Figure
1.
The simulation of the linear part $ {\eta ^{\left( 1 \right)}}(x,t) $ which contains 150 wave elevation points.
| Citation: | YANG Zhitong, LUO Yiyong. Low-frequency variability of the shallow meridional overturning circulation in the South China Sea[J]. Acta Oceanologica Sinica, 2016, 35(3): 10-20. doi: 10.1007/s13131-016-0826-9
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Ocean wave energy refers to the harnessing of the herculean power of ocean waves. Ocean waves hold a gargantuan amount of untapped energy, some of which we can use to power at least a portion of the world’s everyday electricity. The ocean wave energy has the advantages of being highly predictable, renewable and eco-friendly. An engineering device utilized for exploiting the ocean wave power is called a wave energy converter (WEC). In order to successfully design a wave energy converter, accurately simulating the random ocean waves in the WEC dynamic analysis process is of uttermost importance. However, until present, the majority of the people in the worldwide wave energy research community have applied simple linear irregular waves in their WEC dynamic simulation processes (Manuel et al., 2018; Sirnivas et al., 2016; Tom et al., 2016, 2018, 2019). The linear irregular wave model has the disadvantages that it can only generate unrealistic waves with horizontal symmetries, i.e., the generated waves have statistically symmetric wave crests and troughs. This linear wave model is only suitable for approximately simulating random waves from a very mild sea state in a very deep sea. However, real world ocean waves will become statistically asymmetric (i.e., having sharper and higher crests but smoother and shallower troughs) in a harsh deep sea or at a shallow water coastal site.
Fernandes and Fonseca (2013) has pointed out that most of the proposed WECs will be installed and operated in shallow water coastal sites where the water depths are less than 90 m. Because the ocean waves in these shallow water sites will become statistically asymmetric, the linearly simulated statistically symmetric waves obviously should not be used as the inputs in the analysis and design of most of the proposed WECs.
In order to generate shallow water random waves with statistical asymmetries, Lindgren (2015) and Wang (2019) applied quasi-linear wave models for simulating the movements of individual water particles. However, the random wave simulations presented in Lindgren (2015) and Wang (2019) assumed that wave energy is traveling in only one direction (considered the same direction as the wind). That is to say, Lindgren (2015) and Wang (2019) had respectively performed their stochastic wave simulations based on a uni-directional wave spectrum. Similarly, when Wang (2018a, b) and Wang and Wang (2018) used nonlinear wave models for studying the power performances of wave energy converters, they also applied uni-directional wave spectra during their stochastic simulation of asymmetric waves.
In the real world, however, wind-generated ocean wave energy does not necessarily propagate in the same direction as the wind; instead, the ocean wave energy usually spreads over various directions. Therefore, for an accurate description of random seas, it is necessary to clarify the spreading status of energy. The wave spectrum representing energy in a specified direction is called the directional spectrum, denoted by
Motivated by the aforementioned facts, in this paper the power performances of a WEC operating in a nonlinear random sea characterized by a multi-directional spectrum
The reminder of this paper will be organized as follows: In Section 2 the theories behind the second order random wave simulation based on a multi-directional spectrum will be elucidated, and the measured wave elevation data from a multi-directional coastal sea will be utilized to validate the accuracies of the second order nonlinear random wave simulation method. In Section 3 the theoretical background of the nonlinear dynamic filter of a wave energy converter will be provided. In Section 4 the calculation results of some specific calculation examples will be presented and discussed, with concluding remarks finally summarized in Section 5.
The fluid region is described by using the 3D Cartesian coordinates (x, y, z), with x the longitudinal coordinate, y the transverse coordinate, and z the vertical coordinate (positive upwards). Time is denoted by t. The location of the free surface is at z = η(x, y, t) at a specific time of t.
