First-principle description of acoustic radiation of shear flows

Abstract

As a methodology complementary to acoustic analogy, the asymptotic approach to aeroacoustics seeks to predict aerodynamical noise on the basis of first principles by probing into the physical processes of acoustic radiation. The present paper highlights the principal ideas and recent developments of this approach, which have shed light on some of the fundamental issues in sound generation in shear flows. The theoretical work on sound wave emission by nonlinearly modulated wavepackets of supersonic and subsonic instability modes in free shear flows identifies the respective physical sources or emitters. A wavepacket of supersonic modes is itself an efficient emitter, radiating directly intensive sound in the form of a Mach wave beam, the frequencies of which are in the same band as those of the modes in the packet. By contrast, a wavepacket of subsonic modes radiates very weak sound directly. However, the nonlinear self-interaction of such a wavepacket generates a slowly modulated mean-flow distortion, which then emits sound waves with low frequencies and long wavelengths on the scale of the wavepacket envelope. In both cases, the acoustic waves emitted to the far field are explicitly expressed in terms of the amplitude function of the wavepacket. The asymptotic approach has also been applied to analyse generation of sound waves in wall-bounded shear flows on the triple-deck scale. Several subtleties have been found. The near-field approximation has to be worked out to a sufficiently higher order in order just to calculate the far-field sound at leading order. The back action of the radiated sound on the flow in the viscous sublayer and the main shear layer is accounted for by an impedance coefficient. This effect is of higher order in the subsonic regime, but becomes a leading order in the transonic and supersonic regimes.

This article is part of the theme issue ‘Frontiers of aeroacoustics research: theory, computation and experiment’.

1. Introduction

Unsteady shear flows, whether laminar or turbulent, almost always radiate sound waves to large distances. The fluctuations in the main shear region (referred to as the near field) are usually complex and energetic, consisting of small- as well as large-scale components and carrying most of kinetic energy of the flow. When propagating to the far field where the mean flow is nearly uniform, the fluctuations acquire a simple character of sound waves, which obey the acoustic dispersion relation, and carry only a tiny fraction of the total energy. The primary aims of aeroacoustics are to predict the far-field sound waves and seek effective means to reduce noise or mitigate its impact, both requiring adequate understanding of sound-generation mechanisms. Obviously, the sound waves would be obtained if one is able to solve numerically in a sufficiently large domain the full compressible Navier–Stokes (N-S) equations, which govern both the near- and far-field motions. This is what direct computational aeroacoustics intends to do. However, that task is extremely challenging even for the most powerful computers available now and in the feasible future because of the need to resolve the fluctuations across the vast range of length, time and energy scales [1]. Theoretical approaches based on reduced equations are thus indispensable.

The main approach to aeroacoustics has been the acoustic analogy. Since the original work of Lighthill [2], several different forms have been proposed [3–5], but they all share the common idea and practice: rewrite the N-S equations into a linear inhomogeneous equation, or system of equations

$$\mathcal{L}\hspace{0.17em}\mathbf{s}=\mathcal{S},$$ 1.1

where on the left-hand side, $\mathcal{L}$ is a linear operator that reduces to the acoustic wave operator in the far field. On the basis of such a rearrangement, the dependent variable s is considered to represent sound waves, while the right-hand side $\mathcal{S}$ is interpreted as the equivalent ‘source’. Implied in this approach are (a) the fundamental premises that a distinction between the acoustic and non-acoustic fluctuations may be made in the entire flow field, and (b) the equivalent source may be pre-determined before the sound is found.

The recently developed generalized acoustic analogy [6] differs significantly from, and has advantages over, those mentioned above. While the rearranged equations remain of the same form as (1.1), the vector s consists of a component that is a nonlinear combination of the primitive variables. The rearrangement is made such that the Reynolds stresses and energy flux appear as the sources, which allows the spectrum of the far-field sound to be expressed in terms of the auto-covariances of the Reynolds stresses and energy flux. This is deemed to be the essential requirement because otherwise certain statistic quantities characterizing the sources would not exist.

Although the rearrangement is exact, no gain or progress would be possible unless approximations are made to the source and the boundary conditions (see below). In typical applications, the sources are modelled empirically, using the data from experiments, Reynolds-averaged N-S calculations and/or large-eddy simulations. With such input and boundary conditions, the acoustic analogy equation (1.1) can be solved to predict the noise generated by turbulent flows of technological importance [7]. The linear nature of system (1.1) makes it more amenable to analytical and numerical treatments than the original N-S equations. The success of the generalized analogy has been reported by Karabasov et al. [8], Leib & Goldstein [9] and references therein.

On the other hand, the acoustic analogy, in general, is by no means complete because of several inherent problematic issues, which are generally recognized. The predesignation of the source and sound, made without considering the nature of the fluctuations, is rather arbitrary. The arbitrariness could only be alleviated, or eliminated by requiring the ‘sources’ to have certain attributes as was done in the generalized analogy [6]. In wall-bounded flows, there is an issue concerning the boundary condition on the wall. Since the operator $\mathcal{L}$ is typically taken to describe inviscid dynamics, the no-penetration condition, ∂p_s/∂n = 0, is often imposed. This is however inappropriate because the acoustic region and the wall is sandwiched by a viscous shear layer, which acts a cushion. The effect of the latter was parametrized by the so-called ‘acoustic impedance’ Λ, leading to the ‘soft’ condition,

$$\frac{\mathrm{\partial }{p}_s}{\mathrm{\partial }n}=\mathit{\Lambda }{p}_s.$$ 1.2

Usually, Λ was provided empirically, although analytical expressions for Λ were derived by adopting a simple physical model for the flow near the wall [10]. In principle, specification of appropriate boundary conditions requires a detailed analysis of the hydrodynamic fluctuation in the shear layer, but that is precisely what the acoustic analogy intends to avoid. Thus although the rearrangement of the N-S equations is exact, the analogy as a whole is not because it is impossible to provide appropriate initial and boundary conditions to warrant the complete equivalence of an acoustic analogy solution to the true acoustic field.

Since the acoustic analogy treats the sound waves as a byproduct of the hydrodynamic motion, any back effect of the former on the latter is completely ignored, unless the ‘sources’ are determined by experimental data accurate enough to have included the effect of back action. Such back actions can however be significant and even crucial in many important practical situations, e.g. the aerofoil tonal noise [11] and screech of supersonic jets [12], where the radiated sound waves interfere simultaneously with the flow to form a closed-loop interaction, leading to self-sustained oscillations at discrete frequencies.

