The lattice Boltzmann method with applications in acoustics

An introduction is given to the lattice Boltzmann method and its background, with a view towards acoustic applications of the method.To make a larger range of acoustic applications possible, a point source method is proposed.This point source is applied to simulate cylindrical waves and plane waves, and is shown to give a very good numerical result compared with analytic solutions of viscously damped cylindrical and plane waves.Good results are found for simulations of Doppler effect, diffraction, and viscously damped standing waves.It is concluded that the lattice Boltzmann method could be suitable for simulating acoustics in complex flows, at ultrasound frequencies and very small spatial scales.The lattice Boltzmann method is shown to be unfeasible at lower frequencies or for larger systems. Abstract in NorwegianEn introduksjon gis til lattice Boltzmann-metoden og dens bakgrunn, med henblikk på akustiske anvendelser av metoden.For å muliggjøre flere forskjellige akustiske anvendelser foreslås en punktkildemetode.Denne punktkilden anvendes for å simulere sylinder-og planbølger, og det vises at den gir et svært bra numerisk resultat sammenlignet med analytiske løsninger av viskøst dempede sylinder-og planbølger.Gode resultater finnes for simuleringer av Dopplereffekt, diffraksjon og viskøst dempede stående bølger.Det konkluderes med at lattice Boltzmann-metoden kan være passende for å simulere akustikk i komplekse strømninger, ved ultralydfrekvenser og svært små romlige skalaer.Lattice Boltzmann-metoden vises å være upraktisk ved lavere frekvenser og for større systemer.vii viii NTNU's Department of Energy and Process Engineering.Joris has been very helpful in explaining the finer points of the lattice Boltzmann method to me.I would like to thank the community at LBMethod.org, in particular Jonas Lätt and Orestis Malaspinas.This site has been a very useful repository of knowledge for me, and I have received help at its forums when it was needed.Jonas has also been very helpful with questions about his PhD thesis, which has been one of my main sources.Finally, I would like to thank my advisors, for discussions and follow-up questions, and for letting me write about such an interesting subject.Lattice-Gas Cellular Automata and Lattice Boltzmann Models by D. Wolf-Gladrow, 2005.[3] This book is part of Springer's "Lecture Notes in Mathematics" series.It gives approximately equal weight to lattice gas automata and the lattice Boltzmann method, and focuses on theoretical analysis of the method.Lattice Boltzmann Modeling: An Introduction for Geoscientists and Engineers by M. C. Sukop and D. T. Thorne, 2006.[4] This book gives a very simple introduction to the lattice Boltzmann method, with later chapters describing how to implement more complex fluid models.Its scope is unfortunately somewhat limited, but it is an easily digested introduction to the field. Review articles:In addition to these books, a number of review articles for the lattice Boltzmann method have been written, [5-9] with more to be published soon. [10] CommunityThere is an active community around the documentation project at http://lbmethod. org.This is a wiki containing overviews of the lattice Boltzmann models, details of selected lattice Boltzmann aspects, lattice Boltzmann galleries, example codes in several * Note that the code used in simulations in this project is not based on OpenLB, as the author thought writing his own code would facilitate his understanding of the lattice Boltzmann method.* Unfortunately, there is no universally accepted definition of a tensor.It is worth mentioning, though, that this definition of a n-dimensional tensor corresponds to another common definition of a rank 1 tensor of order n.* Other derivations can be found in for instance section 2.3 of Methods of Theoretical Physics, [19] and section 6.4 of Theoretical Acoustics.[20]* The property of determinism is spoiled by the stochastic nature of force simulation in an LGA, if such a force simulation is performed.Still, it might be possible to change the particles' velocity in such a pseudo-random manner so that time reversal is still possible.* The reason for this is that the fourth vector component in the 4D lattice, which is removed in the projection to 3D, can have two different values: 1 and -1.* In the language of statistical mechanics, this can be formulated more briefly by saying that we take the average of all micro-states that share the same macro-state. *The tradeoffs of this are related to the fact that fi is now a real number instead of a boolean.Real numbers take much more memory storage space and are subject to round-off errors.* Locality is, as mentioned earlier, the property of being limited to single nodes. † This also applies for the Inamuro boundary condition. [36]* It is the most-cited method of those mentioned in this section, as determined through ISI Web of Knowledge.* Note that the specific acoustic impedance is a complex number, since phase is included in p and u.* This is named after the two physicists Sydney Chapman and David Enskog, who in the period of 1916-1917 independently used such an expansion to find a solution to the Boltzmann equation.[59]

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