For real fluid flows in the sea surface, it is reasonable to neglect the effects of viscosity. Meanwhile, sea water has a low compressibility, and therefore normally sea water flows are incompressible. Considering the aforementioned facts, it is reasonably accurate to assume that the fluids on the sea surface are ideal (i.e., incompressible and inviscid). Furthermore, rotation of a fluid particle can be caused only by a torque applied by shear forces on the sides of the particle. Since shear forces are absent in an ideal fluid, the flow of ideal fluids is essentially irrotational. Then, the velocity potential
| $${\nabla ^2}\varPhi = 0,$$ | (1) |
| $$\frac{{\partial \varPhi }}{{\partial t}} + \frac{1}{2}{\left( {\nabla \varPhi } \right)^2} + gz = 0,\;\;\begin{array}{*{20}{c}} {z = \eta \left( {x,y,t} \right)} . \end{array}$$ | (2) |
| $$\frac{{\partial \eta }}{{\partial t}} + \frac{{\partial \varPhi }}{{\partial x}}\frac{{\partial \eta }}{{\partial x}} + \frac{{\partial \varPhi }}{{\partial y}}\frac{{\partial \eta }}{{\partial y}} - \frac{{\partial \varPhi }}{{\partial z}} = 0,\;\;\begin{array}{*{20}{c}} {z = \eta \left( {x,y,t} \right)} . \end{array}$$ | (3) |
| $$\frac{{\partial \varPhi }}{{\partial z}} = 0,\;\;\begin{array}{*{20}{c}} {z = - d} . \end{array}$$ | (4) |
In this study the following expansion is utilized to solve the system (1)–(4):
| $$\left\{ \begin{aligned} \varPhi = {\varPhi ^{\left( 1 \right)}} + {\varPhi ^{\left( 2 \right)}} + ... \\ \eta = {\eta ^{\left( 1 \right)}} + {\eta ^{\left( 2 \right)}} + ... \\ \end{aligned} \right.\qquad\!\!\!\!\!\!\!\!,\quad {\frac{{{\varPhi ^{\left( {n + 1} \right)}}}}{{{\varPhi ^{\left( n \right)}}}} = \frac{{{\eta ^{\left( {n + 1} \right)}}}}{{{\eta ^{\left( n \right)}}}}} O\left( \varepsilon \right),$$ | (5) |
where ε is a small parameter which is typically proportional to the wave steepness. For an irregular sea state characterized by a specific wave spectrum
| $${\varPhi ^{\left( 1 \right)}}(x,t) = {\rm{Re}} \sum\limits_{n = 1}^N {\frac{{{\rm{i}}g{c_n}}}{{{\omega _n}}}} \frac{{\cosh {k_n}\left( {z + d} \right)}}{{\cosh {k_n}d}}{{\rm{e}}^{{{i}}({\omega _n}t - {k_n}x + {\varepsilon _n})}},$$ | (6) |
| $${\eta ^{\left( 1 \right)}}(x,t) = {\rm{Re}} \sum\limits_{n = 1}^N {{c_n}} {{\rm{e}}^{{{i}}({\omega _n}t - {k_n}x + {\varepsilon _n})}},$$ | (7) |
where
| $${\omega _n}^2 = g{k_n}\tanh ({k_n}d),$$ | (8) |
where
Equation (7) describes an ideal linear irregular sea model and this model has been derived under the assumption that the wave heights are small compared to the wave length. Then, the surface elevation resulting from irregular ocean waves can be approximated as a superposition of multiple harmonic waves with different amplitudes and phases. Furthermore, in this ideal linear irregular sea model the random phase angles,
| $$\begin{split} {\varPhi ^{\left( 2 \right)}}(x,t) = & 2{\rm{Re}} \sum\limits_{n = 1}^N \sum\limits_{m = - 1}^N {{\rm{i}}{c_n}{c_m}} P({\omega _n},{\omega _m})\frac{{\cosh \left( {{k_n} + {k_m}} \right)\left( {z + d} \right)}}{{\cosh \left( {{k_n} + {k_m}} \right)d}}\times\\ &{{\rm{e}}^{{{i}}({\omega _n}t - {k_n}x + {\varepsilon _n})}} {{\rm{e}}^{{{i}}({\omega _m}t - {k_m}x + {\varepsilon _n})}}+ { 2\sum\limits_{n = 1}^N {\frac{{{c_n}^2g{k_n}}}{{\sinh 2{k_n}h}}} t} , \end{split} $$ | (9) |
| $$\begin{split} &{\eta ^{\left( 2 \right)}}(x,t) = {\rm{Re}} \sum\limits_{m = 1}^N \sum\limits_{n = 1}^N {{c_m}} {c_n}\times\\ &{\left( {r_{mn}{{\rm{e}}^{{{i}}\left( {{\omega _m}t - {k_m}x + {\varepsilon _m} + {\omega _n}t - {k_n}x + {\varepsilon _n}} \right)}} + {q_{mn}}{{\rm{e}}^{{{i}}\left( {{\omega _m}t - {k_m}x + {\varepsilon _m} - {\omega _n}t + {k_n}x - {\varepsilon _n}} \right)}}} \right)}. \end{split} $$ | (10) |
