Second order wave generation in the OEB II

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DOCUMENTATION PAGE

REPORT NUMBER

TR-2006-27

NRC REPORT NUMBER DATE

December 2006

REPORT SECURITY CLASSIFICATION

Unclassified

DISTRIBUTION

Unlimited

TITLE

SECOND ORDER WAVE GENERATION IN THE OEB-II AUTHOR(S)

Hasanat Zaman and Lawrence Mak

CORPORATE AUTHOR(S)/PERFORMING AGENCY(S)

Institute for Ocean Technology, National Research Council, St. John’s, NL

PUBLICATION

SPONSORING AGENCY(S)

Institute for Ocean Technology, National Research Council, St. John’s, NL

IOT PROJECT NUMBER

42_2103_10 NRC FILE NUMBER

KEY WORDS

Second order wave, shallow water, steepness

PAGES iii, 9, App. 1-4 FIGS. 3 TABLES3 SUMMARY

When first-order natural waves are reproduced in the laboratory using the first order wave generation technique, the primary waves and their locked bounded waves are generated along with some unwanted free waves. Those free waves are evidently generated and propagate towards the test model and reflect from the boundaries. The free waves, having the same frequency of the bounded wave are reproduced, as the boundary conditions of the wave paddle are not properly satisfied up to second-order. The other two types of free waves are due to the wave paddle displacement and local disturbances. These so-called free waves cause an amplification of low frequency and/or high frequency wave phenomena, such as harbour resonance and oscillations of moored ships and, propagation and breaking of waves on floating or fixed structures that may exert huge hydrodynamic loads. The so-called second order wave generation technique could eliminate these alleged free waves. In this experiment second order wave generation technique is successfully used to reproduce the correct bounded waves along with elimination of the unwanted free waves from the wave profiles. This experiment is implemented by means of compensating free waves imposed on the system by second-order paddle motion. The control signal for this motion has to be introduced along with the primary waves.

ADDRESS National Research Council

Institute for Ocean Technology Arctic Avenue, P. O. Box 12093 St. John's, NL A1B 3T5

National Research Council Conseil national de recherches Canada Canada Institute for Ocean Institut des technologies

Technology océaniques

SECOND ORDER WAVE GENERATION IN THE OEB-II

TR-2006-27

Hasanat Zaman and Lawrence Mak December 2006

TABLE OF CONTENTS ABSTRACT... 1 INTRODUCTION ... 1 THEORY ... 3 EXPERIMENTS ... 5 Software ... 5 Experimental setup... 5

Case Study-I: Regular wave... 6

Methodology ... 8 RESULTS ... 8 CONCLUSIONS... 9 ACKNOWLEDGEMENT ... 9 REFERENCES ... 9 APPENDIX-I: ... 10 APPENDIX-II:... 41 APPENDIX-III: ... 72 APPENDIX-IV: ... 103 LIST OF FIGURES Figure 1: A regular primary wave and its locked bounded high frequency wave frequencies[H=0.2m and f1 =0.833 Hz] . ... 1

Figure 2: A primary wave group formed by two different waves frequencies [H=0.2m and f1 =0.833 Hz, and H=0.28m and f2 =0.926 Hz] ... 2

