Noise reduction by perforated cascades in annular ducts

Zihan Shen 1

School of Energy and Power Engineering, Beihang University Beijing 100191, China

Xiaoyu Wang 2

Research Institute of Aero-Engine, Beihang University Beijing 100191, China

Xiaofeng Sun School of Energy and Power Engineering, Beihang University Beijing 100191, China

ABSTRACT One of the latest trends in noise control relating to aeroacoustics is to mimic the silent flight capability of owls. Particularly, porosity is most often applied on cascades in ducts with axial flows, such as stator structures in an aero-engine. However, current acoustic scattering models of perforated cascades are based on two-dimensional methods without including the three-dimensional e ff ects. In this paper, we present a fully three-dimensional acoustic scattering model for perforated cascades based on the lifting surface theory in which the dominant sound source reduces to dipoles alone under the thin aerofoil assumption. Accordingly, the acoustic scattering of perforated cascades with single-mode incident wave was studied and obvious noise reduction was observed. The optimum Rayleigh conductivity and the maximum noise-reducing capability of the porosity varied substantially with di ff erent incident duct modes. With a background flow, the Kutta condition also plays an important role in the acoustic scattering and the unsteady vortex shedding at the trailing edge o ff ers extra sound energy dissipation mechanism. It is observed that the soft boundary on vanes will greatly influence such mechanism and could enhance the acoustic energy dissipated by shed vortical waves even when there is no direct dissipation at the perforations. Therefore, the implementation of porosity on cascades is much di ff erent to the design of a traditional acoustic liner.

1. INTRODUCTION

Aero-engine noise has always been a critical issue for civil aviation, and one novel application of noise reduction is to apply porosity on cascades to create sound-absorbing soft boundaries [1–4]. It was shown by theoretical models that considerable noise reduction can be achieved by porous

1 ps2206@buaa.edu.cn

2 bhwxy@buaa.edu.cn

a slaty. inter.noise 21-24 AUGUST SCOTTISH EVENT CAMPUS O ¥, ? GLASGOW

cascades for rotor-stator interaction noise [4, 5]. Nevertheless, the perforated cascade can also lead to noise reduction during a pure acoustic scattering process, resulting in lower transmission and reflection coe ffi cients compared to a hard-wall cascade, as indicated in [4]. Such structure may also be applied in other duct systems with a mean axial flow. However, previous models concerning the noise reduction of a perforated aerofoil [6, 7] or cascade [4] are all based on two-dimensional methods, lacking consideration of the important three-dimensional e ff ects inside a circular / annular duct. Consequently, in this research we extend the three-dimensional model lately established by the authors [5] to study the acoustic-wave scattering of a perforated cascade. Another problem which is unclear is the details of how the incident acoustic energy is reduced and dissipated by the perforations on vanes during its interaction with the cascade. When there exists a mean flow, there are two mechanisms by which acoustic energy may be dissipated on a perforated cascade. One is the sound dissipation due to the viscous e ff ects near the edges of the perforation apertures described by the Rayleigh conductivity [8]. The other mechanism is the vortex shedding from the trailing edge of the vanes [8,9]. This is controlled by viscosity inside the boundary layers of the cascade vanes, and such viscous e ff ects is usually implemented by the unsteady Kutta condition whenever the condition is valid. The first mechanism was studied for sound incident on a homogeneously perforated screen, without the influence of the Kutta condition or edge scatterings. It was found that the sound absorption for a perforated screen is related to the imaginary part and the magnitude of its Rayleigh conductivity, and no dissipation will occur when the Rayleigh conductivity is real [8]. Yet few discussions was made to the wake shedding at the trailing edge for a perforated plate, and there is no detailed information about how the two mechanisms will interact with each other. For the vortex shedding phenomenon, the scattering of the trailing edge of a single solid flat plate was studied and it was discovered that the unsteady process will also interact with the steady mean flow, such that energy could either be injected to or extracted from the mean flow [8,10]. In certain circumstances there can be a net acoustic energy gain and that the scattered sound energy will exceed that of the incident waves. For the scattering of solid stator cascades, though, only dissipation of the acoustic energy was observed [11–14]. However, there was no further detail about the energy carried by the scattered vortical waves. Consequently, there was little study on the energy relation during the scattering of a perforated cascade with a mean flow. More importantly, it is unclear how the above two mechanism would interact with each other, especially how the soft boundary on vane will a ff ect the vortex shedding and the scattered vortical waves. In this research we study the acoustic scattering of a perforated cascade and its noise reduction e ff ect, and then take a glance at the energy of the shed vortical waves induced by a purely acoustic incident disturbance. The layout of the remaining contents is as follows. In section 2 we briefly introduce our lately established model [5] extended from the lifting surface method. We then show how to estimate the energy carried by the convected vortical waves scattered from the cascade in section 3. Results are presented in section 4 about the overall noise reduction by perforations as well as the detailed energy flux of both scattered acoustic waves and vortical waves. Finally, in section 5 we summarise our work.

