Evaluation of an eddy viscosity type wave breaking model for intermediate water depths

European Journal of Mechanics / B Fluids 78 (2019) 115–138

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European Journal of Mechanics / B Fluids journal homepage: www.elsevier.com/locate/ejmflu

Evaluation of an eddy viscosity type wave breaking model for intermediate water depths S.A. Hasan 1 , V. Sriram

∗,2

, R. Panneer Selvam 3

Department of Ocean Engineering, IIT Madras, Chennai, India

highlights • • • • •

Tian–Barthelemy wave breaking model extended to second order wavemaking in HOS. The model preserves suppression of spurious free waves with Schäffer correction. Onset of wave breaking parameter studied in context of multiple wave breaking. Extended model is calibrated for intermediate depth water wave groups. The onset parameter shows considerable scatter in intermediate water depths.

article

info

Article history: Received 25 December 2018 Received in revised form 7 June 2019 Accepted 8 June 2019 Available online 20 June 2019 Keywords: Focused waves High-order spectral method Wave breaking Tian–Barthelemy model Eddy viscosity

a b s t r a c t This paper examines the results in case of breaking wave groups reported in Sriram et al. (2015) against a wave breaking model called Tian–Barthelemy model (Seiffert and Ducrozet, 2017). In this paper, Tian–Barthelemy model is extended to incorporate (Schäffer, 1996) corrections to linear wave making signals for generating focused wave groups and implemented in an open source computer code for high order spectral (HOS) method based numerical wave tank (NWT) viz., HOS-NWT. The developed model is then studied in the context of wave breaking in dispersively focused intermediate water wave groups with particular focus on multiple wave breaking events. The adapted model performs well in conformity with experimental observations in Sriram et al. (2015). A range of kinematic thresholds are studied with this model to calibrate the right threshold for the model against the experiments. We find strong evidence of a non-breaking wave group in intermediate depths crossing the kinematic threshold of 0.85 (stated in Seiffert and Ducrozet (2017)), implying that this threshold is not universal for intermediate depths. It is also shown that the model does not interfere with the spurious free wave suppression. However, various dynamic and kinematical aspects of this study reveal that suppression of spurious free wave makes the short waves dominant in the lead up to breaking. Hence, the model detects numerous threshold exceedences implying plausibly more wave breaking instances. It is also shown that this model in general violates the spectral signature of breaking in a dispersively focused wave. © 2019 Elsevier Masson SAS. All rights reserved.

1. Introduction

Extreme wave events usually have been studied within the paradigm of irregular seas. Since these events contribute to high

∗ Corresponding author. E-mail addresses: [email protected] (S.A. Hasan), [email protected] (V. Sriram), [email protected] (R.P. Selvam). 1 Research Scholar, OED, IIT Madras.

2 Associate Professor, OED, IIT Madras. 3 Professor, OED, IIT Madras. https://doi.org/10.1016/j.euromechflu.2019.06.005 0997-7546/© 2019 Elsevier Masson SAS. All rights reserved.

loads to ocean structures and vessels, a rich stream of literature has been generated towards studying these issues. Consequently, various probabilistic [1] as well as deterministic methodologies [2] have been adopted for these purposes. Another class of extreme wave events are known as ‘freak’ or ‘rogue’ waves in literature [3]. These extreme wave events have been described as a steep wave or a group, that are not necessarily associated with rough seas. As such, their origins have remained a mystery. Two major mechanism responsible for their generation and propagation can be classified as a focusing-type and modulational-type as reported in literature. In this paper, wave groups generated by the focusing methodology are studied in a numerical wave tank. Specifically, the theoretical focusing is determined by the phase speed method [4]

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where in a set of frequencies and amplitudes are selected and then their phase differences are set up for constructive interference. Although, the linear wave making is most popular method, while generating focused waves the paddle motions can become high enough such that non linear effects become important. It has previously been demonstrated that in such cases spurious free waves are generated [5]. Moreover, when studying focused wave groups, one is confronted with the problem of wave breaking as apriori a marginal wavegroup cannot be classified as breaking or non breaking. Schäffer [6], has, therefore, provided second order correcting signals that suppress these spurious waves. These second order correcting signals (henceforth, called Schäffer signals) were used by Sriram et al. [7] to generate a dataset of focusing wave groups based upon phase speed in deep and intermediate water depths. Later on, Hasan et al. [8] studied the same data set for non breaking wave groups in a numerical wave tank using high order spectral method based open source code HOS-NWT [9]. In this paper, the wave breaking is not depth-induced and therefore, we confine ourselves with the spilling and plunging-type breakers obtained by a focusing mechanism with respect to 2D wave groups. Two major issues concerning the physics of water wave breaking have traditionally been-determining a threshold for inception/ onset of wave breaking and describing the post-breaking dynamics in which the question of breaking strength/intensity prominently figures. In laboratory conditions, focusing wave groups have been the preferred way of generating wave breaking events. The experiments by Rapp and Melville [4] were amongst the first to meticulously document wave breaking attained from a focusing methodology. However, this study emphasised the breaking during the developing and post-breaking stage. For the onset of wave breaking (also known as the incipient stage) which is the independent stage prior to the developing stage, it has been hypothesised that a universal threshold [10] exists to identify this stage, at least for breaking that is not depth-induced. In this respect, the study of steep/focused waves has led to insights towards incipient breaking from which now a coherent story towards the question as to just how does a wave crest breaks and related phenomenon, can be studied. As a wave group (those generated by phase speed method) focuses, the crest steepens and the local phase speed slows down by as far as 20% of the corresponding Stokes wave, causing the actual focusing location further away from the theoretical focusing point [11]. The steep crest invokes a local Higgins–Tanaka [12] type crest instability (as opposed to global class I and II instabilities [13,14]), which would lead to incipient breaking geometry of the crest having sufficient steepness. Due to relatively weak surface tension effects, this leads to a pointed jet geometry leading to a plunging crest. On the other hand, strong surface tension leads to a bulge on the front face of the breaker which eventually becomes the source of inception of a spilling type breaker [15]. This common mechanism, therefore, beholds the promise of finding a universal threshold for wave breaking during incipient stages. The question of finding an appropriate threshold can be classified into three main approaches — geometrical, kinematical and dynamical. The geometry and related kinematics of the incipient stage have both numerically as well as empirically studied by many researchers. However, a universal threshold parameter that would specify the onset has so far been not so forthcoming. Earlier studies employed notion of a limiting steepness criterion similar to the Stokes breaking wave limit (crest angle 120 deg) [16] or asymmetry of the breaking wave [17,18]. However, while criterion based on geometric steepness show a large scatter, wave asymmetry parameter seems better suited these towards quantifying strength of breaking. The reader is referred to Perlin et al.

[19] for a thorough review. As such, onset of breaking criterion of kinematic and dynamic type have received a lot of attention. However, compared with geometric criterion, these are not easy to compute particularly in field/laboratory. The idea of a kinematic criterion is to find a threshold based on the ratio of the horizontal fluid particle velocity (U) at the breaking crest to the crest velocity (C ) itself. The crest velocity is often substituted for measures of local wave phase speed [19, 20]. Computing U in laboratory/field conditions is a non-trivial task. On top of that, even the local wave phase speed can be described in myriad forms, for example, phase speed based on linear theory, crest tracking velocity [21], Fedele’s formula [22] or local wavenumber based upon Hilbert transform of surface elevation [23]. It is, however, agreed upon that U /C ≥ 1 will signify onset of wave breaking. However, this may be dependent upon the definition of C . Moreover, it has been reported that such a threshold is even less than 1 [23] implying that the threshold exceeding 1 is merely a sufficient, but not necessary criterion for the onset of wave breaking. Dynamic/energy-based criteria, on the other hand, have been followed mainly on two different yet equivalent formalisms. The first being based on the growth rate of local energy [24] while the other measures the local energy flux [21]. The former postulates a local dimensionless energy growth rate parameter with a threshold value, relatively insensitive to number of wave groups, wind forcing and surface shear. This parameter is a function of the local wave number and local energy density at the envelope maxima of the wave groups. The evaluation and validation of this parameter was extensively performed in many laboratory studies [25]. Barthelemy et al. [21] provided a universal energy-flux type breaking criterion for deep and intermediate water wave groups based on their numerical studies. In absence of wind, this criterion degenerates into the kinematic criterion at the free surface i.e. Bx = U /C . This parameter was extensively verified for laboratory-generated deep water wave groups with a threshold value of (0.84 ± 0.016) in absence of wind by Saket et al. [26]. On the other hand, the calculations of Tian et al. [27] give the value of this parameter close to breaking as 0.836 but without claims towards this being the onset. Seiffert and Ducrozet [28] demonstrated that the instead of crest tracking approach, the crest velocity can be replaced by local phase velocity according to Hilbert transforms Stansell and MacFarlane [23]. They gave the threshold of 0.84 ∼ 0.86, very close to the value suggested by Barthelemy et al. [21]. Moreover, although deep water breaking using this criterion has extensively been studied for a wide range of modulational-type and focusing-type wave groups [29], the question around intermediate water depths remains unsettled. It is interesting to note that wave breaking experiments induced via sloshing in shallow waters also follow the general characteristics of Barthelemy’s energy criterion viz,. deceleration of wave crest prior to breaking [30]. Following Seiffert et al. [29], Hasan et al. [8] applied this criterion dataset of SR15 and found that largely breaking and nonbreaking groups can be separated by the Barthelemy criterion, for both intermediate and deep water wave groups (classification based upon central frequency). However, for one of their non-breaking intermediate wave groups the criterion exceeded 0.88. Thus, they concluded that the sufficient condition for wave breaking onset for intermediate water wave groups is Bx > 0.90. The breaking strength is also one of the related questions post the onset of wave breaking. The strength of breaking is largely characterised by the energy dissipation that occurs in the developing and residual breaking stage following the onset. Energy dissipation from wave breaking largely results from entrainment of air bubbles and, the generation of currents and turbulence [31]. Many laboratory studies have focused upon estimation and parameterisation of total energy dissipation [4,25,27,32–34]. In this

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Fig. 1. Wavemaker and coordinate system for numerical set up. Note that for a piston-type wavemaker used here, displacement of the piston is free of y-dependence i.e. X (y, t) = X (t).

