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

Abstract

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.

Keywords

Edge tone DNS jet motion aerodynamic sound air-jet instrument

Introduction

The edge tone is generated from an oscillating jet colliding with an edge and is one of the aerodynamic sounds observed in a low Mach number region.^1–3 The elucidation of the mechanism of the edge tone has been a long standing problem in the fields of aeroacoustics, fluid dynamics and acoustics,² and its mechanism is not completely understood yet. Furthermore, the study of the edge tone is an important subject in the field of musical acoustics. The edge tone is the sound source of air-jet driven instruments, such as flue organ pipe, flute, recorder, ocarina and shakuhachi, and the study of the edge tone leads us to improving the understanding of the sound mechanism of air-jet driven instruments.^4–6

In the long history of study of the edge tone, some important phenomenological and theoretical formulae, which describe basic properties of the edge tone, have been proposed.^1,2,7–13 A very useful formula was introduced by Brown¹ based on his experimental results. According to Brown’s formula, the frequency of the edge tone linearly increases with the jet velocity until a mode transition arises. At the mode transition, the hydrodynamic wave of the jet discontinuously changes to a higher mode wave and a change of the tone is observed. The mode transition is hysteretic, that is, the upward and downward transitions follow different paths. The frequencies of individual modes are given by different linear functions of the jet velocity.

The source of the edge tone is the oscillation of the jet colliding with the edge. The jet oscillation is sustained by a feedback cycle: the interaction with the edge induces a feedback to the upstream jet flow emanating from the flue exit. This feedback mechanism has been studied by many authors,^7–10,13 Powell introduced a semi-empirical formula for the feedback cycle based on experiments, which he called the irrotational feedback mechanism.⁸ Holger et al. developed the formula focusing on the fluid mechanism of the vortex street and predicted the oscillation frequencies.^9,10 By using a linear analytical model of the incompressible fluid with the help of a trailing-edge Kutta condition and the Wiener-Hopf method, Crighton obtained an almost fully analytic explanation for the feedback cycle and gave a formula that is similar to Brown’s equation¹³

Powell and Holger et al. studied the sound intensity of edge tones experimentally and theoretically.^8–10 According to their experiments, the edge tone is generated by a dipole-like source, and the intensity of the edge tone increases according to the 6th-power of the jet velocity (6th-power law) as expected from the theory of aerodynamics sound.^2,3 The experiments reported observation of multiple higher modes.^1,8 The intensity of each mode of the edge tone obeys a different 6th-power law.⁸ Thus, the sound intensity at a given jet velocity depends on the mode selected. Kaykayoglu and Rockwell indicated that a pair of effective pressure sources are located slightly downstream of the edge tip on both sides of the edge plate.¹¹

In spite of all the above efforts, the generation mechanism of the edge tone has not been adequately explained in terms of the aerodynamics sound theory, yet.^2,3,14-16 As an alternative approach, we can rely on numerical methods for compressible fluids, which reproduce the hydrodynamic and acoustic motions simultaneously. In recent years, the edge tone has been numerically studied using several different methods.^17–24 Dougherty et al. adopted a compressible fluid simulation and partially reproduced the structure of the mode-transition region.¹⁷ Paál and Vaik reported that the mode transitions of the jet motion are reproduced by an incompressible fluid simulation in a laminar flow regime for a jet with a top-hat inlet profile and the dynamical transition is observed at low Reynolds numbers in the mode transitions region, though the underlying mechanism of the dynamical transition was not explained.²¹ The properties of mode transitions change depending on whether the inlet jet velocity profile is a top-hat profile or a parabolic profile, which was also confirmed experimentally.²³ The relationship between the Mach number (jet velocity) and the frequency of the reproduced acoustic wave was discussed in a numerical study with compressible fluid simulations.^19,20 A 3 D edge tone was calculated with a hybrid method, which couples a 2 D fluid simulation with a 3 D acoustic simulation.²² Many types of hybrid methods are widely used for the analysis of aerodynamic sound, because they reduce the computational cost.²⁵ However, the hybrid methods cannot reproduce acoustic feedback to fluid motion, which should not be ignored for the edge tone and air-jet instruments. To the best of the authors’ knowledge, no simulation method has reproduced the 6th-power law of both the first mode and a higher mode together in one model.

The most rigorous numerical method is the method known as the compressible Direct Numerical Simulation (DNS) method.²⁶ This method obeys the governing equations of the compressible fluid without any artificial viscosity terms. It reproduces the interaction between the fluid and acoustic fields and acoustic wave radiation with high accuracy, although it requires a huge amount of computer resources. Dougherty et al. partially reproduced mode transitions but did not fully reproduce acoustic waves due to a small mesh size.¹⁷ Paál and Vaik reproduced mode transitions in a laminar flow regime with an incompressible fluid simulation, but they relied on a hybrid method to reproduce acoustic waves.^21,22 A previous work²⁴ compared the result of compressible Large Eddy Simulation (LES) with that of compressible DNS for a 2 D edge tone model. The first and second modes reproduced by the compressible DNS with extremely small numerical grids,²⁶ well follow Brown’s equation. On the other hand, the compressible LES can also reproduce the first and second modes, when taking a relatively small numerical grid, but its accuracy is low: the frequency of the second mode is obviously higher than that predicted by Brown’s equation. These results indicate that the compressible DNS is necessary to truly reproduce the edge tone.

Our ultimate goals are to clarify the underlying generation mechanism of the edge tone by directly evaluating the generation process of aerodynamic sound from the oscillating jet and to comprehensively understand the sounding mechanism of air-jet instruments driven by the edge tone. In this paper, as the first step, we study a 2 D model of the edge tone with the compressible DNS,²⁶ to check the reliability of the numerical method and to discover new features related to the jet motion and the edge tone generation mechanism. Specifically, we consider a jet with a parabolic inlet profile, because it is more stable than a jet with a top-hat inlet profile.^4,23 In particular, we study the following problems. First, we consider the problem of how the intensity of the edge tone changes with the jet velocity for the first and second oscillation modes, namely the power law of the sound intensity, and we explore the mechanism of the dynamical transition observed in the mode transition region. Next, the spatial profiles of the velocity and the velocity variance of the oscillating jet are investigated for each mode over a range of the jet velocity including the mode transition regime. In particular, we introduce a novel measure based on the jet velocity variance profile as a measure of fluctuation strength that scales with jet inlet velocity. We also discuss the difference in the jet motions of the first and second modes.

The organization of this paper is as follows.

In ‘Sounding mechanism of edge tone’ section, we briefly explain the basic properties of the edge tone by using Brown’s equation,¹ We also explain that the edge tone radiated from a dipole-like source follows a 6th-power law, and the power index of the radiation is decreased by one for the 2 D systems, namely a 5th-power law is observed for the 2 D edge tone.^2,27

In ‘Model and numerical method’ section, we introduce the 2 D model of the edge tone studied in this paper and explain the numerical method, compressible DNS,²⁶ together with the numerical set up.

In ‘Numerical results: Dynamical mode transition and sound intensity’ section, we present results of the investigation of jet oscillation modes and their dependence on jet velocity. First, we consider pressure and velocity distributions. Then, we consider the frequency of oscillations, comparing with Brown’s formula. Finally, we consider the dependence of sound intensity on jet velocity.

