International Journal of Offshore and Polar Engineering (ISSN 1053-5381) http://www.isope.org/publicationsCopyright © by The International Society of Offshore and Polar EngineersVol. 25, No. 1, March 2015, pp. 1–7

Numerical Simulation of Long Wave Run-up for Breaking and Non-breaking Waves

Mostafa S. ShadlooLHEEA Laboratory, Ecole Centrale de Nantes

Nantes, France

Robert WeissDepartment of Geosciences, Virginia Polytech Institute and State University

Blacksburg, Virginia, USA

Mehmet YildizFaculty of Engineering and Natural Sciences, Sabanci University

Istanbul, Turkey

Robert A. DalrympleDepartment of Civil Engineering, Johns Hopkins University

Baltimore, Maryland, USA

Tsunamis produce a wealth of quantitative data that can be used to improve tsunami hazard awareness and to increase thepreparedness of the population at risk. These data also allow for a performance evaluation of the coastal infrastructure andobservations of sediment transport, erosion, and deposition. The interaction of the tsunami with coastal infrastructures and withthe movable sediment bed is a three-dimensional process. Therefore, for runup and inundation prediction, three-dimensionalnumerical models must be employed. In this study, we have employed Smoothed Particle Hydrodynamics (SPH) to simulatetsunami runup on idealized geometries for the validation and exploration of three-dimensional flow structures in tsunamis. Wemake use of the canonical experiments for long-wave run-up for breaking and non-breaking waves. The results of ourstudy prove that SPH is able to reproduce the runup of long waves for different initial and geometric conditions. We havealso investigated the applicability and the effectiveness of different viscous terms that are available in the SPH literature.Additionally, a new breaking criterion based on numerical experiments is introduced and its similarities and differences withexisting criteria are discussed.

INTRODUCTION

Large traveling water waves over the ocean, usually caused byearthquakes, submarine landslides, or volcanic eruptions, are knownas tsunamis. Tsunamis have been causing considerable widespreaddamage and loss of human lives. Since tsunamis are characterizedas water waves with long periods and wavelengths, it is practical toconsider them as solitary waves for the research application. Thesewaves near the coastal area are usually investigated analytically byusing either the Boussinesq or the shallow water wave equation.The Boussinesq approximation is valid for weakly nonlinear andlong water waves. The set of equations for the latter approach canbe directly derived from the former one by neglecting the dispersioneffects and the vertical accelerations. Both sets of equations arecharacterized by the high wavelength-to-water depth ratio. Shallowwater waves have been studied in laboratory experiments fordecades (Synolakis, 1987; Pedersen and Gjevik, 1983). Also,elegant analytical solutions of the shallow water equations weredeveloped by Synolakis (1987) and Pedersen and Gjevik (1983).

As in the case of any other wave, the solitary waves can break.Although breaking and nonbreaking waves have been extensivelystudied in the laboratory (Synolakis, 1986, 1987; Pedersen and

Received September 18, 2014; revised manuscript received by the editorsNovember 10, 2014. The original version was submitted directly to theJournal.

KEY WORDS: Smoothed Particle Hydrodynamics (SPH), particle methods,CUDA GPU, solitary wave, tsunami, Sub Particle Scales (SPS) turbulence.