Fig. 2. Geometry of the breaking wave used for computing eddy viscosity [27].

respect, Rapp and Melville [4] have reported that about 10% of initial energy can be dissipated owing to spilling events while for plunging events it can be more than 25%. Approximately 90% of this dissipation happens within 3–4 wave periods while rest decays as t −1 , where t is time from wave breaking inception. Several studies in support [35] and extension of these observations can be found elsewhere [36–39]. Since Rapp and Melville [4], spectral distribution of energy and its dissipation around the spectral peak with respect to wave groups arising out of modulational instabilities [40,41] and dispersive focusing [7,42,43] has also received a lot of attention. It seems that during the developing stage, the breaking process retains a memory about the process onsetting it [10]. Naturally, a large body of studies exists that tries to correlate onset of breaking criterion with the strength of breaking [25,32,44]. Moreover, breaking wave groups generated by dispersive focusing have a spectral signature whereby low frequencies are largely unaffected and peak spectral shift does not occur post-breaking, unlike those generated by modulational instability. Building on these studies, efforts have been made to parameterise the wave breaking process in a mathematical model based on these empirical findings. In this way, such models can be incorporated in potential flow based numerical models that cannot simulate wave breaking. In this respect, an eddy viscosity-type wave breaking onset have proven to be quite effective. Similar efforts in shallow water waves [45,46] have already been made. For deep water breaking [27] formulated an eddy viscosity model based on Ruvinsky et al. [47] and their own empirical findings. Chalikov and Sheinin [48] also used a similar approach but with a slightly different methodology which in addition to using a coefficient of diffusion, relied upon filtering energy at higher frequencies. In addition to these models, Brocchini & coworkers have proposed a three layer model & its variants for turbulence-induced dissipation at the post-breaking stage [30, 49,50]. However, Seiffert and Ducrozet [51] show that in practical terms Tian et al.’s model remains robust and simpler to implement. Tian et al.’s model must be combined with an appropriate threshold that triggers the wave breaking whenever the threshold is crossed. This was first done by Tian et al. [52] by using a local steepness criterion in a pseudo-spectral model. Later on, the kinematic criterion determined by Barthelemy et al. was incorporated by Seiffert and Ducrozet to study dispersive as well

as modulational type breaking wave groups. Henceforth, this approach is labelled as Tian–Barthelemy model [29]. In these discussions, it must be noted that air–sea interaction models already incorporate sink type functions to simulate wave breaking [54]. However, these are not temporally or spatially local. This paper extends the results of Hasan et al. [8] to numerically study the breaking wave groups in the Sriram et al. [7] dataset (henceforth, called SR15). The Tian–Barthelemy approach is adopted to study these models in HOS-NWT. Specifically, we apply this model with Schäffer corrected wave groups in order to compare them with the non-corrected ones in intermediate water depths. The aim is to study the nature of breaking in absence of spurious free waves. Another closely related aim is to gauge the performance of the Tian–Barthelemy model for various criteria with particular focus on multiple wave breaking events. This is then used to calibrate the right kinematic threshold based on the performance for the wave breaking model in the light of experimental observations. One of the questions we try to investigate is regarding the ‘‘universality’’ of Barthelemy’s parameter in intermediate depths& whether local events like multiple wave breaking/global characteristics like spectral bandwidth apart from water depth give a scatter in this parameter. This article is structured as follows. Section 2 discusses the extension of formulations by Tian et al. [27] and Seiffert and Ducrozet [51] to the second order of wavemaking in order to consistently apply Schäffer signals. The application of Schäffer signals in this study, as opposed to Hasan et al. [8], has been modified and can achieve full suppression of the spurious free waves (Section 3.1). It is shown that according to methodology suggested by Drazen et al. [32], all of the breaking wave groups in SR15 dataset should be classified as intermediate water wave groups (Section 3.2). In Sections 3.4, 3.5, 3.6 and 3.7, the Tian–Barthelemy wave breaking model is extensively explored to recommend appropriate threshold criteria in Section 3.8. Finally, conclusions are drawn up in Section 4. 2. Mathematical formulations 2.1. Numerical wave tank Fig. 1 describes the domain of the numerical wave tank considered. The x-direction is defined length wise. At x = 0 is a piston-type paddle that acts as a wavemaker, whereby wavemaking signals are provided to generate a focusing wave group with the theoretical focusing point at x = xt . This focusing point is based on phase speed method [55] for dispersively focused waves. The end of the tank is modelled as vertical wall at x = L, amid a constant water depth (h) throughout the length of the tank. The still water level is taken as y = 0. The method of providing Schäffer signal for second order correction is described in Hasan et al. [8].

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Fig. 3. Algorithm for applying eddy viscosity model.

The fluid domain in the numerical tank is based on velocity potential φ (x, y, t) and follows the Laplace’s equations neglecting surface tension effects. For the purpose of modelling the waves in a numerical wave tank, the high order spectral tank (HOST) formulation by Ducrozet et al. [9] is adopted. This means decomposing the total velocity potential into spectral and additional potential viz., φ = φspec + φadd . Herein, the spectral potential (φspec ) behaves as the classical HOS potential while additional potential (φadd ) is involved in wavemaking. The Tian–Barthelemy wave breaking model is then applied to the spectral potential in form of a diffusion term in the free surface boundary conditions as detailed in Section 2.3. 2.2. Onset of wave breaking – Barthelemy parameter The onset of wave breaking in HOS model is given by Barthelemy parameter (Bx ) exceeding a given threshold. This parameter can be computed as Bx (x, t) = U(x, t)/C (x, t), where C = ωl /kl is the local phase speed and U is the local horizontal velocity of the particle at the crest. The computation of the local frequency (ωl ) and local wave number (kl ) can be determined from the Hilbert transform (H[η]) of the spatial waveform as follows [23] kl (x, t) =

1

η2 + H2 [η]

(

η

∂ ∂ H[η] − H[η] η ∂x ∂x

) (1)

ω2 = gkl tanh (kl h)

(2)

For a given discretisation of the numerical domain, the Barthelemy parameter (Bx ) is computed at every point of the free surface. The computation of phase velocity based on Hilbert transforms suffer from the problem of phase reversals when high energetic short waves ride upon relatively low energetic long waves. In the presence of these phase reversals [56] near local maxima/minima of the wave amplitude, the computation of Bx using Hilbert transform can give negative wavenumbers or very short waves leading to false exceedences. While negative wavenumbers are easily filtered out, short wave exceedences have been kept as is as in most cases the computed eddy viscosity is close to zero. Unlike Seiffert and Ducrozet [53], the computed wave surface is not stored in a spline form before computing Bx . Rather, the discretised spatial domain is made finer and Bx computed at each of these discretised points after taking care of short waves. For deep water wave breaking, Seiffert and Ducrozet [53] have suggested a threshold value of Bx ∼ 0.85 for onset of wave breaking based on formulae (1) and (2) to be which quite close to that suggested by Barthelemy et al. [21] using crest tracking. In this paper, we revise the threshold from Hasan et al. [8] for intermediate water wave groups to 0.9015 (see Table 1), but this value is only a sufficient condition. Moreover, as discussed earlier in Section 1, it is plausible that the breaking onset parameter has

S.A. Hasan, V. Sriram and R.P. Selvam / European Journal of Mechanics / B Fluids 78 (2019) 115–138

( +2

∂ 2 φadd ∂ x∂ y

)(

∂η ∂x

)

119

,

(5)

wherein φ s = φspec (x, y = η, t). In the above Eq. (5), the additional potential is exploited in the space domain coordinates i.e. (x, y), rather than free surface coordinates in line with HOST computations. In the second term, ∂2φ Laplace equation has exploited for obtaining the term ∂ yadd 2 .

Fig. 4. An example of the nomenclature adopted for this study. Cf. Tables 1 and 2.

a scatter dependent on local as well as global characteristics of the respective wave groups. 2.3. Second order wave breaking model The study by Seiffert and Ducrozet [53] used the HOST formulations [9] for Tian et al. [27] wave breaking model with linear wavemaking conditions. However, in order to accommodate Schäffer’s second order corrected signals in the generation of focused wave groups, a second order extension of this wave breaking model is required. This can be achieved in a very simple manner as follows. In their paper [27], Tian et al. start with the formulations of Ruvinsky et al. [47], whereby the molecular viscosity in Ruvisnksy et al. for gravity-capillary waves is replaced by an eddy viscosity term for gravity waves. As such, the extension of Tian et al.’s formulation also starts at Tian et al. [27, equations (4.10) and (4.11)] and Seiffert et al. [29, equations (3) and (4)]. The kinematic and dynamic free surface boundary conditions at y = η(x, t) are the same as Seiffert and Ducrozet [51]

[ ( )2 ] ( )( ) ∂η ∂η ∂φ ∂˜ φ ∂η ∂ 2η = 1+ − + 2ν 2 (3) ∂t ∂x ∂y ∂x ∂x ∂x [ ] )2 ( ( )2 ( )2 1 1 ∂ 2˜ φ ∂η φ ∂˜ φ ∂ 2˜ ∂φ + = −g η − 1 + + 2 ν , ∂t 2 ∂ x2 2 ∂x ∂y ∂ x2 (4) where the total free surface potential (˜ φ = φ (x, y = η, t)) is introduced along with eddy viscosity coefficient (ν ) and acceleration due to gravity (g). The free surface derivative in Eq. (4) can further be expanded at the free surface as

∂ 2φs ∂ 2˜ φ = + ∂ x2 ∂ x2

(

∂ 2 φadd ∂ y2

)(

) ( )( 2 ) ∂φadd ∂ η ∂ 2η − 1 + 2 ∂x ∂y ∂ x2

Ducrozet et al. [9]’s HOST formulation is a hybrid Stokes-HOS type in which the wavemaking additional potential is resolved by solving the wavemaker boundary condition in the stokesian form rather than HOS type expansions. Hence, for second order wavemaking, it is only required to obtain the complete total potential (φ ) correct to first order stokesian problem of wavemaker boundary condition along with free surface boundary conditions. For the second order problem only the second order wavemaking condition for additional potential need to be solved for HOST formulations. The reader is referred to Ducrozet et al. [9] for the details. Thus, the first order stokesian free surface boundary conditions for the total potential can then be written as

∂η(1) ∂φ (1) ∂ 2 η(1) − = 2ν ∂t ∂y ∂ x2 (1) ∂φ ∂ 2 φ (1) + g η(1) = −2ν ∂t ∂ y2

(6) (7)

Along with the first order wave maker boundary condition, the first order stokesian problem is thus completely posed. Having solved the first order stokesian problem, the additional potential for the second order can be easily obtained by solving the wavemaker boundary conditions at second order in the stokesian sense. Since only the second order additional potential is required, complete solution of free surface boundary conditions at second order is not required. Because of the nature of spectral expansions used for HOST formulations in Ducrozet et al. [9], the (2) contributions from φspec at second order of wavemaking at x = 0 vanishes, leading to a simple explicit equation i.e. (2) ∂φadd ∂ X (2) ∂ 2 φ (1) ∂ X (1) ∂φ (1) = − X (1) + 2 ∂x ∂t ∂x ∂y ∂y

at x = 0

(8)

2.4. Computation of Eddy viscosity The eddy viscosity (ν ) is used in Eqs. (3), (4), (6) and (7) to model the wave breaking process. Since wave breaking is spatially and temporally intermittent process, the eddy viscosity is a local function in space and time. In the model implemented with HOSNWT, the eddy viscosity term is triggered when the wave group crosses a breaking threshold given by Barthelemy parameter (Bx ). The wave breaking process is then applied for a pre-determined

Fig. 5. Comparison of free wave suppression with SR15 results (i.e. the removal of positive lobe of the spurious wave) at x − xt = 0.5640. Left: Case f068c10030 without Schaffer signals; Right: Case f068c20030 with Schaffer signals. See also, Table 1. The solid lines represent probe readings without any filtering. The dashed lines represent reconstructed scaled up (by a factor of 5) readings for lower frequencies [7, cf. figure 10].

