Asymptotic modeling of the wave-propagation over acoustic liners - - PowerPoint PPT Presentation
Asymptotic modeling of the wave-propagation over acoustic liners Adrien Semin Technische Universit at Berlin, Institut f ur Mathematik Joint work with Kersten Schmidt (TU Berlin) and B erang` ere Delourme (Paris 13) Research Center M
Technische Universit¨ at Berlin, Institut f¨ ur Mathematik Joint work with Kersten Schmidt (TU Berlin) and B´ erang` ere Delourme (Paris 13)
Research Center MATHEON Mathematics for key technologies
Analysis and Numerics of Acoustic and Electromagnetic Problems, October 17th-22nd
Courtesy of A. Th¨
2 / 34
Bias flow liner in an acoustic channel (DUCT-C) at the institute of Institute of Propulsion Technology at DLR Berlin (courtesy of F. Bake)
3 / 34
1
2
3
4
4 / 34
1
2
3
4
5 / 34
Θ x− O h(δ) x+ O η(δ) δ x− O Θ x+ O
Γ Figure: On the left: the exact domain Ωδ,η. On the right: the limit domain Ω0.
6 / 34
Θ x− O h(δ) x+ O η(δ) δ x− O Θ x+ O
Γ Figure: On the left: the exact domain Ωδ,η. On the right: the limit domain Ω0.
6 / 34
Θ x− O h(δ) x+ O η(δ) δ x− O Θ x+ O
Γ Figure: On the left: the exact domain Ωδ,η. On the right: the limit domain Ω0.
6 / 34
7 / 34
7 / 34
7 / 34
0pδ = div F,
t pδ = 0,
O of the periodic layer, the macroscopic part is corrected by a
O )/δ).
8 / 34
0pδ = div F,
t pδ = 0,
O of the periodic layer, the macroscopic part is corrected by a
O )/δ).
8 / 34
0pδ = div F,
t pδ = 0,
O of the periodic layer, the macroscopic part is corrected by a
O )/δ).
8 / 34
0pδ = div F,
t pδ = 0,
O of the periodic layer, the macroscopic part is corrected by a
O )/δ).
8 / 34
0pδ = div F,
t pδ = 0,
O of the periodic layer, the macroscopic part is corrected by a
O )/δ).
8 / 34
x−
O
x+
O
O might not be done in a
9 / 34
x−
O
x+
O
O might not be done in a
9 / 34
x−
O
x+
O
O might not be done in a
9 / 34
x−
O
x+
O
O might not be done in a
9 / 34
x−
O
x+
O
O might not be done in a
9 / 34
Γ x−
O
x+
O
O might not be done in a
9 / 34
Γ x−
O
x+
O
O might not be done in a
9 / 34
Γ x−
O
x+
O
O might not be done in a
9 / 34
1
2
3
4
10 / 34
Θ x− O h(δ) x+ O η(δ) δ x− O Θ x+ O
Γ
t pδ = 0
11 / 34
1
2
3
4
12 / 34
0pδ = div F,
t pδ = 0,
0(x) + δpδ 1(x) + . . . ,
1
13 / 34
0pδ = div F,
t pδ = 0,
0(x) + δpδ 1(x) + . . . ,
1
13 / 34
0pδ = div F,
t pδ = 0,
0(x) + δpδ 1(x) + . . . ,
1
13 / 34
0pδ = div F,
t pδ = 0,
0(x) + δpδ 1(x) + . . . ,
1
0 satisfies
0 + k2 0pδ 0 = div F,
0 = 0,
13 / 34
0pδ = div F,
t pδ = 0,
0(x) + δpδ 1(x) + . . . ,
1
1 satisfies
1 + k2 0pδ 1 = 0,
1 = −(1 + ı)ζ∂2 t pδ 0,
1
1
x1 + k2 0)
13 / 34
1
1
x1 + k2 0)
t D = 0,
X2→±∞ D(X1, X2) − (X2 ± D∞) = 0
14 / 34
1
1
x1 + k2 0)
t D = 0,
X2→±∞ D(X1, X2) − (X2 ± D∞) = 0
14 / 34
1
1
x1 + k2 0)
t D = 0,
X2→±∞ D(X1, X2) − (X2 ± D∞) = 0
14 / 34
1
1
x1 + k2 0)
t D = 0,
X2→±∞ D(X1, X2) − (X2 ± D∞) = 0
14 / 34
1
2
3
4
15 / 34
x−
O
x+
O
Θ Γ x−
O
x+
O
Θ
0)pδ 0 = div F,
0)pδ 1 = 0,
0 = 0,
1 = −(1 + ı)ζ∂2 t pδ 0,
1
1
x1 + k2 0)
0 and pδ 1 that hare H1 regularity away from the corners.
