martien oppeneer supervisors: sjoerd rienstra and bob ... · 4/7/2010 · sound propagation in a...

Sound propagation in a lined duct with flow

Martien Oppeneer

supervisors: Sjoerd Rienstra and Bob Mattheij

CASA dayEindhoven, April 7, 2010

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Outline

1 Introduction & Background

2 Modeling the problem

3 Numerical implementation

4 (Numerical) results

5 Future plans

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Introduction & Background

Outline

1 Introduction & BackgroundProject motivationAcoustic linersGeneral project goalBrush-up: modes

2 Modeling the problemPridmore-Brown equation (ODE)Boundary conditions

3 Numerical implementationMethod 1: bvp4c / BVP SOLVERMethod 2: COLSYSContinuation in Z

4 (Numerical) resultsImpedance wall, no flowImpedance wall, uniform mean flowSome numerical problems

5 Future plans3 / 47

Introduction & Background Project motivation

Project motivation

APU: Auxiliary Power Unit

produces power when main enginesare switched offto start main engines, AC, ...major source of ramp noise

Goal: APU noise reduction Figure: APU onan Airbus A380.

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Introduction & Background Acoustic liners

Acoustic liners

Figure: Locally reacting liner(impedance wall).

Figure: Metallic foam (bulk absorber).

Figure: Spiralling (non-locally reacting)liner.

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Introduction & Background General project goal

General project goal

hard wall resistive sheet

liner cavity

cool air inlet

exhaust

mean flow profile u(r)

Figure: APU geometry.

Model sound propagation /attenuation

sheared flownon-locally reacting linerssegmented / non-uniformlinersstrong temperature gradients(swirling flow)(annular hub)(varying duct radius)

Sufficiently fast for liner designcalculations⇒ semi-analytical model, basedon modes

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Introduction & Background Brush-up: modes

Modes

Motivation: Direct Navier-Stokes(DNS) not practical / feasible (esp.for design)

Eigensolution of a BVP

Characterized by:

eigenfunction Pmµ(r)‘eigenvalue’ kmµ ∈ C

Traveling waves of the form:

pmµ(r) =Pmµ(r) exp(−iωt+ikmµx+imθ)

Total field is superposition (orintegral) of modes

Figure: ω = 20, m = 1, Z = 3 + 3i,µ = 3.

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Modeling the problem

Outline







Duct geometry

d

h x

r

θ

Figure: Duct geometry, velocity components: v = ux+ vr + wθ.

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Modeling outline

BVP

No viscosityNo heat conduction

Cylindrical coordinates

Navier-Stokeseqns

LinearizedEuler eqns

Time-harmonicsolutions

Boundaryconditions

Eulereqns

Smallperturbations

ODE forP (r)

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Modeling the problem Pridmore-Brown equation (ODE)


Cylindrical coordinatesNavier-Stokeseqns



Eulereqns

Smallperturbations

ODE forP (r)

Navier-Stokes (conservation laws)

∂

∂tρ+∇ · (ρv) = 0

∂

∂t(ρv) +∇ · (ρvv) = −∇p+∇ · τ

∂

∂t(ρE) +∇ · (ρEv) = −∇ · q −∇ · (pv) +∇ · τv

inviscid: ∇ · τ = 0 non-heat-conducting: ∇ · q = 0

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Eulereqns

Smallperturbations

ODE forP (r)

Euler Equations (in primary variables)

∂

∂tρ+∇ · (ρv) = 0

ρ

(∂v

∂t+ v · ∇v

)= −∇p

∂p

∂t+ v · ∇p+ γp∇ · v = 0

Ideal gas: p = ρRT

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Small perturbations

Sound is due to small pressure perturbationsAssumptions:

main sound source: turbine engine (rotor / stator interaction)

negligible sound source: turbulence

⇒ Total field = mean flow + perturbations:

(u, v, w, ρ, p) = (u, v, w, ρ, p) + (u, v, w, ρ, p)

Linearize: neglect quadratic terms (since perturbations are small)

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Mean flow

Mean flow assumptions:

independent of x: ∂u∂x = 0, ∂v∂x = 0, ∂w∂x = 0

radial velocity v = 0circumferential velocity independent of θ: ∂w

∂θ = 0

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Eulereqns

Smallperturbations

ODE forP (r)

Linearized Euler Equations

∂ρ

∂t+ u

∂ρ

∂x+

1r

∂(rρv)∂r

+w

r

∂ρ

∂θ+ ρ

(1r

∂w

∂θ+∂u

∂x

)= 0

ρ

(∂v

∂t+ u

∂v

∂x+w

r

∂v

∂θ− 2w

rw

)− w2

rρ = −∂p

∂r

ρ

(∂w

∂t+ u

∂w

∂x+w

r

∂w

∂θ+dw

drv +

w

rv

)= −1

r

∂p

∂θ

ρ

(∂u

∂t+ u

∂u

∂x+w

r

∂u

∂θ+ v

∂u

∂r+w

r

∂u

∂θ

)= −∂p

∂x

∂p

∂t+ u

∂p

∂x+w

r

∂p

∂θ+ρw2

rv + γp

(1r

∂(rv)∂r

+1r

∂w

∂θ+∂u

∂x

)= 0

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Eulereqns

Smallperturbations

ODE forP (r)

