Sound generation by a pair of co-rotating vortices using spectral acoustic analogy

Journal Pre-proof Sound generation by a pair of co-rotating vortices using spectral acoustic analogy Feng Feng, Xiannan Meng, Qiang Wang

PII:

S0022-460X(19)30683-2

DOI:

https://doi.org/10.1016/j.jsv.2019.115120

Reference:

YJSVI 115120

To appear in:

Journal of Sound and Vibration

Received Date: 22 March 2019 Revised Date:

21 November 2019

Accepted Date: 25 November 2019

Please cite this article as: F. Feng, X. Meng, Q. Wang, Sound generation by a pair of co-rotating vortices using spectral acoustic analogy, Journal of Sound and Vibration (2020), doi: https://doi.org/10.1016/ j.jsv.2019.115120. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.

Sound Generation by a Pair of Co-rotating Vortices Using Spectral Acoustic Analogy Feng Fenga , Xiannan Mengb∗ , Qiang Wanga , b School

a China Academy of Aerospace Aerodynamics, 100074, Beijing, P.R. China of Mathematics and Manchester centre for Nonlinear Dynamics, University of Manchester, M13 9PL, Manchester, UK

Abstract The spectral acoustic analogy (AA) theory is used to study the acoustic generation by a pair of co-rotating vortices from the decomposition of wave packets. The use of a two-dimensional spectral AA formulation is based on the data obtained by conducting a direct simulation of unsteady compressible Navier-Stokes equations. By way of the spectral AA theory and convolution filtering operations, the role played by the wave packets of the vortices in the generation of the sound are thoroughly discussed. It is found that the far-field sound is generated by the linear superimposition of the interactions between the base flow and the remaining wave packets. Moreover, a notable wave packet of the co-rotating vortices, called the acoustic-related wave packet herein, is identified in the spectral domain, which not only contains all radiating components but also contributes to a discernible acoustic source. It potentially bridges the gap between the near-field wave packets and the far-field sound. The present methodology based on the spectral AA theory shows promise in providing insights into the mechanisms of sound generation in turbulent or unsteady vortical flows. Keywords: spectral acoustic analogy, sound generation, wave packets, co-rotating vortices, acoustic-related wave packet

1. Introduction In aeroacoustics, acoustic analogy (AA) is a competent fundamental theory used to account for the physics of sound generation for turbulent or unsteady vortical flows. The pioneering AA theory proposed by Lighthill [1] presumes that the acoustic source in turbulent flow is comprised of a spatial-temporal 5

correlation of velocity fluctuations, entropy fluctuations, and viscous stress. It is a challenge in Lighthill’s source to clarify the dynamics of sound generation in turbulent flows in a physically concise way. After Lighthill’s work, researchers have been devoting great efforts to understand acoustic sources with AA. For example, the vortex sound theories introduced by Powell [2] and Howe [3] emphasize the role of vorticity as the source of the sound. The vortex sound formula derived by M¨ohring [4] considers that

10

the far-field sound is generated by the vorticity alone for low-Mach-number flows. A generalized AA proposed by Goldstein [5] identifies physical acoustic sources by employing non-radiating components and then excluding refractive and other propagation properties from the acoustic sources. It is necessary ∗ Corresponding

author Email address: [email protected] ()

Preprint submitted to Journal of Sound and Vibration

November 21, 2019

to note that the intrinsic dynamics of the sound generation in turbulent flows still remain unclear, though insights have been gained into acoustic sources in the aforementioned studies. It is therefore imperative 15

to develop a more effective approach to unravel the underlying physical mechanism of acoustic sources, which is a necessary and key step to understand sound generation in practical complex flows. The AA theory formulated in the spectral space is an alternative approach to gain insights into the acoustic-source physics of unsteady flows in addition to the conventional AA theory. The spectral AA originates from Ffowcs Williams in (1963) [6] and Crighton in (1975) [7] who separated the radiating

20

source from the nominal acoustic source of the acoustic analogy by applying a spatial-temporal Fourier transform to Lighthill’s equation. A direct equivalence between the far-field acoustic pressure and the radiating source can be therefore formulated in the wavenumber-frequency space, which demonstrates that only the acoustic source satisfying relation | k |=| ω/c0 | (k: wavenumber, ω: angular frequency and c0 : ambient speed of sound) can contribute to the far-field sound. Additionally, the spectral AA theory

25

shows it is capable of analyzing the contribution of the radiating source to the acoustic directivity of jet noise ([8, 9, 10]), estimating the effective source of acoustic radiation for turbulent channel flow [11] and mixing-layer flow [12], and effective at explaining the acoustic emission from wave packets geometrically [13]. However, it is noted that in the previous studies spectral AA has principally been employed in the qualitative study of unsteady flows and was not used in quantitative study of spectral AA. Hence, this

30

has been the main motivation for us to conduct this research. In complex unsteady flows, e.g., shear and jet flows, a number of coherent structures can be simplified by a two-dimensional vortex model under certain conditions. Consequently, significant research has been conducted to explore the mechanisms of sound generation on the basis of the two-dimensional vortexsound model, see [14, 15, 16, 17, 18, 19, 20]. However, many ambiguities exist in terms of the relation

35

between the acoustic components and the acoustic source. For example, a unified interpretation on the acoustic source for high-frequency acoustic components radiated by two-dimensional co-rotating vortex pairs has not been reached. Tang and Ko [21] demonstrated that the high-frequency sound is generated by the deformation of the vortex core, whereas Eldredge [22] found that the dynamics of the vortex filaments is responsible for the high-frequency sound. Such ambiguities that exist in the two-dimensional

40

problems of vortex and sound could pose a challenge to the full understanding of the mechanism of sound generation in complex practical flows. This motivates us to develop a new approach to explore these vortex-sound problems. This paper develops a new analysis methodology and applies spectral AA theory to study the mechanism of sound generation in a pair of co-rotating vortices. The following two aspects are highlighted: The

45

spectral AA is employed to quantitatively predict sound in our study, unlike previous studies which only qualitatively employed spectral AA theory. Additionally, the definite correlations between the acoustic components and the source components, which is unclear for sound generation in unsteady flows and a challenging problem for conventional AA theory, can now be examined in this study by spectral AA. Specifically, we explore the physical mechanism of the acoustic sources of the co-rotating vortices during

50

the four stages in terms of the evolution of the vortices [23]. Wave packets that are defined as the spectra clustered around single wavenumber/frequency values in the spectral domain, are extracted to construct

2

the sources of the spectral AA. Then, the contribution of each wave packet to the sound generation is interpreted by predicting the far-field sound of the wave-packet sources. Note that in contrast to obtaining wave packets by forcing the flow with the most amplified instability waves, which is often utilized to 55

study jet noise [24, 25], a direct spatial-temporal Fourier transform of the flow field is performed herein to obtain the wave packets in the spectral space. Furthermore, we investigate the spatial-temporal evolution of an acoustic-related wave packet, which potentially builds a connection between the near-field non-radiating components and the far-field sound. The present paper is organized as follows: The two-dimensional spectral AA formulation, the numer-

60

ical model, and the approach used to compute the wave packets’ contribution to the acoustic source are described in Section 2. Then, we validate the spectral AA quantitatively, interpret the roles that the wave packets played in the generation of the sound, and scrutinize the acoustic-related wave packet in Section 3.1-3.3, respectively. Finally, conclusion is presented in Section 4.

