Localisation of Rayleigh-Bloch waves and stability of resonant loads on arrays of bottom-mounted cylinders with respect to positional disorder

Transcription

1 Localisation of Rayleigh-Bloch waves and stability of resonant loads on arrays of bottom-mounted cylinders with respect to positional disorder Luke Bennetts University of Adelaide, Australia

2 Collaborators Malte Peter Uni Augsburg, Germany Fabien Montiel Uni Otago, NZ

3 Resonant loads on an array of cylinders y Plan view ψ Plane incident wave z x C 1 C 2 C N d Cross-sectional view 2a x Load on C4 Maniar & Newman (1997, JFM) ψ = 0 N = 9 d/a = 4 C 1 C 2 C N kd/π

4 Resonant loads on an array of cylinders Loads kd = 2π 3 Neumann trapped modes kd = kd = π kd = Cylinder index

5 Since Maniar & Newman (1997) 177 citations, including... Porter & Evans (1998, Quarterly Journal on Mechanics & Applied Mathematics) Porter & Evans (1999, Journal of Fluid Mechanics) Evans & Porter (1999, Journal of Engineering Mathematics) Utsunomiya & Eatock-Taylor (1999, Journal of Fluid Mechanics) Linton & McIver (2002, Journal of Fluid Mechanics) Walker & Eatock-Taylor (2005, Ocean Engineering) Thompson, Linton & Porter (2007, Quarterly Journal on Mechanics & Applied Mathematics)

6 Since Maniar & Newman (1997) 177 citations, including... Porter & Evans (1998, Quarterly Journal on Mechanics & Applied Mathematics) Porter & Evans (1999, Journal of Fluid Mechanics) Evans & Porter (1999, Journal of Engineering Mathematics) Utsunomiya & Eatock-Taylor (1999, Journal of Fluid Mechanics) Linton & McIver (2002, Journal of Fluid Mechanics) Walker & Eatock-Taylor (2005, Ocean Engineering) Thompson, Linton & Porter (2007, Quarterly Journal on Mechanics & Applied Mathematics)

7 Mathematical model: solitary cylinder Linear potential-flow theory: { g velocity field = Re Φ(x, y, z)e iωt} iω where φ = velocity potential; ω = frequency; g = gravity. z x

8 Mathematical model: solitary cylinder Linear potential-flow theory: { g velocity field = Re Φ(x, y, z)e iωt} iω where φ = velocity potential; ω = frequency; g = gravity. z x 2 Φ = 0 Φ z = 0 z = h

9 Mathematical model: solitary cylinder Linear potential-flow theory: { g velocity field = Re Φ(x, y, z)e iωt} iω where φ = velocity potential; ω = frequency; g = gravity. z Φ z = ω 2 Φ/g x 2 Φ = 0 Φn = 0 Φ z = 0 z = h

10 Vertical eigenfunctions z x Φ = φ(x, y) cosh k(z + h) + φ n (x, y) cos k n (z + h) n=1 where k tanh(kh) = ω 2 /g

11 Vertical eigenfunctions z x φ inc (x, y : ψ) cosh k(z + h) Φ = φ(x, y) cosh k(z + h) + φ n (x, y) cos k n (z + h) n=1 where k tanh(kh) = ω 2 /g

12 Vertical eigenfunctions z x φ inc (x, y : ψ) cosh k(z + h) Φ = φ(x, y) cosh k(z + h)

13 Problem in horizontal plane y 2 φ + k 2 φ = 0 x φ n = 0

14 Problem in horizontal plane y 2 φ + k 2 φ = 0 φ inc φ n = 0 x

15 Problem in horizontal plane y 2 φ + k 2 φ = 0 φ = φ inc + φ sca φ inc φ n = 0 x

16 Horizontal eigenfunctions y r Local polar coordinates (r, θ) θ x Bessel-Fourier Expansions Incident Field φ inc (r, θ) = b m J m (kr) e imθ m= Scattered Field φ sca (r, θ) = Z m b m H m (kr) e imθ m= J m and H m are Bessel and Hankel functions of 1st kind, resp. Z m = J m(ka)/h m(ka)

17 Multiple cylinders: general setting Scattered waves force (become incident) on other cylinders. E.g. below: φ (1) inc = φ amb + φ (2) sca + φ (3) sca + φ (4) sca φ amb C 2 C 3 C 1 C 4 Assign local polars (r p, θ p ) to C p. Map scattered to incident: φ (p) sca = Z j b (p) j H j (kr p ) e ijθp = j= j= Z j b (p) j m= g (p1) jm J m(kr 1 ) e imθ 1 for known g (p1) jm