The terms
| $$ P({\omega _n},{\omega _m}) = \left( {1 - {\text{δ} _{ - n,m}}} \right) \dfrac{{\dfrac{{{g^2}{k_n}{k_m}}}{{2{\omega _n}{\omega _m}}} - \dfrac{1}{4}\left( {{\omega _n}^2 + {\omega _m}^2 + {\omega _n}{\omega _m}} \right) + \dfrac{{{g^2}}}{4}\dfrac{{\left( {{\omega _n}{k_m}^2 + {\omega _m}{k_n}^2} \right)}}{{{\omega _n}{\omega _m}\left( {{\omega _n} + {\omega _m}} \right)}}}}{{\left( {{\omega _n} + {\omega _m}} \right) - g\dfrac{{{k_n} + {k_m}}}{{\left( {{\omega _n} + {\omega _m}} \right)}}\tanh \left( {({k_n} + {k_m})d} \right)}},$$ | (11) |
| $$\begin{split} r_{mn}^{} =& - \left( {\dfrac{1}{g}} \right)\left( {\dfrac{{\left( {\dfrac{1}{{4{\omega _m}{\omega _n}}}} \right)2\left( {{\omega _m} + {\omega _n}} \right)\left( {{\omega _n}^2{\omega _m}^2 - {k_n}{k_m}{g^2}} \right) + {\omega _n}\left( {{\omega _m}^4 - {g^2}{k_m}^2} \right) + {\omega _m}\left( {{\omega _n}^4 - {g^2}{k_n}^2} \right)}}{{{{\left( {{\omega _m} + {\omega _n}} \right)}^2}\cosh \left( {({k_m} + {k_n})d} \right) - g({k_m} + {k_n})\sinh \left( {({k_m} + {k_n})d} \right)}}} \right)\times \\ &\left( {{\omega _m} + {\omega _n}} \right)\cosh \left( {({k_m} + {k_n})d} \right) - \left( {\dfrac{1}{{4g{\omega _m}{\omega _n}}}} \right)\left( {{k_m}{k_n}{g^2} - {\omega _n}^2{\omega _m}^2} \right) + \left( {\dfrac{1}{{4g}}} \right)\left( {{\omega _m}^2 + {\omega _n}^2} \right) , \end{split} $$ | (12) |
| $$\begin{split} q_{mn}^{} =& - \left( {\dfrac{1}{g}} \right)\left( {\dfrac{{\left( {\dfrac{1}{{4{\omega _m}{\omega _n}}}} \right)2\left( {{\omega _m} - {\omega _n}} \right)\left( {{\omega _n}^2{\omega _m}^2 + {k_n}{k_m}{g^2}} \right) - {\omega _n}\left( {{\omega _m}^4 - {g^2}{k_m}^2} \right) + {\omega _m}\left( {{\omega _n}^4 - {g^2}{k_n}^2} \right)}}{{{{\left( {{\omega _n} - {\omega _m}} \right)}^2}\cosh \left( {\left| {{k_m} - {k_n}} \right|d} \right) - g\left| {{k_n} - {k_m}} \right|\sinh \left( {\left| {{k_n} - {k_m}} \right|d} \right)}}} \right)\times \\ &\left( {{\omega _n} - {\omega _m}} \right)\cosh \left( {\left| {{k_n} - {k_m}} \right|d} \right) - \left( {\dfrac{1}{{4g{\omega _m}{\omega _n}}}} \right)\left( {{k_m}{k_n}{g^2} + {\omega _n}^2{\omega _m}^2} \right) + \left( {\dfrac{1}{{4g}}} \right)\left( {{\omega _m}^2 + {\omega _n}^2} \right) , \end{split} $$ | (13) |
where the Kroenecker delta (
| $$\begin{split} \eta (x,t) =& {\eta ^{\left( 1 \right)}}(x,t) + {\eta ^{\left( 2 \right)}}(x,t) = {\rm{Re}} \sum\limits_{n = 1}^N {{c_n}} {{\rm{e}}^{{{i}}({\omega _n}t - {k_n}x + {\varepsilon _n})}} + \\ & {\rm{Re}} \sum\limits_{m = 1}^N \sum\limits_{n = 1}^N {{c_m}} {c_n}\big( {r_{mn}}{{\rm{e}}^{{{i}}\left( {{\omega _m}t - {k_m}x + {\varepsilon _m} + {\omega _n}t - {k_n}x + {\varepsilon _n}} \right)}} + \\ & {q_{mn}}{{\rm{e}}^{{{i}}\left( {{\omega _m}t - {k_m}x + {\varepsilon _m} - {\omega _n}t + {k_n}x - {\varepsilon _n}} \right)}} \big). \end{split} $$ | (14) |
The proposed second order random wave simulation method starts with taking a multi-directional spectrum
| $${c_n} = \sqrt {2S\left( \omega \right)\Delta \omega }, $$ | (15) |
where
As we know, a wave spectrum is actually obtained from the observed time series of water elevation. The transformation from water elevation to wave spectrum is based on the assumption that irregular waves can be treated as a combination of linear waves with different amplitudes, different frequencies and different phases. It would be better to give an explanation for how to reasonably consider the nonlinearity of waves in the inverse transformation (i.e., the transformation from the wave spectrum to the nonlinear water elevation). In the following, we will raise a specific calculation example of this inverse transformation.
