## Digital, statistical and wavelet study of turbulence flow structure in laboratory plunging water waves.

##### Abstract

This dissertation presents an experimental investigation set out to study the evolution of turbulence in
laboratory generated breaking water waves. The waves propagate and break as plunging waves on a 1:20
sloping beach. The aim of the study was to investigate the spatial and temporal evolution of the velocity
field and subsequent turbulence structures induced by these plunging waves as they propagate along the
flume. Experimental parameters measured included free surface elevations, wave height, mean water
levels, wave phase velocity and synoptic measurements of fluid velocities for five accretive runs in the surf
zone. Experimental data were analyzed using digital, statistical as well as wavelet approaches, in order to
derive other turbulence quantities associated with the flow.
The first set of experiments, performed in the vicinity of the break point, involved measuring the external
flow characteristics of the breaker. This was done to get prior information about the breaking behaviour
of the wave in terms of surface elevation, wave heights and wave velocities. 0.4 Hz monochromatic waves
were generated in a glass-walled flume. A set of capacitive wave gauges were calibrated and then employed
to record time series measurements of free surface elevations. Mean water levels, wave heights and phase
velocities were then determined from the water level time series. Results show that as the wave propagates
from deep water towards shallow water, there is an increase in the wave height, reaching a maximum height
at the break point, and then decreases sharply thereafter. Cross correlation techniques were then used to
determine the phase difference between the wave near the generator and the wave at various points along
the flume. The local wave velocity was obtained by taking the phase difference between two points spaced
0.2 m apart. A comparison of the measured wave phase velocity, with linear shallow water and modified
linear wave velocity approximations, is made at various points along the flume.
The second set of experiments involved measuring internal flow parameters of the fluid. The experimental
setup for this included a progressive scan digital camera (connected to a frame grabber inserted inside
a computer) that was used to capture images of the breaking wave. The flow was seeded with neutrally
buoyant, white polystyrene beads. The wave cycle was imaged using twenty fields of view or phases and
100 image pairs of the flow captured at each phase. A pair of strobe lights was used to illuminate the
flow when capturing a pair of images which are spaced a few milliseconds apart. Thus, image pairs of the
particle image field are captured with a set time interval. A video technique called digital correlation image
velocimetry (DCIV) was used to analyze the images. With this method, image pairs were cross-correlated
to determine particle displacements and thereby the instantaneous particle velocities. 100 instantaneous
velocity flow fields spanning the entire water column including the aerated region were obtained at each
phase. This enabled the quantification of the temporal and spatial evolution of the various turbulence
parameters associated with the flow. Measurements were taken at five stations across the entire surf
zone. Two-dimensional velocity flow fields are presented for phases where turbulence was observed to be
predominant. The instantaneous velocities measured are up to two times the wave phase speed.
The instantaneous velocity fields were then processed using phase-ensemble averaging to estimate the
phase-averaged horizontal and vertical velocity fields and their corresponding fluctuating parts. Mean
flow fields obtained by averaging 100 instantaneous flow fields at each phase, show the evolution of a
shear layer between the nearly stagnant underlying fluid and the fast moving crest flow. Evolution of the
stream-wise and along-shore profiles of the mean and turbulent quantities such as turbulence intensity,
turbulent kinetic energy and vorticity of the flow are also presented. For this breaker, peak phase-ensemble
averaged horizontal velocities were observed to be of the order of 250 cm/s while the vertical was about
50 cm/s. Measurements of the forward and reverse mass fluxes indicate a mean relative density for this
plunging breaker to be around 0.44.
Further statistical analysis yielded time-averaged mean horizontal and vertical velocities, root-mean-square
(r.m.s) fluctuating velocities, turbulent kinetic energy and vorticity. Evolution results of these flow pa-
rameters are also presented in the form of contour plots. Vertical and cross-variation of these turbulence
characteristics are also presented for each of the chosen wave phases. Results show that most turbulence
parameters appear to rise steadily from the trough, then rapidly, reaching peak values just above the still
water line. They also show that a relatively large turbulent motion is initially organized in the crest of
the breaking wave region. As the wave crest passes, this turbulent structure will then stretch downward to
the lower interior region of the water column. Measurements of the time-averaged turbulence intensities
and kinetic energy reveal that vertical profiles of these parameters increase from the flume bed up to a
normalized elevation, z/h = 1.0. Thereafter, these parameters begin to decrease towards the crest. Peak
turbulence parameters were observed near the front part of the wave crest with peak values of 0.11 and
0.06 for normalized horizontal and normalized vertical turbulence intensities, respectively. Froude-scaled
turbulent kinetic energy was observed to increase almost linearly from the flume bed up to elevation z/h =
1.0. Both normalized, time-averaged turbulence intensity components and the Froude-scaled time-averaged
turbulent kinetic energy results show nearly exponential decay towards the shore.
Colour contour plots were used to visualize the evolution of vorticity as flow progressed. Instantaneous
vorticity fields were observed to be characterized by patches of counter-rotating eddies. These pairs are
generated at the free surface and translated as units in the direction of the flow. Eventually these high
vorticity patches are observed to diffuse to the bottom of the flume reaching the flume bed after the crest has
passed. While the phase speed is of the order of 150 cm/s, vortex structures were estimated to propagate
with a speed of about 6 cm/s. Vorticity of the mean flow revealed a large vortex of peak value of around
150 s−1 developing around the shear boundary layer. The phenomenon of vortex shedding was observed
in the evolution of mean flow vorticity where the tail of the initially strong shear layer vortex disintegrates
as flow progresses.
Finally, a continuous, one-dimensional, complex Morlet wavelet transform was applied to synthesized test
signals comprising of three sinusoids of different frequencies. This was done to determine the relationship
between the wavelet spatial scales and the period of the signal, from which a one-to-one relationship was
obtained. The amount of energy available at each scale of the synthetic signals was then obtained and
compared against the expected root-mean-square energy. This was done to calibrate the wavelet algorithm
before it was used to extract the turbulent energy of the wave available in the velocity fluctuations. The
wavelet energy was compared against the statistically calculated turbulent kinetic energy, and showed very
good agreement at each phase, and at each elevation. The wavelet scales in the velocity fluctuations were
then subdivided into three bands that are herein referred to as the micro, mid and macro scales. The
wavelet energy spectrum of the flow shows that for the early phases of the flow, which correspond to the
approach of the crest, up to 80 % of the wave energy is confined in the micro scales. For the remainder
of the flow, micro scales contribute almost uniformly, about half of the maximum (40 %) shedding off
excess energy which appears in the macro-scales. Mid scales were observed to contain 15 % of the total
energy, almost uniformly throughout the wave cycle. Further energy analysis was performed to examine
the variation of the total wavelet energy with scale at different phases of the flow. Results indicate that
for phases corresponding to the approach of the crest, most of the energy lies in the micro-scales between
5 cm and 15 cm. The average wavelet energy at each phase of the flow was computed over several cycles
at each elevation. Results show that for the three elevations considered, the micro-scales contain most
of the energy while mid-scales have the least. The available energy of the wave decreases towards the
shore.