Figure 3: Top view of the experimental setup in the OEB ... 5

LIST OF TABLES Table 1 Location of the wave probes in the OEB... 6

Table 2 Experimental wave parameters for 0.6m water depth ... 7

Table 3 Experimental wave parameters for 0.8m water depth ... 7

ABSTRACT

When the interaction between frequencies is considered then the second order solution of Laplace equation produces two extra components: (i) bounded sub harmonics or bounded low frequency waves and, (ii) bounded super harmonics or bounded high frequency waves and they travel locked with their generating / fundamental wave components. These nonlinear components play a predominant role in the intermediate and shallow water region. So when doing any test it is important to make sure that those nonlinear wave components are properly reproduced in the basin along with the primary components. When first-order natural waves are reproduced in the laboratory using the first order wave generation technique, the primary waves and their locked bounded waves are generated along with some unwanted free waves. Those free waves are evidently generated and propagate towards the test model and reflect from the boundaries. The free waves, having the same frequency of the bounded wave are reproduced, as the boundary conditions of the wave paddle are not properly satisfied up to second-order. The other two types of free waves are due to the wave paddle displacement and local disturbances. These so-called free waves cause an amplification of low frequency and/or high frequency wave phenomena, such as harbour resonance and oscillations of moored ships and, breaking of waves on floating or fixed structures. The so-called second order wave generation technique could eliminate these alleged free waves. In this experiment second order wave generation technique is successfully used to reproduce the correct bounded waves along with elimination of the unwanted free waves from the wave profiles. This experiment is implemented by means of compensating free waves imposed on the system by second-order paddle motion. The control signal for this motion has to be introduced along with the primary waves.

INTRODUCTION

Recently, correct generation of the second order waves and the reproduction of group-induced second order low and high frequency waves have been considered essential for physical model test in the laboratory to understand the effects of the wave-action phenomena on, for instance, offshore structures, mooring system, floating vessels, harbour resonance, loadings, etc..

η

Primary wave 2 f1

Figure 1: A regular primary wave and its locked bounded high frequency wave frequencies [H=0.2m and f1 =0.833 Hz].

For the case of a regular wave there will be only one frequency (f1). In this case the

primary wave and its locked bounded high frequency (2 f1) wave are generated along

with unwanted free waves as mentioned before. Figure 1 shows a view of the primary wave and its locked bounded high frequency wave.

When there are two waves of frequencies f1 and f2 formed a group, the group-induced

second order low and high frequency waves are generated along with other unwanted free waves. Figure 2 shows an example of a wave group and its locked bounded high and low frequency waves.

Primary wave group f1-f2

η

f1+f2

Figure 2: A primary wave group formed by two different waves frequencies. [H=0.2m and f1 =0.833 Hz, and H=0.28m and f2 =0.926 Hz]

A low frequency wave or long wave will be produced due to the difference (f1 - f2) of

the frequencies and a high frequency or short wave would be generated due to the summation (f1 + f2) of the frequencies. The profile of the long wave having frequency (f1 -

f2) is generally termed as the set-down in the larger wave zone and the set-up in the

smaller wave zone (see Figure 2). These set-down and set-up phenomena were first investigated and reported by Longuet-Higgins and Stewart (1961, 1962, 1963 and 1964) in a series of papers. They introduced the radiation stress concept, which explains that in a wave group individual wave components exert an internal compressive force in the direction of wave propagation. To balance this force the mean water level goes down in the region of larger waves known as set-down and goes up in the region of smaller waves known as set-up. Bowen et al (1968) later explained the set-down and set-up phenomena with experimental data.

In this experiments first order and second order wave generation techniques are employed and the obtained data are compared with the relevant predicted data. In the experiment second order wave generation technique is successfully used to reproduce the correct bounded waves and to eliminate the unwanted free waves from the wave profiles. This experiment is implemented by means of compensating free waves imposed on the

system by second-order paddle motion. The control signal for this motion has to be introduced along with the primary waves.

THEORY

A wave group would be generated with the presence of at least two frequencies. Difference of these two frequencies would generate a long period bound wave with a period equal to the period of the wave group. This long wave is also known as ‘set-down’.

A pair of regular waves with frequencies fn and fm and, surface elevations ηn(t) and

ηm(t) respectively would constitute a wave group as follows (Sand 1982):

m n nm t η η η ( )= + ) sin( ) cos( ) sin( ) cos( t k₁x₁ b t k₁x₁ a t k₁x₁ b t k₁x₁ a_{n n} − + _{n n} − + _{m m} − + _{m m} − = σ σ σ σ

Here, σ is the wave angular frequency, k the wave number, t the time and, a and b are the Fourier coefficients.

