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On the accurate and efficient calibration of a 3D wave basin
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Ocean Engineering 35 (2008) 763–773 www.elsevier.com/locate/oceaneng
On the accurate and efficient calibration of a 3D wave basin S.R. Masterton1, C. Swan Department of Civil and Environmental Engineering, Imperial College, London SW7 2AZ, UK Received 10 August 2007; accepted 11 February 2008 Available online 10 March 2008
Abstract This paper describes the calibration procedure adopted for the new 3D wave basin located in the Hydrodynamics Laboratory at Imperial College London. Unlike traditional calibrations, based on observations of regular wave trains, the method described herein uses a focused wave approach. Such waves, produced by the constructive interference of freely propagating wave components, have led to a number of recent advances in theoretical wave modelling in which it was essential to know the underlying linear components. In the context of a laboratory study, similar advantages can be realised provided the linear wave components generated by the wave paddles are well defined. This, in turn, can only be achieved if the calibration is sufficiently accurate. The current study provides a calibration based upon a realistic JONSWAP spectrum, describes the details of the methodology employed, and highlights how the application of focused wave techniques eliminates spurious calibration effects due to unwanted reflections from the boundaries of the basin. The final calibration is verified through the generation of test cases, involving linear and nonlinear, unidirectional and directionally spread waves. These confirm both the accuracy of the calibration and the suitability of the methods employed. r 2008 Elsevier Ltd. All rights reserved. Keywords: Laboratory wave generation; Wave basin calibration; Directional waves; Focused waves
1. Introduction Despite a succession of theoretical advances, physical model testing remains an effective method of investigating problems involving ocean waves. Indeed, if the problem includes waves interacting with a structure model testing represents the only reliable method, particularly where the full nonlinearity of the problem must be retained. In itself, the use of scaled models does not represent a new art. However, recent advances in wave modelling (particularly relating to extreme wave groups) necessitate the use of more sophisticated testing techniques. In the context of model testing, the use of these improved methods requires the input wave conditions, or more specifically the underlying linear wave spectrum generated at the wave paddles, to be precisely controlled. If this is to be achieved, the fundamental issue becomes that of accurate calibration. Corresponding author. Tel.: +44 207 594 5999; fax: +44 207 594 5991.
E-mail address: [email protected] (C. Swan). Now at: Shell International Exploration and Production, 2288 GS, Rijswijk, The Netherlands. 1
0029-8018/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2008.02.002
To achieve the most realistic representation of a given wave field, directionality must be included and this necessitates the use of a 3D wave basin. Compared to 2D wave flumes, these facilities are larger, more sophisticated and considerably more expensive. Consequently, there are relatively few examples of 3D wave basins operating worldwide, and these tend to employ ‘traditional’ testing methods; the sea state being specified in terms of some target spectrum and quantified (iteratively) on the basis of a long time-domain simulation at a single point in space. Using these techniques the control of directionality is inherently very poor, principally due to the build up of reflections, with calibrations usually based upon a coarse regular wave approach. The generation of focused wave groups in a 3D wave basin is relatively rare, not least because of the difficulty of obtaining an accurate transfer function. However, one notable exception is provided by Johannessen and Swan (2001). The purpose of this study was to quantify the nonlinear interactions arising during the evolution of nonlinear, directionally spread wave groups. As such, the authors state it was: ‘‘critically important to know the
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subscript n nth wave component subscript p peak value ðx; y; zÞ Cartesian co-ordinates z ¼ 0 still water level t time tfoc focal time Tf transfer function S ZZ wave spectrum g peak enhancement factor
Notation A T f o k y Z f
wave amplitude wave period frequency ðT1 Þ wave frequency ð2p TÞ wavenumber direction of propagation surface elevation phase angle
amplitude, frequency, phasing and direction of individual components generated at the wave paddles’’. In seeking to satisfy this goal they achieved the first focused wave calibration, albeit based upon an idealised tophat spectrum, but the details of the process went unreported. The present paper seeks to correct this omission, describing the procedure used to calibrate a new 3D wave basin located in the Hydrodynamics Laboratory at Imperial College London. The paper continues in Section 2 with a brief review of calibration methods. The wave basin and experimental setup are described in Section 3. Sections 4 and 5 describe the calibration process and its subsequent verification. The paper closes with Section 6, highlighting the advantages of the proposed techniques and offering some wider conclusions. 2. Review of calibration methods In those instances where a precise definition of the generated wave conditions is not essential, particularly if it involves long-crested or uni-directional waves, a theoretical calibration based on potential wavemaker theory may be appropriate. Several authors, including Dean and Dalrymple (1992), give linear solutions for a wide variety of paddle types; the results usually expressed in terms of a wave height vs. paddle stroke relationship (see, for example, their Figure 6.2 on p. 172). However, if precise control of the wave generation is required, as is usually the case for research purposes or detailed model testing, the calibration becomes more complicated. From a theoretical perspective, the avoidance of spurious freely propagating wave modes dictates the use of second-order theory. However, depending on the type of wave paddle, there is limited evidence of the success of this approach, particularly when applied to directional wave generation. Furthermore, if the control of Frequency Amplitude Phase Direction
Tf (, )
the paddle (or paddles) is not based upon position alone, and there are good practical reasons why this should be so (Spinneken and Swan, 2008), the appropriate theoretical calibrations do not exist. In such cases accurate wave generation must be based on empirical calibration. 2.1. General approach—regular wave calibration A transfer function, T f , is defined from the relationship between the required or demand output from a paddle bank and the waves actually generated in the wave basin. The transfer function is applied to the demand signal before it is sent to the paddle bank and corrects for both the electrical and mechanical scaling and the inertia of the wave paddles. It is a function of both the wave frequency, o, and the direction of wave propagation, y, and is specified by a gain and a phase. For any given wave component, the output from the paddles should always be equal to the demand signal. This is illustrated in Fig. 1. It is important to note at this stage that the present study is only interested in a linear transfer function. Consequently, T f is not a function of the amplitude, A, and the calibration process is entirely concerned with small linear waves. This approach is consistent with a calibration appropriate to experiments in which the isolation of nonlinear effects represents an important part of the problem to be solved. Owing to its complexity, the transfer function can only be defined empirically and is calculated in an iterative manner. The basic procedure is to generate a series of regular waves which are measured at a known distance from the paddles. The transfer function gain is then corrected by multiplying by the ratio of the demand amplitude divided by the measured amplitude. Clearly, once the measured amplitude approaches the demand
Paddle Bank
Frequency Amplitude Phase Direction Measured Output
Demand Output
Fig. 1. The role of the transfer function, T f .