2. MATHEMATICAL FORMULATION

We consider an annular stator cascade of V vanes inside an infinite hard-walled duct with a uniform subsonic axial mean flow of inviscid perfect gas, as shown in figure 1. The cascade is of hub radius R h , tip radius R d and chord length b , with a background flow of axial velocity U and no swirling flow. It is noted that although we take the annular duct as an example to establish our model, a circular duct can be easily studied by replacing all radial characteristic functions with that of a circular duct. The stator vanes are assumed to be identical and evenly spaced zero-thickness perforated plates with zero stagger angle and no camber. Accordingly, the unsteady Kutta condition is applied in the form of zero pressure jump at the trailing edge and an integrable pressure singularity at the leading edge.

Figure 1: Schematic of an annular perforated cascade [5].

The disturbances are assumed to be small and isentropic, such that we may use a linearized theory to describe the whole cascade scattering process.

2.1. Scattering Fields In this paper we restrict the incident disturbance to be acoustic waves in the form of duct modes, such that the disturbance and the resulting unsteady pressure loading on vanes both have circumferential periodicity. It was already shown in [5] that there is negligible monopole source on a thin perforated vane. Further ignoring the unimportant quadrupole source [15], the only source on vane surfaces is the dipole sources. Therefore, for an incident sound wave of circumferential mode number σ and frequency ω s , the resulting scattering pressure field of the cascade may be derived as [5],

+ ∞ X

+ ∞ X

p ′ ( ⃗ x , t ) = V e − i ω s t

n = 1 φ m ( k mn r )e i m ϕ Z

m κ nm r ′ φ m ( k mn r ′ )e − i m ϕ ′ ∆ P s ( r ′ , z ′ , ϕ ′ )

4 π

S 1 ( τ )

q = −∞

× n H ( z − z ′ )e i α 1 ( z − z ′ ) + H ( z ′ − z )e i α 2 ( z − z ′ ) o d S ( ⃗ y ) , m = σ − qV . (1)

Here the generalized Lighthill equation and the Green’s function for an infinite hard-walled duct is used [15]. The residue theorem is applied for the solution of the infinite integration of axial wave number α with causality condition applied, and accordingly the scattering sound field is related to the unsteady loading ∆ P s e − i ω s t on one arbitrarily selected vane due to the circumferential periodicity of the disturbance and cascade response. H( z − z ′ ) represents the Heaviside function. With the linearized inviscid momentum equation, we may further obtain the circumferential perturbation velocity induced by the unsteady loading on vanes as

+ ∞ X

+ ∞ X

v ′ ϕ ( ⃗ x , t ) = − V e − i ω s t

r φ m ( k mn r )e i m ϕ Z

m

m r ′ φ m ( k mn r ′ )e − i m ϕ ′ ∆ P s ( r ′ , z ′ , ϕ ′ )

2 πρ 0 U

S 1 ( τ )

q = −∞

n = 1

× ( H ( z − z ′ ) " M β 2 e i α 1 ( z − z ′ )

2 κ nm ( M κ nm − k 0 ) + M 2 e i α 3 ( z − z ′ )

#

k 2 0 + M 2 k 2 mn

− H ( z ′ − z ) M β 2 e i α 2 ( z − z ′ )

) d S ( ⃗ y ) , m = σ − qV . (2)

2 κ nm ( M κ nm + k 0 )

