Coastal Engineering 73 (2013) 167–177

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Coastal Engineering

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Boundary layer approach in the modeling of breaking solitary wave runup

Mohammad Bagus Adityawan a,b,⁎, Hitoshi Tanaka b, Pengzhi Lin c

a Water Resources Engineering Research Group, Institut Teknologi Bandung, Jalan Ganesha 10, Indonesiab Department of Civil Engineering, Tohoku University, 6-6-06 Aoba, Sendai 980-8579, Japanc State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu, Sichuan 610065, China

⁎ Corresponding author at: Department of Civil EngineeAoba, Sendai 980-8579, Japan. Tel./fax: +81 22 795 7453

E-mail address: bagus@kasen1.civil.tohoku.ac.jp (M.

0378-3839/$ – see front matter © 2012 Elsevier B.V. Allhttp://dx.doi.org/10.1016/j.coastaleng.2012.11.005

a b s t r a c t

a r t i c l e i n f o

Article history:Received 13 February 2012Received in revised form 9 November 2012Accepted 20 November 2012Available online 20 December 2012

Keywords:Breaking waveSolitary waveWave runupBoundary layerBed stressSimultaneous coupling method

The boundary layer is very important in the relation between wave motion and bed stress, such as sedimenttransport. It is a known fact that bed stress behavior is highly influenced by the boundary layer beneath thewaves. Specifically, the boundary layer underneath wave runup is difficult to assess and thus, it has not yetbeen widely discussed, although its importance is significant. In this study, the shallow water equation (SWE)prediction of wave motion is improved by being coupled with the k–ω model, as opposed to the conventionalempirical method, to approximate bed stress. Subsequently, the First Order Center Scheme and Monotonic Up-stream Scheme of Conservation Laws (FORCE MUSCL), which is a finite volume shock-capturing scheme, is ap-plied to extend the SWE range for breaking wave simulation. The proposed simultaneous coupling method(SCM) assumes the depth-averaged velocity from the SWE is equivalent to free stream velocity. In turn, freestream velocity is used to calculate a pressure gradient, which is then used by the k–ω model to approximatebed stress. Finally, this approximation is applied to the momentum equation in the SWE. Two experimentalcases will be used to verify the SCM by comparing runup height, surface fluctuation, bed stress, and turbulent in-tensity values. The SCM shows good comparison to experimental data for all before-mentioned parameters. Fur-ther analysis shows that the wave Reynolds number increases as the wave propagates and that the turbulencebehavior in the boundary layer gradually changes, such as the increase of turbulent intensity.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

The boundary layer approach in approximating bed stress underwave motion is crucial, especially in bed stress related analyses, i.e.sediment transport and scouring. It is highly important in relevance tocoastal morphology changes. An extreme example of coastalmorpholo-gy changes is given by the effect of a tsunami wave such as was shownin the recent Great East Japan Tsunami, 2011 and the Great IndianOcean Tsunami, 2004. Studies on bed stress behaviors under waverunup may provide better understanding of this phenomenon withrespect to future disaster.

The studies of tsunami effects on coastal regions are normallyconducted by field assessment, modeling, or experiment. The soli-tary wave approach is commonly used in the study of tsunamis.One of the leading studies on solitary wave runup is given bySynolakis (1986, 1987) in which he conducted experiments and ananalytical solution for runup height. The work has been used as abenchmark for other various models. The popularity of a modelingapproach in wave runup study is continuously increasing. Currenttrends in wave runup modeling emphasize on travel time, runupheight or inundated area. However, studies emphasizing on bed

ring, Tohoku University, 6-6-06;B. Adityawan).

rights reserved.

stress and boundary layer, especially under wave runup, are notcommon yet. Boundary layer beneath the wave motion is essential,especially in the coastal morphology changes. The sediment trans-port process under wave motion is closely related to the bed shearstress, which is influenced by the boundary layer beneath the waveitself (Vittori and Blondeaux, 2008).

There are very limited resources regarding boundary layer for soli-tary waves, especially in open-channel flumes. Measurement of turbu-lent behavior requires multiple wave cycles with the same initialconditions of still water level. It is considered to be difficult and timeconsuming to accomplish these conditions in open-channel flumes.Studiesmainly use closed-channel flumes,whichmay resemble the sol-itary profile to some extent. Liu et al. (2007) have reported that the bedstress changes its sign in the deceleration phase to the opposite direc-tion of the free stream velocity. Sumer et al. (2010) investigated andproposed Reynolds number criteria for a boundary layer under solitarywaves. Tanaka et al. (2011) developed a new generationmethod for in-vestigating the boundary layer under solitarywaves. These studies haveprovided valuable information on the boundary layer under solitarywave motion. However, the boundary layer under wave runup hasnot been investigated widely since the closed-channel flumes experi-ment neglects the effect of nonlinearity. Recently, Sumer et al. (2011)conducted breaking solitary wave experiments in an open channel.Several measurements were performed, including the surface profile,the bed stress and its fluctuation. The experiment was conducted on a

START

INITIAL CONDITIONt = 0

t = 1

t = 2

FINISH

k- modelu, v,

SWE modelh, U

k- modelu, v,

SWE modelh, U

Fig. 1. Computation flow chart for SCM.

