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Interaction of Two Solitary Waves of Large Amplitude Hua Liu Benlong Wang Shanghai Jiao Tong University [email protected] SCSTW-2008, Shanghai, China

Interaction of Two Solitary Waves of Large Amplitude

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SCSTW-2008, Shanghai, China. Interaction of Two Solitary Waves of Large Amplitude. Hua Liu Benlong Wang Shanghai Jiao Tong University [email protected]. Outline. Motivation A high order Boussinesq equation Propagation and reflection of a solitary wave - PowerPoint PPT Presentation

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Page 1: Interaction of Two Solitary Waves of Large Amplitude

Interaction of Two Solitary Waves

of Large Amplitude

Hua Liu Benlong Wang

Shanghai Jiao Tong University

[email protected]

SCSTW-2008, Shanghai, China

Page 2: Interaction of Two Solitary Waves of Large Amplitude

Shanghai Jiao Tong University Outline

Motivation

A high order Boussinesq equation

Propagation and reflection of a solitary wave

Head on collision of two solitary waves

Overtaking of two solitary waves

Concluding remarks

Page 3: Interaction of Two Solitary Waves of Large Amplitude

Shanghai Jiao Tong University Motivation

Validation of the high order Boussinesq equations

check the flow field of a solitary wave of large amplitude and the force acting on a vertical wall

Overtaking of two solitary waves

check if the critical ratio of wave amplitude varies with wave amplitude?

Page 4: Interaction of Two Solitary Waves of Large Amplitude

Shanghai Jiao Tong University A high order Boussinesq equation

Definition of velocity variables

( , , , )x y tu u),,,(~ tyxww

( , , , )b x y h t u u

),,,( thyxwwb

),ˆ,,(ˆ tzyxww ˆ ˆ( , , , )x y z tu u

),0,,(0 tyxww 0 ( , ,0, )x y tu u

Madsen, Bingham & Liu (2002)

Page 5: Interaction of Two Solitary Waves of Large Amplitude

Shanghai Jiao Tong University

Irrotational flows

0~~~

iii xx

wVwt

0)1(~2

1)

~(

2

1~

22

jjiii

i

xxw

xV

xxg

t

V

——Zakharov(1968) , Witting(1984), Dommermuth & Yue (1987)

w~~~uV

Page 6: Interaction of Two Solitary Waves of Large Amplitude

Shanghai Jiao Tong University

0)~~(~

wwt

V

0)1(2

~~ 2

w

gt

V

w~~~uV

0b bw h u

4 equations, 6 unknowns ( , )wu ( , )b bwu

Page 7: Interaction of Two Solitary Waves of Large Amplitude

Shanghai Jiao Tong University

Exact solution of Laplace equation

00 )sin()cos(),,,( wzztzyx uu

00 )sin()cos(),,,( u zwztzyxw

000 ),(),( z

ww uu

n

n

nn

n2

0

2

)!2()1()cos(

12

0

12

)!12()1()sin(

n

n

nn

n

——L. Rayleigh 1876 On waves

Page 8: Interaction of Two Solitary Waves of Large Amplitude

Shanghai Jiao Tong University

Velocity solution formulation in terms as the velocity defined at an arbitrary level of depth

