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Numerical modelling of deformation of multi-directional random waves over a varying topography
ARTICLE IN PRESS
Ocean Engineering 34 (2007) 337–342 www.elsevier.com/locate/oceaneng
Numerical modelling of deformation of multi-directional random waves over a varying topography Filipa Simo˜es Brito Ferreira Oliveira Laborato´rio Nacional de Engenharia Civil; Av. do Brasil 101, 1700-066 Lisboa, Portugal Received 17 February 2005; accepted 11 August 2005 Available online 18 April 2006
Abstract Two numerical formulations of the breaking phenomenon were implemented in a numerical model for random wave propagation based on the elliptic formulation of the mild-slope equation. The randomness of the wave field was simulated based on a spectral component method, in which the 3-D spectrum is discretised in components of equal energy. One of the breaking process formulations is based on the concept of breaking each independent spectral component. The other is based on the distribution of the local amount of energy dissipated through the independent spectral components. The model based on the concept of breaking each independent spectral component produces the best estimates of the wave field, when the numerical results are compared with laboratory data. r 2006 Elsevier Ltd. All rights reserved. Keywords: Random wave propagation; Random wave breaking; Mild-slope equation.
1. Introduction Despite the vast application of monochromatic wave propagation models to estimate the nearshore wave field, either to obtain extreme values for coastal design, harbour planning and beach protection works, or average values to estimate sediment transport and morphodynamical evolution, it is know that this approach leads to inaccurate results. Goda (1985) compared results obtained for monochromatic and irregular wave transformation over a spherical shoal and in the vicinity of a breakwater. The computation of random wave refraction has the effect of smoothing the spatial variation in the wave height due to the presence of various directional and frequency components. Diffraction is badly simulated when considering regular waves because wave height is underestimated in the sheltered area behind the breakwater, resulting in bad design and eventually failure of the structure. The diffraction process is particularly sensitive to the characteristics of the wave spectra, especially to the directional spreading of wave energy. Although these results were Tel.: +351 21844 3457; fax: +351 21844 3016.
E-mail address: [email protected]. 0029-8018/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2005.08.010
obtained from a theoretical approach, i.e., without data to verify them, they generally agree with observations done later by other authors like Vincent and Briggs (1989), who demonstrated, through laboratory experiments of random wave propagation over an elliptic shoal, that: the differences between monochromatic and random wave propagation models are more pronounced where bathymetric features lead to strong wave convergence; monochromatic waves overestimate the maximum wave amplification of random waves in convergence zones and underestimate the wave height in shadow zones up to a factor of two; and that wave height pattern is dependant on the incident frequency spectrum and on the breadth of the directional spreading function. Regarding the influence of the randomness of the sea on the wave breaking phenomenon, it is known that the assumption that the sea state is represented by a monochromatic statistic wave leads to erroneous estimates of the wave field, despite nonlinear wave interactions, that influence the breaking process, are not enough understood and much of the knowledge derived from investigating the breaking phenomenon is empirical. The objective of the present work was to implement two different formulations of the breaking phenomenon in
ARTICLE IN PRESS F.S.B.F. Oliveira / Ocean Engineering 34 (2007) 337–342
338
a numerical model for random wave propagation based on the elliptic formulation of the mild-slope equation. The model simulates the processes of random wave transformation due to the phenomena of shoaling, refraction, diffraction and breaking. The randomness of the wave field is approached based on the spectral component method, which consists on dividing the spectrum into a finite number of independent components in the frequency domain over a certain range of directions. The process of random wave breaking was implemented in the model through two different methodologies: one based on the concept of breaking each independent spectral component; the other based on the calculation of the total amount of energy dissipated at each point and its distribution through the independent spectral components. The verification of the two numerical approaches was performed based on the comparison of the model results with laboratory data.
The directional spectrum was discretised based on an equal energy components method because, when compared with a constant step method, is more efficient for a small number of directional components (Grassa, 1990). The spectral components characterisation parameters wave frequency, f i ¼ oi =2p, and wave direction were obtained from the discretisation of the directional spectrum, through: Z fi m0 Sðf Þ qf ¼ ði 0:5Þ (6) N f f min
2. Numerical modelling of the sea randomness
The phenomenon of wave dissipation by breaking has never been implemented successfully in a numerical model based on the mild-slope equation for nearshore random wave transformation, despite the attempt of Chawla et al. (1998) to introduce it, using a statistical breaking model, in a numerical model (REF/DIF S) based on the parabolic formulation of the mild-slope equation (Kirby, 1986). The present work proposes two different approaches to implement the phenomenon in a numerical model based on the elliptic formulation of the mild-slope equation, for which the sea surface elevation is assumed to be the sum of linear wave components. One approach is based on the concept of breaking each individual wave component. The other approach is based on the estimation of the amount of energy dissipated at each location and its distribution, based on a criterion, through the individual wave components. It is assumed, in both formulations, that dissipation does not interact with other processes affecting the wave evolution, including triad interactions, and that the total energy dissipation is distributed over the spectrum, in such a manner that it does not influence the local rate of evolution of the spectral shape. In the first random wave breaking formulation, the energy dissipated by each spectral wave component is estimated as !6 m0 ð2pf p Þ4 ð2pf i Þ4 Sdsij ¼ c0 Sðf i ; aj Þ, (8) g2 ð2pf p Þ3
The randomness of the sea is simulated through the assumption that the three dimensional (3-D) directional spectrum: Sðo; aÞ ¼ SðoÞGðo; aÞ,
(1)
where S(o) is the frequency spectrum and G(o, a) is the spreading function, is represented by a group of n linear and independent spectral components. The sea surface is represented by a linear superposition of harmonics progressing in various directions: Zðx; y; tÞ ¼
n X
ai cos½oi tfi ki ðx cos ai þ y sin ai Þ,
(2)
i¼1
where ai ¼ H i =2 is the wave amplitude of each individual wave component with wave height Hi, angular frequency oi, direction of propagation ai and wave number ki, determined from the linear dispersion relation for Stokes waves. Thus, the total wave energy, E, equal to the variance of the surface elevation, (m0), or zero-order moment of the frequency spectrum, considered for the range of integration in the azimuth set [p/2, p/2]: Z 1 Z p=2 E ¼ m0 ¼ Sðo; aÞ qo qa (3) 0
p=2
can be calculated as the summation of a finite number of harmonics, n ¼ N f N a , where Nf and Na are the number of frequency and directional spectra components, respectively, as follows: E¼
Nf X Na X
Eðoi ; aj Þ ¼
i¼1 j¼1
Nf X Na X
Sðoi ; aj ÞDoi Daj .
