3dmodelingandmechanismanalysisofbreakingwave-induced

Research Article3DModeling andMechanism Analysis of BreakingWave-InducedSeabed Scour around Monopile

Xiaojian Liu123 Cheng Liu 1 Xiaowei Zhu1 Yong He1 Qisong Wang1

and Zhiyuan Wu 3

1Pearl River Hydraulic Research Institute Pearl River Water Resources Commission of the Ministry of Water ResourcesGuangzhou Guangdong 510611 China2School of Civil Engineering Sun Yet-Sen University Guangzhou Guangdong 510611 China3Key Laboratory of Water-Sediment Sciences and Water Disaster Prevention of Hunan Province ChangshaHunan 410114 China

Correspondence should be addressed to Cheng Liu jacklc2004163com

Received 2 September 2019 Revised 1 December 2019 Accepted 6 January 2020 Published 17 March 2020

Academic Editor Mostafa S Shadloo

Copyright copy 2020 Xiaojian Liu et al -is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Breaking wave-induced scour is recognized as one of the major causes of coastal erosion and offshore structure failure whichinvolves in the full 3D water-air-sand interaction raising a great challenge for the numerical simulation To better understand thisprocess a nonlinear 3D numerical model based on the open-source CFD platform OpenFOAMregwas self-developed in this study-e NavierndashStokes equations were used to compute the two-phase incompressible flow combining with the finite volumemethod(FVM) to discretize calculation domain a modified VOF method to track the free surface and a k minus ε model to closure theturbulence -e nearshore sediment transport process is reproduced in view of shear stress suspended load and bed load inwhich the terms of shear stress and suspended load were updated by introducing volume fraction -e seabed morphology isupdated based on Exner equation and implemented by dynamic mesh technique -e mass conservative sand slide algorithm wasemployed to avoid the incredible vary of the bed mesh Importantly a two-way coupling method connecting the hydrodynamicmodule with the beach morphodynamic module is implemented at each computation step to ensure the fluid-sediment in-teraction -e capabilities of this model were calibrated by laboratory data from some published references and the advantagesdisadvantages as well as proper recommendations were introduced Finally nonbreaking- and breaking wave-induced scouraround the monopile as well as breaking wave-induced beach evolution were reproduced and discussed -is study would besignificantly helpful to understand and evaluate the nearshore sediment transport

1 Introduction

As ocean wave gets closer to the coastal region the freesurface deforms and wave steepens with increasing waveheight and may induce unstable and break waves Wavebreaking and the resulting current are the causes of manycomplex phenomena in the surf zone such as the coastlanddrift beach erosion sediment transport and so on-ey willerode the soil-supporting pile foundations [1] which wereconstructed within aforementioned coastal high-hazardareas for practical reasons [2] -erefore assessing thehydrodynamic and sediment transport processes are

important for inferring the strength of potential destructionduring the extreme environment (eg Wu et al [3 4])

Over the last few decades some research studies wereperformed to improve the understanding of above processesField surveys were performed by analyzing sediment di-ameter distribution instrument data and field videos (egParis et al [5] Szczucinski et al [6] FEMA [2] and USGS[7]) which provided some substantial cognitions such as themajority of the deposits are from beach and not from deepocean floor (eg Morton et al [1] and Jaffe et al [8])However the field research studies raise considerably lim-ited information due to frequent lack of the wave condition

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 1647640 17 pageshttpsdoiorg10115520201647640

and topography especially facing the complex three-di-mension high-speed water roller Additionally theoreticalapproaches for studying this process are still inadequate [9]Hence numerical simulations and physical experimentsproviding effective controlling factors become the preferredmethods

Traditionally physical experiments are clearly helpful toassess this process Many research studies have been performedin the aspect of cross-shore sediment transport on sandy beach(eg Kobayashi and Lawrence [10] Young et al [11] Jiang et al[12] and Daghighi et al [13]) and the scour around offshorestructure foundation (eg Kato et al [14 15] Tonkin et al [16]and Kuswandi et al [17]) which improve our understanding ofbeach evolution and its response to structure by waveWith thedevelopment of computer technology the numerical modelprovided us with another way to understand the mechanismswhich solves the limited capability used in measurement de-vices In the past decades modeling sandy beach evolution hasbeen studied with depth-averaged equations such as shallowwater equations (eg Simpson and Castelltort [18] andPritchard and Dickinson [19]) and Boussinesq-type equations(eg Shimozono et al [20] and Xiao et al [21]) However thisprocess is highly nonlinear as well as local but strong tur-bulence near the free surface and the seabed needs to beconsidered Recently Nakamura and Yim [22] Li et al [23]and Jacobsen et al ([24ndash26]) as well as Liu et al [27] in-troduced a volume of fluid-type model to study this processwhich was based on the generalized NavierndashStokes equationswith a turbulence closure solver computing incompressibleviscous multiphase flows Aforementioned research studies notonly substantially improved our understanding of breakingwave-induced longshore sediment transport mechanisms butalso gave us some inspiration to explore the more complexproblems such as sediment-wave-structure interaction in somecoastal areas Pan and Huang [28] are the pioneers to nu-merically investigate the interactions among wave monopileand sandy beach to access the effectiveness of a linked 2Dhydrodynamic and sediment scour model in this processmodeling However some limitations existed such as the 2Dmodel could only simulate vertically averaged currents insteadof 3D currents in this study in particular about the verticalvelocity component [28] Huang et al [29] pointed out that theapproximation of the 3D scour phenomenon around the pilestructure to a 2D problem may be a contribution to errorsbetween the model predictions and observations To the au-thorsrsquo knowledge the research of 3Dmodeling in this questionis still limited so far and yet to be further investigated

To address the aforementioned issues a 3D numericalmodel was developed based on the open-source Open-FOAM platform in this study supporting two-phasedincompressible flow -e wave dynamic is achieved by itssolver interFoam combined with the active wave generationand absorption boundary conditions developed byHiguera et al [30] Unfortunately the official version ofOpenFOAM lacks the function of sediment transportsimulation and the third-party sediment module could notbe open accessed Additionally similar sediment solvingmodules did not necessarily apply to current research casesdue to the complexities of sediment transport mechanisms

under different sea environments -erefore this studyaimed to overcome those gaps by self-extending the sed-iment functionality based on an effective usage of the betterparts from published sediment modules -is developedmodel will be further used to simulate the breaking wave-induced scour around the monopile as well as investigateits difference with the scenario of monopile on a flat seabedand the potential scour mechanisms -e generation ad-vection and dissipation of turbulence in the fluid model arealso important for sediment transport For breaking wave-driven sediment motion several turbulence models havebeen employed namely the RANS model (Li et al [23] andJacobsen [25 26]) and the LES model (Christensen andDeigaard [31] and Christensen [32]) Generally LES wasunder consideration as the turbulence model because of aconsiderable part of the mixing is resolved However whenusing the LES the stochastic nature of the flow reflectsnonlinearly onto the sediment transport and results inlarger spatial gradients hence the allowable calculationstability or time step will be lowered considerably-us theRANS turbulence model was considered in this studybecause it is more preferable to predict the scour profilethan the LES turbulence model [33 34]

-e rest of this paper is organized as follows -enumerical methods including the hydrodynamic modeland beach morphodynamic model as well as corre-sponding boundary conditions and numerical schemesare introduced in Sections 2ndash4 respectively -e vali-dations of the main modules are conducted in Section 5-e model applications and discussions are given inSection 6 -e main conclusions of this study are shownin Section 7

2 Hydrodynamic Model

3D RANS equations were applied to simulate the two-phaseincompressible viscous fluids and the governing equationsare as follows

Continuity equation

nabla middot u 0 (1)

Momentum equation

zρuzt

+ nabla middot ρuuT1113960 1113961 minusnablaplowast minus g middot xnablaρ + nabla middot μnablau + ρτlowast1113858 1113859

+ σTκcnablac

(2)

where x (xi xj xk) is the Cartesian coordinate systemu (ui uj uk) is the velocity vector ρ is the fluid density plowastis the pressure in excess of hydrostatic and g is the gravi-tational acceleration

-e last term of equation (2) is used to describe thesurface tension in which σT is the surface tension coefficientκc is the surface curvature and c is a scale field used to trackthe fluid movement τlowast is the Reynolds stress tensor given by

2 Mathematical Problems in Engineering

τlowast 2ρμS minus

23

kI (3)

where μ is the eddy viscosity S (nablau + (nablau)T)2 is the rateof strain tensor k is the turbulent kinetic energy and I is theKronecker delta function confirmed as 1 if i equal to j and as0 if i unequal to j

-e second order standard k minus ε turbulent model wasemployed in this study to closure the set of RANS equations

k 12u middot uT

ε C075μ k15

l

(4)

where Cμ is the dimensionless coefficient l is the turbulencelength scale and ε is the turbulent dissipation rate

-e transport equations of k and ε are as followszk

zt+ nabla(ku) nabla

]t

σk

nablak1113888 1113889 + 2]t

ρ|nablau|

2minus ε

zεzt

+ nabla(εu) nabla]t

σεnablaε1113888 1113889 + 2C1ε]t|nablau|

2εk

minus C2εε2

k

(5)

where ]t is the turbulence kinematic viscosity and C1ε C2εσk and σε are the empirical coefficients confirmed as 144192 1 and 13 respectively

-e free-surface motions were captured by the modifiedVOF method and defined as

zαzt

+ nabla middot uα + nabla middot urα(1 minus α) 0 (6)

where the air and water are described by volume fraction (α)ranging from 0 (air) to 1 (water) Compared with the tra-ditional VOF method the extra compression term wasintroduced in the last term of equation (6) to limit theexcessive numerical diffusion and the smearing of the in-terface where ur is the relative velocity

3 Beach Morphodynamic Model

In this study the details on each component of the beachmorphodynamic module are selected and tested accordingto a sequence of classic examples and meanwhile someproper parameters are recommended which is a compre-hensive usage of some previous research studies

-e bed shear stress linking the hydrodynamic featuresand sediment transport is estimated by the method pro-posed by Arzani et al [35]

τ τt minus τt middot n( 1113857n (7)

where τt is the wall traction confirmed as σ middot n n is the unitnormal vector that is perpendicular to the bed surface meshand σ is the stress tensor given by

σ minuspI + 2μS (8)

where the first term is the pressure-induced stress term p isthe pressure and the second term is the viscous stress termcontrolled by the bottom fluid motion

-e volume fraction was introduced into equation (7) toidentify the bed shear stress of the beach surface

τ ατ (9)

-e bed load transport was described by the formulaproposed by Engelund and Fredsoslashe [36]

qb 1874 θ minus θc( 1113857 θ05 minus θ05

c1113872 1113873Rgd50

1113968d50 if θgt θc

0 if θlt θc

⎧⎪⎨

⎪⎩

θ τ

ρgRd50

R ρsed minus ρ

ρ

(10)

where qb is the bed load transport rate per unit width d50 isthe median sediment diameter R is the relative density of thesediment θ is the Shields number and θc is the criticalShields number considering the effect of seabed slope (Allen[37]) and defined as

θc

θco

cos β +sin βtanφ

(11)

Equation (11) indicates that θc is adjusted to a lowervalue for the sediment moving down the slope (ie negativeslope) and a higher value for the sediment moving up theslope (ie positive angle) and φ is the sediment reposeangle θco is threshold shields number under flat bed whichcould be calculated by [38]

θco 03

1 + 12Dlowast+ 0055 1 minus exp minus002Dlowast( 11138571113858 1113859 (12)

where Dlowast is the dimensionless sediment size

Dlowast gR

v21113874 1113875

13d50 (13)

-e bed load transport rates in different directions are

qbi qb

τi

|τ|minus C qb

11138681113868111386811138681113868111386811138681113868

zηzxi

(14)

where η is the bed elevation and C is used to reveal the effectof bed slope on the sediment flux and is specified as 15 inthis study

-e suspended load transport can be describedaccording to the classical convection-diffusion equation asfollows

zc

zt+ nabla middot u + ws

g|g|

1113888 1113889c nabla middotvd

σc

nablac1113888 1113889 (15)

where c is the concentration of the suspended sediment vd isthe sediment diffusivity equaling to the turbulence eddy

Mathematical Problems in Engineering 3

viscosity σc is the turbulent Schmidt number which isrelated to eddy viscosity and sediment diffusivity and istaken to be 08 and ws is the sediment setting velocity whichwill be affected by the suspension concentration

ws (1 minus c)ςws0 (16)

where ζ is the suspended sediment size related constant andis specified to be 50 in the present research according to thestudy of Liang et al [33] and ws0 is the settling velocity inclear water given by [39]

ws0 v

d5010362 + 1049D

3lowast1113872 1113873

12minus 10361113876 1113877 (17)

As far as we know the coastal zone is the interface area ofwater air and sediment resulting in a frequent interactionTo keep the sediment to drop out immediately when acci-dentally left in the air the volume fraction was introduced inequation (15) as follows

zc

zt+ nabla middot αu + ws

g|g|

1113888 1113889c nabla middot α] + vd

σc

nablac1113888 1113889 (18)

where the velocity is multiplied by α and the inclusion of v

in equation (18) is used for numerical stability -is stabilityproblem occurs when vd becomes 0 hence the presence of v

ensures that an advection-diffusion problem is solved locallyrather than an advection problem where the latter is moredifficult to solve numerically [39]

According to sediment continuity the Exner equationwas used to describe the seabed evolution

zz

zt

11 minus n

minusnablaqb + D minus E( 1113857 (19)

where z is the seabed elevation n is the beach porosity andqb is the bed load transport rate and its components aregiven by equation (14) D is deposition rate solved by

D csws (20)

where cs is the sediment concentration at bed defined as thevalue at the nearest cell center of beach in this study

Additionally the sediment concentration at a referencelevel Δb is calculated by the following formula

cb 0015d50T

15

ΔbD03lowast

(21)

where T is the dimensionless excess shear stress calculated by

T μscττcr

minus 11113888 1113889

15

(22)

in which τcr is the critical shear stress and μsc is the effectiveshear stress coefficient -e details could be found in vanRijn [40]

E is the entrainment rate which could be calculated bythe following formula

E v + vt

σc

middot nablac (23)