2. Aeroacoustic Theory and Numerical Model 65

This section presents a formulation of the two-dimensional spectral AA theory and its subsequent numerical method. We follow a similar method to that of Crighton (1975) [7] in our formulation of the spectral AA theory, but the expression derived herein retains a phase difference during the simplification of the two-dimensional Green’s function, and therefore, it includes more details on the terms of the acoustic source. Consequently, the two-dimensional spectral AA formulation can readily be validated

70

through the prediction of sound in vortical flows. 2.1. Two-dimensional spectral AA formulation The two-dimensional Lighthill equation [1] is given by   2 ∂ 2 Tij ∂ ∂2 2 − c ρ= , 0 2 ∂t ∂yi ∂yj ∂yi ∂yj

75

(1)

where Tij = ρui uj + (p − c20 ρ)δij − τij , (i, j = 1, 2), and we note that Einstein’s summation convention is used in the present work. As usual, t denotes time, yi source coordinate, c0 ambient speed of sound, ρ density, Tij Lighthill stress tensor, ui velocity in the yi direction, p pressure, τij viscous stress tensor, and δij Kronecker delta. Applying a Fourier transform to both sides of eq. (1) then leads to an inhomogeneous Helmholtz

80

equation 

ω c0

2

 ∂2 1 + ρe = − 2 qe, ∂yi ∂yj c0

(2)

where ω is angular frequency and qe(y, ω) = 85

∂ 2 Teij (y, ω) . ∂yi yj

(3)

It should be noted that variables with a tilde ∼ above them represent variables in the frequency domain. R∞ Physical quantities in time and frequency domains satisfy f (y, t) = −∞ fe(y, ω)exp(−iωt)dω, where f denotes an arbitrary physical quantity. For a two-dimensional unbounded compact flow, it is natural to 3

(1)

(1)

introduce the free-space Green’s function G(ˆ r) = H0 (ωˆ r/c0 )/4i, where H0 90

is a zeroth-order Hankel

function of the first kind and rˆ is the distance between the source and the observer. Using the Green’s function, one can find the solution of eq. (2) in the form of   Z r i (1) ωˆ qe(y, ω)H0 dy, ρe(x, ω) = 2 4c0 c0

(4)

where variable x = (x1 , x2 ) represents the observer’s coordinates, and the observer and source coordinates 95

satisfy rˆ =| x − y |. Applying the asymptotic relation r 2 (1) H0 (z) ≈ exp[i(z − π/4)] πz 0

0

(5)

0

0

to eq. (4) for z → ∞ and using relation p = c20 ρ (p : pressure fluctuation and ρ : density fluctuation) in the far field, we obtain pe(x, ω) =

100

i 4

r

Z

qe(y, ω)

  2c0 ωˆ r π exp i − dy. πωˆ r c0 4

(6)

Equation (6) has a similar form to that of the three-dimensional AA in the frequency domain [7, 13], but they are essentially different. The differences are characterized by the phase difference and coefficients √ of harmonics, as well as the damping function that takes the form of 1/ rˆ for two-dimensional waves, 105

and the form of 1/ˆ r for three-dimensional waves. The root of these differences is in the distinct responses to the acoustic fields. As two-dimensional waves spread out radially as sharp pulses, their sharp fronts are maintained but they leave a wake behind them after they have passed. As a result, the acoustic perturbations continue after the crest has passed. In contrast, as three-dimensional waves spread out radially, the amplitude diminishes but their sharp fronts and rears are maintained. They leave no wake 0

110

behind them after they have passed. Following the work of [13], we introduce a new coordinate y1 in the 0

0

0

0

x direction and y2 is normal to x. As shown in Fig. 1, the origin O y1 y2 is fixed at the center of the 0

noise source q and the observer in new coordinates is located at y = (r, 0). The distance rˆ between the observer and the source is approximated by 0

rˆ ≈ r − y1 ,

115

(7)

which retains parts of the phase difference between the sound waves generated by the components of the source distribution at different locations in a compact source (λa  l, where λa is the acoustic wavelength p p and l is the source size). Substituting eq. (7) into eq. (6) and approximating 1/ˆ r with 1/r, which

120

introduces an amplitude error O(l/2r) but no phase error, we obtain r   Z  0  0 0 i ωr π ωy1 2c0 pe(x, ω) = exp i − qe (y , ω)exp − i dy, 4 πωr c0 4 c0 0

(8)

0

where qe (y , ω) is the noise source in the new coordinate system. R∞ 0 0 0 0 0 0 0 The spatial Fourier transform is defined by qˆ (k , ω) = −∞ qe (y , ω)exp(−ik y )dy . Using this

125

definition, the integral formula (8) can be represented as r   0 0 i 2c0 ωr π pe(x, ω) = exp i − qˆ (k , ω), 4 πωr c0 4

4

(9)

 

     

x

   

Observer 

     

y2’ 

 

r 

 

q 

y2

 

y1’ 

^ r 

λa 

   

O’ 

 

Sound 

Source 

l 

   

O 

y1

 

Figure 1: Sketch of the source coordinates and the sound propagation. 0

0

 

0

at the point k = (ω/c0 , 0) in the wavenumber plane. The noise source qˆ (k , ω) can be further represented by the Lighthill source in the generalized coordinates. Applying the spatial Fourier transform to both 0

0

0

0

sides of eq.(3) and making use of the property that ∂ 2 /∂yi ∂yj translates to −ki kj this then yields the 130

far-field pressure fluctuation i pe(x, ω) = 4

r

   0 0 2c0 ωr π ω2 exp i − 2 Tˆ11 (k , ω), − πωr c0 4 c0

(10)

0

in the spectral space, where k = (ω/c0 , 0) holds. 0

0

Given that T11 (y , t) is the element of Tij (y, t) in the observer direction, eq. (10) can therefore be 135

rewritten as i pe(x, ω) = 4

r

  2c0 ωr π exp i (−ki kj )Tˆij (k, ω), − πωr c0 4

(11)

0 0 by applying the transform Tˆ11 (k , ω) = xi xj / | x ¯ |2 Tˆij (k, ω), where wavenumber is

  140

k=

ω x . c0 | x |

(12)

Equations (11) and (12) show that only those acoustic source components, which lie on the acoustic p radiation circle k12 + k22 =| ω/c0 |, emit sound waves into the far field in two-dimensional cases. Finally, the pressure solution in the time domain can be obtained using the inverse Fourier transform p(x, t) = R∞ pe(x, ω)exp(−iωt)dω. −∞ 145

2.2. Flow model The dynamics of two Gaussian vortices rotating clockwise around one another (see Fig. 2) is considered. Initially, the two vortices are aligned along the y2 -axis, and hence the orientation of the vortex pair θ, i.e. the angle between the line crossing the centers of the vortices and the y2 -axis, is zero degree. Additionally, the separation distance b of the two vortices initially equals 2R and the tangential velocity for each vortex

5

r0  b y2 O

Г0 

θ  y1 

U0  Г0 

 

Figure 2: Sketch of the vortex pair.

