Prediction of jet mixing noise in flight from static tests

Prediction of jet mixing noise in flight from static tests Ulf Michel ∗ CFD Software Entwicklungs- und Forschungsgesellschaft mbH, Berlin, Germany The flight effect on jet mixing noise can be predicted with a precision of about one decibel with scaling relations that are derived solely based on the Lighthill equation using relatively simple equations and without any arbitrary constant. The prediction formulas are based on measurements of jet mixing noise of static jets with the same nozzles and identical flow temperatures. The flight-effect theory of Michalke and Michel for jet noise is revisited. The flight stream yields a Doppler amplification into the forward arc while the interference within the non-compact and wavelike jet-noise sources results in a rear arc amplification best known for static jets. An important influence on the radiation of jet noise is the stretching of the flow field of the jet under the influence of the flight stream, which increases the source volume, the coherence length scales of the sources in the mean-flow direction and the frequencies in flight. The stretching of the jet is the primary cause for the surprisingly large noise levels in cruise. The influence of the interference in the azimuthal direction of the source region must also be considered. A comparison is made with the experimental A´erotrain “flyover” data and the results show an agreement within ±1 dB of the OASPL for a large range of emission angles for which shock-free static data are available.

Nomenclature A a0 D Df Dj f Fqq Gqq Jm k krc Lxc ,Lxs Lr m Me Mf Mj Ms Mp n OASPL p, p0 p0 Q q, qq , qd re rs ∗ Senior

stretching parameter rp ambient sound speed SPL jet nozzle diameter St Doppler factor t nozzle diameter tr frequency Uf axial interference integral for sources Ui interference integral for sources Uj Bessel function first kind of order m ui , vi circular wave number u?p Helmholtz number for azimuthal interference Up axial coherence/source length scale of sources Wpp radial length scale of sources Wpp,m relative velocity exponent, Wqq azimuthal component number xc , xs effective jet Mach number ∆Uj /Df γs flight Mach number Uf /a0 ∆Uj acoustic jet Mach number Uj /a0 ηi acoustic jet Mach number of static jet ηx phase (convection) Mach number Up /a0 θ, θp , θe Mach number exponent, static jet σ overall sound-pressure level ρ pressure and its fluctuation ϕc , ϕs , ϕp ambient pressure ψr source quantity for acoustic far field ψs source quantity ξc , ξs wave-normal distance radius of ring source

polar observer position sound-pressure level Strouhal number f D/∆Uj time retarded time flight speed velocity vector of flight stream jet speed fluctuating velocities normalized phase speed of disturbances phase speed of disturbances power-spectral density of pressure contribution of m-th azimuthal component power-, cross-spectral density of sources axial source position coherence between source positions Uj − Uf separation vector between source positions xc − xs polar observer angle jet stretching factor density azimuthal source/observer location phase difference due to retarded time phase of source cross-spectral density normalized axial source separation

Aeroacoustics Consultant, [email protected], Member AIAA and DGLR

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I.

Introduction

H e prediction of the influence of a flight stream on the emission of jet noise is of great interest for the estimation of community noise of turbofan powered aircraft during take-off and approach. Equally important is the prediction of the noise in cruise since jet noise including its broadband shock noise component is a major contributor to the noise in the rear cabin of aircraft with wing-mounted engines. The flight effect on jet mixing noise has the following surprising properties: • The noise is much louder than to be expected from the reduced relative velocity. • The frequency is not reduced in flight as to be expected from the reduced relative velocity. • The frequency for observers on the ground is Doppler shifted into the forward arc, which is not explainable with the moving source model of the traditional jet noise theories. • The noise level is almost unchanged in the forward arc for take-off conditions and may even be amplified there for high-speed jets. The causes of all these observations are explained in this paper. The sound emission of a jet in a uniform flight stream is treated theoretically with the convective Lighthill equation.1 The wave model of Michalke is used to describe the jet noise sources.2, 3 The jet flow field is assumed to be stretched in flight by a factor σ, which increases the source volume, the axial coherence length scales, and the normalized frequencies. The consequence of stretching is that the sound intensity of the jet is increased proportionally to the third power of σ. Since σ ≈ 4 for the fan stream of future ultra-high bypass ratio engines in cruise the stretching alone accounts for an increase of the sound-pressure level in cruise by ∆SPL ≈ 18 dB. The industry standard for modeling the effect of flight speed on the mean square sound pressure p˜2 of jet noise is to use the relation4, 5, 6 m(θe )  Uj − Uf (1) p˜2 (Uj , Uf , re , θe ) = p˜2 (Uj , 0, re , θe ) Uj

T

where m(θe ) is a relative velocity exponent, which depends on the emission angle θe . re is the wave-normal distance between nozzle and observer, which is equal to the distance at emission time in the flyover case. The noise reduction ∆OASPL in dB caused by the flight speed is then given by ∆OASPL(θe ) = m(θe ) 10 log10

Uj Mj = m(θe ) 10 log10 , Uj − Uf Mj − Mf

(2)

Uj and Uf are the jet and flight speeds, respectively. Mj = Uj /a0 is the acoustic jet Mach number with respect to the sound speed a0 in the ambience. The relative velocity exponents are generally determined in open-jet wind tunnels and it turns out that the values for θe = 90◦ vary from tunnel to tunnel in the range m = 3 . . . 5.5.7 Michel8 suggested the decorrelation of the acoustic waves in the jet shear layer as explanation for these differences. Equation (2) yields much too low sound-pressure levels for cruise speeds, which led to the wrong conclusion that broadband shock noise is the only significant contribution in cruise. Concerning the frequency of jet noise in flight it was originally assumed, based on the moving eddy model9 that the frequencies for θe = 90◦ scale with the Strouhal number f D/(Uj − Uf ), which would have resulted in a reduction of the frequencies in flight. However, experimental data indicate that the frequency remains almost unchanged in flight.10, 7 Michalke and Hermann11 conclude from their instability analysis of a jet in a flight stream that the Strouhal number is increased by the stretching factor, which explains the experimental findings and confirms the importance of instability waves in the generation of jet noise. The scaling relations developed in this paper based on the work of Michalke & Michel1 are validated against the experimental data that were acquired with the A´erotrain, which was a tracked hovercraft propelled by a GE J85 turbojet engine. In order to greatly reduce internal engine noise, the intake and exhaust of the engine were carefully lined and it was demonstrated by Hoch & Berthelot12 that compressor noise and core noise were no issues except for high frequencies at the smallest jet Mach number tested. The noise data were reported by Drevet, Duponchel & Jacques.13 The relative velocity exponents from these tests for θe = 90◦ were found to be between m = −0.5 and m = 2, decreasing with increasing acoustic jet Mach number, and thus were much lower than those of the tunnel tests. In particular, jet noise was found to be amplified in the flight direction (forward arc) for angles θe ≤ 70◦ relative to the flight direction. The theory of jet noise subjected to an external flight stream is presented in section II following the procedures developed in recent papers on the influence of source interference on jet mixing noise.14, 15 The A´erotrain tests are dis-

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cussed in section III and the flight effect scaling relations are presented in section IV and compared with experimental data of the A´erotrain. The cruise condition is finally discussed in section V.

II.

The acoustic analogy in a uniform flight stream

The sound emission of a jet in a flight stream shall be investigated in terms of time-averaged quantities, such as mean square sound pressure, power-spectral density, overall sound-pressure level, and one third octave levels. This is best achieved with the convective wave equation using a nozzle-fixed coordinate system, which has the following advantages over a coordinate system with a moving nozzle in still air: • The integration boundaries are stationary. • The source field is stationary random. • The emitted acoustic field is stationary random. The solution of the convective Lighthill equation in the acoustic and geometric far field of a jet and the time averaged result in form of the power-spectral density satisfying above conditions was presented and discussed by Michel14 based on earlier work1, 16, 17, 18 but is repeated here for convenience. The solution considers the influence of source interference, which explains many properties of jet noise including broadband shock noise.19 II.A.

