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

Acoustics in lined ducts with sheared mean flow, with applications for aircraft noise

Sjoerd Rienstra & Martien Oppeneer with major contributions from Pieter Sijtsma, Bob Mattheij, Werner Lazeroms TU/e, 31 March 2015

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Summary

Summary Acoustic modes in flow ducts, with radial mean flow and temperature gradients.

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Summary

Summary Acoustic modes in flow ducts, with radial mean flow and temperature gradients. Boundary condition (Helmholtz resonator-type) varies axially.

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Summary

Summary Acoustic modes in flow ducts, with radial mean flow and temperature gradients. Boundary condition (Helmholtz resonator-type) varies axially. Mode matching works well for uniform flow, but here?

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Summary

Summary Acoustic modes in flow ducts, with radial mean flow and temperature gradients. Boundary condition (Helmholtz resonator-type) varies axially. Mode matching works well for uniform flow, but here? Slowly-varying mode approximation works well if Z not passing resonance.

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Summary

Summary Acoustic modes in flow ducts, with radial mean flow and temperature gradients. Boundary condition (Helmholtz resonator-type) varies axially. Mode matching works well for uniform flow, but here? Slowly-varying mode approximation works well if Z not passing resonance. Efficient and accurate new Mode-Matching method based

• n exact integrals of modes.

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Summary

Summary Acoustic modes in flow ducts, with radial mean flow and temperature gradients. Boundary condition (Helmholtz resonator-type) varies axially. Mode matching works well for uniform flow, but here? Slowly-varying mode approximation works well if Z not passing resonance. Efficient and accurate new Mode-Matching method based

• n exact integrals of modes.

Published in: AIAA-2011-2871, AIAA-2013-2172.

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Outline

1

Background & motivation

2

Pridmore-Brown modes

3

Options for varying Z

4

WKB for slowly varying Z

5

New mode-matching method

6

Conclusions

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Outline

1

Background & motivation

2

Pridmore-Brown modes Model & equations Numerical method: COLNEW and path-following

3

Options for varying Z

4

WKB for slowly varying Z

5

New mode-matching method 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

6

Conclusions Epilogue

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Background / motivation

APU: Auxiliary Power Unit

produces power when main engines are switched off to start main engines, AC, ... major source of ramp noise APU on an Airbus A380

Study sound propagation & attenuation in APU exhaust duct.

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Modelling assumptions

Modelling assumptions

hard wall resistive sheet liner cavity cool air inlet exhaust temperature profile T0(r) mean flow velocity profile u0(r)

straight, circular, hollow exhaust duct

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Modelling assumptions

Modelling assumptions

hard wall resistive sheet liner cavity cool air inlet exhaust temperature profile T0(r) mean flow velocity profile u0(r)

straight, circular, hollow exhaust duct non-uniform parallel mean flow (axially varying)

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Modelling assumptions

Modelling assumptions

hard wall resistive sheet liner cavity cool air inlet exhaust temperature profile T0(r) mean flow velocity profile u0(r)

straight, circular, hollow exhaust duct non-uniform parallel mean flow (axially varying) strong temperature gradients (axially varying)

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Modelling assumptions

Modelling assumptions

hard wall resistive sheet liner cavity cool air inlet exhaust temperature profile T0(r) mean flow velocity profile u0(r)

straight, circular, hollow exhaust duct non-uniform parallel mean flow (axially varying) strong temperature gradients (axially varying) segmented liner ⇒ slowly varying or mode-matching Euler eqn. & perfect gas: p = ρRT, c2 = γRT

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Outline

1

Background & motivation

2

Pridmore-Brown modes Model & equations Numerical method: COLNEW and path-following

3

Options for varying Z

4

WKB for slowly varying Z

5

New mode-matching method 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

6

Conclusions Epilogue

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Ordinary & Generalised Pridmore-Brown equation

For perturbations p1, ρ1, v1 of a parallel mean flow v0 = u0(y, z)ex, ρ0 = ρ0(y, z), c0 = c0(y, z), p0 = const

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Ordinary & Generalised Pridmore-Brown equation

For perturbations p1, ρ1, v1 of a parallel mean flow v0 = u0(y, z)ex, ρ0 = ρ0(y, z), c0 = c0(y, z), p0 = const the Linearised Euler equations can be reduced to: Generalised Pridmore-Brown equation (arbitrary cross-section) For modes of the form p1(x, y, z, t) = P(y, z) eikx−iωt: ∇· c2 Ω2 ∇P

• +
• 1 − k2c2

Ω2

• P = 0,

Ω = ω − ku0

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Ordinary & Generalised Pridmore-Brown equation

For perturbations p1, ρ1, v1 of a parallel mean flow v0 = u0(y, z)ex, ρ0 = ρ0(y, z), c0 = c0(y, z), p0 = const the Linearised Euler equations can be reduced to: Generalised Pridmore-Brown equation (arbitrary cross-section) For modes of the form p1(x, y, z, t) = P(y, z) eikx−iωt: ∇· c2 Ω2 ∇P

• +
• 1 − k2c2

Ω2

• P = 0,

Ω = ω − ku0 Ordinary Pridmore-Brown equation (circular cross-section) For u0(r), ρ0(r), c0(r) and p1(x, r, θ, t) = P(r) eikx−iωt+imθ: P ′′ + 1 r + 2c′ c0 + 2ku′ Ω

