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.

We investigate the energy transfer between the fluid field and acoustic field caused by a jet driven by an acoustic particle velocity field across it, which is the key to understanding the aerodynamic sound generation of flue instruments, such as the recorder, flute, and organ pipe.Howe's energy corollary allows us to estimate the energy transfer between these two fields.For simplicity, we consider the situation such that a free jet is driven by a uniform acoustic particle velocity field across it. We improve the semi-empirical model of the oscillating jet, i.e., exponentially growing jet model, which has been studied in the field of musical acoustics, and introduce a polynomially growing jet model so as to apply Howe's formula to it.It is found that the relative phase between the acoustic oscillation and jet oscillation, which changes with the distance from the flue exit, determines the quantity of the energy transfer between the two fields. The acoustic energy is mainly generated in the downstream area, but it is consumed in the upstream area near the flue exit in driving the jet. This theoretical examination well explains the numerical calculation of Howe's formula for the two-dimensional flue instrument model in our previous work [ Fluid Dyn. Res. 46, 061411 (2014) ] as well as the experimental result of Yoshikawa et al.

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.

Two-demensional (2D) models of a flue organ pipe are studied with compressible fluid simulation, specifically compressible Large Eddy Simulation, focusing on the influence of the geometry of the flue and the foot on the jet motion and acoustic oscillation in the pipe. When the foot geometry is fixed, the models having a flue with chamfers show good performances in stabilizing the acoustic oscillation in the steady state. Furthermore, we find that the foot chamber works as a Helmholtz resonator. If the frequency of the acoustic oscillation in the pipe is higher than the resonance frequency of the Helmholtz resonator by almost the full-width at half-maximum, anti-phase synchronization between the acoustic oscillation in the pipe and that in the foot chamber occurs. In this case, the acoustic oscillation in the pipe grows rapidly in the attack transient and is stabilized in the steady state.

In this talk, we discuss how to estimate the acoustic energy generation of “air-jet” instruments with numerical simulation. To attack this problem, we use Howe’s energy corollary, with which we can estimate energy transfer between unsteady flow, i.e., oscillating jet and acoustic field. To calculate Howe’s formula, we need solenoidal velocity of the flow and its vorticity together with acoustic particle velocity separated from the whole velocity of compressible fluid. Recently, a method, which allows us to approximately calculate Howe’s formula, was developed in experiments by Bamberger and Yoshikawa et al., and it can be applied for the numerical calculation. We apply the method for the numerical calculation of a flue organ pipe model. We also introduce a toy model of the oscillating jet to investigate the mechanism of sound generation from the oscillating jet in detail. The acoustic energy is mainly generated in the downstream of the oscillating jet near the edge of the mouth opening, but it is consumed in the upstream near the flue exit to synchronize the jet motion with it. Our results are in good agreement with the experimental result by Yoshikawa et al. as well as Howe’s theoretical prediction.