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amplitude, the correction factor approaches unity and no further iteration is necessary. A similar correction routine can be applied to the phase, having taken into account the phase change associated with: (i) the distance between the measuring location and the paddle bank, k~ x (where k is the wave number and ~ x is the distance from the paddles), and, (ii) the time delay allowed for the wave to propagate to the measuring location, ot (where o is the wave frequency and t is the time). The above description is slightly simplified in that it assumes that the system is entirely linear, with no possibility of experimental errors. In practice this is not the case and, as a result, it is necessary to generate a number of amplitudes, typically around five, for each wave component (o; y). In this manner, a calibration curve, a plot of demand vs. measured amplitude can be drawn and, assuming the line is linear, the gradient of a linear fit used to correct the transfer function gain. Using the same data, the mean phase difference from the five amplitude runs is used to correct the phase of the transfer function. The regular wave approach described above is the most basic method of calculating a transfer function. However, if the gain and phase do not vary smoothly with both the wave frequency and the direction of wave propagation, an accurate definition of the transfer function will be enormously time consuming. This is particularly true if the frequency range is realistically broad (a requirement of the present calibration) and, as a result, a large number of wave components must be calibrated for. This problem is exacerbated by the fact that convergence for a single wave component will often take more than one iteration, where each iteration requires approximately five runs. Clearly, a calibration using this method in a directional wave basin, where each direction must be dealt with separately, becomes entirely impractical. 2.2. Focused wave calibration An alternative to a regular wave calibration is to use transient focused wave groups. Here, a large number of wave components are generated simultaneously and are appropriately phase shifted so that, at a specified time, all the wave components come into phase at the measuring location. Further details of the generation of a focused wave group in a 2D wave flume are given by Baldock et al. (1996) and Baldock and Swan (1996); while Johannessen and Swan (2001) describe the generation of directionally spread focused wave groups in a 3D wave basin. When a focused wave approach is applied in a directional wave basin a separate unidirectional calibration is required for each angle or direction of wave propagation. This is necessary because a single point measurement cannot resolve the direction of propagation. The alternative, a calibration using directionally spread groups is impractical and would require either a large array of wavegauges or some form of spatial measuring technique (Tzivanaki, 2006); the latter being extremely complicated
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to both setup and interpret. Furthermore, for such small waves it would not necessarily be expected to yield a more accurate result. While the time saving is an important motivation, the use of focused wave groups has other significant advantages over a regular wave calibration. First, because wave basins are generally shorter than 2D wave flumes, reflections from the lateral boundaries and the beach must be considered more carefully. The use of transient wave groups limits the effects of reflection by reducing the amplitudes of the waves at the basin boundaries. This is in stark contrast with the regular wave approach where reflections are more of an issue, particularly with regard to the spatial variability of the wave field. Second, the dispersion of the wave group away from the focus location, particularly in directionally spread wave fields, ensures that the waves generated at the paddles are always small. This significantly reduces any potential problems associated with nonlinear wave generation; the latter requiring nonlinear (usually second-order) input signals to avoid the growth of unwanted spurious wave modes. 2.3. Previous work As part of their study of directionally spread, focused wave groups Johannessen and Swan (2001) provide the first evidence of the accurate calibration of a 3D wave basin, although the details of the method employed went unreported. This calibration was undertaken in the wave basin (or widetank) at the University of Edinburgh; a facility that has since been decommissioned. The purpose of the calibration was identical to the present study; the emphasis being placed on the identification of the wave components generated at the wave paddles rather than matching some target statistical properties at a given location. The calibration was initially carried out using a regular wave method. However, despite the generation of 270 different regular wave forms the resolution was poor, with just five frequency components considered every 10 . However, of greater significance was the amplitude variation caused by the regular wave approach, observed to be up to 15%. This was directly attributed to an inadequate control of wave reflections within the wave basin. To overcome these difficulties the first focused wave calibration was undertaken using a narrow-banded tophat laboratory spectrum in which all the frequency components travelling in a specified direction were assumed to be of equal amplitude. 3. Experimental facility and setup The wave basin used in the present study is shown in Fig. 2. It is located in the basement of the Civil and Environmental Engineering Department at Imperial College London and was commissioned in 2003 as part of a major laboratory refurbishment. The facility is 20 m by
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Fig. 2. The 3D wave basin at Imperial College London.
13 m with an operating water depth of 1.5 m. There is also a 3 m deep pit in the centre which is presently covered. The short sides of the wave basin are constructed from glass for visual access and a moving gantry, hung from the ceiling, spans the shortest length of the wave basin. Wavegauges (see below) are fixed to this gantry thereby avoiding any vibration from the paddle bank. Wave generation is achieved by a bank of 56 identical, bottom hinged, flap type wavemakers. These are installed along the longest side of the wave basin. Individual paddles are 0.35 m wide and 0.7 m deep, and are constructed from a lightweight glassfibre. The space behind the wavemakers is kept dry by means of a thin polyurethane-impregnated nylon membrane between paddles. This membrane is sufficiently slack that each individual wave paddle can describe its full range of motion independently of its neighbours. This system incorporates no sliding parts and therefore friction effects are minimal. Hydrostatic pressure is balanced by a system of springs and pulleys attached to a toothed drive belt between each paddle and its high torque servomotor. Individual paddles incorporate a force feedback mechanism, allowing the active absorption of reflected waves that would otherwise build up in the wave basin. This design was pioneered at The University of Edinburgh and works by monitoring the force via a loadcell that is placed between the drive system and the paddle. In this configuration the total force exerted by the water pressure over the front face of the paddle can be incorporated into the control loop, in addition to the usual position and velocity monitoring. Detailed discussion of the force control mechanism lies beyond the scope of the present paper; the interested reader being directed to Spinneken and Swan (2008). Wave generation is controlled via a central PC. The control software assumes the paddle motion is periodic over a long repeat time, which for computational efficiency must be 2n s where n is any positive integer. In the present tests this repeat time was 64 s (see below). Individual wave
components, corresponding to specific frequencies, are defined in terms of so-called wave front numbers; the latter representing the harmonics of the fundamental repeat period. With a repeat time of 64 s, the nth front number corresponds to waves with a period of T n ¼ 64=n; a front number of 64 corresponding to a 1.0 s wave, while a front number of 32 corresponds to a 2.0 s wave. Although the control software allows the efficient generation of wave groups, providing a simple interface with the wavemaking hardware, in practice it is merely a convenience (albeit a significant one) and the calibration process outlined below may be adopted in any wave basin. At the ‘downstream’ end of the wave basin, opposite the wave paddles, the propagating wave energy is absorbed on a smooth, non-perforated beach. This is 2.5 m long, parabolic in shape, and extends from 0.065 m above still water level (SWL) to 0.5 m below. To improve its efficiency, a 0.1 m gap is provided between the top of the beach and the end wall. This ensures that the majority of the wave run-up flows over the beach, returning to the main water body behind the beach. This limits any washdown, helping to ensure that the reflection coefficient is no more than 5% over a broad range of frequencies. Although beach design is an essential element of all wave basins, its role in the present tests is limited due to the use of focused wave groups; the desired wave profile being generated at the measuring section before any unwanted reflections can arise at this point. Within the wave basin surface elevations were measured using standard resistance wavegauges. Each gauge consists of a pair of parallel stainless steel rods, diameter 2 mm, located 15 mm apart. Gauges extend vertically through the water surface to below the level of the deepest wave trough. Previous observations have shown that they produce no disturbance of the flow field, allowing the surface elevation to be measured with an accuracy of 0:5 mm. During the calibration just two wavegauges were used; both were placed on the centreline of the wave basin, one at the measuring location 5.5 m downstream of the paddles and
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4. Calibration 4.1. General procedure With earlier work having shown that the accuracy required for focused wave generation cannot be achieved using a regular wave calibration, the use of this technique was kept to a minimum. Indeed, regular waves were only used in the early stages of the calibration as a starting point for the 0 direction case involving waves propagating perpendicular to the wave paddles. The results of this preliminary work are not presented here. A focused wave calibration was used to converge onto the correct transfer function. Once this had been achieved, the neighbouring angles could be considered using the previous converged solution as a starting point.
sufficiently long that the shortest wave components have time to propagate into the measurement area and are present for the entire sampling interval. A repeat period of 64 s was chosen so that the resolution of the amplitude 1 Hz. spectrum in the frequency domain becomes Df ¼ 64 30 Furthermore, an input frequency range of 64 of o 150 64 , coupled with a peak period of T p ¼ 1:2 s (corresponding to a frequency of approximately 53 64 Hz) was chosen as the calibration case. Finally, the effective spectral bandwidth must also be defined. For the JONSWAP spectrum this is described by the peak enhancement factor, g, which has a lower value of 1, but is usually around 2.5. Fig. 3(a) shows the effect of increasing g; contrasting the spectral shape for g ¼ 1, 2, 3, 4
0.08
0
With the objective of the present calibration being the accurate generation of a wide range of realistic wave conditions, a target JONSWAP spectrum (Hasselmann et al., 1973) was adopted 2 !4 3 0:0081g2 5 f 5gexp½ðf f p Þ2 =2s2 f 2p , S ZZ ðf Þ ¼ exp4 4 5 4 f ð2pÞ f p for f pf p ;
0:09
for f 4f p ;
(1)
where f is the frequency (Hz) of the wave component, f p is the peak frequency of the spectrum defined as 1=T p (where T p is the peak spectral period) and g is the peak enhancement factor. Within the present tests the spectrum was truncated at approximately three times the spectral peak and the amplitude of the last five frequency components was reduced linearly to zero to avoid any discontinuity in the tail of the spectrum. To describe the target spectral shape using a finite number of discrete frequency components, the method of wave description first requires the definition of an appropriate repeat period. This has nothing to do with the repeated generation of a desired wave group, but is used in conjunction with the so-called front numbers to define the individual frequency components as outlined in Section 3. On the basis of this description, it is clear that the chosen repeat period must provide a balance between maintaining an appropriate level of discretisation of wave components whilst ensuring that individual amplitudes are of a realistic size. The repeat period must also be
0.5
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(
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the other, acting as a reference gauge, approximately 2 m from the paddles. To avoid systematic measurement errors, the wavegauges were calibrated each time data for a new transfer function angle was collected, typically every 10 runs.