A methodology different from the acoustic analogy is the asymptotic approach, which defines the sound wave as it is: the far-field asymptote, or ‘ripples’, of the near-field hydrodynamic fluctuations. This approach probes into the true physical process of radiation by analysing how the hydrodynamic fluctuations evolve to acquire the character of sound as the distance from the flow field increases. According to their large-distance asymptotic behaviour, the true sources of sound are identified and its governing equations determined without the arbitrariness associated with pre-designation of ‘source’ and ‘sound’. Generation of sound is thus naturally formulated as a singular perturbation problem with the acoustic far field appearing as the distinguished asymptotic region where the expansion for the hydrodynamic near field becomes invalid. This approach was initiated by Obermeier [13] and Crow [14] to calculate the sound waves radiated by predominantly inviscid compact vortex motions at small Mach numbers, and numerous applications in this context were to appear subsequently. Crow [14] realized that the asymptotic approach could allow some fundamental issues in aeroacoustics to be addressed. Using inviscid compact vortex motion as a vehicle, he was able to clarify the approximation in the acoustic analogy of Lighthill [2]. The idea of this approach may be applied, and its formalism extended, to shear flows. Notable progress in that direction was made by Tam & Morrison [15] and Tam & Burton [16] for the sound waves radiated by linear subsonic and supersonic instability modes in compressible free shear flows, respectively.

Since the asymptotic approach relies upon the large-distance asymptotic behaviours of the near-field fluctuations, further theoretical developments are possible for flows that are amenable to analysis. In this paper, we review the recent progress on acoustic radiation in two broad settings, which are relatively simple but of practical interest also.

The first concerns sound waves emitted by nonlinearly evolving instability modes in subsonic and supersonic shear flows, where the progress was built on high-Reynolds-number nonlinear instability theories, which were reviewed by Goldstein [17] and recently by Wu [18]. The second concerns acoustic radiation of wall-bounded flows on the triple-deck scale, where the well-established results on hydrodynamics, i.e. the base flows [19] and their instabilities [20], provide the foundation for further investigation of their radiation properties. The works to be reviewed demonstrate how the asymptotic approach sheds light on, or even resolve, those fundamental issues that the acoustic analogy is incapable of addressing. These include identification of sources, the role of impedance and acoustic feedback.

The pursuit of physical insights and mathematical consistency, which we take as the priority, comes at a price, namely, a set of small parameters is required. The key one is the amplitude of wavepackets or more generally unsteady fluctuations. This parameter can be at our disposal in some (of course not all) laminar flows of technological relevance, but its availability is not always warranted in turbulent flows. This can be considered as a major drawback of the asymptotic approach. Nevertheless, it appears that the approach may be extended to investigate acoustic radiation of coherent structures (CS) in turbulent shear flows in view of recent developments in modelling the latter. Further research avenues in that direction will be highlighted.

2. Acoustic radiation of instability waves in free shear flows

Noise emitted from fully compressible free shear flows such as jets has been a subject of extensive experimental, theoretical and more recently, computational study. Such flows are inviscidly unstable so that instability modes undergo linear amplification, nonlinear saturation and final decay. Interestingly, even in turbulent state, these flows exhibit a fair degree of order, featured primarily by large-scale coherent structures. It was suggested that coherent structures may be approximated by instability modes of the turbulent mean flow. The early experimental evidence was provided by Crow & Champagne [21] for a circular jet, and by Gaster et al. [22] for a mixing layer, and more recent and compelling evidence came from Suzuki & Colonius [23].

On the other hand, it has long been proposed that wavetrains of instability modes, or more generally coherent structures in turbulent flows, may be important, even dominant, sources of noise in supersonic [24,25] and subsonic [21,26] regimes. This proposition is significant as CS are likely to be more amenable to mathematical description or modelling than random turbulence, and they are controllable by external excitation [27–29] thereby opening up new avenues to suppress noise [30]. That prospect has attracted a great deal of research activities, which were revitalized recently [31].

There have been many studies of acoustic radiation of wavepackets. Often instability modes were used to evaluate the source terms appearing in the acoustic analogy [32–34]. In generalized forms of acoustic analogy, instability modes affect the calculation of acoustics waves in a way different from contributing to the sources. There a decision has to be made about whether one opts for a bounded Green function by discarding instability modes, or for a casual Green function, in which case instability modes must be included. This issue was discussed by Goldstein & Leib [35]. In this framework, instability modes appear to be a purely mathematical object, and can only be linear due to the nature of the formulation. Our focus will be on the progress made in developing an integrated approach, in which nonlinear evolution of instability modes as a physical entity, and their acoustic radiation are both described on the basis of first principles. Experiments reveal that noise is emitted primarily as the wavepackets undergo growth and attenuation [36–38]. The emission is significantly enhanced by the jittering or intermittency of the wavepackets. This feature, which was observed also in numerical simulations (e.g. [39]), has been captured by simple (acoustic-analogy based) models (e.g. [40]). In our integrated approach, jittering or intermittency arises as a result of nonlinear spatial–temporal modulation rather than being introduced on a phenomenological basis.

(a). Acoustic radiation of supersonic modes

In the supersonic regime, there exist supersonic instability modes, which propagate supersonically relative to the ambient flow. They radiate intensive highly directional sound waves in the form of Mach wave beams when the Mach number is sufficiently high. The first study of acoustic radiation using matched asymptotic expansion was made by Tam & Burton [16]. The mode was assumed to be linearly evolving. They showed that the expansion for the hydrodynamic field breaks down at large transverse distances so that an acoustic far field must be introduced. However, rather than pursuing the asymptotic approach fully, Tam & Burton resorted to a numerical procedure to calculate the acoustic field. Wu [41] noted that by taking advantage of the scale disparity between the envelope and wavefront of the wavepacket, the far-field acoustic field can be determined fully by asymptotic matching. The main ideas and mathematical steps are summarized below for the case where the disturbance consists of a pair of helical modes with the same frequency ω but opposite azimuthal wavenumbers ± m.

In the bulk of the shear flow, the pressure of the disturbance can be expressed as

$$p=ϵA(\overline{x},\overline{t}){\hat{p}}_0(r){\mathrm{e}}^{\mathrm{i}(\alpha x-\omega )t}\cos m\theta +c.c.+\dots .$$ 2.1

in a cylindric coordinate system (x, r, θ), where A is the amplitude function of the slow space (axial) and time variables, $\overline{x}$ and $\overline{t}$. By employing the well-developed nonlinear critical-layer theory [17,18], the amplitude equation A was derived as

$$\frac{\mathrm{\partial }A}{\mathrm{\partial }\overline{x}}+c_g^{-1}\frac{\mathrm{\partial }A}{\mathrm{\partial }\overline{t}}=\sigma \overline{x}A+l\mathcal{N},$$ 2.2

where the nonlinear term $\mathcal{N}$ depends on the form of the modes and the asymptotic regime, which is determined by the size of ϵ relative to R^−1/3 (with R being the Reynolds number based on the nozzle diameter). Wu [41] considered the equilibrium regime, and for the special case of an axisymmetric mode (m = 0), ϵ = R^−11/12 and $\mathcal{N}=A|A{|}^2$. However, for a pair of helical modes, ϵ = R^−7/6 and $\mathcal{N}$ is non-local and turns out to be the same as that for a pair of oblique modes in planar shear flows [42,43]. It now transpires that a composite amplitude equation, valid in both the equilibrium and non-equilibrium regimes, can be derived as in Wu & Huerre [44]. Then $\mathcal{N}$ reads

