Proceedings of the 34th International Conference on Ocean, Offshore and Arctic EngineeringOMAE 2015

May 31–June 5, 2015, St. John’s, NL, Canada

OMAE2015-41807

NUMERICAL SIMULATION OF OBLIQUE WAVE IN WATER BASIN

Yu Chen∗, Yali Zhang, Mimi Gao, Johan Gullman-StrandLloyd’s Register GTC1 Fusionopolis Way

Connexis (North Tower) #17-14Singapore 138632

Rajeev Kumar JaimanDepartment of Mechanical Engineering

National University of SingaporeSingapore 117575

MyHa Dao, Lup Wai ChewInstitue of High Performance Computing

A*STAR, Singapore1 Fusionopolis Way

Connexis (North Tower) #16-16Singapore 138632

ABSTRACTSimulation of more realistic ocean conditions in wave basins

is becoming important for offshore industry. As spreading wavehas become more desirable, the capability of reproducing obliqueplanar wave train is critical for a wave basin’s performance. Inthis study, the oblique waves have been simulated in a numeri-cal wave basin based on the volume of fluid (VOF) method andNavier-Stokes solver. Multi-element bottom hinged flap motionusing the conventional snake principle has been simulated bymoving boundary dynamic mesh in OpenFOAM. Finite lengthof the entire wave maker and finite width of each paddle causedconsiderable spatial variations in wave height and wave prop-agating direction. Beaches with slope have been implementedto minimize the sidewall reflection and improve the uniformityof the oblique wave field. The generated linear oblique waveshave been compared with the analytical solutions for validationin terms of uniformity and wave height. The time history of thesurface elevation at different locations have been computed to in-vestigate the uniformities of general wave fields. In addition, the

∗Address all correspondence to this author. Email: Yu.Chen@lr.org

free surface along the wave propagating direction has also beeninvestigated to show the quality of the generated wave.

INTRODUCTIONIn the past three decades, the offshore oil exploitation of nat-

ural resources and the demand for offshore platforms have con-tinued to increase. However, the ocean environment can posemany dangers to offshore structures, risk assessment process isrequired for these offshore working units in order to ensure thesafety under extreme wind and wave conditions. Presently withinthe offshore industry, analytical methods and scaled model testsby laboratory experiments are the main methods for establishingwave-in-deck loads [1–3]. Generally, the analytical methods arebased on simplified fluid dynamics theory and although they areeasy to implement they have low reliability and accuracy. Thus,experimental tests in small wave basins are quite common in thefield of marine engineering. These basins have currents, windand waves generators to try to simulate the marine environment.However, testing in basins also has its disadvantages, e.g. it isrelatively expensive and scaling effects may influence the accu-

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racy especially when there is air entrainment [4–6]. As an alter-native approach, numerical simulation based on computationalfluid dynamics (CFD) has attracted a lot of attention recently dueto advances in numerical methods and the availability of highperformance computers. CFD can provide a useful tool to pre-dict impact forces with high accuracy for the full scale deck, aswell as the loading produced by breaking waves. Detailed insightinto the flow and resultant loadings can also be resolved by CFDwhich is not possible with analytical methods and very difficultwith experimental methods. CFD simulations can also be treatedas a set of preliminary tests of experiments in wave basin sinceand configuration and design of platforms can be easily changedinside, and the wave basin experiments can be used as a finalverification.

Recently, CFD has been widely employed to investigate en-vironmental loads for offshore structures, i.e. wave-in-deck prob-lems [7–9]. A new CFD tool, ComFLOW, based on local grid re-finement, VOF and a blended turbulence model has been utilizedby [10] to study extreme wave impact in offshore and coastalengineering. However, most of these works are restricted to uni-directional regular waves, e.g. 5th order Stokes wave or othersimilar kind of waves, which are significantly different from thedirectionally spreading waves in real ocean conditions. O’deaand Newman [11] conducted a numerical study of directionally-spread waves in WAMIT [12]. However, WAMIT is completelybased on the boundary element method and potential flow theory,so the effect of viscosity and air entrapment cannot be. Thus, it isimportant to generate the spreading waves in the numerical wavebasin based with CFD based on Navier-Stokes equations. Asknown, spreading waves are composed of group of wave com-ponents which travels in different directions, thus generating sin-gle oblique wave precisely is essential for generating spreadingwave. Actually generating oblique waves to adjust angle betweencurrent and wave propagation direction has been widely imple-mented in some physical wave basins in the world, so validatingthe oblique wave numerically generated can be seemed as a pre-liminary stage of numerical spreading wave and also can guidethe design and operation of experimental wave basins.