• P ′ +

Ω2 c2 − k2 − m2 r2

• P = 0

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Boundary conditions

hard wall facing sheet / vortex sheet liner: duct: u0 = c∞M(r) LPridBrown(P) = 0 u0 = 0 centerline r = d r = 0 honeycomb liner: Z = Z(ω) bulk absorber: Z = Z(ω, k)

Ingard-Myers boundary condition for slipping flow: −iω(v1·n)Z = (−iω + v0·∇)p1

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Boundary value problem

Pridmore-Brown equation P ′′ + 1 r + 2c′ c0 + 2ku′ Ω

• P ′ +

Ω2 c2 − k2 − m2 r2

• P = 0

+

Boundary conditions iωZP ′ = −ρ0Ω2P at r = d, P is regular at r = 0

=

Eigenvalue Problem in k Countable set of modal solutions: Pmµ(r) eikmµx−iωt+imθ eigenfunctions: Pmµ(r) eigenvalue (modal axial wavenumber): kmµ

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Boundary value problem

Pridmore-Brown equation P ′′ + 1 r + 2c′ c0 + 2ku′ Ω

• P ′ +

Ω2 c2 − k2 − m2 r2

• P = 0

+

Boundary conditions iωZP ′ = −ρ0Ω2P at r = d, P is regular at r = 0

=

Eigenvalue Problem in k Countable set of modal solutions: Pmµ(r) eikmµx−iωt+imθ eigenfunctions: Pmµ(r) eigenvalue (modal axial wavenumber): kmµ Non-uniform parallel flow: modes are found numerically

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Numerical solution: COLNEW

Write eigenvalue problem as boundary value problem: Add k to solution vector by adding equation k′ = 0 Fix P(r) at r = d

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Numerical solution: COLNEW

Write eigenvalue problem as boundary value problem: Add k to solution vector by adding equation k′ = 0 Fix P(r) at r = d COLNEW (NL-BVP software package available on netlib): Collocation at Gaussian points Runge-Kutta monomial basis representation Automatic meshing Damped Newton solver

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Numerical solution: COLNEW

Write eigenvalue problem as boundary value problem: Add k to solution vector by adding equation k′ = 0 Fix P(r) at r = d COLNEW (NL-BVP software package available on netlib): Collocation at Gaussian points Runge-Kutta monomial basis representation Automatic meshing Damped Newton solver Path-following/predictor-corrector/automatic step-size strategy from known solution to desired solution.

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Numerical approach: example of path-following

Example of path-following in Z

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Numerical results: eigenfunctions & eigenvalues

0.2 0.4 0.6 0.8 1 −4 −3 −2 −1 Re(P) r 0.2 0.4 0.6 0.8 1 −3 −2 −1 Im(P) r

(a) µ = 1

0.2 0.4 0.6 0.8 1 −4 −2 2 4 Re(P) r 0.2 0.4 0.6 0.8 1 −2 −1 1 2 Im(P) r

(b) µ = 2

0.2 0.4 0.6 0.8 1 −4 −2 2 4 Re(P) r 0.2 0.4 0.6 0.8 1 −2 −1 1 2 Im(P) r

(c) µ = 3

0.2 0.4 0.6 0.8 1 −4 −2 2 4 Re(P) r 0.2 0.4 0.6 0.8 1 −2 −1 1 2 Im(P) r

(d) µ = 4

0.2 0.4 0.6 0.8 1 −4 −2 2 4 Re(P) r 0.2 0.4 0.6 0.8 1 −2 −1 1 Im(P) r

(e) µ = 5

0.2 0.4 0.6 0.8 1 −4 −2 2 4 Re(P) r 0.2 0.4 0.6 0.8 1 −1 −0.5 0.5 1 Im(P) r

(f) µ = 6

Eigenfunctions for upstream-running modes, ωd/c∞ = 25, m = 5, Z/ρ∞c∞ = 2 − i, u0/c∞ = 2

3(1 − 1 2r2), uniform temperature.

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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 r2

r1

• γ(r, k) dr = (n − 1

2)π,

n = 1, 2, . . .

µ kQC 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

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Outline

1

Background & motivation

2

Pridmore-Brown modes Model & equations Numerical method: COLNEW and path-following

3

Options for varying Z

4

WKB for slowly varying Z

5

New mode-matching method 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

6

Conclusions Epilogue

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Options for varying Z

General solution by sum over modes pm(r, x) =

µmax

• µ=1
• A+

mµP + mµ(r) eik+

mµx +A−

mµP − mµ(r) eik−

mµx

Classic option for (piecewise) varying Z is Mode Matching.

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Options for varying Z

General solution by sum over modes pm(r, x) =

µmax

• µ=1
• A+

mµP + mµ(r) eik+

mµx +A−

mµP − mµ(r) eik−

mµx

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 (Pmµ = Jm Bessel functions), exact modal inner products (integrals) at interfaces.

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Options for varying Z

General solution by sum over modes pm(r, x) =

µmax

• µ=1
• A+

mµP + mµ(r) eik+

mµx +A−

mµP − mµ(r) eik−

mµx

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 (Pmµ = Jm 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?

