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The run-up on a cylinder in progressive surface gravity waves: harmonic components
Applied Ocean Research 26 (2004) 98–113 www.elsevier.com/locate/apor
The run-up on a cylinder in progressive surface gravity waves: harmonic components M.T. Morris-Thomas*, K.P. Thiagarajan School of Oil and Gas Engineering, The University of Western Australia, Crawley, WA 6009, Australia Received 16 June 2004; revised 5 November 2004; accepted 10 November 2004 Available online 9 February 2005
Abstract Wave run-up is the vertical uprush of water that occurs when an incident wave impinges on a partially immersed body. In this present work, wave run-up is studied on a fixed vertical cylinder in plane progressive waves. These progressive waves are of a form suitable for description by Stokes’ wave theory whereby the typical energy content of a wave train consists of one fundamental harmonic and corresponding phase locked Fourier components. The limitations of canonical wave diffraction theory—whereby the free-surface boundary condition is treated by a Stokes expansion—in predicting the harmonic components of the wave run-up are discussed. An experimental campaign is described and the choice of monochromatic waves is indicative of the diffraction regime for large volume structures where the assumption of potential flow theory is applicable, or more formally A=a! Oð1Þ (A and a being the wave amplitude and cylinder radius, respectively). The wave environment is represented by a parametric variation of the scattering parameter ka and wave steepness kA (where k denotes the wave number). The zeroth-, first-, second- and third-harmonics of the wave run-up are examined to determine the importance of each with regard to local wave diffraction and incident wave nonlinearities. It is shown that the complete wave run-up is not well accounted for by second-order diffraction theory. This is, however, dependent upon the coupling of ka and kA. In particular, whilst the modulus and phase of the second-harmonic are moderately predicted, the mean set-up is not well predicted by a second-order diffraction theory. Experimental evidence suggests this to be caused by higher than second-order nonlinear diffraction. Moreover, these effects most noticeably operate at the first-harmonic in waves of moderate to large steepness when ka/1. q 2005 Elsevier Ltd. All rights reserved. Keywords: Surface gravity waves; Wave run-up; Perturbation theory; Stokes waves
1. Introduction During the preceding four decades the offshore oil and gas industry has paid much attention to the scattering of water waves by large volume offshore structures. Historically, the industry has been dominated by fixed, transparent, offshore structures comprising tubular members, where diffraction effects are considered less important compared to drag dominated or viscous effects. However, with the advent of oil exploration and exploitation into increasing water depths, floating structures are an attractive alternative. These floating structures typically comprise a deck supported by large diameter columns that penetrate * Corresponding author. E-mail address: [email protected] (M.T. MorrisThomas). 0141-1187/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.apor.2004.11.002
the free-surface. Under continuous wave action, these columns create significant diffraction and radiation effects in the surrounding fluid domain. These disturbances are usually associated with a considerable magnification of the local free-surface elevation such that two principal localised free-surface effects are important to platform designers. The first is the localised wave enhancement due to intensified fluid interaction on the free-surface between the platform’s columns. This effect is usually associated with the ‘air gap’ design of the structure and is described by Eatock Taylor and Sincock [1]. The allowable air gap, as measured between the underside of the deck structure and the mean water level, has traditionally been conservative. For offshore structures it is typically based on the highest predicted storm wave during the highest gravitational tide. The increased cost of raising the height of supporting columns was not thought sufficient to necessitate greater
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accuracy in the prediction of the greatest undisturbed wave height. However, this philosophy is often contentious for weight-sensitive floating structures, such as Tension Leg Platforms and semi-submersibles, because the increased air gap can not only increase fabrication costs but adversely affect the platform’s stability. More specifically, an increase in deck height requires a larger buoyant hull, which can increase vertical wave loads and create much difficulty in maintaining tendon tension. The second effect is wave run-up, which is exclusively associated with the columns of an offshore platform. When an incident wave impinges on a partially immersed body the free-surface of the wave undergoes a violent transformation where some portion of the incident wave’s momentum is directed vertically upward. To conserve energy, this momentum flux results in a rapid amplification of the waveform at the free-surface-body interface, see Fig. 1. This highly nonlinear phenomenon is generally known as wave run-up or up-rush and is particularly important for air gap design. Theoretical and experimental studies by numerous researchers have largely concentrated wave run-up investigations on piles, lighthouses, breakwaters, artificial islands, sloped beaches, and columns of fixed and floating offshore structures. Of particular interest to platform designers is the wave run-up on the forward vertical legs of both fixed and floating
Fig. 1. Illustration of the wave run-up resulting from an incident wave impinging on a vertical surface piercing column.
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offshore structures. In severe ocean environments the amplification of the incident wave may give rise to pressure impulse loads on the underside of the deck structuresometimes referred to as ‘slamming’ in the offshore structure community. A pressure impulse event occurs when a horizontal element, such as the platform deck or a member suspended from it, is impacted by a discrete volume of water rushing up the weather side of a platform column. While not posing a threat to the overall structural integrity of the platform, wave run-up is generally associated with localised structural damage. Based on experimental evidence for a circular cylinder in plane progressive waves, some researchers have shown that wave run-up, in extreme conditions, can be more than 1.5 times the incident wave amplitude [2,3]. Understanding wave run-up is fundamental to the correct estimation of a platform’s air gap requirements and to the minimisation of localised pressure impulse events. The extent of the wave run-up is largely dependent upon the transfer of energy to the harmonics of the wave field and the nature of this transfer is largely dependent upon two important parameters: the incident wave steepness kA; and the so-called scattering or slenderness parameter ka. It is understood that k is the wave number and a is the radius of the column. In the limit as kA/0, the slenderness parameter ka can be thought of as one that decides the normalised magnitude of each harmonic component of the vertical displacement of the free-surface. When kA is introduced, it essentially decides the importance of each harmonic contribution to the wave run-up. The ratio of these two parameters produces the free-surface Keulegan-Carpenter number, KcZA/a. For A/a less than of order unity, the flow around the circular cylinder will not separate and the fluid domain can be described by potential theory. To predict the wave run-up around a circular column, the theoretical treatment is generally based on potential theory whereby an idealised fluid domain is assumed and the well-known Laplace equation is solved with applied boundary conditions to yield a velocity potential. For wave motions, this is commonly referred to as diffraction theory. The free-surface elevation around the column may be obtained by application of the unsteady Bernoulli equation and velocity potential at the free-surface position. In solving for the velocity potential, one must deal with the complexity of the free-surface boundary condition. For deep water progressive surface gravity waves this is usually treated by one of two schemes. The first is a fully nonlinear numerical treatment in the time domain in the manner of Longuet–Higgins and Cokelet [4], and the second involves a mathematical approximation in the form of a Stokes expansion [5] whereby the free-surface displacement is assumed small in comparison to the characteristic wavelength. In this latter technique, the velocity potential and free-surface elevation are
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represented by the infinite series fðx; t; 3Þ Z
N X nZ1
3n fðnÞ ðx; tÞ;
zðx; y; t; 3Þ Z
N X
3n zðnÞ ðx; y; tÞ;
nZ1
(1) where the perturbation quantity, 3/1, is understood to be the wave steepness kA. The treatment of wave run-up in the form of a Stokes expansion will be studied here. Specifically, with the aid of experiment, we shall examine whether the wave run-up around a circular column, and the first few harmonic components of it, can be sufficiently predicted by the first two terms of a diffraction theory solution in the form (1). Of the many experimental investigations involving wave run-up on circular cylinders in plane progressive waves, the first study directly applicable to the regime of A=a! Oð1Þ was undertaken by Chakrabarti and Tam [6]. However, these authors were not concerned with the magnitude of the free-surface elevation but rather whether the incident wave direction could be determined from the symmetry exhibited by the pressure distribution on the column. Perhaps, the first experimental study to consider the inadequacy of linear diffraction theory (discussed in Section 2) as a measure of the wave run-up was documented by Isaacson [2]. This inadequacy is to such an extent where Isaacson [2] suggests that a factor of 2 should be applied in order to obtain a better representation of the wave run-up. However, this pragmatic approach gives no account for wave steepness and would presumably be conservative for small kA. The benchmark experiments of Kriebel [3] demonstrated that second-order diffraction theory (discussed in Section 2) provides an improvement over linear theory and compares favourably with the measured run-up for values of kA approximately 30% less than the wave breaking limit. The first experimental investigation that considered the harmonic components of the wave run-up was documented by Mercier and Niedzwecki [7]. Their experiments concerned a fixed vertical truncated circular cylinder in monochromatic waves such that 0.147%ka%0.915 for three distinct wave steepnesses of kAZ0.05, 0.01 and 0.15. Although the actual wave run-up was not investigated, the measured first- and second-harmonic components of it were compared with first- and second-order diffraction theory predictions. The measurements of Mercier and Niedzwecki [7] demonstrate an increasing modulus of each harmonic with increasing ka. Moreover, a negligible influence of kA on the first-harmonic wave run-up, and an unsatisfactory correspondence with theory for large ka was shown. On the other hand, the modulus of the measured second-harmonic was found to agree with predicted values for small ka. However, the effect of incident wave nonlinearities, through the parameter kA, on the measured harmonics was not discussed. It should be noted that the numerical results of Mercier and Niedzwecki [7] appear to contain errors and have clearly not converged for large
scattering parameters. This, perhaps, would explain the poor correspondence with measured results. It is widely understood that linear diffraction theory generally under-predicts wave run-up in all but small wave steepnesses [2,3]. However, what has not been sufficiently investigated is the extent to which the linear diffraction solution predicts the first-harmonic of the wave run-up. This is of fundamental importance and is paramount to the validity of wave diffraction theory with the free-surface boundary condition treated by a Stokes expansion scheme. An investigation on the limitations of perturbation theory for wave forces on a circular cylinder has been presented by Huseby and Grue [8], however, such detailed comparisons have yet to be undertaken for the corresponding wave runup and/or free-surface elevations. In particular, the zerothharmonic component of the wave run-up has yet to be isolated for detailed analysis. At this point it is unclear as to whether, and to what extent, a potential flow model with Stokes’ treatment of the free-surface boundary condition can indeed predict the constituent harmonic components of the wave run-up. More specifically, its ability to do so for the zeroth-, first- and second-harmonics for a Stokes expansion correct to Oð32 Þ is questionable. The purpose of the present paper is to investigate the wave run-up, and the harmonic components of it, on a fixed vertical cylinder in monochromatic progressive waves. A brief theoretical account is given of the scattering of a plane progressive wave by a circular cylinder in Section 2. An experimental campaign is outlined in Section 3. The comparison of theory with measured results is presented in Sections 4 and 5 for the harmonics and wave run-up, respectively. Finally, conclusions are given in Section 6.
2. Scattering of a plane progressive wave For the purpose of this present paper we will consider plane progressive waves in deep water. According to the theory of Stokes (see Lamb [9], Art. 250), the profile of a surface gravity wave travelling in a positive x-direction can be described by 1 zðx; tÞ Z A cosðut K kxÞ C kA2 cos 2ðut K kxÞ 2 3 C k2 A3 cos 3ðut K kxÞ; 8
(2)
correct to third-order in wave steepness where uZu(k) defined as u2Zgk(1Ck2A2), and g is the acceleration due to gravity. In general, only the first two terms of this expansion will be considered, however, it is instructive to reveal the third term in view of third-harmonic contributions to the wave run-up. When the progressive wave described by the first two terms of (2) impinges on a partially immersed cylinder in a fluid domain of an infinite horizontal extent,
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the resulting free-surface elevation can be described by qÞ C
(3)
(1)
in a cylindrical coordinate system where z is of Oð3Þ while z and z(2) are of Oð32 Þ and represent the mean and second-harmonic components, respectively, from the interaction of (2) with the cylinder. This canonical representation is typical of potential flow diffraction theory. The particular form of z(1) is quite fundamental and was first derived by Havelock [10] as ( ) N X ð1Þ Km 3m i Cm ðkrÞ cos mq ; (4) z ðr; qÞ Z < KA mZ0
with Cm ðkrÞ Z Jm ðkrÞ K
Jm0 ðkaÞ 0
Hmð2Þ ðkaÞ
Hmð2Þ ðkrÞ;
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leading-order solution. Moreover, it is interesting to note from (4) that an asymptotic value of z(1)w2A results when the limit of ka/N is considered. This agrees with the linear solution to a progressive wave incident on an infinitely long vertical wall (see Lamb [9], Art. 228–229). The forms of z and z(2) are complicated and will not be reproduced here. They were, however, first given by Hunt and Baddour [12] by means of a Weber integral transform technique and subsequently extended to water of finite depth by Hunt and Williams [13]. For the purpose of evaluation in this present work the second-order radiation/diffraction analysis code WAMIT [14,15] is utilised to numerically evaluate the contributions of z and z(2) in (3). In doing so great care was taken through convergence studies, the details of which can be found in Morris-Thomas [16].
(5)
where 3m denotes the Neumann symbol: 30Z1 and 3mZ2 for mO0, Jm is the Bessel function of order m, Hmð2Þ is a Hankel function of the second kind of order m, the prime denotes differentiation with respect to the argument, and the wave number is given by the dispersion relation kZu2/g. The linear diffraction solution (4) has been simply extended by MaCamy and Fuchs [11] to water of arbitrary depth. Although extending the Stokes expansion of the free-surface boundary condition to odd integer powers of 3 will indeed produce further contributions that operate at the firstharmonic, much information can still be obtained from this
3. Experimental campaign Experiments were conducted in the towing tank of the Ship Hydrodynamics Centre at the Australian Maritime College, Tasmania. The towing tank measures 60 m in length, 3.55 m in width and has a water depth of 1.5 m. A schematic of the experimental configuration is illustrated in Fig. 2. Waves were generated by a computer controlled bottom-hinged flat hydraulic paddle. At the downstream end of the flume, a corrugated beach with multiple baffles, of length 4 m, was installed. In addition to this passive absorption, longitudinal, pneumatically controlled beaches
Fig. 2. Schematic of the test set-up at the Australian Maritime College; wave probes are denoted C; PWP and RWP represent phase wave probe and repeatability wave probe, respectively.
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were installed to absorb water disturbances after the completion of each test run. 3.1. Model configuration The experiments were restricted to fixed cylinders as greater insight into wave run-up can first be gained by investigating this simpler problem. Model dimensions were scaled from a typical offshore structure configuration and chosen so that details of the wave–structure interaction were clearly visible. The cylindrical model dimensions of 30 cm in width and 87 cm in height were selected. Consequently, the wave tank width to model diameter ratio was approximately 11.8 so that side wall reflections, which may produce parasitic wave–wave interactions, should therefore be negligible during data analysis. Moreover, a corner radius variation similar to a Lewis conformal mapping [17] was selected to produce five cylindrical models of varying cross-sectional dimensions, whereby rc/a2 [0, 0.25, 0.5, 0.75 and 1] where rc denotes corner radius. In this paper, we concentrate only on the results pertaining to a circular cross-section where rc/aZ1 (the remaining cross-sections are discussed in a forthcoming paper [18]). The models were constructed with 10 mm marine grade plywood using a CNC Router with an expected error of G0.5 mm over the given dimensions. External surfaces were coated with yellow enamel paint and finished to a smoothness of G21 mm. A 25 mm square grid was scribed on each model for the purpose of free-surface measurements. Rather than prevent water seepage, which is often a difficult and expensive task during model construction, water entry was allowed at the base of each model. Consequently, the cylinder walls were supported by multiple internal baffles, coated with epoxy, to ensure rigidity and the avoidance of internal sloshing modes that could lead to erroneous model deflections. 3.2. Instrumentation and measurement The free-surface elevations were recorded at select locations (see Fig. 2 or the tabulated co-ordinates in Table 1 for the exact locations) with wire probes consisting of two parallel 1.6 mm diameter conductors spaced 10 mm apart. The time series free-surface elevation was digitised at 100 Hz throughout the experimental campaign. The array of Table 1 Positions of the 6 near field wave probes relative to the cylinder surface Wave probe
r(m)
q (radians)
A1 A2 A3 A4 B1 C1
0.001 0.050 0.100 0.150 0.001 0.001
p p p p 3p/4 p/2
wire probes—of 50 mm spacing—located on the up-wave side were assessed for possible signal interaction under wave excitation before testing commenced. No signal interaction was evident in the results. Two additional wire probes were used during each test; one was located midway between the tank wall and the centreline of the cylinder position to obtain phase information; and another 18 m from the wave maker to assess the repeatability of the incident waves. In addition to the wave probe configuration, each run was recorded by two video cameras (cf. Fig. 2). These visual measurements validated readings from the wave probes and assisted in examining the existence of surface tension effects associated with the wave probes adjacent to the cylinder surface. Surface tension was examined by testing the configuration under a variety of wave conditions and correlating the recorded wave elevations from the wave probes with visual observations of the wave-up through the video cameras. The agreement between video recordings and wave probe measurements was excellent. Surface tension effects are thought to be insignificant in the results to follow. 3.3. The wave environment The wave environment was restricted to monochromatic progressive waves, with frequencies and wave heights chosen to produce a parametric variation of both kA and ka. Five distinct wave periods and a range of have heights— denoted T and H, respectively—were selected as shown in Table 2. A scaling ratio of 1:55.5 was chosen to reduce prototype regular wave conditions to those of model scale. No frequencies corresponding to cross-tank resonant modes of u2r;m Z gpm=l, where l is the tank width and mZ1,2,., were used. The largest adopted wave steepness approximates to 63% of the wave breaking limit in deep water. The water depth was held constant at 1.5 m. With the model removed, wave repeatability was investigated by running the test matrix and correlating the results with a calibration wave probe at the removed model position and a repeatability wave probe 3.2 m from the central model position. Less than approximately 0.5% wave amplitude decay was noted between these two probes. For each set of wave conditions, the water surface was initially quiescent and the wave paddle started from the vertical position. Data acquisition commenced after the first few transient surface elevations passed the model, and lasted for approximately 14–30 wave periods. Measurements were recorded before any reflected waves of the fundamental frequency, u, arrived back at the testing area, which was 21.2 m from the vertical position of the paddle face.
4. Results and discussion The maximum wave amplitudes from each wire probe, denoted A(r, q), were found by taking an average over each
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Table 2 The wave environment for the circular cylinder, illustrating the parametric variation of both wave steepness kA and cylinder slenderness ka Prototype
Model
Dimensionless parameters
T (s)
H (m)
T (s)
H (m)
ka
kA
A/a
4.92 6.93 8.96 10.9 12.7
0.75–3.94 1.56–7.27 2.32–11.0 2.87–14.1 4.55–20.6
0.66 0.93 1.20 1.46 1.70
0.01–0.07 0.03–0.13 0.04–0.20 0.05–0.25 0.08–0.37
1.386 0.698 0.417 0.283 0.208
0.041–0.206 0.062–0.284 0.057–0.263 0.047–0.228 0.055–0.233
0.029–0.149 0.089–0.407 0.137–0.630 0.166–0.805 0.264–1.124
wave cycle in the steady state time series. The actual wave run-up R, measured relative to the quiescent frees–urface zZ0, was obtained from A(a, p). All measurements presented here exhibit a small standard deviation in wave amplitude over the time series’ and therefore, resemble progressive waves of permanent form. Harmonic components were extracted by a discrete Fourier transform algorithm. Of particular interest is the zeroth-, first-, second-, and third-harmonics of the wave profiles. These harmonics are denoted R(un) and A(un) for the wave run-up and incident wave, respectively, where nZ0,1,2,.,. Before the Fourier transform was taken the zeros of the time series were found and, in general, an integer number of wave periods were extracted for the Fourier analysis. This is important, not only for capturing the zeroth-harmonic but for minimising the bandwidth of each wave harmonic to avoid energy leakage into neighbouring components of the frequency space. An example of the wave profile and its harmonic components is illustrated in Fig. 3. The wave amplitude A, and fundamental circular wave frequency, u, are defined as the modulus and frequency of the first-harmonic Fourier component of the incident wave. Consequently, it is important to note that the wave steepness kA is based on the first-harmonic of the carrier wave. Since khO2.08 for all test conditions, the water depth is assumed infinite so that the wave run-up caused purely by wave–structure interaction is examined without the additional influence of a nondecaying horizontal fluid velocity and finite depth effects on wave dispersion.
4.1. Relative harmonic contributions As a preliminary, and to gain some insight into the magnitudes of the first few harmonics relative to the actual wave run-up, the ratio jR(un)j/R was investigated, where jR(un)j defines the modulus of the nth harmonic wave runup component. The numerical values of this ratio versus kA for five different values of ka are presented in Fig. 4. In short waves, kaZ1.386 and 0.698, the first harmonic component appears invariable as kA increases. However, in longer waves, kaZ0.208 for instance, jR(u1)j/R decreases with increasing kA and is coupled with an increase in contributions from both jR(u2)j/R and jR(u3)j/R. The thirdharmonic component only appears to contribute in long waves, kaZ0.208, when the wave steepness approaches around 50% of the deep water breaking limit-defined by kAmaxZ0.14p (see Mei [19], equation 10.4.4). A feature that has not been examined in great detail throughout the literature is the zeroth-harmonic or mean component, R(u0), and its contribution is clearly illustrated in Fig. 4. The measurements indicate that jR(u0)j/R linearly increases with increasing kA to an extent where it accounts for approximately 25% of the wave run-up in the region of kAz0.2–0.25. This increase and eventual peak of jR(u0)j/R at approximately 0.25R for kAz0.2–0.25 appears to be independent of ka (cf. Fig. 4). Apart from the fundamental harmonic, the measurements R(u0) provides the dominant contribution to R in short waves. Interestingly though, for kaZ0.417, jR(u0)j and jR(u2)j contribute equally to wave run-up. Consequently, in the region of ka less than
Fig. 3. Time series (right) and corresponding spectral plot (left) for a typical wave train produced by the wave maker, kAZ0.231 (TZ12.7 s full scale). The fundamental, second- and third-harmonic components of the wave train are clearly visible in the spectral plot.
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Fig. 4. The ratio of the modulus of each harmonic component and the wave run-up, denoted: (C), zeroth-harmonic; (C), first-harmonic; (,), second-harmonic and (B), third-harmonic. (a) kaZ1.386; (b) kaZ0.698; (c) kaZ0.417; (d) kaZ0.283 and (e) kaZ0.208.
approximately 0.417, the second-harmonic becomes the dominant bound component of the wave run-up. To further illustrate the relative contributions of the wave run-up, we shall consider one particular test condition, where kaZ0.208 and kAZ0.254. The wave steepness corresponds to approximately 60% of the wave breaking limit. The first-harmonic is moderately amplified by 1.2A while the second-harmonic is greatly influenced by the presence of the cylinder to an extent where jR(u2)j is almost 3A(u2). For the wave run-up the second-harmonic is approximately 40% of R(u1) whereas the mean and thirdharmonic are roughly 20 and 8%, respectively. Despite the long wavelength, where amplifications from diffraction effects are expected to be small, this example coupled with the results of Fig. 4 clearly illustrates the importance of kA
which gives rise to large contributions from both the mean and second-harmonic components. It is interesting to note that for kaZ1.386, the measurements suggest that approximately 60% of the wave run-up is accounted for by jR(u1)j. Given that the other major contribution is from jR(u0)j, this leaves much of the wave run-up unaccounted for. This is explained as follows: for the measurements pertaining to kaZ1.386, the free-surface profile at A(a, p) exhibited a distinct aperiodic nature, such that the wave run-up peaked every 3-4 wave periods despite an incident wave of permanent form. This effect has also been observed by LonguetHiggins and Drazen [20] during experiments involving waves meeting a vertical wall. The two cases could be related but without further exploratory experiments, no
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conclusions can be made. In any event, the results pertaining to kaZ1.386 will not be discussed further here. 4.2. Harmonic measurements and theory compared We shall now investigate the correspondence between second-order diffraction theory and measurements for the first few harmonics of the wave run-up at the position (r, q)Z(a, p). In doing so it is convenient to normalise the measured and predicted harmonics by jRðun ÞjamK1 =Am ;
jzðnÞ jamK1 =Am ;
respectively, where m is the Stokes perturbation order and n is the harmonic component. According to second-order theory, the mean wave run-up is of Oð32 Þ and is therefore, normalised accordingly. The correspondence between theory and measurement for the modulus and phase of each harmonic is illustrated in Figs. 5–8. 4.2.1. Zeroth-harmonic The measured zeroth-harmonic of the wave run-up is not particularly well predicted by theory (see Fig. 5). Generally speaking, the measured results of the zeroth-harmonic exhibit scatter which appears to decrease for decreasing ka. However, in the region of this decreased scatter R(u0) is reasonably well predicted by second-order diffraction theory for small kA. This observation is particularly evident for waves of kaZ0.208. In longer waves, the measured zeroth-harmonic increases with increasing wave steepness (cf. Fig. 5 for kaZ0.208 and kaZ0.283). This indicates that R(u0) is influenced by higher than second-order wave steepness effects in large values of kA. In terms of a Stokes expansion of the freesurface boundary condition, this dependence implies that, at the least, a fourth-order kA expansion is required to recover the dependence of kA on R(u0). Alternatively, a fully nonlinear treatment of the free-surface boundary condition may produce the desired result. It should be pointed out that the undisturbed incident waves contained a small zeroth-harmonic component. In complex form the zeroth-harmonic contained a phase of either Gp. Although this mean component is obviously physical, its origin is unclear since according to the Bernoulli theorem applied at the free-surface position the mean wave component should be zero for a free wave with corresponding locked harmonic components. Although it is possible that the datum for each wave probe drifted, this seems unlikely considering that the datum was recorded before each test run commenced. Correcting for this nonzero mean did not affect the recorded free-surface elevations substantially and the discrepancies observed in Fig. 5 remain. Recently, observation by Stansberg and Braaten [21] have confirmed the existence of a large zerothharmonic component that is not well predicted by secondorder diffraction computations.
Fig. 5. The normalised modulus of mean component of the wave run-up jR(u0)j plotted against wave steepness kA. Measured results, (C) and WAMIT, (—).
4.2.2. First-harmonic Previous authors have noted large discrepancies when comparing the linear diffraction solution of MacCamy and Fuchs [11] to the complete wave run-up [2,22]. Here however, the linear diffraction solution, which contains only the fundamental wave frequency, is compared only to the first-harmonic component of the wave run-up. In doing this, and distinguishing between the coupling of ka and kA, the results indicate that the linear diffraction solution performs reasonably well by capturing both the modulus and phase of R(u1) (cf. Fig. 6). An interesting observation is seen for kaZ0.208 when the wave steepness reaches approximately 50% of
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Fig. 6. The normalised modulus jR(u1)j and phase arg(R(u1)) of first-harmonic component of the wave run-up plotted against wave steepness kA. Measured results, (C) and the linear diffraction solution, (—).
the wave breaking limit. In this region, the measured results exhibit a relatively consistent increase in jR(u1)j for kAO0.2. This is coupled with a decrease in the phase angle. These effects are moderately replicated for the results of kaZ0.283. Although, this observation corresponds to the longest wavelength examined, it is not thought to be caused by amplitude dispersion to an extent where finite depth effects have magnified the wave run-up but rather higher that second-order wave steepness effects operating at the first-harmonic. Such nonlinear effects do not occur for kaZ0.208 in waves of small steepness and would suggest that a first-order prediction of R(u1) is adequate in this region.
4.2.3. Second-harmonic The theoretical prediction of jR(u2)j compares well with the measured results for kaZ0.283 and 0.208 (cf. Fig. 7). However, a discrepancy exists for wave of small steepness which is to be expected since the measured secondharmonic is relatively small. Consequently, its modulus is sensitive to the position of the time window selected within the wave group for the Fourier transform. The theoretical predictions consistently over estimate the measured modulus of the second-harmonic in shorter wavelengths (cf. Fig. 7 for kaZ0.698 and 0.417). It is not possible to attribute this to higher-order effects because in the limit of kA/0, one would expect the measured results
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Fig. 7. The normalised modulus jR(u2)j and phase arg(R(u2)) of second-harmonic component of the wave run-up plotted against wave steepness kA. Measured results, (C); and second-order calculations, (—).
to agree with the theoretical values. For instance, the measured results indicate a reasonable agreement for kaZ0.698 but certainly not for kaZ0.417. A possible explanation could be that the measured second-harmonic exhibits a small degree of variability in these shorter waves and has affected the resolution of the Fourier transform. The phasing of the second-harmonic component is shown to be invariable with wave steepness (cf. Fig. 7), which suggests that the second-harmonic phase is not influenced by higher than second-order in kA effects. However, since second-order theory appears to under predict the phase by a constant amount in all but one case (kaZ0.208), this does suggest that higher-order diffraction
effects are important. One exception is noted at kaZ0.698 whereby the phase appears to decrease with increasing kA. 4.2.4. Third-harmonic Difficulty arose in obtaining an accurate extraction of the third-harmonic as it is considerably smaller than both the first- and second-harmonics. Due to this variability, only measurements of the third-harmonic pertaining to kAO0.125 are shown (cf. Fig. 8). When the onset of viscous effects is considered (A/aZ1), the measured results for kaZ0.208 indicate that jR(u3)j is around 0.1A. On the other hand, for A/aZ0.5 the measured results indicate that jR(u3)j will be around 0.025A. This suggests that only when
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Fig. 8. The normalised modulus jR(u3)j and phase arg(R(u3)) of third-harmonic component of the wave run-up plotted against wave steepness kA. Measured results, (C).
the incident wave amplitude is comparable to the cylinder radius does third-harmonic effects become important to the wave run-up. Although the modulus of the third-harmonic component is small, a fairly constant contribution for kAO0.125 is exhibited by the measured results. Moreover, the phase of the third-harmonic is reasonably invariant with both kA and ka. (see Fig. 8 for kaZ0.417 and kaZ0.283). The consequence of this is that one would expect the thirdharmonic to increase the wave run-up in longer waves due to the discrepancy in phasing with the first-harmonic component (cf. kaZ0.698 and kaZ0.208 in Figs. 6 and 8). This observation can be easily verified by summation of the complex harmonics in the time domain.
5. The wave run-up We shall now consider the measured wave run-up normalised by AZjA(u1)j. A comparison of the diffraction theory computations and measurements is illustrated in Fig. 9 for a range of wave steepnesses and three particular values of ka—a value of 1 on the ordinate indicates no amplification of the incident wave. The second-order wave run-up is defined as the complex addition of the components in (3) evaluated at the position (r, q)Z(a, q). Since A/a!1 for all but one of the selected test conditions, potential flow theory should be applicable and the comparison valid.
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Fig. 9. The wave run-up versus the wave steepness kA for three different scattering parameters, ka. Measured results, (C); linear diffraction theory (– – –) and second-order calculations, (—).
Separating the wave run-up into its dependence on ka and kA clearly illustrates the influence of diffraction and incident wave nonlinearity effects (cf. Fig. 9). For constant ka, the results indicate that the normalised wave run-up increases for increasing kA. For small kA, local diffraction effects are clearly dominant and the amplification of the incident wave generally increases with increasing ka. However, despite large incident wave amplifications for large ka, the actual wave run-up will essentially be governed by kA. It is the intermediate to long wavelengths which are most important since the limiting value for the wave height is governed by kAmaxZ0.14p for water of infinite depth. With this in mind, it is the long waves that will produce the highest vertical water displacements on the column surface. Over the complete range of (ka, kA) combinations considered the linear diffraction solution of Havelock [10] does not perform well-this observation is not new [2,23]. The first-order solution demonstrates no kA dependence.
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Consequently, the error in the first-order prediction monotonically increases for increasing kA. For example, this error is around 20 and 100% in large values of kA for kaZ0.698 and 0.208 respectively. On the other hand, the first-order prediction does realise a reasonable agreement with the measured results for very small kA (kAz0.05 for instance). This indicates that at such a small wave steepness incident wave nonlinearities are not important and that local wave diffraction is confined to the first-harmonic. The decomposition of the wave run-up into harmonic proportions (cf. Section 4.1) suggests this to be so. It is reasonable to suggest that the wave run-up, regardless of ka, approaches the linear diffraction solution as kA/0. This is an important and fundamental result. The under prediction by linear diffraction theory is somewhat arrested by the inclusion of second-order nonlinearities. Perhaps this is best illustrated by the results pertaining to kaZ0.698, where the second-order prediction agrees favourably with the measured wave run-up up to, and including, kAz0.25. However, as the incident wave becomes longer nonlinear effects associated with the incident wave steepness occur at increasingly smaller values of kA. This phenomenon could be described as a ‘threshold’ for extreme, or higher than second-order, nonlinear effects associated with kA. At this threshold the normalised measured values are seen to depart from the second-order predictions (cf. Fig. 9 for kaZ0.417 and 0.208). The measured values are then seen to follow a nth degree polynomial trend. This possibly indicates the existence of higher-order contributions to the wave run-up and would suggest that a fully nonlinear representation of the freesurface elevation would be a more efficacious approach to predict the wave run-up in long steep waves. Furthermore, it also suggests that the wave steepness limit for Stokes second—order wave theory—consult Fenton [24] is essentially not applicable for second-order wave diffraction. When examining the actual wave run-up (on r/aZ1), it is also important to consider measurements of the free-surface elevation in the fluid domain surrounding the cylinder. In particular, along the ray q/pZ1. This accumulation of water build-up is presented in Figs. 10–12 for each value of ka and kA. In these figures the radial distance is normalised by the cylinder radius, a, so that a value of r/aZ2 corresponds to 1 cylinder radius away from the cylinder. Moreover, along with the measured results, linear and second-order numerical results are given. Further evidence of the occurrence of this non-linear threshold is provided in Figs. 10–12. For kaZ0.208 in particular, the free-surface elevation along the ray q/pZ1 is generally well predicted by the second-order model. However, for kAO0.2 the measured results tend to deviate away from the second-order diffraction theory predictions as r/aZ1 is approached. So much so that for kAZ0.233 the error of R/A is approximately 30% at r/aZ1. In contrast, this nonlinear threshold does not appear in the measured results for kaZ0.698 along the ray q/pZ1 as
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Fig. 10. The normalised free-surface elevation at q/pZ1 versus r/a for kaZ0.698. Measured values are given by the dashed line with (C); first-order calculations, (.) and second-order calculations, (—).
r/a/1 (cf. Fig. 10). Interestingly, both the wave run-up and the amplification of the free-surface are reasonably well predicted by second-order diffraction theory for small values of kA. On the other hand, for kAO0.212, although the trend of the free-surface is captured, the second-order model slightly over predicts the free-surface elevations at all values of q/pO1. This could be attributed to a mean set-up that becomes important when the wave steepness becomes large. Despite this, the wave run-up at (r/a, ka)Z(1, 0.698) in large kA is well predicted by the second-order model.
6. Conclusions Wave run-up on fixed surface piercing cylinders in the presence of deep water monochromatic progressive waves has been investigated in the diffraction dominated regime where the assumption of potential flow theory is applicable. Firstly, the harmonic components of the wave run-up were examined to determine the importance of each with
regards to local wave diffraction and incident wave steepness. This investigation has found that the contribution from the first-harmonic decreases with increasing kA. This is coupled with an increase from both zeroth- and higher harmonic components. Experimental evidence suggests that, of the nonlinear contributions, the mean set-up is the dominant contributor for ka!0.42 while above this limit the second-harmonic dominates. The mean set-up accounts for up to 25% of the wave run-up for large wave steepnesses. The third-harmonic only becomes important in long waves when kAO0.2. In this region the modulus of R(u3) is approximately 8% of R. The majority of the wave run-up consists of zeroth-, first-, and second-harmonic components. The next objective was to compare the normalised harmonics of the wave run-up with potential flow numerical predictions. This revealed that both the modulus and phase of the first-harmonic wave run-up are generally well predicted by first-order diffraction theory. However, in long waves, distinct nonlinear effects-greater than Oð32 Þ in Stokes perturbation-operating at the first-harmonic are
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Fig. 11. The normalised free-surface elevation at q/pZ1 versus r/a for kaZ0.417. Measured values are given by the dashed line with (C); first-order calculations, (.) and second-order calculations, (—).
clearly evident for kAO0.2. Consequently, these effects are not predicted by linear diffraction theory and a higherorder description of the free-surface is required. Measurements of the zeroth- and second-harmonic components generally compared unfavourably with calculated values. In particular, the zeroth-harmonic is under predicted by up to 200% in certain cases while the modulus and phase of the second-harmonic is perhaps best predicted for longer wavelengths. The actual wave run-up, containing all harmonic contributions, at r/a and qZp, has been examined. The wave run-up is shown to be distinctly dependent upon both ka and kA. A definite kA threshold is observed where extreme nonlinear effects, that are greater than second-order in a Stokes expansion, commence. The exact value of kA where these nonlinear effects occur is dependent upon ka. For instance, when kaZ0.618 these nonlinear effects were unnoticeable and a Stokes expansion to second-order adequately predicts the measured results. However, for
kaZ0.208 this nonlinear threshold occurs in the region 0.15!kA!0.2, above which, extreme amplifications occur that are as much as twice the incident wave amplitude. Results presented here suggest that linear diffraction theory is insufficient for wave run-up estimation. This confirms conclusions made by previous authors. However, it does appear that in the limit as kA/0, linear diffraction theory performs adequately as a fundamental solution. A Stokes expansion to second-order, when employed in the diffraction problem and as incorporated in WAMIT, produces reasonable estimates of wave run-up provided the nonlinear threshold is not breached. However, for longer wavelengths the phasing of the third-harmonic is such that below the kA threshold the third-harmonic reduces the wave run-up whilst above the kA threshold the third-harmonic increases the wave run-up. This distinct kA dependence of the third-harmonic compromises correspondence between measured and Stokes second-order predictions for the longest waves examined here.
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Fig. 12. The normalised free-surface elevation at q/pZ1 versus r/a for kaZ0.208. Measured values are given by the dashed line with (C); first-order calculations, (.) and second-order calculations, (—).
Acknowledgements The authors gratefully acknowledge the support of MARINTEK, DNV and the Australian Research Council. The first author was partially supported by a George Samaha Research Scholarship during portions of this work. Moreover, the authors would like to thank Gregor Macfarlane and the laboratory staff of the Ship Hydrodynamics Centre at the Australian Maritime College.
References [1] Eatock Taylor R, Sincock P. Wave upwelling effects in TLP and semisubmersible structures. Ocean Eng 1989;16(3):281–306. [2] Isaacson, M de St Q. Wave runup around a large circular cylinder. J Waterway, Port, Coastal and Ocean Div 1978;104(1):69–79. [3] Kriebel DL. A second-order diffraction theory for wave runup and wave forces on a vertical circular cylinder, PhD Thesis, University of Florida; 1987.
[4] Longuet-Higgins MS, Cokelet ED. The deformation of steep surface waves on water I. A numerical method of computation. Proc R Soc Lond A 1976;350:1–26. [5] Schwartz LW. Computer extension and analytic continuation of Stokes’s expansion for gravity waves. J Fluid Mech 1974;62: 553–78. [6] Chakrabarti SK, Tam WA. Wave height distribution around a vertical cylinder. J Waterways Harbors and Coastal Eng Div 1975;101(2): 225–30. [7] Mercier RS, Niedzwecki JM. Experimental measurements of secondorder diffraction by a truncated vertical cylinder in monochromatic waves. BOSS’94 1994;2:265–87. [8] Huseby M, Grue J. An experimental investigation of higher-harmonic wave forces on a vertical cylinder. J Fluid Mech 2000;414:75–103. [9] Lamb H. Hydrodynamics. 6th ed. New York: Dover Publications; 1932. [10] Havelock TH. The pressure of water waves upon a fixed obstacle. Proc R Soc Lond A 1940;175:409–21. [11] MacCamy RC, Fuchs RA. Wave force on piles: A diffraction theory. Beach Erosion Board Office of the Chief Engineers, Department of the Army, Technical Memorandum no. 69; 1954 1–17. [12] Hunt JN, Baddour RE. The diffraction of a nonlinear progressive wave by a vertical cylinder. Quart J Mech Appl Math 1981;34:69–88.
M.T. Morris-Thomas, K.P. Thiagarajan / Applied Ocean Research 26 (2004) 98–113 [13] Hunt JN, Williams AN. Nonlinear diffraction of Stokes water waves by a circular cylinder for arbitrary uniform depth. J Me´c The´or Appl 1982;1(3):429–49. [14] Hess JL, Smith AMO. Calculation of nonlifting potential flow about arbitrary three-dimensional bodies. J Ship Res 1964;8:22–44. [15] Lee CH. WAMIT theory manual, Report 95-2, Massachusetts Institute of Technology; 1995. [16] Morris-Thomas MT. An investigation into wave run-up on vertical surface piercing cylinders in monochromatic waves, PhD Thesis, The University of Western Australia; 2003. [17] Lewis FM. The inertia of water surrouding a vibrating ship, 37th general meeting of SNAME 1929; 1–20. [18] Morris-Thomas MT, Thiagarajan KP. The run-up on a cylinder in progressive surface gravity waves: Geometry effects, in preparation; 2005.
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[19] Mei CC. The applied dynamics of ocean surface waves. Advanced Series on Ocean Engineering. Vol. 1. Singapore: World Scientific; 1989. [20] Longuet-Higgins MS, Drazen DA. On steep gravity waves meeting a vertical wall: a triple instability. J Fluid Mech 2002;466:305–18. [21] Stansberg CT, Braaten H. Nonlinear wave disturbance around a vertical circular column, Proc 21st Intl Conf Offshore Mech. and Arctic Eng, Oslo, Norway; 2002, paper 28620. [22] Niedzwecki JM, Duggal AS. Wave runup and forces on cylinders in regular and random waves. J Waterway, Port, Coastal, Ocean Eng 1992;118(6):615–34. [23] Kriebel DL. Non-linear wave interaction with a vertical circular cylinder. Part 2: wave run-up. Ocean Eng 1992;19(1):75–99. [24] Fenton JD. Nonlinear wave theories. The Sea 1990;9:3–25.
The run-up on a cylinder in progressive surface gravity waves: harmonic components M.T. Morris-Thomas*, K.P. Thiagarajan School of Oil and Gas Engineering, The University of Western Australia, Crawley, WA 6009, Australia Received 16 June 2004; revised 5 November 2004; accepted 10 November 2004 Available online 9 February 2005
Abstract Wave run-up is the vertical uprush of water that occurs when an incident wave impinges on a partially immersed body. In this present work, wave run-up is studied on a fixed vertical cylinder in plane progressive waves. These progressive waves are of a form suitable for description by Stokes’ wave theory whereby the typical energy content of a wave train consists of one fundamental harmonic and corresponding phase locked Fourier components. The limitations of canonical wave diffraction theory—whereby the free-surface boundary condition is treated by a Stokes expansion—in predicting the harmonic components of the wave run-up are discussed. An experimental campaign is described and the choice of monochromatic waves is indicative of the diffraction regime for large volume structures where the assumption of potential flow theory is applicable, or more formally A=a! Oð1Þ (A and a being the wave amplitude and cylinder radius, respectively). The wave environment is represented by a parametric variation of the scattering parameter ka and wave steepness kA (where k denotes the wave number). The zeroth-, first-, second- and third-harmonics of the wave run-up are examined to determine the importance of each with regard to local wave diffraction and incident wave nonlinearities. It is shown that the complete wave run-up is not well accounted for by second-order diffraction theory. This is, however, dependent upon the coupling of ka and kA. In particular, whilst the modulus and phase of the second-harmonic are moderately predicted, the mean set-up is not well predicted by a second-order diffraction theory. Experimental evidence suggests this to be caused by higher than second-order nonlinear diffraction. Moreover, these effects most noticeably operate at the first-harmonic in waves of moderate to large steepness when ka/1. q 2005 Elsevier Ltd. All rights reserved. Keywords: Surface gravity waves; Wave run-up; Perturbation theory; Stokes waves
1. Introduction During the preceding four decades the offshore oil and gas industry has paid much attention to the scattering of water waves by large volume offshore structures. Historically, the industry has been dominated by fixed, transparent, offshore structures comprising tubular members, where diffraction effects are considered less important compared to drag dominated or viscous effects. However, with the advent of oil exploration and exploitation into increasing water depths, floating structures are an attractive alternative. These floating structures typically comprise a deck supported by large diameter columns that penetrate * Corresponding author. E-mail address: [email protected] (M.T. MorrisThomas). 0141-1187/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.apor.2004.11.002
the free-surface. Under continuous wave action, these columns create significant diffraction and radiation effects in the surrounding fluid domain. These disturbances are usually associated with a considerable magnification of the local free-surface elevation such that two principal localised free-surface effects are important to platform designers. The first is the localised wave enhancement due to intensified fluid interaction on the free-surface between the platform’s columns. This effect is usually associated with the ‘air gap’ design of the structure and is described by Eatock Taylor and Sincock [1]. The allowable air gap, as measured between the underside of the deck structure and the mean water level, has traditionally been conservative. For offshore structures it is typically based on the highest predicted storm wave during the highest gravitational tide. The increased cost of raising the height of supporting columns was not thought sufficient to necessitate greater
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accuracy in the prediction of the greatest undisturbed wave height. However, this philosophy is often contentious for weight-sensitive floating structures, such as Tension Leg Platforms and semi-submersibles, because the increased air gap can not only increase fabrication costs but adversely affect the platform’s stability. More specifically, an increase in deck height requires a larger buoyant hull, which can increase vertical wave loads and create much difficulty in maintaining tendon tension. The second effect is wave run-up, which is exclusively associated with the columns of an offshore platform. When an incident wave impinges on a partially immersed body the free-surface of the wave undergoes a violent transformation where some portion of the incident wave’s momentum is directed vertically upward. To conserve energy, this momentum flux results in a rapid amplification of the waveform at the free-surface-body interface, see Fig. 1. This highly nonlinear phenomenon is generally known as wave run-up or up-rush and is particularly important for air gap design. Theoretical and experimental studies by numerous researchers have largely concentrated wave run-up investigations on piles, lighthouses, breakwaters, artificial islands, sloped beaches, and columns of fixed and floating offshore structures. Of particular interest to platform designers is the wave run-up on the forward vertical legs of both fixed and floating
Fig. 1. Illustration of the wave run-up resulting from an incident wave impinging on a vertical surface piercing column.
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offshore structures. In severe ocean environments the amplification of the incident wave may give rise to pressure impulse loads on the underside of the deck structuresometimes referred to as ‘slamming’ in the offshore structure community. A pressure impulse event occurs when a horizontal element, such as the platform deck or a member suspended from it, is impacted by a discrete volume of water rushing up the weather side of a platform column. While not posing a threat to the overall structural integrity of the platform, wave run-up is generally associated with localised structural damage. Based on experimental evidence for a circular cylinder in plane progressive waves, some researchers have shown that wave run-up, in extreme conditions, can be more than 1.5 times the incident wave amplitude [2,3]. Understanding wave run-up is fundamental to the correct estimation of a platform’s air gap requirements and to the minimisation of localised pressure impulse events. The extent of the wave run-up is largely dependent upon the transfer of energy to the harmonics of the wave field and the nature of this transfer is largely dependent upon two important parameters: the incident wave steepness kA; and the so-called scattering or slenderness parameter ka. It is understood that k is the wave number and a is the radius of the column. In the limit as kA/0, the slenderness parameter ka can be thought of as one that decides the normalised magnitude of each harmonic component of the vertical displacement of the free-surface. When kA is introduced, it essentially decides the importance of each harmonic contribution to the wave run-up. The ratio of these two parameters produces the free-surface Keulegan-Carpenter number, KcZA/a. For A/a less than of order unity, the flow around the circular cylinder will not separate and the fluid domain can be described by potential theory. To predict the wave run-up around a circular column, the theoretical treatment is generally based on potential theory whereby an idealised fluid domain is assumed and the well-known Laplace equation is solved with applied boundary conditions to yield a velocity potential. For wave motions, this is commonly referred to as diffraction theory. The free-surface elevation around the column may be obtained by application of the unsteady Bernoulli equation and velocity potential at the free-surface position. In solving for the velocity potential, one must deal with the complexity of the free-surface boundary condition. For deep water progressive surface gravity waves this is usually treated by one of two schemes. The first is a fully nonlinear numerical treatment in the time domain in the manner of Longuet–Higgins and Cokelet [4], and the second involves a mathematical approximation in the form of a Stokes expansion [5] whereby the free-surface displacement is assumed small in comparison to the characteristic wavelength. In this latter technique, the velocity potential and free-surface elevation are
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represented by the infinite series fðx; t; 3Þ Z
N X nZ1
3n fðnÞ ðx; tÞ;
zðx; y; t; 3Þ Z
N X
3n zðnÞ ðx; y; tÞ;
nZ1
(1) where the perturbation quantity, 3/1, is understood to be the wave steepness kA. The treatment of wave run-up in the form of a Stokes expansion will be studied here. Specifically, with the aid of experiment, we shall examine whether the wave run-up around a circular column, and the first few harmonic components of it, can be sufficiently predicted by the first two terms of a diffraction theory solution in the form (1). Of the many experimental investigations involving wave run-up on circular cylinders in plane progressive waves, the first study directly applicable to the regime of A=a! Oð1Þ was undertaken by Chakrabarti and Tam [6]. However, these authors were not concerned with the magnitude of the free-surface elevation but rather whether the incident wave direction could be determined from the symmetry exhibited by the pressure distribution on the column. Perhaps, the first experimental study to consider the inadequacy of linear diffraction theory (discussed in Section 2) as a measure of the wave run-up was documented by Isaacson [2]. This inadequacy is to such an extent where Isaacson [2] suggests that a factor of 2 should be applied in order to obtain a better representation of the wave run-up. However, this pragmatic approach gives no account for wave steepness and would presumably be conservative for small kA. The benchmark experiments of Kriebel [3] demonstrated that second-order diffraction theory (discussed in Section 2) provides an improvement over linear theory and compares favourably with the measured run-up for values of kA approximately 30% less than the wave breaking limit. The first experimental investigation that considered the harmonic components of the wave run-up was documented by Mercier and Niedzwecki [7]. Their experiments concerned a fixed vertical truncated circular cylinder in monochromatic waves such that 0.147%ka%0.915 for three distinct wave steepnesses of kAZ0.05, 0.01 and 0.15. Although the actual wave run-up was not investigated, the measured first- and second-harmonic components of it were compared with first- and second-order diffraction theory predictions. The measurements of Mercier and Niedzwecki [7] demonstrate an increasing modulus of each harmonic with increasing ka. Moreover, a negligible influence of kA on the first-harmonic wave run-up, and an unsatisfactory correspondence with theory for large ka was shown. On the other hand, the modulus of the measured second-harmonic was found to agree with predicted values for small ka. However, the effect of incident wave nonlinearities, through the parameter kA, on the measured harmonics was not discussed. It should be noted that the numerical results of Mercier and Niedzwecki [7] appear to contain errors and have clearly not converged for large
scattering parameters. This, perhaps, would explain the poor correspondence with measured results. It is widely understood that linear diffraction theory generally under-predicts wave run-up in all but small wave steepnesses [2,3]. However, what has not been sufficiently investigated is the extent to which the linear diffraction solution predicts the first-harmonic of the wave run-up. This is of fundamental importance and is paramount to the validity of wave diffraction theory with the free-surface boundary condition treated by a Stokes expansion scheme. An investigation on the limitations of perturbation theory for wave forces on a circular cylinder has been presented by Huseby and Grue [8], however, such detailed comparisons have yet to be undertaken for the corresponding wave runup and/or free-surface elevations. In particular, the zerothharmonic component of the wave run-up has yet to be isolated for detailed analysis. At this point it is unclear as to whether, and to what extent, a potential flow model with Stokes’ treatment of the free-surface boundary condition can indeed predict the constituent harmonic components of the wave run-up. More specifically, its ability to do so for the zeroth-, first- and second-harmonics for a Stokes expansion correct to Oð32 Þ is questionable. The purpose of the present paper is to investigate the wave run-up, and the harmonic components of it, on a fixed vertical cylinder in monochromatic progressive waves. A brief theoretical account is given of the scattering of a plane progressive wave by a circular cylinder in Section 2. An experimental campaign is outlined in Section 3. The comparison of theory with measured results is presented in Sections 4 and 5 for the harmonics and wave run-up, respectively. Finally, conclusions are given in Section 6.
2. Scattering of a plane progressive wave For the purpose of this present paper we will consider plane progressive waves in deep water. According to the theory of Stokes (see Lamb [9], Art. 250), the profile of a surface gravity wave travelling in a positive x-direction can be described by 1 zðx; tÞ Z A cosðut K kxÞ C kA2 cos 2ðut K kxÞ 2 3 C k2 A3 cos 3ðut K kxÞ; 8
(2)
correct to third-order in wave steepness where uZu(k) defined as u2Zgk(1Ck2A2), and g is the acceleration due to gravity. In general, only the first two terms of this expansion will be considered, however, it is instructive to reveal the third term in view of third-harmonic contributions to the wave run-up. When the progressive wave described by the first two terms of (2) impinges on a partially immersed cylinder in a fluid domain of an infinite horizontal extent,
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the resulting free-surface elevation can be described by qÞ C
(3)
(1)
in a cylindrical coordinate system where z is of Oð3Þ while z and z(2) are of Oð32 Þ and represent the mean and second-harmonic components, respectively, from the interaction of (2) with the cylinder. This canonical representation is typical of potential flow diffraction theory. The particular form of z(1) is quite fundamental and was first derived by Havelock [10] as ( ) N X ð1Þ Km 3m i Cm ðkrÞ cos mq ; (4) z ðr; qÞ Z < KA mZ0
with Cm ðkrÞ Z Jm ðkrÞ K
Jm0 ðkaÞ 0
Hmð2Þ ðkaÞ
Hmð2Þ ðkrÞ;
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leading-order solution. Moreover, it is interesting to note from (4) that an asymptotic value of z(1)w2A results when the limit of ka/N is considered. This agrees with the linear solution to a progressive wave incident on an infinitely long vertical wall (see Lamb [9], Art. 228–229). The forms of z and z(2) are complicated and will not be reproduced here. They were, however, first given by Hunt and Baddour [12] by means of a Weber integral transform technique and subsequently extended to water of finite depth by Hunt and Williams [13]. For the purpose of evaluation in this present work the second-order radiation/diffraction analysis code WAMIT [14,15] is utilised to numerically evaluate the contributions of z and z(2) in (3). In doing so great care was taken through convergence studies, the details of which can be found in Morris-Thomas [16].
(5)
where 3m denotes the Neumann symbol: 30Z1 and 3mZ2 for mO0, Jm is the Bessel function of order m, Hmð2Þ is a Hankel function of the second kind of order m, the prime denotes differentiation with respect to the argument, and the wave number is given by the dispersion relation kZu2/g. The linear diffraction solution (4) has been simply extended by MaCamy and Fuchs [11] to water of arbitrary depth. Although extending the Stokes expansion of the free-surface boundary condition to odd integer powers of 3 will indeed produce further contributions that operate at the firstharmonic, much information can still be obtained from this
3. Experimental campaign Experiments were conducted in the towing tank of the Ship Hydrodynamics Centre at the Australian Maritime College, Tasmania. The towing tank measures 60 m in length, 3.55 m in width and has a water depth of 1.5 m. A schematic of the experimental configuration is illustrated in Fig. 2. Waves were generated by a computer controlled bottom-hinged flat hydraulic paddle. At the downstream end of the flume, a corrugated beach with multiple baffles, of length 4 m, was installed. In addition to this passive absorption, longitudinal, pneumatically controlled beaches
Fig. 2. Schematic of the test set-up at the Australian Maritime College; wave probes are denoted C; PWP and RWP represent phase wave probe and repeatability wave probe, respectively.
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were installed to absorb water disturbances after the completion of each test run. 3.1. Model configuration The experiments were restricted to fixed cylinders as greater insight into wave run-up can first be gained by investigating this simpler problem. Model dimensions were scaled from a typical offshore structure configuration and chosen so that details of the wave–structure interaction were clearly visible. The cylindrical model dimensions of 30 cm in width and 87 cm in height were selected. Consequently, the wave tank width to model diameter ratio was approximately 11.8 so that side wall reflections, which may produce parasitic wave–wave interactions, should therefore be negligible during data analysis. Moreover, a corner radius variation similar to a Lewis conformal mapping [17] was selected to produce five cylindrical models of varying cross-sectional dimensions, whereby rc/a2 [0, 0.25, 0.5, 0.75 and 1] where rc denotes corner radius. In this paper, we concentrate only on the results pertaining to a circular cross-section where rc/aZ1 (the remaining cross-sections are discussed in a forthcoming paper [18]). The models were constructed with 10 mm marine grade plywood using a CNC Router with an expected error of G0.5 mm over the given dimensions. External surfaces were coated with yellow enamel paint and finished to a smoothness of G21 mm. A 25 mm square grid was scribed on each model for the purpose of free-surface measurements. Rather than prevent water seepage, which is often a difficult and expensive task during model construction, water entry was allowed at the base of each model. Consequently, the cylinder walls were supported by multiple internal baffles, coated with epoxy, to ensure rigidity and the avoidance of internal sloshing modes that could lead to erroneous model deflections. 3.2. Instrumentation and measurement The free-surface elevations were recorded at select locations (see Fig. 2 or the tabulated co-ordinates in Table 1 for the exact locations) with wire probes consisting of two parallel 1.6 mm diameter conductors spaced 10 mm apart. The time series free-surface elevation was digitised at 100 Hz throughout the experimental campaign. The array of Table 1 Positions of the 6 near field wave probes relative to the cylinder surface Wave probe
r(m)
q (radians)
A1 A2 A3 A4 B1 C1
0.001 0.050 0.100 0.150 0.001 0.001
p p p p 3p/4 p/2
wire probes—of 50 mm spacing—located on the up-wave side were assessed for possible signal interaction under wave excitation before testing commenced. No signal interaction was evident in the results. Two additional wire probes were used during each test; one was located midway between the tank wall and the centreline of the cylinder position to obtain phase information; and another 18 m from the wave maker to assess the repeatability of the incident waves. In addition to the wave probe configuration, each run was recorded by two video cameras (cf. Fig. 2). These visual measurements validated readings from the wave probes and assisted in examining the existence of surface tension effects associated with the wave probes adjacent to the cylinder surface. Surface tension was examined by testing the configuration under a variety of wave conditions and correlating the recorded wave elevations from the wave probes with visual observations of the wave-up through the video cameras. The agreement between video recordings and wave probe measurements was excellent. Surface tension effects are thought to be insignificant in the results to follow. 3.3. The wave environment The wave environment was restricted to monochromatic progressive waves, with frequencies and wave heights chosen to produce a parametric variation of both kA and ka. Five distinct wave periods and a range of have heights— denoted T and H, respectively—were selected as shown in Table 2. A scaling ratio of 1:55.5 was chosen to reduce prototype regular wave conditions to those of model scale. No frequencies corresponding to cross-tank resonant modes of u2r;m Z gpm=l, where l is the tank width and mZ1,2,., were used. The largest adopted wave steepness approximates to 63% of the wave breaking limit in deep water. The water depth was held constant at 1.5 m. With the model removed, wave repeatability was investigated by running the test matrix and correlating the results with a calibration wave probe at the removed model position and a repeatability wave probe 3.2 m from the central model position. Less than approximately 0.5% wave amplitude decay was noted between these two probes. For each set of wave conditions, the water surface was initially quiescent and the wave paddle started from the vertical position. Data acquisition commenced after the first few transient surface elevations passed the model, and lasted for approximately 14–30 wave periods. Measurements were recorded before any reflected waves of the fundamental frequency, u, arrived back at the testing area, which was 21.2 m from the vertical position of the paddle face.
4. Results and discussion The maximum wave amplitudes from each wire probe, denoted A(r, q), were found by taking an average over each
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Table 2 The wave environment for the circular cylinder, illustrating the parametric variation of both wave steepness kA and cylinder slenderness ka Prototype
Model
Dimensionless parameters
T (s)
H (m)
T (s)
H (m)
ka
kA
A/a
4.92 6.93 8.96 10.9 12.7
0.75–3.94 1.56–7.27 2.32–11.0 2.87–14.1 4.55–20.6
0.66 0.93 1.20 1.46 1.70
0.01–0.07 0.03–0.13 0.04–0.20 0.05–0.25 0.08–0.37
1.386 0.698 0.417 0.283 0.208
0.041–0.206 0.062–0.284 0.057–0.263 0.047–0.228 0.055–0.233
0.029–0.149 0.089–0.407 0.137–0.630 0.166–0.805 0.264–1.124
wave cycle in the steady state time series. The actual wave run-up R, measured relative to the quiescent frees–urface zZ0, was obtained from A(a, p). All measurements presented here exhibit a small standard deviation in wave amplitude over the time series’ and therefore, resemble progressive waves of permanent form. Harmonic components were extracted by a discrete Fourier transform algorithm. Of particular interest is the zeroth-, first-, second-, and third-harmonics of the wave profiles. These harmonics are denoted R(un) and A(un) for the wave run-up and incident wave, respectively, where nZ0,1,2,.,. Before the Fourier transform was taken the zeros of the time series were found and, in general, an integer number of wave periods were extracted for the Fourier analysis. This is important, not only for capturing the zeroth-harmonic but for minimising the bandwidth of each wave harmonic to avoid energy leakage into neighbouring components of the frequency space. An example of the wave profile and its harmonic components is illustrated in Fig. 3. The wave amplitude A, and fundamental circular wave frequency, u, are defined as the modulus and frequency of the first-harmonic Fourier component of the incident wave. Consequently, it is important to note that the wave steepness kA is based on the first-harmonic of the carrier wave. Since khO2.08 for all test conditions, the water depth is assumed infinite so that the wave run-up caused purely by wave–structure interaction is examined without the additional influence of a nondecaying horizontal fluid velocity and finite depth effects on wave dispersion.
4.1. Relative harmonic contributions As a preliminary, and to gain some insight into the magnitudes of the first few harmonics relative to the actual wave run-up, the ratio jR(un)j/R was investigated, where jR(un)j defines the modulus of the nth harmonic wave runup component. The numerical values of this ratio versus kA for five different values of ka are presented in Fig. 4. In short waves, kaZ1.386 and 0.698, the first harmonic component appears invariable as kA increases. However, in longer waves, kaZ0.208 for instance, jR(u1)j/R decreases with increasing kA and is coupled with an increase in contributions from both jR(u2)j/R and jR(u3)j/R. The thirdharmonic component only appears to contribute in long waves, kaZ0.208, when the wave steepness approaches around 50% of the deep water breaking limit-defined by kAmaxZ0.14p (see Mei [19], equation 10.4.4). A feature that has not been examined in great detail throughout the literature is the zeroth-harmonic or mean component, R(u0), and its contribution is clearly illustrated in Fig. 4. The measurements indicate that jR(u0)j/R linearly increases with increasing kA to an extent where it accounts for approximately 25% of the wave run-up in the region of kAz0.2–0.25. This increase and eventual peak of jR(u0)j/R at approximately 0.25R for kAz0.2–0.25 appears to be independent of ka (cf. Fig. 4). Apart from the fundamental harmonic, the measurements R(u0) provides the dominant contribution to R in short waves. Interestingly though, for kaZ0.417, jR(u0)j and jR(u2)j contribute equally to wave run-up. Consequently, in the region of ka less than
Fig. 3. Time series (right) and corresponding spectral plot (left) for a typical wave train produced by the wave maker, kAZ0.231 (TZ12.7 s full scale). The fundamental, second- and third-harmonic components of the wave train are clearly visible in the spectral plot.
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Fig. 4. The ratio of the modulus of each harmonic component and the wave run-up, denoted: (C), zeroth-harmonic; (C), first-harmonic; (,), second-harmonic and (B), third-harmonic. (a) kaZ1.386; (b) kaZ0.698; (c) kaZ0.417; (d) kaZ0.283 and (e) kaZ0.208.
approximately 0.417, the second-harmonic becomes the dominant bound component of the wave run-up. To further illustrate the relative contributions of the wave run-up, we shall consider one particular test condition, where kaZ0.208 and kAZ0.254. The wave steepness corresponds to approximately 60% of the wave breaking limit. The first-harmonic is moderately amplified by 1.2A while the second-harmonic is greatly influenced by the presence of the cylinder to an extent where jR(u2)j is almost 3A(u2). For the wave run-up the second-harmonic is approximately 40% of R(u1) whereas the mean and thirdharmonic are roughly 20 and 8%, respectively. Despite the long wavelength, where amplifications from diffraction effects are expected to be small, this example coupled with the results of Fig. 4 clearly illustrates the importance of kA
which gives rise to large contributions from both the mean and second-harmonic components. It is interesting to note that for kaZ1.386, the measurements suggest that approximately 60% of the wave run-up is accounted for by jR(u1)j. Given that the other major contribution is from jR(u0)j, this leaves much of the wave run-up unaccounted for. This is explained as follows: for the measurements pertaining to kaZ1.386, the free-surface profile at A(a, p) exhibited a distinct aperiodic nature, such that the wave run-up peaked every 3-4 wave periods despite an incident wave of permanent form. This effect has also been observed by LonguetHiggins and Drazen [20] during experiments involving waves meeting a vertical wall. The two cases could be related but without further exploratory experiments, no
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conclusions can be made. In any event, the results pertaining to kaZ1.386 will not be discussed further here. 4.2. Harmonic measurements and theory compared We shall now investigate the correspondence between second-order diffraction theory and measurements for the first few harmonics of the wave run-up at the position (r, q)Z(a, p). In doing so it is convenient to normalise the measured and predicted harmonics by jRðun ÞjamK1 =Am ;
jzðnÞ jamK1 =Am ;
respectively, where m is the Stokes perturbation order and n is the harmonic component. According to second-order theory, the mean wave run-up is of Oð32 Þ and is therefore, normalised accordingly. The correspondence between theory and measurement for the modulus and phase of each harmonic is illustrated in Figs. 5–8. 4.2.1. Zeroth-harmonic The measured zeroth-harmonic of the wave run-up is not particularly well predicted by theory (see Fig. 5). Generally speaking, the measured results of the zeroth-harmonic exhibit scatter which appears to decrease for decreasing ka. However, in the region of this decreased scatter R(u0) is reasonably well predicted by second-order diffraction theory for small kA. This observation is particularly evident for waves of kaZ0.208. In longer waves, the measured zeroth-harmonic increases with increasing wave steepness (cf. Fig. 5 for kaZ0.208 and kaZ0.283). This indicates that R(u0) is influenced by higher than second-order wave steepness effects in large values of kA. In terms of a Stokes expansion of the freesurface boundary condition, this dependence implies that, at the least, a fourth-order kA expansion is required to recover the dependence of kA on R(u0). Alternatively, a fully nonlinear treatment of the free-surface boundary condition may produce the desired result. It should be pointed out that the undisturbed incident waves contained a small zeroth-harmonic component. In complex form the zeroth-harmonic contained a phase of either Gp. Although this mean component is obviously physical, its origin is unclear since according to the Bernoulli theorem applied at the free-surface position the mean wave component should be zero for a free wave with corresponding locked harmonic components. Although it is possible that the datum for each wave probe drifted, this seems unlikely considering that the datum was recorded before each test run commenced. Correcting for this nonzero mean did not affect the recorded free-surface elevations substantially and the discrepancies observed in Fig. 5 remain. Recently, observation by Stansberg and Braaten [21] have confirmed the existence of a large zerothharmonic component that is not well predicted by secondorder diffraction computations.
Fig. 5. The normalised modulus of mean component of the wave run-up jR(u0)j plotted against wave steepness kA. Measured results, (C) and WAMIT, (—).
4.2.2. First-harmonic Previous authors have noted large discrepancies when comparing the linear diffraction solution of MacCamy and Fuchs [11] to the complete wave run-up [2,22]. Here however, the linear diffraction solution, which contains only the fundamental wave frequency, is compared only to the first-harmonic component of the wave run-up. In doing this, and distinguishing between the coupling of ka and kA, the results indicate that the linear diffraction solution performs reasonably well by capturing both the modulus and phase of R(u1) (cf. Fig. 6). An interesting observation is seen for kaZ0.208 when the wave steepness reaches approximately 50% of
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Fig. 6. The normalised modulus jR(u1)j and phase arg(R(u1)) of first-harmonic component of the wave run-up plotted against wave steepness kA. Measured results, (C) and the linear diffraction solution, (—).
the wave breaking limit. In this region, the measured results exhibit a relatively consistent increase in jR(u1)j for kAO0.2. This is coupled with a decrease in the phase angle. These effects are moderately replicated for the results of kaZ0.283. Although, this observation corresponds to the longest wavelength examined, it is not thought to be caused by amplitude dispersion to an extent where finite depth effects have magnified the wave run-up but rather higher that second-order wave steepness effects operating at the first-harmonic. Such nonlinear effects do not occur for kaZ0.208 in waves of small steepness and would suggest that a first-order prediction of R(u1) is adequate in this region.
4.2.3. Second-harmonic The theoretical prediction of jR(u2)j compares well with the measured results for kaZ0.283 and 0.208 (cf. Fig. 7). However, a discrepancy exists for wave of small steepness which is to be expected since the measured secondharmonic is relatively small. Consequently, its modulus is sensitive to the position of the time window selected within the wave group for the Fourier transform. The theoretical predictions consistently over estimate the measured modulus of the second-harmonic in shorter wavelengths (cf. Fig. 7 for kaZ0.698 and 0.417). It is not possible to attribute this to higher-order effects because in the limit of kA/0, one would expect the measured results
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Fig. 7. The normalised modulus jR(u2)j and phase arg(R(u2)) of second-harmonic component of the wave run-up plotted against wave steepness kA. Measured results, (C); and second-order calculations, (—).
to agree with the theoretical values. For instance, the measured results indicate a reasonable agreement for kaZ0.698 but certainly not for kaZ0.417. A possible explanation could be that the measured second-harmonic exhibits a small degree of variability in these shorter waves and has affected the resolution of the Fourier transform. The phasing of the second-harmonic component is shown to be invariable with wave steepness (cf. Fig. 7), which suggests that the second-harmonic phase is not influenced by higher than second-order in kA effects. However, since second-order theory appears to under predict the phase by a constant amount in all but one case (kaZ0.208), this does suggest that higher-order diffraction
effects are important. One exception is noted at kaZ0.698 whereby the phase appears to decrease with increasing kA. 4.2.4. Third-harmonic Difficulty arose in obtaining an accurate extraction of the third-harmonic as it is considerably smaller than both the first- and second-harmonics. Due to this variability, only measurements of the third-harmonic pertaining to kAO0.125 are shown (cf. Fig. 8). When the onset of viscous effects is considered (A/aZ1), the measured results for kaZ0.208 indicate that jR(u3)j is around 0.1A. On the other hand, for A/aZ0.5 the measured results indicate that jR(u3)j will be around 0.025A. This suggests that only when
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Fig. 8. The normalised modulus jR(u3)j and phase arg(R(u3)) of third-harmonic component of the wave run-up plotted against wave steepness kA. Measured results, (C).
the incident wave amplitude is comparable to the cylinder radius does third-harmonic effects become important to the wave run-up. Although the modulus of the third-harmonic component is small, a fairly constant contribution for kAO0.125 is exhibited by the measured results. Moreover, the phase of the third-harmonic is reasonably invariant with both kA and ka. (see Fig. 8 for kaZ0.417 and kaZ0.283). The consequence of this is that one would expect the thirdharmonic to increase the wave run-up in longer waves due to the discrepancy in phasing with the first-harmonic component (cf. kaZ0.698 and kaZ0.208 in Figs. 6 and 8). This observation can be easily verified by summation of the complex harmonics in the time domain.
5. The wave run-up We shall now consider the measured wave run-up normalised by AZjA(u1)j. A comparison of the diffraction theory computations and measurements is illustrated in Fig. 9 for a range of wave steepnesses and three particular values of ka—a value of 1 on the ordinate indicates no amplification of the incident wave. The second-order wave run-up is defined as the complex addition of the components in (3) evaluated at the position (r, q)Z(a, q). Since A/a!1 for all but one of the selected test conditions, potential flow theory should be applicable and the comparison valid.
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Fig. 9. The wave run-up versus the wave steepness kA for three different scattering parameters, ka. Measured results, (C); linear diffraction theory (– – –) and second-order calculations, (—).
Separating the wave run-up into its dependence on ka and kA clearly illustrates the influence of diffraction and incident wave nonlinearity effects (cf. Fig. 9). For constant ka, the results indicate that the normalised wave run-up increases for increasing kA. For small kA, local diffraction effects are clearly dominant and the amplification of the incident wave generally increases with increasing ka. However, despite large incident wave amplifications for large ka, the actual wave run-up will essentially be governed by kA. It is the intermediate to long wavelengths which are most important since the limiting value for the wave height is governed by kAmaxZ0.14p for water of infinite depth. With this in mind, it is the long waves that will produce the highest vertical water displacements on the column surface. Over the complete range of (ka, kA) combinations considered the linear diffraction solution of Havelock [10] does not perform well-this observation is not new [2,23]. The first-order solution demonstrates no kA dependence.
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Consequently, the error in the first-order prediction monotonically increases for increasing kA. For example, this error is around 20 and 100% in large values of kA for kaZ0.698 and 0.208 respectively. On the other hand, the first-order prediction does realise a reasonable agreement with the measured results for very small kA (kAz0.05 for instance). This indicates that at such a small wave steepness incident wave nonlinearities are not important and that local wave diffraction is confined to the first-harmonic. The decomposition of the wave run-up into harmonic proportions (cf. Section 4.1) suggests this to be so. It is reasonable to suggest that the wave run-up, regardless of ka, approaches the linear diffraction solution as kA/0. This is an important and fundamental result. The under prediction by linear diffraction theory is somewhat arrested by the inclusion of second-order nonlinearities. Perhaps this is best illustrated by the results pertaining to kaZ0.698, where the second-order prediction agrees favourably with the measured wave run-up up to, and including, kAz0.25. However, as the incident wave becomes longer nonlinear effects associated with the incident wave steepness occur at increasingly smaller values of kA. This phenomenon could be described as a ‘threshold’ for extreme, or higher than second-order, nonlinear effects associated with kA. At this threshold the normalised measured values are seen to depart from the second-order predictions (cf. Fig. 9 for kaZ0.417 and 0.208). The measured values are then seen to follow a nth degree polynomial trend. This possibly indicates the existence of higher-order contributions to the wave run-up and would suggest that a fully nonlinear representation of the freesurface elevation would be a more efficacious approach to predict the wave run-up in long steep waves. Furthermore, it also suggests that the wave steepness limit for Stokes second—order wave theory—consult Fenton [24] is essentially not applicable for second-order wave diffraction. When examining the actual wave run-up (on r/aZ1), it is also important to consider measurements of the free-surface elevation in the fluid domain surrounding the cylinder. In particular, along the ray q/pZ1. This accumulation of water build-up is presented in Figs. 10–12 for each value of ka and kA. In these figures the radial distance is normalised by the cylinder radius, a, so that a value of r/aZ2 corresponds to 1 cylinder radius away from the cylinder. Moreover, along with the measured results, linear and second-order numerical results are given. Further evidence of the occurrence of this non-linear threshold is provided in Figs. 10–12. For kaZ0.208 in particular, the free-surface elevation along the ray q/pZ1 is generally well predicted by the second-order model. However, for kAO0.2 the measured results tend to deviate away from the second-order diffraction theory predictions as r/aZ1 is approached. So much so that for kAZ0.233 the error of R/A is approximately 30% at r/aZ1. In contrast, this nonlinear threshold does not appear in the measured results for kaZ0.698 along the ray q/pZ1 as
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Fig. 10. The normalised free-surface elevation at q/pZ1 versus r/a for kaZ0.698. Measured values are given by the dashed line with (C); first-order calculations, (.) and second-order calculations, (—).
r/a/1 (cf. Fig. 10). Interestingly, both the wave run-up and the amplification of the free-surface are reasonably well predicted by second-order diffraction theory for small values of kA. On the other hand, for kAO0.212, although the trend of the free-surface is captured, the second-order model slightly over predicts the free-surface elevations at all values of q/pO1. This could be attributed to a mean set-up that becomes important when the wave steepness becomes large. Despite this, the wave run-up at (r/a, ka)Z(1, 0.698) in large kA is well predicted by the second-order model.
6. Conclusions Wave run-up on fixed surface piercing cylinders in the presence of deep water monochromatic progressive waves has been investigated in the diffraction dominated regime where the assumption of potential flow theory is applicable. Firstly, the harmonic components of the wave run-up were examined to determine the importance of each with
regards to local wave diffraction and incident wave steepness. This investigation has found that the contribution from the first-harmonic decreases with increasing kA. This is coupled with an increase from both zeroth- and higher harmonic components. Experimental evidence suggests that, of the nonlinear contributions, the mean set-up is the dominant contributor for ka!0.42 while above this limit the second-harmonic dominates. The mean set-up accounts for up to 25% of the wave run-up for large wave steepnesses. The third-harmonic only becomes important in long waves when kAO0.2. In this region the modulus of R(u3) is approximately 8% of R. The majority of the wave run-up consists of zeroth-, first-, and second-harmonic components. The next objective was to compare the normalised harmonics of the wave run-up with potential flow numerical predictions. This revealed that both the modulus and phase of the first-harmonic wave run-up are generally well predicted by first-order diffraction theory. However, in long waves, distinct nonlinear effects-greater than Oð32 Þ in Stokes perturbation-operating at the first-harmonic are
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Fig. 11. The normalised free-surface elevation at q/pZ1 versus r/a for kaZ0.417. Measured values are given by the dashed line with (C); first-order calculations, (.) and second-order calculations, (—).
clearly evident for kAO0.2. Consequently, these effects are not predicted by linear diffraction theory and a higherorder description of the free-surface is required. Measurements of the zeroth- and second-harmonic components generally compared unfavourably with calculated values. In particular, the zeroth-harmonic is under predicted by up to 200% in certain cases while the modulus and phase of the second-harmonic is perhaps best predicted for longer wavelengths. The actual wave run-up, containing all harmonic contributions, at r/a and qZp, has been examined. The wave run-up is shown to be distinctly dependent upon both ka and kA. A definite kA threshold is observed where extreme nonlinear effects, that are greater than second-order in a Stokes expansion, commence. The exact value of kA where these nonlinear effects occur is dependent upon ka. For instance, when kaZ0.618 these nonlinear effects were unnoticeable and a Stokes expansion to second-order adequately predicts the measured results. However, for
kaZ0.208 this nonlinear threshold occurs in the region 0.15!kA!0.2, above which, extreme amplifications occur that are as much as twice the incident wave amplitude. Results presented here suggest that linear diffraction theory is insufficient for wave run-up estimation. This confirms conclusions made by previous authors. However, it does appear that in the limit as kA/0, linear diffraction theory performs adequately as a fundamental solution. A Stokes expansion to second-order, when employed in the diffraction problem and as incorporated in WAMIT, produces reasonable estimates of wave run-up provided the nonlinear threshold is not breached. However, for longer wavelengths the phasing of the third-harmonic is such that below the kA threshold the third-harmonic reduces the wave run-up whilst above the kA threshold the third-harmonic increases the wave run-up. This distinct kA dependence of the third-harmonic compromises correspondence between measured and Stokes second-order predictions for the longest waves examined here.
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Fig. 12. The normalised free-surface elevation at q/pZ1 versus r/a for kaZ0.208. Measured values are given by the dashed line with (C); first-order calculations, (.) and second-order calculations, (—).
Acknowledgements The authors gratefully acknowledge the support of MARINTEK, DNV and the Australian Research Council. The first author was partially supported by a George Samaha Research Scholarship during portions of this work. Moreover, the authors would like to thank Gregor Macfarlane and the laboratory staff of the Ship Hydrodynamics Centre at the Australian Maritime College.
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