This study shows as an example to use Eq. (7) to simulate the linear Gaussian wave time histories of a sea state with a JONSWAP spectrum with a significant wave height
Figure 1 shows our simulation of the linear part wave elevation time series
Figure 2 shows our simulation of the second order correction part wave elevation time series
In Fig. 3 the blue curve shows our simulation of the entire
In this sub-section, the accuracy of the proposed second order random wave simulation method will be validated by a calculation example. Specifically, the proposed second order random wave simulation method will be applied for calculating the wave crest amplitude exceedance probabilities of a sea state with a multi-directional wave spectrum based on the measured surface elevation data at the coast of Yura. The measured surface elevation data at the coast of Yura were obtained at a location 3 km off Yura fishing harbor facing the Sea of Japan. The observations were carried out during the period from 11:10 am to 2:08 pm on November 24, 1987 by the Ship Research Institute, Ministry of Transport of Japan. Temporal sea surface elevations were measured with ultrasonic-type wave gages installed at three points in 42 m water depth. The sampling time interval during the measurement was 1 s. Based on these measured Yura coast surface elevation data, a multi-directional wave spectrum was estimated by using the Maximum Likelihood Method and is shown in Fig. 4. Fig. 5 shows a part of the measured wave elevation time series by the mid-point wave gage at this site.
In the field of ocean engineering, based on a specific wave spectrum, an empirical (or theoretical) model or numerical simulation methods can be used to calculate the exceedance probabilities of the wave crest amplitudes (which is directly related to the occurrence probability of the extreme waves). In the following, taking the multi-directional wave spectrum in Fig. 4 as a calculation example, this study will test the proposed second order random wave simulation method to calculate the exceedance probabilities of the wave crest amplitudes, and compare its accuracy with those of a theoretical model or a linear simulation method.
Figure 6 shows the calculated wave crest amplitudes exceedance probabilities based on the multi-directional spectrum shown in Fig. 4 and the measured Yura coast wave data. The continuous green curve in Fig. 6 represents the calculation results of the wave crest amplitudes exceedance probabilities directly obtained from the measured Yura coast wave data that contains 10700 wave elevation points. This continuous green curves based on the measured Yura coast wave data are used as the benchmark against the accuracy of the results from the various numerical simulation methods and from an existing theoretical wave crest amplitudes model which is checked.
The continuous red curve in Fig. 6 represents the results of the wave crest amplitudes exceedance probabilities obtained from using the theoretical Rayleigh distribution model expressed as follows:
| $$P\left( {{A_\rm{c}} > h} \right) = {\rm{exp}}\left( { - 8{{\left( {\frac{h}{{{H_{\rm{S}}}}}} \right)}^2}} \right),$$ | (16) |
where h represents the wave crest amplitude,
In order to more accurately calculate the wave crest amplitudes exceedance probabilities, the proposed second order random wave simulation method was tried to use. In Fig. 6, a pink * represents the result of the wave crest amplitudes exceedance probability obtained from using the proposed second order random wave simulation method based on the multi-directional spectrum shown in Fig. 4. The proposed second order random wave simulation process started with taking the multi-directional spectrum shown in Fig. 4 and integrating the energy over all directions to give the total energy at each frequency. The obtained equivalent frequency spectrum was then utilized to generate a nonlinear wave elevation time series of 800000 points by applying Eqs (8) and (14). Next, the wave crest amplitudes time series were extracted from these 800000 wave elevation points. Finally, the wave crests amplitudes exceedance probabilities were obtained by statistical and mathematical processing the extracted wave crest amplitudes time series. Note that in the figure legend the phrase “from nonlinear simulation” exactly means “from second order random wave simulation”. The wave crest amplitudes exceedance probabilities obtained from using the proposed second order random wave simulation method fit quite well with the corresponding benchmark results obtained directly from the measured Yura coast wave data. The accuracy of the proposed second order random wave simulation method is therefore convincingly validated.