We seek time-harmonic solutions:

(u, v, w, ρ, p) = (U, V,W,R, P ) exp(−iωt+ ikx+ imθ)

⇒ ODE in P (r):

P ′′ + β(r, k)P ′ + γ(r, k)P = 0, on h ≤ r ≤ d

where β(r, k) and γ(r, k) are functions of:

mean flow parameters: u(r), w(r), ρ(r), p(r)m,ω (given)

r, k

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Eulereqns

Smallperturbations

ODE forP (r)

Simplifications:

No swirl: w(r) = 0, p(r) constantρ(r) constant

Pridmore-Brown equation

P ′′ + β(r, k)P ′ + γ(r, k)P = 0, on h ≤ r ≤ dwhere

β(r, k) =1r

+2ku′

ω − ku

γ(r, k) =(ω − ku)2

c2− k2 − m2

r2

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Modeling the problem Boundary conditions

Three types of conditions

We need 3 types of conditions

1 Impedance wall BC at r = d (and r = h when h 6= 02 Regularization condition at r = 0 (when h = 0)

3 Normalization condition

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1. Impedance wall BC

Assume: locally reacting liner with impedance Z

Due to vanishing mean-flow boundary layer:

−iωvn =(−iω + u

∂

∂x+w

r

∂

∂θ

) (p

Z

)

(Ingard-Myers condition)

Resulting boundary condition for locally reacting liner:

P ′ + κh(r, k)P = 0 at r = h

P ′ + κd(r, k)P = 0 at r = d

where

κh(k) =iρ (ω − ku)2

ωZh, κd(k) = − iρ (ω − ku)2

ωZd.

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2. Regularization BC

No mean flow (u(r) = 0):Pridmore-Brown → Bessel’s equation

P ′′+1rP ′+

(α2 − m2

r2

)P = 0, α2 =

ω2

c2−k2

General solution:

P = AJm(αr) +BYm(αr)

Note: Ym(αr) is singular at r = 0.⇒ Make sure P (r) <∞ at r = 0

P ′(0) = 0, for m 6= 1P (0) = 0, for m = 1

0 5 10 15 20−0.5

0

0.5

1

012

(a) Jm(x)

0 5 10 15 20−1

−0.5

0

0.5

1

012

(b) J ′m(x)

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3. Normalization

General solution:

P = AJm(αr) +BYm(αr)

Every solution P (r) can be scaled

P (r) can become 0 at r = 0

⇒ Choose P (r) = 1 at r = d

0 5 10 15 20−0.5

0

0.5

1

012

(c) Jm(x)

0 5 10 15 20−1

−0.5

0

0.5

1

012

(d) J ′m(x)

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Numerical implementation

Outline






Numerical implementation

Numerical solution of BVP

Why numerics? Sheared flow / temperature gradients

Important: good initial guess for k and P (r)Handle singularity at r = 0Handle unknown parameter k

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Numerical implementation Method 1: bvp4c / BVP SOLVER

bvp4c / BVP SOLVER

Based on

Runge-Kutta (MIRKDC)(damped) Newton root-finderMesh adaptation based on error estimation(⇒ more refinement for boundary layers)

Can handle parameters

Can handle 1/r type singularities

bvp4c: Matlab, BVP SOLVER: Fortran

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Transformation to remove 1/r2 singularity

By introducing:P (r) = rmφ(r),

Pridmore-Brown transforms into:

rm[φ′′ + φ′

(2m+ 1

r+ β(r, k)

)+ φ

(mrβ(r, k) + γ(r, k)

)]= 0,

where

β(r, k) =2ku′

ω − ku

γ(r, k) =(ω − ku)2

c2− k2

Convert to first order system, φ(r) = φ1(r) and φ′(r) = φ2(r):[φ1

φ2

]′=

1r

[0 0−βm −(2m+ 1)

] [φ1

φ2

]+

[0 1−γ −β

] [φ1

φ2

]

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Handeling the 1/r singularity

First order system:

φ′(r) =1rSφ(r) +Aφ(r)

Use:

limr→0

Sφ(r)− φ(0)

r − 0= Sφ′(0)

Make sure that Sφ(0) = 0, then:

φ′(0) = Sφ′(0) +Aφ(0)

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Numerical implementation Method 2: COLSYS

COLSYS

Robust BVP solver: COLNEW / COLSYS (Fortran) [1] [2]

Based on

collocation at Gaussian points (⇒ no evaluation in singular pointr = 0)B-splines (piecewise polynomial functions)(damped) Newton root-finderMesh adaptation based on error estimation(⇒ more refinement for boundary layers)

rr = 0 ︸︷︷︸

subinterval

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Numerical implementation Method 2: COLSYS