168 M.B. Adityawan et al. / Coastal Engineering 73 (2013) 167–177

slopingbeachwith 1/14 slopewith an incomingwave Reynolds numberof 54,000. Based on criteria for solitary wave from Sumer et al. (2010),this condition falls in the laminar region. However, the criterionwas de-rived from a closed flume experiment. In this experiment, it was foundthat the Reynolds number increases as thewave travels to the shore andmay reach as high as 300,000 with significant turbulence observed.

Study on the boundary layer under solitary waves by Suntoyo andTanaka (2009) has shown good accuracy of bed stress approximationfrom the boundary layer using a numerical model. Two equationmodels are often used to assess the boundary layer properties withk-ε and k–ω being the most common. The k–ω model has the abilityto accommodate the roughness effect of the bed boundary condition,and is considered to be more accurate in assessing the boundarylayer properties (Adityawan and Adityawan, 2011). Adityawan andTanaka (in press) proposed the simultaneous coupling method(SCM) to assess boundary layer under non-breaking solitary waverunup. They developed the SCM that couples the SWE with the k–ωmethod. The basic idea is to obtain an efficient model such as theSWE yet capable of assessing the boundary layer beneath the waveitself. However, the wave Reynolds number in the experiment isvery low; hence, there was no significant turbulence activity ob-served. Nevertheless, they have made it clear that bed stress assess-ment using the boundary layer approach provides information onknown bed stress behaviors under wave motion (i.e. phase shiftand sign change), which are not accessible when using the empiricalManning approach.

The modeling of breaking solitary wave runup has been widelystudied through various different approaches. An accurate reproduc-tion of breaking waves requires a 2D vertical system to simulate thedissipation such as given by NEWFLUME (Lin et al., 1999) andCADMAS SURF (Isobe et al., 1999). The breaking wave simulation inthe SWE and other depth-averaged models are not able to accuratelyrepresent breakingwaves. The Boussinesqmodel requires a breakingterm to be included, which is determined by a calibration processwith experimental or field data. Zelt (1991) conducted a detailedlaboratory experiment and developed numerical models based onthe Boussinesq type of model, accommodating the constant frictioncoefficient and artificial dissipation for breaking waves. However, itwas found that the constant friction coefficient value was not agood solution and should be adjusted in time and space. The SWEbased model, on the other hand, is relatively flexible to modify and toaccommodate various treatments. Implementation of certain finite dif-ference numerical schemes in the SWE enhances its capability inmodeling the breaking waves. The Leapfrog scheme performs well insolving the SWE due to the nature of the scheme that provides diffusiveeffect (Imamura, 1995). Thus, it is widely used in far field tsunami sim-ulations. Other finite difference numerical schemes, such as the MacCormack scheme, were used to investigate runup of a uniform bore ona sloping beach (Vincent et al., 2001). Conventional finite differencemethods suffer from high oscillation under shock. Artificial dissipation,i.e. Hansen (1962), or changing to a more dissipative scheme is com-monly used to reduce the high oscillation. Nevertheless, these stepsmust be takenwith care. Implementation of a strongdissipation schememay lead to unrealistic results, such as the rapid decay of the wave.Additionally, a weak dissipation scheme may lead to numerical errorswhen dealing with abrupt changes. Moreover, artificial dissipationmay require determination based on a trial and error procedure. Appli-cation of the Mac Cormack finite difference in combination with artifi-cial dissipation is given for the 2004 Tsunami, Banda Aceh (Kusuma etal., 2008), which requires further enhancement of the method.

Finite volume schemes may provide robust ways to handle shockin the SWE model. Li and Raichlen (2002) developed their modelbased on the SWE without friction and verified their simulationusing experimental data from Synolakis (1986). The breaking wavein their model was treated using the Weighted EssentiallyNon-Oscillatory (WENO). WENO schemes achieve higher order

approximation by a linear combination of lower order fluxes or recon-struction that provides a high order accuracy and non-oscillatory prop-erty near discontinuities. They concluded that the model is simple yetreasonably suited for estimating solitary wave runup height. Modifica-tion of the Godunov-type scheme leads to a second order accuracy inspace such as Monotonic Upstream Scheme of Conservation Laws(MUSCL) scheme (Toro, 1996). Combining it with the First OrderCentered Scheme (FORCE) (Toro, 2001) and Total Variation Diminished(TVD) Runge-Kutta (Mahdavi and Talebbeydokhti, 2009) further en-hanced the method. Employment of such scheme efficiently enhancesthe SWE capability for breaking wave simulations.