The second order long wave ηlnm(t) generated by the above wave group would be as

follows: ⎢ ⎣ ⎡ ∆ − ∆ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = cos( ) ) ( ₂ 1 2 x k t h b b a a h G t nm nm m n m n nm nm l σ η ⎥ ⎦ ⎤ ∆ − ∆ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + ₂ sin( t k x1) h b a b a nm nm m n n m σ

Here, h is the water depth. ∆σnm = σn -σm, ∆knm = kn -km and ∆fnm = fn -fm. The transfer

function Gnm can be given as follows:

m n nm n m m n m n nm nm m n nm nm m n nm D D h k D k D k D D h k h k k k h h k k D hD h G 2 ) coth( ) )( ( ) cosh( ) ( cosh ) cosh( 4 1 2 ₊ ∆ 2 − + ∆ ⎢ ⎢ ⎣ ⎡ ∆ − + ∆ ∆ = π

] [

D D k h k h

]

h k D Dn − m −∆ nm n − m ∆ nm −∆ nm −2π2( )2 /4π2( )2coth( )

where, Dn h g fn and ; g is the acceleration due to gravity. 5 . 0 ) / ( = Dm h g fm 5 . 0 ) / ( =

The first order control signal for only one frequency can be derived as follows [see also Barthel et al (1983)]: ) cos sin ( ) ( sinh 2 ) sinh( ) cosh( 2 1 0 a t b t h k h k h k h k X n n n n n n n n + σ ₋ σ = 3

The second order control signal for the above wave group having only two frequencies is given by the following expression:

(

)(

) (

)

[

a b a b F F a a b b F

]

t X₀2 = _{n m} − _{m n 11} + ₁₂ + _{n m}− _{n m 23} cos∆σ_nm

+

[

(

a_na_m +b_nb_m

)(

F₁₁ +F₁₂

) (

+ a_mb_n −a_nb_m

)

F₂₃

]

sin∆σ_nmt

The transfer functions F11, F12 and F23 are related to the wave generation and

propagation phenomena.

The function F11 is the contribution to the control signal required for the bounded long

wave and used to eliminate the parasitic long waves:

(

)

[

]

) sinh( ) sinh( ) ( 2 ) sinh( ) ( ) sinh( 2 2 2 2 11 h k h k h k h k h h k h k h k h k h k h k h k h k h k h G F f nm f nm f nm f nm f nm f nm f nm ∆ ∆ ∆ − ∆ ∆ − ∆ ∆ + ∆ + ∆ + ∆ ∆ − ∆ ∆ =

The function F12 is required to eliminate the free long wave generated due to the

displacement of the wave peddle from its original position at x=0:

(

)

[

]

) tanh( ) sinh( ) sinh( ) ( 8 ) sinh( ) sinh( 1 1 2 2 2 2 12 h k h k h k h k h k f h k h k h k h k G h hk k f h F n m f f m m m m m n m f m ∆ ∆ − ∆ + + ∆ = − + + − _{δ δ δ} δ

(

)

[

]

) tanh( ) sinh( ) sinh( ) ( 8 ) sinh( ) sinh( 1 1 2 2 2 2 h k h k h k h k h k f h k h k h k h k G h hk k f h _{n f f n m} n n n n m n f n ∆ ∆ − ∆ + + ∆ + − + + − _{δ δ δ} δ

The quantities and, and are obtained, respectively from the following relationships: f k ∆ ± m k δ ± n k δ

(

∆σnm

)

2 =g∆kf tanh(∆kfh) m or n i k k ki = i ±∆ f = ± : δ

To eliminate the free second order long waves produced from the local disturbances, a third transfer function F23 is essential. This function can be described as follows:

) ( 2 23h F h Fm Fn F = − Here,

( )

k h

( )

k h h G G h k F m n m n f tanh tanh 8 ) 1 )( 1 ( 23 + + ∆ =

(

)