Gjevik, 1983), the theoretical understanding of breaking solitarywaves is incomplete because of the limiting boundary and initialconditions that are necessary to find a meaningful analyticalsolution. Even though the shallow-water type of equations can beincorporated with higher-order derivatives to simulate dispersionand other nonlinearities, their results are mainly limited by somecritical assumptions such as two-dimensionality. In the last twodecades, the SPH method has become an important tool in outreachefforts, in testing future engineering designs, and in tsunamiresearch. Extensive SPH simulations have been conducted in orderto study the dynamic behavior of such waves (Landrini et al., 2007;Khayyer et al., 2008). Nevertheless, most of the available SPHsimulations in the literature are two-dimensional, and the effects ofviscosity and/or turbulent viscosity are neglected. The simulation ofnonbreaking and breaking solitary waves with three-dimensionalnumerical models offers an alternative approach to explore thelinear and nonlinear physical processes occurring during near-shorepropagation, runup, and the withdrawal of solitary waves. Recentdevelopments in computer technology and a better understandingof numerical methods have provided the opportunity to carry outmassively parallel simulations of fluid mechanics on a very smallscale. Hence, it is possible to solve the fully three-dimensionalNavier-Stokes equations. We have employed a three-dimensionalLagrangian approach to simulate the dynamics of breaking andnonbreaking solitary waves, thereby introducing some new insightsinto the behavior of these waves.

2 Numerical Simulation of Long Wave Run-up for Breaking and Non-breaking Waves

GOVERNING EQUATIONS

The governing equations employed in our modeling efforts arethe conservation of mass and momentum equations, which arewritten in the Lagrangian form as:

D�

Dt= −�ï · Eu (1)

DEu

Dt= −

ïp

�+ vï 2

Eu+ Eg (2)

where Eu is the velocity vector, p is the pressure, t is the time, �is the density, Eg is the gravitational acceleration vector, v is thelaminar kinematic viscosity, D/Dt = ¡/¡t+ Eu, ï is the materialtime derivative operator.

Nonlinearities in the fluid flow generate hydrodynamic insta-bilities that cause the generation of coherent structures and otherturbulent features. To adequately take into account turbulence andthe dissipation of turbulent energy across a wide spectrum of spatialscales, two avenues can be taken, namely Direct Numerical Simula-tions (DNS) and averaging techniques to dissipate turbulent energythat is of a subgrid spatial size. One example of an averagingtechnique is Large Eddy Simulation (LES), which features a filtertechnique for subgrid turbulence, but is capable of resolving larger-scale turbulent features. For the SPH method, a spatial filter isapplied to Eqs. 1 and 2. Then the governing equations for the Parti-cle Scale (PS) in the Lagrangian representation, which is similar tothe grid scale in the Eulerian representation, can be introduced as:

D�

Dt= −�ï · Eu (3)

D Eu

Dt= −

ïp

�+ Eg + vï 2 Eu+

ï · �∗

�(4)

where the overbar symbol ‘-’ denotes a mean or particle scalingcomponent and

�∗= �

(

2vt S −23

tr4S5I)

−23�C Iã

2 I (5)

is the turbulent stress tensor representing the interaction of theunresolved small motions or the Sub Particle Scales (SPSs) on theresolved large particle scales. In Eq. 5, ã is the initial particlespacing and I is the identity tensor. The eddy viscosity vt iscalculated from the standard Smagorinsky model as:

vt = 4Csã52�¯S� (6)

where �S� = 42S 2 S5005 is the local strain rate and S is the deforma-tion rate tensor defined as:

S =12

(

ï Eu+ 4ï Eu5T)

(7)

CI and Cs are empirical constants with CI = 606 × 10−4 andCs = 0012 (Blin et al., 2002; Dalrymple and Rogers, 2006).

SMOOTHED PARTICLE HYDRODYNAMICS

By being successful in simulating various fluid mechanicsapplications within the last decade, SPH has received increasedattention among the meshless approaches. Owing to its Lagrangiannature, it has unique advantages in dealing with fast flow dynamicsproblems, i.e., no convective term is in the momentum equation.Additionally, since SPH is a member of the meshless particle family,

fluid flows with large deformations, and interfaces and free surfacescan be treated inherently in a relatively easy manner (Zainali etal., 2013; Shadloo et al., 2013). In this method, particles refer tointegration points, which carry all hydrodynamic properties and canmove freely. The hydrodynamic properties of a given particle arecalculated from the weighted contributions of neighboring particlesby using a weighting/kernel function. Neighboring particles includethose that are in the environs of the base particle, called the compactsupport domain. The integral estimate or the kernel approximationfor an arbitrary function f 4Eri5 can be introduced as (Monaghan,1992):