The convective Lighthill equation

The convective form of the Lighthill equation for the pressure p is given by1 1 a20



∂ ∂ + Ui ∂t ∂xi

2 p−

∂2p =q. ∂x2i

(3)

xi describes the position relative to a fixed point in the source region, for example the center of the jet nozzle. Ui is the uniform mean velocity vector in the ambient flow in this coordinate system. The source term q in equation (3) is defined by ∂2 (ρui uj − τij ) − q= ∂xi ∂xj



∂ ∂ + Ui ∂t ∂xi

2   p ρ− 2 , a0

(4)

where ui is the fluctuating velocity relative to the mean velocity Ui in the ambience. u i = vi − U i

(5)

It is now assumed that the entropy of a fluid element remains constant during its motion. This means that the influences of friction and heat conduction on the sound generation are neglected. Equation (4) can then be approximated according to Morfey20 (see also1 ) by q=

∂ 2 qij ∂xi ∂xj

∂qi ∂xi

+

.

(6)

Dipole sources

Quadrupole sources

qij and qi are abbreviations for     p0 ρo qij = ρo ui uj 1 + − 1 − p0 δij , ρo a20 ρ and qi = p0

∂ ∂xi



ρo ρ

(7)

 .

(8)

p0 = p − p0 is the difference between the local pressure p and the pressure p0 in the ambience. In addition, terms of
2 the order p02 / ρ0 a20 were neglected. Since the first term on the right hand side of equation (6) is the result of a second spatial derivative, the term describes a quadrupole source distribution. The second term describes a dipole distribution because of its first spatial

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derivative. The dipole term can only be significant if the local density ρ differs from the density ρ0 in the ambience, which is the case for heated jets. II.B.

Solution of the convective Lighthill equation

If the source region is surrounded by an infinite space the integral solution of (3) is given by the corresponding Green function, which is the solution for the point source q = f (t)δ(xi − yi ) .

(9)

The Green function for the convective wave equation and radiation into free space can be described in a very compact form in terms of emission coordinates (re , θe ).1 p(xi , t) =

1 f (t − re /a0 ) 4πre Df

(10)

with the Doppler factor Df = 1 − Mf cos θe ,

(11)

where the angle θe is defined relative to the flight direction.

Figure 1. Relation between the source position and the observer position in a uniform flow (wind tunnel coordinate system).

The emission distance re and the emission angle θe can be computed from the geometric distance r and the observer angle θ as follows (compare figure 1). re = q and cos θe = cos θ

r

(12)

1 − Mf2 sin2 θ − Mf cos θ

q

 1 − M2f sin2 θ − Mf cos θ + Mf .

(13)

The integral solution for the pressure fluctuation p0 (xi , t) = p(xi , t) − p0 in an arbitrary field point xi is given by Z Z 1 ∂2 qij (xi , yi , tr ) 1 ∂ qi (xi , yi , tr ) p0 (xi , t) = dV (yi ) + dV (yi ) (14) 4π ∂xi ∂xj re Df 4π ∂xi re Df V Quadrupole sources

V Dipole sources

The pressure p0 (xi , t) is inversely proportional to the emission distance re and Doppler amplified in the flight direction according to 1/Df (see equation (11)). The retarded time tr is calculated with the wave-normal (emission) distance re tr = t − re /a0 . (15)

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The solution (14) is valid everywhere, even inside the source region. The two integrals are finite, even for re → 0. It can be concluded that the pressure fluctuation p0 in a position xi inside the source region is closely related to the source terms qij and qi in the vicinity of xi . II.C.

Integral solution in the acoustic far field

Equation (14) simplifies in the acoustic far field of the source. The spatial derivatives in equation (14) can then be substituted by time derivatives as shown by Proudman21 for the static jet. The results for the convective wave equation are 1 ∂ 2 qq ∂ 2 qij = 2 2 ∂xi ∂xj a0 Df ∂t2

(16)

∂qi 1 ∂qd = ∂xi a0 Df ∂t

(17)

The new source quantities qq and qd are defined by.1 p0 ρ0 a20



∂ qd (yi , θe , t) = p ∂yre



qq (yi , θe , t) = ρ0 u2re

 1+

0

  ρ0 − 1− p0 ρ ρ0 ρ

(18)

 (19)

For small values |p0 | inside the source region and for ρ ≈ ρ0 one can describe the quadrupole source quantity qq approximately by qq ≈ ρ0 u2re , (20) which means that the quadrupole source strength ∂ 2 qq /∂t2 is dominated by the second time derivative of the square of the velocity fluctuations in the source element dV (yi ) in the direction θe toward the observer position xi . The dipole source strength ∂qd /∂t is determined by the time derivative of the product between the local pressure fluctuations p0 and the spatial derivative of the inverse of the density gradient in the source element dV (yi ) toward the observer position xi under the angle θe . The solution of the convective Lighthill equation (3) for the sound pressure p0 = p − p0 in an observer position xi in the acoustic far field and in free space is then given by Z Z 1 Qd (yi , θe , tr ) 1 Qq (yi , θe , tr ) 0 p (xi , t) = dV (yi ) + dV (yi ). (21) 3 4πa20 4πa re Df re Df 2 0 V

V

The sound pressure is composed of two integrals over the whole volume V that is occupied by the turbulent flow. The quadrupole source term Qq and the dipole source term Qd for the acoustic far field are abbreviations for Qq (yi , θe , t) =

∂ 2 qq (yi , θe , t) , ∂t2

(22)

∂qd (yi , θe , t) . (23) ∂t It is important to note that Mf has a very strong influence on the sound radiation into the forward arc via the Doppler factor Df as defined in equation (11). The resulting forward arc amplification in equation (21) is higher for the quadrupole sources Qq than for the dipole sources Qd . This effect is especially important for cruise Mach numbers. Qd (yi , θe , t) =

II.D.

Power-spectral density of jet noise in a flight stream

One is generally not interested in the time-dependent solution given by equation (21) but in time averages, like meansquare pressures, one-third octave spectra, or power-spectral densities. Source interference effects of non-compact sources can only be studied in the frequency domain, hence with the power-spectral density Wpp (f, xi ) of the pressure

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fluctuations in the observer position xi in the acoustic far field as function of the frequency f . Wpp (f, xi ) = Wppqq (f, xi ) + Wppqd (f, xi ) + Wppdq (f, xi ) + Wppdd (f, xi ) | | | {z } {z } {z } {z } | Quadrupole sources

Quadrupole and dipole

Dipole and quadrupole

(24)

Dipole sources

This equation was already defined in reference22 but must now be evaluated with consideration of a flight stream. Wppqq (f, xi ) and Wppdd (f, xi ) are the power-spectral densities in the far field due to the quadrupole and dipole sources, respectively, Wppqd (f, xi ) and Wppdq (f, xi ) are the contributions due to the cross-spectral densities between the quadrupole and dipole sources. The imaginary parts of the cross spectra have opposite signs and cancel out. The cross-spectral densities between quadrupole and dipole contributions cannot be neglected since both source terms are partly coherent since they are both dominated by the influence of the same instability waves in the jet’s shear layer. The correlation between the two sources was already observed by Tanna, Dean & Fisher23 by analyzing experimental data. The resulting Wpp is a real function, which can be written as Wpp (f, xi ) = Wppqq (f, xi ) + 2 0. It may be noted that the integration volumes Vs and Vc of equations (26) and (29) are stationary in the nozzle-fixed coordinate system as well as all terms in the integrands are stationary random. This removes any time dependence from the integral. In the geometric far field we can neglect the influence of the detailed source position on the emission distance re and Doppler factor Df and move the terms in front of the integral. Z Z p p 1 1 W Wqqcc γsm cos (ψs + ψr ) dVc dVs (30) Wppqq (f, xi ) = qqss 2 6 2 2 (4πa0 ) re Df Vs Vc | {z } Integral over coherence volume Vc

|

{z

Integral over source volume Vs

}

Equation (30) can alternatively be described by 1 1 Wppqq (f, xi ) = (4πa20 )2 re2 Df6