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Outline

1

Background & motivation

2

Pridmore-Brown modes Model & equations Numerical method: COLNEW and path-following

3

Options for varying Z

4

WKB for slowly varying Z

5

New mode-matching method 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

6

Conclusions Epilogue

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WKB for slowly varying Z

Slowly varying modes:

Assumptions Z(x) has an inherent length scale L ≫ d, no sudden changes. We rewrite Z := Z(εx) = Z(X), ε = d L ≪ 1. X = εx. No modal interaction (reflection, cut-on/cut-off, etc) Mode, slowly varying in axial direction (WKB Ansatz) ˜ pm(r, X) = P(r, X) exp i ε X κ(η)dη

• Eigenfunction P(r, X) and wave number κ(X) to be found.

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WKB for slowly varying Z

Expand in ε P(r, X) = P0(r, X) + εP1(r, X) + O(ε2) To leading order, the slowly varying mode of order m, µ P0(r, X) = N(X)ψmµ(r, X), with κ = κmµ(X) where ψmµ(r, X) and κmµ(X) are modal solutions per X N(X) is found from solvability condition for P1, eventually leading to N(X)2 = N2

0 exp

• −

X f(η) g(η) dη

• where f(X), g(X) are complicated but explicit functions of

X, ω, u0, ρ0, c0, Z(X), ψmµ, and κmµ.

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Numerical results: linear Z(X)

Linear Z(X)

constant impedance 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.05 0.1 0.15 BAHAMAS 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.05 0.1 0.15 WKB 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.05 0.1 0.15

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)

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Numerical results: non-uniform flow velocity

Non-uniform velocity

BAHAMAS 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.05 0.1 0.15

(a) Uniform mean flow velocity with u0/c∞ = 0.3.

WKB 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.05 0.1 0.15

(b) u0/c∞ = 0.3 · 4

3(1 − 1 2r2)

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

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Numerical results: Helmholtz resonator (no resonance)

Helmholtz resonator (no resonance)

BAHAMAS 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.05 0.1 0.15 WKB 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.05 0.1 0.15

ωd/c∞ = 6, m = 2, µ = 1, uniform mean flow velocity u0/c∞ = 0.3

0.2 0.4 0.6 0.8 1 −3 −2 −1 1 x (m) Im(Z/(ρ0 c0)) WKB BAHAMAS

⇒ No resonance and ε ≈ 0.3: BAHAMAS and WKB show good agreement

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Numerical results: Helmholtz resonator (passing resonance)

Helmholtz resonator (passing resonance)

BAHAMAS 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.05 0.1 0.15 WKB 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.05 0.1 0.15

ωd/c∞ = 10, m = 2, µ = 1, uniform mean flow velocity u0/c∞ = 0.3.

0.2 0.4 0.6 0.8 1 −15 −10 −5 x (m) Im(Z/(ρ0 c0)) WKB BAHAMAS

⇒ Resonance: WKB assumptions not valid (Z(x) not slowly varying, intermodal scattering)

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Numerical results: strong temperature gradient

Realistic APU exhaust: strong temperature gradient

hard wall resistive sheet liner cavity cool air inlet exhaust temperature profile ¯ T(r) mean flow velocity profile ¯ u(r)

(a) APU exhaust duct geometry with cool air inlet.

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

(b) Temperature profile T/T∞ = 1

4 + 5 8

• 1 + tanh
• 50( 3

4 − r)

• .

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Numerical results: strong temperature gradient

WKB 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.05 0.1 0.15

(a) WKB, µ = 1.

WKB 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.05 0.1 0.15

(b) WKB, µ = 2.

ωd/c∞ = 10, m = 2, uniform velocity u0/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

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Outline

1

Background & motivation

2

Pridmore-Brown modes Model & equations Numerical method: COLNEW and path-following

3

Options for varying Z

4

WKB for slowly varying Z

5

New mode-matching method 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

6

Conclusions Epilogue

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Classical mode-matching

Mode-Matching Basics

a+

l

a−

l

b+

l

a−

l+1

b−

l+1

a+

l+1

a−

l−1

a+

l−1

a−

l+2

a+

l+2

xl xl+1 xl−1

Total field in segment l: sum of left- and right-running waves pl(x, r) =

∞

• µ=1
• a+

l,µP + l,µ(r) eik+

l,µ(x−xl−1) +a−

l,µP − l,µ(r) eik−

l,µ(x−xl)

(same for velocity)

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Classical mode-matching

Mode-Matching Basics

a+

l

a−

l

b+

l

a−

l+1

b−

l+1

a+

l+1

a−

l−1

a+

l−1

a−

l+2

a+

l+2

xl xl+1 xl−1

At the interface at x = xl: pl(r) =

µmax

• µ=1
• b+

l,µP + l,µ(r) + a− l,µP − l,µ(r)

• .