f 4Eri5û �f 4Eri5� ≡

∫

ìf 4Eri5W4rij 1 h5d

3rj (8)

where W4rij1h5, hereafter represented by Wij , is a smoothingor kernel function, the angle bracket � � denotes the kernelapproximation, and the length h defines the supporting domain ofthe particle of interest. Apparently, the type of kernel functionand the smoothing length are two important input parameters thatcontrol the accuracy and computational costs of the SPH method.rij is the length of the distance vector (Erij = Eri − Erj ) between theparticle of interest i and its neighboring particles j and Eri and Erjare the position vectors for particles i and j , respectively.

By replacing the integration in Eq. 8 with the SPH summationover N neighboring particles j and by setting d3r =mi

/

�i, we canwrite the SPH interpolation for an arbitrary field fi as:

fi = f 4Eri5=∑

j

mj

�j

fjWij (9)

The SPH approximation for the gradient of the same function canbe introduced as:

ïfi =∑

j

mj

�j

4fj − fi5ïiWij (10)

or in the conservative way as:

ïfi =∑

j

mj

(

fi�2i

+fj

�2j

)

ïiWij (11)

It is noted that the above equation is asymmetric with respect toparticles i and j . Following Monaghan (1992), the laminar viscousterm in the linear momentum balance equation is represented by:

vï 2 Eui =∑

j

mj

4v�i +�j

Erij ·ïiWij

r2ij

Euij (12)

Throughout the present simulations, the compactly supportedthree-dimensional Wendland kernel function is used, which is givenin the form of (Wendland, 1995):

Wij = �

(

1 −q

2

)4

42q + 151 q < 2 (13)

where �= 15/416�h35 for 3D simulations and q is defined asq = rij/h.

International Journal of Offshore and Polar Engineering, Vol. 25, No. 1, March 2015, pp. 1–7 3

By applying discrete SPH formulations to the governing equations,we can express the continuity and momentum balance equations as:

D�i

Dt=∑

j

mj4 Eui −Euj5 ·ïiWij (14)

D Eui

Dt=∑

j

mj

(

pi

�2i

+pj

�2j

)

ïiWij + Eg

+∑

j

mj

4v�i +�j

Erij ·ïiWij

r2ij

Euij +∑

j

mj

(

�∗i

�2i

+�∗j

�2j

)

·ïiWij (15)

The numerical scheme used here is the predictor-corrector schemeintroduced by Monaghan (1989). In the current work, an open sourcemassively parallel computing C++ code, so-called GPUSPH, is used.GPUSPH (www.gpusph.org) computes the three main componentsof the SPH method, namely the neighbor list construction, forcecomputation, and integration of the equation of motion, on aGraphical Processing Unit (GPU) using the Compute Unified DeviceArchitecture (CUDA) developed by Nvidia. Further details on thediscretization of model equations and their CUDA implementationcan be found in the work of Herault et al. (2010).

PROBLEM SETUP

The geometrical setup for the long-wave experiments features aconstant depth section that is followed by a sloping beach. Figure 1depicts the geometric setting of the long-wave runup problem.Parameter l1 represents the length of the constant depth part, whileparameter l2 denotes the projected length of the sloping beach. Theslope is �. The length of the setup is L= l1 + l2. The height ischosen in such a way that the maximum runup R does not exceedthe height of the setup. (The runup is estimated with runup lawsprovided later.) The length L is a function of the slope angle and thewater depth D. The width of the computational domain is constantfor all test cases at W = 004 (m). The origin of the coordinatesystem (x = 0, y = 01 and z= 0 (m)) is located at the far left end.

