Study of interaction between wave-current and the horizontal cylinder located near the free surface

Applied Ocean Research 67 (2017) 44–58

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Study of interaction between wave-current and the horizontal cylinder located near the free surface Junli Bai a,b,c , Ning Ma a,b,c,∗ , Xiechong Gu a,b,c a b c

State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai, China Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration (CISSE), Shanghai, China School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, China

a r t i c l e

i n f o

Article history: Received 20 November 2016 Received in revised form 21 June 2017 Accepted 22 June 2017 Keywords: Wave-current-cylinder interaction Submerged horizontal cylinder Wave reflection Wave blockage

a b s t r a c t In this paper, the authors experimentally and numerically study the interaction between wave-current and the horizontal cylinder near the free surface. The experiments are conducted in the Circulating Water Channel with varying axis depths of cylinder. Considering the same wave height and different wave lengths, two regular waves are generated combined with current using a flap-type wave maker. Also, the numerical model based on the RANS equations is solved by the finite volume method, in which the RNG k − ε model is adopted to simulate the turbulence while the VOF method is used to capture the free surface. In this study, the free surface deformation due to wave reflection and blockage is investigated firstly. Then, the typical features of the wave-current force on the cylinder with various axis depths are studied. The peak values of the force are also discussed by comparison with those calculated by the modified Morison’s equation. Besides, the vorticity field around the fully submerged cylinder is discussed in detail. It is found that the wave-current force value is affected by both wave reflection and wave blockage under lower cylinder submergence. To be detailed, the force value increases due to wave reflection while decreases because of wave blockage. With regard to the partially submerged cylinder, wave reflection plays a dominant role compared with wave blockage. Therefore, the measured force value is larger than the theoretical one by modified Morison’s equation. However, for the fully submerged cylinder, its wave blockage is more notable in contrast with dramatically reduced wave reflection, resulting in a lower measured force value compared with theoretical one. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction In real seas, wave-current interaction is a common phenomenon. So offshore structures in operation most probably encounter these combined real seas. While for offshore structures, horizontal cylinders such as offshore pipelines and oil platforms are popular components and will interact with such seas. Therefore, the research on its interaction with wave-current is of great engineering significance to the design of offshore structures. Over the past years, extensive investigations have been made concerning hydrodynamic forces on horizontal cylinders. For totally submerged horizontal cylinder, the well known Morison’s equation (Morison et al. [1]) has been widely used in calculating wave forces. By introducing a varying immersed volume of cylin-

∗ Corresponding author at: State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai, China. E-mail address: [email protected] (N. Ma). http://dx.doi.org/10.1016/j.apor.2017.06.004 0141-1187/© 2017 Elsevier Ltd. All rights reserved.

der and considering the buoyancy effect, Dixon et al. [2] modified Morison’s equation to the calculation of the force on partially submerged horizontal cylinder. However, in this modified Morison’s equation the interaction between cylinder and wave effects, such as wave reflection or wave blockage, was not taken into account. Moreover, the effect of wave steepness was also ignored in calculating the immersed volume of the cylinder. In this way, the modified Morison’s equation would become inapplicable, and this results in bad agreements between the theoretical forces value and measured ones as the wave height and wave steepness increased. For partially submerged horizontal cylinder, Chen et al. [3] developed a numerical model, which demonstrated that the maximum relative errors between their numerical results and the modified Morison’s equation went up to 50% for both horizontal and vertical wave forces. This meant that the interaction between wave and the horizontal cylinder close to the free surface was complicated. Therefore it is necessary to further investigate this issue, especially when the current is taken into consideration.

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In recent years, several studies have been conducted regarding the interaction between wave and the horizontal cylinder close to the free surface. Oshkai and Rockwell [4] used particle image velocimetry (PIV) to investigate the patterns of vorticity in the interaction between regular wave and a submerged horizontal cylinder. The effect of the submergence depth of cylinder was discussed. This study revealed the patterns of the vorticity around the cylinder in detail. Koh and Cho [5] performed fully nonlinear potential flow simulations of the inviscid drift force caused by nonlinear waves on a submerged horizontal cylinder. It indicated that both the magnitude of negative drift force and the higher-harmonic amplitudes averaged over the transmitted wave region became larger with the decrease of submergence depth of the cylinder. Bihs and Ong [6] and Ong et al. [7] performed two-dimensional (2D) numerical simulations using of the Unsteady Reynolds-Averaged Navier-Stokes equations to investigate the flows past partially submerged circular cylinders in free surface waves. The free surface elevations and vorticity fields around the horizontal cylinders have been investigated. Bozkaya and Kocabiyik [8] conducted 2D numerical simulation to investigate the free surface wave interaction with an oscillating cylinder. The effect of the submergence depth on the lift coefficient and the corresponding equivorticity patterns are discussed in detail. According to the simulated results, two new locked-on states of vortex formation were observed in the near wake region. In recently years, numerical studies about solitary wave (Xiao et al. [9]) and focused wave (Gao et al. [10]) interaction with horizontal cylinders near the free surface have been carried out by solving the Reynolds-averaged Navier-Stokes (RANS) equations. For partially submerged cylinder, the free surface tracking is one of the most important steps in RANS simulation of its interaction with wave. To do this work, some typical methods have been employed, such as the volume of fluid method (Xiao et al. [9]), level set method (Ong et al. [7]) and SPH method (Wen et al. [11]). However, for the complex interaction between wave and cylinder, problems remain, such as the disruption of the free-surface and airwater mixing, adequate surface boundary conditions, as described by Brocchini and Peregrine [12] and Brocchini [13]. And these factors should be added in the equation for better simulation of the turbulence. Above researches show that the submergence depth of cylinder is an important parameter in the interaction between wave and the horizontal cylinder close to the surface. Also, so does the interaction between steady current and cylinder near the free surface. Recently, several studies involving the influence of submergence depth on the interaction between current and horizontal cylinder have been carried out. Liang et al. [14] investigated the force and vortex shedding generated by a 2D horizontal cylinder beneath a free surface using the mesh-free viscous discrete vortex method. As the horizontal cylinder approached the free surface, it was found that the center line of the wake vorticity moved downwards and diffusion occurred for a low gap ratio. The effect of the gap ratio on drag coefficient and lift coefficient were discussed in this study. Ozdil and Akilli [15] investigated the flow characteristics around a horizontal cylinder located at different elevations between bottom and free surface using PIV technique. Reichl et al. [16] performed numerical simulation to study the 2D flow past a cylinder close to a free surface at a Reynolds number of 180. Based on the simulation results, the wake behaviors for gap ratios between 0.1 and 5.0 were investigated. Lin and Huang [17] applied a Lagrangian numerical framework to study 2D free surface flow induced by a submerged moving cylinder. A series of computations were carried out to investigate the effects of Froude number, the depth of submergence and still water depth on the free-surface deformation and the wake formation. Obviously, the submergence depth can exert significant influences on the interaction between wave-current and the horizontal