150

is vθ =

Γ0 (1 − exp(−1.25(ˇ r/r0 )2 )), 2πˇ r

(13)

where the circulation is Γ0 = 2π(0.7)−1 U0 r0 and rˇ represents the distance away from the center of the vortex core. The tangential velocity achieves a maximum velocity U0 at a radius r0 away from the center 155

of each vortex core. We follow [16] to stipulate U0 = 0.56c0 and r0 = 0.15R, which implies a maximum Mach number M0 = 0.56 and a Reynolds number Re = Γ0 /ν = 7500. The fluid is assumed to be perfect gas with a ratio of specific heats γ = 1.4 and a Prandtl number of 0.7. The flow is initially specified as solenoidal and homentropic, given that this choice of initial conditions can reduce the magnitude of the initial acoustic transient ([16]). For a homentropic solenoidal

160

flow, the initial pressure and density satisfy the following Poisson equation   γ − 1 ∂ui ∂uj ∂2 p =− ∂yi2 ρ γ ∂yj ∂yi

(14)

where the pressure and density are assumed to be related by ρ/ρ0 = (γp/(ρ0 c20 ))1/γ . The initial revolution speed of the co-rotating vortices Ur , based on the initial velocity field, is Ur = 165

0.06c0 , and the initial period of rotation is τ1 = 104.7R/c0 . This implies that the fundamental wavelength λa of the sound is λa = 52.4R due to the quadrupole nature of the acoustic radiation. Given λa  R, the condition for a compact source is satisfied here. 2.3. Numerical method The flow field and the far-field sound of the co-rotating vortices are computed directly by solving the

170

two-dimensional unsteady compressible Navier-Stokes equations. The ambient speed of sound c0 , the ambient density ρ0 , and half of the separation distance of the vortices R are selected as the characteristic velocity, density, and length to non-dimensionalize the governing equations. A seven-point fourth-order Dispersion-Relation-Preserving (DRP) finite difference scheme, documented in [26], is employed for spatial discretization, which achieves high resolution of the acoustic propagation of the co-rotating vortices. A

175

six-stage second-order low-storage Runge-Kutta scheme in [27] is used to reduce the dispersion and 6

0.02 513 × 513 1025 × 1025 2049 × 2049

ΘR/a0

0.01

0

-0.01

-0.02

0

50

100

150

200

tc0/R Figure 3: Dilatation rate at (y1 , y2 ) = (0, 1.2R).

dissipation errors for the temporal integration. A selective damping is used each iteration to filter out spurious numerical short waves with an artificial stencil Reynolds number Rs = 20 in the entire computational domain, and an additional selective damping is applied in the stretch-mesh zone with Rs = 5 (see [28]). The radiation boundary condition is implemented using DRP non-centered schemes 180

at three ghost points in the boundary region of the computational domain. A square computational domain of size −60R ≤ y1 , y2 ≤ 60R with 1025×1025 grid points is employed. To inspect if such a resolution of mesh is sufficient for a convergent solution, we examine this problem with 513×513, 1025×1025 and 2049×2049 grid points, respectively. The computation results of the dilatation rates at a point (y1 , y2 ) = (0, 1.2R), obtained by employing different grid points, are shown in Fig. 3. It

185

is found that an oscillatory behavior appears in the early stage for the coarse mesh (513 × 513), whereas the grid points (1025 × 1025) can provide the same accurate results as the grid points (2049 × 2049). Further, the initial co-rotating vortices are located at the center of the domain where a uniform grid is set within the bounds of −10R ≤ y1 , y2 ≤ 10R with there being 513 points in each coordinate direction. The mesh is stretched outside of the central domain with a maximum spacing ∆y1 = ∆y2 = 0.56R at the

190

outmost boundary of the grid. This indicates roughly 20 grid points initially across each vortex core and at least 93 grid points across the wavelength λa . The time step is set to ∆t = 0.03R/c0 . The numerical simulation runs until non-dimensional time t = 750R/c0 and takes 25000 time-steps. The first 20480 time-steps (i.e., until t = 614.4R/c0 ) are used for the spectral analysis. 2.4. General approach of the wave packet decomposition

195

Once the density and velocity are determined from numerical simulation, the Lighthill stress tensor Tij can be obtained. It is approximated as Tij ≈ ρui uj , given that the viscous term τij is generally negligible because of its extremely inefficient octupole nature as a noise source, and the entropy-like term (p − c20 ρ)δij can be omitted as well because the temperature: inhomogeneities are not strong. Further, 7

we implement the spatial-temporal fast Fourier transform (FFT) for the Lighthill stress tensor Tij (y, t) 200

to compute Tˆij (k, ω) on the acoustic radiation circle defined by | k |=| ω/c0 |. The far-field pressure fluctuation pe(x, ω) of the vortex pair can be then predicted by using formula (11). Additionally, many aspects are worth illustrating. The problem of interest here is essentially a diverging base flow, which implies that the hydrodynamics of the vortices is the strongest in the regions near the acoustic source and this strength decays spatially and radially until reaching a quiescent state in regions sufficiently far

205

away from the acoustic source. We conduct spatial FFTs on the source terms in a sufficiently large region (−10R × 10R). The values of the quantities on the boundaries of this region are the same. In this case, we can impose a periodicity over the whole domain smoothly with the spatial period 20R and obtain accurate spectra in the spatial FFTs. In addition, the temporal periodic nature overall holds for the problem of co-rotating vortices. Given the explanation of Sinayoko et al. (2011) [29] on diverging

210

base flows that there is no requirement to perform windowing operations, we do not therefore apply any window operation explicitly for the purpose of avoiding contaminating data which could affect the acoustic prediction of the spectral AA. It is known that the finite time-domain Fourier transform of non-strictly periodic quantities can introduce a rectangular window implicitly through the time domain, which leads to the Gibbs phenomenon in Fourier analysis. It is therefore necessary to discuss the influence of the

215

implicit windowing operation on the identification of wave packets, which will be explained in detail later. The spatial-temporal FFT is also applied to the density and velocity and then a filtered acoustic source term ρ¯ui uj is reconstructed. The source term ρ¯ui uj can be used to predict the corresponding far-field sound by substituting it into eq. (11). The contributions of wave packets to the radiating source can be analyzed accordingly. Specifically, the detailed procedure is given as follows:

220

1. conduct the spatial-temporal FFT of the density ρ and velocity u in the acoustic source region to obtain the density ρˆ(k, ω) and the velocity component u ˆi (k, ω), (i = 1, 2), 2. multiply ρˆ(k, ω) and u ˆi (k, ω) by the filter window w(k, ω) to obtain the filtered variables ρ¯ˆ(k, ω) ¯ and u ˆi (k, ω), ¯ˆi (k, ω) to obtain the filtered variables ρ¯ and u 3. conduct the inverse FFT of ρ¯ ˆ(k, ω) and u ¯i in the

225

physical space, 4. reconstruct the filtered acoustic source term ρ¯u ¯i u ¯j , 5. conduct the spatial-temporal FFT of ρ¯u ¯i u ¯j on the radiation circle | k |=| ω/c0 | to obtain the radiating components of the filtered acoustic source term, 6. and predict the far-field sound by substituting the radiating components of the filtered acoustic

230

source term into eq. (11).

3. Results and discussion 3.1. Computing far-field sound using spectral AA Equation (11) indicates that the Lighthill stress tensor is the source term in the generation of sound. Consequently, the behavior of the Lighthill stress tensor in the wavenumber-frequency space is examined 235

first. Without loss of generality, we shall focus on the far-field sound on the y2 -axis for the purpose 8

ω = k c0

= Ur

ω

=

-k 0.3

^ |T22|/(ρ0c20)

r

ω

ω = - k c0

kU

0.6

1

ωR/c0

ω = ω1

0.82

0

0.64

ω = - ω1 -0.3

0.46 0.28 0.1

-0.6 -12

-8

-4

0

4

8

12

k2R Figure 4: Fourier amplitude of the acoustic source term | Tˆ22 (k, ω) |/(ρ0 c20 ) in wavenumber-frequency space. The contour levels are made from Tˆ22 /(ρ0 c20 )= 0.1 to Tˆ22 /(ρ0 c20 )= 1 with interval ∆ | Tˆ22 |/(ρ0 c20 )= 0.18.

of comparing the current results with those published in [16] and [19]. It implies k1 = 0 and only Tˆ22 leads to a non-trivial pressure fluctuation on the y2 -axis. The flow simulated here is sampled every 20 time-steps (20∆t) from the beginning of the simulation, with 1024 time samples in all including the initial transient being used to compute the FFT spectra in this section. As explained above, none of the windows 240

are explicitly applied to the data obtained by direct simulation, before the FFTs in space and time are conducted. Additionally, given that the flow is not strictly periodic, it is necessary to access the influence of the implicit windowing operation introduced by applying the temporal Fourier transform of the source term. The length of the implicit rectangular window, i.e. the difference of the temporal integration bounds, equals t = 614.4R/c0 herein which leads to a spurious angular frequency ωs = 2π/t ≈ 0.01c0 /R