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0 −0.5 −1 −5
0
5
t (s) Fig. 3. Choice of target spectrum: (a) effect of peak enhancement factor g, (b) JONSWAP vs. NewWave (T p ¼ 1:2, g ¼ 2:5), (c) corresponding water surface elevations at the focus position ðx ¼ 0Þ. - - - JONSWAP, — NewWave.
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and 5. Given typical field observations, a g ¼ 2:5 was chosen for the present calibration. The other aspect affecting the bandwidth is the method by which the amplitude spectrum is calculated. In a traditional random wave simulation, the amplitude of a wave component is proportional to the square root of the variance or the energy associated with that component. However, an alternative approach is provided by the NewWave model (Tromans et al., 1991) which describes the most probable shape of a large ocean wave in terms of a focused wave group. In this case, the amplitude spectrum of the focused event is scaled directly from the energy spectrum leading to a more narrow-banded process. Fig. 3(b) contrasts the amplitude spectrum for a focused wave based upon a JONSWAP spectrum and the corresponding NewWave event as would be generated in the wave basin with the frequency range defined above and with an input amplitude of 1. The corresponding time-histories of the water surface elevation, ZðtÞ, are provided in Fig. 3(c). Given the purpose of the present measurements (calibration) and, particularly, the need for the results to be appropriate to other spectral shapes (involving differing T p and g), it is important that the chosen spectrum is sufficiently broad-banded so that the amplitudes of the majority of components are suitably large. Accordingly, the focused wave based upon the JONSWAP amplitude spectrum was adopted. Finally, the directional spread was addressed by considering unidirectional wave groups propagating at 5 intervals within the range 45 p yp þ 45 , giving a total of 19 calibration points. 4.3. Fourier analysis With the target spectrum and repeat time defined, the demand amplitude is specified for each wave component. To correct the transfer function the measured amplitude spectrum is required. Typically, this involves the application of standard Fourier analysis to a measured timehistory. However, in practice this is more complicated since in order to match the resolution of the demand spectrum, 1 Df ¼ 64 Hz, 64 s of measured data is required. This is problematic because: (i) sufficient delay must be allowed for the shortest wave components to arrive at the measuring location before the analysis can be applied, and (ii) following the passage of the focused crest the presence of reflections may contaminate the measured data. The only option available is to window the focused event and pad the record to make up a 64 s record. This approach can be justified because for most of the surface record the waves are very small and out of phase. As a result, the measured data are dominated by residual noise. The balance was to keep as much useful data in the window as possible, while at the same time eliminate the small unwanted reflections. In the present calibration a window of 5 s either side of the focused event was chosen. This was also convenient since the elevation at 5 s is very close to zero (Fig. 3(c)). A variety of padding techniques and
filters were applied. However, the most effective method was to pad the remaining record with zeros. This method was chosen because it was the simplest and eliminated any concerns that excessive handling and manipulation of the data may lead to a loss of accuracy. In adopting the above techniques, one further modification was required to satisfactorily interpret the calibration data. By windowing a small section of data and padding with zeros it was noted that there were small differences between the original spectrum and the spectrum generated by the FFT. This effect was thoroughly examined using numerically generated data. First, it was confirmed that the spectrum could be calculated from the complete timehistory. With the application of a progressively smaller window, the amplitude spectrum generated from the FFT of the windowed data begins to oscillate around the original spectrum. This is a direct effect of the box-shaped window applied to the data; the former leading to significant side lobes in the frequency domain. This effect could have been reduced by applying a smooth window function (Harris, 1978). However, since the amount of data is small it was decided that the maximum signal strength was more important than the imposition of a smooth window. The solution was to revise the definition of the ‘target’ spectrum. This involved generating a linear timehistory from the demand amplitude spectrum. This timehistory was then processed in exactly the same manner as the measured data, i.e. with the same window function and padding applied. The new target spectrum was then obtained from the FFT of this time-trace. The correction to the transfer function is subsequently based on a comparison between the measured and the new modified target spectra; both having been processed identically. At the end of the procedure the effectiveness of the method should be judged by a comparison of the measured and predicted wave profiles. As long as the measured profile equals the demand profile, the calibration will be considered accurate. Finally, the phase of individual wave components should be corrected by comparing the difference between the measured and the target spectrum. In doing this it is important to ensure that the phase output from the FFT is evaluated at the focal time, since some FFT algorithms give the phase relative to the first point. In this case the phase adjustment for the nth component is simply on tfoc , where on the wave frequency of the nth component and tfoc the focal time. 4.4. Calibration curves Fig. 4 shows a selection of the calibration curves generated during the calibration process. The plots (a), (b) and (c) correspond to angles y ¼ 0 , 40 and 40 . For 78 each angle three frequencies are shown, f ¼ 47 64, 64 and 101 64 Hz. All the calibration curves are well described by a linear fit, with a negligible offset. Using these data the correction factor for the next iteration of the transfer
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4.5. Transfer function gain 2
x 10−4
Fig. 5 provides a plot of the final transfer function gain after calibration for angles y ¼ 0 , 40 and 40 . The general trend is that for the low-frequency range the transfer function correction factor is higher. The smoothest line corresponds to the 0 case. At angles of 40 and 40 there is more variation at the high-frequency end. Furthermore, the transfer function is not symmetric.
Measured An
θ = 0° 1
0
−1
5. Verification 0
1
2 x 10−4
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x 10−4 θ = 40°
1
5.1. Linear cases using the target spectra 0
−1 0
2
Measured An
In an attempt to verify the transfer function, a large number of test cases were run. A selection of both the linear and nonlinear cases is presented in the following section; the data being representative of a much wider range of cases considered.
1
2 x 10−4
x 10−4 θ = −40°
Fig. 6 provides details of a unidirectional linear wave ðA ¼ 12 mmÞ generated at an angle of 0 off the paddles; the measured data being compared with linear theory. Parts (a)–(c) correspond to the water surface elevation, the amplitude spectrum and the phase of the frequency components, respectively; where the latter two are calculated from an FFT of a padded time-history (as discussed earlier). All the comparisons show excellent agreement. Indeed, the only significant departures between the measured and demand data are seen in the phase, Fig. 6(c).
1
160
0
140
120
−1 1 Demand An
2 x 10−4
Fig. 4. Calibration curves (a) 0 , (b) 40 , (c) 40 . f ¼ 47 64 Hz, &&& 101 f ¼ 78 64 Hz, BBB f ¼ 64 Hz.
100 Tf Gain
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function was calculated. For the final calibration, a best fit line through the origin of the calibration curve was found to give marginally better results and was therefore adopted. This was in turn used to correct the transfer function gain for the next iteration until convergence was achieved. During the preliminary stages of the calibration, while the optimum calibration method was sought, the y ¼ 0 direction was iterated a large number of times. It is not therefore possible to report precisely how many iterations were required before convergence was achieved in this direction. However, for all other angles satisfactory convergence was achieved after two iterations.
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Fig. 5. Transfer function gain, variation with frequency — 0 , - - - 40 , 40 .
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Fig. 6. Verification of the calibration: unidirectional linear wave group, A ¼ 12 mm and y ¼ 0 . (a) Water surface elevation, (b) amplitude measured data, - - - linear theory. spectrum, (c) phase variation.
Fig. 7. Verification of the calibration: directional linear wave group, A ¼ 12 mm and sy ¼ 30 . (a) Surface elevation, (b) amplitude spectrum, (c) phase variation. measured data, - - - linear theory.
However, these departures correspond to those regions where the amplitude of the demand signal was effectively zero and therefore have no significance. Fig. 7 provides a similar set of plots relating to a linear, directionally spread, wave group. The degree of spreading was normally distributed with a standard deviation sy ¼ 30 . This is a more rigorous test of the calibration since it involves contributions from all directions, the amplitudes of which are significantly smaller than the range over which they were originally calibrated. Parts (a)–(c) again correspond to the water surface elevation, the amplitude spectrum and the phase angle, respectively. For this case the comparison between the measured and the demand signals remains in excellent agreement.
5.2. Linear cases based on the NewWave model A second and more rigorous validation may be achieved through the generation of wave groups based on an amplitude spectrum that differs from the original target spectrum used for the calibration. A variety of linear cases are considered in Fig. 8. Sub-plots (a) and (b) corresponding to an A ¼ 12 mm, T p ¼ 1:2 s, g ¼ 2:5 NewWave simulation, providing water surface elevation and amplitude spectra, respectively. This wave group represents a significantly more narrow-banded wave group than that used for the calibration. Nevertheless, the agreement remains excellent. Sub-plots (c) and (d) concern the surface elevation and amplitude spectra for an A ¼ 12 mm T p ¼
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5 0 −5 −10
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Fig. 8. Verification of the calibration: off-target spectra (a) surface elevation T p ¼ 1:2 s, g ¼ 2:5, (b) amplitude spectrum T p ¼ 1:2 s, g ¼ 2:5, (c) surface elevation T p ¼ 1:4 s, g ¼ 2:5, (d) amplitude spectrum T p ¼ 1:4 s, g ¼ 2:5. measured, - - - demand.