In the practice ocean engineering, the power performances prediction of a WEC is usually carried out by inputting simulated ocean waves in a nonlinear dynamic filter and performing subsequent time domain simulations. The mathematical equations of the WEC nonlinear dynamic filter are written as follows:
| $$\begin{split} {{{M}}_{{\rm{RB}}}}{\ddot{ x}}\left( t \right) =& - {{A}}\left( \infty \right){\ddot{ x}}\left( t \right) - \int_0^t {{{\bar {{K}}}}} \left( {t - \tau } \right){\dot{ x}}\left( \tau \right){\rm{d}}\tau + \\ & {{{P}}_{{\rm{wave}}}}\left( t \right) + {{{P}}_{{\rm{ext}}}}\left( t \right) + {{{P}}_{{\rm{visc}}}}\left( t \right) - {{{P}}_{{\rm{hs}}}}, \end{split}$$ | (17) |
where
In the field of ocean engineering, the WEC hydrodynamic memory effects are captured in Eq. (17) by the convolution integral term that is a function of
However, up until present, the wave excitation load
Our specific calculation examples will be carried out regarding a heaving two-body point absorber wave energy converter. Figure 7 shows this specific wave energy converter modeled in WEC-Sim (
It should be emphasized that this two-body point absorber WEC is installed in a coastal sea area with a water depth of only 49.5 m. Obviously, the random waves in this shallow water area will be nonlinear according to the nonlinear random wave theory in Ochi (1998).
In most wave energy exploitation engineering projects that met in the real world, the only information knew beforehand regarding a sea state usually will be a specific wave spectrum. In Section 2, the accuracy of the proposed second order random wave simulation method was already demonstrated in generating irregular waves based on a measured multi-directional wave spectrum. In the following calculation examples, the proposed second order random wave simulation method was utilized for generating nonlinear irregular waves for calculating the wave excitation load
| $$S\left( {\omega ,\theta } \right) = S\left( \omega \right)D\left( \theta \right),$$ | (18) |
where
The mathematical expression for the frequency spectrum
| $$\begin{split} S(\omega ) =& 5.061\frac{{{g^2}}}{{{\omega ^5}}}\frac{{{H_{\rm{S}}}^2}}{{{T_p}^4}}\left( {1 - 0.287\ln (\gamma )} \right)\times\\ &\exp \left( { - \frac{5}{4}{{\left( {\frac{{\omega {T_p}}}{{2{\text π} }}} \right)}^{ - 4}}} \right){\gamma ^{{\rm{exp}} \left( { - 0.5\left( {\frac{{{{\left( {\frac{{\omega {T_p}}}{{2{\text π} }} - 1} \right)}^2}}}{{{\text{δ} ^2}}}} \right)} \right)}}, \end{split}$$ | (19) |
where
| $$\gamma = \exp ( {3.484( {1 - 0.197\;5( {0.036 - 0.005\;6{T_p}/\sqrt {{H_{\rm{S}}}} } ){T_p}^4/{H_{\rm{S}}}^2} )} ).$$ |
The mathematical expression for the spreading function
| $$D\left( \theta \right) = \frac{{\varGamma \left( {s + 1} \right)}}{{2\sqrt {\text {π}} \varGamma \left( {s + 1/2} \right)}}{\cos ^{2s}}\left( {\frac{\theta }{2}} \right),$$ | (11) |
where the spreading parameter s=15.
Figure 10 is a corresponding polar plot of the multi-directional JONSWAP spectrum as shown in Fig. 9.