$$\begin{array}{rl}\mathcal{N}& ={\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}K(\xi ,\eta ;\overline{s})\overline{A}(\overline{x}-\xi ,\overline{t}-\frac{\xi }{c})\\ & \times \overline{A}(\overline{x}-\xi -\eta ,\overline{t}-\frac{\xi }{c}-\frac{\eta }{c}){\overline{A}}^{\ast }(\overline{x}-2\xi -\eta ,\overline{t}-\frac{2\xi }{c}-\frac{\eta }{c})\mathrm{d}\xi \mathrm{d}\eta ,\end{array}$$ 2.3

where K is given by (3.85) of Wu et al. [43], but with $\overline{s}=({\alpha }^2{\overline{U}}_c^{\mathrm{\prime }2}/3)R^{1/2}$ and ${\overline{U}}_c^{\mathrm{\prime }}$ denoting the gradient of the mean axial velocity at the critical level r_c, where U(r_c) = c. For a supersonic mode, its eigenfunction has the far field asymptote that as r≫1,

$${\hat{p}}_0\to \frac{{\mathcal{C}}_{\mathrm{\infty }}}{\sqrt{r}}{\mathrm{e}}^{\mathrm{i}\alpha qr}\text{with}\hspace{0.17em}q={(\frac{c^2}{c_a^2}-1)}^{1/2},$$ 2.4

where c_a is the speed of sound in the ambient fluid, and constant ${\mathcal{C}}_{\mathrm{\infty }}$ is to be fixed by normalization. Result (2.4) indicates that a mode is radiating if it travels faster than c_a.

In the far field of the hydrodynamic region, where $\overline{r}=R^{-1/2}r=O(1)$, which will be referred to the near-field acoustic region, the pressure expands as [41]

$$p=ϵR^{-1/4}[{\overline{p}}_0(\overline{x},\overline{r},\overline{t})+R^{-1/2}{\overline{p}}_1+\dots ]{\mathrm{e}}^{\mathrm{i}\alpha (x+qr-ct)}\cos (m\theta )+c.c.$$ 2.5

Consideration of the next term ${\overline{p}}_1$ gives the appropriate solution for ${\overline{p}}_0$ [41],

$${\overline{p}}_0=\frac{{\mathcal{C}}_{\mathrm{\infty }}}{\sqrt{\overline{r}}}A(\overline{\xi },\overline{\eta })\text{with}\hspace{0.17em}\overline{\xi }=\overline{x}-\frac{\overline{r}}{q},\overline{\eta }=\overline{t}-\left(\frac{M_a^{2}c}{q}\right)\overline{r},$$ 2.6

so that the instantaneous field of the radiated Mach wave is

$$p_0=\frac{{\mathcal{C}}_{\mathrm{\infty }}}{\sqrt{\overline{r}}}A(\overline{x}-\frac{\overline{r}}{q},\overline{t}-\frac{M_a^{2}c}{q\hspace{0.17em}\overline{r}}){\mathrm{e}}^{\mathrm{i}\alpha (x+qr-ct)}\cos (m\theta )+c.c.$$ 2.7

The solution facilitates the simple interpretation of the radiation process: the acoustic energy flux r|p₀|², or the envelope, propagates along the characteristics $\overline{\xi }=\text{constant}$ and $\overline{\eta }=\text{constant}$. This generalizes the classical ‘wavy analogy’ to a modulated wavy wall.

Now turn to the far field of the Mach wave beam, corresponding to $\overline{r}=O(R^{1/2})$ and $\overline{\xi }=O(1)$, where the appropriate solution may be sought by introducing the variable

$$\tilde{r}=R^{-1/2}\overline{r}=R^{-1}r.$$ 2.8

The large-$\overline{r}$ asymptote of p₀ suggests that the solution for the pressure should expand as

$$p=ϵR^{-1/2}[{\tilde{p}}_0(\overline{\xi },\overline{\eta },\tilde{r})+R^{-1}{\tilde{p}}_1(\overline{\xi },\overline{\eta },\tilde{r})+\dots ]{\mathrm{e}}^{\alpha (x+qr-ct)}\cos m\theta +c.c.,$$

as shown by Wu [41]. The secular condition for ${\tilde{p}}_1$ yields the equation governing ${\tilde{p}}_0$:

$$2\mathrm{i}\alpha q\frac{\mathrm{\partial }{\tilde{p}}_0}{\mathrm{\partial }\tilde{r}}+q^{-2}{M}_a^2\{c^2\frac{{\mathrm{\partial }}^2{\tilde{p}}_0}{\mathrm{\partial }{\overline{\xi }}^2}+2c\frac{{\mathrm{\partial }}^2{\tilde{p}}_0}{\mathrm{\partial }\overline{\xi }\mathrm{\partial }\overline{\eta }}+\frac{{\mathrm{\partial }}^2{\tilde{p}}_0}{\mathrm{\partial }{\overline{\eta }}^2}\}+\frac{\mathrm{i}q}{\tilde{r}}{\tilde{p}}_0=0,$$ 2.9

where the second-order derivatives represent the dispersion effect, due to which the beam forms a tip. In order to match with (2.6), ${\tilde{p}}_0$ must satisfy

$${\tilde{p}}_0\to \frac{{\mathcal{C}}_{\mathrm{\infty }}}{\sqrt{\tilde{r}}}A(\overline{\xi },\overline{\eta })\text{as}\hspace{0.17em}\tilde{r}\to 0.$$ 2.10

In the case of a single wave, for which A is a function of $\overline{x}$ only, the solution to (2.9) and (2.10) is found to be

$${\tilde{p}}_0(\overline{\xi },\tilde{r})=\frac{{\mathrm{e}}^{-(\pi /4)\mathrm{i}}}{\tilde{r}}{\left(\frac{\alpha q^3}{2\pi M_a^2{c}^2}\right)}^{1/2}{\mathcal{C}}_{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}A(\zeta )\exp \left\{\frac{\mathrm{i}\alpha q^3}{2M_a^2{c}^2\tilde{r}}{(\overline{\xi }-\zeta )}^2\right\}\mathrm{d}\zeta .$$ 2.11

For the case of a modulated wavetrain, solving (2.9) and (2.10) by Fourier transform with respect to $\overline{\eta }$, we obtain

$$\hat{p}(\overline{\xi },\tilde{\omega },\tilde{r})=\frac{{\mathrm{e}}^{-(\pi /4)\mathrm{i}}}{\tilde{r}}{\left(\frac{\alpha q^3}{2\pi M_a^2{c}^2}\right)}^{1/2}{\mathcal{C}}_{\mathrm{\infty }}\hat{Q}(\overline{\xi },\tilde{\omega },\tilde{r})\exp \{-\frac{\mathrm{i}{M}_a^2{\tilde{\omega }}^2\tilde{r}}{2\alpha q^3}\}$$ 2.12

for $\hat{p}$, the Fourier transform of ${\tilde{p}}_0$, where

$$\hat{Q}(\overline{\xi },\tilde{\omega },\tilde{r})={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\hat{A}(\zeta ,\tilde{\omega })\exp \left\{\frac{\mathrm{i}\alpha q^3}{2M_a^2{c}^2\tilde{r}}{(\overline{\xi }-\zeta +\frac{M_a^{2}c\hspace{0.17em}\tilde{\omega }\tilde{r}}{\alpha q^3})}^2\right\}\hspace{0.17em}\mathrm{d}\zeta .$$ 2.13