(4)
i¼1 j¼1
Each component is associated with a regular wave through: Sðoi ; aj ÞDoi Daj ¼
a2ij 2
.
(5)
for f i 2 ½f min ; f max and i ¼ 1; . . . ; N f and Z aij 1 Gðf i ; aÞ qa ¼ ðj 0:5Þ N a amin for aij 2 ½amin ; amax ;
i ¼ 1; . . . ; N f
and
(7) j ¼ 1; . . . ; N a .
3. Numerical approaches of the breaking process
where c0 ¼ 1:2 109 and fp is the peak frequency, as proposed by Hasselman and Hasselman (1983) for the energy dissipation for wave growth in shallow water. The dissipation coefficient of each component has a strong dependency on its frequency and on the spectral shape. The wave height of each component is estimated as a2ij 2
¼ ½Sðf i ; aj Þ Sdsij Df i Daj .
(9)
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339
In the second random wave breaking formulation, the spatial distribution of the energy dissipation is calculated based on the wave energy equation under the assumption that the breaking wave resembles a bore (hydraulic jump). The probability function for the lower, non-broken waves, is the same as in the absence of breaking, i.e., follows a Rayleigh distribution. It is assumed that broken waves carry on propagating with the maximum wave height, Hm, possible to occur at each depth, which is calculated as 0:88 gkp h Hm ¼ tanh , (10) kp 0:88
formulations based on the mild-slope equation have been suggested either to increase computational speed, like parabolic approximations as derived by Radder (1979), or to include other phenomena than wave refraction, wave diffraction and wave reflection, like energy dissipation by wave breaking, as derived by Booij (1981), and wave deformation by currents, as derived by Kirby (1984). In the numerical model applied, the original governing equation was
where kp is the wave number correspondent to the peak frequency, g is a Miche type parameter (Miche, 1944), here considered the constant value of 0.78. The energy dissipation is
where f is the wave velocity potential and the effective wave number, kc, is defined by
fp 3 1 rg H Q , (11) 4 h m b as derived by Battjes and Janssen (1978), and the probability of occurrence of broken waves is given by Qb, as follows: 1 Qb H rms 2 ¼ . (12) ln Qb Hm
¼B
The probability of occurrence of Hm, at each location, is estimated substituting pffiffiffipffiffiffiffiffiffi (13) H rms ¼ 8 m0 in Eq. (12), which results in a nonlinear equation solved by the Newton Raphson method. The energy dissipation factor, fD, per unit surface and unit time, is calculated at each location by H2 ¼ 2Qb f p m2 , E H where fD ¼
(14)
H2 (15) 8 and H is the significant wave height, H1/3. The empirical factor B, used to represent the bore conditions, was considered as unity to simulate a fully developed bore. The energy dissipated is distributed through the spectral components based on
E ¼ rg
f Dij ¼ f D
o4i Sðf i ; aj Þ . N Na Pf P 4 oi Sðf i ; aj Þ i¼1 j¼1
(16)
4. Numerical models The application of the mild-slope equation to simulate numerically monochromatic wave propagation over a varying bathymetry has been vastly experienced since its derivation by Berkhoff (1972), as described in Oliveira and Anastasiou (1998). A great number of mathematical
r2 f þ k2c f ¼ 0,
k2c ¼ k2
(17)
r2 ðCC g Þ1=2 ðCC g Þ1=2
,
(18)
where k is the local wave number, C ¼ w=k is the phase velocity and C g ¼ qo=qk is the group velocity. The governing equation of the model where the first breaking approach was implemented remains the original Eq. (17), however, in the model where the second breaking approach was implemented, the dissipation term iof Dij f
(19)
CC g was introduced in Eq. (18), as follows: k2c ¼ k2
r2 ðCC g Þ1=2 1=2
ðCC g Þ
þ
iof Dij CC g
.
