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Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work

Boundary Layer Thickness Effects ofHydrodynamic Instability along an Impedance

Mirela Darau

3rd November 2010

Outline

1 Background

2 The Model

3 Stability Analysis

4 Regularized Boundary Condition

5 Conclusions and Future Work

Outline

1 Background

2 The Model

Aircraft Engine Noise

aircraft certificationaircraft enginethe fanacoustic liners

Ingard-Myers Boundary Conditionfor the mean flow the wallis solid: ((U0,0)·n) = 0 aty = 0for the acoustic field thewall is softat the wall:

p = −vZ , Z ∈ C.

h λtypical, so usually h ↓ 0 is taken (I-M cond.). For apoint near the wall but still (just) inside the mean flow(

iω + U0∂

)p = −iωZ v , at y = 0+.

p = −vZ , Z ∈ C.

iω + U0∂

)p = −iωZ v , at y = 0+.

p = −vZ , Z ∈ C.

iω + U0∂

)p = −iωZ v , at y = 0+.

p = −vZ , Z ∈ C.

iω + U0∂

)p = −iωZ v , at y = 0+.

Problem: unstable in time-domain (numerical experiment:)

Conjectures, Conclusions and Questions

no instabilities seen in practice (except for two isolatedcases: Auregan, Ronneberger)conjecture: flow is stable for small but finite h: h > hc > 0since there is no other length scale in the problem, hc

depends on the impedance Z ;Is it true that in industrial practice hc is much smaller thanprevailing boundary layer thicknesses?Can we modify the Ingard-Myers condition such that itreflects the correct physical behavior?

Outline

1 Background

2 The Model

Linearized Euler Equations

Assumptions: inviscid, isentropic→ ddt p = c2

0ddt ρ.

Linearizing around (U0(y),0,p0, ρ0) and eliminating ρ:

1ρ0c2

(∂p∂t+ U0

∂p∂x

)+ ∂u∂x+ ∂ v∂y= 0

∂u∂t+ U0

∂u∂x+ vU

′0 +

∂p∂x= 0

∂ v∂t+ U0

∂ v∂x+ 1ρ0

∂p∂y= 0.

We consider waves of the type:

p(x, y , t) = 14π2

∫∫ ∞−∞

p(y;α, ω) eiωt−iαx dαdω, similar for u, v

Equations eventually reduce to:

d2pdy2 +

2αU′0

ω − αU0

dpdy+[(ω − αU0)

− α2

]p = 0.

Boundary condition at y = 0: − pv = Z (ω); exponential decay for

y →∞

Piecewise Linear Incompressible Shear Layer

Incompressible limit: ωα, U∞ c0.

System reduces to:

d2pdy2 +

2αU ′0ω − αU0

dpdy− α2p = 0.

A piecewise linear velocity profile:

U0(y) =

U∞ for 0 6 y 6 h

U∞ for h 6 y <∞

Solutions

for y ≥ h:

p = A e−|α|y , where |α| = ±α if Re(α) >< 0.

in the shear layer region (0,h):

p(y) = C1 eαy (hω − αyU∞ + U∞)+ C2 e−αy (hω − αyU∞ − U∞)

u(y) = αhρ0(C1 eαy +C2 e−αy )

v(y) = iαhρ0(C1 eαy −C2 e−αy ).

Solutions

for y ≥ h:

p = A e−|α|y , where |α| = ±α if Re(α) >< 0.

in the shear layer region (0,h):

p(y) = C1 eαy (hω − αyU∞ + U∞)+ C2 e−αy (hω − αyU∞ − U∞)

u(y) = αhρ0(C1 eαy +C2 e−αy )

v(y) = iαhρ0(C1 eαy −C2 e−αy ).

The Dispersion Relation

at the interface y = h: continuity of pressure and particledisplacementimpedance boundary condition at y = 0

yield the dispersion relation

D(α, ω) =

Z (ω)− iρ0

p2 eαh(|α|p1 + αq)+ p1 e−αh(|α|p2 − αq)eαh(|α|p1 + αq)+ e−αh(|α|p2 − αq)

q = ωh − αhU∞, p1 = q + U∞, p2 = q − U∞.

Outline

1 Background

2 The Model

Spatio-Temporal Instabilities

G(x,t)

ABSOLUTELY

UNSTABLE

CONVECTIVELY

UNSTABLE

G(x,t)

Integration Contours

Impulse response: 9(x, y , t) = 1(2π)2

∫Fα

∫Lω

ϕ(y)D(α, ω)

eiωt−iαx dωdα.