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cylinder close to the surface. However, previous researches on this issue have concentrated mostly on deeply submerged cylinders, while the effects of wave reflection and blockage are usually neglected. To investigate the forces on horizontal cylinders in waves and currents, Chaplin and Subbiah [18] conducted some experiments in which the still water depth of the water channel was 5 m and the axis depth of the cylinder was more than 1.75 m. Then Venugopal et al. [19] carried out an experimental investigation to measure the combined wave and current loads on horizontally submerged square and rectangular cylinders. Their experiments were conducted at a water depth of 2.2 m and the center of the cylinder was located at 0.47 m from the still water level. Furthermore, Li and Lin [20] developed a two-dimensional numerical tank to compute the hydrodynamic forces and coefficients induced by waves and currents on a submerged cylinder. The drag and inertia coefficients induced by waves and wave-current flows were compared and the effect of current on the coefficients was discussed as well. Later, a 2D fully nonlinear numerical wave flume was developed by Ning et al. [21] to investigate higher harmonics induced by a submerged horizontal cylinder in the presence of uniform current. The effects of current and cylinder’s submergence on harmonics waves were discussed in detail, proving that the second free harmonics were significantly enhanced by the opposing current and small submergence. The horizontal cylinders used in above researches were all fully submerged in water. According to the studies on the interaction between wave (or current) and cylinder, axis depth becomes an important parameter only under the condition that the cylinder is close to the free surface. Nevertheless, few efforts have been made so far to investigate the interaction between wave-current and the horizontal cylinder near the free surface. Therefore, a deeper understanding is needed to learn how the axis depth affects the interaction between wave-current and horizontal cylinder. In this paper, the interaction between wave-current and a horizontal cylinder near the free surface is experimentally and numerically studied. Meanwhile, the influences of wave reflection and blockage are concerned on the interaction between wavecurrent and cylinders with varying axis depths. Also, the free surface deformation, wave-current forces with various axis depths and the vortex field around the cylinder are discussed in detail.

2. Experimental arrangement The experiments are carried out in a Circulating Water Channel at Shanghai Jiao Tong University. For this channel, the length and height are 24.6 m and 7.4 m respectively. And its working section is 8.0 m in length and 3.0 m in width, and is operated with a still water depth of 1.6 m. Besides, a oscillating flap type wave maker is located at the upstream of the channel, while a honeycomb type wave absorber is positioned at the another end. The wave absorber can absorb the energy of incident waves and allow uniform current to pass through with very low energy loss. The overview of the experimental arrangement is provided in Fig. 1. With a diameter of 0.06 m and a length of 2.98 m (0.02 m shorter than the width of the channel), the cylinder used in this experiment is made of aluminum and positioned horizontally near the free surface, extending along the entire width of the channel. In addition, the distance between the cylinder and the wave maker is 4.0 m. The cylinder is fixed on a steel bracket that is bolted to a sixaxis load cell, while the load cell is fastened on the support system located above the wave channel. Fig. 2 presents the schematic diagram of the experimental arrangement. The distance between the center of the cylinder and the still water line is defined as the axis depth, d. A resistance wire wave gauge is located at the distance of 0.3 m from the cylinder in the experiment.

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J. Bai et al. / Applied Ocean Research 67 (2017) 44–58 Table 1 Experimental conditions. (C = 0.2m/s, D = 6 cm and H = 3 cm for all the cases; d’: Relative axis depth, L’: Relative wavelength).

Fig. 1. Experimental scene in the Circulating Water Channel.

Case

d (cm)

d’ = d/D

SC0 SC1 SC2 SC3 SC4 SC5 SC6 SC7 SC8 SC9 LC0 LC1 LC2 LC3 LC4 LC5 LC6 LC7 LC8 LC9

0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9

0 0.167 0.333 0.5 0.667 0.833 1.0 1.167 1.333 1.5 0 0.167 0.333 0.5 0.667 0.833 1.0 1.167 1.333 1.5

L’ = L/D Partially 12.75 submerged (H + D)/2 − d > 0 Fully submerged (H + D)/2 − d < 0 Partially 26.0 submerged (H + D)/2 − d > 0 Fully submerged (H + D)/2 − d < 0

Kc number 3.90 3.78 3.67 3.56 3.46 3.38 3.29 3.22 3.15 3.08 3.43 3.39 3.35 3.31 3.27 3.23 3.20 3.16 3.13 3.09

case SC0, SC1, SC2 is 1.875%, 2.5% and 3.125%, respectively. For cases SC3–SC9 the blockage ratio is fixed at 3.75%. 3. Theory and model validation 3.1. Numerical model Fig. 2. Schematic diagram of the experiment of wave-current interaction with horizontal cylinder at various axis depths and definition of principal parameters (H, L and T are wave height, wave length and period of the undisturbed incident wave, respectively).