(same for velocity)

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Classical mode-matching

Mode-Matching Basics

a+

l

a−

l

b+

l

a−

l+1

b−

l+1

a+

l+1

a−

l−1

a+

l−1

a−

l+2

a+

l+2

xl xl+1 xl−1

Continuity of pressure at x = xl leads to pl(xl, r) = pl+1(xl, r)

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Classical mode-matching

Mode-Matching Basics

a+

l

a−

l

b+

l

a−

l+1

b−

l+1

a+

l+1

a−

l−1

a+

l−1

a−

l+2

a+

l+2

xl xl+1 xl−1

Continuity of pressure at x = xl leads to

µmax

• µ=1
• b+

l,µ P + l,µ

+ a−

l,µ P − l,µ

• =

µmax

• µ=1
• a+

l+1,µ P + l+1,µ

+ b−

l+1,µ P − l+1,µ

• 27 / 47

Classical mode-matching

Mode-Matching Basics

a+

l

a−

l

b+

l

a−

l+1

b−

l+1

a+

l+1

a−

l−1

a+

l−1

a−

l+2

a+

l+2

xl xl+1 xl−1

inner products with suitable test functions Ψν, e.g. = Jm(ανr)

µmax

• µ=1
• b+

l,µ(P + l,µ, Ψν) + a− l,µ(P − l,µ, Ψν)

• =

µmax

• µ=1
• a+

l+1,µ(P + l+1,µ, Ψν) + b− l+1,µ(P − l+1,µ, Ψν)

• 27 / 47

Classical mode-matching

Mode-Matching Basics

a+

l

a−

l

b+

l

a−

l+1

b−

l+1

a+

l+1

a−

l−1

a+

l−1

a−

l+2

a+

l+2

xl xl+1 xl−1

inner products with suitable test functions Ψν, e.g. = Jm(ανr)

µmax

• µ=1
• b+

l,µ(P + l,µ, Ψν) + a− l,µ(P − l,µ, Ψν)

• =

µmax

• µ=1
• a+

l+1,µ(P + l+1,µ, Ψν) + b− l+1,µ(P − l+1,µ, Ψν)

• Similar for continuity of axial velocity.

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Classical mode-matching

Mode-Matching Basics

a+

l

a−

l

b+

l

a−

l+1

b−

l+1

a+

l+1

a−

l−1

a+

l−1

a−

l+2

a+

l+2

xl xl+1 xl−1

Results in linear system to be solved A+ A− C+ C− b+

l

a−

l

• =

B+ B− D+ D− a+

l+1

b−

l+1

• .

27 / 47

Computing inner products

Matrix entries are inner products A±

νµ = (P ± l,µ, Ψν) =

d P ±

l,µ(r)Ψν(r)r dr

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Computing inner products

Matrix entries are inner products A±

νµ = (P ± l,µ, Ψν) =

d P ±

l,µ(r)Ψν(r)r dr

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 Ψν are oscillatory ⇒ numerical problems

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Computing inner products

Matrix entries are inner products A±

νµ = (P ± l,µ, Ψν) =

d P ±

l,µ(r)Ψν(r)r dr

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 Ψν are oscillatory ⇒ numerical problems

Problem Computing inner products numerically is expensive / less accurate

28 / 47

Computing inner products

Matrix entries are inner products A±

νµ = (P ± l,µ, Ψν) =

d P ±

l,µ(r)Ψν(r)r dr

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 Ψν 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 A±

νµ = (P ± l,µ, Ψν) =

d P ±

l,µ(r)Ψν(r)r dr

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 Ψν 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 A±

νµ = (P ± l,µ, Ψν) =

d P ±

l,µ(r)Ψν(r)r dr

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 Ψν 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 A±

νµ = (P ± l,µ, Ψν) =

d P ±

l,µ(r)Ψν(r)r dr

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 Ψν 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!

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From Classical to a New Mode-matching method

Summary of new matching method Classical → new mode-matching (Pµ, Ψν) → F µ, Ψ ν

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From Classical to a New Mode-matching method

Summary of new matching method Classical (CMM) → new (BLM) mode-matching (Pµ, Ψν) → F µ, Ψ ν with Ψν = Jm(ανr) 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 (Pµ, Ψν) → F µ, Ψ ν = d PµΨνr dr → = d

• w1PµPν + w2UµPν

+w3(VµVν + WµWν)

• r dr

with Ψν = Jm(ανr) 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 (Pµ, Ψν) → F µ, Ψ ν = d PµΨνr dr → = d

• w1PµPν + w2UµPν

+w3(VµVν + WµWν)

• r dr

quadrature → = id kµ − kν PνVµ − VνPµ Ων

• r=d

with Ψν = Jm(ανr) 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 (Pµ, Ψν) → F µ, Ψ ν = d PµΨνr dr → = d

• w1PµPν + w2UµPν

+w3(VµVν + WµWν)

• r dr

quadrature → = id kµ − kν PνVµ − VνPµ Ων

• r=d

with Ψν = Jm(ανr) with Ψ ν = F ν, F = [P, U, V, W] expensive less accurate → cheap accurate

29 / 47

Closed form integrals of 2D eigenmodes

Prototype example of Generalised Prid-Brown : Helmholtz eqn ∇2ψ + β2ψ = 0

• n arbitrarily shaped cross-section A

30 / 47

Closed form integrals of 2D eigenmodes

Prototype example of Generalised Prid-Brown : Helmholtz eqn ∇2ψ + β2ψ = 0 ∇2φ + α2φ = 0

