,t x, 1) + ∂u ∂t ∂ 2 2 = = = ) u/∂x + ! in u u u x u ∂x ∂u x 1 0 n 2 2 2 + + eoti neuto ihmlil terms multiple with equation an obtain we u ( = r n n w esa in as less two and one are 6 + sasur nerbefnto in function integrable square a is u u = 1 − u 2 x x ( 3 − n + x 3 + ∂x ∂p +1) 12 + u 2 u x + 3 p 2 u x n ρν + +1 4 2 x · · · n u ∂x ∂ 2 + npicpe h series the principle, In . 3 2 · · · x u ρν 2 x 3 n · · · ( n reads: iuainresult. simulation e +1)( qaina toy a as equation s n et eoein- become ients e r relevant. are ies . Limitations 2.6 aeyassume safely n tctdadwe and sticated riaydiffer- ordinary u ieelements. nite n respectively. , +2) e ecan we hen os Nu- ions. u n x (2.41) (2.40) (2.43) (2.44) (2.42) +2 with ) x 29 of n

2 2 Lattice Boltzmann - state of the art

The equation for un depends on both un+1 and un+2. The term un+1 can be eliminated because it is multiplied with u0. Since the Navier Stokes equation is Galilean invariant we can choose the origin of the velocity coordinate system at will. We can as well select it to fulfill u0 = 0. However the influence of un+2 cannot be eliminated. The only thing we could do in order to eliminate un+2 was to set ν = 0. Then we obtained the Euler equation. This shows us one of the fundamental differences between the Euler and the Navier Stokes equation. The Navier Stokes equation is intrinsically unclosed for all finite approximations. In order to compute un we always need to know un+2. Our only hope is that u gets small for n . In fact, we know when u is n → ∞ n getting negligible: if its features are below the Kolmogorov scale. For finite n we conclude that the system is closed if the features of un+2 are below the Kolmogorov scale and that it is not closed if its features are larger than the Kolmogorov scale. The Kolmogorov scale is a particularly small scale. We can think of it as the scale that solves the Reynolds number to one [22]. That means for the Knudsen number Kn:

λ Ma Kn = = (2.45) η Re =1 with λ being the mean free path and η|{z}being the Kolmogorov scale. The Machnumber Ma is the ratio of the ﬂuid velocity v and the speed of sound cs:

v Ma = (2.46) cs The speed of sound is the root of the variance of the speed of the particles in the system. At molecular distances v c and therefore λ η. Thus, in ∼ s ∼ principle we need to resolve the molecular scale to close the system. The above problem becomes even more severe if we consider discretization schemes with finite support like finite elements. The analysis then holds for one set of compact ansatz functions at a time. (A compact ansatz function is non-zero only in a finite domain.) The point is that we loose two coefficients for each element in the domain. The accuracy to which we know the gradients and second gradients of the velocity fields are significantly lower than the accuracy to which we know the flow field itself. This is the reason why spectral elements enjoy such a popularity in computational fluid dynamics [25]. Spec- tral elements are particularly large elements with high order ansatz functions. However, the closure problem remains even for a global spectral discretization.

30 n yial o aienivrat oseti econside we this see To invariant. Galilean equation: not i a functional typically entropy maximizing the and entropy of there uses choice that the stability) However, fact of constraint. terms the (in from succe system seen most the the be date, To can operators. as stabl w scattering ”equivalent” space even be syst unclosed parameter to the is own seem close automaton its not to Boltzmann lattice does able numbe The itself be moments. infinite expansion principle, Enskog an in Chapman should, adding the we by equi node infinity the T each to from to expansion. expansion deviations automaton, moment the small Boltzmann the only extended lattice of allows the truncation expansion in o the Enskog principle present form basic be originates a to it is has turbulence it of that problem closure the entropy That with equations under-determined app Closing an in 2.6.4 closed manif not and is problem system the the Howe if restating system. instabilities by the numerical removed of freedom be di of to cannot degrees due problem explicit lost be get to that need coefficients not The schemes. numerical the x incntan.Atpclcoc o h nrp functiona entropy the for choice typical entropy: Boltzmann A constraint. tion 1 Equation ewn omxmz entropy: maximize to want We tsol entdta h lsr rbe osntso pdi up show not does problem closure the that noted be should It and x 2 ecncoei ysplmnigi iha nrp maximiza entropy an with it supplementing by it close can We . 2.47 sucoe eas ti nyoeeuto o w variables two for equation one only is it because unclosed is S = = ln( x x x 1 1 1 ln( ln( ) − y x x ln( 1 1 = ( + ) + ) y x 1 − x y + 2 ∂x ∂S x − ln( x x 1 1 1 0 = ) 2 x x 1 2 ln( ) 0 = = ) y y 2 − x 1 ) su ehdt close to method ssful ,i a,arbitrary way, a in s, oraeway. ropriate steadditive the is l supplementary s . Limitations 2.6 auemeans nature f eetaindo fferentiation e,teclosure the ver, h following the r iru.I we If librium. m However, em. t epc to respect ith eChapman he ssisl in itself ests o higher for e o.There too. fspeeds of r r several are etyin rectly (2.47) (2.49) (2.50) (2.52) (2.51) (2.48) 31 -

2 2 Lattice Boltzmann - state of the art

The maximal entropy constraint tells us to choose x1 = y/2 and hence x2 = y/2. This seems to be reasonable. Now imagine x1 and x2 would represent velocity space occupations in orthogonal directions such like the links of a lattice Boltzmann automaton. There is no special resting velocity and the choice of the coordinate system in velocity space is arbitrary. We could as well use a transformed system with x1 = xa + a and repeat the calculation:

y = xa + a + x2 (2.53) S = x ln(x ) + (y a x ) ln(y a x ) (2.54) a a − − a − − a ∂S = 0 (2.55) ∂xa y a x = − (2.56) a 2 Let us transform this to the original coordinate system:

y + a x = (2.57) 1 2 y a x = − (2.58) 2 2 The result is not the same as for the original coordinate system! Closing the lattice Boltzmann automaton with entropic auxiliary constraints breaks Galilean invariance. Is this good or bad? The Navier Stokes equation is Galilean invariant. However, we could argue that Galilean invariance has to be broken anyway to guarantee positivity of the occupation numbers in order to obtain a stable solution. Thus the entropic ansatz might not be perfect but it has the virtue to always deliver a stable solution. Still, we should ask for the justiﬁcation of auxiliary constraints like equation 2.48. Entropy has a profound physical meaning and using the word in the way we did here could imply that there was a deeper physical truth buried somewhere in the assumption. However, this is clearly not so. Equation 2.48 comes out of the blue. Its justiﬁcation is that it does not contradict other constraints and that it closes the system which was otherwise underdetermined.

2.6.5 Entropy and closure By talking about entropy we should be aware that the modeled entropy in the entropic lattice Boltzmann method is actually the opposite of its physical coun-

32 hti sspoe osoe hsi utaohrwyt state to way another just is This me enough store. not to has it supposed resolved is under is it automaton what the e aut increase if to cellular that fail contai simulations is the it Boltzmann lattice all of that destroy memory not an we is is holographic memory there holographic the Thus, the in overloading information. T entropy of memory. amount the finite finite with for a computer for hold finite history only a whole can on simulation its the record run to we automaton Boltzmann lattice rdal n ti oecmltl when completely gone is it and gradually ssol spsil.W a uetepoeso ogtigw forgetting of me process the the So tune can constant We possible. as possible. as information poi slowly much as floating as in retain rounding However, to beca coded course, designed time. of over is, automaton: memory it rem its cellular and erase automaton impossible the be Boltzmann should lattice in This the history! stored state, data current its of into amount the on rsn fe ahcliinmlile with multiplied collision each after present tt after state rcso aatps h precision arith The reversible perfect types. used data we precision if even anymore reversible Boltzmann lattice the that note Equation we t reversible. this, of time instability see the To for n reason to simulations. the interesting actually is is It entropy ical) content. pr cont information a smallest information is the equilibrium highest has the the automaton, maximall of Boltzmann were lattice state states tropic the particle is equilibri the equilibrium global that The of mean state would the that in world highest is Entropy terpart. rmteps-olso state: post-collision the from ubln eairi bandat obtained is behavior Turbulent h osblt ornteatmtnbcwrshsaprofoun a has backwards automaton dir the particle run the to inverting possibility by The undone trivially is Streaming h esnfrti sta h noigsaeo h hl aut whole the of state incoming the that is this for reason The ω For . t iesesis: steps time ω 1 = equation 2.1 a eivre ooti h r-olso state pre-collision the obtain to inverted be can s i 2.59 P = = s a iglrt n h uoao snot is automaton the and singularity a has i post | (1 P ω 1 − − − → owihw nwaotaprevious a about know we which to ω P ω ωs ) 2 t rp eo ahn precision. machine below drops ystetting By . (1 | i eq − ω ) h eoyi erased is memory The . eltieBoltzmann lattice he toy h problem The ntropy. m ntephysical the In um. t,ta h (phys- the that ote, s on-fferrors round-off use ω dfie tt and state edefined ei n infinite and metic ections. t h relaxation the ith s h problem The ns. mesiswhole its embers n.Frteen- the For ent. 2 = . Limitations 2.6 vr However, ever. toeain is operations nt oyi erased is mory randomized. y oyt store to mory uoao is automaton implication d mtn By omaton. pe bound upper i memory his h closure the etl the tell we mtnis omaton somehow (2.59) (2.60) 33

2 2 Lattice Boltzmann - state of the art problem. The information content (entropy) in turbulent ﬂow is too large. Any approach to simulation of turbulent ﬂow needs to destroy entropy in one way or another. The optimal solution would retain a maximum of entropy that does not overload the computer memory.

2.7 Endnotes

This chapter gave a rudimentary overview of the state of the art of the lattice Boltzmann automaton without focusing on specialties since we want to concentrate on the basic principles in this thesis. We shall see in the forthcoming chapters that there is an astonishingly width space for improvement for the basic principles of the lattice Boltzmann automaton. Especially the application of the Chapman Enskog procedure is not rigorously justified and some of its assumptions are dubious. In conjunction with the Navier Stokes equation the Champamn Enskog expansion gives no constraints for terms higher than second order in velocity. The problem is that the lattice Boltzmann stencils would support higher order terms and without additional constraints some degrees of freedom must be chosen arbitrarily. However, it is quite evident that the choice of the undetermined degrees of freedom has a strong influence on the accuracy and the stability of the model. It is clearly an unsatisfactory condition that our current understanding of the lattice Boltzmann automaton does not provide any clues on how to fix this undetermined degrees of freedom. We could even go so far as to question the validity of the Chapman Enskog expansion in general. To be considered ”good” a theory has to fulfill some constraints. It has to explain what is known, it has to provide testable implications on what is not known, and it should be simpler than other theories which explain the same things and make the same testable predictions. Seen in this light, the Chapman Enskog expansion has yet to show its status as a good theory. It can be used to derive the Euler and the Navier Stokes Equa- tion from the Boltzmann transport equation. However, both the Euler and the Navier Stokes equation are significantly longer known to science than the Boltzmann transport equation. Hence, the Chapman Enskog expansion ex- plains what is already known. But a good theory should also make predictions on what is not known. That is to say, we should obtain better hydrodynamic equations, if we truncated the expansion at higher orders in Mach or Knudsen number. To date, it still seems unclear whether this is the case or not. Either way, we can only benefit from other entries to the theory of lattice Boltzmann automata. In the next chapter we derive the lattice Boltzmann automaton

34 etr ilb eni h atta tdtrie l degree all undetermined. determines leaves diﬀerenc it expansion certain that Enskog to fact Chapman the lead exp the in also Ens Enskog seen Chapman will be Chapman the it will as the but feature results aspects to same many the related in to not sion lead is will method that new method The new a with ffedmwhich freedom of s s prominent A es. . Endnotes 2.7 o expan- kog ansion. 35

2 36 ieto and direction h tt ftesse ewe w usqettm tp su is steps time subsequent two between system the of state The repcieo hi tt npaesae hti osy ti cons say, with to increases is and That quantified is space. Time phase relativistic. in not state their of irrespective h oetmo h atcei unie st lo nycer only allow to as quantified is particle the of momentum The iia iei model kinetic Minimal 3 oaino atcei pc hne ntm codn oi to m according carry time to in allowed changes is space in particle particle A a of ”particle”. location called is quan is mass everything of that assumes momentu moment automata of and Boltzmann coefficient lattice mass (diffusion of viscosity provide given diffusion to with and stance is drift automaton for Boltzmann model lattice mathematical the of purpose The principles Basic 3.1 principle. first edsrb h tt ftesse yteocpto fstate of occupation the by system di particle the specific of a state address the never momen describe in shall specifi We states we in offer occupied. be Nodes be to nodes. can allowed called are are particles locations These as insofar quantified is hr hudntb hrceitcrsig( resting characteristic a be not should There Time with nti hpe edrv iia iei atc Boltzman lattice kinetic minimal a derive we chapter this In pc st h etpsil prxmto stoi n ho and isotropic approximation possible best the to is Space ri Schrödinger Erwin – bu everybo seen, which yet that has about one though, no yet sees. what has see nobody to what much think so to not is, task the Thus, x t ( t sagoa oooosicesn aibevldfralpar all for valid variable increasing monotonous global a is ) en h oiino h particle, the of position the being m en t as h eoiyo h atceis particle the of velocity The mass. its being x ( t ∆ + t = ) x ( t + ) π x /m ∆ t = v π x x 0 = ( t en t oetmin momentum its being + ) eeec on.Space point. reference ) v x ∆ t ) h hoyof theory The m). atse size step tant ie.Tequant The tified. ei Newtonian is me smomentum: ts oain only. locations c mnu.The omentum. ety Instead, rectly. u pc that space tum anvelocities. tain oe from model n nexecutable an ndefined. tnodes at s mo sub- a of um v x mogenous. = dy t π ticles x (3.1) /m ∆ 37 x t x - . .

3 3 Minimal kinetic model and in time t. The utilization of nodes is in conflict with the requirement that there should not be a resting reference point. This conflict cannot be completely resolved. However, its consequences can be minimized. The state of any point in space other than a node is undefined. The occupation of states is also quantified according to the nature of the data type we apply for the occupation numbers. This could be integers (constant quant) or floating point numbers (variable quant). Exact conservation of mass and momentum is guaranteed if we choose integers. Numerical data types have a final range so that a state could be ”full”. Here we assume that this range is large enough to avoid overflow in practice. Particle interactions are modeled by collisions. Collisions are processes which are instantaneous in time and depend only on local information. The outcome of a collision is hence completely determined by the knowledge of the state of the given node at the given time.

3.2 1D prototype

We seek the simplest possible model for transport of mass and momentum in one dimension. Mass and momentum are conserved quantities for a given instant in time and space. A system that describes a non-trivial evolution of mass and momentum at a given point in space needs more than two state variables at every node. The minimal set of state variables for a lattice Boltzmann automaton in one dimension is hence three. Since there is no justiﬁcation for assuming a special direction in momentum space, the arrangement of states must be symmetric. Three is an odd number. The only possibility to arrange an odd number of objects symmetrically in one dimension is to place one in the middle. The remaining two states are placed at equal distances from the center state in positive and in negative direction. Thus, the arguable simplest lattice Boltzmann model in one dimension looks as follows: Particles are allowed to move with velocity c, c, and zero along a line. The state set of a − lattice node has three elements and we call such a lattice D1Q3 (one dimension, three speeds). Nodes are separated on the line by distance ∆l and the cellular automaton is updated in time steps ∆t =∆l/c. Deterministic particle motion is accounted for by the streaming step. In the streaming step, particles in velocity state c on the node with index m move to node m + 1 and occupy the state c. Particles in velocity state c on node m move to velocity state c − − on node m 1. And particles in velocity state 0 on node m remain in velocity − state 0 on node m. All particles on all nodes stream instantaneously. Parti-

38 with lsarvn rmnihoigndsfidepysae which states. states empty other find overwriting without nodes neighboring from arriving cles epciey h osraincntansdtrietwo determine constraints conservation The respectively. ntesatrn process: scattering the in µ invariants two fixed interaction. already particle have prohibit latt we would this invariant and for variables requested three be only cannot moment laws and conservation conservation mass Further spac are momentum model in happens this local for Scattering scatter constraints not on is constraints it updating. non-conservative but and of conservative space time and time the in at instance nodes lattice on atce nvlct state velocity in particles evdquantities: served iru” cteigi upsdt eadsiaieeffect. dissipative a be to a the supposed determine is that Scattering constraints librium”. those are category first The Equilibrium 3.2.1 const equilibrium categories: two into straints. split further be can atce.Det h euaiyo h ellratmtna automaton cellular the of regularity the velocity to Due particles. 0 h w nainsaeas h rttommnso h distri the of moments two first the also are invariants two The h usqetsatrn tpacut o h interactio the for accounts step scattering subsequent The h ereo freedom of degree The e sdnt h oetmsaevco o node for vector state momentum the denote us Let ednt by denote We = r ρ and en h ubro atce nvlct state velocity in particles of number the being c rdspacing grid , µ 1 = v K u oteivrat,teei nyoedge ffreedom of degree one only is there invariants, the to Due . h lmnaycliinwihi rhgnlt h con- the to orthogonal is which collision elementary the k ∆ − sdtrie ynncnevtv osrit which constraints non-conservative by determined is v ρ c l and , n iestep time and = = K µ µ s 0 1 e = ← = ( = a nes)teprilsi eoiystate velocity in particles the east) in (as    s + r − e + 1 1 k − 2 K e w    ∆ + ) t /rho w atce r laslocated always are particles , m by 0 n.Teconservative The ing. , s ansadrt con- rate and raints olso invariants: collision m w hti osy the say, to is That . Dprototype 1D 3.2 c ic ehave we since ice tatro ”equi- or ttractor mconservation. um = dtecoc of choice the nd a nws)the west) in (as a eoccupied be can uinfunction bution ewe the between n n further Any . .W impose We e. nasingular a in e w r (3.4) (3.3) (3.2) (3.5)

39 c T ,

3 3 Minimal kinetic model memory of pre-collision state is erased and the state vector evolves into the direction of the statistically most likely distribution of particles in momentum space. First we have to be aware that the lattice gas ansatz is physically wrong insofar as it restricts the particles to move on a ﬁxed lattice. If arbitrary speeds were allowed it would not be justiﬁed to assume that the variance of the particle velocities could depend on the average speed of the particles v. The variance of the momentum distribution function should hence be independent of velocity. Physically, the variance of the momentum distribution is the square of the speed of sound:

c2 = µ µ2 (3.6) s 2 − 1

−1 Where µ2 = (w + e)ρ is the second statistical moment of the particle distribution function in momentum space on node m. Further constraints arise from symmetry considerations. We cannot justify the existence of a preferred direction in space. The equilibrium distribution function must hence be symmetric with respect to the mean of the distribution (since the mean is the only special point in the distribution). In statistical terms that means, that all odd order (unsymmetric) central moments must be zero. For the given velocity set we have only one remaining parameter (cs) which has to be chosen independent of velocity. So we weaken the symmetry constraint and require only that the third order central moment κ3 (or skewness) of the momentum distribution should be small.

κ = µ 3µ µ + 2µ3 (3.7) 3 3 − 1 2 1 = µ c2 3µ c2 µ3 (3.8) 1 − 1 s − 1

Note that the statistical raw moments µ1 and µ3 are aliases of each other for this speciﬁc lattice. In fact all odd raw moments are aliases of each other 2m and all even raw moments are aliases of each other on this lattice (µ1+nc = + µ1+n+2m n,m Z ). It is not possible to eliminate skewness exactly due to 3 ∀ 3 ∈ 2 2 the µ1 = v term. However, the linear term vanishes if cs = c /3. In the reminder of this thesis we shall assume c = 1, ∆t = 1, and ∆l = 1 for simplicity.

40 φ atce ntend.I scnein oepestescatte the express ( to events convenient scattering is two It between time mean node. the on particles pe frlxto a nydpn nti distance this on depend equilib the only to can the is-state relaxation for the of invari measure from invariant speed scale distance scale a the only choose is The to equilibrium have scattering. we for means law This equilibrium. from nodrt banadsrbto odrv ttsia mome statistical derive to distribution a obtain to order In the dutbe e sdnt h cteigrt by rate scattering and the parameter denote material us a Let essence, in adjustable. is, rate scattering The rate Scattering 3.2.2 nain ea utb xoeta ea ihacntn de constant a with decay exponential ω be must decay invariant normalized: ˜ 1 ( : hr r boueprmtr ntepyia ol,sc as such world, physical the in parameters absolute are There t cteighpesa iceetm steps time discrete at happens Scattering ehv ojsicto o suigaseicmti o the for metric specific a assuming for justification no have We h eaainconstant relaxation The h nerlcnb simplified: be can integral The hc ih as hsasmto ob invalid. be to assumption this cause might which ) k stefato fprilswihhv o ensatrdat scattered been not have which particles of fraction the as htdie h cteigit equilibrium: into scattering the drives that φ ( ω t = ) φ ˜ losu osatra rirr rcino the of fraction arbitrary an scatter to us allows ( t s k (1 = ) φ ˜ eq ← ( t K ) s / = + − Z s ωk eq 0 ω ∞ ) eq − τ ⌊ ∆ φ t/ ˜ .Frti edfietefunction the define we this For ). K s ( t ∆ x ewe which between t ) ⌋ dx ω n let and 1 hti osy scale say, to is That . h ieo h molecules, the of size the imsaeadthe and rium-state nts, . Dprototype 1D 3.2 k tsol eleft be should it igrt sthe as rate ring eq φ eiainfrom derivation ˜ ( a parameter cay φ ˜ t n evolution ant edfie as defined be ) ( time t sconstant. is ) a obe to has distance t (3.11) (3.10) (3.12) : (3.9) 41

3 3 Minimal kinetic model

∞ ∞ n∆t φ˜(x)dx = φ˜(x)dx (3.13) Z0 n=1 Z(n−1)∆t X∞ = (1 ω)n−1 (3.14) − n=1 X 1 = (3.15) ω

Hence, the distribution reads:

φ(t) = (1 ω)⌊t/∆t⌋ω (3.16) − The mean or expected value τ is obtained by integrating over φ(t)t:

∞ τ = φ(t)tdt (3.17) Z0 ∞ n∆t = (1 ω)n−1ωtdt (3.18) − n=1 Z(n−1)∆t X∞ t2 n∆t ∂ t2 = (1 ω)n−1ω n∆t (1 ω)n−1ω dt (3.19) 2 − |(n−1)∆t − ∂t − 2 n=1 Z(n−1)∆t X =0 ∞ 1 = ∆t n (1 ω)n−1ω | {z } (3.20) − 2 − n=1 X 1 1 =∆t (3.21) ω − 2 Thus we have a simple exact relationship between the scattering or relaxation constant ω and the mean free time between two scattering events. Note that the relationship is surjective. That is to say, we can obtain arbitrary mean free times τ by manipulating ω. There might not be much physical sense in choosing negative times. Interesting is that we can obtain mean free times which are arbitrarily shorter than the time step ∆t. In fact, this is the only reason why lattice Boltzmann automata represent tractable models for fluid flow in the first place. Lattice Boltzmann automata had little advantage over molecular dynamics if we had to choose the time step of the order of τ.

42 h aea h itiuino ttsi atws directio east-west in states of velocity north-south- with distribution the states the in as states same of the distribution the example, For eedn ftepril’ tt in state particle’s the of dependent (hereafter aei hr sa n-iesoa atc otmn autom Boltzmann lattice one-dimensional an is there if case in states of distribution the then velocity, special spe no no is is there that stating assumption invariance Galilean ieBlzanatmtna rmtv rmwihw r abl dimensionality. are arbitrary we of which automata from one Boltzmann primitive prob the lattice a use fluidic as can all automaton we nearly Boltzmann But tice since two-dimensional. least useful at very are not interest is Boltzmann lattice It minimal a dimension. developed we dimension section last one the In than more to Generalization 3.3 eas utalwsae ihvelocities with states allow must also we osre uniis(asand (mass quantities conserved tahdt ahsaeo h n-iesoa atc Boltz lattice one-dimensional the y of state each to attached iesoa atc.Hnew have we Hence lattice. dimensional ihal9cmiain of combinations 9 all with reo,btadn iesoscudb trigpitt d to point starting inte a methods.) modeling be these could for dimensions methods adding boun efficient but periodic freedom, more and certainly node freedom single are of a (There d of degrees thickness higher molecular the to with internal it mension model extend easily to could process inductive we example, same auto For the Boltzmann applied freedom. lattice we of dime one-dimensional degrees three conserving internal than energy fake more However, for useful dimension. be o third no the is beyond There go automaton. to Boltzmann automata pro lattice Boltzmann could one-dimensional lattice We dimensional states. higher discrete adding 27 by in results dimensions three a eetne otredmnin yuiga one-dimensio an in using automata by mann dimensions three to extended be can uoaat ahsaein state each to automata oa rirr ubro iesos h esni htther that is reason The dimensions. of number arbitrary an to drcin hsrslsi w-iesoa atc Bolt lattice two-dimensional a in results This -direction. easm pc ob stoi.Ta st a,teei ospe no is there say, to is That isotropic. be to space assume We h lmnaycliin r o oesl rnfrdfrom transfered easily so not are collisions elementary The y drcin.I diint h stoyasmto eappl we assumption isotropy the to addition In -direction). z − drcinadadn w-iesoa atc Boltzmann lattice two-dimensional adding and -direction c , 0 and , − z drcin obnn h states the Combining -direction. c , . eeaiaint oeta n dimension one than more to Generalization 3.3 0 c d and , nes-etdrcin(hereafter east-west-direction in mmna but -momenta) y 3 drcin hscnovosyol ethe be only obviously can This -direction. d c − nthe in d − − c , 1 0 lmnaycliin rdegrees or collisions elementary x and n nthe in and - 3 d c oetmsae na on states momentum nnorth-south-direction in x drcinhst ein- be to has -direction ilvlct.I there If velocity. cial anatmtnin automaton mann tnin aton mn automaton zmann .Snew allow we Since n. ieto utbe must direction aao n had and mataton eda infinitum ad ceed uoao none in automaton y dmninllat- -dimensional eso practical of lems a atc Boltz- lattice nal vosrao to reason bvious rv n justify and erive ayconditions. dary a eue an used we had drcin This -direction. ahsaeo an of state each yadn di- a adding by − nldgesof degrees rnal then imensions n dimension one oconstruct to e c ildirection. cial , soscould nsions x are e 0 x -direction) and , -direction d the y c 1 + 43 d in -

3 3 Minimal kinetic model of freedom. There is always an arbitrary number of possible bases for any multi-dimensional state space. However, the choice for the basis collisions is not at all arbitrary from the physical point of view since we have to determine 3d d 1 relaxation constants corresponding to the same number of trans- − − port phenomena. It is not reasonable to assume that all transport phenomena evolve on the same time scale (that all elementary collisions have the same frequency). Hence we have to determine a set of elementary collisions that distinguishes between diﬀerent physical phenomena. In order to derive this basis set from ﬁrst principle we assume that each elementary collision represents a proper physical law. Criteria for proper physical laws are dimensional consistency, homogeny, and rotational invariance.

