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/centre for analysis, scientific computing and applications
Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
Boundary Layer Thickness Effects ofHydrodynamic Instability along an Impedance
Wall
Mirela Darau
3rd November 2010
/centre for analysis, scientific computing and applications
Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
Outline
1 Background
2 The Model
3 Stability Analysis
4 Regularized Boundary Condition
5 Conclusions and Future Work
/centre for analysis, scientific computing and applications
Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
Outline
1 Background
2 The Model
3 Stability Analysis
4 Regularized Boundary Condition
5 Conclusions and Future Work
/centre for analysis, scientific computing and applications
Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
Aircraft Engine Noise
aircraft certificationaircraft enginethe fanacoustic liners
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Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
Aircraft Engine Noise
aircraft certificationaircraft enginethe fanacoustic liners
/centre for analysis, scientific computing and applications
Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
Aircraft Engine Noise
aircraft certificationaircraft enginethe fanacoustic liners
/centre for analysis, scientific computing and applications
Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
Aircraft Engine Noise
aircraft certificationaircraft enginethe fanacoustic liners
/centre for analysis, scientific computing and applications
Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
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∂
∂x
)p = −iωZ v , at y = 0+.
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Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
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∂
∂x
)p = −iωZ v , at y = 0+.
/centre for analysis, scientific computing and applications
Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
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∂
∂x
)p = −iωZ v , at y = 0+.
/centre for analysis, scientific computing and applications
Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
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∂
∂x
)p = −iωZ v , at y = 0+.
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Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
Problem: unstable in time-domain (numerical experiment:)
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Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
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?
/centre for analysis, scientific computing and applications
Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
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?
/centre for analysis, scientific computing and applications
Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
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?
/centre for analysis, scientific computing and applications
Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
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?
/centre for analysis, scientific computing and applications
Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
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?
/centre for analysis, scientific computing and applications
Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
Outline
1 Background
2 The Model
3 Stability Analysis
4 Regularized Boundary Condition
5 Conclusions and Future Work
/centre for analysis, scientific computing and applications
Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
Linearized Euler Equations
Assumptions: inviscid, isentropic→ ddt p = c2
0ddt ρ.
Linearizing around (U0(y),0,p0, ρ0) and eliminating ρ:
1ρ0c2
0
(∂p∂t+ U0
∂p∂x
)+ ∂u∂x+ ∂ v∂y= 0
∂u∂t+ U0
∂u∂x+ vU
′0 +
1ρ0
∂p∂x= 0
∂ v∂t+ U0
∂ v∂x+ 1ρ0
∂p∂y= 0.
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Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
Modes
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)
2
c20
− α2
]p = 0.
Boundary condition at y = 0: − pv = Z (ω); exponential decay for
y →∞
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Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
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) =
yh
U∞ for 0 6 y 6 h
U∞ for h 6 y <∞
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Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
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 ).
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Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
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 ).
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Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
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
α
p2 eαh(|α|p1 + αq)+ p1 e−αh(|α|p2 − αq)eαh(|α|p1 + αq)+ e−αh(|α|p2 − αq)
= 0
where
q = ωh − αhU∞, p1 = q + U∞, p2 = q − U∞.
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Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
Outline
1 Background
2 The Model
3 Stability Analysis
4 Regularized Boundary Condition
5 Conclusions and Future Work
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Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
Spatio-Temporal Instabilities
x
t
G(x,t)
ABSOLUTELY
UNSTABLE
x
t
CONVECTIVELY
UNSTABLE
G(x,t)
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Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
Integration Contours
Impulse response: 9(x, y , t) = 1(2π)2
∫Fα
∫Lω
ϕ(y)D(α, ω)
eiωt−iαx dωdα.
