Boundary Conditions modeling Acoustic Linings in Flow

Transcription

1 Boundary Conditions modeling Acoustic Linings in Flow Dr Ed Brambley Mathematics Institute, and Warwick Manufacturing Group, University of Warwick SAPEM, 6th December 217, Le Mans, France p. 1

2 Aeroacoustics SAPEM, 6th December 217, Le Mans, France p. 2

3 Acoustic linings SAPEM, 6th December 217, Le Mans, France p. 3

4 Compliant The impedance of a surface Permeable ξ v Suppose a boundary with velocity v = ξ/ t obeys d 2 ξ = Kξ R ξ t2 t +T 2 ξ ξ x 2 B 4 x 4 +p. If p = pexp{iωt ikx} and v = ṽexp{iωt ikx}, ) p (dω ṽ = Z = R+i Kω Tk2 ω Bk4 ω Setting bending stiffness B and tension T to zero gives a mass spring damper model. No k dependence: locally reacting. For the Extended Helmholtz Resonator (EHR) model (Rienstra, 26 AIAA Paper), Z = R+idω iνcot(ωl iε/2). SAPEM, 6th December 217, Le Mans, France p. 4

5 Flow over an impedance surface M y ṽ, p δ x ṽ, p U(y) y = Z = p /ṽ SAPEM, 6th December 217, Le Mans, France p. 5

6 Flow over an impedance surface M y x ṽ, p ṽ, p U(y) δ y = Z = p /ṽ If ṽ = i(ω Uk) ξ, then d p dy = O(δ), d ξ dy = Q T i(ω Uk)T ikq u (ω Uk) 2 +O(δ). For inviscid flow (Eversman & Beckemeyer 1972 JASA; Tester 1973 JSV; Myers 198 JSV), or at high frequency (Aurégan et al. 21 JASA, Brambley 211 JFM), expect p ṽ = p i(ω Mk) ξ = ωz ω Mk SAPEM, 6th December 217, Le Mans, France p. 5

7 Sound in a cylinder with uniform flow r θ x M Linearized Euler equations (nondimensionalized so c 2 = 1): u = Me x + φ D 2 φ Dt 2 = 2 φ. p p = ρ ρ = Dφ Dt Solution: φ = AJ m (αr)exp{iωt ikx imθ} p = i(ω Mk)φ α 2 = (ω Mk) 2 k 2 Myers impedance boundary condition: D(k,ω) iωzαj m(α) (ω Mk) 2 J m (α) =. SAPEM, 6th December 217, Le Mans, France p. 6

8 Boundary value problem (given ω, findk) m ω 2 m ω 5 Surface Modes M =.5, ω = 1, Z = 2 i. Surface modes: Rienstra (23, WM), Brambley & Peake (26, WM). SAPEM, 6th December 217, Le Mans, France p. 7

9 Initial value problem (given k, findω) Frequency, Re(ω(k)) Growth rate, Im(ω(k)) k k Unbounded exponential growth rate Im(ω(k)) k 1/2. Mass spring damping boundary Z = idω ib/ω +R. This behaviour can be predicted in general (Brambley, 29 JSV). SAPEM, 6th December 217, Le Mans, France p. 8

10 Modified Boundary condition (1st order correction in δ) For uniform flow, φ = AJ m (αr)exp{iωt ikx imθ} where α 2 = (ω Mk) 2 k 2. Myers boundary condition (asymptotically correct to o(δ)): iωzαj m (α) ( ω Mk ) 2 Jm (α) =. SAPEM, 6th December 217, Le Mans, France p. 9

11 Modified Boundary condition (1st order correction in δ) For uniform flow, φ = AJ m (αr)exp{iωt ikx imθ} where α 2 = (ω Mk) 2 k 2. Myers boundary condition (asymptotically correct to o(δ)): iωzαj m (α) ( ω Mk ) 2 Jm (α) =. New modified boundary condition (asymptotically correct to o(δ 2 )): i ( ω U(1)k ) Z [ αj m(α) ( k 2 +m 2) δi 1 J m (α) ] (ω Mk) 2[ J m (α) αj m(α)δi ] =. SAPEM, 6th December 217, Le Mans, France p. 1

