Hydrodynamics of the Semi-Immersed Cylinder by Forced Oscillation Model Testing

In this paper, the hydrodynamic coefficients of a horizontal semi-immersed cylinder in steady current and oscillatory flow combining with constant current are obtained via forced oscillation experiments in a towing tank. Three nondimensional parameters (Re, KC and Fr) are introduced to investigate their effects on the hydrodynamic coefficients. The experimental results show that overtopping is evident and dominates when the Reynolds number exceeds 5×105 in the experiment. Under steady current condition, overtopping increases the drag coefficient significantly at high Reynolds numbers. Under oscillatory flow with constant current condition, the added mass coefficient can even reach a maximum value about 3.5 due to overtopping while the influence of overtopping on the drag coefficient is minor.


Introduction
Severe environments may threaten the safety of the floating collar of the fish cage in the deep and open sea, exposed to a combination of waves and currents. The floating collar can be regarded as a combination of many circular cylinders, and accurate prediction of the hydrodynamic loads of floating cylindrical structures has become one of the key issues for the structural design.
For fully immersed cylinders, Morison equation (Morison et al., 1950) has been widely used to estimate its wave force. Fourier series approach (Keulegan and Carpenter, 1956) is widely used in determining the drag and inertia coefficients of a fully immersed cylinder. Mercier (1973) measured the transverse force, drag force and mass force of the oscillating cylinder in still water to obtain the drag coefficient C D and added mass coefficient C M , which had a good agreement with the results of Keulegan and Carpenter (1956). Sarpkaya (1976) found that KC and Re were the key parameters that have a great influence on the drag and inertia coefficients in the oscillatory flow.
The pioneering theoretical work related to wave in-duced effects on cylinders in free surface was made by Ursell (1949). He studied the heave added mass and damping of a semi-submerged circular cylinder, and presented a general expression of the potential flow. Vugts (1968) conducted a thorough experimental study on the two-dimensional hydrodynamic coefficients for horizontal cylinders in the free surface due to forced sway, heave and roll motion. Doynov (1998) used linearized radiation theory to compute the dynamics of free-floating horizontal cylinders subjected to waves, and then verified its accuracy through comparison against experimental results. Based on Morison equation accounting for the nonlinear characteristics of hydrodynamic forces, Li et al. (2007) obtained the hydrodynamic coefficients of the floating pipe and examined the hydrodynamic behavior of a floating straight pipe model under wave conditions. Kristiansen (2010) conducted a model test in the wave tank, with a fixed cylinder on the free surface and waves generated by a wavemaker. As overtopping occurred, he found that the second order component of the vertical force became very important. However, the values of KC in these experiments were much smaller than those in the real ocean circumstance. This paper will focus on the hydrodynamic characteristics of a semi-immersed cylinder through a forced oscillation experiment. Three non-dimensional parameters (Re, KC and Fr) are introduced to investigate their effects on the drag coefficient and the added mass coefficient. Hydrodynamics coefficients of the semi-immersed cylinder in steady current are compared with those of a fully immersed cylinder. The effects of overtopping on hydrodynamic coefficients at high Reynolds numbers would be particularly emphasized.

Experimental method
Owing to the limited capacity of the wave maker in the laboratory, it is almost impossible to obtain as large KC as in the real sea state via traditional wave making methods. Therefore, forced-oscillation experiments have often been conducted to obtain hydrodynamic coefficients for the cylinder in oscillatory flow (Chakrabarti, 1987). The obtained coefficients can then be applied to study the hydrodynamics of fixed cylinders in waves.
In this experiment, the forced oscillation serves to simulate the water particle motions around the cylinder in waves, and the forward towing of the cylinder will be used to simulate current. Based on the measured time histories of hydrodynamic force, the least square method is applied to extract the added mass and drag coefficient (Fu et al., 2013).
Three non-dimensional parameters (Re, KC and Fr) are introduced to investigate the hydrodynamic characteristics of the semi-immersed cylinder. The definitions are listed below: (2) where u max is the amplitude of the oscillatory velocity, U is the current velocity, D is the diameter of the cylinder, ν is the kinematic viscosity coefficient, T is the oscillating period, and g is the gravitational acceleration. If the cylinder diameter is constant, the Reynolds number will be only determined by (u max +U), which is the maximum relative velocity between the cylinder and water particles during a period. It can be deduced that overtopping will be evident and dominate when the Reynolds num-ber reaches a threshold. The parameter KC is introduced to reflect the influence of the oscillation period, since it will be only determined by T if (u max +U) is kept constant. As for the parameter Fr, when (u max +U) stays the same, a larger Fr means a higher U/u max , i.e. the ratio of the current velocity to the maximum oscillatory velocity.

Experimental apparatus
A series of experiments have been conducted in a towing tank, which is 192 m in length, 10 m in width and 4.2 m in depth. The experimental apparatus contains the towing carriage, the forced oscillation apparatus and a cylinder model, as shown in Fig. 1.
The horizontal length of the track is 3.5 m, and the diameter of the cylinder is 0.25 m. The 2 m-long smooth cylinder model is manufactured with polypropylene material and installed on the forced oscillation apparatus. The model makes a designed harmonic oscillation motion in the horizontal direction driven by a servo-motor on the forced oscillation apparatus. Three-component transducers are used to measure the forces on two ends of the cylinder. Oscillating displacements and velocities are measured by the encoders of the servo-motor. More detailed descriptions of the experiment can be referred to Fu et al. (2013).

Experimental cases
Experimental cases are listed in Table 1, and the relationship between KC and Re of the test case are plotted in Fig. 2.