The capacity of the six-axis load cell is 70 N and 7 Nm. In this experiment all the three forces and three moments are measured simultaneously, whereas only the data of horizontal and vertical forces, Fx and Fz, are used in this study. The force along the width direction of the channel, Fy, is almost zero and the three moments are irrelevant to this research. Moreover, the force data are collected at 200 Hz. The surface elevation data are collected by wave gauge at 100 Hz and the accuracy of the wave gauge is 0.5%. At the beginning of the experiment, uniform current is generated alone and lasts for a while to establish a steady flow. Then waves begin to be generated by the wave maker. The six-axis load has been calibrated by hanging known weights showing a linear behavior within 0.3% accuracy, and the wave gauge has been calibrated before the experiment. Table 1 shows the experimental conditions, in which the current velocity, C, is 0.2 m/s for all the cases. In this experiment, two kinds of waves with different wave lengths and same wave height were determined. The cases titled by SC are the shortest wave cases while LC means the longest wave cases. The relative wavelength, L’, and relative axis depth, d’, are dimensionless parameters obtained through dividing the cylinder diameter. L’ is 12.75 for shortest wave and 26.0 for longest wave. The axis depth ranges from 0 to 9 cm below the still water line, and the wave heights are 3 cm for all the cases. The Keulegan-Carpenter (Kc ) number is defined as Um T/D, where T is the wave period, D is the cylinder diameter, Um is the maximum particle velocity at the initial cylinder center and calculated from the undisturbed incident wave based on the second-order Stokes wave theory. Also, the blockage ratio (BR) is introduced to estimate the blockage effect. BR is defined as D/h for fully submerged cylinder and (d + D/2)/h for partially submerged cylinder, respectively. Consequently, the blockage ratio for

To investigate the free surface deformation and the flow field of the interaction between wave-current and the horizontal cylinder, two-dimensional numerical simulations were performed using the commercial software package FLUENT. The RANS equations are solved by the finite volume method (FVM), and used to calculate the velocities and pressure in the flow field. The governing equations are:

∂ui =0 ∂xi

(1)

du ∂p ∂  i =− + dt ∂xi ∂xj

  

∂ui ∂uj + ∂xj ∂xi

 +

∂ ∂xj



−ui uj



+ ␳gi (2)

where ui is the Reynolds averaged velocity component, u’i is the turbulence velocity, ␳ is the density of water, p is the pressure, ␮ is the dynamic viscosity and gi is the gravitational acceleration. Based on the eddy-viscosity assumption, the Reynolds stress term,−␳u’i u’j , can be calculated as: −␳u’i u’j = ␯t (

∂ui ∂uj 2 + ) − ␳␦ij k 3 ∂ xj ∂ xi

(3)

where ␯t is the eddy viscosity, ıij is the Kronecker delta functions and k is the turbulent kinetic energy: k=

1 ’ ’ uu 2 i i

(4)

In the present model the Reynolds stress term is modeled by the RNG k − ε turbulence model (Yakhot et al. [22]; Choudhury [23]):



∂ (k) ∂ (kui ) ∂ ∂k = ˛k ( + ␮t ) + ∂t ∂ xi ∂xj ∂xj +Gk + Gb −  ␧ − YM



(5)

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Fig. 3. The computational domain of the water channel.



∂ (␧) ∂ (␧ui ) ∂ ∂␧ = ˛␧ ( + ␮t ) + ∂t ∂xi ∂ xj ∂xj +C1␧

3.2. Validation of the numerical model



␧ ␧2 − R␧ (Gk + C3␧ Gb ) − C2␧ ␳ k k

(6)

k2 ␧

(7)

␮t = ␳C␮

where ␧ is the dissipation rate of the turbulent kinetic energy. Gb = 0 and YM = 0 as the water is incompressible. The expressions of Gk and Rε are



Gk = ␮t

∂ui ∂uj + ∂xj ∂xi





Rε = C1ε − where ␩ =

∂ui ∂ xj

␩ 1 − /␩0

(8)

(9)

1 + ␤␩3



2Eij Eij ␧k and Eij =

1 2



∂u ∂ui + j ∂xj ∂ xi

 . The values of other

constant terms in above equations are adopted as: C2ε = 1.68,

C␮ = 0.0845,

␩0 = 4.377,

␣k = ␣ε = 1.39,

␤ = 0.012,

C1ε = 1.42.

The spatial discretization chosen is the Second Order Upwind scheme. The pressure-velocity coupling is evaluated by using the SIMPLE (Semi-Implicit Method for Pressure Linked Equations) algorithm described by Patankar [24]. The volume of fluid (VOF) method is used to track the free water surface. The diagram of the computational domain of the wave channel is shown in Fig. 3. This computation domain is built based on reasonable simplification of the real Circulating Water Channel. The origin of coordinates is at the center of horizontal cylinder when it is half submerged. The depth of still water, d, is 1.6m, which is same with the value in the experiment. The dynamic grid technique is used to simulate the motion of the wave maker, and the incident regular waves are traveling from left to right. According to the real Circulating Water Channel, the wave absorber is modeled by several parallel horizontal rigid plates with no thickness. Uniform velocity boundary conditions with C = 0.2 m/s are applied at inlet and outlet. The boundary condition of pressure outlet is applied on the top of the computational domain, and the no-slip boundary condition is applied on other solid boundaries. In the generation of regular waves propagating on the current, the angular velocity ␪˙ of the wave maker’s motion can be expressed as follows: S ␪˙ = ␻sin (␻t) 2

(10)

where S is the stroke of the wave maker, ␻ = 2␲/T is the angular frequency of the motion of wave maker and T is the period.