245

and wavenumber ks = ωs /c0 ≈ 0.01/R. The spurious wavenumber is much smaller than the cutoff wavenumber (kc = 2π/20R = 0.314/R) that was introduced by the spatial spectral resolution. The wavenumbers below the cutoff wavenumber will be not considered. Fig. 4 illustrates the Fourier amplitude of Tˆ22 (k, ω). It is found that, in contrast to the continuous spectra of the turbulent jet flows [9, 10], the spectra of the co-rotating vortices in the spectral space

250

are represented by discrete wave packets. The acoustic components of Tˆ22 (k, ω) lie on the dashed lines (k1 , k2 ) = (0, ±ω/c0 ) in the wavenumber-frequency space. The radiating sources on the radiation line ω = k2 c0 propagate in the positive direction of the y2 -axis, while the sources on the radiation line ω = −k2 c0 propagate in the negative direction of the y2 -axis. The Fourier wave packets corresponding to the radiation lines are symmetrical with respect to the ω-axis, which indicates that the sound intensities,

255

radiated by the acoustic sources into the positive and negative directions of the y2 -axis, are the same. In addition, it is found that non-radiating components principally lie on the convection lines ω = ±k2 Ur represented by the dash-dotted lines in Fig. 4, where Ur denotes the dominant phase velocity of the sound 9

02

30 Mitchell et al. Eldredge et al. Direct simulation AA Spectral AA

0 1.5

p’/(ρ0c20)

τa1

Direct simulation

× 104 01

τa2 0

-1.5 0

AA Spectral AA

τa1

p’/(ρ0c20)

× 104

τa2 0

0

100

200

300

400

500

-1 0

600

0

100

tc0/R

200

300

400

500

600

tc0/R

(a)

(b)

Figure 5: Evolution of the far-field pressure fluctuations. Panel (a) describes pressure at (y1 , y2 ) = (0, λa /2) and panel (b) at (y1 , y2 ) = (0, λa ).

source along the y2 -axis. It is found that Ur = 0.06c0 in the initial stage, which is due to the fact that the dominant motion of the vortical flow is the co-rotation of the vortices and hence one can periodically 260

observe a phase velocity along the y2 -axis that equals the revolution speed of the vortices. Moreover, as indicated by the dash-dot-dotted lines in Fig. 4, dominant radiating components and dominant convection components possess the same angular frequency, ω1 = 0.1227c0 /R, which corresponds to the initial corotating period (τ1 ) of the vortex pair. As expected, the physically dominant angular frequency ω1 and its harmonic frequencies (2ω1 , 3ω1 , etc.) are dramatically different from the spurious angular frequency

265

ωs that was implicitly introduced by the temporal Fourier transform. This implies that the implicit windowing operation plays a trivial role in the present Fourier analysis. Fig. 5 shows the time histories of the far-field pressure fluctuations at (y1 , y2 ) = (0, λa /2) and (y1 , y2 ) = (0, λa ). During the process of analyzing pressure fluctuations, in order to minimize the possibility of interpolation errors we conduct the spatial-temporal FFT of the Lighthill stress component Tˆ22 (y, t)

270

directly on the radiation line ω = k2 c0 instead of employing numerical interpolation from the nearest grid points in the spectral space. The present results are comparable to those of the previous studies ([16] and [19]). Fig. 5(a) demonstrates that the mergence of vortices takes place after around 3.5 revolutions in the present simulation. It occurs after around three revolutions in [16] and after around 4.5 revolutions in [19]. The predicted magnitude and the phase of the pressure at the first two revolutions match the results

275

in [16] and [19] well. When compared with the results obtained by conducting a direct computation of the Navier-Stokes equations, the spectral AA predicts the pressure fluctuation quite accurately. Furthermore, the results of the two-dimensional M¨ ohring equation [16], which is named as AA in the present work, is also shown in Fig. 5. The remarkable agreement of the predictions between the spectral AA and the AA is observed both at (y1 , y2 ) = (0, λa /2) and (y1 , y2 ) = (0, λa ). It indicates that applying the asymptotic

280

solution of the Hankel function and ignoring the transverse phase difference of the source in the derivation of the spectral AA do not affect its accuracy. Besides, a non-trivial advantage of the spectral AA against AA is worth mentioning. As Mitchell et al. (1995) [16] noted, the area integral of the Lighthill source 10

= ω

= ω

Ur

0

0.64

ω = - ω1 -0.3

1

ω = ω1

0.82

ωR/c0

Ur

0.3

^ |ρ|/ρ 0

r

r

ω = k c0

-k

-k

1

ω = ω1

ωR/c0

ω = - k c0

=

= 0.3

2 ^ |p|/(ρ 0c0)

kU

0.6

ω = k c0

ω

ω

ω = - k c0

kU

0.6

0.82

0

0.64

ω = - ω1

0.46

-0.3

0.28

0.28

0.1

8

-4

0

4

k2R

(a)

(b)

0.6

ω = - k c0

ω = k c0

0.3

3.2

-0.3

|u^ 2|/c0

=

0.2

ω = ω1

4.1

ω = - ω1

12

ω

Ur

Ur

ω

=

-k

-k

5

0

8

=

=

|u^ 1|/c0

ω

ω = k c0

ω = ω1

ωR/c0

-8

k2R

ω = - k c0

ω 0.3

-0.6 -12

12

r

4

kU

0

0.1

ωR/c0

0.6

-4

r

-8

kU

-0.6 -12

0.17

0

0.14

ω = - ω1

2.3

-0.3

1.4

-8

-4

0

4

8

0.11 0.08

0.5

-0.6 -12

0.46

0.05

-0.6 -12

12

-8

-4

0

k2R

k2R

(c)

(d)

4

8

12

Figure 6: Fourier amplitude of pressure | pˆ(k, ω) |/(ρ0 c20 ) (panel (a)), density | ρˆ(k, ω) |/ρ0 (panel (b)), velocity component |u ˆ1 (k, ω) |/c0 (panel (c)) and velocity component | u ˆ2 (k, ω) |/c0 (panel (d)).

term T22 diverges in AA theory. However, this area integral is not required in the spectral AA. 3.2. Source Components of Co-rotating Vortices 285

This section begins with briefly scrutinizing behaviors of the Fourier amplitude of the density ρ, pressure p, and velocity components ui to gain insights into the wave packets behind these physical quantities. Then, we investigate the contributions of the wave packets to the sound generation. The Fourier amplitude of density ρ, pressure p and velocities ui in the (k2 , ω)-plane is shown in Fig. 6. The acoustic-related spectra lie on radiation lines ω = ±k2 c0 and the dominant convective spectra lie

290

on convection lines ω = ±k2 Ur in the distributions of Fourier amplitudes of density ρ and pressure p. This is consistent with those observed in Fig. 4 with respect to the acoustic source. The Fourier amplitude of the velocity u1 characterized by dash-dotted lines in Fig. 6(c) represents the non-radiating convective spectrum of the flow, and the Fourier amplitude of the velocity u2 characterized by dashed lines in Fig. 6(d) represents acoustic radiating components. The spectral distribution of the synthesis

11

16π 2880

1.2 1

Merged diffusive stage

Convective stage Second diffusive stage

0.4

θ (rad)

b / b0

2160 12π

First 0.8 diffusive stage 0.6

8π 1440

4π 720

0.2

t2

t1 0 200

t3 0

300

400

500

tc0/R Figure 7: Evolution of the normalized distance of the maximum vorticity magnitude (b/b0 , solid line) and the pair orientation (θ, dash-dotted line) versus time. The separation distance of the vortex centres keeps constant in the first stage (t ≤ t1 = 255R/c0 ), while the separation distance decreases rapidly in the second stage (t1 < t < t2 = 384R/c0 ). The separation distance decreases gently in the third stage (t2 ≤ t < t3 = 465R/c0 ) compared with that in the third stage and the separation distance vanishes in the fourth stage.