Next, the calculation results of the absorbed power of the aforementioned wave energy converter were placed in an ideal linear sea versus in a multi-directional nonlinear random sea. Figure 11 shows the predicted WEC absorbed power time series under the sea state of linear irregular waves based on a multi-directional JONSWAP wave spectrum with
| Statistical measures significant wave height/m | Mean value under sea state of linear waves/W | Mean value under sea state of second order nonlinear waves/W | Standard deviation value under sea state of linear waves/W | Standard deviation value under sea state of second order nonlinear waves/W | Sum value under sea state of linear waves/W | Sum value under sea state of second order nonlinear waves/W |
| 1 | 138437.04116 | 138511.45857 | 147967.47721 | 148037.48374 | 3.59798×108 | 3.59991×108 |
| 2 | 183201.45543 | 183729.92515 | 219208.08886 | 219533.04823 | 4.76141×108 | 4.77514×108 |
| 3 | 199926.33733 | 199983.50274 | 249739.80807 | 251762.62593 | 5.19609×108 | 5.19757×108 |
| 4 | 274600.57560 | 276111.71603 | 403700.40894 | 408900.02558 | 7.13687×108 | 7.17614×108 |
| 5 | 315077.75210 | 316133.23968 | 454553.94521 | 454063.30863 | 8.18887×108 | 8.21630×108 |
| 6 | 433808.09451 | 444136.33628 | 726265.21959 | 769908.28318 | 1.12747×109 | 1.15431×109 |
| 7 | 490131.33153 | 503258.63700 | 776567.66295 | 809812.45690 | 1.27385×109 | 1.30797×109 |
| 8 | 594029.94690 | 621211.75487 | 978890.28927 | 1.02946×106 | 1.54388×109 | 1.61453×109 |
| 9 | 796963.82665 | 874346.20999 | 1.34261×106 | 1.53251×106 | 2.07131×109 | 2.27243×109 |
| 10 | 767280.02013 | 825077.57589 | 1.17164×106 | 1.30227×106 | 1.99416×109 | 2.14438×109 |
| 11 | 1.1492×106 | 1.2581×106 | 2.21824×106 | 2.68345×106 | 2.98680×109 | 3.26980×109 |
| 12 | 1.0986×106 | 1.1918×106 | 2.53972×106 | 2.64326×106 | 2.85525×109 | 3.09748×109 |
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By carefully comparing and analyzing the calculation results, the mean value of the 1 200 s WEC absorbed power under the sea state of ideal linear irregular waves is smaller than the corresponding mean absorbed power value when inputting nonlinear irregular waves. In order to investigate the influences of choosing different significant wave height values on the power performances of wave energy converters, the aforementioned WEC absorbed power values were sequently calculated under 11 other sea states characterized with a multi-directional JONSWAP wave spectrum with the following parameters respectively:
In order to further investigate the influences of using different random seed numbers for generating irregular waves on the power performances of wave energy converters, the aforementioned WEC absorbed power values were sequently calculated when inputting 11 different irregular wave elevation time series simulated based on the same multi-directional JONSWAP wave spectrum (
| Statistical measures significant wave height/m | Mean value under sea state of linear waves/W | Mean value under sea state of second order nonlinear waves/W | Standard deviation value under sea state of linear waves/W | Standard deviation value under sea state of second order nonlinear waves/W | Sum value under sea state of linear waves/W | Sum value under sea state of second order nonlinear waves/W |
| 12 | 1.18485×106 | 1.38075×106 | 2.01713×106 | 2.76839×106 | 3.07941×109 | 3.58857×109 |
| 12 | 1.12270×106 | 1.22233×106 | 2.04328×106 | 2.41229×106 | 2.91791×109 | 3.17685×109 |
| 12 | 1.15923×106 | 1.27871×106 | 2.22910×106 | 2.63901×106 | 3.01285×109 | 3.32336×109 |
| 12 | 1.47223×106 | 1.58350×106 | 2.69001×106 | 3.28096×106 | 3.82632×109 | 4.11550×109 |
| 12 | 1.18207×106 | 1.31431×106 | 2.10716×106 | 2.70589×106 | 3.07219×109 | 3.41588×109 |
| 12 | 1.06410×106 | 1.12036×106 | 1.74052×106 | 1.80526×106 | 2.76560×109 | 2.91183×109 |
| 12 | 1.32898×106 | 1.44481×106 | 2.55403×106 | 2.81290×106 | 3.45403×109 | 3.75506×109 |
| 12 | 1.12296×106 | 1.31056×106 | 1.97489×106 | 2.46568×106 | 2.91857×109 | 3.40615×109 |
| 12 | 1.40660×106 | 1.57163×106 | 2.67732×106 | 3.01967×106 | 3.65575×109 | 4.08466×109 |
| 12 | 1.10365×106 | 1.18066×106 | 1.76090×106 | 2.01755×106 | 2.86838×109 | 3.06853×109 |
| 12 | 1.27946×106 | 1.48130×106 | 2.27087×106 | 2.72554×106 | 3.32531×109 | 3.84990×109 |
| 12 | 1.04726×106 | 1.13799×106 | 1.62224×106 | 1.92767×106 | 2.72184×109 | 2.95764×109 |
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In the present study the power performances of a two body point absorber WEC operating in a nonlinear multi-directional random sea have been rigorously investigated. The absorbed power of the WEC Power-Take-Off system has been predicted by incorporating a second order random wave model into a nonlinear dynamic filter. This is a new approach that is uniquely proposed to ocean wave energy research community. It has been demonstrated in this paper that the second order random wave model can be utilized to accurately simulate nonlinear irregular waves in a multi-directional sea. This will help us to avoid the inaccuracies resulting from using a first order linear wave model in the WEC simulation process. The predicted results in this paper have been systematically analyzed and compared, and the advantages of using our proposed new approach have been convincingly substantiated. The research findings in this paper highlight the vital importance of using the nonlinear irregular waves simulated based on a multi-directional wave spectrum when studying the power performances of wave energy converters. In the future if enough funding is secured, the proposed second order random wave simulation method will also be compared with the related WEC model physical test data in order to fully verify the accuracy of the model.