Waves are generated by a piston wavemaker located near the leftedge. To generate long waves with the shape of

�4x1 t5=H2

sech6k4Ct − xo571 (16)

we employ the wavemaker function:

xo4t5=H

kDtanh6k4Ct − xo571 (17)

which is similar to the one suggested by Goring (1978). Herek =

√

3H/4D3, C =√

g4D+H5, and H is the desired waveamplitude. The sample position and velocity of the piston forthe relative wave height H/D = 001 and the still water level ofD = 004 (m) are shown in Fig. 2. It can be seen in this figure thatthe trajectory of the paddle is such that it can produce a perfectsolitary wave with a desirable wave height.

Fig. 1 2D sketch of three-dimensional numerical simulations

2 3 4 5 6 7 8 90

0.05

0.1

0.15

0.2

0.25

0.3

0.35

t (s)

x (m

)

2 3 4 5 6 7 8 90

0.05

0.1

0.15

0.2

0.25

t (s)

u (m

/s)

Fig. 2 Sample position (up) and velocity (down) of the paddle

RESULTS

In this study, the density and the kinematic viscosity of thefluid are set to � = 1000 (kg/m3) and v = 10−6 (m2/s), respec-tively. The gravitational force Eg acts only in the downward direc-tion (z direction) on all particles with the numerical value ofg = 9081 (m/s2). The slope � ranges from 2088� to 30�. TheSPS turbulent model is employed in all test cases unless statedotherwise.

The boundaries are treated as solid walls, and the no-slip andzero pressure gradient boundary conditions are imposed using theMonaghan-Kajtar method (Monaghan and Kajtar, 2009). In theSPH method, particles may tend to readjust their initial positions,giving rise to spurious currents generally in regions where thekernel is truncated, i.e., near solid boundaries and free surfaces(Monaghan, 1994; Colagrossi et al., 2012). At initial time steps,this situation may result in unwanted disturbances in the watercolumn and the free surface. To circumvent such effects, we waitedfor three seconds before the wavemaker started to move.

We performed a limited parameter study comprising morethan 50 individual simulations. To investigate the sensitivity ofthe numerical solutions to the number of particles, the relativemaximum runup, R/D for a slope of �= 20� with H/D = 001 andD = 004 (m) is computed for the very coarse (ã= 0003 (m)), coarse(ã= 00025 (m)), medium (ã= 0002 (m)), fine (ã= 00015 (m)),and very fine (ã= 00010 (m)) particle spacing. In this work, themaximum run-up height is defined as the y-position of a particlewith the largest x-position provided that the particle in question hasa prescribed number of neighbors, thereby excluding free surfaceparticles completely isolated from the fluid body. This predefinedneighbor number is dependent on simulation parameters such asthe beach angle and wave height, among others; i.e., in this study,

4 Numerical Simulation of Long Wave Run-up for Breaking and Non-breaking Waves

Very coarse Coarse Medium Fine Very fine

Initial particle spacing, ã (m) 0.03 0.025 0.02 0.015 0.01Total number of particles 24976 40080 75435 164322 505419Simulation time (s) 1088 × 102 2076 × 102 5047 × 102 1067 × 103 8015 × 103

Maximum relative wave runup, R/D 0.2775 0.3132 0.3221 0.3287 0.3245Error with respect to the last column, % 14.48 3.47 0.77 1.31 —Error with respect to the next succeeding column, % 11.41 2.72 2.05 1.31 —

Table 1 Sensitivity of the numerical solutions to the particle resolution evaluated based on runup

the neighbor number is around 10 particles, which is determinedby conducting several test simulations and visually monitoring therun-up distance.

Table 1 shows the convergence study based on the relativemaximum wave runup. It is found that for a very coarse spacing,the runup over offshore depth ratio, R/D, has an approximately15% error in comparison to the one obtained by the very fineparticle spacing. This error decreases to 2% for the medium particlespacing. The amount of time needed to simulate one second ofreal time in the model is also given in the table. There is anorder of magnitude difference between the very coarse and thefinest particle spacing test cases. It seems that the medium spacingrepresents a good trade-off between accuracy and computationalcost. Therefore, the intermediate particle number is chosen fornumerical simulations presented in this study, i.e., ã= 0002 (m).It is noted that simulations are performed on a single NvidiaTesla M2075 device with a 448 CUDA core and a Linux (64-bit)operating system.