• n arbitrarily shaped cross-section A

30 / 47

Closed form integrals of 2D eigenmodes

Prototype example of Generalised Prid-Brown : Helmholtz eqn φ

• ∇2ψ + β2ψ
• = 0

ψ

• ∇2φ + α2φ
• = 0
• n arbitrarily shaped cross-section A

30 / 47

Closed form integrals of 2D eigenmodes

Prototype example of Generalised Prid-Brown : Helmholtz eqn φ

• ∇2ψ + β2ψ
• = 0

ψ

• ∇2φ + α2φ
• = 0
• n arbitrarily shaped cross-section A

Subtract (α2 − β2) φψ = φ∇2ψ − ψ∇2φ

30 / 47

Closed form integrals of 2D eigenmodes

Prototype example of Generalised Prid-Brown : Helmholtz eqn φ

• ∇2ψ + β2ψ
• = 0

ψ

• ∇2φ + α2φ
• = 0
• n arbitrarily shaped cross-section A

Subtract and integrate over A (α2 − β2)

• A

φψ dS =

• A

(φ∇2ψ − ψ∇2φ) dS

30 / 47

Closed form integrals of 2D eigenmodes

Prototype example of Generalised Prid-Brown : Helmholtz eqn φ

• ∇2ψ + β2ψ
• = 0

ψ

• ∇2φ + α2φ
• = 0
• n arbitrarily shaped cross-section A

Subtract and integrate over A (α2 − β2)

• A

φψ dS =

• A

∇·(φ∇ψ − ψ∇φ) dS

30 / 47

Closed form integrals of 2D eigenmodes

Prototype example of Generalised Prid-Brown : Helmholtz eqn φ

• ∇2ψ + β2ψ
• = 0

ψ

• ∇2φ + α2φ
• = 0
• n arbitrarily shaped cross-section A

Subtract and integrate over A

GAUSS

↓ (α2 − β2)

• A

φψ dS =

• A

∇·(φ∇ψ − ψ∇φ) dS

30 / 47

Closed form integrals of 2D eigenmodes

Prototype example of Generalised Prid-Brown : Helmholtz eqn φ

• ∇2ψ + β2ψ
• = 0

ψ

• ∇2φ + α2φ
• = 0
• n arbitrarily shaped cross-section A

Subtract and integrate over A

GAUSS

↓ (α2 − β2)

• A

φψ dS =

• Γ

(φ∇ψ·n − ψ∇φ·n) dℓ

30 / 47

Closed form integrals of 2D eigenmodes

Prototype example of Generalised Prid-Brown : Helmholtz eqn φ

• ∇2ψ + β2ψ
• = 0

ψ

• ∇2φ + α2φ
• = 0
• n arbitrarily shaped cross-section A

Subtract and integrate over A (α2 − β2)

• A

φψ dS =

• Γ

(φ∇ψ·n − ψ∇φ·n) dℓ 2D inner-product for Helmholtz eigenfunctions

• φ, ψ

= 1 α2 − β2

• Γ

(φ∇ψ·n − ψ∇φ·n)dℓ, 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 = Jm(αr) eimθ, ψ = Jm(βr) e−imθ

31 / 47

Closed form integrals of 1D eigenmodes

Circular duct: Helmholtz equation → Bessel equation Substitute into 2D inner-product: φ = Jm(αr) eimθ, ψ = Jm(βr) e−imθ 1D inner-product of Bessel functions Jm(αr), Jm(βr) = 1 Jm(αr)Jm(βr) rdr = 1 α2 − β2

• βJm(α)J′

m(β) − αJ′ m(α)Jm(β)

• 31 / 47

Closed form integrals of 1D eigenmodes

Circular duct: Helmholtz equation → Bessel equation Substitute into 2D inner-product: φ = Jm(αr) eimθ, ψ = Jm(βr) e−imθ 1D inner-product of Bessel functions Jm(αr), Jm(βr) = 1 Jm(αr)Jm(βr) rdr = 1 α2 − β2

• βJm(α)J′

m(β) − αJ′ m(α)Jm(β)

• 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 =

• A

1

• Ω
• u0

ρ0c2 +

• k

ρ0 Ω

• PP + ω
• Ω
• PU − ρ0u0(

V V + WW)

• dS

= i k − k

• Γ
• P(V ny + Wnz) − (

V ny + Wnz)P

• Ω

dℓ,

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 =

• A

1

• Ω
• u0

ρ0c2 +

• k

ρ0 Ω

• PP + ω
• Ω
• PU − ρ0u0(

V V + WW)

• dS

= i k − k

• Γ
• P(V ny + Wnz) − (

V ny + Wnz)P

• Ω

dℓ, 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±imθ 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±imθ 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±imθ 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

• Ω
• u0

ρ0c2 +

• k

ρ0 Ω

• P

P + ω

• Ω

U P − ρ0u0(V V + W W)

• r dr

= id k − k PV − V P

• 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) F , F =

• id

PP (k − k) Ωω

• Ω

Z −

• Ω
• Z
• r=d

34 / 47

Some special cases

Some special cases

With Ingard-Myers condition (slipping flow) F , F =

• id

PP (k − k) Ωω

• Ω

Z −

• Ω
• Z
• r=d

For hard walls: “orthogonal”:

• F ,

F = 0 F , F = 0

34 / 47

Some special cases

Some special cases

With Ingard-Myers condition (slipping flow) F , F =

• id

PP (k − k) Ωω

• Ω

Z −

• Ω
• Z
• r=d

For hard walls: “orthogonal”:

• F ,

F = 0 F , F = 0 In case of no-slip flow, u0(d) = 0: F , F =

• id

PP (k − k)ω 1 Z − 1

• Z
• r=d

34 / 47

Some special cases

Some special cases

With Ingard-Myers condition (slipping flow) F , F =

• id

PP (k − k) Ωω

• Ω

Z −

• Ω
• Z
• r=d

For hard walls: “orthogonal”:

• F ,

F = 0 F , F = 0 In case of no-slip flow, u0(d) = 0: F , F =

• id

PP (k − k)ω 1 Z − 1

• Z
• r=d

For hard walls, or same impedance Z = Z: “orthogonal”:

• F ,

F = 0 F , F = 0

34 / 47

Bilinear map-based mode-matching

Classic mode-matching (CMM)

µl

• µ=1

b+

l,µ(P + l,µ, Ψν) + a− l,µ(P − l,µ, Ψν)

=

µl+1

• µ=1

a+

l+1,µ(P + l+1,µ, Ψν) + b− l+1,µ(P − l+1,µ, Ψν)

(same for velocity) with test functions (for example) Ψν = Jm(ανr)

35 / 47

Bilinear map-based mode-matching

Classic mode-matching (CMM)

µl

• µ=1

b+

l,µ(P + l,µ, Ψν) + a− l,µ(P − l,µ, Ψν)

=

µl+1

• µ=1

a+

l+1,µ(P + l+1,µ, Ψν) + b− l+1,µ(P − l+1,µ, Ψν)

(same for velocity) with test functions (for example) Ψν = Jm(ανr) Quadrature required for (Pµ, Ψν) terms (non-uniform flow)

35 / 47

Bilinear map-based mode-matching

Bilinear map-based (BLM) mode-matching

µl

• µ=1

b+

l,µF + l,µ, Ψ ν + a− l,µF − l,µ, Ψ ν

=

µl+1

• µ=1

a+

l+1,µF + l+1,µ, Ψ ν + b− l+1,µF − l+1,µ, Ψ ν

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

µl

• µ=1

b+

l,µF + l,µ, Ψ ν + a− l,µF − l,µ, Ψ ν

=

µl+1

• µ=1

a+

l+1,µF + l+1,µ, Ψ ν + b− l+1,µF − l+1,µ, Ψ ν

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

µl

• µ=1

b+

l,µF + l,µ, Ψ ν + a− l,µF − l,µ, Ψ ν

=

µl+1

• µ=1

a+

l+1,µF + l+1,µ, Ψ ν + b− l+1,µF − l+1,µ, Ψ ν

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 µmax = 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 Temperature T0/T∞ = 1 T0/T∞ = 1 T0/T∞ = 2 log(2)(1 − r2

2 )

Mean flow u0/c∞ = 0.5 · (1 − r2) u0/c∞ = 0.3 · 4

3(1 − r2 2 )

u0/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

x (m) r (m) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.05 0.1 0.15 −1 −0.5 0.5 1

(a) Classical mode-matching.

x (m) r (m) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.05 0.1 0.15 −1 −0.5 0.5 1

(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.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −1.5 −1 −0.5 0.5 1 1.5 x(m) Re(P) (dimless) Re(P) (BLM), r=0.035m Re(P) (CMM), r=0.035m Re(P) (BLM), r=0.075m Re(P) (CMM), r=0.075m Re(P) (BLM), r=0.15m Re(P) (CMM), r=0.15m

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.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 x(m) Re(U) (dimless) Re(U) (BLM), r=0.035m Re(U) (CMM), r=0.035m Re(U) (BLM), r=0.075m Re(U) (CMM), r=0.075m Re(U) (BLM), r=0.15m Re(U) (CMM), r=0.15m

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.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 x(m) Re(V) (dimless) Re(V) (BLM), r=0.035m Re(V) (CMM), r=0.035m Re(V) (BLM), r=0.075m Re(V) (CMM), r=0.075m Re(V) (BLM), r=0.15m Re(V) (CMM), r=0.15m

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.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −1.5 −1 −0.5 0.5 1 1.5 x(m) Re(P) (dimless) Re(P) (BLM), r=0.035m Re(P) (CMM), r=0.035m Re(P) (BLM), r=0.075m Re(P) (CMM), r=0.075m Re(P) (BLM), r=0.15m Re(P) (CMM), r=0.15m

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.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 x(m) Re(U) (dimless) Re(U) (BLM), r=0.035m Re(U) (CMM), r=0.035m Re(U) (BLM), r=0.075m Re(U) (CMM), r=0.075m Re(U) (BLM), r=0.15m Re(U) (CMM), r=0.15m

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.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 x(m) Re(V) (dimless) Re(V) (BLM), r=0.035m Re(V) (CMM), r=0.035m Re(V) (BLM), r=0.075m Re(V) (CMM), r=0.075m Re(V) (BLM), r=0.15m Re(V) (CMM), r=0.15m