In this study, the SCM is enhanced using the FORCE MUSCLshock-capturing scheme for breakingwave simulations. Two case stud-ies of breaking solitary wave runup are used to verify the model. Theboundary layer assessment is verified with the latest open channel ex-periment by Sumer et al. (2011). This case is currently the only studythat provides detailed measurement on bed stress and turbulenceunder solitary wave runup. The runup height estimation is verifiedwith the well-known canonical problems by Synolakis (1986). Thiscase has been widely used as numerical model benchmark for solitarywave runup.

2. Methodology

2.1. Governing equations

The SWE consists of the continuity equation and the momentumequation as follows:

∂h∂t þ

∂ Uhð Þ∂x ¼ 0 ð1Þ

∂U∂t þ U

∂U∂x þ g

∂ hþ zbð Þ∂x þ τ0

ρh¼ 0 ð2Þ

where h is the water depth, U is depth averaged velocity, t is time, g isgravity, zb is the bed elevation, ρ is fluid density and τ0 is the bedstress. The Manning equation is commonly used to assess bed stress.

h,U (SWE)

(Momentum)

dx

dy

k- domain

SWE domain

intial condition in moving boundary for k- linear extrapolation from behind grids

Mild slope Boundary layer develop in y - axis

Bed

(2 – 3 times of BL thickness)

Bed

Fig. 2. Model domain for SCM.

169M.B. Adityawan et al. / Coastal Engineering 73 (2013) 167–177

The bed stress relation in the conventional Manning method is as-sumed linear to the square of velocity as shown below:

τ0ρ

¼ gn2 U Uj jRh

1=3 ð3Þ

where Rh is the hydraulic radius and n is the Manning roughness.However, the SCM applied in this study approximate bed stressfrom the boundary layer approach using the k–ω model.

The boundary layer is assessed using the k–ω model. This model ischosen because it has been shown to perform better in assessing bound-ary layer properties (Sana et al., 2009). Additionally, the model's capabil-ity to accommodate rough bed boundary condition (Suntoyo et al., 2008)is important in the future development of the SCM. The governing

V

MUSCL

V

FORCE

F

TVD

V

Central difference

V

Begint = 0

Endt = T

loopt = t+dt

a) computational flow chart

Fig. 3. Calculation metho

equation for the k–ωmodel is based on the Reynolds-averaged equationsof continuity and momentum:

∂ui

∂xi¼ 0 ð4Þ

ρ∂ui

∂t þ ρuj∂ui

∂xj¼ ∂P

∂xiþ 2μsij−ρu′

iu′j

� �ð5Þ

where ui and xi denotes the velocity in the boundary layer and location inthe grid, ui′ is the fluctuating velocity in the x (i=1) and y (i=2) direc-tions, P is the static pressure, ν is the kinematics viscosity, − ρu′

iu′j is

b) FORCE MUSCL discretization

t

i-1 i+1i

i+1/2

i+2

i+1+1/2

i+1/2(-)

i+1/2(+)

Delta Delta

i+1/2

t

i-1/2

MUSCL

FORCE

Cell i

d for breaking wave.

ToeInitial Shorelinex=0

5.6

Wave breaks HydraulicJump

Measurement section

8 7 6 5 4 3 2 1

Swash Zone

Swash Zone

0.4

Solitary Wave

H

0.150.15

a) Case 1 (Sumer et al., 2011), unit: meter

R

tan1/20

z

x

10 xxx=0

(x,t) H

hoh(x)

b) Case 2 (Synolakis, 1986)

Fig. 4. Solitary wave runup on a sloping bottom sketch.

Table 2

170 M.B. Adityawan et al. / Coastal Engineering 73 (2013) 167–177

theReynolds stress tensor, and Sij is the strain-rate tensor from the follow-ing equation.

Sij ¼12

∂ui

∂xjþ ∂uj

∂xi

!ð6Þ

The Reynolds stress tensor is given through eddy viscosity byBoussinesq's approximation (Boussinesq, 1897):

−u′iu

′j ¼ vt

∂ui

∂xjþ ∂uj

∂xi

!−2

3kδij ð7Þ

Table 1Measurement point location (Case 1).

Section x (m) h (t=0) (m)

Toe 5.6 0.401 0.97 0.072 0.91 0.073 0.73 0.054 0.49 0.045 0.25 0.026 0.01 0.007 −0.05 0.008 −0.25 0.00

where k is the turbulent kinetic energy and δij is the Kronecker delta.The turbulent kinetic energy and specific dissipation rate, ω in thek–ω model, equation is given as follows:

∂k∂t þ uj

∂k∂xj

¼ τij∂ui

∂xj−β � kω þ ∂

∂xjvþ σ � vtð Þ ∂k∂xj

" #ð8Þ

∂ω∂t þ uj

∂ω∂xj

¼ αωkτij

∂ui

∂xj−βω2 þ ∂

∂xjvþ σvtð Þ ∂ω∂xj

" #ð9Þ

Data source and parameters.

