NPGNonlinear Processes in GeophysicsNPGNonlin. Processes Geophys.1607-7946Copernicus GmbHGöttingen, Germany10.5194/npg-22-737-2015Steep unidirectional wave groups – fully nonlinear simulations vs. experimentsShemerL.shemer@eng.tau.ac.ilhttps://orcid.org/0000-0003-0158-1823EeB. K.School of Mechanical Engineering, Tel Aviv University, Tel Aviv 69978, IsraelL. Shemer (shemer@eng.tau.ac.il)4December20152267377471July201524July201518November201526November2015This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://npg.copernicus.org/articles/22/737/2015/npg-22-737-2015.htmlThe full text article is available as a PDF file from https://npg.copernicus.org/articles/22/737/2015/npg-22-737-2015.pdf
A detailed quantitative comparison of fully nonlinear computations with the
measurements of unidirectional wave groups is presented. Computational
results on evolving wave groups were compared with previous available
experiments. The local surface elevation variation, the
evolution of envelope shapes, the velocity of propagation of the steepest
crests in the group and their relation to the height of the crests were
obtained numerically and experimentally. Conditions corresponding to
incipient wave breaking were investigated in greater detail. The results shed
additional light on mechanisms leading to the breaking of steep waves, as
well as on the crucial importance of exact matching between initial
conditions in computations and experiments.
Introduction
Over the past few decades, rogue, or freak, waves have attracted considerable
interest due to their destructive impact on offshore structures and ships
. A number of possible mechanisms for rogue
wave generation have been explored. Wave–current and wave–bathymetry
interactions may result in appearance of rogue waves .
Extremely steep waves in the ocean are thus usually affected by the
directional characteristics of the wave field. Nevertheless, considerable
effort has been invested in recent decades to study unidirectional wave
fields. The accumulated results clearly demonstrate that investigation of
both deterministic and random unidirectional waves can lead to a better
understanding of mechanisms leading to appearance of rogue waves in the
presence of directional spreading as well. Experimental studies of 2-D wave
fields in wave basins require large and expensive facilities and are subject
to numerous limitations on the wave parameters. Generation of unidirectional
wave groups in long tanks by a computer-controlled wavemaker offers
significant advantages in terms of availability and versatility of
operational conditions. Extremely steep waves can be generated due to
constructive interference of numerous harmonics. While this focusing
mechanism is basically linear, it is strongly affected by nonlinearity
. An alternative, essentially nonlinear, mechanism is
related to the specific properties of the governing equations. To that end,
the nonlinear Schrödinger (NLS) equation ,
applicable for description of diverse nonlinear physical phenomena, is the
simplest theoretical model describing the evolution of narrow-banded wave
groups in deep and intermediate depth water. This equation attracts special
interest in rogue waves studies since it admits analytical solutions such as
the so-called Kuznetsov–Ma breathing solitons . These
solutions of the NLS equation present spatially localized patterns that
oscillate in time. The closely related Akhmediev breather
is periodic in space. When the periodicity in time and space tends to
infinity, both these types of solution tend to a simple Peregrine breather
(PB) . It is localized in time and space, breathes only
once and attains a maximum crest height that exceeds that of the background
wave train by a factor of 3. For this reason, the Peregrine and other
breather-type solutions of the NLS equation have been proposed as rogue wave
prototypes .
conducted a series of experiments on the evolution of the
Peregrine breather (PB) along a wave tank. They demonstrated that the
experimental results diverge from Peregrine's solution of the NLS equation.
Notable asymmetry of the crest was observed, in agreement with many earlier
studies of extremely steep waves . The discrepancy between the
fully symmetric NLS solution and experiments manifests itself mainly in
significant asymmetric widening of the wave spectrum, as well as in notably
slower amplification of the wave height than predicted by the PB solution.
Moreover, these experiments suggested that, contrary to the behavior of the
PB, there would be no return to the initial nearly monochromatic wave train.
Similar conclusions based on fully nonlinear simulations of PB evolution in
time were reached by .
demonstrated that the modified nonlinear Schrödinger (MNLS, or Dysthe)
() equation was advantageous in describing the PB evolution
along the laboratory tank as compared to the NLS equation. The improved
performance of the Dysthe model was attributed in to the
additional fourth-order terms in this equation that account for the finite
spectral width . It was thus demonstrated that the
statement made by and elsewhere that PB has been
observed in water wave experiments is essentially unsubstantiated.
demonstrated that the spatial evolution patten of PB
does not differ from that of any initially narrow-banded modulated wave
train.
took advantage of the fact that the spectral widening,
being an essentially nonlinear process, occurs at slow spatial and temporal
scales. Hence, it was found that the wave train behavior with background
steepness of about 0.1 is still described by the PB solution of the NLS
equation with reasonable accuracy, as long as the surface elevation spectrum
remains sufficiently narrow and the maximum wave height in the train remained
below approximately twice that of the background. This observation enabled
to utilize the available PB analytic solution to design
experiments with PB in which the height of the steepest wave in the train at
a prescribed measuring location can be controlled, thus facilitating
quantitative studies of the incipient wave breaking. Their study was
motivated by an earlier attempt by to examine the kinematics
of the steep wave on the verge of breaking using the Zakharov equation
. By comparing the computational results with experimental
observations reported in , the conclusion was reached that wave
breaking may occur when the horizontal liquid velocity at the crest becomes
sufficiently high . These computations also showed that the
maximum negative vertical Lagrangian acceleration seems to remain
significantly below the acceleration of gravity g, so that the Phillips
dynamic breaking criterion cannot be satisfied .