Based on the above numerical model, four cases (SC0, SC2, SC4 and LC0) in Table 1 are simulated. At the beginning of the simulation, the wave maker remains stationary and only the current is simulated. It will take about 60 s for the flow field to become steady. In order to ensure that the numerical simulations can exactly mimic the experiments, wave maker begins to oscillate and generate waves from t = 100 s for all the cases in this paper. The simulation results are compared with the measured data for the validation of the model. Fig. 4 shows the comparisons between the measured wave surface elevations and the simulation results at X = −0.3 m for cases SC0, SC2, SC4 and LC0, respectively. The origin of the horizontal coordinate is the time when wave maker begins to generate waves. The root-mean-square errors (RMSE) of these four cases are 2.12 mm, 2.71 mm, 4.42 mm and 5.22 mm, respectively. The irregular surface elevation in case SC0 is caused by the notable wave reflection due to the interaction between wave-current and the partially submerged cylinder. Fig. 5 shows the comparisons between the measured horizontal/vertical forces on the cylinder and the simulation results for cases SC0, SC2, SC4 and LC0, respectively. The simulated forces on the cylinder are obtained through integrating the pressure and shear force on the cylinder surface. The RMSE of the simulated Fx in cases SC0, SC2, SC4 and LC0 are 0.82 N/m, 1.24 N/m, 0.76 N/m and 0.63 N/m, respectively. The RMSE of the simulated Fz in cases SC0, SC2, SC4 and LC0 are 1.16 N/m, 1.38 N/m, 0.96 N/m and 1.32 N/m, respectively. Based on the comparisons presented in Figs. 4 and 5, the numerical model proposed in this study can deal with the simulation of the interaction between wave-current and the horizontal cylinder, and obtain reasonable results of the water surface elevations and hydrodynamic forces on the cylinder. 3.3. Theoretical model: modified Morison’s equation The theoretical wave forces on the horizontal cylinder can be calculated by the modified Morison’s equation. For the fully submerged cylinder the equations are (Morison et al. [1]): Fx = Fz =

1 ␳DCD ux u2x + u2z + ␳V0 CM ax 2

1 ␳DCD uz u2x + u2z + ␳V0 CM az 2

(11) (12)

For the partially submerged cylinder (Dixon et al. [2], Chen et al. [3]): Fx =

1 ␳CD Ax (t) ux u2x + u2z + ␳CM V (t) ax 2

Fz = ␳CM V (t) az + ␳g [V (t) − V0 ]

(13) (14)

where Fx and Fz are the horizontal and vertical components of the wave-current forces; ux and uz are the horizontal and vertical components of the velocity of the undisturbed wave including the contributions due to the current; ax and az are the horizontal and vertical components of the acceleration of the undisturbed wave. The velocities and accelerations are all calculated at the position of the center of the cylinder if it is lower than the instantaneous water

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Fig. 4. Comparison of the water surface elevation at X = −0.3 m for cases SC0, SC2, SC4 and LC0.

surface, otherwise calculated at the water surface. The secondorder Stokes wave theory is adopted to calculate the velocity and acceleration. Ax (t) is the instantaneous immersed projected area of the cylinder in the horizontal direction; V (t) and V0 are the instantaneous and initial immersed volumes of the cylinder. The effect of wave steepness is neglected when Ax (t) and V (t) are estimated. CD and CM are the drag and inertia coefficients. Similar to the treatment in previous study (Chen et al. [3]), the widely used values of CD = 1.2 and CM = 2.0 are adopted in this paper. For convenience of analysis, the wave forces are normalized by the static buoyancy of the fully submerged cylinder. 4. Results and analysis 4.1. Free surface deformation in the interaction between wave-current and horizontal cylinder It is known that wave reflection and wave blockage will occur due to the existence of cylinder. When the cylinder is deeply submerged in water, the flow can pass smoothly through the areas above and below the cylinder. Therefore, both the wave reflection and blockage are weak. However, when the cylinder gets closer to the free surface, the gap between the free surface and the cylinder become narrower and the effects of wave reflection and blockage turn out to be obvious. Both wave reflection and blockage can give rise to the free surface deformation, which can be observed easily through comparing the surface profiles at the upstream and downstream of the cylin-

der. Fig. 6 illustrates the comparison of the simulated time series of the free surface elevation at a distance of −2D (upstream) and 2D (downstream) from the cylinder for cases SC0, SC2, SC4 and LC0, LC2, LC4. Furthermore, Fourier analysis on the time series of the free surface elevation is conducted and the comparison of the amplitude spectra of the surface elevation for cases SC0, SC2, SC4 and LC0, LC2, LC4 is given in Fig. 7. Firstly, the cases SC0, SC2 and SC4 with shortest wave are discussed. Based on the comparison results in Figs. 6 and 7, the free surface elevations in the upstream and downstream of the cylinder are quite different in partially submerged cases (SC0 and SC2). Specifically, the wave heights in the upstream of the cylinder are much larger than that those in the downstream implying the occurrence of notable wave reflections in these cases. The wave reflection coefficients of cases SC0 and SC2 are 0.78 and 0.61, respectively. However, it can be seen clearly from Fig. 6 that the wave reflection reduces dramatically in case SC4. It is known that the cylinder in SC4 is completely submerged in still water initially and only a few parts of its top surface emerge when wave troughs are passing through it, indicating that the wave reflection decreases rapidly when the cylinder is completely submerged in still water. The wave reflection coefficient of case SC4 is reduced to 0.05. Subsequently, cases LC0, LC2 and LC4 with the longest wave are discussed. From Fig. 7(a), both the upstream wave amplitudes of cases LC0 and LC2 are obviously lower than those of cases SC0 and SC2, suggesting that the wave reflection in cases with the longest wave is lower than that in cases with the shortest wave when the