3.4 Admissible operations

The minimal one-dimensional model had only one degree of freedom or elementary collision. The collision vector K was determined up to a constant factor. Higher dimensional lattice Boltzmann automata have more degrees of freedom and an inﬁnite choice of possible collision vectors. If Ki are legal collision vectors we can build arbitrary new vectors:

K´ = αiKi (3.22) Xi with arbitrary αi. However, it turns out that we cannot use just any set of collision vectors. To understand this we have to develop a theory for admissible operations.

3.4.1 The generic collision

Let us assume that the change in an arbitrary scalar state variable q due to collision is given as:

∆q = ωP (3.23) − where ω is an arbitrary constant and P is an potential deﬁned as the devi- ation of q from its equilibrium.

P = q q (3.24) − e

44 htflosw bancntanson constraints obtain we follows what Let operator commutative generic de The to 3.4.2 need we ﬁrst But rotation. and theory. certa under translation invariant notably is that most collision a understand we that osdrdquantities considered Let translation under Invariance 3.4.3 vector r sn n uthneb nain ne rnlto.Thu translation. under invariant be hence must and using are Equation Equation h rnlto vector translation The h olso sivratunder invariant is collision The A A [ = x · rprcliin utb needn ftecodnt sy coordinate the of independent be must collisions Proper . ] T eagnrcoeao hc omtswtetecliinope collision the withe commutes which operator generic a be eatasaino h eoiycodnt ytmb narbi an by system coordinate velocity the of translation a be 3.25 3.23 A T stegnrccliinoeainvldfrteeouino evolution the for valid operation collision generic the is sipsdon imposed is [ [ q q − − q ω ω ehv o e eemndwhat determined yet not have We . ( ( A T q q x T − − [ [ C C sabtaybtcntn.S ecnwrite: can we So constant. but arbitrary is T [ C q C q T [ [ A e e q q [ [ [ ]= )] ]= )] q q ]= ]] ]= ]] [ q q [ q e = ] C ]= ]] = ] = ] yacliinoperator collision a by [ A · q ]= ]] [ − · A T A T ] q q q C ω n utfulﬁll: must and C [ [ [ [ e othat so q q q q + ( [ q + [ q ] ] ] ] A + ] − − − − g − g ( [ e x ω ω ω ω · q ( g ) ]] e x ( ( ( ( c ) T A T A ( 3.25 ) x [ [ [ [ ) q q q q . disbeoperations Admissible 3.4 ] ] ] ] T − A − T − A − sapoe olso.By collision. proper a is C [ [ [ [ q [ q q q ntransformations, in · e e e e ]) ]) ]) ]) ] : q ehave: we s ok ie In like. looks eo some velop tmwe stem (3.27) (3.25) (3.33) (3.32) (3.31) (3.30) (3.28) (3.29) (3.26) rator: trary all f 45

3 3 Minimal kinetic model

A translation by the zero vector is the identity vector and the additional terms must vanish in this case:

g(0) = gc(0) = ge(0) = 0 (3.34) Together we have:

[q]+ g (x)= q + g(x) ω(q + g(x) q g (x)) (3.35) C c − − e − e Since equation 3.25 must still be true we can subtract it. That leaves us with:

g (x)= g(x) ω(g(x) g (x)) (3.36) c − − e Sine equation 3.36 has to be true for any ω it must also be true for a speciﬁc ω. For ω = 1 equation 3.36 is true only if gc(x) = ge(x). Hence we have for arbitrary ω:

g (x) g(x)= ω(g (x) g(x)) (3.37) c − c − which is true only if gc(x) = g(x). Putting the pieces together we see that for the collision to be admissible it has to fulﬁll the following transformation constraint for any translation vector x:

[q] = q + g(x) (3.38) T [ [q]] = [q]+ g(x) (3.39) T C C [q ] = q + g(x) (3.40) T e e 3.4.4 Invariance under rotation Let = be an arbitrary but constant rotation of the coordinate system. A A R proper collision is invariant under rotation:

[ [q]] = [q] ω( [q] [q ]) (3.41) R C R − R − R e [q ω(q q )] = [q] ω( [q] [q ]) (3.42) R − − e R − R − R e Since the quantity q might have directional properties (it could be a component of the velocity vector in a speciﬁc direction, for example), it is now given as a weighted sum of other quantities:

46 o igewih.S ehv o arbitrary for have we So weight. single a for nyo h oain oehrw get: we Together rotation. the on only to constant ation quantity a aesr htohrcliin onteffect not do collisions other that sure make nurltdcliinw nesadacliinta cso acts that collision quantity a understand we collision unrelated an u atcntan el ihivrac of invariance with deals constraint last Our collisions under Invariance 3.4.5 sam the share hence must and law same constant the by evolve must tities prtrwihi neae oteeouinof evolution the to unrelated is which operator hslosvr iia oteoiia equation original the to similar very looks This ic hshst etu o n obnto fwihsi has it weights of combination any for true be to has this Since easm umto vrindex over summation assume We ydfiiinw have: we definition By oee,i eea hr ol ea error an be could there general in However, q ω ⋆ . q fw suethat assume we If . i a etasomdt nte quantity another to transformed be can C ω ⋆ sntabtayayoe h esnt elandhr sif is here learned be to lesson The anymore. arbitrary not is [ q − ω w ( C i q C ⋆ − [ [ C C q R q i [ [ = ] e q q R ]= )] [ ]= ]] i C R C = ] C q ⋆ [ w [ ⋆ q [ q [ vle u oasnl olso ehv to have we collision single a to due evolves q e C [ i ]= ]] q q q = ] = ] [ i i = ] i q The . − − ]= ]] C C ⋆ ⋆ ω ω q [ [ q q ( ( + w w w q w C ] ] i − − w i i i q i E [ q C q − q q i i ei i ne nurltdcliin By collision. unrelated an under i ω ω ] : [ r osatwihsdepending weights constant are q q E − q ( ( i q ei C C Let . ] gi ehave: we Again . w ) ⋆ ⋆ u otecollision: the to due i [ [ q q q . disbeoperations Admissible 3.4 3.25 ei ] ] q C − C − ) yrtto ohquan- both rotation by A oee,terelax- the However, . ⋆ ⋆ nte observable another n = [ [ q q e e C ]) ]) ⋆ eacollision a be relaxation e ob true be to (3.47) (3.46) (3.45) (3.44) (3.43) (3.51) (3.49) (3.50) (3.48) 47

3 3 Minimal kinetic model

For the equilibrium state we make the following ansatz:

[q ]= q + C (3.52) C⋆ e e Together we have:

[q]= q + E ω(q + E q C) (3.53) C − − e − Since equation 3.25 is still valid we can subtract it to see:

C = E E/ω (3.54) − Hence we have:

[ [q]] = [q]= [q] ω(q q + E/ω) (3.55) C⋆ C C C⋆ − − e The lesson learned here is that if a collision which is unrelated to the C⋆ evolution equation for q adds an error E to q the error can be eliminated by adding E/ω to the equilibrium term q . − e 3.5 Elementary collisions in two dimensions

We apply the theory of the last section to derive concrete collision vectors for the two-dimensional lattice Boltzmann automaton. Let the momentum state of a node be determined by the vector s:

T s = r nw w sw s se e ne n (3.56) Here r denotes the occupation of the resting link while the other links are named after the cardinal direction they are pointing at counterclockwise from northwest to north. There are three conserved quantities for this lattice: The zeroth order and the two ﬁrst order moments.

3.5.1 Moments Let us have a closer look at the deﬁnition of moments. We deﬁne the function f˜(x,y) as:

f˜(x,y) = rδ(x)δ(y)+ nwδ(x + c)δ(y c)+ wδ(x + c)δ(y) (3.57) − +swδ(x + c)δ(y + c)+ sδ(x)δ(y + c)+ seδ(x c)δ(y + c) − +eδ(x c)δ(y)+ neδ(x c)δ(y c)+ nδ(x)δ(y c) − − − −

48 at htdpn on depend that parts n oain rnlto en htw d nostto offset an add we that means Translation rotation. and moments endt eteierlover inegral the be to defined ( ecnapytebnmnltermt pi h rnfre m transformed the split to theorem binominal the apply can We x eaepriual neetdi h eairo oet un moments m of of behavior terms the in in interested expressed particularly be are also We can quantities physical Other o h aeo lans ewrite: we clearness of sake the For h eohodrmmn a eitrrtda h asadthe and mass the as interpreted be can moment order zeroth The nodrt banadsrbto enormalize we distribution a obtain to order In With te oet r endas: defined are moments other + x 0 δ ) m µ en h ia et itiuin h eohodrmoment order zeroth The distribution. Delta Dirac the being x ( y and + µ ˜ x y m 0 µ y ) y n n r h w opnnso h eoiyvector: velocity the of components two the are = µ = = x x m Z 0 X

i =0 m −∞ y and x µ ∞ n m 0 m = i Z = !( f + ! −∞ y Z m x ( f ∞ 0 ˜ Z ,y x, X −∞ i m ( =1 m −∞ ∞ n napr htde o eedo them: on depend not does that part a in and ,y x, − ( − ∞ µ . lmnaycliin ntodimensions two in collisions Elementary 3.5 µ µ x i = ) Z x i y 0 m x i + )! Z ) −∞ !( 0 i −∞ ! ∞ m x x ∞ X = = = j f =0 m ˜ 0 n x ( ) − − f ,y x, m m ˜ ρ v v i j ( n i ( x y ,y x, y x !( )! y ! n 0 i y ) n f + /µ ! n ) − ( − dxdy ,y x,   y 0 j f 0 j ˜ y y ) ( )! 0 j n ,y x, n ) f dxdy + ( ,y x, ) X j =1 ohv nerlunity: integral have to n ) x dxdy j n !( ! and y n n e translation der − − oments. j y j y mnsinto oments . )! 0 j rtorder ﬁrst   (3.64) (3.63) (3.61) (3.62) (3.65) (3.66) (3.60) (3.58) (3.59) µ 0 49 is

3 3 Minimal kinetic model

m m!xm−ixi g (x ,m)= 0 (3.67) x 0 i!(m i)! i=1 − Xn n!yn−jyj g (y ,n)= 0 (3.68) y 0 j!(n j)! Xj=1 − With this we have:

m n m n m n (x + x0) (y + y0) = x y + x gy(y0,n)+ y gx(x0,m)+ gx(x0,m)gy(y0,n) (3.69) Note that gx(0,m) = gy(0,n) = 0. Now we can rewrite the transformed moments (we drop the integration limits):

m n µ˜xmyn = (x + gx(x0,m))(y + gy(y0,n))f(x,y)dxdy (3.70) ZZ = xmynf(x,y)dxdy+ (3.71) ZZ m n (x gy(y0,n)+ y gx(x0,m)+ gx(x0,m)gy(y0,n))f(x,y)dxdy ZZ = µxmyn + g(x0,y0,m,n) (3.72)

Note again that g(0, 0,m,n) = 0. In conclusion we see that the moments comply with the translational invariance constraint developed in the last section. The function g(x0,y0,m,n) has some interesting properties. It is a multi-linear function in moments of order lower than the order of µxmyn . By order we denote the sum of all exponents. So µxmyn has order m + n and all moments appearing in g(x0,y0,m,n) have lower order. Hence, an arbitrary weighted sum of moments of the same order also comply with the translational invariance constraint and we need a further constraint to determine which is admissible and which is not. In what follows we will use rotational invariance for that. We stick to the two-dimensional case for clearness. However, the methodology is general enough for any number of dimensions. Let us introduce a new rotated orthonormal coordinate system with coordinates ξ and ζ. The relation to the original coordinate system is given as:

50 h aeodr( order same the osrit moe ycnevdqatte.Frec norm each For vector a quantities. ident is freedo by conserved there start of by We imposed degrees constraints automaton. admissible rot Boltzmann derive and lattice translational to dimensional of moments theory of the apply evolution we following the In vectors Collision 3.5.2 condition. rhnraiyimplies: Orthonormality o h osre uniiste read: they quantities conserved the For ec,temoment the Hence, otemmnsi h e oriaesse are: system coordinate new the in moments the So µ M ξ M M m m ζ x M y ρ n + ⋆ = = = = ihteproperty: the with n Z Z .Ti locmle ihtetasainlinvariance translational the with complies also This ). µ ξ m 0 1 0 0 1 1 1 1 1 1 1 1 1 ζ ( n x ξ x x − sse ob u fmmnsin moments of sum a be to seen is x ζ ξ ξ ξ 1 x y + ρµ ξ ζ . lmnaycliin ntodimensions two in collisions Elementary 3.5 = = x x x y y + + − ξ 2 ξ ξ 2 ζ 2 2 ⋆ ξ − y + + + 1 + x = y x x ) ζ ξ 1 m x ξ ζ y y y y y s x x − ξ ζ ζ ζ 2 2 2 2 ζ ζ ( · x − + M 0 1 1 1 0 1 + ζ 1 1 = 1 = 1 = 1 = 0 = 0 = x y y ⋆ ξ ζ + y y − y 1 1 0 1 ζ y ) n f (

,y x, T ) dxdy

toa invariant ational T T lzdmoment alized o h two the for m figcollision ifying x and (3.77) (3.73) (3.74) (3.75) (3.85) (3.83) (3.84) (3.76) (3.82) (3.81) (3.78) (3.79) (3.80) y 51 of

3 3 Minimal kinetic model

Admissible collision vectors are orthogonal to Mρ, Mx, and My. For second order moments we check the rotational constraint:

µξζ = (xξx + yξ)(xζ x + yζy)f(x,y)dxdy (3.86) ZZ = x x (µ µ )+ µ (x y + x y ) (3.87) ξ ζ xx − yy xy ξ ζ ζ ξ

According to the theory on rotational invariance, the quantities µxy and µ µ must follow the same evolution equation since they can be trans- xx − yy formed into each other by rotation. The trace of the second order moments is isotropic as can be veriﬁed by direct computation:

2 2 µξξ + µζζ = ((xξx + yξ) + (xζ x + yζy) )f(x,y)dxdy (3.88)

ZZ 2 2 2 2 = µxx (xξ + xζ ) +µyy (yξ + yζ ) +2µxy (xξyξ + xζ yζ) (3.89) =1 =1 =0 Thus, we have identiﬁed| three{z quantities} | for{z second} order| moments{z of} which two must follow the same evolution equation. The following collision vectors are linearly independent of the vectors corresponding to all lower order moment and they are linearly independent of each other:

T K = 0 1 0 10 10 1 0 (3.90) xy − − T K = 001 0 1 01 0 1 (3.91) xx−yy − − T K = 4 2 1 2 1 2 1 2 1 (3.92) xx+yy − − − − − There are only two independent moments of order 3 and symmetr y constraints them to follow the same evolution equation. So we chose for the third order collision vectors:

T K = 0 1 0 1 2 1 0 1 2 (3.93) xxy − − − T K = 0 1 2 1 0 1 2 1 0 (3.94) xyy − − −

Only one degree of freedom is left for the fourth order moment µxxyy:

K = 4 1 2 1 2 1 2 1 2 (3.95) xxyy − − − −

52 h eei vlto qainrqie neulbimval equilibrium an requires equation evolution generic The Equilibria 3.5.3 iesoa case. dimensional tity rbblt itiuin utfactor: must distributions probability distribut velocity in equilibrium ity The homogenous. x be to posed drcini needn ftevlct in velocity the of independent is -direction iulrpeetto fteeeetr olsosi giv is collisions elementary the of representation visual A ute oet antb cone o.Tefia scatterin final The for. accounted be cannot moments Further nadto hr r ie oet.Temmnu usaei subspace momentum The moments. mixed are there addition In h aems etu naydirections: any in true be must same The K q = earayswteeulbimfrtesnl aibei th in variable single the for equilibrium the saw already We . x n in and - M ρ K M = y x drcinaetoidpnetrno aibe n their and variables random independent two are -direction               0 2 1 1 1 1 0 1 1 1 0 1 1 1 0 0 1 0 1 1 1 M y − − − 1 1 0 2 1 1 0 1 K xx − − − µ µ 1 1 0 2 1 0 2 1 1 + xx eq yy eq µ yy . lmnaycliin ntodimensions two in collisions Elementary 3.5 2 eq − − − − − 1 2 0 0 1 1 4 0 0 0 0 4 1 0 0 1 1 = = = K c xx − − s 2 c c − 0 2 0 1 0 1 + s 2 s 2 yy + + µ − − 1 2 µ µ 1 1 1 1 1 K y 2 x 2 y drcin ec,teveloc- the Hence, -direction. xy − − − 1 1 1 1 0 2 K − − − xxy 1 1 1 1 2 ue − − − − K ni figure in en q 2 2 2 2 xyy e o ucinin function ion               o ahquan- each for arxis: matrix g K xxyy (3.100) sup- s (3.97) (3.96) (3.99) (3.98) 3.1 one e . 53

3 3 Minimal kinetic model

Figure 3.1: The six elementary collisions of the D2Q9 cascaded lattice Boltz- mann automaton. From top left to bottom right: Kxx+yy, Kxx−yy, Kxy, Kxxy, Kxyy, and Kxxyy. The volume of the spheres indicate the amount of particles added or withdrawn.

54 qiiraare: equilibria hoeaseilcodnt ytm h qiiraaeinde choose are Apparent we equilibria The if system. only system. coordinate coordinate special any o a moments for influence choose order this not eliminate high system, could on We probl coordinate moments system. the order coordinate to low the adds on of This influence an system. exists coordinate These the subspace. of momentum independent in origin arbitrary an to relative ihtema ftedsrbto.Teeulbi o centra for equilibria The distribution. the of mean the with aldcnrlmmns(eoe by (denoted moments central called qiirafrcnevdqatte r h uniisthe quantities the are quantities conserved for Equilibria h qiiradpn o-ieryo h udvlct whi velocity fluid the on non-linearly depend equilibria The µ x eq m x y 0 n = µ µ = = = = xx eq xx eq − µ − + µ x Z Z Z Z µ µ µ xxyy eq x eq µ µ µ Z µ µ µ xxy eq xyy eq and m xy eq yy eq yy eq 0 eq x eq y eq µ x x x y eq m n y . lmnaycliin ntodimensions two in collisions Elementary 3.5 m m = = = ( = ( = = = 2 = ( = f 0 y y x κ eq = n n eas h oriaesse coincides system coordinate the because ) ( f f ρ v v v v x c c c x − x y x x c 2 eq eq ) s 2 s 2 s 2 s 2 v dx − µ ( ( y + + + + ,y x, x y v ) h orsodn oet are moments corresponding The . v v v v f y 2 x y x 2 2 2 x 2 y ) eq ) )( ) dxdy + v v Z ( c x y y s 2 v ) y y 2 + dxdy n f v y eq y 2 ) ( y ) dy edn fvelocity of pendent oet are: moments l sle.Alnine All mselves. qiiraaenot are equilibria l o special a for nly hi measured is ch y ene to need we ly, mta there that em depending (3.107) (3.101) (3.102) (3.103) (3.104) (3.105) (3.106) (3.109) (3.113) (3.112) (3.108) (3.110) (3.111) 55

3 3 Minimal kinetic model

eq κ0 = ρ (3.114) eq κx = 0 (3.115) eq κy = 0 (3.116) eq eq 2 κxx + κyy = 2cs (3.117) eq eq κxx κyy = 0 (3.118) − eq κxy = 0 (3.119) eq κxxy = 0 (3.120) eq κxyy = 0 (3.121) eq 4 κxxyy = cs (3.122)

With the usage of central moments we introduce an apparent contradiction to the requirement that all operations must be independent of the choice of the coordinate system. However, we are forced to choose a special origin for the coordinate system. The apparent contradiction is resolved by realizing that the mean of the momentum distribution function is the only possible choice for an origin. For example, zero velocity relative to the resting lattice is an arbitrary choice for an origin, at least from the physical point of view. Choosing the center of mass (mean) of the colliding particles is not an arbitrary choice. In fact, the mean is the only characteristic point of the system. Independence of the coordinate system is formally obtained as follows: The mean is determined from an arbitrary coordinate system. The coordinate system is then moved to the mean where the collisions are performed. The mean can be found from any coordinate system and is always the same point. Evolution equations which include a transformation to a characteristic coordinate system are independent of the original coordinate system and are therefore admissible physical laws with respect to translational invariance. We could further argue that choosing any other point than the mean as an origin would corrupt isotropy because any other point would be in a certain direction from the mean. However, there is no special direction in space and the only point which is not in any special direction from the only characteristic point of the collision is the characteristic point itself.

3.6 The collision cascade

In this section the actual algorithm for the collision is derived.

56 r oet rsm fmmnso h eoiydsrbto f distribution order velocity of the moments of system, moments coordinate of arbitrary sums or moments are fodrlre than larger order of and re yodr esatwt h eododrmmnsadsol and moments order second re the we with Instead start We for equations. order. of equations by set order derive a can solving as we straightforward equilibria their to and re mle than smaller order a ocoeaseilcodnt ytmwt h ena nor an as mean choice. the admissible with only system the coordinate is special this a chose to had otedsrtns ftelattice: the s of integral discreteness The the function. to distribution collision post the of tutrdfsin re yodr h asadvlct m velocity and mass introduced. unl The is equation gravity evolution like no order. perform require to by and m quantities natural order conserved order hence fashion, low is effect structured it not a crosstalk does this moments for order compensate high of evolution the eswta h uniisfrwihw edt rt evolutio write to need we which for quantities the that saw We nodrt eieteeouineutosw opt h cen the compute we equations evolution the derive to order In Here h igend olso prto ih ewitnas: written be might operation collision node single The h vlto flwodrmmnseet ihrodrmomen order higher effects moments order low of evolution The with k κ y = drcinrsetvl.B etn h eutn central resulting the setting By respectively. -direction yy eq c = ix k κ κ ρ x post and yy post m y k n x for c = iy k m X k y r h eoiycmoet fprilso link on particles of components velocity the are m i xx eq safnto fteoii ftecodnt ytm We system. coordinate the of origin the of function a as oee,mmnso order of moments However, . ( + k s xx yy i ( + + and yy K · k k s k xx eq xx ) ← k k k i − − x y ρ )( yy yy s c ix : + 0 = 0 = 0 = − k K xy v · x k ) m k xxy ( m c k iy ontdpn nmoments on depend not do oee,ti sntas not is this However, . − k m . h olso cascade collision The 3.6 xyy v eedo oet of moments on depend y mlfist u due sum a to implifies ) n /ρ k s xenlforcing external ess adtemoments the gard xxyy h olsosin collisions the oet equal moments nto.I an In unction. ve rlmoments tral

gnbecause igin mns To oments. equations n T mnsare oments κ xx eq swhile ts = (3.127) (3.123) (3.128) (3.125) (3.126) (3.124) i in κ xx post 57 x -

3 3 Minimal kinetic model

keq = (ρ(v2 + v2) e n s w (3.129) xx+yy x y − − − − 2(se + sw + ne + nw ρ/3))/12 − − keq = (n + s e w + ρ(v2 v2))/4 (3.130) xx−yy − − x − y These are the values to drive the corresponding central moments into equilibrium. The distance from the incoming state to equilibrium represents the potential that has to be multiplied with a relaxation parameter ω⋆. In addition, an optional truncation operation is introduced that allows us to ⌊ · ⌋ perform the collision in integer space and thereby avoiding round-oﬀ errors in the conserved quantities:

k = ω (ρ(v2 + v2) e n s w (3.131) xx+yy ⌊ xx+yy x y − − − − 2(se + sw + ne + nw ρ/3))/12 − − ⌋ k = ω (n + s e w + ρ(v2 v2))/4 (3.132) xx−yy ⌊ xx−yy − − x − y ⌋ Similarly we solve:

κeq = κpost = (s + (K k) )(c v )(c v )/ρ (3.133) xy xy i · i ix − x iy − y Xi Adding a relaxation constant and the truncation operation yields:

k = ω ((ne + sw nw se) v v ρ)/4 (3.134) xy ⌊ xy − − − x y ⌋ The next central moments are third order moments and they are not independent of the lower order central moments. That is to say the k⋆ which eq post were already ﬁxed appear in the equations to come. From κxxy = κxxy and eq post κxyy = κxyy we obtain:

keq = ( ((se + sw ne nw 2v2v ρ + v (ρ n s r))/4 (3.135) xxy − − − − x y y − − − +v /2(ne nw se + sw))) v /2( 3k k ) 2v k x − − − y − xx+yy − xx−yy − x xy keq = ( ((sw + nw se ne 2v2v ρ + v (ρ w e r))/4 (3.136) xyy − − − − y x x − − − +v /2(ne + sw se nw))) v /2( 3k + k ) 2v k y − − − x − xx+yy xx−yy − y xy As shown by equation 3.55 the terms introduced by unrelated collisions must not be multiplied with the relaxation constant:

58 rmﬁs rnilsi hscatr h eiaino h t the of derivation The chapter. two-dimension this in in principles automaton ﬁrst Boltzmann from lattice cascaded The Endnotes 3.7 h otcliin tt ste bandfo equation from obtained then is state collisions post The ytesm ruetw banfrom obtain we argument same the By k k xxy xyy + + = = k v v ⌊ ⌊ xxyy y x ω ω / / xyy xxy 2( 2( nw + − − − 2( + 4 + = sw ne ne ne ( ( se w − − n r e s v (3 2 v ⌊ + (( − (( y x k v 2 2 ω ← ← ← ← ← ← ← ← ← / v (3 ( x xx xxyy sw se n x 2 sw nw v v ne e n s se w sw nw r ( k + y + ne x 2 + xx ( yy − − 4( + + − nw − ρ − (1 − 2 + 2 + ne + 2 + sw 2 + k k − − − nw k k / se yy se xx xx xx 4( xx − − + k nw e 2 k k − k + + − − xx + v xx + ρ/ − + k − xx ne xx nw yy yy x yy xx yy ne + k nw + k + sw + 9 se ne + yy xx xyy + − − yy + − yy − yy − + se − yy ))) − ))) − − + 2 2 se + 2 + 2 ne yy + s − nw k k k − + ne k nw k xxyy xxyy − k + − xxyy − / k xxyy k 2 xxyy − xxyy sw k 2)( xxyy xxyy v − − v se v xxyy sw y − x y nw + ) k + − v 2 2 / / − κ + + ) xxy + x se 2 v v 2( 2( − − xxyy eq + ) k k y 2 x 2 k − + k sw − k − v v xx xx v k xx − k xx k y xy x 4 + y xy se v 3 3 xy xy − − ( ρ ρ ) − sw − y 2 k k = ne yy yy + yy ) + + − xx xx yy − + − v ⌋ κ x 2 + − − + k + + 2 + v v k sw − k xxyy post v k xxy x y yy yy xxy xxy y xxy 2 w nw ( ( 2 k ρ ρ k k k ))) + − k xy xyy xxy : − xxy − − − xyy + + − k k k n k w xx xx 3.123 k k se xyy xyy xyy xyy − − − − hree-dimensional − yy yy s e a derived was s rexplicitly: or , ) ) − sw . Endnotes 3.7 − − − r r )) )) 2 2 )) v v / / y x 4 (3.147) (3.137) (3.139) (3.148) (3.138) 4 (3.146) (3.145) (3.144) (3.143) (3.142) (3.141) (3.140) k k xy xy ⌋ ⌋ 59

3 3 Minimal kinetic model cascaded lattice Boltzmann automaton is conceptually identical to what is shown here but not as concise due to the requirement of 27 links and 25 elementary collisions plus equilibria. The results are presented in an appendix. In this chapter we deliberately avoided the usage of the Navier Stokes equation, the Boltzmann transport equation, and concepts like entropy. The result of the presented methodology is not only a new kind of lattice Boltzmann automaton with improved stability and accuracy properties but also the knowledge that the cascaded lattice Boltzmann automaton represents a model for hydrodynamic fluid flow in its own right. No step in the derivation of the cascaded lattice Boltzmann automaton requires the Navier Stokes equation as an justification. The cascaded lattice Boltzmann automaton can hence be seen as a completely independent model which is supposed to be equivalent to the Navier Stokes equation by virtue of describing the same phenomena. The qualitative difference from a cellular automaton that was derived in order to solve a specific equation is that each discrepancy between the cellular automaton and the modeled equations has to be considered an error of the automaton in the later case. If an equation and a cellular automaton describe the same phenomenon by different means and do not depend on each other, the discrepancies between their results could be an error of the equation as well as one of the cellular automaton. The tenet that a discretized model is necessarily inferior to the differential equation does not hold in this case. We introduced optional truncation operations for the k⋆ in order to perform the scattering with integers and thereby guarantee exact conservation of the chosen invariants. From the theoretical point of view, this is an important property of the cascade lattice Boltzmann automaton. It might, however, not be very useful for its actual application due to the simple fact that common hardware and compilers are designed for high performance floating point computation. A floating point implementation of the cascaded lattice Boltzmann automaton can perform superior to the pure integer version shown here. Hence it is stressed that the truncation is optional. Finally, it has to be mentioned that the usage of higher than second order terms in velocity for the equilibria provoked several objections from specialists in the field. It has been argued [26] that the given lattice is insufficient for a complete cancelation of third order errors. That is true but it concerns only longitudinal terms of third order. The result is that the cascaded lattice Boltz- mann automaton with the D2Q9 lattice is not stable for small bulk viscosities. That is to say, the cascaded lattice Boltzmann model derived here should not be applied to simulate flow with shocks or strong acoustical components. Oth- ers [27, 28] argued that the polynomial expansion of the Maxwell-Boltzmann

60 ie re.Tefut re em o h 29ltiewoul lattice D2Q9 the includ for terms typically order in would fourth di expansion The The polynomial order. order. given A second of n crucial. equilibria did but we dimensional since one distributio here of Maxwell-Boltzmann presented cation the model from the terms to order resul apply higher That not number conserving. does Mach momentum it and in but mass order not are third which than equilibria higher to distribution per nteeulbimaoei h n oigfo h mu the order: from second coming of one equilibria the one-dimensional two is above the equilibrium of the in appears nldd u ewr ee epe oicueohrtrs T solution. terms. correct other four the include of to to naturally terms us tempted other led never here if were described derived we But be could included. equilibrium stable no osrittececet ob constants. be to coefficients the constraint e sasm htteewsabte qiiru ucinth function equilibrium better a was with there that assume us Let qiiru ucin hr shnen on nuigequil using in point with no polynomials identi hence the be is other would There polynomial the function. algorithm, equilibrium the of purpose the ttsprnd.Terao stefloig h oetmst momentum o D The number A following: the space. the momentum as is in velocity points reason discrete in The at coefficients node. of per number states same the with a dnia otebte qiiru ucina the at function equilibrium better the to identical was ngnrl ecnsyta h qiiru ucinms ea be must function equilibrium the that say can we general, In v x 4 , q v offiins ecudte n oyoilwith polynomial a find then could We coefficients. x 3 v y , v x 2 v y 2 , v x v y 3 and , v q y 4 offiinsi eoiy oeta hsde not does this that Note velocity. in coefficients oee,teol emo orhodrwhich order fourth of term only the However, . d Q q atc has lattice q v eeatpit.For points. relevant x 2 q v u ymultipli- by but n q y 2 offiinswhich coefficients eec ssubtle is fference ti rethat true is It . bimfunctions ibrium a otebetter the to cal napolynomial a an l em fa of terms all e ttsprnode. per states nld terms include d si unstable in ts hodrwere order th ih etrue be might . Endnotes 3.7 tdrv the derive ot t sgiven is ate polynomial ltiplication emethod he velocity f 61

3 62 eitosfo qiiru hra h eta oet me as moments directly. expansion assumption central Enskog this the Chapman on whereas me relay the r moments equilibrium that at central from is is with deviations thesis) fluid the this and the in expansion shown if Boltzm Enskog true lattice the Chapman only of the derivation is the that of However, difference distrib principle equilibrium versions. resting same MRT the and and sound of speed same lsr rbe ntrso eouin hti osy h r the say, to is That resolution. of pi terms Stokes in Navier problem (the closure equilibrium a the from deviations small rpriso h acddlattice cascaded the of Properties 4 nhmgnossaedsrbto nvlct pc implie space D3Q1 velocity and in D3Q15, need distribution D3Q13, state D2Q7, homogeneous D2Q6, inhomogen An the with as lattices M such use SRT, distributions to for strin more forbids used invariance our also However, Galilean is automata. which automat Boltzmann lattice Boltzmann lattice D2Q9 tropic lattice the obtained original chapter we the last case the with in properties derived models sic automaton Boltzmann Boltzmann lattice lattice The other to differences Basic 4.1 W model. numeri SRT and the theoretically t to aliasing of them and properties viscosity compare numerical numerical and the automaton investigate Boltzmann we tice chapter this In itnusal ttsi igedimension. single a in states distinguishable h lsr rbe a efruae namliueo ways. of multitude a in formulated be can problem closure The h acd eso fteDQ atc otmn automaton Boltzmann lattice D2Q9 the of version cascade The otmn automaton Boltzmann q ublnei h otipratusle rbe fclassi Feynman of –Richard problem unsolved important most physics. the is Turbulence D peswhere speeds D stenme fdmnin and dimensions of number the is n uoao with automaton ann cally. osvlct state velocity eous q htw always we that s etdfiiinof definition gent to steSRT the as ution tr)w obtain we cture) n o h 2D the For on. stenme of number the is ecsae lat- cascaded he hdde not does thod hrsteba- the shares yassuming By ue small sumes slto of esolution Taden- and RT lattices. 9 s.The est. hd(as thod cal a the has study e 63

4 4 Properties of the cascaded lattice Boltzmann automaton the grid has to be so high that each grid point is in near equilibrium with its neighboring grid points. We need to resolve the Kolmogorov scale. The central moments approach reformulates the problem. We do not assume closeness to a local equilibrium and do not relay on the same assumptions (and limitations) as the Navier Stokes equation. However, using Galilean invariance directly results in an infinite number of constraints. That is just another kind of closure problem, although one that does not require to resolve the Kolomgorov scale. But did we win anything after all? In essence, we need to solve the closure problem by some kind of truncation. The Navier Stokes / Chapman Enskog picture requires a truncation in physical space while the central moments ansatz requires a truncation in moment space. Both truncations are invalid due to the fact that the evolution of the fluid really depends on the states of all molecules in the system and is sensitive to the smallest of perturbations. However, gross statistical properties could be relatively independent of small perturbations. The central moments ansatz is the more statistically one. In both cases it is not possible to simply ignore the closure problem. Instead we have to make sure that there is little ”energy” left in the last resolved scales or moments. In the Navier Stokes / Chapman Enskog picture this would mean that high frequencies in the flow field have to be filtered out. This would actually require a spectral representation of this flow field. Whether spectral filtering can be applied easily or not depends on the size and and the complexity of the simulation domain. An alternative solution is to use filters with finite support. The central moments ansatz, on the other hand, is intrinsically local. Filtering in moment space means that moments which are higher than the hydrodynamic moments relax so fast that their influence is small and they can be ignored. Fast relaxation of high order moments is considerably simpler than spatial filtering. The closure problem is solved by choosing appropriate relaxation constants.

4.1.1 Transport coefficients The only transport coefficient we are actually interested in is the kinematic shear viscosity ν. It can be obtained from the speed of sound and the mean free flight time between two binary collisions:

2 ν = csτbinary (4.1)

Binary collisions are those related to the central moment κxy and the corresponding central moments obtained from the rotational invariance constraint. The relation for the mean free time depending on the relaxation constant ωxy

64 3.8 yloiga h cteigagrtm ftesadr n t and standard fact: prominent the two of recognize algorithms we scattering automaton Boltzmann the at looking By differences Algorithmic 4.1.2 justificatio physical no has t and stressed grounds be numerical should on It made is chapter. reaso for this moments throughout resoled highest understand to order the In equilibrate to model. have minimal we a by provided m order be higher cannot match quantities stat to flexibility the the by However, imposed turbulence. difficulties the considering interesting a bandi equation in obtained was sisl fscn re.TeCamnEso xaso sig is eq expansion Enskog Stokes Chapman Navier The the terms. to order order. higher link second formal of a itself is provides term order higher uses second model new at the while velocity in order second o fipeetdwt otn on rtmtcadtecas the e and round-off with arithmetic implemented arithmetic. Bolt point if lattice automaton floating cascade Boltzmann with lattice the implemented called if therefore ton is distinguish prominent which an most model before the new is recognized cascade been scattering never the the of of has best relationship depen the To moment this conserved. any edge, not of or equilibrium conserved The moments, order mome equilibria. other of of take cascade equilibria to non compute a need to that We account assume into state. not states equilibrium close sion do their a we to is method close are new This moments the moments. In other again. of and assumption, state alone off-equilibrium quantities conserved the the on from computed are tomata idn eun ausfrteohrrlxto osat w constants relaxation other the for values genuine Finding is,tesadr atc otmn uoao a oterm no has automaton Boltzmann lattice standard the First, h eodfc sta h qiirafrtesadr latti standard the for equilibria the that is fact second The obe to c s 1 = / √ 3 ec,w banteuulrlto o viscosity: for relation usual the obtain we Hence, . . ai ieecst te atc otmn models Boltzmann lattice other to differences Basic 4.1 3.21 h pe fsudcnb xdfo equation from fixed be can sound of speed The . ν = 3 1 ω 1 xy − 2 1 sia rprisof properties istical n rpryo the of property ing mnst physical to oments swihw r to try we which ns t.Ti requires This nts. eBlzanau- Boltzmann ce .Tetruncation The s. h application the d a hsdecision this hat mn automa- zmann ls h system the close rrfe integer free rror n. hi otcolli- post their -to-equilibrium uhr knowl- authors so l lower all on ds ontdepend not do enwlattice new he oato any of norant udb most be ould ihrthan higher s ae digital caded ainwhich uation -conserved (4.2) 65

4 4 Properties of the cascaded lattice Boltzmann automaton

4.2 Eﬀects of high order corrections on stability

Before we leave our theoretical analysis of the lattice Boltzmann automaton and show computational evidence of the superiority of the cascaded variant we shall develop a more hands-on understanding of the instabilities which plague the original lattice Boltzmann automaton.

4.2.1 A thought experiment

Let us start this section with a discussion on how viscosity is implemented in lattice Boltzmann automata, since this is crucial and turns out to be a bit counterintuitive. The evolution of the lattice Boltzmann automaton can be split up into a scattering step and a streaming step. Since we associate viscosity with dissipation and since we adjust viscosity with a relaxation scheme in the scattering step we could wrongfully assume that viscosity originates from scattering. However, this is not so and can easily be seen by remembering that viscosity would be maximal if we skipped scattering altogether1:

ω 0 ν (4.3) → ⇒ →∞ So where does viscosity come from? It is a result of streaming. Momentum diffuses by streaming of particles without scattering. For an ideal gas viscosity is proportional to the mean free flight path. The fewer collisions, the higher viscosity. Scattering is basically the lattice Boltzmann variant of an anti-diffusion algorithm. For zero diffusion we choose the relaxation constant ω = 2. That means that the non-equilibrium part of the node distribution function is inverted. The non-equilibrium part of the distribution corresponds to a gradient of density or velocity in the fluid. Inverting this gradient means returning the mass or momentum to where it comes from under the constrained of Galilean invariance, as Galilean invariance is understood in the Chapman Enskog picture for the standard lattice Boltzmann automaton. The understanding of Galilean invariance in the Chapman Enskog picture turns out to be the crucial point here. The Navier Stokes equation is typically derived for the lattice Boltzmann automaton by the Chapman Enskog

1 Unfortunately this does not work for producing solid bodies in lattice Boltzmann simulations since the Knudsen number grows with decreasing ω and the hydrodynamic theory holds only for small Knudsen numbers.

66 v orcasso oet o hsmdl hs hc r conse can are We which Those freedom. unique of model: this is degrees for nine it moments only but of classes has moments, four it of since number moments infinite nine an course, of has, e,teeulbi ftidodrmmnsdpn nyo firs on only depend moments order third considere of ( only equilibria are the invariance with repre same Galilean moment der, the the the use saying for to is beneficial corrections that is However, it understanding basis. our link For per u a automaton on stand equilibri Boltzmann fined The the lattice SRT to Our ”equilibria”. stick explicitly. attractors provided We these call moments. and adjustable ture all for attractors ain ( variant fmmnswihaentidpnetyajsal.W shall We adjustable. independently not ments are which moments of eoiymoments velocity o hsnt eGlla nain ( invariant Galilean be to chosen not imvleo h oetadisrlxto constant. relaxation its st and pre-com moment the the the For of to value velocity. corresponds rium of ”behavior” ”beha th independent automaton the of is that Boltzmann mean moments means the invariance central by Galilean g order displaced order the moment Second on a particles tion. moment, the central of velocity called mean so to momentu quant For respect integral function. with be distribution moment particle to the understood of ”proper are moments its Properties if invariant velocity. Galilean of be to said is distribution iuain.W tr ihavr ipeeape osdrj Consider example. simple very a with start We simulations. nyo as( mass on only and o.I Dw aetreidpnetscn re eta mom central order second independent three have we 2D In ton. esoda eododrGlla naine l ihro higher pi All irrelevant. Enskog invariance. be Galilean to Chapman order considered the second Gal in a equation invariance as Stokes derstood Galilean Navier that dist lattice that means equilibrium out order the This turns second to It this applied make number. be to Mach to have in modifications order cial second to up expansion µ y ,toewihaeidpnetyajsal n hsnt be to chosen and adjustable independently are which those ), xxy eq o e sivsiaewehrti sago oe o turbul for model good a is this whether investigate us let Now o oceees e scnie u 29STltieBoltz lattice SRT D2Q9 our consider us let concreteness, For e snwudrtn htscn re aieninvariance Galilean order second what understand now us Let κ xy = µ κ = ρv n eta moments central and xx y κ , / yx 3 κ , hs r hsnt eGlla nain.Tedistributio The invariant. Galilean be to chosen are Those . µ yy µ xxyy eq and , xyy eq µ x = and = κ ρ/ xy ρv 9 µ ,toewihaeidpnetyajsal u are but adjustable independently are which those ), x ). y / h orhodrmmn sspoe odepend to supposed is moment order fourth The . 3 . ffcso ihodrcretoso stability on corrections order high of Effects 4.2 n hsol eas hyaerltdt the to related are they because only this and ) κ h eaaintm prxmto requires approximation time relaxation The . µ xxy , µ xyy and , µ xxyy is r independent are ties” r o necessarily not are a ,adteifiieset infinite the and ), pt eodor- second to up d drmmnsare moments rder te,frexample for ities, itiuin a distributions m iuini order in ribution e qiirade- equilibria ses etto.Since sentation. vd( rved or fsecond of voir” la invariant. ilean anautoma- mann r nomencla- ard s n ieof line one ust aerwmo- raw name nadlattice andard ents: ue equilib- puted vnnd sa is node iven re terms order t ydfie by defined ly n udflow fluid ent te words. other aienin- Galilean tr sun- is cture distinguish distribu- e en.A means. oespe- some ρ κ , xx v x , and , κ yy 67 n ,

4 4 Properties of the cascaded lattice Boltzmann automaton nodes in x-direction with periodic boundary conditions in y-direction so that the northern neighbor of the node is the node itself. As initial conditions we choose a singular velocity pulse in y-direction at a single node. All other velocity components, especially those in x-direction, are zero. The relaxation rate ω is chosen close to two in order to achieve a high Reynolds number. Let us start with a scattering step. Scattering redistributes momentum on the single node. The result is symmetrical since the initial condition was symmetrical. After scattering, part of the momentum is stored in the central link (the difference between the occupation of the north and the occupation of the south channel) and part of it is stored in the diagonal links (northwest minus southwest and northeast minus southeast). Now comes the streaming step. The y-momentum stored in the diagonal links moves to the neighboring nodes. Streaming is where diffusion of momentum takes place. After streaming, the distribution of momentum in spatial space has increased. In the subsequent scattering step the non-equilibrium part of the distribution on the left and right neighboring nodes are inverted in order to return most of the momentum to where it came from. We stop the thought experiment at this stage to investigate a simulation which is just a little bit more complex. Again we start with a single pulse in y-direction on one line of nodes. This time we add an homogeneous velocity field in x-direction in order to see the difference to the two time step simulation we have just investigated. In the first scattering step we recognize a contribution from the Galilean correction in the second order central moment κxy which is triggered through the concurrence of velocity components in x- and y-direction on a single node. The effect of this correction is a rearrangement of particles in momentum space as to move the mean of the velocity pulse with the flow. In the previous simulation the mean of the y-velocity pulse stayed where it was. Now it is moving. Interestingly, the second order correction influences only the links with contributions in x and y-direction simultaneously. That is to say, momentum is redistributed from northwest-southwest to northeast-southeast. The amount of momentum on the central axis (north-south) is not affected. This is so because we said that the variance of y-momentum in x-direction, namely the moment µxxy, is independent of velocity vx. If we now look at the distribution of y-momentum over the three nodes after the streaming step we recognize the following: The momentum on the central node is unchanged as compared with the simulation without advection in x-direction. The mean of the velocity distribution has been shifted Galilean invariantly in accordance with the second order correction from κxy. This was achieved by lowering the amount of momentum on one side of the central node and increasing it on the other side of the central

68 opstv haigin shearing positive to ed eel h eedneo icst nvlct.I fig In Superimposing velocity. on wave. viscosity shear of dependence sinusoidal the a reveals of fields decay the ulating mle nerrof error an implies oei codnewt h eurmn httevrac of variance the that requirement the with accordance in node sicesn tcno rpblwzero. below drop cannot one it au qualitative a increasing Boltzmann also is but lattice difference quantitative cascaded a the only th not on equilibrated increasing of case slightly latt the cascaded is (in the viscosity that of Bolt Note viscosity lattice stable tomaton. SRT the the with for comparison viscosity in in drop cubic anticipated velocity. with au change Boltzmann not lattice does cascaded variance the the with simulated setup same moment order advectio third to the subject that is is flow t this anti-diffusio the the for Boltzmann when in lattice anti-diffusion coefficients The more wit diffusion duces distribution the equal. that a not obvious of are r quit variance examples Now is of It d rate advection). the growth mean. (with of the case mean is second the diffusion the th to be in respect should tha smaller with distribution assume variance obviously the should the of We However, mean method. the cases. the to in respect flaw with a ance obviously is here But x u acddltieBlzanatmtn(index automaton Boltzmann lattice cascaded our emdpnigon depending term atydffrn nteteCamnEso ae uoao (i automaton based Enskog Chapman the the in different cantly namr udmna ento fGlla naine Fig and invariance. expansion Galilean Enskog of Chapman definition the fundamental cascad more on The a rely moments. on not the order does higher after method of truncated canno mann effects is This ill it all invariant. since obscuring Galilean expansion Enskog be Chapman to the chosen than rather grid drcinwt epc otecnrlnd skp constant kept is node central the to respect with -direction h qiiru for equilibrium The h aineo the of variance The O v x 2 ( µ v v µ xxy CLB y µ x 2 y y xxy CE mmnu in -momentum ) h qiiru o h moment the for equilibrium The . xxy CE drcin fequation If -direction. nvsoiy h icst a emaue ysim- by measured be can viscosity The viscosity. in sse ob ytmtclytolwwt respect with low too systematically be to seen is = = . ffcso ihodrcretoso stability on corrections order high of Effects 4.2 v ρ 3 y v |{z} ≈ y κ ρ/ xx 3 x +2 µ drcini oial hr order third a nominally is -direction xxy v x |{z} κ sajse ihrsett the to respect with adjusted is ≈ xy 0 4.5 + CLB v srgtte equation then right is x 2 v y ): ic h viscosity the Since . r re moments) order ird c otmn au- Boltzmann ice automaton zmann ure µ oao.Ti is This tomaton. ure loih pro- algorithm n xxy dltieBoltz- lattice ed y epc oits to respect h sefigure (see oao.Here, tomaton. ese from seen be t aei both in same e mmnu in -momentum .Tereason The n. ndex mme that emember oconsidered wo 4.3 eodorder, second ieetflow different 4.2 srbto is istribution ok signifi- looks esethe see we h vari- the t hw the shows CE sbased is and ) (4.5) (4.4) 4.1 4.4 69 ).

4 4 Properties of the cascaded lattice Boltzmann automaton

0.1 vx=0.0 vx=0.1

0.08

0.06

vy 0.04

0.02

-0.02 0 2 4 6 8 10 12 14 x Figure 4.1: Transport of a singular velocity peak in y-direction on the D2Q9 SRT lattice Boltzmann automaton with a periodic domain of 15 1 × nodes and ω = 2 for shear velocity vx = 0 and vx = 0.1, respectively. Shown is the state after two time steps. The type of Galilean invariance which comes from the Chapman Enskog expansion effects only the mean of the distribution. The mean moves with the flow by setting the speed on neighboring nodes to opposing values. The original velocity peak is still located at its origin and does not move with the flow. Diffusion is growth of variance in time. The variances in the two plots are different and so is their diffusion coefficient. The diffusion coefficient for the plot with vx = 0.1 is in fact negative and the corresponding simulation represents the simplest example of an unstable setup for the conventional lattice Boltzmann method.

70 iue42 h aesmltosa nﬁgure in as simulations same The 4.2: Figure

vy -0.02 0.02 0.04 0.06 0.08 0.1 0 atc otmn ehdwith method Boltzmann lattice enmv ihteflwbttevrac sas etconstant. kept also is variance stable. does the is only but setup Not simulation flow This the invariance. meth with Galilean moments move of central mean orders The higher unity. obtains being constants relaxation 0 vx=0.0 vx=0.1 2 4 . ffcso ihodrcretoso stability on corrections order high of Effects 4.2 6 x ω xy 8 4.1 = eetdwt h cascaded the with repeated ω xx 10 − yy 2 = 12 n h other the and 14 the od 71

4 4 Properties of the cascaded lattice Boltzmann automaton

We should be aware of the fact that the Galilean invariance error in viscosity has nothing to do with the single relaxation time approximation. It comes from the Chapman Enskog procedure and is build into the equilibrium function. It is not possible to get ride of this problem by just following a diﬀerent path to the same equilibrium. With this lesson learnt we can get a little bit more abstract and ask the general question what makes lattice Boltzmann simulation unstable? We could say that all instabilities go through or result in a violation of the CFL condition. That means the following: Information is allowed to travel one lattice spacing per time step in a lattice Boltzmann simulation. The limiting factor turns out the be the speed of sound. Sound waves must be allowed to travel on the lattice. Since they are supposed to be Galilean invariant, the speed of sound added to the velocity on the node must be smaller than one lattice spacing per time step.

v = 1 c (4.6) CFL − s

If v > vCFL no physical meaningful result can be obtained from the simulation. Violation of the CFL condition might have two reasons. Firstly and trivial, the initial or boundary conditions might be chosen in a way that results in flow conditions which are too fast for the lattice Boltzmann automaton. This can be corrected by lowering the differential input pressure or the input velocity and the viscosity simultaneously in order to obtain the same Reynolds number. Physically this translates to shorter time steps. However, lowering viscosity does not work ad infinitum. The problem with low viscosity can be understood from the example given above. Momentum transport is hardwired into the streaming step. The lattice Boltzmann automaton applies an anti- diffusion algorithm to undo the transport of momentum. This works quit well as long as the errors in viscosity are small compared to the nominal viscosity. The anti-diffusion algorithm is an antagonist to the transport algorithm (streaming). That is also true for the errors. In an anti-diffuison algorithm, advection tents to introduce additional anti-diffusion. It is, however, necessary that transport out-weights anti-diffusion. Negative viscosity leads, perfectly in line with mathematics and physics, to a growth of velocity at points with high velocity at the cost of points with lower velocity (or velocity in the opposite direction). This is not a ”numerical” instability due to finite floating point precision. It is a perfectly correct behavior of a fluid with negative viscosity (irrespective of the fact that negative viscosity does not exist in nature since it was not in accordance with the second law of thermodynamics).

