Coupling LBM and Digital Waveguides COUPLING LATTICE BOLTZMANN MODELS TO DIGITAL WAVEGUIDES FOR WIND INSTRUMENT SIMULATIONS

The lattice Boltzmann method (LBM) is a valuable numerical tool in musical acoustics, particularly when modeling wind instruments for which the interaction between the flow and the acoustic field is important. This paper describes an approach for addressing two limitations of LBM, namely computational cost and the inability to directly impose acoustical boundary conditions at open boundaries. The technique consists in simplifying the system by representing the parts where complex fluidacoustic interaction takes place with LBM, whereas regions that are well approximated by linear wave propagation are represented with a digital waveguide (DWG). The example provided consists of a clarinet-like system whose mouthpiece is represented by a LBM model coupled to a DWG, which represents the instrument's bore. The junction implementation between the LBM model and the waveguide, as well as the conversion of physical variables into wave variables, is detailed. INTRODUCTION The behavior of wind instruments is strongly dependent on the interaction between the flow and the acoustic field. These interactions explain several non-linear phenomena such as self-sustained oscillations in jet instruments, edge tones and nonlinear dissipation due to vortex shedding at the instrument discontinuities. The lattice Boltzmann method (LBM) is a useful tool to represent these systems due to its ability to resolve in a single time step the different scales associated with the flow and the acoustic field, thus facilitating the representation of the nonlinear phenomena previously mentioned. A significant number of LBM studies in music acoustics have been conducted beginning with the pioneering work of Skordos (1995), who represented the interaction between the fluid flow and the acoustic field in organ pipes. Buick et al. (1998, 2000) simulated the propagation of linear sound waves and later, simulated the formation of shock waves using different boundary condition schemes. Kuehnelt (2003) investigated the mechanisms of sound production in the flute using a three-dimensional LBM model. Atig (2004) represented the vortex shedding at duct terminations and Neal (2002) simulated flow aspects in lip-mouthpiece systems of brass instruments. Da Silva and Scavone (2007) proposed an axisymmetric LBM model to predict the acoustic radiation ISMA 2007 Coupling LBM and Digital Waveguides of ducts. More recently, da Silva et al. (2007) derived a single-reed mouthpiece model involving a moving boundary based on a distributed model of the reed. One important drawback of LBM in acoustics is the inability to specify an arbitrary boundary condition at a discontinuity. This problem is encountered when representing the radiation impedances at bore apertures, such as toneholes and open ends. An accurate but rather expensive solution for this problem involves including in the LBM model the radiation domain around the open end (da Silva and Scavone, 2007). This paper presents a technique to simplify the representation of radiation impedances at open boundaries by connecting the LBM grid to a low-order digital filter through a digital waveguide (DWG). As an example, we use a clarinet-like system whose mouthpiece is represented by LBM and connected to a waveguide representing the bore. The radiation impedance in the end of the waveguide is approximated by a low order digital filter, as proposed by Scavone (1999). OVERVIEW OF THE NUMERICAL TECHNIQUES Lattice Boltzmann Method (LBM) The lattice Boltzmann (LB) is classified as a nonequilibrium method whereby the fluid domain is investigated at a particle level. It was derived from the cellular automata method by implementing a simplification of the Boltzmann equation to describe simple collision rules to conserve mass and momentum. A full description of the lattice Boltzman theory can be found in Wolf-Gladrow (2000) and Succi (2001). In this paper we use the D2Q9 model, after Qian et al. (1992). This model is represented by a two-dimensional squared lattice with 9 sites. Each site connects to a neighbor lattice by a unity vector ! c i , where i = 1, 2, ..., 8, indicates the site number, with the exception of the rest site i = 0, represented by the null velocity vector ! c 0 . The discrete Bolzmann equation uses the simplified collision function known as LBGK defined with a single relaxation time τ, and given by ! fi(x + ci,n +1) " f i(x,n) = " 1 # ( fi " f i M ) (1) where ! fi are distribution functions associated with the vectors