- Email: [email protected]
Open rotor broadband interaction noise
Journal of Sound and Vibration 332 (2013) 3956–3970
Contents lists available at SciVerse ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Open rotor broadband interaction noise Michael J. Kingan n Institute of Sound and Vibration Research, University of Southampton, Southampton, SO17 1BJ, UK
a r t i c l e in f o
abstract
Article history: Accepted 8 March 2013 Handling Editor: P. Joseph
A theoretical model is presented for calculating the broadband noise produced by the interaction of an open rotor with the wake from either an upstream contrarotating rotor or a stationary pylon. The model is used to investigate the dependence of the radiated noise on parameters such as pylon–rotor gap and the polar and azimuthal directivity of the noise field. A simple model is also presented which assumes that the unsteady loading on adjacent blades is uncorrelated. It is shown that the simple model can be used to calculate broadband interaction noise for most practical open rotor geometries. & 2013 Elsevier Ltd. All rights reserved.
1. Introduction Broadband noise produced by contra-rotating open rotors can be a significant contributor to the total noise level produced by these engines [1,2]. For an isolated open rotor, broadband noise can be produced by several sources, namely; (1) boundary layer self-noise, which is produced by the scattering of boundary layer turbulence by the trailing edge of the rotor blades, (2) vortex self-noise, which is produced by the scattering of the turbulence in the tip vortex by the rotor blades, (3) atmospheric turbulence ingestion noise and (4) rotor–rotor interaction broadband noise which is produced when the turbulent wake of the upstream rotor impinges on the downstream, contra-rotating rotor blades. When a pylon is installed upstream of the rotor, the wake from this pylon will interact with the open rotors to produce both tone and broadband noise. Tones are produced by the interaction of the rotor blades with the steady component of the flow, whilst broadband noise is produced by the interaction of the rotor blades with the turbulent component. An upstream pylon will be required to attach the open rotor to the fuselage when the open rotor is mounted at the rear of the aircraft in a ‘pusher’ configuration. This particular configuration is currently being investigated by aircraft engine and airframe manufacturers. Two photographs showing a test of the Rolls-Royce 1/6th scale contra-rotating open rotor rig with and without an upstream pylon in the DNW low speed wind tunnel is shown in Fig. 1. This paper will describe a theoretical model for predicting rotor–rotor broadband interaction noise, although the reader should note that this problem has been treated previously by Blandeau and Joseph [3]. The rotor–rotor broadband noise model will then be extended so that it can be used to predict the broadband noise produced by the interaction of a turbulent pylon wake with an open rotor. 2. Formulation In this section, formulae are derived which can be used to calculate the broadband noise produced by the interaction of the wakes from the contra-rotating upstream rotor blades with the downstream rotor. The analysis is very similar to that n
Tel.: +44 2380592946. E-mail address: [email protected]
0022-460X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jsv.2013.03.014
M.J. Kingan / Journal of Sound and Vibration 332 (2013) 3956–3970
3957
Fig. 1. (a) Test of an isolated open rotor rig, (b) test of an open rotor rig with upstream pylon. Pictures courtesy of Rolls-Royce Plc.
presented in Blandeau and Joseph [3], however it includes the effect of axial flow and corrects a number of minor errors in their final result1 . The special case of the interaction of a pylon wake with a downstream rotor is then analysed by considering the interaction of the wake from a stationary, one-bladed upstream rotor with a downstream rotor. 2.1. Problem definition A cylindrical coordinate system with axial, radial and tangential coordinates {x, r, ϕ} will be utilised in the analysis. The upstream and downstream rotors both rotate about the x-axis in the negative and positive ϕ directions respectively. Each rotor has blade count, B, and rotational speed, Ω rad/s. Note that upstream and downstream rotor parameters are denoted by the subscripts 1 and 2 respectively. The formulation presented in this paper can be used to calculate the sound pressure spectrum produced by a stationary open rotor situated in a uniform flow of velocity Ux and corresponding Mach number Mx in the positive x-direction. The observer is assumed to be stationary and located within the flow. This sound pressure spectrum is equivalent to that which would be measured using an inflow microphone mounted adjacent to an open rotor situated in a wind tunnel. Alternatively, this formulation could also be used to calculate the pressure spectrum produced by an open rotor translating through a stationary medium with the pressure being measured using a microphone which translates with the open rotor. 2.2. Acoustic formulation The acoustic pressure, p, at observer location xo and time t due to the force per unit area, f, exerted on the air at point x and time τ by the downstream rotor is given by ([4] Eq. 4.13) Z Te Z pðxo ; tÞ ¼ ∇Gðx; τjxo ; tÞ⋅fðx; τÞdSðxÞdτ; (1) −T e
SB ðτÞ
where −Te≤τ≤Te is the interval of time over which sound is emitted and which will be allowed to become infinitely large, SB is a closed surface corresponding to the blade surfaces and G is a free-field Green's function which is defined as [5] Gðx; τjxo ; tÞ ¼
δðt−τ−so =c0 Þ ; 4πSo
(2)
where, in the farfield So Re ð1−M x cosθe Þ; so Re þ
(3)
xcosθe rsinθe cosðϕ−ϕo Þ − : 1−M x cosθe 1−M x cosθe
(4)
Note that the observer position has been expressed in terms of Re and θe which are the observer emission radius and polar angle and are related to the physical observer radius and polar angle by the following relationships (see for example [6]): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 xo;e ¼ x −M xo 2 þ ð1−M 2x Þðyo 2 þ zo 2 Þ ; yo;e ¼ yo ; zo;e ¼ zo ; o x 1−M 2x xo;e ¼ −Re cosθe ;
yo;e ¼ Re sinθe cosϕo ;
zo;e ¼ Re sinθe sinϕo :
1 The errors in Ref. [3] are listed in this footnote. (1) The effect of wake skew on the ‘mean wake profile’ was not properly accounted for (see Eqs. (38 and 39) of this paper). (2) The final formulation contained an extra factor of 2π due to an inconsistent Fourier transform convention. (3) There was an error in the wavenumber contained in the blade response function g (see Appendix 1 of this paper). (4) There were a number of errors in the ‘acoustically weighted lift function’, ΨL, which is defined in Appendix 1 of this paper.
3958
M.J. Kingan / Journal of Sound and Vibration 332 (2013) 3956–3970
Fig. 2. Propeller geometry (a), local blade geometry (b).
Making use of the properties of the delta function, the Green's function can be written Z ∞ 1 expfiωðt−τ−so =c0 Þgdω: Gðx; τjxo ; tÞ 2 8π Re ð1−Mx cos θÞ −∞ Substituting Eq. (4) into Eq. (5) and making use of the Jacobi–Anger expansion n ∞ π o ; exp izcosðϕ−ϕo Þ ¼ ∑ J n ðzÞexp in ϕ−ϕo þ 2 n ¼ −∞
(5)
(6)
yields
∞
Z
∑
n ¼ −∞
∞ −∞
Jn
1 Gðx; τjxo ; tÞ 8π2 Re ð1−M x cos θÞ
κrsinθe π κxcosθe exp iωðt−τÞ−iκRe þ in ϕ−ϕo þ −i dω; 2 1−M x cosθe 1−M x cosθe
(7)
where κ ¼ω/c0. The interaction of the upstream rotor blade wakes with a downstream rotor blade will induce an unsteady lift force on the blade. Provided that the blade is relatively thin and reasonably flat, the integration over the closed blade surface in ðmÞ Eq. (1) can be approximated by an integration over the blade planform area on which Y ðmÞ 2 ¼ 0 for −c2 =2≤X 2 ≤c2 =2, Rh≤r≤Rt, ðmÞ where Rh is the hub radius, Rt is the blade tip radius, c2 is the downstream blade chord-length and fX ðmÞ ; Y 2 2 g are chordwise/ chord-normal coordinates of the mth blade on the downstream rotor which are shown in Fig. 2. The loading exerted by the blade on the air must also be replaced by the vector sum of the loading exerted by the upper and lower blade surfaces, which from inspection of Fig. 2 can be written ^ cosα2 ΔpðmÞ ðX ðmÞ ; r; τÞ; fðx; τÞ-½x^ sinα2 þ ϕ 2
(8)
where α2 and Δp(m) are, respectively, the stagger angle and pressure jump across the mth downstream rotor blade. Note that ^ are unit vectors in the axial and tangential directions. x^ and ϕ A point on the mth downstream rotor blade is located at ϕ ¼ Ω2 τ−X ðmÞ 2
sinα2 2π þ m; B2 r
x ¼ X ðmÞ 2 cosα2 : Substituting Eqs. (8)–(10) into Eq. (1) and summing over all B2 blades on the downstream rotor yields Z ∞ ~ o ; ωÞexpfiωtgdω; pðxo ; tÞ ¼ pðx
(9) (10)
(11)
−∞
where ~ o ; ωÞ ¼ pðx
Z Rt Z 1 B2 −1 ∞ −i κr sinθ e Δp~ ðmÞ ðX ðmÞ ∑ ∑ 2 ; r; ω−nΩ2 ÞJ n 1−M x cosθe S 4πRe ð1−M x cosθe Þ n ¼ −∞ m ¼ 0 Rh −1
π 2π ðmÞ ðmÞ −ϕo þ m −ikc X 2 exp −iκRe þ in dX 2 dr; 2 B2
(12)
M.J. Kingan / Journal of Sound and Vibration 332 (2013) 3956–3970
3959
ðmÞ
and X 2 ¼ 2X ðmÞ 2 =c2 is a dimensionless chordwise coordinate, Z Te 1 ΔpðmÞ ðX ðmÞ Δp~ ðmÞ ðX ðmÞ 2 ; r; ωÞ ¼ 2 ; r; τÞexpf−iωτgdτ; 2π −T e
(13)
S¼
κcosθe sinα2 n c2 ; − cosα2 1−M x cosθe r 2
(14)
kc ¼
κcosθe cosα2 n c2 : þ sinα2 r 1−M x cosθe 2
(15)
An instantaneous pressure spectrum can be defined Z ∞ 1 Spp ðxo ; ω; tÞ ¼ Rpp ðxo ; t 1 ; tÞexpf−iωt 1 gdt 1 ; 2π −∞
(16)
where Rpp is a non-stationary correlation function Rpp ðxo ; t 1 ; tÞ ¼ E½pðxo ; tÞpðxo ; t þ t 1 Þ:
(17)
Averaging the instantaneous spectrum over the time period −T≤t≤T (where T-∞), yields a time-averaged spectrum level Z T 1 Spp ðxo ; ωÞ ¼ lim Spp ðxo ; ω; tÞdt: (18) T-∞ 2T −T Substituting Eq. (11) into Eq. (17) and the result into Eq. (16) and then Eq. (18) yields Z TZ ∞ 1 ~ o ; ωÞexpfiðω−ω′Þtgdω′dt; Spp ðxo ; ωÞ ¼ lim E½p~ n ðxo ; ω′Þpðx T-∞ 2T −T −∞
(19)
which simplifies to Spp ðxo ; ωÞ ¼ lim
π n ~ o ; ωÞ : E p~ ðxo ; ωÞpðx
T-∞ T
Making use of Eq. (12) yields
B2 −1 B2 −1 ~ o ; ωÞ ¼ ∑ ∑ E p~ n ðxo ; ω′Þpðx
∞
∑
n o exp iðκ′−κÞRe þ iðn′−nÞ ϕo − 2π þ i 2π B2 ðnm−n′m′Þ
∞
∑
ð4πRe Þ2 ð1−M x cosθe Þ2
κ′r′sinθe κrsinθe S′SJ n′ Jn 1−M x cosθe 1−M x cosθe
m ¼ 0m′ ¼ 0n ¼ −∞n′ ¼ −∞
Z Z
1
−1
Z
Rt Rh
1
−1
Z
Rt Rh
′
ðm0 Þ
ðmÞ
ðm′Þ
ðmÞ
~ ðmÞ ðX ðmÞ E½Δp~ ðm′Þ ðX ðm′Þ 2 ′; r′; ω′−n′Ω2 ÞΔp 2 ; r; ω−nΩ2 Þexpfikc X 2 ′−ikc X 2 gdX 2 ′dX 2 dr′dr;
(20)
where
~ ðmÞ ðX ðmÞ Δp~ ðm′Þ ðX ðm′Þ 2 ′; r′; ω′−n′Ω2 ÞΔp 2 ; r; ω−nΩ2 Þ;
(21)
is the only term which is, as yet, undetermined. Note that in Eq. (20), the integration over r′ can be truncated at r′¼r 7Δr/2 where Δr is somewhat larger than the correlation length of the unsteady loading on the downstream rotor blades. Eq. (20) becomes n o exp iðκ′−κÞRe þ iðn′−nÞ ϕo − 2π þ i 2π ∞
n B2 −1 B2 −1 ∞ B2 ðnm−n′m′Þ ~ o ; ωÞ ¼ ∑ ∑ ∑ E p~ ðxo ; ω′Þpðx ∑ ð4πRe Þ2 ð1−M x cosθe Þ2 m ¼ 0m0 ¼ 0n ¼ −∞n0 ¼ −∞
Z rþΔr Z Rt κ′rsinθe κrsinθe S′SJ n′ Jn 1−M x cosθe 1−M x cosθe Rh r−Δr Z 1Z 1 ðm′Þ ðmÞ ðm′Þ ðmÞ ′ ~ ðmÞ ðX ðmÞ E½Δp~ ðmÞn ðX ðm′Þ (22) 2 ′; r′; ω′−n′Ω2 ÞΔp 2 ; r; ω−nΩ2 Þexpfikc X 2 ′−ikc X 2 gdX 2 ′dX 2 dr′dr: −1
−1
′
It has also been assumed that the variation of r′ in S′, kc and the argument of the Bessel function Jn′ can be neglected. This may be an acceptable assumption provided that Δr is not too large. This is equivalent to assuming that all sound generated within a ‘strip’ radiates from the same radius and that the rotor geometry and mean flow parameters do not vary significantly over this strip. Although this crude assumption is used in the method presented in this paper, it should be possible to extend the method so that radial non-compactness effects are correctly taken into account by using Eq. (20) rather than Eq. (22).