12 Modified Boundary condition (1st order correction in δ) For uniform flow, φ = AJ m (αr)exp{iωt ikx imθ} where α 2 = (ω Mk) 2 k 2. Myers boundary condition (asymptotically correct to o(δ)): iωzαj m (α) ( ω Mk ) 2 Jm (α) =. New modified boundary condition (asymptotically correct to o(δ 2 )): i ( ω U(1)k ) Z [ αj m(α) ( k 2 +m 2) δi 1 J m (α) ] (ω Mk) 2[ J m (α) αj m(α)δi ] =. SAPEM, 6th December 217, Le Mans, France p. 11

13 Modified Boundary condition (1st order correction in δ) For uniform flow, φ = AJ m (αr)exp{iωt ikx imθ} where α 2 = (ω Mk) 2 k 2. Myers boundary condition (asymptotically correct to o(δ)): iωzαj m (α) ( ω Mk ) 2 Jm (α) =. New modified boundary condition (asymptotically correct to o(δ 2 )): i ( ω U(1)k ) Z [ αj m(α) ( k 2 +m 2) δi 1 J m (α) ] (ω Mk) 2[ J m (α) αj m(α)δi ] =. where δi = 1 1 ( ω U(r)k ) 2R(r) (ω Mk) 2 dr, δi 1 = 1 1 (ω Mk) 2 ( ω U(r)k ) 2R(r) dr. SAPEM, 6th December 217, Le Mans, France p. 12

14 Modified Boundary condition (1st order correction in δ) For uniform flow, φ = AJ m (αr)exp{iωt ikx imθ} where α 2 = (ω Mk) 2 k 2. Myers boundary condition (asymptotically correct to o(δ)): iωzαj m (α) ( ω Mk ) 2 Jm (α) =. New modified boundary condition (asymptotically correct to o(δ 2 )): i ( ω U(1)k ) Z [ αj m(α) ( k 2 +m 2) δi 1 J m (α) ] (ω Mk) 2[ J m (α) αj m(α)δi ] =. where δi = 1 1 ( ω U(r)k ) 2R(r) (ω Mk) 2 dr, δi 1 = 1 1 (ω Mk) 2 ( ω U(r)k ) 2R(r) dr. See Brambley (211 AIAA J) for details. SAPEM, 6th December 217, Le Mans, France p. 13

15 Stability of the modified boundary condition Frequency, Re(ω(k)) Growth rate, Im(ω(k)) Numeric New BC Myers k k m =, Z = 3+.15iω 1.15i/ω. U(r) M tanh ( (1 r)/δ ) with M =.5 and δ = SAPEM, 6th December 217, Le Mans, France p. 14

16 Experimental evidence of instability 1: compressor, 2: flowmeter, 3: anechoic terminations, 4: microphones, 5: static pressure measurement, 6: acoustical source, 7: lined wall. Taken from Aurégan & Leroux (28, JSV). SAPEM, 6th December 217, Le Mans, France p. 15

17 Accuracy (from Gabard, 213 JSV) ω = 28, Z = 5 i, M =.55, δ =.14 ωδ =.39. SAPEM, 6th December 217, Le Mans, France p. 16

18 Accuracy (from Gabard, 213 JSV) θ Solid: Exact. Dashed: Myers. Blue: Brambley (211 AIAA J). Green & Red: Rienstra & Darau (211, JFM). SAPEM, 6th December 217, Le Mans, France p. 16

19 Accuracy (from Gabard, 213 JSV) Exact Myers Brambley (211 AIAA J) Rienstra & Darau (211, JFM) SAPEM, 6th December 217, Le Mans, France p. 16

20 Im(k) Im(k) Im(k) Im(k) Im(k) Im(k).17 1 Accuracy (Khamis & Brambley, JFM 216) Re(k) Re(k) Re(k) Re(k) Re(k) Re(k) Myers BC Modified BC 2nd order BC.5.6. Error is given by min ( Z 1 Z 2, 1/Z 1 1/Z 2 ). SAPEM, 6th December 217, Le Mans, France p. 17

21 Viscosity Aurégan, Starobinski & Pagneux (21, JASA): ( p = 1 (1 β v ) Mk v ω + β t = β v = 1 M 1 T T() ( ) T T() )β t T() U y (y)exp { y iρ()ω } dy 1 Z T y (y)exp { y iρ()prω } dy High frequency Low frequency β v and β t gives continuity of normal displacement. β v 1 and β t 1 gives continuity of normal mass flux. δ = U y () M ωρ()/2 SAPEM, 6th December 217, Le Mans, France p. 18