Computations of steep wave kinematics accurate up to the third order in the
wave steepness demonstrated, though, that this approximation, while largely
adequate for determination of the shape of the surface elevation, was
insufficient for the accurate characterization of the kinematics of steep
waves . In order to overcome this limitation, the kinematic
parameters of the steepest wave in the PB-like wave train were determined
experimentally in simultaneously with estimates of the
propagation velocity of the steepest crest. To this end, two synchronized
video cameras were used to image the wave field. The maximum possible
horizontal Lagrangian velocities and accelerations at the surface of steep
water waves were measured by particle tracking velocimetry (PTV) for
gradually increasing crest heights, up to the inception of a spilling
breaker. Actual crest and phase velocities were estimated from video-recorded
sequences of the instantaneous wave shape as well as from surface elevation
measurements by wave gauges. The slowdown of the crest as it grows steeper
was observed. It was suggested in that the inception of a
spilling breaker is associated with the horizontal velocity of water
particles at the crest attaining that of the crest, thus confirming the
kinematic criterion for the inception of breaking.
In the present study, we aim to extend the numerical analysis of the
conditions prevailing at the inception of breaking of the steepest crest in
the PB-like wave train by carrying out fully nonlinear simulations. The
simulations were performed using the conformal mapping method approach
developed by (hereafter referred to as the CS numerical
scheme). A somewhat different implementation of this approach was suggested
by . Recently, reported on direct numerical
simulations based on the volume of fluid method to solve the two-phase
Navier–Stokes equations. In their study, Peregrine breather dynamics was
investigated up to the initial stages of wave breaking.
In Sect. , the difference between the spatial and temporal
evolution of the wave field is discussed. In Sect. , we give
details about the solver that is based on the conformal mapping method and
stress that the resulting numerical solution describes the temporal evolution
starting from an initial spatial distribution. In Sect. , the
computational results are discussed for both the temporal evolution problem
and then for the spatial evolution case. The corresponding experimental
results are also presented and compared directly with the numerical
simulations. In Sect. , the numerical and experimental results are
discussed and the conclusions are drawn.
Wave parameters: temporal vs. spatial evolution cases
The CS numerical scheme enables computation of a unidirectional wave field in
time, starting from a given spatial distribution as an initial condition,
under the assumption of potential flow. In laboratory experiments, however,
waves are generated by a wavemaker usually placed at one end of the
experimental facility. The experimental data are commonly accumulated only at
preselected fixed locations within the tank where the sensors are placed.
Quantitative comparison of numerical computations with experimental results
gained in those experiments may thus constitute a complicated task. For a
narrow-banded wave field this problem has been considered by
. For such wave fields, the envelope equations like the NLS
and Dysthe models often provide adequate results. In the narrow-banded models
the temporal t and spatial x coordinates are related by the group
propagation velocity, cg, thus enabling modification of the
governing temporal evolution equations to a spatial form. The spatial form of
the Dysthe model was presented by . Numerical computations
based on the Dysthe model for unidirectional wave groups propagating in a
long wave tank indeed provided good agreement with experiments
. The spatial version of the Dysthe equation was also
derived from the spatial form of the Zakharov equation
that is free of any restrictions on the spectrum
width .
As demonstrated in , the availability of the spatial form
of the evolution model is insufficient to pose the initial conditions for the
numerical simulations that correspond exactly to those in experiments. In the
present work, the temporal evolution is computed by a fully nonlinear solver
of the 2-D potential equations in finite water depth. Following earlier works
, the solution of the spatial version of the
NLS equation is used to set the initial conditions. For a narrow-banded deep
water wave group, the spatial and temporal variations of the surface
elevation z at the leading order can be presented as
ζ(x,t)=Rea(x,t)⋅ei(k0x-ω0t).
Here the radian frequency ω0=2π/T0,T0 being the carrier wave
period, and the wavenumber k0=2π/λ0, λ0 being the
carrier wavelength, satisfy the finite water depth dispersion relation
ω02=gk0tanh(k0h). In Eq. (), a is the slowly
varying complex group envelope. The wave steepness is defined as ϵ=a0k0, where a0 is the characteristic wave amplitude. The wave train
given by Eq. () propagates with the group velocity
cg=∂ω∂kk=k0.