Finally the model coupling between water and sedimentis achieved bymoving the computational mesh in such a waythat the bottom mesh is conformal with the seabed which ismainly based on the method proposed by Jasak and Tukovic[41] Meanwhile to ensure that the bed slope does notexceed the sediment repose angle the mass conservativesand slide algorithm given by Khosronejad et al [42] wasemployed

zbp + Δzbp1113872 1113873 minus zbi + Δzbi( 1113857

Δlpi

tanϕ (24)

where zbp and zbi are the bed elevations at point P and its ithneighbor (see Figure 1) Δlpi is the horizontal distance be-tween the two cell centers and Δzbp and Δzbi are the cor-responding corrections imposed to satisfy the sedimentrepose angle which could be obtained by balancing the cellmass as follows

Abp middot Δzbp minus 1113944i

Abi middot Δzbi 0 (25)

where Abp and Abi are the projections of the cell P and its ithneighbor

4 Boundary Conditions andNumerical Schemes

41 Boundary Conditions Proper boundary conditions arethe key factors to copy laboratory experiments In this studythe active wave generation and the absorption boundarydeveloped by Higuera et al [30] were employed Comparedwith the traditional method proposed by Jacobsen et al [24]this method does not need the relaxation zone so that itcould significantly reduce the computational domain

-e other boundary conditions are set as follows-e topboundary is described as free to the atmosphere and theseabed and the pile surface are set as the no-slip condition-e symmetric condition is applied at the two-side faces ofthe numerical tank to reduce the computational cost in viewof the symmetric distribution in the streamwise direction

42 Numerical Schemes In this study the finite volumemethod (FVM) and Euler scheme are used to address thespace and time term respectively -e PIMPLE algorithm isemployed for the pressure-velocity solver which is a mixturebetween PISO (pressure implicit with the splitting of op-erators) and the SIMPLE (semi-implicit method for pres-sure-linked equations) method -e MULES (multi-dimensional universal limiter for explicit solution) scheme isapplied to maintain the boundedness of volume fraction-e Gauss linear corrected scheme and Gauss linear schemeare used to solve the Laplacian term and gradient termrespectively -e sediment module is addressed as follows-e bed load is solved firstly by shear stress that is derivedfrom the hydrodynamic model Meanwhile the suspendedload is transported as the fluid motion based on the con-vection-diffusion equation and this equation is also solved


by FVM Subsequently the two-dimensional Exner equationis applied to update bed elevation along with the finite areamethod (FAM) addressing the problem of fluid informationmapped from the three-dimensional space to the two-di-mensional space (Tukovic [43]) -e FAM is a variant of theFVM operating on two-dimensional curved surfaces in thethree-dimensional space which makes the seabed mor-phology suitable for arbitrary curved surfaces In the layoutthe FAM follows the structure of the Finite Volume dis-cretization library sharing the basic field algebra linearsystem support boundary condition settings and dis-cretization techniques Subsequently the sand slide proce-dure is implemented to modify the problem of bed slopesexceeding the sediment repose angle

-e self-developed model in this study is a two-waycoupling scheme between water motion and sediment trans-port and the computational procedure is implemented inFigure 2 Firstly the hydrodynamic parameters such asunminus 1 pnminus 1 vnminus1

t αnminus 1 are computed using the hydrodynamicmodule Second above parameters are used to drive suspendedload and bed load transport resulting in the seabed changebased on the Exner equation and the sand slide equation at thesame (n minus 1)th time step ie znminus 1 -ird the hydrodynamicparameters are also affected by the resulting seabed changeFourthly the values of hydrodynamic at the next nth time stepare calculated by the hydrodynamic module from the previousresults based on the updated bed elevation Fifthly the sedi-ment transport and seabed change at the nth time step areperformed using not only the previous sediment environmentat (n minus 1)th time step but also the hydrodynamic results at thenth time step Finally this process is repeated until the cal-culated time reaches the preassigned time During the com-putation the time step was automatically adjusted to ensure theCourant number cr (cr Δt times max(|Δu|)min(|Δx|) inwhich max(|Δu|) and min(|Δx|) were the maximum velocityand the minimum grid size respectively and Δt is the timestep) is always less than one

5 Validations of the Main Modules

51WaveModule Reliable hydrodynamic is the foundationto ensure the accuracy of sediment results -erefore thestability of wave propagation over a flat tank was first tested-e expression of the solitary wave proposed by Lee et al[44] was used to reproduce the free surface

η H sec h2

3H

4h3

1113970

X1113890 1113891 (26)

where η is the free surface elevation H is the wave height h

is the water depth X x minus ct and c g(h + H)

1113968is the

wave celerity Based on one typical solitary wave condition(Hh 03) seven wave gauges with a constant interval048m were placed along the tank to obtain the wave surfaceelevation and the results are shown in Figure 3 -e goodagreement between numerical results and theoretical resultswas found In addition the wave surface was almost noattenuation in the numerical domain indicating the nu-merical model could better model the wave dispersion and

nonlinearity and could be employed in the followingresearch

52 Sand Slide Module In this section a constant beachslope of 45deg with sediment repose angle of 30deg was used tovalidate the robustness of the sand slide module -e beachmorphology before and after employing the sand slidemodel is shown in Figure 4 respectively We could find thatthe beach slope is less than or equal to the sediment reposeangle which indicates the sand slide module can work wellonce the beach slopes exceed the sediment repose angle

53 Suspended Sediment Module -e transport capacity ofthe suspended sediment was tested by the net entrainmentexperiment of van Rijn [40] -e sand bed was laid after arigid bed in this experiment in which x 0 is the startingposition of the sand bed and the inlet of the model is free ofsediment (see Figure 5) -e flow is stable with a water depthof 025m and a velocity of 067ms and the representativesediment diameter is 02mmwith the corresponding settlingvelocity of 0022ms -e roughness height ks is taken to be001m suggested in accordance with the research of van Rijn[45] and Wu et al [46]

-e concentration profiles at x 4H 10H 20H and 40Hwere chosen to validate the simulated results In this modelthe entire computational domain was discretized using astructured mesh and the gird sizes of 0002m were keptconstant in the x-direction y-direction and z-directionFigure 6 shows that the calculated results match well with theexperimental data and the sediment concentration increaseswith the decreasing height above the bed surface

54 Bed Morphodynamic Module -e experiment of Mao[47] was employed to validate the capacity of sedimenttransport with the morphodynamic module -e submarinepipeline with diameter ofD 01m was laid on the sand bedwater depth is h 035m sediment diameter isd50 000036m sediment density is ρsed 2650 kgm3 flowvelocity is u 035ms and critical shields parameter is θc

0048 To avoid the divergence of numerical results at thejunction of the seabed and the pipeline caused by the severedeformation of grids a sinusoidal hole with 01D was usedunder the pipeline inspired by the research studies such asLiang et al [33] Broslashrs [48] and David et al [49] -e meshresolutions around pipelines were increased to solve theproblem of mesh distortion as the scour hole broadens anddeepens as shown in Figure 7

-e scour profiles at 10min and 30min were chosen toverify the seabed scour morphologies as shown in Figure 8Overall the good agreement between the simulation andexperiment was found in which the sediment accretion inthe former behind the pipe was slightly overpredicted thanthat in the latter -is is mainly attributed to the fact that theRANS model tends to smooth out the fluctuations producedby the vortex shedding and therefore underestimates theinteraction between the vortices shed with the seabed andthe similar conclusion was raised by Liang et al [33]


Nevertheless the sediment module established in this studycould well reproduce the morphological change of theseabed

6 Model Applications and Discussions

61 Nonbreaking Wave-Induced Scour around Monopile

611 Model Setup -e water-sediment transport capacityof the model had been verified well by the above calculationsettings Furthermore this model would be used to show itsrobustness of copying the scouring process around a pile-e numerical setup for the model validation is the same asthe experiment by Sumer et al [50]-e flume is 280m long20m wide and 10m high and the pile diameter D 010mfixed in the center of the sand bed as shown in Figure 9 -esediment diameter is d50 000018m sediment density isρsed 2700 kgm3 critical shields parameter is θc 0047and water depth is h 040m A periodic wave with waveperiod of T 45 s and wave height of H 012m was gen-erated to propagate over the flat sand bed Figure 9(c) showsthe model grids and the grids around the pile were refined

612 Verification and Discussion Figure 10 shows thecomputed and theoretical wave surface elevation proposedby Svendsen et al [51] as well as the computed and mea-sured dimensionless maximum scour depth (SD) changearound the pile in which S was presented without dis-tinguishing between specific locations such as behind orbeside the pile -e simulated wave crests and troughs arealmost equal and in of the phase compared to the theoreticaldata Meanwhile it could be seen that the scour depthdeveloped with the periodic wave action is well capturedNevertheless a small difference between the simulated andthe experimental scour topography could be found -e

measured scour depth fluctuated frequently and evenexhibited an obvious disparity at two adjacent moments butthe variation of simulated results is rather regular whichshould be attributed to the turbulence model based on aRANS method used in this study that tends to smooth outthe fluctuations produced by the vortex shedding

Figure 11 shows the scouring process and it is seen thatthe seabed at the upstream side of the pile was eroded afterthe wave impacting the pile which is related to thestrengthened circling flow -en the suspended sedimentwas deposited downstream As the wave continue impactingthe pile the scour depth at the upstream side continuouslyincreased and the sedimentation at the downstream side also

i i

i

P

(a)

i

i

i

i

P

(b)

Figure 1 Schematic shows the cells of the (a) unstructured and (b) structured bed mesh involved in the sand slide correction algorithm

Hydrodynamic

(n ndash 1)th step

(n + 1)th step

nth step

Sediment transport

vtnndash1 αnndash1

unndash1 pnndash1

un+1 pn+1vt

n+1 αn+1

un pn vtn αn Bed elevation

zn

Bed elevationzn+1

Bed elevationznndash1

Figure 2 Two-way coupling procedure of hydrodynamic and thesediment module


0 1 2 3 4 5t (s)

ndash020

02040608

112

ηH

NumTheory

Figure 3 -e stability of wave propagation

(a) (b)

Figure 4 Beach morphology (a) before employing the sand slide module (b) after employing the sand slide module

Velocity profile

H

x = 0Sediment bed

x

y

Concentrationprofiles

Rigid bed

Figure 5 Schematic view of the net entrainment experiment

0

005

01

015

z (m

)

xh = 4

ndash500 0 500 1000 1500 2000 2500 3000c (mgl)

(a)

xh = 10

0

005

01

015

z (m

)

ndash500 0 500 1000 1500 2000 2500 3000c (mgl)

(b)

Figure 6 Continued


exhibited the same tendency Finally the simulated maxi-mum scour depth took place at the upstream side of the pilewhich follows the popular rule-erefore the current modelis capable of modeling the interaction between the periodicwave and the pile

62 Breaking Wave-Induced Sandy Beach Evolution

621 Model Setup -e water-sediment transport capacityof the model had been verified well by the above calculationsettings In this section the numerical evolution of sandybeach was calibrated by the experimental data conducted byKobayashi and Lawrence [10] -e experiment setup isshown in Figures 12(a) and 12(b) with the beach slope of 1 12 sediment diameter of 018mm porosity of 04 and waterdepth of 08m In this experiment a solitary wave with waveheight of 0216m was generated to propagate over sandybeach Another solitary wave with the identical height wasgenerated on the evolved beach after this wave subsided andthat cycle repeats Finally the experimental results of beachprofile wave surface and velocity at the end of the fourthwave were selected to test the robustness of the modelFigure 12(c) shows the model grids for sandy beach and thegrid near the seabed was refined

622 Verification and Discussion Figure 13 shows thecomparison of wave surface elevations from the numericalmodel and physical experiment Good agreements betweenthem could be found not only the free-surface elevations butalso the arrival times

Figures 14(a) and 14(b) show the evolved beach profileand profile change We could found that the numericalresults in the offshore zone agree well with the measureddata However the beach profile near the coastline could notbe predicted accurately which may be due to an overesti-mation of the turbulent dissipation because of the under-estimation of the backwash velocity Meanwhile the timeseries of streamwise velocity (u) at the ADV location isshown in Figure 14(c) -e numerical model satisfactorilyreproduced the streamwise velocity despite there are stillsome issues to be solved in underestimating the backwashvelocity and the associated beach profile evolution

After completing the model validation the wave fieldsuspended load transport and the associated beach profileare shown and discussed in Figure 15 Figures 15(a)ndash15(c)show that the ocean wave got steepened and skewed withclose to the coastal zone and then the wave broke as aplunging breaker due to wave shoaling -e breaking wavesinject nearshore immediately forming a complex wave fieldentraining air on the free surface picking up amounts of

ndash500 0 500 1000 1500 2000 2500 3000c (mgl)

0

005

01

015z (

m)

xh = 20

(c)

0

005

01

015

z (m

)

ndash500 0 500 1000 1500 2000 2500 3000c (mgl)

xh = 40

(d)

Figure 6 Comparison of the predicted (solid lines) and measured (circles) sediment concentrations at four different locations

Figure 7 Refined grids around the pipeline


sediment from sandy beach into suspension Subsequentlythe suspended sediment further runup along with thebreaking wave-induced current -e beach profile have notobviously changed in this stage which should be attributedto the upwash direction opposite with the sediment gravity

and bottom friction which can also be found in the labo-ratory by Young et al [11]

After upwash reaching the maximum height along thebeach it will start to reverse as shown in Figures 15(d)ndash15(f ) A hydraulic jump is formed when the retreating water

ndash03 ndash02 ndash01 00 01 02 03 04 05x (m)

ndash006

0

006

012z (

m)

10 min

NumExp

(a)


ndash006

0

006

012

z (m

)

30 min

NumExp

(b)

Figure 8 -e scour profile at two typical times

2m

28m

D = 01m

(a)

04m

28m

(b)

Model grid

Refined grid

(c)

Figure 9 Experimental setup (a) top view (b) side view (c) model mesh

0 2 4 6 8 10 12t (s)

ndash005

0

005

01

015

η (m

)

TheoryNumerical

(a)

0 200 400 600 800 1000 1200t (s)

0

005

01

015

02

SD

MeasuredNumerical

(b)