Basically, the wave generator consists of the actuator andthe power system to move it. The key of successfully gener-ating a correct oblique wave in a basin is the transfer functionbetween wave height and flap stroke and proper delay among theflaps. Thus it requires a mathematical formulation for the wa-ter waves propagation and numerical models simulating multi-phase problems with free surface, which determine the aim ofthe current paper, e.g. validating the capability of numerical mod-els and transfer functions in generating oblique waves, in termsof wave height, oblique angle and uniformity in the test region.So, the theory of transfer functions are briefly introduced firstlyin Section “Mathematical Model”, then followed by the numeri-cal methodology implemented (i.e. Volume of Fluid, VOF). Theconfiguration of the numerical wave basin and the results are then

presented and discussed.

MATHEMATICAL MODELIn this section, the mathematical connection between the de-

sired waves and the motion of wave maker (e.g. transfer function)is briefly introduced. The transfer function is normally deter-mined based on the potential flow theory for water waves, andhas been studied in the last century [13–15], which is introducedfirstly. A detailed account of oblique wave generation has beenpresented by Frigaard et al. [16].

Velocity Potential“Les Appareils Generateurs de Houle en Laboratoire” [13]

presented by Biesel and Suquet discussed and solved the analyt-ical problems concerning a number of different wave generatortypes. For each wave maker type the paper presented the trans-fer function between wave maker displacement and wave ampli-tude in these cases where the analytical problem could be solved.The article therefore represented a giant step in wave generationtechniques and found the basis for today’s wave generation inhydraulics laboratories.

The simplest velocity potential is that obtained assumingsmall amplitude waves or very small wave slope since linear the-ory can be applied. Under such assumptions the so-called ve-locity potential (φ ) should be determined because it allows thederivation of all the desired wave characteristics. Once the ve-locity potential is known the transfer function and hydrodynamicreaction can be calculated. The configuration of the wave basinand properties of waves are presented in Figure 1.

FIGURE 1. SCHEME OF WAVE MAKER, ADOPTED FROM [17].

The displacement of paddles is described in [16]

x = e(z)sin(ωt) =S (z)

2sin(ωt) . (1)

2 Copyright c© 2015 by ASME

where e(z) is the maximum displacement of different parts ofpaddles, and S(z) is distance between two extreme position ofdifferent part of paddles (see Figure 1 and Eq. 2). For bottomhinged (flap) type wave maker, the movement of the paddle canbe expressed as

S(z) = S0h+ z

h(2)

and the transfer function is

HS0

=2sinh(kh)(1− cosh(kh)+ khsinh(kh))

kh(sinh(kh)cosh(kh)+ kh)(3)

where k is the wave number of the generated sinusoidal wave,and it is the solution to the dispersion relation:

ω2 = kg tanh(kh), (4)

and H is the height of wave generated.

Oblique WaveIn this section, techniques to generate oblique waves will

be discussed. Considering a wave generating system where thegenerator front consists of a number of very small paddles. Aoblique regular wave can then be generated using Huygen’s prin-ciple [16], by introducing a suitable delay between the wave pad-dles as illustrated in Figure 2. It is evident that the required delayof the individual wave paddles will lead to a sinusoidal shape ofthe front of the wave generator.

FIGURE 2. HUYGENS’ PRINCIPLE IN GENERATING OBLIQUEWAVES, REGULAR WAVE TRAVELLING IN θ -DIRECTION,ADOPTED FROM [16].

Phase Delay in Neighbouring PaddleIf the wave length of the generated wave is λ , and the wave

length from maximum to maximum of the sinusoidal front of

the wave generator is l = λ/sinθ . Thus the delay, ϕp, betweenneighbouring wave paddles of width lp is lp ·2π/l or

ϕp = lp2π sinθ

λ(5)

If the wave paddle displacement is calculated for a regularwave with the wave length λ travelling in the x-axis direction, thedelay between the ith and the 0th wave paddle when generatingthe same regular wave travelling in the θ -direction, is

ϕpi = i · lp2π sinθ

λ(6)

Transfer Function for Oblique WavesConsidering the energy propagation, the transfer function of

oblique wave should be corrected as

F3 = F2/cosθ (7)

where F2 is the transfer functions defined in equation 3.

