Morison coefficients for a circular cylinder oscillating with dual frequency in still water: an analysis using independentflow form of Morison’s equation
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Abstract
In this study, a set of experimental results for wave forces acting on a cylinder oscillating with dual frequency in still water are reported. The experiment was designed to mimic a cylinder slowly oscillating in regular waves, with the highfrequency oscillation representing the wave motion and the lowfrequency oscillation, the slow drift motion. The inline forces acting on the cylinder were analyzed using the independentflow form of Morison’s equation. Our experimental results showed that it was not appropriate to simply use in the independentflow form of Morison’s equation the addedmass and drag coefficients obtained for the cylinder oscillating with a single frequency in still water. A new dimensionless parameter was introduced to describe each of the wave force coefficients used in the independentflow form of Morison’s equation, and empirical expressions for the wave force coefficients were proposed using the new dimensionless parameters.
Keywords
Wave force Morison’s equation Dualfrequency oscillation Keulegan–Carpenter number Reduced velocity Drag Inertia1 Introduction
When studying the responses of offshore structures to ocean waves, the wave forces acting on slender members are frequently modeled by Morison’s equation, which has been widely used to model wave loadings on cylinders fixed in waves or cylinders oscillating in still water or waves (e.g., Sarpkaya and Storm 1985; Najafian et al. 1995; Liu and Bergdahl 1996; DNV 2010). Even though the original Morison’s equation was proposed for regular waves, it has been widely used for random waves in practice (Najafian et al. 1995; Burrows et al. 1997). When performing frequencydomain analysis using Morison’s equation, the quadratic drag force needs to be linearized (e.g., Gudmestad and Connor 1983). Recent evaluations of the effectiveness of Morison’s equation using the data from two smallscale field tests can be found in Boccotti et al. (2012, 2013).
For problems involving two frequencies, two forms of Morison’s equation have been proposed: (1) the relativevelocity form of Morison’s equation (e.g., Chakrabarti 1987) and (2) the independentflow form of Morison’s equation (e.g., Laya et al. 1984). In the relativevelocity form of Morison’s equation, which is the most commonly used one, the drag force is quadratically related to the velocity difference between the ambient flow and the cylinder. In the independentflow form of Morison’s equation, which is used less often, the total force is the sum of the wave force due to the ambient flow field acting on a fixed cylinder and the force on the cylinder vibrating in an otherwise still water.
Koterayama (1984) studied the wave force coefficients for a circular cylinder moving with a constant speed in regular waves; the Keulegan–Carpenter numbers for the wave motion ranged from 1.3 to 100 and the reduced velocity defined by the constant speed and the wave period ranged from 0 to 60. For a circular cylinder oscillating sinusoidally with a very low frequency in regular waves and a cylinder oscillating with dual frequency in still water (the ratio of the two frequencies was an integer), Koterayama and Nakamura (1988) measured and analyzed the wave forces using the method similar to that of Koterayama (1984); their Keulegan–Carpenter numbers for the wave motion ranged from 1.1 to 15.7 and the reduced velocity ranged from 0.5 to 5.2. In the data analysis of both Koterayama (1984) and Koterayama and Nakamura (1988), the inline force was first written in terms of the relative velocity form, and then a harmonic analysis was applied to the drag force to obtain a drag coefficient for the steady motion and a pair of drag and inertia coefficients for each frequency component. All the results were presented as functions of the reduced velocity, but the data points were widely scattered in some results. Even though the ranges of the Keulegan–Carpenter number and reduced velocity covered a wide range in Koterayama (1984) and Koterayama and Nakamura (1988), but the definitions of the addedmass and drag coefficients used in these two studies are different from those in the independentflow form of Morison’s Equation: their drag coefficients include the contributions from nonlinear interactions among different frequencies.
2 Experimental setup and test conditions
2.1 Integrated cylindrical force model
All experiments in this study were conducted in a wave flume located in the Hydraulics Modelling Laboratory in Nanyang Technological University, Singapore. The wave flume was 0.55m wide, 0.6m deep and 36m long. The model was placed in the middle section of the flume to minimize any possible influence of the two ends.
The test section was chemically treated (coating) to prevent the metal from reacting with water so that its surface was always kept smooth during the experiment. We stress that the surface roughness may affect the wave force coefficients (Wolfram and Naghipour 1999). The upper dummy section was 300mm long and the length of lower dummy section was adjustable. All these three sections were carefully assembled so that their center lines were aligned. For the details of this instrumented cylinder model, the reader is referred to Yuan and Huang (2010) and Yuan and Huang (2011), who used the same cylindrical force model in their study of hydrodynamic forces on a cylinder fixed in regular waves or oscillating transversely in regular waves.
Each individual component in the experimental system was estimated to have a nominal error bound less than 3 %. Since the overall system error bound cannot be known as a priori, the reliability of the test results can only be assured by comparing our force measurements with some theoretical values and experimental data published in the literature.
2.2 Data analysis
2.3 Verification of the integrated cylindrical force model
2.4 Test conditions
Test conditions
Test code  \(T_L\) (s)  \(T_H\) (s)  \(A_H\) (cm)  \(A_L\) (cm)  \(\beta _L\)  \(\beta _H\)  \({KC}_H\)  \({KC}_L\) 

