Transcript Stokes Wave Train

Chapter 3.1 Weakly Nonlinear Wave Theory
for Periodic Waves (Stokes Expansion)
Introduction
The solution for Stokes waves is valid in deep
or intermediate water depth.
It is assumed that the wave steepness is much
smaller than one.
  ak
1
kh  O(1)
where k is the wavenumber and h is the
water depth which is assumed constant.
Nondimensional Variables
X  xk ,
Z  zk ,
Y  yk ,
t  t ,
   / a,

C
gk

, C , D
,
ag
ag
2
where x, z, t , h,  and C are dimensional
variables and X , Z , t ,  ,  and C are
corresponding nondimensional variables.
•Nondimensional Governing Equation &
Boundary Conditions
 2
2

 2
2

 2
2
0
 kh  Z  
(3.1.1)
X
Y
Z

0
at Z   kh
(3.1.2)
Z


  D h    h  D
at Z  
(3.1.3)
Z
t

1
2
    D    C
at Z  
(3.1.4)
2
t
where  and  h stand for gradient and horizontal
gradient, respectively.
Perturbation (Stokes Expansion)
Assuming the wave train is weakly nonlinear
(  ak  1), its potential and elevation can
be perturbed in the order of  :
1
2


2  3
        
 
1
C C
 
1
 2
C
 
 2
2
 3
 C
2


 3


j

  
j 1
j 1
 

 j
j 1

C
 j

Hierachy Equations
Using the Taylor expansion, the free-surface
boundary conditions (Equations (3.1.3) and (3.1.4)
are expanded at the still water level (Z = 0).
Then we substitute perturbation forms of potential
and elevation into the Laplace Equation, bottom
and free-surface boundary conditions. The equations
are sorted and grouped according to the order in
wave steepness  (j) . The governing equations for j - th
order solutions is given by:
2 ( j)
 

0
j

 j  

t
 j
 hk  Z  0
(3.1.5)
j    j 1  j 1 

 P 
,
at
Z

0
(3.1.6)



j



j    j 1  j 1 

D
 Q 
,
at
Z

0
(3.1.7)


Z
t


j


Z
0
at Z  kh
(3.1.8)
( j)
( j)
where the P and Q can be derived in terms
of the solutions for the potential and elevation of
order ( j -1) or lower. Therefore, the above
hierarchy equations must be solved sequentially
from lower to higher order until the required
accuracy is reached. To derive the third-order
solution for a Stokes wave train, it is adequate
to truncate the equations at j  3.
Up to j  3, P ( j ) and Q ( j ) are given below.
(1)
1
P   C
2
P    
and
1
Q   0
1
1  2  
2
(2)
D
1



C
2
Z t
1
1  2  
1
2
1


Q
 D
 D h 
  h
2
Z
1   D
3
1
2
1



P
  D
 

  
2
Z  2


Q
 3
 D
1
 2   2 
2
   


t 

(3)
1  1   3  
 

C

2
Z t
 Z 2 t
1
 2   2 
Z 2
2
1
1
2
D  1   3  



2 

Z 3
1
 2 
2
1

 D   h      h
  h      h


2
1     
1 
1

+D
 h
  h
Z 

Z


• Solving the non-dimensional Equations from lower
order (j=1) to higher order (j=3) for the non-dimensional
solutions (wave advances in the x-direction).
cosh( kh  Z )
1



sin( X  t ),

1
cosh kh
 cos( X  t ),
C
1
0
3 cosh(2kh  2 Z )
2



sin(2 X  2t )
3
8 sinh kh cosh kh

 2
C
 2


4
(3 2  1) cos(2 X  2t )
1

2 sinh 2kh
1
3



( 2  1)( 2  3)(9 2  13)
64
cosh(3kh  3Z )
sin(3 X  3t )
cosh kh
3 4
 3

  (  3 2  3) cos( X  t ) 
8
3
(8 6  ( 2  1) 2 ) cos(3 X  3t )
64
C
 3
0

1 


2
9

2
2 
D   1      1    

8



where   coth kh
1
2
•The non-dimensional solutions are then transferred
back to the dimensional form.
First-order:

(1)

(1)
cosh[k ( z  h)]
A
sin 
cosh(kh)
 a cos 
where
  kx   t  
and
aA

g
.
Second-order:
3akA 2
2
 
(  1) cosh[2k ( z  h)]sin 2
8
2 1
2
2
   (3  1)a k cos 2
4
1 2
2
Bernoulli Constant: Co 
a kg (  1)
4
(2)
Third-order:
1
(3)
2
2
2
 
