Ocean Engineering 31 (2004) 1083–1092
www.elsevier.com/locate/oceaneng
Determination of wave energy dissipation
factor and numerical simulation of wave height
in the surf zone
Y.H. Zheng a,, Y.M. Shen b, X.G. Wu b, Y.G. You a
a
b
Guangzhou Institute of Energy Conversion, Chinese Academy of Sciences, Guangzhou 510070,
People’s Republic of China
State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology,
Dalian 116023, People’s Republic of China
Received 26 June 2003; accepted 23 October 2003
Abstract
Based on the time-dependent mild slope equation including the effect of wave energy dissipation, an expression for the energy dissipation factor is derived in conjunction with the
wave energy balance equation. The wave height of regular and irregular waves is numerically
simulated by use of the parabolic mild slope equation considering the energy dissipation due
to wave breaking. Comparison of numerical results with experimental data shows that the
expression for the energy dissipation factor is reasonable. The effects of the wave breaking
coefficient on the breaking point and the distribution of wave height after breaking are discussed through the study of a specific experimental topography.
# 2004 Elsevier Ltd. All rights reserved.
Keywords: Wave energy dissipation; Water waves; Breaking wave height; Numerical simulation
1. Introduction
As we know in the process of wave propagation from deep to shallow water,
wave energy dissipation will occur and four kinds of wave energy loss due to the
influence of air resistance on the free surface, internal energy loss, bottom friction
and sea bed percolation, may happen. In addition, if the wave breaks, much of the
wave energy will be dissipated. Wave breaking is a common and complex phenom
Corresponding author. Fax: +86-20-87057597.
E-mail address: zhengyh@ms.giec.ac.cn (Y.H. Zheng).
0029-8018/$ - see front matter # 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.oceaneng.2003.10.013
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Y.H. Zheng et al. / Ocean Engineering 31 (2004) 1083–1092
enon of wave transformation in coastal areas. The study of this phenomenon
shows theoretical and practical significance. It is important, for example, to analyze wave conditions, production of turbulence and energy dissipation in the surf
zone for a description of longshore currents, sedimentation and erosion in coastal
areas (Schäffer et al., 1993). Up to now, the variation of wave height due to wave
breaking is studied mostly by the wave energy balance equation (Battjes and Janssen, 1978; Thornton and Guza, 1983; Stive, 1984; Svendsen, 1984; Battjes and
Stive, 1985; Dally and Dean, 1985) or by the Boussinesq-type equations (Schäffer
et al., 1993; Madsen et al., 1997; Skotner and Apelt, 1999). The wave energy balance equation is effective when it is used to the wave transformation over a simple
topography. Its shortcomings are that it cannot effectively take the wave reflection
into account and that the wave angle, an unknown variable in the wave energy balance equation, should be determined by other equations. It is difficult to determine
the wave angle over a complicated topography, so the application of the wave
energy balance equation to a complicated topography encounters some limitations.
The Boussinesq-type equations are prospectively mathematical models for breaking
waves, but at present the application of it to the practical problems of wave breaking is relatively few due to its complex solution procedure, low computational
efficiency and limited computational areas compared with the mild slope equations.
The mild slope equations are effective wave models for the simulation of wave
transformation such as refraction, diffraction, and reflection. The solution of them
is much simpler than that of the Boussinesq-type equations and the wave angle
need not be determined by other equations, so it is widely used in practical engineering. However, application of the mild slope equations to the problems of wave
breaking is very rare till now, so an expression for the energy dissipation factor is
derived in conjunction with the wave energy balance equation and only the energy
losses due to wave breaking and bottom friction are concerned. The variation of
the wave heights of regular and irregular waves is numerically simulated by use of
the parabolic mild slope equation. Comparison of numerical results with experimental data shows that the method for breaking waves simulated by use of the
mild slope equation is successful. Furthermore, the effects of the wave breaking
coefficient on the breaking point and the distribution of wave height after breaking
are discussed through the study of a specific example of regular wave transformation.
2. Mathematical model
Most of the mathematical formulations and derivations presented in this section
can be found in Zheng et al. (2001), they are repeated here, however, for the completeness and convenience for further interpretation and discussion.
