6.0 m () 1)- INTRODUCTION 8.5 m 0.7 m nα l n i k l k n 1 k n...
1
Low Frequency Wave Generation in a Spectral Model Low Frequency Wave Generation in a Spectral Model Hoda M. El Safty 1 ([email protected]), and James M. Kaihatu 2 ([email protected]) Paper Number: OS13D-1228 Abstract Reference Number: 7080 1)- INTRODUCTION The nearshore areas of the coast often have considerable low frequency energy, which serves to modulate breaking events and can be linked to the formation of larger-scale bottom features. One likely mechanism for the generation of this energy is subharmonic interactions, in which high frequency waves interact to augment energy in frequencies below the spectral peak. The performance of most phase-resolved frequency domain models has not been 1 Graduate Student, Texas A&M University, 2 Assistant Professor, Texas A&M University The model is capable of predicting the energy transfer from low/high frequencies to high/low frequencies on mild slope in deep and intermediate water depths as shown from Delft Data but deviates from data in very shallow water. The optimum values for the free parameters of Thornton and Guza (1983) was 4)- DISCUSSION AND CONCLUSION Random waves (C-3 series) Bichromatic waves (A-5 series) This data set was used to investigate the applicability of Thornton and Guza (1983) breaking mechanism for this bottom slope. A sensitivity study were carried out by using different values for the free parameters (B & γ). The model was performing well in the deep and intermediate water depths but deviated in the very shallow water., likely due to the implementation of breaking in the model. performance of most phase resolved frequency domain models has not been verified for subharmonic interactions, despite the fact that the necessary terms for these interactions are present in the models. In this study we investigate the generation of low frequency energy in two laboratory experiments and discuss the suitability of frequency domain phase resolved models for predicting this energy transfer. Of particular interest is the rate of energy transfer from spectral peak to subharmonic frequencies, and further down to very low frequencies. It has been surmised that energy in these very low frequency found to be B=1 and γ=0.8 based on the sensitivity study and the error analysis. These values of B & γ were used in the comparison to U. Michigan data. however, the smaller number of wave gages make it hard to completely evaluate the model performance over the steeper reef face and a need for a new breaking mechanism was recommended. The new approach is to keep the higher order of bottom slope and obtain the new nonlinear interaction coefficients Random waves (C 3 series) Bichromatic waves (A-5 series) 10 -5 10 -4 10 -3 10 -2 10 -1 s(f) C-3 series spectrum at location x =26.3m b=1.5,γ=0.4 b=1.5, γ =0.6 b=1.5, γ =0.8 b=1,γ=0.4 b=1,γ=0.6 b=1,γ=0.8 experiment 10 -5 10 -4 10 -3 10 -2 10 -1 s(f) C-3 series spectrum at location x =18.5m b=1.5,γ=0.4 b=1.5, γ =0.6 b=1.5,γ=0.8 b=1, γ =0.4 b=1,γ=0.6 b=1, γ =0.8 experiment 0 0.5 1 1.5 2 2.5 10 10 10 10 10 10 10 frequency (HZ) s(f) C-3 series spectrum at location x =13.5m b=1.5, =0.4 b=1.5, =0.6 b=1.5,=0.8 b=1,=0.4 b=1, =0.6 b=1,=0.8 experiment 0 0.5 1 1.5 2 2.5 10 10 10 10 10 10 10 frequency (HZ) s(f) C-3 series spectrum at location x =8.5m b=1.5,=0.4 b=1.5,=0.6 b=1.5,=0.8 b=1, =0.4 b=1, =0.6 b=1,=0.8 experiment 10 10 10 s(f) a-5 series