______________________________________
* Corresponding Author: [email protected] (M. S. Parvin)
App. Math. and Comp. Intel., Vol. 3 (1) (2014) 259–272 http://amci.unimap.edu.my
© 2014 Institute of Engineering Mathematics, UniMAP
Mass Transport Velocity in a Two-dimensional Wave Tank
M. S. Parvina*, M. S. Alam Sarkera, and M. S. Sultanaa
aDepartment of Applied Mathematics, University of Rajshahi, Rajshahi-6205, Bangladesh
Received: 13 July 2013; Accepted: 23 November 2013
Abstract: Waves occurred on the surface of water in two-dimensional wave tank are studied. Deriving vorticity equation from incompressible Navier-Stokes equation, the boundary condition
on the free surface, water bottom and flap surface are discussed. Using separation of variable
method, velocity potential takes the general form in terms of eigen function. To obtain the values of
horizontal and vertical velocity components, the boundary conditions on the water bottom and the
flap surface are used and then mass transport velocity has been derived. Finally, time average of
horizontal velocity component at mean water level is generalized.
Keywords: Mass Transport Velocity, Navier-Stokes Equation, Stream Function.
1. Introduction
260 M.S. Parvin et al.
[5] also constructed Lagrangian asymptotic solutions of the non-linear water waves. Iskandarani and Liu [6] discussed on mass transport in two-dimensional wave tank. Many Lagrangian asymptotic solutions of the non-linear water waves have been developed such as Buldakov et al [7], Clamond [8] and Constantin [9]. Boufermel et al. [10] formulated velocity of mass transport taking as variable to model acoustic streaming. Frode [11] studied mass transport velocity in shallow water waves reflected at right angles from an infinite and straight coast in a rotating ocean. In this paper, mass transport velocity has been formulated using boundary conditions at water bottom and the flap surface. Finally, we see that the time average of horizontal velocity component is zero at mean water level which is same as Dean and Dalrymple [12].
2. Mathematical Formulation
2.1. Governing Equation
The motion of the water surface consists in general regular waves for local disturbance progressing on a constant water depth
(
z
=
h
)
as shown in Fig.1.Figure1: Wave Train
Mass transport velocity (
ε
2U
1,
ε
2W
1), defined as the average of the Lagrangian velocity which is in terms of particle velocity and stream function can be written as.
2 1 2 1 1
2 1 1
1
z
z
dt
x
z
x
dt
z
u
U
∂
Ψ
∂
=
∂
∂
∂
∂
−
∂
∂
∂
∂
∂
+
=
∫
ψ
ψ
∫
ψ
ψ
(1)and
Waves
h z
App. Math. and Comp. Intel., Vol. 3 (1), 2014 261
.
1 2 1 2
1 2 1 1
1
x
z
x
dt
x
x
dt
z
w
W
∂
Ψ
∂
−
=
∂
∂
∂
∂
∂
+
∂
∂
∂
∂
−
=
∫
ψ
ψ
∫
ψ
ψ
(2)For the mass transport velocity
U
, the stream functionΨ
can be written as.
1 1 2
∫
∂
∂
∂
∂
+
=
Ψ
x
dt
z
ψ
ψ
ψ
(3)The Navier-Stokes equation for incompressible flow is
.
1
2u
F
p
u
z
w
x
u
t
=
−
∇
+
+
∇
∂
∂
+
∂
∂
+
∂
∂
υ
ρ
(4)Where
u
and
w
denote the horizontal and vertical velocity components respectively,p
is the pressure gradient,F
is the external body force andυ
denotes the kinematic coefficient ofviscosity where,
ρ
µ
υ
=
.Taking curl on both sides of Eq. (4), we have
ξ
υ
ξ
2∇
=
∂
∂
+
∂
∂
+
∂
∂
z
w
x
u
t
, since, vorticityξ
=
∇
×
u
(5)(
)
=
+
−
∇
×
∇
∇
−
=
−
=
,
φ
0
ρ
φ
ρ
ρ
g
and
p
F
Q
Now, the vorticity equation from the above equation is in the following
.