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.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −1.5 −1 −0.5 0.5 1 1.5 x(m) Re(P) (dimless) Re(P) (BLM), r=0.035m Re(P) (CMM), r=0.035m Re(P) (BLM), r=0.075m Re(P) (CMM), r=0.075m Re(P) (BLM), r=0.15m Re(P) (CMM), r=0.15m

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.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 x(m) Re(U) (dimless) Re(U) (BLM), r=0.035m Re(U) (CMM), r=0.035m Re(U) (BLM), r=0.075m Re(U) (CMM), r=0.075m Re(U) (BLM), r=0.15m Re(U) (CMM), r=0.15m

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.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 x(m) Re(V) (dimless) Re(V) (BLM), r=0.035m Re(V) (CMM), r=0.035m Re(V) (BLM), r=0.075m Re(V) (CMM), r=0.075m Re(V) (BLM), r=0.15m Re(V) (CMM), r=0.15m

Perfect match between BLM and CMM results

40 / 47

Numerical results — Energy balance

Energy balance (Myers’ Energy Corollary) vs µmax for conf. I

5 10 15 20 25 30 35 40 45 50 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 mumax log10 of normalized energy balance test1 (BLM) test1 (CMM)

41 / 47

Numerical results — Energy balance

Energy balance (Myers’ Energy Corollary) vs µmax for conf. I

5 10 15 20 25 30 35 40 45 50 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 mumax log10 of normalized energy balance test1 (BLM) test1 (CMM)

Energy balance better with more µ-modes.

41 / 47

Numerical results — Energy balance

Energy balance (Myers’ Energy Corollary) vs µmax for conf. I

5 10 15 20 25 30 35 40 45 50 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 mumax log10 of normalized energy balance test1 (BLM) test1 (CMM)

Energy balance better with more µ-modes. BLM performs better than CMM!

41 / 47

Numerical results — Energy balance

Energy balance (Myers’ Energy Corollary) vs µmax for conf. I

5 10 15 20 25 30 35 40 45 50 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 mumax log10 of normalized energy balance test1 (BLM) test1 (CMM)

Energy balance better with more µ-modes. BLM performs better than CMM! Why?

41 / 47

Numerical results — Convergence of modal amplitudes

Edge Condition (a posteriori)

42 / 47

Numerical results — Convergence of modal amplitudes

Edge Condition (a posteriori) It is reasonable to assume that for some p < 0 the amplitudes An = O(np) for n → ∞ so log |An| = p log n + O(1).

42 / 47

Numerical results — Convergence of modal amplitudes

Edge Condition (a posteriori) It is reasonable to assume that for some p < 0 the amplitudes An = O(np) for n → ∞ so log |An| = p log n + O(1). Then pn = log |An| log n → p for n → ∞

42 / 47

Numerical results — Convergence of modal amplitudes

Edge Condition (a posteriori) It is reasonable to assume that for some p < 0 the amplitudes An = O(np) for n → ∞ so log |An| = p log n + O(1). Then pn = log |An| log n → p for n → ∞ At the interface, at the wall (edge): boundary cond. discontinuous. Field may be singular, but Power Flux must vanish at edge.

42 / 47

Numerical results — Convergence of modal amplitudes

Edge Condition (a posteriori) It is reasonable to assume that for some p < 0 the amplitudes An = O(np) for n → ∞ so log |An| = p log n + O(1). Then pn = log |An| log n → p for n → ∞ At the interface, at the wall (edge): boundary cond. discontinuous. Field may be singular, but Power Flux must vanish at edge. It can be shown that: p < −1 ⇒ uniform convergence of modal series ⇒ edge condition satisfied

42 / 47

Numerical results — Convergence of modal amplitudes

Do we have p < −1 for numerical solutions?

43 / 47

Numerical results — Convergence of modal amplitudes

Do we have p < −1 for numerical solutions? Convergence of amplitudes (BLM and CMM), for conf. I, II and III

10 20 30 40 50 −3 −2.8 −2.6 −2.4 −2.2 −2 −1.8 −1.6 −1.4 −1.2 −1 n pn BLM CMM 10 20 30 40 50 −3 −2.8 −2.6 −2.4 −2.2 −2 −1.8 −1.6 −1.4 −1.2 −1 n pn BLM CMM 10 20 30 40 50 −3 −2.8 −2.6 −2.4 −2.2 −2 −1.8 −1.6 −1.4 −1.2 −1 n pn BLM CMM

43 / 47

Numerical results — Convergence of modal amplitudes

Do we have p < −1 for numerical solutions? Convergence of amplitudes (BLM and CMM), for conf. I, II and III

10 20 30 40 50 −3 −2.8 −2.6 −2.4 −2.2 −2 −1.8 −1.6 −1.4 −1.2 −1 n pn BLM CMM 10 20 30 40 50 −3 −2.8 −2.6 −2.4 −2.2 −2 −1.8 −1.6 −1.4 −1.2 −1 n pn BLM CMM 10 20 30 40 50 −3 −2.8 −2.6 −2.4 −2.2 −2 −1.8 −1.6 −1.4 −1.2 −1 n pn BLM CMM

p ≈ −2 ⇒ edge condition satisfied ✓

43 / 47

Numerical results — Convergence of modal amplitudes

Do we have p < −1 for numerical solutions? Convergence of amplitudes (BLM and CMM), for conf. I, II and III