spectrum at location x =25.4m b=1.5, =0.4 b=1.5, =0.6 b=1.5, =0.8 b=1, =0.4 b=1, =0.6 b=1, =0.8 experiment 10 10 10 s(f) a-5 series spectrum at location x =18.5m b=1.5, =0.4 b=1.5, =0.6 b=1.5, =0.8 b=1, =0.4 b=1, =0.6 b=1, =0.8 experiment 0 0.5 1 1.5 2 2.5 10 10 10 10 10 frequency (HZ) s(f) a-5 series spectrum at location x =13.5m b=1.5, =0.4 b=1.5, =0.6 b=1.5, =0.8 b=1, =0.4 b=1, =0.6 b=1, =0.8 experiment 0 0.5 1 1.5 2 2.5 10 10 10 10 10 frequency (HZ) s(f) a-5 series spectrum at location x =8.5m b=1.5, =0.4 b=1.5, =0.6 b=1.5, =0.8 b=1, =0.4 b=1, =0.6 b=1, =0.8 experiment bands have potential for coastal flooding during storm events over reef systems. new nonlinear interaction coefficients. 2)- MODEL MATHEMATICAL FORMULATION ( ) ( ) 2 = + + n n n n g nx g nx A A kCC kCC A α 3.2)- UNIVERSITY OF MICHIGAN DATA (Demirbilek 2007) 5)- CURRENT AND FUTURE WORK This equation was derived with retaining the higher orders terms of bottom slope in both one dimensional and two dimensional versions. A new breaking mechanism is required to account for the plunging breaking on the steep slope 0 0.5 1 1.5 2 2.5 10 -6 10 -4 10 -2 frequency (HZ) s(f) C-3 series spectrum at location x =32.3m b=1.5,γ=0.4 b=1.5,γ=0.6 b=1.5,γ=0.8 b=1,γ=0.4 b=1,γ=0.6 b=1,γ=0.8 experiment 0 0.5 1 1.5 2 2.5 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 frequency (HZ) s(f) C-3 series spectrum at location x =28.7m b=1.5,γ=0.4 b=1.5,γ=0.6 b=1.5,γ=0.8 b=1,γ=0.4 b=1,γ=0.6 b=1,γ=0.8 experiment 0 0.5 1 1.5 2 2.5 10 10 10 frequency (HZ) 0 0.5 1 1.5 2 2.5 10 10 10 frequency (HZ) 0 0.5 1 1.5 2 2.5 10 10 10 10 10 frequency (HZ) s(f) a-5 series spectrum at location x =32.6m b=1.5,=0.4 b=1.5,=0.6 b=1.5,=0.8 b=1,=0.4 b=1,=0.6 b=1,=0.8 experiment 0 0.5 1 1.5 2 2.5 10 10 10 10 10 frequency (HZ) s(f) a-5 series spectrum at location x =28.4m b=1.5,=0.4 b=1.5,=0.6 b=1.5,=0.8 b=1,=0.4 b=1,=0.6 b=1,=0.8 experiment 0 0.5 1 1.5 2 2.5 10 10 frequency (HZ) 0 0.5 1 1.5 2 2.5 10 10 frequency (HZ) ( ) ( ) ( ) . 2 8 1 1 1 1 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∫ + ∫ ⋅ − ∑ ∑ − = − = − − + − + − + − n l n N l dx k k k i l n l dx k k k i l n l n g n l l n n n l e A SA e A RA kCC i ( ) ( ) 2 3 3 4 5 6 7 8 9 10 1 2 { } 2 g nx nx n x n n n g n n N n xx x kCC A f kf fh fh fh fh f f A A kCC i h h α − − + + + + + + + − + + = ⎡ ⎤ ∫ ∫ steep slope. Additionally the bottom roughness effect needs to be included to realize better prediction of energy transfer and decay of wave propagation over reef. A n is complex amplitudes of free surface, C & Cg are the phase speed and group velocity (fully dispersive linear theory) R and S are the nonlinear coefficients Approximately 80 tests were conducted for different combinations of wave only, wind only, and combined wind-wave tests. Runup gage 1:10.6 1 18 8 1:12 0.93 0.54 0.58 2.17 2.17 3.34 0.33 0.19 Wave gages 3)- EXPERIMENTAL DATA The data records for the water surface elevations were collected from 80 different locations with a sample rate of 25 Hz for different wave conditions (random and bichromatic waves). The different locations were distributed every 0.5 m up to 23m and then every 0.3 m after that to the end of the flume. ( ) ( ) ( ) 1 1 1 2 1 1 2 1 ( ) 2 ( ) . 