(20)
Together with first-order radiation boundary conditions the governing equations were discretised by a finite differences technique resulting in a linear system of equations, solved by the Stabilised Bi-Conjugate Gradient Method (Bi-CGSTAB) (Oliveira and Anastasiou, 1998). 5. Verification of the numerical models The correctness and accuracy of both breaking approaches implemented in the model were verified with results from laboratory experiments performed by Vincent and Briggs (1989), for random wave propagation over an elliptic shoal laying in a region of constant water depth, in Fig. 1. The sea state was obtained using a TMA spectrum (Bouws et al., 1985) given by ( 4 fp 4 5 2 Sðf Þ ¼ ag ð2pÞ f exp 1:25 þ ðln gÞ f " #) ðf f p Þ2 exp Fðf ; hÞ, ð21Þ 2s2 f 2p where a is the Phillips constant; fp is the peak frequency, assumed as 1/1.05Tm for T m ¼ 1:3 s; g is the peak enhancement factor, assumed as 2 and 20, for the broad
ARTICLE IN PRESS F.S.B.F. Oliveira / Ocean Engineering 34 (2007) 337–342
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Fig. 1. Test case bottom geometry: elliptic shoal over constant depth.
and narrow frequency spectrum; s is the shape parameter, assumed as ( 0:07 if f of p ; 0:09 if f Xf p ; and F is a factor that incorporates the effect of depth, h, 8 2 for oh o1; > < 0:5ðoh Þ 2 F ¼ 1 0:5ð2 oh Þ for 1poh p2; > : 1 for oh 42; pffiffiffiffiffiffiffiffi for oh ¼ 2pf h=g. The lower and upper limits of the frequency spectrum considered were 0.5 and 2.0 Hz. The directional spreading function derived from a Fourier series is J 1 1X ðjsm Þ2 þ Gðf ; aÞ ¼ exp 2p p j¼1 2 cos jða am Þ,
ð22Þ
Table 1 Parameters of the tested spectra Case identification
Period, Tm (s)
Height, H0s (cm)
Phillips constant, a
Peak enhancement factor, g
Spreading parameter, sm
B5 N5
1.30 1.30
19.00 19.00
0.08650 0.02620
2 20
30 10
where sm is the spreading parameter, assumed as 101 and 301, for a narrow and broad spreading function, respectively; am is the mean wave direction, assumed as 01; and J is an arbitrary number of harmonics chosen to represent the Fourier series, assumed as 20. The azimuth is within the range 451oao451 and 901oao901, for the narrow and broad spreading function, respectively. Two types of spectrum were simulated (Table 1): with broad frequency spreading and broad directional spreading, identified as B5; and with narrow frequency spreading and narrow directional spreading, identified as N5. Eight
ARTICLE IN PRESS F.S.B.F. Oliveira / Ocean Engineering 34 (2007) 337–342
Normalised wave height
(a)
Normalised wave height
3.00 2.50 2.00 1.50 1.00 0.50 0.00 70
1st formulation 2nd formulation Lab. data
80 90 100 110 120 130 j co-ordinate along section 4 N5 1 st formulation 2 nd formulation
80 90 100 110 120 130 j co-ordinate along section 5
The present study presents two numerical models developed to simulate the process of multi-directional random wave propagation in nearshore regions, where the main phenomena of wave deformation are shoaling, refraction, diffraction and breaking. The randomness of the wave field was simulated through an equal energy spectral components method. Two breaking formulations, in which the energy dissipated by each spectral component is highly dependent on frequency, were established, implemented and experimented in the elliptic formulation of the mild-slope equation. The breaking formulation based on the concept of breaking each individual wave
Normalised wave height
80 90 100 110 120 130 j co-ordinate along section 3 N5
3.00 2.50 2.00 1.50 1.00 0.50 0.00 70
(b)
(c)
1 st formulation 2 nd formulation
6. Conclusions and suggestions
3.00 2.50 2.00 1.50 1.00 0.50 0.00 102
(d) Normalised wave height
N5 3.00 2.50 2.00 1.50 1.00 0.50 0.00 70
of energy convergence immediately after the shoal (along section 7), particularly for case B5, in which the laboratory results show a good agreement outside the area of energy convergence (section 4). A probable cause for this result is that Battjes and Janssen (1978) model, applied in the second breaking formulation, was derived for waves breaking when approaching a beach (slope) and not for the case of a shoal, with a fast varying topography. The amount of energy dissipated was estimated based on onedimensional breaking waves, i.e., did not take into account directional effects.
