Interaction of Two Solitary Waves
of Large Amplitude
Hua Liu Benlong Wang
Shanghai Jiao Tong University
SCSTW-2008, Shanghai, China
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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
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?
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)
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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
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0)~~(~
wwt
V
0)1(2
~~ 2
w
gt
V
w~~~uV
0b bw h u
4 equations, 6 unknowns ( , )wu ( , )b bwu
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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
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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
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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
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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
)(
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Linear dispersion
Nonlinearity
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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
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Propagation of a solitary wave
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Shanghai Jiao Tong University End-wall reflection of a solitary wave
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Head-on collision of two solitary waves
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Overtaking of two solitary waves
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12
21 /
Wang, Zhang & Liu (2007, PRE)
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0
x
0
2
2
x
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KdV
mKdV
Full potential theory
32
53
3
142.3
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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
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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.
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Thank you for your attention.
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