Here and above

α 1 = − Mk 0 + κ nm

β 2 , α 2 = − Mk 0 − κ nm

β 2 , α 3 = ω s

U = k 0

M (3)

| ea

and



q

k 2 0 − β 2 k 2 mn , if k 2 0 > β 2 k 2 mn

κ nm =

β 2 k 2 mn − k 2 0 , if k 2 0 < β 2 k 2 mn . (4)

i q

The e i α 1 ( z − z ′ ) and e i α 2 ( z − z ′ ) terms correspond to the upstream and downstream pressure waves, and the e i α 3 ( z − z ′ ) term corresponds to the vortical waves convected downstream. Similarly, the induced velocity in the other two direction of the cylindrical coordinate illustrated in figure 1 can be derived, and the scattered vortical waves can be expressed as

⃗ w = ( w ′ r , w ′ ϕ , w ′ z ) , (5)

where

+ ∞ X

+ ∞ X

w ′ r ( ⃗ x , t ) = i V e − i ω s t

n = 1 k mn φ ′ m ( k mn r )e i m ϕ Z

m r ′ φ m ( k mn r ′ )e − i m ϕ ′ ∆ P s ( r ′ , z ′ , ϕ ′ )

2 πρ 0 U

S 1 ( τ )

q = −∞

× H ( z − z ′ ) M 2 e i α 3 ( z − z ′ )

k 2 0 + M 2 k 2 mn d S ( ⃗ y ) , m = σ − qV , (6)

+ ∞ X

+ ∞ X

w ′ ϕ ( ⃗ x , t ) = − V e − i ω s t

r φ m ( k mn r )e i m ϕ Z

m

m r ′ φ m ( k mn r ′ )e − i m ϕ ′ ∆ P s ( r ′ , z ′ , ϕ ′ )

2 πρ 0 U

S 1 ( τ )

q = −∞

n = 1

× H ( z − z ′ ) M 2 e i α 3 ( z − z ′ )

k 2 0 + M 2 k 2 mn d S ( ⃗ y ) , m = σ − qV , (7)

+ ∞ X

+ ∞ X

w ′ z ( ⃗ x , t ) = − V e − i ω s t

n = 1 φ m ( k mn r )e i m ϕ Z

m r ′ φ m ( k mn r ′ )e − i m ϕ ′ ∆ P s ( r ′ , z ′ , ϕ ′ )

2 πρ 0 U

S 1 ( τ )

q = −∞

× H ( z − z ′ ) k 0 M e i α 3 ( z − z ′ )

k 2 0 + M 2 k 2 mn d S ( ⃗ y ) , m = σ − qV . (8)

Here φ ′ m ( · ) denotes the direct derivative of the radial characteristic function φ ′ m ( · ) such that d φ m ( k mn r ) / d r = k mn φ ′ m ( k mn r ).

2.2. Establishment of the Integral Equation For a cascade of perforated vanes, there allows a normal seepage velocity across the vane. Therefore, we apply the commonly used model of Rayleigh conductivity to describe the boundary condition on vane surfaces, which is the same as in [4] and [5]. In this paper, cascade vanes are simplified as rigid straight plates in the axial direction with circular apertures, thus the normal induced velocity of the cascade at vane surfaces is v ′ ϕ , as shown in Equation 2. Moreover, the vane is aligned along radial direction, such that for equations above, inside the integration of source surface R

S 1 ( τ ) the circumferential coordinate ϕ ′ is the same and may be taken as zero in the calculation. The unsteady loading ∆ P s is also simplified as only a function of radial and axial coordinate, denoted as ∆ P s ( r ′ , z ′ ). Accordingly, the normal velocity boundary condition on vane surfaces reduces to

v ′ ϕ + v d = ˜ v R = α H

π R 2 − i K R

ω s ρ 0 ∆ P s e − i ω s t . (9)

Here v d is the disturbance velocity normal to the vane surface (the circumferential component of the incident disturbance velocity), ˜ v R is the mean flux velocity induced by the unsteady pressure

di ff erence across vane, K R is the Rayleigh conductivity of the apertures on vane, R is the radius of the aperture and α H is the local fractional open area. An integral equation is thus established based on this boundary condition on vane surfaces with a given incident wave velocity v d . The only unknown is the unsteady pressure loading distribution ∆ P s ( r ′ , z ′ ) on the cascade vane. For an incident cut-on acoustic wave of duct mode number ( m in , n in ) propagating upstream / downstream, the pressure fluctuation is expressed as

p in = A d φ m ( k mn r )e i( m ϕ + α 1 , 2 z − ω s t ) , (10)

where A d is the complex amplitude coe ffi cient of the incident mode. Integrating over the annular-duct cross-section, the acoustic power propagating upstream / downstream is [16]:

W d = π | A d | 2

(1 − M 2 ) 2 k 0 κ nm

( ± k 0 + M κ nm ) 2 . (11)

ρ 0 c 0

This equation can also be used to calculate the power of the reflected and transmitted acoustic waves. In addition, in our case the corresponding normal disturbance velocity on vane is

v d = 1 ρ 0 U m r ( α 1 , 2 − α 3 ) A d φ m ( k mn r )e i( m ϕ + α 1 , 2 z − ω s t ) (12)

2.3. Solution To solve the above integral equation, a finite radial mode expansion [17] is first applied. Then a collocation method is used to numerically solve the integral equation. Hereafter the time dependence term e − i ω s t is dropped since at each time a single frequency is studied. Following similar procedures as in [5], the integral equation is rewritten as

v ′ ϕ − ˜ v R = − v d . (13)

The unsteady pressure loading ∆ P s is expanded using the basic functions suggested in [17], expressed as

I X

J X

 , (14)

 A 1 j ′ cot ξ ′

!

i ′ = 2 A i ′ j ′ sin(( i ′ − 1) ξ ′ )

j ′ = 1 ψ ( ∞ ) j ′ ( r ′ )

∆ P s ( r ′ , z ′ ) =

2

with I axial terms and J radial terms. It is noted that the Glauert’s transformation is made to axial coordinates for both the source position z ′ and the observation location z as is usual in a thin aerofoil theory, which is

2 (1 − cos ξ ′ ) , z = b

z ′ = b

2 (1 − cos ξ ) , z , z ′ ∈ [0 , b ] , ξ, ξ ′ ∈ [0 , π ] . (15)

This ensures a proper leading edge and trailing edge behaviour corresponding to the unsteady Kutta condition [18]. Using evenly spaced collocation points similar to that of [19], Equation 13 can be written in a discrete matrix form [5] and the unknown coe ffi cients A i ′ j ′ can be solved numerically. The scattered pressure waves and vortical waves by the cascade can correspondingly be obtained by Equation 1, 6, 7 and 8.

3. ESTIMATION OF VORTICAL WAVE ENERGY

Inspired by the analysis of Myers [20], for the pure hydrodynamic vortical waves ⃗ w , its energy convected downstream could be estimated using the classical definition of flow energy flux, which is the same as in a conserved form of the Euler equation. This requires a basic assumption that for an inviscid flow inside a hard-walled duct, the propagation of acoustic waves and vortical waves is

Figure 2: Illustration of the energy fluxes upstream and downstream of the perforated cascade.

fully decoupled, such that the energy carried by the vortical waves can be estimated by the vortical disturbances along. This should be valid since in our linearized model there exist no interaction between acoustic and vortical waves during the propagation in a hard-walled duct without acoustic components. Therefore, downstream of the cascade, the variation of the flow energy flux after the axial background flow is superimposed with the shed vortical waves is

∆ ⃗ W = ⃗ m ˆ H − ⃗ m 0 ˆ H 0 , (16)

where ⃗ m = ρ⃗ u is the mass flux vector and ˆ H = h + ( u 2 / 2) = e + ( p /ρ ) + ( u 2 / 2) is the specific stagnation enthalpy. Subscript 0 represent the quantity of the background mean flow. The flow velocity vector containing vortical waves is ⃗ u = ⃗ u 0 + ⃗ w = U ⃗ e z + ⃗ w , (17)

and the flow density is unchanged since vortical waves only involve velocity disturbances. Consequently, if we use ϵ to denote the small order of the vortical wave amplitude | ⃗ w | / U , in our linearized model ⃗ m only contains first order terms of O ( ϵ ) and ˆ H contains disturbances of the first and second order. Thus, the energy flux ∆ ⃗ W has small terms up to the third order O ( ϵ 3 ). However, to estimate the vortical energy power convected downstream, the axial energy flux ( ∆ ⃗ W ) z must be integrated and averaged over one oscillation period. Due to the time periodicity of ⃗ w , the time integral terms that has non-zero value are only those of the second order small terms. The averaged axial energy flux is thus