3.00 2.50 2.00 1.50 1.00 0.50 0.00 102
(e) Normalised wave height
Normalised wave height
frequency components, N f ¼ 8, and eight directional components, N a ¼ 8, were used to discretise the frequency spectrum and the directional spreading function. A total number of 64 components were used to simulate the multidirectional random waves. The results, in Fig. 2, show that the numerical model describes adequately the effects of frequency and directional spreading. Comparing the normalised wave height for both cases, N5 and B5, in the six sections, 3–8, it can be observed that wider frequency and directional spreading have the effect of reducing the disturbance on the wave deformation process, caused by the presence of the shoal. The energy gradient in the x and y directions is lower for case B5 than for case N5. Nevertheless, both models (each corresponding to a breaking formulation) overpredict the wave energy immediately after the shoal, in the area of energy convergence, due to their nature of being based on a linear superposition of monochromatic wave components, therefore unable to predict wave–wave interactions. These interactions lead do the growth of higher harmonics in sea, and become more pronounced with increased nonlinearity. In fact, studies on frequency spectrum have shown the presence of secondary peaks formed due to these higher harmonics that are not reproduced by the models. The first breaking formulation is more efficient than the second breaking formulation in the reduction of the peak
341
(f)
N5 1 st formulation 2 nd formulation
112 122 132 142 152 162 i co-ordinate along section 6 N5 1 st formulation 2 nd formulation
112 122 132 142 152 162 i co-ordinate along section 7 N5
3.00 2.50 2.00 1.50 1.00 0.50 0.00 102
1 st formulation 2 nd formulation
112 122 132 142 152 162 i co-ordinate along section 8
Fig. 2. Computational results along Sections 3–8, for case N5, in (a)–(f), and case B5, in (g)–(l).
ARTICLE IN PRESS
B5 3.00 2.50 2.00 1.50 1.00 0.50 0.00 70
Normalised wave height
(h)
(i)
80 90 100 110 120 130 j co-ordinate along section 3
3.00 2.50 2.00 1.50 1.00 0.50 0.00 70
1st formulation 2nd formulation Lab Data
80 90 100 110 120 130 j co-ordinate along section 4 B5 1 st formulation 2 nd formulation
80 90 100 110 120 130 j co-ordinate along section 5
3.00 2.50 2.00 1.50 1.00 0.50 0.00 102
(j) Normalised wave height
Normalised wave height
(g)
1 st formulation 2 nd formulation
3.00 2.50 2.00 1.50 1.00 0.50 0.00 102
(k)
B5 1 st formulation 2nd formulation
112 122 132 142 152 162 i co-ordinate along section 6 B5 1st formulation 2nd formulation
112 122 132 142 152 162 i co-ordinate along section 7
Normalised wave height
Normalised wave height
B5 3.00 2.50 2.00 1.50 1.00 0.50 0.00 70
Normalised wave height
F.S.B.F. Oliveira / Ocean Engineering 34 (2007) 337–342
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B5 3.00 1 st formulation 2.50 2 nd formulation 2.00 1.50 1.00 0.50 0.00 102 112 122 132 142 152 162 ico-ordinate along section 8 (l)
Fig. 2. (Continued)
component proved to be more efficient than the breaking formulation based on distributing the total energy dissipated at each location through the individual wave components. However, none of the models computes accurately the distribution of the spectral energy in convergence areas because nonlinear cross spectral transfers of energy and phase modifications are not considered. The results obtained suggest that two improvements should be tested in the future: one is the introduction of directional effects on the energy dissipation coefficient; the other is the introduction of interactions between the harmonic components. References Battjes, J.A., Janssen, J.P.F.M., 1978. Energy loss and set-up due to breaking of random waves. In: Proceedings of the 16th ICCE, vol. 1. Hamburg, ASCE, pp. 569–587. Berkhoff, J.C.W., 1972. Computation of combined refraction-diffraction. In: Proceedings of the 13th ICCE. Vancouver, ASCE, pp. 471–490. Booij, N., 1981. Gravity waves on water with non-uniform depth and current. Ph.D. Dissertation, Technical University of Delft, Delft, Netherlands. Bouws, E., Gunther, H., Rosenthal, W., Vincent, C., 1985. Similarity of the wind wave spectrum in finite depth water. Journal of Geophysical Research 90, 975–986.
Chawla, A., Ozkan-Haller, H.T., Kirby, J.T., 1998. Spectral model for wave transformation and breaking over irregular bathymetry. Journal of Waterway, Port, Coastal and Ocean Engineering 124 (4), 189–198. Goda, Y., 1985. Random Seas and Design of Maritime Structures. University of Tokyo Press, Tokyo, Japan. Grassa, J.M., 1990. Directional random wave propagation on beaches. In: Proceedings of the 22nd ICCE, vol. 1. The Netherlands, ASCE, pp. 798–811. Hasselman, S., Hasselman, K., 1983. Integration of the spectral transport equation with exact and parametrical computational of the nonlinear energy transfer. In: Proceedings of the Symposium on Wave Dynamics and Ratio Probing of the Ocean Surface. Miami, Plenum Press. Kirby, J.T., 1984. A note on linear surface wave current interaction over slowly varying topography. Journal of Geophysical Research 89 (C1), 745–747. Kirby, J.T., 1986. Higher-order approximations in the parabolic equation for water waves. Journal of Geophysical Research 91, 933–952. Miche, M., 1944. Mouvements ondulatoires de la mer en profonder constant on decroissante. Annales des Ponts et Chausses. Oliveira, F.S.B.F., Anastasiou, K., 1998. An efficient computational model for water wave propagation in coastal regions. Applied Ocean Research 20 (5), 263–271. Radder, A.C., 1979. On the parabolic equation method for water wave propagation. Journal of Fluid Mechanics 95 (1), 159–176. Vincent, C.L., Briggs, M.J., 1989. Refraction-diffraction of irregular waves over a mound. Journal of Waterway, Port, Coastal and Ocean Engineering 15 (2), 269–284.
