Acoustics in lined ducts with sheared mean flow, with applications - PowerPoint PPT Presentation

Numerical results: further tests Test case borrowed from quantum-mech. potential well problem: Pridmore-Brown equation: P ′′ + β ( r, k ) P ′ + γ ( r, k ) P = 0 Quantisation condition based on high-freq. approximation � r 2 � γ ( r, k ) d r = ( n − 1 2 ) π, n = 1 , 2 , . . . r 1 µ k QC k 1 -60.470038 -60.4392 2 -55.761464 -55.7281 3 -51.134207 -51.0980 - 0.0000i 4 -46.605323 -46.5659 - 0.0003i 5 -42.195790 -42.1422 - 0.0212i 6 -37.931052 -37.5622 - 0.3254i k ’s for upstream-running modes. High-freq. approx. & numerical result: excellent agreement 14 / 47

Outline Background & motivation 1 Pridmore-Brown modes 2 Model & equations Numerical method: COLNEW and path-following Options for varying Z 3 WKB for slowly varying Z 4 New mode-matching method 5 Mode-matching basics Closed-form integrals of Helmholtz modes Closed-form integrals of radial Pridmore-Brown modes Mode-matching based on closed form integrals of PB modes Numerical results: comparing CMM and BLM Conclusions 6 Epilogue 15 / 47

Options for varying Z General solution by sum over modes µ max � mµ x � � mµ ( r ) e i k + mµ x + A − A + mµ P + mµ P − mµ ( r ) e i k − p m ( r, x ) = µ =1 Classic option for (piecewise) varying Z is Mode Matching . 16 / 47

Options for varying Z General solution by sum over modes µ max � mµ x � � mµ ( r ) e i k + mµ x + A − A + mµ P + mµ P − mµ ( r ) e i k − p m ( r, x ) = µ =1 Classic option for (piecewise) varying Z is Mode Matching . This is efficient and well-established (BAHAMAS ◭ NLR) for no-flow and uniform flow conditions, mainly because exact solutions of PB equation ( P mµ = J m Bessel functions), exact modal inner products (integrals) at interfaces. 16 / 47

Options for varying Z General solution by sum over modes µ max � mµ x � � mµ ( r ) e i k + mµ x + A − A + mµ P + mµ P − mµ ( r ) e i k − p m ( r, x ) = µ =1 Classic option for (piecewise) varying Z is Mode Matching . This is efficient and well-established (BAHAMAS ◭ NLR) for no-flow and uniform flow conditions, mainly because exact solutions of PB equation ( P mµ = J m Bessel functions), exact modal inner products (integrals) at interfaces. Questions for non-uniform mean flow: What can we do with a slowly varying impedance? Can we improve the efficiency of the mode-matching? 16 / 47

Outline Background & motivation 1 Pridmore-Brown modes 2 Model & equations Numerical method: COLNEW and path-following Options for varying Z 3 WKB for slowly varying Z 4 New mode-matching method 5 Mode-matching basics Closed-form integrals of Helmholtz modes Closed-form integrals of radial Pridmore-Brown modes Mode-matching based on closed form integrals of PB modes Numerical results: comparing CMM and BLM Conclusions 6 Epilogue 17 / 47

WKB for slowly varying Z Slowly varying modes: Assumptions Z ( x ) has an inherent length scale L ≫ d , no sudden changes. We rewrite ε = d Z := Z ( εx ) = Z ( X ) , L ≪ 1 . X = εx. No modal interaction (reflection, cut-on/cut-off, etc) Mode, slowly varying in axial direction (WKB Ansatz) � i � X � p m ( r, X ) = P ( r, X ) exp ˜ κ ( η )d η ε 0 Eigenfunction P ( r, X ) and wave number κ ( X ) to be found. 18 / 47

WKB for slowly varying Z Expand in ε P ( r, X ) = P 0 ( r, X ) + εP 1 ( r, X ) + O ( ε 2 ) To leading order, the slowly varying mode of order m, µ P 0 ( r, X ) = N ( X ) ψ mµ ( r, X ) , with κ = κ mµ ( X ) where ψ mµ ( r, X ) and κ mµ ( X ) are modal solutions per X N ( X ) is found from solvability condition for P 1 , eventually leading to � � X � f ( η ) N ( X ) 2 = N 2 0 exp − g ( η ) d η 0 where f ( X ) , g ( X ) are complicated but explicit functions of X , ω , u 0 , ρ 0 , c 0 , Z ( X ) , ψ mµ , and κ mµ . 19 / 47