3960
M.J. Kingan / Journal of Sound and Vibration 332 (2013) 3956–3970
2.3. Aerodynamic formulation In the previous section, an expression was derived for calculating the far-field pressure spectral density produced by the interaction of the turbulent component of the front rotor wakes with the downstream rotor blades. The only unknown quantity in this expression is given by Eq. (21). This section is devoted to deriving a set of equations for estimating this quantity. The derivation makes use of ‘strip theory’ which assumes that aerodynamic interactions along a radial ‘strip’ of the rotor blade and centred at a particular radius can be approximated by considering an equivalent two-dimensional problem. Such an approach is common in turbo-machinery noise analysis and has been used in many previous studies. For example, the approach was used by Blandeau and Joseph [3], to calculate open rotor broadband interaction noise, Shlinker and Amiet [7] and Blandeau and Joseph [8] for calculating rotor self-noise and Ventres et al. [9] and Nallasamy et al. [10] to calculate turbo-fan rotor–stator interaction noise. The method involves formulating an equivalent two-dimensional problem by ‘unwrapping’ the rotor at radius, r, as shown in Fig. 3 below. Note that for each blade row, two coordinate systems will be utilised. A chordwise/chord-normal coordinate system {X, Y} and a ‘blade-locked’ axial/tangential coordinate system {x, y}. Note that the blades are assumed to be aligned with the local flow-direction and thus the stagger angle of each blade, α, is defined by tan α ¼ Ωr=U x . The effect of blade sweep and lean is neglected in this analysis. Also, a subscript 1 or 2 is respectively used to denote the upstream and downstream rotors, whereas a superscript (m) is used to denote that the origin of that coordinate system is at the mid-chord of the mth downstream rotor blade. The incident flow field and unsteady loading on the downstream rotor blades is then calculated as if the interaction was taking place between the wakes from an upstream cascade of two-dimensional rotor blades with a cascade of downstream rotor blades. It is assumed that it will be possible to express the fluctuating component of velocity normal to the mth downstream rotor blade in terms of its spatio-temporal Fourier transform Z ∞Z ∞Z ∞Z ∞ ~ u¼ (23) uðk; ωÞexpfiωτ−ik⋅xðmÞ 2 gdkdω −∞
−∞
−∞
−∞
~ The pressure jump due to a single Fourier component of this upwash, uðk; ωÞexpfiωτ−ik⋅xðmÞ 2 g, can be estimated using two-dimensional strip theory [11] which yields ðmÞ ~ ΔpðmÞ ðX ðmÞ 2 ; r; τÞ ¼ 2πρU r 2 uðk; ωÞgðX 2 ; k; ωÞexpfiωτ−ikr rg;
(24)
gðX ðmÞ 2 ; k; ωÞ
where is a ‘blade response function’ which will be defined later. The total pressure jump on the downstream rotor blade is thus given by Z ∞Z ∞Z ∞Z ∞ ~ uðk; ωÞgðX ðmÞ (25) ΔpðmÞ ðX ðmÞ 2 ; r; τÞ ¼ 2πρU r2 2 ; k; ωÞexpfiωτ−ikr rgdkdω: −∞
−∞
−∞
−∞
Taking the Fourier transform of this expression and substituting the result into Eq. (21) yields
~ ðmÞ ðX ðmÞ E½Δp~ ðm′Þ ðX ðm′Þ 2 ′; r′; ω′−n′Ω2 ÞΔp 2 ; r; ω−nΩ2 Þ Z ∞Z ∞Z ∞Z ∞Z ∞Z ∞ 2 ~ ω−nΩ2 Þ ¼ ð2πρU r2 Þ E½u~ n ðk′; ω′−n′Ω2 Þuðk; −∞
−∞
−∞
−∞
−∞
−∞
′
ðmÞ g n ðX ðm′Þ 2 ′; k′; ω′−n′Ω2 ÞgðX 2 ; k; ω−nΩ2 Þexpfikr r′−ikr rgdk′dk:
Fig. 3. ‘Unwrapped’ blade geometry and coordinate systems.
(26)
M.J. Kingan / Journal of Sound and Vibration 332 (2013) 3956–3970
3961
The only unknown remaining is the term in the expected value brackets in the integrand of Eq. (26) which is defined by Eq. (27)
1 RRRRRRRR ðmÞ ~ E u~ n ðk′; ω′−n′Ω2 Þuðk; ω−nΩ2 Þ ≡ ð2πÞ E½un ðxðm′Þ 8 2 ′; τ′Þuðx2 ; τÞ (27) ðm′Þ ðmÞ ðm′Þ expfiðω′−n′Ω2 Þτ′−iðω−nΩ2 Þτ þ ik⋅xðmÞ 2 −ik′⋅x 2 ′gdx 2 dx2 ′dτdτ′: Note that the limits of integration in Eq. (27) are allowed to approach 7∞. It will be convenient to convert to a coordinate system rotating with the front rotor (and with its origin located at the mid-chord of the reference blade on the front rotor). By inspection of Fig. 3 we have xðmÞ 2 ¼ x1 −g r ; yðmÞ 2 ¼ y1 −ðΩ1 þ Ω2 Þrτ−
(28) 2πr m: B2
Substituting Eqs. (28) and (29) into Eq. (27) yields
1 RRRRRRRR ~ ω−nΩ2 Þ ≡ ð2πÞ E½un ðx′1 ; τ′Þuðx1 ; τÞ E u~ n ðk′; ω′−n′Ω2 Þuðk; 8 8 9 < −iðω−nΩ2 Þτ−iky ðΩ1 þ Ω2 Þrτ þ iðω′−n′Ω2 Þτ′ þ ik′y ðΩ1 þ Ω2 Þrτ′ þ ikr r−ik′r r′ = dx dx′ dτdτ′: exp 2πr : −ik′x x′1 þ ikx x1 −ikx g r þ ik′x g r þ iky y1 −ik′y y′1 þ ik′y 2πr ; 1 1 B2 m′−iky B2 m
(29)
(30)
Note that in Eq. (30) the list of dependent variables associated with u has changed. Providing that the rotor wakes are well separated when they impinge on the down stream rotor, it is convenient to assume that the upwash has the following form: uðx1 ; τÞ ¼ f ðx1 Þwðx1 −U1 τÞ;
(31)
where f essentially represents the (normalised) distribution of the rms turbulence intensity in the wake of the front rotor and w is a random function which corresponds to the turbulent velocity produced by frozen, stationary, isotropic and homogeneous turbulence with an rms velocity equal to that at the centerline of each wake. U1 is the air velocity relative to the front blade row and thus has components in the axial, tangential and radial directions, U x1 ¼ U x , U y1 ¼ Ω1 r and Ur ¼ 0 respectively. Note that swirl and induced axial velocity are neglected which is a common assumption for rotors with low solidity. Therefore we have n
E½un ðx′1 ; τ′Þuðx1 ; τÞ ¼ f ðx′1 Þf ðx1 ÞE½wn ðx′1 −U1 τ′Þwðx1 −U1 τÞ:
(32)
The function f is periodic in the y1 direction (see Fig. 3) and can thus be expressed using Poisson's summation formula
∞ 2πr B1 ∞ ~ kB1 kB1 ; r exp i y1 ; k; r ¼ f ðx1 Þ ¼ ∑ f 0 x1 ; y1 þ (33) ∑ f 0 x1 ; B1 r k ¼ −∞ r r k ¼ −∞ where f0 is the normalised distribution of the rms turbulence intensity in the wake of one rotor blade and f~ 0 is defined by
Z ∞ kB1 1 kB1 ;r ¼ y1 dy1 : f 0 ðx1 ; y1 ; rÞexp −i (34) f~ 0 x1 ; 2π −∞ r r In this study, f0 is defined by the following Gaussian profile which corresponds to that of the mean streamwise velocity deficit in the wake of a two-dimensional aerofoil (from Parry [12] and [13]) (
) Y1 2 f 0 ðx1 ; y1 ; rÞ ¼ exp −lnð2Þ ; (35) b1=2 b1=2 ¼
1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C D1 c1 Lw : 4
(36)
Note that Y 1 ¼ y1 cosα1 −x1 sinα1 is the chord-normal coordinate of the front rotor reference blade and b1/2 is the ‘halfwidth’ of the wake produced by the front rotor reference blade. C D1 ðrÞ and c1(r) are the front rotor blades drag coefficient and chord length, while Lw(X1,r) is the helical length of the wake. The wake develops relatively slowly and therefore Lw(X1,r) may be replaced by its value at the leading edge of the downstream rotor blade Lw ðrÞ f evaluates as pffiffiffi ∞ π B1 b1=2 kB1 kB1 y1 −i tanα1 x1 ; f ðx1 Þ ¼ pffiffiffiffiffiffiffiffiffiffiffi (37) ∑ Gðkb Þexp i r r lnð2Þ2πrcosα1 k ¼ −∞ where
(
) 2 kb Gðkb Þ ¼ exp − ; 4lnð2Þ
(38)
3962
M.J. Kingan / Journal of Sound and Vibration 332 (2013) 3956–3970
and kb ¼
kB1 b1=2 : rcosα1
(39)
Now, E½wn ðx′1 −U1 τ′Þwðx1 −U1 τÞ is effectively a two-point turbulent velocity correlation, which will be denoted Rww(η) where η ¼ x1 −x′1 −U1 ðτ−τ′Þ. If the unit vectors in the axial, tangential and radial directions are respectively denoted e^ x , e^ y and e^ r , then η ¼ e^ x ηx þ e^ y ηy þ e^ r ηr where ηx ¼ x1 −x′1 −U x ðτ−τ′Þ;
ηy ¼ y1 −y′1 −Ω1 rðτ−τ′Þ;
ηr ¼ r−r′:
(40)
The two-point velocity correlation Rww(η) is periodic in ηy, with period 2πr, and can be rewritten using Poisson's summation formula Z ∞ n q o ∞ 1 ∑ Rww;0 ðe^ x ηx þ e^ y ξ þ e^ r ηr Þexp i ðξ−ηy Þ dξ; (41) Rww ðηÞ ¼ 2πr q ¼ −∞ −∞ r where Rww,0(η) is the two-point correlation of w between −πroηy oπr. Substituting Eqs. (32), (37) and (41) into Eq. (30) yields (after some manipulation) ∞ ∞
π B1 b1=2 2 ∞ ′ ~ ω−nΩ2 Þ ¼ U x rlnð2Þ ∑ ∑ ∑ Gðkb ÞGðkb Þ E u~ n ðk′; ω′−n′Ω2 Þuðk; 2πrcosα1 k ¼ −∞k′ ¼ −∞q ¼ −∞
Ω1 q 2πr ′ 1 ′ Φww kx −kB1 ðky m′−ky mÞ δ ky − q−kB1 e^ x þ e^ y þ kr e^ r exp iðkx −kx Þg r þ i r B2 r Ux
ω−nΩ2 þ ky Ω2 r B1 Ω1 B1 ′ ′ ′ δ kx − kx − ðk−k′Þ δ ky − ky − ðk′−kÞ δðkr −kr Þδ kx − Ux Ux r ′
δðω′−½n′Ω2 þ ky Ω1 r−ky ðΩ1 þ Ω2 Þr þ kx U x Þ; where Φww ðkÞ ¼
Z
1 3
ð2πÞ
∞ −∞
Z
∞ −∞
Z
∞
−∞
(42)
Rww;0 ðηÞexpfik⋅ηgdη;
is a turbulent velocity spectrum. Substituting this result into Eq. (26) yields ρU B b 2 ∞ h i r 2 1 1=2 π ~ ðmÞ ðX ðmÞ E Δp~ ðm′Þ ðX ðm′Þ ∑ 2 ′; r′; ω′−n′Ω2 ÞΔp 2 ; r; ω−nΩ2 Þ ¼ rU x lnð2Þ rcosα1
(43)
∞
∑
∞
′
∑ Gðkb ÞGðkb Þ
k ¼ −∞k′ ¼ −∞q ¼ −∞
Ω1 2π δðω− ω′ þ ðn−n′ÞΩ2 þ ðk−k′ÞB1 ðΩ1 þ Ω2 Þ Þexp −iB1 ðk−k′Þg r þ i ðq−k′B1 Þm′−ðq−kB1 Þm B2 Ux Z ∞ ðmÞ Φww ðK X 2 e^ X 2 þ K Y 2 e^ Y 2 þ kr e^ r Þg n ðX ðm′Þ 2 ′; K X 2 ′; kr ; ω′−n′Ω2 ÞgðX 2 ; K X 2 ; kr ; ω−nΩ2 Þexpf−ikr ðr−r′Þgdkr ;
(44)
−∞
where e^ X 2 and e^ Y 2 are, respectively, unit vectors which are aligned with the chordwise and chord-normal directions of the downstream blade row ω−nΩ2 ; U r2
(45)
ω−nΩ2 þ ðk′−kÞB1 ðΩ1 þ Ω2 Þ ; U r2
(46)
ω−nΩ2 −kB1 ðΩ1 þ Ω2 Þ ; U r2
(47)
K X2 ¼ K X2 ′ ¼
K X2 ¼ and
K Y 2 ¼ K X 2 tanα2 þ
q : rcosα2
Note that the list of dependent variables associated with the blade response function g has changed. 2.4. Broadband noise Substituting Eq. (44) into Eq. (22) and then the result into Eq. (19) yields ∞ ∞ ∞ ∞ ∞ exp iðκ′−κÞRe þ iðn′−nÞ ϕo − 2π Spp ðxo ; ωÞ ¼ ∑ ∑ ∑ ∑ ∑ I t I m I m′ ð4πRe Þ2 ð1−M x cosθe Þ2 n ¼ −∞n′ ¼ −∞k ¼ −∞k′ ¼ −∞q ¼ −∞
(48)
M.J. Kingan / Journal of Sound and Vibration 332 (2013) 3956–3970
Z Z
Rt
Rh ∞ −∞
π rU x lnð2Þ
Φww ðK X 2 e^ X 2 þ K Y 2 e^ Y 2
3963
2
ρU r2 B1 b1=2 Ω1 κ′rsinθe κrsinθe ′ Gðkb ÞGðkb Þexp iB1 ðk′−kÞg r S′SJ n′ Jn rcosα1 Ux 1−M x cosθe 1−M x cosθe Z rþΔr=2 ′ n þ kr e^ r Þ expf−ikr ðr−r′Þgdr′Ψ L ðK X 2 ′; kr ; ω′−n′Ω2 ; kc ÞΨ L ðK X 2 ; kr ; ω−nΩ2 ; kc Þdkr dr;
(49)
r−Δr=2
where ω′¼ω−(n−n′)Ω2−(k−k′)B1(Ω1+Ω2), 1 T-∞ 2T
I t ¼ lim
Z
T
−T
expfi½ðn−n′ÞΩ2 þ ðk−k′ÞB1 ðΩ1 þ Ω2 Þtgdt;
(50)
B2 −1 2πm ðq−kB1 −nÞ ; I m ¼ ∑ exp −i B2 m¼0
(51)
2πm′ ðq−k′B1 −n′Þ ; ∑ exp i B2 m0 ¼ 0
(52)
I m′ ¼
B2 −1
and Z Ψ L ðK X 2 ; kr ; ω; kc Þ ¼
1
−1
ðmÞ
ðmÞ
gðX ðmÞ 2 ; K X 2 ; kr ; ωÞexpf−ikc X 2 gdX 2
(53)
