interaction of two solitary waves of large amplitude
DESCRIPTION
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 PresentationTRANSCRIPT
![Page 1: Interaction of Two Solitary Waves of Large Amplitude](https://reader035.vdocuments.mx/reader035/viewer/2022062217/56814a67550346895db783aa/html5/thumbnails/1.jpg)
Interaction of Two Solitary Waves
of Large Amplitude
Hua Liu Benlong Wang
Shanghai Jiao Tong University
SCSTW-2008, Shanghai, China
![Page 2: Interaction of Two Solitary Waves of Large Amplitude](https://reader035.vdocuments.mx/reader035/viewer/2022062217/56814a67550346895db783aa/html5/thumbnails/2.jpg)
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](https://reader035.vdocuments.mx/reader035/viewer/2022062217/56814a67550346895db783aa/html5/thumbnails/3.jpg)
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](https://reader035.vdocuments.mx/reader035/viewer/2022062217/56814a67550346895db783aa/html5/thumbnails/4.jpg)
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](https://reader035.vdocuments.mx/reader035/viewer/2022062217/56814a67550346895db783aa/html5/thumbnails/5.jpg)
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](https://reader035.vdocuments.mx/reader035/viewer/2022062217/56814a67550346895db783aa/html5/thumbnails/6.jpg)
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](https://reader035.vdocuments.mx/reader035/viewer/2022062217/56814a67550346895db783aa/html5/thumbnails/7.jpg)
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](https://reader035.vdocuments.mx/reader035/viewer/2022062217/56814a67550346895db783aa/html5/thumbnails/8.jpg)
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](https://reader035.vdocuments.mx/reader035/viewer/2022062217/56814a67550346895db783aa/html5/thumbnails/9.jpg)
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](https://reader035.vdocuments.mx/reader035/viewer/2022062217/56814a67550346895db783aa/html5/thumbnails/10.jpg)
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](https://reader035.vdocuments.mx/reader035/viewer/2022062217/56814a67550346895db783aa/html5/thumbnails/11.jpg)
Shanghai Jiao Tong University
Linear dispersion
Nonlinearity
![Page 12: Interaction of Two Solitary Waves of Large Amplitude](https://reader035.vdocuments.mx/reader035/viewer/2022062217/56814a67550346895db783aa/html5/thumbnails/12.jpg)
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](https://reader035.vdocuments.mx/reader035/viewer/2022062217/56814a67550346895db783aa/html5/thumbnails/13.jpg)
Shanghai Jiao Tong University
Propagation of a solitary wave
![Page 14: Interaction of Two Solitary Waves of Large Amplitude](https://reader035.vdocuments.mx/reader035/viewer/2022062217/56814a67550346895db783aa/html5/thumbnails/14.jpg)
Shanghai Jiao Tong University
![Page 15: Interaction of Two Solitary Waves of Large Amplitude](https://reader035.vdocuments.mx/reader035/viewer/2022062217/56814a67550346895db783aa/html5/thumbnails/15.jpg)
Shanghai Jiao Tong University End-wall reflection of a solitary wave
![Page 16: Interaction of Two Solitary Waves of Large Amplitude](https://reader035.vdocuments.mx/reader035/viewer/2022062217/56814a67550346895db783aa/html5/thumbnails/16.jpg)
Shanghai Jiao Tong University
![Page 17: Interaction of Two Solitary Waves of Large Amplitude](https://reader035.vdocuments.mx/reader035/viewer/2022062217/56814a67550346895db783aa/html5/thumbnails/17.jpg)
Shanghai Jiao Tong University
![Page 18: Interaction of Two Solitary Waves of Large Amplitude](https://reader035.vdocuments.mx/reader035/viewer/2022062217/56814a67550346895db783aa/html5/thumbnails/18.jpg)
Shanghai Jiao Tong University
![Page 19: Interaction of Two Solitary Waves of Large Amplitude](https://reader035.vdocuments.mx/reader035/viewer/2022062217/56814a67550346895db783aa/html5/thumbnails/19.jpg)
Shanghai Jiao Tong University
![Page 20: Interaction of Two Solitary Waves of Large Amplitude](https://reader035.vdocuments.mx/reader035/viewer/2022062217/56814a67550346895db783aa/html5/thumbnails/20.jpg)
Shanghai Jiao Tong University
Head-on collision of two solitary waves
![Page 21: Interaction of Two Solitary Waves of Large Amplitude](https://reader035.vdocuments.mx/reader035/viewer/2022062217/56814a67550346895db783aa/html5/thumbnails/21.jpg)
Shanghai Jiao Tong University
Overtaking of two solitary waves
![Page 22: Interaction of Two Solitary Waves of Large Amplitude](https://reader035.vdocuments.mx/reader035/viewer/2022062217/56814a67550346895db783aa/html5/thumbnails/22.jpg)
Shanghai Jiao Tong University
![Page 23: Interaction of Two Solitary Waves of Large Amplitude](https://reader035.vdocuments.mx/reader035/viewer/2022062217/56814a67550346895db783aa/html5/thumbnails/23.jpg)
Shanghai Jiao Tong University
12
21 /
Wang, Zhang & Liu (2007, PRE)
![Page 24: Interaction of Two Solitary Waves of Large Amplitude](https://reader035.vdocuments.mx/reader035/viewer/2022062217/56814a67550346895db783aa/html5/thumbnails/24.jpg)
Shanghai Jiao Tong University
0
x
0
2
2
x
![Page 25: Interaction of Two Solitary Waves of Large Amplitude](https://reader035.vdocuments.mx/reader035/viewer/2022062217/56814a67550346895db783aa/html5/thumbnails/25.jpg)
Shanghai Jiao Tong University
KdV
mKdV
Full potential theory
32
53
3
142.3
![Page 26: Interaction of Two Solitary Waves of Large Amplitude](https://reader035.vdocuments.mx/reader035/viewer/2022062217/56814a67550346895db783aa/html5/thumbnails/26.jpg)
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](https://reader035.vdocuments.mx/reader035/viewer/2022062217/56814a67550346895db783aa/html5/thumbnails/27.jpg)
Shanghai Jiao Tong University
![Page 28: Interaction of Two Solitary Waves of Large Amplitude](https://reader035.vdocuments.mx/reader035/viewer/2022062217/56814a67550346895db783aa/html5/thumbnails/28.jpg)
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](https://reader035.vdocuments.mx/reader035/viewer/2022062217/56814a67550346895db783aa/html5/thumbnails/29.jpg)
Shanghai Jiao Tong University
Thank you for your attention.
![Page 30: Interaction of Two Solitary Waves of Large Amplitude](https://reader035.vdocuments.mx/reader035/viewer/2022062217/56814a67550346895db783aa/html5/thumbnails/30.jpg)
Shanghai Jiao Tong University
![Page 31: Interaction of Two Solitary Waves of Large Amplitude](https://reader035.vdocuments.mx/reader035/viewer/2022062217/56814a67550346895db783aa/html5/thumbnails/31.jpg)
Shanghai Jiao Tong University