Ultrasound Scattering from a Rigid Sphere Using the Discontinuous Galerkin Method in the Time-Explicit Domain
Ultrasound technology is deployed in a variety of scientific and engineering applications, ranging from the non-invasive retrieval of system information to the stirring of molten metals. While ultrasonic sound waves found a wide range of applications in liquid media, the use of those in gaseous environments has not yet reached the same level of maturity – especially in the context of industrial measurements. This calls for fundamental investigations that focus on the propagation properties of airborne acoustic waves and their interaction with solids.
In this study, we examine the propagation of ultrasound in air and its scattering properties from rigid spheres. Four different sphere radii are examined, being 0.001, 0.01, 0.1 and 1 times the wavelength λ. The geometry is set as a 2D-axisymmetric rectangular enclosure with the sphere, represented as a half-circle, placed in the center of the z-axis. The model is based on the Convected Wave Equation, Time Explicit interface which features the discontinuous Galerkin method to solve the set of governing equations due to its capability to handle large-distance wave propagation problems in a memory-efficient way.
The numerical scheme is further tailored to the particular needs of a rigid sphere scattering by means of a mesh study. Due to the direct coupling of wavelength and maximum mesh element size by λ/2, the larger radii 0.1·λ and 1·λ are examined numerically, whereas the smaller radii, 0.001·λ and 0.01·λ, are investigated analytically. The mesh study sheds light on limiting cases in which the CFL condition leads to unreasonably small time steps due to the fine elements on the curved sphere boundary.
The simulation results of this study are compared to the analytical solution of rigid scattering phenomena based on spherical Bessel functions and Legendre polynomials and show good agreement between the corresponding radiation patterns. In the final manuscript, the analysis will be described in detail with discussion on the suitability of capturing scattering phenomena with the Ultrasonics physics within the Acoustics Module. This lays the fundament for improved acoustic simulations that include loss mechanisms.