Parameter Case 1 Case 2

Sumer et al. (2011) Synolakis (1987)

h0 (m) 0.4 0.13H/h0 0.175 0.3Slope 1/14 1/20Incoming wave Re 54000 18000Data source acquired for comparison

Spatial Re OBreaking wave sequence OWater level fluctuation O DimensionalSurface profile O (Non-dimensional)Bed stress and k value O DimensionalRun up height O (Non-dimensional)

0

250000

500000

750000

1000000

1250000

-10 -5 0 5 10 15

x*

Re

Critical Reynolds Number2x105<Re<5x105(Sumer, 2010)

Section 1

Fig. 5. Spatial Reynolds number (Case 1).

171M.B. Adityawan et al. / Coastal Engineering 73 (2013) 167–177

with ν being the kinematic viscosity of the fluid and the eddy viscosity(νt) is given by

vt ¼kω

ð10Þ

The values of the closure coefficients are given byWilcox (1988) asβ=3/40, β*=0.09, α=5/9, and σ=σ*=0.5. The boundary conditionat the bottom is a no-slip boundary, hence a zero value of turbulent ki-netic energy and velocity as well as the dissipation rate gradient. At thefree stream, it is assumed that the velocity gradient, turbulent kineticenergy gradient and the dissipation rate gradient are zero.

2.2. Simultaneous coupling method (SCM)

There are two governing equations in the SCM as mentioned in theprevious section. They are the SWE and the k–ω equation. The equa-tions are calculated separately at each time step, however their resultsare connected, allowing for simultaneous calculation. The basic idea be-hind the calculation is to upgrade the SWEmodel by replacing theMan-ning method with a more accurate method to approximate the bedstress termwithin the momentum equation. The commonly used Man-ning approach will be replaced by a direct approach of bed stress in theregion near the bed using a k–ω model.

Calculation begins with an initial condition of the parameters. Aninitial value of friction coefficient is stated for bed stress calculation inthe SWE model (Tanaka and Thu, 1994). The velocity obtained fromthe SWE model is applied as the free stream velocity boundary condi-tion in the k–ω model as given below:

−∂P∂x ¼ ∂U

∂t þ U∂U∂x ð11Þ

0

100000

200000

300000

400000

500000

-10 -5 0

Re

Fig. 6. Spatial Reynolds

where U is the obtained free steamvelocity fromSWE, and P is the pres-sure applied in assessing thin boundary layer thickness as comparedwith its water depth. Furthermore, the bed stress obtained from thek–ω model is applied in the momentum equation of SWE model.

τ0ρ

¼ vþ vtð Þ ∂u∂y ð12Þ

The process continues until the end of simulation time as shown inFig. 1.

A grid system is developed to allow both models to be coupled si-multaneously. The grid system for the method does not require a hori-zontal and vertical grid system to cover the whole domain from bed tosurface. The vertical grid is only required in the near bottom area to as-sess the boundary layer for bed stress calculation. The grid system limitsthe model to simulating a beach with a slope of less than 1/5. In addi-tion, a steep slope causes the boundary layer approach in the y-axis tono longer apply since the boundary layer develops perpendicular tothe bed.

The water depth becomes very thin at the wave front. In this region,the boundary layer may develop up to the surface. The boundary layerthickness is defined as the location where the upper boundary velocityratio to free stream is larger than 99%. The limit for the region in whichthe boundary layer does not develop up to the surface is defined as thelocationwhere the boundary layer thickness andwater depth ratio doesnot exceed 33%. This limit acts as amoving boundary for the simulation,whichwill change location at each time step. The valueswere chosen toensure the stability and the efficiency of the simulation. A higher valuewill be used to cover larger domains; however, the boundary willchange rapidly in each calculation step. Other values of threshold maybe applied but must not exceed 50% as shown by Tanaka et al. (1999).Outside this region the bed stress is calculated using the momentumequation in the SWE as proposed by Elfrink and Fredsøe (1993). The

5 10 15 20

x*

Critical Reynolds Number

2x105<Re<5x105(Sumer, 2010)

number (Case 2).

Sumer et al., (2011)

Sumer et al., (2011)

Sumer et al., (2011)

a) Shoaling and wave breaking

b) Runup

c) Rundown and hydraulic jump

d) Trailing wave

Sumer et al., (2011)

Fig. 7. Breaking solitary wave runup sequence.

172 M.B. Adityawan et al. / Coastal Engineering 73 (2013) 167–177

model domain definition and treatments are shown in Fig. 2. Outsidethe boundary layer thickness (δ), the gradient of k, ω and u to they-axis is zero (∂F/∂y=0).

2.3. Breaking wave treatment

SCM applies a shock-capturing scheme, FORCE-MUSCL (Mahdaviand Talebbeydokhti, 2009), to extend the SWE capability for breakingwave computation. This scheme was chosen since it was mainly de-veloped to handle shock for wave breaking.