8 l n n nl l n n N n i k k k dx i k k k dx l nl l nl l l g n i RA A R Re SA A S S e kCC − + − − + − − − − + = = ⎡ ⎤ ∫ ∫ − ⋅ − + + − + ⎢ ⎥ ⎣ ⎦ ∑ ∑ 3.1)- DELFT UNIVERSITY DATA (Van Norloos 2003) A n is complex amplitudes of free surface, C & Cg are the phase speed and group velocity, as before. R and S are the nonlinear coefficients terms of Kaihatu & Kirby (1995), This data set was used to investigate the model performance on steep slope. It became apparent that the mild-slope formulation of the model precluded good performance when initiated at the offshore gage. To investigate the nonlinear behavior over the reef top, the model was initiated from gage-7 on the horizontal bottom. A need for new breaking All Units in meters 0.26 1.11 1.30 2.14 1.35 4.80 0.58 1:5 1:18.8 0.37 0.50 Reference gage 6.0 m 1 0 m every 0.5 m up to 23m and then every 0.3 m after that to the end of the flume. The data used here is series (A-5) for bichromatic waves and series (C-3) for the random waves. See Van Norloos 2003 for more details R 1 ,R 2 ,S 1 ,S 2 , are new nonlinear terms which retain bottom slope, f 3 ,f 4 ,f 5 ,f 6 ,f 7 ,f 8 ,f 9 ,f 10 : are the linear terms which retain curvature and slope squared terms Test-16 (H s =5.2 m & T p =1.5 sec) Test-45 model was initiated from gage 7 on the horizontal bottom. A need for new breaking mechanism to deal with the sudden increase in wave energy in the high frequencies was clear. This work was supported by the Surge and Wave Island Modeling Studies under the Coastal Field Data Collection Program of the Coastal and Hydraulics Laboratory, U.S. Army Engineer Research and Development Center. We would like to thank Dr. Ap van Acknowledgments 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Distance (m) Frequency(Hz) Test-16 Energy Gradient -4 -3 -2 -1 0 1 2 3 4 x 10 0 0.5 1 1.5 2 10 10 10 10 10 Frequency (HZ) s(f) Test-16 at location x=4.34m b=1,gamma=0.8 experiment 0 0.5 1 1.5 2 10 10 10 10 10 Frequency (HZ) s(f) Test-16 at location x=2.17m b=1,gamma=0.8 experiment 1.2 1.4 1.6 1.8 y(Hz) Test-45 Energy Gradient 0 0.5 1 1.5 x 10 10 10 Test-45 at location x=4.34m b=1,gamma=0.8 experiment 10 10 Test-45 at location x=2.17m b=1,gamma=0.8 experiment Demirbilek, Z., & Nwogu, O. G. & Ward. D.L. (2007). Laboratory study of wind effect on Runup over Fringing Reefs. ERDC/CHL TR-07-4. Kaihatu, J. M., & Kirby, J. T. (1995). Nonlinear transformation of waves in finite water depth. Physical Fluids, 7, 1903-1914. Thornton, E.B., & Guza, R.T.,(1983). Transformation of wave height distribution. Journal of Geophys. Research, Vol. 88, no. C10, pp. 5925-5938. Van Norloos, J.C., (2003). Energy transfer between short wave groups and bound long waves on a plane slope. M.S. Thesis,pp.68,Delft Uni. Of Tech., Delft, Netherlands. 0.7 m 8.5 m 33.0 m 40.0 m 1.0 m paddle (H s =6.1 m & T p =1.0 sec) Test-48 (H s =7.5 m & T p =1.5 sec) Army Engineer Research and Development Center. We would like to thank Dr. Ap van Dongeren for the Van Norloos (2003) data, and Dr. Jane M. Smith for the Demirbilek et al. (2007)data. 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1 Distance (m) Frequency -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 10 10 10 Frequency (HZ) s(f) 0 0.5 1 1.5 2 10 10 10 Frequency (HZ) s(f) 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Distance (m) Frequency(Hz) Test-48 Energy Gradient -4 -3 -2 -1 0 1 2 3 x 10 0 0.5 1 1.5 2 10 10 10 10 10 Frequency (HZ) s(f) Test-48 at location x=4.34m b=1,gamma=0.8 experiment 0 0.5 1 1.5 2 10 10 10 10 10 Frequency (HZ) s(f) Test-48 at location x=2.17m b=1,gamma=0.8 experiment
Transcript of 6.0 m () 1)- INTRODUCTION 8.5 m 0.7 m nα l n i k l k n 1 k n...