is an ‘acoustically weighted lift function’ which is independent of the summation index m. Now, following Blandeau et al. [3], assuming that Δr is sufficiently large, the integration over r′ can be approximated as follows: Z rþΔr=2 expf−ikr ðr−r′Þgdr′≈2πδðkr Þ: (54) r−Δr=2
This yields the following expression for the time averaged pressure spectral density: ∞ ∞ ∞ ∞ ∞ exp iðκ′−κÞRe þ iðn′−nÞ ϕo − 2π Spp ðxo ; ωÞ ¼ ∑ ∑ ∑ ∑ ∑ 2πI t I m I m′ ð4πRe Þ2 ð1−M x cosθe Þ2 n ¼ −∞n′ ¼ −∞k ¼ −∞k′ ¼ −∞q ¼ −∞
Z Rt 2 ρU r2 B1 b1=2 π Ω1 κ′rsinθe κrsinθe ′ Gðkb ÞGðkb Þexp iB1 ðk′−kÞg r S′SJ n′ Jn rU x lnð2Þ rcosα1 Ux 1−M x cosθe 1−M x cosθe Rh ′
Φww ðK X 2 e^ X 2 þ K Y 2 e^ Y 2 ÞΨ L n ðK ′X 2 ; 0; ω′−n′Ω2 ; kc ÞΨ L ðK X 2 ; 0; ω−nΩ2 ; kc Þdr
(55)
3. Rotor–rotor interaction noise For rotor–rotor interaction noise, provided that the ratio of the rotor shaft speeds is an irrational number, then it is relatively straightforward to show that It ¼δk,k’δn,n′. Substitution of this result into Eq. (55) yields Im ¼Im′ ¼ B2 and the summation over q is replaced by a summation over j where q ¼jB2+kB1+n. Thus, the equation for the time-averaged rotor– rotor broadband pressure spectrum simplifies to
2
Z Rt ∞ ∞ ∞ ρU r2 B1 b1=2 2πB2 2 π κrsinθe Spp ðxo ; ωÞ ¼ ∑ ∑ ∑ G2 ðkb ÞS2 J n 2 2 2 rU x lnð2Þ rcosα1 1−M x cosθe n ¼ −∞k ¼ −∞j ¼ −∞ ð4πRe Þ ð1−M x cosθ e Þ Rh jB2 þ kB1 þ n (56) Φww K X 2 e^ X 2 þ K X 2 tanα2 þ e^ Y 2 Ψ L ðK X 2 ; 0; ω−nΩ2 ; kc Þ2 dr; rcosα2 where G(kb) is given by Eqs. (38) and (39), S is given by Eq. (14), K X 2 , is defined by Eq. (47), K X 2 , is defined by Eq. (45) and kc, is defined by Eq. (15). The acoustically weighted lift functions ΨL are defined in Appendix 1 and a von Karman turbulence spectrum will be utilised which is defined by 2
2
ðK −K Y 2 Þ 55 Γð5=6Þ w2rms 1 Φww ðK X 2 e^ X 2 þ K Y 2 e^ Y 2 þ K r e^ r Þ ¼ pffiffiffi 9 π Γð1=3Þ k5e 4π ð1 þ ðK=ke Þ2 Þ17=6 ; 2
2
2
2
K ¼ K X2 þ K Y 2 þ K r ; ke ¼
pffiffiffi π Γð5=6Þ : Λ1 Γð1=3Þ
(57)
(58) (59)
Note that Λ1 is the integral lengthscale of the front rotor wake turbulence which is estimated using the expression given by Jurdic and Joseph [9] Λ1 0.42b1/2. Also, wrms 0.404udef is estimated using an empirical expression due to Blandeau [14]
3964
M.J. Kingan / Journal of Sound and Vibration 332 (2013) 3956–3970
who derived this relationship by analysing the experimental data presented in Wygnanski et al. [15] (N.B. it has been assumed that the helical path-length of the front rotor wake is sufficiently large for these relationships to be valid). udef is the wake centerline velocity deficit which is calculated using Eq. (65) below which relies on the front rotor drag coefficient C D1 rffiffiffiffiffiffiffiffiffiffiffi
lnð2Þ c1 C D1 0:5 udef ¼ 2U r1 (60) π Lw Note that the summation index j only appears in the turbulence spectrum, i.e. Eq. (56) contains a factor
∞ kB1 þ n B2 þj e^ Y 2 : ∑ Φww K X 2 e^ X 2 þ K X 2 tanα2 þ rcosα2 rcosα2 j ¼ −∞
(61)
Recalling that Φww is defined as the spatial Fourier transform of the two-point correlation function, Eq. (61) can be rewritten Z ∞Z ∞Z ∞ ∞ 1 ∑ Rww;0 ðηX 2 e^ X 2 þ ηY 2 e^ Y 2 þ ηr e^ r Þ 3 −∞ −∞ −∞ j ¼ −∞ ð2πÞ kB1 þ n B2 j ηY 2 þ i ηY 2 dηX 2 dηY 2 dηr : exp iK X 2 ηX 2 þ iK X 2 tanα2 ηY 2 þ i rcosα2 rcosα2
(62)
Noting that
∞ B2 j rcosα2 ∞ rcosα2 ; ηY 2 ¼ 2π ∑ δ ηY 2 −2πj ∑ exp i rcosα2 B2 j ¼ −∞ B2 j ¼ −∞ Eq. (61) reduces to the following expression: Z ∞Z ∞ rcosα2 ∞ 1 2 ^ ^ Rww;0 ηX 2 e^ X 2 þ 2πj rcosα ∑ B2 eY 2 þ ηr er 2 B2 j ¼ −∞ ð2πÞ −∞ −∞ kB1 þ n 2πrcosα2 exp iK X 2 ηX 2 þ ij K X 2 tanα2 þ dηX 2 dηr rcosα2 B2
(63)
If the distance 2πrcosα2 =B2 is much greater than the correlation length of the wake turbulence, then only the j¼0 term contributes to the summation and the factor becomes Z ∞Z ∞ rcosα2 1 rcosα2 ð2Þ Rww;0 ðηX 2 e^ X 2 þ ηr e^ r ÞexpfiK X 2 ηX 2 gdηX 2 dηr ¼ Φww ðK X 2 e^ X 2 Þ; (64) B2 ð2πÞ2 −∞ −∞ B2 ^ X 2 þ K r e^ r Þ is the two-wavenumber turbulent velocity spectrum which is defined by where Φð2Þ ww ðK X 2 e ð2Þ Φww ðK X 2 e^ X 2 þ K r e^ r Þ ¼
ðK X 2 =ke Þ2 þ ðK r =ke Þ2 4 w2rms : 9π k2 ð1 þ ðK X 2 =ke Þ2 þ ðK r =ke Þ2 Þ7=3 e
(65)
Fig. 4. Effect of rotor–rotor spacing on rotor–rotor broadband interaction noise for a representative take-off case. Ten decibel per vertical gridline.
M.J. Kingan / Journal of Sound and Vibration 332 (2013) 3956–3970
Thus, provided that there is no blade–blade correlation, the time-averaged pressure spectral density is given by
2
Z Rt ∞ ∞ ρB1 b1=2 2πB2 πU r2 κrsinθe 2 2 2 Spp ðxo ; ωÞ ¼ ∑ ∑ G ðk ÞS J b n 2 2 lnð2Þ rcosα1 1−Mx cosθe n ¼ −∞ k ¼ −∞ ð4πRe Þ ð1−M x cosθ e Þ Rh ð2Þ 2 Φww ðK X 2 e^ X 2 ÞjΨ L ðK X 2 ; 0; ω−nΩ2 ; kc Þj dr:
3965
(66)
A similar form of this simplified expression was originally presented by Blandeau [14]. A realistic 1/6th scale open rotor configuration, operating at a representative take-off condition, was used to generate all of the noise predictions which are presented in this paper. There are 12 blades on the front rotor and 9 on the rear. The axial flow Mach number was 0.23 and the front and rear blade-tip Mach numbers were 0.63 and 0.47 respectively. The aerodynamic and geometric data for each rotor was supplied by Rolls-Royce plc. Eq. (66) was used to calculate the effect of increasing rotor–rotor spacing on broadband interaction noise for the open rotor operating at a representative take-off condition. The results of this calculation are shown in Fig. 4 which plots the onethird octave band sound power level spectrum. The calculations were performed for 6 different axial distances between the front and rear blade mid-chord at the rotor hub (gr in Fig. 3). As originally shown by Blandeau [14], as rotor–rotor spacing is increased, the front rotor wake width increases while the centerline velocity deficit decreases. The increase in wake width causes the turbulence correlation length within the wake to increase which produces higher levels of low frequency noise and moves the spectral peak towards lower frequencies. Also, as rotor–rotor separation is increased, there is a reduction in w2rms which should tend to reduce the level of broadband noise produced. This effect does not however appear to significantly influence the calculated noise levels. 4. Pylon–rotor interaction noise For Pylon–rotor interaction noise, only the interaction of the wake with the front rotor is considered. The rotor–rotor interaction noise model developed in the previous sections is applied with the pylon modelled as a stationary, one-bladed upstream rotor and noise being produced solely by the downstream (front) open rotor. Ricouard et al. [16] show that the tone noise produced by the interaction of a pylon wake with a contra-rotating open rotor is dominated by interaction noise from the front rotor. Thus it is possibly reasonable to assume that the same will be true for broadband interaction noise. For pylon–rotor interaction noise, B1 ¼1 and Ω1 ¼0 and therefore Z T 1 I t ¼ lim expfi½n−n′ þ k−k′Ω2 tgdt (67) T-∞ 2T −T Thus, It ¼δn′,n+k−k′. This yields the following expression for the radiated pressure spectral density:
2 Z Rt ∞ ∞ ρb1=2 2πB22 πU r2 Spp ðxo ; ωÞ ¼ ∑ ∑ Φww ðK X 2 e^ X 2 þ K Y 2 e^ Y 2 ÞjAn j2 dr; r ð4πRe Þ2 ð1−M x cosθe Þ2 n ¼ −∞ j ¼ −∞ Rh rcosα2 lnð2Þ where An ¼
(68)
n ∞ π o κrsinθe Gðkb ÞSJ n−k Ψ L ðK X 2 ; 0; ω−ðn−kÞΩ2 ; kc Þ ∑ exp ik ϕo − 2 1−M x cosθe k ¼ −∞
(69)
ω þ ðk−nÞΩ2 ; U r2
(70)
ω−nΩ2 ; U r2
(71)
K X2 ¼
K X2 ¼
K Y 2 ¼ K X 2 tanα2 þ
n þ jB2 ; rcosα2
(72)
S¼
κcosθe sinα2 ðk−nÞ c2 cosα2 ; þ r 1−M x cosθe 2
(73)
kc ¼
κcosθe cosα2 ðk−nÞ c2 sinα2 ; − r 1−Mx cosθe 2
(74)
(
) 2 kb Gðkb Þ ¼ exp − ; 4lnð2Þ kb ¼
kb1=2 : r
(75)
(76)
3966
M.J. Kingan / Journal of Sound and Vibration 332 (2013) 3956–3970
Assuming no blade to blade correlation, the expression can be simplified as before to yield
2 Z Rt ∞ ρb1=2 2πB2 πU r2 ð2Þ Spp ðxo ; ωÞ ¼ Φww ðK X 2 e^ X 2 ÞjAn j2 dr: ∑ r ð4πRe Þ2 ð1−M x cosθe Þ2 n ¼ −∞ Rh lnð2Þ
(77)
Note that the blade response function described in Appendix 1 is a piecewise function which contains a switch between a low and high frequency approximation. Thus, sound generated at a particular frequency will contain contributions from both the low and high frequency blade loading components and in particular, calculated noise levels at low frequencies will contain a significant contribution from both the low- and high-frequency blade loading models. This switch point has been selected to minimise the discontinuity between the response calculated for both frequency regimes, whilst ensuring that each model is applied at frequencies where it is reasonably valid. However, the discontinuity in the blade response model does introduce discontinuities in the calculated noise spectrum at very low frequencies. For this reason, predictions of pylon wake-rotor broadband interaction noise which are presented in this paper are only for frequencies higher than (approximately) the first blade passing frequency of the front rotor which do not appear to be significantly affected by this discontinuity. A solution to this discontinuity problem would be to calculate the blade response using a more sophisticated method such as the numerical collocation technique used in the code LINSUB [17]. The sound pressure level produced by the pylon–rotor broadband interaction noise source is highly directional and varies both with polar emission angle θe and azimuthal angle ϕo. This highly directional sound field is due to several effects. One effect is caused by the different phasing of the sound generated by the thrust and torque components of the unsteady lift force as the polar emission angle changes. The contribution of the thrust component of loading can be calculated by setting the second term in the square brackets in Eq. (73) equal to zero. Similarly, the contribution of the torque component to the
Fig. 5. SPL at different azimuthal positions. ϕo ¼ 01 (solid black with+markers), ϕo ¼901 (solid black) and ϕo ¼ −901 (dashed). Torque loading noise (plots (a), (d) and (g)), thrust loading noise (plots (b), (e) and (h)), total noise (plots (c), (f) and (i)). θe ¼451 (plots (a)–(c)), θe ¼901 (plots (d)–(f)), θe ¼ 1351(plots (g)–(i)). Twenty decibel per vertical tick.