Figure 10 Time series of (a) free-surface elevation and (b) the maximum scour depth


tongue collided with the relatively still massive body ofwater but there is no obvious sediment resuspension Asupstream retreating water continues to inject into the hy-draulic jump zone the still water body turns to a complexwater-air mixing zone that led to a turbulent roll forming arecirculation region Meanwhile an intensive sedimenttransport in the form of sheet flowwas found on the landsideof the hydraulic jump point resulting in net erosion of theonshore region In contrast a mass of sediments were de-posited on the seaside of the hydraulic jump point due to thesudden deceleration of sediment-rich seaward flow as wellas the long particle residence time induced by the

recirculation region Consequently the bed morphology ofnet erosion in the nearshore zone and net deposition in theoffshore zone were formed which was consistent with themeasurements by Kobayashi and Lawrence [10] and Younget al [11]

63 Breaking Wave-Induced Scour around the Monopile

631 Model Setup Considering a more complex conditionthe response of beach evolution to the structure would bediscussed -e experiment of local scour around a pile on

Scour depth (cm)

12 09 06 03 0 ndash03 ndash06 ndash09 ndash12

t = 0s t = 300s t = 600s

t = 1500st = 1200s t = 900s

zy xzy x

zy x

zy xzy x

zy x

Figure 11 -e simulated scouring processes around the single pile

1 2 3 4 5 6 7 8

11280cm

Wave maker

Beach

Side view

SWL

(a)

115cm

35m30m

175mPlan view

24m

(b)

Refined grid

(c)

Figure 12 Experimental setup (a) side view (b) plan view (c) model grid for sandy beach


sandy beach given by Tonkin et al [16] was applied tovalidate the model robustness In this experiment the waterdepth was 245m wave height was 022m beach slope was1 20 sediment diameter was 035mm and sediment densitywas 2643 kgm3 A single pile with the diameter of 05m wasplaced upright on the location of still water level as shown inFigures 16(a) and 16(b) -e wave gauges were applied tomeasure water surface elevation at the locations of stationsA C and D An electromagnetic flowmeter placing atstation B was employed to measure the streamwise velocity-e scour depths at the front side and back of the pile iestations C D and E were measured using digital cameratechnology More details could be found in Tonkin et al [16]-e numerical model was setup in the same dimension withthis experiment and the mesh is shown in Figure 16(c) withthe refined grids around the pile

632 Verification and Discussion Figure 17 shows the timeseries of free-surface elevations at stations A C and D andthe time series of streamwise velocity at station B fromnumerical results and measured data Overall the predictedresults were in good agreement with the measured data withthe exception of the velocity since t 13 s after the initialwave impact Numerical and measured differences of Figure17(b) after t 13 s should be attributed to during the mostturbulent part of the drawdown at which the water level istoo low to give a valid velocity measurement the sensor isnot fully submerged at this point (Tonkin et al [16])Fortunately the current model is better to make up thisdeficiency and shows the subsequent velocity process untilabout t 20 s Nevertheless due to the further reduction ofwater level the time series of velocity start to exhibit theintense numerical oscillation

ndash04

0

04η

(m) G1

40 45 50 55 60 65 70t (s)

NumExp

G2

ndash04

0

04

η (m

)

40 45 50 55 60 65 70t (s)

NumExp

(a)

ndash04

0

04

η (m

) G3

40 45 50 55 60 65 70t (s)

NumExp

G4

ndash04

0

04

η (m

)

40 45 50 55 60 65 70t (s)

NumExp

(b)

ndash04

0

04

η (m

) G5

40 45 50 55 60 65 70t (s)

NumExp

G6

ndash04

0

04η

(m)

40 45 50 55 60 65 70t (s)

NumExp

(c)

40 45 50 55 60 65 70t (s)

ndash04

0

04

η (m

) G7

NumExp

ndash04

0

04

η (m

)

40 45 50 55 60 65 70t (s)

G8

NumExp

(d)

Figure 13 Time series of the free-surface elevation


4 6 8 10 12 14x (m)

ndash04ndash02

00204

z (m

) ADV

NumExp

(a)

4 6 8 10 12 14x (m)

ndash01

0

01

Δz (m

)

NumExp

(b)

40 45 50 55 60 65 70t (s)

ndash1

0

1u

(ms

)

ADV Location

NumExp

(c)

Figure 14 -e comparison of predicated results and measured data during the fourth wave for (a) beach profile (b) beach profile changeand (c) streamwise velocity at the ADV location

Breaking wavet = 4656sRunup

x (m)

z (m

)

75 8 85 9 95 10 10506

08

1

12

0 20 40 60 80 100C (kgm3)

2ms

(a)

High concentration

t = 4686s

z (m

)

75 8 85 9 95 10 10506

08

1

12

x (m)

0 20 40 60 80 100C (kgm3)

2ms

(b)

t = 4691sHigh concentration

z (m

)

75 8 85 9 95 10 10506

08

1

12

x (m)

0 20 40 60 80 100C (kgm3)

2ms

(c)

z (m

)

75 8 85 9 95 10 10506

08

1

12

Hydraulic jumpt = 5166sDrawdown

x (m)

0 20 40 60 80 100C (kgm3)

2ms

(d)

z (m

)

75 8 85 9 95 10 10506

08

1

12

High concentration

t = 5636 s

x (m)

0 20 40 60 80 100C (kgm3)

2ms

(e)

z (m

)

75 8 85 9 95 10 10506

08

1

12t = 5686s

Obvious profile change

x (m)

0 20 40 60 80 100C (kgm3)

2ms

(f)

Figure 15 (a)ndash(c) Predicted wave field suspended concentration and the resulting beach profile change during runup process (d)ndash(f )Predicted wave field suspended concentration and the resulting beach profile during drawdown process


Figure 18 shows the time series of the local scourdepth around a single pile We could find that the seabedmorphology at the front and side of the pile (ie stationsC and E) was sharply eroded of approximately 10 cmwhen the moment of wave impacting the pile which isrelated to the stronger velocity (see Figure 17(b)) -enthe scour depth decreased as the net deposition occurredduring the drawdown -e seabed morphology behind

the pile (ie station D) did not exhibit an obvious changeat the moment of the wave impacting the pile and thenthe scour depth continuously increased from t 12 s tot 18 s During the latter stage of the backwash the scourdepth decreased to some extent due to the sedimentsettlement After about t 20 s the seabed around the pilewas relatively in equilibrium Finally we noted that thepredicted maximum scour depth is slight lesser than the

245m120 slope

50 60 70 80 90 100 1100

1

2

3

x (m)

z (m

)

Cylinder

(a)

Wave directionCylinder

A

B

C D

E035m

075m2m

3m

y (m

)

x (m)

(b)

(c)

Figure 16 Experiment setup (a) side view (b) plan view (c) refined grids around the pile

2242832

ndash5 0 5 10 15 20 25t (s)

η (m

)

NumExp

(a)

ndash5 0 5 10 15 20 25t (s)

ndash3

0

3

u (m

s) Shallow water

NumExp

(b)

2242832

ndash5 0 5 10 15 20 25t (s)

η (m

)

NumExp

(c)

ndash5 0 5 10 15 20 25t (s)

2242832

η (m

)

NumExp

(d)

Figure 17 Time series of the free-surface elevation and streamwise velocity (a) Station A (b) Station B (c) Station C (d) Station D


measured data which may be caused by seepage flow (orgroundwater flow) as described in the Tonkin et al [16]but the current model was incapable of reproducing it

To more visually present the topography evolution ofsandy beach during the interaction between the wave and the

monopile the three-dimensional scour patterns are shown inFigure 19 It is obvious that the remarkable erosion took placeat the front of the pile when the wave impacting the pilewhich is different with the aforementioned scenario of the pileplaced on a horizontal seabed As upstream retreating water

ndash20ndash10

010

z (cm

)

ndash5 0 5 10 15 20 25t (s)

NumExp

(a)

ndash20ndash10

010

z (cm

)

ndash5 0 5 10 15 20 25t (s)

NumExp

(b)

ndash5 0 5 10 15 20 25t (s)

ndash20ndash10

010

z (cm

)

NumExp

(c)

Figure 18 Time series of the local scour depth around a single pile (a) Seaward (b) Side (c) Leeside

t = 05s

Runup

(a)

t = 45s

Runup

(b)

t = 145s

Drawdown

(c)

t = 245s

Drawdown

z

y x

(d)

Figure 19 -e local 3D scour patterns around a single pile


the scour depth on the seaside of the pile increased quickly-evorticity fields which are related to the sediment transport(Kobayashi and Oda [52]) were chosen to reveal the potentialcauses of above phenomena as shown in Figure 20 At therunup stages the flow separated and the vortex motion formedand developed mainly located at the area less than 05D behindthe pile Following the wave drawdown the wake regionappeared before the pile-e generated vortex was transportedfurther upstream by advection and the wake region wasstretched with a length scale more than 2D-ismay be relatedto the backwash velocity larger than the uprush velocity whichis showed in Section 62 Overall it is acceptable that backwash-induced scour around a pile is larger than the uprush

7 Conclusions

A nonlinear three-dimensional coupled model which wascomposed of the hydrodynamic module and the self-de-veloped beach morphodynamic module was developedbased on the CFD tool OpenFOAM in this study To bettersimulate the breaking wave-induced seabed scour aroundthe monopile an innovative combination of sedimentcomponents was proposed which is different with publishedpapers and its robustness was verified by a series of tests-eresults showed that this self-developed model was able tosatisfactorily reproduce the wave transmission suspendedload and bed load motion and the seabed morphology

1086420ndash2ndash4ndash6ndash8ndash10

t = 27s

0

05

1

15

2

y (m

)

48 485 49 495 50 505475x (m)


t = 34s

0

05

1

15

2Velocity Velocity

Velocity Velocity

Velocity Velocity

y (m

)

48 485 49 495 50 505475x (m)

1086420ndash2ndash4ndash6

ndash10ndash8

t = 28s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 35s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 29s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 36s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)

Figure 20 Vorticity distribution characteristics during wave runup and drawdown


evolution as well as proved the sand slide module workswell

-e 3Dmodel developed here was first applied to exploretwo preliminary scenarios one with the nonbreaking wave-induced scour around the monopile and the other withbreaking wave-induced sandy beach evolution In bothcases the calculated results were basically in agreement withthe measurements and meanwhile showed us that the ob-vious scour occurred in the upstream side of the pile underthe nonbreaking wave as well as the backrush-inducedbeach deformation is larger than the uprush Subsequentlythe more complicated case of wave-induced scour aroundthe monopile on sandy beach was simulated and its mainfeatures were well reproduced by this model-e remarkableerosion took place at the front of the pile when the waveimpacting on the pile during the wave runup which isdifferent with the scenario of the monopile placed on ahorizontal seabed Following the wave drawdown scourdepth at the upstream side of the pile increased quickly andthis should be attributed to the high-speed backwash andstrengthen the wake region Finally we remark that furtherresearch studies are needed to investigate the effect of theoffshore structures on beach morphological evolution basedon the current study which would be significantly helpful toevaluate the nearshore sediment transport

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is study was supported financially by the National NaturalScience Foundation of China (grant no 51779280) the OpenFoundation of Key Laboratory of Water-Sediment Sciencesand Water Disaster Prevention of Hunan Province (grantnos 2020SS02 and 2020SS06) and the Guangdong Scienceand Technology Plan Program (grant no 2013B020200008)

References

[1] R A Morton G Gelfenbaum and B E Jaffe ldquoPhysicalcriteria for distinguishing sandy tsunami and storm depositsusing modern examplesrdquo Sedimentary Geology vol 200no 3-4 pp 184ndash207 2007

[2] FEMA Guidelines for Design of Structures for VerticalEvacuation from tsunamis FEMA P646 Applied TechnologyCouncil FEMA Washington DC USA 2008

[3] Z Wu C Jiang B Deng J Chen and X Liu ldquoSensitivity ofWRF simulated typhoon track and intensity over the SouthChina Sea to horizontal and vertical resolutionsrdquo Acta Oce-anologica Sinica vol 38 no 7 pp 74ndash83 2019

[4] Z Wu J Chen C Jiang et al ldquoNumerical investigation ofTyphoon Kai-tak (1213) using a mesoscale coupled WRF-ROMS model-Part II wave effectsrdquo Ocean Engineeringvol 196 pp 1ndash14 2020

[5] R Paris F Lavigne P Wassmer and J Sartohadi ldquoCoastalsedimentation associated with the december 26 2004 tsunamiin Lhok Nga west Banda aceh (sumatra Indonesia)rdquo MarineGeology vol 238 no 1ndash4 pp 93ndash106 2007

[6] W Szczucinski M Kokocinski M Rzeszewski et al ldquoSedi-ment sources and sedimentation processes of 2011 tohoku-okitsunami deposits on the sendai plain Japan-insights fromdiatoms nannoliths and grain size distributionrdquo SedimentaryGeology vol 282 no 1 pp 40ndash56 2012

[7] USGS Tsunami Observations of Tsunami Impact USGSReston VA USA 2010

[8] B Jaffe G Gelfenbaum D Rubin et al ldquoIdentification andinterpretation of tsunami deposits from the June 23 2001Peru tsunamirdquo in Proceedings of International Conference onCoastal Sediments pp 1ndash13 Clearwater Beach FL USA May2003

[9] C C Mei M Stiassnie and D K P Yue Ieory and Ap-plications of Ocean Surface Waves World Scientific Pub-lishing Singapore 2005

[10] N Kobayashi and A R Lawrence ldquoCross-shore sedimenttransport under breaking solitary wavesrdquo Journal of Geo-physical Research vol 109 no C3 pp 1ndash13 2004

[11] Y L Young H Xiao and T Maddux ldquoHydro-and morpho-dynamic modeling of breaking solitary waves over a fine sandbeach Part I experimental studyrdquo Marine Geology vol 269no 3-4 pp 107ndash118 2010

[12] C Jiang J Chen Y Yao J Liu and Y Deng ldquoStudy onthreshold motion of sediment and bedload transport bytsunami wavesrdquo Ocean Engineering vol 100 pp 97ndash1062015

[13] N Daghighi A H N Chegini M Daliri and D HedayatildquoExperimental assessment of sediment transport and bedformation of sandy beaches by tsunami wavesrdquo InternationalJournal of Environmental Research vol 9 no 3 pp 795ndash8042015

[14] F Kato S Tonkin H Yeh S Sato and K Torii ldquo-e grain-size effects on scour around a cylinder due to tsunami run-uprdquo in Proceedings of the International Tsunami Symposiumpp 905ndash917 Seattle WA USA August 2001