18 Multiple cylinders: general setting Scattered waves force (become incident) on other cylinders. E.g. below: φ (1) inc = φ amb + φ (2) sca + φ (3) sca + φ (4) sca φ amb C 2 C 3 C 1 C 4 Assign local polars (r p, θ p ) to C p. Map scattered to incident: Z j b (p) j H j (kr p ) e ijθp φ (p) sca = = j= m= j= g (p1) jm Z j b (p) j J m(kr 1 ) e imθ 1

19 Multiple cylinders: general setting Scattered waves force (become incident) on other cylinders. E.g. below: φ (1) inc = φ amb + φ (2) sca + φ (3) sca + φ (4) sca Then express ambient field C 2 C 3 φ amb = m= c (1) m J m (kr 1 ) e imθ 1 φ amb C 1 C 4 Matching modal amplitudes gives b (1) m = c (1) m + 4 p=2 j= for m = 0, ±1, ±2,.... Z j b (p) j g (p1) jm

20 Multiple cylinders: general setting Scattered waves force (become incident) on other cylinders. E.g. below: φ (1) inc = φ amb + φ (2) sca + φ (3) sca + φ (4) sca φ amb C 2 C 3 C 1 C 4 Repeat for C 2, C 3 and C 4 and solve. But, computational limit is O(100) cylinders.

21 Infinite line of eqispaced cylinders y C 2 C 1 C 0 C 1 C 2 x x = 0 x = d x = 2d Quasi-periodicity implies there exists Q such that Hence b m = c (?) m + where σ n (Qd) = b (p) m = e iqdp b m. p=1 j= Z j b j σ j m (Qd) { ( 1) n e ipqd e ipqd} H n (kp)

22 Infinite line of eqispaced cylinders y C 2 C 1 C 0 C 1 C 2 x x = 0 x = d x = 2d Quasi-periodicity implies there exists Q such that Hence b m = c (?) m + where σ n (Qd) = b (p) m = e iqdp b m. p=1 j= Z j b j σ j m (Qd) { ( 1) n e ipqd e ipqd} H n (kp)

23 Forced problem T 0 e ik(x cos ψ 0+y sin ψ 0 ) T 1 e ik(x cos ψ 1+y sin ψ 1 ) φ amb = eik(x cos ψ+y sin ψ) Incident amplitudes Transmitted field (similar for reflected) φ = c (p) m = e i2kdp cos ψ c m Q = 2k cos ψ n= ik(x cos ψn+y sin ψn) T n e where cos ψ n = cos ψ 0 + nπ/kd ψ 0 = ψ

24 Forced problem T 0 e ik(x cos ψ 0+y sin ψ 0 ) T 1 e ik(x cos ψ 1+y sin ψ 1 ) φ amb = eik(x cos ψ+y sin ψ) Incident amplitudes Transmitted field (similar for reflected) φ = c (p) m = e i2kdp cos ψ c m Q = 2k cos ψ n= ik(x cos ψn+y sin ψn) T n e where cos ψ n = cos ψ 0 + nπ/kd ψ 0 = ψ

25 Forced problem T 0 e ik(x cos ψ 0+y sin ψ 0 ) T 1 e ik(x cos ψ 1+y sin ψ 1 ) φ amb = eik(x cos ψ+y sin ψ) Incident amplitudes Transmitted field (similar for reflected) φ = c (p) m = e i2kdp cos ψ c m Q = 2k cos ψ n= ik(x cos ψn+y sin ψn) T n e ψ n R travelling waves; ψ n C decaying waves

26 Unforced problem (e.g. Porter & Evans, 1999) Rayleigh-Bloch wave modes Incident amplitudes c m = 0 Seek eigenvalues e iqrbd and associated eigenfunctions RB modes bound to array (In general) solutions exist for kd < Q RB d < π Incident wave cannot excite them Peter & Meylan (JFM, 2007)

27 Semi-infinite problem (e.g. Peter & Meylan, 2007) Solution method Write where b (p) m = e i2kdp cos ψ b ψ m + αe iqrb dp b RB m + b (p) m b ψ m are amplitudes for forced problem with incident angle ψ. are amplitudes for unforced problem and α is a constant. b (p) m 0 as p. b RB m φ amb C 1 C 2 C 3