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Table 1 Calculated parameters for the simulated non-breaking wave groups with π Ga = S. (*) (Bx )max Locations are revised from Hasan et al. [8, table 2] with Nx = 2049, M = 5 and output frequency 50 Hz with numerical wave making order as 2. Case f068a10019 f068a20019 f068b10028 f068b20028 f068c10020 f068c20020 f068c10030 f068c20030 f108a10010 f108a20010 f108a10015 f108a20015 f108b10020 f108b20020 f108c10030 f108c20030

Corrected signal

Central frequency (fc ) (Hz)

Primary bandwidth δ f /fc

Primary components (Nf )

Gain (Ga )

(Bx )max

N Y N Y N Y N Y N Y N Y N Y N Y

0.68 0.68 0.68 0.68 0.68 0.68 0.68 0.68 1.08 1.08 1.08 1.08 1.08 1.08 1.08 1.08

0.50 0.50 0.75 0.75 1.00 1.00 1.00 1.00 0.50 0.50 0.50 0.50 0.75 0.75 1.00 1.00

32 32 32 32 32 32 32 32 40 40 40 40 40 40 40 40

0.0019 0.0019 0.0028 0.0028 0.0020 0.0020 0.0030 0.0030 0.0010 0.0010 0.0015 0.0015 0.0020 0.0020 0.0030 0.0030

0.3009 0.3053 0.6932 0.7206 0.3414 0.3406 0.9015 0.8852 0.1596 0.1599 0.3109 0.3114 0.5170 0.5165 0.7062 0.7066

(Bx )max location* (m)

SR 15 Focusing location (m)

22.08 21.75 22.61 22.50 21.48 21.54 22.67 22.88 21.16 21.16 21.43 21.43 21.43 21.43 21.70 21.75

21.5 21.5 22.0 22.0 – – 22.0 22.0 – – 21.0 21.0 21.2 21.2 21.5 21.5

Table 2 Details of the simulated breaking wave groups with π Ga = S reproduced from Sriram et al. [7]. Case

Corrected signal

Central frequency (fc ) (Hz)

Primary bandwidth δ f /fc

Primary components (Nf )

Gain (Ga )

SR 15 Breaking location (m)

SR15 description

f068a10029 f068a20029 f068b10034

N Y N

0.68 0.68 0.68

0.50 0.50 0.75

32 32 32

0.0029 0.0029 0.0034

24.0 24.0 23.0

f068b20034 f068c10038 f068c20038 f108b10040 f108b20040 f108c10061

Y N Y N Y N

0.68 0.68 0.68 1.08 1.08 1.08

0.75 1.00 1.00 0.75 0.75 1.00

32 32 32 40 40 40

0.0034 0.0038 0.0038 0.0040 0.0040 0.0061

23.5 22.0 23.5 21.0 21.0 20.8

f108c20061

Y

1.08

1.00

40

0.0061

20.8

Plunging Plunging Plunging with initial mild spilling before Plunging Plunging Plunging Spilling Spilling Plunging with Initial spilling Plunging with Initial spilling

wave breaking length (Lbr ) and wave breaking time (Tbr ) starting from moment a given crest exceeds the breaking threshold (see Fig. 3). A suitable ramp is provided for smooth transition between wave breaking and inviscid regions of the spatial domain. kb = π /Lc

(9)

Sb = kb (2Hc + Ht1 + Ht2 )/4

(10)

R b = L2 / Lc

(11)

kb Lbr = 24.3Sb − 1.5

(12)

kb Hbr = 0.87Rb − 0.3

(13)

ωb Tbr = 18.4Sb + 1.4

(14)

The breaking wave height (Hbr ) along with Tbr and Lbr are determined from the empirically determined Eqs. (9) to (14) [27], where kb , Sb and Rb are local wave number, local wave steepness and wave asymmetry parameter, respectively which are determined according to the pre-wave breaking geometry as shown in Fig. 2. The eddy viscosity is thus given by

ν=α

Hbr Lbr Tbr

,

(15)

where α is a proportionality constant to include the effect of air entrainment in the wave breaking model. The value of α as provided by Tian et al. [52] is taken as 0.02 for most cases. But it turns out inadequate for heavy breaking particularly in temporally close multiple breaking events, wherein it has been taken as 0.035 in this study(see Section 3.4 for details). This selection of proportionality constant is, thus a subjective tuning

parameter in the breaking model. A better model for α based on parameterisation of air entrainment in breaking events is, thus, needed. This is, however, beyond the scope of this study and will be attempted in future. Looking at above equations viz., (12) and (13), it is amply clear that the sufficient condition for eddy viscosity (ν ) to be non-negative is that Sb > 0.062 and Rb > 0.345, which can be interpreted as specifying a sufficient threshold for a wave group to be non breaking. In case of phase reversals (Section 2.2), the condition Sb > 0.062 acts as another filter, whereby Lbr is set to zero if the condition is not met. This point is further elaborated in Section 3.1. 3. Results and discussions In this section, the results are validated against the laboratory experiments of Sriram et al. [7] referred to as SR15 dataset. The reader is referred to Sriram et al. [7] for the details of experimental set up. The non-breaking wave groups were validated for HOS-NWT in a previous study [8]. However, lingering questions remained about the spatial location of maximum Bx being far away from actual recorded location of wave breaking in SR15. It will be shown that these are the wave groups where multiple breaking events occur. It must be noted that the SR15 dataset was used to demonstrate the suppression of spurious free waves and pre-mature wave breaking for wave groups where such a suppression was not enforced [7]. The breaking locations and their types were, however, based on visual observations. The

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Fig. 6. Exponential fitting for viscous losses in non-breaking wave groups. Left - f068b20028 ; Right- f068c20030.

nomenclature of the wave groups adopted in this study is illustrated in Fig. 4 and can be understood in combination with Tables 1 and 2. 3.1. Numerical set up, methodology and convergence The length and water depth of the wave tank is taken as the same as of SR 15 i.e. 110 m and 0.7 m, respectively. In line with SR15 a piston paddle, with mean position x = 0, is used to generate both linear and second order corrected breaking wave groups. The theoretical breaking location is set to xt = 21 m. Table 2 shows the details of the simulated breaking wave groups of constant global steepness (S) in this paper and can be compared to Hasan et al. [8]. In Table 1 we reproduce the results for the non-breaking wave groups in SR15 dataset with the number of modes for HOSNWT in x-direction and y-direction are taken as Nx = 2049 and Ny = 65 with an output frequency of 50 Hz while HOS order M = 5. The maximum Bx observed for a non-breaking wave group is 0.9015. This apparent increase when compared to the value of 0.882 from Hasan et al. [8] is mainly due to the output results being sampled for finer grid size. Moreover, for second order corrected wavemaking [6] for this non-breaking wave group (Ga = 0.0030), Fig. 5 shows complete suppression of spurious wave component in line with SR15 results. For breaking wave groups, again, the number of modes for HOS-NWT in x-direction and y-direction are taken as Nx = 2049 and Ny = 65, but with an output frequency of 150 Hz. The Nx modes and output frequency are more than the parameters required for numerical convergence taken in the related study [8]. The choice of higher parameters here is dictated by Eq. (13) for breaking wave elevation which is sensitive with respect to wave asymmetry. The HOS order M is again taken as 5 while wave making order is set to 2. Only for second order corrected wave groups is the Schaffer’s second order signal applied to suppress the spurious wave (e.g. case f068c20030 in Fig. 5). Keeping with the breaking wave groups, an unhealthy consequence of higher number of Nx modes used in this study compared with Hasan et al. [8] is that there are larger number of false exceedences of Bx due to small wave resolution in absence of filtering. Moreover, the determination of eddy viscosity (ν ) as in Section 2.4 dictates determination of turning points around the location of breaking crest. Thus, it becomes imperative to filter out small waves for such computations. As such, for the

computation of Bx and ν , the wave profile is passed through a simple low pass filter eliminating contributions from modes below 25 cm. It must be noted that this is done only for the computation of parameters Bx and ν on a separate array in which the wave elevations (η) are extracted. Computations regarding the free surface boundary conditions are not effected by this extraction and retain the contribution from higher frequencies. The above procedure for computing Bx = U /C leads to a slight anomaly since U is still computed from free surface boundary conditions retaining the contribution from higher frequencies. This is, however, checked across Table 1 for usage with and without the low pass filters. The values of Bx , thus calculated do not show large relative deviations. As such, the methodology with low pass filters is accepted as adequate. Inspite of these measures, false exceedences due to locally short waves may remain present. Most of these exceedences are, however, inconsequential and lead to ν ∼ 0. Some of these are due to exceedences at locations with negative elevations that are close to the mean water level, others may be due to Sb < 0.062 as discussed in Section 2.4. Any sort of exceedences happening below the mean water level (η < 0) is also filtered out. In this paper for the purpose of computing statistical and spectral properties of the probe readings a windowing method is adopted. A suitable window size of specified seconds is selected straddling a particular time in the time series recorded by the probe. The choice of this particular time depends upon the type of comparison between SR15 experiments and HOS-NWT simulations. In the simulations, the computed time series can suffer from a phase difference with respect to the experiments. This means that there is a difference in the group velocities from the experiments and the simulations. However, in some situations it is warranted that the effect of phase difference is not accounted for and arguments are made completely for energy comparisons represented by the time series. In such cases, for example, the compared spectra between the experiments and HOS-NWT is computed by taking the particular time of the windowing process as the time of the respective peaks in the two recorded timeseries (henceforth, referred to as first variant of windowing). On the other hand, if a measure of the difference in the group velocities is to be retained, only one particular time (viz., the peak of the respective empirical series) is used for the straddling window for a given location of probe, both for SR15 as well as HOS-NWT (henceforth, referred to as second variant of windowing).

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Fig. 7. Comparison of averaged steepness of breaking wave geometry in HOS-NWT (2|kl | H[η]2 + η2 ) versus Tian–Barthelemy model ( kb Hbr ) for all cases.

√

Fig. 8. Geometry of the breaking wave (ν > 10−3 m2 /s) for case f068b20034 for thresholds (a) Bx > 0.80 (b) Bx > 0.85 (c) Bx > 0.90 (d) Bx > 0.92. Dashed vertical lines of corresponding colour show respective breaking crest locations.