10 20 30 40 50 −3 −2.8 −2.6 −2.4 −2.2 −2 −1.8 −1.6 −1.4 −1.2 −1 n pn BLM CMM 10 20 30 40 50 −3 −2.8 −2.6 −2.4 −2.2 −2 −1.8 −1.6 −1.4 −1.2 −1 n pn BLM CMM 10 20 30 40 50 −3 −2.8 −2.6 −2.4 −2.2 −2 −1.8 −1.6 −1.4 −1.2 −1 n pn BLM CMM

p ≈ −2 ⇒ edge condition satisfied ✓ Convergence of pn reveals inaccuracies of CMM amplitudes: BLM amplitudes smoother than CMM as n → ∞: no quadrature inaccuracies for BLM. Explains energy behaviour.

43 / 47

Outline

1

Background & motivation

2

Pridmore-Brown modes Model & equations Numerical method: COLNEW and path-following

3

Options for varying Z

4

WKB for slowly varying Z

5

New mode-matching method 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

6

Conclusions Epilogue

44 / 47

Conclusions

The Pridmore-Brown equation was solved numerically Using standard BVP solver COLNEW Path-following/predictor-corrector with automatic step size Favourable comparison with high-frequency approximation

45 / 47

Conclusions

The Pridmore-Brown equation was solved numerically Using standard BVP solver COLNEW Path-following/predictor-corrector with automatic step size Favourable comparison with high-frequency approximation Slowly varying mode-approximation applied to typical APU duct Small enough ε: favourable comparison with BAHAMAS (mode matching) WKB fails when Helmholtz liner passes resonance Strong effects of temperature and mean flow gradients. The need for Mode Matching was clear

45 / 47

Conclusions

Classic mode-matching (CMM):

46 / 47

Conclusions

Classic mode-matching (CMM): Uniform flow & temp:

Mode shapes are Bessel functions Inner products are available in closed form

46 / 47

Conclusions

Classic mode-matching (CMM): Uniform flow & temp:

Mode shapes are Bessel functions Inner products are available in closed form

Parallel (non-uniform) flow & temp:

Mode shapes are Pridmore-Brown solutions (determined numerically) Inner products require numerical quadrature → expensive & less accurate

46 / 47

Conclusions

Classic mode-matching (CMM): Uniform flow & temp:

Mode shapes are Bessel functions Inner products are available in closed form

Parallel (non-uniform) flow & temp:

Mode shapes are Pridmore-Brown solutions (determined numerically) Inner products require numerical quadrature → expensive & less accurate

Bilinear map-based mode-matching (BLM):

46 / 47

Conclusions

Classic mode-matching (CMM): Uniform flow & temp:

Mode shapes are Bessel functions Inner products are available in closed form

Parallel (non-uniform) flow & temp:

Mode shapes are Pridmore-Brown solutions (determined numerically) Inner products require numerical quadrature → expensive & less accurate

Bilinear map-based mode-matching (BLM): Parallel (non-uniform) flow & temp:

Mode shapes are Pridmore-Brown solutions (determined numerically) Closed form expressions for “inner-products” cheaper Solutions in very good agreement with CMM BLM amplitudes more accurate

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Epilogue

Epilogue

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Epilogue

Epilogue

The success of the BLM matching method is, in a way, too

• good. At least far better than expected, because the

“inner-product” is not a proper inner-product (unless u0 = 0

• r uniform) and we can’t be sure that it is able to single out

each modal contribution.

47 / 47

Epilogue

Epilogue

The success of the BLM matching method is, in a way, too

• good. At least far better than expected, because the

“inner-product” is not a proper inner-product (unless u0 = 0

• r uniform) and we can’t be sure that it is able to single out

each modal contribution. Nevertheless, from the success we can only conclude that it must be “almost” an inner-product. The modes are all “seen” and distinguished.

47 / 47

Epilogue

Epilogue

The success of the BLM matching method is, in a way, too

• good. At least far better than expected, because the

“inner-product” is not a proper inner-product (unless u0 = 0

• r uniform) and we can’t be sure that it is able to single out

each modal contribution. Nevertheless, from the success we can only conclude that it must be “almost” an inner-product. The modes are all “seen” and distinguished. Possibly related to this is the fact that the set of discrete modes is not complete, i.e. not sufficient to construct any possible solution. There is a “continuous” spectrum at the locus of ω − ku0(r) = 0. From the energy result we can conclude that this part is very small.

47 / 47

Epilogue

Epilogue

The success of the BLM matching method is, in a way, too

• good. At least far better than expected, because the

“inner-product” is not a proper inner-product (unless u0 = 0

• r uniform) and we can’t be sure that it is able to single out

each modal contribution. Nevertheless, from the success we can only conclude that it must be “almost” an inner-product. The modes are all “seen” and distinguished. Possibly related to this is the fact that the set of discrete modes is not complete, i.e. not sufficient to construct any possible solution. There is a “continuous” spectrum at the locus of ω − ku0(r) = 0. From the energy result we can conclude that this part is very small. A fine task in functional analysis remains . . .

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