Z Wqqss Gqq dVs (yi )

(31)

Vs

|

{z

}

integral over source volume

with the abbreviation for the inner integral Z Gqq = Vc

s

Wqqcc Wqqss | {z }

γsm cos(ψs + ψr ) dVc (ηi ) . |{z} | {z }

(32)

coherence source interference

rel. source strength

|

{z

integral over coherence volume

}

Gqq describes the sound radiation of a volume element dVs with source strength Wqqss = 1, by including its interference with the whole surrounding source volume Vc , as far as the coherence γsm > 0. Wqqss is the powerspectral density of Qq in the source position yi and Wqqcc the corresponding value in the second position yi + ηi (compare figure 2). Both values describe the strength of the source term Qq as function of the frequency f with respect to the observer position xi . Wqqcc /Wqqss is the relative source strength, where the source strength in position yi + ηi is normalized with the strength in position yi . γsm indicates that only the coherent part of the fluctuations in the two positions contributes to the sound in the far field and this contribution can be additive or subtractive depending on the phase ψs + ψr . The source interference term in equation (32) describes the phase relationship between the contributions from different source positions. The phase ψs considers the effect of the wavelike source motion in the flow and ψr the influence of the retarded times. The product of coherence function times source interference function in equation (32) can have very large effects on the sound radiation of the source field and explains many features of the directivity of jet mixing noise like convective amplification in the rear arc and Mach wave radiation, which occurs for ψs + ψr = 0.14, 15, 22

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The integration over the coordinate ηi in equation (32) needs only be carried out over the region in which the coherence γsm is not negligible. This region is termed coherence volume in the corresponding under-brace of equation (32). II.E.

Source interference

Jet noise is likely caused by the growth and breakup of instability waves in the jet shear layer. The resulting large scale structures yield non-compact noise sources, for which the influence of source interference was shown to be important.24, 25, 14, 15 In the following, the influences of source interference in the azimuthal and axial directions shall be considered. II.E.1.

Decomposition into azimuthal components

As proposed by Michalke2, 3 the source region shall be described in cylindrical coordinates (xs , rs , ϕs ), while the observer is described in polar coordinates (rp , θp , ϕp ). Two source rings are shown in figure 3 with the coordinates (xs , rs ) and (xc , rc ) equivalent to the source positions shown in figure 2. Since the source cross-spectral density

Figure 3. Coordinate system with observer in polar coordinates (rp , θp , ϕp ) and sources in cylindrical coordinates (xs , rs , ϕs ) on source ring and (xc , rc , ϕc ) on coherent source ring. ϕs is defined relative to the azimuthal angle of the plane defined by the observer location and the jet axis.

Wqqsc between two source locations (xs , rs , ϕs ) and (xc , rc , ϕc ) is periodic in 2π in the azimuthal direction with respect to ∆ϕs = ϕc − ϕs , it can be expanded into azimuthal Fourier components m. By further assuming that the time averaged mean and fluctuating quantities of the jet flow are axisymmetric, the source field depends only on ∆ϕs but is independent on ϕs and ϕc . ∞ X Wqqsc (∆ϕs ) = Wqqsc,m eim∆ϕs (33) m=−∞

In addition, the cross-spectral densities Wqqsc,m of the sources with negative m are conjugate complex for non-swirling jets. The power-spectral density Wpp in any observer location in the geometric far field can then be described in terms of a sum of contributions from azimuthal components m of the source field as follows.3 Wpp =

1 1 2 2 (4π) re Df6

Z Z ∞ X m=−∞

Wqqsc,m e−im∆ϕs eiψr dVc dVs

(34)

Vs Vc

Since the far field is periodic in 2π it can be described as a sum of azimuthal components, as well. Wpp =

∞ X

Wpp,m

(35)

m=−∞

Each azimuthal component Wpp,m in the far field is related to the azimuthal component Wqqsc,m of the source field with the same order number m.3

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II.E.2.

Azimuthal interference

The volume elements are defined in cylindrical coordinates by dVs = dxs drs rs dϕs and similarly dVc . Z Z Z Z Z Z 1 1 Wqqsc,m eiψr rs rc eim(ϕc −ϕs ) drc drs dxc dxs dϕc dϕs Wpp,m = (4π)2 re2 Df6

(36)

ϕs ϕc xs xc rs rc

The integration of the integrals over ϕs and ϕc can be carried out analytically as shown by Michalke for the static jet.3 The solution for each azimuthal component of the sound pressure in the geometric far field in a flight stream is then described by Z Z Z Z 1 1 Wpp,m = (37) Wqqsc,m rc Jm (krc sin θe )eψr drc dxc drs dxs rs Jm (krs sin θe ) 4 re2 Df6 x c rc

x s rs

with the phase ψr due to retarded time differences ψr = k

xc − xs cos θe . Df

(38)

Wqqsc,m can be described by the power-spectral densities Wqqss,m and Wqqcc,m in the two ring source locations (xs1 , rs1 ) and (xs2 , rs2 ), the coherence γsm and the phase ψsm in analogy to equation (27). p Wqqsc,m = Wqqss,m Wqqcc,m γsm eiψsm (39) γsm and ψsm may depend on the azimuthal component number m. γsm describes the decay of the coherence of the source strength with increasing axial and radial separation between the source rings. ψsm depends on the phase speed of the mth azimuthal component of the source strength and is an outcome of the wave-like motion of the sources, which also likely depends on m. Equation (37) can be abbreviated to Z Z 1 1 2 Wqqss,m rs2 Jm (krs sin θe )Gqqm drs dxs (40) Wpp,m = 4 re2 Df6 xs rs

with the interference integral Z Z s Gqqm =

Wqqcc,m rc Jm (krc sin θe ) γsm cos(ψsm + ψr ) drc dxc Wqqss,m rs Jm (krs sin θe )

(41)

x c rc

Jm is the Bessel function of the first kind and order m and describes the effect of the azimuthal interference. II.E.3.

Radial interference

We now assume that the acoustic sources are compact in the radial direction. The sources are concentrated in a thin circular cylinder of thickness Lr with radius rs = rc = Dj /2 and it is assumed that the source strength is independent of r. This means that radial interference effects are neglected in the following. The radial direction of the integrals (40) and (41) can then be solved approximately. Z 1 1 2 Wpp,m = Lr Lxs 2 6 rs2 Jm (krs sin θe ) Wqqss,m Fqqm dξs (42) 4 re Df ξs

and

Z s Fqqm = Lr Lxc

Wqqcc,m γsm cos(ψsm + ψr ) dξc Wqqss,m

(43)

ξc

ξs = xs /Lxs and ξc = (xc − xs )/Lxc are normalized axial positions with the axial source length scale Lxs and axial coherence length scale Lxc .

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II.E.4.

Axial interference

If it is further assumed that the axial range of the coherent source field is sufficiently small such that the ratio of the source strength Wqqcc,m /Wqqss,m ≈ 1, we obtain Z (44) Fqqm = Lr Lxc γsm cos(ψsm + ψr ) dξc . |{z} | {z } ξc coherence

interference

|

{z

integral over coherence volume

}

where the axial coherence length scale Lxc is defined by Z∞ Lxc = 2

γsm (xc ) dxc .

