Non-linear hydrodynamics
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Transcript of Non-linear hydrodynamics
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7/24/2019 Non-linear hydrodynamics
1/13
NAME: RATNAKAR GADI
STUDENT NUMBER: 4519515
ASSIGNMENT FOR OE4630-3(as part ! MTM1410"
B#$s %$&st'#:
& ass$)& t*at t*& +a,&s a# .r&a/ a# t*at a# .& &sr'.& . a# aapt&
!r&& s$r!a& #'t'#:
tt+t+gz=0,on z=0, t*& !r&& s$r!a&
I# +*'* 's t*& ps't',& 'ss'pat'# -&2't
D&r',& *+ a pa#& prr&ss',& +a,& +$ &a as a !$#t'# ! 'sta#&
A#s"
upositive dissipative coefficient
T*& ,&'t ptt'a ! a# '#'t +a,& 's & .:
=igA
e
iteikx
ekz
t=iAg
( i ) eiteikx ekz
2
t2=iAg
(2) eiteikx ekz
z=iAg
( k) eiteikx ekz
S$.st't$t'# '# t*& ', !r&& s$r!a& &7$at'# +& &t:
2
+iugk=0
*'* '&s t*& !+'# r&at'#:
k=
2iwu
g
& /#+ t*& +a,& &&,at'# 's:
=A eiwt
eikx
1
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7/24/2019 Non-linear hydrodynamics
2/13
NAME: RATNAKAR GADI
STUDENT NUMBER: 4519515
S$.st't$t'# a# +r't'# t*& &7$at'# .a/ '# t*'s !r):
=A eiwt
ei
2
gx
e
wux
g
T*& +a,& &as &8p#t'a +'t* spa&(8" as s*+# '# &&p +at&rs
Fr s*a+ +at&rs t*& !r&& s$r!a& &7$at'# '&s t*& !+'# #'t'#:
h (kh)=gktan
2iwu
As ta#*(/*"/*
& &t:
k=( 2iwu
gh )0.5
A!t&r s)& a&.ra +& &t:
k=
gh (1 iu
2)
Aa'# s$.st't$t'# &,&r t*'#:
=A eiwt
ei
ghx
e
ux
2gh
a,& Dr'!t Fr& !r a Sp*&r&:%$&st'# I AD&r',& t*& &8pr&ss'# !r t*& )&a# +a,& r'!t !r& # t*& +a !r +a,&
a#&s t*at ar& #t ;&r
A#s"
i= cos (tkxcosky sin )
r= cos (t+kxcosky sin )
T*& rr&sp#'# ptt'as ar& ', .:
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7/24/2019 Non-linear hydrodynamics
3/13
NAME: RATNAKAR GADI
STUDENT NUMBER: 4519515
i=g
sin ( cos (tkx cosky sin )) ekz
r=g sin ( cos (t+kx cosky sin ) ) ekz
T*& r&s$t'# ptt'a .&)&s:
2g
cos ( kxcos ) sin (tky sin ) ekz
p=gz
t
2( del ( ))2
t=2gcos (kxcos ) cos ( wtkysin ) ekz
x=2 sin kx cos sin ( wtkysin ) ekz
y=2 cos (kx cos )cos ( wtkysin ) ekz
z=2cos (kxcos ) sin (wtky sin )ekz
At 80=
x=0
2
2
y+
2
2
z=4
2
2 (sin2 cos (wtky sin )2+(sin ( wtkysin ))2 )e2kz
p0=gz
p1=2gcos (tky sin ) ekz
p2= 2
2
e2kz (1+sin2 )
p3= 2
2e
2 kzcos (2wt2kysin ) cos2
T*& !r& 's ', .:
3
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7/24/2019 Non-linear hydrodynamics
4/13
NAME: RATNAKAR GADI
STUDENT NUMBER: 4519515
F=p n ds
N(100" Ds1;
F=inf
0
p (z !t) dz0
(
t)p (z ! t) dz
F=F1+F
2
N&&t'# *rstat' pr&ss$r&:
F1=
g 2 (1+sin2 )
2
F2=
g ( ( t))22
(t)=2 cos (wtky sin )
F2=g 2g
2cos (2 (wtky sin ))
T*& s #tr'.$t&s ;&r t t*& '#t&ra:
F=F1+F2
g
2cos
2
2
T*'s !r& 's 'r&t& t+ars t*& +a
%$&st'# I B
Fr) t*at r&s$t &r',& t*& *'* !r&7$ ')'t ! t*& +a,& !r&7$ ')'t !
t*& +a,& r'!t !r& t t*& sp*&r&
4
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7/24/2019 Non-linear hydrodynamics
5/13
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7/24/2019 Non-linear hydrodynamics
6/13
NAME: RATNAKAR GADI
STUDENT NUMBER: 4519515
T*'s 's 'r&t& t+ars t*& sp*&r&
Fy=F n ds
T*& #r)a ,&tr '# -'r&t'# 's sin =
T*& !r& '# t*'s as& 's ', .:
F=g
2sin
2
2
0
"g
2sin
3d
2
S,'# t*'s ',&s:
F=2
3g
2r
%$&st'# II
Tas/-I Mr'# sst&) st'>#&ss a# #at$ra !r&7$
11" ?a$at& t*& s+a #at$ra p&r' ! t*& F@SO
Mass1044t##&s
?
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7/24/2019 Non-linear hydrodynamics
7/13
NAME: RATNAKAR GADI
STUDENT NUMBER: 4519515
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7/24/2019 Non-linear hydrodynamics
8/13
NAME: RATNAKAR GADI
STUDENT NUMBER: 4519515
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7/24/2019 Non-linear hydrodynamics
9/13
NAME: RATNAKAR GADI
STUDENT NUMBER: 4519515
T*& MATHAB &:
a
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7/24/2019 Non-linear hydrodynamics
10/13
NAME: RATNAKAR GADI
STUDENT NUMBER: 4519515
If amplitudes are doubled, then the mean wave drift wave is four
timesI! a)p't$&s ! .t* t*& +a,&s ar& *a,& t*& )a8')$) r'!t +a,& !r&
's ', . 7$art&r&
T*& )&a# +a,& r'!t !r&:
F1= aii2' (i ! i )
S t*'s )&a#s $.& t*& a)p't$& t*& )&a# +a,& r'!t !r& 's !$r
t')&s
Ma8')$) r'!t !r& !r t+ r&$ar +a,&s:
F= aii2'ii+aii a++('i++'+i)cos ( i+) t+aii a++(,i++, +i) sin (i+ ) t
T*$s half the wave amplitudes in regular waves, one-fourth is the
maximum wave drift force.
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7/24/2019 Non-linear hydrodynamics
11/13
NAME: RATNAKAR GADI
STUDENT NUMBER: 4519515
.2
A(-4
s )
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7/24/2019 Non-linear hydrodynamics
12/13
NAME: RATNAKAR GADI
STUDENT NUMBER: 4519515
pt(+E"=
t*& )&a# +a,& r'!t !r& 's 533309KN
a,& r'!t !r& Fr&7$ 0050 010 0150 0