M.J. Kingan / Journal of Sound and Vibration 332 (2013) 3956–3970
3967
sound field can be determined by setting the first term in the square brackets of the same equation equal to zero. The same open rotor design which was described in the previous section and which was operating at the same representative ‘take-off’ condition, was used to generate the noise predictions which are presented in this section. However, in this configuration a realistic pylon was installed upstream of the front rotor. Only the noise produced by the interaction of the pylon wake with the front rotor is calculated. Fig. 5 plots the ‘1 Hz bandwidth sound pressure level’ versus frequency at a number of different observer positions. The three rows correspond respectively to polar emission angles θe ¼451(plots (a)–(c)), θe ¼901 (plots (d)–(f)) and θe ¼1351 (plots (g)–(i)). The left-hand column (plots (a), (d) and (g)) shows the predicted sound pressure level including only the torque component of loading, the middle column (plots (b), (e) and (h)) contains only the thrust component while the righthand column (plots (c), (f) and (i)) shows the total sound field containing both the thrust and torque components. Each plot contains three curves corresponding to an observer azimuthal angle of ϕo ¼01(solid black line with+markers), ϕo ¼ 901(solid black line) and ϕo ¼−901 (dashed line). Note that the pylon is located at ϕ¼01. Firstly, considering only the left-hand column which plots the contribution from the torque component of loading. At all three polar emission angles, the sound pressure level at ϕo ¼901 is significantly higher than that at ϕo ¼−901. Note that the blade is moving towards the observer position at ϕo ¼901 and away from the observer position at ϕo ¼ −901 when it passes through the pylon wake which occupies a narrow range of angles close to ϕ ¼01. For the torque component there is maximum radiation towards ϕo ¼901, slightly lower radiation towards ϕo ¼−901 and minimum radiation close to the ϕo ¼01 and ϕo ¼1801 directions. This radiation pattern is clearly observed in Fig. 6 which shows the azimuthal directivity of the thrust and loading components of SPL at 1 kHz, 5 kHz and 10 kHz and at a polar emission angle of 701. It is observerd that both the torque and thrust components of noise radiate with maximum level towards ϕo ¼901, with lower levels at ϕo ¼−901. It is hypothesised that the torque component has a dipole like radiation pattern due to the loading only occurring over a small range of azimuthal angles. This is not observed for the thrust component of loading as the thrust force acts in the same direction relative to the observer, regardless of the azimuthal location of the blade. Now considering the middle column of Fig. 5 which plots the contribution from the thrust component of loading. At θe ¼ 451 and θe ¼1351, the same trend is observed. There is maximum radiation in the direction of blade motion through the pylon wake (ϕo ¼901) and minimum radiation in the opposite direction (ϕo ¼−901). The radiation at ϕo ¼01 appears to be at a level somewhere between these two cases. Note that there is no thrust noise contribution at θe ¼901. Turning our attention now to the right hand column of Fig. 5 which corresponds to the total sound pressure level produced by the pylon–rotor broadband noise source. At θe ¼ 901 the thrust component of loading makes no contribution to the overall sound field and thus, the total field is dominated by the torque loading component. However, at θe ¼ 451
Fig. 6. Plot of Spp in decibels as a function of ϕo at θe ¼ 701 and at a frequency of 6kHz. Twenty decibel per radial gridline. (a) Torque loading, (b) thrust loading.
3968
M.J. Kingan / Journal of Sound and Vibration 332 (2013) 3956–3970
inspection of Eq. (73) shows that the thrust component has the opposite sign to that at θe ¼1351, while the torque component remains the same. Thus, there is a change in the phase between the thrust and torque components of loading as the observer moves from θe ¼451 to θe ¼1351. This change in sign of the torque component significantly alters the directivity of the total field in the forward and rearward polar arcs. Fig. 7 plots the sound pressure level produced by the pylon–rotor interaction broadband noise source at a particular observer location. The front rotor blade number is allowed to vary between 3 and 36. Note that only blade number is altered— no change to the blade geometry or aerodynamics are taken into account. Both the full (Eq. (68)) and simple (Eq. (77)) models are used to calculate the radiated sound pressure level. Note that the simple model is in excellent agreement with the full model at all frequencies for blade numbers less than, or equal to, 12 and thus the pressure spectral density scales in proportion to the number of blades on the front rotor. However, at higher blade numbers, the correlation of the unsteady loading on adjacent rotor blades means that the full model must be used and the level does not simply scale with blade number.
Fig. 7. Effect of blade number on radiated noise level. B1 ¼3, 6, 12, 18, 36. Simple model (solid line with markers), full model (solid line). Five decibel per vertical tick spacing.
Fig. 8. Effect of pylon–rotor gap on radiated noise level. gr ¼g0 [0.4, 0.5, 1, 5]. Simple model (solid line with markers), full model (solid black line). Five decibel per vertical tick spacing.
M.J. Kingan / Journal of Sound and Vibration 332 (2013) 3956–3970
3969
Fig. 8 plots the variation of sound pressure level at a particular observer location as the pylon–rotor gap is increased relative to the original pylon–rotor gap which is denoted g0. As before, both the simple and full models are used to predict the noise level. As the gap increases, the wake width increases and thus the correlation length of the turbulence incident on the front rotor increases. This shifts the spectral peak towards lower frequencies increasing low frequency noise but decreasing high frequency noise. Note that if the pylon–rotor gap becomes too large, then the correlation length of the turbulence within the wake causes the loading on adjacent blades to become correlated in which case the full model is required to calculate the radiated noise field. 5. Conclusion A theoretical model for the prediction of rotor–rotor interaction broadband noise has been developed and extended so that it can be applied to predict broadband noise produced by the interaction of the turbulent wake from an upstream pylon with an open rotor. A brief parameter study has been conducted which shows the effect of varying parameters such as pylon–rotor and rotor–rotor spacing and blade number. The directivity of the sound field for the case of pylon–rotor interaction has also been investigated. A simple model, which assumes no correlation between the loading on adjacent blades has also been presented. This model is much simpler than the full model.
Acknowledgements The author gratefully acknowledges the continuing support of Rolls-Royce plc. by way of the UTC in gas turbine noise at the University of Southampton and also funding from SAGE. The author is also grateful for discussions with Vincent Blandeau. Appendix 1. Acoustically weighted lift functions The acoustically weighted lift function is defined Z Ψ L ðK X 2 ; 0; ω; kc Þ ¼
1 −1
ðmÞ
ðmÞ
gðX ðmÞ 2 ; K X 2 ; 0; ωÞexpf−ikc X 2 gdX 2 :
For low frequency gusts the blade response function is given by Amiet [18] vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ( ) u ðmÞ h i 1 u ðmÞ t 1−X 2 S s exp i s M2 X ðmÞ þ f ðM Þ ; gðX 2 ; K X 2 ; 0; ωÞ ¼ r 2 r2 2 πβr2 1 þ X ðmÞ β2r β2r2 2 2
where s ¼ K X 2 c2 =2, βr2 ¼
(A1)
jsjM r2 π o : 4 β2r2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1−M 2r2 ;
(A2)
(A3)
f ðM r2 Þ ¼ ð1−βr2 ÞlnðM r2 Þ þ βr2 lnð1 þ βr2 Þ−lnð2Þ
(A4)
and SðxÞ ¼
1 : ix½K 0 ðixÞ þ K 1 ðixÞ
(A5)
is the well-known Sears' function. For high frequency gusts the blade response function is given by Landhal [19] and Goldstein [4] n sM o ðmÞ r exp −i 1þM2r ð1 þ X 2 Þ þ is jsjMr2 π ðmÞ 2 4 : gðX 2 ; K X 2 ; 0; ωÞ ¼ pffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 ðmÞ 4 β r 2 π is πð1 þ M r2 Þð1 þ X 2 Þ
(A6)
Substituting and evaluating the integral yields ( ) !
1 s s Ψ L ðK X 2 ; 0; ω; kc Þ ¼ S 2 exp i 2 f ðM r2 Þ J 0 ðη′Þ−iJ 1 ðη′Þ ; βr2 βr2 βr2
jsjM r2 π o : 4 β2r2
(A7)
where η′ ¼
sM 2r2 β2r2
−kc ;
(A8)
3970
and
M.J. Kingan / Journal of Sound and Vibration 332 (2013) 3956–3970
pffiffiffi 2 pffiffiffiffi 2 expfikc þ isg Ψ L ðK X 2 ; 0; ω; kc Þ ¼ pffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffi En pffiffiffi μ′ ; π μ′ π 1 þ M r2 is
jsjM r2 π 4 : 4 β2r2
(A9)
where μ′ ¼
sM r2 þ kc ; 1 þ M r2
(A10)
and E[] denotes a complex Fresnel integral. References [1] A. Parry, M.J. Kingan, B.J. Tester Relative importance of open rotor tone and broadband noise sources, Proceedings of the 17th AIAA/CEAS Aeroacoustics Conference, Portland, US, 2011. [2] V.P. Blandeau, P.F. Joseph, B.J. Tester, Broadband noise prediction from rotor-wake interaction in contra-rotating propfans, Proceedings of the 15th AIAA/ CEAS Aeroacoustics Conference, Miami, USA, 2009. [3] V.P. Blandeau, P.F. Joseph, Broadband noise due to rotor-wake/rotor interaction in contra-rotating open rotors, AIAA Journal 48 (11) (2010) 2674–2686. [4] M.E. Goldstein, Aeroacoustics, McGraw-Hill, 1976. [5] D.B. Hanson, D.J. Parzych Theory of noise of propellers in angular inflow with parametric studies and experimental verification, Proceedings of NASA CR 4499, 1993. [6] D.B. Hanson, Sound from a propeller at angle of attack: a new theoretical viewpoint, Proceedings of the Royal Society of London A 449 (1995) 315–328. [7] R.H. Schlinker, R.K. Amiet Helicopter rotor trailing edge noise, Proceedings of NASA TR 3470, 1981. [8] V.P. Blandeau, P.F. Joseph, On the validity of Amiet's model for propeller trailing-edge noise, AIAA Journal 49 (2011) 1057–1066. [9] C.S. Ventres, M.A. Theobald, W.D. Mark, Turbofan Noise Generation, Volume 1: Analysis, Contractor Report CR-167952, NASA, 1982. [10] M. Nallasamy, E. Envia, Computation of rotor wake turbulence noise, Journal of Sound and Vibration 282 (2005) 649–678. [11] R.K. Amiet Noise produced by turbulent flow into a rotor: theory manual for noise calculation, Proceedings of NASA CR 181788, 1989. [12] A.B. Parry Theoretical prediction of counter-rotating propeller noise, PhD Thesis, University of Leeds, 1988. [13] A.B. Parry, Modular prediction scheme for blade-row interaction noise, Journal of Propulsion and Power 13 (1997) 334–341. [14] V.P. Blandeau Aerodynamic broadband noise from contra-rotating open rotors, PhD Thesis, University of Southampton, 2011. [15] I. Wygnanski, F. Champagne, B. Marasli, On the large-scale structures in two-dimensional, small-deficit, turbulent wakes, Journal of Fluid Mechanics 168 (1986) 31–71. [16] J. Ricouard, E. Julliard, M. Omais, V. Regnier, A.B. Parry, S. Baralon Installation effects on contra-rotating open rotor noise, Presented at 16th AIAA/CEAS Aeroacoustics Conference, Stockholm, AIAA Paper. 2010-3795, 2010. [17] D.S. Whitehead, Classical two-dimensional methods. In manual on aeroelasticity in axial-flow turbomachines, Unsteady Turbomachinery Dynamics 1 (1987) 1–22. [18] R.K. Amiet, Compressibility effects in unsteady thin-airfoil theory, AIAA Journal 12 (1974) 253–255. [19] M.J. Landahl, Unsteady transonic flow, Pergamon Press, New York, 1961.