0 ∈ H1(Ω), can we find pδ 1 ∈ H1(Ω) satisfying these transmission conditions?
0 ∈ H1(Ω)?
16 / 34
x−
O
x+
O
Θ Γ x−
O
x+
O
Θ
0)pδ 0 = div F,
0)pδ 1 = 0,
0 = 0,
1 = −(1 + ı)ζ∂2 t pδ 0,
1
1
x1 + k2 0)
0 and pδ 1 that hare H1 regularity away from the corners.
0 ∈ H1(Ω), can we find pδ 1 ∈ H1(Ω) satisfying these transmission conditions?
0 ∈ H1(Ω)?
16 / 34
x−
O
x+
O
Θ Γ x−
O
x+
O
Θ
0)pδ 0 = div F,
0)pδ 1 = 0,
0 = 0,
1 = −(1 + ı)ζ∂2 t pδ 0,
1
1
x1 + k2 0)
0 and pδ 1 that hare H1 regularity away from the corners.
0 ∈ H1(Ω), can we find pδ 1 ∈ H1(Ω) satisfying these transmission conditions?
0 ∈ H1(Ω)?
16 / 34
x−
O
x+
O
Θ Γ x−
O
x+
O
Θ
0)pδ 0 = div F,
0)pδ 1 = 0,
0 = 0,
1 = −(1 + ı)ζ∂2 t pδ 0,
1
1
x1 + k2 0)
0 and pδ 1 that hare H1 regularity away from the corners.
0 ∈ H1(Ω), can we find pδ 1 ∈ H1(Ω) satisfying these transmission conditions? No.
0 ∈ H1(Ω)? No.
16 / 34
x−
O
x+
O
Θ Γ x−
O
x+
O
Θ
0)pδ 0 = div F,
0)pδ 1 = 0,
0 = 0,
1 = −(1 + ı)ζ∂2 t pδ 0,
1
1
x1 + k2 0)
0 and pδ 1 that hare H1 regularity away from the corners.
0 ∈ H1(Ω), can we find pδ 1 ∈ H1(Ω) satisfying these transmission conditions? No.
0 ∈ H1(Ω)? No.
◮ corner singularities due to the transmission conditions (two next slides), ◮ corner singularities coming from the matching to the near field (slides beneath), 16 / 34
x−
O
x+
O
Θ Γ x−
O
x+
O
Θ
0)pδ 0 = div F,
0)pδ 1 = 0,
0 = 0,
1 = −(1 + ı)ζ∂2 t pδ 0,
1
1
x1 + k2 0)
0 and pδ 1 that hare H1 regularity away from the corners.
0 ∈ H1(Ω), can we find pδ 1 ∈ H1(Ω) satisfying these transmission conditions? No.
0 ∈ H1(Ω)? No.
◮ corner singularities due to the transmission conditions (two next slides), ◮ corner singularities coming from the matching to the near field (slides beneath), 16 / 34
Θ Hδ δ ρδ x−
O
Θ x−
O
0(x) + δpδ 1(x) + . . . ,
q(x) under the form
q(x) = p0,q(x) + δ
π Θ p1,q(x) + δ 2π Θ p2,q(x) + . . .