2 ρ 0 Uw 2 ⟩ = ⟨ 1

2 ρ 0 Uw ′ 2 r + 1

2 ρ 0 Uw ′ 2 ϕ + 3

⟨ ∆ ( ⃗ W ) z ⟩ = ⟨ ρ 0 Uw ′ 2 z + 1

2 ρ 0 Uw ′ 2 z ⟩ , (18)

where ⟨·⟩ denotes the time averaged value. After we drop the time dependence e − i ω s t of the disturbances, the time averaged value over one period can be simply calculated by the complex variable notation such that ⟨ ab ⟩ = ab ∗ , where a and b are arbitrary time-varying variable of the same frequency and □ ∗ denotes the complex conjugate of the variable. Accordingly, the power convected by the shed vortical waves can be calculated by integration over an annular cross-section at cascade downstream:

Z r = R d

W v = Z 2 π

r = R h ⟨ ∆ ( ⃗ W ) z ⟩ r d r d ϕ. (19)

ϕ = 0

The spatial distribution of vortical disturbance velocity is given in Equation 6, 7 and 8. The vortical waves shed by a cascade do not decay along the axial direction, such that we may take any position downstream of the cascade to perform the integration. It is noted that in Equation 7 the circumferential vortical-wave velocity has a singularity when ϕ approaches the circumferential coordinates where cascade vanes are positioned, i.e. approaches the

O O O O O O O

wake position of the vanes. This singularity can be processed the same as in [5] or [17], by separating the expression in Equation 7 to a regular part and a singular part, where the singular part equals zero outside the wake surfaces. The resultant regular part can then be calculated and integrated. The other two velocity component of vortical waves, shown in Equation 6 and 8 do not have such singularity problem and their summations of infinite series q and n also uniformly converge. Therefore, we may well calculate the integration in Equation 19 numerically and estimate the kinetic energy carried by the scattered vortical waves, with some simplifications using the properties introduced by the finite radial mode expansion [17]. Details are not presented here for simplicity. The acoustic energy for reflected ( p r ) and transmitted ( p t ) sound waves can also be calculated as in Equation 11. The energy relation within the scattering of a perforated cascade can be illustrated as in Figure 2 and a control volume enclosing the cascade is taken to estimate the axial power flux. It should be noted that the scattered acoustic modes which are cut-o ff have no axial energy flux and do not dissipate sound either, thus play no role in our stable-state power analysis. Accordingly, the power that goes into the control volume carried by incident sound wave should ultimately go out, either as cut-on scattered sound waves (reflected and transmitted) or as shed vortical waves, unless there is energy transmission between the unsteady flow and the background mean flow.

4. RESULTS

We choose the geometry and mean-flow setup similar to our previous study [5] and the input parameters are listed in Table 1. The perforation parameters chosen are moderate and are acceptable in engineering practice. For di ff erent circumferential incident mode m with same radial mode number n = 1, we first calculated the total scattered energy, i.e. the sum of the acoustic power reflected by and transmitted through the perforated cascade, varying with di ff erent perforation Rayleigh conductivity. Only downstream propagating incident waves are illustrated, and an upstream incident wave with the same mode number will result in a similar contour shape. In addition, a circumferential mode with opposite sign will lead to an exactly same result, since the cascade is circumferentially symmetric with no stagger or leaned and the background flow also contains no swirl. The Rayleigh conductivity K R is non-dimensionalized by dividing 2 R , and it is convenient to use its non-dimensionalized form