In ‘Numerical results: Velocity profile and velocity variance of oscillating jet’ section, we consider the change of properties of the jet motion with the jet velocity. First, we investigate the temporal modulation of spatial profiles of velocity and introduce the skewness of the profile to characterize the temporal modulation. Next, we study the velocity variance of the jet fluctuation that is defined as the square of the standard deviation of the temporal fluctuation of velocity and is a function of the coordinate. The amplitude of the velocity variance profile, named the fluctuation strength, is introduced to characterize the strength of the jet fluctuation. In particular, we consider the problem of how the fluctuation strength changes with the jet velocity for the first and second oscillation modes, namely the power law of the fluctuation strength. We also introduce a theoretical model of the jet velocity profile, a skew-Bickley profile model, to support our numerical observations.

The final section is devoted to the summary and discussion.

Sounding mechanism of edge tone

Mode transition of edge tone

In this subsection, we review basic properties of the edge tone. As shown in Figure 1, a jet emitted from a flue collides with an edge and causes vortices. This interaction with the edge produces a feedback toward the flue exit, which influences the jet to oscillate in the vertical direction. This cycle of events is repeated periodically and the jet oscillation is sustained.^8–10,13 As a result, the aerodynamic sound, known as the edge tone, is created by the oscillating jet together with the vortices.^2,8

Figure 1.

2 D model of the edge tone and schematic picture of the jet. The parameters are taken as l = 5 mm, w_edge = 1.2 mm, w_nozzle = 2 mm, d = 1 mm, l_edge = 3 mm and L = 12 mm. The origin of the polar coordinates ( $r, θ$ ) is put at the tip of the edge, while the origin of the Cartesian coordinates (x, y) is taken at the center of the inlet.

The relation between the oscillation frequency of the edge tone and the jet injection velocity was experimentally obtained by Brown, i.e., Brown’s equation¹

f = 0.466 j (100 U - 40) (1 / (100 l) - 0.07),

(1)

where f [Hz], U [m/s] and l [m] are the oscillation frequency, jet velocity and distance between the flue exit and the edge, i.e., stand-off distance, respectively. Since the variables of f, U and l are in SI units, the two constants with units are 40 m/s and 0.07 m^– ¹, while the other constants are dimensionless. The parameter j changes depending on the oscillation mode of the jet and takes values j = 1.0, 2.3, 3.8 and 5.4. The value j = 1.0 corresponds to the first hydrodynamic mode and the other values correspond to higher hydrodynamic modes (see Figure 2). The frequency of the first mode increases linearly with increasing U. But the oscillation normally jumps to the next mode when U exceeds a threshold value. As long as it is allowed, this process, i.e., the transition from the n-th mode to the n + 1-th mode, is repeated with increasing U. As shown in Figure 2, the transition is hysteretic, namely the downward transition occurs at a different threshold value, which is less than that of the upward transition. Let us call the region between these upward and downward transition threshold values ’the mode transition region’. For the case of l = 5 mm, the transition between the first and second modes occurs in the range from U = 12.0 to 17.5 m/s. Hereafter, we concentrate on this case. From the view point of nonlinear dynamics,²⁸ the system becomes bistable, namely there are two attractors, in the mode transition region and, when the jet velocity U is fixed, which mode, the first or second mode, is excited depends on the initial condition and boundary condition.

Figure 2.

Mode transition of the edge tone.

Acoustic power radiated from Lighthill’s source

The generation mechanism of aerodynamic sound is formally explained by the Lighthill acoustic analogy.³ The aerodynamic sound obeys an inhomogeneous wave equation called the Lighthill equation, whose inhomogeneous term plays the role of sound source and is regarded as a quadrupole source. From the properties of the quadrupole source generated by turbulence flow, Lighthill showed that the acoustic energy generated by it is proportional to the 8th power of mean fluid velocity in transonic and subsonic ranges.³ Since the 8th-power radiation rapidly decreases with decreasing flow velocity, there is a possibility that a power law of different order dominates the sound generation in a range less than subsonic: 4th-power law of monopole-like radiations or 6th-power law of dipole-like radiations. Note that in the case of 2 D systems, the power index of the radiation is usually decreased by one, for example 4th and 6th-power laws change to 3rd and 5th-power laws, respectively (see Table 1).^2,27

Table 1.

Power law of sound source.

Sound source	Monopole	Dipole	Quadrupole
3 D	4th-power law	6th-power law	8th-power law
2 D	3rd-power law	5th-power law	7th-power law

The pressure distribution of the edge tone is antisymmetric with respect to the edge plate. It is known that the edge tone is radiated from a dipole-like source and obeys a 6th-power law for 3 D systems, that is, a 5th-power law for 2 D systems.⁸ From experimental observation for 3 D systems, we expect that it deviates from a 5th-power law in the mode transition region due to the instability of oscillations.

Model and numerical method

Figure 1 shows the 2 D model of the edge tone studied in this paper. The distance between the nozzle and the tip of the edge and the width of the nozzle are respectively taken as l = 5 mm and d = 1 mm. For other dimensions of the model see the figure caption. The ratio $l / d = 5$ is a typical value for air-jet instruments. We take a nozzle plate of semi-infinite length, which is extended to negative infinity. We do not incorporate a real physical flue pipe to avoid the disturbance of high frequency acoustic-resonances induced by a small tube of the flue. A jet flow is directly injected at the end of the nozzle plate. The velocity distribution at the end of the nozzle is given by the Poiseuille flow, i.e., a parabolic inlet profile, and U denotes the mean velocity at the nozzle end. In this case, the jet emitted from the inlet soon converges to a velocity profile well approximated by the Bickley velocity profile.^4,29

In this paper, we use the compressible DNS for calculating the edge tone, which is the most reliable scheme without any artificial viscosity terms.²⁶ To handle the boundary condition on the solid surface embedded in a flow field, we adopt the volume penalization (VP) method, which is one of the immersed boundary methods. In the VP method, penalization terms are added to the system of governing equations instead of imposing no-slip boundary conditions at the solid surface:

\frac{\partial ρ}{\partial t} + \frac{\partial}{\partial x_{j}} (ρ u_{j}) = - (\frac{1}{ϕ} - 1) χ \frac{\partial}{\partial x_{j}} (ρ u_{j})

(2)

\frac{\partial ρ u_{i}}{\partial t} + \frac{\partial}{\partial x_{j}} (ρ u_{i} u_{j}) = - \frac{\partial p}{\partial x_{i}} + \frac{\partial τ_{i j}}{\partial x_{j}} - \frac{χ}{η} u_{i} - u_{i} (\frac{1}{ϕ} - 1) χ \frac{\partial}{\partial x_{j}} (ρ u_{j})

(3)

\frac{\partial e}{\partial t} + \frac{\partial}{\partial x_{j}} ((e + p) u_{j}) = κ \frac{\partial^{2} T}{\partial x_{j} \partial x_{j}} + \frac{\partial}{\partial x_{j}} (τ_{i j} u_{i}) - \frac{χ}{η_{T}} (T - T_{0}) - \frac{χ}{η} u_{j} - \frac{1}{2} u_{i} u_{i} (\frac{1}{ϕ} - 1) χ \frac{\partial}{\partial x_{j}} (ρ u_{j})

(4)

where ρ is the density of the fluid, u_i (i = 1, 2) are the horizontal and vertical components of velocity, p is the pressure,

τ_{i j} = μ (\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}} - \frac{2}{3} \frac{\partial u_{l}}{\partial x_{l}} δ_{i j})

is the viscous stress tensor, e is the total energy, T is the temperature, T₀ is the temperature of the rigid bodies,

ϕ

is the porosity, η is the viscous permeability, and η_T is the thermal permeability. We consider an ideal gas, so the following equation is used in the simulation,

p = ρ R T = (γ - 1) (e - \frac{1}{2} ρ u_{i} u_{i})