Numerical results: linear Z ( X ) Linear Z ( X ) constant impedance 0.15 0.1 0.05 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 BAHAMAS 0.15 0.1 0.05 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 WKB 0.15 0.1 0.05 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Z/ρ ∞ c ∞ varies linearly from 1 . 5 − i to 1 . 5 + i . BAHAMAS: 10 segments. ⇒ x -dependency of Z is important ⇒ BAHAMAS and WKB agree well ( ε ≈ 0 . 2 ) 20 / 47

Numerical results: non-uniform flow velocity Non-uniform velocity BAHAMAS 0.15 0.1 0.05 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (a) Uniform mean flow velocity with u 0 /c ∞ = 0 . 3 . WKB 0.15 0.1 0.05 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (b) u 0 /c ∞ = 0 . 3 · 4 3 (1 − 1 2 r 2 ) ωd/c ∞ = 10 , m = 2 , µ = 1 , Z/ρ ∞ c ∞ varies linearly from 1 . 5 − i to 1 . 5+i so ε ≈ 0 . 2 . ⇒ Non-uniformity of mean flow velocity is important 21 / 47

Numerical results: Helmholtz resonator (no resonance) Helmholtz resonator (no resonance) BAHAMAS 0.15 0.1 0.05 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 WKB 0.15 0.1 0.05 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ωd/c ∞ = 6 , m = 2 , µ = 1 , uniform mean flow velocity u 0 /c ∞ = 0 . 3 1 0 ⇒ No resonance and ε ≈ 0 . 3 : BAHAMAS Im(Z/( ρ 0 c 0 )) −1 and WKB show good agreement −2 WKB −3 BAHAMAS 0 0.2 0.4 0.6 0.8 1 22 / 47 x (m)

Numerical results: Helmholtz resonator (passing resonance) Helmholtz resonator (passing resonance) BAHAMAS 0.15 0.1 0.05 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 WKB 0.15 0.1 0.05 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ωd/c ∞ = 10 , m = 2 , µ = 1 , uniform mean flow velocity u 0 /c ∞ = 0 . 3 . 0 ⇒ Resonance: WKB assumptions not valid Im(Z/( ρ 0 c 0 )) −5 ( Z ( x ) not slowly varying, intermodal WKB BAHAMAS scattering) −10 −15 0 0.2 0.4 0.6 0.8 1 x (m) 23 / 47

Numerical results: strong temperature gradient Realistic APU exhaust: strong temperature gradient cool air inlet hard wall resistive sheet 1.6 1.4 exhaust 1.2 1 0.8 0.6 liner cavity 0.4 temperature mean flow velocity profile ¯ 0.2 T ( r ) profile ¯ u ( r ) 0 0.2 0.4 0.6 0.8 1 (a) APU exhaust duct geometry (b) Temperature profile T/T ∞ = 1 4 + 5 50( 3 with cool air inlet. � 1 + tanh � 4 − r ) �� . 8 24 / 47

Numerical results: strong temperature gradient WKB 0.15 0.1 0.05 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (a) WKB, µ = 1 . WKB 0.15 0.1 0.05 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (b) WKB, µ = 2 . ωd/c ∞ = 10 , m = 2 , uniform velocity u 0 /c ∞ = 0 . 3 , Z ( x ) /ρ ∞ c ∞ linear: 1 . 5 − i to 1 . 5 + i . ⇒ 2 different sound speeds: 2 concentric ducts Sound refracts from warm to cold Enhances effect of lining 25 / 47

Outline Background & motivation 1 Pridmore-Brown modes 2 Model & equations Numerical method: COLNEW and path-following Options for varying Z 3 WKB for slowly varying Z 4 New mode-matching method 5 Mode-matching basics Closed-form integrals of Helmholtz modes Closed-form integrals of radial Pridmore-Brown modes Mode-matching based on closed form integrals of PB modes Numerical results: comparing CMM and BLM Conclusions 6 Epilogue 26 / 47

Classical mode-matching Mode-Matching Basics a + a + b + a + a + l − 1 l l l +1 l +2 a − a − b − a − a − l − 1 l l +1 l +1 l +2 x l − 1 x l x l +1 Total field in segment l : sum of left- and right-running waves � l,µ ( x − x l ) � � ∞ l,µ ( r ) e i k + l,µ ( x − x l − 1 ) + a − l,µ ( r ) e i k − a + l,µ P + l,µ P − p l ( x, r ) = µ =1 (same for velocity) 27 / 47