Ocean Engineering 34 (2007) 337–342 www.elsevier.com/locate/oceaneng
Numerical modelling of deformation of multi-directional random waves over a varying topography Filipa Simo˜es Brito Ferreira Oliveira Laborato´rio Nacional de Engenharia Civil; Av. do Brasil 101, 1700-066 Lisboa, Portugal Received 17 February 2005; accepted 11 August 2005 Available online 18 April 2006
Abstract Two numerical formulations of the breaking phenomenon were implemented in a numerical model for random wave propagation based on the elliptic formulation of the mild-slope equation. The randomness of the wave field was simulated based on a spectral component method, in which the 3-D spectrum is discretised in components of equal energy. One of the breaking process formulations is based on the concept of breaking each independent spectral component. The other is based on the distribution of the local amount of energy dissipated through the independent spectral components. The model based on the concept of breaking each independent spectral component produces the best estimates of the wave field, when the numerical results are compared with laboratory data. r 2006 Elsevier Ltd. All rights reserved. Keywords: Random wave propagation; Random wave breaking; Mild-slope equation.
1. Introduction Despite the vast application of monochromatic wave propagation models to estimate the nearshore wave field, either to obtain extreme values for coastal design, harbour planning and beach protection works, or average values to estimate sediment transport and morphodynamical evolution, it is know that this approach leads to inaccurate results. Goda (1985) compared results obtained for monochromatic and irregular wave transformation over a spherical shoal and in the vicinity of a breakwater. The computation of random wave refraction has the effect of smoothing the spatial variation in the wave height due to the presence of various directional and frequency components. Diffraction is badly simulated when considering regular waves because wave height is underestimated in the sheltered area behind the breakwater, resulting in bad design and eventually failure of the structure. The diffraction process is particularly sensitive to the characteristics of the wave spectra, especially to the directional spreading of wave energy. Although these results were Tel.: +351 21844 3457; fax: +351 21844 3016.
E-mail address: [email protected]. 0029-8018/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2005.08.010
obtained from a theoretical approach, i.e., without data to verify them, they generally agree with observations done later by other authors like Vincent and Briggs (1989), who demonstrated, through laboratory experiments of random wave propagation over an elliptic shoal, that: the differences between monochromatic and random wave propagation models are more pronounced where bathymetric features lead to strong wave convergence; monochromatic waves overestimate the maximum wave amplification of random waves in convergence zones and underestimate the wave height in shadow zones up to a factor of two; and that wave height pattern is dependant on the incident frequency spectrum and on the breadth of the directional spreading function. Regarding the influence of the randomness of the sea on the wave breaking phenomenon, it is known that the assumption that the sea state is represented by a monochromatic statistic wave leads to erroneous estimates of the wave field, despite nonlinear wave interactions, that influence the breaking process, are not enough understood and much of the knowledge derived from investigating the breaking phenomenon is empirical. The objective of the present work was to implement two different formulations of the breaking phenomenon in
ARTICLE IN PRESS F.S.B.F. Oliveira / Ocean Engineering 34 (2007) 337–342
338
a numerical model for random wave propagation based on the elliptic formulation of the mild-slope equation. The model simulates the processes of random wave transformation due to the phenomena of shoaling, refraction, diffraction and breaking. The randomness of the wave field is approached based on the spectral component method, which consists on dividing the spectrum into a finite number of independent components in the frequency domain over a certain range of directions. The process of random wave breaking was implemented in the model through two different methodologies: one based on the concept of breaking each independent spectral component; the other based on the calculation of the total amount of energy dissipated at each point and its distribution through the independent spectral components. The verification of the two numerical approaches was performed based on the comparison of the model results with laboratory data.
The directional spectrum was discretised based on an equal energy components method because, when compared with a constant step method, is more efficient for a small number of directional components (Grassa, 1990). The spectral components characterisation parameters wave frequency, f i ¼ oi =2p, and wave direction were obtained from the discretisation of the directional spectrum, through: Z fi m0 Sðf Þ qf ¼ ði 0:5Þ (6) N f f min
2. Numerical modelling of the sea randomness
The phenomenon of wave dissipation by breaking has never been implemented successfully in a numerical model based on the mild-slope equation for nearshore random wave transformation, despite the attempt of Chawla et al. (1998) to introduce it, using a statistical breaking model, in a numerical model (REF/DIF S) based on the parabolic formulation of the mild-slope equation (Kirby, 1986). The present work proposes two different approaches to implement the phenomenon in a numerical model based on the elliptic formulation of the mild-slope equation, for which the sea surface elevation is assumed to be the sum of linear wave components. One approach is based on the concept of breaking each individual wave component. The other approach is based on the estimation of the amount of energy dissipated at each location and its distribution, based on a criterion, through the individual wave components. It is assumed, in both formulations, that dissipation does not interact with other processes affecting the wave evolution, including triad interactions, and that the total energy dissipation is distributed over the spectrum, in such a manner that it does not influence the local rate of evolution of the spectral shape. In the first random wave breaking formulation, the energy dissipated by each spectral wave component is estimated as !6 m0 ð2pf p Þ4 ð2pf i Þ4 Sdsij ¼ c0 Sðf i ; aj Þ, (8) g2 ð2pf p Þ3
The randomness of the sea is simulated through the assumption that the three dimensional (3-D) directional spectrum: Sðo; aÞ ¼ SðoÞGðo; aÞ,
(1)
where S(o) is the frequency spectrum and G(o, a) is the spreading function, is represented by a group of n linear and independent spectral components. The sea surface is represented by a linear superposition of harmonics progressing in various directions: Zðx; y; tÞ ¼
n X
ai cos½oi tfi ki ðx cos ai þ y sin ai Þ,
(2)
i¼1
where ai ¼ H i =2 is the wave amplitude of each individual wave component with wave height Hi, angular frequency oi, direction of propagation ai and wave number ki, determined from the linear dispersion relation for Stokes waves. Thus, the total wave energy, E, equal to the variance of the surface elevation, (m0), or zero-order moment of the frequency spectrum, considered for the range of integration in the azimuth set [p/2, p/2]: Z 1 Z p=2 E ¼ m0 ¼ Sðo; aÞ qo qa (3) 0
p=2
can be calculated as the summation of a finite number of harmonics, n ¼ N f N a , where Nf and Na are the number of frequency and directional spectra components, respectively, as follows: E¼
Nf X Na X
Eðoi ; aj Þ ¼
i¼1 j¼1
Nf X Na X
Sðoi ; aj ÞDoi Daj .