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Fig. 5. Comparison of the horizontal forces, Fx, and vertical forces, Fz, on the cylinder for cases SC0, SC2, SC4 and LC0.

cylinder is partially submerged. However, the wave amplitude in LC4 is larger than that in SC4. Therefore, the wave reflection in LC4 is larger than that in SC4, which means that the decrease of wave reflection with the drop of cylinder in cases with the longest wave is slower than that in cases with the shortest wave. In fact, the reflection coefficients of cases LC0, LC2 and LC4 are 0.49, 0.42 and 0.31, respectively.

The wave profile at X = −2D (upstream) is the superposition of incident wave and reflected wave. As is known, wave reflection is dominated by first order thus the composition of incident and reflected wave are almost monochromatic. Fig. 6 demonstrates that the wave profiles at X = −2D are similar to the sinusoidal curves. However, when the waves pass over the cylinder wave profiles at X = 2D change significantly compared with that at X = −2D due

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Fig. 6. Comparison of the surface elevation in upstream (x = −2D) and downstream (x = 2D) of the cylinder for cases SC0, SC2, SC4 and LC0, LC2, LC4.

to the wave-cylinder interaction, especially for the partially submerged cases. The existence of the cylinder causes wave blockage and changes the flow field around the cylinder remarkably. From Fig. 7(b), the downstream wave amplitudes in longest wave cases are all larger than that in shortest wave cases. This is because the longer the wave length is, the easier the wave passes over the horizontal cylinder. Next, the free surface deformation in a typical partially submerged case is discussed in detail. Fig. 8 shows the typical free surface profiles of case SC1 in one wave cycle. The red lines and

yellow lines denote the upper and lower limits of the surface elevations in the upstream and downstream of the cylinder, respectively. According to Fig. 8 the wave height in the upstream of the cylinder is much larger than that in the downstream. This behavior is similar to that observed in Fig. 6. Fig. 8(a) demonstrates the instantaneous free surface profile corresponding to the instant when the wave trough just passes the cylinder. Based on Fig. 8(a), water line on left side of the cylinder surface attains its minimum elevation. After that, the wave surface begins to rise and then the reflected wave is generated at the lower

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Fig. 7. Comparison of the amplitude spectra of the surface elevation for cases SC0, SC2, SC4 and LC0, LC2, LC4: (a) upstream, x = −2D, (b) downstream, x = 2D.

left surface of the cylinder, which can be found in Fig. 8(b). The reflected waves are marked by the red arrows in Fig. 8(b). Fig. 8(c) and (d) obviously reveal the propagation of the reflected wave. Fig. 8(e) shows the instantaneous free surface profile corresponding to the instant when the superimposed wave in the upstream of the cylinder attains its maximum elevation. It is shown in Fig. 8 that when the cylinder is partially submerged, the water line on the left side of the cylinder surface is usually not equal to that on the right side because of the wave steepness. Therefore, there is a height difference between the two water lines. Besides, the wave reflection and blockage change the wave heights upstream and downstream of the cylinder, thus leading to larger height differences. According to Fig. 8, there is about one wave-height difference between the two water lines and in Fig. 8(a) this difference attains. Obviously, the height difference will cause additional force and affect the peak values of the force on the cylinder.

4.2. Wave-current forces on horizontal cylinder In this section, the overall features of the wave-current forces in a wave cycle under different axis depths are considered. For convenience, the wave forces are normalized by the static buoyancy of the fully submerged cylinder. The measured horizontal components of wave-current forces, Fx, on the horizontal cylinder for various axis depths are shown in Fig. 9. In addition, the partially submerged cases and the fully submerged cases are presented respectively in Fig. 9(b) and (c) in order to facilitate the analysis. Meanwhile, the force of the fully submerged case SC4 is also shown in Fig. 9(b) for the need of comparing it with the forces given in Fig. 9(b) and (c). Similarly, the vertical components of the wave-current forces on the cylinder, Fz, are provided in Fig. 10. The forces in Figs. 9 and 10 are phase-averaged and based on the data of the first six steady waves, so that representative profiles of forces in a wave cycle can be yielded. All corresponding dimensionless free surface as well as

Fig. 8. Typical free surface profiles of case SC1 in one wave cycle (red lines: upper and lower limits of the surface elevation in the upstream of the cylinder; yellow lines: upper and lower limits of the surface elevation in the downstream of the cylinder; red arrows: reflected wave). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

horizontal and vertical acceleration components of the undisturbed wave are presented in Fig. 9(a). At first, horizontal and vertical forces in fully submerged cases are discussed. According to the analysis in previous section, the wave reflection in fully submerged cases is relatively weak, and thus the Morison’s equation is used to analyze the forces. In Morison’s equation, the forces are calculated as the sum of the drag force (the first term on the right side of Eqs. (11) and (12)) and inertial force (the second term on the right side of Eqs. (11) and (12)). Besides, the drag force and inertial force are determined by the velocity (ux , uz ) and acceleration (ax , az ) of the water particle,

