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

7/24/2019 Non-linear hydrodynamics

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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& ', .:

7/24/2019 Non-linear hydrodynamics

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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

7/24/2019 Non-linear hydrodynamics

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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

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

?

7/24/2019 Non-linear hydrodynamics

7/13

NAME: RATNAKAR GADI

STUDENT NUMBER: 4519515

7/24/2019 Non-linear hydrodynamics

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NAME: RATNAKAR GADI

STUDENT NUMBER: 4519515

7/24/2019 Non-linear hydrodynamics

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NAME: RATNAKAR GADI

STUDENT NUMBER: 4519515

T*& MATHAB &:

a

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.

7/24/2019 Non-linear hydrodynamics

11/13

NAME: RATNAKAR GADI

STUDENT NUMBER: 4519515

.2

A(-4

s )

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