(4)
i¼1 j¼1
Each component is associated with a regular wave through: Sðoi ; aj ÞDoi Daj ¼
a2ij 2
.
(5)
for f i 2 ½f min ; f max and i ¼ 1; . . . ; N f and Z aij 1 Gðf i ; aÞ qa ¼ ðj 0:5Þ N a amin for aij 2 ½amin ; amax ;
i ¼ 1; . . . ; N f
and
(7) j ¼ 1; . . . ; N a .
3. Numerical approaches of the breaking process
where c0 ¼ 1:2 109 and fp is the peak frequency, as proposed by Hasselman and Hasselman (1983) for the energy dissipation for wave growth in shallow water. The dissipation coefficient of each component has a strong dependency on its frequency and on the spectral shape. The wave height of each component is estimated as a2ij 2
¼ ½Sðf i ; aj Þ Sdsij Df i Daj .
(9)
ARTICLE IN PRESS F.S.B.F. Oliveira / Ocean Engineering 34 (2007) 337–342
339
In the second random wave breaking formulation, the spatial distribution of the energy dissipation is calculated based on the wave energy equation under the assumption that the breaking wave resembles a bore (hydraulic jump). The probability function for the lower, non-broken waves, is the same as in the absence of breaking, i.e., follows a Rayleigh distribution. It is assumed that broken waves carry on propagating with the maximum wave height, Hm, possible to occur at each depth, which is calculated as 0:88 gkp h Hm ¼ tanh , (10) kp 0:88
formulations based on the mild-slope equation have been suggested either to increase computational speed, like parabolic approximations as derived by Radder (1979), or to include other phenomena than wave refraction, wave diffraction and wave reflection, like energy dissipation by wave breaking, as derived by Booij (1981), and wave deformation by currents, as derived by Kirby (1984). In the numerical model applied, the original governing equation was
where kp is the wave number correspondent to the peak frequency, g is a Miche type parameter (Miche, 1944), here considered the constant value of 0.78. The energy dissipation is
where f is the wave velocity potential and the effective wave number, kc, is defined by
fp 3 1 rg H Q , (11) 4 h m b as derived by Battjes and Janssen (1978), and the probability of occurrence of broken waves is given by Qb, as follows: 1 Qb H rms 2 ¼ . (12) ln Qb Hm
¼B
The probability of occurrence of Hm, at each location, is estimated substituting pffiffiffipffiffiffiffiffiffi (13) H rms ¼ 8 m0 in Eq. (12), which results in a nonlinear equation solved by the Newton Raphson method. The energy dissipation factor, fD, per unit surface and unit time, is calculated at each location by H2 ¼ 2Qb f p m2 , E H where fD ¼
(14)
H2 (15) 8 and H is the significant wave height, H1/3. The empirical factor B, used to represent the bore conditions, was considered as unity to simulate a fully developed bore. The energy dissipated is distributed through the spectral components based on
E ¼ rg
f Dij ¼ f D
o4i Sðf i ; aj Þ . N Na Pf P 4 oi Sðf i ; aj Þ i¼1 j¼1
(16)
4. Numerical models The application of the mild-slope equation to simulate numerically monochromatic wave propagation over a varying bathymetry has been vastly experienced since its derivation by Berkhoff (1972), as described in Oliveira and Anastasiou (1998). A great number of mathematical
r2 f þ k2c f ¼ 0,
k2c ¼ k2
(17)
r2 ðCC g Þ1=2 ðCC g Þ1=2
,
(18)
where k is the local wave number, C ¼ w=k is the phase velocity and C g ¼ qo=qk is the group velocity. The governing equation of the model where the first breaking approach was implemented remains the original Eq. (17), however, in the model where the second breaking approach was implemented, the dissipation term iof Dij f
(19)
CC g was introduced in Eq. (18), as follows: k2c ¼ k2
r2 ðCC g Þ1=2 1=2
ðCC g Þ
þ
iof Dij CC g
.