Classical mode-matching Mode-Matching Basics a + a + b + a + a + l − 1 l l l +1 l +2 a − a − b − a − a − l − 1 l l +1 l +1 l +2 x l − 1 x l x l +1 At the interface at x = x l : µ max � � � b + l,µ P + l,µ ( r ) + a − l,µ P − p l ( r ) = l,µ ( r ) . µ =1 (same for velocity) 27 / 47

Classical mode-matching Mode-Matching Basics a + a + b + a + a + l − 1 l l l +1 l +2 a − a − b − a − a − l − 1 l l +1 l +1 l +2 x l − 1 x l x l +1 Continuity of pressure at x = x l leads to p l ( x l , r ) = p l +1 ( x l , r ) 27 / 47

Classical mode-matching Mode-Matching Basics a + a + b + a + a + l − 1 l l l +1 l +2 a − a − b − a − a − l − 1 l l +1 l +1 l +2 x l − 1 x l x l +1 Continuity of pressure at x = x l leads to µ max � � � b + l,µ P + + a − l,µ P − l,µ l,µ µ =1 µ max � � � a + l +1 ,µ P + + b − l +1 ,µ P − = l +1 ,µ l +1 ,µ µ =1 27 / 47

Classical mode-matching Mode-Matching Basics a + a + b + a + a + l − 1 l l l +1 l +2 a − a − b − a − a − l − 1 l l +1 l +1 l +2 x l − 1 x l x l +1 inner products with suitable test functions Ψ ν , e.g. = J m ( α ν r ) µ max � � � b + l,µ ( P + l,µ , Ψ ν ) + a − l,µ ( P − l,µ , Ψ ν ) µ =1 µ max � � � a + l +1 ,µ ( P + l +1 ,µ , Ψ ν ) + b − l +1 ,µ ( P − = l +1 ,µ , Ψ ν ) µ =1 27 / 47

Classical mode-matching Mode-Matching Basics a + a + b + a + a + l − 1 l l l +1 l +2 a − a − b − a − a − l − 1 l l +1 l +1 l +2 x l − 1 x l x l +1 inner products with suitable test functions Ψ ν , e.g. = J m ( α ν r ) µ max � � � b + l,µ ( P + l,µ , Ψ ν ) + a − l,µ ( P − l,µ , Ψ ν ) µ =1 µ max � � � a + l +1 ,µ ( P + l +1 ,µ , Ψ ν ) + b − l +1 ,µ ( P − = l +1 ,µ , Ψ ν ) µ =1 Similar for continuity of axial velocity. 27 / 47

Classical mode-matching Mode-Matching Basics a + a + b + a + a + l − 1 l l l +1 l +2 a − a − b − a − a − l − 1 l l +1 l +1 l +2 x l − 1 x l x l +1 Results in linear system to be solved � A + � � b + � � B + � � a + � A − B − l l +1 = . C + C − a − D + D − b − l l +1 27 / 47

Computing inner products Matrix entries are inner products � d A ± νµ = ( P ± P ± l,µ , Ψ ν ) = l,µ ( r )Ψ ν ( r ) r d r 0 28 / 47

Computing inner products Matrix entries are inner products � d A ± νµ = ( P ± P ± l,µ , Ψ ν ) = l,µ ( r )Ψ ν ( r ) r d r 0 Note that for non-uniform flow: P ± l,µ is determined numerically All inner-products have to be determined at all interfaces by quadrature P ± l,µ and Ψ ν are oscillatory ⇒ numerical problems 28 / 47

Computing inner products Matrix entries are inner products � d A ± νµ = ( P ± P ± l,µ , Ψ ν ) = l,µ ( r )Ψ ν ( r ) r d r 0 Note that for non-uniform flow: P ± l,µ is determined numerically All inner-products have to be determined at all interfaces by quadrature P ± l,µ and Ψ ν are oscillatory ⇒ numerical problems Problem Computing inner products numerically is expensive / less accurate 28 / 47

Computing inner products Matrix entries are inner products � d A ± νµ = ( P ± P ± l,µ , Ψ ν ) = l,µ ( r )Ψ ν ( r ) r d r 0 Note that for non-uniform flow: P ± l,µ is determined numerically All inner-products have to be determined at all interfaces by quadrature P ± l,µ and Ψ ν are oscillatory ⇒ numerical problems Problem Computing inner products numerically is expensive / less accurate e 1.000.000 question Can we find closed-form expressions for the inner-product? 28 / 47