Contents lists available at SciVerse ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Open rotor broadband interaction noise Michael J. Kingan n Institute of Sound and Vibration Research, University of Southampton, Southampton, SO17 1BJ, UK
a r t i c l e in f o
abstract
Article history: Accepted 8 March 2013 Handling Editor: P. Joseph
A theoretical model is presented for calculating the broadband noise produced by the interaction of an open rotor with the wake from either an upstream contrarotating rotor or a stationary pylon. The model is used to investigate the dependence of the radiated noise on parameters such as pylon–rotor gap and the polar and azimuthal directivity of the noise field. A simple model is also presented which assumes that the unsteady loading on adjacent blades is uncorrelated. It is shown that the simple model can be used to calculate broadband interaction noise for most practical open rotor geometries. & 2013 Elsevier Ltd. All rights reserved.
1. Introduction Broadband noise produced by contra-rotating open rotors can be a significant contributor to the total noise level produced by these engines [1,2]. For an isolated open rotor, broadband noise can be produced by several sources, namely; (1) boundary layer self-noise, which is produced by the scattering of boundary layer turbulence by the trailing edge of the rotor blades, (2) vortex self-noise, which is produced by the scattering of the turbulence in the tip vortex by the rotor blades, (3) atmospheric turbulence ingestion noise and (4) rotor–rotor interaction broadband noise which is produced when the turbulent wake of the upstream rotor impinges on the downstream, contra-rotating rotor blades. When a pylon is installed upstream of the rotor, the wake from this pylon will interact with the open rotors to produce both tone and broadband noise. Tones are produced by the interaction of the rotor blades with the steady component of the flow, whilst broadband noise is produced by the interaction of the rotor blades with the turbulent component. An upstream pylon will be required to attach the open rotor to the fuselage when the open rotor is mounted at the rear of the aircraft in a ‘pusher’ configuration. This particular configuration is currently being investigated by aircraft engine and airframe manufacturers. Two photographs showing a test of the Rolls-Royce 1/6th scale contra-rotating open rotor rig with and without an upstream pylon in the DNW low speed wind tunnel is shown in Fig. 1. This paper will describe a theoretical model for predicting rotor–rotor broadband interaction noise, although the reader should note that this problem has been treated previously by Blandeau and Joseph [3]. The rotor–rotor broadband noise model will then be extended so that it can be used to predict the broadband noise produced by the interaction of a turbulent pylon wake with an open rotor. 2. Formulation In this section, formulae are derived which can be used to calculate the broadband noise produced by the interaction of the wakes from the contra-rotating upstream rotor blades with the downstream rotor. The analysis is very similar to that n
Tel.: +44 2380592946. E-mail address: [email protected]
0022-460X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jsv.2013.03.014
M.J. Kingan / Journal of Sound and Vibration 332 (2013) 3956–3970
3957
Fig. 1. (a) Test of an isolated open rotor rig, (b) test of an open rotor rig with upstream pylon. Pictures courtesy of Rolls-Royce Plc.
presented in Blandeau and Joseph [3], however it includes the effect of axial flow and corrects a number of minor errors in their final result1 . The special case of the interaction of a pylon wake with a downstream rotor is then analysed by considering the interaction of the wake from a stationary, one-bladed upstream rotor with a downstream rotor. 2.1. Problem definition A cylindrical coordinate system with axial, radial and tangential coordinates {x, r, ϕ} will be utilised in the analysis. The upstream and downstream rotors both rotate about the x-axis in the negative and positive ϕ directions respectively. Each rotor has blade count, B, and rotational speed, Ω rad/s. Note that upstream and downstream rotor parameters are denoted by the subscripts 1 and 2 respectively. The formulation presented in this paper can be used to calculate the sound pressure spectrum produced by a stationary open rotor situated in a uniform flow of velocity Ux and corresponding Mach number Mx in the positive x-direction. The observer is assumed to be stationary and located within the flow. This sound pressure spectrum is equivalent to that which would be measured using an inflow microphone mounted adjacent to an open rotor situated in a wind tunnel. Alternatively, this formulation could also be used to calculate the pressure spectrum produced by an open rotor translating through a stationary medium with the pressure being measured using a microphone which translates with the open rotor. 2.2. Acoustic formulation The acoustic pressure, p, at observer location xo and time t due to the force per unit area, f, exerted on the air at point x and time τ by the downstream rotor is given by ([4] Eq. 4.13) Z Te Z pðxo ; tÞ ¼ ∇Gðx; τjxo ; tÞ⋅fðx; τÞdSðxÞdτ; (1) −T e
SB ðτÞ
where −Te≤τ≤Te is the interval of time over which sound is emitted and which will be allowed to become infinitely large, SB is a closed surface corresponding to the blade surfaces and G is a free-field Green's function which is defined as [5] Gðx; τjxo ; tÞ ¼
δðt−τ−so =c0 Þ ; 4πSo
(2)
where, in the farfield So Re ð1−M x cosθe Þ; so Re þ
(3)
xcosθe rsinθe cosðϕ−ϕo Þ − : 1−M x cosθe 1−M x cosθe
(4)
Note that the observer position has been expressed in terms of Re and θe which are the observer emission radius and polar angle and are related to the physical observer radius and polar angle by the following relationships (see for example [6]): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 xo;e ¼ x −M xo 2 þ ð1−M 2x Þðyo 2 þ zo 2 Þ ; yo;e ¼ yo ; zo;e ¼ zo ; o x 1−M 2x xo;e ¼ −Re cosθe ;
yo;e ¼ Re sinθe cosϕo ;
zo;e ¼ Re sinθe sinϕo :
1 The errors in Ref. [3] are listed in this footnote. (1) The effect of wake skew on the ‘mean wake profile’ was not properly accounted for (see Eqs. (38 and 39) of this paper). (2) The final formulation contained an extra factor of 2π due to an inconsistent Fourier transform convention. (3) There was an error in the wavenumber contained in the blade response function g (see Appendix 1 of this paper). (4) There were a number of errors in the ‘acoustically weighted lift function’, ΨL, which is defined in Appendix 1 of this paper.
3958
M.J. Kingan / Journal of Sound and Vibration 332 (2013) 3956–3970
Fig. 2. Propeller geometry (a), local blade geometry (b).
Making use of the properties of the delta function, the Green's function can be written Z ∞ 1 expfiωðt−τ−so =c0 Þgdω: Gðx; τjxo ; tÞ 2 8π Re ð1−Mx cos θÞ −∞ Substituting Eq. (4) into Eq. (5) and making use of the Jacobi–Anger expansion n ∞ π o ; exp izcosðϕ−ϕo Þ ¼ ∑ J n ðzÞexp in ϕ−ϕo þ 2 n ¼ −∞
(5)
(6)
yields
∞
Z
∑
n ¼ −∞
∞ −∞
Jn
1 Gðx; τjxo ; tÞ 8π2 Re ð1−M x cos θÞ
κrsinθe π κxcosθe exp iωðt−τÞ−iκRe þ in ϕ−ϕo þ −i dω; 2 1−M x cosθe 1−M x cosθe
(7)
where κ ¼ω/c0. The interaction of the upstream rotor blade wakes with a downstream rotor blade will induce an unsteady lift force on the blade. Provided that the blade is relatively thin and reasonably flat, the integration over the closed blade surface in ðmÞ Eq. (1) can be approximated by an integration over the blade planform area on which Y ðmÞ 2 ¼ 0 for −c2 =2≤X 2 ≤c2 =2, Rh≤r≤Rt, ðmÞ where Rh is the hub radius, Rt is the blade tip radius, c2 is the downstream blade chord-length and fX ðmÞ ; Y 2 2 g are chordwise/ chord-normal coordinates of the mth blade on the downstream rotor which are shown in Fig. 2. The loading exerted by the blade on the air must also be replaced by the vector sum of the loading exerted by the upper and lower blade surfaces, which from inspection of Fig. 2 can be written ^ cosα2 ΔpðmÞ ðX ðmÞ ; r; τÞ; fðx; τÞ-½x^ sinα2 þ ϕ 2
(8)
where α2 and Δp(m) are, respectively, the stagger angle and pressure jump across the mth downstream rotor blade. Note that ^ are unit vectors in the axial and tangential directions. x^ and ϕ A point on the mth downstream rotor blade is located at ϕ ¼ Ω2 τ−X ðmÞ 2
sinα2 2π þ m; B2 r
x ¼ X ðmÞ 2 cosα2 : Substituting Eqs. (8)–(10) into Eq. (1) and summing over all B2 blades on the downstream rotor yields Z ∞ ~ o ; ωÞexpfiωtgdω; pðxo ; tÞ ¼ pðx
(9) (10)
(11)
−∞
where ~ o ; ωÞ ¼ pðx
Z Rt Z 1 B2 −1 ∞ −i κr sinθ e Δp~ ðmÞ ðX ðmÞ ∑ ∑ 2 ; r; ω−nΩ2 ÞJ n 1−M x cosθe S 4πRe ð1−M x cosθe Þ n ¼ −∞ m ¼ 0 Rh −1
π 2π ðmÞ ðmÞ −ϕo þ m −ikc X 2 exp −iκRe þ in dX 2 dr; 2 B2
(12)
M.J. Kingan / Journal of Sound and Vibration 332 (2013) 3956–3970
3959
ðmÞ
and X 2 ¼ 2X ðmÞ 2 =c2 is a dimensionless chordwise coordinate, Z Te 1 ΔpðmÞ ðX ðmÞ Δp~ ðmÞ ðX ðmÞ 2 ; r; ωÞ ¼ 2 ; r; τÞexpf−iωτgdτ; 2π −T e
(13)
S¼
κcosθe sinα2 n c2 ; − cosα2 1−M x cosθe r 2
(14)
kc ¼
κcosθe cosα2 n c2 : þ sinα2 r 1−M x cosθe 2
(15)
An instantaneous pressure spectrum can be defined Z ∞ 1 Spp ðxo ; ω; tÞ ¼ Rpp ðxo ; t 1 ; tÞexpf−iωt 1 gdt 1 ; 2π −∞
(16)
where Rpp is a non-stationary correlation function Rpp ðxo ; t 1 ; tÞ ¼ E½pðxo ; tÞpðxo ; t þ t 1 Þ:
(17)
Averaging the instantaneous spectrum over the time period −T≤t≤T (where T-∞), yields a time-averaged spectrum level Z T 1 Spp ðxo ; ωÞ ¼ lim Spp ðxo ; ω; tÞdt: (18) T-∞ 2T −T Substituting Eq. (11) into Eq. (17) and the result into Eq. (16) and then Eq. (18) yields Z TZ ∞ 1 ~ o ; ωÞexpfiðω−ω′Þtgdω′dt; Spp ðxo ; ωÞ ¼ lim E½p~ n ðxo ; ω′Þpðx T-∞ 2T −T −∞
(19)
which simplifies to Spp ðxo ; ωÞ ¼ lim
π n ~ o ; ωÞ : E p~ ðxo ; ωÞpðx
T-∞ T
Making use of Eq. (12) yields
B2 −1 B2 −1 ~ o ; ωÞ ¼ ∑ ∑ E p~ n ðxo ; ω′Þpðx
∞
∑
n o exp iðκ′−κÞRe þ iðn′−nÞ ϕo − 2π þ i 2π B2 ðnm−n′m′Þ
∞
∑
ð4πRe Þ2 ð1−M x cosθe Þ2
κ′r′sinθe κrsinθe S′SJ n′ Jn 1−M x cosθe 1−M x cosθe
m ¼ 0m′ ¼ 0n ¼ −∞n′ ¼ −∞
Z Z
1
−1
Z
Rt Rh
1
−1
Z
Rt Rh
′
ðm0 Þ
ðmÞ
ðm′Þ
ðmÞ
~ ðmÞ ðX ðmÞ E½Δp~ ðm′Þ ðX ðm′Þ 2 ′; r′; ω′−n′Ω2 ÞΔp 2 ; r; ω−nΩ2 Þexpfikc X 2 ′−ikc X 2 gdX 2 ′dX 2 dr′dr;
(20)
where
~ ðmÞ ðX ðmÞ Δp~ ðm′Þ ðX ðm′Þ 2 ′; r′; ω′−n′Ω2 ÞΔp 2 ; r; ω−nΩ2 Þ;
(21)
is the only term which is, as yet, undetermined. Note that in Eq. (20), the integration over r′ can be truncated at r′¼r 7Δr/2 where Δr is somewhat larger than the correlation length of the unsteady loading on the downstream rotor blades. Eq. (20) becomes n o exp iðκ′−κÞRe þ iðn′−nÞ ϕo − 2π þ i 2π ∞
n B2 −1 B2 −1 ∞ B2 ðnm−n′m′Þ ~ o ; ωÞ ¼ ∑ ∑ ∑ E p~ ðxo ; ω′Þpðx ∑ ð4πRe Þ2 ð1−M x cosθe Þ2 m ¼ 0m0 ¼ 0n ¼ −∞n0 ¼ −∞
Z rþΔr Z Rt κ′rsinθe κrsinθe S′SJ n′ Jn 1−M x cosθe 1−M x cosθe Rh r−Δr Z 1Z 1 ðm′Þ ðmÞ ðm′Þ ðmÞ ′ ~ ðmÞ ðX ðmÞ E½Δp~ ðmÞn ðX ðm′Þ (22) 2 ′; r′; ω′−n′Ω2 ÞΔp 2 ; r; ω−nΩ2 Þexpfikc X 2 ′−ikc X 2 gdX 2 ′dX 2 dr′dr: −1
−1
′
It has also been assumed that the variation of r′ in S′, kc and the argument of the Bessel function Jn′ can be neglected. This may be an acceptable assumption provided that Δr is not too large. This is equivalent to assuming that all sound generated within a ‘strip’ radiates from the same radius and that the rotor geometry and mean flow parameters do not vary significantly over this strip. Although this crude assumption is used in the method presented in this paper, it should be possible to extend the method so that radial non-compactness effects are correctly taken into account by using Eq. (20) rather than Eq. (22).