Following and , in intermediate water
depth the spatial NLS equation for the complex normalized envelope
Q=a(x,t)/a0 is given by
-iQX+αQTT+β|Q|2Q=0,
where the scaled dimensionless temporal and spatial coordinates are T=ϵω0(x/cg-t) and X=ϵ2k0x,
respectively. The coefficients in the NLS equation have the following
dimensionless form:
α=-ω022k0cg3∂cg∂k,β=1ncosh(4k0h)+8-2tanh2(k0h)16sinh4(k0h)-12sinh2(2k0h)(2cosh2(k0h)+n2)k0htanh(k0h)-n2,
where the parameter n=cg/cp represents the ratio of
group and phase velocities and is given by
n=121+2k0hsinh(2k0h).
For the deep water case, k0h→∞ and α=β=1. The Peregrine breather solution of the NLS equation for intermediate
water depth (Eq. ) is
Q=-2αβe-2iαX1-4(1-4iαX)1+4T2+16(αX)2.
Equations () and () provide variation of the
instantaneous surface elevation envelope in time and in space, with focusing
corresponding to T=X=0. In and , the
wavemaker driving signal as a function of time was chosen using the
deep-water version of Eq. () and the prescribed focusing distance
from the wavemaker, x0, by substituting X=X0=-ϵ2k0x0 into
Eq. (). The relative height of the initial hump in the wave
amplitude distribution at the wavemaker in exceeded
10 % above the background.
The present study has been carried out in the 18 m long, 1.2 m wide and 0.9 m deep wave
tank (water depth h=0.6m). More details about the experimental
facility are given in . The carrier wave period
T0=0.8s was selected, corresponding to the carrier wavelength
λ0=1.0m and the dimensionless water depth k0h=3.77. For
these parameters, both coefficients in the NLS equation (Eq. )
given by Eqs. () and () in fact differ from unity:
α=1.078 and β=0.711. The carrier wave amplitude of
ζ0=2αβa0=0.026m was used,
corresponding to the nonlinearity ϵ=k0a0=0.094.
Since the nonlinear numerical solver in the present study requires an initial
condition as a given spatial distribution at a certain instant, the following
procedure to determine the appropriate initial spatial distribution was
adopted.
Using Eq. (), a value of x0 is specified at which the prescribed
maximum crest of the PB is to be located. Note that in physical terms, this
initial condition corresponds to a situation in which the whole wave train is
placed upstream of the wavemaker. Due to the focusing properties of the NLS
equation in sufficiently deep water (k0h>1.36), the maximum wave height
in the train increases in the course of the evolution. In
, the amplification at the wavemaker was about
20 %. To obtain a similar evolution pattern in the present study,
the initial height of the PB hump has to be significantly smaller than that.
The amplification corresponding to 5 % was selected. The temporal
variation of the surface elevation at x=x0, ζ(x0,t), can be
calculated using T(x0,t) and X(x0). The maximum amplification occurs at
T=0 corresponding to the dimensional time
tmax=-x0/cg; the resulting ζ(t) is
symmetric with respect to tmax. Note that at the instant
t=tmax, the spatial distribution ζ(x,tmax) is not
fully symmetric with respect to x0 due to the presence of term
e-2iαX in Eq. (). It should be stressed that in
experiments as well as in numerical simulations, the actual extent of the
wave group is necessarily finite. In the experiments of
and , the wave train with a duration of 70 carrier wave
periods was generated by the wavemaker. The duration of the wave train of
70T0 is long enough to eliminate the effect of truncation onto the central
part of the train where the hump is located and which is of particular
interest, and on the other hand sufficiently short to prevent contamination
of the measured surface elevation by possible reflection from the far end of
the tank. In the spatial domain, this duration of the wave train corresponds
to 35λ0. The numerical method applied in the present study assumes
spatially periodic boundary conditions. Since the initial spatial
distribution is not periodic, the periodicity was enforced by applying a
linear tapering window over wavelength 2λ0 at the edges of the wave
train. As a result, the effective, undisturbed by tapering, wave group
extended initially for about 32λ0. To allow evolution of the wave
train unaffected by boundaries, the computational domain was selected as
[-32λ0,32λ0], with x=0 corresponding to the location of
the maximum crest at the initial instant of the computations, t=0. The
temporal initial condition adopted in the study is plotted in the top panel
of Fig. . The corresponding spatial variation of
the surface elevation with the same maximum crest height is plotted in the
bottom panel of the same figure.