1:4 s g ¼ 2:5, NewWave simulation. The difference in peak period having no effect on the quality of the waves generated. Finally, for each of the two cases mentioned above, Figs. 9(a) and (b), respectively, show the effect of directionality. Each plot presents the results of four repeated generations at sy ¼ 0 (unidirectional), 10 , 20 and 30 . Both plots show that directionality has no effect on the accuracy of wave generation, even when this involves an off-target wave spectrum.
−3
Fig. 9. Verification of the calibration: variation with directionality in offtarget spectra. (a) T p ¼ 1:2 s, g ¼ 2:5, (b) T p ¼ 1:4 s, g ¼ 2:5. unidirectional, - - - sy ¼ 10 , sy ¼ 20 , sy ¼ 30 .
Finally, sub-plots (b) and (c) provide time-histories of the water surface elevation recorded at positions x ¼ 5:5 m, and y ¼ 1:5 m, respectively; the incident wave conditions being the same as those described above. In each case the focus position has been linearly adjusted so that the waves focus at these locations. This wave generation, being offtarget, directional, and focused away from the original calibration location, demonstrates not only the consistency of the calibration but also the quality of the wave paddles. 5.4. Nonlinear cases
5.3. Spatial verification of the calibration The final verification of the calibration addresses the spatial variation of the wave profile. This is considered in Fig. 10 and concerns a linear, off-target (T p ¼ 1:4 s, g ¼ 2:5) NewWave simulation with sy ¼ 30 . Sub-plot (a) shows the variation in crest elevation measured at 9 points along a section that is transverse to the mean wave direction and hence parallel to the wave paddles. The figure shows the measured crest elevation at the focal time, t ¼ 0, and makes direct comparisons with predictions based on linear theory; very good agreement being achieved. These results confirm that the directionality of the desired wave field is also successfully reproduced.
Fig. 11 shows several highly nonlinear wave cases each with an input amplitude of A ¼ 58 mm. They all relate to a JONSWAP spectrum with T p ¼ 1:2 and g ¼ 2:5, and concern four directional spreads corresponding to sy ¼ 0 (unidirectional), 10 , 20 and 30 . In each case the maximum crest elevation is larger than that predicted by linear theory (shown by the dashed line). This arises due to nonlinear wave–wave interactions and the corresponding energy shifts that occur during the evolution of a large wave event. The results are in agreement with both the experimental observations of Johannessen and Swan (2001) and the numerical predictions of Johannessen and Swan (2003); the largest nonlinear increase occurring in the unidirectional case, the effect decreasing with increasing
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0.1 0.015
η (m)
0.01
0.05
0 −2
−1
0 y (m)
1
2
η (m)
0.005
0
x 10−3
η (m)
10 5
−0.05
0
−2
−1
0
1
2
t (s)
−5
−5
0
5
t (s)
Fig. 11. Verification of the calibration: nonlinear wave case based upon a JONSWAP spectrum (A ¼ 58 mm, T p ¼ 1:2, g ¼ 2:5) including directionality. linear, unidirectional, - - - sy ¼ 10 , sy ¼ 20 , sy ¼ 30 .
x 10−3
6. Conclusions 10
η (m)
5 0 −5 −5
0
5
t (s)
Fig. 10. Verification of the calibration: spatial verification T p ¼ 1:4 s, g ¼ 2:5, NewWave simulation. (a) Spatial variation in the transverse direction at t ¼ 0, (b) surface elevation ZðtÞ with the wave measured and focused at x ¼ 5:5 m, y ¼ 1:5 m, (c) surface elevation ZðtÞ with the wave measured and focused at x ¼ 5:5 m, y ¼ 1:5 m. —, measured, - - demand.
directional spread. A physical explanation of the complex nonlinear wave–wave interactions responsible for these changes being provided by Gibson and Swan (2007). A second effect that confirms the successful generation of these nonlinear wave forms relates to the downstream shift in the focus position. Although this has been removed from the data presented in Fig. 11 (to facilitate comparisons), the data confirm that this is largest in the unidirectional case (typically of the order of 1–1.5 m), reducing with increasing directional spread. This is fully consistent with Johannessen and Swan (2001, 2003).
A previously unreported procedure for the calibration of a 3D wave basin using transient focused wave groups is described. It is reported that although the present method is at odds with traditional wave basin calibration methods, the use of such techniques allows a significant improvement in the description of the waves generated by the paddles. As such, the method may be described as a ‘wavemaker’ calibration rather than a ‘wave basin’ calibration. The calibration has been verified for linear and nonlinear wave cases, both with and without directional spreading. For the linear cases the agreement is excellent, particularly when directionally spread linear wave groups are considered. The nonlinear groups confirm the quality of the calibration, not least because they are consistent with earlier numerical calculations. The motivation for adopting this more complicated approach is driven by recent advances in wave modelling, particularly the nonlinear evolution of extreme wave groups. Rigorous comparison with theories is only possible if the underlying linear spectrum is confidently known. Regardless of the facility used, this can only be achieved if the present techniques are employed. Furthermore, if future work is to examine the effect of these extreme waves on structures, both fixed and floating, and in so doing provide new physical insights into the applied nonlinear loading and the occurrence of high-frequency scattering, it is essential that such studies are again based upon a sound understanding of the underlying linear wave
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components generated at the wave paddles. Once this is achieved the local and rapid spectral changes identified in the vicinity of these extreme events (Gibson and Swan, 2007) will be accurately reproduced in the wave basin (Johannessen and Swan, 2001) and their significance in terms of the local wave–structure and wave–vessel interactions can be fully assessed. References Baldock, T.E., Swan, C., 1996. Extreme waves in shallow and intermediate water depths. Coastal Engineering 27 (1–2), 21–46. Baldock, T.E., Swan, C., Taylor, P.H., 1996. A laboratory study of nonlinear surface waves on water. Philosophical Transactions of the Royal Society of London A 354, 649–676. Dean, R.G., Dalrymple, R.A., 1992. Water Wave Mechanics for Engineers and Scientists. Advanced Series on Ocean Engineering, vol. 2, second ed. World Scientific, Singapore. Gibson, R.S., Swan, C., 2007. The evolution of large ocean waves: the role of local and rapid spectral changes. Proceedings of the Royal Society, Series A 463, 21–48.
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Harris, F., 1978. On the use of windows for harmonic analysis with the discrete Fourier transform. Proceedings of the IEEE 66, 51–83. Hasselmann, K., Barnet, T.P., Bouws, E., Carlson, H., Cartwright, D.E., Enke, K., Ewing, K., Ewing, J.A., Gienpp, H., Hasselmann, D.E., Kruseman, P., Meerburg, A., Mu¨ller, P., Olbes, D.J., Richter, K., Sell, K., Walden, H., 1973. Measurements of Wind-Wave Growth and Swell Decay during the Joint North Sea Wave Project (JONSWAP). Deutsches Hydrographisches Institut. Johannessen, T.B., Swan, C., 2001. A laboratory study of the focusing of transient and directionally spread surface water waves. Proceedings of the Royal Society of London, Series A 457, 1–36. Johannessen, T.B., Swan, C., 2003. On the nonlinear dynamics of focused wave groups in two and three dimensions. Proceedings of the Royal Society of London, Series A 459, 1021–1052. Spinneken, J., Swan, C., 2008. A second-order theory for wave paddles with force-feedback. Technical Report, Imperial College London. Tromans, P.S., Anaturk, A., Hagemeijer, P., 1991. A new model for the kinematics of large ocean waves—application as a design wave. In: Proceedings of 1st Offshore and Polar Engineering Conference (ISOPE), vol. 3, Edinburgh, pp. 64–71. Tzivanaki, K., 2006. Simultaneous measurements of surface water waves in space and time. Ph.D. Thesis, Imperial College London.
Ocean Engineering 35 (2008) 763–773 www.elsevier.com/locate/oceaneng
On the accurate and efficient calibration of a 3D wave basin S.R. Masterton1, C. Swan Department of Civil and Environmental Engineering, Imperial College, London SW7 2AZ, UK Received 10 August 2007; accepted 11 February 2008 Available online 10 March 2008
Abstract This paper describes the calibration procedure adopted for the new 3D wave basin located in the Hydrodynamics Laboratory at Imperial College London. Unlike traditional calibrations, based on observations of regular wave trains, the method described herein uses a focused wave approach. Such waves, produced by the constructive interference of freely propagating wave components, have led to a number of recent advances in theoretical wave modelling in which it was essential to know the underlying linear components. In the context of a laboratory study, similar advantages can be realised provided the linear wave components generated by the wave paddles are well defined. This, in turn, can only be achieved if the calibration is sufficiently accurate. The current study provides a calibration based upon a realistic JONSWAP spectrum, describes the details of the methodology employed, and highlights how the application of focused wave techniques eliminates spurious calibration effects due to unwanted reflections from the boundaries of the basin. The final calibration is verified through the generation of test cases, involving linear and nonlinear, unidirectional and directionally spread waves. These confirm both the accuracy of the calibration and the suitability of the methods employed. r 2008 Elsevier Ltd. All rights reserved. Keywords: Laboratory wave generation; Wave basin calibration; Directional waves; Focused waves
1. Introduction Despite a succession of theoretical advances, physical model testing remains an effective method of investigating problems involving ocean waves. Indeed, if the problem includes waves interacting with a structure model testing represents the only reliable method, particularly where the full nonlinearity of the problem must be retained. In itself, the use of scaled models does not represent a new art. However, recent advances in wave modelling (particularly relating to extreme wave groups) necessitate the use of more sophisticated testing techniques. In the context of model testing, the use of these improved methods requires the input wave conditions, or more specifically the underlying linear wave spectrum generated at the wave paddles, to be precisely controlled. If this is to be achieved, the fundamental issue becomes that of accurate calibration. Corresponding author. Tel.: +44 207 594 5999; fax: +44 207 594 5991.
E-mail address: [email protected] (C. Swan). Now at: Shell International Exploration and Production, 2288 GS, Rijswijk, The Netherlands. 1
0029-8018/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2008.02.002
To achieve the most realistic representation of a given wave field, directionality must be included and this necessitates the use of a 3D wave basin. Compared to 2D wave flumes, these facilities are larger, more sophisticated and considerably more expensive. Consequently, there are relatively few examples of 3D wave basins operating worldwide, and these tend to employ ‘traditional’ testing methods; the sea state being specified in terms of some target spectrum and quantified (iteratively) on the basis of a long time-domain simulation at a single point in space. Using these techniques the control of directionality is inherently very poor, principally due to the build up of reflections, with calibrations usually based upon a coarse regular wave approach. The generation of focused wave groups in a 3D wave basin is relatively rare, not least because of the difficulty of obtaining an accurate transfer function. However, one notable exception is provided by Johannessen and Swan (2001). The purpose of this study was to quantify the nonlinear interactions arising during the evolution of nonlinear, directionally spread wave groups. As such, the authors state it was: ‘‘critically important to know the
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subscript n nth wave component subscript p peak value ðx; y; zÞ Cartesian co-ordinates z ¼ 0 still water level t time tfoc focal time Tf transfer function S ZZ wave spectrum g peak enhancement factor
Notation A T f o k y Z f
wave amplitude wave period frequency ðT1 Þ wave frequency ð2p TÞ wavenumber direction of propagation surface elevation phase angle
amplitude, frequency, phasing and direction of individual components generated at the wave paddles’’. In seeking to satisfy this goal they achieved the first focused wave calibration, albeit based upon an idealised tophat spectrum, but the details of the process went unreported. The present paper seeks to correct this omission, describing the procedure used to calibrate a new 3D wave basin located in the Hydrodynamics Laboratory at Imperial College London. The paper continues in Section 2 with a brief review of calibration methods. The wave basin and experimental setup are described in Section 3. Sections 4 and 5 describe the calibration process and its subsequent verification. The paper closes with Section 6, highlighting the advantages of the proposed techniques and offering some wider conclusions. 2. Review of calibration methods In those instances where a precise definition of the generated wave conditions is not essential, particularly if it involves long-crested or uni-directional waves, a theoretical calibration based on potential wavemaker theory may be appropriate. Several authors, including Dean and Dalrymple (1992), give linear solutions for a wide variety of paddle types; the results usually expressed in terms of a wave height vs. paddle stroke relationship (see, for example, their Figure 6.2 on p. 172). However, if precise control of the wave generation is required, as is usually the case for research purposes or detailed model testing, the calibration becomes more complicated. From a theoretical perspective, the avoidance of spurious freely propagating wave modes dictates the use of second-order theory. However, depending on the type of wave paddle, there is limited evidence of the success of this approach, particularly when applied to directional wave generation. Furthermore, if the control of Frequency Amplitude Phase Direction
Tf (, )
the paddle (or paddles) is not based upon position alone, and there are good practical reasons why this should be so (Spinneken and Swan, 2008), the appropriate theoretical calibrations do not exist. In such cases accurate wave generation must be based on empirical calibration. 2.1. General approach—regular wave calibration A transfer function, T f , is defined from the relationship between the required or demand output from a paddle bank and the waves actually generated in the wave basin. The transfer function is applied to the demand signal before it is sent to the paddle bank and corrects for both the electrical and mechanical scaling and the inertia of the wave paddles. It is a function of both the wave frequency, o, and the direction of wave propagation, y, and is specified by a gain and a phase. For any given wave component, the output from the paddles should always be equal to the demand signal. This is illustrated in Fig. 1. It is important to note at this stage that the present study is only interested in a linear transfer function. Consequently, T f is not a function of the amplitude, A, and the calibration process is entirely concerned with small linear waves. This approach is consistent with a calibration appropriate to experiments in which the isolation of nonlinear effects represents an important part of the problem to be solved. Owing to its complexity, the transfer function can only be defined empirically and is calculated in an iterative manner. The basic procedure is to generate a series of regular waves which are measured at a known distance from the paddles. The transfer function gain is then corrected by multiplying by the ratio of the demand amplitude divided by the measured amplitude. Clearly, once the measured amplitude approaches the demand
Paddle Bank
Frequency Amplitude Phase Direction Measured Output
Demand Output
Fig. 1. The role of the transfer function, T f .
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amplitude, the correction factor approaches unity and no further iteration is necessary. A similar correction routine can be applied to the phase, having taken into account the phase change associated with: (i) the distance between the measuring location and the paddle bank, k~ x (where k is the wave number and ~ x is the distance from the paddles), and, (ii) the time delay allowed for the wave to propagate to the measuring location, ot (where o is the wave frequency and t is the time). The above description is slightly simplified in that it assumes that the system is entirely linear, with no possibility of experimental errors. In practice this is not the case and, as a result, it is necessary to generate a number of amplitudes, typically around five, for each wave component (o; y). In this manner, a calibration curve, a plot of demand vs. measured amplitude can be drawn and, assuming the line is linear, the gradient of a linear fit used to correct the transfer function gain. Using the same data, the mean phase difference from the five amplitude runs is used to correct the phase of the transfer function. The regular wave approach described above is the most basic method of calculating a transfer function. However, if the gain and phase do not vary smoothly with both the wave frequency and the direction of wave propagation, an accurate definition of the transfer function will be enormously time consuming. This is particularly true if the frequency range is realistically broad (a requirement of the present calibration) and, as a result, a large number of wave components must be calibrated for. This problem is exacerbated by the fact that convergence for a single wave component will often take more than one iteration, where each iteration requires approximately five runs. Clearly, a calibration using this method in a directional wave basin, where each direction must be dealt with separately, becomes entirely impractical. 2.2. Focused wave calibration An alternative to a regular wave calibration is to use transient focused wave groups. Here, a large number of wave components are generated simultaneously and are appropriately phase shifted so that, at a specified time, all the wave components come into phase at the measuring location. Further details of the generation of a focused wave group in a 2D wave flume are given by Baldock et al. (1996) and Baldock and Swan (1996); while Johannessen and Swan (2001) describe the generation of directionally spread focused wave groups in a 3D wave basin. When a focused wave approach is applied in a directional wave basin a separate unidirectional calibration is required for each angle or direction of wave propagation. This is necessary because a single point measurement cannot resolve the direction of propagation. The alternative, a calibration using directionally spread groups is impractical and would require either a large array of wavegauges or some form of spatial measuring technique (Tzivanaki, 2006); the latter being extremely complicated
765
to both setup and interpret. Furthermore, for such small waves it would not necessarily be expected to yield a more accurate result. While the time saving is an important motivation, the use of focused wave groups has other significant advantages over a regular wave calibration. First, because wave basins are generally shorter than 2D wave flumes, reflections from the lateral boundaries and the beach must be considered more carefully. The use of transient wave groups limits the effects of reflection by reducing the amplitudes of the waves at the basin boundaries. This is in stark contrast with the regular wave approach where reflections are more of an issue, particularly with regard to the spatial variability of the wave field. Second, the dispersion of the wave group away from the focus location, particularly in directionally spread wave fields, ensures that the waves generated at the paddles are always small. This significantly reduces any potential problems associated with nonlinear wave generation; the latter requiring nonlinear (usually second-order) input signals to avoid the growth of unwanted spurious wave modes. 2.3. Previous work As part of their study of directionally spread, focused wave groups Johannessen and Swan (2001) provide the first evidence of the accurate calibration of a 3D wave basin, although the details of the method employed went unreported. This calibration was undertaken in the wave basin (or widetank) at the University of Edinburgh; a facility that has since been decommissioned. The purpose of the calibration was identical to the present study; the emphasis being placed on the identification of the wave components generated at the wave paddles rather than matching some target statistical properties at a given location. The calibration was initially carried out using a regular wave method. However, despite the generation of 270 different regular wave forms the resolution was poor, with just five frequency components considered every 10 . However, of greater significance was the amplitude variation caused by the regular wave approach, observed to be up to 15%. This was directly attributed to an inadequate control of wave reflections within the wave basin. To overcome these difficulties the first focused wave calibration was undertaken using a narrow-banded tophat laboratory spectrum in which all the frequency components travelling in a specified direction were assumed to be of equal amplitude. 3. Experimental facility and setup The wave basin used in the present study is shown in Fig. 2. It is located in the basement of the Civil and Environmental Engineering Department at Imperial College London and was commissioned in 2003 as part of a major laboratory refurbishment. The facility is 20 m by
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Fig. 2. The 3D wave basin at Imperial College London.
13 m with an operating water depth of 1.5 m. There is also a 3 m deep pit in the centre which is presently covered. The short sides of the wave basin are constructed from glass for visual access and a moving gantry, hung from the ceiling, spans the shortest length of the wave basin. Wavegauges (see below) are fixed to this gantry thereby avoiding any vibration from the paddle bank. Wave generation is achieved by a bank of 56 identical, bottom hinged, flap type wavemakers. These are installed along the longest side of the wave basin. Individual paddles are 0.35 m wide and 0.7 m deep, and are constructed from a lightweight glassfibre. The space behind the wavemakers is kept dry by means of a thin polyurethane-impregnated nylon membrane between paddles. This membrane is sufficiently slack that each individual wave paddle can describe its full range of motion independently of its neighbours. This system incorporates no sliding parts and therefore friction effects are minimal. Hydrostatic pressure is balanced by a system of springs and pulleys attached to a toothed drive belt between each paddle and its high torque servomotor. Individual paddles incorporate a force feedback mechanism, allowing the active absorption of reflected waves that would otherwise build up in the wave basin. This design was pioneered at The University of Edinburgh and works by monitoring the force via a loadcell that is placed between the drive system and the paddle. In this configuration the total force exerted by the water pressure over the front face of the paddle can be incorporated into the control loop, in addition to the usual position and velocity monitoring. Detailed discussion of the force control mechanism lies beyond the scope of the present paper; the interested reader being directed to Spinneken and Swan (2008). Wave generation is controlled via a central PC. The control software assumes the paddle motion is periodic over a long repeat time, which for computational efficiency must be 2n s where n is any positive integer. In the present tests this repeat time was 64 s (see below). Individual wave
components, corresponding to specific frequencies, are defined in terms of so-called wave front numbers; the latter representing the harmonics of the fundamental repeat period. With a repeat time of 64 s, the nth front number corresponds to waves with a period of T n ¼ 64=n; a front number of 64 corresponding to a 1.0 s wave, while a front number of 32 corresponds to a 2.0 s wave. Although the control software allows the efficient generation of wave groups, providing a simple interface with the wavemaking hardware, in practice it is merely a convenience (albeit a significant one) and the calibration process outlined below may be adopted in any wave basin. At the ‘downstream’ end of the wave basin, opposite the wave paddles, the propagating wave energy is absorbed on a smooth, non-perforated beach. This is 2.5 m long, parabolic in shape, and extends from 0.065 m above still water level (SWL) to 0.5 m below. To improve its efficiency, a 0.1 m gap is provided between the top of the beach and the end wall. This ensures that the majority of the wave run-up flows over the beach, returning to the main water body behind the beach. This limits any washdown, helping to ensure that the reflection coefficient is no more than 5% over a broad range of frequencies. Although beach design is an essential element of all wave basins, its role in the present tests is limited due to the use of focused wave groups; the desired wave profile being generated at the measuring section before any unwanted reflections can arise at this point. Within the wave basin surface elevations were measured using standard resistance wavegauges. Each gauge consists of a pair of parallel stainless steel rods, diameter 2 mm, located 15 mm apart. Gauges extend vertically through the water surface to below the level of the deepest wave trough. Previous observations have shown that they produce no disturbance of the flow field, allowing the surface elevation to be measured with an accuracy of 0:5 mm. During the calibration just two wavegauges were used; both were placed on the centreline of the wave basin, one at the measuring location 5.5 m downstream of the paddles and
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4. Calibration 4.1. General procedure With earlier work having shown that the accuracy required for focused wave generation cannot be achieved using a regular wave calibration, the use of this technique was kept to a minimum. Indeed, regular waves were only used in the early stages of the calibration as a starting point for the 0 direction case involving waves propagating perpendicular to the wave paddles. The results of this preliminary work are not presented here. A focused wave calibration was used to converge onto the correct transfer function. Once this had been achieved, the neighbouring angles could be considered using the previous converged solution as a starting point.
sufficiently long that the shortest wave components have time to propagate into the measurement area and are present for the entire sampling interval. A repeat period of 64 s was chosen so that the resolution of the amplitude 1 Hz. spectrum in the frequency domain becomes Df ¼ 64 30 Furthermore, an input frequency range of 64 of o 150 64 , coupled with a peak period of T p ¼ 1:2 s (corresponding to a frequency of approximately 53 64 Hz) was chosen as the calibration case. Finally, the effective spectral bandwidth must also be defined. For the JONSWAP spectrum this is described by the peak enhancement factor, g, which has a lower value of 1, but is usually around 2.5. Fig. 3(a) shows the effect of increasing g; contrasting the spectral shape for g ¼ 1, 2, 3, 4
0.08
0
With the objective of the present calibration being the accurate generation of a wide range of realistic wave conditions, a target JONSWAP spectrum (Hasselmann et al., 1973) was adopted 2 !4 3 0:0081g2 5 f 5gexp½ðf f p Þ2 =2s2 f 2p , S ZZ ðf Þ ¼ exp4 4 5 4 f ð2pÞ f p for f pf p ;
0:09
for f 4f p ;
(1)
where f is the frequency (Hz) of the wave component, f p is the peak frequency of the spectrum defined as 1=T p (where T p is the peak spectral period) and g is the peak enhancement factor. Within the present tests the spectrum was truncated at approximately three times the spectral peak and the amplitude of the last five frequency components was reduced linearly to zero to avoid any discontinuity in the tail of the spectrum. To describe the target spectral shape using a finite number of discrete frequency components, the method of wave description first requires the definition of an appropriate repeat period. This has nothing to do with the repeated generation of a desired wave group, but is used in conjunction with the so-called front numbers to define the individual frequency components as outlined in Section 3. On the basis of this description, it is clear that the chosen repeat period must provide a balance between maintaining an appropriate level of discretisation of wave components whilst ensuring that individual amplitudes are of a realistic size. The repeat period must also be
0.5
1
1.5
2
2.5
1.5
2
2.5
f (Hz) 0.06
0.04 An (m)
0:07
0
0.02
0 0
0.5
1 f (Hz)
1 0.5 η (m)
s¼
0.04 0.02
4.2. Target amplitude spectrum
(
γ=1 γ=2 γ=3 γ=4 γ=5
0.06 An (m)
the other, acting as a reference gauge, approximately 2 m from the paddles. To avoid systematic measurement errors, the wavegauges were calibrated each time data for a new transfer function angle was collected, typically every 10 runs.
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0 −0.5 −1 −5
0
5
t (s) Fig. 3. Choice of target spectrum: (a) effect of peak enhancement factor g, (b) JONSWAP vs. NewWave (T p ¼ 1:2, g ¼ 2:5), (c) corresponding water surface elevations at the focus position ðx ¼ 0Þ. - - - JONSWAP, — NewWave.
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and 5. Given typical field observations, a g ¼ 2:5 was chosen for the present calibration. The other aspect affecting the bandwidth is the method by which the amplitude spectrum is calculated. In a traditional random wave simulation, the amplitude of a wave component is proportional to the square root of the variance or the energy associated with that component. However, an alternative approach is provided by the NewWave model (Tromans et al., 1991) which describes the most probable shape of a large ocean wave in terms of a focused wave group. In this case, the amplitude spectrum of the focused event is scaled directly from the energy spectrum leading to a more narrow-banded process. Fig. 3(b) contrasts the amplitude spectrum for a focused wave based upon a JONSWAP spectrum and the corresponding NewWave event as would be generated in the wave basin with the frequency range defined above and with an input amplitude of 1. The corresponding time-histories of the water surface elevation, ZðtÞ, are provided in Fig. 3(c). Given the purpose of the present measurements (calibration) and, particularly, the need for the results to be appropriate to other spectral shapes (involving differing T p and g), it is important that the chosen spectrum is sufficiently broad-banded so that the amplitudes of the majority of components are suitably large. Accordingly, the focused wave based upon the JONSWAP amplitude spectrum was adopted. Finally, the directional spread was addressed by considering unidirectional wave groups propagating at 5 intervals within the range 45 p yp þ 45 , giving a total of 19 calibration points. 4.3. Fourier analysis With the target spectrum and repeat time defined, the demand amplitude is specified for each wave component. To correct the transfer function the measured amplitude spectrum is required. Typically, this involves the application of standard Fourier analysis to a measured timehistory. However, in practice this is more complicated since in order to match the resolution of the demand spectrum, 1 Df ¼ 64 Hz, 64 s of measured data is required. This is problematic because: (i) sufficient delay must be allowed for the shortest wave components to arrive at the measuring location before the analysis can be applied, and (ii) following the passage of the focused crest the presence of reflections may contaminate the measured data. The only option available is to window the focused event and pad the record to make up a 64 s record. This approach can be justified because for most of the surface record the waves are very small and out of phase. As a result, the measured data are dominated by residual noise. The balance was to keep as much useful data in the window as possible, while at the same time eliminate the small unwanted reflections. In the present calibration a window of 5 s either side of the focused event was chosen. This was also convenient since the elevation at 5 s is very close to zero (Fig. 3(c)). A variety of padding techniques and
filters were applied. However, the most effective method was to pad the remaining record with zeros. This method was chosen because it was the simplest and eliminated any concerns that excessive handling and manipulation of the data may lead to a loss of accuracy. In adopting the above techniques, one further modification was required to satisfactorily interpret the calibration data. By windowing a small section of data and padding with zeros it was noted that there were small differences between the original spectrum and the spectrum generated by the FFT. This effect was thoroughly examined using numerically generated data. First, it was confirmed that the spectrum could be calculated from the complete timehistory. With the application of a progressively smaller window, the amplitude spectrum generated from the FFT of the windowed data begins to oscillate around the original spectrum. This is a direct effect of the box-shaped window applied to the data; the former leading to significant side lobes in the frequency domain. This effect could have been reduced by applying a smooth window function (Harris, 1978). However, since the amount of data is small it was decided that the maximum signal strength was more important than the imposition of a smooth window. The solution was to revise the definition of the ‘target’ spectrum. This involved generating a linear timehistory from the demand amplitude spectrum. This timehistory was then processed in exactly the same manner as the measured data, i.e. with the same window function and padding applied. The new target spectrum was then obtained from the FFT of this time-trace. The correction to the transfer function is subsequently based on a comparison between the measured and the new modified target spectra; both having been processed identically. At the end of the procedure the effectiveness of the method should be judged by a comparison of the measured and predicted wave profiles. As long as the measured profile equals the demand profile, the calibration will be considered accurate. Finally, the phase of individual wave components should be corrected by comparing the difference between the measured and the target spectrum. In doing this it is important to ensure that the phase output from the FFT is evaluated at the focal time, since some FFT algorithms give the phase relative to the first point. In this case the phase adjustment for the nth component is simply on tfoc , where on the wave frequency of the nth component and tfoc the focal time. 4.4. Calibration curves Fig. 4 shows a selection of the calibration curves generated during the calibration process. The plots (a), (b) and (c) correspond to angles y ¼ 0 , 40 and 40 . For 78 each angle three frequencies are shown, f ¼ 47 64, 64 and 101 64 Hz. All the calibration curves are well described by a linear fit, with a negligible offset. Using these data the correction factor for the next iteration of the transfer
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4.5. Transfer function gain 2
x 10−4
Fig. 5 provides a plot of the final transfer function gain after calibration for angles y ¼ 0 , 40 and 40 . The general trend is that for the low-frequency range the transfer function correction factor is higher. The smoothest line corresponds to the 0 case. At angles of 40 and 40 there is more variation at the high-frequency end. Furthermore, the transfer function is not symmetric.
Measured An
θ = 0° 1
0
−1
5. Verification 0
1
2 x 10−4
Measured An
2
x 10−4 θ = 40°
1
5.1. Linear cases using the target spectra 0
−1 0
2
Measured An
In an attempt to verify the transfer function, a large number of test cases were run. A selection of both the linear and nonlinear cases is presented in the following section; the data being representative of a much wider range of cases considered.
1
2 x 10−4
x 10−4 θ = −40°
Fig. 6 provides details of a unidirectional linear wave ðA ¼ 12 mmÞ generated at an angle of 0 off the paddles; the measured data being compared with linear theory. Parts (a)–(c) correspond to the water surface elevation, the amplitude spectrum and the phase of the frequency components, respectively; where the latter two are calculated from an FFT of a padded time-history (as discussed earlier). All the comparisons show excellent agreement. Indeed, the only significant departures between the measured and demand data are seen in the phase, Fig. 6(c).
1
160
0
140
120
−1 1 Demand An
2 x 10−4
Fig. 4. Calibration curves (a) 0 , (b) 40 , (c) 40 . f ¼ 47 64 Hz, &&& 101 f ¼ 78 64 Hz, BBB f ¼ 64 Hz.
100 Tf Gain
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function was calculated. For the final calibration, a best fit line through the origin of the calibration curve was found to give marginally better results and was therefore adopted. This was in turn used to correct the transfer function gain for the next iteration until convergence was achieved. During the preliminary stages of the calibration, while the optimum calibration method was sought, the y ¼ 0 direction was iterated a large number of times. It is not therefore possible to report precisely how many iterations were required before convergence was achieved in this direction. However, for all other angles satisfactory convergence was achieved after two iterations.
40
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2
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Fig. 5. Transfer function gain, variation with frequency — 0 , - - - 40 , 40 .
ARTICLE IN PRESS S.R. Masterton, C. Swan / Ocean Engineering 35 (2008) 763–773
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Fig. 6. Verification of the calibration: unidirectional linear wave group, A ¼ 12 mm and y ¼ 0 . (a) Water surface elevation, (b) amplitude measured data, - - - linear theory. spectrum, (c) phase variation.
Fig. 7. Verification of the calibration: directional linear wave group, A ¼ 12 mm and sy ¼ 30 . (a) Surface elevation, (b) amplitude spectrum, (c) phase variation. measured data, - - - linear theory.
However, these departures correspond to those regions where the amplitude of the demand signal was effectively zero and therefore have no significance. Fig. 7 provides a similar set of plots relating to a linear, directionally spread, wave group. The degree of spreading was normally distributed with a standard deviation sy ¼ 30 . This is a more rigorous test of the calibration since it involves contributions from all directions, the amplitudes of which are significantly smaller than the range over which they were originally calibrated. Parts (a)–(c) again correspond to the water surface elevation, the amplitude spectrum and the phase angle, respectively. For this case the comparison between the measured and the demand signals remains in excellent agreement.