For practical reasons, the wave runup is an important measurefor solitary waves on a sloping beach. Depending on the waterheight, wave amplitude, and angle of the sloping beach, the runupis different for breaking and nonbreaking waves. Synolakis (1987)derived the following runup law for nonbreaking waves:

R/D = 2083 cot4�5005 4H/D51025 (18)

Figure 3 shows the relative maximum runup of solitary wavesclimbing up on different beaches versus the normalized wave height.The numerical results of the current simulations plotted in thisfigure are summarized in Table 2. We observed close agreementbetween the runup law equation (Eq. 18), the selected experimentaldata (originally collected in Table T3.2 by Synolakis (1986)), the

10−2

10−1

100

101

10−2

10−1

100

101

2.83 cot(β )1/2(H/D)5/4

R/D

Runup law Eq. 18ExperimentNumericSPH

Fig. 3 Simulation results for nonbreaking waves

numerical calculations of Pedersen and Gjevik (1983), Heitner andHousner (1970), and Kim et al. (1983), and the current simulationresults.

Increasing the wave height changes the flow regime from thenon-breaking solitary wave to the breaking ones. Pedersen andGjevik (1983) suggested that the waves break when

H/D = 008183 cot4�5−10/9 (19)

Synolakis (1986) later developed the following weaker restrictionbased on nonlinear analysis:

H/D = 00479 cot4�5−10/9 (20)

Equation 19 differs from Eq. 20 due to the fact that the formerindicates the border for the wave height at which a solitary wavebreaks during the washback, while the latter shows the limit atwhich a solitary wave first breaks during the runup. Stated otherwise,the wave that was not broken during the runup might get brokenduring the washback. However, both criteria indicate that with anincrease of the beach angle � and/or the still water height D, thesystem will have more nonbreaking waves.

H/D � D (m) R/D

0.019 2.884 0.31 0.06030.021 2.884 0.2914 0.12460.1 10 0.4 0.30770.15 10 0.4 0.48370.2 10 0.4 0.66370.1 15 0.4 0.29470.15 15 0.4 0.43570.2 15 0.4 0.60120.05 20 0.4 0.17450.1 20 0.4 0.32210.15 20 0.4 0.45820.2 20 0.4 0.61720.25 20 0.4 0.77720.3 20 0.4 0.90630.35 20 0.4 1.15520.4 20 0.4 1.35310.1 20 0.2 0.27810.2 20 0.2 0.46450.3 20 0.2 0.68350.1 25 0.4 0.28610.15 25 0.4 0.46730.2 25 0.4 0.61720.1 30 0.4 0.25830.15 30 0.4 0.45830.2 30 0.4 0.6515

Table 2 Reported test cases for nonbreaking waves

International Journal of Offshore and Polar Engineering, Vol. 25, No. 1, March 2015, pp. 1–7 5

Based on laboratory beach findings, Synolakis (1987) reportedthat washback waves break at H/D> 00044 and a break duringthe runup occurs when H/D> 00055, which are different fromthe values calculated for H/D using Eqs. 19 and 20, respectively.The main reason for the differences between the theoretical andexperimental results is that the analytical solution for modelingrunup is based on shallow water wave formulas that involveseveral simplifications. Furthermore, Synolakis mentioned that theasymptotic result from the runup law (Eq. 18) is also valid for allwaves that first break during the washback. However, for the runupbreaking waves, he reported the correlation

