applied sciences
Article Hydrodynamic Forces of a Semi-Submerged Cylinder in an Oscillatory Flow
Haojie Ren 1,2, Shixiao Fu 1,2,*, Chang Liu 3 , Mengmeng Zhang 1,2, Yuwang Xu 1,2 and Shi Deng 4
1 State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, China; [email protected] (H.R.); [email protected] (M.Z.); [email protected] (Y.X.) 2 Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai 200240, China 3 Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218, USA; [email protected] 4 Department of Marine Technology, Norwegian University of Science and Technology, 7052 Trondheim, Norway; [email protected] * Correspondence: [email protected]
Received: 26 August 2020; Accepted: 10 September 2020; Published: 14 September 2020
Featured Application: The present work highlights a novel empirical formula to estimate both mean and fluctuating lift force on a semi-submerged cylinder under oscillatory flow. It greatly avoids the complicated and time-consuming hydrodynamic calculation problem and will be helpful to calculate the hydrodynamic force in the design of relevant marine structures, such as floating collar of fish farming.
Abstract: This work experimentally investigated the performance of hydrodynamic forces on a semi-submerged cylinder under an oscillatory flow. To generate the equivalent oscillatory flow, the semi-submerged cylinder is forced to oscillate in several combinations of different periods and amplitudes. The mean downward lift force was observed to be significant and the fluctuating lift forces show dominant frequency is twice that of oscillatory flow and amplitude that is the same as the mean lift force. Based on this main hydrodynamic feature, a novel empirical prediction formula for the lift forces on semi-submerged cylinder under oscillatory flow is proposed where the lift forces expression is proportional to the square of oscillatory flow velocity. This novel empirical formula directly assigns the fluctuating lift force with frequency twice of oscillatory flow and the amplitude that is the same as the mean lift force. This assignment of empirical lift force formula reduces parameters required to determine a dynamic lift force but is demonstrated to well predict the fluctuating lift force. The lift coefficient can reach 1.5, which is larger than the typical value 1.2 of the drag coefficient for a fully submerged cylinder with infinite depth. Moreover, relationships among hydrodynamic coefficients, Keulegan-Carpenter (KC) number, Stokes number and Froude number are studied. With the increase of KC number, the Froude number has a more significant influence on the distribution of hydrodynamic coefficients. As Froude number is increasing, the drag coefficient shows a nonlinear decay (KC < 20) but a linear increase (KC > 20), while the added mass coefficients show a nonlinear (KC < 20) and a linear (KC > 20) increase trend. The present work can provide useful references for design of the relevant marine structures and serve as the useful guideline for future research.
Keywords: semi-submerged cylinder; oscillatory flow; hydrodynamic force; free surface; empirical prediction
Appl. Sci. 2020, 10, 6404; doi:10.3390/app10186404 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 6404 2 of 17
1. Introduction Marine structures including oil containment booms, floating tunnel, pipelines and fish-farming float collars cannot avoid the effects of an oscillatory flow derived from the wave and relative structure motions. These marine structures can be viewed as a semi-submerged cylinder under oscillatory flow. The surface distortion and water fragmentation will result in a complicated response and hydrodynamic loads. This problem is still not fully understood and threatens the safety of the relevant marine structures. Thus, it is essential to study the hydrodynamic features of semi-submerged cylinder in an oscillatory flow for the marine structures design. The current research of flow past a cylinder contains mainly three categories including cylinders submerged in infinite depth water, cylinders near the free surface and cylinders partially submerged under the water surface. In the oscillatory flow, many focuses are on the problem of flow past a fully immersed cylinder with infinite depth. Morison et al., (1953) [1]; Morison et al., (1950) [2] first raised a formula to calculate the in-line (IL) component of hydrodynamic force, which is now known as the Morison formula. The drag coefficient (CD) and added mass coefficient (Cm) in the Morison formula were found to be respectively of value 1.7 and 0.4 through an experimental study of stationary rigid cylinders under wave actions. The values of CD and Cm for a cylinder under the action of waves were further determined by Keulegan (1958) [3] through Fourier analysis and it was found that the Stokes parameters have a significant influence on these coefficients. Mercier (1973) [4] measured the force in the transverse direction, drag force and added-mass force (inertia) of the fully immersed cylinder in the oscillatory flow. The drag coefficient and added mass coefficient were witnessed to have good agreements with results of Keulegan (1958) [3]. Justesen (1989) [5] and Sarpkaya (2006) [6] experimentally studied the effects of the Re number at low Keulegan-Carpenter (KC) numbers on a rigid stationary cylinder under this oscillatory flow. Moreover, Sarpkaya (1976) [7] carried out a more comprehensive experimental studies on the forces of cylinders under oscillatory flows in sinusoidal form that was generated through an oscillatory U-shaped tube. The results were systematically reported that the hydrodynamic coefficients closely related with the Re number, Keulegan-Carpenter (KC) number and the relative roughness of the cylinder