72 iue43 oprsno h eoiydpnec fviscos of dependence velocity the of Comparison 4.3: Figure

nu 0.00105 0.00065 0.00075 0.00085 0.00095 0.0011 0.0007 0.0008 0.0009 0.001 akdCB a bandwt vrrlxto ftidorde third of over-relaxation with ( obtained moments was CLB* marked is rmvlct.Tepo akdwt L a bandwt equ ( with moments obtained order was third CLB of with libartion marked plot The velocity. from hra h eaieerro h R ehdi destabilizin is method sta SRT hence the of and error (dissipative) negative positive the resultin always whereas The is fast. viscosity equilibrate pr to in closure moments The unfortuna higher errors. comes, the aliasing This quires destabilizing of further. cost the results mom at the order improve third to of seems Over-relaxation sensitiv fields. less flow considerably superimposed is automaton Boltzmann lattice o ed r ueipsdin superimposed are fields flow h aevco spitn in pointing is vector wave the eidcdmi.Tevlct etrpit in points vector spaci velocity lattice The of 30 domain. decay of wavelength periodic the a a measuring with wave from shear obtained sinusoidal was a Boltz data lattice The cascaded the automaton. and automaton Boltzmann lattice 0 ν CLB* 0 = SRT CLB . 001 0.05 ω xxy a rirr au)adspoe ob independent be to supposed and value) arbitrary (an = . ffcso ihodrcretoso stability on corrections order high of Effects 4.2 0.1 ω xyy 1 = 0.15 . 9999 vx x x drcin h oia viscosity nominal The -direction. .Tevsoiyo h cascaded the of viscosity The ). drcin eea homogenous Several -direction. ω 0.2 xxy = ω 0.25 xyy 1 = y hl h plot the while ) drcinwhile -direction t nteSRT the on ity 0.3 be re- oblem othe to e bilizing error g 0.35 g in ngs mann g. tely, ents 73 i- r

4 4 Properties of the cascaded lattice Boltzmann automaton

With what we know now we can draw up the following scenario for instabilities in standard lattice Boltzmann simulations at high Reynolds numbers: We have chosen the viscosity to be small and velocity to be high in order to obtain a high Reynolds number. The simulation runs for some time without noticeable problems and we see formation of vortex structures. As the vortex structures become more and more convoluted, diffusion of momentum is more and more affected by the flow speed in different directions. We saw that the anti-diffusion becomes stronger in regions were velocity is higher. Viscosity must hence drop in this regions. This increases the Reynolds number in those regions further. The more turbulent the flow gets the more the lattice Boltz- mann automaton has the tendency to lower viscosity. At some point viscosity becomes negative, resulting in an exponential growth of the shear rate and velocity, making viscosity even smaller. Finally, the flow speed exceeds vCFL at some point and no physical result is possible anymore. There is an obvious solution to this problem. If there is a negative error in viscosity we can write the real viscosity ν in terms of the input viscosity νset and the error viscosity νerr(x):

ν = ν ν (x) (4.7) set − err We can add a so called eddy viscosity νeddy(x) which is adjusted locally with some kind of turbulence model.

νset =ν ˜set + νeddy(x) (4.8) ν =ν ˜ + ν (x) ν (x) (4.9) set eddy − err It seems that it should be possible to implement a turbulence model that choses νeddy(x)= νerr(x). Unfortunately that does not work properly because viscosity is isotropic while the error in viscosity is unisotropic (there are two relevant third order moments µxxy and µxyy). Furthermore, the error viscosity depends on the wave number while the eddy viscosity does not.

4.2.2 Aliasing From ﬁgure 4.3 we would draw the conclusion that high over-relaxation of third order moments is necessary for accurate viscosities. In fact, the graph for over-relaxed third order moments looks rather perfect compared to the SRT model and to the cascaded lattice Boltzmann model with equilibrated third order moments. Unfortunately, the cascaded lattice Boltzmann automaton

74 uhri o wr faysseai td fi o h latti the for it of study systematic any omnipres of and automaton. aware severe not a the is is in aliasing author i or though, diff problems measurement Even physically of a two resentation. class in that broad indistinguishable understand a are we to quantities aliasing belongs By It aliasing. one. complicated a is ti tefntsal o rirr o icste a lea (at viscosities low arbitrary stabl for version). stable a equilibrated not has itself lat moments is cascaded order it The third over-relaxed b numbers. with wave and automaton Th higher viscosity and increases field. velocities always velocity higher it superimposed since a error with stabilizing viscosity of increase si hsrseta vrielzdts.Frequilibrated For test. prom over-idealized an ( its wa respect shear hold this The in not simulations. is relevant does more physically moments to order comes third over-relaxed with µ laigerrfrom error aliasing omo laigwihsest ersosbefrstability for o mix responsible concentrate in be shall simulations. to we aliasing Boltzmann seems and which cross i subject aliasing have broad of freedom, even form a of We is degree Aliasing each space. tions. in momentum aliasing and have space, We error. aliasing apigapyia o edbttk iuainrslswhi results s frequency”? Boltzman simulation place grid take first lattice ”with but the field the flow in physical in present a be happening sampling t wavelength resolved with fact, th non and in can to wavelength how is, go long This should at which plitude. surface Energy would frequencies frequencies. high low which for frequencies mistaken higher sampl any be is contain field not flow must the and is of frequency problem velocity similar A The simulations. problem. this Boltzmann avoid remov al to the sampling are to requires frequency prior processing sampling signal the Digital half than frequencies. st higher theorem are Nyquist-Shannon which The cies processing. signal digital ω xxx eecutrdaisn led nequation in already aliasing encountered We h rbe hc rssi ed o qiirt ihrorde higher equilibrate not do we if arises which problem The laigi ototnascae ihtesrmln fwav of scrambling the with associated often most is Aliasing ttrsotta h tt faltieBlzannd contai node Boltzmann lattice a of state the that out turns It xxy = = µ ω x xyy c 2 n hs pcfi pe fsudt euetecorrespondin the reduce to sound of speed specific a chose and 1 = h acddltieBlzanatmtnsosaslight a shows automaton Boltzmann lattice cascaded the ) O ( Ma ) to . ffcso ihodrcretoso stability on corrections order high of Effects 4.2 O ( Ma 3 ) u hsi yn en h only the means no by is this But . 3.8 hr ehdtealiasing the had we where tnta tbea the as stable as not st lo ihfrequencies high of al hr re moments order third cmssrne for stronger ecomes rbesi lattice in problems tsta frequen- that ates hwr computed were ch edcyproblem decay ve rnmrclrep- numerical ir n rbe,the problem, ent rsn nlattice in present rn observable erent l naspecific a on nly ae ihlower with iased ol otherwise would ehd But method. n dwt h grid the with ed si perfectly a is is ieBoltzmann tice ie spatial time, n new r not are we ince sabgda of deal big a ns icst but viscosity e ewogam- wrong he eBoltzmann ce toue by ntroduced ubr in numbers e dissipative e sswe it when ises moments r ddirec- ed 75 g

4 4 Properties of the cascaded lattice Boltzmann automaton subgrid information. Spatial velocity gradients, which were necessary to solve the Navier Stokes equation, are replaced by moments of the distribution function. The second order central moment κxy is a replacement for the derivative of vy in x-direction. It is only a replacement since it is not possible to compute the actual gradient due to several aliasing effects. First we have the aliasing κxy = κyx. The gradient of vx in y is indistinguishable from the gradient of vy in x. In two dimensions, this problem could possibly be solved since we saw that the difference of two other central moments can be transformed into such a shear moment by rotation. The second degree of freedom was then κ κ . Two unknown gradients and two independent degrees of freedom. xx − yy Looks good. Now we consider the same problem in three dimensions. We want to replace the gradients of vx in y and z, of vy in x and z, and of vz in x and y. This are six gradients and in analogy to the two-dimensional case we would represent them by six central moments: κ , κ , κ , κ κ , xy xz yz xx − yy κ κ , and κ κ . Aliasing strikes once more. The last expression is not xx − zz yy − zz independent of the the others: κ κ = κ κ (κ κ ). So we have yy − zz xx − zz − xx − yy actually only five independent degrees of freedom to represent 6 gradients. In addition to this we cannot distinguish gradients of velocity from gradients of pressure. Even though, our knowledge of velocity gradients is incomplete it is still present and good enough for the lattice Boltzmann method to work. What does the knowledge on velocity gradients imply for subgird wavelength contributions to the velocity field? The Nyquist-Shannon theorem assumes that we have only knowledge of the value of the sampled field. The knowledge of gradients at the same sampling points doubles the information content and we can distinguish frequencies up to the sampling frequency, provided that the knowledge of the gradients was perfect (which it is not). Lattice Boltzmann goes even a step further since we also know higher moments like κxxy which is, in principle, a replacement for the second derivative of velocity vy in direction x. Remember that neglecting κxxy causes the diffusion of momentum (viscosity) to be wrong. In the framework of differential equations a diffusion operator represents a second derivative. The analogy between third order moments and second derivatives is genuine, the same thing expressed in a different language, but still not translatable due to aliasing (note that κxxy could also be taken for a third derivative of density or for the second order mixed derivative of vx in x and y). Hence we have also knowledge of higher spatial gradients and with this we have implicit knowledge of velocity fields at subgrid resolution. We know the value, the first and the second derivative at the sampling points. The Nyquist-Shannon theorem appears to be irrelevant for us. However, our knowledge is too implicit. We do not know about any methodology to extract

76 ih eipsil npoiiyt onaynodes. boundary o to amount proximity considerable in d a impossible applied over be filter fi sampling might sharp The require comparable would a size. field while filter flow time compact a at super node a single with implemen frequency it cut-off and exactly sharp momentum conserves also it simpler, hr pta le sapidt eoetehg aenumbe wave high the remove to [ larg applied field from is flow different filter actual completely spatial is a This where ac the derivatives. effect first not s does Boltzmanits first, moments lattice order that the third is on equilibrating thing that entities nice different The are moments fields. flow order subgrid de scattering all non-zero each out after a filter moments order implies the higher moments Only the considered equilibrate automaton: soluti the Boltzmann the lattice However, of the equilibrium belong. t for know not they trivial not does is because do it lem dangerous we where hence but to and fields energy flow their aliased subgrid are about fields know a Subgrid velocity we the Ga that sample is is we system since point the frequency that spacing is grid problem at thi corres Another only in a like. how address look possibilities not might several field will are we there aliasing which to problem due refinement grid the to h ugi o edfo ihodrmmns hswudbe would This moments. order high from field flow subgrid the h eoddrvtv fawv qainipista h effe the that number implies wave equation the wave with a quadratically of h derivative increase frequencies second higher and seen The since dissipative hence frequencies general higher is in for curvature (thir is stronger the curvature It of new misa removal fi effects: is sampled The flow following field is the removed. tooth velocity as and saw long when found the as moves, and it fine lattice Once is s the around. This tooth move saw introduc not field. a the does flow since the implies frequency) an onto curvature value grid superimposed of velocity (beyond removal numbers the The wave while high point same. (sec each curvature the at the main field removes velocity moments the order of third of libration h on sta atc otmn osntwr nFuirs Fourier in work not does Boltzmann lattice that viscosity is numerical point positive The additional the does where But 22 .Teeulbaino ihrodrmmnsi o only not is moments order higher of equilibration The ]. ∂ 2 sin( . ffcso ihodrcretoso stability on corrections order high of Effects 4.2 ∂x 2 kx ) k = : − k 2 sin( kx ) v ihrcurvature. higher ave odn ugr flow subgird ponding h icst.I is It viscosity. the s vn eeffectively we event dysimulation eddy e rdpit.The points. grid t ulflwfil and field flow tual uoao and automaton n saconsiderably a ts cn,adthird and econd, nt hsprob- this to on oefrom? come eiainfrom derivation n derivative) ond rcl nothe onto irectly h lp re- slope the d ieninvariant lilean iaie fwe If rivative. tsol grow should ct oet)is moments) d n orelease to end oe which nodes f trat na on acts lter hss But thesis. s ga snow is ignal ino very of tion the ae Equi- pace. sfo the from rs ohv the have to indwith ligned l itself eld solution enough. (4.10) 77

4 4 Properties of the cascaded lattice Boltzmann automaton

The dissipation increases with the speed at which the flow field moves around since the misalignment of the saw tooth field with the grid after streaming depends on velocity.

4.2.3 Over-relaxation of third order moments We might ask whether equilibration of third order moments is really necessary if it causes additional errors in viscosity. What are the ill eﬀects of the over- relaxation of third order moments? We saw that second order central moments like κxy are measures for the gradient of the velocity vy in x-direction. Over-relaxation with ωxy = 2 leads to an inversion of the central moment κxy:

κpost = κpre (4.11) xy − xy But if κxy measures the gradient of vy in x and we invert it, did we then also invert the velocity gradient? The answer is no. In fact, we had inverted the velocity gradient if we had not inverted κxy. The reason is that κxy is anti-symmetric in x. It measures the y-momentum coming from the west minus the y-momentum coming from the east. Note that the occupation of the eastern channels comes from western neighbors and the occupation of the western channels comes from eastern neighbors. If we want no diffusion of momentum we have to return the momentum to where it came from. But this is obviously impossible because if we return particles to where they came from they must have the opposite momentum (since they move in the opposite direction). The solution to this problem lies in the background pressure. There are always particles moving everywhere and we can move negative momentum to a neighboring node by sending less particles back. This is exactly what happens if κxy is being over-relaxed. Since κxy is anti-symmetric, we can undo the deterministic drift of momentum by inverting κxy. The same logic does not hold for the third order moment κxxy which is a measure for the curvature of vy in x. It is symmetric in x and inverting it has a completely different effect. In principle κxxy measures the difference of the y-momentum at a node and on its neighbors in western and eastern directions. Inverting κxxy means that we invert the curvature at this point. If there was more momentum on the neighboring nodes in the last time step, less momentum will be returned to them in the next time step. This leads to a flipping-over of the curvature of the velocity field at very short spatial wavelength and at half the time step frequency as can be seen in figure 4.4. That is to say, a concave corner in the velocity field will flip to a convex corner and vice versa. It is

78 al ops t(e figure latti (see standard it the pass that see to to fails instructive is it test, alized eoiisi h extended the in velocities 4.4 n osntapyayaeaigi ie ffcslk hs se these like Effects time. in s averaging time any la with apply much field step not at flow time does more steady the and or samples be four algorithm only over Boltzmann averaged can we lattice field if flow example a for scales, such that evident quite oased tt ihasal eoiypa.Ised h os the Instead, peak. velocity does stable automaton a The with time. state noise. steady a with to nodes other all contaminated h aiiyo h eoiyfil neeytm tp n over Any step. time every like in moments field order velocity third the of validity the sfrhrielzdb hoigteatmtnt eperiodi be to automaton the the in choosing ze thick from by node different whi pe idealized velocity on has a further node automaton in is one Boltzmann peak and beha lattice zero velocity a the velocity single have such understand a for we viscosi of this test evolution By zero idealized time most at say, The the i to vestigate way. nor is dissolve any neither in That s should Turbulent structures vortex) fluid. a friction. like the (such from into structure free ”frozen” is be fluid then the viscosity zero rest At at peak velocity Frozen-in 4.3.1 la of difficulties algorithmic some at simulations. point to motivated chosen physically not are are They here shown l results examples cascaded the The the compare tions. and of tests behavior idealized some the with investigate automaton we section this In tests Simple 4.3 whi automaton t assumption. Boltzmann instructive time lattice relaxation be single entropic would the the It for viscosities. means of b this highest that the Note rule but are allowed. all approximations not time compreh is relaxation intuitively single moments is conclusion order it third grounds of this relaxation On field. velocity the r iatosadcno etlrtdb nagrtmta d that algorithm an by tolerated be cannot and disastrous are y drcin h eoiypa onsin points peak velocity The -direction. x 4.5 κ drcinaezr.Ee huhti savr ide- very a is this though Even zero. are -direction xxy .Atrafwtm tp h eoiypa has peak velocity the steps time few a After ). ilcueflpigoe flclcraueof curvature local of flipping-over cause will eBlzanmethod Boltzmann ce u npicpefor principle on out d trc ihremote with nteract y drcinwieall while -direction netgt what investigate o nil htover- that ensible tieBoltzmann attice ihSTsimula- SRT with iltosgo in grow cillations ecmr tests. benchmark rcue should tructures o h problem The ro. . ipetests Simple 4.3 tc Boltzmann ttice .Hwvr the However, s. n nyone only and c has applies also ch idcdomain. riodic rwn this drawing y e frequency tep rlxto of -relaxation iri oin- to is vior velocity a ty o converge not halnodes all ch ni figure in en pnson epends grtime rger 79

4 4 Properties of the cascaded lattice Boltzmann automaton

0.12 t=2 t=4 0.1 t=6

0.08

0.06

vy 0.04

0.02

-0.02

-0.04 0 2 4 6 8 10 12 14 x Figure 4.4: The same simulation set-up as in figure 4.2 (without superimposed flow field) with over-relaxed third order moments. Shown are only even time steps since the effect of interest happens with a frequency of two time steps as explained in the text. The over-relaxation of third order moments leads to a flipping-over of the local curvature every two time steps. Concave corners are turned into convex corners and vice versa. This artifact leads to intolerable contam- inations of the flow field at high frequencies. The local curvature grows with the square of the wave number. The effect might hence be relatively weak if no high frequency components are present in the flow field. However, under-resolved turbulent flow simulations always imply the existence of high frequency components. These high frequency components get hopelessly scrambled if over- relaxation of third order moments is allowed.

80 iue45 ieeouino igevlct ekin peak velocity single a of evolution Time 4.5: Figure

vy -0.02 0.02 0.04 0.06 0.08 0.12 0.1 0 dcltieBlzanatmtno dimension of automaton Boltzmann lattice odic ihteSTagrtm icst sstt eo( zero state. steady to a set to is converge Viscosity not does automaton algorithm. SRT the with 0 5 10 15 20 x 25 30 y drcino peri- a on -direction 35 t=1000 t=100 45 . ipetests Simple 4.3 t=2 × ω 40 1 2 = simulated .The ). 45 81

4 4 Properties of the cascaded lattice Boltzmann automaton

0.12 t=2 t=100 t=1000 0.1

0.08

0.06 vy 0.04

0.02

-0.02 0 5 10 15 20 25 30 35 40 45 x Figure 4.6: Same setup as in ﬁgure 4.5 simulated with the cascaded lattice Boltzmann method and ωxy = ωxx−yy = ωxxy = ωxyy = 2. The result looks similar to the SRT simulation and is not stable.

The cascaded lattice Boltzmann automaton behaves similar if third order moments are maximally over-relaxed together with second order moments (ωxy = ωxx−yy = ωxxy = ωxyy = 2 and other relaxation parameter set to one, see ﬁgure 4.6).

If we include the equilibration of third order moments (ωxxy = ωxyy = 1) the oscillations vanish completely. The velocity peak broadens just a little bit and reaches a steady state after a few time steps (see figure 4.7). Only close by nodes are affected by the initial blur of the peak. The equilibration of third order moments does not cause any additional viscosity in this case. The shape of the velocity profiles at time step 100 and at timesep 1000 are indistinguishable.

82 iue47 aestpa nﬁgure in as setup Same 4.7: Figure

vy -0.02 0.02 0.04 0.06 0.08 0.12 0.1 0 eoiypa ssilnal ssapa t nta conditi Th initial its indistinguishable. as are sharp fe as 1000 nearly a and still after 100 is state step peak steady velocity Time a steps. reaches time automaton The disappeared. ω 0 xyy 1 = 5 h siltosse ntepeiu iuain have simulations previous the in seen oscillations The . 10 15 4.6 20 with x 25 ω xy = 30 ω xx − yy 35 t=1000 t=100 2 = . ipetests Simple 4.3 t=2 40 and on. ω xxy 45 83 = w e

4 4 Properties of the cascaded lattice Boltzmann automaton

4.3.2 Frozen-in velocity peak in motion

To make the test from the last subsection just a little bit more difficult, we add an homogenous velocity field to all nodes with vx = 0.1 in lattice units. The difficulty here is that we try to move the peak to a place in between two lattice points. This is an test for the aliasing properties of the automaton and one for Galilean invariance. Since the SRT model failed this test even without the superimposed velocity it is no surprise that it shows the same oscillation in the moving case (see figure 4.8). The same is true for the cascaded lattice Boltzmann automaton without equilibration of third order moments as seen in figure 4.9. If we turn the equilibration of third order moments on, the oscillations are much weaker (see figure 4.10). But we also see that an de- aliasing viscosity comes into play. The peak dissolves over time. This is, of course, an error. But it is an error that removes energy from the simulation and is therefore stabilizing. Note that the velocity vx = 0.1 is already rather high as it is nearly a quarter of the maximally allowed velocity vCLF .

4.3.3 Step advection with ﬁnite viscosity

The next test example is included because it is one of the few idealized problems for which simulation results from the entropic lattice Boltzmann methods are available in the literature. The experiment presented in [29] is similar to the one in the last subsection. On a one dimensional periodic domain with 800 lattice notes a step profile of a passive scalar is initialized from node 200 to 400. The diffusion coefficient (equivalent to our kinematic viscosity) is set to ν = 5 10−5 in lattice units. A homogenous flow field with v = 0.1 is applied × x and the resulting profile is plotted after 3000 time steps. The article implies that this setup is comparable to a fluidic entropic lattice Boltzmann automaton. We compare the results in [29] to the standard lattice Boltzmann method (also included in the paper) and our cascaded lattice Boltzmann automaton with equilibrated high order moments. Again we use the D2Q9 lattice with a thickness of only one cell in y-direction and periodic boundary conditions. In- stead of a passive scalar we advect momentum in y-direction. That our setup is reasonable close to the simulations presented in [29] is confirmed by visually comparing the results for our SRT model (figure 4.11) with the corresponding result in the paper (figure 4 in [29]). Note however, that a rigorous comparison is not possible since the two experiments in the paper were obviously not performed at the same Mach number as can be seen by the differences in the replacement of the steps after 3000 time steps. Here we present only

84 iue48 aestpa nﬁgure in as setup Same 4.8: Figure

vy -0.03 -0.02 -0.01 0.01 0.02 0.03 0.04 0.05 0.06 0 togoclain otmnt h euto hsSTlatt SRT simulation. this mann of result the contaminate oscillations Strong 0 5 10 15 4.5 20 ihhmgnu velocity homogenous with x 25 30 35 t=100 . ipetests Simple 4.3 40 v c Boltz- ice x 0 = 45 . 85 1 .

4 4 Properties of the cascaded lattice Boltzmann automaton

0.06 t=100 0.05

0.04

0.03

0.02 vy 0.01

-0.01

-0.02

-0.03 0 5 10 15 20 25 30 35 40 45 x

Figure 4.9: Same setup as in ﬁgure 4.6 (CLB with ωxy = ωxx−yy = ωxxy = ωxyy = 2) with homogenous velocity vx = 0.1. By over-relaxing the third order central moments we obtain a result which is as noisy as the SRT simulation.

86 iue41:Sm eu si ﬁgure in as setup Same 4.10: Figure

vy -0.03 -0.02 -0.01 0.01 0.02 0.03 0.04 0.05 0.06 0 0 ω ihtegvnvlct h ektaesdtecci domain cyclic the apart. traversed steps peak time the times. 450 velocity measure ve given are the the proﬁles of With ﬂow dissolution de-al shown and the The a step in However, peak. time results which each reduced. appears in drastically viscosity moments are order oscillations third the the equilibrate we xxy 5 = ω xyy 10 1 = ihhmgnu velocity homogenous with ) 15 4.7 20 CBwith (CLB x 25 30 ω xy 35 = t=100 t=550 v ω . ipetests Simple 4.3 x xx 0 = − 40 yy . 2 = 1 Now . iasing 45 locity and 87 10

4 4 Properties of the cascaded lattice Boltzmann automaton simulations performed at the same Mach number as in figure 6 of the paper which shows the result of the entropic lattice Boltzmann automaton. The data points in the figure in [29] were extracted electronically and the plot is repro- duced in figure 4.12. By comparing this to the result from the cascaded lattice Boltzmann automaton at the same viscosity (figure 4.13) one recognizes two differences: First, simulated with the entropic lattice Boltzmann automaton the oscillations at the edges of the step are smaller (but still visible) when compared to the result from the cascaded lattice Boltzmann automaton. The second difference is that the entropic lattice Boltzmann method must have a considerably larger numerical viscosity as compared to the cascaded lattice Boltzmann method. A simple possibility to access this data from figure 4.12 is to count the ticks (or nodes) on the slope of the step in the plot after 3000 time steps. The number of ticks on the slope is a measure for its steepness and the decrease of steepness in time is a measure for the viscosity. It is evident that the entropic lattice Boltzmann automaton introduces a considerable amount of numerical viscosity in this example. The author counts 12 ticks on both slopes in the entropic simulation (figure 4.12). The number of ticks on each slope in figures 4.11 and 4.13 is between 5 and 7, depending on whether the overshooting is considered a part of the slope or not. The number of ticks in the SRT simulation in [29] is about seven on each slope, comparable with the results shown here. A more accurate method to compare the steepness of the slopes is to apply numerical differentiation to the data. We apply central differentiation to the array of values v in x:

v v grad(v )= n+1 − n−1 (4.12) n 2 The values from figure 4.12 have to be divided by 3370 in order to obtain the same magnitude as in figure 4.13. The gradient is a direct measure for the steepness of the slope. Figures 4.14 and 4.15 show close-ups of the gradients of the entropic and the cascaded simulation, respectively. It is clearly seen that the steepness of the step in the entropic lattice Boltzmann simulation is much lower than the steepness of the step in the corresponding cascaded lattice Boltzmann simulation. This implies a much higher numerical viscosity of the entropic lattice Boltzmann automaton. To get a rough estimate of the actual value of the viscosity we can try to find a nominal viscosity for which the steepness of the step in the cascaded lattice Boltzmann method is comparable to the steepness seen in figure 4.14. Figure 4.16 shows the steepness of the slope obtained from the cascaded lattice Boltzmann method with ν = 10−3

88 iue41:Tasoto eoiyse rﬁein proﬁle step velocity a of Transport 4.11: Figure

vy -0.02 0.02 0.04 0.06 0.08 0.12 0.1 0 0 00aeson h icst is viscosity The shown. are 3000 otmn uoao fdimension of automaton Boltzmann moe ooeosflwfield flow homogenous imposed 100 200 300 400 x v x ν 0 = 5 = 500 800 y . × drcindet super- a to due -direction 1 × naDQ R lattice SRT D2Q9 a on 10 1 600 iesesoeand one steps Time . − 5 . . ipetests Simple 4.3 700 800 89

4 4 Properties of the cascaded lattice Boltzmann automaton

1.6

1.4

1.2

0.8

0.6

0.4

0.2

-0.2 0 100 200 300 400 500 600 700 800 x Figure 4.12: Transport of a step profile of a passive scalar on a one-dimensional entropic lattice Boltzmann automaton with a superimposed flow field vx = 0.1. The plot is a reproduction of figure 6 in [29]. Data points were extracted electronically from the plot and might have the wrong magnitude. No label for the y-axis is provided by the authors. The article [29] implies that the presented setup is equivalent to the lattice Boltzmann simulations shown in figures 4.11 and 4.13 and should have the same transport coefficient.