28 Semi-infinite problem (e.g. Peter & Meylan, 2007) Solution method Write where b (p) m = e i2kdp cos ψ b ψ m + αe iqrb dp b RB m + b (p) m b ψ m are amplitudes for forced problem with incident angle ψ. are amplitudes for unforced problem and α is a constant. b (p) m 0 as p. b RB m Scattered field φ sca Linton et al. (2007)

29 Finite problem (Thompson et al., 2007) C 1 C 2 C N φ amb Solution method b (p) m = e i2kdp cos ψ b ψ m + α + e iqrb dp b RB m + b (p)+ m + where b (p)+ m 0 as p N. b (p) m 0 as p 1. b (p) m + α e iqrb dp ( 1) m b RB m

30 Finite problem (Thompson et al., 2007) C 1 C 2 C N φ amb Resonance requires 1 Strong excitation of Rayleigh-Bloch mode. 2 Strong reflection of Rayleigh-Bloch mode at ends. 3 Coherence of Rayleigh-Bloch modes.

31 A different approach Motivation Plane wave with amplitude A and at angle χ ik(x cos χ+y sin χ) ϕ(x, y : χ) Ae is a solution of ϕ xx + ϕ yy + k 2 ϕ = 0 Where are the plane waves in the direct formulation? J n (kr)e inθ = ( i)n 2π H n (kr)e inθ = ( i)n π π π γ+θ e inχ ϕ(χ) dχ e inχ ϕ(χ) dχ γ χ C

32 A different approach Motivation Plane wave with amplitude A and at angle χ ik(x cos χ+y sin χ) ϕ(x, y : χ) Ae is a solution of ϕ xx + ϕ yy + k 2 ϕ = 0 Where are the plane waves in the direct formulation? J n (kr)e inθ = ( i)n 2π H n (kr)e inθ = ( i)n π π π γ+θ e inχ ϕ(χ) dχ e inχ ϕ(χ) dχ γ χ C

33 A different approach Motivation Plane wave with amplitude A and at angle χ ik(x cos χ+y sin χ) ϕ(x, y : χ) Ae is a solution of ϕ xx + ϕ yy + k 2 ϕ = 0 Where are the plane waves in the direct formulation? J n (kr)e inθ = ( i)n 2π H n (kr)e inθ = ( i)n π π π γ+θ e inχ ϕ(χ) dχ e inχ ϕ(χ) dχ γ χ C

34 A different approach Motivation Plane wave with amplitude A and at angle χ ik(x cos χ+y sin χ) ϕ(x, y : χ) Ae is a solution of ϕ xx + ϕ yy + k 2 ϕ = 0 Where are the plane waves in the direct formulation? J n (kr)e inθ = ( i)n 2π H n (kr)e inθ = ( i)n π π π γ+θ e inχ ϕ(χ) dχ e inχ ϕ(χ) dχ γ χ C

35 A different approach Hankel functions H n (kr)e inθ = ( i)n π { π 2 π 2 π + π 2 2 i } π 2 + e inχ ϕ(χ) dχ π 2 +i if x 0 i.e. right travelling/decaying waves (real/complex branches) H n (kr)e inθ = ( i)n π { 3π 2 π 2 3π + 3π 2 2 i π } + 2 π 2 +i e inχ ϕ(χ) dχ if x 0 i.e. left travelling/decaying waves (real/complex branches)

36 Decompose geometry d

37 Single cylinder φ inc = π 2 i π 2 +i c (χ)ϕ(χ) dχ φ + sca = π 2 i π 2 +i b + (χ)ϕ(χ) dχ φ sca = 3π 2 i b (χ)ϕ(χ) dχ π 2 +i φ + inc = 3π 2 i c + (χ)ϕ(χ) dχ π 2 +i c ± (χ) are incident amplitude functions b ± (χ) are scattered amplitude functions

38 Single slab Introduce scattering kernels R and T : b (χ) = b + (χ) = π 2 i π 2 +i R(χ ψ)c (ψ) dψ + π 2 i π 2 +i T (χ ψ)c (ψ) dψ + 3π 2 i T (χ ψ)c + (ψ) dψ π 2 +i 3π 2 i R(χ ψ)c + (ψ) dψ π 2 +i Truncate complex branches and sample (discretise) χ-space. Gives array expressions: b = Rc + T c + and b + = T c + Rc + Associated transfer matrix (left-to-right map) ( T RT P = 1 R RT 1 T 1 R T 1 ) calculating transfer matrix solving problem.