(20)
Together with first-order radiation boundary conditions the governing equations were discretised by a finite differences technique resulting in a linear system of equations, solved by the Stabilised Bi-Conjugate Gradient Method (Bi-CGSTAB) (Oliveira and Anastasiou, 1998). 5. Verification of the numerical models The correctness and accuracy of both breaking approaches implemented in the model were verified with results from laboratory experiments performed by Vincent and Briggs (1989), for random wave propagation over an elliptic shoal laying in a region of constant water depth, in Fig. 1. The sea state was obtained using a TMA spectrum (Bouws et al., 1985) given by ( 4 fp 4 5 2 Sðf Þ ¼ ag ð2pÞ f exp 1:25 þ ðln gÞ f " #) ðf f p Þ2 exp Fðf ; hÞ, ð21Þ 2s2 f 2p where a is the Phillips constant; fp is the peak frequency, assumed as 1/1.05Tm for T m ¼ 1:3 s; g is the peak enhancement factor, assumed as 2 and 20, for the broad
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340
Fig. 1. Test case bottom geometry: elliptic shoal over constant depth.
and narrow frequency spectrum; s is the shape parameter, assumed as ( 0:07 if f of p ; 0:09 if f Xf p ; and F is a factor that incorporates the effect of depth, h, 8 2 for oh o1; > < 0:5ðoh Þ 2 F ¼ 1 0:5ð2 oh Þ for 1poh p2; > : 1 for oh 42; pffiffiffiffiffiffiffiffi for oh ¼ 2pf h=g. The lower and upper limits of the frequency spectrum considered were 0.5 and 2.0 Hz. The directional spreading function derived from a Fourier series is J 1 1X ðjsm Þ2 þ Gðf ; aÞ ¼ exp 2p p j¼1 2 cos jða am Þ,
ð22Þ
Table 1 Parameters of the tested spectra Case identification
Period, Tm (s)
Height, H0s (cm)
Phillips constant, a
Peak enhancement factor, g
Spreading parameter, sm
B5 N5
1.30 1.30
19.00 19.00
0.08650 0.02620
2 20
30 10
where sm is the spreading parameter, assumed as 101 and 301, for a narrow and broad spreading function, respectively; am is the mean wave direction, assumed as 01; and J is an arbitrary number of harmonics chosen to represent the Fourier series, assumed as 20. The azimuth is within the range 451oao451 and 901oao901, for the narrow and broad spreading function, respectively. Two types of spectrum were simulated (Table 1): with broad frequency spreading and broad directional spreading, identified as B5; and with narrow frequency spreading and narrow directional spreading, identified as N5. Eight
ARTICLE IN PRESS F.S.B.F. Oliveira / Ocean Engineering 34 (2007) 337–342
Normalised wave height
(a)
Normalised wave height
3.00 2.50 2.00 1.50 1.00 0.50 0.00 70
1st formulation 2nd formulation Lab. data
80 90 100 110 120 130 j co-ordinate along section 4 N5 1 st formulation 2 nd formulation
80 90 100 110 120 130 j co-ordinate along section 5
The present study presents two numerical models developed to simulate the process of multi-directional random wave propagation in nearshore regions, where the main phenomena of wave deformation are shoaling, refraction, diffraction and breaking. The randomness of the wave field was simulated through an equal energy spectral components method. Two breaking formulations, in which the energy dissipated by each spectral component is highly dependent on frequency, were established, implemented and experimented in the elliptic formulation of the mild-slope equation. The breaking formulation based on the concept of breaking each individual wave
Normalised wave height
80 90 100 110 120 130 j co-ordinate along section 3 N5
3.00 2.50 2.00 1.50 1.00 0.50 0.00 70
(b)
(c)
1 st formulation 2 nd formulation
6. Conclusions and suggestions
3.00 2.50 2.00 1.50 1.00 0.50 0.00 102
(d) Normalised wave height
N5 3.00 2.50 2.00 1.50 1.00 0.50 0.00 70
of energy convergence immediately after the shoal (along section 7), particularly for case B5, in which the laboratory results show a good agreement outside the area of energy convergence (section 4). A probable cause for this result is that Battjes and Janssen (1978) model, applied in the second breaking formulation, was derived for waves breaking when approaching a beach (slope) and not for the case of a shoal, with a fast varying topography. The amount of energy dissipated was estimated based on onedimensional breaking waves, i.e., did not take into account directional effects.