(5)

where γ and R are the ratio of the specific heats and the gas constant, respectively. We also assume that the viscosity μ and the thermal conductivity κ are constant parameters and are related with Prandtl number as

P r = c_{p} μ / κ

, where the isobaric specific heat c_p is given by

c_{p} = γ R / (γ - 1)

When the speed of the sound c₀ and the mean density ρ₀ are given, the mean pressure and temperature are respectively given as $p_{0} = c_{0}^{2} ρ_{0} / γ$ and $T_{0} = p_{0} / (R ρ_{0})$ , so that the temperature of the rigid bodies is the mean temperature. To represent rigid bodies in VP terms, the mask function χ is set to

χ (x) = {\begin{array}{l} 1 if x is in a rigid body, \\ 0 otherwise \end{array}

(6)

For spatial derivatives, the sixth-order-accurate compact scheme is used except for the penalization terms for which the fourth-order central scheme is used.^26,30 For time evolution, the second-order implicit method is used for the penalization terms, while the second-order Adams-Bashforth method is used for the other terms. To suppress aliasing errors due to nonlinear terms and oscillations due to the penalization terms, the sixth-order compact scheme for filtering is also used at each time step.^26,30

We adopt a dimensionless system for numerical calculations, in which the speed of sound c₀, the mean density ρ₀ and the width of the nozzle d are taken as units of non-dimensionalization. For details, see the previous work.²⁶ To compare our results with those of the real physical experiments and numerical calculations in the previous works, we fix some of the physical quantities and dimensions of the model, and translate dimensionless quantities to dimensional quantities. In dimensional units, we take $c_{0} = 340$ m/s, $ρ_{0} = 1.2$ kg/m³ and d = 1.0 mm. Since the speed of sound c₀, coefficient of kinetic viscosity $ν (= μ / ρ_{0})$ and nozzle width d are fixed, the Reynolds number Re and Mach number M are proportional to the jet velocity U. Table 2 shows the correspondences between the values of jet velocity U, Reynolds number Re and Mach number M for the representative values of U in this work.

Table 2.

Values of jet velocity U, Reynolds number Re and Mach number M.

U[m/s]	3.3	4.6	6.7	8.0	10.0	12.0	13.3	15.0	16.7	17.5	18.3	20.0
Re	220	307	447	533	667	800	887	1000	1113	1167	1220	1333
M	0.0097	0.0135	0.0197	0.0235	0.0294	0.0353	0.0392	0.0441	0.0491	0.0515	0.0538	0.0588

Figure 3 shows a mesh of DNS near the operating portion of the edge tone, where the grid interval takes the minimum value, for the case of U = 3.3 m/s with time step $Δ t = 1.8 \times 10^{- 9}$ s. To accurately reproduce the behavior of the compressible fluid including acoustic waves, the minimum grid interval $Δ g_{\min}$ should be smaller than the smallest scale of the flow. Indeed, the minimum grid size is set as almost a tenth part of the boundary layers of the solid walls with handling sigmoid functions (see the next paragraph), where the boundary layers are estimated as $d / \sqrt{R e}$ . At U = 3.3 m/s, it is taken as $Δ g_{\min} / d = 6.4 \times 10^{- 3}$ . The minimum grid interval $Δ g_{\min}$ and time step $Δ t$ decrease with increasing U, because the Reynolds number is increased. For example, $Δ g_{\min} / d = 2.5 \times 10^{- 3}$ at U = 20 m/s with time step $Δ t = 1.2 \times 10^{- 9}$ s. Then, the minimum grid sizes are one-tenth smaller than those of the previous works.^17,21,23,24

Figure 3.

Mesh of DNS: the mesh near the operating portion of the edge tone at U = 3.3 m/s. The lines are drawn every 10 grid intervals.

Figure 4 shows the whole mesh of the computational domain in the dimensionless units, which is divided into four regions, i.e., boundary layer region R_bl, flow region R_f, sound region R_s and buffer region R_bf. The grid interval $Δ g$ takes the minimum value in R_bl and the maximum value at the outer circumference of R_bf; thus, it is stepwisely but smoothly increased from R_bl to R_bf through R_f and R_s by using sigmoid functions as shown in Figure 5. Thus, the numbers of cells at U = 3.3 and 20 m/s are 1549 × 1276 and 3582 × 2902, respectively.

Figure 4

The whole mesh of the computational domain in the dimensionless units. Dotted lines indicate the vertical and horizontal center lines.

Figure 5.

Grid interval $Δ g$ and the limit of the reproducible sound frequencies f_lim as functions of the grid index n_G on the vertical center line (y-axis) of the whole mesh at U = 3.3 m/s.

The limit of reproducible sound frequencies f_lim is determined by the grid width: the formal minimum wavelength $c_{0} / f_{\lim}$ corresponds to two grid intervals $2 Δ g$ , but a practical limit of reproducible wavelengths is considered to be several times $c_{0} / f_{\lim}$ , even though the sixth-order-accurate compact scheme is used. Figure 5 also shows the limit of reproducible sound frequencies $f_{\lim} = c_{0} / 2 Δ g$ as a function of the grid index n_G. In sound region R_f, the limit of reproducible frequencies is around 16.6 kHz. The dynamical variables are sampled in the region indicated by R_samp and the limit of reproducible frequencies of this region is around 126.3 kHz. The largest frequency of the edge tone treated in this paper, namely, the second mode at U = 20 m/s, is estimated as $f \approx 4$ kHz and is much less than f_lim of the region R_samp.

To remove reflection at the outer boundary of the computational domain, the non-reflecting boundary condition introduced by Poinsot and Lele is used.^26,31 However, this method does not work completely. To suppress motions in areas near the outer boundary, the computational domain is taken wide enough and the filtering effect is enhanced by taking grid sizes larger.^26,30 Thus, the reflection from the outer boundary is almost completely removed.

For calculations, we use the supercomputer, Fujitsu PRIMERGY CX2550/CX2560 M4 (3456 GFLOPS/node). For the cases of U = 3.3 and 20.0 m/s, the calculations of 0.01 sec using 1 node with 36 cores (36 parallel threads) take about 5 and 51 days, respectively. Therefore, a huge calculation time is required to obtain a stationary oscillation in time evolution, which is longer than 0.01 s. It is practically impossible to reproduce the hysteresis of mode transition with sweeping the jet velocity U adiabatically, very slowly. Thus, the jet velocity U is fixed at representative values in the range of 3.3 to 20.0 m/s. In the mode transition region, we observe which mode, the first or second mode, is excited depending on U.

Numerical results: Dynamical mode transition and sound intensity

In this section we present and discuss the results of the numerical study on the dynamical mode transition, frequencies of the edge tone and sound intensity.

Pressure and velocity distributions

First, in this subsection, we consider the properties of pressure and velocity distributions to check the reliability of the numerical scheme and to explore the mechanism of the dynamical transition observed in the mode transition region.

Figure 6(a) and (b) show the distributions of the absolute values of velocity at U = 8.0 and 20.0 m/s, respectively. Figure 7(a) and (b) show the corresponding sound pressure distributions at U = 8.0 and 20.0 m/s, respectively. Note that the sound pressure p_s is defined as $p_{s} \equiv p - p_{0}$ , where p is pressure and p₀ is pressure in the equilibrium state. The hydrodynamic oscillations of the first and second modes are observed in Figure 6(a) and (b), respectively. Thus, the mode transition is reproduced numerically. As shown in Figure 7, the pressure distributions in the upper and lower half regions seem to oscillate in anti-phase for both fundamental and second modes. Namely, antisymmetric waves are radiated to the upper and lower sides from the active portion of the edge tone. As pointed out by Kaykayoglu and Rockwell, the active potion can be reduced to a pair of effective pressure sources in anti-phase that are slightly downstream of the edge tip and radiate a pressure wave that is antisymmetric with respect to the horizontal axis.¹¹ In this sense, this radiation can be thought of as dipole-like radiation.