295

of the velocity components u ˆ1 and u ˆ2 is consistent with the spectral distributions of the flow quantities, e.g., ρˆ(k, ω) and pˆ(k, ω). The Fourier wave packets of the source term Tˆ22 (k, ω) are generated by the nonlinear interaction between wave packets of the density ρˆ(k, ω) and the velocity component u ˆ2 (k, ω). Provided that the contributions of these wave packets to the source term are known, the roles played by these wave packets

300

in the sound generation can be detected accordingly. The detailed procedure is presented in section 2.4, in which the density ρ is selected as the proxy to illustrate the wave packets of the vortices because it requires less computational burden as compared to the way in which the velocity component ui (i = 1, 2) is selected. In the following, the contributions of wave packets to the sound generation are presented during the

305

first three stages (the 1st diffusive stage, the convective stage, and the 2nd diffusive stage), which are divided according to the evolution of the separation distance of the vortex centres b(t). b(t) has distinct characters during the four stages observed in the near field ([23]) as Fig. 7 exhibits. The fourth stage in which the vortices have already merged into one vortex is acoustically of no interest and we do not therefore discuss sound generation in this stage. The four stages above are defined in relation to the

310

acoustic source and there exist three characteristic timescales t1 , t2 and t3 (see Fig. 7). We particularly pay attention to the far-field sound and a delayed time is therefore required to be added onto t1 , t2 and t3 when considering the far-field sound. (i) Contributions of wave packets in the 1st diffusive stage We collect 256 time samples of the density in the source region from t = 102R/c0 to t = 255R/c0 for

315

the spatial-temporal FFT to illustrate spectral distribution in the first stage. Because the initial acoustic

12

0.6

0.6

0.6

^ |ρ|/ρ 0

^ |ρ|/ρ 0

0.5

0.3

0.5

0.3

0.3

0.5

I.6

I.3

0.332

0

0.248

0.416

ωR/c0

0.332

0

0.416

ωR/c0

ωR/c0

0.416

^ |ρ|/ρ 0

I.5 I.4 I.7 I.3

0.332

0

I.2

0.248

I.3 -0.3

0.164

-0.3

0.164

0.08

-0.6 -12

-8

-4

0

4

8

12

0.164

I.1

0.08

-0.6 -12

-8

-4

0

4

8

-0.6 -12

12

0.248

I.3

-0.3

-8

-4

0.08

0

k1R

k12R

k2R

(a)

(b)

(c)

4

8

12

Figure 8: Fourier amplitude of the density field in (a) (k1 , ω)-plane, (b) (k12 , ω)-plane and (c) (k2 , ω)-plane in the 1st diffusive stage. In panel (b) k12 represents the axis of k1 = k2 .

transient of the co-rotating vortices is not physically interesting, the sampling procedure begins from t = 102R/c0 when the effect of the initial acoustic transient has vanished. The characteristic scales of the co-rotating vortices in each azimuthal direction are identical in this stage because of symmetry, and the wave packets are presumably axisymmetric around the ω-axis in the spectral space. The spectral 320

distribution of wave packets in the (k1 , k2 , ω)-space is examined. Three slices of the wavenumber-frequency (k1 , k2 , ω) density spectra (Fourier amplitude) are extracted and shown in Fig. 8. The wave packets in the (k1 , ω)-, (k12 , ω)- and (k2 , ω)-planes demonstrate an identical distribution in this stage, which implies that the axisymmetry of the flow spectra holds. The wave packets can be identified simply in an arbitrary plane, e.g., in the (k2 , ω)-plane. In Fig. 8(c), the dashed ellipses are used to denote the Fourier amplitude of the wave packets.It is shown that the physically dominant frequency ω1 = 0.1227c0 /R is very different (1)

from the spurious angular frequency ωs

= 0.04c0 /R implicitly introduced by the temporal Fourier

transform. For the sake of brevity the wave packets are only marked in the first quadrant. We define the base flow by I.1. The labels I.3, I.4 and I.5, which go along the convection lines defined by ω = ±k2 Ur , represent the convective wave packets. The label I.7 represents the acoustic-related wave packet, given

0.6 ^ |ρ|/ρ 0

I.4 I.5 I.7 I.3

1

0.3

I.6 0.82

ωR/c0

325

0.64

0

I.2

0.46

-0.3

0.28

I.1 -0.6 -12

-8

-4

0.1

0

4

8

12

k2R Figure 9: Fourier amplitude of the filtered density field in the (k2 , ω)-plane based on the spectral distribution of the 1st diffusive stage.

13

330

the fact that the radiation lines pass through this wave packet. To isolate the physically important components the filtering operation is applied to the three stages of the vortices. To this end, a convolution filter is introduced and it is taken in the form [30, 29]: ¯ ˆ Q(k, ω) = w(k, ω)Q(k, ω),

335

(15)

ˆ where Q(k, ω) is the Fourier coefficient of a physical quantity of interest Q(y, t) and w(k, ω) is a filter window. As Fig. 8(c) exhibits, the wave packets are mainly distributed near convection lines due to the dominant co-rotating motion of the vortices. To capture those wave packets, the filter-line clusters satisfying slopes Ur = ±0.06c0 in the spectrum    0,   w(k, ω) = 0,     1,

340

domain are designed. A bandpass filter in the form 0

if | k1 |< (| ω | −lω )/Ur , 0

if | ω |< Ur (| k1 | −lk ),

(16)

otherwise,

is selected, where the parameters lω = 0.107c0 /R and lk = 2/R are specifically chosen to ensure that I.7 can be captured, and I.2 and I.6 which are less dominant compared to I.3-I.5 can be eliminated in the 0

filtering process. Additionally, k1 in eq. (16) satisfies q 0 | k1 |= k12 + k22 345

(17)

due to the axisymmetric condition of the spectra. The convolution filtering window w(k, ω) = 0 is required on top of (16) to separate out the dominant wave packets I.1, I.3, I.4, I.5 and I.7 one by one, and then study the contributions of each individual wave packet to the sound generation. The Fourier amplitude of the density after being filtered is shown in Fig. 9 to visualize the results after the filtering operation.

350

The density and velocity wave packets in the spectral domain can be filtered by utilizing the aforemenˆi (k, ω), which are further transformed tioned filter operation to derive the filtered quantities ρˆ(k, ω) and u into the physical space by conducting an inverse FFT. The polar coordinate system is chosen to interpret the characteristics of the wave packets. Fig. 10 shows wave packets with instantaneous density, in which the wave packets are classified based on their distribution configurations in physical space. For example,

355

I.1 represents the base flow, I.3-I.5 the azimuthal convective wave packets from lower to higher order, I.2 and I.6 the less dominant wave packets, and I.7 the acoustic-related wave packet. Particularly, the acoustic-related wave packet is a representation of a wave-packet structure in the near-acoustic field, or alternatively it is termed the irrotational hydrodynamic pressure field, the pseudosound region or the entrainment region, see [31]. The azimuthal convective wave packets I.3-I.5 rotate around the origin with

360

constant co-rotation velocity Ur . The angular velocity of the wave packets I.3, I.6 and I.7 is half of the angular frequency, i.e. ω1 /2. An identical filtering operation is applied to the velocity component u2 in the spectral domain. Then, the resulting spectra of the density and velocity are employed to reconstruct the acoustic source term ρ¯u ¯2 u2 of the fluctuation pressure and subsequently to predict the far-field sound. The temporal evolution