Computing inner products Matrix entries are inner products � d A ± νµ = ( P ± P ± l,µ , Ψ ν ) = l,µ ( r )Ψ ν ( r ) r d r 0 Note that for non-uniform flow: P ± l,µ is determined numerically All inner-products have to be determined at all interfaces by quadrature P ± l,µ and Ψ ν are oscillatory ⇒ numerical problems Problem Computing inner products numerically is expensive / less accurate e 1.000.000 question Can we find closed-form expressions for the inner-product? No 28 / 47

Computing inner products Matrix entries are inner products � d A ± νµ = ( P ± P ± l,µ , Ψ ν ) = l,µ ( r )Ψ ν ( r ) r d r 0 Note that for non-uniform flow: P ± l,µ is determined numerically All inner-products have to be determined at all interfaces by quadrature P ± l,µ and Ψ ν are oscillatory ⇒ numerical problems Problem Computing inner products numerically is expensive / less accurate e 1.000.000 question Can we find closed-form expressions for other ‘inner-product’? 28 / 47

Computing inner products Matrix entries are inner products � d A ± νµ = ( P ± P ± l,µ , Ψ ν ) = l,µ ( r )Ψ ν ( r ) r d r 0 Note that for non-uniform flow: P ± l,µ is determined numerically All inner-products have to be determined at all interfaces by quadrature P ± l,µ and Ψ ν are oscillatory ⇒ numerical problems Problem Computing inner products numerically is expensive / less accurate e 1.000.000 question Can we find closed-form expressions for other ‘inner-product’? Yes! 28 / 47

From Classical to a New Mode-matching method Summary of new matching method Classical → new mode-matching ( P µ , Ψ ν ) → � F µ , Ψ ν � 29 / 47

From Classical to a New Mode-matching method Summary of new matching method Classical (CMM) → new (BLM) mode-matching ( P µ , Ψ ν ) → � F µ , Ψ ν � with Ψ ν = J m ( α ν r ) with Ψ ν = F ν , F = [ P, U, V, W ] 29 / 47

From Classical to a New Mode-matching method Summary of new matching method Classical (CMM) → new (BLM) mode-matching ( P µ , Ψ ν ) → � F µ , Ψ ν � � d � d � = P µ Ψ ν r d r = w 1 P µ P ν + w 2 U µ P ν → � 0 0 + w 3 ( V µ V ν + W µ W ν ) r d r with Ψ ν = J m ( α ν r ) with Ψ ν = F ν , F = [ P, U, V, W ] 29 / 47

From Classical to a New Mode-matching method Summary of new matching method Classical (CMM) → new (BLM) mode-matching ( P µ , Ψ ν ) → � F µ , Ψ ν � � d � d � = P µ Ψ ν r d r = w 1 P µ P ν + w 2 U µ P ν → � 0 0 + w 3 ( V µ V ν + W µ W ν ) r d r � P ν V µ − V ν P µ � i d = quadrature → k µ − k ν Ω ν r = d with Ψ ν = J m ( α ν r ) with Ψ ν = F ν , F = [ P, U, V, W ] 29 / 47

From Classical to a New Mode-matching method Summary of new matching method Classical (CMM) → new (BLM) mode-matching ( P µ , Ψ ν ) → � F µ , Ψ ν � � d � d � = P µ Ψ ν r d r = w 1 P µ P ν + w 2 U µ P ν → � 0 0 + w 3 ( V µ V ν + W µ W ν ) r d r � P ν V µ − V ν P µ � i d = quadrature → k µ − k ν Ω ν r = d with Ψ ν = J m ( α ν r ) with Ψ ν = F ν , F = [ P, U, V, W ] expensive cheap → less accurate accurate 29 / 47

Closed form integrals of 2D eigenmodes Prototype example of Generalised Prid-Brown : Helmholtz eqn ∇ 2 ψ + β 2 ψ = 0 on arbitrarily shaped cross-section A 30 / 47

Closed form integrals of 2D eigenmodes Prototype example of Generalised Prid-Brown : Helmholtz eqn ∇ 2 ψ + β 2 ψ = 0 ∇ 2 φ + α 2 φ = 0 on arbitrarily shaped cross-section A 30 / 47

Closed form integrals of 2D eigenmodes Prototype example of Generalised Prid-Brown : Helmholtz eqn � � ∇ 2 ψ + β 2 ψ φ = 0 � � ∇ 2 φ + α 2 φ ψ = 0 on arbitrarily shaped cross-section A 30 / 47