surface. Moreover, Williamson (1985) [8] and Tatsumo and Bearman (1990) [9] conducted a comprehensive flow visualization of the flow past cylinder, which provided a better understanding of the underlying fluid mechanism. Therefore, the investigation of hydrodynamic loads on the fully immersed cylinder with infinite depth in an oscillatory flow is relatively comprehensive. Different from flow past the fully submerged cylinder with infinite depth, there are few studies focusing on cylinders near the free surface and cylinders partially submerged below the water surface in an oscillatory flow. As for close to free surface cylinders, the existing research mainly focuses on the steady flow field. Roshko (1975) [10] investigated the drag and transverse forces and discussed the relationships of the drag force and the lift force respectively with a submerged gap ratio. Bearman and Zdravkovich (1978) [11] further observed that the free surface can greatly influence the response frequency and the corresponding Strouhal number decreases quickly with the submerged gap ratio decreasing. Lei et al. (1999) [12] experimentally investigated a cylinder under steady flow and highlighted the strong effects of the boundary layer development on this lift force. Different from the experimental studies, computational fluid dynamics (CFD) provides more opportunities to perform comprehensive studies on the underlying mechanisms of complicated fluid-structure interactions (FSI) problems. Reichl et al. (1997) [13] and Sheridan et al. (1995) [14] conducted numerical simulation of cylinders near a free surface under steady flow and studied corresponding motion behavior. The jet state and metastable behavior were noted by these researchers, at the free surface under certain gap ratio and Froude (Fr) number. The configuration of a partially submerged cylinder under flow is an unavoidable and actually widely observed problem for float collars of the fish farming structures and numerous studies were conducted in the literature of the flow past float collars. For example, Triantafyllou and Dimas (1989) [15] studied the stability of a semi-submerged cylinder under flow, where results are influenced by the Fr number. At all downstream locations, convective instability Appl. Sci. 2020, 10, 6404 3 of 17 of the wake is also observed based on the computation of potential flow theory. However, potential flow theory precludes the effect of the free surface and fluid viscosity, which cannot be ignored for a partially submerged cylinder. As a result, it is questionable whether traditional diffraction and radiation theories will provide a complete answer for this problem. Kristiansen and Faltinsen (2008) [16] and Kristiansen (2010) [17] experimentally measured the sway and heave motion of a semi-submerged circular cylinder under action of waves. Fu et al. (2013) [18] experimentally investigated hydrodynamic loads of semi-submerged cylinders that are employed in fish farming systems. It is reported that the effects of free surface are critically important for the hydrodynamic coefficients including drag coefficients and added mass coefficients in the IL direction; the direction parallel to the oscillatory flow. However, hydrodynamic forces and coefficients in the cross-flow (CF) direction have not yet been revealed and considered into models and this relationship between hydrodynamic loads and Froude number has not been studied. Recently, Ren et al. (2019) [19] conducted experiments of the partially submerged cylinder with different submerged depths in steady flow. Surprisingly, the mean downward transverse force is witnessed to be significant, which catches more and more attentions from both academia and industry. The lift coefficient was found to be greatly affected by Froude number. However, the understanding of hydrodynamic load in the CF direction on the semi-submerged cylinder under the oscillatory flow is still blank and potential influence parameters were not studied comprehensively. Therefore, it is urgent to investigate the hydrodynamic features of the semi-submerged cylinder in an oscillatory flow, especially for the lift coefficient and providing a further understanding for relevant structural design. In this study, model tests of semi-submerged cylinders in an oscillatory flow were carried out in a towing tank. The rigid cylinder was enforced to oscillate harmonically under numerous combinations of periods and amplitudes to generate an equivalent oscillatory flow. The KC number was designed to range from 6.3 to 37.7. The Stokes number and corresponding Froude number varied from 5434.7 to 11,363.6 and from 0.12 to 1.55, respectively. The hydrodynamic force in both IL and CF directions and the displacement in the IL direction were measured. We also establish the method for identifying the drag and added mass coefficients in the IL direction. Based on the observations of the lift force, a novel empirical prediction method of lift force is first proposed. The features of hydrodynamic loads, the relationships between hydrodynamic coefficients and influential parameters are further studied and discussed. The remainder of this paper is organized as what follows. Section2 describes the experimental set-up, test arrangement and definition of the key parameters such as Froude number, Stokes number and KC number. In Section3, the identification of the drag and added mass coe fficient are presented. Then, main results including hydrodynamic load features, the relationships among the drag, added mass, lift coefficients of the semi-submerged cylinder and influential parameters as well as a novel empirical prediction method of lift force are discussed in Section4. The main conclusions and future work directions are summarized in Section5.