3.00 2.50 2.00 1.50 1.00 0.50 0.00 102
(e) Normalised wave height
Normalised wave height
frequency components, N f ¼ 8, and eight directional components, N a ¼ 8, were used to discretise the frequency spectrum and the directional spreading function. A total number of 64 components were used to simulate the multidirectional random waves. The results, in Fig. 2, show that the numerical model describes adequately the effects of frequency and directional spreading. Comparing the normalised wave height for both cases, N5 and B5, in the six sections, 3–8, it can be observed that wider frequency and directional spreading have the effect of reducing the disturbance on the wave deformation process, caused by the presence of the shoal. The energy gradient in the x and y directions is lower for case B5 than for case N5. Nevertheless, both models (each corresponding to a breaking formulation) overpredict the wave energy immediately after the shoal, in the area of energy convergence, due to their nature of being based on a linear superposition of monochromatic wave components, therefore unable to predict wave–wave interactions. These interactions lead do the growth of higher harmonics in sea, and become more pronounced with increased nonlinearity. In fact, studies on frequency spectrum have shown the presence of secondary peaks formed due to these higher harmonics that are not reproduced by the models. The first breaking formulation is more efficient than the second breaking formulation in the reduction of the peak
341
(f)
N5 1 st formulation 2 nd formulation
112 122 132 142 152 162 i co-ordinate along section 6 N5 1 st formulation 2 nd formulation
112 122 132 142 152 162 i co-ordinate along section 7 N5
3.00 2.50 2.00 1.50 1.00 0.50 0.00 102
1 st formulation 2 nd formulation
112 122 132 142 152 162 i co-ordinate along section 8
Fig. 2. Computational results along Sections 3–8, for case N5, in (a)–(f), and case B5, in (g)–(l).
ARTICLE IN PRESS
B5 3.00 2.50 2.00 1.50 1.00 0.50 0.00 70
Normalised wave height
(h)
(i)
80 90 100 110 120 130 j co-ordinate along section 3
3.00 2.50 2.00 1.50 1.00 0.50 0.00 70
1st formulation 2nd formulation Lab Data
80 90 100 110 120 130 j co-ordinate along section 4 B5 1 st formulation 2 nd formulation
80 90 100 110 120 130 j co-ordinate along section 5
3.00 2.50 2.00 1.50 1.00 0.50 0.00 102
(j) Normalised wave height
Normalised wave height
(g)
1 st formulation 2 nd formulation
3.00 2.50 2.00 1.50 1.00 0.50 0.00 102
(k)
B5 1 st formulation 2nd formulation
112 122 132 142 152 162 i co-ordinate along section 6 B5 1st formulation 2nd formulation
112 122 132 142 152 162 i co-ordinate along section 7
Normalised wave height
Normalised wave height
B5 3.00 2.50 2.00 1.50 1.00 0.50 0.00 70
Normalised wave height
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B5 3.00 1 st formulation 2.50 2 nd formulation 2.00 1.50 1.00 0.50 0.00 102 112 122 132 142 152 162 ico-ordinate along section 8 (l)
Fig. 2. (Continued)
component proved to be more efficient than the breaking formulation based on distributing the total energy dissipated at each location through the individual wave components. However, none of the models computes accurately the distribution of the spectral energy in convergence areas because nonlinear cross spectral transfers of energy and phase modifications are not considered. The results obtained suggest that two improvements should be tested in the future: one is the introduction of directional effects on the energy dissipation coefficient; the other is the introduction of interactions between the harmonic components. References Battjes, J.A., Janssen, J.P.F.M., 1978. Energy loss and set-up due to breaking of random waves. In: Proceedings of the 16th ICCE, vol. 1. Hamburg, ASCE, pp. 569–587. Berkhoff, J.C.W., 1972. Computation of combined refraction-diffraction. In: Proceedings of the 13th ICCE. Vancouver, ASCE, pp. 471–490. Booij, N., 1981. Gravity waves on water with non-uniform depth and current. Ph.D. Dissertation, Technical University of Delft, Delft, Netherlands. Bouws, E., Gunther, H., Rosenthal, W., Vincent, C., 1985. Similarity of the wind wave spectrum in finite depth water. Journal of Geophysical Research 90, 975–986.
Chawla, A., Ozkan-Haller, H.T., Kirby, J.T., 1998. Spectral model for wave transformation and breaking over irregular bathymetry. Journal of Waterway, Port, Coastal and Ocean Engineering 124 (4), 189–198. Goda, Y., 1985. Random Seas and Design of Maritime Structures. University of Tokyo Press, Tokyo, Japan. Grassa, J.M., 1990. Directional random wave propagation on beaches. In: Proceedings of the 22nd ICCE, vol. 1. The Netherlands, ASCE, pp. 798–811. Hasselman, S., Hasselman, K., 1983. Integration of the spectral transport equation with exact and parametrical computational of the nonlinear energy transfer. In: Proceedings of the Symposium on Wave Dynamics and Ratio Probing of the Ocean Surface. Miami, Plenum Press. Kirby, J.T., 1984. A note on linear surface wave current interaction over slowly varying topography. Journal of Geophysical Research 89 (C1), 745–747. Kirby, J.T., 1986. Higher-order approximations in the parabolic equation for water waves. Journal of Geophysical Research 91, 933–952. Miche, M., 1944. Mouvements ondulatoires de la mer en profonder constant on decroissante. Annales des Ponts et Chausses. Oliveira, F.S.B.F., Anastasiou, K., 1998. An efficient computational model for water wave propagation in coastal regions. Applied Ocean Research 20 (5), 263–271. Radder, A.C., 1979. On the parabolic equation method for water wave propagation. Journal of Fluid Mechanics 95 (1), 159–176. Vincent, C.L., Briggs, M.J., 1989. Refraction-diffraction of irregular waves over a mound. Journal of Waterway, Port, Coastal and Ocean Engineering 15 (2), 269–284.