Numerical solution methodology
The temporal evolution of the initial wave field presented in the bottom
panel of Fig. is obtained by solving potential
flow equations following the fully nonlinear numerical approach developed by
. The CS numerical scheme is known to be robust and does not
have limitations in terms of wave steepness. It has been extensively and
successfully used for numerical simulations of numerous problems related to
evolution of nonlinear waves. The conformal mapping method is applied to
solve Laplace's equation for the velocity potential. Surface tension effects
are neglected. The principal equations are re-written in a surface-following
coordinate system (ξ and ζ) and reduced to two time-evolutionary
equations for the surface elevation z and velocity potential
ϕs at the surface. The evolutionary equations representing
the kinematic and dynamic boundary conditions at the free surface are written
in terms of the Fourier coefficients of z and ϕs. This
enables the reduction of the evolutionary equations into a system of
time-dependent ordinary differential equations for 4M+2 Fourier
coefficients of z and ϕs, coupled with appropriate initial
conditions. Here M refers to the truncation number of the Fourier series.
The initial surface elevation is given as a function of the physical variable
x. To solve the problem in the mapped space, this initial condition has to
be converted into a function of equally spaced ξ. This is done by an
iteration procedure.
The prescribed initial variation of the surface elevation η in
the temporal (top) and spatial (bottom) domains for the carrier wave period
T0=0.8s and background carrier amplitude
ζ0=0.026m; calculated for X=-2.613 (x0=31m) in Eq. ().
For time integration, a fourth-order Runge–Kutta scheme was used. We refer
the reader to for further details. In the present computations,
the dimensional spatial discretization interval was λ0/ 256, so
that the total number of spatial points N=17 920; the truncation number
for the Fourier series is M=N/ 9. This value of M effectively means
that waves with wavelengths shorter than 1.5cm where capillary
effects become dominant are disregarded. The dimensional integration step in
time is dt=3.125×10-6 s.
The CS numerical scheme allows for the computation of the velocity potential
as a function of two parameters: ξ and ζ; the velocity potential
for the entire domain can thus be calculated at any instant. In view of the
focus of the present study, the output parameters of the numerical
integration are the surface elevation z, velocity potential at the surface
ϕs and the physical spatial coordinate x, which are all
functions of ξ and t. In order to record the data for future use, the
results for surface elevation, the coordinates x and non-dimensional
velocity potential are saved at every 3.125 ms. Note also that the
spline interpolation procedure is needed to obtain values of the surface
elevation z and the velocity potential ϕs at equally
spaced values of x.
have also employed the conformal mapping method to investigate
the unsteady evolution of 2-D fully nonlinear free surface gravity–capillary
solitary waves for infinite depth. Though their numerical approach is similar
to that of CS, certain differences between the methods exist. The numerical
approach of was implemented in our computations as well. No
significant differences with the results based on the CS numerical scheme
were obtained, thus further demonstrating the robustness of the present
results.
Numerical and experimental results
In Fig. , the spatial instantaneous wave surface
profile is plotted for several characteristic selected instants. As mentioned
above, the origin of the frame of references x=0 corresponds to the
location of the maximum crest in the initial spatial distribution. The
simulations demonstrate that abnormally high waves appear at both edges of
the wave train as a result of truncation and tapering of the infinite wave
train defined by Eqs. () and () as specified in the
previous section. A similar phenomenon was observed in experiments with
truncated wave trains reported in earlier works
. The effect of truncation, however,
apparently does not extend to the central part of the wave train even at
relatively long times, as can be seen from the upper curves in this figure.
The effect of nonlinear focusing on the behavior of this central part of the
train in the vicinity of the hump is of principal interest in this study. The
dashed lines in Fig. originate at the leading
edge, the center and the trailing edge of the initial wave train and
correspond to the location of the point propagating with the group velocity
cg=0.63ms-1. It transpires from the figure that
the leading edge of the train indeed propagates with the speed that is close
to cg, while the trailing edge seems to move somewhat faster.
The propagation velocity of the steepest crest, however, exceeds notably the
group velocity cg, in agreement with the experimental
observations and the numerical simulation based on the Dysthe equation in
.
The temporal evolution of the surface elevation η in a fixed
reference frame; wave parameters as in Fig. . The
vertical line marks the location of the wavemaker at x=xwm=25.273m; broken lines correspond to propagation with the group
velocity cg.
The temporal evolution of the horizontal fluid velocity in a moving
reference frame; wave parameters as in Fig. .
The spatial variation of the velocity of the fluid at the surface
uh=∂ϕs/∂x is plotted in
Fig. at the same instants as in
Fig. , but in a frame of reference that moves with
a group velocity cg. Only the central part of the wave train is
shown. The horizontal fluid velocity at the steepest crest increases notably
during the focusing process. At the upper curve in
Fig. , the fast increase in the horizontal velocity at
the crest is clearly seen. Note that in earlier experiments by
, wave breaking was indeed observed at comparable
distances from the wavemaker. The individual waves in
Fig. manifest variable left–right asymmetry.