(45)

xc =xs

With this definition, the integral in (44) achieves a maximum value of 1, when the phase ψsm + ψr = 0. The phase ψr is defined in equation (38). The phase ψsm is determined by the phase speed Upm of the wave-like motion of the disturbances between the rings at xs and xc . ψsm = 2πf

a0 xc − xs a0 = k(xc − xs ) = kLxc ξc Upm Upm Upm

(46)

The phase speed of the wavelike disturbances in a jet with ambient flight stream may be defined by Upm = Uf + u?pm (Uj − Uf )

(47)

with the flight speed Uf and the jet speed Uj . The factor u?pm = 0.7 is a common estimate for unheated jets. In reality it is related to the phase speed of the wavelike motion of the growing and decaying instability waves in the shear layer. This depends theoretically on the mode number m of the instability wave and on a Strouhal number based on the thickness of the shear layer.11 The phase speed also depends on the density ratio between jet and its ambience. Heated jets have smaller phase speeds26 and may require a smaller factor, e.g., u?pm = 0.6. The axial interference effect can then be described by Z∞ Fqqm = Lr Lxc −∞

II.F.

    f Lxc Upm cos θe 1+ ξc dξc . γsm (ξc ) cos 2π Upm a0 Df

(48)

Normalization of equations

Scaling relations can best be derived with non-dimensional equations. With the nozzle diameter Dj as reference length, ∆Uj = Uj − Uf as reference speed, Dj /∆Uj as reference time the quadrupole source term Qq of equation (22) can be made dimensionless as follows Q?q (yi , xi , t) = Qq (yi , xi , t)

Dj2 , ρ0 ∆Uj4

(49)

where the star indicates a non-dimensional quantity, which will be assumed to be independent of Uj , Uf , Dj , ρ0 in a first order approximation. However, Q?q will likely depend on the temperature of the jet. The corresponding dimensionless form of the dipole source term Qd of equation (23) is Q?d (yi , xi , t) = Qd (yi , xi , t) II.F.1.

Dj2 . ρ0 ∆Uj3

(50)

Jet stretching in flight

A complication in the flight situation is that the jet flow field is stretched in the flow direction. This means that the growth rate of the free shear layer thickness is decreased by the flight stream, which yields a lengthening of the flow

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field of the jet. Michalke & Michel1 considered this lengthening in first order by proposing a jet stretching factor σ due to flight speed, σ = 1 + A Uf /∆Uj (51) by assuming a stretching parameter A = 1.5 . . . 3.0. The stretching was later verified theoretically by an instability analysis of Michalke & Hermann11 for a jet in an external flight stream. However, they concluded that the stretching parameter should be chosen closer to A = 1.4. The constant A may be related to the normalized phase speed u?pm of equation (47) via A = 1/u?pm . (52) The stretching is illustrated in figure 4. The time averaged mean and fluctuating flow properties in the ring position (xs , rs ) of the static jet will be found in position (σxs , rs ) for the jet exposed to a flight stream. The coherence of any fluctuating property between positions (xs , rs ) and (xs +ηx , rs ) is replicated in positions (σxs , rs ) and (σxs +σηx , rs ) in the jet in an external flight stream.

(a) static

(b) flight

Figure 4. Influence of jet stretching on the flow field of the jet. (a) Two source positions in static jet. (b) Corresponding positions in the same jet exposed to a flight stream.

The normalized axial length scales are then defined by L?xs =

Lxs σDj

(53)

L?xc =

Lxc . σDj

(54)

L?r = Lr /Dj

(55)

and

The radial length scale is unaffected by stretching

if the comparison is made for equal normalized axial locations x?s =

xs . σDj

(56)

Fqqm . σL?xc L?r

(57)

The normalized axial interference function is given by ? Fqqm =

Michalke and Hermann11 also showed that the normalized frequency of the instability waves in the shear layer is increased in flight proportionally to σ, yielding f? = f II.F.2.

Dj . σ∆Uj

(58)

Scaling relations

The power-spectral density of the source term Qq in the source position xs is then given by Wqqs =

ρ20 ∆Uj7 ? Wqqs . σDj3

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(59)

With a20 = κp0 /ρ0 (κ is the isentropic exponent) we finally obtain Wppqq,m =

 κp 2  D 2 0

j

4π

re

(Mj − Mf )7

σ2 2 J (krs sin θe )L?xs L?xc L?r Df6 m

Z

? ? Wqqs Fqqm dξs? ,

(60)

ξs?

? where the non-dimensional axial interference integral Fqqm is defined by

? Fqqm

Z∞ = −∞



f Lxc γsm (ξc ) cos 2π Upm

   Upm cos θe 1+ ξc dξc . a0 Df

(61)

The Strouhal number f Lxc /Upm based on the coherence length scale Lxc and the phase speed Upm is rather constant over a large frequency range as can be concluded from the experimental data of Harper-Bourne.27 Integration of equa? tion (60) over the whole frequency range (0, fmax ) yields for the mean square of the sound pressure due to quadrupole th noise from the m azimuthal component p˜2qq,m

 κp 2  D 2  M − M 8 j j f 0 σ 3 Df2 G?qqm = 4π re Df

(62)

with the interference integral ?

G?qqm

Zfmax =

2 Jm (krs sin θe )L?xs L?xc L?r

f ? =0

Z

? ? Wqqs Fqqm dξs? df ? .

(63)

ξs?

The Helmholtz number krs is defined by krs = k

Dj f Dj =π = πf ? σ(Mj − Mf ) . 2 a0

(64)

An inspection of equation (60) shows that the directivity of a static jet (Doppler factor Df = 1) can be caused by a ? ? , of the sources, by a directivity of the axial interference function Fqqm directivity of the power-spectral density Wqqs 14, 15, 22 2 that many and by the directivity of the azimuthal interference function Jm . It was shown in previous papers ? and by the features of the directivity of jet noise can be explained with source interference effects expressed by Fqqm azimuthal interference. The corresponding equations for the quadrupole-dipole noise are p˜2qd,m =

 κp 2  D 2  M − M 7 0 j j f σ 3 Df2 G?qdm 4π re Df

(65)

?

G?qdm

Zfmax =

2 Jm (krs

sin θe )L?xs L?xc L?r

f ? =0

Z

? ? Wqds Fqdm dξs? df ? .

(66)

ξs?

and for dipole noise p˜2dd,m

 κp 2  D 2  M − M 6 j j f 0 σ 3 Df2 G?ddm = 4π re Df

(67)

?

G?ddm =

Zfmax

2 Jm (krs sin θe )L?xs L?xc L?r

f ? =0

Z

? ? Wqqs Fddm dξs? df ? .

(68)

ξs?

The main differences between the three equations (62), (65), and (67) are the powers for the relative Mach number Mj − Mf and for the Doppler amplification. The same equations are valid for integrals with constant relative bandwidth, e.g., one-third octave spectra or one-twelfth octave spectra, provided the frequencies of the static jets satisfy equation (58).17, 18

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Concerning the normalized interference integrals G?m defined in equations (63), (66) and (68) we will assume for scaling purposes that all three are identical. The reason for this assumption is that the quadrupole and dipole sources are outcome of the same instability process in the jet’s shear layer. G?m = G?qqm = G?qdm = G?ddm

(69)

For the same reason we assume that the three axial interference functions are identical, ? ? ? ? Fm = Fqqm = Fqdm = Fddm ,

(70)

requiring that the decay of the coherence function γsm (ξc ) is identical for all source types. Similarly the phases ψsm due to wave motion of the sources are identical. With respect to the normalized phase speed u?pm and the normalized length scales f Lxc /Upm , these values may differ for heated and unheated jets. II.G. II.G.1.

Influence of flight stream on sound radiation Convective amplification into forward arc

It can be concluded from equations (62), (65) and (67) that the flight stream yields a Doppler amplification into the −5 forward arc, which is proportional to D−6 f for the quadrupole noise sources, Df for the quadrupole-dipole noise and D−4 f for the dipole noise. These Doppler amplifications are a direct outcome of the sound emission from the jet sources described in a nozzle-fixed coordinate system into an external flight stream. II.G.2.

