Since its origins, lattice-Boltzmann methods have been restricted to rectangular coordinates, a fact which jeopardises the applications to problems with cylindrical or spherical symmetries and complicates the implementations with complex geometries. However, M. Mendoza [1] 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, trumpets and tori. 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.

Rayleigh-Taylor and Kelvin-Helmholtz hydrodynamic instabilities are frequent in many natural and industrial processes, but their numerical simulation is not an easy challenge. This work simulates both instabilities by using a lattice Boltzmann model on multiphase fluids at a liquid-vapour interface, instead of multicomponent systems like the oil-water one. The model, proposed by He, Chen and Zhang (1999) [1] was modified to increase the precision by computing the pressure gradients with a higher order, as proposed by McCracken and Abraham (2005) [2]. The resulting model correctly simulates both instabilities by using almost the same parameter set. It also reproduces the relation γ ∝ √ A between the growing rate γ of the Rayleigh-Taylor instability and the relative density difference between the fluids (known as the Atwood number A), but including also deviations observed in experiments at low density differences. The results show that the implemented model is a useful tool for the study of hydrodynamic instabilities, drawing a sharp interface and exhibiting numerical stability for moderately high Reynolds numbers.

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The acoustic waves generated by moving bodies and the movement of bodies by acoustic waves are central phenomena in the operation of musical instruments and in everyday's experiences like the movement of a boat on a lake by the wake generated by a propelled ship. Previous works have successfully simulated the interaction between a moving body and a lattice-Boltzmann fluid by immersed boundary methods. Hereby, we show how to implement the same coupling in the case of a Lattice-Boltzmann for waves, i.e. a LBGK model that directly recovers the wave equation in a linear medium, without modeling fluids. The coupling is performed by matching the displacement at the medium-solid boundary and via the pressure, which characterizes the forces undergone by the medium and the immersed body. The proposed model simplifies the preceeding immersed boundary methods and reduces the calculations steps. The method is illustrated by simulating the movement of immersed bodies in two dimensions, like the displacement of a two-dimensional disk due to an incoming wave or the wake generated by a moving object in a medium at rest. The proposal consitutes a valuable tool for the study of acoustical waves by lattice-Boltzmann methods.