: i m or n h k h k G i i i 2 sinh 2 = =

( )

[

( ) (

)

( )

]

( ) ( )

[

]

: i m or n h k h k h k h k h k h k h k h k h k h k h k h k f f F j j f j j j j f f j j j j i i cos sin ) ( cos coth sin sin 2 1 2 2 2 2 _{+∆ +} = ∆ + ∆ ∆ =

∑

∞ = 4

The term kjh in the above equation can be evaluated within the

(

j− 2

)

π <kjh< jπ

1 _limit

from the following equation: ) tanh( 4π2hfm2 =−gkjh kjh

EXPERIMENTS

In the present experiment two different water depths were used, 0.6m and 0.8m. The experiments were carried out for second order regular waves, second order bichromatic waves and unidirectional irregular second order waves.

Software

We have two FORTRAN codes here in IOT to generate second order waves in the tank. The code SPWNW is designed to generate primary wave(s) with their super-harmonic bounded wave(s). This code has the option to remove the unwanted free waves. SPWNW code can also predict the individual wave components in the tank at different location. The code SPWNW would be used for the primary wave(s) with super-harmonic second order wave.

Experimental setup

Ten (10) wave probes were employed to record the experimental data in the OEB. Figure 3 shows the experimental setup and red circles are the locations of the wave probes. The distances of the wave probes from the wave paddle are shown in Table 1.

10 9 8 6 7 5 4 1 2 3

Figure 3: Top view of the experimental setup in the OEB

Table 1 Location of the wave probes in the OEB No of the probe Distance from the paddle

(m)

Distance from the south wall (m) 1 15.000 13.0 2 15.576 13.0 3 16.280 13.0 4 17.112 13.0 5 19.000 13.0 6 30.000 13.0 7 30.576 13.0 8 31.280 13.0 9 32.112 13.0 10 34.000 13.0

The bottom of the basin was flat and the blanking plates were deployed to cover the north beach.

Case Study-I: Regular wave

In this study, we mainly concentrated on the generation of regular wave (i.e. fn = fm).

The aim of this study was to investigate the possibility of accurate generation of the second order wave in the OEB. In this category of experiments our focus would be to understand and use the second order wave generation technique to generate correct primary and bounded waves in the basin.

When a regular wave in shallow water is generated in the wave basin using the Ist order generation method then parasitic free waves having same frequency as the bounded waves are involuntarily generated. The parasitic free waves, free waves due to displacement of the wave paddles and free waves due to local disturbance have to be eliminated to ensure the correct reproduction of the wave in the basin for model test. The so-called second order wave generation technique was employed for this purpose.

In the experiments two different water depths were used. For both cases four different wave steepness were utilized. The experimental parameters for 0.6m and 0.8m water depths are shown in Table 2 and Table 3, respectively.

Table 2 Experimental wave parameters for 0.6m water depth Depth D(m) Wave period T(s) Wave height H(m) Wave length L(m) Steepness (H/L) Relative depth (d/L) 0.6 1.5 0.0299 2.99039 0.01 0.200642726 0.6 2.0 0.0436 4.36198 0.01 0.137552213 0.6 2.5 0.0567 5.67309 0.01 0.105762468 0.6 3.0 0.0695 6.95175 0.01 0.086309203 0.6 3.5 0.0821 8.21166 0.01 0.073066834 0.6 1.5 0.0598 2.99039 0.02 0.200642726 0.6 2.0 0.0872 4.36198 0.02 0.137552213 0.6 2.5 0.1135 5.67309 0.02 0.105762468 0.6 3.0 0.1390 6.95175 0.02 0.086309203 0.6 3.5 0.1642 8.21166 0.02 0.073066834 0.6 1.5 0.0897 2.99039 0.03 0.200642726 0.6 2.0 0.1309 4.36198 0.03 0.137552213 0.6 2.5 0.1702 5.67309 0.03 0.105762468 0.6 3.0 0.2086 6.95175 0.03 0.086309203 0.6 3.5 0.2463 8.21166 0.03 0.073066834