0
2 2
=
∇
∇
−
∂
∂
+
∂
∂
+
∂
∂
ψ
υ
z
w
x
u
t
(6)where,
∂
∂
−
∂
∂
−
∂
∂
∂
∂
=
∂
∂
∂
∂
−
∂
∂
∂
∂
=
∂
∂
−
∂
∂
=
×
∇
=
z
z
x
x
x
z
z
x
z
u
x
w
u
φ
φ
ψ
ψ
ξ
.
2 2
2
2 2
ψ
ψ
=
∇
∂
∂
+
∂
∂
=
z
x
Now,.
1 4 1
2
∫
∇
=
∇
ψ
υ
ψ
dt
(7) [taking first-order term of Eq. (6)]and
.
2 4 1 2 1
1
∇
ψ
=
υ
∇
ψ
∂
∂
+
∂
∂
z
w
x
262 M.S. Parvin et al.
[taking time average of second-order term of Eq. (6)] Using Eq. (7) in Eq. (8), we have
.
1 4 1
1 2 4
dt
z
w
x
u
∫
∇
∂
∂
+
∂
∂
=
∇
ψ
ψ
(9)From Eq. (3), the stream function for the mass transport velocity
U
can be written as.
1 1 4 1 4 1
1 4
∫
∫
∇
+
∇
∂
∂
∂
∂
∂
∂
+
∂
∂
=
Ψ
∇
x
dt
z
dt
z
w
x
u
ψ
ψ
ψ
(10)Assuming
ψ
1 satisfies Laplace equation, Eq. (10) which is considered as the boundary layer theory can be rewritten as.
1 1 4 4
∫
∂
∂
∂
∂
∇
=
Ψ
∇
x
dt
z
ψ
ψ
(11)
According to Longuet-Higgins [1], boundary condition for
Ψ
on the bottom of water is given by ( ) ( ).
4
3
5
*1 1
∞ ∞ ∞
=
∂
∂
−
=
∂
Ψ
∂
s s n
q
s
q
i
i
n
ω
(12)where
q
s( )∞1 represents the first order tangential velocity at outer edge of the boundary layer at the fixed wall and superscript * represents the complex conjugate of the value and the boundary condition at free surface is( ) ( )
.
4
0*1 0 1 2
2
s n n
q
s
q
s
i
n
∂
∂
∂
∂
=
∂
Ψ
∂
∞
=
ω
(13)
2.2. Linear Water Wave Theory
Linear water waves are of small amplitude for which we can linearise the equations of motion. The water wave motion is represented by a velocity potential
Φ
(
x
,
z
,
t
)
which is considered as(
x
,
z
,
t
)
=
Re
{
φ
(
x
,
z
)
e
iωt}
.
Φ
(14) We assume that the bottom surface is of constant depth atz
=
−
h
.The water surface is at0
=
z
and the region of interest is−
h
<
z
<
0
. Now the equations are in the following0
=
+
zz xxφ
φ
,−
h
<
z
<
0
.
(15)
=
=
∂
∂
−
=
=
∂
∂
.
0
,
,
,
0
z
K
z
h
z
z
φ
φ
φ
App. Math. and Comp. Intel., Vol. 3 (1), 2014 263 The last expression can be obtained from combining of the following two equations
=
=
=
±
=
∂
∂
.
0
,
,
0
,
z
g
i
z
i
z
η
ωφ
ωη
φ
m
(17)
where
...
,...
2
,
1
,
tan
tanh
2
=
−
=
=
=
n
h
k
k
kh
k
g
K
n n
ω
(18)
Eq. (15) is solved by the separation of variable method. So the velocity potential
φ
are taken as(
x
,
z
)
=
U
( ) ( ).
x
P
z
φ
(19) Substituting this value in Eq. (15), we have.
1
1
2 2
2 2
dz
P
d
P
dx
U
d
U
=
−
(20) The L.H.S of Eq. (20) is only a function ofx
and the R.H.S of Eq. (20) is only a function ofz
. So each term is constant. Therefore,.
1
1
22 2
2 2
k
dz
P
d
P
dx
U
d
U
=
−
=
±
(say) (21)Taking positive sign, we have from Eq. (21),
.