90 iue41:Sm eu si ﬁgure in as setup Same 4.13: Figure

vy -0.02 0.02 0.04 0.06 0.08 0.12 0.1 0 0 otmn automaton. Boltzmann 100 200 300 4.11 400 iuae ihtecsae lattice cascaded the with simulated x 500 600 . ipetests Simple 4.3 700 800 91

4 4 Properties of the cascaded lattice Boltzmann automaton which is twenty times the nominal viscosity of the entropic lattice Boltzmann simulation. The steepness is now comparable to the entropic simulation albeit it is still slightly higher. Note that the cascaded lattice Boltzmann automaton introduces less noise to the result for the same effective viscosity. The cascaded lattice Boltzmann automaton itself is not free of numerical viscosity in this setup due to the superimposed flow field. However, numerical viscosity is nearly absent if vx = 0. Figure 4.17 shows the steepness of the slope obtained from the same setup without superimposed flow field. We see that the cascaded as well as the entropic lattice Boltzmann automaton introduce a considerable amount of numerical viscosity. The magnitude of the numerical viscosity in the cascaded method is much lower. A comment on the presented comparison between the entropic and the cascaded method has to be added. Unfortunately, the two examples do not describe exactly the same physical problem. The entropic automaton conserves only mass while the velocity is kept constant. The two primary authors of [29], Ilya Karlin and Santosh Ansumali, consider this comparison unjustified since the underlying physical problems are qualitatively different. They argue that the velocity component in y-direction introduces a pressure wave in x- direction with the result that the velocity vx is not constant anymore. Their position should be acknowledged and respected. However, the author is of the opinion that the effects of pressure waves on the result should be negligible. There is no qualitative or severe quantitative difference in the their single relaxation time simulation and the one shown here. The strength of pressure waves is significantly higher in the SRT simulation as compared to cascaded lattice Boltzmann simulations since the bulk viscosity is decoupled from the shear viscosity in the cascaded method and kept high by equilibration of the trace of the second order velocity moments. If the effect is not visible in the SRT simulation it should not be visible in the cascaded lattice Boltzmann simulation. The presence of pressure waves is by no means a sufficient explication for one order of magnitude more numerical viscosity in the entropic method as compared to the cascaded method.

4.4 Measurement of the viscosity error

In this section we investigate the error introduced by the equilibration of higher order moments. In ﬁgure 4.3 we see a slight increase of viscosity with Mach number for the cascaded lattice Boltzmann automaton with equilibrated higher order moments. In order to quantify this error we simulate the time

92 iue41:Tesepeso h lpo h tppol nﬁg in proﬁle step the of slop the of steepness The 4.14: Figure

grad(v) -0.035 -0.025 -0.015 -0.005 0.005 -0.04 -0.03 -0.02 -0.01 0.01 0 550 h rdeto n fteflnso h tpatr30 ieste time 3000 after o step the close-up of a flanks shows the the of dir figure for one allow This of as gradient to value simulations. the order same our in the to simulation to comparison Boltzmann scaled was lattice data cascaded the of magnitude iuain srvae ynmrclydffrnito in differentiation numerically by revealed is simulation) 600 . esrmn ftevsoiyerror viscosity the of Measurement 4.4 650 x 700 ure 4.12 (entropic x 750 The . ect ps. 93 f

4 4 Properties of the cascaded lattice Boltzmann automaton

0.01

0.005

-0.005

-0.01

-0.015 grad(v) -0.02

-0.025

-0.03

-0.035

-0.04 550 600 650 700 750 x Figure 4.15: Gradient of the data shown in figure 4.13. This plot reveals the steepness of the flank in the cascaded lattice Boltzmann simulation with ν = 5 10−5. Even though, the nominal transport × coefficient is the same as in the entropic simulation shown in figure 4.14 the slope here is considerably steeper. The numerical viscosity is hence lower in this simulation.

94 iue41:Ti ltsostegain ftesoeo h s the of slope the of gradient the shows plot This 4.16: Figure

grad(v) -0.035 -0.025 -0.015 -0.005 0.005 -0.04 -0.03 -0.02 -0.01 0.01 0 550 rfiea sdi figures in used as profile tlat2 ie shg stenmnlviscosity nominal the as high as simula times entropic 20 the in least viscosity at numerical the that implies vi nominal a with run was cosity simulation Boltzmann lattice caded nrpcltieBlzansmlto eni figure in in slope seen the simulation of Boltzmann steepness the lattice to entropic comparable now is viscosity) oao snwcnieal esniyta h euto the of result the than a noisy Boltzmann less lattice simulation. cascaded considerably entropic the now of is result tomaton the that also Note ν 10 = − 600 3 h tens ftesoe(n ec h actual the hence (and slope the of steepness The . 4.11 . esrmn ftevsoiyerror viscosity the of Measurement 4.4 , 650 x 4.12 , 4.13 , 4.14 700 and , m nta step initial ame 4.15 ν 5 = 4.14 h cas- The . × This . inis tion 750 10 the − 95 u- 5 s- .

4 4 Properties of the cascaded lattice Boltzmann automaton

0.01

0.005

-0.005

-0.01

-0.015 grad(v) -0.02

-0.025

-0.03

-0.035

-0.04 250 300 350 400 450 x Figure 4.17: This plot shows the gradient of the slope of a step proﬁle at rest after 3000 time steps and with ν = 5 10−5. The setup is × physically identical to the simulations shown in ﬁgures 4.11-4.15. Numerical viscosity is very low for vx = 0. This plot shows the correct steepness for the given viscosity.

96 esrn h iedcyo iuodlwv ihkonwave known with wave sinusoidal a of decay time the measuring echoose We viscosity numerical the Determining 4.4.2 viscosity. the determine can xoetcnb eemndb tigtedt otepwrlaw power the to data the fitting by determined be can exponent ns udai eedneo hskn sipidb equat by implied is kind this of dependance quadratic A ings. lt ftevsoiyi oaihi cl gis velocit erroneo against hence scale (figure logarithmic is tion in experiment viscosity decay the wave of plots shear the in measured xml ic ti n ftevr e ae o hc h Navi grad the ( pressure which frame without for material velocity), cases the mean few In the b very with important solution. moves the an analytical is system of an domain one has periodic is a equation it in wave since shear example a of decay The experiment decay wave Shear 4.4.1 vector. superimp wave the different of with direction wave the shear in sinusoidal pointing a of evolution for ayn nyin only varying otemtra rm,teNve tkseutossimplifie equations Stokes Navier the [ frame, material the to 30 Where So emk h ansatz the make We ]: a and v sin( w b kx r itdi table in listed are ω stewvlnt ftets aegvni utpe fgi s grid of multiples in given wave test the of wavelength the is 4.18 xy exp( ) x ∂t ∂ oe a eairfrln aelnt seiet The evident. is length wave long for behavior law power a ) = drcin n ihalohrvlcte en eowt re with zero being velocities other all with and -direction, av v − ω sin( sin( xx νk − kx kx v 2 yy t y ) exp( ) exp( ) = saslto fteNve tkseuto n by and equation Stokes Navier the of solution a is 2 = v ν 4.1 sin( numberic at at ∂v n hence and ∂t h eedneo aenme em to seems number wave on dependence The . a = ) = ) kx y exp( ) = = . esrmn ftevsoiyerror viscosity the of Measurement 4.4 ν = ν − − ∂ ∂x ∂x 2 at k k w ∂ a v 2 2 2 ν 2 2 y ) νv ν 2 : v v 0 = x b sin( sin( n eann viscosity remaining Any . kx kx exp( ) exp( ) nwv etrdirec- vector wave in y at at otefollowing the to s ) ) ion sdflwfields flow osed h coordinate the et,with ients, s rmthe From us. number : 4.10 enchmark rStokes er Values . (4.17) (4.15) (4.14) (4.16) (4.13) spect k pac- we 97 v y

4 4 Properties of the cascaded lattice Boltzmann automaton

Fit parameter of the numerical viscosity wavelength w a b 30 4.1202 3.8309 60 4.8639 3.9841 120 4.9177 3.9962 200 5.22453 4.0494

Table 4.1: Fit parameter of the viscosity measurement shown in ﬁgure 4.18 to 2 b the function νnumeric = a/w vx obtained with the software Gnuplot. The exponent b is close to 4.

be a bit less strong than predicted but the tendency is clearly correct. The exponent b converges to 4 for large wavelength. The numerical viscosity is hence an (v4k2) error with k being the wave number. The viscosity is accu- O x rate to third order in Mach number. This is a reasonable result. Given that the highest explicitly considered central moment κ is, in fact, in (v2) we xxy O x could expect only second order accuracy. However, a third order error in vx 3 would break the symmetry of the problem since vx is an odd expression.

We have reason to think that the (Ma3) accuracy of viscosity is the final O word for this type of lattice Boltzmann automata and that higher accuracies can only be obtained with larger velocity sets. It is interesting to note that the numerical viscosity seams to vanish when we apply over-relaxation to the third order moments. But we have already seen that this causes aliasing errors with disastrous results. The aliasing properties are the final limit for the accuracy to which we can resolve the flow field.

It should be noted that the remaining numerical viscosity is very small. In SRT and MRT lattice Boltzmann automata it is diﬃcult to obtain stable simulation results for ν < 10−3 [11, 31] and v > 0.2. For a relatively short wavelength of 30 lattice spacings we measure a numerical vicosity of ν 10−6 ∼ at the same Mach number and the simulation is stable. This is a three orders of magnitude improvement in Reynolds number! And it is getting better for longer wavelength and lower Mach number. Note that we gain another four orders of magnitude by reducing the velocity to the more reasonable value of 0.02.

98 iue41:Maueeto h ueia icst nteD the in viscosity numerical the of Measurement 4.18: Figure

nu 0.0001 1e-12 1e-11 1e-10 1e-09 1e-08 1e-07 1e-06 1e-05 0.001 0.01 h lp t esnbewl o4 h ueia icst i viscosity numerical The ( number Mach 4. in effect to order well fourth a reasonable hence fits slope The ac ntesprmoe eoiyi aevco direction vector wave dep in law velocity power superimposed plot a th the the implying in on length lines used wave dance straight were larger parallel spacings) For to lattice converge experiment. 200 decay and w wave different 120, shear with waves 60, Test (30, zero. length to set mom viscosity order nominal third and equilibrated with method Boltzmann lattice L=200 L=120 L=60 L=30 . esrmn ftevsoiyerror viscosity the of Measurement 4.4 0.1 vx ν numeric Q cascaded 2Q9 = O ( Ma 1 ents 4 ave en- ) 99 x ). e s s .

4 4 Properties of the cascaded lattice Boltzmann automaton

4.5 Speed bounds

The cascaded lattice Boltzmann automaton has superior stability properties as compared to the SRT model. It is, however, not unconditionally stable like the entropic lattice Boltzmann automaton since it is still sensible to the violation of the CFL condition. This problem turns out to be more severe than one might think. First and foremost we should recognize that the cascaded lattice Boltzmann automaton is certainly less sensitive to high Mach number flows than the SRT model under the same (namely laminar) conditions. But with the ability to reduce viscosity arbitrarily comes the desire to simulate turbulent flow. Turbulent flow has no natural speed limitation. In a laminar flow simulation we would assume a simple relationship between the maximal flow velocity and the boundary conditions (pressure difference or input velocity). The relationship would be of the kind that allows us to decrease the maximal velocity if it was found to be too high. This does not hold for turbulent flow and the reason is easily understood form common sense. Regard the atmosphere of our home planet. Nowhere on earth are severe pressure or temperature gradients which would explain (in a linear sense) the acceleration of air to several hundreds of kilometers per hour and yet we observe such strong winds. On average the atmosphere is at rest compared to the earth but turbulence allows - actually implies - strong deviations from the average. There is no reason for fluids to be slow. Slowness can only be defined with respect to a resting coordinate system in momentum space. But we put our efforts in making lattice Boltzmann independent from the coordinate system. Un- fortunately, convoluted turbulent flow fields favor the development of sudden confined bursts of high velocity which are unrelated to boundary conditions. The velocity in this bursts can easily exceed the inflow velocity by several orders of magnitude. However, the fluid velocity must not exceed vCFL. In a way we have developed a faulty model. We required Galilean invariance irrespective of the finite speed with which information is allowed to travel on the lattice. In fact, we built our theory upon an outdated assumption. Galilean invariance is not considered a physical fact anymore. The system should actually be Lorenz invariant. That is to say, the velocity should increase non-linearly with the momentum and should be bounded by a maximal velocity: the speed of information, the speed of light. Such a Lorenz invariant lattice Boltzmann automaton was in fact likely to be unconditionally stable. It could also be regarded as nearly incompressible since the speed of sound was 57.7% of the speed of light. However, a Lorenz invariant lattice Boltz- mann model with the speed of light being one lattice spacing per time step is

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4 4 Properties of the cascaded lattice Boltzmann automaton cade lattice Boltzmann automaton but it would not work for the SRT or the entropic lattice Boltzmann automat, at least not without modifications. The assumption of a linear relationship between momentum and velocity is hard- coded into the SRT model. Changing velocity would also change momentum. For the cascaded lattice Boltzmann automaton and the MRT model this is not true. In fact, we could choose a bunch of different ways to compute velocity. Taking the mean of the momentum distribution function is just picking the most likely value. It would also be reasonable to pick a random value with a likelihood scaled with the distribution function. In the cascaded lattice Boltz- mann method this would not change the momentum because nothing changes the momentum. The scattering matrix is orthogonal to the momentum vectors. The cut-off velocity bound described here is introduced only to handle a rarely occurring problem. The cascaded lattice Boltzmann automaton does not depend on it for the simulation of turbulent flow. It is not unconditionally stable without it. The author implemented the velocity bound only in the floating point version of the cascaded lattice Boltzmann automaton. The integer version, cascaded digital lattice Boltzmann, was implemented without velocity bound. Both simulators work fine up to Reynolds numbers of over one million without velocity bounds. Fatal velocity burst appeared at Reynolds numbers greater than one billion and only after billions of scattering events. In what follows it will be explicitly mentioned if a velocity bounds was used.

4.6 Cascaded digital lattice Boltzmann

This chapter focused mainly on the comparison between the cascaded lattice Boltzmann automaton and the SRT scheme so far. The presented cascaded lattice Boltzmann automaton uses double precision floating point numbers as link data type just like the SRT model does in most implementations. A comparison between two schemes that use different numerical precision might lead to faulty conclusions. The SRT model dictated the data type in this case. The cascaded lattice Boltzmann automaton is not so stringent when it comes to the choice of data types. In fact, we introduced a optional truncation operation in the last chapter that allows the automaton to run entirely digital with integers as occupation numbers. From the practical point of view there might be little justification for doing so. Hardware and compilers are designed for the efficient use of floating point arithmetic. Even though, integer arithmetic is in principle simpler than floating point arithmetic it might still be slower given

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4 4 Properties of the cascaded lattice Boltzmann automaton it does not break completely. The kind of integer arithmetic used in the last chapter has some interesting features which should be summarized here. The interdependence of the different scattering operators in the lattice Boltzmann automaton is complicated and one might asked why we should bother to solve the equation analytically in the first place since there are robust methods for solving a set of linear equations numerically. The answer is, that this is true only for approximative solutions, not for integers. But the set of equations has no integer solution anyway and that is where the collision cascade comes into play. By applying the lowest scattering operator we certainly make numerical errors of some kind. We need to compensate for the application of the low order scattering operator in all higher order scattering operators. It certainly makes some difference whether we compensate ”analytically” to what the low order scattering operators should look like, or whether we compensate for what they actually are. The difference is that if we compensate for the scattering of low order moments ”as is” we hinder the numerical errors to propagate to the next stage. Numerical errors are hence cleanly confined in the way the scattering cascade works for integers. This confinement allows us to start each calculation with fresh unworn ”tires”. Suppose that we decided to use 32 bit integers as occupation numbers and we would do all intermediate calculations with 64 bit floating point numbers. For the computation of each scattering operator we would start with numbers in which nearly half of the digits were unused. The numerical wear during the relatively short calculation for just one scattering operator should be negligible under this conditions. In fact, the whole wear is concentrated in the controlled truncation at the end of the calculation. The truncation is an error but it is an error that does not propagate to other moments and it is an error that does not change the conserved quantities. In the cascaded digital lattice Boltzmann automaton the conserved quantities are hence conserved exactly in the very sense of what exactness means. (Unfortunately, some people refer to a numeric solution with an error of the order of 10−16 as being exact since this is exact up to machine precision. Here we shall understand exactness in the sense that the error is identically zero.) Note that there is an alternative to the truncation operation which was not tested by the author. The information stored in the discarded digits can be restored in a statistical sense by a process called dittering [32]. Dittering is a random rounding process weighted with the rational part of the number. For example, if we applied a dittered conversion of the number 1.67 to integer we would draw a random number which was with likelihood 0.67 one and with Z likelihood 1 0.67 zero. The dittered result was then 1.67 + . Compared − ⌊ ⌋ Z

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4 106 h omgrvter ftruetflw[ flow turbulent of theory Kolmogorov The turbulence Under-resolved 5.1 flow. lat turbulent cascaded developed three-dimensional fully of the simulations apply to we automaton chapter this In flow turbulent of Simulation 5 edt hoeagi pcn fteodrof order the r of all spacing resolve to grid order a In choose to one. need is scale number Klomogorov Reynolds the the which of for size the has feature spatial eouini nrcal o l oeeal uueirres future of hardware. foreseeable kind computer all this in for to intractable approach is computational b resolution direct around a flow A and around billion. airplanes flow one around flow the like like Problems applications million. engineering in relevance eovdsmlto.Wehrti stu rntms certai must basis. not case or by true case is a this obta prope be Whether statistical could simulation. gross and fluctuations resolved that small hope of could independent were we field Then e co zero. of we to However, independent aged statistically system incorrect. were the potentially fluctuation from unresolved be energy will withdraw result Neglecting we lation problem. that the means of non-linearity contributions the to due coupled oee,ti osntse ob refrtecsae lattic cascaded the for true be to n seem not not typically does are this simulations However, Boltzmann lattice resolved eoiyflcutosa l aeeghdw oteKolmogor the to down scale? wavelength Kolmogorov all the at resolve to fluctuations need Velocity we do why But oepatclrao o o goigteKlooo scal Kolmogorov the ignoring not for reason practical more A yrslsaentpoe,te r re n hsi uhmore much is Kolmogorov this Nikolaevich and –Andrei true, are they proved, important. not are results My 22 η ttsta h mletrelevant smallest the that states ] enlsnmeso practical of numbers Reynolds . η hc stelnt scale length the is which etv fteprogress the of pective c te n aver- and other ach l oeta the that hope uld mrclystable. umerically l ejde on judged be nly hr wavelength short ndfo under- from ined lvn clswe scales elevant rbesa full at problems te fteflow the of rties ieBoltzmann tice a xedone exceed car sta under- that is e idnsexceed uildings Boltzmann e h simu- The . vsaeare scale ov 107

5 5 Simulation of turbulent ﬂow automaton. Figure 4.18 implies that we could use grid spacing which are orders of magnitude larger than the Kolmogorov scale. If we assume a numerical viscosity of 5/w2v4 as the limiting factor for viscosity, we could hope for ∼ lattice spacing Reynolds number of:

1v w2 Re1 = 3 (5.1) νnumeric ≈ 5v

For wavelength w = 15 and velocity v = 0.1 this would translate to a Reynolds number of 45000 per grid spacing! Is this an artifact of the over- idealized shear wave decay test or does the cascaded lattice Boltzmann method hold its promise even in scientiﬁcally relevant setups? To answer this question we examine a few results of three-dimensional simulations with the cascaded lattice Boltzmann automaton.

5.2 Turbulent wake

In our first setup we use the D3Q27 cascaded digital lattice Boltzmann algorithm without cut-off speed limiter to simulate the wake behind a rectangular obstacle at Reynolds number Re = 1400000 (see figure 5.1). The inflow velocity is implemented by a differential pressure between front and end of a domain with 400 120 120 nodes and is measure to be v = 0.097. Boundary × × conditions in the y and z-directions are periodic. The nominal viscosity is chosen to be ν = 2 10−6. The obstacle has dimensions 20 30 nodes and × × the Mach number is Ma = 0.168. The Reynolds number with respect to the grid spacing is Re1 = v/ν = 48500. The simulation is hence under-resolved by a factor of 48500. A small perturbation is added to the initial conditions to break the symmetry of the setup. (This procedure was found to be indispensable for turbulent simulations. It could be avoided if we used dittering.) The simulation is run for 16600 time steps. The result shown here was first presented by the author in [33]. Using the method described in appendix B we can compute physically equivalent scales. Taking the viscosity of air and letting the grid spacing be one cen- timeter, we obtain a velocity of 69m/s and a time step duration of 14 10−6s. × The simulated object measures hence 30 20 centimeters and the duration of × the whole simulation is 0.2324s. Alternatively, letting the grid spacing be 10 centimeters results in a velocity of 6.9m/s and a duration of 23.24s.

108 bev ntecsaeltieBlzansmlto.Figur simulation. is Boltzmann This lattice cascade turbulence. the the of in origin observe the of independent number) iue51 ubln eoiyfil ntewk farectangu a of wake the in field velocity Turbulent 5.1: Figure ieto ntewk fteosal.Sailaeaigwas averaging Spatial obstacle. the of perpendic wake measured the energy in kinetic direction of spectrum dimensional like behave fields velocity turbulent of energy srldotb h hoi aueo h rbe.Amnmlju Boltzma minimal lattice A same the problem. with the it compare of to nature was chaotic result the the by for whether or experiments out on to clue re ruled comparison no never is direct have will since We not and experiment. or wil times) similar reasonable observable which a wake in in never unsteady state same (or an state of this snapshot to single a represents it ed a nyb endi ttsia es.W nwfo t from [ know We experiments from sense. as statistical well a as betw in theory Similarity defined be state. only steady can a fields reach never tur since and difficult chaotically is data experimental to comparison Direct spectrum Energy 5.2.1 o edlk h n hw nfigure in shown one the like field flow A study Convergence 5.2.2 behin spacings lattice 40 starting domain simulation the of oia enlsnumber Reynolds nominal a ehhas mesh 0 . 168 u hog h eta li sshown. is plain central the through cut A . 400 × 120 × 120 22 – oe n h ahnme is number Mach the and nodes 24 5.1 httesailsetu fkinetic of spectrum spatial the that ] Re si eea o eyueu since useful very not general in is 1400000 = E ∼ k − 5 / 3 h computational The . (where . ubln wake Turbulent 5.2 uetflw behave flows bulent h obstacle. the d e ubln flow turbulent een e te simulations other 5.2 xcl htwe what exactly lrt h flow the to ular ple to applied c xcl the exactly ach eKolmogorov he a btceat obstacle lar ee return never l nsimulation nn k hw one- a shows h eutis result the stewave the is stification Ma 5 / 109 12 =

5 5 Simulation of turbulent ﬂow

0.01 k-5/3 wake L=120

0.001

0.0001 E(k)

1e-05

1e-06

1e-07 1 10 k Figure 5.2: The one-dimensional spectrum of the kinetic energy of the simulation shown in ﬁgure 5.1. Units are arbitrary. The solid line is a guide to the eye to indicated that the spectrum follows a k−5/3 power law as predicted by the Kolmogorov theory and known from measurements of turbulent ﬂow.