3.2. Wave group classification for SR15 Table 3 shows the properties of the empirical SR15 wave groups, if classified according to central frequency (fc ) and wave number (kc ), according to linear dispersion. Along with them are tabulations for spectral weighting of the constant steepness spectra according to procedure laid down in Drazen et al. [32] and Tian et al. [27]. The spectrally weighted frequency is calcu2

∑

f a lated as fs = ∑n na n2 , while ks is the corresponding wave number n n

for fs according to linear dispersion. Similar procedure is adopted to central group velocity (Cgc ) vis-a-vis spectrally weighted group velocity (Cgs ). The procedure according to Tian et al. [27] is followed, whereby a windowed time series faraway from breaking location (x0 − xt = −16.165 m) is chosen for these computations. The summation is taken over for frequencies less than 10 Hz. Sriram et al. [7] have classified the constant steepness-type SR15 wave groups according to their central frequencies. In such a case, wave groups with central frequency 0.68 Hz have been

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Fig. 9. Geometry of the breaking wave (ν > 10−3 m2 /s) for case f108c20061 for thresholds (a) Bx > 0.80 (b) Bx > 0.85 (c) Bx > 0.90 (d) Bx > 0.92. Dashed vertical lines of corresponding colour show respective breaking crest locations.

Fig. 10. Waterfall plot comparison of probe readings for case f068c20038 & threshold set at Bx = 0.85. Black - SR15. Red - HOS-NWT. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

∑

a ∼ 0.34) for constant steepness-

classified as intermediate wave groups while those of 1.08 Hz as

the limiting steepness (k

deep water wave groups. As demonstrated by Drazen et al. [32],

type spectra identified in Tian et al. [27] as inferred from the

for a constant wave steepness spectrum, it is better to spectrally

computation of ks

weigh the spectrum for a better comparison with constant am-

remarks on their breaker type. The marked deviations of k

plitude spectrum. In this paradigm, both type of spectra qualify

breaking wave groups from Drazen et al. (0.32–0.35) & Tian et al.

as intermediate water wave groups (0.1π ≤ ks

(0.34) gives confidence that the threshold for intermediate water

∑

an ≤ π ).

According to Table 3, all wave groups are close to or exceed

∑

an . This can be compared to Table 2 for SR15

depths is different & may suffer from a scatter.

∑

a for

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Fig. 11. Comparison of probe readings between SR15 experiments and HOS-NWT. Left - case f068b10034; Right case f068b20034 at x − xt = (a,b) −0.2360 (c,d) 1.5980 (e,f) 4.0530. Dashed lines show the scaled up (by a factor of 5) reconstructed long waves below the primary frequency range [7]. First variant of windowing used.

3.3. Energy loss for non-breaking waves Energy loss in non-breaking waves follows from the viscous losses at the boundary layers in the free surface, sidewalls and bottom. Following Tian et al. [27], the spatial decay of energy will then follow an exponential distribution. Thus Ei = E0 exp (−σ ∆x), where Ei is measure of energy at the ith probe location while E0 is the measure at a given reference probe location (x0 = 4.835 m from wavemaker) from the theoretical focusing point (xt = 21

m ). Moreover, ∆x is the distance between the reference point & the probe location with σ as the decay rate. However, to obtain a good fitting with the exponential decay, a large number of probes are required along the length of the tank both in the pre & post focusing regions. Unfortunately, in case of SR15 dataset, only locations close to focusing have been recorded. This means that for wave groups with significant oscillations in the measure of energy near the focusing location do not follow this decay. For the few cases that show limited oscillations the

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Fig. 12. Comparison of energy spectral density (ESD) of probe readings between SR15 experiments and HOS-NWT. Left - case f068b10034; Right case f068b20034 at x − xt = (a,b) −0.2360 (c,d) 1.5980 (e,f) 4.0530. Second variant of windowing used.

decay factor σ is O(0.01), although the exact values show slight scatter (Fig. 6). 3.4. Choice of appropriate threshold In order to tune the Tian–Barthelemy wave breaking model for an appropriate choice of Bx in HOS-NWT, the non-corrected breaking linear wave groups were tested for a choice of threshold from 0.80 to 0.92 in steps of 0.10. This is done to test whether the threshold value of 0.85 suggested in [29] is uniform across all the wave groups tested.

The hypothesis here is that if the right threshold is selected for each wave group simulated, the right one will give the closest match. But this turns out to be inadequate as the model performs largely the same for a large range of thresholds with mixed results. Moreover, one does not find a ‘‘single right metric’’ that identifies this closest match. As such, the whole exercise is limited to identifying the best calibration thresholds for Tian–Barthelemy wave breaking in terms of a variety of measures. Table 4 documents the location and time of breakdown of computations when no such thresholds are set. The maximum Bx noticed before breakdown of computations in HOS-NWT exceeds

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Fig. 13. (Bx )max vs. time for cases f068b10034 (left) and f068b20034 (right). Note that total number of exceedences in this plot should be counted as sum total of exceedences in Tables 8 and 9, respectively. Dashed vertical lines show timing of dominant breaking events.

Fig. 14. Time evolution of total energy in wave groups. Left-f068b10034; Right-f068b20034. Dashed vertical lines show timing of dominant breaking events.

Table 3 Comparison of wave group properties according to central frequency and according to spectrally weighted properties [27,32]. Case

fc (Hz)

kc (m−1 )

kc h

S = ka

Cgc (m/s)

fs (Hz)

ks m−1

ks h

ks

f068a10029 f068a20029 f068b10034 f068b20034 f068c10038 f068c20038 f108b10040 f108b20040 f108c10061 f108c20061

0.68 0.68 0.68 0.68 0.68 0.68 1.08 1.08 1.08 1.08

2.08 2.08 2.08 2.08 2.08 2.08 4.71 4.71 4.71 4.71

1.45 1.45 1.45 1.45 1.45 1.45 3.29 3.29 3.29 3.29

0.0091 0.0091 0.0107 0.0107 0.0119 0.0119 0.0126 0.0126 0.0192 0.0192

1.36 1.36 1.36 1.36 1.36 1.36 0.73 0.73 0.73 0.73

0.66 0.64 0.61 0.59 0.57 0.51 0.90 0.89 0.77 0.76

2.01 1.92 1.76 1.70 1.60 1.38 3.31 3.28 2.50 2.45

1.40 1.34 1.23 1.19 1.12 0.97 2.32 2.30 1.75 1.72

0.29 0.28 0.31 0.30 0.34 0.29 0.40 0.40 0.51 0.50

Table 4 Calculated parameters for the simulated breaking wave groups without setting a threshold Bx in HOS-NWT. Nx = 2049, M = 5 and output frequency 150 Hz with numerical wave making order as 2. Case

Tstop (s)

Time at breakdown (s)

Location of breakdown (m)

Bx at breakdown

f068a10029 f068a20029 f068b10034 f068b20034 f068c10038 f068c20038 f108b10040 f108b20040 f108c10061 f108c20061

101.00 101.00 89.00 89.00 74.00 74.00 76.00 76.00 66.00 66.00

67.89 62.88 48.39 48.43 38.00 38.56 50.29 50.29 35.55 35.64

27.81 20.46 22.07 21.91 21.43 21.97 20.41 20.36 19.28 19.34

30.193 32.527 16.867 43.730 8.6144 29.477 22.575 20.279 8.658 23.712

∑

an

Cgs (m/s) 1.43 1.41 1.57 1.54 1.68 1.64 1.02 1.02 1.27 1.27

the values in the range 0.80–0.92. As such it is expected that all thresholds will eventually be crossed making the choice for the right threshold difficult. Moreover, difficulty in the explanations of breaking events are compounded by their multiplicity, as would be shown later. It is also noted that for wave groups with Schaffer signals applied the computed Bx at the breakdown exceeds for those where it is not applied. This is because suppression of a spurious long wave makes the short waves dominant and the denominator in the parameter Bx = U /C thus decreases. In general, the suppression of spurious long waves makes the short waves dominant and therefore the exceedences for corrected wave groups outnumber those when correction is not applied. The value of α = 0.02 for wave groups with central frequency 1.08 Hz turns out ∑ to be entirely inadequate with increased strength of breaking (ks an ≥ 0.40). As such, in these cases

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Fig. 15. Comparison of probe readings between SR15 experiments and HOS-NWT. Left - case f108c10061; Right case f108c20061 at x − xt = (a,b) −1.2360 (c,d) 0.5640 (e,f) 3.3240. Dashed lines show the scaled up (by a factor of 5) reconstructed long waves below the primary frequency range [7]. First variant of windowing used.

(Table 3) α is taken as 0.035. This value is supposed to be representative of the air entrainment effects not accounted for [27]. However, in absence of a parameterised model for α , this value is subjective to the results. Even with these modifications, for threshold Bx = 0.87 for case f068c20061, the computations breakdown due to insufficient viscosity computed from Eq. (13), although they do not in case of adjacent values viz. 0.86 and 0.88. As such threshold Bx = 0.87 is not discussed for comparisons among cases f068c10061 and f068c20061.

For the case f068b20029, the computations breakdown for all threshold Bx > 0.83 as opposed to case f068b10029 in which all thresholds in the range 0.80 ≤ Bx ≤ 0.92 are crossed. The reason being that due to the suppression of spurious wave in f068a20029 short waves dominate during exceedences. As the threshold is increased beyond 0.83, the undamped short waves race down the forward slope of the crest to the trough wherein the exceedance is detected. Since η < 0 in this case the algorithm filters these instances out and eventually the computations numerically break down near the trough.

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Fig. 16. Comparison of energy spectral density (ESD) of probe readings between SR15 experiments and HOS-NWT. Left - case f108c10061; Right case f108c20061 at x − xt = (a,b) −1.2360 (c,d) 0.5640 (e,f) 3.3240. Second variant of windowing used.

For other cases, however, all of the thresholds are crossed and simulations proceed towards end of simulation (Tstop ) with varying results. This makes the appropriate choice of threshold Bx difficult to discern. It is not clear whether a single metric is enough for obviating other choices of threshold. In Fig. 7, averaged wave breaking steepness over breaking

breaking geometry for HOS-NWT parameter is banded (1.0–3.0) for all breaking strengths predicted by Tian–Barthelemy model with an average of 2.0. As shown in Figs. 8 and 9 the selected breaking geometries (ν > 10−3 m2 /s ) are dominated primarily by breaking events that have similar pointed geometries at the breaking instant. Thus, indeed, it seems that based on a measure

geometry (2|kl | H[η]2 + η2 ) for HOS-NWT is compared against wave breaking strength from Tian–Barthelemy model (kb Hbr ) for dominant wave breaking events (defined as ν > 0) in all cases, where the overline represents averaging over a domain of length Lbr . Barring a few outliers, averaged wave breaking steepness over

of local wave steepness (2|kl | H[η]2 + η2 ) there exists a breaking criterion that is banded with a considerable scatter across all the breaking wave groups, confirming the hypothesis that the underlying instability mechanism in the lead up to breaking remains the same.