Measurements in a wave tank are routinely performed using wave gauges spread
along the facility. To facilitate the direct comparison between numerical and
experimental results, we need to first determine the location of the
wavemaker in our numerical simulations. Then, we examine “vertical” cross
sections of the data as presented in Fig. at fixed
locations relative to the adopted coordinate of the wavemaker. The location
of the wavemaker is identified by the vertical line in the latter figure
where x=xwm=25.273m. The temporal variation of the
surface elevation is plotted in Fig. at selected
locations that cover the range of the wave gauge positions in the experiment,
starting with that at xwm or x*=0, the variable x*
denoting the distance from the wavemaker. The disturbances at the leading
edge of the truncated wave group are disregarded here. The growth of the
maximum crest height with the distance is obvious, albeit non-monotonic. In
the uppermost curve in Fig. the relative crest
amplification exceeds the factor of 2, as in at a
comparable distance from the wavemaker. Here again, the broken line that
corresponds to the propagation velocity of cg clearly shows
that the steepest crests in the train propagate at velocities exceeding
cg.
A closer look at the surface elevation variation with time is presented in
Fig. ; the time here is shifted at each position by
delay that would occur if the hump in the group indeed propagated with the
group velocity. Actually, the maximum crest is invariably observed earlier.
Both the vertical (trough–crest) and horizontal (right–left) asymmetries of
steep waves are clearly visible in the plotted records.
In Fig. , we follow the highest crests in the
instantaneous snapshots of wave trains. The elevations of the highest crests
at each instant are compared with the propagation velocity of those crests,
vcr. To enable comparison of parameters with different
dimensions, crest heights are normalized by the background wave amplitude
ζ0, while crest propagation velocities are normalized by the carrier
wave phase velocity cp. Note that this figure corresponds to
evolution times at which the hump's amplification is still relatively modest.
Nevertheless, the crests propagate with time-dependent velocities
vcr that may be notably different from cp. It was
demonstrated in experiments of that even for waves in the
train that are far away from the hump and thus seem to be essentially
monochromatic, the mean crest propagation velocity is somewhat higher than
cp due to the presence of the exponential term in
Eq. ().
Surface elevation variation at fixed values of x* vs. time t.
The group velocity is cg=0.63ms-1; x*
refers to the wave gauge distance from the wavemaker at x*=0.
The computed temporal variation of the surface elevation at various
locations relative to the wavemaker in a moving (with cg) reference frame.
The computed variation with time of the normalized maximum crest
elevation relative to the background wave amplitude ζ0 and the
velocity of the highest crest propagation (vcr) relative to the
phase velocity cp=1.248ms-1.
It transpires from the comparison of the two curves in
Fig. that the higher the crests are, the lower
their propagation velocity is. The
minima in the instantaneous maximum crest heights correspond to the local
maxima in their instantaneous propagation velocities. The averaged highest
crest propagation velocity in Fig. is
1.253ms-1, slightly above cp.
The experiments were carried out with the goal of enabling quantitative
comparison of the numerical results with experiments. The wavemaker driving
signal was designed to generate surface elevation variation in time
corresponding to the lowest curve in Fig. .
Measurements were performed by multiple (up to four) resistance-type wave
gauges placed on a bar in the center of the tank and connected to a
computer-controlled carriage. The spacing between adjacent gauges was
0.4 m. At each run, the position of the carriage was set by computer.
Each successive run was initiated only after any disturbance of the water
surface from the previous run had fully decayed. Measurements performed in
different runs at fixed locations demonstrated excellent repeatability of
results. Thus, the data collected at different locations obtained in various
runs could be compared using the initiation of the wavemaker driving signal
as a common temporal reference. Multiple experimental runs with different
carriage positions provided experimental records of the temporal variation of
the surface elevation in the wave train propagating along the whole tank with
spacing that did not exceed 0.2 m; denser measurements were carried
out in the vicinity of the locations where inception of breaking was detected
in visual observations.
An example of the sequence of the experimentally recorded wave trains for
6.6≤x≤7.8m is presented in Fig. .
Modifications of the wave train shape between the adjacent locations are
relatively minor. The variation along the tank of the location and height of
the steepest crest in the central part of the train can be easily followed
from these records. Note that the highest crest at x=7.6m ceases
to be such at x=7.8m, where the following wave in the train
becomes the steepest one. Such a transition of the highest crest in the train
from one wave to another causes discontinuity in the velocity of propagation
of the steepest crest; see e.g. .
It is impractical to carry out direct comparison of the fast varying surface
elevation records measured in the experiments as presented in
Fig. , with the corresponding numerical results. In
order to compare the computed and measured results, the corresponding
envelopes were computed; the absolute values of the measured and simulated
wave train envelopes are presented in Fig. for
various distances from the wavemaker. To calculate the envelopes of the wave
train in both simulations and experiments, the records were first band-pass
filtered in the domain 0.5ω0≤ω≤1.5ω0. This
procedure leaves only the “free” waves, while the higher-order “bound”
waves that cause vertical asymmetry of the records are removed. The envelopes
of the filtered signals were then computed using the Hilbert transform. For
more details, see e.g. .
Measured surface elevation η at various locations.
Comparison between the envelopes of the measured and numerical
surface elevations η at various locations.