R/D = 10109 4H/D500583 (21)

for the relative maximum runup. In reference to Fig. 4, ournumerical simulations indeed generate a behavior similar to the oneelaborated above, but the values of the relative wave amplitudesobtained numerically for breaking waves are somewhat higher thanthose obtained in the experiment. These values are H/D > 002 andH/D > 006 for the waves that break during the washback and runup,respectively. These discrepancies can be attributed to differences inthe angle of the sloping beach for the experimental and currentnumerical studies. Furthermore, it can be seen in Fig. 4 that thenumerically calculated R/D values are quite well represented bythe run-up law in Eq. 18 up to H/D < 006. (Recall that the waveswith H/D< 006 do not break during the runup.) We also foundthat the correlation

R/D = 30149 4H/D50051 (22)

well represents runup breaking waves (i.e., H/D > 006) on thenumerical beach. Since the current numerical simulations have botha higher beach angle � and a higher still water height D, theycontain more nonbreaking waves. Therefore, waves for numericalexperiments first break during the runup at higher relative waveamplitudes H/D, confirming the analytical solutions in Eqs. 19and 20.

Although a change in slope appears at a higher relative waveamplitude H/D, it can be seen from Fig. 4 that the two correlationsfor the breaking waves (Eqs. 21 and 22) are almost parallel to eachother and the curves follow the same behavior, meaning that thisslope change appears at the region of transition from nonbreaking

10−3

10−2

10−1

100

101

10−3

10−2

10−1

100

101

H/D

R/D

Eq. 18 for β =2.884o

Breaking waves Eq. 21Exp. (Synolakis, 1987)

Eq. 18 for β =20o

Breaking waves Eq. 22Current SPH

Fig. 4 Relative maximum runup as a function of the relative waveamplitude for both breaking and nonbreaking waves

to breaking waves. Figure 5 illustrates the simulated runup andwashback of a solitary wave with the relative wave amplitude ofH/D = 0065 and D = 004 (m). As seen in this figure, as the waveis reaching the inclined beach, a fluid jet starts to appear on thehighest part of it. (Refer to dimensionless times t∗ = 180161 andt∗ = 190047). This jet impinges on the tail of the front wave bodyand breaks first during the runup. (Refer to dimensionless timest∗ = 190932 and t∗ = 200818). At this time step, a strong transportof momentum takes place and a turbulent mixing of energy occurs.Afterwards, fluid continues to the runup. The breaking of wavesduring the runup and the turbulent diffusion of energy due to theeddy viscosity are the main reasons for the slope change shownin Fig. 4. Since some portions of energy are diffused, the wavehas less energy compared to the initial energy induced to theflow by the wavemaker, so it climbs less high than the runup lawpredicts.

In later times, a steep front develops during the washback. (Referto dimensionless times t∗ = 270905 and t∗ = 280791.) Due to thehigh kinetic energy, the wash-backed fluid pushes the bulk fluidback near the plate and creates a rollup that entraps air inside it andcollapses after the second impinging. (Refer to dimensionless timest∗ = 290677 and t∗ = 300653.) Meanwhile and until the end of thebackwash, the flow feels the intense turbulent mixing of energy.(Refer to dimensionless times t∗ = 310499 and t∗ = 320335.)