Figure 6.

Velocity distributions in the stationary states. The origin of the Cartesian coordinates (x, y) is taken at the center of the inlet. (a) U = 8 m/s. (b) U = 20 m/s.

Figure 7.

Sound pressure distributions in the stationary states. (a) U = 8 m/s. (b) U = 20 m/s.

In order to consider the properties of the sound wave, pressure oscillations are observed at the points of $(r / d = 100, θ = \pm π / 2 rad)$ , where the origin of the coordinate is at the tip of the edge (see Figure 1). Figure 8(a) and (b) show fluctuations of sound pressure at the observation points for U = 8.0 and 20.0 m/s, respectively. The stationary oscillations of the first mode are observed after t = 0.02 s at U = 8.0 m/s, while those of the second mode appear after t = 0.007 s at U = 20.0 m/s. For both cases, the oscillations at the two observation points are in antiphase with each other.

Figure 8.

Time evolution of sound pressure p_s at the observation points $(r / d = 100, θ = \pm π / 2 rad)$ : the red and blue lines denote the sound pressure at $θ = + π / 2$ and $- π / 2$ , respectively. (a) U = 8 m/s. (b) U = 20 m/s.

We show results in the mode transition region, where the first and second modes may appear in the course of time evolution. Namely, transitions between the first and second modes can be observed in time evolution; thus, even if one of the modes is excited, it may change to the other. In our simulations, the jet is sometimes destabilized by vortices, which can trigger a transition.

For example, Figure 9 shows the sound pressure fluctuations at the observation points for U = 16.7 m/s. The transition from the first to second modes is observed in time evolution. Even in this case, the fluctuations of sound pressure at the two observation points are in antiphase to each other. As shown in Figure 10(a) and (b), in the case of U = 16.7 m/s, the first and second hydrodynamic modes of the jet are observed at t = 0.0072 and 0.0142 s, respectively. There are often observed vortices with relatively high velocities near the jet in the mode transition region. They are shed by the interaction of the jet with the edge and drift near the jet. Such strong vortices are often observed in experiments of the edge tone and air-jet instruments.^{23, 32} Figure 11(a) and (b) show sound pressure radiations of the first mode at t = 0.0072 s and the second mode at t = 0.0142 s in the case of U = 16.7 m/s, respectively. Like those in Figure 7, the sound pressure distribution seems to be antisymmetric with respect to the horizontal line. In this sense, even in the mode transition region, the oscillating jet behaves as a dipole-like source.

Figure 9.

Time evolution of sound pressure p_s at the observation points for U = 16.7 m/s.

Figure 10.

Velocity distributions in the stationary states at U = 16.7 m/s. (a) The first mode at t = 0.0072 s. (b) The second mode at t = 0.00142 s.

Figure 11.

Sound pressure distributions in the stationary states at U = 16.7 m/s. (a) The first mode at t = 0.0072 s, (b) The second mode at t = 0.00142 s.

Let us consider the mechanism of the mode transition. For the case of U = 16.7 m/s, the pressure oscillations are destabilized in the time interval ( $0.0085 < t < 0.0125$ s), because the jet motion is influenced by vortices being closer to it. This interval is nearly 6 cycles of the first mode. Indeed, as shown in Figure 12, the vortices surrounded by the red circle are far from the jet at t = 0.0072 s but come closer to it at t = 0.0125 s; then, the oscillation of the jet becomes unstable. In the recovery process to a stationary state, the jet oscillation does not go back to the first mode, but changes to the second mode. Then, the frequency of the edge tone also changes from the first mode to the second mode. As another example, a transition from the second mode to the first mode is observed at U = 15.0 m/s. As shown in Figure 13, the transition starts at t = 0.0066 s and the recovery process ends at 0.0138 s. Thus, this interval is nearly 9.5 cycles of the first mode. The observations indicate that the system becomes bistable in the mode transition region, even if the one of the modes is first excited, and the disturbance due to the approach of vortices can induce the mode transition to the other mode. Vortices such as shown in Figure 10 are often observed near the jet in the mode transition region. Thus, in long time evolution, the disturbances would occur many times and intermittent behavior between the first and second modes should be observed. Typically, the intervals of the recovery process seem to be less than 10 cycles of the first modes.

Figure 12.

Vorticity distributions at U = 16.7 m/s. (a) t = 0.0072 s. (b) t = 0.011 s.

Figure 13.

Time evolution of sound pressure p_s at the observation points for U = 15.0 m/s.

Disturbance due to the approach of vortices is expected in the whole range of U, though the probability and impact may depend on the value of U. When U is out of the mode transition region, the probability of the disturbance seems to become low and the jet motion should return to the original mode in a recovery process, even if disturbed by vortices. Actually, at U = 6.7 and 10.0 m/s in the first mode range, the jet motions are disturbed by vortices, but they always return to oscillations of the first mode. For the case of U = 10.0 m/s, see Figure 20(a).

Let us comment on similarities and differences compared with other reports of mode transitions. Paál and Vaik found with an incompressible fluid simulation of a two dimensional model that dynamical transitions are often observed at low Reynolds numbers and a superposition of several modes occurs at high Reynolds numbers for a top-hat inlet profile,²¹ while a dynamical transition was not observed and each mode was rather isolated even in mode transition regions for a parabolic inlet profile.^8,23 They mainly treated the case of $l / d = 10$ , while our case is $l / d = 5$ . Thus, mode transitions at $l / d = 10$ occur at relatively lower Reynolds numbers than those at $l / d = 5$ and vortex shedding should occur less often. Furthermore, in their simulations, a tiny background flow (about 1% of the exit velocity) was added to stabilize the flow. We also suppose that a tiny background flow is difficult to exclude from experimental conditions. Since a jet with a top-hat inlet profile is more unstable than that with a parabolic inlet profile,⁴ we suspect that dynamical transitions could be observed, even though the jet is not disturbed by vortices. Overall, we hypothesize that such a tiny background flow washed out vortices near the jet out and the jet did not suffer the disturbance of vortices. In this sense, we treat an ideal case, in which the jet flow is well approximated by a two dimensional model and is not affected by any influence of a background flow.

Sound frequency compared with Brown’s equation

For comparison with Brown’s equation (1), we calculate the frequency of the pressure oscillation of the edge tone at the observation points. The calculation times of edge tones with DNS are not long enough to attain acceptable resolutions in frequency by FFT. Therefore, we measure the mean frequency of oscillation as follows. If $T'$ is a time interval of N cycles of the pressure fluctuation, the frequency f is approximated as $f = N / T'$ . The results are shown in Figure 14. When the jet velocity U takes values less than 12 m/s (Re = 800), the frequency of pressure oscillation follows Brown’s equation (1) with j = 1.0, namely the first mode. When U > 17.5 m/s (Re = 1667), it follows Brown’s equation (1) with j = 2.3, i.e., the second mode. In the mode transition region, the frequencies of the first mode are slightly higher than those of Brown’s equation, while the frequencies of the second mode, if they are observed, are somewhat lower than those of Brown’s equation.

Figure 14.