Rear arc amplification due to wave-like motion

Jet noise has a well known rear arc amplification due to the wave-like motion of the noise sources, which is caused ? by instability waves in the shear layer. This amplification is described by the normalized interference function Fm , which is described in equation (61) for the quadrupole sources but as explained in equation (70) shall be assumed to be identical for all source terms. The amplification is influenced by the flight stream via its influence on the phase ? = 1 when the term speed Upm of the wave-like motion. The axial interference function reaches a maximum of Fm in the brackets of the argument of the cosine in equation (61) vanishes, which is the case for the Mach wave radiation angle θe,MW defined by 1 1 (71) cos θe,MW = − =− ? Mp − Mf upm (Mj − Mf ) For this angle all sources of the jet have the same phase for any value of ξ ? . The flight stream increases the Mach wave radiation angle as can be seen in figure 5. It also increases the lowest acoustic jet Mach number with Mach wave radiation. ? < 1 due to the canceling effect of the oscillating sign of the cosine For all angles θe 6= θe,MW the integral Fm function when ξc is varied. It can be concluded that the “rear arc amplification” is in fact a forward arc attenuation and this attenuation depends according to equation (61) on the shape of the coherence function γsm (ξc ), on the value of the normalized axial length scale 2πf Lxc /Upm and on the phase Mach number Upm /a0 . II.G.3.

Modification of azimuthal interference

2 The azimuthal interference as expressed by the term Jm is also influenced by the flight stream because the wave number k = 2πf /a0 is increased by a factor of σ(Mj − Mf )/Mj , yielding

fflight Mf = 1 + (A − 1) fstatic Mj

(72)

For A = 1.4, Mf = 0.26, and Mj = 1.0 we obtain fflight /fstatic = 1.11, a frequency increase of about one half of a one-third octave. This influence is larger in cruise for a future engine, which may have an acoustic jet Mach number Mj = 1.25 in cruise at Mf = 0.85 yielding a frequency ratio fcruise /fstatic = 1.272, roughly one one-third octave. Larger frequencies result in smaller values for the radiation effectiveness due to azimuthal interference. However, since the frequency increase is small even in cruise, its influence on the flight effect scaling shall be neglected in this paper.

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Figure 5. Mach wave radiation angle as function of acoustic jet Mach number Mj for u? pm = 0.7 with flight Mach number Mf as parameter. The axial ? interference function reaches a maximum of Fm = 1 for this angle.

III.

The A´erotrain tests

The theory shall be compared with some results of the noise tests carried out with the A´erotrain.13 III.A.

The data set

The overall sound-pressure levels (OASPL) for aerodynamically subsonic jets are replotted in figure 6(a) as polar directivities with respect to the flight direction. The polar distance of re = 50 m equals 170 nozzle diameters. The data points are shown as symbols while the lines are smoothing splines. The acoustic jet Mach numbers Mj = Uj /a0 of these jets are supersonic. The temperatures were 760 K, 790 K, and 825 K (Tj /T0 = 2.64, 2.74, 2.86) for the jet speeds of 370 m/s, 440 m/s, and 505 m/s, respectively. It was not mentioned in the reference if these temperatures were total or static. It may be noted that all directivities in the flyover data show a forward arc amplification for roughly θe ≤ 70◦ . This might be related to the high acoustic jet Mach numbers in these tests. The Mach number exponent n of the static experiments is defined by n(θe ) =

OASPL(Uj ) − OASPL(Uj,ref ) lg(Uj /Uj,ref )

(73)

A least-square fit to the three static directivities yields the result shown in figure 6(b). The curve values are very similar to those reported by Viswanathan28 for a total temperature ratio Tt /T0 = 3.3. III.B.

Normalized overall sound pressure levels

The next figures show normalizations of the measured overall sound pressures of the jets based on the normalizations used in equation (62) for the mth component of quadrupole noise and in equation (67) for dipole noise. In terms of the overall sound-pressure levels OASPL we obtain OASPLnorm,q = OASPL − 80 lg(Mj − Mf ) + 60 lg Df − 30 lg σ

(74)

assuming quadrupole noise scaling and OASPLnorm,d = OASPL − 60 lg(Mj − Mf ) + 40 lg Df − 30 lg σ

(75)

for dipole noise. Equations (62) and (67) are defined for a single azimuthal component, yet the normalized sound pressures include all azimuthal components m as well as contributions from quadrupole, quadrupole-dipole and dipole sources.

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(a) Measured OASPL

(b) Mach number exponent

Figure 6. (a) Measured noise data of Drevet et al.13 for subsonic jets with supersonic acoustic jet Mach numbers Mj . The static tests are compared with the results obtained with a flyover Mach number Mf = 0.24. The data were digitized from the paper and recalculated for a constant polar distance of 50 m. (b) Mach number exponent determined with a least-square fit of the static data for three acoustic jet Mach numbers. n = 5.1 for θe = 90◦ .

The normalized directivities of the static and flyover data are compared in figures 7, 8, and 9 for the three jet Mach numbers Mj = 1.09, 1.29, 1.49, respectively. The left subfigures show the scaling based on quadrupole noise and the right subfigures based on dipole noise. It can be seen that the scaling based on dipole noise yields better results 2 is because the levels at 90◦ are almost uninfluenced by the flight Mach number. The azimuthal interference term Jm almost unchanged because the frequency and thus the parameter krs is hardly changed by the flight stream (compare equation (58)). The only differences are the smaller slopes of the directivity of OASPLnorm,d in flight in figures 7(b), 8(b), and 9(b). The fall-off of the directivity values for angles θe > 150◦ is caused by refraction.22

(a) Quadrupole scaling

(b) Dipole scaling

Figure 7. Influence of flight Mach number on interference integral for Mj = 1.09.

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(a) Quadrupole scaling

(b) Dipole scaling

Figure 8. Influence of flight Mach number on interference integral for Mj = 1.29.

(a) Quadrupole scaling

(b) Dipole scaling

Figure 9. Influence of flight Mach number on interference integral for Mj = 1.49.

The influence of the acoustic jet Mach number is evaluated in figure 10. The dipole scaling shown in figure 10(b) does no longer show a perfect agreement for θe = 90◦ . The normalized sound pressure levels are reduced by roughly 2 dB when the acoustic jet Mach number is increased. This can be explained by the azimuthal interference effect to be discussed later with figure 12. The increased phase speed due to the larger jet speed primarily changes the slope of the normalized directivity. The same influence can be observed in figure 11 for the flyover cases. From the good agreement between the three cases with dipole normalization at 90◦ of figure 7(b), 8(b), and 9(b) it can be concluded that the noise emission of the A´erotrain is dominated by dipole noise.

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(a) Quadrupole scaling

(b) Dipole scaling

Figure 10. Influence of jet Mach number on interference integral for static tests.

(a) Quadrupole scaling

(b) Dipole scaling

Figure 11. Influence of jet Mach number on interference integral for flyover tests.

III.C.

Influence of azimuthal interference

The Mach number exponent n = 5.1 for θe = 90◦ is smaller than n = 6 to be expected for dipole noise according to equation (67). It shall now be shown that this is likely caused by azimuthal interference. The normalized interference integrals in figures 10(b) and 11(b) show an influence of the Mach number at θe = 90◦ . It must be concluded that the normalized interference function G?ddm of equation (68) depends on the Mach 2 number. This is in fact the case since the term Jm (krs sin θe ) in equation (68) depends on the Mach number because 2 the wave number k increases proportionally to the acoustic Mach number. An evaluation of Jm (krs sin θe ) as function of emission angle is shown in figure 12 for the three acoustic Mach numbers of the A´erotrain tests and it can be concluded that the radiation efficiency for θe = 90◦ decreases with increasing Mach number. This explains the reduced noise levels in figures 10(b) and 11(b) for increasing jet Mach number. Since the relative amplitudes of the azimuthal components m in the engine jet are not known they are arbitrarily selected as 0.32, 0.22, 0.16, 0.11, 0.08, 0.06, 0.05, respectively, for m = 0 . . . 6. It can be seen that the azimuthal interference effects are largest at θe = 90◦

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and that this effect can be as large as 1 dB for St = 0.25 and 2 dB for St = 0.56 when the Mach number is increased from Mj =1.09 to 1.49. Therefore, this influence must be included for precise flight effect predictions.