[15] F Kato S Sato and H Yeh ldquoLarge-scale experiment ondynamic response of sand bed around a cylinder due totsunamirdquo in Proceedings of the International Conference onCoastal Engineering pp 1848ndash1859 ASCE Sydney AustraliaMarch 2000

[16] S Tonkin H Yeh F Kato and S Sato ldquoTsunami scouraround a cylinderrdquo Journal of Fluid Mechanics vol 496pp 165ndash192 2003

[17] Kuswandi R Triatmadja and Istiarto ldquoVelocity around acylinder pile during scouring process due to tsunami con-gress of the asia Pacific division of the international associ-ation for hydro environment engineering and researchrdquo AtColombo Srilanka vol 20 pp 1ndash8 2016

[18] G Simpson and S Castelltort ldquoCoupled model of surfacewater flow sediment transport and morphological evolutionrdquoComputers amp Geosciences vol 32 no 10 pp 1600ndash1614 2006

[19] D Pritchard and L Dickinson ldquoModelling the sedimentarysignature of long waves on coasts implications for tsunamireconstructionrdquo Sedimentary Geology vol 206 no 1ndash4pp 42ndash57 2008

[20] T Shimozono S Sato and Y Tajima Numerical Study ofTsunami Run-Up over Erodible Sand Dunes Coastal Sedimentspp 1089ndash1102 ASCE Reston VA USA 2007

[21] H Xiao Y L Young and J H Prevost ldquoHydro-and morpho-dynamic modeling of breaking solitary waves over a fine sand


beach Part II numerical simulationrdquo Marine Geologyvol 269 no 3-4 pp 119ndash131 2010

[22] T Nakamura and S C Yim ldquoA nonlinear three-dimensionalcoupled fluid-sediment interaction model for large seabeddeformationrdquo Journal of Offshore Mechanics and Arctic En-gineering vol 133 no 3 pp 1ndash14 2011

[23] J Li M Qi and D R Fuhrman ldquoNumerical modeling of flowand morphology induced by a solitary wave on a slopingbeachrdquo Applied Ocean Research vol 82 pp 259ndash273 2019

[24] N G Jacobsen D R Fuhrman and J Fredsoslashe ldquoA wavegeneration toolbox for the open-source CFD library Open-Foamregrdquo International Journal for Numerical Methods inFluids vol 70 no 9 pp 1073ndash1088 2012

[25] N G Jacobsen J Fredsoe and J H Jensen ldquoFormation anddevelopment of a breaker bar under regular waves Part 1model description and hydrodynamicsrdquo Coastal Engineeringvol 88 pp 182ndash193 2014

[26] N G Jacobsen and J Fredsoe ldquoFormation and developmentof a breaker bar under regular waves Part 2 sedimenttransport and morphologyrdquo Coastal Engineering vol 88pp 55ndash68 2014

[27] C Liu X Liu C Jiang et al ldquoNumerical investigation ofsediment transport of sandy beaches by a tsunami-like solitarywave based on NavierndashStokes equationsrdquo Journal of OffshoreMechanics and Arctic Engineering vol 141 pp 1ndash13 2019

[28] C Pan andW Huang ldquoNumerical modeling of tsunami waverun-up and effects on sediment scour around a cylindricalpierrdquo Journal of Engineering Mechanics vol 138 no 10pp 1224ndash1235 2012

[29] W Huang Q Yang and H Xiao ldquoCFD modeling of scaleeffects on turbulence flow and scour around bridge piersrdquoComputers amp Fluids vol 38 no 5 pp 1050ndash1058 2009

[30] P Higuera J L Lara and I J Losada ldquoRealistic wave gen-eration and active wave absorption for Navier-Stokesmodelsrdquo Coastal Engineering vol 71 pp 102ndash118 2013

[31] E D Christensen and R Deigaard ldquoLarge eddy simulation ofbreaking wavesrdquoCoastal Engineering vol 42 no 1 pp 53ndash862001

[32] E D Christensen ldquoLarge eddy simulation of spilling andplunging breakersrdquo Coastal Engineering vol 53 no 5-6pp 463ndash485 2006

[33] D Liang L Cheng and F Li ldquoNumerical modeling of flowand scour below a pipeline in currentsrdquo Coastal Engineeringvol 52 no 1 pp 43ndash62 2005

[34] N G Jacobsen and J Fredsoslashe A Full Hydro-And Morpho-dynamic Description of Breaker Bar Development TechnicalUniversity of Denmark Lyngby Denmark 2011

[35] A Arzani A M Gambaruto G Chen and S C ShaddenldquoLagrangian wall shear stress structures and near-walltransport in high-Schmidt-number aneurysmal flowsrdquoJournal of Fluid Mechanics vol 790 pp 158ndash172 2016

[36] F Engelund and J Fredsoslashe ldquoA sediment transport model forstraight alluvial channelsrdquo Hydrology Research vol 7 no 5pp 293ndash306 1976

[37] J R L Allen ldquoSimple models for the shape and symmetry oftidal sand waves (1) statically stable equilibrium formsrdquoMarine Geology vol 48 no 1-2 pp 31ndash49 1982

[38] R L Soulsby and R J S Whitehouse ldquo-reshold of sedimentmotion in coastal environmentsrdquo in Proceedings of the PacificCoasts and Ports 1997 Conference pp 149ndash154 ChristchurchNew Zealand September 1997

[39] R J Leveque Finite VolumeMethods for Hyperbolic ProblemsCambridge University Press Cambridge UK 2007

[40] L C van Rijn ldquoSediment transport part II suspended loadtransportrdquo Journal of Hydraulic Engineering vol 110 no 11pp 1613ndash1641 1984

[41] H Jasak and Z Tukovic ldquoAutomatic mesh motion for theunstructured finite volume methodrdquo Transactions ofFAMENA vol 30 pp 1ndash20 2006

[42] A Khosronejad S Kang I Borazjani and F SotiropoulosldquoCurvilinear immersed boundary method for simulatingcoupled flow and bed morphodynamic interactions due tosediment transport phenomenardquo Advances in Water Re-sources vol 34 no 7 pp 829ndash843 2011

[43] Z Tukovic Finite volume method on domains of varying shape(in Croatian) PhD thesis Faculty of Mechanical Engineeringand Naval Architecture University of Zagreb Zagreb Cro-atia 2005

[44] J J Lee J E Skjelbreia and F Raichlen ldquoMeasurmment ofvelocities in solitary wavesrdquo Journal of the Waterway PortCoastal and Ocean Division vol 108 pp 200ndash218 1982

[45] L C van Rijn ldquoMathematical modeling of suspended sedi-ment in Nonuniform flowsrdquo Journal of Hydraulic Engineer-ing vol 112 no 6 pp 433ndash455 1986

[46] W Wu W Rodi and T Wenka ldquo3D numerical modeling offlow and sediment transport in open channelsrdquo Journal ofHydraulic Engineering vol 126 no 1 pp 4ndash15 2000

[47] Y Mao Ie Interaction between a Pipeline and an ErodibleBed Technical University of Danmark Kgs Lyngby Dan-mark 1986

[48] B Broslashrs ldquoNumerical modeling of flow and scour at pipelinesrdquoJournal of Hydraulic Engineering vol 125 no 5 pp 511ndash5231999

[49] D R Fuhrman C Baykal B Mutlu Sumer N G Jacobsenand J Fredsoslashe ldquoNumerical simulation of wave-induced scourand backfilling processes beneath submarine pipelinesrdquoCoastal Engineering vol 94 no 7 pp 10ndash22 2014

[50] B M Sumer J Fredsoslashe and N Christiansen ldquoScour aroundvertical pile in wavesrdquo Journal ofWaterway Port Coastal andOcean Engineering vol 118 no 1 pp 15ndash31 1992

[51] I A Svendsen Introduction to Nearshore HydrodynamicsVol 24 World Scientific Singapore 2006

[52] T Kobayashi and K Oda ldquoExperimental study on developingprocess of local scour around a vertical cylinderrdquo CoastalEngineering vol 2 pp 1284ndash1297 1994


and topography especially facing the complex three-di-mension high-speed water roller Additionally theoreticalapproaches for studying this process are still inadequate [9]Hence numerical simulations and physical experimentsproviding effective controlling factors become the preferredmethods

Traditionally physical experiments are clearly helpful toassess this process Many research studies have been performedin the aspect of cross-shore sediment transport on sandy beach(eg Kobayashi and Lawrence [10] Young et al [11] Jiang et al[12] and Daghighi et al [13]) and the scour around offshorestructure foundation (eg Kato et al [14 15] Tonkin et al [16]and Kuswandi et al [17]) which improve our understanding ofbeach evolution and its response to structure by waveWith thedevelopment of computer technology the numerical modelprovided us with another way to understand the mechanismswhich solves the limited capability used in measurement de-vices In the past decades modeling sandy beach evolution hasbeen studied with depth-averaged equations such as shallowwater equations (eg Simpson and Castelltort [18] andPritchard and Dickinson [19]) and Boussinesq-type equations(eg Shimozono et al [20] and Xiao et al [21]) However thisprocess is highly nonlinear as well as local but strong tur-bulence near the free surface and the seabed needs to beconsidered Recently Nakamura and Yim [22] Li et al [23]and Jacobsen et al ([24ndash26]) as well as Liu et al [27] in-troduced a volume of fluid-type model to study this processwhich was based on the generalized NavierndashStokes equationswith a turbulence closure solver computing incompressibleviscous multiphase flows Aforementioned research studies notonly substantially improved our understanding of breakingwave-induced longshore sediment transport mechanisms butalso gave us some inspiration to explore the more complexproblems such as sediment-wave-structure interaction in somecoastal areas Pan and Huang [28] are the pioneers to nu-merically investigate the interactions among wave monopileand sandy beach to access the effectiveness of a linked 2Dhydrodynamic and sediment scour model in this processmodeling However some limitations existed such as the 2Dmodel could only simulate vertically averaged currents insteadof 3D currents in this study in particular about the verticalvelocity component [28] Huang et al [29] pointed out that theapproximation of the 3D scour phenomenon around the pilestructure to a 2D problem may be a contribution to errorsbetween the model predictions and observations To the au-thorsrsquo knowledge the research of 3Dmodeling in this questionis still limited so far and yet to be further investigated

To address the aforementioned issues a 3D numericalmodel was developed based on the open-source Open-FOAM platform in this study supporting two-phasedincompressible flow -e wave dynamic is achieved by itssolver interFoam combined with the active wave generationand absorption boundary conditions developed byHiguera et al [30] Unfortunately the official version ofOpenFOAM lacks the function of sediment transportsimulation and the third-party sediment module could notbe open accessed Additionally similar sediment solvingmodules did not necessarily apply to current research casesdue to the complexities of sediment transport mechanisms

under different sea environments -erefore this studyaimed to overcome those gaps by self-extending the sed-iment functionality based on an effective usage of the betterparts from published sediment modules -is developedmodel will be further used to simulate the breaking wave-induced scour around the monopile as well as investigateits difference with the scenario of monopile on a flat seabedand the potential scour mechanisms -e generation ad-vection and dissipation of turbulence in the fluid model arealso important for sediment transport For breaking wave-driven sediment motion several turbulence models havebeen employed namely the RANS model (Li et al [23] andJacobsen [25 26]) and the LES model (Christensen andDeigaard [31] and Christensen [32]) Generally LES wasunder consideration as the turbulence model because of aconsiderable part of the mixing is resolved However whenusing the LES the stochastic nature of the flow reflectsnonlinearly onto the sediment transport and results inlarger spatial gradients hence the allowable calculationstability or time step will be lowered considerably-us theRANS turbulence model was considered in this studybecause it is more preferable to predict the scour profilethan the LES turbulence model [33 34]

-e rest of this paper is organized as follows -enumerical methods including the hydrodynamic modeland beach morphodynamic model as well as corre-sponding boundary conditions and numerical schemesare introduced in Sections 2ndash4 respectively -e vali-dations of the main modules are conducted in Section 5-e model applications and discussions are given inSection 6 -e main conclusions of this study are shownin Section 7

2 Hydrodynamic Model

3D RANS equations were applied to simulate the two-phaseincompressible viscous fluids and the governing equationsare as follows

Continuity equation

nabla middot u 0 (1)

Momentum equation

zρuzt

+ nabla middot ρuuT1113960 1113961 minusnablaplowast minus g middot xnablaρ + nabla middot μnablau + ρτlowast1113858 1113859

+ σTκcnablac

(2)

where x (xi xj xk) is the Cartesian coordinate systemu (ui uj uk) is the velocity vector ρ is the fluid density plowastis the pressure in excess of hydrostatic and g is the gravi-tational acceleration

-e last term of equation (2) is used to describe thesurface tension in which σT is the surface tension coefficientκc is the surface curvature and c is a scale field used to trackthe fluid movement τlowast is the Reynolds stress tensor given by



23

kI (3)



k 12u middot uT

ε C075μ k15

l

(4)



zt+ nabla(ku) nabla

]t

σk

nablak1113888 1113889 + 2]t

ρ|nablau|

2minus ε

zεzt



2εk

minus C2εε2

k

(5)



zαzt











τ ατ (9)



c1113872 1113873Rgd50


0 if θlt θc

⎧⎪⎨

⎪⎩

θ τ

ρgRd50

R ρsed minus ρ

ρ

(10)


θc

θco

cos β +sin βtanφ

(11)


θco 03



Dlowast gR

v21113874 1113875

13d50 (13)


qbi qb

τi

|τ|minus C qb

11138681113868111386811138681113868111386811138681113868

zηzxi

(14)



zc


g|g|


σc

nablac1113888 1113889 (15)






ws0 v

d5010362 + 1049D

3lowast1113872 1113873

12minus 10361113876 1113877 (17)


zc


g|g|


σc

nablac1113888 1113889 (18)





zz

zt

11 minus n



D csws (20)



cb 0015d50T

15

ΔbD03lowast

(21)


T μscττcr

minus 11113888 1113889

15

(22)



E v + vt

σc

middot nablac (23)



Δlpi

tanϕ (24)















η H sec h2

3H

4h3

1113970

X1113890 1113891 (26)



1113968is the

















i i

i

P

(a)

i

i

i

i

P

(b)