Figure demonstrates that essential similarity
exists between the shapes of the measured wave trains at different distances
from the wavemaker and those obtained in the numerical simulations. The
propagation velocities of the leading edge of the wave train, as well as of
the steepest crest, are also quite close in simulations and in experiments.
The agreement between the numerical solution and the experimental results is,
however, not perfect; the differences cannot be attributed to experimental
errors only.
The computed temporal variation of the maximum crest heights, of the
velocities of the steepest crests (vcr), and of maximum
horizontal fluid velocity (uhmax).
Variation along the tank of the maximum crest heights, propagation
velocity of the highest crests, and the water particle velocity at those
crests. Red lines denote numerical results, blue symbols experiments. The
locations where breaking was observed in the experiments are marked.
Important parameters of the wave train in the course of its propagation along
the tank obtained in the simulations are plotted in
Figs. and . In
Fig. , the temporal variations of the computed
velocities of the highest crests at each instant, vcr, and of the
fluid velocity at those crests, uhmax, are presented at
late stages of the evolution, up to the apparent breakdown of computations at
t/T0≈74. However, the maximum crest height amplification
exceeding 3 was observed at t/T0≈62. The maximum crest
elevations are also plotted in this figure for comparison. To enable
comparison, all data are rendered dimensionless by normalizing them by their
appropriate characteristic values. The fluid velocities increase with crest
heights, while the crest propagation velocities decrease. At final stages the
fluid velocity at the crest seems to exceed the crest velocity. The
corresponding spatial variations are plotted in Fig. .
In this figure, whenever available, the related experimental results are
plotted as well.
The evolution of the steepest crest heights along the tank, as plotted in
Fig. , in simulations and in experiments exhibit
qualitative and to some extent quantitative similarity. The steepest crest
heights have a tendency to grow along the tank; this growth is essentially
non-monotonic in computations as well as in measurements. At distances from
the wavemaker beyond 7 m the measured steepest crest heights may
exceed the background by a factor of 2.5; the amplification factor in
simulations is somewhat higher than that. The propagation velocity of the
steepest crest, vcr, varies to a certain extent in experiments
and in computations, remaining close to the phase velocity of the carrier
wave, cp. The discontinuity in the steepest crest propagation
velocity obtained in the experiments is related to the transition of the
steepest crest in the train from one wave to another, as discussed with
relation to Fig. . The spatial resolution of the
determination of vcr is obviously much better in the numerical
simulations than in the experiments. For that reason, oscillations of the
measured steepest crest velocity are less pronounced in the experimental
results. As discussed with respect to the temporal variation of
ccr in Fig. , the crest propagation
velocity decreases when crests become higher. This feature is more visible in
the results of simulations as compared to the measurements due to their
better resolution.
The bottom curve in Fig. represents the variation
along the tank of the instantaneous water particle velocity at the steepest
crest, computed as
uhmax=∂ϕs/∂x at
the crest. This velocity varies in accordance with the variation of the crest
height; as the crest becomes higher, the values of
uhmax grow and may exceed notably the group velocity
cg. Nevertheless, for the whole domain of computations, the
horizontal liquid velocity at the crest remains lower than the computed
vcr. Note that the computed temporal variations of
vcr and uhmax plotted in
Fig. demonstrate that the values of
vcr may decrease below the local maximum of
uhmax; however, this does not occur simultaneously. No
measurements of uhmax were carried out in this study;
however, detailed results on the Lagrangian kinematics at the wave crest
approaching breaking obtained using particle tracking velocimetry were
presented for the identical carrier wave parameters and somewhat different
wavemaker driving signal in .
At distances exceeding about 7 m from the wavemaker, the pattern of
variation of the steepest crests' height and of their propagation velocity
vcr plotted in Fig. becomes less
organized. In experiments, inception of the spilling breaker was observed at
those distances; see the video in the Supplement. In order to obtain more
accurate estimates of vcr in this region, measurements of the
surface elevation were performed every 0.1 m. The resulting steepest
crest propagation velocities are plotted in Fig. using
different symbols. These results demonstrate that at the locations where the
spilling breakers were observed, the measured vcr may indeed fall
below the computed water surface velocity at the crest,
uhmax.
It was suggested in that spilling breakers appear when
the horizontal water particle velocity at the steep crest
uhmax attains the instantaneous crest propagation
velocity vcr. While spilling breakers were clearly seen in the
experiments at distances of about 7.5–8 and 8.5–9 m from the
wavemaker (see the video in the Supplement), in computations the values of
uhmax, while increasing at steep crests, remain
consistently lower by about 10 % than the computed vcr,
although extremely low steepest crest propagation velocities were
occasionally obtained numerically; see Fig. . The
experimentally determined values of crest propagation velocity may indeed
fall below the computed water particle velocity uhmax.