(  1)(  3)(9  13) 
64
cosh[3k ( z  h)] 2 2
a k A sin 3
cosh 3kh
3 4
(3)
   (  3 2  3)a3k 2 cos  
8
3
(8 6  ( 2  1)2 )a3k 2 cos 3
64
Nonlinear Dispersion Relation:


2


9


2
2 2
2
2 
  gk tanh(kh) 1  k a    1    

8



Convergence
For the fast convergence of the perturbed coefficient, ,
must be much smaller than unity, which is consistent with
weakly nonlinear assumption. However, when the ratio of
depth to wave length is small, the Stokes perturbation may
not be valid.
Convergence rate:
 (2)
R 

mag
(1)
,
mag
R is the ratio of the potential magnitude of
second-order to that of first order solution at
z  0.
3 ( 2  1) 2
R 
.
8
For fast convergence, R should be << 1. This is
true when kh ~ O(1). When kh  1, we have :
3
3 
 ~ (kh) , hence R ~ O   (kh) 
8

R may be much greater than unity
1
a 1

Ursell number U r 
=
h (kh) 2 (kh)3
8
For R 1, then U r
.
3
A few striking features of a nonlinear wave train
can be described for the above equation:
• The crests are steeper and troughs are flatter;
(see applet (Nonlinear Wave Surface)).
• Phase velocity increases with the increase in
wave steepness.
• Non-closed trajectories of particles movement.
(see applet (N-Trajectory)).
• Nonlinear wave characteristics (up to 2nd order).
Wave advancing in the x-direction
Particle velocity
V  iu  kw
akg cosh[ k ( z  h)]
3 a 2 k 2 g cosh[2k ( z  h)]
u
cos  
cos 2
3

cosh kh
4  sinh kh cosh kh
akg sinh[k ( z  h)]
3 a 2 k 2 g sinh[2k ( z  h)]
w
sin  
sin 2
3

cosh kh
4  sinh kh cosh kh
Acceleration
a  iax  kaz
(1)
u
cosh[ k ( z  h)]
(1)
ax 
 V u  akg
sin  
t
cosh kh
1
 3 cosh[2k ( z  h)]

a2k 2 g 

sin 2
3

 2 sinh kh cosh kh sinh 2kh 
(1)
w
sinh[k ( z  h)]
az 
 V w(1)   akg
cos  
t
cosh kh
1

2 2  3 sinh[2 k ( z  h)]
a k g

cos 2
3

 2 sinh kh cosh kh sinh 2kh 
Particle Trajectory
Denoting the mean position of a particle by ( x, z ) , and its
instantaneous displacement from the mean position by ( ,  ),
the Lagrangian velocities of the particle are hence
u ( x   , z   ) and w( x   , z   ), they are related to the
Eulurian velocities through a Taylor Expansion:
(1)
(1)

u

u
u ( x   , z   )  u (1) ( x, z )  u (2) ( x, z ) 
 (1) 
 (1)  O( 2 )u (1)
x
z
(1)
(1)

w

w
w( x   , z   )  w(1) ( x, z )  w(2) ( x, z ) 
 (1) 
 (1)  O( 2 ) w(1)
x
z
where u (1) ( x, z ), u (2) ( x, z ), w(1) ( x, z ) and w(2) ( x, z ) are first- and second-
order horizontal and vertical velocities.
( ,  ) are calculated by integrating the related Lagrangian velocities.
t
t
0
0
 (t )   u ( x   , z   , )d   0 ;  (t )   w( x   , z   , )d   0
We intend to compute ( ,  ) up to second order in wave steepness
(2)
 (t )   ( x, z, t )   ( x, z.t )   ( x, z )  O( 2 ) (1)
(1)
(2)
 (t )   ( x, z, t )   ( x, z, t )  O( )
where superscripts stand for orders and overbar denotes a secular term.
At leading-order, the solution is the same as that in LWT,
t
cosh[k ( z  h)]
(1)
(1)
(1)
   u ( x   , z   , )d   0  a
sin 
0
sinh kh
t
sinh[k ( z  h)]
(1)
(1)
   w ( x   , z   , )d   0  a
cos
0
sinh kh
(1)
(2)
2
(1)
u (1) (1) u (1) (1) a 2 k 2 g  cosh 2 [k ( z  h)] 2 sinh 2 [k ( z  h)] 2 
 
 
sin  
cos  

x
z
  sinh kh cosh kh
sinh kh cosh kh

cosh  2k ( z  h) 
a2k 2 g 
1

cos 2 


  sinh 2kh
sinh 2kh 
w(1) (1) w(1) (1)
 
 0
x
z
(1)
(1)
2 2

u

u
a
k g  3 cosh[2k ( z  h)]
1 
(2)
(1)
(1)
u ( x, z ) 
 
