Applicability of Wave Models in Shallow Coastal Waters(2)

Dbr

Hrms γ2 1 Qbγ1α2

= Qb ρ g Hmax , = , Hmax=tanh k d

ln Qbk4 Hmax γ1

2

(12)

in which D br is the mean dissipation rate per area, Q b is the fraction of breaking waves, ρ the

water density, H rms is the root mean square, H max is the maximum wave height and α, γ 1, γ 2 areadjustable coefficients.

In HISWA and SWAN the total dissipation rate D is assigned to the dissipation rate for eachspectral component (Booij et al., 1985, and Ris , 1997):HISWA:SWAN:

Sds,br(x,y,θ)= Dbr Sds,br(x,y,σ,θ)= Dbr

E(x,y,θ)Etot(x,y)Etot(x,y)

, where Etot(x,y)=∫E(x,y,θ)dθ , where Etot(x,y)=∫∫E(x,y,σ,θ)dσdθ

(13)(14)

E(x,y,σ,θ)

In MIKE 21 EMS the dissipation rate is used to calculate the factor of energydissipationeb∝Dbr/E.

2.4 Generation of Wave Energy due to Wind

The wind input of wave energy Sin(x,y,σ,θ) is calculated within SWAN using the first-generationmode (Ris, 1997):Sin(x,y,σ,θ)

ρ =A+B E(x,y,σ,θ), where B=max0,

4 ρw

2 CD,10 U10

28cos (15)θ θw) 1 σ( c

and A=

0.0015

2 π g2

(

2

CD,10 U10 max0,cos(θ θw)

(

))

4

4σ (16)exp

. g 2 π 013 2 CU 28 1010D,

in which C D,10 is the drag coefficient, U 10 is the wind velocity, ρ A and ρ W are the air and water

density and θW is the wind direction.

3. NUMERICAL SIMULATIONS COMPARED TO PHYSICAL MODELING

The experimental data on which the model test presented in this paper is based were collectedduring an investigation on the influence of summer dikes, i.e. submerged dikes, on wavespropagating along a foreland.

For this reason a model of a foreland with summer dike of typical height and width was build atprototype scale in the large wave tank (324 m length, 5 m width and 7 m depth) of theFORSCHUNGSZENTRUM KÜSTE. See for details at Mai (1998). The graph at the bottom ofFigure 1 shows a cross section of the foreland profile. The height of the foreland in the wave tankwas approximately 1.5 m corresponding to a height of 2.0 m above German datum. The crestheight of the summer dike was 3.0 m or 3.5 m above German datum respectively. Its crest widthwas 3.0 m. The slope was approximately 1:7. The foreland length in seeward direction of thesummer dike is approximately 40 m. The polder length in landward direction was about 70 m. Thesummer dike consists of a sand core protected from erosion by a concrete filled geotextile mattresssimulating a clay cover with grass as applied in nature while the foreland was build of sandwithout any cover. The boundary conditions were varied in a range typical at the German North-Sea coast (water-level: 3.5 m to 4.5 m, i.e. 4.0 m to 5.0 m above German datum, significant waveheight of incoming waves: 0.6 m to 1.2 m, peak period of incoming waves: 3.5 s to 8.0 s). Thewave parameter were measured at 27 locations along the flume.

The numerical models HISWA, SWAN and MIKE 21 EMS were applied for the same bathymetryand boundary conditions. The three graphs at the top of Figure 1 show the significant wave heightcalculated by the numerical models in comparison to the wave height determined experimentally.Both physical and numerical modelling show a large decrease of wave height at the summer dikewhile the decline in the polder area is only very small. An increase of the significant wave heightdue to shoaling above the foreland can also be found in all models but the amount of increaseseems to be underestimated by the numerical models HISWA and SWAN compared to theexperiment. This may be contributed to the reflection of waves at the summer dike. The results ofMIKE 21 EMS shown in figure 1 were averaged in wave direction using a running filter. Theaveraging is necessary since MIKE 21 EMS calculates the wave propagation only for regularwaves. This leads to a standing wave directly in front of the summer dike which is not reasonable

for irregular waves. The filter width is chosen equal to the wave length of a regular wave with aperiod T p.

significant wave height H [m]

S

1.25

1.00

0.75

0.50

01.25

25

50

75

100

125

150

175

200

significant wave height H S [m]

1.00

0.75

0.50

25

50

75

100

125

150

175

200

significant wave height H S

[m]

1.25

1.00

0.75

0.50

05.004.003.002.001.000.00

25

50

75

100

125

150

175

200

125

150

water

level z [m]

175

200

75100

x [m]

Figure 1

Comparison of the measured significant height of waves propagating in a wave tankwith results of the models HISWA, SWAN, MIKE 21 EMS.

The parameters of the numerical models describing bottom friction and wave breaking wereadjusted in order to give the best agreement of experimentally and numerically derivedtransmission coefficients of foreland and summer dike. The transmission coefficient is defined as

trans

the quotient of the transmitted significant wave height Hs(x=185m) and the incoming significant

wave height Hins(x=50m):

cT=

trans

HS(x=185m)inHS(x=50m)

(15)

The best agreement was achieved using the model parameter listed in table 1.Dissipation

process

Wave breaking

HISWA

α

= 0.95

γ

1

= 0.85

γ

2

= 0.95

C

fw

= 0.01 (using eq.

8)

Numerical model

SWAN

α

= 1.45

γ

= 0.75

KN=0.02 (using eq.9)

MIKE 21 EMS

α

= 1.0 (not adjustable)

γ

1 = 1.05

γ

2

= 0.85

KN = 0.03

Bottom friction

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