0
2 2 2
=
+
k
P
dz
P
d
(22)
and
.
0
2 2 2
=
−
k
U
dz
U
d
(23)
Solving Eq. (22), we obtain
( )
z
Ae
ikzBe
ikzP
=
+
− , where A and B are arbitrary constants. Therefore, Eq. (19) becomes(
x
,
z
)
=
(
Ae
ikz+
Be
−ikz)
U
( )
x
.
φ
(24)Applying boundary condition
|
=
0
∂
∂
=h z
z
φ
in Eq. (24), we get
A
=
Be
−2ikh.
Hence,
( )
z
D
cos
k
(
z
h
)
.
P
=
−
=
=
−ikhBe
264 M.S. Parvin et al.
(
x
,
z
)
=
D
cos
k
(
z
−
h
) ( )
U
x
.
φ
(25) Again, solving Eq. (23), we havekx kx
e
B
e
A
U
=
1+
1 − , whereA
1and
B
1 are arbitrary constants.If we set
A
1=
0
and
B
1=
1
, thenU
=
e
−kx.
Therefore, Eq. (25) becomes
(
x
,
z
)
=
D
cos
k
(
z
−
h
)
e
−kx.
φ
(26) Using Eq. (17) in Eq. (26), we have0
,
sin
=
−
=
−
=
∂
∂
−z
i
khe
Dk
z
kx
ωη
φ
.
0
,
sin
=
=
⇒
−z
i
khe
Dk
kxωη
(27) This expression forφ
and
η
become more convenient if we writeη
=
ae
−kx, Eq. (27) can be written as.
sin
kh
k
ai
D
=
ω
Substituting the value of
D
in Eqs. (24) and (26), we have(
)
cos
(
)
.
sin
,
kxe
h
z
k
kh
k
ai
z
x
=
ω
−
−φ
(28)and
( )
cos
(
).
sin
kh
k
z
h
k
ai
z
P
=
ω
−
(29)The boundary condition at the free surface gives
ω
2=
−
gk
tan
kh
, which is the dispersion relation for a free surface.The above equation is not really the dispersion relation for a free surface, it would be better to refer to it as a transcendental equation. If we solve for all roots in the complex plane, we find that the first root is a pair of imaginary roots. We denote the imaginary solutions of this equation by
ik
k
0=
±
and real solution byk
n,
n
≥
1
. Thek
of the imaginary solution is the wave number.So we put
ω
2=
gk
tanh
kh
.Finally, we define the function
P
( )
z
as( )
cos
(
),
0
.
sin
−
≥
=
k
z
h
n
h
k
k
i
a
z
P
nn n
App. Math. and Comp. Intel., Vol. 3 (1), 2014 265 as the vertical eigen function in the open water region.
Also, we can write
U
=
e
−knx.
Hence, the velocity potential eigen function expansion can be written as
(
)
cos
(
)
.
sin
,
n k xn n
n
k
z
h
e
nh
k
k
i
a
z
x
=
ω
−
−φ
(30)Using Eqs. (18) in (30), we have
(
)
cos
(
)
.
cos
,
n k xn
n
k
z
h
e
nh
k
K
i
a
z
x
=
−
ω
−
−φ
(31)Similarly, taking negative sign, we have from Eq. (20),
.
0
2 2 2
=
−
k
P
dz
P
d
(32)
and
.
0
2 2 2
=
+
k
U
dz
U
d
(33)
Solving Eq. (32), we have
( )
kz kze
B
e
A
z
P
=
2+
2 − , whereA
2and
B
2 are arbitrary constants. Therefore, Eq. (18) becomes(
x
,
z
)
=
(
A
2e
kz+
B
2e
−kz)
U
( )
x
.
φ
(34)Applying boundary condition
|
=
0
∂
∂
=h z
z
φ
in Eq. (34), we get
A
2=
B
2e
−2kh.
Hence,
(
z
h
)
k
D
P
=
1cosh
−
, where,D
1=
2
B
2e
−kh=
constant. Therefore, Eq. (34) becomes(
x
,
z
)
=
D
1cosh
k
(
z
−
h
) ( )
U
x
.
