Environmental Engineering Reference
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of the surface topography h(x, y) is made by a finite difference approximation for
spatial derivatives of the surface gradient.
With respect the implementation of Fourier Transform Profilometry an opaque
liquid surface is required. This is achieved by adding a concentrated white dye to
the water. The fringe pattern projected on the free surface covers an area of 28 cm
×
50 cm, which is sufficient to investigate wave field before and after the Huygens
cusp. On the other hand, the distance from projector to liquid surface is L
=
1.14m,
the distance D is 0.30 cm and wavelength of fringe pattern was p
0.003 m (3mm).
In order to avoid the appearance of undesirable bright spot the image produced by
the videoprojector was shifted with no deformation (a feature available in newer
equipment) and additionally two crossed polarizers have been put on the lenses of
camera and videoprojector. As stated before, we use a digital camera capable of
recording in raw format. A further conversion of images to a standard format (tif
images of 16 bits per color) was made and finally processing was made with routines
written in matlab.
=
4 Numerical Method
The surface waves are governed by the Navier-Stokes and continuity equations. In
recent decades many researches of surface waves were made through numerical
codes, however there are some difficulties in its use, for instance, the domain of
integration changes in time. In this work we present numerical results for surface
waves in the shallow water approximation, that is, when liquid depth is much lower
than the wavelength
). The choice of this approximation was made on the
basis that the system remains 2D but at the same time non linearity is retained. The
equations to solve are:
ʻ
(
H
ʻ
v
2
r
=−
∂
u
u
∂
u
v
r
∂
u
∂ʸ
−
g
∂
h
t
+
r
+
r
,
(13)
∂
∂
∂
∂
v
u
∂
v
v
r
∂
v
∂ʸ
−
uv
r
=−
g
r
∂
h
∂ʸ
,
t
+
r
+
(14)
∂
∂
∂(
∂(
∂
)
+
∂(
)
)
+
∂
h
1
r
rhu
hv
H
r
ru
v
∂ʸ
t
+
=−
,
(15)
∂
∂
r
∂ʸ
∂
r
where
u
and
v
are the horizontal components of the velocity field, h is the free surface
deformation and H is the depth of the liquid layer. In the deduction of these equations,
the viscosity was neglected and the continuity equation and the kinematical condition
have been used. The numerical method used for solving Eqs. (
13
)-(
15
)involvesa
centered second order finite differences for radial coordinate, a backward second
order finite difference for time and a spectral code for
coordinate. The numerical
code cannot predict the wave breaking because this process implies that variable h
ʸ
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