Hydrodynamic

(n ndash 1)th step

(n + 1)th step

nth step

Sediment transport

vtnndash1 αnndash1

unndash1 pnndash1

un+1 pn+1vt

n+1 αn+1


zn

Bed elevationzn+1




0 1 2 3 4 5t (s)

ndash020

02040608

112

ηH

NumTheory


(a) (b)


Velocity profile

H

x = 0Sediment bed

x

y


Rigid bed


0

005

01

015

z (m

)

xh = 4

ndash500 0 500 1000 1500 2000 2500 3000c (mgl)

(a)

xh = 10

0

005

01

015

z (m

)

ndash500 0 500 1000 1500 2000 2500 3000c (mgl)

(b)

Figure 6 Continued








ndash500 0 500 1000 1500 2000 2500 3000c (mgl)

0

005

01

015z (

m)

xh = 20

(c)

0

005

01

015

z (m

)

ndash500 0 500 1000 1500 2000 2500 3000c (mgl)

xh = 40

(d)








ndash006

0

006

012z (

m)

10 min

NumExp

(a)


ndash006

0

006

012

z (m

)

30 min

NumExp

(b)


2m

28m

D = 01m

(a)

04m

28m

(b)

Model grid

Refined grid

(c)


0 2 4 6 8 10 12t (s)

ndash005

0

005

01

015

η (m

)

TheoryNumerical

(a)

0 200 400 600 800 1000 1200t (s)

0

005

01

015

02

SD

MeasuredNumerical

(b)







Scour depth (cm)


t = 0s t = 300s t = 600s

t = 1500st = 1200s t = 900s

zy xzy x

zy x

zy xzy x

zy x


1 2 3 4 5 6 7 8

11280cm

Wave maker

Beach

Side view

SWL

(a)

115cm

35m30m

175mPlan view

24m

(b)

Refined grid

(c)





ndash04

0

04η

(m) G1

40 45 50 55 60 65 70t (s)

NumExp

G2

ndash04

0

04

η (m

)

40 45 50 55 60 65 70t (s)

NumExp

(a)

ndash04

0

04

η (m

) G3

40 45 50 55 60 65 70t (s)

NumExp

G4

ndash04

0

04

η (m

)

40 45 50 55 60 65 70t (s)

NumExp

(b)

ndash04

0

04

η (m

) G5

40 45 50 55 60 65 70t (s)

NumExp

G6

ndash04

0

04η

(m)

40 45 50 55 60 65 70t (s)

NumExp

(c)

40 45 50 55 60 65 70t (s)

ndash04

0

04

η (m

) G7

NumExp

ndash04

0

04

η (m

)

40 45 50 55 60 65 70t (s)

G8

NumExp

(d)



4 6 8 10 12 14x (m)

ndash04ndash02

00204

z (m

) ADV

NumExp

(a)

4 6 8 10 12 14x (m)

ndash01

0

01

Δz (m

)

NumExp

(b)

40 45 50 55 60 65 70t (s)

ndash1

0

1u

(ms

)

ADV Location

NumExp

(c)



x (m)

z (m

)

75 8 85 9 95 10 10506

08

1

12

0 20 40 60 80 100C (kgm3)

2ms

(a)

High concentration

t = 4686s

z (m

)

75 8 85 9 95 10 10506

08

1

12

x (m)

0 20 40 60 80 100C (kgm3)

2ms

(b)


z (m

)

75 8 85 9 95 10 10506

08

1

12

x (m)

0 20 40 60 80 100C (kgm3)

2ms

(c)

z (m

)

75 8 85 9 95 10 10506

08

1

12


x (m)

0 20 40 60 80 100C (kgm3)

2ms

(d)

z (m

)

75 8 85 9 95 10 10506

08

1

12

High concentration

t = 5636 s

x (m)

0 20 40 60 80 100C (kgm3)

2ms

(e)

z (m

)

75 8 85 9 95 10 10506

08

1

12t = 5686s


x (m)

0 20 40 60 80 100C (kgm3)

2ms

(f)





245m120 slope

50 60 70 80 90 100 1100

1

2

3

x (m)

z (m

)

Cylinder

(a)


A

B

C D

E035m

075m2m

3m

y (m

)

x (m)

(b)

(c)


2242832

ndash5 0 5 10 15 20 25t (s)

η (m

)

NumExp

(a)

ndash5 0 5 10 15 20 25t (s)

ndash3

0

3

u (m

s) Shallow water

NumExp

(b)

2242832

ndash5 0 5 10 15 20 25t (s)

η (m

)

NumExp

(c)

ndash5 0 5 10 15 20 25t (s)

2242832

η (m

)

NumExp

(d)






ndash20ndash10

010

z (cm

)

ndash5 0 5 10 15 20 25t (s)

NumExp

(a)

ndash20ndash10

010

z (cm

)

ndash5 0 5 10 15 20 25t (s)

NumExp

(b)

ndash5 0 5 10 15 20 25t (s)

ndash20ndash10

010

z (cm

)

NumExp

(c)


t = 05s

Runup

(a)

t = 45s

Runup

(b)

t = 145s

Drawdown

(c)

t = 245s

Drawdown

z

y x

(d)




7 Conclusions



t = 27s

0

05

1

15

2

y (m

)

48 485 49 495 50 505475x (m)


t = 34s

0

05

1

15

2Velocity Velocity

Velocity Velocity

Velocity Velocity

y (m

)

48 485 49 495 50 505475x (m)


ndash10ndash8

t = 28s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 35s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 29s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 36s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)





Data Availability




Acknowledgments


References

























































23

kI (3)



k 12u middot uT

ε C075μ k15

l

(4)



zt+ nabla(ku) nabla

]t

σk

nablak1113888 1113889 + 2]t

ρ|nablau|

2minus ε

zεzt



2εk

minus C2εε2

k

(5)



zαzt











τ ατ (9)



c1113872 1113873Rgd50


0 if θlt θc

⎧⎪⎨

⎪⎩

θ τ

ρgRd50

R ρsed minus ρ

ρ

(10)


θc

θco

cos β +sin βtanφ

(11)


θco 03



Dlowast gR

v21113874 1113875

13d50 (13)


qbi qb

τi

|τ|minus C qb

11138681113868111386811138681113868111386811138681113868

zηzxi

(14)



zc


g|g|


σc

nablac1113888 1113889 (15)






ws0 v

d5010362 + 1049D

3lowast1113872 1113873

12minus 10361113876 1113877 (17)


zc


g|g|


σc

nablac1113888 1113889 (18)





zz

zt

11 minus n



D csws (20)



cb 0015d50T

15

ΔbD03lowast

(21)


T μscττcr

minus 11113888 1113889

15

(22)



E v + vt

σc

middot nablac (23)



Δlpi

tanϕ (24)















η H sec h2

3H

4h3

1113970

X1113890 1113891 (26)



1113968is the

















i i

i

P

(a)

i

i

i

i

P

(b)


Hydrodynamic

(n ndash 1)th step

(n + 1)th step

nth step

Sediment transport

vtnndash1 αnndash1

unndash1 pnndash1

un+1 pn+1vt

n+1 αn+1


zn

Bed elevationzn+1




0 1 2 3 4 5t (s)

ndash020

02040608

112

ηH

NumTheory


(a) (b)


Velocity profile

H

x = 0Sediment bed

x

y


Rigid bed


0

005

01

015

z (m

)

xh = 4

ndash500 0 500 1000 1500 2000 2500 3000c (mgl)

(a)

xh = 10

0

005

01

015

z (m

)

ndash500 0 500 1000 1500 2000 2500 3000c (mgl)

(b)

Figure 6 Continued








ndash500 0 500 1000 1500 2000 2500 3000c (mgl)

0

005

01

015z (

m)

xh = 20

(c)

0

005

01

015

z (m

)

ndash500 0 500 1000 1500 2000 2500 3000c (mgl)

xh = 40

(d)








ndash006

0

006

012z (

m)

10 min

NumExp

(a)


ndash006

0

006

012

z (m

)

30 min

NumExp

(b)


2m

28m

D = 01m

(a)

04m

28m

(b)

Model grid

Refined grid

(c)


0 2 4 6 8 10 12t (s)

ndash005

0

005

01

015

η (m

)

TheoryNumerical

(a)

0 200 400 600 800 1000 1200t (s)

0

005

01

015

02

SD

MeasuredNumerical

(b)







Scour depth (cm)


t = 0s t = 300s t = 600s

t = 1500st = 1200s t = 900s

zy xzy x

zy x

zy xzy x

zy x


1 2 3 4 5 6 7 8

11280cm

Wave maker

Beach

Side view

SWL

(a)

115cm

35m30m

175mPlan view

24m

(b)

Refined grid

(c)





ndash04

0

04η

(m) G1

40 45 50 55 60 65 70t (s)

NumExp

G2

ndash04

0

04

η (m

)

40 45 50 55 60 65 70t (s)

NumExp

(a)

ndash04

0

04

η (m

) G3

40 45 50 55 60 65 70t (s)

NumExp

G4

ndash04

0

04

η (m

)

40 45 50 55 60 65 70t (s)

NumExp

(b)

ndash04

0

04

η (m

) G5

40 45 50 55 60 65 70t (s)

NumExp

G6

ndash04

0

04η

(m)

40 45 50 55 60 65 70t (s)

NumExp

(c)

40 45 50 55 60 65 70t (s)

ndash04

0

04

η (m

) G7

NumExp

ndash04

0

04

η (m

)

40 45 50 55 60 65 70t (s)

G8

NumExp

(d)



4 6 8 10 12 14x (m)

ndash04ndash02

00204

z (m

) ADV

NumExp

(a)

4 6 8 10 12 14x (m)

ndash01

0

01

Δz (m

)

NumExp

(b)

40 45 50 55 60 65 70t (s)

ndash1

0

1u

(ms

)

ADV Location

NumExp

(c)



x (m)

z (m

)

75 8 85 9 95 10 10506

08

1

12

0 20 40 60 80 100C (kgm3)

2ms

(a)

High concentration

t = 4686s

z (m

)

75 8 85 9 95 10 10506

08

1

12

x (m)

0 20 40 60 80 100C (kgm3)

2ms

(b)


z (m

)

75 8 85 9 95 10 10506

08

1

12

x (m)

0 20 40 60 80 100C (kgm3)

2ms

(c)

z (m

)

75 8 85 9 95 10 10506

08

1

12


x (m)

0 20 40 60 80 100C (kgm3)

2ms

(d)

z (m

)

75 8 85 9 95 10 10506

08

1

12

High concentration

t = 5636 s

x (m)

0 20 40 60 80 100C (kgm3)

2ms

(e)

z (m

)

75 8 85 9 95 10 10506

08

1

12t = 5686s


x (m)

0 20 40 60 80 100C (kgm3)

2ms

(f)





245m120 slope

50 60 70 80 90 100 1100

1

2

3

x (m)

z (m

)

Cylinder

(a)


A

B

C D

E035m

075m2m

3m

y (m

)

x (m)

(b)

(c)


2242832

ndash5 0 5 10 15 20 25t (s)

η (m

)

NumExp

(a)

ndash5 0 5 10 15 20 25t (s)

ndash3

0

3

u (m

s) Shallow water

NumExp

(b)

2242832

ndash5 0 5 10 15 20 25t (s)

η (m

)

NumExp

(c)

ndash5 0 5 10 15 20 25t (s)

2242832

η (m

)

NumExp

(d)






ndash20ndash10

010

z (cm

)

ndash5 0 5 10 15 20 25t (s)

NumExp

(a)

ndash20ndash10

010

z (cm

)

ndash5 0 5 10 15 20 25t (s)

NumExp

(b)

ndash5 0 5 10 15 20 25t (s)

ndash20ndash10

010

z (cm

)

NumExp

(c)


t = 05s

Runup

(a)

t = 45s

Runup

(b)

t = 145s

Drawdown

(c)

t = 245s

Drawdown

z

y x

(d)




7 Conclusions



t = 27s

0

05

1

15

2

y (m

)

48 485 49 495 50 505475x (m)


t = 34s

0

05

1

15

2Velocity Velocity

Velocity Velocity

Velocity Velocity

y (m

)

48 485 49 495 50 505475x (m)


ndash10ndash8

t = 28s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 35s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 29s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 36s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)





Data Availability




Acknowledgments


References



























































ws0 v

d5010362 + 1049D

3lowast1113872 1113873

12minus 10361113876 1113877 (17)


zc


g|g|


σc

nablac1113888 1113889 (18)





zz

zt

11 minus n



D csws (20)



cb 0015d50T

15

ΔbD03lowast

(21)


T μscττcr

minus 11113888 1113889

15

(22)



E v + vt

σc

middot nablac (23)



Δlpi

tanϕ (24)















η H sec h2

3H

4h3

1113970

X1113890 1113891 (26)



1113968is the

















i i

i

P

(a)

i

i

i

i

P

(b)


Hydrodynamic

(n ndash 1)th step

(n + 1)th step

nth step

Sediment transport

vtnndash1 αnndash1

unndash1 pnndash1

un+1 pn+1vt

n+1 αn+1


zn

Bed elevationzn+1




0 1 2 3 4 5t (s)

ndash020

02040608

112

ηH

NumTheory


(a) (b)


Velocity profile

H

x = 0Sediment bed

x

y


Rigid bed


0

005

01

015

z (m

)

xh = 4

ndash500 0 500 1000 1500 2000 2500 3000c (mgl)

(a)

xh = 10

0

005

01

015

z (m

)

ndash500 0 500 1000 1500 2000 2500 3000c (mgl)

(b)

Figure 6 Continued








ndash500 0 500 1000 1500 2000 2500 3000c (mgl)

0

005

01

015z (

m)

xh = 20

(c)

0

005

01

015

z (m

)

ndash500 0 500 1000 1500 2000 2500 3000c (mgl)

xh = 40

(d)








ndash006

0

006

012z (

m)

10 min

NumExp

(a)


ndash006

0

006

012

z (m

)

30 min

NumExp

(b)


2m

28m

D = 01m

(a)

04m

28m

(b)

Model grid

Refined grid

(c)


0 2 4 6 8 10 12t (s)

ndash005

0

005

01

015

η (m

)

TheoryNumerical

(a)

0 200 400 600 800 1000 1200t (s)

0

005

01

015

02

SD

MeasuredNumerical

(b)