For a wavepacket with a frequency bandwidth Δ⁻¹, the ‘time averaged’ acoustic intensity can be expressed as

$$\overline{{({\tilde{p}}_0)}^2}=\frac{1}{{\tilde{r}}^2}\left(\frac{\alpha q^3|{\mathcal{C}}_{\mathrm{\infty }}{|}^2}{4{\pi }^2{M}_a^2{c}^2\mathrm{\Delta }}\right){\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|\hat{Q}(\xi ,\tilde{\omega },\tilde{r}){|}^2\hspace{0.17em}\mathrm{d}\tilde{\omega }.$$ 2.14

Explicit analytic relations have therefore been established between the Mach wave beam and the amplitude function of the instability mode wavepacket. A similar expression was given for the planar case where the ambient velocity is non-zero [41].

Figure 1a shows the Mach wave field radiated by a linear axisymetric mode, and the theoretical prediction is found to be in quantitative agreement with the DNS data [25]. Figure 1b displays the pressure contours of the Mach wave field radiated by a pair of nonlinearly interacting helical modes. The structure of the beam mimics closely the measurement of [46].

Figure 1.

Mach wave field. (a) $x|{\overline{p}}_0|$ versus ϕ = tan⁻¹(r/x) at different axial locations. The theoretical and DNS results [45] are presented by solid lines and symbols, respectively: triangles, x = 40; circles, x = 70; diamonds, x = 90. (b) Pressure contours of the Mach waves radiated by a pair of helical modes, predicted by (2.11) using the nonlinear solution (solid lines) and linear approximation (dotted lines) for A. Reproduced from Wu [41].

(b). Acoustic radiation of subsonic modes

In subsonic shear flows, only subsonic instability modes, which propagate subsonically relative to the ambient stream, exist, but in the supersonic regime they may co-exist with supersonic modes. The eigenfunction of a subsonic mode in a parallel flow exhibits exponential decay in the transverse direction everywhere including at the neutral position, and a purely sinusoidal mode thus emits no sound wave. However, under the combined effects of nonlinearity and mean-flow spreading, instability waves actually undergo amplification followed by saturation and decay over a long length scale. Such a spatially modulated wave contains a supersonic Fourier component, and hence emits a sound wave with the same frequency as that of the instability mode. This mechanism, which may be termed ‘direct radiation’, was demonstrated by Tam & Morris [15] in the case of a subsonic jet, and by Crighton & Huerre [47] for a simple specified wavepacket. This linear radiation mechanism is rather weak, and the emitted acoustic field exhibits super-directivity, i.e. is confined in the small angles to the axial direction.

Wu & Huerre [44] considered sound generated by a nonlinear mechanism, specifically by the interaction of a pair of helical modes in a circular jet. The disturbance is of the same form as (2.1). As the modes evolve nonlinearly, their amplitude is governed by the amplitude equation (2.2) with the nonlinear term $\mathcal{N}$ being given by (2.3). An important fact is that the nonlinear interaction drives a spanwise dependent mean-flow distortion, the streamwise velocity of which exhibits a jump ${\mathcal{J}}_u$ across the critical layer, through the jump ${\mathcal{J}}_u$ (whose expression is given in (2.16)), a mean-flow distortion, modulated slowly in both space and time, is generated in the main layer with the streamwise velocity being of O(ϵ) [42], and the pressure is ϵR^−2/3p_mcos(2mθ), where p_m is found to satisfy the equation,

$$\{\frac{\mathrm{\partial }}{\mathrm{\partial }\tilde{t}}+\overline{U}\frac{\mathrm{\partial }}{\mathrm{\partial }\tilde{x}}\}\{\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{r}^2}+\frac{1}{r}\frac{\mathrm{\partial }}{\mathrm{\partial }r}-\frac{4m^2}{r^2}+\frac{{\overline{T}}^{\mathrm{\prime }}}{\overline{T}}\frac{\mathrm{\partial }}{\mathrm{\partial }r}\}{p}_m-2{\overline{U}}^{\mathrm{\prime }}\frac{{\mathrm{\partial }}^2{p}_m}{\mathrm{\partial }\tilde{x}\mathrm{\partial }r}=0.$$ 2.15

With the second-order derivative with respect to $\tilde{x}$ being absent, this is the long-wavelength limit of the compressible Rayleigh equation. Across the critical layer, the pressure p_m is continuous, but its radial gradient exhibits a jump

$$p_m^{\mathrm{\prime }}(r_c^{+},\overline{x},\overline{t})-p_m^{\mathrm{\prime }}(r_c^{-},\overline{x},\overline{t})=-\frac{{\overline{R}}_c}{{\overline{U}}_c^{\mathrm{\prime }}}{(\frac{\mathrm{\partial }}{\mathrm{\partial }\overline{t}}+c\frac{\mathrm{\partial }}{\mathrm{\partial }\overline{x}})}^2{\mathcal{J}}_u(\overline{x},\overline{t})\equiv {\mathcal{J}}_p(\overline{x},\overline{t}),$$ 2.16

where a prime indicates a derivative with respect to r, and

$${\mathcal{J}}_p(\overline{x},\overline{t})=j_0{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-2s{\eta }^3}{\left|\overline{A}(\overline{x}-\eta ,\overline{t}-\frac{\eta }{c})\right|}^2\mathrm{d}\eta (j_0=\frac{16\pi {\overline{R}}_c{\overline{U}}_c^{\mathrm{\prime }}{\left(m/r_c\right)}^4}{(\alpha c)}).$$ 2.17

The jump ${\mathcal{J}}_p$ acts as a radially compact, low-frequency physical source embedded in a shear flow. Unlike the eigen modes, which decay exponentially in the radial direction, the nonlinearly induced mean flow attenuates algebraically. For the pressure,

$$p_m\sim \frac{\mathcal{B}(\overline{x},\overline{t})}{r^{2m}}\text{as}\hspace{0.17em}r\to \mathrm{\infty },$$ 2.18

where ${\mathcal{C}}_0(\overline{x},\overline{t})$ and $\mathcal{B}(\overline{x},\overline{t})$ are functions to be determined numerically. Obviously, among helical modes those with azimuthal wavenumbers m = ± 1 are most efficient in radiating sound; the axisymmetric mode (m = 0) is a special case, whose nonlinear dynamics and acoustic radiation requires a separate investigation.