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J. Bai et al. / Applied Ocean Research 67 (2017) 44–58

Fig. 9. Measured horizontal forces, Fx, on the horizontal cylinder for various axis depths, (b): partially submerged cases, (c): full submerged cases. (a): The corresponding dimensionless surface elevation, horizontal and vertical acceleration components of the undisturbed wave.

respectively. To a certain extent, Fx in Fig. 9(c) is in phase with ax , while Fz in Fig. 10(b) is in phase with az , indicating that the inertial force makes the dominant contribution to the loading, and the drag force is relatively small compared with the inertial force. From Fig. 10(b), the variation of axis depths exerts relatively small influence on the phase of the force in all fully submerged cases. Fx and Fz are appropriately symmetric, and such symmetry increases due to the smaller effects of reflection and blockage under deeper cylinder axis. Then, the horizontal forces on the cylinder in partially submerged cases are studied. As shown in Fig. 9(b), the horizontal forces are very sensitive to the slight changes in axis depths and the horizontal forces in all partially submerged cases are asymmetric. In Fig. 9(b), the positive peak value is about to occur when t = 0.08 T and the demonstration when the positive peak value occurs in case

SC1 is provided in Fig. 11(a). This figure indicates that the surface elevation of the superposition wave reaches its peak value during the propagation of reflected wave along the upstream direction. Also, the corresponding times when the negative peak values occur are gradually deviating forwards from case SC0 to SC3 with the drop of the cylinder depth according to Fig. 9(b). Subsequently, the vertical forces in partially submerged cases (Fig. 10(a)) are investigated. Different from the horizontal force, the vertical force is determined by the combination of varying immersed volume effect and inertial force. As illustrated in Fig. 8, the increase of buoyancy results from the rise of immersed volume when a wave crest passes, while the decrease of buoyancy is caused by the drop of immersed volume when a wave trough passes. From Fig. 10(a), the vertical force in case SC0, which can be defined as water-surface-type force, is approximately in phase with the sur-

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Fig. 10. Measured vertical forces, Fz, on the cylinder for various axis depths. (a): partially submerged cases, (b): full submerged cases.

Fig. 11. Free surface of case SC1 ((a): the instant corresponding to the positive peak value of horizontal force occurs; (b): the instant corresponding to the negative peak value of the vertical force occurs).

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Fig. 12. Definition of the positive peak value, Fx+, and negative peak value Fx−, of the dimensionless horizontal force.

face elevation, meaning that buoyancy plays a dominant role in vertical force. In Fig. 10, the phases of fully submerged cases represented by cases SC4–SC9 are approximately opposite to that of case SC0. According to previous analysis, buoyancy keeps constant during the entire wave cycle under fully submerged cases. As a result, the vertical force can be considered as acceleration-type force in fully submerged cases due to it being dominated by inertia force. Between the above cases there is a transition region within cases SC1 and SC3, in which the intermediate-type force occurs. In the intermediate-type force, both surface elevation and inertia force play important roles in the contribution of the vertical force. In the transition region, the influence of buoyancy decreases while that of inertia force increases on vertical force with the drop of cylinder. In addition, such features can also be recognized through the curves of vertical forces under different axis depths. According to Fig. 10, the vertical force curves gradually reverse from phase synchronization with free surface curves to phase synchronization with vertical acceleration curves as the cylinder drops (case SC0–case SC5). As Fig. 10(a) shows, the absolute values of negative peaks are larger than those of positive peaks for the vertical forces in partially submerged cases. Fig. 11(b) illustrates the free surface of case SC1 at the instant when negative peak value of vertical force occurs, suggesting that the vertical force reaches its minimum value when the cylinder is at the position near the wave trough because of the minimum buoyancy. The transition region also exists in both theoretical and experimental cases correspondingly. Compared with the theoretical vertical force, the trough value of measured vertical force is lower due to the lower water elevation caused by wave reflection, which indicates that the measured force in transition region is largely affected by the wave reflection. More discussion of peak values of the measured forces on the cylinder are discussed in the next section. 4.3. Peak values of the wave-current forces on the cylinder In this section, the peak values of the horizontal and vertical forces on the cylinder for various axis depths are taken into consideration. Based on the results in Figs. 9 and 10, the profiles of horizontal and vertical wave-current forces are all asymmetrical. Consequently, the positive peak value and negative peak value of the force are different. Therefore, it is necessary to investigate them separately. Fig. 12 shows the definitions of the positive peak value, Fx+, and negative peak value, Fx−, of the dimensionless horizontal force on the cylinder. Similar definitions are adopted for the positive peak value, Fz+, and negative peak value, Fz−, of the dimensionless vertical force. Fig. 13 presents the positive and negative peak values of the horizontal and vertical wave-current forces on the horizontal cylinder obtained from the measured force data for various axis depths. Besides, the theoretical values of the horizontal and vertical wave-current forces calculated by Eqs. (11)–(14) are also shown in Fig. 13 for comparisons.