√

√

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Fig. 17. (Bx )max vs. time for cases f108c10061 (left) and f108c20061 (right). Two breaking events in close proximity for 37 s ≤ t ≤ 40 s show up as doubly peaks. Note that total number of exceedences in this plot should be counted as sum total of exceedences in Tables 10 and 11, respectively. Dashed vertical lines show timing of dominant breaking events.

Fig. 18. Time evolution of total energy in wave groups. Left-f108c10061; Right-f108c20061. Dashed vertical lines show timing of dominant breaking events.

3.5. Validation of /Tian–Barthelemy model Tables 6–11 summarise the computations done for the respective wavegroups. In many cases, because of prevalence of short waves numerous exceedences can be observed, particularly at the end of the simulations. Only a few of them are dominant events (ν > 0). Even among dominant events, only a few register the expected eddy viscosity of O(10−3 ) m2 /s. As mentioned before, the spurious free wave suppression infact makes the short wave dominant. Thus the number of false exceedences (and plausibly multiplicity of breaking events) increases in such cases. Thus, with respect to computation of Bx , the suppression does not seem beneficial. It is expected as threshold is increased in these cases, either the multiplicity of breaking events will reduce or the simulation will breakdown. This expectation is more often than not violated. Firstly, during focusing Bx rises very fast such that in order to detect whether the threshold is crossed, the temporal sampling rate of computed Bx must be very high. Secondly, as remarked in Section 3.1 the strength of breaking is sensitive to the asymmetry parameter Rb which means that as the threshold is raised assuming asymmetry increases, the strength of breaking also increases. On the other hand, a missed detection due to a higher threshold can also mean multiple short breaking events later. The situation is complicated further if the domain Lbr includes areas of waveform with negative curvature. The plethora of these scenarios can make the interpretation of the simulation with respect to the set threshold Bx very difficult to discern. As a result only a few general statements about the whole range of thresholds

(0.80 ≤ Bx ≤ 0.92) can be made with respect to validity of the computations with the experiments. As documented in Figs. 11 and 12, not only do the probe readings and energy spectral density (ESD) show a good match (case f068b10034 vs f068b20034), the model also keeps the suppression of spurious free waves intact. The rise of (Bx )max with time (Fig. 13) show two clearly separated lobes which signify two breaking events depending upon the threshold value set. For Fig. 13, beyond time greater than 65 s, the abrupt spikes in (Bx )max are due to phase reversals when short waves ride the longer modes near the mean water level. The accompanying evolution of energy is shown in Fig. 14. Whereas, the comparison between the model and experimental results show moderate to poor performance for central frequency of 1.08 Hz, as the strength of breaking (Figs. 15 and 16) increases in addition to temporally close proximity of the breaking events (Figs. 17 and 18). It is observed that as the wave group progresses away from the region of breaking a phase difference develops between the probe readings of SR15 and HOSNWT simulations, which get progressively larger as the breaking strength increases (see Fig. 10). In contrast to Table 1 where location of maximum computed Bx is in line with focusing locations of SR15, Tables 6–11 show a poor match between breaking locations with SR15 (Table 2). This leads to fluctuating differences in phase locations i.e. the simulated wave group decelerate & accelerate in pre-breaking & post-breaking stages at positions different from SR15 (see figure for case f068c20038 & threshold set at Bx = 0.85). However, in most cases, one observes the wave elevations with respect to experimental observations comparable.

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Fig. 19. Comparison of spectral distribution of variance (left to right) for low frequency range (f /fc < 0.53), high frequency range (f /fc > 1.47) and total spectrum [7]. Second variant of windowing of size 60 s used. Dashed lines show the location of dominant wave breaking events for case f068b20034.

Fig. 20. Comparison of respective energy in probe energies for varying threshold levels for Bx in HOS-NWT relative to observed ones in SR15 along the locations of the probes. Case: f068b20034 . The vertical dashed line represents the dominant breaking locations (Table 9).

Fig. 21. Comparison of respective energy in probe energies for varying threshold levels for Bx in HOS-NWT relative to observed ones in SR15 along the locations of the probes. Case: f108c20061 . The vertical dashed line represents the dominant breaking locations (Table 11).

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Fig. 22. Waterfall plot of probe readings along the domain of the wave tank at various thresholds compared to SR15 dataset (case f068b20034). The dashed slope represents Cgc while solid slope corresponds to Cgs . The maxima of the envelope is represented by an asterisk.

3.6. Spectral distribution of energy It has been observed that breaking events in dispersive focusing follow a spectral signature whereby the energy accumulated in higher frequencies prior to breaking is dissipated, while the longer modes appear undisturbed [10]. On these lines, Fig. 19 show the variances of probe readings windowed using the second variant of size 60 s and normalised by the variance at location x − xt = −16.165 along the length of the domain for the case f068b20034. The variances as a measure of energy of the signal are divided into total, lower and higher frequency ranges according to Sriram et al. [7] and compared with the experiments. Overall the total energy pre and post breaking follows a trend in line with SR15 experiments. However, the Tian–Barthelemy model applies across spectrum and as such dampens the long

modes as well, which remain relatively unaffected in the experiments. The energy on the higher end of the spectrum closely follows the experimental results, however. Thus, the usual spectral signature of breaking for longer modes in dispersively focused groups is violated. However, as an improvement one can preserve the signature by first passing the diffusion term in Eqs. (3)–(7) through a high pass filter before solving these equations. This is however, not attempted in this study. In order to measure the flux of energy along the domain for Tian–Barthelemy model vis-a-vis experimental values the ratio of variances (simulation to experiments) as a measure of total energy at each probe location are reported in two ways. Using the first variant of windowing (see Section 3.1), the respective ratio i of variance ((σhos /σsri )2 for the ith probe) as representative of the relative energy of the time series is compared against the location

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Fig. 23. Waterfall plot of probe readings along the domain of the wave tank at various thresholds compared to SR15 dataset (case f108c20061). The dashed slope represents Cgc while solid slope corresponds to Cgs . The maxima of the envelope is represented by an asterisk.

of probes. This assumes that energy is fluxed across the domain with the same velocity in the experiments as well as HOS-NWT. In order to compensate for that, spectrally weighted group velocities for each of the probe in HOS-NWT as well as SR15 experiments are computed in a fashion similar to as outlined in Section 3.2, but using the second variant of windowing and using non linear corrections ( ) to the group velocities [27, Appendix A]. Their ratios

hos iC ˆ gs sr iC ˆ gs

are then multiplied to ratio of the variances

and plotted against the locations of the respective probes (Figs. 20 and 21). Since the latter measure combines both the group velocity estimation as well energy, it was expected that such a comparison will throw up the right threshold candidate among the various

choices of Bx available. However, this exercise does not show any candidate that can be suitably judged to be the right threshold. This is also hampered by the unavailability of probe readings at more locations. At best, it seems that the ratio observed using this combination of group velocity and energy comparisons invariably oscillates about a zero mean. This aspect remains rather poorly understood. 3.7. Other kinematic properties Another measure of the performance of the Tian–Barthelemy model can be discerned from a measure of group velocity. For this a waterfall plot of the maxima of Hilbert envelope along with probe readings is shown in Figs. 22 and 23. It is apparent that

S.A. Hasan, V. Sriram and R.P. Selvam / European Journal of Mechanics / B Fluids 78 (2019) 115–138 Table 5 Appropriate Bx threshold values for calibrating Tian–Barthelemy model against SR15 breaking wave groups. Case

Threshold Bx

f068a10029 f068a20029 f068b10034 f068b20034 f068c10038 f068c20038 f108b10040 f108b20040 f108c10061 f108c20061

0.80–0.92 0.83–0.84 0.88–0.92 0.88–0.92 0.80–0.92 0.83–0.90 0.80–0.92 0.80–0.92 0.80–0.92 0.80–0.92

before focusing the crest slows down and then accelerates a little after each breaking event towards frequency weighted estimate of the group velocity (Cgs in Table 3) rather than central group velocity(Cgc ). This can also be used as a metric to calibrate the right threshold Bx . 3.8. Appropriate threshold revisited Based on various evaluations done on the basis of predicting multiplicity of breaking event, comparison to elevations and spectra, efficacy of predicted viscosity and kinematic properties

133

vis-a-vis SR15 experiments, appropriate thresholds Bx for Tian– Barthelemy model are summarised in Table 5. Although a large set of thresholds perform more or less similar fashion in some cases values close to 0.90 perform better. Note that this evaluation is solely on the basis of simulations and regardless of value suggested by Barthelemy et al. [21], Saket et al. [26] and Seiffert and Ducrozet [28]. In this respect Barthelemy et al.’s assertion on a hard stop value of Bx ∼ 0.85 should be weakened to include some scatter in the thresholds for intermediate water depths. 4. Conclusions In this study, an extension of Tian–Barthelemy wave breaking model [27,52,53] has been mathematically formulated for HOST formulations [9] and implemented in HOS-NWT. Due to this extension of the breaking model to second-order wave making in HOST formulations by the authors, Schäffer [6] signals can be used to suppress the spurious free wave commonly seen in focusing wave groups generated in wave flumes. Thereafter, this model is used to test simulation results against phase speed focused wave groups studied in SR15 [7]. It is shown that based on Drazen et al. [32] weighted average, the SR15 wave groups should be classified as intermediate depth wave groups. Various kinematical properties point in this direction. The appropriateness of Barthelemy-type [21] threshold (Bx = U /C ) to breaking in intermediate depth water wave groups has

Table 6 Calculated parameters from Tian–Barthelemy model (Fig. 3) with α = 0.02 for case f068a10029. Threshold Bx

Exceedences

0.80

24

Dominants 2

x − xt (m) 1.72

−2.79 0.81

25

2

1.72

−2.68 0.82

32

2

1.83

−2.79 0.83

31

2

1.83

−2.79 0.84

31

3

1.83

−5.64 −2.79 0.85

30

2

1.88

−2.79 0.86

29

2

1.88

−2.79 0.87

32

2

1.88

−2.79 0.88

25

2

1.88

−2.79 0.89

37

6

1.99

−5.16 −2.79

0.90

36

6

0.91

34

6

0.92

34

6

12.89 11.23 −9.08 1.99 −5.16 −2.79 12.89 11.23 −9.08 1.99 −5.16 −2.79 12.89 11.23 17.56 1.99 −5.16 −2.79 12.89 11.23 17.56

t (s)

ν x103

Lbr (m)

Hbr (m)

Tbr (s)

kb (m−1 )

wb

Sb

Rb

(m2 s−1 )