Sound frequencies f obtained from the results of DNS vs. Brown’s equation (1): the blue and red points are frequencies of the first and second modes, respectively.

In Figure 15, the Strouhal number $(S t = f d / U)$ is presented as a function of the Reynolds number Re. For each of the first and second modes, the Strouhal number slightly increases at the beginning and almost converges to that estimated by the Brown’s equation, but in the first mode, it slightly decreases, after the second mode appears, as reported in the previous works.^8,21,23 Note that in the numerical calculation by Vaik et al., mode transitions were not well reproduced in the range Re > 600 (U > 9 m/s) for the case of a parabolic inlet profile at $l / d = 10$ .²³

Figure 15.

Strouhal number St as a function of Reynolds number Re.

Sound intensity

In this subsection, we consider the change of sound intensity I with the jet velocity U. The sound intensity I is defined as

I = \frac{1}{ρ_{0} c_{0}} p_{a}^{2}

(7)

where p_a is the mean amplitude of the sound pressure fluctuation at an observation point in a stationary state.

Figure 16 shows the radiation pattern, the sound pressure amplitude p_a at $r / d = 100$ as a function of radiation angle for representative values of the jet velocity U. As shown in Figure 16(a), the sound pressure amplitude of the first mode increases with U and always takes the maximum value at the angle $θ = 11 π / 12$ rad. In the second mode shown in Figure 16(b), the sound pressure amplitude also increases with U, but it takes the maximum value at $θ = π / 2$ rad. Therefore, the directivities of the first and second modes are different. In Figure 17, the sound pressure amplitudes of the first mode are compared with those of the second mode at U = 15 and 16.7 m/s in the mode transition region. At almost all angles, the sound pressure oscillations of the first mode are larger in amplitude than those of the second mode; thus, the radiation power of the first mode is larger than that of the second mode.

Figure 16.

Changes of the sound pressure amplitude at $r / d = 100$ with angle θ. (a) First mode. (b) Second mode.

Figure 17.

Changes of the sound pressure amplitude at $r / d = 100$ of the first mode and second modes with the angle for U = 15 and 16.7 m/s.

The shape of the radiation pattern of the first mode shown in Figure 16(a) seems to be close to a cardioid. Ffowcs Williams and Hall discussed the case that a solid half plane is embedded in a turbulence flow, which can be applied to a two dimensional situation because they supposed uniformity in the direction of z. They showed that the sound intensity, which is proportional to the square of the sound pressure, changes as a cardioid as a function of the angle θ and obeys the 5th-power law U.^5,16 Even though an additional finite length edge plate exists in our case, their formula seems to be applicable to our results. The shape of the sound pressure radiation pattern of the first mode is similar to that of the square root of the cardioid function, $| \sin θ / 2 |$ (see the black dotted line in Figure 16(a)). However, that of the second mode is distinctly different from a cardioid shape (see the black dotted line in Figure 16(b)). Ffowcs Williams and Hall supposed a uniform distribution of aerodynamic sound sources near the solid plane. We hypothesize that the distribution of aerodynamic sound sources is not uniform at the second mode, which should affect the directivity of sound radiation. Indeed, the directivities of sound radiation change depending on many factors, for example the configuration of sound sources, their frequencies and the geometry of solid walls near the sound sources.⁴ The directivities also change between the two- and three-dimensional systems. Actually, it was experimentally observed that the radiation in the vertical direction is the strongest even for the first mode, where the nozzle palate is not a half plate and it cannot be reduced to a two-dimensional system.⁸ Note that we calculate the model with an edge plate of a finite length - the directivities should change for a model with a semi-infinite edge plate.

Figure 18 shows the changes of the sound intensity of the first and second modes as functions of the jet velocity U, where each sound intensity is calculated from the maximum value of the sound pressure amplitude p_a. The black lines are the slopes of 5th power. For both first and second modes, the sound intensity follows a 5th-power law as expected for dipole-like radiations. However, the intensity of the second mode is less than that of the first mode. The sound intensity of the first mode is 2.36 times larger than that of the second mode. Note that the sound intensities observed in experiments each obey a 6th-power law, but different modes have different intensity values.^8–10 Therefore, our numerical results correspond to a 2 D analog of the dipole-like radiation observed for the edge tone. Furthermore, in the mode transition region, the intensity of the first mode is suppressed and deviates from the upper black line while that of the second mode is also suppressed slightly. In addition, in a low velocity range, e.g., U = 3.3 m/s, the intensity drops off and deviates from the 5th-power law.

Figure 18.

Changes of the sound intensity I for the first and second modes with the jet velocity U. The dimensionless sound intensity I_n is defined as $I_{n} \equiv I / (ρ_{0} c_{0}^{3})$ . The black lines have slopes corresponding to the 5th-power law.

Numerical results: Velocity profile and velocity variance of oscillating jet

In this section, we study the properties of the jet motion in detail. Specifically, we explore how the transverse spatial profiles of jet motion and jet fluctuation change with increasing jet velocity. We also discuss the difference in the jet motions of the first and second modes.

Jet motion and velocity profile

In this subsection, we investigate transverse profiles of the oscillating jet. To do this, we fix the observation line parallel to the vertical axis at a distance of 3.0 mm from the nozzle ( $x / l = 0.6$ ), where the jet oscillation is stable and its amplitude is relatively large.

Figure 19 shows snapshots of normalized velocity profiles $q (y, t) / U$ in the stationary states at U = 6.7, 8.0, 10.0 and 12.0 m/s, where q(y, t) denotes the absolute value of the jet velocity (u₁, u₂), i.e., $q = \sqrt{u_{1}^{2} + u_{2}^{2}}$ , for fixed value of x. Note that in the range from U = 6.7 to 12.0, the sound intensity well follows the 5th-power law. The peak heights of the normalized velocity profiles q/U slightly increase with U and the right and left slopes of the distributions become steeper with increasing U, although the growth of the heights are relatively suppressed from U = 10 m/s to U = 12 m/s. At each value of U, the velocity profile sways almost periodically due to the oscillation of the jet. At the center position (see broken lines), the profile is almost symmetric and forms a bell shape. The profiles at the left and rightmost positions are almost in reflection symmetry to each other with respect to y = 0, but each profile is asymmetric: at the rightmost position, the right slope of the distribution is steeper than the left slope and vice versa. Note that such asymmetric velocity profiles are observed experimentally.³³

Figure 19.

Changes of the normalized velocity profile $q (y, t) / U$ in the stationary states at U = 6.7, 8.0, 10.0, 12.0 m/s. The solid lines denote the profiles at the left and rightmost positions and broken lines denote those at the center position.

To quantify the degree of asymmetricity of velocity profiles, we calculate the skewness s_U of the profile. Details are shown in the Appendix 1. Table 3 shows the skewness s_U of velocity profiles in Figure 19. From the definition given by equation (15), the skewness s_U is related with the difference in gradient between the right and left slopes of the normalized velocity profile q/U. At the center positions, $| s_{U} |$ takes relatively smaller values because the velocity profiles are almost symmetric. As the height of q/U increases with U, the difference in gradient between the left and right slopes increases and the skewness $| s_{U} |$ tends to increase with U at left and rightmost positions. At the rightmost positions, the absolute values of s_U increase with U, but the values at U = 10.0 and 12.0 m/s are almost the same. At the leftmost positions, the values of s_U do not increase monotonically. The value at U = 6.7 m/s is larger than that at U = 8.0 m/s and the value at U = 10.0 m/s is slightly larger than that at U = 12.0 m/s. The oscillations at U = 6.7 and 10.0 m/s are slightly unstable and asymmetric due to the influence of vortices, as in the case of the jet oscillation at U = 10 m/s in Figure 20(a), and so the velocity profiles at the left and rightmost positions are not completely in reflection symmetry. Note however that the sum of the absolute values of the skewness at the left and rightmost positions monotonically increases with U, though it is suppressed from U = 10 m/s to 12 m/s because the growth of q/U slows down.