Table 3 Experimental wave parameters for 0.8m water depth Depth d(m) Wave period T(s) Wave height H(m) Wave length L(m) Steepness (H/L) Relative depth (d/L) 0.8 1.5 0.0322 3.2171 0.01 0.248671163 0.8 2.0 0.0485 4.8494 0.01 0.164968862 0.8 2.5 0.0640 6.3996 0.01 0.125007813 0.8 3.0 0.0790 7.9011 0.01 0.101251724 0.8 3.5 0.0937 9.3740 0.01 0.085342437 0.8 1.5 0.0643 3.2171 0.02 0.248671163 0.8 2.0 0.0970 4.8494 0.02 0.164968862 0.8 2.5 0.1280 6.3996 0.02 0.125007813 0.8 3.0 0.1580 7.9011 0.02 0.101251724 0.8 3.5 0.1875 9.3740 0.02 0.085342437 0.8 1.5 0.0965 3.2171 0.03 0.248671163 0.8 2.0 0.1455 4.8494 0.03 0.164968862 0.8 2.5 0.1920 6.3996 0.03 0.125007813 0.8 3.0 0.2370 7.9011 0.03 0.101251724 0.8 3.5 0.2812 9.3740 0.03 0.085342437 7

Methodology

The Fortran code SPWNW was used to generate correct second order wave in the basin. The surface elevations from which the required drive signal would be produced is consisted of the following components surface elevations:

PW = Primary wave

BW = Bounded wave

FPW = Free parasitic wave

FPDW = Free wave due to wave paddle displacement FLDW = Free wave due to local disturbance

CFPW = Correction for FPW

CFPDW = Correction for FPDW

CFLDW = Correction for FLDW

Following the above abbreviations the surface elevation for the correctly generated wave should be: * * * ) (t =PW+BW+FPW+FPDW+FLDW−CFPW −CFPDW −CFLDW η (1)

The quantities with * sign have to be subtracted from the resulted wave profile to eliminate the unwillingly generated free waves components. The numerical code SPWNW implemented the above equation. This code produces the above η(t) and the code DWREP2 used this η(t) to produce the required drive signal to generate the wave in the basin. Then the deployed wave probes would acquire the wave data in the basin. The acquired data could be split up into its contributing components.

RESULTS

In this experiments two different water depths (d) were used. In both cases wave period T was varied from 1.5s to 3.5s, wave steepness (H/L) was varied from 1% to 3% and relative water depths (d/L) was varied from shallow to intermediate water depth limits.

Both First-Order and Second-Order wave generation techniques were employed in the present experiments. In the experiment 0.6m and 0.8m water depths were considered. In all cases the wave steepness were varied from 1% to 3%.

In the case of First-Order wave generation technique, Appendix-I show the comparisons of the measured and numerically predicted wave profiles for 0.6m and Appendix-II show the comparisons of the measured and numerical wave profiles for 0.8m water depth.

In contrast, for the Second-Order wave generation technique, Appendix-III show the comparison of the measured and predicted wave profiles for 0.6m water depth and Appendix-IV show the comparisons of the wave profiles for 0.8m water depth.

APPENDIX-I : results for first order wave generation with 0.6m water depth. APPENDIX-II : results for first order wave generation with 0.8m water depth. APPENDIX-III : results for second order wave generation with 0.6m water depth. APPENDIX-IV : results for second order wave generation with 0.8m water depth.

CONCLUSIONS

In this experiment the water depths were 0.6m and 0.8m. In the entire test wave steepness were varied from 1% to 3%. In the case of 0.6m water depth, it was observed that the bottom of the basin plays a predominant role on the propagation of the wave in the OEB. In this case the wave profile looses its stability with moderate steep waves as well as with longer wave periods. On the other hand, the 0.8m water depth can accommodate more steep waves than 0.6m water depth. This is because the same wave feels bottom more in 0.6m water depth than that of 0.8m water depth. Experimental and predicted wave profiles are compared for both first order and second order wave cases for both water depths and presented in the appendix.

