Environmental Engineering Reference
In-Depth Information
3 Optical Methods to Study Surface Waves
In the last decade two optical methods have been developed to detect the deformation
of the free surface in liquids. Both exploit the emergence of high definition digital
cameras. The first one, known as synthetic Schlieren (Moisy et al. 2009 ), is based
in the refraction of light and the second one, named Fourier transform profilometry
(Cobelli et al. 2009 ; Maurel et al. 2009 ), is based in the reflection of light. In the two
cases a full reconstruction of the free surface topography is realized, but they differ in
the range of wave amplitudes they canmeasure. The underlying principle of synthetic
Schlieren is the same used to detect density fluctuations inside a transparent fluid, that
is, the change in the light trajectory due to variations in the refraction index. Consider
a ray that starts at the bottom of a liquid layer and moves to the liquid-air interface.
The trajectory followed by the ray satisfies the Snell law. If the surface is deformed,
incidence angle is modified and consequently refraction angle is also modified. For
the implementation of this method a set of dots randomly distributed is put at the
bottom of the fluid. In a first step a snapshot of the dots pattern is recorded when free
surface is flat, this is called the reference image. In a second step, an image is taken
when the wave progresses. Due to the modification of the incidence angle, related to
the deformation of the liquid-air interface, an apparent displacement of dots appear
when we compare first and second images. If we assume that deformations are small
(compared with wavelength) and if we remain in the paraxial approximation, the
apparent displacement
ʴ
r is proportional to the gradient of the free surface, namely
(Moisy et al. 2009 ):
=− ʴ
r
h ,
h
(11)
1
h
1
1
where
L , H is the depth layer, L is the distance to the camera to the
bottom ow liquid layer and
=
ʱ
H
ʱ
is the related to the ratio of the refractive indices, that
n n
ʱ =
ʴ
is
r is performed with a PIV software. To this
end the digital image is divided in small cells, where a cross correlation is made
between actual image and the reference one. The reconstruction of the topography
of free surface is made through the integration of the gradient field. The number of
equations is twice the number of unknowns, so the system is overdetermined. For
this reason, solution is made with a technique of least square. This method works
well for small deformations. This fact limits the use of the method for cases where
non linearity are still weak, however it allows the investigation of phenomena like
diffraction or the appearance of dislocations in the dark side of caustics. A method
better suited for the study of non linear waves is the Fourier transform profilometry
(PTF). This procedure is based on light reflection. If we are interested in the study of
waves in a fluid, the liquid must remain opaque to produce diffuse reflection. In order
to implement it a pattern of fringes is projected on the liquid surface with the aid of
a high definition video projector. The size of images is 1,920
1
. The determination of
1,080 pixels, it has
depth of 12 bits per color and its intensity is 2,000 lumens. Images of the fluid surface
are recorded with a digital camera using a raw format to avoid lost of information
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