New asymmetric gravity-capillary and flexural waves Jean-Marc - - PDF document

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New asymmetric gravity-capillary and flexural waves Jean-Marc - - PDF document

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New asymmetric gravity-capillary and flexural waves Jean-Marc Vanden-Broeck University College London ICERM , April 2017 COWORKERS Zhan Wang Tao Gao Paul Milewski Emilian Parau Olga Trichtchenko 1 2 3 4 5 6 inviscid,

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New asymmetric gravity-capillary and flexural waves

Jean-Marc Vanden-Broeck University College London ICERM , April 2017

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COWORKERS Zhan Wang Tao Gao Paul Milewski Emilian Parau Olga Trichtchenko

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  • inviscid, incompressible, irrotational
  • gravity
  • surface tension
  • steady

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NON SYMMETRIC WAVES....in two and three dimensions

  • Periodic waves
  • Solitary waves
  • Generalised Solitary waves

flexural waves (thursday.....Olga....). stability

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PART 1 TWO-DIMENSIONAL FLOWS

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FORMULATION GRAVITY-CAPILLARY WAVES φxx + φyy = 0 φy = φxζx

  • n

y = ζ(x) 1 2(φ2

x + φ2 y) + gy−T

ρ κ = B

  • n

y = ζ(x) φy = 0

  • n

y = −h FLEXURAL WAVES D ρ (∂2

s κ + 1

2κ3) T = surface tension, D = flexural rigidity κ = ζxx (1 + ζ2

x)3/2

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PERIODIC and SOLITARY waves Gravity waves

ental educational concepts of linear and non-linear ions of nonlinear equations from the book High-Temperature he Nonlinear Mechanism and Tunneling Measurements (Klu s, Dordrecht, 2002, pages 101-142) is given. There are a few

  • tons. For example, there are topological and
  • ns. Independently of the topological nature of solitons,

Craig W. and Sternberg P. (1988)

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Gravity-capillary solitary waves

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NUMERICAL METHODS boundary integral equation methods, series trun- cation methods or ANY OTHER METHODS....

  • 1. Iterations by using Newton’s method
  • 2. Continuation methods
  • 3. INITIAL GUESS: bifurcations, symmetry

breaking...

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Dimensioless variables: ( T

ρg)1/2 (reference length),

( T

ρg3)1/2 (reference time)

amplitude: A phase velocity: c energy: E E = 1 2

∞

−∞

η

−∞(φ2 x + φ2 y)dydx + 1

2

∞

−∞ η2dx

+

∞

−∞(

  • 1 + η2

x − 1)dx

Boundary integral equation, Newton iterations, continuation

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Gravity capillary solitary waves infinite depth

  • 15
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1.24 1.26 1.28 1.3 1.32 1.34 1.36 1.38 1.4 1.4142 0.05 0.1 0.15 0.2

(0)

1.395 1.405 −0.8 −0.6 −0.4 −0.2

Speed (0) B (b) C D A E F D C (a)

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1.24 1.26 1.28 1.3 1.32 1.34 1.36 1.38 1.4 1.4142 1 2 3 4 5 6

Speed Energy (c)

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1.24 1.26 1.28 1.3 1.32 1.34 1.36 1.38 1.4 1.4142 0.05 0.1 0.15 0.2

(0)

1.395 1.405 −0.8 −0.6 −0.4 −0.2

Speed (0) B (b) C D A E F D C (a)

Zufiria (1987), Buffoni, Champneys and Toland (1996), Yang and Akylas (1997), Champneys and Groves (1997).......

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1.38 1.385 1.39 1.395 1.4 1.405 1.41 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

C

  • D
  • A

B

  • Speed

Energy

1.37875 1.37879 1.405 1.4065 1.408 0.5 0.54 0.58 0.62 0.66

C

  • D
  • A
  • B
  • (a)

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1.375 1.38 1.385 1.39 1.395 1.4 1.405 1.41 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

5

  • 6
  • 3
  • 4
  • 1

2

  • 1.3787

1.3788 1.3845 1.3855

5

  • 6
  • 1.405

1.406 1.407 1.408 1.409 0.5 0.55 0.6 0.65

3

  • 4
  • 1
  • 2
  • (a)

−0.2 −0.15 −0.1 −0.05 0.05 0.1 0.15 −40 −20 20 40 60 80 −0.25 −0.2 −0.15 −0.1 −0.05 0.05 0.1

(b)

−0.2 −0.15 −0.1 −0.05 0.05 0.1 0.15 −80 −60 −40 −20 20 40 60 80 −0.25 −0.2 −0.15 −0.1 −0.05 0.05 0.1 0.15

(c)

−0.3 −0.2 −0.1 0.1 0.2 −80 −60 −40 −20 20 40 60 −0.3 −0.2 −0.1 0.1 0.2

(d)

−0.15 −0.1 −0.05 0.05 0.1 0.15 −100 −80 −60 −40 −20 20 40 60 80 100 −0.2 −0.15 −0.1 −0.05 0.05 0.1

(e)

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1.24 1.26 1.28 1.3 1.32 1.34 1.36 1.38 1.4 1.4142 0.05 0.1 0.15 0.2

