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
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
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
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
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
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
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
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
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
Helmholtz resonators Figure: Classical (left); with extended neck (right). 10/22
Helmholtz resonator, classical TL (db) 60 50 40 30 20 TL modèle mixte TL simplifié TL Ingard Z 0.3 0.25 0.2 0.15 0.1 0.05 Zero 100 modes 1 mode 2 modes 10 modes 50 modes 0 10 0.05 0 60 70 80 90 100 110 120 Frequency (Hz) 0.1 60 70 80 90 100 110 120 Frequency (Hz) 11/22
Helmholtz resonator, with extended neck 50 45 TL modèle mixte TL simplifié 40 35 30 25 20 15 10 5 0 60 70 80 90 100 110 120 12/22
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
Herschel-Quincke resonator 14/22
Herschel-Quincke resonator 80 70 60 TL simplifié TL tube droit TL modèle mixte 50 TL (db) 40 30 20 10 0 0 100 200 300 400 500 600 700 800 Frequency (Hz) 15/22
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
Fan noise Actual configuration... Collecteur Soufflante Tube HQ Redresseurs Model... B + A + A 17/22
Validation on a small size configuration 0.1 z (m) 0.05 0 0.05 0.1 0 0.1 0.2 0.3 x (m) 0.4 0.05 0 0.05 0.1 y (m) TL (db) 40 35 30 25 20 15 10 5 0 0 500 1000 1500 2000 2500 3000 Frequency (Hz) Matrix size CPU time (Matlab) Our model 500 1 h 50 min FEM 82 000 31 h 15 min 18/22
Optimal configuration (36 HQ tubes) 100 12 98 10 96 8 SPL (db) 94 92 TL (db) 6 4 90 88 2 86 0 5 10 15 20 25 30 35 40 45 ka 0 0 5 10 15 20 25 30 35 40 45 ka Incident Liner-HQ system 19/22
What does the liner do? 18 45 16 40 14 35 Im(β0,n) 12 10 8 6 4 2 Im(β20,n) 30 25 20 15 10 5 0 0 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 Re(β0,n) Re(β20,n) Surface wave modes at 1530 Hz : 20/22
Influence of the number of tubes on the first BPF (iso-surface) Modal power (db) 90 70 50 30 (7,1) (4,2) (8,1) (2,3) Modes (-5,2) (-9,1) (-3,3) 0 (Incident) 0 (Liner) 36 30 45 4037 Number of tubes 72 6755 21/22
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