φ
(35) Again, solving Eq. (33), we haveikx ikx
e
B
e
A
U
=
3+
3 − , whereA
3and
B
3 are arbitrary constants. If we setA
3=
0
and
B
3=
1
, thenU
=
e
−ikx.
Therefore, Eq. (35) becomes
(
x
,
z
)
=
D
1cosh
k
(
z
−
h
)
e
−ikx.
266 M.S. Parvin et al.
.
0
,
sinh
1
=
=
=
∂
∂
−z
i
khe
k
D
z
ikxωη
φ
.
0
,
sinh
1
=
=
⇒
−z
i
khe
k
D
ikxωη
This expression for
φ
and
η
become more convenient if we writeη
=
ia
0e
−ikx, we get.
sinh
0 1kh
k
a
D
=
−
ω
Hence, from Eq. (36), we have
(
)
cosh
(
)
.
sinh
,
0k
z
h
e
ikxkh
k
a
z
x
=
−
ω
−
−φ
(37)Using Eq. (18) in Eq. (37), we have
(
)
cosh
(
)
.
cosh
,
0 ikxe
h
z
k
kh
K
a
z
x
=
−
ω
−
−φ
Therefore,
(
)
(
)
cos
(
)
.
cos
cosh
cosh
,
1 0∑
∞ = − −−
−
−
−
=
n x k n n nikx
k
z
h
e
nh
k
K
i
a
e
h
z
k
kh
K
a
z
x
ω
ω
φ
[Using eigen function expansion]
In general form, the velocity potential can be expressed as
(
)
(
)
( )(
)
−
−
−
−
=
Φ
∑
∞ = − − − 1 0cos
cos
cosh
cosh
Re
,
,
n t i x k n n n t kx ie
e
h
z
k
h
k
K
i
a
e
h
z
k
kh
K
a
t
z
x
ω
ωω
n ω(
)
(
)
( )(
)
−
+
−
−
=
Φ
⇒
∑
∞ = − − − 1 0cos
cos
cosh
cosh
Re
,
,
n t i x k n n n n t kxi
k
z
h
e
e
h
k
k
i
a
e
h
z
k
kh
k
a
t
z
x
ω
ωω
n ω[
Q
from Eq. (18),n
k
k
K
=
=
−
](
)
(
)
( )(
)
−
+
−
=
Φ
∑
∞ = − − − 1 0 2cos
cos
cosh
cosh
Re
,
,
n t i x k n n n n t kx ie
e
h
z
k
h
k
k
i
a
e
h
z
k
kh
k
a
i
t
z
x
ω
ωω
n ω(
)
(
)
( )cos
(
)
.
cos
cosh
cosh
Re
,
,
1 0 0 0 2
−
+
−
=
Φ
⇒
∑
∞ = − − − n t i x k n n n n t kx ie
e
h
z
k
h
k
k
K
i
a
e
h
z
k
kh
k
K
a
i
t
z
x
θ
ω
ωθ
ω
n ω[Assuming
K
1
0
=
App. Math. and Comp. Intel., Vol. 3 (1), 2014 267
(
x
,
z
,
t
)
=
Re
{
i
ωθ
0φ
1(
x
,
z
)
e
iωt}
.
Φ
∴
(38) where,(
)
(
)
cos
(
)
.
cos
cosh
cosh
,
1 0 1∑
∞ = − −+
−
−
=
n x k n n n nikx
k
z
h
e
nh
k
k
Ka
e
h
z
k
kh
k
i
Ka
z
x
φ
(39)2.3. Formulation of mass transport velocity
The first order potential
φ
1(
x
,
z
)
can be written by the eigen function expansion,(
)
(
)
(
)
k xn n n n
n
ikx
e
nh
k
h
z
k
k
K
a
e
kh
h
z
k
k
K
ia
z
x
− ∞ = −−
+
−
=
∑
cos
cos
cosh
cosh
,
1 0 1φ
, [From Eq. (39)](
)(
)
(
)
.