π Θ p1,0(x) + δp0,1(x) + δ 2π Θ p2,0(x) + δ π Θ +1p1,1(x) + . . .
O in polar coordinates as
Θ (k0r) cos π
17 / 34
Θ Hδ δ ρδ x−
O
Θ x−
O
0(x) + δpδ 1(x) + . . . ,
q(x) under the form
q(x) = p0,q(x) + δ
π Θ p1,q(x) + δ 2π Θ p2,q(x) + . . .
π Θ p1,0(x) + δp0,1(x) + δ 2π Θ p2,0(x) + δ π Θ +1p1,1(x) + . . .
O in polar coordinates as
Θ (k0r) cos π
17 / 34
Θ Hδ δ ρδ x−
O
Θ x−
O
0(x) + δpδ 1(x) + . . . ,
q(x) under the form
q(x) = p0,q(x) + δ
π Θ p1,q(x) + δ 2π Θ p2,q(x) + . . .
π Θ p1,0(x) + δp0,1(x) + δ 2π Θ p2,0(x) + δ π Θ +1p1,1(x) + . . .
O in polar coordinates as
Θ (k0r) cos π
17 / 34
Θ x−
O
Θ (k0r) cos π
x1 + k2 0) p0,0Γ .
Θ −1(k0r).
Θ −1(k0r)ψ1(θ) such that
0)φ1 = 0 and a numeric constant C1 such that
O
18 / 34
Θ x−
O
Θ (k0r) cos π
x1 + k2 0) p0,0Γ .
Θ −1(k0r).
Θ −1(k0r)ψ1(θ) such that
0)φ1 = 0 and a numeric constant C1 such that
O
18 / 34
Θ x−
O
Θ (k0r) cos π
x1 + k2 0) p0,0Γ .
Θ −1(k0r).
Θ −1(k0r)ψ1(θ) such that
0)φ1 = 0 and a numeric constant C1 such that
O
18 / 34
Θ H 1 ρ x−
O
O )/δ:
π Θ P1,0(X−) + δP0,1(X−) + δ 2π Θ P2,0(X−) + δ π Θ +1P1,1(X−) + . . .
π Θ cos π
Θ(θ − π) towards infinity.
ΓS = 0,
π Θ cos π
19 / 34
Θ H 1 ρ x−
O
O )/δ:
π Θ P1,0(X−) + δP0,1(X−) + δ 2π Θ P2,0(X−) + δ π Θ +1P1,1(X−) + . . .
π Θ cos π
Θ(θ − π) towards infinity.
ΓS = 0,
π Θ cos π
19 / 34
Θ H 1 ρ x−
O
O )/δ:
π Θ P1,0(X−) + δP0,1(X−) + δ 2π Θ P2,0(X−) + δ π Θ +1P1,1(X−) + . . .
π Θ cos π
Θ(θ − π) towards infinity.
ΓS = 0,
π Θ cos π
19 / 34
Θ H 1 ρ x−
O
O )/δ:
π Θ P1,0(X−) + δP0,1(X−) + δ 2π Θ P2,0(X−) + δ π Θ +1P1,1(X−) + . . .
π Θ cos π
Θ(θ − π) towards infinity.