The long-wavelength Rayleigh equation (2.15) governing the slowly modulating mean flow is no longer valid in the far field corresponding to r = O(R^1/3), because the transverse and streamwise length scales become comparable. We thus introduce the radial variable

$$\tilde{r}=R^{-1/3}r.$$ 2.19

For $\tilde{r}=O(1)$, the instability modes and their harmonics all have diminished completely owing to exponential transverse attenuation. By contrast, the slowly breathing ‘mean field’, which decays algebraically, acquires the character of sound, and the solution expands as

$$(u,v,w,p,\theta ,\rho )=ϵR^{-4/3}({\tilde{u}}_s,{\tilde{v}}_s,{\tilde{w}}_s,{\tilde{p}}_s,{\tilde{\theta }}_s,{\tilde{\rho }}_s)+\dots .$$

As expected, functions ${\tilde{u}}_s$, ${\tilde{v}}_s$, etc., satisfy the standard linearized equations for an acoustic perturbation in a uniform background flow, $\overline{U}=0$ and $\overline{T}=T_a$. Specifically, the governing equation for ${\tilde{p}}_s$ is

$$M_a^2\frac{{\mathrm{\partial }}^2{\tilde{p}}_s}{\mathrm{\partial }{\overline{t}}^2}-\{\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\tilde{r}}^2}+\frac{1}{\tilde{r}}\frac{\mathrm{\partial }}{\mathrm{\partial }\tilde{r}}-\frac{4m^2}{{\tilde{r}}^2}+\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\overline{x}}^2}\}{\tilde{p}}_s=0,$$ 2.20

where the acoustic Mach number M_a = M/T^1/2_a with T_a being the ambient air temperature. For $\tilde{r}\ll 1$, ${\tilde{p}}_m\sim \mathcal{B}(\overline{x},\overline{t})/{\tilde{r}}^{2m}$ in order to match with (2.18). This indicates that $\mathcal{B}(\overline{x},\overline{t})$ acts as the apparent acoustic source, which will be expressed in terms of the physical source ${\mathcal{J}}_p$. The source is non-compact in the axial direction because its spatial extent is comparable with the wavelength of the emitted sound. On taking the Fourier transform with respect to both $\overline{x}$ and $\overline{t}$, equation (2.20) reduces to the Helmholtz equation

$$\{\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\tilde{r}}^2}+\frac{1}{\tilde{r}}\frac{\mathrm{\partial }}{\mathrm{\partial }\tilde{r}}+(M_a^2{\omega }^2-k^2-\frac{4m^2}{{\tilde{r}}^2})\}{\hat{p}}_s=0,$$ 2.21

subject to the matching condition that ${\hat{p}}_s\to \hat{\mathcal{B}}(k,\omega )/{\tilde{r}}^{2m}$ as $\tilde{r}\to 0$, where $\hat{\mathcal{B}}(k,\omega )$ is the Fourier transform of $\mathcal{B}(\overline{x},\overline{t})$. The solution for ${\hat{p}}_s$ is then found as

$${\hat{p}}_s=\frac{2^{2m}\pi }{(2m)!}{\mathcal{K}}^{2m}(k,\omega )\hat{\mathcal{B}}(k,\omega )H_{2m}^{(1)}(\mathcal{K}\tilde{r}),$$ 2.22

where H⁽¹⁾_2m denotes the first-kind Hankel function of order 2m, and $\mathcal{K}(k,\omega )={(M_a^2{\omega }^2-k^2)}^{1/2}$. The acoustic pressure in physical space follows from the inversion of the Fourier transform, but of primary interest is the far field of the acoustic region, corresponding to $\tilde{R}={({\tilde{r}}^2+{\overline{x}}^2)}^{1/2}\gg 1$. We then approximate H⁽¹⁾_2m by its asymptotic expansion. Using the stationary phase method, the instantaneous pressure in the far field is found as

$${\tilde{p}}_s\sim \frac{2^{2m+1}\pi }{(2m)!\tilde{R}}{\mathrm{e}}^{-\mathrm{i}(m\pi +\pi /2)}{(\sin \theta )}^{2m}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{(M_a\omega )}^{2m}\hat{\mathcal{B}}(-M_a\omega \cos \theta ,\omega ){\mathrm{e}}^{\mathrm{i}\omega (M\tilde{R}-\overline{t})}\mathrm{d}\omega ,$$ 2.23

where $\theta ={\tan }^{-1}(\tilde{r}/\overline{x})$.

To obtain $\hat{\mathcal{B}}(-M_a\omega \cos \theta ,\omega )$, one has to solve (2.15). The Fourier transform of this equation depends on ω/k with k = k_s = − M_aωcosθ, which is a function of θ but independent of ω. It follows that we may write

$$\hat{\mathcal{B}}(-M_a\omega \cos \theta ,\omega )=\mathcal{T}(\theta ){\hat{\mathcal{J}}}_p(-M_a\omega \cos \theta ,\omega ),$$ 2.24

where $\mathcal{T}(\theta )$ is determined by the following boundary-value problem

$$\begin{array}{r}\{\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{r}^2}+\frac{1}{r}\frac{\mathrm{\partial }}{\mathrm{\partial }r}-\frac{4m^2}{r^2}+(\frac{{\overline{T}}^{\mathrm{\prime }}}{\overline{T}}-\frac{2{\overline{U}}^{\mathrm{\prime }}}{\overline{U}-1/(M_a\cos \theta )})\frac{\mathrm{\partial }}{\mathrm{\partial }r}\}{\hat{p}}_m=0,\\ {\hat{p}}_m(r_c^{+})-{\hat{p}}_m(r_c^{-})=0,{\hat{p}}_m^{\mathrm{\prime }}(r_c^{+})-{\hat{p}}_m^{\hspace{0.17em}\mathrm{\prime }}(r_c^{-})=1,\\ \hat{p}\sim r^{2m}\text{as}\hspace{0.17em}\tilde{r}\to 0,\hat{p}\to \frac{\mathcal{T}(\theta )}{r^{2m}}\text{as}\hspace{0.17em}r\to \mathrm{\infty }.\end{array}\}$$ 2.25

As $\mathcal{T}(\theta )$ relates the forcing ${\hat{\mathcal{J}}}_p$ to the output $\hat{\mathcal{B}}$, it is referred to as a transfer function. It depends on the velocity and temperature profiles of the base flow, and the critical level r_c, which in turn depends on the carrier-wave frequency. Clearly, $\mathcal{T}(\theta )$ characterizes the base-flow refraction (shielding and amplification) effects (cf. [7,30]).