cos
cos
sin
cos
cosh
cosh
1 0 x k n n n n n ne
h
k
h
z
k
k
K
a
kx
i
kx
kh
h
z
k
k
K
ia
− ∞ =−
+
−
−
=
∑
(40)Substituting this value in Eq. (38), we get
(
)
(
) (
)
(
)
−
+
−
−
=
Φ
− ∞ =∑
t i x k n n n n ne
e
h
k
h
z
k
k
K
a
kx
i
kx
kh
h
z
k
k
K
ia
i
t
z
x
n ωωθ
cos
cos
sin
cos
cosh
cosh
Re
,
,
1 0 0(
)(
)
(
)
(
)
+
−
+
−
−
−
=
− ∞ =∑
e
t
i
t
h
k
h
z
k
k
K
a
i
kx
i
kx
kh
h
z
k
k
K
a
x k n n n n n nω
ω
ωθ
cos
sin
cos
cos
sin
cos
cosh
cosh
Re
1 0 0(
) (
{
) (
)
}
(
)
(
)
−
−
−
+
−
+
+
−
−
=
− ∞ = − ∞ =∑
∑
e
t
h
k
h
z
k
k
K
a
t
e
h
k
h
z
k
k
K
a
i
t
kx
t
kx
i
t
kx
t
kx
kh
h
z
k
k
K
a
x k n n n n n x k n n n n n nn
ω
ω
ω
ω
ω
ω
ωθ
sin
cos
cos
cos
cos
cos
cos
sin
sin
cos
sin
sin
cos
cos
cosh
cosh
Re
1 1 0 0(
)
(
)
(
)
(
)
(
)
(
)
. sin cosh cosh cos cos cos sin cos cos cos cosh cosh Re 0 1 1 0 0 − − − − + − − − − − = − ∞ = − ∞ =∑
∑
t kx kh h z k k K a t e h k h z k k K a i t e h k h z k k K a t kx kh h z k k K a x k n n n n n x k n n n n n n nω
ω
ω
ω
ωθ
(
)
(
)
(
)
(
)
sin
.
cos
cos
cos
cosh
cosh
,
,
1 0 00
e
t
h
k
h
z
k
k
K
a
t
kx
kh
h
z
k
k
K
a
t
z
x
k xn n n n n n
ω
ω
θ
ω
ω
θ
∞ −268 M.S. Parvin et al.
(
)
(
)
(
)
.
sin
cos
cos
sin
cosh
cosh
1 0 00
e
t
h
k
h
z
k
K
a
t
kx
kh
h
z
k
K
a
x
u
kxn n n n n
ω
ω
θ
ω
ω
θ
∞ −=
−
+
−
−
=
∂
Φ
∂
=
∑
(42) Now, at the bottom surface,(
)
sin
.
cos
1
sin
cosh
1
1 0 00
e
t
h
k
K
a
t
kx
kh
K
a
x
u
kxn n n h z h z n
ω
ω
θ
ω
ω
θ
∞ −= = =
=
−
+
∑
∂
Φ
∂
=
(
)
sin
.
cos
1
cos
cosh
1
1 0 00
e
t
h
k
Kk
a
t
kx
kh
Kk
a
x
u
kxn n n n h z n
ω
ω
θ
ω
ω
θ
∞ −= =
=
−
−
∑
∂
∂
and(
)
(
)
−
−
+
−
=
∂
∂
− ∞ = − ∞ = =∑
∑
t
e
h
k
Kk
a
t
kx
kh
Kk
a
t
e
h
k
K
a
t
kx
kh
K
a
x
u
u
x k n n n n x k n n n h z n nω
ω
θ
ω
ω
θ
ω
ω
θ
ω
ω
θ
sin
cos
1
cos
cosh
1
sin
cos
1
sin
cosh
1
1 0 0 0 1 0 0 0(
)
(
)
sin
.
cos
1
cos
cosh
1
sin
cos
1
sin
cosh
1
1 0 1 0 2 2 2 0
−
−
+
−
=
− ∞ = − ∞ =∑
∑
t
e
h
k
k
a
t
kx
kh
k
a
t
e
h
k
a
t
kx
kh
a
K
x k n n n n x k n n n n nω
ω
ω
ω
ω
θ
(43)At
t
=
0
,.