The FORCE-MUSCL scheme is based on a finite volume scheme. Itis basically a combination of several schemes, which covers bothhigh and low dissipation schemes (Lax Friedrich and Lax Wendroff),with the application of a slope limiter. The grid reconstruction forconservative variables in the SWE utilizes MUSCL, which is common-ly used in finite volume. It handles shock by applying a slope limiterfunction. This method adopted the Superbee-type-non-linear slopelimiter (Toro, 2001). Moreover, to ensure its stability, time deriva-tion is solved using the TVD Runge Kutta scheme. Thus, the timestep for the computation is not constant. It has to follow the Courrantnumber criteria. It is found that the Courrant number range of 0.4-0.8provides an accurate result with efficient computation time. The flowchart for the computation is shown in Fig. 3.

2.4. Simulated cases

The model is used to simulate two experimental cases of breakingsolitary wave runup on a sloping bottom from previous studies. Thefirst one is the case of solitary wave runup by Sumer et al. (2011),from here on referred to as Case 1, in which verification and analysisare conductedmostly for the boundary layer. The second experimentis the case of solitary wave runup by Synolakis (1986), from here onreferred to as Case 2, in which verification and analysis areconducted for the water surface profile and runup height. Bothcases have a different bed slope value and incoming wave ratio tostill water depth. Case 1 provided measurement data for the timevariation of bed stress, bed stress fluctuation, and surface fluctuationat measured point. Additionally, this case also provides illustration ofthe breaking wave sequence. Parameters in Case 1 are given in theirdimensional form. On the other hand, Case 2 provided detailed mea-surements of the water level profile along the channel and the corre-sponding runup height, based on analytical and empirical methods.The parameters in Case 2 are given in a non-dimensional form. Dueto the nature of its experiment, Case 2 has been widely used as a nu-merical model benchmark. The illustration for case 1 and case 2 isgiven in Fig. 4(a) and (b), respectively.

The experiment in Case 1 was conducted on a sloping beach with 1/14 slope. The incoming wave height ratio to the still water level (H/h0)was 0.1775. Detailedmeasurementwas conducted at 8 points. The loca-tion of these points is given in Fig. 4(a) and Table 1. The incomingwavein Case 2 is given by H/h0=0.3, which corresponds to breaking wavesimulation. The beach slope was 1/20.

Case 2 is generally presented in non-dimensional parameter as fol-lows. Linear dimension of horizontal distance (x) and vertical distance(y) are divided by the still water depth (h0), giving non-dimensionalcoordinates as follows:

x� ¼ x=h0 ð13Þ

h� ¼ h=h0 ð14Þ

the relative free surface elevation (η*) is given by:

η� ¼ η=h0 ð15Þ

the non-dimensional velocity (U*) is given by:

U� ¼ U=Uc ð16Þ

Uc ¼ Hffiffiffiffiffiffiffiffiffiffiffiffiffiffig=h0ð Þ

qð17Þ

and the non-dimensional time (t*) is given by:

t� ¼ t g=h0ð Þ0:5 ð18Þ

A summary of these cases, along with the acquired parameters forverification of the SCM, is given in Table 2.

173M.B. Adityawan et al. / Coastal Engineering 73 (2013) 167–177

The SCM adopts an adaptive time step interval ensuring the sta-bility of the computation for any grid size in horizontal direction.The grid size in the horizontal direction for both cases is chosen tocapture the runup movement as accurately as possible. In thisstudy, the value is taken to be 0.1x*. On the other hand, the verticalgrid size is determined to ensure that the model can fully capturethe boundary layer thickness. The mesh size of the calculation forCase 1 was given by 0.04 m horizontally and 0.0002 m vertically.Themesh size of the calculation for Case 2 was given by 0.013 m hor-izontally and 0.0005 m vertically.

3. Results and discussion

3.1. Spatial Reynolds number

Verification of the SCM range of applicability is given by conductinganalysis on the spatial wave Reynolds (Re) number along the domain.

-5

0

5

10

-2 0 2 4 6 8 10 12 14

t (s)

(cm

)

Sumer et al. (2011)

SCM

a) Toe

Toe5.6 m

-5

0

5

10

0 2 4 6 8 10 12 14

t (s)

Sumer et al. (2011)

SCM

c) Section 3

Section5

Toe

Section5

0.25 m5.6 m

-5

0

5

10

0 2 4 6

e) S

Toe

Section8

-0.25 m

Section8

5.6 m

(cm

)

(cm

)

Fig. 8. Water level com

Here the wave Re is based on the half stroke of particle displacementas in Sumer et al. (2011):

Re ¼ Um2

ω1υð19Þ

in which Um is the maximum free stream velocity. Here ω is given asfollow:

ω1 ¼ffiffiffiffiffiffiffiffiffiffi34gH

r1h0

ð20Þ

where H is the incoming wave height and h0 is the initial normal waterdepth.