110 iue53 ubln eoiyfil ntewk farectangu a of wake the in field velocity Turbulent 5.3: Figure efre nacasrgi.I h w eut eefudt b to figure found compare change were we not would results if more two But even the grid further. the If refining that grid. reason could coarser a on performed banadffrn eut(e figure (see result different a obtain ecmae oec te.Teaeaeflwfil ih lob o be also might average field like flow values average extract can The we since other. o interest each technical experiments to different sta compared and The be deterministic fields. is re flow no field the and do of turbulent we nodes averages if statistical more in different but are times the snapshots things However, eight in features. has discrepancy re more simulation slightest same times the the finer obtain had the even we Second, not would if we resolution and chaotic same is field flow the only eouino h rbe hc sacpal ntecs th at case the wo in it acceptable loos is since we which here averaging problem out temporal the ruled In of hence resolution whic information. is over geometrical averaging dimension any Spatial The lost. averaging. is ensemble and averaging, hr r tlattreknso vrgn:sailaveragi spatial averaging: of kinds three least at are There 200 × 60 h icst a cldto scaled was viscosity The oia enlsnumber Reynolds nominal a ehhas mesh hscltm hc consfrhl h ubro iestep time simulation. of finer number the the half to compared for accounts which time physical rbe si figure in as problem × 60 oe n ihhl h iesesadhl h icst we viscosity the half and steps time the half with and nodes 200 × 60 × 60 5.1 oe n h ahnme is number Mach the and nodes 5.1 h itr a bandatrtesame the after obtained was picture The . 5.3 otesm iuainpromdwith performed simulation same the to .Ti a w biu esn:First, reasons: obvious two has This ). ν Re 10 = 1400000 = 6 ooti h aephysical same the obtain to h computational The . . ubln wake Turbulent 5.2 rgfo it. from drag itclaeaeof average tistical iuain can simulations r r ocompare to try t h temporal the e ta condition. itial n.T retain To and. a btceat obstacle lar h eut any results the l o retain not uld eaverage we h g temporal ng, Ma iia we similar e ovseight solves ut tthe at sults 0 = higher f . 168 as s 111 .

5 5 Simulation of turbulent flow spatial and temporal resolution we had to repeat the simulation for many times with slightly different initial conditions to obtain ensembles of the flow field, which was very costly. Figure 5.4 compares the two temporally averaged flow fields. Even though, both simulations have not yet completely converged (they are not yet symmetric) large scale similarities are certainly evident. It is also possible to investigate higher temporal moments of the velocity field. Here we show only the variance (figure 5.5) as a measure of the turbulent energy. Turbulence is strongest in a certain distance behind the obstacle and gets weaker later on. The two simulations predict similar shapes of the turbulent cloud. However, the magnitude of the variance is larger in the fine simulations since more fluctuations are actually resolved.

5.3 Turbulence induced by Kelvin Helmholtz instability

The next numerical experiment deals with the free decay of turbulence in a periodic domain without boundary conditions. Turbulence is first induced by two layers of constant flow in opposing directions and a small random perturbation. In between the two shear layers is a sudden inflection of the velocity which is not stable. The fluid is subject to a Kelvin Helmholtz instability which is known from ocean waves developing when the speed of the wind above water is different from the speed of the water. The presented setup does not not imply different densities for the fluids since that is not strictly necessary. For this simulation we use the floating point cascaded lattice Boltzmann algorithm with cut-off velocity vlimit = 0.4 (a velocity value which will not be reached in the simulation anyway). We chose a relatively coarse grid of 64 64 64 × × nodes and a nominal viscosity ν = 10−12. The velocity in the two shear layers are initialized with v = 0.01427 and v = 0.01427 respectively. Visible x1 x2 − roll-up of vortices in the inflection layer appears after 6000 time steps (see figure 5.6). The flow gets completely unstable over time and we observe free decay of turbulence which seems to have completely forgotten about the initial conditions (see figure 5.7). In order to investigate the effects of resolution we run the same simulation on a finer grid with 128 128 128 nodes and a viscosity ν = 2 10−12 to × × × obtain the same physically problem. The results are, however, not the same. The time needed for the roll-up to start is pretty independent of viscosity. We see the roll-up after about 6000 time steps again irrespective of the fact that one time step in the coarse simulations corresponds to to two time steps in the fine simulation. But we have to remember that the inflection layer is now

112 iue54 h vrgdtruetvlct edi h wake the in field velocity turbulent averaged The 5.4: Figure btcea hw nfigures in shown as obstacle ae 6tmstetm ftecas simulation. coarse simula the fine of the time that the Note times simulat samples 16 available. coarse of takes the easily number for were taken The were they samples apart. Sam because More steps 99. time and started. an 85 60 were 6160 averaging and for 120 before run taken respectively to were allowed steps were time simulations 3080 coarse the and fine h eoiyfil ftecas ( coarse the of field velocity the otmpcuesostefie( fine the shows picture bottom . ublneidcdb evnHlhlzinstability Helmholtz Kelvin by induced Turbulence 5.3 5.3 200 400 and × × 5.1 60 120 h o itr shows picture top The . × × 60 120 iuainadthe and simulation ) iuain The simulation. ) farectangular a of tion ples 113 ion d

5 5 Simulation of turbulent ﬂow

Figure 5.5: The variance of the temporal variations in the flow fields shown in figure 5.4 is a measure of the strength of the turbulence. Part of the flow field is not affected by turbulence. Strong turbulence is present in some proximity behind the obstacle. The higher resolved simulation at the bottom shows a much higher intensity of turbulent fluctuations.

114 iue56 olu fvrie nteifcinlyrbetwe layer infection the in vortices of Roll-up 5.6: Figure oe.Tesontm tp ag rm60 o10800. to 6000 from range steps time shown The nodes. hc r oigi poiedrcin.Tegi has grid The directions. opposite in moving are which . ublneidcdb evnHlhlzinstability Helmholtz Kelvin by induced Turbulence 5.3 ntoﬂi layers ﬂuid two en 64 × 64 × 115 64

5 5 Simulation of turbulent ﬂow

Figure 5.7: The same simulation as shown in ﬁgure 5.6 from time step 12300 to 36300. The ﬂow is now completely turbulent and has forgotten about its initial conditions.

116 iue58 olu fvrie nteiflcinlyrbetwe layer inflection the in vortices of Roll-up 5.8: Figure oedti rsn nti iuaina oprdt h pre the to compared as simulation one. this in present detail more hc r oigi poiedrcin.Tegi has grid The directions. opposite in moving are which orsod otesm hscltm pna h eut sho results the as span time physical figure in same the to corresponds 128 oe.Tesontm tp ag rm50 o130 This 15300. to 5700 from range steps time shown The nodes. 5.6 . ublneidcdb evnHlhlzinstability Helmholtz Kelvin by induced Turbulence 5.3 h olu cuserirhr.Teei significantly is There here. earlier occurs roll-up The . ntofli layers fluid two en 128 × 128 vious 117 wn ×

5 5 Simulation of turbulent flow physically thiner and the initial fluctuations are physically larger. Figure 5.8 shows the same physical time span as figure 5.6. The simulation results are qualitatively similar but both contain features down to their grid resolutions. Neglecting short wavelength contributions to the flow field is hence not always possible. Figure 5.9 shows the spectra of kinetic energy of the simulations1 which comply with the Kolmogorov theory. The energy was normalized. The magnitude of the energy, measured after the same physical time, is different in the simulations since the decay started earlier in the fine simulation. It must be noted that the Kolmogorov theory is statistical in nature and states that the energy law holds on average. Any concrete flow condition contains structured vortex features which do not comply with this theory. Each individual measurement is hence necessarily noisy.

5.3.1 Resolution of turbulent ﬂow We are interested in the eﬀective resolution of the cascaded lattice Boltzmann automaton in the under-resolved case. In the previous simulations, the lattice spacing was much bigger than the Kolomogorov scale. All resolved wave numbers are nominally in the inertial range and should follow the energy spectrum law:

2π −5/3 E(k)= E k−5/3 = E (5.2) 0 0 w Where E0 is a constant and w is the wavelength in multiples of grid spacings. We are interested in the smallest wavelength which complies with the energy spectrum law. We deﬁne compensated energy as E(k)k5/3. If our result complies with the Kolmogorov theory, compensated Energy should be constant:

5/3 −5/3+5/3 E(k)k = E0k = E0 (5.3) Figure 5.10 shows the compensated energy spectra from the free decay of turbulence induced by Kelvin Helmholtz instability versus wavelength. The compensated spectra should be horizontal. From the plots we would judge that the inertial range of turbulence goes down to six lattice spacings for both resolutions. The energy content at smaller scales is obviously too low for this Reynolds number.

1 For technical reasons, the simulations had to be repeated and were not completely identical

118 iue59 h n-iesoa nrysetu ftesimu the of spectrum energy one-dimensional The 5.9: Figure

E(k) 0.0001 1e-07 1e-06 1e-05 0.001 0.01 0.1 nﬁgures in oresmlto a esrdatr200tm tp n th and th steps for the time on data 24000 simulation The the after for superposition. measured data and good was range a a same simulation to follows the coarse scaled to spectrum was scaled the were energy that numbers the wave indicated The to law. eye power the to guide fe 80 iesescrepnigt h aepyia ti physical same the to corresponding steps time 48000 after 1 . ublneidcdb evnHlhlzinstability Helmholtz Kelvin by induced Turbulence 5.3 5.7 and 5.8 nt r rirr.Tesldln sa is line solid The arbitrary. are Units . 128 k 10 × 128 × 128 w=128 rdwsmeasured was grid w=64 k -5/3 ain shown lations me. k − 119 5 / e e 3

5 5 Simulation of turbulent ﬂow

0.1

0.01 5/3 Ek

0.001

w=64 w=128 0.0001 1 10 100 wavelength/grid spacing Figure 5.10: Compensated spectra of kinetic energy of the decaying Kelvin Helmholtz instability (same data as in ﬁgure 5.9) versus wavelength in lattice spacings. Energy was normalized. The Fourier transform is ignorant of wavelength shorter than two lattice spacings due to the Nyquist-Shannon theorem. According to the Kol- mogorov theory energy should depend on the wave number like E(k) k−5/3. Consequently, E(k)k5/3 should be a constant and ∼ the plots should be horizontal in the inertial range, the range of turbulence which should span the whole measurable spectrum. Here it is the case for wavelength longer than six grid spacings. Six grid spacings could therefor be taken as the minimal wavelength at which we observe turbulent behavior.

120 otie from (obtained sefigures (see hdigsat fe bu 00 iesesi h simulati the in steps an as time fact, 20900 in about are, results after The generatio starts steps. vortex shedding time the 100000 trigger for run to were order in conditions initial ubr r compared are numbers with unc exac the The beyond is with flow. square, domain or the a choose round in We is cylinder lattice. li it a single whether singl of a obstacle, a approximation the of for sufficient consisting numbers a obstacle Reynolds is an a higher whether Boltzmann much investigate lattice obtain we cascaded to the trivial With is 40. number minim Reynolds at latt at conditions [ cascaded flow known the laminar is whether It capture investigate also we does example automaton last our In flow laminar Under-resolved 5.4 efr ne-eovdsmltoswtota okturbul hock bec ad it without automaton simulations Boltzmann under-resolved lattice perform cascaded the irrespect compone With scale and small tions. the future neglect foreseeable to hardware an all in scientific improvements for of us problems requires dynamic terest fluid of intractability rule hitherto The it were However, simulations Accura under-resolved simulations. highly under-resolved that number. highly Reynolds in possible issue an any aut Boltzmann at lattice behavior cascaded the turbulent that saw we conclusion In Endnotes 5.5 beh flow. dynamic laminar fluid in correct spacing lattic the lattice cascaded capture single to The a seen obstacle. hence is the tomaton n behind a directly is nodes simulation the the this at in expect representation would computational we Their as simulation, the in present viously 1 × h eodtm u osihl ieetiiilconditions initial different slightly this. to by due affected time second the 1 × Re 30 19 = oe n h no eoiyis velocity inflow the and nodes 5.11 24 . 9 htlmnrvre hdigbhn iclrclnesst cylinders circular behind shedding vortex laminar that ] ν sefigure (see and 2 = × 5.12 10 Re hl ovre hdigocr ntesimulation the in occurs shedding vortex no while ) − 5.13 19 = 4 .Tesm rsueaymtyi ple othe to applied is asymmetry pressure same The ). .Nvrhls,tosainr otcsaeob- are vortices stationary two Nevertheless, ). . 9 100 otie from (obtained × 50 v in × 0 = 50 . ne-eovdlmnrflow laminar Under-resolved 5.4 oe.Teosal measures obstacle The nodes. . 0159 ν 8 = h cetfi euti not is result scientific The . w ieetReynolds different Two . × u yinstabilities. by out d v falimaginable all of ive .Bt simulations Both n. 10 nemdln and modeling ence hudb stressed be should gtv eoiyat velocity egative enlsnumber. Reynolds eo oi nodes solid of ne iiae.Vortex ticipated. t norsimula- our in nts niern in- engineering d nwith on otmn au- Boltzmann e − mtncaptures omaton mspsil to possible omes 4 c Boltzmann ice ranyo the of ertainty yi certainly is cy and ) oe Here node. e vo onto down avior lresolution. al tmtnit utomaton hp of shape t Re Re 79 = 79 = arts 121

5 5 Simulation of turbulent ﬂow

Figure 5.11: Velocity isosurface plots of the onset of vortex shedding behind an obstacle measuring 1 1 30 lattice spacings in a domain with × × 100 50 50 nodes. The Reynolds number is 79. Shown are time × × steps 20000 to 44000. Even though the object has the size of a single grid spacing, qualitatively correct ﬂuid dynamic behavior is observed.

122 iue51:Cttruhtevlct edo h aesimula same the of ﬁeld velocity the through Cut 5.12: Figure ﬁgure 79). 5.11 Dmnin fobstacle of (Dimensions 1 × 1 × 30 enlsnumber Reynolds , ina hw in shown as tion . Endnotes 5.5 123

5 5 Simulation of turbulent ﬂow

Figure 5.13: Flow field behind an obstacle measuring 1 1 30 lattice spacings × × at Reynolds number 19.9 after 100000 time steps. Two stationary vortices should form behind the obstacle at this Reynolds number. The physical size of the vortices is subgrid scale. Yet they are present in the simulations as can be seen from the fact that there is a back flow of fluid directly behind the obstacle. The blue isosurface indicates a velocity v = 10−5 (opposite to the main x − flow direction). The isosurface is thinner than one single lattice spacing.

124 apndi h ih ag fRyod ubr teeatpo investigated). exact (the further numbers lamin not Reynolds to of was flow ar range right steady obstacle the laminar minimal in of happened the transition of the that shape flo out that the turned laminar and and for dimensions spacing suitable exact grid that also the the is showed sim to automaton 19.9 The Boltzmann down and lattice behavior possible. 79 fluidic is numbers correct this Reynolds the that at out obstacle turned supposed minimal th simply was in It automaton Boltzmann assume resolved. lattice explicitly cascaded we the the did of that Nowhere accuracy high automaton. applic the sayi from Boltzmann of without benefit field goes also main It should the simulations automaton. become Boltzmann will lattice this cascaded that assume should we s vnthough Even ws. rused flow unsteady ar resolved that ng acddlattice cascaded n ftransition of int ltoso the of ulations neti,it uncertain, e ornunder- run to . Endnotes 5.5 h cascaded the to o the for ation derivation e eobtain we 125

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6 6 Conclusions loosing relevant geometrical information. The more realistic conclusion is that the Reynolds number and the Kolmogorov scale are not the limiting factors for the applicability of the lattice Boltzmann automaton anymore. However, it is still instructive to discuss what this would mean for the comparison with MRT simulations. Given that the wake simulation from the last chapter took about 14 days on a single processor for 17000 time steps and used 5760000 nodes (exact values depend on implementation details and hardware and are not of interest for an order of magnitude calculation). If we used the MRT model and were forced to choose a grid Reynolds number of 40 we had to break up each node to 109 MRT nodes and would need 1000 times more time steps. This would be 5.76 1015 nodes and 17000000 time steps. Now let us apply × the pessimistic assumption that the additional operations in the scattering cascade would make a single scattering event 100 times slower in the cascaded model as compared to the MRT model. Then the MRT model would still need 14 1010 days for the simulation given the absurd assumption that the mem- × ory management associated with 1015nodes 15links 8bytes 1017bytes, for × × ∼ storing the current state of the system only, would not cause any additional overhead. The 100000 terabyte machine had to work for 380 million years to solve this problem. To get hold of this time span we might take a look at the Devonian era 380 million years ago. This is the time when life claimed land. The first amphibian evolved from fish. The first insects and spiders appear and the earliest seed-bearing plants spread on land. Another 150 million years had to pass before the first dinosaur emerged. The span of 380 million years is the hole history of dry land life on earth. The 100000 terabytes computer would need just this time for the wake simulation which the cascaded lattice Boltzmann automaton performed in 14 days.

6.2 Turbulent ﬂow

The subject of under-resolved turbulent fluid simulations has been addressed by other means and the question will arise on how the cascaded lattice Boltz- mann automaton compares to established methods of turbulence modeling. The subject of turbulence modeling is broad and we cannot discuss the pros and cons of every model here (see [22] for a thorough discussion on turbulence modeling). The intrinsic difficulty of the topic might already be anticipated from the number of different models in use. Was one of this models superior to the others in most respects, the others would be obsolete. From the theoretical point of view, turbulence models are in general not very good justified. Simple

128 oetisfra ejsie oigo hsclgons Th t grounds. note physical should on we noting But justified we as modeling. insofar turbulence honest about said was what under-reso tu unstable if otherwise especially stabilize to questionable, turbu used least for I primarily at justifications stable. is seeking more grounds that even wit physical author it and make this unstable actually of not should opinion general to system a in contradiction from is in m ergy Nature be field flow should thermodynamics. the mechanism of in a components Such frequency mech stabilities. high physical of any discarding exists the there whether modeling turbulence h o edi ag dysmltosi nykondw oth to no down single known a only to is applied simulations is eddy filter. (it large the compact in super field Bolt flow is lattice The cascaded filter the the of that benefit A effectivelyis works scales. moments subgrid the order for third fluctuatio wavelength of filter short equilibration spatial remove that to a sim order There, some in also field simulation. velocity are eddy There large assumi called equilibrium. by method problem in closure are the casc moments solves the velocity it to since similarities automaton certain mann has R equation ave The Stokes unsisotropy. Reynolds the Navier a the for like isotropic account approach is models equation viscosity Stokes complicated eddy More while numbers. number wave wave unis the are instabilities on cause depend that errors hav numerical We the viscosities. that eddy algebraic apply models turbulence sudrorcnrli hscircumstances. viscos this st the in that flow control said our and honestly under gradients, be is cannot velocity It severe numbers. zero, wave high to close very encoun n typically are wave we low simulations This and turbulent velocities, under-resolved model. low our in viscosities, implic accurately high we viscosity relatively grounds adjust to physical able on are we problem models The viscosity eddy simulations. justify actually under-resolved is co viscosity of the eddy of stabilization that just physical use recognize a we make However, about which speculate cept. to models tempting all is It for viscosity. true especially is This esol o xld h acddltieBlzanautoma Boltzmann lattice cascaded the exclude not should We tsol entdta ottruec oesaedissipati are models turbulence most that noted be should It nteoiino hsato ti n ftems motn qu important most the of one is it author this of opinion the In toi nntr and nature in otropic mes oee,in However, umbers. vdsimulations. lved fiainfrti con- this for ification blnemdl are models rbulence e icste which viscosities ter led discussed already e . ubln flow Turbulent 6.2 ddltieBoltz- lattice aded sdfrnumerical for used ec oeigon modeling lence mn automaton zmann recognized We ns. eei hti we if that is here acddlattice cascaded e sasailfilter spatial a as ih etu for true be might yod averaged eynolds tyasm that assume itly sapidt the to applied is utrsa very at ructures ns ywhich by anism t normodel our in ity h eodlaw second the gta higher that ng lrte othe to ilarities ea time). a at de dat nall on acts nd cp feddy of ncept shnethe hence is t daigen- hdrawing ei nature. in ve ae Navier raged ycuein- cause ay a ewere we hat it of width e sin in estions o from ton 129

6 6 Conclusions

Boltzmann automaton is an idealized model. Physics was not involved in the derivation. We were not looking for physical truth but for a model that works out of pure logic. We might take it as a sign of virtue of the underlying theory that the cascaded lattice Boltzmann automaton solves the problem of turbulent flow simulations out of the box. However, that does not mean that some kind of turbulence model might not be beneficial in some cases. The role of this turbulence model was not one of stabilization but one of modeling the influence of the unresolved wave numbers on the resolved ones. In particular, it should be investigated whether it makes sense to feed the kinetic energy, which is lost due to the equilibration of high order moments, back into the system. An in-depth comparison of established turbulence models with the cascaded lattice Boltzmann automaton is certainly desired. This should be done by someone how is not preoccupied with one of the methods. The inventor of a certain scheme will always argue in favor for his own work.

6.3 Complexity

The complexity of the cascaded lattice Boltzmann automaton is certainly prob- lematic. The derivation of the scattering matrix and the collision cascade is a demanding task. Even worse is the computational effort which has to be put in the evaluation of the scattering operator. It might become difficult to apply the presented methodology to problems which include other effects like energy conservation, transport of passive or active scalars, multi-fluids and so on. The method of central moments is in principle simple but its results scale exponentially with the number of dimensions or included effects. The reason is the strong emphasis on mixed moments which grow very fast in number if new dimensions are added. It has not been investigated which mixed moments are negligible. We accepted the result of the central moments method as it was. However, the contribution of the high order terms might be very small. There is possibly a large potential for simplifications. Here it was the objective to show that the cascaded lattice Boltzmann automaton could be derived completely from first principle with all degrees of freedom constraint by Galilean invariance. However, the errors introduced by weakening some of the constraints might be acceptable. It should be emphasized that the cascaded lattice Boltzmann automaton is still simple and based upon a small number of basic concepts. The length

130 oriaesse oteltiecodnt system. the coordinate transform lattice to the requirement to the system from coordinate comes equations the of ftecsae atc otmn uoao.W i o comp not did We automaton. imp param most Boltzmann fit the lattice without of cascaded one model therefore, the a mod is, of for Ptolemaic parameters call the fit would for from said freedom razor be Occam’s also can if motion. that results, But correct produce correctly. might set mod they larg course, turbulence a with Of of typically to rameters. basis come back physical models Coming turbulence the that and recognize reasoning. flow simple turbulent of by ing parameters Ptolemaic the the c in obtain unde like compile we our parameters, to model on need our we light into whether servations some consideration t sheds behaves under it that problem However, system the idealized system. an natural build We numeric some why place. reason first only wron the the actually, in completely adjust is, is of This number which enough. the large theory that given a predictions that have correct should is produce we predictive thing model One the Ptolemaic them. From the between difference systems. no was coordinate there different of from descriptions seen phenomenological completely where models n rdcin ocrigtepaeaymotions planetary deriving the by concerning other the predictions no against to ing theory interesting one is test It to simpler. possibility was t univ it in centered as completely sun were esthetic motion scientific a planetary for q of t favor mechanisms sun is the in invariance when the argument Galilean nor only of s earth The point in s The Neither necessary. debates universe. the other question. the or many wrong of so earth center the As whether on universe. on argument the confront debate an of Galilei the center with the place. sents time first his the of in community forward put was variance 1 aie’ nih a httehael oiswr o what not were bodies heavenly the that was insight Galilei’s ti ntutv oivsiaetecs o hc h notio the which for case the investigate to instructive is It otepeitoso h tlmi oe.I shneunders hence is It contemporaries. model. his hea Ptolemaic convince the the to of motion able of the predictions of predictions the Galilei’s the to with improve not even did unknown him still by were motion planetary the nhsosrain ftepae fvns[ venus of sys phases Ptolemaic the the of that observations un proof his the The of in center question. the this sunsp not to was moon, earth unrelated the that on evidence as mountains qualify of not existence the However, time. 34 .Tepithr sta h ehnssof mechanisms the that is here point The ]. 1 hti obcueboth because so is That . e a rn st eseen be to is wrong was tem vre hs at r quite are facts These iverse. nih.Temdlfavored model The insight. hysee ob tthis at be to seemed they adbeta ewsnot was he that tandable el oisa compared as bodies venly t,adjva on do moons jovian and ots, eta aie a no had Galilei that te etbecontradict- testable h aaeesare parameters the ubro tpa- fit of number e a etri not is center a hat oe,o whether or model, beprmtr is parameters able lmdln works modeling al lntr motion planetary fGlla in- Galilean of n dtescientific the ed . Complexity 6.3 esm a as way same he edr,wsits was dark, he ratfeatures ortant lo planetary of el re tatime a at erse, nuiso ob- of enturies etro mass of center on fview, of point l ehv to have we els ine twas it cience, ere from learned ih still might g sadn of rstanding h model- the aiya a as ualify tr The eter. l empir- ile nrepre- un 131

6 6 Conclusions ical observations into a phenomenological model ignorant of the underlying mechanisms. We started with the knowledge of the underlying mechanisms and were forced by the nature of the cellular automaton to express them in a coordinate system which is bounded to the lattice. The conversion of the coordinate system results in a complicated model with many parameters. But we see that even if the equations of the cascaded lattice Boltzmann automaton remind us of complicated polynomial ﬁts which are often used in engineering sciences to comply with empirical observations, each term in the equations is justiﬁed on theoretical basis.