NUMERICAL MODELThe open-source CFD code OpenFOAM, which employs a

nonlinear Navier-Stokes equation solver, has been used to con-duct the numerical simulation of the wave basin. Specifically,the volume of fluid (VOF) method was employed to simulate thewater-air interface. This includes the viscous dissipation duringwave breaking and slamming which cannot be simulated by the-oretical methods or potential flow based methods.

“interFoam”, one of the multi-phase solvers in OpenFoam,has been used in the current numerical study. It solves the 3D N-S equations for two incompressible phases using a finite volumediscretization and the volume of fluid (VOF) method.

In a two-phase flow simulation with fluid densities ρl andρg, viscosities µl and µg and surface tension coefficient σ , theflow is governed by the following momentum equations [18]:

∂ρU∂ t

+∇ · (ρU⊗U) =−∇p+[∇ · (µ∇U)+∇U ·∇µ]

+ρg+∫

Γ

σκδ (x− xs)ndΓ(xs)(8)

where Γ is the air-water interface, δ (x − xs) is the three-dimensional (3D) Dirac delta function and the viscous term ∇ ·[µ(∇U +∇UT

)]has been rewritten as ∇ ·(µ∇U)+∇U ·∇µ . In

interFoam, the continuum surface force (CSF) model of Brack-bill et al. [19] has been employed. Readers are referred to [19]

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for detailed discussion of the integration of surface tension termand approximation of the local interfacial curvature κ .

The solver employs a modified pressure (pd) rather thansolving for total pressure p and their relationship is given by

pd = p−ρg ·x,∇pd = ∇p−ρg−g ·x∇ρ (9)

with both the surface tension term and the substitution of pd forpressure, the volume integral of equation (8) over an arbitrarycell is solved in interFoam by constructing a predicted velocityfield and then correcting it using the Pressure Implicit with Split-ting of Operators (PISO) [20] implicit pressure correction proce-dure to time advance the pressure (pd) and velocity fields. If thecell in question is identified by subscript P, and the PISO itera-tion procedure is indexed by m, with m = 0 corresponding to theinitial step and pertaining to the present time level tn, then, wemay consider the discrete version of equations (8) neglecting forthe moment pressure term, which yields an explicit expressionfor the predicted velocity field U r

P, namely

ρn+1P U r

P− (ρU)nP

∆t|ΩP|+∑ f∈∂Ωi

(ρ f Φ f

)U r′

f

=∑ f∈∂Ωi

[µ

n+1f

(∇⊥f U)r′ ∣∣S f

∣∣]+∇UnP ·∇µ

n+1P |ΩP|

+

[(σκ)n+1

f

(∇⊥f γ

)n+1]∣∣S f

∣∣−[(g ·x)n+1f ∇

⊥f ρ

n+1]∣∣S f

∣∣(10)

where the variables with subscription f are interpolated valuesat cell faces. There is wide range of interpolation schemes withor without limiters implemented in OpenFOAM for this purpose(centred, upwind, TVD/NVD). The associated volume flux Φ f =(U r

P) f ·S f .∣∣S f∣∣ and |ΩP| are magnitude of face surface area and

cell volume respectively. In this expression, the fluid density ρ

and viscosity µ are obtained by

ρ = γρl +(1− γ)ρg,µ = γµl +(1− γ)µg (11)

and volume fraction function γ is solved by a separate equationand is known at time level tn and tn+1 when solving momentumequations, the density and viscosity fields in equation (11) arealso known. By employing the techniques described in [21] toapproximate the velocity face value U r′

f and face normal gradient(∇⊥f U

)r′, equation (10) can be re-arranged to a linear system

A · x = B (12)

where A is a large sparse matrix, x the dependent variables vec-tor and B the constant source vector. Equation (12) can be solved

using the preconditioned conjugate gradient (PCG) method. Be-sides PCG, OpenFOAM provides various other options [22] suchas preconditioned bi-conjugate gradient, generalized geometricalgebraic multi-grid and smooth Solver, which uses a smootherfor convergence. Equations (10–12) completes the velocity pre-dictor step, and now the pressure equation can be formulated byincluding the pressure contribution (which is not accounted inderiving Equation (11)) and enforcing the continuity (mass con-servation) for an incompressible medium. This results in

∑ f∈∂Ωi

[(1

AP

)f

(∇⊥f pm+1

d

)∣∣S f∣∣]= ∑ f∈∂Ωi

Φ f (13)

where quantity AP is the computed coefficient before unknownU r

P after rearranging Equation 10. Equation (13) again leads toa linear system for pm+1

d , and can be solved using one of thetechniques introduced above. Equation (13) completes the briefintroduction to the solution of the incompressible Navier-Stokessystem.