The incoming wave Re for Case 1 was 54,000. Based on criteria forsolitary wave from Sumer et al. (2010), this condition falls in the lami-nar region. However, it was found in the experiment that the Re value atpoint 1 is 300,000.Moreover, bed stressmeasurement shows significant

-5

0

5

10

0 2 4 6 8 10 12 14

t (s)

Sumer et al. (2011)

SCM

b) Section 1

Section1

Toe

Section1

0.97 m 5.6 m

-5

0

5

10

0 2 4 6 8 10 12 14

t (s)

Sumer et al. (2011)

SCM

d) Section 5

Section6

Toe

Section6

0.01 m 5.6 m

8 10 12 14

t (s)

Sumer et al. (2011)SCM

ection 8

(cm

) (

cm)

parison (Case 1).

174 M.B. Adityawan et al. / Coastal Engineering 73 (2013) 167–177

turbulence behavior. The incoming wave condition for Case 2 was18,000. As in Case 1, this condition still falls within laminar conditionsbased on the criteria by Sumer et al. (2010) Nevertheless, as it hasbeen shown by Sumer et al. (2011), higher Re with notable turbulencebehavior may occur in the shallower areas.

The spatial Reynolds number value for Case 1 is shown in Fig. 5.Here, the non-dimensional distance (x*) is used as in Case 2(Fig. 6). Case 2 is provided here to show that the SCM range coverslow and high Reynolds number regimes. The Reynolds number forCase 1 at Section 1 from the simulation agrees with the experimentvalue of approximately 300,000. The Reynolds number increasesdrastically as it gets closer to the shoreline although the incomingReynolds number falls in the laminar region. The highest Reynoldsnumber based on the SCMmay reach up to 1,100,000, located aroundthe initial position of the shoreline. The SCM is able to simulate all re-gimes, including transition and turbulence based on the classifica-tion by Sumer et al. (2010).

Similar behaviors were observed in Case 2 as shown in Fig. 6. Asignificant increase in the Reynolds number also occurred near theshoreline. However, the highest Reynolds number was approximately450,000, at similar locations to Case 1, around the initial position ofthe shoreline. In both cases, a critical value of Re was achieved at theshallow area. Moreover, the Reynolds number values in Case 2exceeded these critical values in Case 1.

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

-5 0 5

x

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

-5 0 5

x

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

-5 0 5

x

c) t* = 25

b) t* = 20

a) t* = 15

Fig. 9. Free surface com

In both figures (Figs. 5 and 6), the Reynolds numbers oscillate themost in shallower area. The Re value is estimated from the maximumvelocity recorded at the corresponding point. The value of the maxi-mumvelocitymay oscillate in shallower area due to the limiter functionthat handles the shock.

3.2. Breaking wave sequence

Breakingwave sequence comparison fromCase 1 is compared to thesequence from the SCM. The sequence comparisons are shown inFig. 7(a)–(d). It is clearly shown that the SCM may reproduce all ofthe sequences as observed in the experiment. The breaking solitarywave runup sequence starts with an increase in the incoming waveheight. This increase occurs along with wave deformation and followsimmediately after wave breaking. During the run down, the flow isstrong enough tomove the shoreline in a seaward direction. This strongflow creates a hydraulic jump-like behavior. The shoreline moves backto its original position creating trailing waves. A similar behavior is ob-served in the SCM results.

3.3. Surface profile and runup height

Comparisons of free surface fluctuation between the SCM data andmeasured data for Case 1 at several points in the domain are given in

10 15 20

*

10 15 20

*

10 15 20

*

BottomExp. (Synolakis, 1986)SWE (Manning-FORCE MUSCL)

SWE (SCM-FORCE MUSCL)

SWE (Manning-Mac Cormack)

NEWFLUME

parison (Case 2).

0.01

0.1

1

0.001 0.01 0.1 1

H/h0

R/h

0

Run Up Law (Synolakis, 1986)SWE (SCM) non-breaking (Aditawan and Tanaka, 2011 b)SWE (Manning) non-breaking (Adityawan and Tanaka, 2011b)SWE (Manning-Mac Cormack)SWE (Manning-FORCE MUSCL)NEWFLUMESWE (SCM-FORCE MUSCL)

Breaking wave

Non breaking wave

Fig. 10. Runup height comparison (slope 1/20).

175M.B. Adityawan et al. / Coastal Engineering 73 (2013) 167–177

Fig. 8(a)–(e). Overall, SCM shows good comparison with measurementdata in all sections. A more detailed surface profile and runup height isgiven in Case 2. Case 2 is commonly accepted as a runup simulationbenchmark. Thus, several other models were compared for verification.They are the SWE (SCM-FORCE MUSCL), SWE (Manning-FORCEMUSCL), SWE (Manning-Mac Cormack) and NEWFLUME (Lin et al.,1999). Here, the Manning roughness (n) value in the SWE is 0.01,which was determined by trial and error to give the highest accuracyof the runup height. This value corresponds well with the experimentcondition.