Closed form integrals of 2D eigenmodes Prototype example of Generalised Prid-Brown : Helmholtz eqn � � ∇ 2 ψ + β 2 ψ φ = 0 � � ∇ 2 φ + α 2 φ ψ = 0 on arbitrarily shaped cross-section A Subtract ( α 2 − β 2 ) φ ∇ 2 ψ − ψ ∇ 2 φ φψ = 30 / 47

Closed form integrals of 2D eigenmodes Prototype example of Generalised Prid-Brown : Helmholtz eqn � � ∇ 2 ψ + β 2 ψ φ = 0 � � ∇ 2 φ + α 2 φ ψ = 0 on arbitrarily shaped cross-section A Subtract and integrate over A �� �� ( α 2 − β 2 ) ( φ ∇ 2 ψ − ψ ∇ 2 φ ) d S φψ d S = A A 30 / 47

Closed form integrals of 2D eigenmodes Prototype example of Generalised Prid-Brown : Helmholtz eqn � � ∇ 2 ψ + β 2 ψ φ = 0 � � ∇ 2 φ + α 2 φ ψ = 0 on arbitrarily shaped cross-section A Subtract and integrate over A �� �� ( α 2 − β 2 ) ∇ · ( φ ∇ ψ − ψ ∇ φ ) d S φψ d S = A A 30 / 47

Closed form integrals of 2D eigenmodes Prototype example of Generalised Prid-Brown : Helmholtz eqn � � ∇ 2 ψ + β 2 ψ φ = 0 � � ∇ 2 φ + α 2 φ ψ = 0 on arbitrarily shaped cross-section A Subtract and integrate over A GAUSS �� �� ↓ ( α 2 − β 2 ) ∇ · ( φ ∇ ψ − ψ ∇ φ ) d S φψ d S = A A 30 / 47

Closed form integrals of 2D eigenmodes Prototype example of Generalised Prid-Brown : Helmholtz eqn � � ∇ 2 ψ + β 2 ψ φ = 0 � � ∇ 2 φ + α 2 φ ψ = 0 on arbitrarily shaped cross-section A Subtract and integrate over A GAUSS �� � ↓ ( α 2 − β 2 ) ( φ ∇ ψ · n − ψ ∇ φ · n ) d ℓ φψ d S = A Γ 30 / 47

Closed form integrals of 2D eigenmodes Prototype example of Generalised Prid-Brown : Helmholtz eqn � � ∇ 2 ψ + β 2 ψ φ = 0 � � ∇ 2 φ + α 2 φ ψ = 0 on arbitrarily shaped cross-section A Subtract and integrate over A �� � ( α 2 − β 2 ) ( φ ∇ ψ · n − ψ ∇ φ · n ) d ℓ φψ d S = A Γ 2D inner-product for Helmholtz eigenfunctions � 1 ( φ ∇ ψ · n − ψ ∇ φ · n )d ℓ, � � φ, ψ � � = α 2 − β 2 Γ for arbitrary boundary conditions on φ and ψ What if α = β and φ = ψ ? Something similar. 30 / 47

Closed form integrals of 1D eigenmodes Circular duct: Helmholtz equation → Bessel equation Substitute into 2D inner-product: φ = J m ( αr ) e i mθ , ψ = J m ( βr ) e − i mθ 31 / 47

Closed form integrals of 1D eigenmodes Circular duct: Helmholtz equation → Bessel equation Substitute into 2D inner-product: φ = J m ( αr ) e i mθ , ψ = J m ( βr ) e − i mθ 1D inner-product of Bessel functions � 1 � J m ( αr ) , J m ( βr ) � = J m ( αr ) J m ( βr ) r d r 0 � � 1 βJ m ( α ) J ′ m ( β ) − αJ ′ = m ( α ) J m ( β ) α 2 − β 2 31 / 47

Closed form integrals of 1D eigenmodes Circular duct: Helmholtz equation → Bessel equation Substitute into 2D inner-product: φ = J m ( αr ) e i mθ , ψ = J m ( βr ) e − i mθ 1D inner-product of Bessel functions � 1 � J m ( αr ) , J m ( βr ) � = J m ( αr ) J m ( βr ) r d r 0 � � 1 βJ m ( α ) J ′ m ( β ) − αJ ′ = m ( α ) J m ( β ) α 2 − β 2 If α = β : something similar. 31 / 47

Closed form integrals for Generalised P-B modes By analogous manipulations . . . 32 / 47

Closed form integrals for Generalised P-B modes By analogous manipulations . . . � � Define vector of shape functions F ( y, z ) = P, U, V, W P solution of Generalised PB equation, U, V, W follow from P 32 / 47