ΓS = 0,
π Θ cos π
19 / 34
Θ H 1 ρ x−
O
ΓS = 0,
π Θ cos π
20 / 34
Θ H 1 ρ x−
O
ΓS = 0,
π Θ cos π
π Θ cos π
π Θ −1ψ1(θ) + O(R π Θ −2 ln R)
Θ cos π
Θ −2,− 2π Θ ) ln R) 21 / 34
Θ H 1 ρ x−
O
ΓS = 0,
π Θ cos π
π Θ cos π
π Θ −1ψ1(θ) + O(R π Θ −2 ln R)
Θ cos π
Θ −2,− 2π Θ ) ln R) 21 / 34
Θ (k0r) cos π
π Θ cos π
π Θ −1ψ1(θ) + L (S)R− π Θ cos π
π Θ term ⇐ comes from the matching with p0,0,
π Θ −1 term ⇒ will be matched by p0,1 (already matched),
Θ term ⇒ will be matched by p2,0: there exists a numeric constant C2 such that
Θ (k0r) cos π
O
22 / 34
Θ (k0r) cos π
π Θ cos π
π Θ −1ψ1(θ) + L (S)R− π Θ cos π
π Θ term ⇐ comes from the matching with p0,0,
π Θ −1 term ⇒ will be matched by p0,1 (already matched),
Θ term ⇒ will be matched by p2,0: there exists a numeric constant C2 such that
Θ (k0r) cos π
O
22 / 34
Θ (k0r) cos π
π Θ cos π
π Θ −1ψ1(θ) + L (S)R− π Θ cos π
π Θ term ⇐ comes from the matching with p0,0,
π Θ −1 term ⇒ will be matched by p0,1 (already matched),
Θ term ⇒ will be matched by p2,0: there exists a numeric constant C2 such that
Θ (k0r) cos π
O
22 / 34
1
2
3
4
23 / 34
1
1
x1 + k2 0)
10−1 100 10−2 10−1 100 101
1
ρ D∞
H = 0.3 H = 0.6
24 / 34
0pδ = div F,
25 / 34
0pδ = div F,
0 satisfies
0 + k2 0pδ 0 = div F,
0 = 0,
25 / 34
0pδ = div F,
0 satisfies
0 + k2 0pδ 0 = div F,
0 = 0,
25 / 34
δ η δ
0 + k2 0pδ 0 = div F,
0 = 0,
0.
26 / 34
δ η δ
0 + k2 0pδ 0 = div F,
0 = 0,
0.
26 / 34
δ η δ
0 + k2 0pδ 0 = div F,
0 = 0,
0.
26 / 34
Corresponds to the experimental results provided by DLR Berlin
(2015) Acta Acust. united Ac., 101 (1).
27 / 34
1
2
3
4
28 / 34
10−3 10−2 10−1 10−6 10−5 10−4 10−3 10−2 10−1 100 101 102 103
1 4/3 2
Distance δ between two consecutive holes Relative macroscopic error
two scale limit u0,0 two scale expansion u0,0 + δu0,1 two scale expansion u0,0 + δu0,1 + δ4/3u2,0
Figure: Numerically computed errors of macroscopic expansions truncated at different orders in dependence of δ for the inviscid model for the porosity ρ = 0.3 and the height h(δ) = 0.6 δ.
29 / 34
10−3 10−2 10−1 10−4 10−3 10−2 10−1 100 101 102 103
∼ 1 ∼ 4/3
Distance δ between two consecutive holes Relative macroscopic error
two scale limit u0,0 two scale expansion u0,0 + δu0,1
Figure: Numerically computed errors of macroscopic expansions truncated at different orders in dependence of δ for the viscous model (ζ = 0.1) for the porosity ρ = 0.3 and the height h(δ) = 0.6 δ.
29 / 34
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10−4 10−3 10−2 10−1 100 Porosity ρ Relative macroscopic error two scale limit u0,0 two scale expansion u0,0 + δuδ
0,1
two scale epxansion u0,0 + δuδ
0,1 + δ
4 3 uδ
2,0
three scale limit u0,0 30 / 34
1
2
3
4
31 / 34
Γ x−
O
x+
O
Θ
O , using different stories.
32 / 34
Γ x−
O
x+
O
Θ
O , using different stories.
32 / 34
Θ H 1 ρ x−
O
Θ M
ΓS = 0,
π Θ cos π
33 / 34
34 / 34