3960
M.J. Kingan / Journal of Sound and Vibration 332 (2013) 3956–3970
2.3. Aerodynamic formulation In the previous section, an expression was derived for calculating the far-field pressure spectral density produced by the interaction of the turbulent component of the front rotor wakes with the downstream rotor blades. The only unknown quantity in this expression is given by Eq. (21). This section is devoted to deriving a set of equations for estimating this quantity. The derivation makes use of ‘strip theory’ which assumes that aerodynamic interactions along a radial ‘strip’ of the rotor blade and centred at a particular radius can be approximated by considering an equivalent two-dimensional problem. Such an approach is common in turbo-machinery noise analysis and has been used in many previous studies. For example, the approach was used by Blandeau and Joseph [3], to calculate open rotor broadband interaction noise, Shlinker and Amiet [7] and Blandeau and Joseph [8] for calculating rotor self-noise and Ventres et al. [9] and Nallasamy et al. [10] to calculate turbo-fan rotor–stator interaction noise. The method involves formulating an equivalent two-dimensional problem by ‘unwrapping’ the rotor at radius, r, as shown in Fig. 3 below. Note that for each blade row, two coordinate systems will be utilised. A chordwise/chord-normal coordinate system {X, Y} and a ‘blade-locked’ axial/tangential coordinate system {x, y}. Note that the blades are assumed to be aligned with the local flow-direction and thus the stagger angle of each blade, α, is defined by tan α ¼ Ωr=U x . The effect of blade sweep and lean is neglected in this analysis. Also, a subscript 1 or 2 is respectively used to denote the upstream and downstream rotors, whereas a superscript (m) is used to denote that the origin of that coordinate system is at the mid-chord of the mth downstream rotor blade. The incident flow field and unsteady loading on the downstream rotor blades is then calculated as if the interaction was taking place between the wakes from an upstream cascade of two-dimensional rotor blades with a cascade of downstream rotor blades. It is assumed that it will be possible to express the fluctuating component of velocity normal to the mth downstream rotor blade in terms of its spatio-temporal Fourier transform Z ∞Z ∞Z ∞Z ∞ ~ u¼ (23) uðk; ωÞexpfiωτ−ik⋅xðmÞ 2 gdkdω −∞
−∞
−∞
−∞
~ The pressure jump due to a single Fourier component of this upwash, uðk; ωÞexpfiωτ−ik⋅xðmÞ 2 g, can be estimated using two-dimensional strip theory [11] which yields ðmÞ ~ ΔpðmÞ ðX ðmÞ 2 ; r; τÞ ¼ 2πρU r 2 uðk; ωÞgðX 2 ; k; ωÞexpfiωτ−ikr rg;
(24)
gðX ðmÞ 2 ; k; ωÞ
where is a ‘blade response function’ which will be defined later. The total pressure jump on the downstream rotor blade is thus given by Z ∞Z ∞Z ∞Z ∞ ~ uðk; ωÞgðX ðmÞ (25) ΔpðmÞ ðX ðmÞ 2 ; r; τÞ ¼ 2πρU r2 2 ; k; ωÞexpfiωτ−ikr rgdkdω: −∞
−∞
−∞
−∞
Taking the Fourier transform of this expression and substituting the result into Eq. (21) yields
~ ðmÞ ðX ðmÞ E½Δp~ ðm′Þ ðX ðm′Þ 2 ′; r′; ω′−n′Ω2 ÞΔp 2 ; r; ω−nΩ2 Þ Z ∞Z ∞Z ∞Z ∞Z ∞Z ∞ 2 ~ ω−nΩ2 Þ ¼ ð2πρU r2 Þ E½u~ n ðk′; ω′−n′Ω2 Þuðk; −∞
−∞
−∞
−∞
−∞
−∞
′
ðmÞ g n ðX ðm′Þ 2 ′; k′; ω′−n′Ω2 ÞgðX 2 ; k; ω−nΩ2 Þexpfikr r′−ikr rgdk′dk:
Fig. 3. ‘Unwrapped’ blade geometry and coordinate systems.
(26)
M.J. Kingan / Journal of Sound and Vibration 332 (2013) 3956–3970
3961
The only unknown remaining is the term in the expected value brackets in the integrand of Eq. (26) which is defined by Eq. (27)
1 RRRRRRRR ðmÞ ~ E u~ n ðk′; ω′−n′Ω2 Þuðk; ω−nΩ2 Þ ≡ ð2πÞ E½un ðxðm′Þ 8 2 ′; τ′Þuðx2 ; τÞ (27) ðm′Þ ðmÞ ðm′Þ expfiðω′−n′Ω2 Þτ′−iðω−nΩ2 Þτ þ ik⋅xðmÞ 2 −ik′⋅x 2 ′gdx 2 dx2 ′dτdτ′: Note that the limits of integration in Eq. (27) are allowed to approach 7∞. It will be convenient to convert to a coordinate system rotating with the front rotor (and with its origin located at the mid-chord of the reference blade on the front rotor). By inspection of Fig. 3 we have xðmÞ 2 ¼ x1 −g r ; yðmÞ 2 ¼ y1 −ðΩ1 þ Ω2 Þrτ−
(28) 2πr m: B2
Substituting Eqs. (28) and (29) into Eq. (27) yields
1 RRRRRRRR ~ ω−nΩ2 Þ ≡ ð2πÞ E½un ðx′1 ; τ′Þuðx1 ; τÞ E u~ n ðk′; ω′−n′Ω2 Þuðk; 8 8 9 < −iðω−nΩ2 Þτ−iky ðΩ1 þ Ω2 Þrτ þ iðω′−n′Ω2 Þτ′ þ ik′y ðΩ1 þ Ω2 Þrτ′ þ ikr r−ik′r r′ = dx dx′ dτdτ′: exp 2πr : −ik′x x′1 þ ikx x1 −ikx g r þ ik′x g r þ iky y1 −ik′y y′1 þ ik′y 2πr ; 1 1 B2 m′−iky B2 m
(29)
(30)
Note that in Eq. (30) the list of dependent variables associated with u has changed. Providing that the rotor wakes are well separated when they impinge on the down stream rotor, it is convenient to assume that the upwash has the following form: uðx1 ; τÞ ¼ f ðx1 Þwðx1 −U1 τÞ;
(31)
where f essentially represents the (normalised) distribution of the rms turbulence intensity in the wake of the front rotor and w is a random function which corresponds to the turbulent velocity produced by frozen, stationary, isotropic and homogeneous turbulence with an rms velocity equal to that at the centerline of each wake. U1 is the air velocity relative to the front blade row and thus has components in the axial, tangential and radial directions, U x1 ¼ U x , U y1 ¼ Ω1 r and Ur ¼ 0 respectively. Note that swirl and induced axial velocity are neglected which is a common assumption for rotors with low solidity. Therefore we have n
E½un ðx′1 ; τ′Þuðx1 ; τÞ ¼ f ðx′1 Þf ðx1 ÞE½wn ðx′1 −U1 τ′Þwðx1 −U1 τÞ:
(32)
The function f is periodic in the y1 direction (see Fig. 3) and can thus be expressed using Poisson's summation formula
∞ 2πr B1 ∞ ~ kB1 kB1 ; r exp i y1 ; k; r ¼ f ðx1 Þ ¼ ∑ f 0 x1 ; y1 þ (33) ∑ f 0 x1 ; B1 r k ¼ −∞ r r k ¼ −∞ where f0 is the normalised distribution of the rms turbulence intensity in the wake of one rotor blade and f~ 0 is defined by
Z ∞ kB1 1 kB1 ;r ¼ y1 dy1 : f 0 ðx1 ; y1 ; rÞexp −i (34) f~ 0 x1 ; 2π −∞ r r In this study, f0 is defined by the following Gaussian profile which corresponds to that of the mean streamwise velocity deficit in the wake of a two-dimensional aerofoil (from Parry [12] and [13]) (
) Y1 2 f 0 ðx1 ; y1 ; rÞ ¼ exp −lnð2Þ ; (35) b1=2 b1=2 ¼
1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C D1 c1 Lw : 4
(36)
Note that Y 1 ¼ y1 cosα1 −x1 sinα1 is the chord-normal coordinate of the front rotor reference blade and b1/2 is the ‘halfwidth’ of the wake produced by the front rotor reference blade. C D1 ðrÞ and c1(r) are the front rotor blades drag coefficient and chord length, while Lw(X1,r) is the helical length of the wake. The wake develops relatively slowly and therefore Lw(X1,r) may be replaced by its value at the leading edge of the downstream rotor blade Lw ðrÞ f evaluates as pffiffiffi ∞ π B1 b1=2 kB1 kB1 y1 −i tanα1 x1 ; f ðx1 Þ ¼ pffiffiffiffiffiffiffiffiffiffiffi (37) ∑ Gðkb Þexp i r r lnð2Þ2πrcosα1 k ¼ −∞ where
(
) 2 kb Gðkb Þ ¼ exp − ; 4lnð2Þ
(38)
3962
M.J. Kingan / Journal of Sound and Vibration 332 (2013) 3956–3970
and kb ¼
kB1 b1=2 : rcosα1
(39)
Now, E½wn ðx′1 −U1 τ′Þwðx1 −U1 τÞ is effectively a two-point turbulent velocity correlation, which will be denoted Rww(η) where η ¼ x1 −x′1 −U1 ðτ−τ′Þ. If the unit vectors in the axial, tangential and radial directions are respectively denoted e^ x , e^ y and e^ r , then η ¼ e^ x ηx þ e^ y ηy þ e^ r ηr where ηx ¼ x1 −x′1 −U x ðτ−τ′Þ;
ηy ¼ y1 −y′1 −Ω1 rðτ−τ′Þ;
ηr ¼ r−r′:
(40)
The two-point velocity correlation Rww(η) is periodic in ηy, with period 2πr, and can be rewritten using Poisson's summation formula Z ∞ n q o ∞ 1 ∑ Rww;0 ðe^ x ηx þ e^ y ξ þ e^ r ηr Þexp i ðξ−ηy Þ dξ; (41) Rww ðηÞ ¼ 2πr q ¼ −∞ −∞ r where Rww,0(η) is the two-point correlation of w between −πroηy oπr. Substituting Eqs. (32), (37) and (41) into Eq. (30) yields (after some manipulation) ∞ ∞
π B1 b1=2 2 ∞ ′ ~ ω−nΩ2 Þ ¼ U x rlnð2Þ ∑ ∑ ∑ Gðkb ÞGðkb Þ E u~ n ðk′; ω′−n′Ω2 Þuðk; 2πrcosα1 k ¼ −∞k′ ¼ −∞q ¼ −∞
Ω1 q 2πr ′ 1 ′ Φww kx −kB1 ðky m′−ky mÞ δ ky − q−kB1 e^ x þ e^ y þ kr e^ r exp iðkx −kx Þg r þ i r B2 r Ux
ω−nΩ2 þ ky Ω2 r B1 Ω1 B1 ′ ′ ′ δ kx − kx − ðk−k′Þ δ ky − ky − ðk′−kÞ δðkr −kr Þδ kx − Ux Ux r ′
δðω′−½n′Ω2 þ ky Ω1 r−ky ðΩ1 þ Ω2 Þr þ kx U x Þ; where Φww ðkÞ ¼
Z
1 3
ð2πÞ
∞ −∞
Z
∞ −∞
Z
∞
−∞
(42)
Rww;0 ðηÞexpfik⋅ηgdη;
is a turbulent velocity spectrum. Substituting this result into Eq. (26) yields ρU B b 2 ∞ h i r 2 1 1=2 π ~ ðmÞ ðX ðmÞ E Δp~ ðm′Þ ðX ðm′Þ ∑ 2 ′; r′; ω′−n′Ω2 ÞΔp 2 ; r; ω−nΩ2 Þ ¼ rU x lnð2Þ rcosα1
(43)
∞
∑
∞
′
∑ Gðkb ÞGðkb Þ
k ¼ −∞k′ ¼ −∞q ¼ −∞
Ω1 2π δðω− ω′ þ ðn−n′ÞΩ2 þ ðk−k′ÞB1 ðΩ1 þ Ω2 Þ Þexp −iB1 ðk−k′Þg r þ i ðq−k′B1 Þm′−ðq−kB1 Þm B2 Ux Z ∞ ðmÞ Φww ðK X 2 e^ X 2 þ K Y 2 e^ Y 2 þ kr e^ r Þg n ðX ðm′Þ 2 ′; K X 2 ′; kr ; ω′−n′Ω2 ÞgðX 2 ; K X 2 ; kr ; ω−nΩ2 Þexpf−ikr ðr−r′Þgdkr ;
(44)
−∞
where e^ X 2 and e^ Y 2 are, respectively, unit vectors which are aligned with the chordwise and chord-normal directions of the downstream blade row ω−nΩ2 ; U r2
(45)
ω−nΩ2 þ ðk′−kÞB1 ðΩ1 þ Ω2 Þ ; U r2
(46)
ω−nΩ2 −kB1 ðΩ1 þ Ω2 Þ ; U r2
(47)
K X2 ¼ K X2 ′ ¼
K X2 ¼ and
K Y 2 ¼ K X 2 tanα2 þ
q : rcosα2
Note that the list of dependent variables associated with the blade response function g has changed. 2.4. Broadband noise Substituting Eq. (44) into Eq. (22) and then the result into Eq. (19) yields ∞ ∞ ∞ ∞ ∞ exp iðκ′−κÞRe þ iðn′−nÞ ϕo − 2π Spp ðxo ; ωÞ ¼ ∑ ∑ ∑ ∑ ∑ I t I m I m′ ð4πRe Þ2 ð1−M x cosθe Þ2 n ¼ −∞n′ ¼ −∞k ¼ −∞k′ ¼ −∞q ¼ −∞
(48)
M.J. Kingan / Journal of Sound and Vibration 332 (2013) 3956–3970
Z Z
Rt
Rh ∞ −∞
π rU x lnð2Þ
Φww ðK X 2 e^ X 2 þ K Y 2 e^ Y 2
3963
2
ρU r2 B1 b1=2 Ω1 κ′rsinθe κrsinθe ′ Gðkb ÞGðkb Þexp iB1 ðk′−kÞg r S′SJ n′ Jn rcosα1 Ux 1−M x cosθe 1−M x cosθe Z rþΔr=2 ′ n þ kr e^ r Þ expf−ikr ðr−r′Þgdr′Ψ L ðK X 2 ′; kr ; ω′−n′Ω2 ; kc ÞΨ L ðK X 2 ; kr ; ω−nΩ2 ; kc Þdkr dr;
(49)
r−Δr=2
where ω′¼ω−(n−n′)Ω2−(k−k′)B1(Ω1+Ω2), 1 T-∞ 2T
I t ¼ lim
Z
T
−T
expfi½ðn−n′ÞΩ2 þ ðk−k′ÞB1 ðΩ1 þ Ω2 Þtgdt;
(50)
B2 −1 2πm ðq−kB1 −nÞ ; I m ¼ ∑ exp −i B2 m¼0
(51)
2πm′ ðq−k′B1 −n′Þ ; ∑ exp i B2 m0 ¼ 0
(52)
I m′ ¼
B2 −1
and Z Ψ L ðK X 2 ; kr ; ω; kc Þ ¼
1
−1
ðmÞ
ðmÞ
gðX ðmÞ 2 ; K X 2 ; kr ; ωÞexpf−ikc X 2 gdX 2
(53)
