2.1 FNV formulation in finite water depth

We consider gravity waves with wavelength $\unicode[STIX]{x1D706}$ , incident on a vertical surface-piercing and non-moving circular cylinder with cross-sectional radius $a$ standing on the sea floor in finite water depth $h$ ; see figure 1. We define a Cartesian coordinate system $Oxyz$ with the $z$ -axis along the cylinder axis and with positive direction upwards. Furthermore, $z=0$ and $z=-h$ correspond to the calm free surface and the sea floor respectively. The positive $x$ -direction is along the incident wave-propagation direction. The water has infinite horizontal extent.

In the original FNV theory, a small parameter $\unicode[STIX]{x1D716}$ was introduced and both the linear incident wave slope $k\unicode[STIX]{x1D701}_{a}$ and the non-dimensional cylinder radius $ka$ were of $O(\unicode[STIX]{x1D716})$ . The total potential was expressed as $\unicode[STIX]{x1D711}=\unicode[STIX]{x1D711}_{I}+\unicode[STIX]{x1D711}_{s}+\unicode[STIX]{x1D713}$ , where $\unicode[STIX]{x1D711}_{D}=\unicode[STIX]{x1D711}_{I}+\unicode[STIX]{x1D711}_{s}$ is a diffraction potential satisfying the two-dimensional Laplace equation in each horizontal plane up to and including the incident free surface and $\unicode[STIX]{x1D713}$ is a third-order potential satisfying the three-dimensional Laplace equation and approximate free-surface conditions to third order. Here, $\unicode[STIX]{x1D711}_{I}$ represents the incident wave potential. These features are all retained in finite water depth.

In § 2.2, the main expressions of the load model are given. A brief outline of the derivation is next provided in § 2.3. All expressions are valid for both deep water and finite water depths. It should be noted, however, that the third-order incident potential is zero in deep water, whereas it is a dominant contributor to the third harmonic load in finite water depth.

2.2 Generalized FNV load model

The total load is obtained by integrating the distributed horizontal hydrodynamic force $F^{\prime }$ due to $\unicode[STIX]{x1D711}_{D}$ along the cylinder up to the incident free surface, and adding the load term due to $\unicode[STIX]{x1D713}$ ,

(2.1) $$\begin{eqnarray}F_{x}=\int _{-h}^{\unicode[STIX]{x1D701}_{I}}F^{\prime }(z,t)\,\text{d}z+F^{\unicode[STIX]{x1D713}}.\end{eqnarray}$$

The distributed load term $F^{\prime }$ due to $\unicode[STIX]{x1D711}_{D}$ is

(2.2) $$\begin{eqnarray}\displaystyle F^{\prime }(z,t)=\unicode[STIX]{x1D70C}\unicode[STIX]{x03C0}a^{2}\left(\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}t}+u\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}x}+w\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}z}\right)+a_{11}\left(\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}t}+w\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}z}\right)\quad \text{for }h\leqslant z\leqslant \unicode[STIX]{x1D701}_{I}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D70C}$ is the mass density of water, $a_{11}=\unicode[STIX]{x1D70C}\unicode[STIX]{x03C0}a^{2}$ is the 2D added mass in surge, $\unicode[STIX]{x1D701}_{I}$ is the incident wave elevation (to any order) at $x=0$ , $\unicode[STIX]{x1D711}_{I}$ is the incident wave potential (to corresponding order), $u=\unicode[STIX]{x2202}\unicode[STIX]{x1D711}_{I}/\unicode[STIX]{x2202}x|_{x=0},\,\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}x=\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D711}_{I}/\unicode[STIX]{x2202}x^{2}|_{x=0}$ and $w=\unicode[STIX]{x2202}\unicode[STIX]{x1D711}_{I}/\unicode[STIX]{x2202}z|_{x=0}$ . The horizontal force due to $\unicode[STIX]{x1D713}$ is, no differently from in the deep water case,

(2.3) $$\begin{eqnarray}\displaystyle F^{\unicode[STIX]{x1D713}}=\unicode[STIX]{x1D70C}\unicode[STIX]{x03C0}a^{2}\frac{4}{g}u^{2}\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}t}, & & \displaystyle\end{eqnarray}$$

where the quantities are evaluated at $x=0$ , $z=0$ . The centre of action of $F^{\unicode[STIX]{x1D713}}$ is at $z=0$ .

The generalized formulation (2.1)–(2.2) is consistent with the original FNV method to third order, as long as the incident wave potential, $\unicode[STIX]{x1D711}_{I}$ , to at least third order is used. If higher-order wave kinematics is used, higher-order load terms appear in the incident wave only. The generalized formulation allows for any regular wave theory, a numerical wave tank and irregular long-crested incident waves to be applied, as long as the local wave slope is not so large that the waves are breaking. The influence of the sea floor topography with small changes in the $x$ -direction can be considered, as long as there are no sea bottom changes in the $y$ -direction. The numerical wave tank must be based on potential-flow theory of incompressible water and can account for exact nonlinear free-surface conditions. Breaking waves can implicitly not be analysed. Essential assumptions in the load formula are also that wave generation by the cylinder and flow separation are secondary and can be neglected. The latter is not generally true, however, as is discussed throughout the paper. The moment is calculated by multiplying with the wanted arm inside the integration of (2.2) and the term (2.3). Computational time makes it realistic to simulate many realizations of different sea states.

The most common version of Morison’s equation within potential-flow theory expresses $F^{\prime }$ as $(\unicode[STIX]{x1D70C}\unicode[STIX]{x03C0}a^{2}+a_{11})\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}t$ . Another version is to use the total advection term $\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}t+u\,\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}x+w\,\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}z$ instead of $\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}t$ . Neither agrees with (2.2). For circular cross-sections, the distributed force terms in Rainey’s model (equation (22) in Manners & Rainey (Reference Manners and Rainey1992)) agree with $F^{\prime }$ in the FNV model, given by (2.2). The only difference between the two load models is therefore the $F^{\unicode[STIX]{x1D713}}$ term.

2.3 Brief derivation

A brief outline of the derivation is provided in the following. The diffraction potential $\unicode[STIX]{x1D711}_{D}$ is Taylor expanded around the cylinder axis to $O(\unicode[STIX]{x1D716}^{3})$ , and is thus expressed as

(2.4) $$\begin{eqnarray}\unicode[STIX]{x1D711}_{D}(x,y,z,t)=\unicode[STIX]{x1D711}_{I}(0,0,z,t)+\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}_{I}}{\unicode[STIX]{x2202}x}(0,0,z,t)(x+\unicode[STIX]{x1D711}_{11})+\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D711}_{I}}{\unicode[STIX]{x2202}x^{2}}(0,0,z,t)\left(\frac{1}{2}x^{2}+\unicode[STIX]{x1D711}_{21}\right).\end{eqnarray}$$

The scattering potentials $\unicode[STIX]{x1D711}_{11}$ and $\unicode[STIX]{x1D711}_{21}$ satisfy the 2D Laplace equation in the horizontal plane, and are easily found as simple harmonic functions by satisfying the body-boundary condition $\unicode[STIX]{x2202}(x+\unicode[STIX]{x1D711}_{11})/\unicode[STIX]{x2202}r=\unicode[STIX]{x2202}(0.5x^{2}+\unicode[STIX]{x1D711}_{21})/\unicode[STIX]{x2202}r=0$ on $r=a$ , where $x=r\cos \unicode[STIX]{x1D703}$ . The distributed force per unit length is found by integrating the pressure, $p$ , as given by the Bernoulli equation,

(2.5) $$\begin{eqnarray}\frac{1}{\unicode[STIX]{x1D70C}}p=-\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}_{D}}{\unicode[STIX]{x2202}t}-\frac{1}{2}\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}_{D}}{\unicode[STIX]{x2202}r}\right)^{2}-\frac{1}{2a^{2}}\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}_{D}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}\right)^{2}-\frac{1}{2}\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}_{D}}{\unicode[STIX]{x2202}z}\right)^{2}\end{eqnarray}$$

along the cylinder wall,

(2.6) $$\begin{eqnarray}F^{\prime }(z,t)=-\int _{0}^{2\unicode[STIX]{x03C0}}p\cos \unicode[STIX]{x1D703}\,a\,\text{d}\unicode[STIX]{x1D703}.\end{eqnarray}$$

Here, the body-normal vector points into the water, such that the component in the $x$ -direction is $n_{x}=\cos \unicode[STIX]{x1D703}$ . The radial derivative term in (2.5) is zero on the cylinder wall. Only terms proportional to $\cos \unicode[STIX]{x1D703}$ contribute to the force (2.6), due to orthogonality. The cross-couplings between the incident potentials ( $x$ and $0.5x^{2}$ ) and scattered potentials ( $\unicode[STIX]{x1D711}_{11}$ and $\unicode[STIX]{x1D711}_{21}$ ) due to the remaining two square terms result in the load terms involving $u\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}x$ and $w\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}z$ . The result is (2.2). The derivation of $F^{\unicode[STIX]{x1D713}}$ is no different in finite water depth from in deep water (Faltinsen et al. Reference Faltinsen, Newman and Vinje1995), and is therefore not outlined here.

Some more elaboration is needed to show that (2.1)–(2.3) are consistent with the original theory (to third order). The FNV theory was originally separately considered in the ranges $-\infty \leqslant z\leqslant 0$ , from $z=0$ to $z=\unicode[STIX]{x1D701}_{I1}$ and from $z=\unicode[STIX]{x1D701}_{I1}$ to $z=\unicode[STIX]{x1D701}$ . Here, $\unicode[STIX]{x1D701}=\unicode[STIX]{x1D701}_{I1}+\unicode[STIX]{x1D701}_{2}$ is the local wave elevation correct to $O(\unicode[STIX]{x1D716}^{3})$ including both the effect of the incident waves and the locally disturbed free surface due to $\unicode[STIX]{x1D711}_{D}$ . It is straightforward to show that this applies also in finite water depth, as outlined in the following. Taylor expanding $F^{\prime }(z,t)$ around $z=0$ , i.e. $F^{\prime }(z,t)=F^{\prime }(0,t)+z\,\unicode[STIX]{x2202}F^{\prime }/\unicode[STIX]{x2202}z(0,t)$ , and inserting into (2.1) results in the original formulation for the load up to, and including, third order,

(2.7) $$\begin{eqnarray}\displaystyle F_{x}=\int _{-h}^{0}F^{\prime }\,\text{d}z+F_{HD}+F_{Q}+F^{\unicode[STIX]{x1D713}}, & & \displaystyle\end{eqnarray}$$

where

(2.8) $$\begin{eqnarray}\displaystyle & \displaystyle F_{HD}=F^{\prime }\unicode[STIX]{x1D701}_{I1}+\frac{1}{2}\unicode[STIX]{x1D701}_{I1}^{2}\frac{\unicode[STIX]{x2202}^{2}u}{\unicode[STIX]{x2202}t\unicode[STIX]{x2202}z}(\unicode[STIX]{x1D70C}\unicode[STIX]{x03C0}a^{2}+a_{11}), & \displaystyle\end{eqnarray}$$

(2.9) $$\begin{eqnarray}\displaystyle & \displaystyle F_{Q}=\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}t}\unicode[STIX]{x1D701}_{I2}(\unicode[STIX]{x1D70C}\unicode[STIX]{x03C0}a^{2}+a_{11}). & \displaystyle\end{eqnarray}$$

In (2.8) and (2.9), all quantities are evaluated at $x=0,z=0$ . Finite water depth, or deep water, wave incident wave kinematics can be used. The terms $F_{HD}$ and $F_{Q}$ represent the two parts of the load originating from integrating from $z=0$ to $z=\unicode[STIX]{x1D701}_{I1}$ and from $z=\unicode[STIX]{x1D701}_{I1}$ to $z=\unicode[STIX]{x1D701}$ respectively.

The subscript $Q$ indicates that this part of the force may be termed ‘quasi-hydrostatic’; that is, the pressure between $\unicode[STIX]{x1D701}_{I1}$ and $\unicode[STIX]{x1D701}_{2}$ is to leading order $p(z>\unicode[STIX]{x1D701}_{I1})=\unicode[STIX]{x1D70C}g(\unicode[STIX]{x1D701}-z)$ . Substituting the expression for $\unicode[STIX]{x1D701}_{2}$ given by the dynamic free-surface condition for the second-order problem, involving $\unicode[STIX]{x1D711}_{D}$ , and integrating the pressure expressed by this from $\unicode[STIX]{x1D701}_{I1}$ to $\unicode[STIX]{x1D701}$ yields the expression for $F_{Q}$ . Only the incident wave, $\unicode[STIX]{x1D701}_{I2}$ , appears in the expression for $F_{Q}$ , since, for a circular cylinder, the diffracted free surface will not contribute to the horizontal load due to orthogonality (or symmetry about the $yz$ -plane).

For deep water, Faltinsen et al. (Reference Faltinsen, Newman and Vinje1995) provide explicit expressions for (2.8) and (2.9). For finite water depth, however, the expressions become too long to be of practical use. Instead, a numerical approach is taken here. We denote by $f$ either $u$ , $w$ , $\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}t$ , $\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}x$ or $\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}z$ . In (2.7)–(2.9), these quantities are substituted by $f=f_{1}+f_{2}+f_{3}$ , referring to the first-, second- and third-order Stokes wave theory respectively. A time series of $F_{x}$ is constructed, and the first, second and third harmonics are extracted by band-pass filtering. A standard Fourier transformation based technique is used here. This is a fast, accurate and robust procedure, and numerical convergence does not require a very high spatial or temporal resolution.

Figure 2.

Illustrative example of force components of the third harmonic load in the generalized FNV model. Water depth $h/a=7.83$ . (a) Force components, where $F_{x}$ denotes the total force (2.7),  $F_{i}=\int _{-h}^{0}F^{\prime }\,\text{d}z$ , with $F^{\prime }$ given by (2.2), while $F_{HD}$ , $F_{Q}$ and $F^{\unicode[STIX]{x1D6F9}}$ are defined by (2.7), (2.8) and (2.9) respectively. (b) Components of $F_{i}$ . It should be noted that, for circular cross-sections, the distributed force terms in Rainey’s model (equation (22) in Manners & Rainey Reference Manners and Rainey1992) agree with $F^{\prime }$ in the FNV model, given by (2.2). The only difference between the two load models is therefore the $F^{\unicode[STIX]{x1D713}}$ term.

2.4 Force term contributions to the $3\unicode[STIX]{x1D714}$ loads

An interesting question relates to the relative magnitudes of the different terms in the load formulation. This is illustrated for water depth $h/a=7.83$ in figure 2. The contributions to the third harmonic of the horizontal force from the terms in (2.7) are compared. The range of the horizontal axis spans from deep water to intermediate, or near shallow, water depth wave conditions. Due to the fact that there is no third-order wave radiation in the FNV method, all of the components are either 0 or $180^{\circ }$ apart. This is indicated by positive and negative sign of the quantities given in the figure.

An important observation is that the term due to the incoming third-order incident wave component, $\unicode[STIX]{x2202}u_{3}/\unicode[STIX]{x2202}t$ , becomes a major contributor in finite water depth. Another observation is that the $F^{\unicode[STIX]{x1D713}}$ term, which accounts for half of the third harmonic force in deep water, contributes the least in intermediate water depth. It should be noted also that the horizontal advection terms provide larger-magnitude forces than the vertical ones, even though the latter are multiplied by a factor of 2, cf. (2.2). This is an effect of finite water depth; the horizontal velocities and accelerations are larger than the vertical ones. Two illustrative examples showing the horizontal velocity $u(z)$ as predicted by Stokes fifth-order wave theory at time steps $T/80$ over one wave period $T$ are provided in figure 3. The kinematics clearly deviates from that predicted by linear wave theory at this water depth. For later discussions related to flow separation, it should be noted that the magnitude of the horizontal velocity is clearly larger at the wave crest than at the wave trough.

Figure 3.

Example of horizontal velocity $u$ at the cylinder axis according to fifth-order Stokes wave theory for two selected relevant regular wave conditions: $h/a=7.83$ , wave steepness $H_{1}/\unicode[STIX]{x1D706}=1/25$ , $ka=0.127$ (a) and $ka=0.105$ (b). Here, $\unicode[STIX]{x1D701}_{a}$ is the linear wave amplitude and $\unicode[STIX]{x1D714}=2\unicode[STIX]{x03C0}/T$ , with $T$ the wave period. The time step between each curve is $T/80$ . It should be noted that for $ka=0.105$ , the magnitude of the velocity is larger at the sea floor in the positive direction than at the free surface in the negative direction.

2.5 Direct numerical integration

A direct approach of evaluating the integral in (2.1) must be used in the case of, for instance, applying a numerical wave tank. In the present work, we present results both from the approach described in § 2.3 and from direct numerical integration. For the direct numerical integration approach, we integrate $F^{\prime }$ up to the instantaneous fifth-order incident free surface using Stokes fifth-order wave theory (see, for instance, the classical paper by Skjelbreia & Hendrickson (Reference Skjelbreia and Hendrickson1960)), and add the $F^{\unicode[STIX]{x1D713}}$ from (2.3). A misprint in their paper pointed out, for instance, by Fenton (Reference Fenton1985) should be noted: the sign ahead of the number 2592 in the expression of $C_{2}$ used in the fifth-order formula for the phase speed is wrong. Consistent with the derivation of the fifth-order Stokes wave theory, we use the following Taylor expansion to evaluate the quantities in (2.2) for $0\leqslant z\leqslant |\unicode[STIX]{x1D701}|$ :

(2.10) $$\begin{eqnarray}\displaystyle f(z) & = & \displaystyle (f_{1}+f_{2}+f_{3}+f_{4}+f_{5})+z\frac{\unicode[STIX]{x2202}(f_{1}+f_{2}+f_{3}+f_{4})}{\unicode[STIX]{x2202}z}\end{eqnarray}$$

(2.11) $$\begin{eqnarray}\displaystyle & & \displaystyle +\,\frac{z^{2}}{2}\frac{\unicode[STIX]{x2202}^{2}(f_{1}+f_{2}+f_{3})}{\unicode[STIX]{x2202}z^{2}}+\frac{z^{3}}{6}\frac{\unicode[STIX]{x2202}^{3}(f_{1}+f_{2})}{\unicode[STIX]{x2202}z^{3}}+\frac{z^{4}}{24}\frac{\unicode[STIX]{x2202}^{4}f_{1}}{\unicode[STIX]{x2202}z^{4}},\end{eqnarray}$$

where $f_{i}$ and derivatives of $f_{i}$ are evaluated at $z=0$ . As described above, a time series of $F_{x}$ is in turn constructed numerically and band-pass filtered to extract the harmonics.

This approach provides higher harmonic loads than $3\unicode[STIX]{x1D714}$ . It should be noted that the scattering potential is consistent to third order only, as in the original FNV method, and the inclusion of terms of higher order than three is only correctly expressed in the incident wave potential.

Figure 4.

Model test set-up of the present experiments performed in a medium-size wave tank at the Marine Technology Centre, NTNU.

Figure 5.

Snapshots from high-speed video with incident wave from right towards left. The time instants are approximately $T/80$ apart and are labelled sequentially from (a) to (t). Water depth $h/a=7.83~\text{m}$ , $ka=0.127$ , $H_{1}/\unicode[STIX]{x1D706}=1/25$ . At a model scale of $1:48$ , the water depth is $h=27~\text{m}$ , the cylinder radius $a=3.45~\text{m}$ , wave period $T=12~\text{s}$ , and the horizontal lines are 1 m apart (numbers given above their corresponding lines). The vertical lines are $30^{\circ }$ apart. The confetti was used for flow visualization. One should note in particular the local run-up at the rear of the model, which breaks laterally and forms a local upstream propagating wave.

3.1 Instrumentation

The measurements included the horizontal force $F_{x}$ and the overturning moment $M_{y}^{m}$ , free-surface elevations by wave gauges, wave maker motion and water temperature. High-speed video was taken, including attempts at flow visualization using confetti on the free surface (cf. the supplementary movie). The moment was measured at a position $z_{m}=-0.317~\text{m}$ below the tank floor (mudline). The mudline moment $M_{y}$ was calculated as $M_{y}=M_{y}^{m}+z_{m}F_{x}$ . The mean temperature varied between 13 and $17~^{\circ }\text{C}$ over the testing period. The wave gauge calibration factors increased by approximately $2\,\%/^{\circ }\text{C}$ . This was corrected for in the post-processing. The force and moment transducer was insensitive to temperature changes, and was found to be very stable in general. The testing took place during several phases with more than one year in between, but the calibration factor of the load sensor varied by less than 1 % over the whole period. This applies to force transducer set-up 1 (see figure 4); two load measuring set-ups were used, denoted as force transducer set-ups 1 and 2. The purpose of force transducer set-up 2 was to investigate the accuracy of the force transducer used in force transducer set-up 1. In force transducer set-up 1, the horizontal force and moment were measured with a commercially available seismic shear force transducer, modified by MARINTEK to also measure the moment. The range of the force transducer was 7500 N, which was two to three orders of magnitude higher than needed for the typical loads in the present tests. Its high stiffness was necessary in order to obtain an adequately high natural frequency of the set-up. The wet natural frequency of the set-up was 23 Hz. The accelerometers at the top of the model were used to check that the cylinder vibrations were negligible. In force transducer set-up 2, 2 $\times$ 400 N shear force transducers from MARINTEK were used, one at the top and one at the bottom of the model. The natural frequency was 30 Hz. The model was clamped onto the transducers, so that moments were not obtained in this set-up. There was clearly less measurement noise in the force signals obtained in force transducer set-up 2, but the main trends of the load harmonics were practically identical to those obtained in force transducer set-up 1 (figure 15).

3.2 Test conditions

One single monopile model was tested in two water depths, $h/a=7.83$ and $h/a=5.51$ . The waves ranged from deep to intermediate depth wave conditions. Our main focus was on relatively long and moderately steep to considerably steep waves up to the breaking limit. The reason is that we observed consistent discrepancies with the theoretical third harmonic of the loads for steep wave conditions.

At an imagined model scale of $1:48$ , the diameter of the monopile is $D=6.9~\text{m}$ , the two water depths are $h=27~\text{m}$ and 19 m and the wave periods are between $T=6~\text{s}$ and 16.5 s by Froude scaling.

The wave steepness $H_{1}/\unicode[STIX]{x1D706}$ ranged from $1/140$ to $1/16$ , where $H_{1}$ is the linear wave height. By linear wave height, $H_{1}$ , we refer to the following. For small-amplitude waves, the wave maker motion was prescribed according to linear wave maker theory. Except for rather minor modifications of the wave maker motion by use of a mechanical transfer function, good agreement between the measurements and the theory was achieved for small-amplitude waves, say for $H_{1}/\unicode[STIX]{x1D706}<1/50$ . When generating waves with higher steepness, the wave maker motion was simply scaled linearly. This means that the first harmonic in the measured wave does not correspond exactly to $H_{1}$ as used herein; it is somewhat smaller. This effect is, however, not dominant, and we use the term $H_{1}/\unicode[STIX]{x1D706}$ to characterize the wave conditions throughout the paper.

Table 1.

Test condition parameters. The leftmost column provides the (model scale) wave periods in the present model tests. The Keulegan–Carpenter numbers ( $KC$ ), Ursell numbers ( $Ur$ ) and Reynolds numbers ( $Re$ ) are given for a wave steepness $H_{1}/\unicode[STIX]{x1D706}=1/25$ . $KC_{1}$ and $KC_{5}$ refer to the maximum wave particle velocity at the wave crest according to linear and fifth-order Stokes theory respectively.

Selected descriptive parameters are provided in table 1. The Keulegan–Carpenter numbers are given by

(3.1a,b ) $$\begin{eqnarray}\displaystyle KC_{1}=\frac{U_{m1}T}{D}=\unicode[STIX]{x03C0}\frac{H_{1}}{D},\quad KC_{5}=\frac{U_{m5}T}{D}, & & \displaystyle\end{eqnarray}$$

where $T=2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D714}$ is the wave period, $U_{m1}=\unicode[STIX]{x1D714}\unicode[STIX]{x1D701}_{a}/\text{tanh}\,kh$ is the amplitude of the horizontal velocity at $z=0$ as predicted by linear wave theory and $U_{m5}$ is the maximum horizontal velocity at the incident wave crest according to fifth-order Stokes theory. In the table, the numbers are given for wave steepness $H_{1}/\unicode[STIX]{x1D706}=1/25$ .

The different Keulegan–Carpenter numbers are illustrated for two relevant wave conditions in figure 6. In the quantities with minus sign as superscript, the maximum velocity in the opposite direction to the wave-propagation direction is used, that is, the horizontal velocity at the incident free surface during a wave trough. Since the effect of flow separation increases with $KC$ , this figure illustrates clearly that flow separation effects are expected to be most prominent in the direction of the wave propagation.

Figure 6.

Example illustrating different $KC$ numbers for wave steepness $H_{1}/\unicode[STIX]{x1D706}$ ranging from 0 to $1/16$ at water depth $h/a=7.83$ , for $ka=0.127$ (a) and $ka=0.105$ (b). Here, $KC_{n}$ means that the horizontal velocity in the incident wave at the wave crest according to $n$ th-order Stokes theory is used. A minus sign as superscript indicates that the velocity at the wave trough is used.

The wavenumber $k$ in $KC_{1}$ and $H_{1}/\unicode[STIX]{x1D706}$ is taken to be that from the linear dispersion relationship. In $KC_{3}$ and $KC_{5}$ , the dispersion relationship is corrected to third order (cf., for instance, Skjelbreia & Hendrickson Reference Skjelbreia and Hendrickson1960).

The wave height in the Ursell number, $Ur$ , is taken as the linear wave height, $H_{1}$ , which is in accordance with the original paper by Ursell (Reference Ursell1963),

(3.2) $$\begin{eqnarray}\displaystyle Ur=\frac{H_{1}\unicode[STIX]{x1D706}^{2}}{h^{3}}. & & \displaystyle\end{eqnarray}$$

The Reynolds number is given by

(3.3) $$\begin{eqnarray}\displaystyle Re=\frac{U_{m1}D}{\unicode[STIX]{x1D708}}. & & \displaystyle\end{eqnarray}$$

Here, $\unicode[STIX]{x1D708}=10^{-6}~\text{m}^{2}~\text{s}^{-1}$ is used as the kinematic viscosity of water, which neglects small variations due to the temperature of fresh water. The Reynolds numbers in the tests suggest laminar boundary layers, which implies a problem when flow separation occurs and the results are scaled to prototype conditions with turbulent boundary layer flow. The reasons are that the flow separation line is affected by the boundary layer condition, and flow separation influences pressure loads. Since viscous stresses do not have important contributions to the hydrodynamic force and moments on the cylinder, the state of the boundary layer is less important with no flow separation.

Figure 7.

Amplitudes of the first three regular wave harmonics (model scale) measured at model position $w8$ (cf. figure 4) from selected wave calibration tests versus the wave period $T$ . The numbers in the legend refer to the wave steepness $H_{1}/\unicode[STIX]{x1D706}$ . Markers: experiments. The solid and dashed curves indicate that $k$ is calculated from the linear (dashed) and third-order (solid) dispersion relationships respectively ( $\unicode[STIX]{x1D701}_{a}=H_{1}/2$ in both cases).

Figure 8.

A representative set of the regular wave conditions tested in the present study plotted in an $H/h$ versus $\unicode[STIX]{x1D706}/h$ diagram, as suggested by Fenton (Reference Fenton1990). Grey markers: waves broke along the tank. Black markers: the three wave periods where detailed data are provided. (a) Water depth $h/a=5.51$ ; (b) water depth $h/a=7.83$ .

3.3 Calibrated waves

The wave gauges were calibrated to have an error within 1 %–2 %. Wave calibration tests without the model were run for a representative range of waves. The amplitudes of the harmonics were extracted using a narrow-band filter based on the fast Fourier transform. The amplitudes of the first three wave harmonics for selected conditions, measured at the model position $w8$ (cf. figure 4), are provided in figure 7. The agreement between theory and experiments is fair.

There are oscillating types of deviations from the theory. We believe that there are two main causes: wave reflections from the beach and parasitic waves generated by the wave maker. There are also other uncertainties related to wave tests in a closed tank, such as, for instance, seiching, transverse waves and mechanical noise of the wave maker.

The longest and most nonlinear waves broke just after the wave maker. The waves that broke are indicated by grey markers in figure 8, where a representative set of the presently tested regular wave conditions are plotted in a chart suggested by Fenton (Reference Fenton1990); an Ursell number of 40 (dashed curve) is suggested to provide an upper limit for fifth-order Stokes theory be valid. Above this limit, he recommends using cnoidal wave theory. We see from figure 8 that basically all of the presently considered non-breaking wave conditions are within the Stokes fifth-order wave theory regime, and in particular the three wavenumbers that are focused on in the results section (black markers). The curve $H/h=0.55$ was suggested by Nelson (Reference Nelson1994) to be an upper practical limit before wave breaking in a very flat bottomed wave tank.

3.3.1 Beach reflections

The wave reflections from the beach are most pronounced for wavelengths that are of equal length or longer than the beach. The reflections cause an oscillatory nature of the reduced data as a function of the non-dimensional wavenumber $ka$ . This is illustrated in figure 9, where the effect of an (erroneous) extreme beach height on the horizontal force is illustrated. By beach height we mean the height of the uppermost point of the beach, relative to still water. The beach height in the test was 1.5 cm (referred to as ‘normal’ in the legend). The results in figure 9 show that the reflection level is reasonably low for waves that are shorter than the beach. Based on observing the deviations in the data from a curve fit (not shown), the relative error is estimated to be of the order of 2 %–3 %.

Figure 9.

Illustration of beach reflection effects on the first three harmonics of the horizontal force $F_{x}$ versus non-dimensional wavenumber $ka$ . The parabolic beach was normally placed with its top position 1.5 cm above the still water line; in the tests denoted as ‘Wrong beach position’, this value was 18 cm. $H_{1}/\unicode[STIX]{x1D706}=1/25$ ; water depth $h/a=5.51$ .

Figure 10.

Illustration of parasitic wave occurrence and reduction in the experiments. The amplitudes of the first three harmonics of the measured waves at wg7 and wg8 in wave calibration tests (cf. figure 4) and the amplitude of the measured wave maker motion, denoted by WM (only first and second harmonics of WM shown), versus wave period $T$ . Wave steepness $H_{1}/\unicode[STIX]{x1D706}=1/30$ and water depth $h/a=5.51$ . (a) The motion of the wave maker contains $\unicode[STIX]{x1D714}$ signal only. (b) The $2\unicode[STIX]{x1D714}$ component of the wave maker motion has been added to reduce the parasitic second-order waves.

Figure 11.

Examples of wave amplitude versus distance $x$ from the wave maker from simulations using a fully nonlinear 2D numerical wave tank based on a Rankine type boundary element method (BEM), compared with Stokes wave theory. (a) Without $2\unicode[STIX]{x1D714}$ parasitic wave correction of the wave maker signal. (b) With $2\unicode[STIX]{x1D714}$ correction signal (amplitude $\unicode[STIX]{x1D701}_{2p}=0.55\unicode[STIX]{x1D701}_{a2}$ based on observation). Dash-dot curve: $\unicode[STIX]{x1D701}_{a1}$ as predicted by linear wave theory. Solid horizontal curves: amplitudes $\unicode[STIX]{x1D701}_{a1}$ , $\unicode[STIX]{x1D701}_{a2}$ , $\unicode[STIX]{x1D701}_{a3}$ predicted by Stokes fifth-order wave theory. Dashed curve: theoretical amplitude of second harmonic along the tank due to interference between bound and parasitic waves with amplitude $\unicode[STIX]{x1D701}_{p2}=0.55\unicode[STIX]{x1D701}_{a2}$ . Water depth $h/a=7.83$ , $ka=0.105$ , $H_{1}/\unicode[STIX]{x1D706}=1/25$ .

3.3.2 Parasitic waves

Parasitic waves were investigated in some detail. Parasitic waves are free (unwanted) waves generated by the wave maker, or by the tank topology. The latter effect is not relevant in the case of constant water depth as in the present study. In all of our tests, due to the finite water depth conditions, there was no available time window prior to the arrival of second-order parasitic waves, as utilized, for instance, by Huseby & Grue (Reference Huseby and Grue2000). Parasitic waves were therefore reduced by adding a $2\unicode[STIX]{x1D714}$ signal to the wave maker motion.