(a) St=0.25

(b) St=0.5

Figure 12. Influence of azimuthal interference on radiated noise level. Amplitudes of Fourier component numbers are arbitrarily chosen as 0.32, 0.22, 0.16, 0.11, 0.08, 0.06, 0.05, respectively, for m = 0 . . . 6. (a) St=0.25, (b) St=0.5. It can be concluded that the azimuthal interference may reduce the power-spectral density (PSD) of the sound pressure at θe = 90◦ by several dB with increasing jet Mach number.

IV.

Flight effect scaling

The objective of this paper is to define static jets that have theoretically a similar sound emission as a jet in a flight stream. Michalke & Michel1 realized that the velocity-dependent terms in front of the integrals in equations (62),(65) and (67) can be written as Men σ 3 Df2 (76) with the effective (acoustic) jet Mach number Me 29 Me =

Mj − M f . Df

(77)

n = 8, 7, 6 for quadrupole, quadrupole-dipole, and dipole noise, respectively. The power 3 (rather than 2) for the jet stretching factor σ was later suggested by Michalke & Hermann11 as an outcome of their instability analysis of a jet in an external flight stream. IV.A.

Procedure of Michalke and Michel

Michalke & Michel1 proposed to carry out static experiments with the effective jet Mach number Me and adjust the results for the influence of the flight stream via σ 3 Df2 . The mean square sound pressures of a single-stream jet in a flight stream with Mach number Mf with any composition of quadrupole and dipole sources can then be related to static jets with identical jet temperature by applying the following flight-effect scaling formula p˜2 (M j, Mf , θe ) = p˜2 (Me , 0, θe ) σ 3 Df2 .

(78)

The equation assumes that the ambient pressure p0 is equal for the static and flight conditions. The overall soundpressure level can then be predicted with OASPL(Mj , Mf , θe ) = OASPL(Me , 0, θe ) + 30 lg(1 + 1.4Mf /(Mj − Mf )) + 20 lg(1 − Mf cos θe )

(79)

by using the definitions (51) for σ and (11) for Df . This scaling procedure simultaneously satisfies the conditions for all three equations (62),(65) and (67), which means the scaling works for cold as well as for hot jets. The procedure implicitly assumes that the interference integrals Gm in the three equations depend only on the effective jet Mach number Me . This is not fully the case and as a result the predicted flyover noise levels deviate slightly from the measured ones. Nevertheless, Rawls30 found that the scheme compares well with measured data as long as Uj /a0 < 1.5. This is also true for one-third octave spectra under the condition that the frequencies of the static jets satisfy

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equation (58). The only discrepancy found by Rawls30 was in the rear arc, where the predicted noise levels appeared to be slightly too low. The disadvantage of a prediction of the flight effect according to equation (79) is that the effective jet Mach number Me depends on the emission angle θe according to equation (77). This requires a static noise test to be performed for each emission angle, a problem that can be circumvented by scaling the static jet noise with a velocity exponent n(θe ) according to Me . (80) OASPL(Me , 0, θe ) = OASPL(Mj,ref , 0, θe ) + 10 n(θe ) lg Mj,ref n(θe ) depends on the temperature of the jet. Examples of velocity exponents n(θe ) for jets with various temperatures can be found in Viswanathan.31 The Mach number exponent of the A´erotrain tests was already shown in figure 6(b) as function of emission angle. This exponent was determined by a least-square fit over the three available acoustic Mach numbers 1.09, 1.29, and 1.49. Predictions of the A´erotrain flyover cases for the three tested jet Mach numbers are shown in figures 13(a), 14(a), and 15(a). The Mach number range of the static data base was extended by inter- and extrapolations shown in figure 16(a). The extrapolations are made with equation (80) with Mj,ref = 1.29. The Mach number exponents are shown in figure 16(b). Predictions for lower Mach numbers are made with the Mach number exponent between the two lower test Mach numbers and higher Mach numbers are predicted with the exponent between the two higher Mach numbers. The prediction for the jet Mach number Mj = 1.49 shown in figure 13(a) reveals an almost perfect agreement with the experimental data with differences of less than 1/2 dB in the angular range where static data are available. Predictions on the basis of extrapolated static data are indicated with a dotted line. The predictions in the far forward arc at θe = 20◦ are 2.5 dB too high and in the far rear arc at θe > 150◦ they are 1.5 dB too low. Reasons could be errors in the extrapolations or incorrect modeling of the interference function G?m defined in equations (69) and (63).

(a) Mj = 1.49

(b) Me

Figure 13. (a) Prediction of the flyover noise from static tests for Mj = 1.49 with the method of Michalke & Michel1 and comparison with test data. Predictions shown in magenta. Dotted line indicates predictions based on extrapolated static data. A = 1.4 is the stretching parameter. (b) Effective jet Mach numbers required for the prediction. The angular ranges θe < 50◦ and θe > 130◦ are predicted with extrapolated data.

The prediction for Mj = 1.29 shown in figure 14(a) is equally good in the angular range, where static data are available, except for θe < 30◦ . The rapid fall-off of the experimental data in the far forward arc may be non-physical. The predictions for θe > 80◦ had to made with extrapolated data for small jet Mach numbers, which may be wrong.

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(a) Mj = 1.29

(b) Me

Figure 14. (a) Prediction of the flyover noise from static tests for Mj = 1.29 with the method of Michalke & Michel1 and comparison with test data. Predictions shown in magenta. Dotted line indicates predictions based on extrapolated static data. A = 1.4 is the stretching parameter. (b) Effective jet Mach numbers required for the prediction. The range θe > 90◦ is predicted with extrapolated data.

The predictions for Mj = 1.09 shown in figure 15(a) are based on static data with lower jet Mach numbers than available for almost the whole range of emission angles as can be concluded from figure 15(b). The deviations are likely due to the required extrapolations for lower Mach numbers shown in figure 16(a), which seem to be non-physical especially in the rear arc.

(a) Mj = 1.09

(b) Me

Figure 15. (a) Prediction of the flyover noise from static tests for Mj = 1.09 with the method of Michalke & Michel1 and comparison with test data. Predictions shown in magenta. Dotted line indicates that static data were extrapolated. A = 1.4 is the stretching parameter. (b) Effective jet Mach numbers required for the prediction. The whole angular range is predicted with extrapolated data, which may explain the differences between predictions and measurements.

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(a) Static interpolated data base

(b) Mach number exponent

Figure 16. (a) Static data base used for the predictions. OASPL data need to be extrapolated above Mj = 1.49 and below Mj = 1.09 with large uncertainty. (b) Mach number exponents used for extrapolations. The blue line is used for predictions Me ≤ 1.29 and the green line for Me > 1.29.

The very good agreement between predictions and experimental data is remarkable, because the prediction formulas are derived only with the Lighthill theory and the scheme does not contain a single arbitrary constant. The only free constant, the stretching parameter A = 1.4 is based on results of an instability theory for unheated jets in a co-flowing flight stream.11 One condition used in the prediction scheme is that the source interference integral G?m (Equation (63)) is only a function of Me . In reality it depends on the azimuthal interference expressed by the Bessel functions Jm and the axial ? source interference integral Fm according to (61). For a correct description of the source interference of a jet in flight ? by a static jet, Gm would have to be identical for the static jet and the jet in the flight stream. The following list of possible reasons for deviations between predictions and measurements includes aerodynamic conditions. • Incorrect modeling of axial interference. • Incorrect modeling of azimuthal interference. • Wrong static data due to extrapolation of static directivities. • Composition of azimuthal components is changed in flight. • Assumptions that the normalized turbulence values are independent of jet and flight Mach number are not valid. These are defined in equations (53), (54), (55), (57), and (58). • Influence of boundary layers about the engine nacelle. It must be expected that boundary layers shield the jet from the full influence of the flight stream and might yield smaller values for the stretching factor σ and larger Me values. IV.B. IV.B.1.