Table 3.

Skewness s_U defined by equation (15) for velocity profiles in Figure 19.

U[m/s]	6.7	8.0	10.0	12.0
s_U (left)	0.695	0.622	0.714	0.712
s_U (center)	$- 7.33 \times 10^{- 2}$	$4.00 \times 10^{- 2}$	$- 4.43 \times 10^{- 2}$	$- 4.875 \times 10^{- 2}$
s_U (right)	-0.537	-0.668	-0.707	-0.710
$\| s_{U}$ (left) $\| + \| s_{U}$ (right) $\|$	1.232	1.390	1.421	1.422

Figure 20.

Oscillations of the maximum point y_max of the velocity profile at $x / l = 0.6$ mm. (a) U = 10.0 m/s (b) U = 16.7 m/s.

Next, we consider the amplitude of the jet oscillation in the transverse y direction. We let y_max represent the value of y at which the velocity profile has maximum value at time t. This maximum point oscillates almost periodically when the system is in stationary mode oscillation. We define the amplitude of jet oscillation in the y-direction as the amplitude of oscillation of the maximum point y_max, and denote it by a_j. Figure 20(a) and (b) show oscillations of y_max at U = 10.0 and 16.7 m/s, respectively. Figure 21 shows the changes of the amplitude of the jet oscillation a_J with U for the first and second modes. The oscillations at U = 10.0 and 16.7 m/s are relatively unstable and have a low-frequency drift even in a duration of a single mode due to the disturbance of vortices compared with more stable cases (see the error bars in Figure 21). Indeed, the sound pressure oscillations at U = 8 and 20 in Figure 8 are much more stable than that at U = 16.7 in Figure 9.

Figure 21.

Changes of the amplitude of the jet oscillation a_J at $x / l = 0.6$ in stationary states with U for the first and second modes. The error bars indicate mean absolute deviations (MAD) of sampled data.

At U = 16.7 m/s in Figure 20(b), a stable oscillation of the first mode is observed in the range $0.006 \leq t \leq 0.008$ s and an oscillation of the second mode appears after t = 0.0125 s. The behavior of the oscillation of y_max is similar to that of the pressure oscillation in Figure 9. The amplitude of the second mode is almost half as large as that of the first mode. This indicates that the change of amplitude of the jet oscillation between the first and second modes could induce the change of the sound intensity. At U = 10 m/s in Figure 20(a), oscillations of the first mode are observed except for an irregular motion in the range $0.015 \leq t \leq 0.02$ s, which is caused by the approach of vortices. The amplitude of the stable oscillation observed in the latter half is almost as large as that of the first mode at U = 16.7 m/s. This indicates that the amplitude of the jet oscillation does not increase with U.

As shown in Figure 21, the amplitude of the first mode increases until U = 6.7 m/s and slightly decreases in the range $6.7 \leq U \leq 10.0$ m/s, but it rather irregularly changes in the mode transition region. The amplitude of the second mode takes almost constant values except at U = 15.0 m/s. In summary, in regard to jet motion and velocity profile, we find that in the range from U = 6.7 to 12.0 m/s, where the sound intensity follows the 5th-power law, the skewness of the velocity profile tends to increase with the jet velocity, while the amplitude of the jet oscillation is saturated. The same tendency is observed for the second mode.

Fluctuation of velocity distributions of the jet

To study the properties of the velocity variation in the transverse vertical line, we introduce the velocity variance profile defined by

Δ q^{2} (y) = \frac{1}{T'} \int_{t_{0}}^{t_{0} + T^{'}} {(q (y, t) - \bar{q} (y))}^{2} d t

(8)

where t₀ and

T'

are starting time and time duration of a stationary state mode oscillation, respectively, and

\bar{q} (y)

denotes the mean value of q(y, t) defined by

\bar{q} (y) = \frac{1}{T'} \int_{t_{0}}^{t_{0} + T^{'}} q (y, t) d t

(9)

When the coordinate y is fixed, $Δ q^{2} (y)$ gives the variance of the temporal fluctuation of q(y, t) at this point, namely the square of the standard deviation of q(y, t), and has a dimension of kinetic energy per unit mass.

Figure 22(a) and (b) show the normalized velocity variance profile $Δ q^{2} (y) / U^{2}$ at representative values of the jet velocity for the first and second modes, respectively. For both first and second modes, $Δ q^{2} (y)$ has two peaks and the distance between the two peaks takes almost constant values, which are slightly less than the width of the nozzle d. Figure 23 shows oscillations of q(y, t) at y = 0 and the right and left peaks of $Δ q^{2} (y)$ in the stationary state for U = 8.0 m/s. The amplitudes of q(y, t) at the two peaks are much larger than that at y = 0. At the positions of the two peaks, the steep-slope part of the velocity profile q(y, t) sways from side to side and the velocity largely fluctuates so that $Δ q^{2} (y)$ takes a large value. At y = 0, the mean velocity $\bar{q} (y)$ takes a large value, but the amplitude of oscillation of q(y, t) is relatively small and $Δ q^{2} (y)$ takes a small value.

Figure 22.

Normalized velocity variance profile $Δ q^{2} (y) / U^{2}$ at $x / l = 0.6$ for the representative values of U. (a) The first mode. (b) The second mode.

Figure 23.

Time evolution of q(y, t) at $y / d = - 0.305$ (left peak of $Δ q^{2} (y)$ ), $y / d = 0.0$ and $y / d = 0.322$ (right peak of $Δ q^{2} (y)$ ) in the stationary state for U = 8.0 m/s.

For the first mode, the two peaks of $Δ q^{2} (y)$ are almost symmetric except for U = 10.0, 15.0 and 16.7 m/s, at which the jet suffers disturbances due to vortices. For the second mode, asymmetric distributions are observed except for U = 20.0 m/s. Thus, the asymmetric distributions are mostly observed in the mode transition region and in a neighborhood of it, because the jet oscillations have a low-frequency drift due to the disturbance of vortices together with their own instability even in an individual stationary state, and also the time duration of the stationary state $T'$ is too short for convergence (see Figure 20). Note that if the jet oscillation was completely symmetric with respect to the edge plate, $Δ q^{2} (y)$ would be symmetric and the two peaks would have the same height. For the first mode, the peaks of $Δ q^{2} (y) / U^{2}$ increase with U in the range $U \leq 12.0$ m/s. This means that $Δ q^{2} (y)$ increases faster than the 2nd power of U. Also, for the second mode, the peaks of $Δ q^{2} (y)$ seem to increase faster than the 2nd power of U. Comparing the results at U = 15.0 and 16.7 m/s in Figure 22(a) with those in Figure 22(b), it can be seen that the peaks of the first mode are always larger than those of the second mode.