5.2. Linear cases based on the NewWave model A second and more rigorous validation may be achieved through the generation of wave groups based on an amplitude spectrum that differs from the original target spectrum used for the calibration. A variety of linear cases are considered in Fig. 8. Sub-plots (a) and (b) corresponding to an A ¼ 12 mm, T p ¼ 1:2 s, g ¼ 2:5 NewWave simulation, providing water surface elevation and amplitude spectra, respectively. This wave group represents a significantly more narrow-banded wave group than that used for the calibration. Nevertheless, the agreement remains excellent. Sub-plots (c) and (d) concern the surface elevation and amplitude spectra for an A ¼ 12 mm T p ¼
ARTICLE IN PRESS S.R. Masterton, C. Swan / Ocean Engineering 35 (2008) 763–773
x 10−3
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Fig. 8. Verification of the calibration: off-target spectra (a) surface elevation T p ¼ 1:2 s, g ¼ 2:5, (b) amplitude spectrum T p ¼ 1:2 s, g ¼ 2:5, (c) surface elevation T p ¼ 1:4 s, g ¼ 2:5, (d) amplitude spectrum T p ¼ 1:4 s, g ¼ 2:5. measured, - - - demand.
1:4 s g ¼ 2:5, NewWave simulation. The difference in peak period having no effect on the quality of the waves generated. Finally, for each of the two cases mentioned above, Figs. 9(a) and (b), respectively, show the effect of directionality. Each plot presents the results of four repeated generations at sy ¼ 0 (unidirectional), 10 , 20 and 30 . Both plots show that directionality has no effect on the accuracy of wave generation, even when this involves an off-target wave spectrum.
−3
Fig. 9. Verification of the calibration: variation with directionality in offtarget spectra. (a) T p ¼ 1:2 s, g ¼ 2:5, (b) T p ¼ 1:4 s, g ¼ 2:5. unidirectional, - - - sy ¼ 10 , sy ¼ 20 , sy ¼ 30 .
Finally, sub-plots (b) and (c) provide time-histories of the water surface elevation recorded at positions x ¼ 5:5 m, and y ¼ 1:5 m, respectively; the incident wave conditions being the same as those described above. In each case the focus position has been linearly adjusted so that the waves focus at these locations. This wave generation, being offtarget, directional, and focused away from the original calibration location, demonstrates not only the consistency of the calibration but also the quality of the wave paddles. 5.4. Nonlinear cases
5.3. Spatial verification of the calibration The final verification of the calibration addresses the spatial variation of the wave profile. This is considered in Fig. 10 and concerns a linear, off-target (T p ¼ 1:4 s, g ¼ 2:5) NewWave simulation with sy ¼ 30 . Sub-plot (a) shows the variation in crest elevation measured at 9 points along a section that is transverse to the mean wave direction and hence parallel to the wave paddles. The figure shows the measured crest elevation at the focal time, t ¼ 0, and makes direct comparisons with predictions based on linear theory; very good agreement being achieved. These results confirm that the directionality of the desired wave field is also successfully reproduced.
Fig. 11 shows several highly nonlinear wave cases each with an input amplitude of A ¼ 58 mm. They all relate to a JONSWAP spectrum with T p ¼ 1:2 and g ¼ 2:5, and concern four directional spreads corresponding to sy ¼ 0 (unidirectional), 10 , 20 and 30 . In each case the maximum crest elevation is larger than that predicted by linear theory (shown by the dashed line). This arises due to nonlinear wave–wave interactions and the corresponding energy shifts that occur during the evolution of a large wave event. The results are in agreement with both the experimental observations of Johannessen and Swan (2001) and the numerical predictions of Johannessen and Swan (2003); the largest nonlinear increase occurring in the unidirectional case, the effect decreasing with increasing
ARTICLE IN PRESS S.R. Masterton, C. Swan / Ocean Engineering 35 (2008) 763–773
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Fig. 11. Verification of the calibration: nonlinear wave case based upon a JONSWAP spectrum (A ¼ 58 mm, T p ¼ 1:2, g ¼ 2:5) including directionality. linear, unidirectional, - - - sy ¼ 10 , sy ¼ 20 , sy ¼ 30 .
x 10−3
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Fig. 10. Verification of the calibration: spatial verification T p ¼ 1:4 s, g ¼ 2:5, NewWave simulation. (a) Spatial variation in the transverse direction at t ¼ 0, (b) surface elevation ZðtÞ with the wave measured and focused at x ¼ 5:5 m, y ¼ 1:5 m, (c) surface elevation ZðtÞ with the wave measured and focused at x ¼ 5:5 m, y ¼ 1:5 m. —, measured, - - demand.
directional spread. A physical explanation of the complex nonlinear wave–wave interactions responsible for these changes being provided by Gibson and Swan (2007). A second effect that confirms the successful generation of these nonlinear wave forms relates to the downstream shift in the focus position. Although this has been removed from the data presented in Fig. 11 (to facilitate comparisons), the data confirm that this is largest in the unidirectional case (typically of the order of 1–1.5 m), reducing with increasing directional spread. This is fully consistent with Johannessen and Swan (2001, 2003).
A previously unreported procedure for the calibration of a 3D wave basin using transient focused wave groups is described. It is reported that although the present method is at odds with traditional wave basin calibration methods, the use of such techniques allows a significant improvement in the description of the waves generated by the paddles. As such, the method may be described as a ‘wavemaker’ calibration rather than a ‘wave basin’ calibration. The calibration has been verified for linear and nonlinear wave cases, both with and without directional spreading. For the linear cases the agreement is excellent, particularly when directionally spread linear wave groups are considered. The nonlinear groups confirm the quality of the calibration, not least because they are consistent with earlier numerical calculations. The motivation for adopting this more complicated approach is driven by recent advances in wave modelling, particularly the nonlinear evolution of extreme wave groups. Rigorous comparison with theories is only possible if the underlying linear spectrum is confidently known. Regardless of the facility used, this can only be achieved if the present techniques are employed. Furthermore, if future work is to examine the effect of these extreme waves on structures, both fixed and floating, and in so doing provide new physical insights into the applied nonlinear loading and the occurrence of high-frequency scattering, it is essential that such studies are again based upon a sound understanding of the underlying linear wave
ARTICLE IN PRESS S.R. Masterton, C. Swan / Ocean Engineering 35 (2008) 763–773
components generated at the wave paddles. Once this is achieved the local and rapid spectral changes identified in the vicinity of these extreme events (Gibson and Swan, 2007) will be accurately reproduced in the wave basin (Johannessen and Swan, 2001) and their significance in terms of the local wave–structure and wave–vessel interactions can be fully assessed. References Baldock, T.E., Swan, C., 1996. Extreme waves in shallow and intermediate water depths. Coastal Engineering 27 (1–2), 21–46. Baldock, T.E., Swan, C., Taylor, P.H., 1996. A laboratory study of nonlinear surface waves on water. Philosophical Transactions of the Royal Society of London A 354, 649–676. Dean, R.G., Dalrymple, R.A., 1992. Water Wave Mechanics for Engineers and Scientists. Advanced Series on Ocean Engineering, vol. 2, second ed. World Scientific, Singapore. Gibson, R.S., Swan, C., 2007. The evolution of large ocean waves: the role of local and rapid spectral changes. Proceedings of the Royal Society, Series A 463, 21–48.
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Harris, F., 1978. On the use of windows for harmonic analysis with the discrete Fourier transform. Proceedings of the IEEE 66, 51–83. Hasselmann, K., Barnet, T.P., Bouws, E., Carlson, H., Cartwright, D.E., Enke, K., Ewing, K., Ewing, J.A., Gienpp, H., Hasselmann, D.E., Kruseman, P., Meerburg, A., Mu¨ller, P., Olbes, D.J., Richter, K., Sell, K., Walden, H., 1973. Measurements of Wind-Wave Growth and Swell Decay during the Joint North Sea Wave Project (JONSWAP). Deutsches Hydrographisches Institut. Johannessen, T.B., Swan, C., 2001. A laboratory study of the focusing of transient and directionally spread surface water waves. Proceedings of the Royal Society of London, Series A 457, 1–36. Johannessen, T.B., Swan, C., 2003. On the nonlinear dynamics of focused wave groups in two and three dimensions. Proceedings of the Royal Society of London, Series A 459, 1021–1052. Spinneken, J., Swan, C., 2008. A second-order theory for wave paddles with force-feedback. Technical Report, Imperial College London. Tromans, P.S., Anaturk, A., Hagemeijer, P., 1991. A new model for the kinematics of large ocean waves—application as a design wave. In: Proceedings of 1st Offshore and Polar Engineering Conference (ISOPE), vol. 3, Edinburgh, pp. 64–71. Tzivanaki, K., 2006. Simultaneous measurements of surface water waves in space and time. Ph.D. Thesis, Imperial College London.