Scour depth (cm)


t = 0s t = 300s t = 600s

t = 1500st = 1200s t = 900s

zy xzy x

zy x

zy xzy x

zy x


1 2 3 4 5 6 7 8

11280cm

Wave maker

Beach

Side view

SWL

(a)

115cm

35m30m

175mPlan view

24m

(b)

Refined grid

(c)





ndash04

0

04η

(m) G1

40 45 50 55 60 65 70t (s)

NumExp

G2

ndash04

0

04

η (m

)

40 45 50 55 60 65 70t (s)

NumExp

(a)

ndash04

0

04

η (m

) G3

40 45 50 55 60 65 70t (s)

NumExp

G4

ndash04

0

04

η (m

)

40 45 50 55 60 65 70t (s)

NumExp

(b)

ndash04

0

04

η (m

) G5

40 45 50 55 60 65 70t (s)

NumExp

G6

ndash04

0

04η

(m)

40 45 50 55 60 65 70t (s)

NumExp

(c)

40 45 50 55 60 65 70t (s)

ndash04

0

04

η (m

) G7

NumExp

ndash04

0

04

η (m

)

40 45 50 55 60 65 70t (s)

G8

NumExp

(d)



4 6 8 10 12 14x (m)

ndash04ndash02

00204

z (m

) ADV

NumExp

(a)

4 6 8 10 12 14x (m)

ndash01

0

01

Δz (m

)

NumExp

(b)

40 45 50 55 60 65 70t (s)

ndash1

0

1u

(ms

)

ADV Location

NumExp

(c)



x (m)

z (m

)

75 8 85 9 95 10 10506

08

1

12

0 20 40 60 80 100C (kgm3)

2ms

(a)

High concentration

t = 4686s

z (m

)

75 8 85 9 95 10 10506

08

1

12

x (m)

0 20 40 60 80 100C (kgm3)

2ms

(b)


z (m

)

75 8 85 9 95 10 10506

08

1

12

x (m)

0 20 40 60 80 100C (kgm3)

2ms

(c)

z (m

)

75 8 85 9 95 10 10506

08

1

12


x (m)

0 20 40 60 80 100C (kgm3)

2ms

(d)

z (m

)

75 8 85 9 95 10 10506

08

1

12

High concentration

t = 5636 s

x (m)

0 20 40 60 80 100C (kgm3)

2ms

(e)

z (m

)

75 8 85 9 95 10 10506

08

1

12t = 5686s


x (m)

0 20 40 60 80 100C (kgm3)

2ms

(f)





245m120 slope

50 60 70 80 90 100 1100

1

2

3

x (m)

z (m

)

Cylinder

(a)


A

B

C D

E035m

075m2m

3m

y (m

)

x (m)

(b)

(c)


2242832

ndash5 0 5 10 15 20 25t (s)

η (m

)

NumExp

(a)

ndash5 0 5 10 15 20 25t (s)

ndash3

0

3

u (m

s) Shallow water

NumExp

(b)

2242832

ndash5 0 5 10 15 20 25t (s)

η (m

)

NumExp

(c)

ndash5 0 5 10 15 20 25t (s)

2242832

η (m

)

NumExp

(d)






ndash20ndash10

010

z (cm

)

ndash5 0 5 10 15 20 25t (s)

NumExp

(a)

ndash20ndash10

010

z (cm

)

ndash5 0 5 10 15 20 25t (s)

NumExp

(b)

ndash5 0 5 10 15 20 25t (s)

ndash20ndash10

010

z (cm

)

NumExp

(c)


t = 05s

Runup

(a)

t = 45s

Runup

(b)

t = 145s

Drawdown

(c)

t = 245s

Drawdown

z

y x

(d)




7 Conclusions



t = 27s

0

05

1

15

2

y (m

)

48 485 49 495 50 505475x (m)


t = 34s

0

05

1

15

2Velocity Velocity

Velocity Velocity

Velocity Velocity

y (m

)

48 485 49 495 50 505475x (m)


ndash10ndash8

t = 28s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 35s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 29s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 36s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)





Data Availability




Acknowledgments


References





























































η H sec h2

3H

4h3

1113970

X1113890 1113891 (26)



1113968is the

















i i

i

P

(a)

i

i

i

i

P

(b)


Hydrodynamic

(n ndash 1)th step

(n + 1)th step

nth step

Sediment transport

vtnndash1 αnndash1

unndash1 pnndash1

un+1 pn+1vt

n+1 αn+1


zn

Bed elevationzn+1




0 1 2 3 4 5t (s)

ndash020

02040608

112

ηH

NumTheory


(a) (b)


Velocity profile

H

x = 0Sediment bed

x

y


Rigid bed


0

005

01

015

z (m

)

xh = 4

ndash500 0 500 1000 1500 2000 2500 3000c (mgl)

(a)

xh = 10

0

005

01

015

z (m

)

ndash500 0 500 1000 1500 2000 2500 3000c (mgl)

(b)

Figure 6 Continued








ndash500 0 500 1000 1500 2000 2500 3000c (mgl)

0

005

01

015z (

m)

xh = 20

(c)

0

005

01

015

z (m

)

ndash500 0 500 1000 1500 2000 2500 3000c (mgl)

xh = 40

(d)








ndash006

0

006

012z (

m)

10 min

NumExp

(a)


ndash006

0

006

012

z (m

)

30 min

NumExp

(b)


2m

28m

D = 01m

(a)

04m

28m

(b)

Model grid

Refined grid

(c)


0 2 4 6 8 10 12t (s)

ndash005

0

005

01

015

η (m

)

TheoryNumerical

(a)

0 200 400 600 800 1000 1200t (s)

0

005

01

015

02

SD

MeasuredNumerical

(b)







Scour depth (cm)


t = 0s t = 300s t = 600s

t = 1500st = 1200s t = 900s

zy xzy x

zy x

zy xzy x

zy x


1 2 3 4 5 6 7 8

11280cm

Wave maker

Beach

Side view

SWL

(a)

115cm

35m30m

175mPlan view

24m

(b)

Refined grid

(c)





ndash04

0

04η

(m) G1

40 45 50 55 60 65 70t (s)

NumExp

G2

ndash04

0

04

η (m

)

40 45 50 55 60 65 70t (s)

NumExp

(a)

ndash04

0

04

η (m

) G3

40 45 50 55 60 65 70t (s)

NumExp

G4

ndash04

0

04

η (m

)

40 45 50 55 60 65 70t (s)

NumExp

(b)

ndash04

0

04

η (m

) G5

40 45 50 55 60 65 70t (s)

NumExp

G6

ndash04

0

04η

(m)

40 45 50 55 60 65 70t (s)

NumExp

(c)

40 45 50 55 60 65 70t (s)

ndash04

0

04

η (m

) G7

NumExp

ndash04

0

04

η (m

)

40 45 50 55 60 65 70t (s)

G8

NumExp

(d)



4 6 8 10 12 14x (m)

ndash04ndash02

00204

z (m

) ADV

NumExp

(a)

4 6 8 10 12 14x (m)

ndash01

0

01

Δz (m

)

NumExp

(b)

40 45 50 55 60 65 70t (s)

ndash1

0

1u

(ms

)

ADV Location

NumExp

(c)



x (m)

z (m

)

75 8 85 9 95 10 10506

08

1

12

0 20 40 60 80 100C (kgm3)

2ms

(a)

High concentration

t = 4686s

z (m

)

75 8 85 9 95 10 10506

08

1

12

x (m)

0 20 40 60 80 100C (kgm3)

2ms

(b)


z (m

)

75 8 85 9 95 10 10506

08

1

12

x (m)

0 20 40 60 80 100C (kgm3)

2ms

(c)

z (m

)

75 8 85 9 95 10 10506

08

1

12


x (m)

0 20 40 60 80 100C (kgm3)

2ms

(d)

z (m

)

75 8 85 9 95 10 10506

08

1

12

High concentration

t = 5636 s

x (m)

0 20 40 60 80 100C (kgm3)

2ms

(e)

z (m

)

75 8 85 9 95 10 10506

08

1

12t = 5686s


x (m)

0 20 40 60 80 100C (kgm3)

2ms

(f)





245m120 slope

50 60 70 80 90 100 1100

1

2

3

x (m)

z (m

)

Cylinder

(a)


A

B

C D

E035m

075m2m

3m

y (m

)

x (m)

(b)

(c)


2242832

ndash5 0 5 10 15 20 25t (s)

η (m

)

NumExp

(a)

ndash5 0 5 10 15 20 25t (s)

ndash3

0

3

u (m

s) Shallow water

NumExp

(b)

2242832

ndash5 0 5 10 15 20 25t (s)

η (m

)

NumExp

(c)

ndash5 0 5 10 15 20 25t (s)

2242832

η (m

)

NumExp

(d)






ndash20ndash10

010

z (cm

)

ndash5 0 5 10 15 20 25t (s)

NumExp

(a)

ndash20ndash10

010

z (cm

)

ndash5 0 5 10 15 20 25t (s)

NumExp

(b)

ndash5 0 5 10 15 20 25t (s)

ndash20ndash10

010

z (cm

)

NumExp

(c)


t = 05s

Runup

(a)

t = 45s

Runup

(b)

t = 145s

Drawdown

(c)

t = 245s

Drawdown

z

y x

(d)




7 Conclusions



t = 27s

0

05

1

15

2

y (m

)

48 485 49 495 50 505475x (m)


t = 34s

0

05

1

15

2Velocity Velocity

Velocity Velocity

Velocity Velocity

y (m

)

48 485 49 495 50 505475x (m)


ndash10ndash8

t = 28s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 35s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 29s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 36s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)





Data Availability




Acknowledgments


References































































i i

i

P

(a)

i

i

i

i

P

(b)


Hydrodynamic

(n ndash 1)th step

(n + 1)th step

nth step

Sediment transport

vtnndash1 αnndash1

unndash1 pnndash1

un+1 pn+1vt

n+1 αn+1


zn

Bed elevationzn+1




0 1 2 3 4 5t (s)

ndash020

02040608

112

ηH

NumTheory


(a) (b)


Velocity profile

H

x = 0Sediment bed

x

y


Rigid bed


0

005

01

015

z (m

)

xh = 4

ndash500 0 500 1000 1500 2000 2500 3000c (mgl)

(a)

xh = 10

0

005

01

015

z (m

)

ndash500 0 500 1000 1500 2000 2500 3000c (mgl)

(b)

Figure 6 Continued








ndash500 0 500 1000 1500 2000 2500 3000c (mgl)

0

005

01

015z (

m)

xh = 20

(c)

0

005

01

015

z (m

)

ndash500 0 500 1000 1500 2000 2500 3000c (mgl)

xh = 40

(d)








ndash006

0

006

012z (

m)

10 min

NumExp

(a)


ndash006

0

006

012

z (m

)

30 min

NumExp

(b)


2m

28m

D = 01m

(a)

04m

28m

(b)

Model grid

Refined grid

(c)


0 2 4 6 8 10 12t (s)

ndash005

0

005

01

015

η (m

)

TheoryNumerical

(a)

0 200 400 600 800 1000 1200t (s)

0

005

01

015

02

SD

MeasuredNumerical

(b)







Scour depth (cm)


t = 0s t = 300s t = 600s

t = 1500st = 1200s t = 900s

zy xzy x

zy x

zy xzy x

zy x


1 2 3 4 5 6 7 8

11280cm

Wave maker

Beach

Side view

SWL

(a)

115cm

35m30m

175mPlan view

24m

(b)

Refined grid

(c)





ndash04

0

04η

(m) G1

40 45 50 55 60 65 70t (s)

NumExp

G2

ndash04

0

04

η (m

)

40 45 50 55 60 65 70t (s)

NumExp

(a)

ndash04

0

04

η (m

) G3

40 45 50 55 60 65 70t (s)

NumExp

G4

ndash04

0

04

η (m

)

40 45 50 55 60 65 70t (s)

NumExp

(b)

ndash04

0

04

η (m

) G5

40 45 50 55 60 65 70t (s)

NumExp

G6

ndash04

0

04η

(m)

40 45 50 55 60 65 70t (s)

NumExp

(c)

40 45 50 55 60 65 70t (s)

ndash04

0

04

η (m

) G7

NumExp

ndash04

0

04

η (m

)

40 45 50 55 60 65 70t (s)

G8

NumExp

(d)



4 6 8 10 12 14x (m)

ndash04ndash02

00204

z (m

) ADV

NumExp

(a)

4 6 8 10 12 14x (m)

ndash01

0

01

Δz (m

)

NumExp

(b)

40 45 50 55 60 65 70t (s)

ndash1

0

1u

(ms

)

ADV Location

NumExp

(c)



x (m)

z (m

)

75 8 85 9 95 10 10506

08

1

12

0 20 40 60 80 100C (kgm3)

2ms

(a)

High concentration

t = 4686s

z (m

)

75 8 85 9 95 10 10506

08

1

12

x (m)

0 20 40 60 80 100C (kgm3)

2ms

(b)


z (m

)

75 8 85 9 95 10 10506

08

1

12

x (m)

0 20 40 60 80 100C (kgm3)

2ms

(c)

z (m

)

75 8 85 9 95 10 10506

08

1

12


x (m)

0 20 40 60 80 100C (kgm3)

2ms

(d)

z (m

)

75 8 85 9 95 10 10506

08

1

12

High concentration

t = 5636 s

x (m)

0 20 40 60 80 100C (kgm3)

2ms

(e)

z (m

)

75 8 85 9 95 10 10506

08

1

12t = 5686s


x (m)

0 20 40 60 80 100C (kgm3)

2ms

(f)





245m120 slope

50 60 70 80 90 100 1100

1

2

3

x (m)

z (m

)

Cylinder

(a)


A

B

C D

E035m

075m2m

3m

y (m

)

x (m)

(b)

(c)


2242832

ndash5 0 5 10 15 20 25t (s)

η (m

)

NumExp

(a)

ndash5 0 5 10 15 20 25t (s)

ndash3

0

3

u (m

s) Shallow water

NumExp

(b)

2242832

ndash5 0 5 10 15 20 25t (s)

η (m

)

NumExp

(c)

ndash5 0 5 10 15 20 25t (s)

2242832

η (m

)

NumExp

(d)






ndash20ndash10

010

z (cm

)

ndash5 0 5 10 15 20 25t (s)

NumExp

(a)

ndash20ndash10

010

z (cm

)

ndash5 0 5 10 15 20 25t (s)