To assess the importance of the turbulent mixing and the eddyviscosity dissipation, it is important to investigate the shortcomingsof the inviscid and/or laminar viscosity formulas if relevant. Table 3compares the results of the relative maximum runup, the maximumvelocities (umax), and the maximum vorticities (�max) obtainedusing three different viscosity formulas, namely SPS turbulenceviscosity (SPS-Vis), artificial viscosity (Art-Vis), and kinematicviscosity (Kin-Vis). It should be noted that the vorticity is computedthroughout the whole computational domain and its maximumvalue is reported in Table 3. For the Kin-Vis model, the values ofthe flow Reynolds number are 56 × 103, 72 × 104, and 18 × 105 forthree different H/D values given in Table 3, respectively, whichare calculated based on the characteristic scales of maximumvelocity and the wave amplitude as Re =Humax/v. It can be seenfrom Table 3 that the Kin-Vis always produces closer resultsto the SPS-Vis in comparison to the Art-Vis in terms of therelative maximum runup. Due to its highly diffusive nature, theArt-Vis formulation (Monaghan, 1992) always predicts the smallestR/D values even for a very small artificial viscosity coefficient.(Here � = 0001 is used.) However, the Kin-Vis appears to beoverpredicting the values of the maximum velocities and maximumvorticities with respect to the SPS-Vis, except for the nonbreakingwave (i.e., H/D = 001). This discrepancy comes from the factthat in the Kin-Vis formulation, the amount of dissipation is lesscompared to the SPS-Vis, which is equal to the turbulent eddyviscosity. The discrepancy becomes more evident for the highervalues of the initial relative wave height, especially for those thatcause air entrainment in the breaking waves. It can be seen fromTable 3 that in comparison to the SPS-Vis, the Art-Vis alwaysunderestimates the maximum values of the velocities and vorticities,while the Kin-Vis overestimates these values. Since the SPSmodeling depends more on the properties of the flow rather thanthe fluid, unlike the Kin-Vis and Art-Vis, one may expect that thismodel increases the accuracy of the SPH method, especially withrespect to higher Reynolds numbers. Finally, it is further noted thatexcept for values with an asterisk (*), all maximum velocities andvorticities occurred during the backwash step of the wave-breakingphenomena.

6 Numerical Simulation of Long Wave Run-up for Breaking and Non-breaking Waves

Fig. 5 2D view of the three-dimensional time evolution of a breaking wave during runup (left) and washback (right), with t ∗ ∗ = t√g/H

R/D umax �max

H/D 0.1 0.25 0.5 0.1 0.25 0.5 0.1 0.25 0.5

SPS-Vis 0.322 0.777 1.817 2.049 3.982 4.793 53.689 120.45 175.18Art-Vis 0.302 0.706 1.597 1.098* 2.812 8.986 18.053 90.448 166.11Kin-Vis 0.314 0.783 1.838 1.396* 7.197 9.139 42.342 170.41 324.13

Table 3 The comparison of maximum relative runup, maximum velocities, and maximum vorticities for three different viscosity formulas

CONCLUSIONS

An SPH method, GPUSPH, has been used to study the longwave runup of breaking and nonbreaking solitary waves. Three-dimensional numerical simulations were performed for numerousbeach angles and initial dimensionless wave heights. The simu-

lation results are observed to be in good agreement with thosecorresponding to the analytical solutions and experimental data interms of the maximum runup. With more nonbreaking waves, itis illustrated that a change in the slope in maximum runup plotsappears at higher values for higher beach steep angles. Addition-ally, the effect of different viscosity terms is investigated. It is

International Journal of Offshore and Polar Engineering, Vol. 25, No. 1, March 2015, pp. 1–7 7

further observed that the use of an appropriate turbulent modelingin violent flows can improve the accuracy of the results, espe-cially for higher wave amplitudes and in turn for higher Reynoldsnumbers.

ACKNOWLEDGEMENTS

The research leading to these results has received funding fromIRT Jules Verne through the chair program "SimAvHy." Thethird author, Mehmet Yildiz, gratefully acknowledges the financialsupport provided by the Scientific and Technological ResearchCouncil of Turkey (TUBITAK) for Project No. 112M721.

REFERENCES

Blin, L, Hadjadj, A, and Vervisch, L (2002). “Large Eddy Simulationof Turbulent Flows in Reversing Systems,” J Turbul, 4, 1–19.http://dx.doi.org/10.1088/1468-5248/4/1/001.

Colagrossi, A, Bouscasse, B, Antuono, M, and Marrone, S (2012).“Particle Packing Algorithm for SPH Schemes,” Comput PhysCommun, 183(8), 1641–1653.http://dx.doi.org/10.1016/j.cpc.2012.02.032.