The radiated sound is of broadband nature. It follows from (2.23) to (2.24) that its normalized spectrum at an arbitrary point $(\tilde{R},\theta )$ (polar coordinates) can be defined as

$$\mathcal{I}(\omega ;\theta )={\omega }^{4m}{|{\hat{\mathcal{J}}}_p(-M_a\omega \cos \theta ,\omega )|}^2,$$ 2.26

while the acoustic pressure intensity, measured by its root-mean-square value, is

$$\sqrt{\overline{{\tilde{p}}_s^2}}=\frac{2^{2m+1}\pi }{(2m)!}{M}_a^{2m}\frac{\mathcal{D}(\theta )}{\tilde{R}},$$ 2.27

with the directivity function $\mathcal{D}(\theta )$ being given by

$$\mathcal{D}(\theta )=\mathcal{T}(\theta ){(\sin \theta )}^{2m}{\left[{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\omega }^{4m}{|{\hat{\mathcal{J}}}_p(-M_a\omega \cos \theta ,\omega )|}^2\mathrm{d}\omega \right]}^{1/2}.$$ 2.28

Figure 2a shows the development of the wavepacket envelope for different initial amplitude a₀. The directivity and spectrum of the radiated acoustic field are displayed in figure 2b,c, respectively. The acoustic field is characterized by a single-lobed directivity pattern beamed at an angle about 45°–60° to the jet axis, and a broadband spectrum centred at a Strouhal number St ≈ 0.07 − 0.2, where St is based on the nozzle diameter. Nonlinear effect enhances the radiation and causes the noise spectrum to broaden. The gross features are broadly in agreement with experimental observations.

Figure 2.

The nonlinear development of a wavepacket (with St = 1.0, Δω = 0.5) and its acoustic field. (a$)\sqrt{\overline{A^2}}$ versus $\hat{x}$. (b) Directivity pattern. (c) Spectrum at θ = 90°. ‘Initial amplitude’ a₀ = 2.53 × 10⁻² (curve 1), a₀ = 2.95 × 10⁻² (curve 2), a₀ = 3.38 × 10⁻² (curve 3). The dashed lines represent the result for a linear wavepacket with a₀ = 2.53 × 10⁻². Reynolds number R = 1800 and σ_s = 0.043. Reproduced from Wu & Huerre [44].

It is of interest to consider how the present nonlinear sound radiation mechanism operates in planar flows. It turns out that significant differences arise. For a pair of oblique modes with spanwise waveumbers ± β, the pressure of the nonlinearly generated three-dimensional mean flow satisfies (2.15) with r being interpreted as the transverse variable y, (1/r)∂/∂r being removed, and most crucially 4m²/r² replaced by 4β². It follows that this part of the disturbance decays exponentially towards the outer edge of the main shear layer if β = O(1), in which case the two-dimensional part of the nonlinearly generated mean flow acts instead as the emitter. Alternatively, a pair of weakly three-dimensional modes with β = O(ϵ^1/2) may interact to radiate low-frequency sound waves. In both cases, the forcing due to the nonlinear interaction in the main shear layer becomes important and forms part of the source as well [48]. Most interestingly, although an equivalent source could be identified, it cannot be predetermined before the acoustic field is obtained, a feature different from what acoustic analogy envisages.

In developing the theories in an asymptotic framework, the evolution equations governing the wavepacket amplitude were derived using the nonlinear critical layer theory [18], the mathematical details of which are typically highly complex. It is important to note that the explicit relationships between acoustic field and the amplitude of the wavepackets are independent of how the amplitude equation is derived. This means that if the amplitude evolutions are obtained by other means, such as nonlinear parabolized stability equations (PSE), DNS or even experiments, they may be substituted into the formulae to calculate the acoustic field. This leads to a hybrid approach, which avoids, if one so wishes, the technical complexity of nonlinear critical layer theory.

3. Sound waves generated by wall-bounded shear flows

Another development of the asymptotic approach to aeroacoustics took place in the area of sound generation in shear flows that can be described by triple-deck theory. The acoustic radiation of such flows has been investigated partly because the problem could be of practical relevance, but primarily because useful light may be shed on a number of rather fundamental issues concerning sound radiation in a wall-bounded shear flow, such as identification and characterization of sources, mean-flow shielding effect, the impedance effect of the boundary layer and the back action of the radiated sound on the source.

The problems considered include: (I) the boundary layer subject to time-harmonic wall suction/blowing [49] and (II) the scattering of a Tollmien-Schlingting (T-S) wave by the localized mean-flow distortion caused by an isolated surface roughness [50]. The suction and roughness are located at a distance L from the leading edge, and their width is assumed to be of R^−3/8L, where R is the Reynolds number based on L. It is convenient to define a small parameter ϵ = R^−1/8, the coordinate $\overline{x}={ϵ}^{-3}x={ϵ}^{-3}(x^{\ast }/L)$ and time variable $\overline{t}={ϵ}^{-2}{t}^{\ast }{U}_{\mathrm{\infty }}/L$ to describe the local steady and unsteady flows.

(a). The procedure of formal systematic expansion

In each problem, the solutions for the near-field hydrodynamic motions can be constructed systematically in the terms of asymptotic series, which takes different forms in the lower, main and upper decks. The acoustic field is deduced by analysing the far-field asymptotic behaviour of the solution, and thus the source can be identified and characterized precisely. A main finding is that the expansion for the hydrodynamic field must be carried out to a sufficient high order in order to obtain the acoustic far field to leading-order accuracy.

The salient feature can be highlighted by considering the pressure, ${\tilde{p}}_s{\mathrm{e}}^{-\mathrm{i}\omega \overline{t}}+c.c.$. In the upper deck, where $\overline{y}={ϵ}^{-3}{y}^{\ast }/L=O(1)$, ${\tilde{p}}_s$ expands as

$${\tilde{p}}_s(\overline{x},\overline{y})={ϵ}_s({\tilde{p}}_0+ϵ{\tilde{p}}_1+{ϵ}^2{\tilde{p}}_2+{ϵ}_3{\tilde{p}}_3+\dots .),$$ 3.1

where ϵ_s depends on the problem considered. The solution may be sought by taking Fourier transform with respect to $\overline{x}$, and the transformed expansion is written as

$${\hat{p}}_s(k,\overline{y})={ϵ}_s({\hat{p}}_0+ϵ{\hat{p}}_1+{ϵ}^2{\hat{p}}_2+{ϵ}^3{\hat{p}}_3+\dots ).$$ 3.2

The near-field hydrodynamic fluctuations comprise primarily of components with k = O(1), for which the higher-order terms in (3.2) merely provide a small correction, that is, provided that the Reynolds number R is sufficiently high, the hydrodynamics is accurately represented by ${\hat{p}}_0$. However, ${\hat{p}}_0$ may not be adequate for predicting even the leading-order far-field acoustics no matter how large R is. This is because in the limit of k≪1, each ${\hat{p}}_j$ typically has the asymptote

$${\hat{p}}_j(k,\overline{y})=g_j{k}^{n_j}+O(k^{n_j+1}),$$ 3.3

with the powers proceeding like n_j = m − j, where the value of m depends on the problem considered. Correspondingly, in physical space, ${\tilde{p}}_j=O({\overline{r}}^{-n_j})$ for n_j≥1 and $O(\ln \overline{r})$ for n_j = 0 as $\overline{r}={({\overline{x}}^2+{\overline{y}}^2)}^{1/2}\gg 1$. It follows that as $\overline{r}\to \mathrm{\infty }$,

$${\tilde{p}}_s={ϵ}_s[g_0{\overline{r}}^{-m}+ϵg_1{\overline{r}}^{-m+1}+\dots +g_m\ln \overline{r}].$$ 3.4