t > 0
t < 0
ωi
ωr
× ×ω1(α)
ω2(α)
Lω
complex ω-plane
x < 0
x > 0
αi
αr
×α−(ω)
×α+(ω)
complex α-plane
Fα
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Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
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
-500
500
1000
ΩI
20 40 60 80 100 120ΑR
20
40
60
ΑI
- 400
- 250
ΩI = - 165
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Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
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
-500
500
1000
ΩI
20 40 60 80 100 120ΑR
20
40
60
ΑI
- 400
- 250
ΩI = - 165
/centre for analysis, scientific computing and applications
Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
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
-500
500
1000
ΩI
20 40 60 80 100 120ΑR
20
40
60
ΑI
- 400
- 250
ΩI = - 165
/centre for analysis, scientific computing and applications
Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
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
-500
500
1000
ΩI
20 40 60 80 100 120ΑR
20
40
60
ΑI
- 400
- 250
ΩI = - 165
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Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
Typical Aeronautical Example
Z (ω) = R + iωm − iρ0c0 cot(ωLc0
)≈ R + iω
(m + 1
3ρ0L)− i
ρ0c20
ωL
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.
-5.e4
-4.e4
-3.e4
-2.e4
-1.e4
0.e
h@ΜmD
ImHΩ*L@s-1D
1.e4 2.e4 3.e4 4.e4
-5.e4
-4.e4
-3.e4
-2.e4
-1.e4
Ω* Î C
1.e4 2.e4 3.e4 4.e4
2.e4
4.e4
6.e4
8.e4
Α* Î C
hc = 10.5 µm, with ω∗ = 1.1 ·104 s−1, α∗ = (0.4+4i) ·103 m−1.
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Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
Asymptotic Behavior
For large resistance and high quality factor
r = Rρ0U∞
1,
√mKR= O(r ),
in D(α, ω) = Dα(α, ω) = 0 with ω is real, then asymptotically
hc = 14
(ρ0U∞
R
)2
U∞
√mK
(This includes the industrially typical cases!)
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Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
Outline
1 Background
2 The Model
3 Stability Analysis
4 Regularized Boundary Condition
5 Conclusions and Future Work
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Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
Asymptotic Behavior for Small αh
For αh→ 0 and = ω − αU∞ we find
Z ' ρ0
i·
2 + |α|(ω+ 13U2∞α
2)h|α|ω + α2h
Non-uniqueness of expansion: multiply by e−|α|hθ / e−|α|hθ
Z ' ρ0
i·
2 + |α|((1− θ)ω2 − (1− 2θ)ωαU∞ + (13 − θ)U2
∞α2)h
|α|ω + α2(− θω)h
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Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
Optimal Agreement for θ = 13
2. 4. 6. 8. 10. 12. 14.
-5.e4
-4.e4
-3.e4
-2.e4
-1.e4
0.e
Zoom In
3.68 3.681
-6033
-6031
h@ΜmD
ImHΩ*L@s-1D
1.e4 2.e4 3.e4 4.e4
-5.e4
-4.e4
-3.e4
-2.e4
-1.e4
Zoom In
15 900 15 902
-9600
-9598
Ω* Î C
1.e4 2.e4 3.e4 4.e4
2.e4
4.e4
6.e4
8.e4
Zoom In
18 000 18 005
37 365
37 370
Α* Î C
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Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
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 ):
Z =(
iω + U∞∂
∂x
)p − hρ0iω
(23 iω + 1
3U∞∂
∂x
)(v·n)
iω(v·n)+ hρ0
∂2
∂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.
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Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
Tentative numerical experiments confirm transition unstable - stable
For smooth U0 ∼ 160 m/s, h ∼ 0.06 m, mρ0= 1 m, L = 3 mm
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Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
Outline
1 Background
2 The Model
3 Stability Analysis
4 Regularized Boundary Condition
5 Conclusions and Future Work
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Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
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.
/centre for analysis, scientific computing and applications
Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
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.
/centre for analysis, scientific computing and applications
Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
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.
/centre for analysis, scientific computing and applications
Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
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.
/centre for analysis, scientific computing and applications
Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
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.
/centre for analysis, scientific computing and applications
Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
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.
/centre for analysis, scientific computing and applications
Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
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
c20
−(α2 + m2
r2
))p = 0.
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Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
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
c20
−(α2 + m2
r2
))p = 0.
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Background The Model Stability Analysis Regularized Boundary Condition Conclusions and Future Work
Thank you for your attention!