Closed form integrals for Generalised P-B modes By analogous manipulations . . . � � Define vector of shape functions F ( y, z ) = P, U, V, W P solution of Generalised PB equation, U, V, W follow from P Similarly to 2D Helmholtz example, it can be found: Closed form integral of parallel flow modes � F , � � F � � = �� � � �� � 1 u 0 k PP + ω � PU − ρ 0 u 0 ( � � V V + � + WW ) d S � ρ 0 c 2 ρ 0 � � Ω Ω Ω A 0 � P ( V n y + Wn z ) − ( � � V n y + � i Wn z ) P = d ℓ, k − � � k Ω Γ 32 / 47

Closed form integrals for Generalised P-B modes By analogous manipulations . . . � � Define vector of shape functions F ( y, z ) = P, U, V, W P solution of Generalised PB equation, U, V, W follow from P Similarly to 2D Helmholtz example, it can be found: Closed form integral of parallel flow modes � F , � � F � � = �� � � �� � 1 u 0 k PP + ω � PU − ρ 0 u 0 ( � � V V + � + WW ) d S � ρ 0 c 2 ρ 0 � � Ω Ω Ω A 0 � P ( V n y + Wn z ) − ( � � V n y + � i Wn z ) P = d ℓ, k − � � k Ω Γ Something similar for k = � k . 32 / 47

Closed form integrals for radial Pridmore-Brown modes Substitute for circular symmetric geometry. . . 33 / 47

Closed form integrals for radial Pridmore-Brown modes Substitute for circular symmetric geometry. . . modes of the form F ( r ) e ± i mθ F ( r ) = [ P ( r ) , U ( r ) , V ( r ) , W ( r )] 33 / 47

Closed form integrals for radial Pridmore-Brown modes Substitute for circular symmetric geometry. . . modes of the form F ( r ) e ± i mθ F ( r ) = [ P ( r ) , U ( r ) , V ( r ) , W ( r )] P solution of the radial Pridmore-Brown equation U, V, W follow from P 33 / 47

Closed form integrals for radial Pridmore-Brown modes Substitute for circular symmetric geometry. . . modes of the form F ( r ) e ± i mθ F ( r ) = [ P ( r ) , U ( r ) , V ( r ) , W ( r )] P solution of the radial Pridmore-Brown equation U, V, W follow from P Exact integrals of radial Pridmore-Brown modes � F , � F � = �� � � � d � 1 u 0 k P + ω P � U � P − ρ 0 u 0 ( V � V + W � + W ) r d r ρ 0 c 2 � ρ 0 � � Ω Ω Ω 0 0 � � � PV − � i d V P = k − � � k Ω r = d Weighted products of Pridmore-Brown eigenfunctions. Something similar for k = � k . 33 / 47

Some special cases Some special cases With Ingard-Myers condition (slipping flow) � � �� i d � � PP Ω Ω � F , � F � = Z − ( k − � k ) � � Ω ω Z r = d 34 / 47

Some special cases Some special cases With Ingard-Myers condition (slipping flow) � � �� i d � � PP Ω Ω � F , � F � = Z − ( k − � k ) � � Ω ω Z r = d For hard walls: � � F , � F � = 0 “orthogonal”: � F , F � � = 0 34 / 47

Some special cases Some special cases With Ingard-Myers condition (slipping flow) � � �� i d � � PP Ω Ω � F , � F � = Z − ( k − � k ) � � Ω ω Z r = d For hard walls: � � F , � F � = 0 “orthogonal”: � F , F � � = 0 In case of no-slip flow, u 0 ( d ) = 0 : � �� � 1 i d � PP Z − 1 � F , � F � = ( k − � � k ) ω Z r = d 34 / 47

Some special cases Some special cases With Ingard-Myers condition (slipping flow) � � �� i d � � PP Ω Ω � F , � F � = Z − ( k − � k ) � � Ω ω Z r = d For hard walls: � � F , � F � = 0 “orthogonal”: � F , F � � = 0 In case of no-slip flow, u 0 ( d ) = 0 : � �� � 1 i d � PP Z − 1 � F , � F � = ( k − � � k ) ω Z r = d For hard walls, or same impedance Z = � Z : � � F , � F � = 0 “orthogonal”: � F , F � � = 0 34 / 47