Steady current
In this part, we mainly discuss the relationship between the drag coefficient and the Reynolds number for a semi-immersed cylinder.  The drag coefficient can be expressed as: is the hydrodynamic force per unit length in the horizontal direction and D is the characteristic length. For comparison with the results of a fully immersed cylinder, the diameter of the cylinder model, rather than its radius or draft, is taken as the characteristic length. Fig. 3 shows three curves of the drag coefficient against the Reynolds number. First of all, it can be found that the value of the drag coefficient for the fully immersed cylinder reaches almost 1.2 when Re<2×10 5 . Comparatively, the drag coefficient of the semi-immersed cylinder starts decreasing at approximately Re=5×10 4 . Besides, the drag coefficients of the semi-immersed cylinder are less than half of those of the fully immersed cylinder when the Re number ranges from 6×10 4 to 3×10 5 . It has already been known that the drag coefficient is mainly affected by the vortex shedding which has great influence on the pressure at the rear side of the fully immersed cylinder (Sumer and Fredsøe, 1997). Meanwhile, for a semi-immersed cylinder, the formation of the vortex-shedding will be greatly affected by the disturbed free surface. For the case of a semiimmersed cylinder, the free surface has a similar effect as an infinitely long splitter plate which causes a clear reduction of the drag coefficient (Faltinsen, 1993).
As the Reynolds number increases (i.e., Re>3×10 5 ), the drag coefficients will be larger than half of the ones of the fully immersed cylinder. Moreover, the drag coefficients of the semi-immersed cylinder keep increasing and become larger than those of the fully immersed cylinder when Re>5×10 5 . The overtopping phenomenon is quite remarkable in this region, which may account for the rising tendency of the drag coefficient. Test pictures of a case with overtopping and a case without overtopping are shown in Fig. 4. Based on the experimental results and the reason previously stated, we take the Reynolds number as a key parameter for overtopping. The overtopping phenomenon is evident and has a dominant effect when Re >5×10 5 .

Oscillatory flow combined with constant current
In this part, we mainly discuss the influences of parameters Re, KC and Fr on the hydrodynamic coefficients of the semi-immersed cylinder under the oscillatory flow combined with constant current. There are totally four current velocities in the experiment, i.e., 0, 0.4 m/s, 1.2 m/s, and 2.0 m/s, and the corresponding values of Fr are 0, 0.255, 0.766 and 1.277, respectively. For each Fr, the drag coefficient C D or the added mass coefficient C M can be regarded as a function of Re and KC, which is shown in Fig. 5 and Fig. 7, respectively. In these figures, the black circular points denote the experimental cases illustrated in Table 1. The hydro- Fig. 2. Test cases in the experiment. Fig. 3. Relationship between the drag coefficient and Reynolds number for the semi and fully immersed cylinder. dynamic coefficients varying with Re or KC for different Fr are shown in Appendix.
In the experiment, overtopping cannot be obviously observed when Re<5×10 5 , while it is evident when Re>5×10 5 . For all cases of Fr=0 and 0.255 in the experiment, the Reynolds number does not exceed 5×10 5 , while it exceeds 5×10 5 for most cases of Fr=0.766 and all cases of Fr=1.277.

Drag coefficient
The contours of the drag coefficient of the semi-im-mersed cylinder are shown in Fig. 5. For Fr=0 (no current), the drag coefficient decreases with Reynolds number to a value of about 0.25 and then increases. The trend is similar to that of a fully immersed cylinder in oscillatory flow (Sarpkaya, 1976 The drag coefficients with Reynolds numbers of 3×10 5 and 6×10 5 for different Fr are presented in Fig. 6. For both values of Reynolds number, a larger value of Fr leads to a smaller drag coefficient. Since (u max +U) is kept constant, a larger value of Fr indicates a larger component of the current velocity. It can be concluded that the drag coefficient decreases due to the current. A similar conclusion has been drawn by Fu et al. (2013).

Added mass coefficient
The contours of the added mass coefficient of the semi-immersed cylinder are shown in Fig. 7. For Fr=0 and Fr=0.255, the added mass coefficient increases with Re and  SONG Chun-hui et al. China Ocean Eng., 2018, Vol. 32, No. 1, P. 110-116 decreases with KC. However, for Fr=1.277, an opposite trend can be found. In addition, the maximum value of the added mass coefficient reaches about 3.5 for Fr=1.277, which is much larger than those for Fr=0, 0.255 and 0.766, indicating that overtopping has a significant effect on the added mass coefficient at a large Fr. The added mass coefficients at Reynolds numbers of 3×10 5 and 6×10 5 for different Fr are presented in Fig. 8

. For
Re=3×10 5 , a larger Fr leads to a smaller added mass coefficient. However, for Re=6×10 5 , the added mass coefficient for Fr=1.277 is much larger than that for Fr=0.766. For Fr=1.277, U/u max is larger and U/u max >1, indicating that the water particle velocity is always along a fixed direction, so the overtopping phenomenon lasts longer and is more obvious during a period. Moreover, the least square method offers us an average value rather than the instantaneous value.

Concluding remarks
In this paper, the hydrodynamic coefficients of a semiimmersed cylinder are obtained by forced oscillation experiment. Three parameters Re, KC and Fr are introduced and their physical meanings are clarified. The experimental results show the overtopping phenomenon is evident and dominates when the Reynolds number exceeds 5×10 5 . Under the steady current condition, the drag coefficient of the semiimmersed cylinder unexpectedly surpasses that of the fully immersed cylinder at high Reynolds numbers due to the overtopping phenomenon. Under the oscillatory flow combining with constant current condition, overtopping has a significant effect on the added mass coefficient but the influence on the drag coefficient is minor. The maximum added mass coefficient reaches a value about 3.5 in the experiment, which should raise people's concern in the design of the floating collar.