Supply Servo controller
Chapter 5 Chapter 5 Presentation and Discussion of Results
5.3 Breaking Solitary Wave Run-Up
5.3.1 Wave Breaking Characteristics
5.3.1.2 Geometry of the Jet
0.4
0.2
0 -
-0.2
-0.4 .-
13.6
- - - -
Experiment Numerical Model Grilli (1997)
13.8
x/ho
H Iho= 0.30 to= 12.52
14.4
Figure 5.17: Detailed comparison of breaking jet obtained from high-speed video and numerical result at time of impact of jet in front of wave for H / ho
=
0.30. The solid line is the experimental results. the dashed line is the numerical results from Grilli et a1. (1997).section that discusses the run-up of breaking and broken waves.
0.6
0.4 I----'""-::..:-==-=-~-~-- - - __
0.2
o
-0.2 -
-0.4
12.8
Experiment Numerical Model Grilli (1997)
I 13.0
---
13.2
---
......
H/ho = 0.45 j ' = 12.73
...
-... ... ... ,
13.4
...
"
" ...
"
"" ... _..>
"
13.6 13.8
Figure 5.18: Detailed comparison of breaking jet obtained from high-speed video and numerical result at time of impact of jet in front of wave for H
/11.
0=
0.45. The solid line is the experimental results. the dashed line is the numerical results from Grilli et al. (1997).the bottom slope was 1: 15. A schematic drawing of the jet of a plunging breaking wave is illustrated in Figure 5.2l. Three parameters were used to define the jet: (i) The trajectory of the tip of the jet. This trajectory will define the motion and loca- tion of the jet and the impact point. The distance between the tip and the breaking point with respect to the constant water depth seaward of the slope. i.e. ((:rb -
:rd /
ho.yd
11.0 ) was used to represent the trajectory. (ii) The length and thickness of the jet before impingement. The length of the jet Ll was defined as the horizontal distance from the tip of the jet to the nearest location of the wave surface which was vertical.as shown in Figure 5.2l. Two parameters were used to define the thickness of the jet:
0.6
0.4 -
0.2
o
-0.2
-0.4
12.8
---
--
...----==-=~ ... ...
...
... ...
H Ih oF 0.45
... ...
"-
...
-.;.=.:====-=.:==-==-->- - - - - -
Experiment \'=12.913 Numerical Model t'=12.73 Grilli (1997)
13.0 13.2
x/ho
13.4 13.6 13.8
Figure 5.19: Detailed comparison of breaking jet from numerical result to experimen- tal shape after shifting the latter by 6t*
=
0.183 at time of impact of jet on front of wave for H /11.0=
0.30. The solid line is the experimental results. the dashed line is the numerical results from Grilli et a1. (1997).one is the thickness of the jet at the wave vertical plane. i.e. L'2' the other is the thick- ness of the jet at half length of the jet. i.e. L:~. These two variables not only describe the thickness of the jet but they also show how the thickness changes at different locations. (iii) The horizontal impinging velocity of the jet. This can describe "the strength of the impingement". i.e .. how strong the momentum exchangE~ happens at the impingement point.
Figure 5.22 shows the trajectory of the tip of the impinging jet. In the Figure 5.22 a curve which is denoted as the "free-falling" curve as simply the trajectory of a free-falling jet is also shown. The initial horizontal velocity of the free-falling jet is
1.5
1.4
1.3
1.2
~ 1.1
:c
1.0
0.9
0.8
0.7 0
... .. .. ..
_ _ Numerical Model HIh,,=O.30 _ _ Numerical Model Hlho=O.45
o Experiment H/ho=O.3() .. Experiment H/ho=O.45
2 4
..
6
~
....
.. ..
.. ..
..
10
o
o o
.. ..
..
12 14
Figure 5.20: Comparison of variation relative wave height on slope H' / H to the relative water depth hoi h from experiments and from numerical results. The circles are the experimental results for H /110 = 0.45. the triangles are the experimental results for H / ho
=
0.30, the dashed line is the numerical results from Grilli et 0.1.(1997) for H/ho
=
0.45. the solid line is the numerical results from Grilli et 0.1. (1997) for H / ho=
0.30.chosen to be the wave celerity in the constant depth region seaward of the slope, and the initial jet tip position was chosen from the experiment. The assumption is made here that the water particle velocity at breaking is essentially equal to the wave speed just before the wave propagates up the slope. Thus. the trajectory can be described as:
. _ _ ~[:J:th - :1:/0]2 '/
l)th - 2
+
.lJtOCth
(5.9) where (1'th ,:IJth) is the theoretical location of the tip according to free-falling aSSllIllp-
1"
L,
x" Tip(x"y,)
Figure 5.21: Definition sketch of the jet produced by the plunging breaking wave tion. (:J'tO.Yw) is the initial jet tip location from experiments. and Cth
=
Jg(ho+
H)is the theoretical wave celerity in the constant depth region. The good agreement. be- tween the experiment al results and the free- falling curve shows that once t.he water jet is propelled from the breaking wave. the trajectory is the same as that of a free-falling jet. until it impinges on the free surface. To verify this result. the horizontal velocity of the jet. tip was also calculated from the high-speed video images. and is shown in Figure 5.23 as a function of the jet location wit.h respect to the breaking point (:1:b -:Tt) / lio· The horizontal velocity was computed by dividing the distance between the :r coordinate of the tip in consecutive images by the time interval between frames.