NumExp

(b)

ndash5 0 5 10 15 20 25t (s)

ndash20ndash10

010

z (cm

)

NumExp

(c)


t = 05s

Runup

(a)

t = 45s

Runup

(b)

t = 145s

Drawdown

(c)

t = 245s

Drawdown

z

y x

(d)




7 Conclusions



t = 27s

0

05

1

15

2

y (m

)

48 485 49 495 50 505475x (m)


t = 34s

0

05

1

15

2Velocity Velocity

Velocity Velocity

Velocity Velocity

y (m

)

48 485 49 495 50 505475x (m)


ndash10ndash8

t = 28s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 35s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 29s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 36s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)





Data Availability




Acknowledgments


References
























































0 1 2 3 4 5t (s)

ndash020

02040608

112

ηH

NumTheory


(a) (b)


Velocity profile

H

x = 0Sediment bed

x

y


Rigid bed


0

005

01

015

z (m

)

xh = 4

ndash500 0 500 1000 1500 2000 2500 3000c (mgl)

(a)

xh = 10

0

005

01

015

z (m

)

ndash500 0 500 1000 1500 2000 2500 3000c (mgl)

(b)

Figure 6 Continued








ndash500 0 500 1000 1500 2000 2500 3000c (mgl)

0

005

01

015z (

m)

xh = 20

(c)

0

005

01

015

z (m

)

ndash500 0 500 1000 1500 2000 2500 3000c (mgl)

xh = 40

(d)








ndash006

0

006

012z (

m)

10 min

NumExp

(a)


ndash006

0

006

012

z (m

)

30 min

NumExp

(b)


2m

28m

D = 01m

(a)

04m

28m

(b)

Model grid

Refined grid

(c)


0 2 4 6 8 10 12t (s)

ndash005

0

005

01

015

η (m

)

TheoryNumerical

(a)

0 200 400 600 800 1000 1200t (s)

0

005

01

015

02

SD

MeasuredNumerical

(b)







Scour depth (cm)


t = 0s t = 300s t = 600s

t = 1500st = 1200s t = 900s

zy xzy x

zy x

zy xzy x

zy x


1 2 3 4 5 6 7 8

11280cm

Wave maker

Beach

Side view

SWL

(a)

115cm

35m30m

175mPlan view

24m

(b)

Refined grid

(c)





ndash04

0

04η

(m) G1

40 45 50 55 60 65 70t (s)

NumExp

G2

ndash04

0

04

η (m

)

40 45 50 55 60 65 70t (s)

NumExp

(a)

ndash04

0

04

η (m

) G3

40 45 50 55 60 65 70t (s)

NumExp

G4

ndash04

0

04

η (m

)

40 45 50 55 60 65 70t (s)

NumExp

(b)

ndash04

0

04

η (m

) G5

40 45 50 55 60 65 70t (s)

NumExp

G6

ndash04

0

04η

(m)

40 45 50 55 60 65 70t (s)

NumExp

(c)

40 45 50 55 60 65 70t (s)

ndash04

0

04

η (m

) G7

NumExp

ndash04

0

04

η (m

)

40 45 50 55 60 65 70t (s)

G8

NumExp

(d)



4 6 8 10 12 14x (m)

ndash04ndash02

00204

z (m

) ADV

NumExp

(a)

4 6 8 10 12 14x (m)

ndash01

0

01

Δz (m

)

NumExp

(b)

40 45 50 55 60 65 70t (s)

ndash1

0

1u

(ms

)

ADV Location

NumExp

(c)



x (m)

z (m

)

75 8 85 9 95 10 10506

08

1

12

0 20 40 60 80 100C (kgm3)

2ms

(a)

High concentration

t = 4686s

z (m

)

75 8 85 9 95 10 10506

08

1

12

x (m)

0 20 40 60 80 100C (kgm3)

2ms

(b)


z (m

)

75 8 85 9 95 10 10506

08

1

12

x (m)

0 20 40 60 80 100C (kgm3)

2ms

(c)

z (m

)

75 8 85 9 95 10 10506

08

1

12


x (m)

0 20 40 60 80 100C (kgm3)

2ms

(d)

z (m

)

75 8 85 9 95 10 10506

08

1

12

High concentration

t = 5636 s

x (m)

0 20 40 60 80 100C (kgm3)

2ms

(e)

z (m

)

75 8 85 9 95 10 10506

08

1

12t = 5686s


x (m)

0 20 40 60 80 100C (kgm3)

2ms

(f)





245m120 slope

50 60 70 80 90 100 1100

1

2

3

x (m)

z (m

)

Cylinder

(a)


A

B

C D

E035m

075m2m

3m

y (m

)

x (m)

(b)

(c)


2242832

ndash5 0 5 10 15 20 25t (s)

η (m

)

NumExp

(a)

ndash5 0 5 10 15 20 25t (s)

ndash3

0

3

u (m

s) Shallow water

NumExp

(b)

2242832

ndash5 0 5 10 15 20 25t (s)

η (m

)

NumExp

(c)

ndash5 0 5 10 15 20 25t (s)

2242832

η (m

)

NumExp

(d)






ndash20ndash10

010

z (cm

)

ndash5 0 5 10 15 20 25t (s)

NumExp

(a)

ndash20ndash10

010

z (cm

)

ndash5 0 5 10 15 20 25t (s)

NumExp

(b)

ndash5 0 5 10 15 20 25t (s)

ndash20ndash10

010

z (cm

)

NumExp

(c)


t = 05s

Runup

(a)

t = 45s

Runup

(b)

t = 145s

Drawdown

(c)

t = 245s

Drawdown

z

y x

(d)




7 Conclusions



t = 27s

0

05

1

15

2

y (m

)

48 485 49 495 50 505475x (m)


t = 34s

0

05

1

15

2Velocity Velocity

Velocity Velocity

Velocity Velocity

y (m

)

48 485 49 495 50 505475x (m)


ndash10ndash8

t = 28s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 35s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 29s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 36s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)





Data Availability




Acknowledgments


References






























































ndash500 0 500 1000 1500 2000 2500 3000c (mgl)

0

005

01

015z (

m)

xh = 20

(c)

0

005

01

015

z (m

)

ndash500 0 500 1000 1500 2000 2500 3000c (mgl)

xh = 40

(d)








ndash006

0

006

012z (

m)

10 min

NumExp

(a)


ndash006

0

006

012

z (m

)

30 min

NumExp

(b)


2m

28m

D = 01m

(a)

04m

28m

(b)

Model grid

Refined grid

(c)


0 2 4 6 8 10 12t (s)

ndash005

0

005

01

015

η (m

)

TheoryNumerical

(a)

0 200 400 600 800 1000 1200t (s)

0

005

01

015

02

SD

MeasuredNumerical

(b)







Scour depth (cm)


t = 0s t = 300s t = 600s

t = 1500st = 1200s t = 900s

zy xzy x

zy x

zy xzy x

zy x


1 2 3 4 5 6 7 8

11280cm

Wave maker

Beach

Side view

SWL

(a)

115cm

35m30m

175mPlan view

24m

(b)

Refined grid

(c)





ndash04

0

04η

(m) G1

40 45 50 55 60 65 70t (s)

NumExp

G2

ndash04

0

04

η (m

)

40 45 50 55 60 65 70t (s)

NumExp

(a)

ndash04

0

04

η (m

) G3

40 45 50 55 60 65 70t (s)

NumExp

G4

ndash04

0

04

η (m

)

40 45 50 55 60 65 70t (s)

NumExp

(b)

ndash04

0

04

η (m

) G5

40 45 50 55 60 65 70t (s)

NumExp

G6

ndash04

0

04η

(m)

40 45 50 55 60 65 70t (s)

NumExp

(c)

40 45 50 55 60 65 70t (s)

ndash04

0

04

η (m

) G7

NumExp

ndash04

0

04

η (m

)

40 45 50 55 60 65 70t (s)

G8

NumExp

(d)



4 6 8 10 12 14x (m)

ndash04ndash02

00204

z (m

) ADV

NumExp

(a)

4 6 8 10 12 14x (m)

ndash01

0

01

Δz (m

)

NumExp

(b)

40 45 50 55 60 65 70t (s)

ndash1

0

1u

(ms

)

ADV Location

NumExp

(c)



x (m)

z (m

)

75 8 85 9 95 10 10506

08

1

12

0 20 40 60 80 100C (kgm3)

2ms

(a)

High concentration

t = 4686s

z (m

)

75 8 85 9 95 10 10506

08

1

12

x (m)

0 20 40 60 80 100C (kgm3)

2ms

(b)


z (m

)

75 8 85 9 95 10 10506

08

1

12

x (m)

0 20 40 60 80 100C (kgm3)

2ms

(c)

z (m

)

75 8 85 9 95 10 10506

08

1

12


x (m)

0 20 40 60 80 100C (kgm3)

2ms

(d)

z (m

)

75 8 85 9 95 10 10506

08

1

12

High concentration

t = 5636 s

x (m)

0 20 40 60 80 100C (kgm3)

2ms

(e)

z (m

)

75 8 85 9 95 10 10506

08

1

12t = 5686s


x (m)

0 20 40 60 80 100C (kgm3)

2ms

(f)





245m120 slope

50 60 70 80 90 100 1100

1

2

3

x (m)

z (m

)

Cylinder

(a)


A

B

C D

E035m

075m2m

3m

y (m

)

x (m)

(b)

(c)


2242832

ndash5 0 5 10 15 20 25t (s)

η (m

)

NumExp

(a)

ndash5 0 5 10 15 20 25t (s)

ndash3

0

3

u (m

s) Shallow water

NumExp

(b)

2242832

ndash5 0 5 10 15 20 25t (s)

η (m

)

NumExp

(c)

ndash5 0 5 10 15 20 25t (s)

2242832

η (m

)

NumExp

(d)






ndash20ndash10

010

z (cm

)

ndash5 0 5 10 15 20 25t (s)

NumExp

(a)

ndash20ndash10

010

z (cm

)

ndash5 0 5 10 15 20 25t (s)

NumExp

(b)

ndash5 0 5 10 15 20 25t (s)

ndash20ndash10

010

z (cm

)

NumExp

(c)


t = 05s

Runup

(a)

t = 45s

Runup

(b)

t = 145s

Drawdown

(c)

t = 245s

Drawdown

z

y x

(d)




7 Conclusions



t = 27s

0

05

1

15

2

y (m

)

48 485 49 495 50 505475x (m)


t = 34s

0

05

1

15

2Velocity Velocity

Velocity Velocity

Velocity Velocity

y (m

)

48 485 49 495 50 505475x (m)


ndash10ndash8

t = 28s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 35s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 29s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 36s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)





Data Availability




Acknowledgments


References




























































ndash006

0

006

012z (

m)

10 min

NumExp

(a)


ndash006

0

006

012

z (m

)

30 min

NumExp

(b)


2m

28m

D = 01m

(a)

04m

28m

(b)

Model grid

Refined grid

(c)


0 2 4 6 8 10 12t (s)

ndash005

0

005

01

015

η (m

)

TheoryNumerical

(a)

0 200 400 600 800 1000 1200t (s)

0

005

01

015

02

SD

MeasuredNumerical

(b)







Scour depth (cm)


t = 0s t = 300s t = 600s

t = 1500st = 1200s t = 900s

zy xzy x

zy x

zy xzy x

zy x


1 2 3 4 5 6 7 8

11280cm

Wave maker

Beach

Side view

SWL

(a)

115cm

35m30m

175mPlan view

24m

(b)

Refined grid

(c)





ndash04

0

04η

(m) G1

40 45 50 55 60 65 70t (s)

NumExp

G2

ndash04

0

04

η (m

)

40 45 50 55 60 65 70t (s)

NumExp

(a)

ndash04

0

04

η (m

) G3

40 45 50 55 60 65 70t (s)

NumExp

G4

ndash04

0

04

η (m

)

40 45 50 55 60 65 70t (s)

NumExp

(b)

ndash04

0

04

η (m

) G5

40 45 50 55 60 65 70t (s)

NumExp

G6

ndash04

0

04η

(m)

40 45 50 55 60 65 70t (s)

NumExp

(c)

40 45 50 55 60 65 70t (s)

ndash04

0

04

η (m

) G7

NumExp

ndash04

0

04

η (m

)

40 45 50 55 60 65 70t (s)

G8

NumExp

(d)



4 6 8 10 12 14x (m)

ndash04ndash02

00204

z (m

) ADV

NumExp

(a)

4 6 8 10 12 14x (m)

ndash01

0

01

Δz (m

)

NumExp

(b)

40 45 50 55 60 65 70t (s)

ndash1

0

1u

(ms

)

ADV Location

NumExp

(c)



x (m)

z (m

)

75 8 85 9 95 10 10506

08

1

12

0 20 40 60 80 100C (kgm3)

2ms

(a)

High concentration

t = 4686s

z (m

)

75 8 85 9 95 10 10506

08

1

12

x (m)

0 20 40 60 80 100C (kgm3)

2ms

(b)


z (m

)

75 8 85 9 95 10 10506

08

1

12

x (m)

0 20 40 60 80 100C (kgm3)

2ms

(c)

z (m

)

75 8 85 9 95 10 10506

08

1

12


x (m)

0 20 40 60 80 100C (kgm3)

2ms

(d)

z (m

)

75 8 85 9 95 10 10506

08

1

12

High concentration

t = 5636 s

x (m)

0 20 40 60 80 100C (kgm3)

2ms

(e)

z (m

)

75 8 85 9 95 10 10506

08

1

12t = 5686s


x (m)

0 20 40 60 80 100C (kgm3)

2ms

(f)





245m120 slope

50 60 70 80 90 100 1100

1

2

3

x (m)

z (m

)

Cylinder

(a)


A

B

C D

E035m

075m2m

3m

y (m

)

x (m)

(b)

(c)


2242832

ndash5 0 5 10 15 20 25t (s)

η (m

)

NumExp

(a)

ndash5 0 5 10 15 20 25t (s)

ndash3

0

3

u (m

s) Shallow water

NumExp

(b)

2242832

ndash5 0 5 10 15 20 25t (s)

η (m

)

NumExp

(c)

ndash5 0 5 10 15 20 25t (s)

2242832

η (m

)

NumExp

(d)






ndash20ndash10

010

z (cm

)

ndash5 0 5 10 15 20 25t (s)