64.653 76.840 64.653 76.940 64.700 76.827 64.700 76.827 64.707 73.953 76.827 64.733 76.827 64.733 76.827 64.740 76.827 64.747 76.827 64.787 74.413 76.827 93.560 96.987 99.647 64.787 74.420 76.827 93.560 96.987 99.647 64.793 74.420 76.827 93.560 96.980 99.640 64.793 74.420 76.827 93.560 96.980 99.640

2.759 0.010 2.759 0.019 2.253 0.065 2.253 0.065 2.787 0.019 0.064 2.763 0.062 2.763 0.062 2.766 0.062 2.781 0.062 2.768 0.013 0.037 0.015 0.007 0.009 2.768 0.009 0.037 0.015 0.007 0.009 2.772 0.008 0.037 0.019 0.041 0.019 2.772 0.008 0.037 0.019 0.041 0.019

2.827 0.009 2.827 0.024 2.842 0.075 2.842 0.075 2.921 0.024 0.072 2.840 0.070 2.840 0.070 2.849 0.070 2.900 0.070 2.856 0.011 0.039 0.018 0.021 0.016 2.856 0.007 0.039 0.018 0.021 0.016 2.868 0.006 0.039 0.025 0.060 0.013 2.868 0.006 0.039 0.025 0.060 0.013

0.081 0.014 0.081 0.009 0.066 0.014 0.066 0.014 0.081 0.009 0.014 0.081 0.014 0.081 0.014 0.081 0.014 0.081 0.014 0.081 0.018 0.014 0.009 0.004 0.005 0.081 0.018 0.014 0.009 0.004 0.005 0.081 0.018 0.014 0.009 0.009 0.014 0.081 0.018 0.014 0.009 0.009 0.014

1.650 0.266 1.650 0.234 1.656 0.316 1.656 0.316 1.688 0.234 0.314 1.655 0.312 1.655 0.312 1.659 0.312 1.679 0.312 1.662 0.306 0.289 0.228 0.254 0.170 1.662 0.304 0.289 0.228 0.254 0.170 1.666 0.304 0.289 0.235 0.267 0.197 1.666 0.304 0.289 0.235 0.267 0.197

2.659 9.748 2.659 14.623 2.659 9.748 2.659 9.748 2.659 14.623 9.748 2.659 9.748 2.659 9.748 2.659 9.748 2.659 9.748 2.659 7.311 9.748 14.623 11.698 29.245 2.659 7.311 9.748 14.623 11.698 29.245 2.659 7.311 9.748 14.623 14.623 19.497 2.659 7.311 9.748 14.623 14.623 19.497

4.986 9.779 4.986 11.977 4.986 9.779 4.986 9.779 4.986 11.977 9.779 4.986 9.779 4.986 9.779 4.986 9.779 4.986 9.779 4.986 8.469 9.779 11.977 10.713 16.938 4.986 8.469 9.779 11.977 10.713 16.938 4.986 8.469 9.779 11.977 11.977 13.830 4.986 8.469 9.779 11.977 11.977 13.830

0.3710 0.0654 0.3710 0.0764 0.3726 0.0917 0.3726 0.0917 0.3813 0.0760 0.0908 0.3724 0.0897 0.3724 0.0897 0.3734 0.0897 0.3790 0.0896 0.3742 0.0649 0.0773 0.0726 0.0721 0.0807 0.3742 0.0639 0.0773 0.0726 0.0721 0.0807 0.3755 0.0637 0.0773 0.0766 0.0976 0.0723 0.3755 0.0637 0.0773 0.0766 0.0976 0.0723

0.5909 0.5000 0.5909 0.5000 0.5455 0.5000 0.5455 0.5000 0.5909 0.5000 0.5000 0.5909 0.5000 0.5909 0.5000 0.5909 0.5000 0.5909 0.5000 0.5909 0.5000 0.5000 0.5000 0.4000 0.5000 0.5909 0.5000 0.5000 0.5000 0.4000 0.5000 0.5909 0.5000 0.5000 0.5000 0.5000 0.6667 0.5909 0.5000 0.5000 0.5000 0.5000 0.6667

(s−1 )

134

S.A. Hasan, V. Sriram and R.P. Selvam / European Journal of Mechanics / B Fluids 78 (2019) 115–138

Table 7 Calculated parameters from Tian–Barthelemy model (Fig. 3) with α = 0.02 for case f068a20029. Threshold Bx

Exceedences

0.80

98

Dominants 2

0.81

84

3

0.82

76

3

0.83

72

2

x − xt (m)

t (s)

ν x103

Lbr (m)

Hbr (m)

Tbr (s)

kb (m−1 )

wb

Sb

Rb

(m2 s−1 )

−3.54

60.713 64.553 60.720 64.547 92.613 60.727 64.573 92.613 60.780 64.580

1.888 2.420 1.731 2.759 0.016 1.730 2.766 0.015 1.728 2.428

2.741 2.856 2.779 2.827 0.011 2.773 2.850 0.010 2.764 2.889

0.056 0.071 0.051 0.081 0.014 0.051 0.081 0.014 0.051 0.071

1.623 1.671 1.631 1.650 0.195 1.628 1.659 0.194 1.624 1.684

2.785 2.785 2.659 2.659 19.497 2.659 2.659 19.497 2.659 2.785

5.124 5.124 4.986 4.986 13.830 4.986 4.986 13.830 4.986 5.124

0.3759 0.3891 0.3658 0.3710 0.0707 0.3651 0.3736 0.0697 0.3641 0.3929

0.5238 0.5714 0.5000 0.5909 0.6667 0.5000 0.5909 0.6667 0.5000 0.5714

1.29 −3.54 1.29 15.25 −3.54 1.34 15.25 −3.44 1.34

(s−1 )

Table 8 Calculated parameters from Tian–Barthelemy model (Fig. 3) with α = 0.02 for case f068b10034. Threshold Bx

Exceedences

0.80

3

Dominants 2

0.81

2

2

0.82

5

2

0.83

2

2

0.84

5

2

0.85

4

2

0.86

3

2

0.87

2

2

0.88

3

2

0.89

4

2

0.90

5

2

0.91

3

2

0.92

5

2

x − xt (m)

t (s)

ν x103 (m2 s−1 )

Lbr (m)

Hbr (m)

Tbr (s)

kb (m−1 )

wb

Sb

Rb

−4.13

44.213 48.213 44.240 48.213 44.267 48.240 44.293 48.220 44.300 48.240 44.347 48.240 44.353 48.267 44.380 48.233 44.400 48.273 44.427 48.260 44.453 48.267 44.480 48.287 44.487 48.287

2.574 3.761 2.573 3.761 2.575 3.769 2.399 4.106 2.399 3.772 2.397 3.772 2.401 3.777 2.067 4.113 2.397 4.130 2.067 3.793 2.066 4.327 2.068 3.800 2.065 3.800

2.758 2.751 2.755 2.750 2.761 2.769 2.737 2.764 2.736 2.774 2.730 2.774 2.744 2.786 2.762 2.777 2.730 2.811 2.764 2.820 2.759 2.859 2.767 2.836 2.754 2.835

0.075 0.110 0.075 0.110 0.075 0.110 0.070 0.120 0.070 0.110 0.070 0.110 0.070 0.110 0.061 0.120 0.070 0.120 0.061 0.110 0.061 0.125 0.061 0.110 0.061 0.110

1.616 1.606 1.615 1.605 1.617 1.612 1.603 1.609 1.603 1.615 1.601 1.614 1.606 1.619 1.618 1.614 1.601 1.627 1.618 1.632 1.616 1.647 1.619 1.638 1.614 1.638

2.543 2.340 2.543 2.340 2.543 2.340 2.437 2.250 2.437 2.340 2.437 2.340 2.437 2.340 2.543 2.250 2.437 2.250 2.543 2.340 2.543 2.340 2.543 2.340 2.543 2.340

4.856 4.615 4.856 4.615 4.856 4.615 4.732 4.502 4.732 4.615 4.732 4.615 4.732 4.615 4.856 4.502 4.732 4.502 4.856 4.615 4.856 4.615 4.856 4.615 4.856 4.615

0.3504 0.3266 0.3501 0.3265 0.3506 0.3283 0.3362 0.3176 0.3361 0.3288 0.3356 0.3288 0.3369 0.3300 0.3508 0.3188 0.3356 0.3220 0.3509 0.3333 0.3504 0.3370 0.3513 0.3347 0.3500 0.3347

0.5652 0.6400 0.5652 0.6400 0.5652 0.6400 0.5417 0.6538 0.5417 0.6400 0.5417 0.6400 0.5417 0.6400 0.5217 0.6538 0.5417 0.6538 0.5217 0.6400 0.5217 0.6800 0.5217 0.6400 0.5217 0.6400

0.81 −4.08 0.81 −4.03 0.86 −3.97 0.81 −3.97 0.86 −3.87 0.86 −3.87 0.91 −3.81 0.86 −3.76 0.91 −3.71 0.91 −3.65 0.91 −3.60 0.97 −3.60 0.97

(s−1 )

Table 9 Calculated parameters from Tian–Barthelemy model (Fig. 3) with α = 0.02 for case f068b20034. Threshold Bx 0.80

Exceedences 3

Dominants 2

0.81

7

2

0.82

6

2

0.83

6

2

0.84

5

2

0.85

5

2

0.86

6

2

0.87

6

2

0.88 0.89 0.90 0.91 0.92

11 14 14 11 10

1 1 1 1 1

x − xt (m)

t (s)

ν x103 (m2 s−1 )

Lbr (m)

Hbr (m)

Tbr (s)

kb (m−1 )

wb

Sb

Rb

−3.71

44.580 48.180 44.633 48.187 44.640 48.173 44.693 48.193 44.747 48.200 44.780 48.193 44.853 48.193 44.913 48.187 48.120 48.127 48.127 48.147 48.147

2.046 3.810 2.043 4.343 1.712 4.153 1.711 4.171 1.710 3.830 1.535 4.175 1.532 4.176 1.201 4.372 4.703 4.703 4.703 4.724 4.724

2.678 2.857 2.665 2.891 2.684 2.857 2.678 2.892 2.673 2.904 2.635 2.902 2.619 2.903 2.620 2.951 2.916 2.916 2.916 2.955 2.955

0.061 0.110 0.061 0.125 0.051 0.120 0.051 0.120 0.051 0.110 0.046 0.120 0.046 0.120 0.036 0.125 0.134 0.134 0.134 0.134 0.134

1.584 1.646 1.579 1.659 1.592 1.644 1.590 1.658 1.588 1.664 1.567 1.661 1.561 1.661 1.566 1.683 1.666 1.667 1.667 1.681 1.681

2.543 2.340 2.543 2.340 2.659 2.250 2.659 2.250 2.659 2.340 2.543 2.250 2.543 2.250 2.659 2.340 2.250 2.250 2.250 2.250 2.250