Discussion of possible improvements for modeling the interference Modeling axial interference

? are identical in the static and flyover A correct modeling of the axial interference means that the values for Fm ? cases. We assume that the shape of the coherence function γ(ξ ) as function of the normalized separation ξ ? remains unchanged in flight as well as the normalized length scale f Lxc /Upm . The condition for identical axial source ? interference Fm then requires that cos θe = 1 + Mp,s cos θe (81) 1 + Mp Df

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where Mp = Mf +0.7(Mj −Mf ) is the phase Mach number in flight while Mp,s = 0.7Ms is the phase Mach number of the static jet that yields the same interference pattern. Using equation (51) for σ, we obtain Ms = σMe =

Mj + 0.4Mf . 1 − Mf cos θe

(82)

We conclude that the static experiment has to be carried out with a jet speed that is σ times larger than Me , requiring very large static jet speeds for predictions in the forward arc. This method may not be usable for cruise Mach numbers, when σ can reach large values up to σ = 4 and the static Mach number Ms = 8 in the forward arc. The prediction formula (79) would have to be modified according to OASPL(Mj , Mf , θe ) = OASPL(Me , 0, θe ) + 30 lg σ + 20 lg Df + 10 lg

Fq? (σMe ) . Fq? (Me )

(83)

The function Fq? (Me ) can in principle be deduced from the static experiments via the results shown in figure 10(b) but in practice the available Mach number range of the experiments my be too small. IV.B.2.

Scaling based on dipole noise only

The analysis of the A´erotrain data in section III has shown that its noise is dominated by dipole noise, which means that we do not have to use the effective Mach number Me for the static experiments as explained in section IV.A. This was only necessary to satisfy the scaling conditions simultaneously for all three contributions to the power-spectral density of the sources (quadrupole-quadrupole, quadrupole-dipole, and dipole-dipole). For dipole noise alone the scaling can be based on equation (67) with (68), leaving more freedom for the selection of the jet Mach number Ms of the static jet. The scaling equation for the prediction of the flyover noise for dipole noise would be 2



2

p˜ (Mj , Mf , θe ) = p˜ (Ms , 0, θe )

M j − Mf Ms

6

σ 3 Df−4

(84)

The overall sound-pressure level can then be predicted with OASPL(Mj , Mf , θe ) = OASPL(Ms , 0, θe ) + 30 lg(1 + 1.4Mf /(Mj − Mf )) Mj − Mf − 40 lg(1 − Mf cos θe ) +60 lg Ms

(85)

Using equation (82) for Ms yields the results shown in figure 17. It is obvious that the predicted sound-pressure levels are too low at θe = 90◦ . This may be explained with the neglect of the influence of the jet speed on the azimuthal 2 (krs sin(θe ) in equation (63). interference in equation (85). This interference is described by the term Jm By comparing figure 17 with the predictions shown in figures 13(a), 14(a), and 15(a) it is obvious that the new predictions shown in figure 17 yield smaller values in the forward arc and possibly larger values in the rear arc. This agrees with the conclusion of Rawls30 for the rear arc. It can be concluded that an improvement of the method of Michalke & Michel would require a correction based on the last term in equation (83) based on a compromise between the jet speeds Me according to equation (77) and Ms according to equation (82). IV.C.

Dual stream jets

Dual stream jets can be predicted with similar procedures as demonstrated by Michalke & Michel.32

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(a) Mj = 1.09

(b) Mj = 1.29

(c) Mj = 1.49

(d) Ms

Figure 17. (a,b,c) Prediction of the flyover noise from static tests for Ms = (Mj + 0.4Mf )/Df and comparison with test data. Predictions shown in magenta. Dotted line indicates predictions based on extrapolated static data. A = 1.4 is the stretching parameter. (d) Static jet Mach numbers required for the prediction.

V.

Scaling for cruise conditions

The scaling of Michalke & Michel according to equation (79) shall now be applied to cruise conditions by assuming an ambient pressure of p0 = 101.3 kPa. The results for three flight Mach numbers Mf = 0.28, 0.56, and 0.84 are shown in figure 18. Predictions are only shown for the range of angles for which data for the equivalent static jets are available. It can be concluded that the OASPL in cruise is larger than for the static jet due to the influences of jet stretching, which amount to ∆SPL = 3.7, 8.0, 13.5 dB for the three flight Mach numbers, respectively. However, the actual sound pressure levels in cruise are smaller due to the smaller ambient pressure at cruising altitudes. For an ambient pressure of one fourth of the standard ambient pressure the sound pressure levels would have to be reduced by 12 dB. The shown predictions suffer from the influences of the interference errors discussed in section IV.B, which likely increase the noise levels in the forward arc and reduce them in the rear arc. Therefore, a more uniform directivity than shown in figure 18 may be expected for a cruise Mach number Mf = 0.84.

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Figure 18. Application of equation (79) to three flight Mach numbers including the cruise Mach number Mf = 0.84.

VI.

Conclusions

The flight effect prediction scheme of Michalke and Michel1 is revisited. Predictions for the overall sound pressure level are compared with the experimental data obtained with the A`erotrain. The good agreement already found in reference1 is confirmed. The predictions in the forward arc yield slightly too high sound pressure levels. An error in the modeling of the axial interference is identified as the likely cause of this deviation. The rear arc prediction is hampered by the availability of static jet noise data for small jet speeds. It is very likely that the rear arc predictions yield slightly too small levels as was suggested by Rawls.30 Predictions are also shown for cruise conditions for which the noise levels are increased substantially through the influence of jet stretching. An improvement for the prediction scheme is proposed but no yet developed, because it requires a larger dataset of static jet noise data for validation. A possible source could be the RAE jet noise data,33 which were already used to demonstrate the Helmholtz number scaling in the rear arc.22 This Helmholtz number scaling is equally important in the flight situation but not yet implemented. One-third octave spectra can be predicted with the same formulas, provided the normalized frequencies of the equivalent static jets are divided by the stretching factor σ according to equation (58). An interesting result of the analysis of the A´erotrain data in section III is that these data are dominated by the dipole noise source contribution. An additional validation of the method with unheated jet data would be desirable. A further result is that azimuthal interference is found as likely cause for the deviation of the Mach number exponent n from the theoretical value n = 6 for dipole noise to the actual value n = 5.1 at θe = 90◦ . There are two problems with the method of Michalke & Michel, (i) the unavailability of required static test data, and (ii) the possible error caused by the assumption that the interference integrals G?qqm , G?qdm , and G?ddm according to equations (63), (66), and (68) scale with the effective jet Mach number Me . A larger list of possible reasons for deviations between predictions and measurements is • Incorrect modeling of axial interference. • Incorrect modeling of azimuthal interference. • Wrong static data due to extrapolation of static directivities. • Composition of azimuthal components is changed in flight. • Assumption is not valid that the normalized turbulence values are independent of jet and flight Mach numbers. These normalizations are defined in equations (53), (54), (55), (57), and (58). • Influence of boundary layers about the engine nacelle. It must be expected that boundary layers shield the jet from the full influence of the flight stream and might yield smaller values for the stretching factor σ and larger Me values. The theory of the flight effect on jet noise is developed based on the convective Lighthill equation for the pressure. The sources are described as wave-like motions, which are caused by the growth and decay of instability waves. The 24 of 26 American Institute of Aeronautics and Astronautics

growth is related to the most unstable frequency of the shear layer and the decay determines the coherence length scale of the sources. The jet is assumed to be stretched in flight by a factor σ, defined in equation (51), which means that the turbulence in a source location xs of the static jet has to be compared with a location σxs of the jet in flight. The good agreement between theory and flight-effect prediction indicates that the assumptions made in section II.F.1 are reasonable. It is assumed that the normalized turbulence values defined in equations (53), (54), (55), (57), and (58) are independent of jet and flight Mach numbers if compared for identical normalized axial source positions. Since the theory is based on the assumption of a large scale motion of the sources, the good agreement may also be considered as an indirect proof that jet noise is dominated by the noise emission of large scales for all emission angles and all frequencies.14, 15 Large scale means in this connection that the scales of the turbulent fluctuations for each frequency are in the order of the acoustic wave length. One interesting result of the A´erotrain results is a forward arc amplification of jet noise, which is replicated by the predictions. It is shown in the paper that two effects are present in a jet in a flight stream. (i) A forward arc amplification is the result of the theory for acoustic sources in a flight stream and is independent of the jet Mach number. (ii) A rear-arc amplification is caused by the wave-like motion in the jet and increases with the flight Mach number. This influence is relatively larger for smaller jet Mach numbers. This may explain why the addition of both effects does not yield a forward arc amplification for modern turbofan engines at take-off flight Mach numbers. Future work will extend this work to broadband shock noise based on the broadband shock noise theory of Michel,19 which is formulated as a variation of jet mixing noise.