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| Statistical measures significant wave height/m | Mean value under sea state of linear waves/W | Mean value under sea state of second order nonlinear waves/W | Standard deviation value under sea state of linear waves/W | Standard deviation value under sea state of second order nonlinear waves/W | Sum value under sea state of linear waves/W | Sum value under sea state of second order nonlinear waves/W |
| 1 | 138437.04116 | 138511.45857 | 147967.47721 | 148037.48374 | 3.59798×108 | 3.59991×108 |
| 2 | 183201.45543 | 183729.92515 | 219208.08886 | 219533.04823 | 4.76141×108 | 4.77514×108 |
| 3 | 199926.33733 | 199983.50274 | 249739.80807 | 251762.62593 | 5.19609×108 | 5.19757×108 |
| 4 | 274600.57560 | 276111.71603 | 403700.40894 | 408900.02558 | 7.13687×108 | 7.17614×108 |
| 5 | 315077.75210 | 316133.23968 | 454553.94521 | 454063.30863 | 8.18887×108 | 8.21630×108 |
| 6 | 433808.09451 | 444136.33628 | 726265.21959 | 769908.28318 | 1.12747×109 | 1.15431×109 |
| 7 | 490131.33153 | 503258.63700 | 776567.66295 | 809812.45690 | 1.27385×109 | 1.30797×109 |
| 8 | 594029.94690 | 621211.75487 | 978890.28927 | 1.02946×106 | 1.54388×109 | 1.61453×109 |
| 9 | 796963.82665 | 874346.20999 | 1.34261×106 | 1.53251×106 | 2.07131×109 | 2.27243×109 |
| 10 | 767280.02013 | 825077.57589 | 1.17164×106 | 1.30227×106 | 1.99416×109 | 2.14438×109 |
| 11 | 1.1492×106 | 1.2581×106 | 2.21824×106 | 2.68345×106 | 2.98680×109 | 3.26980×109 |
| 12 | 1.0986×106 | 1.1918×106 | 2.53972×106 | 2.64326×106 | 2.85525×109 | 3.09748×109 |
DownLoad:
CSV
| Statistical measures significant wave height/m | Mean value under sea state of linear waves/W | Mean value under sea state of second order nonlinear waves/W | Standard deviation value under sea state of linear waves/W | Standard deviation value under sea state of second order nonlinear waves/W | Sum value under sea state of linear waves/W | Sum value under sea state of second order nonlinear waves/W |
| 12 | 1.18485×106 | 1.38075×106 | 2.01713×106 | 2.76839×106 | 3.07941×109 | 3.58857×109 |
| 12 | 1.12270×106 | 1.22233×106 | 2.04328×106 | 2.41229×106 | 2.91791×109 | 3.17685×109 |
| 12 | 1.15923×106 | 1.27871×106 | 2.22910×106 | 2.63901×106 | 3.01285×109 | 3.32336×109 |
| 12 | 1.47223×106 | 1.58350×106 | 2.69001×106 | 3.28096×106 | 3.82632×109 | 4.11550×109 |
| 12 | 1.18207×106 | 1.31431×106 | 2.10716×106 | 2.70589×106 | 3.07219×109 | 3.41588×109 |
| 12 | 1.06410×106 | 1.12036×106 | 1.74052×106 | 1.80526×106 | 2.76560×109 | 2.91183×109 |
| 12 | 1.32898×106 | 1.44481×106 | 2.55403×106 | 2.81290×106 | 3.45403×109 | 3.75506×109 |
| 12 | 1.12296×106 | 1.31056×106 | 1.97489×106 | 2.46568×106 | 2.91857×109 | 3.40615×109 |
| 12 | 1.40660×106 | 1.57163×106 | 2.67732×106 | 3.01967×106 | 3.65575×109 | 4.08466×109 |
| 12 | 1.10365×106 | 1.18066×106 | 1.76090×106 | 2.01755×106 | 2.86838×109 | 3.06853×109 |
| 12 | 1.27946×106 | 1.48130×106 | 2.27087×106 | 2.72554×106 | 3.32531×109 | 3.84990×109 |
| 12 | 1.04726×106 | 1.13799×106 | 1.62224×106 | 1.92767×106 | 2.72184×109 | 2.95764×109 |