4.856 4.615 4.856 4.615 4.986 4.502 4.986 4.502 4.986 4.615 4.856 4.502 4.856 4.502 4.986 4.615 4.502 4.502 4.502 4.502 4.502

0.3420 0.3368 0.3406 0.3401 0.3554 0.3262 0.3547 0.3295 0.3541 0.3414 0.3375 0.3304 0.3359 0.3305 0.3484 0.3459 0.3317 0.3317 0.3317 0.3353 0.3353

0.5217 0.6400 0.5217 0.6800 0.5000 0.6538 0.5000 0.6538 0.5000 0.6400 0.4783 0.6538 0.4783 0.6538 0.4545 0.6800 0.6923 0.6923 0.6923 0.6923 0.6923

0.48 −3.60 0.48 −3.60 0.48 −3.49 0.54 −3.38 0.54 −3.33 0.54 −3.17 0.54 −3.06 0.54 0.43 0.43 0.43 0.48 0.48

(s−1 )

S.A. Hasan, V. Sriram and R.P. Selvam / European Journal of Mechanics / B Fluids 78 (2019) 115–138

135

Table 10 Calculated parameters from Tian–Barthelemy model (Fig. 3) with α = 0.035 for case f108c10061. Threshold Bx = 0.87 is ommited due to computational difficulties. Threshold Bx

Exceedences

0.80

9

Dominants 7

x − xt (m)

t (s)

ν x103

Lbr (m)

Hbr (m)

Tbr (s)

kb (m−1 )

wb

Sb

Rb

(m2 s−1 )

−5.32 −2.15 −1.34

30.487 35.280 37.727 40.993 58.647 60.000 61.387 30.493 35.313 37.727 41.020 58.647 60.000 62.667 30.527 35.287 37.747 40.987 58.660 61.247 30.527 35.287 37.747 41.013 58.700 60.007 30.527 35.287 37.753 41.060 58.707 60.000 62.727 30.527 35.287 37.753 41.087 58.707 60.013 30.533 35.287 37.753 41.087 58.707 60.060 30.573 35.327 37.760 41.120 58.747 60.160 30.573 35.327 37.760 41.120 58.767 61.240 30.580 35.340 37.780 41.040 58.773 60.100 30.580 35.340 37.780 41.040 58.800 61.227 30.620 35.333 37.780 41.153 58.807 60.153

1.323 2.841 5.393 5.177 0.378 0.382 0.415 1.750 2.234 5.413 5.215 0.377 0.383 0.409 1.322 2.590 5.109 5.192 0.560 0.692 1.322 2.590 5.109 5.235 0.383 0.842 1.322 2.590 5.431 5.275 0.329 0.379 0.726 1.322 2.590 5.431 5.286 0.328 0.556 1.297 2.564 5.428 5.288 0.328 0.542 1.333 2.653 5.429 5.316 0.400 0.322 1.333 2.653 5.429 5.316 0.534 0.681 0.848 2.818 4.562 5.249 0.328 0.399 0.848 2.818 4.562 5.249 0.405 0.988 0.814 3.123 5.437 4.804 0.325 0.330

0.798 1.030 2.444 2.127 0.491 0.505 0.626 0.886 1.004 2.473 2.169 0.488 0.506 0.604 0.798 0.971 2.406 2.143 0.660 0.566 0.798 0.971 2.406 2.192 0.509 0.590 0.798 0.971 2.502 2.238 0.588 0.496 0.638 0.798 0.971 2.502 2.252 0.587 0.590 0.762 0.949 2.496 2.254 0.587 0.593 0.813 1.027 2.497 2.288 0.568 0.554 0.813 1.027 2.497 2.288 0.567 0.545 0.910 1.010 2.454 2.209 0.586 0.563 0.910 1.010 2.454 2.209 0.585 0.567 0.822 0.943 2.511 2.349 0.567 0.594

0.038 0.072 0.096 0.095 0.014 0.014 0.014 0.048 0.057 0.096 0.095 0.014 0.014 0.014 0.038 0.067 0.091 0.095 0.019 0.024 0.038 0.067 0.091 0.095 0.014 0.029 0.038 0.067 0.096 0.095 0.012 0.014 0.024 0.038 0.067 0.096 0.095 0.012 0.018 0.038 0.067 0.096 0.095 0.012 0.019 0.038 0.067 0.096 0.095 0.014 0.012 0.038 0.067 0.096 0.095 0.019 0.024 0.023 0.072 0.081 0.095 0.012 0.014 0.023 0.072 0.081 0.095 0.014 0.033 0.023 0.082 0.096 0.086 0.012 0.012

0.802 0.918 1.521 1.372 0.630 0.640 0.732 0.846 0.904 1.534 1.389 0.628 0.642 0.715 0.802 0.882 1.496 1.378 0.783 0.676 0.802 0.882 1.496 1.398 0.643 0.705 0.802 0.882 1.546 1.417 0.723 0.634 0.726 0.802 0.882 1.546 1.422 0.722 0.685 0.781 0.870 1.544 1.423 0.722 0.727 0.811 0.910 1.544 1.437 0.688 0.695 0.811 0.910 1.544 1.437 0.705 0.661 0.867 0.907 1.526 1.405 0.721 0.684 0.867 0.907 1.526 1.405 0.701 0.670 0.816 0.867 1.550 1.466 0.705 0.728

5.849 4.874 3.078 2.659 9.748 9.748 9.748 5.317 4.874 3.078 2.659 9.748 9.748 9.748 5.849 4.499 2.925 2.659 11.698 8.356 5.849 4.499 2.925 2.659 9.748 9.748 5.849 4.499 3.078 2.659 11.698 9.748 8.356 5.849 4.499 3.078 2.659 11.698 7.311 5.849 4.499 3.078 2.659 11.698 11.698 5.849 4.499 3.078 2.659 9.748 11.698 5.849 4.499 3.078 2.659 11.698 8.356 5.849 4.874 3.078 2.659 11.698 9.748 5.849 4.874 3.078 2.659 9.748 7.311 5.849 4.499 3.078 2.785 11.698 11.698

7.572 6.906 5.412 4.970 9.779 9.779 9.779 7.217 6.906 5.412 4.970 9.779 9.779 9.779 7.572 6.629 5.256 4.970 10.713 9.054 7.572 6.629 5.256 4.970 9.779 9.779 7.572 6.629 5.412 4.970 10.713 9.779 9.054 7.572 6.629 5.412 4.970 10.713 8.469 7.572 6.629 5.412 4.970 10.713 10.713 7.572 6.629 5.412 4.970 9.779 10.713 7.572 6.629 5.412 4.970 10.713 9.054 7.572 6.906 5.412 4.970 10.713 9.779 7.572 6.906 5.412 4.970 9.779 8.469 7.572 6.629 5.412 5.109 10.713 10.713

0.2539 0.2683 0.3713 0.2944 0.2586 0.2642 0.3130 0.2557 0.2632 0.3751 0.2990 0.2577 0.2649 0.3042 0.2538 0.2415 0.3513 0.2962 0.3797 0.2564 0.2538 0.2415 0.3513 0.3016 0.2658 0.2984 0.2538 0.2415 0.3786 0.3066 0.3447 0.2608 0.2810 0.2538 0.2415 0.3786 0.3081 0.3442 0.2392 0.2452 0.2374 0.3780 0.3084 0.3442 0.3473 0.2575 0.2518 0.3781 0.3121 0.2898 0.3284 0.2575 0.2518 0.3781 0.3121 0.3345 0.2492 0.2808 0.2642 0.3726 0.3034 0.3439 0.2876 0.2808 0.2642 0.3726 0.3034 0.2964 0.2324 0.2596 0.2363 0.3798 0.3310 0.3345 0.3479

0.6000 0.7500 0.6842 0.6364 0.5000 0.5000 0.5000 0.6364 0.6667 0.6842 0.6364 0.5000 0.5000 0.5000 0.6000 0.6923 0.6500 0.6364 0.6000 0.5714 0.6000 0.6923 0.6500 0.6364 0.5000 0.6667 0.6000 0.6923 0.6842 0.6364 0.5000 0.5000 0.5714 0.6000 0.6923 0.6842 0.6364 0.5000 0.5000 0.6000 0.6923 0.6842 0.6364 0.5000 0.6000 0.6000 0.6923 0.6842 0.6364 0.5000 0.5000 0.6000 0.6923 0.6842 0.6364 0.6000 0.5714 0.5000 0.7500 0.6316 0.6364 0.5000 0.5000 0.5000 0.7500 0.6316 0.6364 0.5000 0.6250 0.5000 0.7692 0.6842 0.6190 0.5000 0.5000

2.15

0.81

9

7

−8.65 −7.89 −7.14 −5.32 −2.15 −1.34 2.20

0.82

8

6

−8.65 −7.89 −6.44 −5.26 −2.15 −1.29 2.15

0.83

6

6

−8.65 −7.25 −5.26 −2.15 −1.29

7

−8.59 −7.89 −5.26 −2.15 −1.29

2.20

0.84

8

2.26

0.85

8

6

−8.59 −7.89 −6.39 −5.26 −2.15 −1.29 2.31

0.86

7

6

−8.59 −7.89 −5.26 −2.15 −1.29

6

−8.59 −7.84 −5.21 −2.09 −1.29

6

−8.59 −7.79 −5.21 −2.09 −1.29

6

−8.54 −7.25 −5.21 −2.04 −1.23

6

−8.54 −7.84 −5.21 −2.04 −1.23

6

−8.54 −7.25 −5.16 −2.04 −1.23

2.31

0.88

9

2.36

0.89

8

2.36

0.90

9

2.26

0.91

8

2.26

0.92

8

2.42

−8.54 −7.79

(s−1 )

136

S.A. Hasan, V. Sriram and R.P. Selvam / European Journal of Mechanics / B Fluids 78 (2019) 115–138

Table 11 Calculated parameters from Tian–Barthelemy model (Fig. 3) with α = 0.035 for case f108c20061. Threshold Bx = 0.87 is omitted due to computational difficulties. Threshold Bx

Exceedences

0.80

9

Dominants 6

x − xt (m)

t (s)

ν x103

Lbr (m)

Hbr (m)

Tbr (s)

kb (m−1 )

wb

Sb

Rb

(m2 s−1 )

−5.21 −2.09 −1.34

30.580 35.373 37.793 40.960 58.780 61.307 30.620 35.380 37.807 40.920 58.780 61.307 30.620 35.380 37.813 40.973 58.833 61.307 30.620 35.380 37.813 40.993 58.833 61.307 30.627 35.407 37.813 41.000 58.840 61.307 30.633 35.393 37.813 41.007 58.853 61.300 30.667 35.400 37.820 41.007 58.867 60.233 61.533 30.673 35.440 37.820 41.007 58.993 60.120 30.680 35.433 37.820 41.040 59.987 62.767 30.720 35.433 37.840 40.987 59.987 62.767 30.720 35.433 37.840 40.993 59.993 62.760 30.733 35.413 37.840 40.993 59.993 62.767