00 )ˆsin()ˆcos(),,(ˆ wzztyx uu

00 )ˆsin()ˆcos(),,(ˆ u zwztyxw

zwzzzztzyx u ˆˆ))ˆsin((ˆ))ˆcos((),,,( uu

zzzwzztzyxw w ˆˆ))ˆsin((ˆ))ˆcos((),,,( u

)ˆ))ˆsin((ˆ))ˆ)(cos((ˆ( wzzzzzzu u

)ˆ))ˆsin((ˆ))ˆ)(cos((ˆ( u zzwzzzzw

Page 9: Interaction of Two Solitary Waves of Large Amplitude

Shanghai Jiao Tong University

Series expansions

zwzzz u ˆ*ˆ))ˆ((*ˆ)1()( 55

33

44

22 uu

zzzwzw w ˆ*ˆ))ˆ((*ˆ)1()( 55

33

44

22 u

Taylor expansion

2

ˆ 2

2)( zz

24

ˆ 4

4)( zz

6

ˆ 3

3)( zz

120

ˆ 5

5)( zz

)ˆ,ˆ(*)*,( ww uu

Page 10: Interaction of Two Solitary Waves of Large Amplitude

Shanghai Jiao Tong University

Series expansions

zwzzz u ˆ*ˆ))ˆ((*ˆ)1()( 55

33

44

22 uu

zzzwzw w ˆ*ˆ))ˆ((*ˆ)1()( 55

33

44

22 u

Pade expansion

18

ˆ

2

ˆ 22

2zzz

)(

504

ˆ

36

)ˆ(ˆ

24

ˆ 4224

4zzzzzz

)(

18

)ˆ(ˆ

6

ˆ 23

3zzzzz

)(

504

)ˆ(ˆ

108

)ˆ(ˆ

120

ˆ 435

5zzzzzzzz

)(

Page 11: Interaction of Two Solitary Waves of Large Amplitude

Shanghai Jiao Tong University

Linear dispersion

Nonlinearity

Page 12: Interaction of Two Solitary Waves of Large Amplitude

Shanghai Jiao Tong University

Numerical aspects Spatial discrectization: 7 point central difference scheme

Time stepping: 5 order Cash-Karp-Runge-Kutta scheme

Smoothing: Savitsky-Golay smoothing method

Relaxed analytic approach for wave generation and absorbing

Page 13: Interaction of Two Solitary Waves of Large Amplitude

Shanghai Jiao Tong University

Propagation of a solitary wave

Page 14: Interaction of Two Solitary Waves of Large Amplitude

Shanghai Jiao Tong University

Page 15: Interaction of Two Solitary Waves of Large Amplitude

Shanghai Jiao Tong University End-wall reflection of a solitary wave

Page 16: Interaction of Two Solitary Waves of Large Amplitude

Shanghai Jiao Tong University

Page 17: Interaction of Two Solitary Waves of Large Amplitude

Shanghai Jiao Tong University

Page 18: Interaction of Two Solitary Waves of Large Amplitude

Shanghai Jiao Tong University

Page 19: Interaction of Two Solitary Waves of Large Amplitude

Shanghai Jiao Tong University

Page 20: Interaction of Two Solitary Waves of Large Amplitude

Shanghai Jiao Tong University

Head-on collision of two solitary waves

Page 21: Interaction of Two Solitary Waves of Large Amplitude

Shanghai Jiao Tong University

Overtaking of two solitary waves

Page 22: Interaction of Two Solitary Waves of Large Amplitude

Shanghai Jiao Tong University

Page 23: Interaction of Two Solitary Waves of Large Amplitude

Shanghai Jiao Tong University

12

21 /

Wang, Zhang & Liu (2007, PRE)

Page 24: Interaction of Two Solitary Waves of Large Amplitude

Shanghai Jiao Tong University

0

x

0

2

2

x

Page 25: Interaction of Two Solitary Waves of Large Amplitude

Shanghai Jiao Tong University

KdV

mKdV

Full potential theory

32

53

3

142.3

Page 26: Interaction of Two Solitary Waves of Large Amplitude

Shanghai Jiao Tong University

0 0.1 0.2 0.3 0.4 0.5 0.6 0.72.5

3

3.5

4

4.5

2

Kodaman eKdVMarchant eKDVFNHD-B

Page 27: Interaction of Two Solitary Waves of Large Amplitude

Shanghai Jiao Tong University

Page 28: Interaction of Two Solitary Waves of Large Amplitude

Shanghai Jiao Tong University Concluding Remarks

The high order Boussinesq model is applied to numerical simulation of a solitary wave reflected by a vertical wall.

Among the three patterns of overtaking of two solitary waves, the critical condition for the flat peak pattern is related with the incoming wave amplitude.

For extremely small wave, the critical relative amplitude approaches to 3, which indicates the various KdV models or bidirectional long wave models give reasonable correct predictions.

With increasing of the wave amplitude, the critical relative amplitude increases and is apparently different from 3. For the incoming solitary wave of extremely large amplitude, e.g. a= 0.6, the critical condition reaches the magnitude of 4.

Page 29: Interaction of Two Solitary Waves of Large Amplitude

Shanghai Jiao Tong University

Thank you for your attention.

Page 30: Interaction of Two Solitary Waves of Large Amplitude

Shanghai Jiao Tong University

Page 31: Interaction of Two Solitary Waves of Large Amplitude

Shanghai Jiao Tong University