is an ‘acoustically weighted lift function’ which is independent of the summation index m. Now, following Blandeau et al. [3], assuming that Δr is sufficiently large, the integration over r′ can be approximated as follows: Z rþΔr=2 expf−ikr ðr−r′Þgdr′≈2πδðkr Þ: (54) r−Δr=2
This yields the following expression for the time averaged pressure spectral density: ∞ ∞ ∞ ∞ ∞ exp iðκ′−κÞRe þ iðn′−nÞ ϕo − 2π Spp ðxo ; ωÞ ¼ ∑ ∑ ∑ ∑ ∑ 2πI t I m I m′ ð4πRe Þ2 ð1−M x cosθe Þ2 n ¼ −∞n′ ¼ −∞k ¼ −∞k′ ¼ −∞q ¼ −∞
Z Rt 2 ρU r2 B1 b1=2 π Ω1 κ′rsinθe κrsinθe ′ Gðkb ÞGðkb Þexp iB1 ðk′−kÞg r S′SJ n′ Jn rU x lnð2Þ rcosα1 Ux 1−M x cosθe 1−M x cosθe Rh ′
Φww ðK X 2 e^ X 2 þ K Y 2 e^ Y 2 ÞΨ L n ðK ′X 2 ; 0; ω′−n′Ω2 ; kc ÞΨ L ðK X 2 ; 0; ω−nΩ2 ; kc Þdr
(55)
3. Rotor–rotor interaction noise For rotor–rotor interaction noise, provided that the ratio of the rotor shaft speeds is an irrational number, then it is relatively straightforward to show that It ¼δk,k’δn,n′. Substitution of this result into Eq. (55) yields Im ¼Im′ ¼ B2 and the summation over q is replaced by a summation over j where q ¼jB2+kB1+n. Thus, the equation for the time-averaged rotor– rotor broadband pressure spectrum simplifies to
2
Z Rt ∞ ∞ ∞ ρU r2 B1 b1=2 2πB2 2 π κrsinθe Spp ðxo ; ωÞ ¼ ∑ ∑ ∑ G2 ðkb ÞS2 J n 2 2 2 rU x lnð2Þ rcosα1 1−M x cosθe n ¼ −∞k ¼ −∞j ¼ −∞ ð4πRe Þ ð1−M x cosθ e Þ Rh jB2 þ kB1 þ n (56) Φww K X 2 e^ X 2 þ K X 2 tanα2 þ e^ Y 2 Ψ L ðK X 2 ; 0; ω−nΩ2 ; kc Þ2 dr; rcosα2 where G(kb) is given by Eqs. (38) and (39), S is given by Eq. (14), K X 2 , is defined by Eq. (47), K X 2 , is defined by Eq. (45) and kc, is defined by Eq. (15). The acoustically weighted lift functions ΨL are defined in Appendix 1 and a von Karman turbulence spectrum will be utilised which is defined by 2
2
ðK −K Y 2 Þ 55 Γð5=6Þ w2rms 1 Φww ðK X 2 e^ X 2 þ K Y 2 e^ Y 2 þ K r e^ r Þ ¼ pffiffiffi 9 π Γð1=3Þ k5e 4π ð1 þ ðK=ke Þ2 Þ17=6 ; 2
2
2
2
K ¼ K X2 þ K Y 2 þ K r ; ke ¼
pffiffiffi π Γð5=6Þ : Λ1 Γð1=3Þ
(57)
(58) (59)
Note that Λ1 is the integral lengthscale of the front rotor wake turbulence which is estimated using the expression given by Jurdic and Joseph [9] Λ1 0.42b1/2. Also, wrms 0.404udef is estimated using an empirical expression due to Blandeau [14]
3964
M.J. Kingan / Journal of Sound and Vibration 332 (2013) 3956–3970
who derived this relationship by analysing the experimental data presented in Wygnanski et al. [15] (N.B. it has been assumed that the helical path-length of the front rotor wake is sufficiently large for these relationships to be valid). udef is the wake centerline velocity deficit which is calculated using Eq. (65) below which relies on the front rotor drag coefficient C D1 rffiffiffiffiffiffiffiffiffiffiffi
lnð2Þ c1 C D1 0:5 udef ¼ 2U r1 (60) π Lw Note that the summation index j only appears in the turbulence spectrum, i.e. Eq. (56) contains a factor
∞ kB1 þ n B2 þj e^ Y 2 : ∑ Φww K X 2 e^ X 2 þ K X 2 tanα2 þ rcosα2 rcosα2 j ¼ −∞
(61)
Recalling that Φww is defined as the spatial Fourier transform of the two-point correlation function, Eq. (61) can be rewritten Z ∞Z ∞Z ∞ ∞ 1 ∑ Rww;0 ðηX 2 e^ X 2 þ ηY 2 e^ Y 2 þ ηr e^ r Þ 3 −∞ −∞ −∞ j ¼ −∞ ð2πÞ kB1 þ n B2 j ηY 2 þ i ηY 2 dηX 2 dηY 2 dηr : exp iK X 2 ηX 2 þ iK X 2 tanα2 ηY 2 þ i rcosα2 rcosα2
(62)
Noting that
∞ B2 j rcosα2 ∞ rcosα2 ; ηY 2 ¼ 2π ∑ δ ηY 2 −2πj ∑ exp i rcosα2 B2 j ¼ −∞ B2 j ¼ −∞ Eq. (61) reduces to the following expression: Z ∞Z ∞ rcosα2 ∞ 1 2 ^ ^ Rww;0 ηX 2 e^ X 2 þ 2πj rcosα ∑ B2 eY 2 þ ηr er 2 B2 j ¼ −∞ ð2πÞ −∞ −∞ kB1 þ n 2πrcosα2 exp iK X 2 ηX 2 þ ij K X 2 tanα2 þ dηX 2 dηr rcosα2 B2
(63)
If the distance 2πrcosα2 =B2 is much greater than the correlation length of the wake turbulence, then only the j¼0 term contributes to the summation and the factor becomes Z ∞Z ∞ rcosα2 1 rcosα2 ð2Þ Rww;0 ðηX 2 e^ X 2 þ ηr e^ r ÞexpfiK X 2 ηX 2 gdηX 2 dηr ¼ Φww ðK X 2 e^ X 2 Þ; (64) B2 ð2πÞ2 −∞ −∞ B2 ^ X 2 þ K r e^ r Þ is the two-wavenumber turbulent velocity spectrum which is defined by where Φð2Þ ww ðK X 2 e ð2Þ Φww ðK X 2 e^ X 2 þ K r e^ r Þ ¼
ðK X 2 =ke Þ2 þ ðK r =ke Þ2 4 w2rms : 9π k2 ð1 þ ðK X 2 =ke Þ2 þ ðK r =ke Þ2 Þ7=3 e
(65)
Fig. 4. Effect of rotor–rotor spacing on rotor–rotor broadband interaction noise for a representative take-off case. Ten decibel per vertical gridline.
M.J. Kingan / Journal of Sound and Vibration 332 (2013) 3956–3970
Thus, provided that there is no blade–blade correlation, the time-averaged pressure spectral density is given by
2
Z Rt ∞ ∞ ρB1 b1=2 2πB2 πU r2 κrsinθe 2 2 2 Spp ðxo ; ωÞ ¼ ∑ ∑ G ðk ÞS J b n 2 2 lnð2Þ rcosα1 1−Mx cosθe n ¼ −∞ k ¼ −∞ ð4πRe Þ ð1−M x cosθ e Þ Rh ð2Þ 2 Φww ðK X 2 e^ X 2 ÞjΨ L ðK X 2 ; 0; ω−nΩ2 ; kc Þj dr:
3965
(66)
A similar form of this simplified expression was originally presented by Blandeau [14]. A realistic 1/6th scale open rotor configuration, operating at a representative take-off condition, was used to generate all of the noise predictions which are presented in this paper. There are 12 blades on the front rotor and 9 on the rear. The axial flow Mach number was 0.23 and the front and rear blade-tip Mach numbers were 0.63 and 0.47 respectively. The aerodynamic and geometric data for each rotor was supplied by Rolls-Royce plc. Eq. (66) was used to calculate the effect of increasing rotor–rotor spacing on broadband interaction noise for the open rotor operating at a representative take-off condition. The results of this calculation are shown in Fig. 4 which plots the onethird octave band sound power level spectrum. The calculations were performed for 6 different axial distances between the front and rear blade mid-chord at the rotor hub (gr in Fig. 3). As originally shown by Blandeau [14], as rotor–rotor spacing is increased, the front rotor wake width increases while the centerline velocity deficit decreases. The increase in wake width causes the turbulence correlation length within the wake to increase which produces higher levels of low frequency noise and moves the spectral peak towards lower frequencies. Also, as rotor–rotor separation is increased, there is a reduction in w2rms which should tend to reduce the level of broadband noise produced. This effect does not however appear to significantly influence the calculated noise levels. 4. Pylon–rotor interaction noise For Pylon–rotor interaction noise, only the interaction of the wake with the front rotor is considered. The rotor–rotor interaction noise model developed in the previous sections is applied with the pylon modelled as a stationary, one-bladed upstream rotor and noise being produced solely by the downstream (front) open rotor. Ricouard et al. [16] show that the tone noise produced by the interaction of a pylon wake with a contra-rotating open rotor is dominated by interaction noise from the front rotor. Thus it is possibly reasonable to assume that the same will be true for broadband interaction noise. For pylon–rotor interaction noise, B1 ¼1 and Ω1 ¼0 and therefore Z T 1 I t ¼ lim expfi½n−n′ þ k−k′Ω2 tgdt (67) T-∞ 2T −T Thus, It ¼δn′,n+k−k′. This yields the following expression for the radiated pressure spectral density:
2 Z Rt ∞ ∞ ρb1=2 2πB22 πU r2 Spp ðxo ; ωÞ ¼ ∑ ∑ Φww ðK X 2 e^ X 2 þ K Y 2 e^ Y 2 ÞjAn j2 dr; r ð4πRe Þ2 ð1−M x cosθe Þ2 n ¼ −∞ j ¼ −∞ Rh rcosα2 lnð2Þ where An ¼
(68)
n ∞ π o κrsinθe Gðkb ÞSJ n−k Ψ L ðK X 2 ; 0; ω−ðn−kÞΩ2 ; kc Þ ∑ exp ik ϕo − 2 1−M x cosθe k ¼ −∞
(69)
ω þ ðk−nÞΩ2 ; U r2
(70)
ω−nΩ2 ; U r2
(71)
K X2 ¼
K X2 ¼
K Y 2 ¼ K X 2 tanα2 þ
n þ jB2 ; rcosα2
(72)
S¼
κcosθe sinα2 ðk−nÞ c2 cosα2 ; þ r 1−M x cosθe 2
(73)
kc ¼
κcosθe cosα2 ðk−nÞ c2 sinα2 ; − r 1−Mx cosθe 2
(74)
(
) 2 kb Gðkb Þ ¼ exp − ; 4lnð2Þ kb ¼
kb1=2 : r
(75)
(76)
3966
M.J. Kingan / Journal of Sound and Vibration 332 (2013) 3956–3970
Assuming no blade to blade correlation, the expression can be simplified as before to yield
2 Z Rt ∞ ρb1=2 2πB2 πU r2 ð2Þ Spp ðxo ; ωÞ ¼ Φww ðK X 2 e^ X 2 ÞjAn j2 dr: ∑ r ð4πRe Þ2 ð1−M x cosθe Þ2 n ¼ −∞ Rh lnð2Þ
(77)
Note that the blade response function described in Appendix 1 is a piecewise function which contains a switch between a low and high frequency approximation. Thus, sound generated at a particular frequency will contain contributions from both the low and high frequency blade loading components and in particular, calculated noise levels at low frequencies will contain a significant contribution from both the low- and high-frequency blade loading models. This switch point has been selected to minimise the discontinuity between the response calculated for both frequency regimes, whilst ensuring that each model is applied at frequencies where it is reasonably valid. However, the discontinuity in the blade response model does introduce discontinuities in the calculated noise spectrum at very low frequencies. For this reason, predictions of pylon wake-rotor broadband interaction noise which are presented in this paper are only for frequencies higher than (approximately) the first blade passing frequency of the front rotor which do not appear to be significantly affected by this discontinuity. A solution to this discontinuity problem would be to calculate the blade response using a more sophisticated method such as the numerical collocation technique used in the code LINSUB [17]. The sound pressure level produced by the pylon–rotor broadband interaction noise source is highly directional and varies both with polar emission angle θe and azimuthal angle ϕo. This highly directional sound field is due to several effects. One effect is caused by the different phasing of the sound generated by the thrust and torque components of the unsteady lift force as the polar emission angle changes. The contribution of the thrust component of loading can be calculated by setting the second term in the square brackets in Eq. (73) equal to zero. Similarly, the contribution of the torque component to the
Fig. 5. SPL at different azimuthal positions. ϕo ¼ 01 (solid black with+markers), ϕo ¼901 (solid black) and ϕo ¼ −901 (dashed). Torque loading noise (plots (a), (d) and (g)), thrust loading noise (plots (b), (e) and (h)), total noise (plots (c), (f) and (i)). θe ¼451 (plots (a)–(c)), θe ¼901 (plots (d)–(f)), θe ¼ 1351(plots (g)–(i)). Twenty decibel per vertical tick.
