Side branch resonators modelling with Green s function methods

Transcription

1 Side branch resonators modelling with Green s function methods E. Perrey-Debain, R. Maréchal, J.-M. Ville Laboratoire Roberval UMR 6253, Equipe Acoustique, Université de Technologie de Compiègne, France July 09-10, 2012 MATHmONDES 2012, Reading, U.-K. 1/22

2 Outline Introduction Impedance matrix and Green s function formalism Simplified models Low-frequency applications : Helmholtz and Herschel-Quincke resonators Medium-frequency applications : HQ-liner system for inlet fan noise reduction Conclusions 2/22

3 Introduction Side branch resonators are commonly used for engine exhaust noise control : (i) low-frequency applications with a single plane wave mode (automotive) (ii) medium-frequency applications and highly multimodal context (aeronautics). P inc P ref P tr Figure: Side branch resonator (cavity Ω) with two openings Γ. Numerical predictions : (i) 1D approximations (with length corrections); (ii) FEM, BEM -> computationally demanding (this includes mesh preparation etc...) especially in a highly multimodal context, lack of physical interpretation. 3/22

4 Impedance matrix In the frequency domain, the acoustic pressure must obey the Helmholtz equation p +k 2 p = 0, k = ω/c, and q = n p = 0 everywhere except on Γ. The Green s function for the rigid-wall cavity is given by the infinite series G Ω (r,r ) = n=0 φ n (r)φ n (r ) λ n λ where λ = ω 2. Eigenfunctions φ n are properly normalized so that application of the Green s theorem in the cavity yields p(r) = G Ω (r,r )q(r )dγ(r ) Γ 4/22

5 Impedance matrix We need to precompute a finite set of eigenfunctions and estimate the truncation error... Consider the eigenvector Φ n, the FE discretization of nth eigenmode φ n : A(λ n )Φ n = 0 where A(λ) = K λm After reduction to the interfacial nodes, we obtain the impedance matrix Z(λ) = I T Γ A 1 (λ)i Γ with Finally, A 1 (λ) = n=0 Φ n Φ T n λ n λ = ΦD(λ)ΦT p = Z(λ) Fq = G Ω q 5/22

6 Truncation Keeping the first N eigenmodes gives Z(λ) = ( ΦD(λ) Φ T ) N +R(λ). The correction term R remains weakly dependent on the frequency, so we can take the first order Taylor expansion R(λ) R(λ)+(λ λ) R λ +... The residual matrices are computed via R = I T Γ A 1 I Γ ( ΦD(λ) Φ T ) N λ R = I T Γ A 1 MA 1 I Γ ( Φ λ D Φ T ) N 6/22

7 Scattering matrix The theory starts by introducing the hard-walled duct Green s function G(r,r ψ l (x,y)ψl ) = (x,y ) e iβ l z z 2iβ l l=0 The transverse eigenmodes ψ l are solution of the boundary value problem ( 2 xx + 2 yy)ψ l +k 2 ψ l = β 2 l ψ l with n ψ l = 0 on the boundary line S. These modes are normalized as ψ l 2 ds = 1 S in particular ψ 0 = 1/ A d where A d is the cross section area of the main duct. For circular ducts : ψ l = N m,n J m (α m,n r)e imθ, β l = k 2 α 2 m,n 7/22

8 Scattering matrix The finite element discretization of the integral equation p(r) = G(r,r )q(r )dγ(r )+p I (r) Γ gives (r i is the FE node location) Ñ p i = G ij q j +pi I with G ij = G(r i,r ) φ j (r )dγ(r ) j=1 Γ The acoustic velocity is deduced from q = (G Ω G) 1 p I Note : i. The computation is not trivial as z G is discontinuous at z = z i. ii. The matrix G Ω G is of small size. 8/22

9 Simplified models : one opening Starting with one opening only, the impedance matrix relation can be averaged to give p = Z(λ) q where Z(λ) = 1 Ñ Ñ Ñ (Z(λ) F) ij i=1 j=1 By the same token, p = 1 W q + pi, where W = 2ikA d A and A denotes the area of the interface. An incident plane wave p I = e ikz produces a transmitted pressure field p = Te ikz with T = 1+ 1 W Z(λ) 1 9/22

10 Helmholtz resonators Figure: Classical (left); with extended neck (right). 10/22

11 Helmholtz resonator, classical TL (db) TL modèle mixte TL simplifié TL Ingard Z Zero 100 modes 1 mode 2 modes 10 modes 50 modes Frequency (Hz) Frequency (Hz) 11/22

12 Helmholtz resonator, with extended neck TL modèle mixte TL simplifié /22

13 Simplified models : two openings We consider a symmetric resonator connected to the main duct via two openings located at z = z 1 and z = z 2. ( ) ( ) ( ) p1 Z = 11 Z 12 q1 p 2 Z 12 Z 11 q 2 Moreover, ( p1 p 2 ) = 1 W ( 1 e ikl e ikl 1 ) ( q1 q 2 ) + ( p I 1 p I 2 ) where L = z 2 z 1. This gives T = 1+ 2W Z 11 2W Z 12 cos(kl)+(e 2ikL 1) (W Z 11 1) 2 (W Z 12 e ikl ) 2 Thus, no acoustic energy is transmitted if Z 2 12 Z 2 11 Asin(kL) ka d Z 12 = 0 13/22

14 Herschel-Quincke resonator 14/22

15 Herschel-Quincke resonator TL simplifié TL tube droit TL modèle mixte 50 TL (db) Frequency (Hz) 15/22

16 Fan noise Perforate sheet The fan noise is one of the dominant components at take-off and landing for aircraft with modern high bypass ratio turbofan engines : broadband noise + Blade Passing Frequency (BPF) tones 16/22

17 Fan noise Actual configuration... Collecteur Soufflante Tube HQ Redresseurs Model... B + A + A 17/22

18 Validation on a small size configuration 0.1 z (m) x (m) y (m) TL (db) Frequency (Hz) Matrix size CPU time (Matlab) Our model h 50 min FEM h 15 min 18/22

19 Optimal configuration (36 HQ tubes) SPL (db) TL (db) ka ka Incident Liner-HQ system 19/22

20 What does the liner do? Im(β0,n) Im(β20,n) Re(β0,n) Re(β20,n) Surface wave modes at 1530 Hz : 20/22

21 Influence of the number of tubes on the first BPF (iso-surface) Modal power (db) (7,1) (4,2) (8,1) (2,3) Modes (-5,2) (-9,1) (-3,3) 0 (Incident) 0 (Liner) Number of tubes /22

22 Conclusions and prospects The proposed Green s function based method allows to reduce the computational effort as only the acoustic velocity at the interface needs to be calculated. A very high number of propagative modes (few hundreds) can be handled easily on a single PC. Gives access to physical interpretation in the low-frequency regime. In prospect : - could be used for designing taylor-made resonators using optimization procedures. - viscosity effects should be included 22/22