Problem formulation for COLSYS

Add dif. eq. for parameter k: k′ = 0

k′ = 0P ′′ = −β(r, k)P ′ − γ(r, k)P

Split into real and imaginary parts

k′R = 0, k′I = 0,P ′′R = −βR(r, kR, kI)P ′R + βI(r, kR, kI)P ′I

− γR(r, kR, kI)PR + γI(r, kR, kI)PI ,P ′′I = −βI(r, kR, kI)P ′R − βR(r, kR, kI)P ′I

− γI(r, kR, kI)PR − γR(r, kR, kI)PI

COLSYS solves for {kR, kI , PR, P ′R, PI , P ′I}Calculate JacobiansSimilarly for BCs

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Numerical implementation Continuation in Z

Hard wall, no flow

No flow: P (r) = AJm(αr), α2 = ω2 − k2

Hard walls: P ′(1) = J ′m(α) = 0

−20 −15 −10 −5 0 5 10 15 20−50

−40

−30

−20

−10

0

10

20

30

40

50

Re(k)

Im(k

)

right runningleft running

Figure: h = 0, d = 1,m = 3, ω = 20.

Here: using p = P (r) exp(+iωt− ikx− imθ) convention29 / 47

Numerical implementation Continuation in Z

Continuation in Z

Good initial guess is important ⇒ continuation

Re

Im i∞

−i∞

Z

Z = R+ iX

Keep R constant

Vary X from −∞ to ∞(from hard wall to hard wall)

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(Numerical) results

Outline






(Numerical) results Impedance wall, no flow

R is large: close to hard wall

−30 −20 −10 0 10 20 30

−25

−20

−15

−10

−5

0

5

10

15

20

25

Figure: Trajectories of k for R = 2, X runs from −∞ to ∞,h = 0, d = 1,m = 3, ω = 20, no mean flow.

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R becomes smaller: trajectories join

−30 −20 −10 0 10 20 30

−25

−20

−15

−10

−5

0

5

10

15

20

25

Figure: Trajectories of k for R = 1.5, X runs from −∞ to ∞,h = 0, d = 1,m = 3, ω = 20, no mean flow.

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R becomes smaller: acoustic surface waves arise

−30 −20 −10 0 10 20 30

−25

−20

−15

−10

−5

0

5

10

15

20

25

Figure: Trajectories of k for R = 1, X runs from −∞ to ∞,h = 0, d = 1,m = 3, ω = 20, no mean flow.

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Surface wave

k far away from hard wall value⇒ magnitude of Im(α) < 0 is largeThen:

|P (r)| =∣∣∣∣Jm(αr)Jm(α)

∣∣∣∣ 'eIm(α)(1−r)√r

⇒ P (r) decays away from wall: surface wave

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(Numerical) results Impedance wall, uniform mean flow

Mean flow: trajectories shift

−40 −30 −20 −10 0 10 20 30

−30

−20

−10

0

10

20

30

Figure: Trajectories of k for R = 2, X runs from ∞ to −∞,h = 0, d = 1,m = 3, ω = 5, u = 0.5.

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Mean flow: poles go to ∞

−40 −30 −20 −10 0 10 20 30 40

−30

−20

−10

0

10

20

30

Figure: Trajectories of k for R = 0.5, X runs from ∞ to −∞,h = 0, d = 1,m = 3, ω = 5, u = 0.5.

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Mean flow: hydrodynamic surface waves arise

−40 −30 −20 −10 0 10 20 30 40

−30

−20

−10

0

10

20

30


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Mean flow: hydrodynamic surface waves arise

−40 −30 −20 −10 0 10 20 30 40

−30

−20

−10

0

10

20

30


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(Numerical) results Some numerical problems

Some numerical problems

bvp4c: no problems, but slow

COLSYS: some convergence problems

BVP SOLVER: currently working on it

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Everything ok

−15 −10 −5 0 5 10 15

−10

−5

0

5

10

intermediatehard−wallsoft−wall

Figure: Paths of wave number k for several modes, where ω = 5, m = 1,Ma = 0.08, and Z = 1 + iZi where Zi runs from -100 to 100.

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Convergence problems

−15 −10 −5 0 5 10 15

−10

−5

0

5

10



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More convergence problems

−20 −15 −10 −5 0 5 10 15

−10

−5

0

5

10

15



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More convergence problems

−25 −20 −15 −10 −5 0 5 10 15 20 25

−20

−15

−10

−5

0

5

10

15

20



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Future plans

Outline






Future plans

Future plans

1 Create fast and robust solver

2 Add non-uniform flow

3 Add non-locally reacting liners

4 Add segmented liners

5 Add temperature gradients

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Future plans

Thank you for your attention

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Appendix

Bibliography I

U. Ascher, J. Christiansen, and R.D. Russel.Collocation software for boundary-value odes.ACM Transaction on Mathematical Software, 7(2):209–222, June1981.

Uri M. Ascher, Robert M.M. Mattheij, and Robert D. Russel.Numerical solution of Boundary Value Problems for OrdinaryDifferential Equations.Computational Mathematics. Prentice Hall, 1988.

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martien oppeneer supervisors: sjoerd rienstra and bob ... · 4/7/2010 · sound propagation in a...

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