Low Frequency Wave Generation in a Spectral ModelLow Frequency Wave Generation in a Spectral ModelHoda M. El Safty1 ([email protected]), and James M. Kaihatu2 ([email protected])
Paper Number: OS13D-1228Abstract Reference Number: 7080
1)- INTRODUCTION
The nearshore areas of the coast often have considerable low frequency energy,which serves to modulate breaking events and can be linked to the formationof larger-scale bottom features. One likely mechanism for the generation ofthis energy is subharmonic interactions, in which high frequency wavesinteract to augment energy in frequencies below the spectral peak. Theperformance of most phase-resolved frequency domain models has not been
1Graduate Student, Texas A&M University,2Assistant Professor, Texas A&M University
The model is capable of predicting the energy transfer from low/high frequencies to high/low frequencies on mild slope in deep and intermediate water depths as shown from Delft Data but deviates from data in very shallow water.
The optimum values for the free parameters of Thornton and Guza (1983) was
4)- DISCUSSION AND CONCLUSION
Random waves (C-3 series) Bichromatic waves (A-5 series)
This data set was used to investigate the applicability of Thornton and Guza (1983) breaking mechanism for this bottom slope. A sensitivity study were carried out by using different values for the free parameters (B & γ). The model was performing well in the deep and intermediate water depths but deviated in the very shallow water., likely due to the implementation of breaking in the model.
performance of most phase resolved frequency domain models has not beenverified for subharmonic interactions, despite the fact that the necessary termsfor these interactions are present in the models. In this study we investigate thegeneration of low frequency energy in two laboratory experiments and discussthe suitability of frequency domain phase resolved models for predicting thisenergy transfer. Of particular interest is the rate of energy transfer fromspectral peak to subharmonic frequencies, and further down to very lowfrequencies. It has been surmised that energy in these very low frequency
found to be B=1 and γ=0.8 based on the sensitivity study and the error analysis.
These values of B & γ were used in the comparison to U. Michigan data. however, the smaller number of wave gages make it hard to completely evaluate the model performance over the steeper reef face and a need for a new breaking mechanism was recommended.
The new approach is to keep the higher order of bottom slope and obtain the new nonlinear interaction coefficients
Random waves (C 3 series) Bichromatic waves (A-5 series)
10-5
10-4
10-3
10-2
10-1
s(f)
C-3 series spectrum at location x =26.3m
b=1.5,γ=0.4b=1.5,γ=0.6
b=1.5,γ=0.8
b=1,γ=0.4
b=1,γ=0.6
b=1,γ=0.8experiment
10-5
10-4
10-3
10-2
10-1
s(f)
C-3 series spectrum at location x =18.5m
b=1.5,γ=0.4b=1.5,γ=0.6
b=1.5,γ=0.8
b=1,γ=0.4
b=1,γ=0.6
b=1,γ=0.8
experiment
0 0.5 1 1.5 2 2.510
-7
10-6
10-5
10-4
10-3
10-2
10-1
frequency (HZ)
s(f)
C-3 series spectrum at location x =13.5m
b=1.5,γ=0.4b=1.5,γ=0.6
b=1.5,γ=0.8
b=1,γ=0.4
b=1,γ=0.6
b=1,γ=0.8experiment
0 0.5 1 1.5 2 2.510
-7
10-6
10-5
10-4
10-3
10-2
10-1
frequency (HZ)
s(f)
C-3 series spectrum at location x =8.5m
b=1.5,γ=0.4b=1.5,γ=0.6
b=1.5,γ=0.8
b=1,γ=0.4
b=1,γ=0.6
b=1,γ=0.8experiment
10-4
10-2
100
s(f)
a-5 series spectrum at location x =25.4m
b=1.5,γ=0.4b=1.5,γ=0.6
b=1.5,γ=0.8
b=1,γ=0.4
b=1,γ=0.6
b=1,γ=0.8experiment
10-4
10-2
100
s(f)
a-5 series spectrum at location x =18.5m
b=1.5,γ=0.4b=1.5,γ=0.6
b=1.5,γ=0.8
b=1,γ=0.4
b=1,γ=0.6
b=1,γ=0.8experiment
0 0.5 1 1.5 2 2.510
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10-6
10-4
10-2
100
frequency (HZ)
s(f)
a-5 series spectrum at location x =13.5m
b=1.5,γ=0.4
b=1.5,γ=0.6
b=1.5,γ=0.8
b=1,γ=0.4
b=1,γ=0.6
b=1,γ=0.8
experiment
0 0.5 1 1.5 2 2.510
-8
10-6
10-4
10-2
100
frequency (HZ)
s(f)
a-5 series spectrum at location x =8.5m
b=1.5,γ=0.4b=1.5,γ=0.6
b=1.5,γ=0.8
b=1,γ=0.4
b=1,γ=0.6
b=1,γ=0.8
experiment
q gy y q ybands have potential for coastal flooding during storm events over reefsystems.