22 Importance of Viscosity (Renou & Aurégan, 211 JASA) SAPEM, 6th December 217, Le Mans, France p. 19

23 Importance of Viscosity (Renou & Aurégan, 211 JASA) SAPEM, 6th December 217, Le Mans, France p. 19

24 Viscous compressible governing equations Nondimensionalize based on a lengthscale (e.g. duct radius), the speed of sound, and the centreline density. At r =, find 1/µ = MRe and 1/κ = MRePr. Dimensionless governing equations: ρ t + (ρu) = ρ Du Dt = p+ σ ( ui σ ij = µ + u ) j + (µ B 23 ) x j x µ δ ij u i ρ DT Dt = Dp Dt +σ ij T = γ γ 1 u i + (κ T ) x j µ,µ B,κ linear in T and independent of p. p ρ SAPEM, 6th December 217, Le Mans, France p. 2

25 Viscosity within the boundary layer M y ṽ, p δ x ṽ, p U(y) y = Z = p /ṽ SAPEM, 6th December 217, Le Mans, France p. 21

26 Viscosity within the boundary layer Linearize about a steady flow and rescale inside boundary layer (δ = µ() 1/2 ): r = 1 δy u = (U+εũ)e x δεṽe r +ε we θ ε = ε exp{iωt ikx imθ} To leading order in δ, linearized Navier Stokes gives p = constant w = unconnected ρ = (γ 1)ρ 2 T i(ω Uk) T +T y ṽ Tṽ y +iktũ =, i(ω Uk)ũ+U y ṽ = (γ 1) 2 T ( Tũ y + TU y )y, [ i(ω Uk) T +T y ṽ = (γ 1) 2 T 1 ) Pr( TT yy + TU ] y 2 +2TU y ũ y. Boundary conditions for a fixed permeable boundary to leading order in δ: ũ() =, ṽ() = ṽ /δ, T() =, ũ(y), ṽ(y) ṽ /δ, T(y), as y. Note: For an impermeable flexible boundary, get ũ() = u y()ṽ iω. For details, see Brambley (211 JFM). SAPEM, 6th December 217, Le Mans, France p. 21

27 But... Re(ω) Im(ω) Myers 2 Viscous 2 3 Inviscid 1st order k k For Myers, Im(ω) = O(k 1/2 ). For viscous leading order, Im(ω) = O(k 1/3 ). See Brambley (211, JFM) for details. SAPEM, 6th December 217, Le Mans, France p. 22

28 Next order Viscous (Khamis & Brambley, 217 JFM) Linearized nondimensionalized governing equations: ( ) i(ω Uk)ˆT + iktû T 2 ˆv y = δ[γi(ω Uk)T p Tˆv imt w], T i(ω Uk)û + U yˆv (γ 1) 2 T(Tû y + U y ˆT)y = δ [i(γ 1)kT p (γ 1) 2 ] T(Tû y + U y ˆT), [ ( µ p y = δ 2(γ 1)(Tˆv y ) y + (γ 1) B µ 2 ) (Tˆv y iktû) y 3 ] i(ω Uk) (γ 1)T ˆv i(γ 1)k(Tû y + U y ˆT), (T w y ) y i(ω Uk) (γ 1) 2 T im w = p + O(δ), γ 1 i(ω Uk)ˆT + T yˆv (γ 1)2 T Pr (T ˆT) yy (γ 1) 2 T(Uy 2 ˆT + 2TU y û y ) = [ δ (γ 1)i(ω Uk)T p (γ 1)2 T Pr (T T) y ], Boundary conditions at O(1) : û () =, ˆT () =, ˆv ()Z = p, w () =, as y û (y), ˆT (y), w w u. Boundary conditions at O(δ) : û 1 () =, ˆT1 () =, ˆv 1 () =, as y û 1 (y) u u, ˆT1 (y) T u. SAPEM, 6th December 217, Le Mans, France p. 23

29 Effects of viscosity 2 Re = Re = Im(ω) 1 Re = Myers condition Sheared inviscid numerics Sheared viscous numerics k M =.5, δ = 5 1 3, m =, Z = 3+i(.15ω 1.15/ω), tanh boundary layer. SAPEM, 6th December 217, Le Mans, France p. 24

30 Effects of viscosity Im(k).3.2 Sheared viscous numerics O(δ) asymptotics.1 High frequency asymptotics Sheared inviscid numerics Myers condition Re(k) ω = 28, M =.5, δ = 2 1 3, Re = 5 1 6, m =, Z = 3+i(.15ω 1.15/ω), tanh boundary layer. SAPEM, 6th December 217, Le Mans, France p. 24