Dalrymple, RA, and Rogers, BD (2006). “Numerical Modelingof Water Waves with the SPH Method,” Coastal Eng, 53(2-3),141–147. http://dx.doi.org/10.1016/j.coastaleng.2005.10.004.

Goring, DG (1978). Tsunamis: The Propagation of Long WavesOnto a Shelf, Report KH-R-38, California Institute of Technology,Pasadena, CA, USA.http://resolver.caltech.edu/CaltechKHR:KH-R-38.

Heitner, K, and Housner, GW (1970). “Numerical Model forTsunami Runup,” J Waterw Harbors Coastal Eng Div, ASCE,96(3), 701–719.

Herault, A, Bilotta, G, and Dalrymple, RA (2010). “SPH on GPUwith CUDA,” J Hydraul Res, 48, 74–79.http://dx.doi.org/10.3826/jhr.2010.0005.

Khayyer, A, Gotoh, H, and Shao, SD (2008). “Corrected Incom-pressible SPH Method for Accurate Water-surface Tracking inBreaking Waves,” Coastal Eng, 55(3), 236–250.http://dx.doi.org/10.1016/j.coastaleng.2007.10.001.

Kim, SK, Liu, PL-F, and Liggett, JA (1983). “Boundary IntegralEquation Solutions for Solitary Wave Generation, Propagationand Run-up,” Coastal Eng, 7(4), 299–317.http://dx.doi.org/10.1016/0378-3839(83)90001-7.

Landrini, M, Colagrossi, A, Greco, M, and Tulin, MP (2007).“Gridless Simulations of Splashing Processes and Near-shoreBore Propagation,” J Fluid Mech, 591, 183–213.http://dx.doi.org/10.1017/S0022112007008142.

Monaghan, JJ (1989). “On the Problem of Penetration in ParticleMethods,” J Comput Phys, 82(1), 1–15.http://dx.doi.org/10.1016/0021-9991(89)90032-6.

Monaghan, JJ (1992). “Smoothed Particle Hydrodynamics,” AnnuRev Astron Astrophys, 30, 543–574.http://dx.doi.org/10.1146/annurev.aa.30.090192.002551.

Monaghan, JJ, and Kajtar, J (2009). “SPH Particle Boundary Forcesfor Arbitrary Boundaries,” Comput Phys Commun, 180(10),1811–1820. http://dx.doi.org/10.1016/j.cpc.2009.05.008

Pedersen, G, and Gjevik, B (1983). “Run-up of Solitary Waves,”J Fluid Mech, 135, 283–299.http://dx.doi.org/10.1017/S0022112083003080.

Shadloo, MS, Zainali, A, and Yildiz, M (2013). “Simulationof Single Mode Rayleigh-Taylor Instability by SPH Method,”Comput Mech, 51(5), 699–715.http://dx.doi.org/10.1007/s00466-012-0746-2.

Synolakis, CE (1986). The Runup of Long Waves, PhD The-sis, California Institute of Technology, Pasadena, CA, USA.http://resolver.caltech.edu/CaltechETD:etd-09122007-111121.

Synolakis, CE (1987). “The Runup of Solitary Waves,” J Fluid Mech,185, 523–545. http://dx.doi.org/10.1017/S002211208700329X.

Wendland, H (1995). “Piecewise Polynomial, Positive Definite andCompactly Supported Radial Functions of Minimal Degree,” AdvComput Math, 4(1), 389–396.http://dx.doi.org/10.1007/BF02123482.

Zainali, A, Tofighi, N, Shadloo, MS, and Yildiz, M (2013). “Numer-ical Investigation of Newtonian and Non-Newtonian MultiphaseFlows Using ISPH Method,” Comput Meth Appl Mech Eng, 254,99–113. http://dx.doi.org/10.1016/j.cma.2012.10.005.