The two-peak variance profile of $Δ q^{2} (y)$ is the normal type of velocity variance profile observed for the conditions studied in this paper. Thus, to quantify the amplitude of the velocity variance profile, we introduce the fluctuation strength F defined by

F = \frac{(Δ q_{1}^{2} + Δ q_{2}^{2})}{2}

(10)

where

Δ q_{1}^{2}

and

Δ q_{2}^{2}

are the heights of the two peaks of

Δ q^{2} (y)

. Figure 24 shows the changes of the fluctuation strength F with the jet velocity U for the first and second modes. For each of the first and second modes, the fluctuation strength F obeys a 3rd-power law (see the black lines with slopes of the 3rd power). However, the fluctuation strength F of the first mode is larger than that of the second mode at the same value of U and the line of the 3rd-power law of the first mode is 4.74 times higher than that of the second mode. The difference in the fluctuation strength F between the first and second modes can be attributed to the difference in the amplitude of the jet oscillation. In a low velocity range, e.g., U = 3.3 m/s, the fluctuation strength F drops off and deviates from the 3rd-power law. In the mode transition region, the fluctuation strength F of the first mode is apparently suppressed and its values deviate from the upper black line, while the values of the second mode are slightly less than the lower black line. Roughly speaking, the fluctuation strength F follows the 3rd-power law in the ranges where the amplitude of the jet oscillation is saturated for each mode. In addition, the properties of the fluctuation strength show the same tendencies as those of sound intensity in Figure 18 with respect to the relative magnitude of the values for the first and second modes, and the deviations from the power law in the lower range of jet velocity and in the mode transition region.

Figure 24.

Changes of the fluctuation strength F for the first and second modes with the jet velocity U. The black lines have slopes corresponding to the 3rd-power law.

Theoretical model for velocity profile

In this subsection we briefly discuss theoretical aspects of the power law behavior. First, we point out that the 3rd-power law behavior of the fluctuation strength F (that is, the amplitude of the velocity variance profile) is not expected from the common lowest-order model of the velocity profile. Then we present a modified model that is consistent with the above observations of both the velocity profile q(y, t) and the velocity variance profile $Δ q^{2} (y)$ . On the basis of this model, we propose that the skewness of the profile is a key factor leading to the 3rd-power law for the fluctuation strength.

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 by³⁴

q (y, t) = U {sech}^{2} [(y - a \sin ω t) / 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

Δ q^{2} (y)

is regarded as a time average of a periodic function. If a function

f (ω t)

is a periodic function of t with a period

T = 2 π / ω

, the time average of

f (ω t)

is given as

\bar{f} = \frac{1}{T} \int_{0}^{T} f (ω t) d t = \frac{1}{2 π} \int_{0}^{2 π} f (θ) d θ

(12)

and is independent of ω. Thus, even though the angular frequency ω increases with U following Brown’s equation (1), the fluctuation

Δ q^{2} (y)

calculated from equation (11) with fixed amplitude a increases in proportion to U², i.e., it obeys the 2nd-power law, and does not obey the 3rd-power law. Therefore, the jet model given by equation (11) is not appropriate for explaining the mechanism of the 3rd-power law. The Bickley profile is symmetric and the skewness s_U is zero. Based on the above simulation results, we propose the hypothesis that the increase of the skewness s_U is the key factor to understand the 3rd-power law. That is, increasing the skewness with U enhances the fluctuation of q(y, t). Actually, comparing Table 3 with Figure 24 we can see that the total skewness

s_{U T} \equiv | s_{U}

(left)

| + | s_{U}

(right)

|

is strongly correlated with the fluctuation strength F; from U = 6.7 m/s to 10.0 m/s, s_UT increases and F follows the 3rd-power law, while from U = 10.0 m/s to 12.0 m/s, the saturation of s_UT seems to induce the dropping off and deviation of F from the 3rd-power law.

To support the hypothesis, we introduce a modified model of velocity profile, which reproduces the 3rd-power law under the condition that the skewness increases with the jet velocity and the amplitude of jet oscillation is saturated. The detail requirements for the model are as follows. When U is fixed, the velocity profile is periodically deformed, but the volume flow is held constant. At the leftmost and rightmost positions, $| s_{U} |$ becomes the maximum, but s_U = 0 at the center position. In the range $6.7 \leq U \leq 12.0$ m/s, the amplitude of the jet oscillation is almost constant and the values of $| s_{U} |$ at the leftmost and rightmost positions increase monotonically.

We rely on the idea of the skew-normal distribution that generalizes the normal distribution to allow for non-zero skewness by using the error function, the antiderivative of the Gaussian function.³⁵ Since $\tanh x$ is the antiderivative of ${sech}^{2} x$ , the Bickley distribution ${sech}^{2} (y / w_{y})$ can be deformed to obtain non-zero skewness by using a hyperbolic-tangent function like the skew-normal distribution. Thus, the model is defined as

q_{b} (y, t) = U (1 + a_{t} \tanh (s_{k} (t) y)) {sech}^{2} (y / w_{y})

(13)

where

s_{k} (t)

and a_t are used for controlling the skewness. Since the

\tanh y

is an odd function, the volume flow, i.e., the integral of

q_{b} (y, t)

over y, is independent of

s_{k} (t)

and a_t. To represent the periodic deformation of the shape, the parameter

s_{k} (t)

is changed periodically as

s_{k} (t) = a_{p} U \sin (ω t) / w_{y}

(14)

where the parameter a_p is introduced to adjust the skewness. At

t = n T / 2

with

T = 2 π / ω, q_{b} (y, t)

becomes symmetric and

s_{U} (t) = 0

. We call this model ’the skew-Bickley profile model’.

To adjust the model to the numerical situation, the parameters are set as follows. The width of the velocity profile w_y is set as $w_{y} / d = 0.45$ . The parameters a_t and a_p are given as $a_{t} = 0.5$ and $a_{p} = 0.22$ s/m for the first mode, and $a_{t} = 0.24$ and $a_{p} = 0.2$ s/m for the second mode.

Figure 25(a) shows the changes of the amplitude of the jet oscillation a_J with increasing U for the first and second modes. The amplitude of the first mode increases in the range U < 10.5 m/s, but it slightly decreases for U > 10.5 m/s. The normalized amplitude $a_{J} / d$ takes values in the range $0.116 \sim 0.134$ for $U \geq 6.7$ m/s, which are relatively less than those of the numerical results, $0.18 \sim 0.23$ , in Figure 21. The normalized amplitude $a_{J} / d$ of the second mode takes values around 0.08 for U > 13 m/s, which are slightly less than those of the numerical results. Figure 25(b) shows the changes of the normalized velocity profile $q_{b} (y, t) / U$ of the first mode at U = 6.7, 8.0 10.0, 12.0 m/s. All normalized velocity profiles $q_{b} (y, t) / U$ coincide at the center potion (see the broken lines). The shapes of the velocity profiles at the leftmost and rightmost positions have reflection symmetry. Table 4 shows values of s_U at the leftmost positions for the cases in Figure 25(b). The values of s_U are relatively smaller than those in Table 3. However, s_U at the leftmost position increases monotonically with U, because the height of $q_{b} (y, t) / U$ increases with U and the difference in gradient between the right and left sides of the profile becomes larger. Therefore, all the requirements for the model are satisfied.

Figure 25.

Properties of the skew-Bickley profile model. (a) Changes of the amplitude of jet oscillation a_J with U for the first and second modes. (b) Changes of the normalized velocity profile $q_{b} (y, t) / U$ at U = 6.7, 8.0, 10.0, 12.0 m/s.

Table 4.

Skewness s_U of velocity profiles at the leftmost positions in Figure 25(b).