The mathematical formulations of the two phase systemwith free surface are closed by solving a separate convectionequation for the evolution of volume fractions (γ). The presentsolver, interFoam, employs a modified approach similar to oneproposed in [23], named High Resolution Surface CapturingScheme (HRIC), relying on a two-fluid formulation of the con-ventional volume-of-fluid (VOF) model in the framework of fi-nite volume method. In this model an additional convective termoriginating from modeling the velocity in terms of weighted av-erage of the corresponding liquid and gas velocities is introducedinto the transport equation for phase fraction, providing a sharperinterface resolution. The model makes use of the two-fluid Eu-lerian model for two-phase flow, where phase fraction equationsare solved separately for each individual phase. For details read-ers are referred to [21]. This completes the mathematical formu-lations for a two phase incompressible system and what remainare the formulations for focused extreme wave generator.

RESULTS AND DISCUSSIONSThe linear wave investigated in the current study was a small

amplitude regular wave, whose properties are listed in Table 1. Itcan be described by the 1st order Stokes wave theory. There area lot of different types of actuators but the flap is one of the mostcommon types in the physical basins, so it was chosen herein.For simplicity but without any loss of generality, a small numer-ical wave basin was chosen in order to reduce the computationalcost, since the aim of the study was to validate the capability ofthe numerical wave basin generating oblique waves. The obliquewave is expected to travel in the 60 degree direction to the x-axisas shown in Figure 4(a). Two side walls of the wave basin areinstalled with the moving paddles, and the other two are wave

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TABLE 1. PARAMETERS OF WAVE IMPLEMENTED ANDWAVE BASIN DIMENSIONS

Wave properties Value

Water depth d (m) 0.7

Wave height H (m) 0.08

Wave period T (sec) 1.303

Wave crest elevation E (m) 0.04, 0.06

Wave length λ (m) 2.5

Basin dimension L×W×D (m) 12×12×0.85

Test section L×W (m) 8×8

Paddle width lp (m) 0.5

Paddle range (m) 7×7

Beach slope 1:5

absorbing beaches as shown in Figure 4(b). More detailed di-mensions of the wave basin, paddles and beaches can be foundin Table 1. The mesh employed for the simulation are illustratedin Figure 3, which is refined near the interface between water andair.

FIGURE 3. MESH FOR THE SIMULATION OF WAVE BASIN.

Four wave gauges were defined inside the wave basin (seeTable 2) to monitor the wave elevation and investigate the obliqueangle and uniformity of the generated waves. These four wavegauges were arranged in a line orthogonal to the oblique wave’spropagation direction (also see Figure 4(a)).

Figure 4 presents the visualization of the wave basin togetherwith the generated oblique wave’s free surface. It can be foundthat qualitatively the generated wave is travelling in the expecteddirection within the test region and then is absorbed in the beach

TABLE 2. WAVE GAUGES’ LOCATIONS INSIDE THE NUMERI-CAL WAVE BASIN

Point X (m) Y (m)

1.00 2.00 7.00

2.00 3.00 5.27

3.00 4.00 3.54

4.00 5.00 1.80

region. It can also be observed that the wave turns its travellingdirection in the beach region, which is a refraction phenomenonof wave in shallow water. Furthermore, shoaling effect can alsobe found in this beach zone.

Figure 5 shows the time history of the surface elevation ofthe generated wave at different locations and it can be seen thatthe the phase of free surface elevation at different locations syn-chronized with each other very well, proving that the obliquewave was travelling in the prescribed angle. In addition, the mag-nitude of the free surface elevation match well with the expectedvalue, 0.04m, thus the transfer function based on the linear poten-tial flow theory has been correctly created. However, the surfaceelevation magnitude of point #1 was found to be slightly higherthan the expected value, which is due to the fact that it is closeto the paddles and beach zone on the side walls. It can also beobserved that it takes about 5-6 cycles for the generated wave tobecome stable after the paddles start to move.