Because of tIl(' limits of the spatial and time ({ccuracy of the high-speed video. the velocity data obtained this way have a relatively large variation. especially when the tip of the jet is close to the free surface. The shape of the tip makes the measurement of the tip location difficult. Large variation also exists at the initial stage of the jet because the jet dimension is very small and the error associated with obtaining the
1.4 r - - - ,
1.3
" "
...
1.2
" Jet Tip Location - Trajectory of Free·falling 1.1
" " "
" "
"
" "
"
" "
"
1.0 ' - -_ _ _ --'-_ _ _ _ _ -"-_ _ _ _ --'-_ _ _ _ ---''--_ _ _ - - - . J
0.0 0.2 0.4 0.6 0.8
(x".x)lh"
Figure 5.22: Trajectory of the tip of the jet produced by the plunging breaking wave.
The tTiangles are the experimental results. the solid line is the fitted free-falling curve.
jet tip location from the video images is relatively large. Nevertheless. it seems that the horizontal velocity of the jet tip is almost constant over most of the jet trajectory.
The theoretical wave celerity eth described above is also shown in the Figure 5.23.
The results suggest that the wave velocity at breaking is of the sallle order as the wave celerity ill the constant depth region offshore. This has been pointed out by other researchers. for example. Skjelbreia (1987).
The water velocity Vm and the angle of impact of the jet trajectory,
em,
at impinge- ment can be derived from the free-falling jet trajectory. If we assume the maximum height of the wave at breaking is Hb measured from the free surface where impact will take place. the wave breaking velocity is Vb ~ Cth. the impingement velocity and2 . 0 , - - - , 1.8
1.6
104
1.2
0.8
\/
0.6
0.4
---- Tip Horizontal Velocity 0.2 - - Offshore Wave Celerity
o.n L -_ _ _ _ - ' - -_ _ _ _ --"--_ _ _ _ - - ' -_ _ _ _ _ l...-_ _ _ ---.l
-0.2 n.o 0.2 0.4 0.6 O.S
Figure 5.23: Horizontal velocity of the tip of the jet produced by the plunging breaking wave. The triangles with the dashed line are the experimental results, the solid line is the theoretical wave celerity of the incident solitary waVE' in the constant depth regIOn.
angle is:
Vm JCZ
h+
2gHu (5.10)em
arctan[ ('til . 1 (5.11)y!2gHb
The length of the jet with respect to the distance between the location where the wave crest breaks_ i.e .. :J:b, and the location of the tip. :ft. is shown in Figure 5.24. It WetS seen that the length of tlw jet increases linearly as the plunging breetking wave and the jet propagates on the slope. A linear curve from linear regression analysis is
0.35 ~---~
0.30
0.25 .. Length 01' the Jet L,
0.20
0.15
0.10
- LinearFit
.. ..
.. .. .. .. .. .. .. .. .. ..
..
..
.. ..
..
0.05 '--_ _ _ ---' _ _ _ _ _ ..L-_ _ _ _ - ' -_ _ _ _ _ ' - -_ _ _ _ -l
0.0 0.2 0.4 0.6 0.8
Figure 5.24: Hori20ntal length of the jet produced by the plunging breaking wave.
The triangles are the experimental results, the solid line is the fitted curve from a lillear regression analysis.
also presented in the figure as the form:
L l' - T
_1
=
0.282['0 . t] _ 0.067ho ho (5.12)
Since the velocity of the water jet tip is constant from above analysis, the wave celerity of the plunging breaking wave is less than the jet tip velocity at the order of incident wave height (0.282) from the linear regression analysis.
The thickness of the jet at t he middle of the jet and the thickness of the jet at the location that the plunging wave surface becomes vertical are shown in Figure 5.25. It can be found from the experiments that both these measurement for the jet thickllcss
0 . 2 0 , - - - . . ,
0.15
0.10 .:;:.-
::l
0.05
..
•.. ..
.. ....
....... . .
.. ..
.. .. ..
.. ...
...
••
• • • • • •---~~---.--~---~-~~----
• • • • •
••
.. Thickness of the .let L2
• Thickness of the .let L,
0.00 L -_ _ _ _ - L _ _ _ _ _ L -_ _ _ _ -L.. _ _ _ _ _ L -_ _ _ _ ....J
-0.2 0.0 0.2 0.4 0.6 0.8
(x"-x)lh,,
Figure 5.25: Thickness of the jet produced hy the plunging breaking wave. The triangles are the experimental data for L2 , the circles are the experimental data for L:,. The solid line and dashed line are the fitting curves from linear regression analysis for L2 , L3 respectively.
are almost constant. The thickness of the jet at the middle length. I.e, L:3 • is about half of that at the base of the jet L2 .
The overall geometry of the impacting jet produced by the plunging breaking wave was measured accurately during the experiments. These geometric parameters which describe the jet associated with the plunging breaker can be used to model the jet impingement process perhaps leading to a better understanding of the air entrainment, and the energy dissipation associated with plunging breaking waves.