new nonlinear interaction coefficients.
2)- MODEL MATHEMATICAL FORMULATION
( )( )2
=++ nnn
ng
nxgnx AA
kCC
kCCA α
3.2)- UNIVERSITY OF MICHIGAN DATA (Demirbilek 2007)
5)- CURRENT AND FUTURE WORK
This equation was derived with retaining the higher orders terms of bottom slope in both one dimensional and two dimensional versions.
A new breaking mechanism is required to account for the plunging breaking on the steep slope
0 0.5 1 1.5 2 2.5
10-6
10-4
10-2
frequency (HZ)
s(f)
C-3 series spectrum at location x =32.3m
b=1.5,γ=0.4b=1.5,γ=0.6b=1.5,γ=0.8b=1,γ=0.4b=1,γ=0.6b=1,γ=0.8experiment
0 0.5 1 1.5 2 2.510
-7
10-6
10-5
10-4
10-3
10-2
10-1
frequency (HZ)
s(f)
C-3 series spectrum at location x =28.7m
b=1.5,γ=0.4b=1.5,γ=0.6
b=1.5,γ=0.8
b=1,γ=0.4
b=1,γ=0.6
b=1,γ=0.8experiment
0 0.5 1 1.5 2 2.510
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10-6
10
frequency (HZ)
0 0.5 1 1.5 2 2.5
10-7
10-6
10
frequency (HZ)
0 0.5 1 1.5 2 2.510
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10-6
10-4
10-2
100
frequency (HZ)
s(f)
a-5 series spectrum at location x =32.6m
b=1.5,γ=0.4b=1.5,γ=0.6
b=1.5,γ=0.8
b=1,γ=0.4
b=1,γ=0.6
b=1,γ=0.8
experiment
0 0.5 1 1.5 2 2.510
-8
10-6
10-4
10-2
100
frequency (HZ)
s(f)
a-5 series spectrum at location x =28.4m
b=1.5,γ=0.4
b=1.5,γ=0.6
b=1.5,γ=0.8
b=1,γ=0.4
b=1,γ=0.6
b=1,γ=0.8experiment
0 0.5 1 1.5 2 2.510
-8
10-6
frequency (HZ)
0 0.5 1 1.5 2 2.5
10-8
10-6
frequency (HZ)
( )( ) ( )
.28
1
1 1
1
⎥⎦
⎤⎢⎣
⎡ ∫+∫⋅− ∑ ∑−
=
−
=
−−
+
−+
−+−
n
l
nN
l
dxkkki
lnl
dxkkki
lnl
ng
nllnnnl eASAeARAkCC
i
( )( )
2 33 4 5 6 7 8 9 10
1
2{ }2
g nxnx n x n n n
g n
n N n
xx x
kCCA f k f f h f h f h f h f f A A
kCC
i
h h α
− −
+ + + + + + + − + + =
⎡ ⎤∫ ∫
steep slope.
Additionally the bottom roughness effect needs to be included to realize better prediction of energy transfer and decay of wave propagation over reef.