The wave profile comparison between the experimental dataand the numerical methods is shown in Fig. 9. The Mac Cormackmethod is not able to provide a realistic wave profile at the top. Itproduces a pointy-shaped wave top, followed by a near flat watersurface which can be seen at t*=15 and 20 in Fig. 9. It also failsto give a realistic profile at the first instances of motion of a break-ing wave over a dry bed. At this location, shock and discontinuityoccur and cannot be accurately simulated with the Mac Cormackmethod. The FORCE MUSCL method gives a better profile for com-parison to the measured data than the Mac Cormack method.Moreover, FORCE MUSCL with the SCM performs better than theconventional Manning method.

The NEW FLUME undeniably provides very realistic and accurateresults (Fig. 9). In addition, the NEWFLUME provides detail on the ki-netic energy especially near the surface. This ability is highly crucial inbreaking wave study and suspended load transport mechanisms.However, the computation time is about 15 times that of the SWEand 2 times that of the SCM (Table 3). Nevertheless, the simulationresult is highly informative (Lin et al., 1999).

Runup height comparison is given in Fig. 10. Here, the modelswere used to simulate various incoming height condition. Imple-mentation of the FORCE MUSCL method gives a much betterrunup height comparison for the SWE-type model (Table 3). Itshould be noted here that additional cases with different magni-tudes of incoming wave were simulated for runup height compar-ison with the runup law (Synolakis, 1986) which were also usedto estimate the root mean square error value (RMSE) for eachmodel. The lowest RMSE value is given by SWE (SCM-FORCEMUSLC), followed by SWE (Manning-FORCE MUSCL). The highesterror is given by SWE (Manning-Mac Cormack). The runup heightprediction in NEWFLUME depends highly on the method used forconstructing the surface. Therefore, this model was not includedin the RMSE comparison. In addition, non-breaking wave simula-tions from a previous study (Adityawan and Tanaka, in press) wasadded in Fig. 10 to show the SCM range of applicability. It shouldbe noted here that Reynolds number for non-breaking wave caseis very low. Therefore, the Manning roughness value for non-SCMmodel was adjusted (n=0.046). The SWE (SCM)model gives a bet-ter estimation of the runup height than SWE (Manning). Details onthe non-breaking wave simulation can be found in the reference(Adityawan and Tanaka, in press).

Table 3Model performance comparison (Case 2).

Method SWE

SCM-FORCE MUSCL Ma

Mc

Governing equation SWE+k–ω SWReal time⁎ Approx. 70 min ApBreaking wave Very good PooRMSE of run up height (R/h0) 0.001 0.0Advantage Breaking wave+boundary

layer–

*All simulated using Core i3 330 m, average of 5 repetitions.

3.4. Bed stress and turbulent intensity

Case 1 provided valuable parameters regarding bed stress and kvalues under solitary wave runup. Here, bed stress is normalized asfollow:

τ0� ¼ τ0ρUc

2 ð21Þ

where τ0 is bed stress.Bed stress comparison between the SCM data and measured data at

several points in the domain is given in Fig. 11. Overall, the SCM resultshows good comparison with the measurement. However, discrepan-cies can be found especially during the down rush. As it was explainedearlier, the SCM employs moving boundary condition threshold, basedon the ratio of the boundary layer thickness to the correspondingwater depth for separating the region where the boundary layer ap-proach is used. During the down rush, there can be sudden change inthe before-mentioned threshold due to the sudden decrease of thewater level resulting in less accurate bed stress estimation.

The measured bed stress fluctuation at the corresponding points iscompared to the turbulent intensity from the SCM. Bed stress

NEWFLUME

nning (n=0.01)

Cormack FORCE MUSCL

E SWE RANS 2DV k-εprox. 15 min Approx. 12 min Approx. 160 minr Very good Excellent32 0.002 –

Breaking wave Reproduces realistic wavebreaking in details

-1

-0.5

0

0.5

1

1.5

0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14

0 2 4 6 8 10 12 140 2 4 6 8 10 12 14

t (s)

τ0*

Sumer et al. (2011)

SCM

a) Section 2

-1

-0.5

0

0.5

1

1.5

t (s)

τ0*

Sumer et al. (2011)

SCM

b) Section 5

Section 2

Toe

Section 2

0.91 m 5.6 m

Section 5

Toe

Section 5

0.25 m 5.6 m

-1

-0.5

0

0.5

1

1.5

t (s)

τ0*

Sumer et al. (2011)

SCM

c) Section 6

-1.5

-1

-0.5

0

0.5

1

1.5

t(s)

τ0*

Sumer et al. (2011)

SCM

d) Section 8

Section 6

Toe

Section 6

0.01 m 5.6 m

Toe

Section 8

-0.25 m

Section 8

5.6 m

Fig. 11. Bed stress comparison.

