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On the Numerical Simulation of a Piston-type Wavemaker

H.B. Gu1,2, D.M. Causon1, C.G Mingham1, L. Qian1, Han-Bao. Chen2

1 Centre for Mathematical Modelling and Flow Analysis

School of Computing, Mathematics and Digital Technology

The Manchester Metropolitan University,

Manchester, UK

2 Tianjin Research Institute for Water Transportation Engineering,

Key Laboratory of Engineering Sediment of Ministry Communications,

Tianjin, China

ABSTRACT

The piston-type wavemaker is a very popular and important piece of

equipment in coastal and ocean engineering physical wave modelexperiments so in this paper we use a numerical model to simulate the

characteristics of the piston-type wavemaker. The numerical model is

based on solving the Navier-Stokes equations, in which a two-fluidwater and air system is adopted. The interface between the water and

air is tracked by a particle level set method and the partial cell

technique combined with the local relatively stationary method is used

to simulate the motion of the wavemaker in the numerical model.Firstly, linear wave maker theory is implemented and tested for regular

waves at various water depths, wave periods and wave heights, to see

which wave making methods give reasonable results for short wave

cases. The method is then extended to simulate irregular waves where

the wave spectrum is specified and the individual wave components aresuperimposed. The output spectrum from a wave gauge in thenumerical flume compared well with the target spectrum. Finally a

focused wave is simulated by the numerical model and compared with

published results. It is shown that the numerical flume can simulate an

experimental flume with a piston-type wavemaker and thus physicaltests can be accurately replicated by numerical modelling.

KEY WORDS: Wavemaker; regular wave; irregular wave; focusedwave; level set; partial cell; local relatively stationary method.

INTRODUCTION

Wavemakers are important pieces of equipment in coastal and oceanengineering laboratory experiments to study wave interaction with

structures such as wharves, breakwaters and any other near-shore or

ocean structures. Generation of waves is one of the most significanttasks in this kind of laboratory.

The most common way in a physical experimental flume to generate

waves is through the movement of a paddle, which is located at one endof the flume. Paddles used in flumes can be a flap, a piston or a wedge

type, of which the piston-type is the most popular permitting simple

generation of shallow water waves according to the velocity pattern

near the paddle.

Havelock (1929) derived an analytical solution for waves generated bypiston and flap wavemakers based on linear wave theory. Ursell et al .

(1960) and Flick and Guza (1980) experimentally verified a pistonwavemaker by using first order wavemaker theory. Ottesen-Hansen et

al. (1980), Sand (1982) and Sand and Donslund (1985) discussed

second-order effects such as long waves in experimental models.

Schaffer (1996) derived a complete mathematical model for pistonwavemakers to second order. The linear wavemaker theory has been

extended to a 3D wave basin (Liu 1994, 1996; Newman 2010). In the

implementation of wave making in the laboratory active absorption is

often applied to avoid spurious reflection (Spinneken and Swan 2009a,2009b).

Because the size of experimental models is limited by the size of wave

tanks leading to scaling effect errors, numerical wave flumes began to

be considered as a possible tool to support the design and regulation of

costal, ocean and near shore structures. Numerical wave models basedon nonlinear shallow water equations (SWE) can be found in Van Gent

(1994), and Hu et al. (2000). Zhang (2005) built a numerical model

based on the Boussinesq equations and compared numerically

generated waves to physical results. As models based on the previous

methods do not show the detailed information in the vertical direction

advanced numerical wave flumes should be based on the Navier-Stokesequations (NSE).

The main challenge in solving NSE for wave generation is how to

locate the free surface. There are two approaches for solving the NSE:

mesh based methods and meshless methods. Meshless methods include

the smoothed particle hydrodynamic (SPH) and moving particle semi-

implicit (MPS) methods. Shao et al. (2006) presented an

incompressible SPH model to investigate wave overtopping in coastalstructures. Koshizuka et al. (1995) proposed the MPS method to solve

the NSE used for the simulation of numerical wave flumes.