This far-field asymptote implies that the higher-order terms in the expansion (3.1) decay less rapidly, and the expansion becomes disordered at large distances where $\overline{r}=O({ϵ}^{-1})$. An outer acoustic region with an O(R^−1/4l) size in both the streamwise and normal directions must be introduced. The pressure in this region is governed by the wave equation in a uniform medium, as opposed to the Laplace or Poisson equation in the upper layer. The formal scheme thus shows how an unsteady viscous motion changes its nature to acquire the character of sound as the distance increases, and allows one to focus on the dominant sources, that is, in seeking the solution for ${\hat{p}}_j$, the terms whose small-k asymptote is smaller than k^n_j+1 can be discarded. The analysis determines the intensity of the sound, ${\tilde{p}}_s=O({ϵ}_s{ϵ}^m)$. The solution in the acoustic field can be determined by matching with the far-field asymptote (3.4). In the region where $\overline{r}\gg {ϵ}^{-2}$, the acoustic pressure is found as

$${\tilde{p}}_s={ϵ}_s{ϵ}^m\frac{\sigma (\omega )}{\sqrt{\overline{r}}}{q}_0\exp \left[\mathrm{i}(\varphi (\overline{x},\overline{y})-\frac{\pi }{4})\right],$$ 3.5

where ϕ, σ and q₀ represent the phase, intensity and directivity of the acoustic field, respectively. It transpires that the power n_j characterizes the multipolar nature, or the radiating efficiency, of the source. Since n_j = m − j, the higher-order terms (i.e. ${\tilde{p}}_j$ with j≥1) represent progressively lower-order acoustic poles, which emit more efficiently. With this understanding, it is perhaps not surprising that acoustic analogy may lack ‘robustness’ with respect to approximating $\mathcal{S}$ [51]. When an exact source $\mathcal{S}$, a quadrupole say, is approximated by ${\mathcal{S}}_0$, the latter may consist of dipole or even monopole components, which may produce sound waves of comparable strength with that emitted by $\mathcal{S}$ unless $(\mathcal{S}-{\mathcal{S}}_0)$ is controlled at an extremely small level.

For case I, it was shown that n₀ = m = 2, and so the leading-order term in (3.1) represents a quadrupole rather than a monopole that is normally associated with suction, while the second and third terms play the role of dipole and monopole, respectively. The quadrupole nature of ${\tilde{p}}_0$ is due to the source cancellation in the long-wavelength limit in spectrum space. The directivity function q₀ is

$$q_0(\theta )=\frac{1}{{(1-M^2{\sin }^2\theta )}^{1/4}}{[1-\frac{M\cos \theta }{{(1-M^2{\sin }^2\theta )}^{1/2}}]}^2.$$ 3.6

Interestingly, the result does not contain any quantity associated with the mean shear. Nevertheless, it has accounted for the shielding effect of the shear flow because a purely inviscid approach with a uniform mean flow gives a directivity different from (3.6) in that the power 2 is reduced to 1. Thus, the mean shear affects the radiation in a rather subtle way. A significant hydrodynamic process is the generation of instability waves. As the present approach treats the hydrodynamics fully, the amplitude of the instability mode generated and the sound wave radiated can both be determined without any ambiguity, whereas in certain (e.g. Lilley's) forms of acoustic analogy uncertainty may arise as to whether the instability wave should be included or discarded [30]. It is worth pointing out that the viscous dynamics in the lower deck plays a leading-order role in sound generation, unlike any acoustic analogy, where the (propagation) operator $\mathcal{L}$ describes inviscid dynamics.

For case II, it was shown that n₀ = m = 3, implying that the leading-order term in (3.1) represents an octupole rather than a quadrupole that is normally expected of stresses, while the ensuing terms act as a quadrupole, dipole and monopole, respectively. Specifically, the mean pressure gradient produces a quadrupole source in ${\tilde{p}}_1$. The octupole nature of ${\tilde{p}}_0$ is again a consequence of the source cancellation in the long-wavelength limit in spectrum space. The directivity function q₀ is

$$q_0(\theta )=\frac{1}{{(1-M^2{\sin }^2\theta )}^{1/4}}{[1-\frac{M\cos \theta }{{(1-M^2{\sin }^2\theta )}^{1/2}}]}^2[[1-\frac{\cos \theta /M}{{(1-M^2{\sin }^2\theta )}^{1/2}}]+{\mathcal{Q}}_p],$$ 3.7

where ${\mathcal{Q}}_p$ is contributed by the mean pressure gradient.

(b). Composite expansion, back action of sound and impedance

In the formal asymptotic expansion, the main and upper decks are quasi-steady; the time-varying terms appear as inhomogeneous terms at higher orders and constitute lower-order acoustic monopoles. The lengthy algebra dealing with these terms can be avoided by retaining the unsteady terms at leading order in both the upper and main layers, as a result of which the upper and acoustic layers merge. This leads to simpler composite expansions. The cases I and II were revisited using this approach [52]. The result turned out to remain of the form (3.5) provided that q₀ is replaced by

$$q(\theta )=\frac{\sin \theta }{\sin \theta +{ϵ}^2\mathit{\Lambda }}{q}_0.$$ 3.8

The Λ appearing in the pre-factor is of interest. It consists of contributions from the actions of the radiated sound wave on the inviscid disturbance in the main layer and on the viscous motion in the lower deck. It was found to play the same role as the impedance in the boundary condition (1.2), and hence Λ characterizes the interplay between the sound wave and the boundary layer sandwiching the acoustic field and wall. The result (3.8) indicates that the back effect is of secondary importance in the majority of the acoustic region with θ = O(1). However, it plays a leading-order role in the regions corresponding to θ = O(ϵ²) and θ − π = O(ϵ²).

Analysis of the expression for Λ shows that the back action becomes stronger in the transonic regime, and that as $M\to 1^{-}$, the sound wave concentrates in a beam centred at θ = π/2. When 1 − M² = O(R^−1/10), the back action influences the main acoustic field in the upstream region (π/2 < θ < π) at leading order, but no the main beam. The latter is influenced at leading also when 1 − M² = O(R^−1/9).

The radiation properties are rather distinct in different regimes. In the subsonic regime, only hydrodynamic fluctuations in the small-wavenumber (long-wavelength) band radiate into the far field. The acoustic source is compact so that the acoustic field radiated can be interpreted as combinations of acoustic multipoles. This is no longer the case in the transonic or supersonic regime, corresponding to 1 − M² = O(R^−1/9) or M > 1, respectively. In these cases, a broadband of hydrodynamic fluctuations radiates, and moreover the unsteady fluctuation in the upper deck has already acquired the character of sound and simultaneously acts on the viscous motion in the lower deck, that is, the back action operates at leading order.

Figure 3 shows that acoustic-field directivity of cases I and II in the subsonic case. In the latter, a favourable pressure gradient enhances the radiation. Clearly, the back action becomes appreciable when M > 0.5. Figure 4 shows the acoustic field for case II in the supersonic regime with the highly directional beams being the main feature.

Figure 3.