× ×ω1(α)

ω2(α)

complex ω-plane

×α−(ω)

×α+(ω)

complex α-plane

Integration contours

find ωmin = minα∈R(ωi); ifωimin > 0→ stable;ωimin < 0→ unstable→continue

ωi is increased⇔ α+ andα− approach each other,and eventually collide→α∗

Im(ω(α∗)) < 0 then abs.instab.Im(ω(α∗)) > 0 then conv.instab.

50 100 150 200ΑR

20 40 60 80 100 120ΑR

ΩI = - 165

50 100 150 200ΑR

20 40 60 80 100 120ΑR

ΩI = - 165

50 100 150 200ΑR

20 40 60 80 100 120ΑR

ΩI = - 165

50 100 150 200ΑR

20 40 60 80 100 120ΑR

ΩI = - 165

Typical Aeronautical Example

Z (ω) = R + iωm − iρ0c0 cot(ωLc0

)≈ R + iω

(m + 1

3ρ0L)− i

ρ0c20

R = 2ρ0c0, L = 3.5 cm, mρ0= 20 mm (U∞ = 60 m/s, ρ0 = 1.2 kg/m3)

2. 4. 6. 8. 10. 12. 14.

h@ΜmD

ImHΩ*L@s-1D

1.e4 2.e4 3.e4 4.e4

Ω* Î C

1.e4 2.e4 3.e4 4.e4

Α* Î C

hc = 10.5 µm, with ω∗ = 1.1 ·104 s−1, α∗ = (0.4+4i) ·103 m−1.

Asymptotic Behavior

For large resistance and high quality factor

r = Rρ0U∞

√mKR= O(r ),

in D(α, ω) = Dα(α, ω) = 0 with ω is real, then asymptotically

hc = 14

(ρ0U∞

(This includes the industrially typical cases!)

Outline

1 Background

2 The Model

Asymptotic Behavior for Small αh

For αh→ 0 and = ω − αU∞ we find

Z ' ρ0

2 + |α|(ω+ 13U2∞α

2)h|α|ω + α2h

Non-uniqueness of expansion: multiply by e−|α|hθ / e−|α|hθ

Z ' ρ0

2 + |α|((1− θ)ω2 − (1− 2θ)ωαU∞ + (13 − θ)U2

∞α2)h

|α|ω + α2(− θω)h

Optimal Agreement for θ = 13

2. 4. 6. 8. 10. 12. 14.

Zoom In

3.68 3.681

h@ΜmD

ImHΩ*L@s-1D

1.e4 2.e4 3.e4 4.e4

Zoom In

15 900 15 902

Ω* Î C

1.e4 2.e4 3.e4 4.e4

Zoom In

18 000 18 005

37 365

37 370

Α* Î C

A Modified Ingard-Myers Boundary Condition

Identifying:

iαp ∼ ∂p∂x; − |α|

iρ0∼ (v · n); |α|(v · n) ∼ ∂

∂n(v · n)

then we have the regularized boundary condition (θ = 13 ):

iω + U∞∂

)p − hρ0iω

(23 iω + 1

3U∞∂

)(v·n)

iω(v·n)+ hρ0

∂x2 p − 13 iωh

∂n(v·n)

which indeed reduces for h = 0 to the Ingard approximation forthin boundary layers but has now the correct stability behavior.

Tentative numerical experiments confirm transition unstable - stable

For smooth U0 ∼ 160 m/s, h ∼ 0.06 m, mρ0= 1 m, L = 3 mm

Outline

1 Background

2 The Model

Conclusions

Stability analysis for incompressible flow with piecewiselinear profile and MSD-liner with R,K ,m > 0.Flow is absolutely unstable for small but finite h:0 < h < hc. Hyper-unstable for h = 0 (Ingard-Myers limit).hc is a property of flow and liner, and has nothing to dowith the acoustic wavelength.In industrial practice hc is much smaller than prevailingboundary layer thicknesses. There may be a convectiveinstability that remains too small to be measured.Explicit approximate formula for hc.Corrected Ingard-Myers condition, including small heffects, which is stable for h > hc.

Conclusions

Future Work

Cooperation with FFT for numerical experimentsCooperation DAMTP: analysis of the critical layer(ω − αU = 0) for a linear velocity profile in a duct(cylindrical coordinates)

d2pdr2 +

(1r+ 2αU ′

ω − αU

)dpdr+((ω − αU(r ))2

−(α2 + m2

))p = 0.

Future Work

Cooperation with FFT for numerical experimentsCooperation DAMTP: analysis of the critical layer(ω − αU = 0) for a linear velocity profile in a duct(cylindrical coordinates)

d2pdr2 +

(1r+ 2αU ′

ω − αU

)dpdr+((ω − αU(r ))2

−(α2 + m2

))p = 0.

Thank you for your attention!

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