 

cos 2
3


x
z
  4 sinh kh cosh kh sinh 2kh 
a 2 k 2 g cosh  2k ( z  h)
+
,

sinh 2kh
(1)
(1)
2 2

w

w
3
a
k g sinh[2k ( z  h)]
(2)
(1)
(1)
w ( x, z ) 
 
 
sin 2
3
x
z
4  sinh kh cosh kh
The leading-order trajectory of a particle is an ellipse of the center at ( x, z )
cosh[k ( z  h)]
sinh[k ( z  h)]
and a major-axis a
and minor-axis a
.
sinh kh
sinh kh
( (1) ) 2
( (1) ) 2

 1.
2
2
 cosh[k ( z  h)]   sinh[k ( z  h)] 
 a sinh kh   a sinh kh 
The secon-order solutions for the displacement are calculated by integrating
the related second-order lagrangian velocities.
1
 3 cosh[2k ( z  h)]

  a k 

sin 2
4
2

sinh kh
4sinh kh 
 8
(2 )
cosh[2k ( z  h)]
2
  a k
t
2
2sinh kh
3 2 sinh[2k ( z  h)]
(2)
  ak
cos 2
4
8
sinh kh
(2)
2
(2)
The secular term ( ) in the horizontal displacement indicates the
particles will continuously move in the wave direction. Hence, the
trajectory of a particle is no longer an ellipse. Becasue the horiztonal
mean position of a particle is not fixed at x but change with time, we
re-define the horizontal mean position by
cosh[2k ( z  h)]
2
x '  x  a k
t and  '  kx '  t.
2
2sinh kh
Correspondingly, the displacement with respect to the instantaneous
mean position ( x ', z ) is given by,
cosh[k ( z  h)]
sinh[k ( z  h)]
(1)
  a
sin  ',   a
cos  ',
sinh kh
sinh kh
1
 3 cosh[2k ( z  h)]

 (2)  a 2 k  

sin 2 ',
4
2

sinh kh
4sinh kh 
 8
3 2 sinh[2k ( z  h)]
(2)
  ak
cos 2 '.
4
8
sinh kh
(1)
The trajectory of a particle based on the solution is plotted in Applet
(N-trajctory). The time average Lagragian velocity of a particle is
(2)
equal to the derivative of the secular term ( ) with respect to time.
cosh[2k ( z  h)]
2
u l  a k
2sinh 2 kh
The integral of the average Lagragian velocity with respect to water
depth renders the average mass flux induced by a periodic wave
train over a unit width.
0
1 2
1 2
Mass flux    u l dz =  a    a kg /   E / C p ,
h
2
2
which is consistent with the result derived using Eulurian approach.
Dynamic Pressure
Using the Bernoulli equation, dynmaic pressure head induced by a
periodic wave train can be calculated up to second-order,
p
1   (1)  (2)  1  (1) 2 
 

     C0 ,


g
g  t
t  2 g 
p (1)
cosh[k ( z  h)]
a
cos  ,
g
cosh(kh)
p (2) 3a 2 k 2 2
a2k

(  1) cosh[2k ( z  h)]cos 2 
 ( 2  1) cos 2 ,
 g 4
4
(2)
p
 2 2
 a k (  1) 1  cosh[2k ( z  h)].
g 4
Radiation Stress
Radiation stress: defined as the time average of excess quasi
momentum flux due to the presence of a periodic wave train.
 S xx
Up to second order, a wave train advancing in the x - axis, S  
 S yx
 p  u dz   p dz    u dz    p  p dz  
Noticing that  p   w     gz  p ,
S xx  

0
2
h
h
0
0
2
h
0
0
h

0
pdz
2
0
0
S xx    (u
h
S yy  

h
(1)2
w
(1)2
)dz  
0
 p   v dz  
S xy  S yx  0
2

0
h
 ga 2 kh
1
p dz 
  ga 2 .
sinh 2kh 4
(1)

p  p0 dz  

h
0
p0 dz  
0
pdz 
 ga 2 kh
2sinh 2kh
S xy 
S yy 
1
 2kh

0 
 sinh 2kh  2
1
2
S E
,
where
E


ga
, the energy density.

kh 
2

0

sinh 2kh 
In deep water
In shallow water
1/ 2 0 
3/ 2 0 
S E
.
S E
.


 0 0
 0 1/ 2
In the case of a wave train having an angle,  , with respect to the x-axis,
 kh

1   3  cos 2  1
2kh  sin 2

 



1 

2
2
 sinh 2kh 2 
 sinh 2kh  4


SE

2kh  sin 2
1   3  cos 2  1 

 kh
 
 

1 


2
2 

 sinh 2kh  4
 sinh 2kh 2 