(a)  4.0  1.2  3.25  3.90\(\)10.40  625  2083  4.1  4.9\(\)13.1 
(b)  4.0  1.2  3.90  3.90\(\)9.15  625  2083  4.9  4.9\(\)11.5 
(c)  4.0  1.6  3.90  3.90\(\)10.40  625  1563  4.9  4.9\(\)13.1 
(d)  4.0  1.6  4.55  3.90\(\)10.40  625  1563  5.7  4.9\(\)13.1 
(e)  6.0  1.2  3.25  5.17\(\)14.30  417  2083  4.1  6.3\(\)18.0 
(f)  6.0  1.2  3.90  3.90\(\)14.30  417  2083  4.9  4.9\(\)18.0 
(g)  6.0  1.6  3.90  3.90\(\)14.30  417  1563  4.9  4.9\(\)18.0 
(h)  6.0  1.6  4.55  3.90\(\)12.97  417  1563  5.7  4.9\(\)16.3 
(i)  8.0  1.2  3.25  3.90\(\)18.20  313  2083  4.1  4.9\(\)22.9 
(j)  8.0  1.2  3.90  3.90\(\)15.59  313  2083  4.9  4.9\(\)19.6 
(k)  8.0  1.6  3.90  3.90\(\)18.20  313  1563  4.9  4.9\(\)22.9 
(l)  8.0  1.6  4.55  3.90\(\)18.20  313  1563  5.7  4.9\(\)22.9 
3 Results
In this section, we will first present wave force coefficients as functions of (\({KC}_L\), \({KC}_H\), \(\beta _L\), \(\beta _H\)). Empirical formulas for the wave force coefficients will be proposed in Sect. 4 when we discuss our results.
3.1 Effects of highfrequency motion on wave force coefficients for lowfrequency motion
3.2 Effects of lowfrequency motion on wave coefficients for highfrequency motion
4 Discussion
When using the independentflow form of Morison’s equation to model the hydrodynamic force acting on a cylinder moving in regular waves, our results showed that it was not appropriate to simply use the addedmass and drag coefficients obtained for a cylinder oscillating with a singlefrequency in still water. For dualfrequency oscillations, a simple dimensionless analysis shows that the wave force coefficients depend on four dimensionless parameters. It is desirable to find a single dimensionless parameter for each wave force coefficient so that a simple empirical equation can be proposed for each wave force coefficient.
 1.
A suitable parameter for \(C_A^H\) is \(V_R\).
 2.
A suitable parameter for \(C_D^H\) is \(V_R+{KC}_H\).
 3.
A suitable parameter for \(C_A^L\) and \(C_D^L\) is \(V_R{KC}_H/2\).
5 Conclusions

A suitable parameter for the addedmass coefficient of the highfrequency component is the reduced velocity.

A suitable parameter for the drag coefficient of the highfrequency component is the sum of the reduced velocity and the Keulegan–Carpenter number for the highfrequency component.

A suitable parameter for both the addedmass and drag coefficients of the lowfrequency component is the reduced velocity minus one half of the Keulegan–Carpenter number for the highfrequency component.
Notes
Acknowledgments
The second author would like to thank the Ministry of Education, Singapore, for an AcRF Tier 1 grant (RG07/7).
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