Bilinear map-based mode-matching Classic mode-matching (CMM) µ l � b + l,µ ( P + l,µ , Ψ ν ) + a − l,µ ( P − l,µ , Ψ ν ) µ =1 µ l +1 � a + l +1 ,µ ( P + l +1 ,µ , Ψ ν ) + b − l +1 ,µ ( P − = l +1 ,µ , Ψ ν ) µ =1 (same for velocity) with test functions (for example) Ψ ν = J m ( α ν r ) 35 / 47

Bilinear map-based mode-matching Classic mode-matching (CMM) µ l � b + l,µ ( P + l,µ , Ψ ν ) + a − l,µ ( P − l,µ , Ψ ν ) µ =1 µ l +1 � a + l +1 ,µ ( P + l +1 ,µ , Ψ ν ) + b − l +1 ,µ ( P − = l +1 ,µ , Ψ ν ) µ =1 (same for velocity) with test functions (for example) Ψ ν = J m ( α ν r ) Quadrature required for ( P µ , Ψ ν ) terms (non-uniform flow) 35 / 47

Bilinear map-based mode-matching Bilinear map-based (BLM) mode-matching µ l � b + l,µ � F + l,µ , Ψ ν � + a − l,µ � F − l,µ , Ψ ν � µ =1 µ l +1 � a + l +1 ,µ � F + l +1 ,µ , Ψ ν � + b − l +1 ,µ � F − = l +1 ,µ , Ψ ν � µ =1 but now as test functions the same modes: Ψ ν = F l,ν 35 / 47

Bilinear map-based mode-matching Bilinear map-based ∗ (BLM) mode-matching µ l � b + l,µ � F + l,µ , Ψ ν � + a − l,µ � F − l,µ , Ψ ν � µ =1 µ l +1 � a + l +1 ,µ � F + l +1 ,µ , Ψ ν � + b − l +1 ,µ � F − = l +1 ,µ , Ψ ν � µ =1 but now as test functions the same modes: Ψ ν = F l,ν ∗ Technically not an inner-product, except for no flow or uniform flow. 35 / 47

Bilinear map-based mode-matching Bilinear map-based ∗ (BLM) mode-matching µ l � b + l,µ � F + l,µ , Ψ ν � + a − l,µ � F − l,µ , Ψ ν � µ =1 µ l +1 � a + l +1 ,µ � F + l +1 ,µ , Ψ ν � + b − l +1 ,µ � F − = l +1 ,µ , Ψ ν � µ =1 but now as test functions the same modes: Ψ ν = F l,ν No extra calculations and � F µ , Ψ ν � in closed form ∗ Technically not an inner-product, except for no flow or uniform flow. 35 / 47

Numerical results Comparing CMM and BLM Test configurations Length: 1m Radius: 15cm hard wall – soft wall, interface at x = 0.5m µ max = 50 modes in both directions Configuration I II III Helmholtz & m ωd/c ∞ = 13 . 86 , m = 5 ωd/c ∞ = 8 . 86 , m = 5 ωd/c ∞ = 15 , m = 5 T 0 /T ∞ = 2 log(2)(1 − r 2 Temperature T 0 /T ∞ = 1 T 0 /T ∞ = 1 2 ) 3 (1 − r 2 u 0 /c ∞ = 0 . 5 · (1 − r 2 ) u 0 /c ∞ = 0 . 3 · 4 Mean flow 2 ) u 0 /c ∞ = 0 . 3 · tanh(10(1 − r )) Impedance Z/ρ ∞ c ∞ = 1 − i Z/ρ ∞ c ∞ = 1 + i Z/ρ ∞ c ∞ = 1 − i Incident mode µ = 1 µ = 1 µ = 2 36 / 47

Numerical results — Conf I: no-slip flow, uniform temp Real part of pressure 0.15 1 0.5 0.1 r (m) 0 0.05 −0.5 −1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x (m) (a) Classical mode-matching. 0.15 1 0.5 0.1 r (m) 0 0.05 −0.5 −1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x (m) (b) Bilinear map-based mode-matching. Perfect match between BLM and CMM results 37 / 47

Numerical results — Conf I: no-slip flow, uniform temp Pressure at r = { 0 . 035 , 0 . 075 , 0 . 15 } m . 1.5 1 0.5 Re(P) (dimless) 0 −0.5 Re(P) (BLM), r=0.035m Re(P) (CMM), r=0.035m Re(P) (BLM), r=0.075m −1 Re(P) (CMM), r=0.075m Re(P) (BLM), r=0.15m Re(P) (CMM), r=0.15m −1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x(m) Perfect match between BLM and CMM results 38 / 47