cos
sin
cosh
1
cos
cosh
1
sin
cosh
1
2 2 2 2 0 2 0 0 0 2 2 2 0kx
kx
kh
k
K
a
kx
kh
k
a
kx
kh
a
K
x
u
u
z hω
θ
ω
θ
=
=
∂
∂
= (44)The governing equation of a stream function for the mass transport of the boundary layer is given by Eq. (11) and the boundary conditions on the free surface, water bottom and the flap surface are summarized as
( )
(
,
0
)
Re
4
.
* 2 2
∂
∂
∂
∂
=
Ψ
∂
∂
x
u
x
w
i
x
z
ω
(45)(
)
(
)
∂
∂
−
=
Ψ
∂
∂
x
u
u
i
i
h
x
z
*4
3
5
Re
,
ω
, where asterisk( )
*
is the complex conjugate.
.
4
3
*x
u
u
∂
∂
−
=
App. Math. and Comp. Intel., Vol. 3 (1), 2014 269 and
(
)
.
4
3
5
Re
,
0
*
∂
∂
−
−
=
Ψ
∂
∂
z
w
w
i
i
z
x
ω
(47)Hence Eq. (46) becomes
(
)
(
)
.
cos
sin
cosh
1
4
3
cos
sin
cosh
1
4
3
,
2 2 2 0 2 0 2 2 2 2 0 2 0kx
kx
kh
k
K
a
kx
kx
kh
k
K
a
h
x
z
ω
θ
ω
θ
ω
−
=
−
=
Ψ
∂
∂
(48)Therefore, Eq. (1) can be written as
.
cos
sin
cos
sin
cosh
1
4
3
2 2 2 0 2 0 1kx
kx
A
kx
kx
kh
k
K
a
U
=
−
=
θ
ω
(49)
where,
kh
k
a
A
02 2cosh
1
4
3
ω
−
=
(50)Again, the velocity component in z-direction in the following form
(
)
(
)
(
)
.
sin
cos
sin
cos
cosh
sinh
1 0 00
e
t
h
k
h
z
k
K
a
t
kx
kh
h
z
k
K
a
z
w
k xn n n n n
ω
ω
θ
ω
ω
θ
∞ −=
−
+
−
−
−
=
∂
Φ
∂
=
∑
Therefore, at the flap surface,
(
)
(
)
.
sin
cos
sin
cos
cosh
sinh
1 0 0 0 0 0t
h
k
h
z
k
K
a
t
kh
h
z
k
K
a
z
w
n n n n xx
θ
ω
ω
θ
ω
ω
−
+
−
−
=
∂
Φ
∂
=
∞ = = =∑
(
)
(
)
.
sin
cos
cos
cos
cosh
cosh
1 0 0 0 0t
h
k
h
z
k
Kk
a
t
kh
h
z
k
Kk
a
z
w
n n n n nx
θ
ω
ω
θ
ω
ω
270 M.S. Parvin et al.
(
)
(
)
(
)
(
)
−
−
−
−
−
−
=
∞ = ∞ =∑
∑
t
h
k
h
z
k
k
a
t
kh
h
z
k
k
a
t
h
k
h
z
k
a
t
kh
h
z
k
a
K
n n n n n n n n nω
ω
ω
ω
ω
θ
sin
cos
cos
cos
cosh
cosh
sin
cos
sin
cos
cosh
sinh
1 0 1 0 2 2 2 0 (51)At
t
=
0
(
)
(
)
(
)
cosh
(
)
.
sinh
cosh
1
cosh
cosh
cosh
sinh
2 2 0 2 2 2 0 0 0 2 2 2 0h
z
k
h
z
k
kh
k
a
K
kh
h
z
k
k
a
kh
h
z
k
a
K
z
w
w
−
−
=
−
−
=
∂
∂
ω
θ
ω
θ
(52)Also Eq. (46) can be written as
(
)
∂
∂
−
−
=
Ψ
∂
∂
∴
z
w
w
i
i
z
x
*4
3
5
Re
,
0
ω
z
w
w
∂
∂
=
*4
3
ω
(
)
cosh
(
)
.