References 1 Michalke,

A. and Michel, U., “Prediction of Jet-Noise in Flight from Static Tests,” J. Sound Vib., Vol. 67, 1979, pp. 341–367.

2 Michalke, A., “A Wave Model for Sound Generation in Circular Jets,” Tech. Rep. Deutsche Luft- und Raumfahrt. Forschungsbericht ; 70-57,

Deutsche Luft- und Raumfahrt, November 1970, available at elib.dlr.de. 3 Michalke, A., “An Expansion Scheme for the Noise From Circular Jets,” Z. Flugwiss., Vol. 20, 1972, pp. 229–237. 4 Cocking, B. J. and Bryce, W. D., “Subsonic jet noise in flight based on some recent wind-tunnel results,” AIAA-Paper-1975-0462, 1975. 5 Packman, A., Ngt, K., and Paterson, R. W., “Effect of Simulated Forward Flight on Subsonic Jet Exhaust Noise,” Journal of Aircraft, Vol. 13, No. 12, December 1976, pp. 1007–1013. 6 Tanna, H. K. and Morris, P. J., “In-flight simulation experiments on turbulent jet mixing noise,” J. Sound Vib., Vol. 53, No. 3, August 1977, pp. 389–405, Presented at the Third AIAA Aeroacoustics Conference, Palo Alto, California, U.S.A. July 1976 as AIAA Paper No. 76-554. 7 Viswanathan, K. and Czech, M. J., “Measurement and Modeling of Effect of Forward Flight on Jet Noise,” AIAA J., Vol. 49, No. 1, January 2011, pp. 216–234. 8 Michel, U., “On the systematic error in measurements of jet noise flight effects using open jet wind tunnels,” AIAA-2015-2996, June 2015, 21st AIAA/CEAS Aeroacoustics Conference, 22-26 June 2015, Dallas, TX. 9 Ffowcs Williams, J. E., “The Noise from Turbulence Convected at High Speed,” Phil. Trans. Roy. Soc., Vol. A 225, 1963, pp. 469–503. 10 Bryce, W. D., “The Prediction of Static-to-Flight Changes in Jet Noise,” AIAA-1984-2358, 1984. 11 Michalke, A. and Hermann, G., “On the inviscid instability of a circular jet with external flow,” J. Fluid Mech., Vol. 114, 1982, pp. 343–359. 12 Hoch, R. and Berthelot, M., “Use of the Bertin A´ erotrain for the investigation of flight effects on aircraft engine exhaust noise,” J. Sound Vib., Vol. 54, 1977, pp. 153–172. 13 Drevet, P., Duponchel, J. P., and Jacques, J. R., “The Effect of Flight on Jet Noise as Observed on the Bertin A´ erotrain,” J. Sound Vib., Vol. 54, 1977, pp. 173–201. 14 Michel, U., “Influence of Source Interference on the Directivity of Jet Noise,” AIAA-2007-3648, 2007, 13th AIAA/CEAS Aeroacoustics Conference, Rome, Italy, May 21-23, 2007. 15 Michel, U., “The role of source interference in jet noise,” AIAA-2009-3377, 2009, 15th AIAA/CEAS Aeroacoustics Conference, Miami, Fl, May 11-13, 2009. 16 Michalke, A. and Michel, U., “Importance of Jet Temperature on the Prediction of Jet-Noise in Flight,” Mechanics of Sound Generation in Flows, edited by E.-A. M¨uller, Springer-Verlag Berlin Heidelberg, 1979, pp. 256–263, Proceedings of the IUTAM/ICA/AIAA-Symposium, G¨ottingen, August 28-31, 1979. 17 Michel, U. and Michalke, A., “Prediction of Flyover Jet Noise Spectra,” AIAA-1981-2025, 1981, AIAA 7th Aeroacoustics Conference, Palo Alto, CA, Oct 5-7, 1981. 18 Michel, U. and Michalke, A., “Prediction of Flyover Jet Noise Spectra from Static Tests,” NASA Technical Memorandum 83219, 1981. 19 Michel, U., “Broadband Shock Noise: Theory Vis-A-Vis Experimental Results,” First Joint CEAS/AIAA Aeroacoustics Conference (16th AIAA Aeroacoustics Conference), DGLR, Deutsche Gesellschaft f¨ur Luft-und Raumfahrt, 1995, pp. 545 – 554, CEAS/AIAA-95-071, DGLRBericht 95-01. 20 Morfey, C. L., “Amplification of Aerodynamic Noise by Convected Flow Inhomogeneities,” J. Sound Vib., Vol. 31, 1973, pp. 391–397. 21 Proudman, I., “The Generation of Noise by Isotropic Turbulence,” Proc. Roy. Soc. London, Vol. A 214, 1952, pp. 119–132. 22 Michel, U. and Ahuja, K. K., “On the Scaling of Jet Noise with Helmholtz Number Close to the Jet Axis,” AIAA-2014-2338, June 2014, 20th AIAA/CEAS Aeroacoustics Conference, Atlanta, Ga, USA. 23 Tanna, H. K., Dean, P. D., and Fisher, “The influence of temperature on shock-free supersonic jet noise,” J. Sound Vib., Vol. 39 Journal o f Sozozd and Vibration (1975) 39(4), 429–460, No. 3, 1975, pp. 429–460. 24 Michalke, A., “On the Effect of Spatial Source Coherence on the Radiation of Jet Noise,” J. Sound Vib., Vol. 55, 1977, pp. 377–394. 25 Michalke, A., “Some Remarks on Source Coherence Affecting Jet Noise,” J. Sound Vib., Vol. 87, 1983, pp. 1–17.

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A., “Survey on jet instability theory,” Progress in Aerospace Sciences, Vol. 21, 1984, pp. 159–199.

27 Harper-Bourne, M., “Jet noise turbulence measurements,” AIAA-2003-3214, 2003, 9th AIAA/CEAS Aeroacoustics Conference and Exhibit,

12-14 May 2003, Hilton Head, South Carolina. 28 Viswanathan, K., “Improved method for prediction of noise from single jets,” AIAA J., Vol. 45, No. 1, 2007, pp. 151–161. 29 Michel, U. and Michalke, A., “Prediction of flyover jet noise spectra from static tests,” 1980, Presented at the 100th Meeting of the Acoust. Soc. of Am., Los Angeles, Nov. 1980. 30 Rawls, Jr., J. W., “Comparison of Forward Flight Effects Theory of A. Michalke and U. Michel with Measured Data,” NASA Contractor Report 3665, 1983. 31 Viswanathan, K., “Scaling Laws and a Method for Identifying Components of Jet Noise,” AIAA J., Vol. 44, No. 10, October 2006, pp. 2074– 2085. 32 Michalke, A. and Michel, U., “Prediction of Flyover Noise from Single and Coannular Jets,” AIAA-1980-1031, June 1980, AIAA 6th Aeroacoustics Conference, June 4-6, 1980, Hartford, CT. 33 Staff of the Noise Test Facility, “Results from an experimental programme on static single-stream jet noise,” 1983, QinetiQ (RAE) 1983 NTF Jet Noise Data.

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