1.299 2.253 5.384 5.250 0.539 0.701 1.329 2.241 4.554 6.047 0.539 0.700 1.329 2.241 5.422 5.219 0.530 0.693 1.329 2.241 5.422 5.268 0.530 0.693 1.564 2.197 5.436 5.261 0.534 0.694 1.388 2.211 5.427 5.230 0.410 0.969 0.997 2.753 5.402 5.224 0.399 0.318 0.627 1.564 2.252 5.155 6.133 0.318 0.548 1.396 2.249 5.422 5.325 0.414 1.208 1.403 2.239 4.563 5.282 0.413 1.208 1.403 2.239 4.563 5.253 0.402 1.251 1.032 2.212 4.555 5.242 0.402 1.233

0.765 1.026 2.431 2.210 0.585 0.584 0.808 1.013 2.440 2.187 0.583 0.583 0.808 1.013 2.487 2.174 0.554 0.569 0.808 1.013 2.487 2.231 0.555 0.569 0.914 0.965 2.508 2.222 0.567 0.570 0.901 0.979 2.495 2.186 0.606 0.543 0.849 0.954 2.458 2.179 0.562 0.534 0.476 0.913 1.025 2.474 2.275 0.533 0.616 0.914 1.022 2.488 2.298 0.622 0.502 0.925 1.010 2.455 2.247 0.620 0.502 0.925 1.010 2.455 2.213 0.573 0.539 0.932 0.981 2.441 2.200 0.574 0.524

0.038 0.057 0.096 0.095 0.019 0.024 0.038 0.057 0.081 0.110 0.019 0.024 0.038 0.057 0.096 0.095 0.019 0.024 0.038 0.057 0.096 0.095 0.019 0.024 0.043 0.057 0.096 0.095 0.019 0.024 0.038 0.057 0.096 0.095 0.014 0.033 0.028 0.072 0.096 0.095 0.014 0.012 0.023 0.043 0.057 0.091 0.110 0.012 0.019 0.038 0.057 0.096 0.095 0.014 0.043 0.038 0.057 0.081 0.095 0.014 0.043 0.038 0.057 0.081 0.095 0.014 0.043 0.028 0.057 0.081 0.095 0.014 0.043

0.783 0.916 1.516 1.405 0.721 0.688 0.808 0.908 1.520 1.396 0.719 0.687 0.808 0.908 1.540 1.391 0.695 0.678 0.808 0.908 1.540 1.414 0.695 0.678 0.881 0.883 1.549 1.410 0.705 0.678 0.862 0.890 1.543 1.396 0.717 0.655 0.841 0.877 1.527 1.393 0.684 0.678 0.613 0.881 0.915 1.525 1.432 0.677 0.746 0.870 0.913 1.540 1.441 0.728 0.627 0.876 0.907 1.526 1.420 0.727 0.627 0.876 0.907 1.526 1.407 0.692 0.650 0.892 0.891 1.520 1.401 0.693 0.641

5.849 4.874 3.078 2.659 11.698 8.356 5.849 4.874 3.078 2.659 11.698 8.356 5.849 4.874 3.078 2.659 11.698 8.356 5.849 4.874 3.078 2.659 11.698 8.356 6.499 4.874 3.078 2.659 11.698 8.356 5.849 4.874 3.078 2.659 9.748 7.311 6.499 4.874 3.078 2.659 9.748 11.698 5.849 6.499 4.874 2.925 2.659 11.698 11.698 5.849 4.874 3.078 2.659 9.748 6.499 5.849 4.874 3.078 2.659 9.748 6.499 5.849 4.874 3.078 2.659 9.748 6.499 6.499 4.874 3.078 2.659 9.748 6.499

7.572 6.906 5.412 4.970 10.713 9.054 7.572 6.906 5.412 4.970 10.713 9.054 7.572 6.906 5.412 4.970 10.713 9.054 7.572 6.906 5.412 4.970 10.713 9.054 7.983 6.906 5.412 4.970 10.713 9.054 7.572 6.906 5.412 4.970 9.779 8.469 7.983 6.906 5.412 4.970 9.779 10.713 7.572 7.983 6.906 5.256 4.970 10.713 10.713 7.572 6.906 5.412 4.970 9.779 7.983 7.572 6.906 5.412 4.970 9.779 7.983 7.572 6.906 5.412 4.970 9.779 7.983 7.983 6.906 5.412 4.970 9.779 7.983

0.2460 0.2676 0.3698 0.3035 0.3435 0.2625 0.2563 0.2649 0.3709 0.3011 0.3424 0.2622 0.2563 0.2649 0.3768 0.2996 0.3284 0.2573 0.2563 0.2649 0.3768 0.3058 0.3288 0.2573 0.3061 0.2553 0.3795 0.3049 0.3346 0.2577 0.2785 0.2581 0.3778 0.3009 0.3050 0.2252 0.2887 0.2531 0.3731 0.3001 0.2873 0.3187 0.1763 0.3059 0.2674 0.3595 0.3106 0.3182 0.3583 0.2817 0.2667 0.3769 0.3132 0.3111 0.1961 0.2845 0.2644 0.3727 0.3075 0.3103 0.1961 0.2845 0.2644 0.3727 0.3039 0.2917 0.2060 0.3110 0.2584 0.3710 0.3024 0.2922 0.2019

0.6000 0.6667 0.6842 0.6364 0.6000 0.5714 0.6000 0.6667 0.6316 0.6818 0.6000 0.5714 0.6000 0.6667 0.6842 0.6364 0.6000 0.5714 0.6000 0.6667 0.6842 0.6364 0.6000 0.5714 0.6667 0.6667 0.6842 0.6364 0.6000 0.5714 0.6000 0.6667 0.6842 0.6364 0.5000 0.6250 0.5556 0.7500 0.6842 0.6364 0.5000 0.5000 0.5000 0.6667 0.6667 0.6500 0.6818 0.5000 0.6000 0.6000 0.6667 0.6842 0.6364 0.5000 0.6667 0.6000 0.6667 0.6316 0.6364 0.5000 0.6667 0.6000 0.6667 0.6316 0.6364 0.5000 0.6667 0.5556 0.6667 0.6316 0.6364 0.5000 0.6667

2.04

0.81

8

6

−8.54 −7.20 −5.16 −2.09 −1.29

6

−8.54 −7.20 −5.16 −2.09 −1.29

6

−8.49 −7.20 −5.16 −2.09 −1.29

6

−8.49 −7.20 −5.16 −2.09 −1.29

6

−8.49 −7.20 −5.16 −2.09 −1.29

7

−8.49 −7.20 −5.10 −2.04 −1.29

1.99

0.82

9

2.04

0.83

8

2.10

0.84

8

2.10

0.85

9

2.10

0.86

8

2.10

0.88

9

6

−8.49 −7.73 −7.04 −5.10 −2.04 −1.29 2.10

0.89

8

6

−8.38 −7.79 −5.10 −2.04 −1.29

6

−7.89 −6.34 −5.05 −2.04 −1.23

6

−7.89 −6.34 −5.05 −2.04 −1.23

6

−7.89 −6.34 −5.05 −2.04 −1.23

2.15

0.90

8

2.10

0.91

7

2.10

0.92

8

2.10

−7.89 −6.34

(s−1 )

S.A. Hasan, V. Sriram and R.P. Selvam / European Journal of Mechanics / B Fluids 78 (2019) 115–138

also been studied. It seems that though the threshold exists around 0.85 [28], there seems to be a considerable scatter for intermediate depth water wave groups, in which some wave groups do not break even around Bx ∼ 0.90 − 0.92 in line with [8]. As such, right threshold in such conditions is not so readily discernable in order to calibrate Tian–Barthelemy model. However, the most appropriate ones are suggested in Table 5 for the studied wave groups based on varied metrics. However, the search for a universal threshold in intermediate depths remains inconclusive. With such uncertainty around the threshold, applying the Tian–Barthelemy model should be a two-step process. First, to identify whether a wave groups break, the model should be run in absence of a threshold to identify whether the computations breakdown. Then, an appropriate threshold be specified for the eddy viscosity model. It is seen that the usage of second order wavemaking to suppress spurious free waves, infact, makes the shorter modes dominant near breaking events making the computation of local phase speed based on Hilbert transforms [23] difficult. Thus, from a computational point of view suppression of spurious waves for breaking wave groups is not beneficial. Inspite of that, Tian–Barthelemy model performs satisfactorily for multiple wave breaking events. A phase difference in the time evolution of simulated wave groups versus SR15 experiments is observed which becomes progressively larger as the strength and multiplicity of the breaking events increase. This has been attributed to the discrepancy in the location of breaking for simulated wave groups vis-a-vis experiments. Tian–Barthelemy wave breaking model applies across the wave frequencies and as such the spectral signature of wave breaking in dispersively focused waves i.e. the slower frequencies also tend to lose energy. This readily suggests an improvement technique in Tian–Barthelemy model whereby the eddy viscosity terms are passed through a high-pass filter. This improvement will be attempted in future studies. A drawback in breaking model [27] is the absence of parameterisation of air entrainment during breaking events which makes the proportionality constant α a subjective tuning parameter. Studies that parameterise this proportionality constant are, therefore, warranted and would significantly improve this aspect. Acknowledgement The authors would like to thank Dr. Guillaume Ducrozet, LHEEA, ECN-Nantes, France for providing the open-source code HOS-NWT which has been modified by the authors for this study. The authors also thank the reviewers for their valuable comments. Appendix. Eddy viscosity calculations See Tables 6–11. References [1] M.K. Ochi, Ocean Waves, the Stochastic Approach, Cambridge University Press, 1998, p. 319. [2] L.A. O’Neill, E. Fakas, M. Cassidy, A methodology to simulate floating offshore operations using a design wave theory, J. Offshore Mech. Arct. Eng. 128 (4) (2006) 304. [3] C. Kharif, E. Pelinovsky, Physical mechanisms of the rogue wave phenomenon, Eur. J. Mech. B/Fluids 22 (6) (2003) 603–634. [4] R.J. Rapp, W.K. Melville, Laboratory measurements of deep-water breaking waves, Phil. Trans. R. Soc. A 331 (1622) (1990) 735–800, URL http://rsta. royalsocietypublishing.org/cgi/doi/10.1098/rsta.1990.0098. [5] R.T. Hudspeth, W. Sulisz, Stokes Drift in two-dimensional wave flumes, J. Fluid Mech. 230 (-1) (1991) 209, URL http://www.journals.cambridge.org/ abstract{_}S0022112091000769.

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