6.4 Future challenges

This thesis has not addressed certain topics which are essential for a successful application of the cascaded lattice Boltzmann automaton to problems of scientific or engineering interest. The most important are solid boundary conditions and local grid refinement. Both topics have been subject of intense studies in conjunction with other lattice Boltzmann automata and the obtained results might also suit the cascaded lattice Boltzmann automaton to some extend. However, there are certainly some challenges. The main concern in the modeling of solid boundary conditions in the standard lattice Boltzmann automaton is to capture boundaries which do not coincide with grid nodes [35–37]. This is a geometrical problem. With the high grid Reynolds numbers accessible with the cascaded lattice Boltzmann problem comes a new problem. The turbulent boundary layer on the surface on solid bodies is obviously much smaller than one lattice spacing. It is therefore necessary to included turbulent effects and surface parameters such as roughness into the boundary conditions. The subject of local grid refinement has been addressed by several authors [38–42]. The most common methods can be classified as finite volume based and interpolation based. The subject has probably not yet settled completely for the standard lattice Boltzmann automaton and it certainly has not sattled for the cascaded lattice Boltzmann automaton. The problem is the same as with the boundary conditions. The existing methods were developed for laminar flow conditions. Laminar flow is relatively smooth at short wavelength. Transfer of data from fine to coarse grids and vice versa is relatively straightforward, if velocity and pressure fields are smooth. Under turbulent conditions, things are quite different. We saw that turbulent effects are present down to the lattice spacing itself. For example, if a single vortex on the finer grid measures only two lattice spacings in diameter, it is simply impossible to transfer

132 r opeeyidpneto ahohradtecorrespond the and other i each reason of The s independent averaging. be completely temporal even computat are or run, more spatial long than requires the cheaper no simply in and might, does It averaging averaging ensemble theory. ensemble However, of the application of improvement The Bolt lattice answer. cascaded could averaged a deterministi ensemble was had the influence we which this points question Whether this flow. At the on thunderstorm. influence po it no flap critical to or wou had This the thunderstorm butterfly the over diverge. locate where flows places identical example, the practi nearly spots, for two vulnerable of which could, lots at We had time provi and and control. certainly turbulence flow would of in nature This averages. the ensemble into fashion. by evolut insights resolved its chapter and spatially last turbulence of and the build-up in the investigate shown could t fields way flow the compli is the a this whether certainly show is can subject giv work The scientific the intense with gird. only complying coarse grid the fine ap on the reasonable moments on a metho gradients maximization find velocity entropy to the some automaton lend Boltzmann from might gri coming lattice we fine By entropic grid destroyed. a fine is form the information gradien transition to which velocity in the coding one since by is moments grid grid coarse order a higher translate on to into node possible single are be There a should into field. it flow However, subgrid statis unique is possibilities. an information to subgrid translated the T be that cannot entities. is latti invariant) approach cascaded this (Galilean automa with the physical Boltzmann as for lattice moments cascaded possible order the be me only new only since conceptually would automaton information a it be subgrid would and the of This coupling exploit orders moments. to many order by able higher gr viscosity the be At in might jump we lower. a However, much experience is would automaton nomina flow the the Boltzman the since lattice for methods cascaded problem other the bigger for a than is automaton this Boltzmann interpolati Again, as of well viscosity. kind averaging as numerical same apply averaging apply However, methods to existing filter. necessary the primitive be of must Some It transition. grid. to coarser a to it ehia hleg st elc h eprladspatial and temporal the replace to is challenge technical A . uuechallenges Future 6.4 ncue additional causes on ac ffiegrid fine of patch a o natemporally a in ion ia nntr and nature in tical mn automaton zmann emi difficulty main he ig odecided to wings s hc ok sa as works which htensembles that s eu ihnew with us de n sbei a is usable and c acddlattice cascaded a applications cal nhge order higher en so h patch the on ts n simulations ing oo not. or go o of proximation high keeps ton h oregrid coarse the db h most the be ld ae n and one cated mlr faster, impler, Boltzmann ce lasseveral always oacoarse a to d eur any require t leigprior filtering nsi space in ints oigfrom coming icst in viscosity l oa power. ional vrgn of averaging dinterfaces id hdo grid of thod sfo the from ds tremendous magnitude. hnwe Then 133

6 6 Conclusions can be run in parallel without any need for communication between the jobs. Temporal averaging, on the other hand, requires one and the same job to run for a long time since turbulent features might be very persistent (the great red spot on Jupiter is such a turbulent feature which is known to exist for several centuries). With the introduction of grid computing into the scientiﬁc workﬂow comes to opportunity to lend a large number of processors for a short time. Ensembles can then be calculated independently and quickly since long simulations for temporal averaging are not necessary anymore.

Finally, there are a bunch of other topics of scientiﬁc interest. The method of central moments must be applied to a velocity set that allows for energy conservation. According to Noether’s theorem [43] it should then be necessary to add invariance under time shift to the symmetry constraints. A minimal model for energy conservation in 1D has four speeds, since there are three conserved quantities. Applying the homogenous velocity set constraint we see that we need 43 = 64 speeds in three dimensions for a model of an ideal gas without internal degrees of freedom. This seems to be quite costly but it would be instructive to see whether the method of central moments could deliver such a model and whether this model turned out to be superior to other models for thermal ﬂows.

It would be desirable to have a cascaded lattice Boltzmann automaton that allows for the transport of passive or active scalars. In principle it is possible to use the computed flow field to move entities by interpolation, but this is not desirable due to the high numerical diffusion associated with interpolation. The cascaded lattice Boltzmann automaton solves the problem of passive transport with a very low numerical viscosity (note the small numerical viscosity we experienced when we moved momentum in y-direction with a flow field in x-direction). The numerical diffusion of any passive value would be the same as the small numerical viscosity if we modeled the new passive entity as a new dimension of the automaton. This would introduce many new mixed moments. It must be investigated which of the mixed moments could be neglected without introducing numerical viscosity.

Other important topics include moving boundaries, multiphase flow, gran- ular flow, non-Newtonian fluids, magneto hydrodynamics and the application to other transport phenomena like electrons and holes in semiconductors, phonons in solids, photons in semi-transparent media and electromagnetic waves interacting with soft material as seen in applications of magnetic reso- nance imaging.

134 ilo,oewudlk ootmz h eino ag build large of design numbers the Reynolds at optimize still to but like t scale would cause smaller one and little cities billion, exa a nearby For to On supplies area. air smog. the constructi fresh in planed vital climate stop micro The T not the on environment. directions. effects the wind negative preferred for have o the importance She to of loads. fit buildin is wind to the severe building that by the firs wants effected design The who negatively climate? not engineer micro is civil the bridge, the to for do pl building interest climate micro the the of the does in does what What buildings and interest: around ing of flow points man two by the are made investigate There be protec to could in which was anything result problem than might larger Wel much which are turbulence. sedimentation which inducing induce p by can and erosion tures build worsen to might expensive s they very protecting place be coast might of of structures exa design erosion and protecting For arrangement the the simulate engineering. optimize to to civil interest with practical conjunction considerable in climates cro xasienme fsinicfilsta ih eetfrom benefit might t that method. impossible fields Boltzmann an scientific lattice is of it and number nature exhaustive in omnipresent is Turbulence applications of Fields 6.5 h hr emsmltosascae ihtewahrfore weather the with associated simulations term short T the techniques. Climat computational Th of intractability. magnitude. improvements tasks the of considerably scientific orders of many outstanding terms by most under-resolved in necessarily the as p well of of as one investigation importance is The applic climate flows. important oceanic the more and in atmospheric somewhat and in exotic seen Less evolut e the of t on properties. implications interior of profound molten has ionosphere The which flow the turbulent extension). some with The require wind for would this example solar turbulence. for of of problems, interaction understanding astrophysical turbulent to of applied lack be our could topics by of number inhibited large from a is benefit on only Progress can methods. which computational interest considerable of subject nitreit rbe fsgicn motnei h sim the is importance significant of problem intermediate An turbulenc that evident is it sciences, pure the Considering . ilso applications of Fields 6.5 oslnsi order in coastlines esm stu for true is same he edvlpetof development he o fismagnetic its of ion cast. ujc requires subject e rhi ujc to subject is arth iuain are simulations e netgtn the investigating mrvmnsin improvements ,freapea example for g, rcue.Coast tructures. nslk bridges like ings eerh(albeit earth he emgtthen might he r pe twsof was it mple, escn point second he wrong the in ut s onm an name to ask npr sciences pure in ot h build- the to do iesandbanks tive n hudnot should ons pe tshould it mple, sil changes ossible sb tefa itself by is e lcdstruc- placed l ntrso its of terms in xedn one exceeding toscnbe can ations lto fmi- of ulation h cascaded the nn phase. aning e method new usinis question t similar A . 135

6 6 Conclusions and the structures on top of them to reduce wind induced oscillations. For underwater structures in proximity or contact to the ground, such as poles of bridges, undersea cables, and pipelines, it is instructive to investigate the turbulence induced by the obstacle in order to optimize their design for a minimum for erosion in the turbulent wake since this could eventually cause the collapse of the supporting ground which might result in the destruction of the structure. But turbulence plays in important role at much smaller scales of engineering interest. It is obvious that automotive design as well as airplane design depends heavily on the availability of efficient and valid models for turbulent flow. This is true for the external as well as for the internal components of vehicles. The efficiency of motors depends on several fluidic components which are all subject to turbulence. Air must be brought to the combustion chamber efficiently. Inside the combustion chamber, turbulence is required for an efficient mixing of fuel and air. The combustion process by itself is turbulent. Finally, the combustion products have to be removed from the combustion chamber efficiently. This is true for the petrol engine of a car, the Diesel engine of a ship and the jet engine of an aircraft. Objects might be optimized for high turbulence or low turbulence. For example we might want to reduce turbulence in gas pipelines since turbulence induces additional pressure drops. But we would certainly try to induce turbulence wherever mixing or cooling is required. In both cases we need efficient and valid models for the simulation of turbulent flow. Without genuine models, the engineer might arrive at a reasonable design based on her or his experience. But engineering experience cannot be automatized. With a trustworthy model for turbulent flow comes the possibility to apply automated methods for optimal design, like shape and topology optimization techniques, to engineering problems which could otherwise only be solved approximately by inspiration or by design iteration in hardware. The low numerical viscosity of the cascaded lattice Boltzmann automaton and the implied high accuracy is also beneficial when no real turbulence is possible like, for example, in micro fluidic applications. Even laminar vortex shedding drives some computational tools to its limits. The cascaded lattice Boltzmann automaton can be used to simulate unsteady flow at a minimal resolution and hence with minimal cost. Optimization of micro fluidic structures, such like micro mixers, will benefit from the quick simulations. With the higher efficiency comes the opportunity to use many iterations of the optimization cycle in order to obtain better designs. Finally it should be noted that there might be applications for the cascaded

136 ehiusue odrv atc otmn automata. substantially Boltzmann differs lattice moments derive central extens to simple used of a techniques method to attributed The be cannot computati theory. they of that s terms large by so in enlarged are improvements accessible been obtained of has The range simulations magnitude. The Boltzmann lattice flow. intri th in turbulent have with bers to of even out simulation turned But the model for new discarded. the be information must subgrid smalle of and scales incomplete on few had is we a spacing information to that b down recognized to captured We was out ings. motion turned fluid further, Turbulent any ones. push mental th not cure could to we ways simple which found accuracy, mechan and the models into m such insight in simulation found instabilities We executable principles. an first obtained from fluids have we conclusions, In Endnotes 6.6 per weak the recognize quickly. and models turbulence applied world real lack turbulence. with However, inhibits rience standards. and viscosity scientific fl artificial at introduce water trivial to be of requireme to accuracy effects The seem ma the els methods. Movie capture computational with to fire communities. want and artistic designers in game fluids computer f turbulent market large for a potentially, tools in is, scientific There neither interest. are gineering which automaton Boltzmann lattice ecie.Tesse ste nain ne transformati under invariant evol then system the is which system o in The characteristics system obtain described. coordinate to transformatio special possible th any a be under on specify invariant must depend not it to is app system, seems system nate be a that might If moments system moments central system. a of central to method of the ce constraints of executable invariance method ne essence derive The the not to is problems. that sense other dynamics and makes fluid models it that which ton noted underst for be of applied subject should lack only and it our pure Finally by inhibited of lence. currently fields is many progress on which implications profound have h viaiiyo necetto o h iuaino tur of simulation the for tool efficient an of availability The utmr aeexpe- have Customers rraitcsimulation realistic or m h iisfor limits The em. facrc tends accuracy of h ytmthat system the f uetflwmight flow bulent sso numerical of isms auenro en- of nor nature hntelattice the than r nigo turbu- of anding o fteknown the of ion n ftecoor- the of ons ftecoordi- the of n sccapabilities nsic omneo the of formance to a obe to has ution vrlodr of orders everal enlsnum- Reynolds t ftemod- the of nts dlfrsimple for odel esa elas well as kers llrautoma- llular nlefficiency onal rmprevious from ut funda- quite e st establish to is . Endnotes 6.6 coordinate e esrl the cessarily discarding e atc spac- lattice w smoke, ow, cecsin sciences idto lied 137

6 6 Conclusions dinate system, if the characteristic coordinate system is unique and can be obtained from any coordinate system. For the lattice Boltzmann automaton the characteristic point is the mean of the velocity distribution, the only special point in the system. Other systems might have other characteristic points and may also include characteristic directions and distances. However, the methodology should be quite universal. As a final remark we should note that the improvements of the lattice Boltz- mann automaton introduced here are mostly trivial and far from being new (a notable exception is the scattering cascade). Others used complete homogenous velocity sets like the D3Q27 lattice. Others tried higher order equilibria. Others equilibrated higher order moments. The difficulties we encountered are related to the non-linear nature of turbulence and the multi-causality of numerical instabilities. The important point here is that none of the introduced corrections lead to a significant improvement one their own. Our analytical mind is used to gradual improvements. We would, for example, compare the SRT model on a D3Q19 lattice to a SRT model on a D3Q27 lattice. If we found that we were able to increase the lattice Reynolds number by one order of magnitude, we might proceed with the D3Q27 lattice. However, in reality we would not find a significant improvement and would instead try to increase the polynomial order for the equilibrium for the D3Q19 lattice since our linear mind had already ruled out any benefits from the D3Q27 lattice. Once again we would not find a significant improvement over the standard model and would abandon this idea, too. What we needed to understand was that each source of instability strikes roughly at the same point, namely when the energy cascaded of turbulence reaches length and time scales which are comparable to the grid spacings and the time steps. Therefore, stability is only obtained by eliminating all sources of instability simultaneously. The difficulty of the subject discussed in this thesis is that we could not proceed by trial and error. There is no path of gradual advancement coming from the standard lattice Boltzmann automaton to the new model. We were blinded by the complexity of the problem and cold logic was our only guide in the maze of non-linearity. The nature of turbulence, the most important unsolved problem of classical physics, remains still in the dark. But at least we can say that we had a first sensation of light in the distance. We found a direction and made the first step.

138 Appendices

139

iia acddlattice cascaded Minimal A edrvdtesm a sw eie h w-iesoa mod two-dimensional the derived we as way dimensions. same three the derived in be automaton Boltzmann lattice cascaded hsapni it h cteigmti n h equilibri the and matrix scattering the lists appendix This otmn uoao n3D in automaton Boltzmann onvnNeumann mo von truly John is – that methods, analytical shoul possible. This with mathematical, attack gradually it. an and to make relationship problems end, intuitive eff of useful, of complex a chance this veloping reasonable in a penetrations is real a there informati broader done, etc., of properly available, basis the must is becoming on this information repeated then If relevant is operation where calculations. str direct certain complex, by name this obtained could in Th one points cases. that gic special indications out of t an strong family by vast however, representative is, too that a are attack, of computational questions calculation direct in a the problems by solved the ’break be that to admitted hope e be some computational must well-planned, be but extensive, might by there deadlock’ conditions these a and Under used. factors, be relevant to the machinery to subjec analytical as the any proper disoriented of at part quite succeeded any having still in not are – penetration loose mathematical too deep still relat like intuitive is our subject That the above: reason to indicated The was prohibitive. as still probably are is moment with s this beset the at is with which approach ties, experience analytical entire purely the The that mecha for. indicates the called m of is understanding turbulence considerable detailed of a a that towards view effort matical the justify considerations These o h minimal the for a h oe can model The ,i the in d, ot.It fforts. othe to s difficul- r are, ere nthen on ionship o this for dthe nd l nythe Only el. ubject ective thing athe- ,we t, right nism ate- de- be re o 141

A A Minimal cascaded lattice Boltzmann automaton in 3D results are shown since the derivation is lengthy and does not provide new theoretical insights. It should be noted that the rotational invariance constraint for higher order moments is trivially met if all moments of a given order relax with the same relaxation constant. This allows us to be less stringent in the choice of the elementary collisions for higher order moments.

A.1 Scattering matrix

The scattering matrix contains the vectors for the conserved moments and the vectors for the elementary collisions in the following order: Mρ, Mx, My, Mz, Kxy, Kxz, Kyz, Kxx−yy, Kxx−zz, Kxx+yy+zz, Kxyy+xzz, Kxxy+yzz, Kxxz+yyz, Kxyy−xzz, Kyzz−xxy, Kxxz−yyz, Kxyz, Kxxyy+xxzz+yyzz, Kxxyy+xxzz−yyzz, Kxxyy−xxzz, Kxxyz, Kxyyz, Kxyzz, Kxyyzz, Kxxyzz, Kxxyyz, and Kxxyyzz. The matrix reads:

142 100000000 −300 0 0 0 0 0 0120 0 0 0 0 0 0 0 −8 2 1 −11010000 8 −1 1 0 −1 −1001110022 −2 0 −2 3 1 −10000011 −114 0 0 0 0 0 0 −4 −40 0 0 0 −40 0 4 6 7 6 1 −1 −1 0 −10 0 0 0 8 −1 −1 0 −110011100 −22 2 0 −2 7 6 7 6 1 0 −10 0 0 0 −1 0 −110 4 0 0 0 0 0 −4 2 −20 0 0 0 −4 0 4 7 6 7 6 1 1 −1010000 8 1 −101100111002 −2 2 0 −2 7 6 7 6 110000011 −11 −400 00 00 −4 −400004004 7 6 7 6 11 1 0 −10000 8 1101 −10011100 −2 −2 −2 0 −2 7 6 7 6 10 10 000 −1 0 −11 0 −40 0 0 0 0 −4 2 −20000404 7 6 7 6 10 1 1 0 0 −100 8 01101 −1 0 1 −2 0 −20 0 0 −2 −2 −2 7 6 7 6 1 −10101000 8 −101101011 −10 2 0 2 0 −2 −2 7 6 7 6 1 0 −1100100 8 0 −1 1 0 −1 −1 0 1 −2020002 −2 −2 7 6 7 6 11 0 1 0 −1000 8 101 −10 1 0 1 1 −1 0 −2 0 −2 0 −2 −2 7 6 7 6 10010000 −1 −11 0 0 −40 0 0 0 −4220000044 7 6 7 6 1 0 1 −100100 8 01 −10 1 1 0 1 −20 2 0 0 0 −2 2 −2 7 6 7 6 1 −1 0 −1 0 −10 0 0 8 −1 0 −1 1 0 −10 1 1 −1 0 −20 2 0 2 −2 7 6 7 6 1 0 −1 −1 0 0 −10 0 8 0 −1 −1 0 −11 0 1 −2 0 −20 0 0 2 2 −2 7 6 7 6 1 1 0 −101000 8 10 −1 −1 0 −10 1 1 −10 2 0 −2 0 2 −2 7 6 7 6 1 0 0 −10 0 0 0 −1 −110 0 4 0 0 0 0 −42200000 −4 4 7 6 1 −1 1 −10000000000001000 −1 1 −1 −1 1 −1 1 7 6 7 6 1 1 1 −100000 0 000000 −10 0 0 −1 −11 1 1 −1 1 7 6 7 6 111100000000000010001111111 7 6 7 6 1 −111000000000000 −10 0 0 1 −1 −1 −11 1 1 7 6 7 6 1 −1 −1 −100000 0 000000 −1000111 −1 −1 −1 1 7 6 7 6 1 1 −1 −100000000000010001 −1 −1 1 −1 −1 1 7 6 7 6 1 1 −11000000000000 −10 0 0 −1 1 −1 1 −1 1 1 7 6 7 4 1 −1 −110000000000001000 −1 −1 1 −1 −1 1 1 5 . cteigmatrix Scattering A.1 143

A A Minimal cascaded lattice Boltzmann automaton in 3D

A.2 Length of collision vectors

For the scattering process we need to compute the length of the scattering vectors. The derivation is the same as for the 2D model. We solve:

post m n l κ m n l = (s + (K k) )(c v ) (c v ) (c v ) /ρ (A.1) x y z i · i ix − x iy − y iz − z Xi for k. The equilibria for the one dimensional second order central moments 2 are κxx = κyy = κzz = cs for normalized density. Here we assume density ρ. The speed of sound for this model is ﬁxed to the same value as in the one-dimensional case (cs = 1/3). The equilibria for the remaining second order central moments are zero. This gives: p

k = ω (ne + neb + nef nw nwb nwf se seb (A.2) xy ⌊ ν − − − − − − sef + sw + swb + swf v v ρ)/4) − x y ⌋ k = ω ( eb + ef neb + nef + nwb nwf (A.3) xz ⌊ ν − − − seb + sef + swb swf + wb wf v v ρ)/4 − − − − x z ⌋ k = ω ( nb neb + nef + nf nwb + nwf + sb + seb (A.4) yz ⌊ ν − − − sef sf + swb swf v v ρ)/4 − − − − z y ⌋

k = ω 1/6( b e f + nb + ne + nf + nw + sb + se (A.5) xx−yy ⌊ ν − − − + sf + sw w + 2( eb ef + n + s wb wf) − − − − − + ρ(v2 2v2 + v2)) z − y x ⌋ k = ω 1/6( e + eb + ef n + nb + nf s + sb + sf w (A.6) xx−zz ⌊ ν − − − − + wb + wf + 2(b + f ne nw se sw) − − − − + ρ(v2 + v2 2v2)) x y − z ⌋ k = ω 1/126( b e f n s w + 2( eb ef (A.7) xx+yy+zz ⌊ BULK − − − − − − − − nb ne nf nw sb se sf sw wb wf) − − − − − − − − − − + 3( neb nef nwb nwf seb sef swb swf) − − − − − − − − + ρ(1 + v2 + v2 + v2)) y z x ⌋ For simpliﬁcation we use only two relaxation constants ων for the kinematic viscosity and ωBULK for the bulk viscosity. The bulk viscosity is typically

144 falhge re oet.Frtidodrmmnsalequi all moments order third For get: we moments. and order remainin higher we all since all for omitted true of are is constants scattering same all The following the moments. order second the of o osdrdsneti ol ml h edfreeg con energy choose for would need we case the incompressible imply nearly would this since considered not k k xxy xyy + + xzz yzz = = ⌊ ⌊ + − − − 1 + + 2( + + − + − − − 1 + + 2( + + − (1 (1 sef v sw v swb f nf swf v sw v sf f ef / / z x / z y / 8( 8( + + eb nb ( ( ( ( 4( 4( nb + − eb − − − − − sb nw + nb n + − − + − ne ne + swb ne nb swb − swb + + neb + neb sf swf ef wb + + neb − − ef neb − sf ne + − ne sw + − + − − − neb neb + se v + + + + + swb y swf − swf neb swf + nef nef wf + ( neb + nef wb se nw b − − nef nf wf + v + 4 + + + nef nef ( + x − + + 4 + 4 + − + e ( + + swf nef sw + b − nwb wf nwb 84 + nef k s − + nf nw k k xz + + wb nf + xy xy k − eb + xx + )) 4 + nw nw + se ) ) − + − k + − nw + − k xx + nwb − nwb nwf nwf nwb sb + xx yy k ω eb + nwb + + ef 1 yz − BULK − yy sw / − + nwb nwb )) yy + 4 . egho olso vectors collision of Length A.2 ne + 84 + + + v + − k se − ))) zz + − x nwf xx nwf nwf ρ seb seb + nwf + + wb 1 − 1 = ( k v / zz nwf nwf sf xx y 2 4 + − + + )))) v + + + + oeulbaetetrace the equilibrate to − y + sef sef seb yy wf ρ sb v seb sueequilibration assume sb + + ( sw z + 2 v )) + zz z + 2 se se + + − − ⌋ evto.I the In servation. + + seb sef swb swb seb + + sef oet.In moments. g iraaezero are libria v w x 2 seb seb + )) + + − sef ⌋ + + swf swf swb sef sef (A.9) (A.8) 145

A A Minimal cascaded lattice Boltzmann automaton in 3D

k = (1/8(eb + nb + sb + wb ef nf (A.10) xxz+yyz ⌊ − − sf wf + v (e + eb + ef + n + nb + nf + s + sb + sf+ − − z w + wb + wf + 84kxx+yy+zz + 2(ne + neb + nef + nw + nwb + nwf + se + seb + sef

+ sw + swb + swf + kxx−zz))) + 1/4( nef + neb + nwb nwf + seb sef + swb swf − − − − v (eb ef + neb nef nwb + nwf + seb sef − x − − − − swb + swf wb + wf + 4k ) − − xz v (nb + neb nef nf + nwb nwf sb seb + sef − y − − − − − + sf swb + swf + 4k )) 1/4v ρ(v2 + v2)) − yz − z x y ⌋

k = (1/8(eb + ef ne + nw se + sw wb wf (A.11) xyy−xzz ⌊ − − − − + v ( b eb ef f + n + ne + nw + s + se x − − − − + sw wb wf + 2(k k ))) − − xx−zz − xx−yy + 1/4(v XZ v XY + ρv (v2 v2))) z − y x z − y ⌋

k = (1/8(ne nb nf + nw + sb se + sf sw (A.12) yzz−xxy ⌊ − − − − + v ( e + b + f + nb ne + nf nw + sb se y − − − − + sf sw w + 2( k 2k ))) − − − xx−yy − xx−zz + 1/4(v XY v Y Z + ρv (v2 v2))) x − z y x − z ⌋

k = (1/8(eb ef nb + nf sb + sf + wb wf (A.13) xxz−yyz ⌊ − − − − + v (e + eb + ef n nb nf s sb sf z − − − − − − + w + wb + wf + 2(kxx−zz + 2kxx−yy))) + 1/4(v Y Z v XZ + ρv (v2 v2))) y − x z y − x ⌋

k = 1/8(neb nef nwb + nwf seb + sef + swb swf (A.14) xyz ⌊ − − − − v Y Z v XZ v XY 2ρv v v ) − x − y − z − x y z ⌋ For fourth order moments we need to consider the non-zero equilibrium eq eq eq eq 2 4 κxxyy = κxxzz = κyyzz = (κxx) = cs. This gives 1/9 in the normalized case (ρ/9 without normalization of the moments). The equilibrium of all other fourth order central moments is zero.