365

of the pressure fluctuations at (y1 , y2 ) = (0, λa ), generated by combinations of wave packets, is shown in Fig. 11, in which the sound predicted by relation (11) which incorporates all the wave packets is taken 14

4

4

ρ/ρ0 0.995

0.02

2

0.965

2

-2

y2/R

0

-2

-4

-2

0

2

-4

4

-4

-2

0

ρ/ρ0

0

-2

ρ/ρ0

2

y2/R

0

-2

-4

-2

0

2

-4

4

-4

-2

0

y1R

(d)

y1/R

(e)

(f) 4

ρ/ρ0

ρ/ρ0 0.996

0.0003

5

2 0.906

y2/R

y2R

-0.0003

0

-5

-10 -10

4

-0.006

y1/R 10

2

ρ/ρ0 0.006

0

-4

4

4

2

-2

0

2

(c) 4

-0.005

0

-2

-2

y1/R

2

-0.01

-4

-4

0.005

y2R

y2/R

-4

4

(b) 4

0.01

-4

2

y1/R

(a)

2

0

-2

y1/R 4

-0.02

-0.006

0

-4

ρ/ρ0

0.020

y2/R

y2/R

2

4

ρ/ρ0

0

-2

-5

0

5

-4

10

-4

-2

0

y1/R

y1/R

(g)

(h)

2

4

Figure 10: Density wave packets of the vortex pair at t = 192R/c0 . Panel (a)-(g) corresponds to I.1–I.7, respectively, and panel (h) describes the whole density field. The trace of the vortex centers is marked by the dashed circle of a radius R for reference in each panel.

as the actual sound and represented by the solid lines. The sound generated in the 1st diffusive stage is indicated by the arrow, where a delayed time, t = 52.4R/c0 , due to the sound’s propagation from the source to the observation point, is considered. By comparing the pressure fluctuations obtained by 370

combinations of different wave packets and the actual sound, the following conclusions can be drawn. 1. Neither the base flow I.1 nor the remaining wave packets (I.2+I.3+I.4+I.5+I.6+I.7) can generate sound independently, see Fig. 11(a). The sound can only be generated by the interaction between the base flow and the remaining wave packets. This finding holds both in the convective and the 2nd diffusive stages.

375

2. The sound generated by the interaction between the base flow I.1 and the azimuthal wave packets (I.3+I.4+I.5+I.7) is approximately equal to the actual sound, except that the amplitude of the

15

02

20 All wave packets I.2+I.3+I.4+I.5+I.6+I.7 I.1

× 104

10

p’/(ρ0c20)

p’/(ρ0c20)

10

0

First diffusive stage

-10

0

100

200

All wave packets I.1+(I.3+I.4+I.5+I.7) I.1+(I.2+I.6)

× 104

0

First diffusive stage

-10

300

400

0

100

200

tc0/R

300

400

tc0/R

(a)

(b)

20 All wave packets I.1+I.3+I.7 I.1+I.3 I.1+I.7

× 104

p’/(ρ0c20)

01

0

First diffusive stage

-10

0

100

200

300

400

tc0/R (c)

Figure 11: Sound generated by the wave packets in the 1st diffusive stage at (y1 , y2 ) = (0, λa ). Panel (a) describes the wave packets in silence, panel (b) the dominant and less dominant wave packets and panel (c) the azimuthal convective wave packets.

sound pressure generated by these wave packets is slightly higher than that of the actual sound during the initial one and a half periods of the sinusoidal acoustic wave (see Fig. 11(b)). The azimuthal convective wave packet I.3 and the acoustic-related wave packet I.7 play crucial roles in 380

the sound generation, see Fig. 11(c), whereas the high-order azimuthal wave packets I.4 and I.5 only slightly enhance the sound intensity. Additionally, the interaction between the less dominant wave packets (I.2 and I.6) and the base flow I.1 generates the low-amplitude and opposite-phase sound during the initial one and a half periods. As a result, the interaction between the less dominant wave packets and the base flow suppresses the sound in the initial phase.

385

3. It is found that the actual sound is generated by the linear superimposition of the interaction between each wave packet and the base flow. For instance, the sound generated by I.1+I.3+I.7 equals the summation of the sound generated by I.1+I.3 and I.1+I.7, see Fig. 11(c). Furthermore, the overall sound generated by I.1+I.2+I.3+I.4+I.5+I.6+I.7 equals the summation of the two kinds

16

sd

kU

^ |ρ|/ρ 0

^

|ρ|/ρ0

ω=

U sd

ω = ω2

II.5

-k

0.3

0.6

ω=

0.6

II.4

II.5 II.4

0.3

0.3

0.802

0

ωR/c0

ωR/c0

0.246 0.192

0

0.604

II.2 0.406

0.138

-0.3

II.2

ω = - ω2

-0.3

0.084

II.1 -0.6 -12

-8

1

II.3

II.3

-4

0.208

II.1

0.03

0

4

8

-0.6 -12

12

k2R

-8

-4

0.01

0

4

8

12

k2R

(a)

(b)

Figure 12: Fourier amplitude of the density field in the(k2 , ω) plane. Panel (a) describes density in the spectral space for the convective stage and panel (b) filtered density over the entire time domain.

of sound separately generated by I.1+I.3+I.4+I.5+I.7 and I.1+I.2+I.6, see Fig. 11(b). This finding 390

holds as well in the convective and the 2nd diffusive stages. 4. An apparent sound is generated by the interaction between I.1 and I.7, which indicates that the acoustic-related wave packet I.7 not only contains the entire radiating acoustic components but also contributes to the sound generation as a discernible acoustic source. According to the theory of Goldstein (2005) [5], the acoustic sources of the subsonic compact flow depend only on the

395

non-radiating components. This implies that I.7 contains non-radiating components. A detailed discussion on the acoustic-related wave packet I.7 is discussed in Sec. 3.3. (ii) Contributions of wave packets in the convective stage The filtering effect in the convective stage can be characterized by taking the density as an example. Fig. 12(a) shows the Fourier amplitude of the density field in the convective stage. The long dash-dotted

400

lines satisfying ω = ±k2 Usd (Usd = 0.0835c0 ) denote the convective motions and the dash-dot-dotted lines represent the dominant frequency ω2 = 0.317c0 /R in the 2nd diffusive stage. To ensure that the identification of wave packets is not affected by the implicit temporal rectangular window introduced by the Fourier transform, we compare spurious angular frequencies with the physical dominant frequencies of the wave packets. The length of the rectangular window for the sample time t = t2 − t1 leads to a (2)

405

spurious angular frequency ωs

= 0.05c0 /R, which is dramatically different from the physically dominant

frequencies (ω1 and ω2 ) as well as their harmonic frequencies. The spectral distribution in this stage is similar to that in the first stage, which means that the dominant convective wave packets II.3 and II.4 are along the convective lines; the acoustic-related wave packet II.5 is on the sonic lines and II.2 is the less dominant wave packet. Based on the spectral 410

distribution in Fig. 12(a), the filter window for the convective stage is similar to the filter (16), except that the convective velocity Ur in the first row of window (16) is replaced by Usd and the parameters lω = 0.2c0 /R and lk = 3/R in order to capture II.7 and eliminate II.2 and II.6 in the filtering process

17

20 All wave packets II.1+II.2+II.3+II.4+II.5 II.1+II.3+II.4+II.5 II.1+II.3+II.4 II.1+II.5

× 104

p’/(ρ0c20)

01

0

-10

Convective stage 300

400

500

600

tc0/R Figure 13: Sound generated by the wave packets in the convective stages at (y1 , y2 ) = (0, λa ).