(0)

1.395 1.405 −0.8 −0.6 −0.4 −0.2

Speed (0) B (b) C D A E F D C (a)

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1.24 1.26 1.28 1.3 1.32 1.34 1.36 1.38 1.4 1.42 1 2 3 4 5 6 7 8 9 10

3 4

  • 5

6

  • 1
  • 2
  • Speed

Energy

1.258 1.261

5

  • 6
  • 1.332

1.342

3

  • 4
  • 1.398

1.402 1.406 0.9 1.1 1.3 1.5

1

  • 2
  • (a)

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1.24 1.26 1.28 1.3 1.32 1.34 1.36 1.38 1.4 1.42 1 2 3 4 5 6 7 8 9 10

C D

  • E

F

  • A
  • B
  • Speed

Energy

1.258 1.261

E

  • F
  • 1.332

1.342

C

  • D
  • 1.398

1.402 1.406 0.9 1.1 1.3 1.5

A

  • B
  • (a)

−0.4 −0.3 −0.2 −0.1 0.1 0.2 −40 −20 20 40 60 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2

(b)

−0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 −60 −40 −20 20 40 60 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2

(c)

−0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 −25 −20 −15 −10 −5 5 10 15 20 25 30 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2

(d) P Q

−0.2 −0.15 −0.1 −0.05 0.05 0.1 0.15 −80 −60 −40 −20 20 40 60 80 −0.25 −0.2 −0.15 −0.1 −0.05 0.05 0.1 0.15

(e)

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HYDROELASTIC WAVES Tao Gao, Zhan Wang

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−150 −100 −50 50 100 150 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8

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GENERALISED SOLITARY WAVES

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Clamond et al, Journal of Fluid Mechanics (2015) 784, pp 664-680

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Wang Z., Parau E.I. , Milewski P.A. and Vdb (2014) Proc. Roy. Soc. A 470

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  • 0.2

(1)

  • 40

40

  • 0.2

(2)

  • 0.2

(1†)

  • 40

40

  • 0.2

(2†)

  • 0.2

(1‡)

  • 40

40

  • 0.2

(2‡) 30

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Non-symmetric PERIODIC gravity-capillary waves Tao Gao and Zhan Wang Zufiria (1987) Shimizu ans Shoji (2012)

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Symmetric waves

  • 1.5
  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 16 18 20

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Non-symmetric waves

  • 7
  • 6
  • 5
  • 4
  • 3
  • 2
  • 1
  • 0.16
  • 0.14
  • 0.12
  • 0.1
  • 0.08
  • 0.06
  • 0.04
  • 0.02

0.02

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Non-symmetric waves

  • 6
  • 5
  • 4
  • 3
  • 2
  • 1
  • 0.2
  • 0.1

0.1 0.2 0.3 0.4 0.5 34

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Non-symmetric waves

  • 7
  • 6
  • 5
  • 4
  • 3
  • 2
  • 1
  • 0.15
  • 0.1
  • 0.05

0.05 0.1 0.15 0.2 0.25

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Non-symmetric waves

  • 6
  • 5
  • 4
  • 3
  • 2
  • 1

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 36

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0.04 0.06 0.08 0.1 0.12 0.14 0.16

b6

0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14

q

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THREE-DIMENSIONAL FLOWS Use Green’s theorem instead of Cauchy inte- gral equation formula. Emilian Parau, Mark Cooker, VdB Olga Trichtchenko, Paul Milewski, Emilian Pa- rau, VdB

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−30 −20 −10 10 20 30 −40 −20 20 40 −1.5 −1 −0.5 0.5

z y x

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−30 −20 −10 10 20 30 10 20 30 40 −1.5 −1 −0.5 0.5

x y z

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−30 −20 −10 10 20 30 10 20 30 40 −1.5 −1 −0.5 0.5

x y z

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NON-SYMMETRIC 3D WAVES Model: Akers and Milewski (2009) ut + √ 2 2 ux − √ 2 4 H[u − uxx − 2uyy] + α(u2)x = 0 Zhan Wang

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Three dimensional flexural waves Olga....

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Conclusions New non-symmetric gravity-capillary waves for the Euler’s equations in 2D (solitary waves) New non-symmetric flexural waves for the Eu- ler’s equations in 2D (solitary waves) New non-symmetric gravity-capillary waves for a model in 3D (solitary waves) New non-symmetric generalised solitary waves in 2D New non-symmetric periodic gravity-capillary waves in 2D

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References

  • 1. Wang Z. , Vanden-Broeck J.-M. and Milewski,

PA., 2014, J. Fluid Mech. 759, R2

  • 2. Gao T., Wang Z. and Vanden-Broeck J.-

M., 2016, J. Fluid Mech. 788, 469-491

  • 3. Gao T., Wang Z. and Vanden-Broeck J.-
  • M. 2016, Proc.

Roy,. Soc. A 472, No. 2194, p. 20160454

  • 4. Gao T., Wang Z. and Vanden-Broeck J.-
  • M. 2016, J. Fluid Mech. 811, 622-641

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