DownLoad:
CSV
| Statistical measures significant wave height/m | Mean value under sea state of linear waves/W | Mean value under sea state of second order nonlinear waves/W | Standard deviation value under sea state of linear waves/W | Standard deviation value under sea state of second order nonlinear waves/W | Sum value under sea state of linear waves/W | Sum value under sea state of second order nonlinear waves/W |
| 1 | 138437.04116 | 138511.45857 | 147967.47721 | 148037.48374 | 3.59798×108 | 3.59991×108 |
| 2 | 183201.45543 | 183729.92515 | 219208.08886 | 219533.04823 | 4.76141×108 | 4.77514×108 |
| 3 | 199926.33733 | 199983.50274 | 249739.80807 | 251762.62593 | 5.19609×108 | 5.19757×108 |
| 4 | 274600.57560 | 276111.71603 | 403700.40894 | 408900.02558 | 7.13687×108 | 7.17614×108 |
| 5 | 315077.75210 | 316133.23968 | 454553.94521 | 454063.30863 | 8.18887×108 | 8.21630×108 |
| 6 | 433808.09451 | 444136.33628 | 726265.21959 | 769908.28318 | 1.12747×109 | 1.15431×109 |
| 7 | 490131.33153 | 503258.63700 | 776567.66295 | 809812.45690 | 1.27385×109 | 1.30797×109 |
| 8 | 594029.94690 | 621211.75487 | 978890.28927 | 1.02946×106 | 1.54388×109 | 1.61453×109 |
| 9 | 796963.82665 | 874346.20999 | 1.34261×106 | 1.53251×106 | 2.07131×109 | 2.27243×109 |
| 10 | 767280.02013 | 825077.57589 | 1.17164×106 | 1.30227×106 | 1.99416×109 | 2.14438×109 |
| 11 | 1.1492×106 | 1.2581×106 | 2.21824×106 | 2.68345×106 | 2.98680×109 | 3.26980×109 |
| 12 | 1.0986×106 | 1.1918×106 | 2.53972×106 | 2.64326×106 | 2.85525×109 | 3.09748×109 |
| Statistical measures significant wave height/m | Mean value under sea state of linear waves/W | Mean value under sea state of second order nonlinear waves/W | Standard deviation value under sea state of linear waves/W | Standard deviation value under sea state of second order nonlinear waves/W | Sum value under sea state of linear waves/W | Sum value under sea state of second order nonlinear waves/W |
| 12 | 1.18485×106 | 1.38075×106 | 2.01713×106 | 2.76839×106 | 3.07941×109 | 3.58857×109 |
| 12 | 1.12270×106 | 1.22233×106 | 2.04328×106 | 2.41229×106 | 2.91791×109 | 3.17685×109 |
| 12 | 1.15923×106 | 1.27871×106 | 2.22910×106 | 2.63901×106 | 3.01285×109 | 3.32336×109 |
| 12 | 1.47223×106 | 1.58350×106 | 2.69001×106 | 3.28096×106 | 3.82632×109 | 4.11550×109 |
| 12 | 1.18207×106 | 1.31431×106 | 2.10716×106 | 2.70589×106 | 3.07219×109 | 3.41588×109 |
| 12 | 1.06410×106 | 1.12036×106 | 1.74052×106 | 1.80526×106 | 2.76560×109 | 2.91183×109 |
| 12 | 1.32898×106 | 1.44481×106 | 2.55403×106 | 2.81290×106 | 3.45403×109 | 3.75506×109 |
| 12 | 1.12296×106 | 1.31056×106 | 1.97489×106 | 2.46568×106 | 2.91857×109 | 3.40615×109 |
| 12 | 1.40660×106 | 1.57163×106 | 2.67732×106 | 3.01967×106 | 3.65575×109 | 4.08466×109 |
| 12 | 1.10365×106 | 1.18066×106 | 1.76090×106 | 2.01755×106 | 2.86838×109 | 3.06853×109 |
| 12 | 1.27946×106 | 1.48130×106 | 2.27087×106 | 2.72554×106 | 3.32531×109 | 3.84990×109 |
| 12 | 1.04726×106 | 1.13799×106 | 1.62224×106 | 1.92767×106 | 2.72184×109 | 2.95764×109 |
DownLoad:
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