in this stage. The Fourier amplitude of the density field of the vortex-pairing process after the filtering operation in the convective stage is performed, is shown in Fig. 12(b). The same filtering operation 415

is conducted for the velocity component u2 , and then the far-field sound can be predicted. Fig. 13 demonstrates the sound generated by the combination of wave packets in the convective stage. The far-field pressure fluctuations generated in the 1st and 2nd diffusive stages are apparently present due to overlap of the frequency spectrums of the source, even though the filtering operation has been performed. Nevertheless, it does not affect the analysis of the noise generation in the convective stage, which is

420

bounded by arrows. Some new mechanisms of sound generation are found in this stage due to the strong non-linearity of the merging dynamics of the co-rotating vortices. The interaction of II.1+II.3+II.4+II.5 can basically recover the actual sound, and the amplitude of the sound generated by these wave packets becomes elevated with time. The agreement between the sound generated by the filtered wave packets, and the

425

actual sound, consolidates the validity of the filtering operation of the flow spectra in the second stage. The interaction between the dominant azimuthal wave packets II.3+II.4 and the base flow II.1 constitutes the main radiating acoustic source; however, the amplitude of the sound generated by II.1+II.3+II.4 is constant with time during the convective stage. Conversely, the sound intensity generated by II.1+II.5 is apparently increasing, which implies that the acoustic-related wave packet II.5 plays a crucial role in

430

enhancing the sound intensity during this stage. (iii) Contributions of wave packets in the 2nd diffusive stage In the second diffusive stage, 128 time samples of the density are collected from t = 388.8R/c0 to t = 465R/c0 . As shown in Fig. 14(a), the spectral distribution in this stage is similar to that in the 1st stage. A dominant phase velocity Usd = 0.0835c0 appears in the convective dominant wave packet III.3.

435

The convective dominant wave packet III.3 and the acoustic-related wave packet III.4 have a dominant frequency ω2 = 0.317c0 /R, which is higher than those in the previous two stages. The dominant frequency (3)

is dramatically different from the spurious angular frequency ωs

= 0.08c0 /R implicitly introduced by the

temporal Fourier transform, which ensures that the identification of wave packets is sufficiently accurate. 18

U sd

ω = ω2

ω=

-k

0.3

0.05

III.2

ωR/c0

III.4

^ |ρ|/ρ 0

1

0.041

0.64

0

0.023

ω = - ω2

-8

-4

0.46

III.1 -0.3

0.014

0.28

0.005

III.1 -0.6 -12

0.82

III.2

0.032

0

-0.3

^ |ρ|/ρ 0

III.3

0.3

ωR/c0

III.4

kU

III.3

sd

0.6

ω=

0.6

0

4

8

0.1

-0.6 -12

12

-8

-4

k2R

0

4

8

12

k2R

(a)

(b)

Figure 14: Fourier amplitude of the density field in (k2 , ω) plane. Panel (a) describes density in the spectral space for the 2nd diffusive stage and panel (b) the filtered density for the entire time domain.

02 All wave packets III.1+III.2+III.3+III.4 III.1+III.3+III.4 III.1+III.3 III.1+III.4

× 104

p’/(ρ0c20)

01

0

Second diffusive stage

-10

400

500

600

tc0/R Figure 15: Sound generated by the wave packets in the 2nd diffusive stage at (y1 , y2 ) = (0, λa ).

The wave packets in this stage can be separated from those of the previous two stages using a filter 440

window based on the frequency thresholds. The filter window is given by,    0,    w(k, ω) = 0,     1,

if lω1 <| ω |< lω2 if | ω |> lω3 ,

(18)

otherwise,

where lω1 = 0.08c0 /R, lω2 = 0.24c0 /R and lω3 = 0.45c0 /R. These values are specifically selected to ensure that the frequency of wave packets III.3 and III.4 ∈ [lω2 , lω3 ] and the frequencies of III.1 and III.2 are below lω1 , and hence the wave packets in this stage can be completely extracted. The Fourier 445

amplitude of the density after the filtering operation is demonstrated in Fig. 14(b). The time histories of the far-field pressure fluctuations, generated by the filtered wave packets in the 2nd diffusive stage, are shown in Fig. 15. The interaction of III.1+III.2+III.3+III.4 can recover the actual 19

sound in this stage. Furthermore, there is very little change in sound intensity when III.2 is removed, which shows that III.2 makes a trivial contribution to the sound. Due to the linear superimposition 450

nature of wave packets in the generation of sound, the sound generated by III.1+III.3+III.4 equals the summation of the sound generated by III.1+III.3 and III.1+III.4. 3.3. Acoustic Wave Packet As explained above, the acoustic-related wave packet of the co-rotating vortices contains the acoustic components characterized in the flow spectra, and contributes to a discernible acoustic source. This

455

implies that the acoustic-related wave packet contains both acoustic components and non-radiating components. There is the prospect of building a connection between the near-field non-radiating wave packets and the far-field sound of the co-rotating vortices, which can be scrutinized by conducting quantitative analysis of the acoustic-related wave packet in the spectral and physical domains. A square domain is chosen to extract the acoustic-related wave packet from the pressure field in

460

the spatial domain. This square domain has the length and width of 60 non-dimensional units. Each coordinate direction is discretized into 256 grid points. The discretized values of the pressure from t = 102 to t = 255 (1st diffusive stage) with time interval ∆t = 0.6R/c0 (256 time samples) are stored and they are used to compute the Fourier coefficients pˆ(k, ω). The physically dominant frequency ω1 = 0.1227c0 /R (1)

in this stage is very different from the spurious angular frequency ωs 465

= 0.04c0 /R. Additionally, a

different cutoff wavenumber (kc = 2π/60R) from that employed in Section 3.2(i), introduced to accurately extract the acoustic-related wave packet, is also dramatically different from the spurious wavenumber ks = 0.04/R. This ensures that the identification and extraction of wave packets are not affected by the implicit temporal rectangular window.

470

A Gaussian filter window, which is defined by    2    exp − ln2 | k | − | ω | , σ2 c0 w(k, ω) =   0, if | ω |< ω , b

if

| ω |≥ ωb ,

(19)

is used to extract the acoustic-related wave packet, where the cutoff frequency ωb = 0.04c0 /R and the Gaussian half width σ = 0.6. The filter window (19) is designed based on the definite acoustic dispersion relation to ensure that ω on the lines | k |=| ω/c0 | is equal to 1 and elsewhere equal to zero. The acoustic-related wave packet is obtained by multiplying the coefficient pˆ(k, ω) by w(k, ω), while the non475

radiating component can be obtained by multiplying the coefficient pˆ(k, ω) by 1 − w(k, ω). Fig. 16(a) shows the temporal evolution of the far-field sound at (y1 , y2 ) = (0, λa ) that was obtained using different σ. It demonstrates that the filtering width, σ = 0.6, is capable of extracting the acoustic components completely. The choice of σ = 0.6 can also get rid of the redundant components of the near-field convective wave packets of the radial pressure fluctuations, see Fig. 16(b). The analysis of the evolution of the radial

480

pressure fluctuations demonstrates that the acoustic-related wave packet is prone to having a nonlinear growth and decay of the wave amplitude around r = 5.5R. Based on the theory of Goldstein (2005) [5], it may be reasonable to deduce that the non-radiating components of the acoustic-related wave packet dominate in the region of r < 5.5R, while radiating components prevail within 5.5R < r < 25R. It is

20

20

× 10

4

p’/(ρ0c20)

10

t , θ

All wave packets σ=0.1 σ=0.2 σ=0.4 σ=0.6 σ=0.8 σ=1.0

0

10

-3

10

-4

σ=0.1 σ=0.2 σ=0.4 σ=0.6 σ=0.8 σ=1.0

r-1/2

10

-5

r = 5.5R -1 0 100

200

300

100

400

101

tc0/R

r = 25.0R 102

r/R

(a)