ACKNOWLEDGEMENT

The authors highly acknowledge the help from Mike Sullivan and Shane Mckay of the Facilities Department at the Institute of Ocean Technology, National Research Council of Canada.

REFERENCES

Barthel V., Mansard, E.P.D., Sand, S.E. and Vis, F.C. (1983) : Group bounded long waves in physical models, Ocen Eng. 10(4).

Bowen, A. J., Inman, D. L. and Simmons, V. P. (1968) : Wave set-down and set-up, J. of

Geophy. Res. 73(8), 2569-2577.

Longuet-Higgins, M. S. and R. W. Stewart (1961) : The changes in amplitude of short gravity waves on steady non-uniform currents, J. Fluid Mech., 10, 529-549.

Longuet-Higgins, M. S. and R. W. Stewart (1962) : Radiation stress and mass transport in gravity waves with application to surf beats, J. Fluid Mech., 13, 481.

Longuet-Higgins, M. S. and R. W. Stewart (1963) : A note on wave set-up, J. Marine

Res., 21, 9.

Longuet-Higgins, M. S. and R. W. Stewart (1964) : Radiation stress in water waves, a physical discussion with application, Deep-Sea Res., 11, 529.

Sand, S. E. (1982) : Long wave problems in laboratory models, Proceedings of the ASCE,

J. of Waterway, Port, coast. And Ocn. Div. 198 (WW4).

APPENDIX - I

Comparisons of the measured and the predicted First Order wave profiles for varying wave periods with varying wave steepness when the water depth is 0.6m.

Reading of the figures and wave conditions:

W_D0P6_T1P5_S1_001---Æ D0P6_T1P5_S1 (Ignore W and 001)

D0P6 = Depth(D) Zero(0) Point(P) Six(6) = > d = 0.6m T1P5 = Period(T) One(1) Point(P) Five(5) = > T = 1.5s S1 = H/ L=1%

S2 = H/L =2% S3 = H/L =3%

APPENDIX – II

Comparisons of the measured and the predicted First Order wave profiles for varying wave periods with varying wave steepness when the water depth is 0.8m.

Reading of the figures and wave conditions:

W_D0P8_T1P5_S1_001---Æ D0P8_T1P5_S1 (Ignore W and 001)

D0P8 = Depth(D) Zero(0) Point(P) Eight(8) = > d = 0.8m T1P5 = Period(T) One(1) Point(P) Five(5) = > T = 1.5s S1 = H/ L=1%

S2 = H/L =2% S3 = H/L =3%

APPENDIX - Ill

Comparisons of the measured and the predicted Second Order wave profiles for varying wave periods with varying wave steepness when the water depth is 0.6m.

Reading of the figures and wave conditions:

W_D0P6_T1P5_S1_001---Æ D0P6_T1P5_S1 (Ignore WB and 001)

D0P6 = Depth(D) Zero(0) Point(P) Six(6) = > d = 0.6m T1P5 = Period(T) One(1) Point(P) Five(5) = > T = 1.5s S1 = H/ L=1%

S2 = H/L =2% S3 = H/L =3%

APPENDIX - IV

Comparisons of the measured and the predicted Second Order wave profiles for varying wave periods with varying wave steepness when the water depth is 0.8m.

Reading of the figures and wave conditions:

W_D0P8_T1P5_S1_001---Æ D0P8_T1P5_S1 (Ignore WB and 001)

D0P8 = Depth(D) Zero(0) Point(P) Eight(8) = > d = 0.8m T1P5 = Period(T) One(1) Point(P) Five(5) = > T = 1.5s S1 = H/ L=1%

S2 = H/L =2% S3 = H/L =3%