Theoretical Estimation of the Acoustic Energy Generation and Absorption Caused by Jet Oscillation

Abstract

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.

Role of the foot chamber in the sounding mechanism of a flue organ pipe

“…This method well reproduces the detail behavior of the jet motion and acoustic oscillation, such as the change of the acoustic frequency with the jet velocity. Furthermore, it can be used for the calculation of Howe's energy corollary and gives essentially the same result as that obtained experimentally [18,19]. Therefore, numerical simulation based on a compressible fluid scheme is an important tool to explore the detail behavior of flue instruments.…”

Nonlinear acoustics of an aperture under grazing flow

“…A recent contribution [20] introduces two alternative approaches to modelling the effects of nonlinearity on the aperture dynamics in presence of mean grazing flow. In particular, model 2 [20, §II.B] relies on a coherent vertical velocity profile, which is deduced from an assumed coherent displacement field along the mean shear layer, analogous to Takahashi et al [37]. Conversely, model 1 [20, §II.A] is based on the linear theory of Howe [33, pp.…”

“…However, we start from the nonlinear Navier-Stokes equations and aim at representing the dominant nonlinear features of the aperture flow, as inspired by the work of Hersh & Rogers [6] and Cummings [7] on the special case of zero mean grazing flow. This approach eliminates the need of modelling short cuts like empirical parameters or velocity profile assumptions as employed by the latter two references and alternative developments on the grazing flow problem [20,37], while still accounting for (some) relevant nonlinear effects. This offers both physical insight and reasonable predictions, at least in terms of general trends, as illustrated in §3(a) by comparison with other theories and in §3(b) by validation of the model predictions against recently reported experimental measurements [24].…”

Numerical study on edge tone with compressible direct numerical simulation: Sound intensity and jet motion

“…The first order approximation of the velocity profile of a static jet is given by a squared hyperbolic-secant function called the Bickley profile. 29 As the lowest order approximation, the velocity profile of the oscillating jet can be represented by 34 qðy; tÞ ¼ Usech 2 ½ðy À asinxtÞ=w y (11) where the parameters w y and a determine the width of the jet and the amplitude of the jet oscillation, respectively. When y is fixed, q(y, t) is a periodic function of t. Thus, the fluctuation Dq 2 ðyÞ is regarded as a time average of a periodic function.…”