An is complex amplitudes of free surface,
C & Cg are the phase speed and group velocity (fully dispersive linear theory)
R and S are the nonlinear coefficients
Approximately 80 tests were conducted for different combinations of wave only, wind only, and combined wind-wave tests.
Runup gage
1:10.6
1 18 8
1:12
0.93 0.54 0.58
2.17 2.17 3.34
0.33 0.19
Wave gages
3)- EXPERIMENTAL DATA
The data records for the water surface elevations were collected from 80different locations with a sample rate of 25 Hz for different wave conditions(random and bichromatic waves). The different locations were distributedevery 0.5 m up to 23m and then every 0.3 m after that to the end of the flume.
( )( ) ( )1
1
1 2 11
21
( ) 2 ( ) .8
l n n n l l nn N n
i k k k dx i k k k dx
l n l l n ll lg n
iRA A R R e SA A S S e
kCC− +
− −+ − − −
− += =
⎡ ⎤∫ ∫− ⋅ − + + − +⎢ ⎥⎣ ⎦∑ ∑
3.1)- DELFT UNIVERSITY DATA (Van Norloos 2003)An is complex amplitudes of free surface,
C & Cg are the phase speed and group velocity, as before.
R and S are the nonlinear coefficients terms of Kaihatu & Kirby (1995),
This data set was used to investigate the model performance on steep slope. It became apparent that the mild-slope formulation of the model precluded good performance when initiated at the offshore gage. To investigate the nonlinear behavior over the reef top, the model was initiated from gage-7 on the horizontal bottom. A need for new breaking
All Units in meters
0.26
1.11 1.30 2.14 1.35 4.80 0.58
1:5
1:18.8
0.37
0.50
Reference gage
6.0 m
1 0 m
every 0.5 m up to 23m and then every 0.3 m after that to the end of the flume.The data used here is series (A-5) for bichromatic waves and series (C-3) forthe random waves. See Van Norloos 2003 for more details
R1,R2,S1,S2, are new nonlinear terms which retain bottom slope,
f3,f4,f5,f6,f7,f8,f9,f10: are the linear terms which retain curvature and slope squared termsTest-16
(Hs=5.2 m & Tp=1.5 sec)
Test-45
model was initiated from gage 7 on the horizontal bottom. A need for new breaking mechanism to deal with the sudden increase in wave energy in the high frequencies was clear.
This work was supported by the Surge and Wave Island Modeling Studies under the Coastal Field Data Collection Program of the Coastal and Hydraulics Laboratory, U.S. Army Engineer Research and Development Center. We would like to thank Dr. Ap van
Acknowledgments0 1 2 3 4
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b=1,gamma=0.8experiment
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b=1,gamma=0.8experiment
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y(H
z)Test-45 Energy Gradient
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0.5
1
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Test-45 at location x=4.34m
b=1,gamma=0.8experiment
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Test-45 at location x=2.17m
b=1,gamma=0.8experiment
Demirbilek, Z., & Nwogu, O. G. & Ward. D.L. (2007). Laboratory study of wind effect on Runup over Fringing Reefs. ERDC/CHL TR-07-4.Kaihatu, J. M., & Kirby, J. T. (1995). Nonlinear transformation of waves in finite water depth. Physical Fluids, 7, 1903-1914.Thornton, E.B., & Guza, R.T.,(1983). Transformation of wave height distribution. Journal of Geophys. Research, Vol. 88, no. C10, pp. 5925-5938.Van Norloos, J.C., (2003). Energy transfer between short wave groups and bound long waves on a plane slope. M.S. Thesis,pp.68,Delft Uni. Of Tech., Delft, Netherlands.
0.7 m
8.5 m33.0 m
40.0 m
1.0 mpaddle
(Hs=6.1 m & Tp=1.0 sec)
Test-48
(Hs=7.5 m & Tp=1.5 sec)
Army Engineer Research and Development Center. We would like to thank Dr. Ap van Dongeren for the Van Norloos (2003) data, and Dr. Jane M. Smith for the Demirbilek et al. (2007)data.
0 1 2 3 40
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0 0.5 1 1.5 2
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10-6
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s(f)
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b=1,gamma=0.8experiment
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s(f)
Test-48 at location x=2.17m
b=1,gamma=0.8experiment