NumExp

(a)

ndash20ndash10

010

z (cm

)

ndash5 0 5 10 15 20 25t (s)

NumExp

(b)

ndash5 0 5 10 15 20 25t (s)

ndash20ndash10

010

z (cm

)

NumExp

(c)


t = 05s

Runup

(a)

t = 45s

Runup

(b)

t = 145s

Drawdown

(c)

t = 245s

Drawdown

z

y x

(d)




7 Conclusions



t = 27s

0

05

1

15

2

y (m

)

48 485 49 495 50 505475x (m)


t = 34s

0

05

1

15

2Velocity Velocity

Velocity Velocity

Velocity Velocity

y (m

)

48 485 49 495 50 505475x (m)


ndash10ndash8

t = 28s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 35s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 29s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 36s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)





Data Availability




Acknowledgments


References




























































Scour depth (cm)


t = 0s t = 300s t = 600s

t = 1500st = 1200s t = 900s

zy xzy x

zy x

zy xzy x

zy x


1 2 3 4 5 6 7 8

11280cm

Wave maker

Beach

Side view

SWL

(a)

115cm

35m30m

175mPlan view

24m

(b)

Refined grid

(c)





ndash04

0

04η

(m) G1

40 45 50 55 60 65 70t (s)

NumExp

G2

ndash04

0

04

η (m

)

40 45 50 55 60 65 70t (s)

NumExp

(a)

ndash04

0

04

η (m

) G3

40 45 50 55 60 65 70t (s)

NumExp

G4

ndash04

0

04

η (m

)

40 45 50 55 60 65 70t (s)

NumExp

(b)

ndash04

0

04

η (m

) G5

40 45 50 55 60 65 70t (s)

NumExp

G6

ndash04

0

04η

(m)

40 45 50 55 60 65 70t (s)

NumExp

(c)

40 45 50 55 60 65 70t (s)

ndash04

0

04

η (m

) G7

NumExp

ndash04

0

04

η (m

)

40 45 50 55 60 65 70t (s)

G8

NumExp

(d)



4 6 8 10 12 14x (m)

ndash04ndash02

00204

z (m

) ADV

NumExp

(a)

4 6 8 10 12 14x (m)

ndash01

0

01

Δz (m

)

NumExp

(b)

40 45 50 55 60 65 70t (s)

ndash1

0

1u

(ms

)

ADV Location

NumExp

(c)



x (m)

z (m

)

75 8 85 9 95 10 10506

08

1

12

0 20 40 60 80 100C (kgm3)

2ms

(a)

High concentration

t = 4686s

z (m

)

75 8 85 9 95 10 10506

08

1

12

x (m)

0 20 40 60 80 100C (kgm3)

2ms

(b)


z (m

)

75 8 85 9 95 10 10506

08

1

12

x (m)

0 20 40 60 80 100C (kgm3)

2ms

(c)

z (m

)

75 8 85 9 95 10 10506

08

1

12


x (m)

0 20 40 60 80 100C (kgm3)

2ms

(d)

z (m

)

75 8 85 9 95 10 10506

08

1

12

High concentration

t = 5636 s

x (m)

0 20 40 60 80 100C (kgm3)

2ms

(e)

z (m

)

75 8 85 9 95 10 10506

08

1

12t = 5686s


x (m)

0 20 40 60 80 100C (kgm3)

2ms

(f)





245m120 slope

50 60 70 80 90 100 1100

1

2

3

x (m)

z (m

)

Cylinder

(a)


A

B

C D

E035m

075m2m

3m

y (m

)

x (m)

(b)

(c)


2242832

ndash5 0 5 10 15 20 25t (s)

η (m

)

NumExp

(a)

ndash5 0 5 10 15 20 25t (s)

ndash3

0

3

u (m

s) Shallow water

NumExp

(b)

2242832

ndash5 0 5 10 15 20 25t (s)

η (m

)

NumExp

(c)

ndash5 0 5 10 15 20 25t (s)

2242832

η (m

)

NumExp

(d)






ndash20ndash10

010

z (cm

)

ndash5 0 5 10 15 20 25t (s)

NumExp

(a)

ndash20ndash10

010

z (cm

)

ndash5 0 5 10 15 20 25t (s)

NumExp

(b)

ndash5 0 5 10 15 20 25t (s)

ndash20ndash10

010

z (cm

)

NumExp

(c)


t = 05s

Runup

(a)

t = 45s

Runup

(b)

t = 145s

Drawdown

(c)

t = 245s

Drawdown

z

y x

(d)




7 Conclusions



t = 27s

0

05

1

15

2

y (m

)

48 485 49 495 50 505475x (m)


t = 34s

0

05

1

15

2Velocity Velocity

Velocity Velocity

Velocity Velocity

y (m

)

48 485 49 495 50 505475x (m)


ndash10ndash8

t = 28s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 35s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 29s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 36s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)





Data Availability




Acknowledgments


References


























































ndash04

0

04η

(m) G1

40 45 50 55 60 65 70t (s)

NumExp

G2

ndash04

0

04

η (m

)

40 45 50 55 60 65 70t (s)

NumExp

(a)

ndash04

0

04

η (m

) G3

40 45 50 55 60 65 70t (s)

NumExp

G4

ndash04

0

04

η (m

)

40 45 50 55 60 65 70t (s)

NumExp

(b)

ndash04

0

04

η (m

) G5

40 45 50 55 60 65 70t (s)

NumExp

G6

ndash04

0

04η

(m)

40 45 50 55 60 65 70t (s)

NumExp

(c)

40 45 50 55 60 65 70t (s)

ndash04

0

04

η (m

) G7

NumExp

ndash04

0

04

η (m

)

40 45 50 55 60 65 70t (s)

G8

NumExp

(d)



4 6 8 10 12 14x (m)

ndash04ndash02

00204

z (m

) ADV

NumExp

(a)

4 6 8 10 12 14x (m)

ndash01

0

01

Δz (m

)

NumExp

(b)

40 45 50 55 60 65 70t (s)

ndash1

0

1u

(ms

)

ADV Location

NumExp

(c)



x (m)

z (m

)

75 8 85 9 95 10 10506

08

1

12

0 20 40 60 80 100C (kgm3)

2ms

(a)

High concentration

t = 4686s

z (m

)

75 8 85 9 95 10 10506

08

1

12

x (m)

0 20 40 60 80 100C (kgm3)

2ms

(b)


z (m

)

75 8 85 9 95 10 10506

08

1

12

x (m)

0 20 40 60 80 100C (kgm3)

2ms

(c)

z (m

)

75 8 85 9 95 10 10506

08

1

12


x (m)

0 20 40 60 80 100C (kgm3)

2ms

(d)

z (m

)

75 8 85 9 95 10 10506

08

1

12

High concentration

t = 5636 s

x (m)

0 20 40 60 80 100C (kgm3)

2ms

(e)

z (m

)

75 8 85 9 95 10 10506

08

1

12t = 5686s


x (m)

0 20 40 60 80 100C (kgm3)

2ms

(f)





245m120 slope

50 60 70 80 90 100 1100

1

2

3

x (m)

z (m

)

Cylinder

(a)


A

B

C D

E035m

075m2m

3m

y (m

)

x (m)

(b)

(c)


2242832

ndash5 0 5 10 15 20 25t (s)

η (m

)

NumExp

(a)

ndash5 0 5 10 15 20 25t (s)

ndash3

0

3

u (m

s) Shallow water

NumExp

(b)

2242832

ndash5 0 5 10 15 20 25t (s)

η (m

)

NumExp

(c)

ndash5 0 5 10 15 20 25t (s)

2242832

η (m

)

NumExp

(d)






ndash20ndash10

010

z (cm

)

ndash5 0 5 10 15 20 25t (s)

NumExp

(a)

ndash20ndash10

010

z (cm

)

ndash5 0 5 10 15 20 25t (s)

NumExp

(b)

ndash5 0 5 10 15 20 25t (s)

ndash20ndash10

010

z (cm

)

NumExp

(c)


t = 05s

Runup

(a)

t = 45s

Runup

(b)

t = 145s

Drawdown

(c)

t = 245s

Drawdown

z

y x

(d)




7 Conclusions



t = 27s

0

05

1

15

2

y (m

)

48 485 49 495 50 505475x (m)


t = 34s

0

05

1

15

2Velocity Velocity

Velocity Velocity

Velocity Velocity

y (m

)

48 485 49 495 50 505475x (m)


ndash10ndash8

t = 28s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 35s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 29s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 36s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)





Data Availability




Acknowledgments


References
























































4 6 8 10 12 14x (m)

ndash04ndash02

00204

z (m

) ADV

NumExp

(a)

4 6 8 10 12 14x (m)

ndash01

0

01

Δz (m

)

NumExp

(b)

40 45 50 55 60 65 70t (s)

ndash1

0

1u

(ms

)

ADV Location

NumExp

(c)



x (m)

z (m

)

75 8 85 9 95 10 10506

08

1

12

0 20 40 60 80 100C (kgm3)

2ms

(a)

High concentration

t = 4686s

z (m

)

75 8 85 9 95 10 10506

08

1

12

x (m)

0 20 40 60 80 100C (kgm3)

2ms

(b)


z (m

)

75 8 85 9 95 10 10506

08

1

12

x (m)

0 20 40 60 80 100C (kgm3)

2ms

(c)

z (m

)

75 8 85 9 95 10 10506

08

1

12


x (m)

0 20 40 60 80 100C (kgm3)

2ms

(d)

z (m

)

75 8 85 9 95 10 10506

08

1

12

High concentration

t = 5636 s

x (m)

0 20 40 60 80 100C (kgm3)

2ms

(e)

z (m

)

75 8 85 9 95 10 10506

08

1

12t = 5686s


x (m)

0 20 40 60 80 100C (kgm3)

2ms

(f)





245m120 slope

50 60 70 80 90 100 1100

1

2

3

x (m)

z (m

)

Cylinder

(a)


A

B

C D

E035m

075m2m

3m

y (m

)

x (m)

(b)

(c)


2242832

ndash5 0 5 10 15 20 25t (s)

η (m

)

NumExp

(a)

ndash5 0 5 10 15 20 25t (s)

ndash3

0

3

u (m

s) Shallow water

NumExp

(b)

2242832

ndash5 0 5 10 15 20 25t (s)

η (m

)

NumExp

(c)

ndash5 0 5 10 15 20 25t (s)

2242832

η (m

)

NumExp

(d)






ndash20ndash10

010

z (cm

)

ndash5 0 5 10 15 20 25t (s)

NumExp

(a)

ndash20ndash10

010

z (cm

)

ndash5 0 5 10 15 20 25t (s)

NumExp

(b)

ndash5 0 5 10 15 20 25t (s)

ndash20ndash10

010

z (cm

)

NumExp

(c)


t = 05s

Runup

(a)

t = 45s

Runup

(b)

t = 145s

Drawdown

(c)

t = 245s

Drawdown

z

y x

(d)




7 Conclusions



t = 27s

0

05

1

15

2

y (m

)

48 485 49 495 50 505475x (m)


t = 34s

0

05

1

15

2Velocity Velocity

Velocity Velocity

Velocity Velocity

y (m

)

48 485 49 495 50 505475x (m)


ndash10ndash8

t = 28s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 35s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 29s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 36s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)





Data Availability




Acknowledgments


References


























































245m120 slope

50 60 70 80 90 100 1100

1

2

3

x (m)

z (m

)

Cylinder

(a)


A

B

C D

E035m

075m2m

3m

y (m

)

x (m)

(b)

(c)


2242832

ndash5 0 5 10 15 20 25t (s)

η (m

)

NumExp

(a)

ndash5 0 5 10 15 20 25t (s)

ndash3

0

3

u (m

s) Shallow water

NumExp

(b)

2242832

ndash5 0 5 10 15 20 25t (s)

η (m

)

NumExp

(c)

ndash5 0 5 10 15 20 25t (s)

2242832

η (m

)

NumExp

(d)






ndash20ndash10

010

z (cm

)

ndash5 0 5 10 15 20 25t (s)

NumExp

(a)

ndash20ndash10

010

z (cm

)

ndash5 0 5 10 15 20 25t (s)

NumExp

(b)

ndash5 0 5 10 15 20 25t (s)

ndash20ndash10

010

z (cm

)

NumExp

(c)


t = 05s

Runup

(a)

t = 45s

Runup

(b)

t = 145s

Drawdown

(c)

t = 245s

Drawdown

z

y x

(d)




7 Conclusions



t = 27s

0

05

1

15

2

y (m

)

48 485 49 495 50 505475x (m)


t = 34s

0

05

1

15

2Velocity Velocity

Velocity Velocity

Velocity Velocity

y (m

)

48 485 49 495 50 505475x (m)


ndash10ndash8

t = 28s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 35s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 29s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 36s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)





Data Availability




Acknowledgments


References



























































ndash20ndash10

010

z (cm

)

ndash5 0 5 10 15 20 25t (s)

NumExp

(a)

ndash20ndash10

010

z (cm

)

ndash5 0 5 10 15 20 25t (s)

NumExp

(b)

ndash5 0 5 10 15 20 25t (s)

ndash20ndash10

010

z (cm

)

NumExp

(c)


t = 05s

Runup

(a)

t = 45s

Runup

(b)

t = 145s

Drawdown

(c)

t = 245s

Drawdown

z

y x

(d)




7 Conclusions



t = 27s

0

05

1

15

2

y (m

)

48 485 49 495 50 505475x (m)


t = 34s

0

05

1

15

2Velocity Velocity

Velocity Velocity

Velocity Velocity

y (m

)

48 485 49 495 50 505475x (m)


ndash10ndash8

t = 28s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 35s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 29s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 36s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)





Data Availability




Acknowledgments


References

























































7 Conclusions



t = 27s

0

05

1

15

2

y (m

)

48 485 49 495 50 505475x (m)


t = 34s

0

05

1

15

2Velocity Velocity

Velocity Velocity

Velocity Velocity

y (m

)

48 485 49 495 50 505475x (m)


ndash10ndash8

t = 28s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 35s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 29s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)


t = 36s

48 485 49 495 50 505475x (m)

0

05

1

15

2

y (m

)





Data Availability




Acknowledgments


References


























































Data Availability




Acknowledgments


References
























































3dmodelingandmechanismanalysisofbreakingwave-induced

Documents