K R = 2 R ( Γ R ( ω ) − i ∆ R ( ω )) , (20)

where Γ R ( ω ) and ∆ R ( ω ) are real valued functions of frequency. The negative sign before the imaginary part ensures that a positive ∆ R corresponds to a sound dissipation. Results are shown in figure 3, and the energy fluxes are non-dimensionalized by the power of the incident acoustic wave. To better illustrate the gradient, non-dimensional total scattered energy greater than 1.1 are compressed in the legend and are represented in one colour, i.e. in yellow. The reflected acoustic power are pretty small for incident mode m = 1 and m = 4, such that the contour map of the transmitted energy variation is very similar to that of the total energy. For m = 8 though, the reflected energy is noticeable under a small range of Rayleigh conductivity, and is therefore illustrated in Figure 3(d). For a small circumferential mode number m = 1, the wave front is nearly parallel to cascade vanes, and little dissipation is observed in the scattering, even for a perforated cascade. Of course, for an m = 0 mode no acoustic reflection or dissipation can be captured by the present linearized model based on the singularity method with axially-placed zero-thickness vanes. For incident circumferential modes with higher mode number, the maximum noise dissipation achieved by the optimum Rayleigh conductivity increases, although in the studied case the real part of the optimum Rayleigh conductivity often lies in the negative area, which is not very achievable for an ordinary perforated plate with tangential background flow. Nevertheless, noise reduction can still be achieved by a Rayleigh conductivity with zero real part and positive ∆ R , which is more applicable for the potential usage of perforated cascades in ducts. For the m = 8 case, the hard-walled cascade case corresponds to a total energy of 0.4721, which alrealy achieve a noise reduction over 3dB through

Table 1: Physical parameters used in the calculation of our dimensional model.

Parameters Symbols Values

Duct outer radius (tip) R d 1.0 m

Duct inner radius (hub) R h 0.5 m

Cascade vane number V 24

Vane chord length b 0.2618 m

Axial Mach number of background flow M 0.5

Sound speed of background flow c 0 340.0 m / s

Density of background flow ρ 0 1.225 kg / m 3

Angular frequency of incident sound wave ω s 4259.5 rad / s

Fractional open area of perforation α H 0.02

Aperture radius of perforation R 0.001 m

scattering, and at a better Rayleigh conductivity of K R = 2 R (0 − 0 . 2i), an additional 3dB of noise reduction is obtained by the perforations. Moreover, the acoustic impedance on vane surfaces corresponding to the optimum Rayleigh conductivity might be achieved by a more delicate structure with back cavities, such as the soft vane structure proposed by NASA [1, 3]. It is also observed that for incident sound waves with lower circumferential mode number m , less noise reduction can be achieved, though this can be partly compensated by increasing the chord length of the cascade. On the other hand, when the imaginary part of the Rayleigh conductivity is positive, i.e. ∆ R < 0, energy will be extracted from the mean flow by the vortex-sound interactions near the apertures such that perforations will behave as a net source of sound [8]. Accordingly, the total scattered energy is increased with a negative ∆ R and will exceed 1, i.e. the sound energy is increased after scattered by the perforated cascade. However, the total energy does not immediately surpass 1 since there exists another sound dissipation mechanism, the vortex shedding at the trailing edge, for a perforated cascade. To better illustrate how the soft boundary on vanes will influence the trailing-edge vortex shedding, we calculated the energy flux of the induced vortical waves at cascade downstream with an acoustic incident wave of mode number ( m , n ) = (8 , 1), and results are shown in Figure 4 and 5. Sound and vortical power are all non-dimensionalized by dividing the acoustic power of the incident wave. As we may expect, in Figure 4 where Γ R = 0, the unsteady loadings on vanes will reduce as ∆ R increases [5] such that the strength of the vortical waves shed by the dipoles will also decrease monotonically. In addition, because the vanes are radially placed with zero leaned angle, the vortical power contributed by the radial oscillation velocity w ′ r is little compared to the other two velocity components. However, sound dissipation has an optimum conductivity around ∆ R = 0 . 17, where the energy dissipated at perforations is large and the power carried by the shed wakes is not too small. The energy reduction due to the first mechanism we mentioned in the introduction can be roughly estimated using the di ff erence between the total scattered energy (carried by acoustic waves calculated using Equation 11 and by vortical waves estimated using Equation 19) and the incident acoustic energy. This is not precise because there can also exist energy transmission between the unsteady flow we studied and the background mean flow [8], although in the specific case we studied, with a solid-vane ( ∆ R = 0) this transmission with steady-flow energy is small (the total scattered energy illustrated by black solid line at ∆ R = 0 is very close to 1) and such e ff ect might be of secondary importance. The e ff ect of altering the boundary condition on vanes on the vortical wave shedding is better