Mesh based methods include the volume of fluid (VOF) method (Hirt& Nichols 1981, Yongs 1982, Ubbink 1997), the level set method (Gu

et al.2009, 2010; Chen 2009) and the marker and cell method (MAC)(Harlow & Welch 1965) to capture or track the free surface together

with a NSE solver such as the projection or SIMPLE method. Another

652

Proceedings of the Twenty-fir st (2011) In ternational Of fshore and Polar Engineeri ng Conference

Maui, Hawaii, USA, June 19-24, 2011

Copyri ght 2011 by the International Society of Of fshore and Polar Engineers (I SOPE)

ISBN 978-1-880653-96-8 (Set); ISSN 1098-6189 (Set); www.isope.org

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0

0.00005

0.0001

0.00015

0.0002

0.00025

0.0003

0.00035

0.0004

0 0.5 1 1.5 2 2.5 3

frequency f

spectrum

density

Targeted

Simulated

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

time (s)

eta

(m)

is a little smaller than the target value. Based on the wave history a new

spectrum can be estimated by autocorrelation analysis, which is calledthe measured wave spectrum shown in Figure 4 in a black solid line.

Comparing the measured and target spectra, the peak frequency and

spectral shape are very close to each other. The estimated spectrum hassome oscillations around the peak frequency. These may be caused by

numerical errors in the simulation and the discretisation of the targetwave spectrum. However errors are small and so the numerical model

can be used to simulate irregular waves.

Figure 4: Wave spectrum targeted and estimated

Focused waves

The mechanism of wave focusing in two dimensions is related to wavedispersion. When rapidly travelling long waves catch up with slowly

travelling short waves, a large amplitude wave can appear at some fixed

location or time due to the superposition of wave components. Then anextreme wave is expressed as

))(2)(cos(),(1

pipi

N

ii ttfxxkatx

f

==

(25)

where xp and tp are focus position and time respectively, Nf is thenumber of wave components with frequency fi and amplitude ai andwave number k

i. The number of wave components is chosen as 50 to

approximate a continuous wave spectrum. Wave components are equi-

spaced across a bandwidth of dfand centered at frequencyfc. The wavedensity distribution is calculated by the modified JONSWAPspectrum (Goda, 1999).

Figure 5: Irregular wave time history

In this section, we use the numerical piston wavemaker to simulate thefocused waves in tests cases conducted by Zhao and Hu (2010). The

schematic of the setup is similar to that of Figure 1. The length of the

flume is 20m, with water depth of 0.4m. The focusing position is set at

5m away from the wave paddle and the focusing time is 10s.Computations are conducted with two different frequency bandwidth df

=0.3 and 0.9 with the peak frequency fc= 0.83. The focusing amplitudeA,which represents the linear sum of the component wave amplitudes(A= ai), is set to 0.06m by adjusting input significant wave height.The simulation ran for 20s.

A wave gauge is set at the target focus position to record the wave

profiles. Figure 6 and Figure 7 show the free surface elevation with

frequency bandwidths 0.3 and 0.9 respectively. Wave profiles are

similar to results from Zhao and Hu (2010). Comparing Figure 6 toFigure 7, the frequency bandwidth has an effect on focused wave

profiles. The wider frequency bandwidth case gives a more clearlydefined extreme wave. Therefore the wave frequency bandwidth should

be carefully selected in practical application.

CONCLUSIONS

Linear wavemaker theory is implemented in the numerical model. Themodel is based on solving the NSE and using a level set method to

track the water free surface in conjunction with the partial cell method

combined with the local relatively stationary method to simulate themotion of the wavemaker. A piston-type wavemaker is used to generateregular waves, irregular waves and focused waves. It is shown that

linear wavemaker theory is suited to simulate short wavelength regular

waves and can also be used to approximate irregular waves and focused

waves. For more accurate wave simulation high order wavemakertheory will be tested in the future.

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

5 6 7 8 9 10 11 12 13 14 15

time (s)

eta/A

Figure 6: Focused wave with df= 0.3,A=0.06m

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

5 6 7 8 9 10 11 12 13 14 15

time (s)

eta/A

Figure 7: Focused wave with df= 0.9,A=0.06m

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