In this respect it should be stressed that the velocities
uhmax and vcr obtained in the present fully
nonlinear numerical simulations, while apparently close to their actual
values as demonstrated in Fig. , cannot be seen as the
exact ones. Although special effort has been made in this study to calculate
the required conditions at the wavemaker that correspond to the numerical
solution, there remains a certain discrepancy between the computed and
measured initial conditions due to an essentially nonlinear character of the
wavemaker transfer function. While the differences are small, they can affect
the exact locations of the observed extremal values of the surface elevation
and the horizontal velocity. The experimental determination of
vcr as performed in the present study is inaccurate mainly due to
the presence of unavoidable low-level noise in the surface elevation records
that limit the precision of defining the instant when the maximum surface
elevation is attained. The PTV-derived results on uhmax
presented in Fig. 7 of show that horizontal surface
velocities as high as 0.8cp, notably higher than the maximum
values of uhmax in Fig. , were
indeed measured at the breaking location. It should be noted that since the
wavemaker is an essentially nonlinear device, it is difficult to adjust the
actual surface elevation variation in the tank to that prescribed by the
computations. It thus can be concluded that the differences between
uhmax and vcr obtained numerically and
those measured in the tank as presented in Fig. stem
from less than perfect matching between the initial conditions in the
experiment and the numerical simulations. The total body of numerical and
experimental results thus provides further support for the validity of the
kinematic breaking criterion according to which the spilling breaker emerges
when the instantaneous liquid velocity at the crest,
uhmax, attains that of the crest, vcr.
Frequency spectra at selected locations. Red line – numerical
simulations; blue line – experimental data.
Wavenumber spectra at selected times.
The amplitude spectra of the wave train are plotted in
Figs. and . In
Fig. , the numerically derived frequency spectra of
η(t) are compared at selected values of x* with the corresponding
experimental results. At the wavemaker (x*=0), the spectrum in
linear–logarithmic coordinates still retains resemblance to the triangular
shape characteristic of the Peregrine breather. Nevertheless, a weak
asymmetry around the dominant frequency ω0 can already be noticed at
this location. Note that at x*=0 the wave train already evolved over
significant duration from its initial PB shape in
Fig. . The non-negligible contribution of
low-frequency as well as second and third bound wave harmonics is also
evident. The spectral asymmetry gets stronger and the spectrum widens with
the distance from the wavemaker. Reasonable agreement is obtained between the
experimental and numerical results. The wavenumber spectrum for the computed
variation of η(x) plotted in Fig. at
selected instants t cannot be compared with the experiment. This is due to
the fact that the spatial extent of the wave train exceeds significantly the
length of the tank; see Fig. . Note that even for
significantly shorter wave trains, the experimental procedure that enables
extraction of wavenumber spectra (as opposed to frequency spectra) is
extremely tedious; see . The temporal evolution of
wavenumber spectra in Fig. is qualitatively
similar to that discussed with respect to Fig. . The initial spectrum is nearly symmetric around the dominant wave number. Then, the spectrum widens with time and becomes more asymmetric towards breaking.
Discussion and conclusions
In the present study, fully nonlinear numerical simulations of the evolution
of a unidirectional nonlinear wave train with an initial shape of a Peregrine
breather were qualitatively and quantitatively compared with the experimental
results. To the best of our knowledge, this is the first attempt to carry out
direct comparison of the results of fully nonlinear simulations of a
deterministic wave train with experiments. The simulations were carried out
using a conformal mapping approach as detailed in . To validate
the accuracy of the code, the computational results were reproduced using an
alternative numerical approach of . These and some other
numerical methods that are often applied to solve wave propagation problems
require complete information on the wave field over the entire computational
range at a certain instant. These initial conditions are unavailable in any
controlled wave experiment in a laboratory facility. The unidirectional wave
field in a tank is in fact prescribed by the wavemaker that is usually
located at one end of the facility and driven by a computer-generated signal.
The initial condition in experiments thus corresponds to surface elevation
variation with time at a prescribed wavemaker location.
sought to reconcile the fundamental difference between the initial spatial
distribution of wave field parameters required for the numerical solution,
and the temporal variation of the surface elevation at the wavemaker
prescribed as the initial condition in the experiments. This approach was
generalized here to a fully nonlinear wave field with an arbitrary spectral
width, thus enabling one to carry out consistent quantitative comparison of
the results of numerical simulations and measurements.
In the present simulations, the initial spatial distribution of the surface
elevation is based on the PB analytical solution. In order to determine in
the numerical solution the measurable temporal variation of the surface
elevation at any given location along the tank, the initial spatial
distribution in the present study was centered upstream of the wavemaker; see
Fig. . The appropriate location of the wavemaker
was determined then by comparing the surface elevation variation in time with
that measured in the experiment. The wavemaker driving signal generates
surface elevation variation in time that corresponds to the bottom curve in
Fig. . This dependence that is very different from
the analytical solution given by PB is obtained as a result of the evolution
of the wave train with an initial shape given in the bottom panel of
Fig. . It enables detailed and quantitative
comparison of the simulations with experiment.