Numerical results — Conf I: no-slip flow, uniform temp Axial velocity at r = { 0 . 035 , 0 . 075 , 0 . 15 } m . 1 0.8 0.6 0.4 Re(U) (dimless) 0.2 0 −0.2 Re(U) (BLM), r=0.035m −0.4 Re(U) (CMM), r=0.035m Re(U) (BLM), r=0.075m −0.6 Re(U) (CMM), r=0.075m Re(U) (BLM), r=0.15m −0.8 Re(U) (CMM), r=0.15m −1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x(m) Perfect match between BLM and CMM results 38 / 47

Numerical results — Conf I: no-slip flow, uniform temp Radial velocity at r = { 0 . 035 , 0 . 075 , 0 . 15 } m . 0.5 0.4 0.3 0.2 Re(V) (dimless) 0.1 0 −0.1 Re(V) (BLM), r=0.035m Re(V) (CMM), r=0.035m −0.2 Re(V) (BLM), r=0.075m Re(V) (CMM), r=0.075m −0.3 Re(V) (BLM), r=0.15m Re(V) (CMM), r=0.15m −0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x(m) Perfect match between BLM and CMM results 38 / 47

Numerical results — Conf II: slipping flow, uniform temp Pressure at r = { 0 . 035 , 0 . 075 , 0 . 15 } m . 1.5 1 0.5 Re(P) (dimless) 0 −0.5 Re(P) (BLM), r=0.035m Re(P) (CMM), r=0.035m Re(P) (BLM), r=0.075m −1 Re(P) (CMM), r=0.075m Re(P) (BLM), r=0.15m Re(P) (CMM), r=0.15m −1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x(m) Perfect match between BLM and CMM results 39 / 47

Numerical results — Conf II: slipping flow, uniform temp Axial velocity at r = { 0 . 035 , 0 . 075 , 0 . 15 } m . 0.8 0.6 0.4 Re(U) (dimless) 0.2 0 −0.2 Re(U) (BLM), r=0.035m Re(U) (CMM), r=0.035m −0.4 Re(U) (BLM), r=0.075m Re(U) (CMM), r=0.075m −0.6 Re(U) (BLM), r=0.15m Re(U) (CMM), r=0.15m −0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x(m) Perfect match between BLM and CMM results 39 / 47

Numerical results — Conf II: slipping flow, uniform temp Radial velocity at r = { 0 . 035 , 0 . 075 , 0 . 15 } m . 0.3 0.2 0.1 Re(V) (dimless) 0 −0.1 −0.2 Re(V) (BLM), r=0.035m Re(V) (CMM), r=0.035m −0.3 Re(V) (BLM), r=0.075m Re(V) (CMM), r=0.075m −0.4 Re(V) (BLM), r=0.15m Re(V) (CMM), r=0.15m −0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x(m) Perfect match between BLM and CMM results 39 / 47

Numerical results — Conf III: bndary layer, non-unif. temp Pressure at r = { 0 . 035 , 0 . 075 , 0 . 15 } m . 1.5 1 0.5 Re(P) (dimless) 0 −0.5 Re(P) (BLM), r=0.035m Re(P) (CMM), r=0.035m Re(P) (BLM), r=0.075m −1 Re(P) (CMM), r=0.075m Re(P) (BLM), r=0.15m Re(P) (CMM), r=0.15m −1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x(m) Perfect match between BLM and CMM results 40 / 47

Numerical results — Conf III: bndary layer, non-unif. temp Axial velocity at r = { 0 . 035 , 0 . 075 , 0 . 15 } m . 0.8 0.6 0.4 Re(U) (dimless) 0.2 0 −0.2 Re(U) (BLM), r=0.035m Re(U) (CMM), r=0.035m −0.4 Re(U) (BLM), r=0.075m Re(U) (CMM), r=0.075m −0.6 Re(U) (BLM), r=0.15m Re(U) (CMM), r=0.15m −0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x(m) Perfect match between BLM and CMM results 40 / 47

Numerical results — Conf III: bndary layer, non-unif. temp Radial velocity at r = { 0 . 035 , 0 . 075 , 0 . 15 } m . 0.8 0.6 0.4 Re(V) (dimless) 0.2 0 Re(V) (BLM), r=0.035m −0.2 Re(V) (CMM), r=0.035m Re(V) (BLM), r=0.075m Re(V) (CMM), r=0.075m −0.4 Re(V) (BLM), r=0.15m Re(V) (CMM), r=0.15m −0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x(m) Perfect match between BLM and CMM results 40 / 47