The time-dependent mild slope equation including the effect of the energy dissipation can be written as (Pan et al., 2000):
@2U
r ðCCg rUÞ þ ðx2 k2 CCg þ l ixF ÞU ¼ 0
@t2
ð1Þ
Y.H. Zheng et al. / Ocean Engineering 31 (2004) 1083–1092
1085
where C is the wave celerity; Cg is the wave group velocity; U is the wave potential
function; k is the wave number; x is the wave angular frequency; l is the nonlinear
factor determined by use of Kirby and Dalrymple’s nonlinear dispersion model
(1986), l ¼ gk½1 þ f1 ðkhÞe2 D
tanh½kh þ f2 ðkhÞe
gktanhðkhÞ; f1 ðkhÞ ¼ tanh5 ðkhÞ;
e ¼ kjAj; f2 ðkhÞ ¼ ½kh=sinhðkhÞ
4 ; D ¼ ½coshð4khÞ þ 8 2tanh2 ðkhÞ
=8sinh4 ðkhÞ; F
is the wave energy dissipation factor, which can be used to take all kinds of energy
loss mentioned above into account if one can give a simple expression for each
kind of energy loss and whose expression will be derived in the following.
The wave potential function U can be expressed by (Kirby, 1984):
U ¼ igReiW
ð2Þ
where g is the gravitational acceleration; R ¼ A=x; A is the real wave amplitude; W
¼ k x xt is the wave phase function; k is the wave number vector, jkj ¼ k; x is
the coordinate vector.
Substituting Eq. (2) into Eq. (1) gives:
@R
Rr kCCg 2CCg k rR xFR
@t
@2R
þ i 2 þ r ðCCg rRÞ lR ¼ 0
@ t
2x
ð3Þ
Considering kCCg ¼ xCg , one can obtain from the real part of the above equation:
2
@R
þ Rr Cg þ 2Cg rR þ FR ¼ 0
@t
ð4Þ
Multiplying R in both sides of the above equation and making some arrangements
yield:
@E
þ r ðECg Þ þ FE ¼ 0
@t
ð5Þ
where E ¼ qgH 2 =8 is wave energy; q is the density of sea water; H is the wave
height.
The wave energy balance equation including the effect of the energy dissipation
due to wave breaking and bottom friction can be expressed as:
@E
þ r ðECg Þ þ Df þ Db ¼ 0
@t
ð6Þ
where Df and Db are the energy dissipation due to the bottom friction and wave
breaking, respectively. Df and Db can be evaluated by use of the following equa-
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tions (Battjes and Stive, 1985; Roelvink, 1993; Van Rij and Wijnberg, 1996):
3
qfw
xH
6p sinhðkhÞ
8
<
xH 2
0:25qga
1
Db ¼
2p
:
0:25qgQb a1 fp Hb2
Df ¼
ð7Þ
regular waves
ð8Þ
irregular waves
where h is the still water depth; a1 is the correction coefficient, whose value
depends on the wave breaking type, sea bed slope and other factors. In this paper,
it is simply taken to be 1 for the geometry of bottom slope 1:40, and calculated by
use of the equation a1 ¼ 1 exp½ðH=chÞn where n ¼ 1, for bottom slope 1:100;
fw is the bottom friction coefficient; fp is the peak frequency of the energy spectrum
of irregular waves; Qb and Hb are the local fraction of breaking waves and the
maximum possible wave height associated with irregular wave breaking,
respectively.
Many researchers have contributed to the determination of fw (such as Jonsson,
1966; Swart and Fleming, 1980; O’Connor and Yoo, 1988; You et al., 1991;
Nielsen, 1992; Tanaka and Thu, 1994; Voulgaris et al., 1995). The simplest way is
to take fw as a constant and generally it equals 0.01–0.02 (Li, 1989). The empirical
equation
"
0:194 #
A
fw ¼ exp 5:977 þ 5:213
ð9Þ
kN
was used by many researchers in the past to consider the effect of bottom friction,
where A is the amplitude of water practice movement at the wave action and kN is
the Nikuradze number of sea bed roughness which can be found in many literatures. Here, the effect of bottom friction is not taken into account for lack of bottom roughness.
Qb and Hb can be determined by the following expressions (Battjes and Stive,
1985):
1 Qb
Hrms 2
¼
ð10Þ
lnQb
Hb
Hb ¼
0:88
tanhðckp h=0:88Þ
kp
ð11Þ
where Hrms is the root mean square (rms) wave height of irregular waves; kp is the
local wave number corresponding to the peak frequency fp of the energy spectrum
of irregular waves; c is the wave breaking coefficient, which can be determined by
the empirical equation
c ¼ 0:5 þ 0:4tanhð33S0 Þ
or by practical conditions; S0 is the wave steepness in deep water.
ð12Þ
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1087
Comparison Eq. (5) with Eq. (6) gives:
F ¼ ðDf þ Db Þ=E
ð13Þ
It should be noted that for regular waves Db in the above equation is equal to 0
before breaking. When the calculated wave height of a point is larger than or equal
to Hb, waves break at that point, and the wave height of the point is recalculated
by considering the effect of Db. For irregular waves, the effect of Db should be
included in the whole computation because Db is dependent on the local fraction of
irregular breaking waves.