146 k xxyy + xxzz + yyzz = ⌊ 2( + + − − 4( + 2( + + − 3 + − − − 2(( + − + − − + − 1 / v nwb nb v ne nwb sef nb n sef 3( eb v sb 12( y z x ρ 2 2 2 − − v − nwf ( ( ( ( − − − − k − − − − x v ρ/ k k − + xx ne v x 2 − − 84 84 ef se 84 xx xx nf neb nf y v 3 − sf seb XY y 2 nwf nwf − − + k k − k yy − − − + xx xx xx zz yy − − − − nwb nw 2( + ) sf 96 + nb + + + v s nw − + − nef x swb 2 yy yy yy − − k − k v − − − ne v eb xx xx + + + z 2 + se seb x − sb zz zz zz sw v ne s v + + − − − − − nef x z sb − yy − zz − X − − − XZ v − nf ef swf − neb − + y ) 2 seb se − sf b e b v + sef zz + nf wb z − 2 − − − − se + − )) / v − )) − . egho olso vectors collision of Length A.2 − neb neb y e eb nwb eb − ⌋ − − v sw − Y nef w sef − y nw − wf − v swb sf + + − − f z − − Z Y ef ef − swf − v wb − − nef wb z nwf − nw Z sw − − nb sw ) − − swf ) f n + − wf − wf − swb swb w − sb wb − − seb − swf wf (A.15) )) )) 147

A A Minimal cascaded lattice Boltzmann automaton in 3D

k = 1/12( eb ef ne nw se sw wb wf (A.16) xxyy+xxzz−yyzz ⌊ − − − − − − − − + 2(nb + nf + sb + sf + vxX + v (ne neb nef + nw nwb nwf se y − − − − − + seb + sef sw + swb + swf − + 2(sf + sb nb nf (2k + 6k ))) − − − xxy+yzz yzz−xxy + v (ef eb + neb nef + nwb nwf z − − − + seb sef + swb swf wb + wf − − − + 2(nb nf + sb sf (2k 6k )))) − − − xxz+yyz − xxz−yyz + v2( b eb ef f n ne x − − − − − − nw s se sw wb wf 84k − − − − − − − xx+yy+zz 2(nb + neb + nef + nf + nwb + nwf + sb + seb − + sef + sf + swb + swf (k + k ))) − xx−yy xx−zz + v2(eb e + ef ne + neb + nef y − − nw + nwb + nwf se + seb − − + sef sw + swb + swf w + wb + wf + 42k − − xx+yy+zz + 2(b + f + nb + nf + sb + sf (k + 3k ))) − xx−yy xx−zz + v2(ne e eb ef + neb + nef z − − − + nw + nwb + nwf + se + seb + sef + sw + swb + swf w wb wf + 42k − − − xx+yy+zz + 2(n + nb + nf + s + sb + sf (k + 3k ))) − xx−zz xx−yy + 4(v (v XY + v XZ)) 8v v Y Z x y z − y z + ( 6v2v2 + 3v2(v2 + v2))ρ) − y z x y z ⌋

148 k xxyy k k k − xxyz xyyz xyzz xxzz = = = = ⌊ ⌊ ⌊ + + + + + + 1 1 1 ⌊ 4 + − 2( + − + + 2( + + − 2( + − + 2( + − + 2( + − + 2( + / / / 1 v v v v vx v 8( 8( 8( / z x y x y 8 s nw v seb v s v sef v sef v 2 2 12( Z XY XZ Y Z z y x z x v v 2 2 2 nwb 2 neb neb − + e eb b eb wb eb ( ( ( ( ( y x p Z Y p n nf ne b nb ne ( v + + + + − sb sb + + − + v + z + + − + + x − − + nb XZ + eb sb eb sef − − − − v − sf sw v XY + − + v e ef ef ef y z wf nef nef v z v n neb n − X neb + + + + Y nb X se se x x v + + ( + + + + n v + v y − − n p ne ef se f ef + z v y − + + + 2 + swb swb 2 + − + C P neb wb wb B P 2 + z − swb − ne nb + − v A P + eb − + seb seb + neb z nwb nwb nef nef 6 nb v Z Y − + + v + v sf − nf v f − w x + + + + z + y x 2 + ( + + ( wf wf − neb neb ( + + YZ XY v swf + ef nef ρ YZ XY swf ρ YZ XY + swf + + − ) − − z 2 ρ 3 sef sef 3 nef ne wb wb 3 v 3( + v sw nwf nwf nwf − − nw nw + + v y 2 z 2 + x v 2 + v + + (2 (2 − + + nef nef x v x + + + + + + nwb − v y v nf v k k wf wb sf sf wf + + − v z x 2 y w v nwb xxz xyy v nwb v ) z sb ) y + y ⌋ x − + ) seb ⌋ seb seb XZ 84 + + − + XZ ( + ⌋ Z Y + (3 + + + − v nf nf xzz . egho olso vectors collision of Length A.2 yyz − sw sw wf z 2 + nwf + − + se k ) + v k + + nwf xx + − k nwf sef sef sef y 6 + − 2 + xx + + − xx v ) − v v nw nw ρ z + 6 x swb swb z yy − k sf XY ) XY k k Z Y yy xx − − − − ⌋ yy + xxz xyy − + + + − − swb swb swb sb 2 + + + zz se ) ) yy 2 ) − nwb − nwb sw k + xzz swf swf yyz )) − xx k − + + seb xx − seb )))) ) + ))) + + swf swf swf zz − 42 + 42 + nwf nwf zz ))) ))) v k k y xx xx Y (A.17) (A.21) (A.20) (A.19) (A.18) + + yy yy 149 + + zz zz

A A Minimal cascaded lattice Boltzmann automaton in 3D

All equilibria of ﬁfth order central moments are zero. This gives:

k = 1/8(nwb neb nef + nwf seb sef + swb + swf (A.22) xyyzz ⌊ − − − − v A 2(v XYZZ + v XYYZ) v2X − x − y z − y n v2X + 2v ( v Y v Z ) − z p x − y p − z n 4v v (XY Z + v Y Z) v (v2PC + v2PB) − y z x − x z y 2(v2v XZ + v v2XY ) 4v v2v2ρ) − y z y z − x y z ⌋ k = 1/8(seb neb nef nwb nwf + sef + swb + swf (A.23) xxyzz ⌊ − − − − v C 2(v XYZZ + v XXYZ) v2Y − y − x z − x p v2Y + 2v ( v X v Z ) − z n y − x n − z p 4v v (XY Z + v XZ) v (v2PB + v2P A) − x z y − y x z 2(v2v Y Z + v v2XY ) 4v v2v2ρ) − x z x z − y x z ⌋ k = 1/8(neb nef + nwb nwf + seb sef + swb swf (A.24) xxyyz ⌊ − − − − v CA 2(v XYYZ + v XXYZ) v2Z − z − x y − x n v2Z + 2v ( v X v Y ) − y p z − x p − y n 4v v (XY Z + v XY ) v (v2PC + v2P A) − x y z − z x y 2(v2v Y Z + v v2XZ) 4v2v2v ρ) − x y x y − x y z ⌋

6 The only sixth order moment that we are able to ﬁt has equilibrium cs (1/27

150 or k xxyyzz osmlf h qain eue oesubstitutions: some used we equations the simplify To ρ/ 27 XY XZ Z Y X Y Z ntenraie n o omlzdcs respectively): case normalized non and normalized the in = = = = ⌊ = = = 5 + + 8 + + 2( + 4( + + + + 8 + 2( + 1 ef nb eb nb eb / 2( + 2( + 2( + + + + − v v swf v v 8( y x z y ρv v k 2 + − + − sf swf sef ne v v v + ( ( v v x xyyzz ρ/ − neb x x x 2 z y 2 v 2 x 2 nef neb neb ef ef ne eb v v ( A P neb X y v 8 + neb 27 − − neb y y v y 2 − n Y YZZ XY z − v − + + + + ) swb neb YZ XY − + + p z 2 + k − − sw 4 + + ) wb nb ne + + xxyyz nef neb nf ⌋ neb nef nef v neb nef nef − nef x + v − v + − + v + x 2 x − nef + z 2 v + swf − swb v )+ )) nf nw + − + − wf X z nw − y − nef − Z nwb v v nef p v nwb nwb nwb x nf 2 z n − nwb x + + nwb 4 + + − v ( 4 + − v v + v z 2 sb − x z 2 nw se z v + + swf C P − sb YYZ Y XY A + + − XY v k z 2 k nwb + + y − v 2 xz nwf nwb − nwb yz + − nwf nwf nwf v y + nwf + sf z Y nwf sw 4 + v se Z + nwb n y 2 + v ) p − − − C − x v 2 − − − − + k + nwf v y − nwf nwf xy + wb + seb XZ y 2 + sf wb seb seb seb v B P seb seb v y nwf + v z 2 − − . egho olso vectors collision of Length A.2 − + + − + z CA + − − wf + X Z XXY + sw sef wf seb sef sef sef v sb seb sef + x sef 8 + Z Y 8 + − 8 + − − se − − − − − seb − sef swb k k ) + sef swb swb swb k ) xxz xxy swb swb xyy seb + − + − + − + − − + sef yyz yzz swb − xzz swf swb swf swf swf swf − 8 + ) ) ) swf k (A.27) xxyzz (A.31) (A.30) (A.29) (A.28) (A.26) (A.25) 151 )

A A Minimal cascaded lattice Boltzmann automaton in 3D

X =eb + ef + neb + nef nwb nwf + seb + sef swb (A.32) n − − − swf wb wf + 4(k k ) − − − xyy+xzz − xyy−xzz X =ne + neb + nef nw nwb nwf + se + seb + sef (A.33) p − − − sw swb swf + 4(k + k ) − − − xyy+xzz xyy−xzz Y =ne + neb + nef + nw + nwb + nwf se seb sef (A.34) n − − − sw swb swf + 4(k k ) − − − xxy+yzz − yzz−xxy Y =nb + neb + nef + nf + nwb + nwf sb seb sef (A.35) p − − − sf swb swf + 4(k + k ) − − − xxy+yzz yzz−xxy Z =nef nb neb + nf nwb + nwf sb seb + sef (A.36) n − − − − − + sf swb + swf + 4(k k ) − xxz+yyz − xxz−yyz Z =nef eb + ef neb nwb + nwf seb + sef swb (A.37) p − − − − − + swf wb + wf + 4(k + k ) − xxz+yyz xxz−yyz

XY Z = neb + nef + nwb nwf + seb sef swb + swf + 8k − − − − xyz (A.38)

P A = 42k e eb ef ne neb nef nw nwb (A.39) − xx+yy+zz − − − − − − − − nwf se seb sef sw swb swf w wb wf − − − − − − − − − − 2(k + k ) − xx−yy xx−zz PB = 42k b eb ef f nb neb nef nf nwb − xx+yy+zz − − − − − − − − − (A.40) nwf sb seb sef sf swb swf wb wf − − − − − − − − − + 2kxx−zz PC = 42k n nb ne neb nef nf nw nwb (A.41) − xx+yy+zz − − − − − − − − nwf s sb se seb sef sf sw swb swf − − − − − − − − − − + 2kxx−yy

152 CA h nlrsl sgvnby: given is result ﬁnal The rexplicitly: or C A nw sw X Z XXY YYZ Y XY YZZ XY r = =4( =4 w s − 8 + − 8 + ← ← ← − − − − − k − ← + ← k k k xxyy 32 4( se neb nwf nw sw r xxyy k k xyy xyy 4 k s xx xx (12 + k k w = = = k xx − xxyy − xxyy + + + + xx + + + neb neb nwb − − − − xzz xzz xxzz seb − yy yy 2 + k xxzz k yy nef k 2 xxyy yy + + + + k xxyy 4 xxyy k − − − + zz zz xxyy xxzz k + xxzz − − − + xxyy xxyy yyzz zz − nef nef k k − 2( + 2( + + k neb sef − − xxy xxy xx xxzz + + − xxzz k − + nwb xxzz + yyzz xxzz xxyy yyzz − xxzz − + − + + − ne xxzz − − − zz − yzz yzz 32 nwb nwb k k + nef sw yyzz 4 + + − + − − 4( + xxyyzz xxyyzz + − + yyzz k xxzz k − + yyzz nwf 4 xx yyzz neb k xxyy − k k − + + xxyy + k k s xxy + − xxyy + swb xy xy − ← yy nwf nwf nwf 4 + k k − yyzz + + − − − + xxyy xxyyz + − + + 30 xxzz + s nef yzz k k sb zz xxzz seb − k xxzz k k k xxyyz xxyyz ) + xyy − + + xx xxyy xxyy − − − − swf − − K xxzz + − − yyzz + seb seb seb + + nb + seb eb 11 yy xzz · + + k yyzz sef nw + + k k xxyyzz + − xxzz xxzz k − + − + xxyy − − − xx k zz k − . egho olso vectors collision of Length A.2 ef − neb xyyzz xyyzz sef sef sef − k 4( + k sef + − + + xyy 11 + xyy nwb yy swb − yyzz yyzz ) xxzz 8 − k + − − − − k − − − 32 xx − + zz xzz xxyyzz xzz nef swb swb swb k − − + sf + k k k xyyzz − xx yyzz yy xyzz xyzz nwf swf + − − k + + − + + − xx k ) zz yy k ) swb ) ) yzz nf yzz swf swf swf + − + − yy zz k − − − wb xxyyzz − xxy xxy − − − nwb swf − 8 8 8 k k k wf xyyz xyzz xxyz ) (A.47) (A.52) (A.51) (A.45) (A.44) (A.46) (A.49) (A.43) (A.50) (A.48) (A.42) 153

A A Minimal cascaded lattice Boltzmann automaton in 3D

se se + k + k + k + k (A.53) ← xxyy+xxzz−yyzz xxyy−xxzz xyy−xzz yzz−xxy + k k + k + k xyy+xzz − xxy+yzz xy xxyy+xxzz+yyzz + 8k + 2( k + k k + k ) xx+yy+zz − xxyyzz xxyyz − xyyzz xyzz

e e 2k 4k 11k + k (A.54) ← − xxyy+xxzz−yyzz − xyy+xzz − xx+yy+zz xx−yy + k 4k + 4(k + k ) xx−zz − xxyy+xxzz+yyzz xyyzz xxyyzz

ne ne + k + k + k + k (A.55) ← xxyy−xxzz xxyy+xxzz−yyzz xyy+xzz xxy+yzz + k k k + k xyy−xzz − yzz−xxy − xy xxyy+xxzz+yyzz + 8k + 2( k k k k ) xx+yy+zz − xxyyzz − xxyyz − xyyzz − xyzz

n n 2k 4k 11k k (A.56) ← − xxyy−xxzz − xxy+yzz − xx+yy+zz − xx−yy 4k + 4(k + k ) − xxyy+xxzz+yyzz xxyyz xxyyzz

nf nf k + k k + k (A.57) ← − xxyy+xxzz−yyzz yzz−xxy − xxz−yyz xxy+yzz + k k + k + 8k xxz+yyz − yz xxyy+xxzz+yyzz xx+yy+zz + 2( k k k k ) − xxyyzz − xxyyz − xxyyz − xxyz

wf wf k + k + k k (A.58) ← − xxyy−xxzz xyy−xzz xxz−yyz − xyy+xzz + kxxz+yyz + kxz + kxxyy+xxzz+yyzz + 8kxx+yy+zz + 2( k + k k + k ) − xxyyzz xyyzz − xxyyz xyyz

sf sf k k k k (A.59) ← − xxyy+xxzz−yyzz − yzz−xxy − xxz−yyz − xxy+yzz + kxxz+yyz + kyz + kxxyy+xxzz+yyzz + 8kxx+yy+zz + 2( k + k k + k ) − xxyyzz xxyyz − xxyyz xxyz

ef ef k k + k + k (A.60) ← − xxyy−xxzz − xyy−xzz xxz−yyz xyy+xzz + k k + k + 8k xxz+yyz − xz xxyy+xxzz+yyzz xx+yy+zz + 2( k k k k ) − xxyyzz − xyyzz − xxyyz − xyyz

f f + 2k + 2k 4k (A.61) ← xxyy+xxzz−yyzz xxyy−xxzz − xxz+yyz 11k k 4k + 4(k + k ) − xx+yy+zz − xx−zz − xxyy+xxzz+yyzz xxyyzz xxyyz

154 b sb − ← wb eb 2( + − ← 11 b nb k 2 + k 2( + − 2( + − ← ← sb xxz xx − k k 8 + + ← eb wb − + k xxz xxz k + − − yy xxyy xxyyzz k yyz k − nb nwf nef k xxy k k − + + + xxyy nwb neb xx xxyyzz xxyyzz zz yyz yyz k + − k − + xxyy + xxyy xxzz − − yzz ← + yy − ← k k − + ← + − − ← xxzz xxyy yz + k ( ( − ( nef k − k k ( − + − nwf − xx − − zz neb k nwb xxzz k xxyyz + xz xz xxzz yyzz k xxyyz k xxyyz − − + k k k 2( + xxyyz xxyyz k yyzz xyyzz xyyzz zz + + xxz xxzz + − xxyy − − + 2 + + k k − k + + k − xxyy xxyy − k + k − xyz k − xyz yyz k 4 − + xyz + k xyy xyz k + + yyzz k xyy k k k xxzz xxyyz xxyyzz k xxyy xxyy k xyyzz k xyyzz + + + k k yzz + − − xyyzz + xyyzz − − xxyyz xxyyz xxzz xxzz xzz + k + xzz k k − − k + xxyz k yz xxyz yyzz − xxy k xxyz xxzz xxyz xxzz − + + − − − yzz − − + − yyzz k yyzz + − k k k k xxyz + + 8 + − k k − xxyyz k k + xxyyz xxyyz xxz − k k 4 + + xxz xxyyz xxyy xxy xxyyz yyzz xyyz xyyz k k k 8 + 8 + . egho olso vectors collision of Length A.2 k xxz xyyz k k ) xyyz − xyyz − k xyyz xx yyz + + ) + ) + xxz yyz ) ) + k k 4( + − + ) + ) + xxzz xx xx k yyz + + yy k + + xxz − k − − k + + xxyyz yyz k + k xxyyzz k k xxyyzz − yy k yy + − k xxyyzz xyzz k k zz xxyyzz xyzz − xyy yyzz xyzz k xyzz xyy + + yyz k xxyyz zz zz xxy + + + xzz xzz k + xxyz yzz + k ) xxyyzz (A.67) (A.70) (A.69) (A.68) (A.66) (A.65) (A.64) (A.63) (A.62) ) 155

A A Minimal cascaded lattice Boltzmann automaton in 3D

swb swb k + k + k + k (A.71) ← − xyz xxyz xyyz xyzz (k + k + k )+ k − xxyyz xyyzz xxyyz xxyyzz

seb seb + k + k k k (A.72) ← xyz xxyz − xyyz − xyzz (k k + k )+ k − xxyyz − xyyzz xxyyz xxyyzz

sef sef k k + k k (A.73) ← − xyz − xxyz xyyz − xyzz ( k k + k )+ k − − xxyyz − xyyzz xxyyz xxyyzz

swf swf + k k k + k (A.74) ← xyz − xxyz − xyyz xyzz ( k + k + k )+ k − − xxyyz xyyzz xxyyz xxyyzz

156 apn atc nt otereal the to units lattice Mapping B nti pedxw ics o h atc nt ftelatti the of units. units world lattice real the to scaled how be discuss can we automaton appendix this In otmn uoao sisma retm ewe iaycol binary between time free mean its is automaton Boltzmann n h pe fsound of speed the and nrdcdvamdlprmtr [ parameters model via introduced ecnwietevsoiyi ieetforms: kin different in appropriate viscosity an errors. the compressibility choosing write avoid by can We to c scale to order physical accuracy, the in of introduce possible terms as prob in considered low praxis, the as best in is condition flow it the incompressible, consider we a linear If capture phys best. conservation Note the energy nature. without to acoustic automata of matched is mann not problem specific typically the is unless sound sound of speed Boltzmann cl nainei udmna rpryo hscllaws physical of property fundamental a is invariance Scale world .e,i h quantities the if num e., dimensionless are equation i. complete a of coefficients The .Buckingham E. – fro it of differentiate interrelations and system. fixed system other the the siz characterize the on on which only depend units but not units do fundamental equation the the of coefficients the units, c s nytema retm sajsal.Telattice The adjustable. is time free mean the Only . Q r esrdb naslt ytmof system absolute an by measured are 44 .Tetobscprmtr ftelattice the of parameters basic two The ]. os h velocity the hoose htltieBoltz- lattice that osi ffcsat effects coustic mtcviscosity. ematic e ob nearly be to lem nti ae we cases this In cl sonly is Scale . eBoltzmann ce lisions clsedof speed ical any m bers, of e the τ binary 157

B B Mapping lattice units to the real world

2 ν = csτbinary (B.1) 1 1 = κ c2∆t (B.2) xx ω − 2 xy 1 1 ∆l 2 = κ ∆t (B.3) xx ω − 2 ∆t xy 1 1 = κ c∆l (B.4) xx ω − 2 xy The kinematic viscosity is hence given in units c∆l. Mapping a real world problem onto the lattice Boltzmann automaton requires choices for the particle speed c and the lattice spacing ∆l. Let us assume we wanted to simulate a car driving with 100 km/h (27.8 m/s) with a resolution of 0.1 m. It is necessary to choose the particle speed much higher than the highest velocity in the simulation. One order of magnitude is the minimum. Say, we decided to choose c = 278m/s and ∆l = 0.1m. The kinematic viscosity of air is ν = 14.23 10−6m2/s = 0.5119 10−6c∆l. The rest is simple algebra. Since × × we know κxx = 1/3 we can solve for ωxy:

1 1 0.5119 10−6c∆l = κ c∆l (B.5) × xx ω − 2 xy ωxy = 1.99999386 (B.6)

The duration of one time step in this case is ∆t =∆l/c = 0.36 10−3s. ×

158 Bibliography

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162 Index

airplane design, 136 closure problem, 29, 33, 63 aliasing, 40, 75, 98 coast protecting structures, 135 test, 84 collision cascade, 65 Ansumali, Santosh, 92 collision invariants, 39 anti-diffusion, 66, 69, 72 collision operator, generic, 45 artistic applications of lattice Boltz- collisions, 38 mann, 137 combustion, 136 atmospheric flow, 135 complexity, 130 automative design, 136 average of turbulent flow field, 111, D1Q3 lattice, 38 113 D2Q7 lattice, 14 ensemble, 133 D2Q9 lattice, 14 D3Q13 lattice, 15 Boltzmann transport equation, 20 D3Q15 lattice, 15 boundary condition D3Q19 lattice, 15 pressure, 23 D3Q27 lattice, 15 wall, 23, 132 Devonian era, 128 Buckingham, E., 157 Burnett equations, 21 eddy viscosity, 74, 129 educated guess, 10 cellular automata, 8, 13 elementary collision central difference, 27 1D, 39 central moments, 55, 67 2D, 52, 54 CFL condition, 24, 72, 100 3D, 142 chaotic flow field, 111 energy conservation, 23, 134 Chapman Enskog expansion, 8, 20, energy spectrum, 109, 110, 118, 119 31, 34, 63 compensated, 118, 120 civil engineering, 135 entropic lattice Boltzmann automata, climate changes, 135 18, 84

163 Index entropy, 18 Kolmogorov and closure, 31, 32 Andrei Nikolaevich, 107 equilibrium, 39 length, 26, 30, 64, 107 local, 105 theory of turbulence, 107, 109 unstable, 61 equivalent scales, 108, 157 Lamb, Horace, 7 erosion, 135 laminar flow, 121 Euler equation, 30 large eddy simulation, 77, 129 exactness, 104 lattice, 14 lattice Boltzmannn automata, 13 Feynman, Richard, 9, 63 lattice gas automata, 13 finite difference, 27 patent, 20 finite elements, 30 Lorenz invariance, 100 fire, 137 flow control, 133 Mach number, 8, 30 Frisch, Uriel, 20 Mandelbrot, Benoit, 127 Maxwell-Boltzmann distribution, 17, Galilean invariance, 43 22, 61 Chapman Enskog picture, 66 mean free path, 30, 66 of entropy, 19, 31 mean free time, 41, 64 of viscosity, 73 micro fluidics, 136 test, 84 model performance, 9 Galilei, Galileo, 131 moment space, 18 grid computing, 134 moments, 48 grid refinement, 132 momentum states, 13 multiple relaxation time operators, Hasslacher, Brosl, 20 18, 127 hexagonal lattice, 14 Navier Stokes equation, 7, 20, 26, ideal gas, 23 65 incompressibility assumption, 24, 100 one-dimensional, 29 instability, 24, 72 Neumann, John von, 141 integer, 17, 60, 102 Noether’s theorem, 134 internal degrees of freedom, 43 Nyquist-Shannnon theorem, 75 isotropy, 14 Occam’s razor, 131 Karlin, Ilya, 92 oceanic flow, 135 Kelvin Helmholtz instability, 112, over-relaxation, 78 115–117 Knudsen number, 8, 20 particle, 13, 37

164 Index polynomial expansion, 61 time, 37 Pommeau, Yves, 20 topology optimization, 136 Popper, Karl, 13 translation, 45 porper physical law, 44 of moments, 49 trustworthiness, 9 resolution of turbulence, 118 turbulence, 25, 100 reversibility, 33 boundary layer, 132 Reynolds average Navier Stokes equa- free decay, 112, 116 tion, 129 in wake, 108 Reynolds number, 26 modeling, 128 per grid spacing, 108 velocity bursts, 100 rotation, 46 of moments, 51 variance of turbulent ﬂow ﬁeld, 114 viscosity, 23, 64, 66 scale invariance, 157 2nd order error, 69 scattering rate, 41 bulk, 60, 92, 144 scattering step, 16, 39 de-aliasing, 77, 84, 92, 98, 99 Schrödinger equation, 9 numerical, 28, 88, 97 Schrödinger, Erwin, 37 vortex shedding, 121–123 shear wave decay experiment, 69, 97 weather forecast, 135 single relaxation time operators, 17 wind induced oscillations, 136 skewness, 40 Wolfram, Stephan, 20 smoke, 137 solar wind, 135 space, 37 spectral elements, 30 speed bounds, 100 speed of sound, 23, 30, 40, 65 square lattice, 14 state vector D2Q9, 21 stencil low pass characteristic, 28 streaming step, 16, 38 symmetry break of, 108 temperature, 23 thermodynamics, second law, 19, 72, 129

165