39 Spectrum of transfer matrix Rayleigh-Bloch mode Eigenvalues Eigenfunction 1 e iqrb d v RB rightward - - leftward Im Re Re(χ)/π Im(χ)

40 Spectrum of transfer matrix Forced mode Eigenvalues Eigenfunction 1 ikd cos(0) e rightward - - leftward Im Re Re(χ)/π Im(χ)

41 Summary of new approach Strengths Solving for infinite, semi-infinite and finite arrays requires linear algebra only. Straightforward to incorporate disorder P 1,N = P N P N 1... P 1. Bounded system size (independent of num. of cylinders). Weaknesses Relies on discretisation and truncation. Involves exponentially growing/decaying terms.

42 Wave localisation Appears in all (?!?) branches on wave science. Disorder can localise waves in space without dissipation. Manifest as exponential attenuation of wave amplitude. Due to interference between scattered waves. Attenuation rate depends on frequency and medium properties. log 10 amplitude distance

43 Positional disorder damps resonance Introduce positional disorder φ amb ɛ 1 d ɛ 2 d ɛ N d Max Load ψ = 0 N = 100 d/a = 4 Perturbations chosen randomly. ɛ n U(0, ɛ) for prescribed ɛ. 0 1 kd/π ɛ = 0; ɛ = 0.1; ɛ = 0.2; ɛ = 0.4

44 Are Rayleigh-Bloch waves localised? Question Does positional disorder localise Rayleigh-Bloch waves? Is this the process that damps the resonant loads? Idea Excite motions with Rayleigh-Bloch waves rather than plane incident wave. Assume Rayleigh-Bloch wave modes dominate. Positional disorder expected to perturb RB modes. Extract Rayleigh-Bloch wavenumber Q RB (ɛ). Q RB (ɛ) C implies localisation. Problem Can t excite rightward/leftward travelling RB wave only.

45 Are Rayleigh-Bloch waves localised? Question Does positional disorder localise Rayleigh-Bloch waves? Is this the process that damps the resonant loads? Idea Excite motions with Rayleigh-Bloch waves rather than plane incident wave. Assume Rayleigh-Bloch wave modes dominate. Positional disorder expected to perturb RB modes. Extract Rayleigh-Bloch wavenumber Q RB (ɛ). Q RB (ɛ) C implies localisation. Problem Can t excite rightward/leftward travelling RB wave only.

46 Are Rayleigh-Bloch waves localised? Question Does positional disorder localise Rayleigh-Bloch waves? Is this the process that damps the resonant loads? Idea Excite motions with Rayleigh-Bloch waves rather than plane incident wave. Assume Rayleigh-Bloch wave modes dominate. Positional disorder expected to perturb RB modes. Extract Rayleigh-Bloch wavenumber Q RB (ɛ). Q RB (ɛ) C implies localisation. Problem Can t excite rightward/leftward travelling RB wave only.

47 Are Rayleigh-Bloch waves localised? Method 1 Solve 2 problems: 1 With forward RB incident wave from left of array. 2 With backward RB incident wave from right of array. 2 Use ansatz for amplitude spectra known ( cn b n ) = α + e iqrbn v RB + + α e iqrb (N+1 n) v RB, unknown through the array n = 0,..., N. 3 Combine solutions to separate rightward and leftward modes.

48 Example results: no disorder kd = ; d/a = 4; N = 100; ɛ = 0 ±Q RB d ±3.104 ± i (Q RB d = ) Im(pQ RB d) Re(pQ RB d) 0 cylinder p cylinder p 100

49 Example results: with disorder kd = ; d/a = 4; N = 100; ɛ = 0.2 Q RB d i Q RB d i Im(pQ RB d) Re(pQ RB d) 0 cylinder p cylinder p 100

50 Example results: ensemble runs ε = 0.2; ensemble size 500 Re(Q RB d) ±2.73 Im(Q RB d) ±

51 Work in progress Localisation of Rayleigh-Bloch waves... Evidence of this.... and stability of resonant loads on arrays of bottom mounted cylinders with respect to positional disorder Disorder damps/eliminates resonance. Still to do Make the link undeniable.

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