( a ) ( b )

m=4 Total Energy

m=1 Total Energy

1.5

1.5

1

1

1

1.05

0.9

0.5

0.5

0.8

1

0

0

0.7

0.95

-0.5

-0.5

0.6

0.9

0.5

-1

-1

0.4

-0.2 0 0.2 0.4 0.6 0.8 1 -1.5

-0.2 0 0.2 0.4 0.6 0.8 1 -1.5

( c ) ( d )

m=8 Reflected Energy

m=8 Total Energy

0.5

1.5

1

0.18

1

0.16

0.8

0.14

0.5

0.12

0.6

0.1

0

0

0.08

0.4

-0.5

0.06

0.04

-1

0.2

0.02

0 0.05 0.1 0.15 -0.5

-0.2 0 0.2 0.4 0.6 0.8 1 -1.5

Figure 3: Total acoustic energy power (reflected and transmitted) in the scattering of a perforated cascade with di ff erent Rayleigh conductivity. Incident wave is propagating downstream with wave modes ( a ) ( m , n ) = (1 , 1), ( b ) ( m , n ) = (4 , 1) and ( c ) ( m , n ) = (8 , 1). The reflected power for the ( m , n ) = (8 , 1) incident wave is shown in ( d ). The acoustic power is non-dimensionalized by that of the incident wave, and the Rayleigh conductivity is non-dimensionalized by dividing 2 R .

2.5

Non-dimensionalized energy

2

1.5

1

0.5

0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Figure 4: Power of the acoustic and vortical waves scattered by an annular perforated cascade with zero Γ R and varying ∆ R . Power is non-dimensionalized by dividing the acoustic power of the incident wave, and the vortical energy contributed by three di ff erent velocity components of ⃗ w is illustrated separately in dashed lines.

Non-dimensionalized energy

1.5

1

0.5

0

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Figure 5: Power of the acoustic and vortical waves scattered by an annular perforated cascade with zero ∆ R and varying Γ R . Power is non-dimensionalized by dividing the acoustic power of the incident wave, and the vortical energy contributed by three di ff erent velocity components of ⃗ w is illustrated separately in dashed lines.

shown in Figure 5, where ∆ R is set to be zero such that there should be no energy dissipation directly through the apertures and that the trailing-edge vortex shedding should be responsible for any noise reduction. Both with positive and negative Γ R there exist ranges where the power of both the acoustic and vortical shed waves oscillates with varying Rayleigh conductivity. At negative Γ R , acoustic power can be further reduced compared to the solid-vane case, and correspondingly the energy flux of the vortical waves convected downstream increases. Additionally, the magnitude of the noise reduction achieved at the trough is comparable to that in Figure 4. Therefore, a soft boundary can greatly change the behavior of the vortex shedding at the trailing edge and may cause much intense energy conversion from the incident sound wave into the kinetic energy of the shed vortical waves, even when the perforations itself o ff ers no dissipation at all. It can also be observed that the total scattered enegy exceeds 1 in a large range. Those extra energy must be extracted from the mean flow, although the details of this transmission is beyond the scope of the present model. It should be noted though, that the e ff ects above are only noticeable when a considerable background mean flow exists.

5. CONCLUSIONS

In this paper we studied the acoustic scattering of a perforated cascade whose soft boundary is modelled by the Rayleigh conductivity. Our method is fully three-dimensional, and results concerning the noise reduction caused by perforations are presented, with special attention paid to the energy carried by the scattered vortical waves convected downstream. It was shown that the energy dissipation at the trailing edges through vortex shedding is also an important noise reduction mechanism, and that a soft boundary on vanes can considerably alter the magnitude of the energy dissipated in such way. Even when perforations o ff er no dissipation at the apertures, which corresponds to a Rayleigh conductivity of real values, they can still reduce noise through enhancing the vortex shedding at the trailing edge. Accordingly, those two dissipation mechanisms, caused directly by the viscosity at perforations and by the trailing-edge vortex shedding, are deeply coupled. This must be considered when designing a cascade with a soft boundary to reduce noise through acoustic scattering.

ACKNOWLEDGEMENTS

The authors acknowledge support from National Natural Science Foundation of China (grant number 51790514) and National Science and Technology Major Project (2017-II-003-0015).

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