Firstly, the forces in shortest wave length cases marked by SC are considered. When the cylinder is fully submerged (d’ < 0.677), all the peak values (Fx+, Fx−, Fz+, Fz−) increase gradually with the rise of the cylinder because the closer the cylinder is to water surface, the larger the velocity and acceleration of the water particles will be. As the rise continues, the cylinder is transferred into partially submerged state due to the emergence of its top surface. Obviously, both Fx+ and |Fx−| increase rapidly with the rise of cylinder attaining their maximum values simultaneously in case SC2. Then, Fx+ and |Fx−| will be reduced with the further rise of the cylinder due to the decrease of the immersed volume. Similarly, |Fz−| increases rapidly with the rise of cylinder and attains its maximum values at case SC1 and subsequently reduces with the further rise. However, the variation of Fz+ differs from other peak values performing to decrease previously and increase later with the rise of the cylinder due to the phase reverse of vertical force curves from case SC5 to SC0, which can be recognized from Fig. 10(a). As shown in Fig. 13, the variation of peak values of forces with the axis depth in longest wavelength cases marked by LC shows similar tendency with the shortest wave case. In general, the horizontal wave-current forces in shortest wave cases are larger than that in the longest wave cases because both the velocity and acceleration of water particles near the free surface in the shortest wave are larger than that in the longest wave. Additionally, due to the effect of the current, Fx+ is much larger than |Fx−| for both types. As is known, the effects of wave reflection and blockage on the force acting on the cylinder are not considered in Morison’s equation. However, the cylinder in this paper is located near the free surface contributing to significant wave reflection and blockage. The effect of wave reflection and blockage can be recognized in detail through comparative analysis between measured peak values of the forces and theoretical ones. Fig. 13 presents the peak values of the forces calculated by the modified Morison’s equation for comparison. Based on the comparison between the theoretical and measured peak values, it is demonstrated that the theoretical prediction underestimates the wave-current force for partially submerged cases but overestimates the wave-current force for fully submerged cases. This phenomenon can be explained as follows. The interaction between the wave-current and horizontal cylinder includes two parts, including wave reflection and wave blockage. On the one hand, the superposition of reflected wave and incident wave enhances the wave height, as well as the peak value of the surface elevation, along the upstream direction of the cylinder in comparison with undisturbed incident wave. According to Fig. 11(a), the peak value of the horizontal force on the cylinder occurs as soon as peak value of the surface elevation under superposition waves is attained, and it is known that the force on the cylinder rises with the increase of the wave height. Therefore, wave reflection will increase the positive peak value of wave-current force. On the other hand, the vertical force reaches its peak value when the wave trough passes. Since buoyancy plays a dominant role among the vertical force for partially submerged cases, |Fz−| rises due to the lower buoyancy which results from the decrease of surface elevation with the drop of wave trough. As shown in Fig. 11(b), reflection lowers the wave trough, thus leading to a larger |Fz−|. In addition, surface elevations are different between the waterlines for the cylinder of both left and right sides, and such difference will cause additional force and increase the force on the cylinder. The wave blockage restricts the water particles from free movements, thus contributing to the decrease of the velocity and acceleration of the water particles around the cylinder, thus reducing the wave force. As previously mentioned, the wave-current force on the horizontal cylinder is affected by the combined action of the wave reflection and wave blockage. For the partially submerged cases,

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Fig. 13. Positive and negative peak values of the horizontal and vertical wave-current forces on the cylinder for various axis depths (SC and LC represent measured data for shortest and longest wave cases, respectively; SCT and LCT represent the theoretical values calculated by Eqs. (11)–(14) for shortest and longest wave cases, respectively).

the effect of the wave reflection is more notable than the wave blockage. Consequently, the measured horizontal forces are larger than the theoretical predicted forces. However, for the fully submerged cases, the wave reflection reduces rapidly and the wave blockage becomes notable in comparison with the wave reflection. As a result, the measured forces are smaller in comparison with the theoretical forces. Above analysis shows the obvious influence of wave reflection and blockage on the hydrodynamic force acting on horizontal cylinder. A question that is of great concern in the engineering is whether it would be possible to further modify Morison’s equation to account for the effect of wave reflection and blockage. There are many factors that can affect wave reflection and blockage, such as the shape of the cylinder’s cross section, the relative submerged depth of the cylinder, the relative wave amplitude, the relative wave length, current velocity, vorticity field around the cylinder and so on. It is conceivable that a complex formulation that involves the above factors needs to be first established to predict the wave reflection and blockage theoretically. Then, the relationship between wave reflection/blockage and their influences on force should be established. Both research aspects above require large amount of further experiments regarding different combinations of cylinder and wave-current parameters. 4.4. Vorticity field Currents and waves are two major factors in wave-currentcylinder interaction. For fully submerged cylinder in steady flow, the vorticity field around the cylinder resembles that of the Karman vortex street, in which two single vortices alternatively shed

from the cylinder, as shown in Fig. 14. When the cylinder is exposed in waves, as per (Oshkai and Rockwell [4]), the orbital nature of the wave motion results in multiple sites of vortex development, i.e., onset of vortices, along the surface of the cylinder, followed by distinctive types of shedding from the cylinder. All of these vortices then exhibit orbital motion about the cylinder. However, when current and wave is considered simultaneously, the development of vortices around the cylinder shows some different features compared with those in wave case (wave-cylinder interaction) and current case (current-cylinder interaction). Fig. 15 shows the vorticity fields around the fully submerged cylinder in wave-current case SC6. It is observed from Fig. 15, there are two major vortices, positive vortex A and negative vortex B, formed along the surface of cylinder in one wave cycle. Two main differences have been observed on the development of the vortex between the wave case and wave-current case. In the wave case (Oshkai and Rockwell [4]), the positive vortex forms at right side of cylinder and moves to its left side during developing process, and then shed after a while; but in wave-current case, the formation and separation of positive vortex A are both limited in its right side due to the current effect. In addition to this, as shown in Fig. 15(f), a new positive vortex A1 is formed at the lower-right side of the cylinder, at the point, where vortex A is shedding. Similarly, a new negative vortex B1 is formed at the upper-right side of the cylinder, where vortex B is shedding as shown in Fig. 15(b). While, for the case of wave (Oshkai and Rockwell [4]), all the vortices move clockwise around the cylinder. However, the range of vortex motion in wave-current case is constrained in the downstream area of cylinder due to the current effect, as illustrated in Fig. 15.