176 M.B. Adityawan et al. / Coastal Engineering 73 (2013) 167–177

fluctuation and turbulent intensity cannot be compared directly.However, they both show the turbulence behavior in the boundarylayer. Thus, similar behavior between the two parameters is expected.It is found that the fluctuation magnitude and occurrence time areclosely related to the estimated turbulent intensity value (Fig. 12).Here, y’ is the non-dimensional height from the bottom with its

1 k*

a) Section 2

0

0.2

0.4

0.6

0.8

t (s)

t (s)

0.0 0

y'

y'

-0.1

0

0.1

0.2

0 2 4 6 8 10 12 14t (s)

0 2 4 6

t (s)0 2 4 6

0 2 4 6 8 10 12 14

Section 2

Toe

Section 2

0.91 m 5.6 m

0.7

b) Se

0

0.2

0.4

0.6

0.8

0

1

-0.1

0

0.1

0.2

Fig. 12. Turbulent intensity and bed

boundary layer thickness (δ) and k* is the non-dimensional turbulentintensity given as,

k� ¼ k

g=h0ð Þ0:5H� �2 ð22Þ

8 10 12 14t (s)

0 2 4 6 8 10 12 14

t (s)0 2 4 6 8 10 12 14

8 10 12 14

ction 5

0

0.2

0.4

0.6

0.8

0.0

Section 5

Toe

Section 5

0.25 m 5.6 m

0.7

-0.1

0

0.1

0.2

k* k*

c) Section 8

y’

1

0

0.7

0.0

Toe

Section 8

-0.25 m

Section 8

5.6 m

stress fluctuation comparison.

177M.B. Adityawan et al. / Coastal Engineering 73 (2013) 167–177

where k is the turbulent intensity. τ'0* refers to the measured bedstress fluctuation, normalized as follow:

τ00� ¼ τ0

0

ρUc2 ð23Þ

It is observed that in the deeper locations during the runup, thereis no significant k value that is confirmed by the low value of bedstress fluctuation. On the other hand, the k value is high at therunup, which is confirmed with a high magnitude of bed stress fluctu-ation. Thus, the bed-generated stress plays an important role in theoverall process. Overall, the similarity of k values from the SCM dataand the measured bed stress fluctuation confirms the SCM's abilityto assess a turbulent boundary layer.

It is interesting to note that turbulence behavior appears earlier asthe wave approaches the shoreline with respects to the wave shape. Ithas been shown by Sumer et al. (2010) that turbulent activities shiftfrom a deceleration phase to an acceleration phase as the Re in-creases. The SCM results show that at Section 2 (Fig. 12(a), at Re=350,000) turbulent activity under the wave runup is low. At Section5 (Fig. 12(b), at Re=400,000) the turbulence behavior under thewave runup occurs around the wave peak. However, turbulence be-havior under the wave runup appears earlier than the wave peak atSection 8 (Fig. 12(c), at Re=1,000,000).

4. Conclusions

The application of the boundary layer approach for breaking soli-tary wave runup simulation has been accomplished. The FORCEMUSCL scheme has been implemented in the SCM. Two cases weresimulated to verify the SCM. In general, the SCM is able to reproducewater surface evolution along with the general sequence of breakingsolitary wave runup. The SCM estimates bed stress directly from theboundary layer, which leads to a higher accuracy. Runup height com-parison shows that the SCM increases the SWE based model accuracyin predicting water surface profile and the runup height.

The SCM data has been verified with the measured bed stress andbed stress fluctuation. The bed stress from the SCM shows good com-parison to the measured value. In addition, the turbulent activity inthe boundary layer beneath the wave corresponds well to the mea-sured bed stress fluctuation. It is noted that as the wave approachesthe shoreline and the wave Reynolds number increases, turbulent ac-tivities may start earlier than the wave peak.

Overall, the SCM is a promising solution for boundary layer analy-sis under wave runup for future practical application. The SCM hasthe simplicity of the SWE, yet it provides more details in terms ofthe boundary layer. Future development may combine the basicidea of the SCM with NEWFLUME. The SCM assumes uniform velocityoutside the boundary layer up to the surface. NEWFLUME may pro-vide a better estimation of vertical velocity profile outside the bound-ary layer with the k-ε model. Therefore, the velocity outside theboundary layer can be accurately estimated. The k–ω model mayuse this calculated velocity and utilize it to assess boundary layer.Thus, the developed model may assess boundary layer as in theSCM, yet be fully capable of simulating flow in detail as in NEWFLUME.Both models were developed under different approaches and concerns.The combination of both will be highly beneficial for coastal sedimenttransport related studies, since it may accurately predict both bed loadand suspended load.

Acknowledgement

The authors would like to thank the financial supports fromGrant-in-Aid for Scientific Research from Japan Society for Promotionof Science (No. 21360230, No. 22360193, and No. 2301367), the River

Environmental Fund (REF) from the Foundation of River andWatershedEnvironmental Management (FOREM), Japan, Open Fund from StateKey laboratory of Hydraulics and Mountain River Engineering(SKLH-OF-0907), Natural Science Foundation of China (51061130547and 51279120). The first author is a Postdoctoral Fellow granted byJSPS (No. P11367).

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