U[m/s]	6.7	8.0	10.0	12.0
s_U	$1.27 \times 10^{- 2}$	$4.06 \times 10^{- 2}$	$7.13 \times 10^{- 2}$	$9.26 \times 10^{- 2}$

Figure 26(a) and (b) show the normalized velocity variance profiles $Δ q_{b}^{2} (y) / U^{2}$ at the representative values of U for the first and second modes, respectively. The distributions of $Δ q_{b}^{2} (y)$ take a value of zero at y = 0, because $q_{b} (y = 0, t) = U$ , and have twin peaks, which increase faster than the 2nd power of U with U for each mode but whose separation is are approximately constant. The peaks of the first mode are 4.6 times higher than those of the second mode at the same U. These properties of $Δ q_{b}^{2} (y)$ are very similar to those of $Δ q^{2} (y)$ in Figure 22. Figure 27 shows the changes of the height of the peaks of $Δ q_{b}^{2} (y)$ , namely the fluctuation strength F, with U for the first and second modes. For the first mode, it obeys the 3rd-power law except for the low velocity range, and for the second mode, it also obeys the 3rd-power law but with a lower offset.

Figure 26.

Normalized velocity variance profile $Δ q_{b}^{2} (y) / U^{2}$ at the representative values of U. (a) The first mode. (b) The second mode.

Figure 27.

Changes of the fluctuation strength F for the first and second modes with U for the skew-Bickley profile model. The black lines have slopes corresponding to the 3rd-power law.

The above analysis of the skew-Bickley profile model indicates that the increase of $| s_{U} |$ with U at leftmost and rightmost positions is the main factor that causes the 3rd-power law of the fluctuation strength F under the condition that the amplitude of jet oscillation is almost constant. Note that we could construct a more realistic model, for example by using a hybrid model of the Bickley profile model and the skew-Bickley profile model, which provides velocity profiles more similar to the numerical ones and also reproduces the 3rd-power law.

Summary and discussion

In this paper, we calculated motions of two-dimensional compressible fluid generating edge tones with a high-precision integration scheme, compressible DNS, when a jet with a parabolic inlet profile is injected. First, a transition of the hydrodynamic mode is observed with increasing jet velocity, namely the transition from the first mode to second mode, which induces the transition of the acoustic mode of the edge tone. The frequencies of the acoustic oscillations agree well with Brown’s equation for both first and second modes.¹ When the jet velocity takes a value in the mode transition region, changes of oscillations from the first mode to the second mode and vice versa are observed in time evolution. Namely, the system in the mode transition region can be regarded as a bistable system with two attractors from the view point of nonlinear dynamics.²⁸ Thus, even if the jet stationarily oscillates in one of the modes, a sudden disturbance due to the approach of vortices would induce a transition to the other mode. This mechanism is different from that observed for a top-hat inlet profile in the previous works.^21,23

The acoustic pressure distribution observed numerically is antisymmetric with respect to the horizontal axis as observed in experiments.⁸ This means that the edge tone is radiated from a dipole-like source. For both first and second modes, the sound intensity well follows a 5th-power law, which corresponds to a 6th-power law in three dimensional systems;⁸ thus, this is another evidence that the edge tone is radiated from a dipole-like source. It was observed that the intensity of the second mode is less than that of the first mode. Furthermore, for the first mode, the intensity is suppressed and deviates from the 5th-power law at both ends of its oscillation range. The second mode shows the same tendency at the lower end of its oscillation range. Similar deviations from the 6th-power law at the ends of the oscillating range of each mode are observed experimentally.⁸ From the above results, it can be seen that our numerical calculations well reproduce the properties of the edge tone in detail and it is safe to say that the compressible DNS is a reliable tool for analyzing the sounding mechanism of the edge tone.

To consider the change of the fluctuation strength of the jet motion with the inlet jet velocity, we studied the spatial profiles of the velocity q(y, t) and the velocity variance of the oscillating jet $Δ q^{2} (y)$ , i.e., the square of the standard deviation of the temporal fluctuation of velocity, for each mode over the range of the jet velocity including the mode transition regime. We also calculated the amplitude of the two-peak velocity variance profile, named the fluctuation strength F, and defined by equation (10) with equation (8). For each of the first and second modes, the fluctuation strength F follows a 3rd-power law. The fluctuation of the second mode is relatively smaller than that of the first mode. This is attributed to the fact that the jet oscillation of the second mode is smaller in amplitude than that of the first mode, which should cause the difference in sound intensity between the first and second modes. For each mode, the amplitude of the jet oscillation a_J is approximately constant except for a low velocity range of the first mode and the shape of the velocity profile is deformed periodically. Furthermore, the absolute value of skewness of the shape $| s_{U} |$ tends to increase with U. A theoretical model using the Bickley profile model without any deformation in shape predicts a 2nd-power law for the fluctuation strength. Thus, we hypothesized that the skewness is a key factor of the 3rd-power law, namely increasing the skewness with U enhances the fluctuation strength F. To clarify this point, we introduced a model with a periodically deformed velocity profile, the skew-Bickley profile model $q_{b} (y, t)$ , and confirmed that increasing the absolute value of skewness of the velocity profile $| s_{U} |$ induces the 3rd-power law. The above numerical observations should be checked by physical experiments.

The change of the fluctuation strength F with U shows the same tendency as that of the sound intensity. This indicates a physical correspondence between the velocity fluctuation of the oscillating jet, the sound source, and the radiated sound. The results of the simulation reported in this paper motivate us to calculate the Lighthill’s acoustic source terms, i.e., the first and second terms, in an accurate manner and to clarify the relation of the source terms with the radiated sound based on the Lighthill acoustic analogy together with the Curle analogy or the Ffowcs Williams-Hawkings analogy.^3,14,15 Furthermore, the demonstrated reliability of the simulation method is encouraging for other detailed numerical studies of 2 D models of air-jet instruments.

Footnotes

Acknowledgements

The authors are very grateful to Dr. P. Davis (Telecognix Co.) for useful discussions and proofreading our English writing.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The present work was supported by Grant-in-Aid for Scientific Research (C) Nos. 16K05477 and 19K03655 from the Japan Society for the Promotion of Science (JSPS), “Joint Usage/Research Center for Interdisciplinary Large-Scale Information Infrastructures” and “High Performance Computing Infrastructure” in Japan (Project IDs: jh190010-MDH and jh200001-MDH), and Kawai Foundation for Sound Technology & Music. Part of the work was carried out under the Collaborative Research Project of the Institute of Fluid Science, Tohoku University.

ORCID iD

Kin’ya Takahashi

Appendix 1. Definition of skewness.

The skewness of velocity profile s_U is defined by (15)

s_{U} (t) \equiv 〈 {((y - 〈 y 〉) / σ)}^{3} 〉 = \frac{\int_{- y_{I}}^{y_{I}} ((y - 〈 y 〉) / σ)^{3} q (y, t) d y}{\int_{- y_{I}}^{y_{I}} q (y, t) d y}

where

\int_{- y_{I}}^{y_{I}} q (y, t) d y

denotes an instantaneous volume flow, and the average position

〈 y 〉

and the standard deviation σ are respectively given by (16)

〈 y 〉 = \frac{\int_{- y_{I}}^{y_{I}} y q (y, t) d y}{\int_{- y_{I}}^{y_{I}} q (y, t) d y}

(17)

σ^{2} = 〈 {(y - 〈 y 〉)}^{2} 〉 = \frac{\int_{- y_{I}}^{y_{I}} {(y - 〈 y 〉)}^{2} q (y, t) d y}{\int_{- y_{I}}^{y_{I}} q (y, t) d y}

To exclude the background flows, the endpoints of the integration are taken as $\pm y_{I} = \pm 1.0$ mm for calculations in Table 3, while $\pm y_{I} = \pm 4.0$ mm in Table 4.

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