31 Re(ũ) Weakly viscous model (Khamis & Brambley, AIAA J 217) Inspired by a turbulent boundary layer (δ 1/ Re), we try δ Re 1/ Numerics Uniform (ũ u ) 1. Main layer (ũ m ) Sublayer (ũ s ) r r r ω = 15, k = 5+2ß, m = 6, M =.5, δ = 7 1 3, Re = Axial velocity SAPEM, 6th December 217, Le Mans, France p. 25

32 Effective impedance Z eff = ω ω Mk Z + (γ 1)T(1) iωre ku r (1) ω Z i ω (ω Mk)2 δi +(S 2 +S 3 )Z 1+i(k 2 +m 2 ) ωz (ω Mk) 2 δi 1 +S 1 Z +O(δ 2 ), where S 1 = S 2 = S 3 = ( (γ 1)T(1) k 2 +m 2 iωre (ω Mk) 2ikU r(1)δi 1 + iω σ ( (γ 1)T(1) ) ( 2 1 I µ iωre T(1) 2 δ 2 + σ 1+σ ( (γ 1)T(1) iωre ) 3 ( kur (1) ωt(1) + 151k3 32ω 3 U r(1) 3 + (2σ2 +4σ +1) ku r (1)T rr (1) (1+σ) 2 1 2U r (1) 2 T(1) ) i (γ 1)+ ωρ(1) (k2 +m 2 ), 5k2 4ω 2U r(1) 2 ), I µ δ k2 8ω 2 U r(1)u rr (1)+ k ω U rrr(1)+ T rrr(1) σ 3 T(1) (7σ +3) ku r (1) 3 (1+σ) 2 2ωT(1) (σ3 +σ 2 2σ 1) 2U r (1)T rr (1) σ(1+σ) 2 ωt(1) ), ωt(1) ρ(r)(ω U(r)k)2 (ω Mk) 2 δi = 1 (ω Mk) 2 dr, δi 1 = 1 ρ(r)(ω U(r)k) 2dr. I 1 ( µ δ 2 = ω 1 ω Uk 2Pr (T2 ) rrr +(TUr) 2 r + kt ) ω Uk (U rt) rr dr 1 SAPEM, 6th December 217, Le Mans, France p. 26

33 Effect on upstream modes.6.5 Myers condition Modified Myers condition Asymptotics Asymptotics, no sublayer LNSE numerics.4 Im(k) Re(k) ω = 28, m =, M =.5, δ = 2 1 3, Re = SAPEM, 6th December 217, Le Mans, France p. 27

34 Plane wave reflection (Khamis & Brambley, JASA 217) log 1 R 1 15 R 2log 1 RLNSE LNSE WV HF ASP LEE SO MM MY π/4 π/2 3π/4 π θ π/4 π/2 3π/4 π θ LNSE = Linearised Navier Stokes equations, WV = weakly viscous two-deck, HF = high frequency viscous, ASP = Aurégan, Starobinski & Pagneux, LEE = Linearised Euler equations (LEE), SO = Second Order Inviscid, MM = first order inviscid, MY = Myers. Re = 5 1 6, ω = 28, δ = , M =.55, Z = 5 i. SAPEM, 6th December 217, Le Mans, France p. 28

35 Other models Non-asymptotic approximate solutions across the boundary layer (as yet all invsicid) (Ko, JSV 1972; Färm, Boij, Glav & Dazel, AA 216; Khamis & Brambley, JFM 216) Weak nonlinearity in the boundary layer (Petrie & Brambley, AIAA 217) Swirling flow (Masson, Mathews, Moreau, Posson & Brambley, JFM 217) Time domain formulation and numerical implementation (Brambley & Gabard, JCP 216) Still needed... Simple time-domain boundary condition, which is: simple (no convolution over space or past history); accurate (correct impedance eduction upstream and downstream); and has the correct stability. SAPEM, 6th December 217, Le Mans, France p. 29

36 Acknowledgements Gonville & Caius College The Royal Society Prof. Nigel Peake Cambridge DAMTP University of Cambridge Dr Doran Khamis Owen Petrie Dr Gwénaël Gabard Zoology DAMTP LAUM University of Oxford University of Cambridge Le Mans Université SAPEM, 6th December 217, Le Mans, France p. 3

37 Thank you for your attention! SAPEM, 6th December 217, Le Mans, France p. 31