(b)

Figure 16: Comparisons of the temporal evolutions of the pressure fluctuations at (y1 , y2 ) = (0, λa ) (panel (a)) and the temporal and azimuthal averages of radial pressure fluctuations in the physical space for different filter widths (panel (b)).

noted that a more sophisticated approach is required to quantitatively decompose the non-radiating and 485

radiating components and account for the physics of the acoustic-related wave packet in the near field. As r > 25R, the acoustic-related wave packet decays in the form of r−1/2 , which is consistent with the decay nature of the two-dimensional pure acoustic wave. The pressure field of the acoustic-related wave packet, non-radiating pressure and overall flow in the 1st diffusive stage are illustrated in the spectral space (left column) and in the physical space (right

490

column), respectively, in Fig. 17. The pressure spectra demonstrate that the acoustic-related wave packet and the remaining non-radiating wave packets are completely separated. The pressure contour, shown in Fig. 17(d), demonstrates that apparently no acoustic wave is present in the non-radiating pressure, while a round quadrupole-like configuration is captured in the near field of the acoustic-related wave packet, see Fig. 17(b). It is therefore natural to conclude that the wave packets of the vortex pair induce a large-

495

scale rotating quadrupole-like configuration in the near field close to r = 5.5R, which is the non-radiating components of the acoustic-related wave packet. The rotating motion of the quadrupole-like non-radiating components leads to the conclusion that the local fluids oscillate periodically in the form of compressed ˙ A portion of the wavenumber spectrum with rarefaction waves with an oscillation frequency, ω = 2θ. the scale λa (wave length of sound) in the radial direction acoustically matches the dispersive relation

500

(12) which then leads to sound leakage. In the region of 5.5R < r < 25R, the acoustic wave is radiated outward from the near-field components, see Fig. 17(b). In addition, the phase lag between the acoustic waves in the radial and azimuthal directions leads to a spiral trajectory of the acoustic-related wave packet. The acoustic-related wave packet can also be used to interpret the sound intensity. In Fig. 18, the 0

505

instantaneous azimuthal average of the radial pressure fluctuations < p >θ of the acoustic-related wave packets in the first three stages are compared. The amplitude of near-field components of the acousticrelated wave packet in the convective stage is higher than that in 1st diffusive stage, which yields that a strong near-field fluctuating component satisfying the dispersive relation | kc0 /ω |= 1 emits intense sound 21

0.6

ω = - k c0

ω = k c0

2 ^ |p|/(ρ 0c0)

50

0.8

0.3

25

0

1.6E-04

0.52

y2/R

ωR/c0

0.66

0.38

-0.3

6.7E-05

0 -2.2E-05

0.24

-25

-1.1E-04

0.1 -2.0E-04

-0.6

-6

-4

-2

0

2

4

6

-50 -50

k2R

-25

(a)

0.6

ω = - k c0

0

25

50

y1/R (b)

ω = k c0

50

^ |p|/(ρ c2) 0 0

p/(ρ0c20)

0.8

0.3

25 0.7143

0

y2/R

ωR/c0

0.66 0.52

0.7141

0

0.38

-0.3

0.7063

0.24

-25

0.6903

0.1 0.6744

-0.6

-6

-4

-2

0

2

4

6

-50 -50

k2R

-25

(c)

0.6

ω = - k c0

25

50

(d)

ω = k c0

2 ^ |p|/(ρ 0c0)

40

p/(ρ0c20)

20

0.7143

0.8

0.3

0

y2/R

0.66

ωR/c0

0

y1/R

0.52

0.7141

0

0.38

-0.3

0.7082

-20

0.24

0.6959 0.1

-40

-0.6

-6

-4

-2

0

2

4

6

0.6836

-40

k2R

-20

0

20

40

y1/R

(e)

(f)

Figure 17: Pressure in the (k2 , ω)-plane (left column) and the physical space (right column) at t = 192R/c0 . Acousticrelated wave packet in (a) (k2 , ω)-plane and (b) physical space, non-acoustic pressure in (c)(k2 , ω)-plane and (d) physical space, overall pressure in (e) (k2 , ω)-plane and physical space. The styles of the lines in the spectral space follow the convention in Fig. 4.

22

10

-3

t = 236.4R/c0

θ

t = 324.6R/c0 t = 424.5R/c0 10-4

10-5

r

r = 4.0R

-1/2

r = 5.5R

100

101

102

r/R 0

Figure 18: Azimuthal average of the radial pressure fluctuations (< p >θ ) in the first three stages.

into the far field. Consequently, the amplitude of near-field components in 2nd diffusive stage significantly 510

reduces, and the acoustic intensity drops markedly. In the merged diffusive stage, the co-rotating vortices eventually merge into a single vortical structure. The pressure field of the vortex evolves into an axisymmetric pattern and the acoustic-related wave packet vanishes. At this point the vortical flow does not emit acoustic waves anymore.

4. Conclusions 515

A spectral AA theory was used to study the sound generated by a compressible co-rotating vortex pair using results from direct simulation of the unsteady compressible Navier-Stokes equations to gain physical insights into the mechanisms of acoustic generation and radiation from the vortical flow induced by hydrodynamic wave packets. Through a comparison with previous results [16, 19], the spectral AA theory shows particular promise in dealing with such problems, which motivates to conduct deeper

520

analysis. The whole analysis is based on the technique of wave-packet decomposition with respect to the four phases of the dynamics of the co-rotating vortices, i.e. the 1st diffusive stage, the convective stage, the 2nd diffusive stage and the merged stage; the latter acoustically being of no interest. The wave packets are separated and extracted in the spectral space by using convolution filters to construct the source of the spectral AA and to explore the role of each wave packet in the generation of sound.

525

Using the spectral AA theory and convolution filtering operations, it is found that the far-field sound can only be generated by the interaction between the base flow and remaining wave packets of the corotating vortices. Precisely, the overall sound is generated by the linear superimposition of the sound generated by the interactions between each wave packet and the base flow. It is interesting to note that in contrast to the concise decomposition of the radiating and non-radiating components in the spectral

530

space, a more sophisticated operation is required to decompose them in the physical space. For instance, Zhong and Zhang [32, 33] used a convection operator to filter out the acoustically inefficient components

23

in the turbulent flows and then obtain acoustic components. The present work shares similarity with the principle of the flow decomposition proposed in [34] and [35] to some degree, which decomposed the flow field into incompressible hydrodynamics (mean flows) and compressible perturbations to calculate the 535

sound field of compressible unsteady flows. The present approach provides insights into the components of the acoustic source of the vortical flows, which might support and extend the predictive models (e.g.,[32, 33, 34, 35]) of the previous computational aeroacoustics. Additionally, it has been demonstrated that the acoustic-related wave packet can potentially bridge the gap between the near-field non-radiating wave packets and the far-field sound. However, deeper

540

analysis is required to interpret the process of energy shift from non-radiating components to acoustic components in the near field of the acoustic-related wave packet.

Acknowledgement This work was supported by the National Natural Science Foundation of China under Grant number 11302215 and the Foundation of the Equipment Development Department of China under Grant number 545

6140206040103. We would like to thank our colleagues Dr. Li Guo, Dr. Yan Liu and Prof. Wubing Yang for many helpful discussions on this work. The first author is especially grateful to Mr. Chuan Tian, who gave much of his time and effort to the initial simulation work.

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Statement of Author Contribution Feng Feng: Methodology, Formal analysis, Software Xiannan Meng: Conceptualization, Writing – Review & Editing Qiang Wang: Project administration, Writing –Review & Editing

Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: