## Abstract

There are a great many studies that investigate the force on bodies in periodic oscillatory motions, but almost no studies that focus on the kinds of fluid loading that are of great relevance to offshore structure designers; namely, the problem of a cylinder subjected to the forces from ocean wave groups, especially those of large amplitude. In this study, we move a vertical circular cylinder in non-periodic horizontal orbital motion through stationary fluid, in a towing tank at *Re*≲1.15×10^{4}. The motion is chosen to represent the relative fluid motion incident upon a horizontal cylinder with its axis aligned parallel to the crest of a large ocean wave group, as defined by the ‘NewWave’ formulation. The vector form of the well-known Morison equation provides a good representation of the measured forces. By measuring the force components in the radial and azimuthal directions, we clearly demonstrate that the presence of vigorous force fluctuations at a higher frequency than the orbital motion are associated with vortex shedding that is otherwise masked by the choice of coordinates. We find vortex frequencies comparable with those for flow past fixed bodies and a ‘transverse’ force magnitude similar to the fixed flow case at the same average speed. Finally, we show that by retaining only an azimuthal (constant) drag coefficient term to represent the fluid loading throughout the wave group orbits, the resolved *X*- and *Y*-force fluctuations agree well with measured forces. This demonstrates that we can obtain a reasonable estimate of time-varying forces using a single term. It is expected that such a simple force representation will become less effective in shallower fluids and for smaller wave amplitudes relative to body size.

## 1. Introduction

Offshore engineers are interested in determining the hydrodynamic forces on structures subjected to wave loading. In particular, predictions are usually required of the peak forces that may occur during the design life of the structure. The extreme structural response is dependent on both the surface elevation frequency spectrum of the underlying sea state and on the localized wave kinematics that can occur within this irregular wave field. For an appropriate wave spectrum, designers must consider peak structural loads induced by local extremes within the irregular wave field and fluctuating forces induced by the underlying random sea state. Several studies have investigated the forces on structural components due to a random wave field, but those due to extremely large waves (typically classified as waves with crest–trough height *H*>2*H*_{s}, where *H*_{s} is the significant wave height of the wave field) have received less attention. At least to a linear approximation, an irregular wave field can be viewed as the summation of multiple component regular waves of different amplitude and frequency. Within such wave fields, surface elevation and flow velocity extremes occur when the crests or troughs of numerous wave components coincide as in the NewWave model (Tromans *et al*. 1991; Taylor & Williams 2004). Based on theoretical work by Lindgren (1970) and Boccotti (1983), it has been shown that this formulation accurately predicts the shape, to first order, of large waves measured in the North Sea (Taylor & Williams 2004).

Numerous semi-empirical models have been developed from which the wave loading within a design event can be estimated. A widely used expression for the force experienced by small bodies is the Morison equation (Morison *et al*. 1950) that accounts for the contribution of viscous drag (*C*_{D}) and added mass (*C*_{M}). Coefficients have been tabulated for a wide range of flow conditions, including regular waves, irregular waves, planar oscillatory flow and one- and two-dimensional cylinder motion within a still fluid (e.g. Sumer & Fredsöe 1997). Forced elliptic orbital motion within a quiescent fluid is kinematically equivalent to the case of a horizontal cylinder subjected to (linear) intermediate depth water waves in which particles prescribe closed orbits. A circular orbit (i.e. orbit with ellipticity, *E*=*A*_{x}/*A*_{y}=1, where *A*_{x} denotes the horizontal radii) represents deep water waves, whereas increasing ellipticity (*E*>1) is observed with reducing water depth towards the shallow water case of planar oscillatory motion. For forced oscillation, both drag and inertia coefficients reduce with decreasing ellipticity (Chaplin 1988), although this reduction is less significant in comparable regular waves, possibly due to the Stokes drift of the wake in the direction of wave propagation.

Large ‘lift’ forces have also been measured normal to the direction of cylinder motion (Ramberg & Niedzwecki 1979; Williamson *et al*. 1998; Oshkai & Rockwell 1999). These forces are associated with both vortex shedding and repeated wake re-encounter where the body passes through the wake formed earlier in the motion. As a result, the forces tend to be somewhat smaller when observed in comparable regular waves (Longoria *et al*. 1991; Chaplin & Subbiah 1997) or irregular waves where wake re-encounter is not exactly repeated. For small amplitude ratios (*A*/*D*<2), an estimate of the peak forces imposed by an irregular wave can be obtained from summation of the forces due to a small number of regular wave components (Li *et al*. 1997), but this would not accurately describe the time-varying force.

Although structural loading in both regular and irregular waves has received much attention, very few studies have been reported regarding the effect of large storm-induced waves on structures, particularly for small diameter horizontal components. Studies have investigated the loading of either large surface piercing cylinders (Swan *et al*. 2002; Walker *et al*. 2007) or large horizontal cylinders (Ogilvie 1963; Skourup & Jonsson 1992; Boccotti 1996), but these generally concern cases where structural loading is dominated by inertial forces (Keulegan–Carpenter numbers, based on a cylinder of diameter *D* oscillating with amplitude *A*). Less work has been reported concerning loading of components in the drag regime (typically *KC*≳20). Sundar *et al*. (1999) measured the pressure distribution on an inclined surface piercing cylinder within random and focused waves and observed qualitatively similar distributions in each case, but larger peak pressures in focused waves. However, the cylinder diameter was relatively large compared with the focused crest amplitude (*A*≤*D*), hence viscous forces are expected to be small. Extreme wave loading of horizontal components has received much less attention; a preliminary numerical study is presented by Yeung *et al*. (1999) but, to our knowledge, experimental measurements of this type of loading have not been published.

In this study, the fluid-dynamic force components experienced by a fixed horizontal cylinder subjected to loading by a focused wave group are simulated by moving a vertical cylinder along a two-dimensional trajectory drawn in the horizontal plane. Somewhat comparable experiments were carried out by Rodenbusch & Kallstrom (1986), at much larger scale, with cylinder motions consistent with linear wave kinematics within a random sea state. By contrast, in this work, the trajectory and velocity of the cylinder are defined by the kinematics of a fluid particle within a (linear) NewWave group, as defined by the scaled autocorrelation function. While the linear fluid particle kinematics and the imposed cylinder motion are chosen to be identical, there are some differences between the flow fields in the two situations. Firstly, the kinematics in a deep water wave decay exponentially with depth, so the velocity fields away from the cylinder are different. However, this is likely to have only a small effect because the wavelength is much greater than the amplitude of motion (*A*), which is also greater than the cylinder diameter (*D*).

It is worth remarking on the scales relevant for large offshore structures exposed to severe storms: with a wave period of 12 s, the wavelength is approximately 200 m; the amplitude of the largest individual waves in the northern North Sea in the winter is approximately 12 m; the diameter of the cylindrical elements within a typical steel jacket structure 2–5 m; and the conductors through which the oil and gas is produced are typically 1 m in diameter. Thus, the practically relevant *A*/*D* ratio falls in the range 3–10, matching the range in this experimental study. In common with all laboratory scale experimental studies, the one important non-dimensional parameter that cannot be matched is the Reynolds number; here, we are short by a factor of approximately 10^{3}.

One complication is that regular waves cause a net transport, or drift, of fluid particles downstream. In cases where wake re-encounter is important, this Stokes drift might play a significant role. We return to this point in §2. Also, for a horizontal cylinder in waves, the moving free surface is a pressure relief condition applied relatively close to the body and its vortex wake. Neither this free-surface effect, nor the simpler Froude–Krylov force (zero here because there are no externally imposed pressure gradients), are included in our experiments. Clearly, the Reynolds number of these experiments is much lower than ocean scale structures, and various complications in real wave-driven flows are not fully modelled. The intent of this study is to investigate the applicability of the well-known Morison equation for modelling the forces resulting from complex time-varying flow around a cylinder. We believe that the complications mentioned above warrant further study, but the present experiments constitute a basis for reference.

## 2. Wave group and cylinder kinematics

The transient surface elevation, *ζ*(*t*), of an irregular sea state may be approximated by the sum of a finite number of independent regular wave components of different phase and amplitude. Consider the superposition of *N* component waves of frequency *ω*_{n}(2.1)where *A*_{n} and *B*_{n} are independent amplitudes for each spectral component whose total variance is equivalent to the variance, *σ*^{2}, of *ζ*(*t*). Note that, for ocean waves in deep water, the standard deviation of the surface elevation and significant wave height are related by *H*_{s}=4*σ*. When the crests of many regular component waves coincide, an extreme wave can be generated that provides a critical design case for offshore structures. Tromans *et al*. (1991) suggested that an extreme wave event be regarded as a localized wave group, whose kinematics can be described by relating the free surface elevation time history to the energy spectrum of random sea motion. The NewWave formulation defines the average shape of a large wave (of crest amplitude *A*) in an underlying linear random sea state as simply the scaled autocorrelation function. For a crest amplitude *A*≳2*σ*, the instantaneous horizontal and vertical displacements (*ξ*, *ζ*) of a particle on the free surface of a NewWave group on deep water are given by linear theory as(2.2)and(2.3)where *S*(*ω*) denotes the energy spectrum of the wave field. Thus, the amplitudes of the spectral components in NewWave are in proportion to the power spectrum of the underlying random sea state. The kinematic velocity and acceleration components are obtained by successive differentiation of equations (2.2) and (2.3) with respect to time. Thus, the Cartesian velocity components, , are approximately proportional to *Aω*_{0} and the acceleration components proportional to , where *ω*_{0} is the dominant frequency of the wave spectrum. Perhaps the most commonly used wave energy spectrum is the JONSWAP spectrum developed by Hasselmann *et al*. (1973)(2.4)where *σ* is a measure of the width of peak enhancement (typically, *σ*=0.07 for *ω*/*ω*_{0}<1 and *σ*=0.09 for *ω*/*ω*_{0}>1, but a constant value *σ*=0.07 is used herein for convenience). The JONSWAP coefficient, *γ*, describes the sharpness of the peak of the spectrum and is typically in the range 1<*γ*<7. The Pierson–Moskowitz spectrum is recovered for *γ*=1, whereas a typical value for a North Sea storm is *γ*=3.3. Figure 1 shows the two-dimensional trajectories (i.e. variation of *ξ*(*t*) and *ζ*(*t*) by equations (2.2) and (2.3)) and the corresponding time-varying velocity of a fluid particle within the NewWaves associated with JONSWAP and Gaussian wave spectra. In each case, the same focused crest-free surface amplitude *A* and peak frequency *ω*_{0} are used. Unlike a regular wave in which fluid particles prescribe circular or elliptical orbits, the same path is not repeated in a focused wave group. Here, the particle trajectory consists of an outward spiral from the origin until the maximum of both vertical displacement and horizontal velocity occur at *t*=0. Subsequently, the particle traverses a mirror image (in the vertical *ζ*-axis) of the outward spiral trajectory to return to its initial position. For simplicity, we neglect the faster depth decay of the high-frequency linear components within the fluid column, which implies that a more deeply submerged cylinder is exposed not only to a weaker wave-induced flow, but also one with a narrower bandwidth.

There are few published studies of the variation of particle trajectories with depth beneath a focused wave group, but the depth variation of maximum horizontal velocity and second-order kinematics have been documented. As alluded to previously, fluid particles undergo a net drift in nonlinear waves, for regular waves known as the Stokes drift. However, the motion in localized wave groups is less well known. Near the free surface, the Lagrangian time-integrated motion is forward in the direction of wave propagation. Deep in the fluid, this Lagrangian motion is backwards because the depth-integrated transport must be zero for isolated groups, and the depth to which this return flow penetrates depends on the physical length of the envelope of the wave group (approx. 1/spectral bandwidth). This implies that there is a vertical position where the net transport is zero, and linear kinematics would be most applicable. This rather complicated behaviour occurs at second order in wave steepness. Calculations by Taylor & Vijfvinkel (1998), among others, demonstrate the magnitude of this second-order return flow. Even for steep waves on deep water, the return current is small, typically less than 10 per cent of the large linear kinematics component close to the free surface, so neglecting it is a reasonable first approximation. Although this is crude, many offshore structures have been designed using ‘Wheeler stretching’ of the linear kinematics profile, see Wheeler (1970), so we consider that the motions defined by equations (2.2) and (2.3) based on linear deep water wave kinematics are an appropriate starting point.

For a narrow-banded Gaussian spectrum (figure 1*a*), all of the component waves with significant amplitude oscillate at a similar frequency. Therefore, the component waves are in phase for several overall wave periods on either side of the focused wave crest that occurs at *t*=0. The resultant wave group contains multiple high-amplitude oscillations that persist for approximately seven wave periods on either side of *t*=0. In this case, fluid particles follow a tight spiral trajectory that approaches circular during the three central orbits. Figure 1*b* represents the NewWave group for a typical North Sea storm with the same dominant frequency (*ω*_{0}) as the Gaussian spectrum. Owing to the wider frequency range of the JONSWAP spectrum, component waves are in phase for a shorter interval than for the Gaussian spectrum, and so the wave height decays rapidly on either side of the focused crest. Thus, fluid particles follow a much more open spiral path when driven according to a JONSWAP spectrum. The high-frequency wave components described by the tail of the JONSWAP spectrum also contribute to an increased maximum horizontal velocity, at least for cylinders close to the free surface. In addition to effects related to the shape of the wave energy spectrum, wave-induced forces are expected to vary with the ratio of focused crest amplitude to cylinder diameter (*A*/*D*, comparable with the Keulegan–Carpenter number) and with the Reynolds number.

### (a) Experimental system

Herein, the position of a vertical cylinder within the horizontal plane (*x*, *y*) of a quiescent fluid is defined as the displacement of a fluid particle within the vertical plane (*ξ*, *ζ*) of a wave field. Maximum horizontal displacement, *y*(*t*)=*A*, occurs at *t*=0 and corresponds to the displacement of a fluid particle beneath the focused wave crest (figure 1). Experiments were conducted in the computer-controlled XY towing tank at Cornell University. The arrangement is illustrated in figure 2 and comprises a vertical cylinder (8) mounted on a pair of transducers (6) and (7) that are aligned with the *x*- and *y*-directions, respectively. These are supported on a carriage (2) that is moved in the *y*-direction by a rotary screw (1) at velocities up to 11 cm s^{−1}. Both the supporting rails and parallel screw rail are mounted within a triangulated cross beam that is moved in the *x*-direction by a stepper motor-controlled tension cable (4), allowing a peak velocity of 30 cm s^{−1}. Two cylinders of diameter *D*_{1}=2.54 cm and *D*_{2}=3.81 cm were employed with an immersed cylinder length *L*=50 cm and the width of the tank was 1 m. With the two cylinders used for the experiments, the closest approach to a side wall of the tank was 6*D*. Given the transient nature of the orbital motion, blockage effects are assumed to be unimportant. During each experiment, a constant endplate separation of 3 mm and edge distance greater than 5*D* was maintained. Prior to each experiment, the water was left undisturbed for a minimum of 45 min to allow dissipation of the vorticity field generated by any earlier cylinder motion. The water surface was also skimmed after each experiment to reduce the influence of surface impurities on vortex breakdown at the free surface.

Cylinder motion was coordinated from a Labview interface installed on a Pentium personal computer. This provides a voltage signal to each stepper motor to produce the cylinder kinematics corresponding to a user-defined wave group (i.e. *u*_{x} and *u*_{y} calculated from specified *S*(*ω*) and *A*). Both velocity signals and the voltage measured from each force transducer were recorded on a second personal computer and synchronized by a third binary signal. Both *x*- and *y*-direction force transducers were calibrated *in situ* by recording the voltage corresponding to a range of applied forces. The combined inertial mass of the cylinder and force transducer were calculated by forced sinusoidal oscillation of an equivalent mass along each axis. This calibration allows the instantaneous force to be accurate to less than 1 per cent of the largest peak-to-peak value. Time-varying forces recorded during each test were passed through a digital Butterworth filter (twice to avoid a phase shift) to eliminate high-frequency noise. Since the peak frequency expected due to physically induced forces is approximately *f*_{max}=*Stu*_{max}/*D*_{1}=0.2×0.11/0.0254<1 Hz (assuming a Strouhal number, *St*∼0.2), a cut-off frequency of 2.5 Hz (5*π* rad s^{−1}) was employed to ensure that physically significant forces were not obscured. Comparison of force–time histories with different cut-off frequencies indicated that this value was suitable in the majority of cases.

Measured time-varying forces (*F*_{k}) are non-dimensionalized with respect to the peak cylinder velocity (*u*_{max}), the cylinder diameter (*D*) and the immersed cylinder length (*L*) by , where the suffix *k* denotes one of the Cartesian axes *x* or *y*. Peak cylinder velocity *u*_{max}=30 cm s^{−1} was specified for planar oscillation in the *x*- and *y*-direction peak velocity, 11 cm s^{−1}, specified for orbital motion. All force measurements from orbital motion are therefore normalized by the same factor. Peak Reynolds numbers are 11 400 for planar oscillations in the *x*-direction and 4800 for orbital motions including the *y*-direction. Drag and inertia coefficients are derived from a measured force time history, *C*_{k}, by a least-squares method (described in §4). Figure 3 shows drag and inertia coefficients (*C*_{D}, *C*_{M}) calculated from the in-line force (*F*_{x} or *F*_{y} depending on the axis of oscillation) measured during sinusoidal oscillation of the cylinder for Keulegan–Carpenter numbers, 11<*KC*<46 (where *KC*=*u*_{max}*T*/*D*). For both axes of motion, in-line force coefficients are in reasonable agreement with numerical and experimental results presented by Obasaju *et al*. (1988).

## 3. Forces due to wave group motion

Two series of tests were conducted to investigate the influence of both the JONSWAP spectrum coefficient γ and the ratio of crest amplitude to cylinder diameter (*A*/*D*) on the peak and time-varying force experienced by a cylinder undergoing non-periodic oscillatory motion. Initially, planar cylinder motions were defined by the horizontal displacement of a fluid particle (given by equation (2.2) only). Neglecting depth variation of wave particle kinematics and flow along the cylinder axis, these one-dimensional cases are relevant to the focused wave loading of a vertical cylinder. Subsequently, two-dimensional orbital motions were studied in which the cylinder velocity is defined as that of a particle beneath a NewWave, as a representation of a wave group interacting with a horizontal cylinder. Reynolds numbers (*Re*=*u*_{max}*D*/*ν*) are limited to less than 11 400 and 4800 for the one- and two-dimensional cases, respectively, due to the attainable *x*- and *y*-direction cylinder velocities.

### (a) Motion in one dimension only

Figure 4 shows the non-dimensional in-line, *C*_{x}, and transverse (lift), *C*_{y}, force–time histories due to planar oscillatory motion at Reynolds numbers of 7620 and 11 400. In both cases, the cylinder motion is geometrically similar, defined by the same JONSWAP spectrum (*γ*=3.3) with crest amplitude *A*=10*D*, but the Reynolds number differs by a factor of 1.5 due to the cylinder diameters used. The time-varying in-line force, *C*_{x}, is similar in both cases, exhibiting a small phase lag with respect to the *x*-direction velocity (*u*_{x}). The peak force occurs close to maximum velocity (*t*=0) when the drag coefficient (*C*_{D}) is approximately 1.5. Initially, the magnitude of the transverse force, *C*_{y}, is small while the cylinder velocity is low, but an unsteady force develops during the third period (*t*/*T*∼−2) due to the development of vortex shedding. Subsequently, several bursts of high-amplitude oscillations of the transverse force occur while the cylinder velocity is high. Unlike the in-line force, the magnitude of the peak transverse force is sensitive to the change in the Reynolds number.

### (b) Motion in two dimensions

The peak force components induced by orbital motion given by NewWave groups defined by JONSWAP spectra are quantitatively similar, irrespective of the value of *γ* in equation (2.4). Figure 5 shows the non-dimensional force components (*C*_{x}, *C*_{y}) plotted against *t*/*T* for a range of crest amplitude ratios but a fixed JONSWAP parameter *γ*=3.3. The phase lag between velocity and force components is small (the peak force in the *x*-direction occurring at very close to *t*=0), indicating that the force is drag dominated. During two-dimensional motion, the peak force components are substantially smaller than those experienced due to the identical planar motion only (as figure 4). As known for periodic motion (Chaplin 1988; Williamson *et al*. 1998), this is because the wake-induced flow velocity acts against the direction of cylinder motion during planar oscillation, but in a similar direction to cylinder motion during orbital motion. Thus, the incident-flow velocity, and hence drag forces, are lower during two-dimensional motion. On either side of the peak velocity (*t*=0), both force components decay more rapidly than the corresponding flow velocity components, suggesting a nonlinear relationship between force and cylinder velocity.

In general, the peak forces as the trajectory spirals outwards are greater than those experienced as the trajectory spirals inwards. (For example, for the typical JONSWAP spectrum.) Similar reductions have been observed for cylinders undergoing periodic orbital motion (e.g. Williamson *et al*. 1998) and can probably be attributed to a reduction in the incident-flow velocity as the cylinder encounters a previously generated vortex wake flow moving in the same direction as the body. Both force component spectra closely match the shape of the spectrum of *qu*, where . This qualitative difference between the force spectra for planar and orbital motion explains the smoother, more sinusoidal shape of the two-dimensional force–time histories in comparison with the more sharply peaked profile of the one-dimensional transient forces.

In addition to the drag and inertial forces that oscillate at a similar frequency to the cylinder motion, smaller amplitude fluctuations in both *C*_{x} and *C*_{y} are also evident at a higher frequency, particularly during the three largest orbits (−0.5<*t*/*T*<1.0 in figure 5 and clearly visible in figures presented in §4). From measurements of cylinder loading in an elliptical flow, Chaplin & Subbiah (1997) and Williamson *et al*. (1998) observe that a vortex force acting on the cylinder could act parallel and normal to the instantaneous velocity vector of the body, and so the resultant force fluctuations would be evident in both Cartesian force components (also reported by Chaplin 1988). In §5, it is shown that these forces are due to a vortex-induced force acting transverse to the direction of cylinder motion. Apart from these higher frequency fluctuations, the shape of the time-varying force history is independent of amplitude ratio (figure 5) and closely follows the time variation of the product of the Cartesian velocity component (e.g. *u*_{x}) and the cylinder speed (*q*). The vector form of the Morison equation, as used by Ramberg & Niedzwecki (1979) and Chaplin & Subbiah (1997), would therefore provide an estimate of the time-varying Cartesian force components.

## 4. Force prediction using the Morison equation

For a body oscillating sinusoidally along a line within an otherwise still fluid, the fluid force per unit length in line with the direction of motion can be estimated by the combined drag and inertia equation of Morison *et al*. (1950)(4.1)where *u*_{x} and *a*_{x} denote the velocity and acceleration parallel to the *x*-axis and *C*_{D} and *C*_{M} denote time-invariant drag and added mass coefficients. (In keeping with much of the literature, throughout this paper, we refer to (4.1) as the ‘Morison equation’, although we recognize the four authors involved.) The first term represents the part of the total force, in the form of a classical drag force (proportional to the square of velocity, in phase with the velocity), while the second term represents the force in phase with the acceleration of the fluid (generally referred to as the added mass). For a body undergoing general two-dimensional motion, equation (4.1) cannot be valid since the net resultant force would be dependent on the orientation of the coordinate system. The only consistent vector form of the Morison equation is based on rewriting the drag term to give(4.2)where and is equivalent to the instantaneous speed of the cylinder. Since *q* is a scalar, the combined forces obtained for a given pair of coefficients are independent of the choice of coordinate system. This vector form has been employed in numerous studies of wave-induced forces including those by Ramberg & Niedzwecki (1979), Chaplin & Subbiah (1997) and Li *et al*. (1997).

In this section, we derive values of both drag and added mass coefficients for each of the time-varying force measurements and investigate how these are affected by the ratio of the amplitude of motion to cylinder diameter (*A*/*D*) and the shape of the power spectra of the NewWave group.

### (a) Constant drag and added mass coefficients

Constant values of drag, *C*_{D}, and added mass, *C*_{M}, coefficients are calculated for each cylinder trajectory for both the *x*- and *y*-directions by a least-squares best fit to the measured force components. For the *x*-direction, the following criterion is minimized:(4.3)where *C*_{x,p} is the *x*-direction non-dimensional component derived from equation (4.2) at instant *t* and *C*_{x,m}(*t*) is the corresponding measured component. Both measured and predicted forces are non-dimensionalized, as described in §2*a*, and a similar expression is applied for the *y*-direction force. To avoid spurious values due to near-zero flow velocities, equation (4.3) is evaluated over the range −4<*t*/*T*<4. We adopt this approach of fitting to a wide part of the main motion (the eight largest orbits) rather than just the largest orbit since the aim is to derive constant coefficients valid for the duration of a wave group. Fits to individual orbits are considered later. We present the force coefficients *C*_{D} and *C*_{M} in the *x*- and *y*-directions in figure 6. The drag coefficient, *C*_{D}, has an average of 0.95 and does not change dramatically over the range of amplitudes studied. On the other hand, the added mass, *C*_{M}, increases roughly linearly, although one may suspect that *C*_{M} will ultimately asymptote to a limiting value based on its character for planar oscillatory flow (Williamson 1985). Note that the forces are drag dominated and so the net contributions of the inertial forces are small. There is no significant effect of either the Reynolds number (over the small range considered) or oscillation direction (*x* or *y*) on either *C*_{D} or *C*_{M}. A much wider regime of the Reynolds number would be needed to quantify its influence.

Figure 7 shows the Cartesian force component (*C*_{x,p}, *C*_{y,p}) time histories calculated by equation (4.2) using drag and added mass coefficients obtained by a least-squares fit to the measured forces for the orbital motion corresponding to a NewWave group for a typical JONSWAP spectrum. For comparison, the residual (Δ*C*_{x}=*C*_{x,m}−*C*_{x,}_{p}) between the measured force and predicted Morison force is plotted in the lower part of the figure. For this motion, defined by a JONSWAP spectrum, the fitted Morison equation provides a remarkably accurate representation of the measured force over the entire motion. While the cylinder velocity is low (*t*/*T*>|1.5|), non-zero values of the force residual can be attributed to experimental inaccuracies, whereas while cylinder speed is close to its maximum (specifically while −1<*t*/*T*<1), the force residual contains several larger amplitude oscillations that can be attributed to a radial force related to vortex shedding. Although the fitted Morison equation does not capture the force oscillations due to vortex shedding, the peak force that occurs during each orbit is closely approximated. Similar agreement is observed for all cylinder trajectories defined by a JONSWAP spectrum, irrespective of the wave amplitude.

Figure 8 shows that the underlying shape of the force–time history for the motion corresponding to a Gaussian NewWave group is also reasonably well estimated by the fitted Morison equation. For this narrow bandwidth Gaussian wave energy spectrum, the residuals of both force components are dominated by high-frequency oscillations. Short bursts of high-amplitude oscillation occur alternately in the *C*_{x} and *C*_{y} force residual time history that are qualitatively similar to the vortex-induced transverse force observed during planar oscillatory motion (figure 4). (For example, bursts of force fluctuations occur in the *x*-direction, while body motion is more aligned with the *y*-direction.) In the *x*-direction, the maximum force is *C*_{x,m}=1.15 and occurs at *t*=0 due to a combination of drag-, inertia- and vortex-induced forces. It appears that, as the bandwidth of the wave energy spectrum is reduced, the measured force–time history becomes asymmetric and peak forces during the increasing radii orbits (before a focused wave crest) are underpredicted, whereas those during decreasing radii orbits are overpredicted (figure 8). These variations are not fully explained by a constant coefficient expression such as equation (4.2).

### (b) Time-varying drag and added mass coefficients

To assess the relative contribution of drag and inertia over the duration of a complete wave group, time-varying drag and added mass coefficients are employed in the following generalized Morison equation:(4.4)Instantaneous values of drag and added mass coefficients are obtained at each instant (*t*) by a least-squares fit to the measured force over a time increment of one-half wave period about this point (i.e. by minimization of equation (4.3) over the interval *t*−*T*/4 to *t+T*/4). The resultant time-varying values of *C*_{D} and *C*_{M} obtained from the measured force for the cylinder trajectory defined by a Gaussian spectrum are plotted in figure 9. Both drag coefficients reduce to *C*_{D}∼0.75 over the interval −1<*t*/*T*<2.5, indicating that incident-flow velocity reduces during successive orbits. For this nearly circular spiral trajectory, both the drag and added mass coefficients change slowly during each orbit. While *t*/*T*<−1.0, both drag coefficients are close to a mean of *C*_{D,x}=*C*_{D,y}=1.0, but undergo an approximately linear reduction thereafter.

At the instant *t*/*T*=0.5, the cylinder crosses its trajectory for the first time (figure 1*a*) and the subsequent orbit has a similar radius to that of the previous orbit. Thereafter, the cylinder spirals inwards through a region of previously disturbed fluid. Hence, this near linear drag reduction can be attributed to a reduction in the relative velocity between the cylinder and fluid.

## 5. High-frequency force fluctuations due to vortex formation

We have seen that employing the Morison equation with constant coefficients (*C*_{D} and *C*_{M}) yields a very good representation of the measured force in the *x*- and *y*-directions. These large-magnitude forces at the wave frequency are perhaps the most significant component of fluid loading for structures subjected to these wave groups, but higher frequency forces are also of importance, particularly for structural fatigue assessment. In this section, we shall show that the higher frequency force fluctuations are induced by the wake vortex formation, as the body progresses around its orbital trajectory.

One may readily observe, in several plots of the force measurements (for example, see figure 8), the presence of higher frequency force fluctuations. By decomposing the forces into radial and azimuthal components, it is clear that these fluctuations are indicative of the forces induced by classical vortex shedding, whose effect is otherwise masked by, not only the choice of force decomposition in the *x*- and *y*-directions, but also by the larger magnitude of low-frequency force components. Radial and azimuthal force components were used initially by Holmes & Chaplin (1978) and by Ramberg & Niedzwecki (1979), in the case of circular orbits, where it is clearly more suitable than Cartesian components. In their studies of elliptic orbits of bodies, Williamson *et al*. (1998) found, as one might expect, that both sets of force decompositions yield important fluid-loading characteristics, dependent on orbit ellipticity. Since the orbits considered here, particularly those corresponding to a Gaussian NewWave group, are approximately circular, we also adopt radial and azimuthal force components as a (very close) approximation to the normal and tangential force components.

In the case of the radial fluid force, the presentation of the measurements is made clearer if we include only the ‘vortex force’ component, subtracting the ideal added mass force (proportional to the body's acceleration) from the total force. It was shown by Lighthill (1986) that the total fluid force (*F*_{total}) can be conveniently decomposed into a potential force component *F*_{potential}, given in this case by the potential added mass force, and a vortex force component (*F*_{vortex}) that is due to the dynamics of what is called the ‘additional vorticity’. The subdivision of such forces proved very useful to interpreting vortex-induced vibration phenomena in Govardhan & Williamson (2000), where further discussion is found. A full knowledge of the vorticity field would yield the vortex force through the concept of vorticity impulse. The radial vortex force can thus be found by subtracting the potential added mass force from the total force. The instantaneous potential added mass force *C*_{potential} acting on the cylinder is given by(5.1)where ** a**(

*t*) is the acceleration vector;

*m*

_{d}(=

*πD*

^{2}

*L*/4) is the displaced fluid mass; and

*C*

_{a}takes on the value 1.0. Removing this potential force reduces the zero offset of the radial force, allowing us to concentrate on the force fluctuations due to vortex shedding. (For the azimuthal force on its own, shown in later figures, we continue to present the azimuthal component of the total force.)

We present both the resultant vortex force vectors, superposed upon the cylinder trajectory, and the radial (vortex) force in figure 10. One can see immediately that there is a radial force (equivalent to a lift force for a fixed cylinder in a free stream) that fluctuates in a manner that one expects from classical vortex shedding for a fixed cylinder. A similar phenomenon is observed for orbits defined by both JONSWAP and Gaussian spectra. The presence of vortex shedding in the type of flow considered here is illustrated by surface particle visualization of the vortex formation in the wake of a cylinder conducting circular orbits, as shown in figure 11. Vigorous vortex dynamics are observed, which lead to the kind of high-frequency radial force fluctuations presently measured during near-orbital motion.

If we now plot the radial force as a function of time, as the cylinder progresses around the orbits, then it can be seen even more clearly (figure 12) that there are distinct lift force fluctuations due to the vortex shedding. By normalizing the radial vortex force using instantaneous azimuthal velocity *u*_{θ}(*t*), instead of *u*_{max}, then the magnitude of the radial force becomes slightly more uniform, yielding a magnitude *C*_{R}∼0.35, during the largest orbits of the trajectory (approx. *t*=0). In general, the radial forces are larger during those cylinder orbits close to, or shortly after, the maximum displacement while the cylinder spirals inwards towards the origin. Based on the peak Reynolds number, a lift force coefficient of *C*_{L}=0.85 may be expected in fixed cylinder flow (fig. 2 and eqn (30) of Norberg 2003). The radial force amplitude *C*_{R}=0.35 measured here corresponds to the lift coefficient in steady flow velocity of 0.875*u*_{max}.

It is also informative to consider the time variation of vortex shedding frequency. Figure 13 shows the variation of the peak frequency of a discrete Fourier transform of the radial force applied over an interval *t*−*T*/4 to *t+T*/4 for each instant *t*. When normalized to the maximum cylinder velocity, the shedding frequency is greatest during the middle portions of the motion where we have the largest orbit size (while body velocity is greatest). However, we find that the normalized frequency (or Strouhal number, *St*) becomes more uniform when based on instantaneous azimuthal velocity *u*_{θ}(*t*) and, in fact, tends to decrease slightly throughout the motion. Since azimuthal velocity does not vary greatly during a half-wave period, similar peak frequency variation (1.2% greater at maximum velocity) is obtained if normalized to the average azimuthal velocity during the interval *t*−*T*/4 to *t*+*T*/4. The Strouhal number during the central orbit of the motion is approximately *St*∼0.18, which is less than the value *St*∼0.21 found for comparable Reynolds numbers in the case of fixed cylinders in steady flow (Norberg 2003). A similar observation can be made for the azimuthal force coefficient, *C*_{θ}(*t*) (figure 14). When normalized to the maximum cylinder velocity, *C*_{θ}(*t*) is maximum during the central orbits, but becomes more uniform, again with a gradual reduction, if normalized with respect to the instantaneous velocity *u*_{θ}(*t*). These reductions could be attributed to the decrease in incident-flow velocity caused by the ‘stirring’ action of the cylinder as the motion progresses. This is particularly valid as the orbits diminish in size, after *t*=0, when extensive wake re-encounter must occur.

Finally, we may use a least-squares best fit value of *C*_{θ}(*t*), evaluated from figure 14, and employ that value to estimate the *X*- and *Y*-force components over the complete duration of the cylinder motion using(5.2)where *α*(*t*)=tan^{−1}[*y*(*t*)/*x*(*t*)] is the instantaneous angle to the cylinder centre from the orbit origin. The comparison with measured forces *C*_{x,m} and *C*_{y,m} is quite close (figure 15), and one may conclude that the use of a single coefficient, *C*_{θ}=0.875, is essentially as effective as the classic use of two coefficients *C*_{D} and *C*_{M} in the Morison equation (figure 8), to represent the principal components of force acting on structures subjected to this type of non-periodic motion.

In shallower water, the orbits would become more elliptic, and such a force representation would inevitably become less appropriate than the classical Morison approach used in §§3 and 4, as found in the case of forces on cylinders in elliptic orbital motion (Chaplin 1988; Williamson *et al*. 1998). Similarly, for much smaller normalized amplitudes of motion (much smaller *KC* number), the use of a single coefficient *C*_{θ} to predict the *X* and *Y* forces will not be as effective, and one might need to include also an inertia force term for effective force representation.

## 6. Conclusions

Experimental measurements are presented of the time-varying force experienced by a cylinder undergoing two-dimensional motion representative of the kinematics of a fluid particle within a linear NewWave group. The study of forces on structures due to wave groups has had very little attention in the past, and one of the principal questions is how best to represent the fluid forces during these non-periodic events. In the Morison force formulation, the drag coefficient is only weakly dependent on the linearized amplitude of the focused wave crest relative to the cylinder diameter (*A*/*D*). By contrast, the added mass coefficient increases gradually with crest amplitude. For all cases studied, the forces on the body are drag dominated, and a reasonable estimate of peak force is obtained by neglecting the inertia term, such that (where *C*_{D}=1.0 in our experiments).

Generally, the time-varying force experienced by a cylinder undergoing two-dimensional motion in a still fluid can be predicted with reasonable confidence using the vector form of the two-parameter Morison equation. However, this does not capture high-frequency force components. When viewed in a radial and azimuthal coordinate system, it is clear that these high-frequency components are due to a fluctuating lift-type force in the radial direction. Clearly, this force is associated with vortex shedding from the cylinder and the Strouhal frequency is close to that observed in steady flow, although the force amplitude is greater. The Strouhal frequency is found to reduce with increasing number of similar radii orbits and this is associated with a reduction in the incident-flow velocity due to wake re-encounter (stirring of the flow). Furthermore, we find that transient forces can be predicted throughout the wave group with similar confidence to the Morison equation by employing a single (constant) azimuthal force coefficient (i.e. where *C*_{θ}≈0.875). Although straightforward to implement, it is expected that such a simple expression for the transient force will become less effective for waves on shallow water or for smaller wave amplitudes relative to body size.

## Acknowledgments

The support for this experimental study was provided by Cornell University, and financial support from the Ocean Engineering Division of the US Office of Naval Research, monitored by Dr Tom Swean, is gratefully acknowledged (ONR Contract nos. N00014-04-1-0031 and N00014-07-1-0303). The first author was also funded by the Engineering and Physical Sciences Research Council under a DPhil studentship, The Royal Academy of Engineering (under ITG03-860), Oxford University and St Edmund Hall, Oxford. In addition, we gratefully acknowledge the enthusiastic help and contributions made by the Cornell Fluid Dynamics Research Laboratory (FDRL) team, especially James Buescher, Jackie Romero and Jon Rupp, as well as Matt Horowitz and Tim Morse.

## Footnotes

- Received August 20, 2008.
- Accepted January 15, 2009.

- © 2009 The Royal Society