sinh
cosh
1
4
3
2 2 2 0 20
k
z
h
k
z
h
kh
k
K
a
−
−
=
θ
ω
(53)Therefore, Eq (2) can be written as
(
)
(
)
(
)
(
)
cosh
(
)
.
sinh
cosh
sinh
cosh
1
4
3
,
0
2 2 2 0 2 0 1h
z
k
h
z
k
A
h
z
k
h
z
k
kh
k
K
a
x
z
W
−
−
=
−
−
−
=
∂
Ψ
∂
−
=
ω
θ
(54)From Eqs. (49) and (54), mass transport velocity has been formulated.
2.4. Time Average of Horizontal Velocity Component
Time average of horizontal velocity u can be expressed as
(
,
0
,
)
1
(
,
0
,
)
.
0
dt
t
x
u
T
t
x
u
T∫
=
(55)Hence Eq. (42) can be written as
(
)
[
a
K
(
kx
t
)
a
Ke
t
]
dt
T
t
z
x
u
kxn n T n
ω
ω
θ
ω
ω
θ
sin
sin
1
,
,
1 0 0 0 0 − ∞ =∑
∫
−
+
=
(
)
T x k nn
Ke
t
a
t
kx
K
a
T
n 0 1 2 0 2 00
cos
cos
1
−
−
=
∞ − =∑
ω
ω
θ
ω
ω
App. Math. and Comp. Intel., Vol. 3 (1), 2014 271 Using eigen function expansion, velocity potential has been derived. Then we see that time average of horizontal velocity component is zero, which is same as Dean and Dalrymple [15], who derived some nonlinear properties for water waves of small amplitude.
3. Conclusion
Mass transport velocity in two dimensional wave tank has been studied. Vorticity equation from incompressible Navier-Stokes equation is solved using boundary layer theory and boundary conditions on the free surface, water bottom and the flap surface. Generalising the velocity potential in terms of eigen function expansion, we obtain the velocity components which are related to stream function. Using the given boundary conditions, mass transport velocity has been formulated. Then we see time average of horizontal velocity component is zero at mean water level.
References
[1] M.S. Longuet-Higgins. The Trajectory of Particles in Steep Symmetric Gravity Waves.
Journal of Fluid Mechanics, 94: 497-517, 1979.
[2] M.S. Longuet-Higgins. Eulerian and Lagrangian Aspects of Surface Waves. Journal of Fluid Mechanics, 173: 683-707, 1986.
[3] M.S. Longuet-Higgins .Lagrangian Moments and Mass Transport in Stokes Waves. Journal of Fluid Mechanics, 179: 547-555, 1987
[4] H.K. Chang, J.C. Liou and M. Y. Su. Particle Trajectory and Mass Transport of Finite-Amplitude Waves in Water of Uniform Depth. Elsevier Masson SAS.2006.
[5] M. Naciri and C. Mei. Evalution of Short Gravity Waves on Long Gravity Waves. Physical of Fluids, A, 5(8):1869-1878, 1993.
[6] Iskandarani, M. and Liu, P.L.F. Mass Transport in Two-dimensional Wave Tank. J. Fluid Mech. 231: 395- 415, 1991.
[7] Buldakov et-al. New Asymptotic Description of Nonlinear Water Waves in Lagrangian Coordinates. Journal of Fluid Mechanics, 562: 431-444, 2006.
[8] Clamond. On the Lagrangian Description of Steady Surface Gravity Waves. Journal of Fluid Mechanics, 589: 433-454, 2007.
[9] Constantin. On the Deep Water Wave Motion. Journal of Physics A: Mathematical General,
34: 1405-1417, 2001.
272 M.S. Parvin et al.
[11] Frode Hoydalsvik. Mass Transport Velocity in Shallow Water Waves Reflected in a Rotating Ocean with a Coastal Boundary. J. Phys. Oceangr. 37, Issue 10, 2429-2445, 2007.