One can write the total signal of the free surface to second order, including the (free) second harmonic parasitic wave component as

(3.4) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D701}=\unicode[STIX]{x1D701}_{a}\sin (\unicode[STIX]{x1D714}t-kx)+\unicode[STIX]{x1D701}_{2a}\sin (2\unicode[STIX]{x1D714}t-2kx)-\unicode[STIX]{x1D701}_{2p}\sin (2\unicode[STIX]{x1D714}t-k_{2}x), & & \displaystyle\end{eqnarray}$$

where $k_{2}$ obeys $(2\unicode[STIX]{x1D714})^{2}=gk_{2}\tanh k_{2}h$ , $\unicode[STIX]{x1D701}_{2a}$ is the amplitude of the bound second harmonic component according to Stokes wave theory and $\unicode[STIX]{x1D701}_{2p}$ is the amplitude of the second-order parasitic wave. The magnitude was determined empirically as $\unicode[STIX]{x1D701}_{2p}=0.55\unicode[STIX]{x1D701}_{2a}$ , based on the results exemplified in figure 10(a). The second-order parasitic waves give rise to an oscillatory nature of the wave harmonic amplitude as a function of $T$ , as expected, based on (3.4). The phase between the bound and parasitic free $2\unicode[STIX]{x1D714}$ components is $\unicode[STIX]{x03C0}$ at the mean wave maker position, independent of the wave frequency (cf. Hughes Reference Hughes1993). An alternative could be to use second-order wave maker theory, as, for instance, described by Schäffer (Reference Schäffer1996).

It is also evident that the second-order parasitic waves influence the first and third harmonics, in particular the latter. With the $2\unicode[STIX]{x1D714}$ correction signal of the wave maker motion, the third harmonic of the waves behaves significantly better as a function of $T$ ; see figure 10(b). There are still irregularities associated with the fact that the parasitic second-order wave is not totally eliminated. Wave reflection from the beach is a further candidate that may have contributed to the irregularities. Still, the main trends are acceptable.

Figure 12.

Experimental time series of normalized horizontal force (low-pass filtered at 5 Hz) for nine wave steepnesses: $H_{1}/\unicode[STIX]{x1D706}=1/60,1/55,1/50,1/45,1/40,1/35,1/30,1/25,1/20$ . Wavenumber $ka=0.105$ ; water depth $h/a=7.83$ .

The existence of third-order parasitic waves is also exemplified in figure 11. Here, results from simulations with a 2D fully nonlinear wave tank based on the boundary element method (BEM) (cf. Kristiansen Reference Kristiansen2009) are presented. The first three harmonics of the wave are presented as functions of the spatial coordinate along the tank. Addition of the correction signal gives a clear improvement for the second-order parasitic waves, consistent with the experiments. The same oscillatory features are clearly seen in other studies where results from simulations of fully nonlinear wave tanks are compared with experimental data; see, for instance, Shao & Faltinsen (Reference Shao and Faltinsen2014).

3.4 Force and moment time series

Examples of time series of normalized experimental horizontal force are provided in figure 12. The wave condition $h/a=7.83$ , $ka=0.105$ is considered at nine wave steepnesses $H_{1}/\unicode[STIX]{x1D706}$ ranging from $1/60$ up to $1/20$ . The signals are low-pass filtered at 5 Hz to remove measurement noise. A linear ramp lasting for five wave periods was used when generating the waves. Except for the $1/20$ wave steepness case, the total time series are reasonably steady after the ramp-up period. There are more irregularities in the time series of each harmonic. These are obtained by narrow-band filters around each frequency. Examples including the first six harmonics are provided in figure 13. For clarity, the Hilbert envelopes, rather than time series, are displayed. The first and second harmonics are rather steady, while the third and higher harmonic signals display appreciable variations. There remain further studies to fully understand the irregularities. Possible explanations are parasitic waves, beach and side-wall reflections, as discussed above. Several time windows were used when extracting reduced data.

Figure 13.

Hilbert envelopes of the first six harmonics of the normalized horizontal force $F_{x}/\unicode[STIX]{x1D70C}ga^{3}$ for nine wave steepnesses: $H_{1}/\unicode[STIX]{x1D706}=1/60$ , $1/55$ , $1/50$ , $1/45$ , $1/40$ , $1/35$ , $1/30$ , $1/25$ and $1/20$ . Same case as in figure 12 ( $h/a=7.83$ , $ka=0.105$ ).

Figure 14.

Example of time series of the total and first three harmonics of the normalized horizontal force. The time series of each harmonic is obtained by narrow band-pass filtering. (a) Experiments. (b) The generalized FNV method with wave kinematics integrated to the fifth-order free surface. It should be noted that the magnitude of the third harmonic is not in agreement, but the phase is in good agreement; $ka=0.105$ , $H_{1}/\unicode[STIX]{x1D706}=1/25$ , water depth $h/a=7.83$ .