The spatial distribution of surface elevation along theoblique wave’s travelling direction is shown in Figure 6 (see line1 in Figure 4(a)). For convenience of comparison, the theoreticalresults based on sinusoidal wave with the expected wave heightand wave length are plotted together with the numerical results.It can be found that the numerical results matches well with theexpected wave within the test region away from the beach zone(distance < 6m). However, the numerical surface elevation ofwave deviate from the theoretical results when the location isclose to the beach zone (distance > 6m), where the incident waveis significantly affected by the reflection from the beach. Espe-cially, it is observed that free surface can rise up very high at theend of the beach (distance > 12m). Based on this observationit can be concluded that the test region with a wave of sufficientquality is small in the current study. However, it is acceptablebecause the aim of the current study was to validate the numeri-cal wave basin’s capability to generate oblique waves. The areaof the region affected by the reflection from beach is almost onlydetermined the wave properties and the configuration of beach(slope, surface curve), so a larger test region can be expectedthrough enlarging the dimension of wave basin in the future. Ithas not been done in the present study is due the fact that solv-

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(a) top view

(b) isometric view

FIGURE 4. CONFIGURATION OF THE NUMERICAL WAVEBASIN AND VISUALIZATION OF THE FREESURFACE OF THEGENERATED WAVE.

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0 5 10 15 20

Surf

ace E

levation [m

]

Time [s]

point #1point #2point #3point #4

FIGURE 5. SURFACE ELEVATION OF WATER AT DIFFERENTLOCATIONS MEASURED BY THE WAVE GAUGES.

ing dynamic mesh is time consuming (seven days to finish thecomputation with one million cells) and the aim of current studyis only to validate the capability of generating oblique waves innumerical wave basin.

-0.1

-0.05

0

0.05

0.1

0 2 4 6 8 10 12

Surf

ace E

levatio

n [

m]

Horizontal Distace [m]

numerical resultstheoretical results

FIGURE 6. SURFACE ELEVATION OF WATER ALONG THEWAVE TRAVELLING DIRECTION.

To further validate the capability of the numerical wavebasin in generating oblique waves, a wave with larger height,0.06m, was investigated. The same wave basin and wave gaugeswere utilized to monitor the quality of the wave generated. InFigure 7, the time history of surface elevation at four wavegauges along the same wave front plane is illustrated to validatethe uniformity and propagating direction. Similar to the casewith lower amplitude 0.04m, it is shown that the phase of waveat different gauges match with each other perfectly. Furthermore,the uniformity of the generated wave is also generally acceptable.

-0.1

-0.05

0

0.05

0.1

0 5 10 15 20

Surf

ace E

levation [m

]

Time [s]

point #1point #2point #3point #4

FIGURE 7. SURFACE ELEVATION OF WATER ALONG THEWAVE TRAVELLING DIRECTION.

6 Copyright c© 2015 by ASME

CONCLUSIONSIn this study, the technique of generating oblique waves in

the numerical wave basin has been studied. The mathematicalmodel and transfer function of generating oblique waves by pad-dles have been established and the numerical model based onthe volume of fluid approach was utilized in this study. Thenumerical wave basin based on the CFD tool “interFoam” ofOpenFOAM and the previously introduced wave theory has beentested for a linear wave travelling in the 60-degree direction. Tominimize the influence of the reflection from the side wall, a kindof simple beach was employed at two sides.

The quality of the generated oblique wave was investigatedin terms of the oblique angle and uniformity. The free surfacegeometry obtained in the numerical wave basin was comparedwith the theoretical results. All these findings indicated thatthe present numerical wave basin based on CFD tools and VOFcan generate oblique waves as expected. It should be noted thatthe transfer function cannot only be employed in physical wavebasins but also is applicable for numerical wave basin.

Due to the time constrains, the computational domain waskept small, which resulted in a small qualified test zone. In thefuture, a larger wave basin can be implemented to minimize theeffect of reflection from beach and extend the test zone.

ACKNOWLEDGMENTFinancial support provided by Lloyd’s Register and

A*STAR for this work is gratefully acknowledged.

DISCLAIMERThe views expressed in this paper are those of the authors

and do not necessarily reflect those of their affiliated companies.

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