The directivity of the acoustic field as shown by plotting q(θ;M) (see (3.8)) in the polar coordinate. (a) Suction/injection and (b) scattering Hartree parameter (β_H = − 0.08). The dashed lines represent q₀(θ;M), the result without accounting for the effect of back action. Reproduced from Wu [52].

Figure 4.

Contours of the acoustic pressure |p_s| in the supersonic regime for the case of local suction/injection with M = 2 and f = 60. Reproduced from Wu [52].

(c). Global acoustic feedback

The asymptotic approach developed was further applied to investigate global upstream–downstream coupling via an acoustic feedback loop, which cannot be treated by the acoustic analogy. Such coupling is an important fundamental mechanism underpinning tonal noise, cavity noise and jet screech. The key process is that the sound wave emitted by an instability wave propagates upstream to re-excite the instability mode. In order to mimic this, a model problem consisting of two well-separated roughness elements was considered [52]. In this model, a T-S wave is scattered by the downstream roughness to produce a sound wave, and the latter propagates upstream and impinges on the upstream roughness to regenerate the T-S wave, which then amplifies downstream. The simple model embodies the three crucial elementary processes: radiation, receptivity and growth of the T-S mode, which are characterized by transfer functions, ${\mathcal{T}}_1$ and ${\mathcal{T}}_2$, and the N-factor, respectively. A closed feedback loop forms when the composite transfer function

$$\mathcal{T}(\omega ;x_d,R)\equiv {\mathcal{T}}_1{\mathcal{T}}_2{\mathrm{e}}^N=1,$$ 3.9

where x_d is the distance between the two roughness elements. It is worth mentioning that both radiation and receptivity coefficients, ${\mathcal{T}}_1$ and ${\mathcal{T}}_2$, are complex quantities, each contributing a phase lag. It follows that the simplistic consideration of mere propagation time is inadequate for establishing the phase relation for a closed feedback loop [53]. Numerical calculations suggest that condition (3.9) can be satisfied for moderate Reynolds numbers and roughness heights with the consequences that the long-range acoustic coupling renders the boundary layer absolutely unstable, and a state of self-sustained oscillations at discrete frequencies is eventually established. The dominant peak frequency is found to exhibit the characteristic ‘ladder structure’, i.e. it jumps from one value to another as the Reynolds number or the distance between the roughness elements is varied gradually [53]. An example is shown in figure 5.

Figure 5.

Frequency switching and ladder structure: (a) tonal frequencies f versus x_d for M = 0.3 and R = 10⁶; (b) tonal frequencies ω_d versus R for M = 0.3 and x_d = 0.5. Reproduced from Wu [52].

The generation of a sound wave when a T-S mode is scattered by surface roughness (protuberance) was investigated experimentally by Kobayashi et al. [54]. In their set-up, the regeneration of the T-S wave was due to leading-edge adjustment, which is a rather weak receptivity mechanism. Furthermore, the Mach number is rather low. As a result, for tonal noise to occur the protuberance height had to be comparable with the boundary-layer thickness, which is much larger than what the theory allows. Nevertheless, the directivity measured was quite similar to the theoretical predictions of Wu & Hogg [50] and Wu [52]. Abo et al. [55] showed that when the boundary layer is subject to an adverse pressure gradient, which enhances the amplification of the instability waves, tonal noise occurs at reduced protuberance heights. Kobayashi & Asai [56] demonstrated that roughness of very small height (about one-tenth of the boundary layer thickness) placed upstream of the protuberance activates a strong receptivity, thereby facilitating the self-sustained oscillations. Their experimental results broadly confirm the theoretical work [52].

The theoretical work summarized in this section assumed that the roughness height is sufficiently small that the mean-flow distortion remains linear. The extension to the case of nonlinear distortion was carried out by Dong & Wu [57].

4. Relevance to acoustic radiation of coherent structures in turbulent shear flows: an outlook

The theoretical efforts have thus far focused on instability modes on laminar flows in order to provide definitive and self-consistent descriptions. The mathematical procedures and even results are, however, applicable to coherent structures in the corresponding turbulent shear flows insofar as the latter can be treated as a wavepacket of instability modes with an amplitude function A. However, two important facts must be noted about turbulent flows. The first is that turbulent mean flows vary more rapidly in the streamwise direction than their laminar counterparts, and thus the resulting stronger non-parallelism must be taken into account. Secondly, the coherent structures are influenced by fine-scale turbulence via the phase averaged small-scale Reynolds stresses [58]. For turbulent flows with a generalized inflection point, the Reynolds stress tensor may be represented by a local gradient type of closure model allowing for a possible phase lag between it and the strain rate tensor of the large-scale structures [59]. With these two effects being taken into account in the nonlinear critical-layer approach, the kernel function in the amplitude equations must be modified [48]. Of course, the amplitude evolution of CS may well be provided by DNS or experiments, in which case the information may be supplied to the formulae.

While the results were derived for free shear flows, the circular jet in particular, they may be applied to instability waves/CS in laminar/turbulent wall-bounded flows. The analysis discussed in §§2b indicates that in subsonic shear flows the most energetic hydrodynamic fluctuations in the main spectrum do not radiate directly. It is long-wavelength/low-frequency components that act as emitters. This is likely to be the case in the supersonic regime of moderate Mach number. The evidence supporting this speculation can be found in the experimental data of Laufer [60] and DNS results of Duan et al. [61], which are reproduced in figure 6. The experimental result shows that the radiated acoustic field contains more/less energy in the low/high wavenumber components than does the near field hydrodynamic fluctuation. The DNS result reveals that the peak of the pressure energy spectrum shifts towards lower frequencies as the free stream is approached. In the high-Mach-number supersonic regime, energetic fluctuations propagate supersonically relative to the ambient flow, emitting Mach waves [60], and so it would be interesting as well to examine if turbulent mean velocity and temperature profiles support supersonic modes.

Figure 6.

Experimental and numerical evidence showing low-frequency radiation. (a) Comparison of the wall and far-field pressure spectra (from [60]). (b) Pressure spectrum at different transverse positions (from [61]).

Effects of surface roughness on noise generation in turbulent boundary layers are of practical importance, but are not well understood. If coherent structures can be represented by wavepackets, as experiments suggest, the scattering process described in §3 would be operating there.

The asymptotic approach and acoustic analogy are clearly distinct, but they share some commonality in the sense that both solve inhomogeneous systems using a certain form of input as forcing. When applied to turbulence generated noise, the input in the acoustic analogy is the correlations of velocity fluctuations, or auto-covariances of the Reynolds stresses and energy flux. These are usually taken to be a wavepacket form in spectral (frequency-wavenumber) space. In the asymptotic approach, the direct input is a wavepacket in physical space, but the corresponding correlations in spectral space are also of wavepacket form, probably not dissimilar to what are assumed and used in calculations based on the acoustic analogy. The above discussion suggests that a link may be established between the two approaches.

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Authors' contributions

X.W., 60; Z.Z. 40.

Competing interests

We declare we have no competing interests.

Funding

Z.Z. was supported by NSFC (grant no. 11472190).