Eq. (1) is an elliptic equation and the solution of it is relatively complex, so a
simplification is made, which yields the following parabolic mild slope equation
including the effect of the energy dissipation:
@A0
F
1 @kCCg 0
i
@
l
¼ i kk
A þ
2kCC
2C
2kCC
2kCC
@y
@x
@x
g
g
g
g
@A0
CCg
ð14Þ
@y
where A0 is the complex wave amplitude. The above equation will be used in the
following to numerically simulate the variation of the wave heights of regular and
irregular waves due to wave breaking.
3. Verification of mathematical model
3.1. Regular wave height associated with breaking
To verify the model proposed, here, the experimental results obtained by Zou et al.,
(1999) for regular waves is used. The experimental topography, which was used to
model the wave breaking on the mild slope beach, is shown in Fig. 1. The experimental and computational parameters are shown in Table 1, where h0 is the still
water depth at the wave maker boundary; H0 is the incident wave height; T is the
wave period; c is the wave breaking coefficient. Fig. 2 shows the experimental and
numerical wave height of four different cases. It can be seen that the experimental
data and the numerical results agree relatively well, which illustrates that the
Fig. 1. Experimental topography.
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Table 1
Experimental and computational parameters of regular waves
Case no.
Bottom slope
h0 (m)
H0 (cm)
T (s)
c
Case 1
Case 2
Case 3
Case 4
1:40
1:40
1:100
1:100
0.325
0.325
0.13
0.13
4.0
9.4
4.9
5.0
1.4
2.0
1.0
1.3
0.81
0.81
0.76
0.71
expression derived for the energy dissipation factor in the mild slope equation is correct.
3.2. Irregular wave height associated with breaking
Here, the rms wave height variation of irregular waves due to breaking over
another typical topography will be simulated. The experiment for irregular waves
associated with breaking over this topography was made by Stive (1985) and the
results from the experiment have been used previously by several researchers to
verify their mathematical models (Stive, 1985; Battjes and Stive, 1985). The experi-
Fig. 2. Calculated wave height and experimental data of regular waves. (u) Experimental data; (—)
numerical results. (a) Case 1; (b) case 2; (c) case 3; (d) case 4.
Y.H. Zheng et al. / Ocean Engineering 31 (2004) 1083–1092
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Table 2
Experimental and computational parameters of the irregular wave
Case no.
Bottom slope
h0 (m)
Hrms0 (m)
fp (Hz)
c
Case 5
1:40
0.70
0.136
0.633
0.84
mental and computational parameters are shown in Table 2, where Hrms0 is rms
wave height of irregular waves at the incident position, fp is the peak frequency of
the energy spectrum, and the meanings of the other parameters are the same as
those in Table 1. Fig. 3 gives the comparison of the calculated wave height by the
mathematical model with the experimental data. It can be seen that numerical
results are in good agreement with the experimental data, showing further that the
method proposed in the present paper is effective.
4. Discussion
The wave breaking coefficient c is a very important parameter which influences
to a large extent the calculation of the position of the breaking point and the
distribution of wave height after breaking. The values of c for case 1 to case 4
shown in Table 1 are chosen to make the calculated results agree relatively well
with the experimental data. To consider the effect of c on the wave heights, we also
use Eq. (12) to calculate c and the values are given in Table 3. The distribution of
wave heights for different cases by use of these values of c are shown in Fig. 4. It
can be seen that the calculated positions of breaking points of all cases agree well
Fig. 3. Numerical wave height and experimental data of irregular waves. (u) Experimental data; (—)
numerical results.
Table 3
Values of c by use of Eq. (12)
Case no.
Case 1
Case 2
Case 3
Case 4
c
0.71
0.79
0.87
0.83
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Fig. 4. Calculated wave height and experimental data of regular waves. (u) Experimental data; (—)
numerical results. (a) Case 1; (b) case 2; (c) case 3; (d) case 4.
with experimental data, but the calculated distribution of wave heights after breaking is in relatively poor accordance with experimental data compared with those
presented in Fig. 2.
5. Concluding remarks
The problem of wave breaking is studied in this paper by use of the mild slope
equation. The expression for the energy dissipation factor is derived in conjunction
with the wave energy balance equation. The wave heights of regular and irregular
waves are numerically simulated by use of the parabolic mild slope equation considering the effect of the energy dissipation due to wave breaking. Comparison of
numerical results with experimental data shows that the expression for the energy
dissipation factor is reasonable, and the mild slope equation including the effect of
the energy dissipation can give satisfactory results of breaking waves in coastal
areas.
Y.H. Zheng et al. / Ocean Engineering 31 (2004) 1083–1092
1091
Acknowledgements
This research is sponsored by the National Science Fund for Distinguished
Young Scholars under Grant No. 50125924 and by the National Natural Science
Foundation of China under Grant Nos. 10332050 and 50379001.
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