Acoustic black hole analogy to analyze nonlinear acoustic wave dynamics in accelerating flow fields

Abstract

We present a physical model and a numerical method based on a space- and time-dependent Galilean-type coordinate transformation to simulate acoustic waves in the presence of an accelerating background flow field with sonic transition. Kinematically, the coordinate transformation is designed so as to maintain the well-posedness of the transformed wave equation, which is solved in a fixed computational domain using standard finite differences. Considering an acoustic black hole analogy, we analyze the nonlinear dynamics of acoustic waves in a stationary but non-uniformly accelerating flow field under the assumption of spherical symmetry. The choice of the acoustic black hole analogy is motivated by the fact that the steady-state spherical sonic horizon allows us to parameterize the wave-flow configuration in terms of a Helmholtz number He=c2/(λagh), which is expressed as a function of the speed of sound c, the emitted wavelength λa, and the flow acceleration at the sonic horizon, that is, the acoustic surface gravity gh. The results of the numerical simulations show that He describes geometrically similar sets of wave characteristics for different combinations of gh and λa. However, we also observe nonlinear variations of the wave amplitude along the wave characteristics, which are attributed to nonlinear Doppler modulations. It appears that these amplitude modulations depend on the acceleration of the flow field and can, therefore, differ for geometrically similar characteristics.