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Ab Initio Derivation of the Cascaded Lattice Boltzmann Automaton

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Ab Initio Derivation of the Cascaded Lattice Boltzmann Automaton 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 kl är 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, ro- tation, and the commutation of the collision operators are sufficient for the unique determination of all model parameters. No other physical or mathe- matical 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 automa- ton. 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 trans-
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