7.2 Phase-ensemble averaged turbulence characteristics
7.2.2 Turbulent kinetic energy
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Phase t/T
u’/ w’
z = −8.0 cm z = −3.0 cm z = +2.0 cm
Figure 7.9: Phase dependence of the ratio of horizontal to vertical turbulence intensity components uw′′, at points : (blue)- near the bed (x, z) = (-238, -8) cm ; (black)- between trough and crest (x, z) = (-238, -3) cm and (red)- near crest (x, z) = (-238, +2) cm.
0 0.2 0.4 0.6 0.8 1
−1.1
−1
−0.9
−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
w’2/u’2
z/h
phase = 0.00 phase = 0.05 phase = 0.10 phase = 0.15 phase = 0.20 phase = 0.25
De Serio−Moosa sect 47 De Serio−Moosa sect 48
Figure 7.10: Variation with depth of the ratio of horizontal to vertical turbulence intensities,w′2/u′2measured atx= -238 cm at each phase of the flow. Also included for comparisons are plunging wave results byDe Serio&Mossa[68].
De Serio&Mossa [68]. Below the trough level, the ratiow′2/u′2has peak values of up to 0.8 for phases most phases.The graph shows there is a reasonably good agreement between results from the represent experiment and those ofDe Serio & Mossa [68] for the elevations considered, particularly during the early phases of the flow.
the fluctuations as :
k′= 1
2 u′2+v′2+w′2
(7.2) where u′ ,v′ and w′ are the horizontal, (x), transverse, (y), and vertical, (z), turbulence intensity components in the indicated directions. In measurements such as the one presented here, where only two componentsu′ andw′ have been measured, and the transversal componentv′, missing, partial turbulent kinetic energy is given by :
k∗=
u′2+w′2 2
(7.3) However, if the transversal component v′ is missing, turbulent kinetic energy in the surf zone can be estimated as suggested bySvendsen [177]:
k′= 4 3
u′2+w′2 2
= 1.33
u′2+w′2 2
(7.4)
This means the transverse component is approximated (Kimmoun &Branger [43]) by :
v′2= 1
3 u′2+w′2
(7.5) A similar estimation was also used byTing & Kirby [30], Chang & Liu [41], Shin &Cox [67],Liiv &
Lagemaa [176],Govender et al.[78]. As stated bySvendsen[177], the value of 1.33 in Eqn. (7.4) is based on the assumption that breaking waves have turbulence characteristics similar to that of plane wakes, whereu′2:v′2:w′2= 0.43 : 0.26 : 0.3, and hencek′/k∗= 1.33. The coefficient does not vary significantly for different turbulent flows, ranging from 1.33 for the plane wake to 1.36, 1.40, and 1.50 for the plane mixing layer, plane jet and homogeneous isotropic turbulence, respectively (Chang &Liu [41]).
Contour plots of turbulent kinetic energy for the six wave phases are shown in Figure 7.11. These results show that the region of highest kinetic energy levels is confined above elevationsz= -5 cm as can be seen for phases 0.00, 0.05, 0.10 and 0.15 with peak values of up to 0.4 m2/s2 . This arises from the dominant contribution tok′ coming from the horizontal turbulence intensity component. Peak values ofk′observed are consistent with results using fiber optic LDV byShin &Cox [67] for regular plunging breakers over a rough rigid bottom.
To get a picture of the evolution of the turbulent kinetic energy with depth as flow progresses, the variation with depth, of turbulent kinetic energy for the six phases are plotted on the same axis as shown in Figure 7.12. Results show thatk′ is concentrated in the wave crest with very negligible energy in the rest of the wave. High turbulent kinetic energy is confined above elevationz/h= -0.5 cm, and decreases with advance of the phase of the flow. After the passage of the crest,k’ values gradually decrease with each phase. Turbulent kinetic energy below elevationz/h= -0.5 cm is observed to increase with phase.
In agreement with the results ofTing &Kirby [30], it is observed that higherk′ values lie on the front face of the wave, showing a peak that decays rapidly after the wave crest passes. A similar distribution of turbulent kinetic energy with depth was observed byYoon &Cox [186]. Misra et al.[72] observed that
−255 −250 −245 −240 −235 −230 −225 −220
−10
−5 0 5 10
Distance from still water mark (cm)
Elevation, z, relative to SWL (cm)
t/T = 0.00
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−10
−5 0 5 10
Distance from still water mark (cm)
Elevation, z, relative to SWL (cm)
t/T = 0.05
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−10
−5 0 5 10
Distance from still water mark (cm)
Elevation, z, relative to SWL (cm)
t/T = 0.10
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−10
−5 0 5 10
Distance from still water mark (cm)
Elevation, z, relative to SWL (cm)
t/T = 0.15
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−10
−5 0 5 10
Distance from still water mark (cm)
Elevation, z, relative to SWL (cm)
t/T = 0.20
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−10
−5 0 5 10
Distance from still water mark (cm)
Elevation, z, relative to SWL (cm)
t/T = 0.25
(m2/s2)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Figure 7.11: Contour plots showing the evolution of turbulent kinetic energy,k′, for the six phases under consideration.
the distribution of both horizontal and vertical turbulence intensities are qualitatively very similar to the turbulent kinetic energy distribution and decay monotonically away from the mean surface. Ting [187]
advocated the notion ofk′in breaking waves being transported to the bottom by convection and turbulent
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
−1
−0.5 0 0.5 1
z/h
Turbulent Kinetic Energy, (m2/s2)
bed position
phase = 0.00 phase = 0.05 phase = 0.10 phase = 0.15 phase = 0.20 phase = 0.25
Figure 7.12: Evolution of the vertical profile of turbulent kinetic energy,k′, as a function of depth, measured atx= -238 cm as flow progresses.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.05 0.1 0.15 0.2 0.25
Phase t/T
Turbulent kinetic energy, (m2 /s2 )
z = −8.0 cm z = −3.0 cm z = +2.0 cm
Figure 7.13: Phase variation of the turbulent kinetic energy at elevationsz= -8 cm , -3 cm and + 2cm as flow progresses.
diffusion, and transported upwards by the undertow.
Figure 7.13 shows the phase variation of turbulent kinetic energy measured at different elevations for each of the twenty phases in the flow. During the early phases of the flow (0.00 - 0.30), turbulent kinetic energy is observed to increase with elevation from the flume bed. At all elevations presented in the figure, significant amounts of turbulent kinetic energy are observed at phases t/T = 0.05 - 0.30. Very little amounts of turbulent kinetic energy are observed to remain as flow progresses (phasest/T = 0.35 - 0.95). At later phases, 0.35 - 0.55, turbulent kinetic energy at elevationz = +2 cm is observed to be the least of the three. This suggests that turbulent kinetic energy in the flow, initially observed to be dominant near the free surface, diffuses to the flume bed when the crest has passed, and eventually gets dissipated.