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Fig. 14. Vorticity field around the fully submerged cylinder in steady current (C = 0.2 m/s, d’ = 1.0).

Fig. 15. Vorticity field around the fully submerged cylinder in wave-current for case SC6 (C = 0.2 m/s, T = 0.7 s, H = 3 cm, d’ = 1.0; a–h are results at the instant corresponding to t = 0 T, 0.25 T, 0.5 T, 0.75 T, 1.0 T, 1.25 T, 1.5 T, 1.75 T,).

Fig. 16 shows the comparison of vorticity fields for the case LC6. As shown in Table 1, KC numbers for LC6 and SC6 are almost equal. Hence, it is inferred that, the pattern of vortices are similar

for Fig. 16 and for Fig. 15. A pair of vortices, positive vortex A and negative vortex B, are formed along the bottom and top surfaces of cylinder, respectively, and shed alternately from the cylinder sur-

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Fig. 16. Vorticity field around the fully submerged cylinder in wave-current for case LC6 (C = 0.2 m/s, T = 1.0 s, H = 3 cm, d’ = 1.0; a–h are results at the instant corresponding to t = 0 T, 0.25 T, 0.5 T, 0.75 T, 1.0 T, 1.25 T, 1.5 T, 1.75 T,).

face. However, the vortex shedding frequency is 1 Hz for LC6, while it is 1.43 Hz (=1/0.7 s) for SC6. In addition to this, the scale of the vorticity concentrations in LC6 is slightly larger than those in SC6, because the vortices in LC6 have more time to develop. According to previous studies (Reichl et al. [16]), for the case of current-cylinder interaction, the vortex shedding frequency depends on the Reynolds number and varies slightly with the submergence depth. Nevertheless, it can be seen from Figs. 15 and 16 that, the vortices shed from the cylinder in every wave cycle. This implies that the vortex shedding frequency in the wave-currentcylinder interaction synchronizes with the wave frequency. In other words, although, the current velocity, C, and submergence depth, d, are same in cases SC6 and LC6, the vortex shedding frequencies are different due to the different wave periods in these two cases. Based on the simulation results, the vortex shedding frequency for the case of current-cylinder interaction, shown in Fig. 14 is about 0.63 Hz; which indicates that the time taken for the full development of vortices before shedding from the cylinder is 1.6 s (=1/0.63 Hz). However, in cases SC6 and LC6, the wave periods are

0.7 s and 1.0 s, respectively, both of which are less than 1.6 s. Thus, sufficient time cannot be guaranteed for the complete development of the vortex in wave-current-cylinder interaction case, since the vortex sheds in every wave cycle. Consequently, the range of the vortex pair shedding from the cylinder in wave-current-cylinder interaction case (Figs. 15 and 16) is much smaller than that in current-cylinder interaction case (Fig. 14). Based on the above analysis, the vortex shedding frequency is determined by the wave frequency; while the motion range of the vortex is dominated by the current in the interaction between wave-current and the fully submerged cylinder. In addition to this, it is emphasized that the vortex pattern around the cylinder is largely controlled by the Kc number. This analysis is based on the simulation results of cases with small Kc number. However, for cases with sufficiently high value of Kc number, more complex features of the development of the vortex will occur, which is beyond the scope of this study. According to the analysis in Section 4.2, inertia force, among the hydrodynamic forces on the cylinder, plays a dominant role

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in fully submerged cases. Moreover, the inertia force is considered to be proportional to the acceleration of the water particle, while the velocity of water particle at the vorticity concentration is smaller than that at the same location, when the water is regarded as inviscid fluid and no vortex occurs around the cylinder. Consequently, the acceleration of water particles around the vorticity concentration decreases during the oscillation motion of the water particles in waves. Therefore, the hydrodynamic force on the cylinder decreases due to the lower acceleration, which is the reason why the measured force in fully submerged cases (as shown in Fig. 13) is lower than the theoretical value. This is one of the mechanisms for wave blockage to decrease the hydrodynamic force. 5. Conclusions To conclude, the interaction between wave-current and a horizontal cylinder located near the free surface is experimentally and numerically studied in this paper. The free surface deformation, force profile and the peak values of the wave-current forces for cases with various axis depths are investigated. Through the comparison between measured wave-current forces and the theoretical forces of Morison’s equation, the effects of wave reflection and blockage on the wave-current force on the cylinder are revealed and the main conclusions of this paper are as follows: 1. The interaction between wave-current and the horizontal cylinder located near the free surface mainly includes two aspects, which are wave reflection and wave blockage. The wave reflection increases the wave height upstream of the cylinder, thus increasing the force acting on the cylinder. The wave blockage reduces the velocity and acceleration of the water particles around the cylinder, thus leads to the decrease of the force. In addition, for partially submerged cases with combination of the wave reflection and blockage, water heights are different between the waterlines for the cylinder of both left and right side. Such difference results in additional force and increases peak value of the force acting on the cylinder. 2. In comparison with wave blockage, the effect of wave reflection on the force is dominant in partially submerged cases. However, as the cylinder falls down and reaches full submergence condition, the wave reflection diminishes dramatically resulting in the relative prominence of the wave blockage effect. 3. The wave-current forces tend to be underestimated on partially submerged cylinders while overestimated on fully submerged cylinders through the modified Morison’s equation due to Conclusion 2. 4. When the cylinder is fully submerged, the vortex shedding frequency synchronizes with the wave frequency, and the motion range of the vortices is limited in the downstream area of the cylinder. The above conclusions are based on the results at small Keulegan-Carpenter number cases (KC < 4).

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