Several important points regarding PB were highlighted in this study. The
solution (Eq. ) of the spatial form of the nonlinear
Schrödinger equation (Eq. ) is aperiodic in space due to the
presence of an exponential term. Similarly, the temporal form of the NLS
equation yields PB that has an asymmetry in time.
The exact shape of the analytical solution (Eq. ) thus cannot be
reproduced either in the experiments or in computations. Note that in the
present study, the actual initial condition for the simulations and the
wavemaker driving signal have been modified and are fundamentally different
from PB. The presented results on steep crests in the wave train are
therefore of a generic nature and applicable beyond the 2-D PB wave packets.
Two different approaches were suggested to deal with the problems outlined in
the previous paragraph. In numerical simulations, the truncated wave train
with the spatial extension that contains an integer number of carrier
wavelengths is often used as the initial condition .
However, imposing a non-zero periodic boundary condition on an essentially
aperiodic function may affect significantly the nature of the solution. It
was thus decided in the present study to follow the experimental approach of
and . The theoretical solution given
by Eq. () was truncated and tapered before being used to determine
the initial condition at the wavemaker. In order to mitigate the effect of
truncation on the central part of the wave train, a sufficiently large number
of wave periods (about 70) was used in those experiments. A similar approach
was adopted in the present study. As can be seen in both experimental and
numerical results (Figs. ,
and ), truncation and
tapering, while indeed satisfying periodic boundary conditions in the
computational domain, cause appearance of abnormally high waves at the
leading and trailing edges of the wave train. The effect of truncation is
apparently limited to the edges of the train, and does not affect the
behavior of the central part of the PB-like wave train with the gradually
amplified, albeit non-monotonically, hump in the envelope. These high waves
do not characterize the wave train proper and therefore were disregarded in
the present study.
The computational results indeed are in a good qualitative and, to a large
extent, quantitative agreement with the current experiments, as well as with
those of . This includes the behavior of the truncated
train edges, the amplification of the hump along the tank, the asymmetric
spectral widening, as well as the variation of the envelope shape along the
tank. Crest slowdown was noted by Johannssen and Swan in fully nonlinear
calculations and experiments . The slowdown of
crests in the PB-like wave train as they grow in height was first observed by
in experiments and NLS solutions and is of particular
interest. More recently, this effect was also stressed in the context of
focusing of 2-D and 3-D nonlinear deep water wave packets by
, and for 2-D nonlinear wave packets by
as well as by , thus providing
additional evidence of the generic nature of the phenomenon. It was noticed
in that the increase in the maximum crest height along
the tank is not monotonic. As the maximum crest height increases, the water
particles at the crest accelerate to higher maximum velocities
uhmax, while the crest propagation speed
vcr decreases. The equality
uhmax=vcr was thus suggested as the kinematic
criterion for wave breaking. A slightly different version of this criterion
was offered by ; they maintain that the maximum
liquid particle velocity uhmax exceeds about
0.8vcr at breaking. If only the simulations are considered, it
seems that this somewhat weaker version of the kinematic breaking criterion
is confirmed. However, the present experimental as well as numerical results,
combined with those obtained experimentally by alternative methods in
, provide strong, albeit not fully conclusive, support for
the conjecture that indeed the particle velocities at the inception of
breaking attain and exceed the crest propagation velocities and thus for the
kinematic breaking criterion in the formulation suggested in that study. This
conjecture is further corroborated by visual evidence as seen in video clips
presented in Supplements to and to the present
study.
A different approach was suggested by Saket et
al., 2015, ArXiv 1508.07702, and Barthelemy et
al., 2015, ArXiv 1508.06002, after this manuscript was
submitted.
This combined numerical and experimental study of nonlinear wave trains also
clarifies the limitations of possible agreement between fully nonlinear
solution and experiment. We note that while the periodicity in the time
domain is possible for propagating and evolving unidirectional waves, they
are, strictly speaking, aperiodic in space. This point adds an additional
aspect to essential differences that exist between the spatial and temporal
formulations of the wave evolution problem, as discussed above. We therefore
believe that all nonlinear solutions based on spatially periodic boundary
conditions, as in the method adopted here, as well as in a variety of
alternative methods that employ spatial discrete Fourier decomposition,
contain intrinsic inaccuracy. These numerical solutions thus can only provide
approximate results and require careful experiments to verify their validity.
The present study shows that the fully nonlinear solution, although flawed,
yields better agreement with experiments than the application of the spatial
version of the modified nonlinear Schrödinger (Dysthe) equation limited
to the third order that does not require spatial periodicity
(.
The Supplement related to this article is available online at doi:10.5194/npg-22-737-2015-supplement.
Acknowledgements
The support of this study by a grant no. 20102019 from the US-Israel
Binational Science Foundation is gratefully acknowledged. The authors wish to
thank Andrey Zavadsky for his valuable assistance. Edited by:
V. Shrira Reviewed by: two anonymous referees
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