Lattice Boltzmann model in curvilinear coordinates for the study of the vibrational modes of a trumpet

Abstract

Since its origins, lattice-Boltzmann methods have been restricted to rectangular coordinates, a fact which jeopardizes the applications to problems with cylindrical or spherical symmetries and complicates the implementations with complex geometries. However, M. Mendoza [2] recently proposed in his doctoral thesis a general procedure (based on Christoffel symbols) to construct lattice-Boltzmann models on curvilinear coordinates, which has shown very good results for hydrodynamics on cylindrical and spherical coordinates. In this work, we construct a lattice-Boltzmann model for the propagation of scalar waves in curvilinear coordinates, and we use it to determine the vibrational modes inside cylinders and trumpets (both problems with axial symmetry). The model correctly reproduces the theoretical expectations for the vibrational modes, and exemplifies the wide range of future applications of lattice-Boltzmann models on general curvilinear coordinates.