The sounding mechanism of a recorder-like air-jet instrument at low Strouhal number is numerically investigated by three-dimensional direct aeroacoustic simulation and acoustic simulation. Howe's energy corollary is applied to estimate the acoustic energy generation and absorption induced by an oscillating jet and vortex shedding. The quantitative results show that the main acoustic energy generation occurs in the jet downstream, and the absorption occurs in the jet upstream. It is found that the region defined by the Q-criterion identifies the main acoustic energy generation (absorption) region in the downstream (upstream) region of the jet. The results indicate that the vortex shedding mainly induced by the jet deflection gives additional contributions to the acoustic energy absorption. The shed vortices affect the temporal structure of the acoustic energy transfer, in particular, the timing of the double peaks with respect to the jet displacement. If we focus only on the air-jet, the dominant peak is observed when the jet crosses the edge from the inside to the outside of the pipe, as reported in previous experimental works. However, when we include the contributions of shed vortices, the dominant peak appears when the jet dives under the edge, which is consistent with the jet-drive model.

A two-dimensional model of the edge tone is studied by a highly accurate and reliable method of direct numerical simulation of the compressible Navier-Stokes equations, and used to verify key features observed in previous experimental and numerical studies, and to discover new features related to the jet motion and the edge tone generation mechanism. The first and second modes of the edge tone that are numerically reproduced agree well with Brown’s equation. In the mode transition region, dynamical mode transition is observed at a fixed jet velocity. For both first and second modes, the pressure distributions are antisymmetric with respect to the edge plate, and the sound intensity is proportional to the fifth power of the jet velocity. These results are consistent with the edge tone being radiated from a dipole-like source. Spatial profiles of the velocity and the velocity variance of the oscillating jet are also investigated for each mode over a range of the jet velocity including the mode transition regime. The amplitude of the velocity oscillation becomes constant with increasing jet velocity, while a measure of the amplitude of the velocity variance profile, which is introduced to characterize the strength of the jet fluctuation and named the ’fluctuation strength’, is proportional to the third power of the jet velocity. Some properties of the fluctuation strength correspond to properties of the sound intensity, including the first mode having larger amplitude than the second mode, and the way of deviating from the power law at smaller values of jet velocity and in the mode transition region. It is proposed that the third-power law exhibited by behavior of the fluctuation strength could be related to the increase of the skewness observed in the velocity profile with increase of jet velocity, and a model calculation is used to support this proposal.

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In this presentation, we will discuss acoustic energy transfer in a recorder-like air-jet instrument at a low Strouhal number. We will share the methods and results for a three-dimensional numerical analysis of acoustic energy transfer estimated using Howe’s energy corollary, based on the results of a large eddy simulation. The results are remarkably stable compared with previous two-dimensional numerical results, allowing us to consider the relative timing of the energy transfer with respect to the jet displacement. We will show that three-dimensional analysis is essential, especially when considering the energy transfer due to vortex shedding above and below the pipe edge (labium).

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