M.J. Kingan / Journal of Sound and Vibration 332 (2013) 3956–3970
3967
sound field can be determined by setting the first term in the square brackets of the same equation equal to zero. The same open rotor design which was described in the previous section and which was operating at the same representative ‘take-off’ condition, was used to generate the noise predictions which are presented in this section. However, in this configuration a realistic pylon was installed upstream of the front rotor. Only the noise produced by the interaction of the pylon wake with the front rotor is calculated. Fig. 5 plots the ‘1 Hz bandwidth sound pressure level’ versus frequency at a number of different observer positions. The three rows correspond respectively to polar emission angles θe ¼451(plots (a)–(c)), θe ¼901 (plots (d)–(f)) and θe ¼1351 (plots (g)–(i)). The left-hand column (plots (a), (d) and (g)) shows the predicted sound pressure level including only the torque component of loading, the middle column (plots (b), (e) and (h)) contains only the thrust component while the righthand column (plots (c), (f) and (i)) shows the total sound field containing both the thrust and torque components. Each plot contains three curves corresponding to an observer azimuthal angle of ϕo ¼01(solid black line with+markers), ϕo ¼ 901(solid black line) and ϕo ¼−901 (dashed line). Note that the pylon is located at ϕ¼01. Firstly, considering only the left-hand column which plots the contribution from the torque component of loading. At all three polar emission angles, the sound pressure level at ϕo ¼901 is significantly higher than that at ϕo ¼−901. Note that the blade is moving towards the observer position at ϕo ¼901 and away from the observer position at ϕo ¼ −901 when it passes through the pylon wake which occupies a narrow range of angles close to ϕ ¼01. For the torque component there is maximum radiation towards ϕo ¼901, slightly lower radiation towards ϕo ¼−901 and minimum radiation close to the ϕo ¼01 and ϕo ¼1801 directions. This radiation pattern is clearly observed in Fig. 6 which shows the azimuthal directivity of the thrust and loading components of SPL at 1 kHz, 5 kHz and 10 kHz and at a polar emission angle of 701. It is observerd that both the torque and thrust components of noise radiate with maximum level towards ϕo ¼901, with lower levels at ϕo ¼−901. It is hypothesised that the torque component has a dipole like radiation pattern due to the loading only occurring over a small range of azimuthal angles. This is not observed for the thrust component of loading as the thrust force acts in the same direction relative to the observer, regardless of the azimuthal location of the blade. Now considering the middle column of Fig. 5 which plots the contribution from the thrust component of loading. At θe ¼ 451 and θe ¼1351, the same trend is observed. There is maximum radiation in the direction of blade motion through the pylon wake (ϕo ¼901) and minimum radiation in the opposite direction (ϕo ¼−901). The radiation at ϕo ¼01 appears to be at a level somewhere between these two cases. Note that there is no thrust noise contribution at θe ¼901. Turning our attention now to the right hand column of Fig. 5 which corresponds to the total sound pressure level produced by the pylon–rotor broadband noise source. At θe ¼ 901 the thrust component of loading makes no contribution to the overall sound field and thus, the total field is dominated by the torque loading component. However, at θe ¼ 451
Fig. 6. Plot of Spp in decibels as a function of ϕo at θe ¼ 701 and at a frequency of 6kHz. Twenty decibel per radial gridline. (a) Torque loading, (b) thrust loading.
3968
M.J. Kingan / Journal of Sound and Vibration 332 (2013) 3956–3970
inspection of Eq. (73) shows that the thrust component has the opposite sign to that at θe ¼1351, while the torque component remains the same. Thus, there is a change in the phase between the thrust and torque components of loading as the observer moves from θe ¼451 to θe ¼1351. This change in sign of the torque component significantly alters the directivity of the total field in the forward and rearward polar arcs. Fig. 7 plots the sound pressure level produced by the pylon–rotor interaction broadband noise source at a particular observer location. The front rotor blade number is allowed to vary between 3 and 36. Note that only blade number is altered— no change to the blade geometry or aerodynamics are taken into account. Both the full (Eq. (68)) and simple (Eq. (77)) models are used to calculate the radiated sound pressure level. Note that the simple model is in excellent agreement with the full model at all frequencies for blade numbers less than, or equal to, 12 and thus the pressure spectral density scales in proportion to the number of blades on the front rotor. However, at higher blade numbers, the correlation of the unsteady loading on adjacent rotor blades means that the full model must be used and the level does not simply scale with blade number.
Fig. 7. Effect of blade number on radiated noise level. B1 ¼3, 6, 12, 18, 36. Simple model (solid line with markers), full model (solid line). Five decibel per vertical tick spacing.
Fig. 8. Effect of pylon–rotor gap on radiated noise level. gr ¼g0 [0.4, 0.5, 1, 5]. Simple model (solid line with markers), full model (solid black line). Five decibel per vertical tick spacing.
M.J. Kingan / Journal of Sound and Vibration 332 (2013) 3956–3970
3969
Fig. 8 plots the variation of sound pressure level at a particular observer location as the pylon–rotor gap is increased relative to the original pylon–rotor gap which is denoted g0. As before, both the simple and full models are used to predict the noise level. As the gap increases, the wake width increases and thus the correlation length of the turbulence incident on the front rotor increases. This shifts the spectral peak towards lower frequencies increasing low frequency noise but decreasing high frequency noise. Note that if the pylon–rotor gap becomes too large, then the correlation length of the turbulence within the wake causes the loading on adjacent blades to become correlated in which case the full model is required to calculate the radiated noise field. 5. Conclusion A theoretical model for the prediction of rotor–rotor interaction broadband noise has been developed and extended so that it can be applied to predict broadband noise produced by the interaction of the turbulent wake from an upstream pylon with an open rotor. A brief parameter study has been conducted which shows the effect of varying parameters such as pylon–rotor and rotor–rotor spacing and blade number. The directivity of the sound field for the case of pylon–rotor interaction has also been investigated. A simple model, which assumes no correlation between the loading on adjacent blades has also been presented. This model is much simpler than the full model.
Acknowledgements The author gratefully acknowledges the continuing support of Rolls-Royce plc. by way of the UTC in gas turbine noise at the University of Southampton and also funding from SAGE. The author is also grateful for discussions with Vincent Blandeau. Appendix 1. Acoustically weighted lift functions The acoustically weighted lift function is defined Z Ψ L ðK X 2 ; 0; ω; kc Þ ¼
1 −1
ðmÞ
ðmÞ
gðX ðmÞ 2 ; K X 2 ; 0; ωÞexpf−ikc X 2 gdX 2 :
For low frequency gusts the blade response function is given by Amiet [18] vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ( ) u ðmÞ h i 1 u ðmÞ t 1−X 2 S s exp i s M2 X ðmÞ þ f ðM Þ ; gðX 2 ; K X 2 ; 0; ωÞ ¼ r 2 r2 2 πβr2 1 þ X ðmÞ β2r β2r2 2 2
where s ¼ K X 2 c2 =2, βr2 ¼
(A1)
jsjM r2 π o : 4 β2r2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1−M 2r2 ;
(A2)
(A3)
f ðM r2 Þ ¼ ð1−βr2 ÞlnðM r2 Þ þ βr2 lnð1 þ βr2 Þ−lnð2Þ
(A4)
and SðxÞ ¼
1 : ix½K 0 ðixÞ þ K 1 ðixÞ
(A5)
is the well-known Sears' function. For high frequency gusts the blade response function is given by Landhal [19] and Goldstein [4] n sM o ðmÞ r exp −i 1þM2r ð1 þ X 2 Þ þ is jsjMr2 π ðmÞ 2 4 : gðX 2 ; K X 2 ; 0; ωÞ ¼ pffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 ðmÞ 4 β r 2 π is πð1 þ M r2 Þð1 þ X 2 Þ
(A6)
Substituting and evaluating the integral yields ( ) !
1 s s Ψ L ðK X 2 ; 0; ω; kc Þ ¼ S 2 exp i 2 f ðM r2 Þ J 0 ðη′Þ−iJ 1 ðη′Þ ; βr2 βr2 βr2
jsjM r2 π o : 4 β2r2
(A7)
where η′ ¼
sM 2r2 β2r2
−kc ;
(A8)
3970
and
M.J. Kingan / Journal of Sound and Vibration 332 (2013) 3956–3970
pffiffiffi 2 pffiffiffiffi 2 expfikc þ isg Ψ L ðK X 2 ; 0; ω; kc Þ ¼ pffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffi En pffiffiffi μ′ ; π μ′ π 1 þ M r2 is
jsjM r2 π 4 : 4 β2r2
(A9)
where μ′ ¼
sM r2 þ kc ; 1 þ M r2
(A10)
and E[] denotes a complex Fresnel integral. References [1] A. Parry, M.J. Kingan, B.J. Tester Relative importance of open rotor tone and broadband noise sources, Proceedings of the 17th AIAA/CEAS Aeroacoustics Conference, Portland, US, 2011. [2] V.P. Blandeau, P.F. Joseph, B.J. Tester, Broadband noise prediction from rotor-wake interaction in contra-rotating propfans, Proceedings of the 15th AIAA/ CEAS Aeroacoustics Conference, Miami, USA, 2009. [3] V.P. Blandeau, P.F. Joseph, Broadband noise due to rotor-wake/rotor interaction in contra-rotating open rotors, AIAA Journal 48 (11) (2010) 2674–2686. [4] M.E. Goldstein, Aeroacoustics, McGraw-Hill, 1976. [5] D.B. Hanson, D.J. Parzych Theory of noise of propellers in angular inflow with parametric studies and experimental verification, Proceedings of NASA CR 4499, 1993. [6] D.B. Hanson, Sound from a propeller at angle of attack: a new theoretical viewpoint, Proceedings of the Royal Society of London A 449 (1995) 315–328. [7] R.H. Schlinker, R.K. Amiet Helicopter rotor trailing edge noise, Proceedings of NASA TR 3470, 1981. [8] V.P. Blandeau, P.F. Joseph, On the validity of Amiet's model for propeller trailing-edge noise, AIAA Journal 49 (2011) 1057–1066. [9] C.S. Ventres, M.A. Theobald, W.D. Mark, Turbofan Noise Generation, Volume 1: Analysis, Contractor Report CR-167952, NASA, 1982. [10] M. Nallasamy, E. Envia, Computation of rotor wake turbulence noise, Journal of Sound and Vibration 282 (2005) 649–678. [11] R.K. Amiet Noise produced by turbulent flow into a rotor: theory manual for noise calculation, Proceedings of NASA CR 181788, 1989. [12] A.B. Parry Theoretical prediction of counter-rotating propeller noise, PhD Thesis, University of Leeds, 1988. [13] A.B. Parry, Modular prediction scheme for blade-row interaction noise, Journal of Propulsion and Power 13 (1997) 334–341. [14] V.P. Blandeau Aerodynamic broadband noise from contra-rotating open rotors, PhD Thesis, University of Southampton, 2011. [15] I. Wygnanski, F. Champagne, B. Marasli, On the large-scale structures in two-dimensional, small-deficit, turbulent wakes, Journal of Fluid Mechanics 168 (1986) 31–71. [16] J. Ricouard, E. Julliard, M. Omais, V. Regnier, A.B. Parry, S. Baralon Installation effects on contra-rotating open rotor noise, Presented at 16th AIAA/CEAS Aeroacoustics Conference, Stockholm, AIAA Paper. 2010-3795, 2010. [17] D.S. Whitehead, Classical two-dimensional methods. In manual on aeroelasticity in axial-flow turbomachines, Unsteady Turbomachinery Dynamics 1 (1987) 1–22. [18] R.K. Amiet, Compressibility effects in unsteady thin-airfoil theory, AIAA Journal 12 (1974) 253–255. [19] M.J. Landahl, Unsteady transonic flow, Pergamon Press, New York, 1961.