2. Experimental Setup
2.1. Test Apparatus The model tests are carried out at the towing tank located at Shanghai Ship and Shipping Research Institute. The main dimension of this towing tank is 192 m 10 m 4.2 m (length width depth). × × × × The test apparatus were mounted under the towing carriage, which consists of the vertical and horizontal tracks, semi-submerged cylinder and end plate. The semi-submerged cylinder model is made of polypropylene and can be regarded as a smooth rigid cylinder. The hydrodynamic diameter D and length L of the model are respectively 0.25 m and 2 m. The vertical tracks driven by serve motors are employed for adjustment of the height of cylinder to the semi-submerged state. The horizontal track drives the semi-submerged cylinder to oscillate with various amplitudes and periods. The axonometric sketch and the actual apparatus are respectively shown in Figures1 and2. Appl. Sci. 2020, 10, 6404 4 of 17
Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 18 Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 18
Horizontal Track Horizontal Track
Vertical Track Vertical Track Cylinder End Plate Dummy Cylinder End Plate Dummy O KC O KC
X(IL) X(IL)
Y(CF) Y(CF)
Figure 1. SketchFigure of the 1. Sketch experimental of the experimental apparatus apparatus for for a a semi semi-submerged-submerged cylinder cylinderunder oscillatory under flow; oscillatory flow; Figure 1. Sketch of the experimental apparatus for a semi-submerged cylinder under oscillatory flow; X and Y are respectively IL and CF directions. X and Y are respectivelyX and Y are respectively IL and CF IL and directions. CF directions.
Figure 2. Photography of the experimental apparatus. Figure 2.FigurePhotography 2. Photography of of thethe experimental experimental apparatus. apparatus. To guarantee a two-dimensional flow and avoid three-dimensional effects arising from the rigid To guarantee a two-dimensional flow and avoid three-dimensional effects arising from the rigid cylinder with the finite length, we adopted two methods to avoid above problem. The first method cylinder with the finite length, we adopted two methods to avoid above problem. The first method To guaranteeis to aassemble two-dimensional two dummy cylinders flow at andboth the avoid model three-dimensionalends. The diameter of the dummy effects cylinder arising is from the rigid is to assemble two dummy cylinders at both the model ends. The diameter of the dummy cylinder is the same as the rigid cylinder model. To avoid the force transducer measuring the force derived from cylinder with thethe same finite as the length, rigid cylinder we adopted model. To avoid two the methods force transducer to avoid measuring above the force problem. derived from The first method dummy cylinders, a small gap (<1 mm) is left between and the rigid cylinder model and two end dummy cylinders, a small gap (<1 mm) is left between and the rigid cylinder model and two end is to assemble twodummy dummy cylinders.cylinders Another method at bothis to assemble the model two circular ends. endplates Thediameter with diameter of of the 3 times dummy cylinder dummy cylinders. Another method is to assemble two circular endplates with diameter of 3 times is the same as thecylinder rigid model cylinder diameter model.at the end To of the avoid dummy the cylinder force so transducer as to restrict measuringthe effects of other the force derived cylinder model diameter at the end of the dummy cylinder so as to restrict the effects of other apparatus on the flow past the cylinder. The force sensors are employed to record the hydrodynamic from dummy cylinders,apparatus on a the small flow past gap the (< cylinder.1 mm) Th ise force left sensor betweens are employed and the to rigidrecord the cylinder hydrodynamic model and two end loads on the rigid cylinder model. loads on the rigid cylinder model. dummy cylinders.To Another validate the method measurement is system to assemble and the smoothness two circular of the cylinder, endplates the cylinder with was diameter first of 3 times To validate the measurement system and the smoothness of the cylinder, the cylinder was first fully immersed in the water and towed in steady flow. The time history of hydrodynamic force in the cylinder modelfully diameter immersed at in the the endwater ofand the towed dummy in steady cylinderflow. The time so history as to restrictof hydrodynamic the eff forceects in of the other apparatus IL direction was recorded and shown in Figure 3. The entire time history contains four phases—initial IL direction was recorded and shown in Figure 3. The entire time history contains four phases—initial on the flow pastzero, the transition, cylinder. stable The stage force and finial sensors zero. areThe mean employed force value to F recordXM in thethe stable hydrodynamic stage in the IL loads on the zero, transition, stable stage and finial zero. The mean force value FXM in the stable stage in the IL direction is approximately 47.3 N/m under flow velocity U of 0.4 m/s. The corresponding drag rigid cylinder model.direction is approximately 47.3 N/m under flow velocity U of 0.4 m/s. The corresponding drag coefficient CD is calculated by, To validatecoefficient the measurement CD is calculated by, system and the smoothness of the cylinder, the cylinder was first FXM fully immersed in the water and towed in steadyCD flow.FXM The2 time history of hydrodynamic(1) force in the CD 1/ 2DLU (1) IL direction was recorded and shown in Figure3.1/ The 2DLU entire2 time history contains four phases—initial The CD is about 1.18, which is consistent with the typical value of 1.2 reported by Achenbach and The CD is about 1.18, which is consistent with the typical value of 1.2 reported by Achenbach and zero, transition,Heinecke stable (1981) stage [20]. andThis result finial demonstrates zero. The our experimental mean force apparatus value is reliable.FXM in the stable stage in the IL direction is approximatelyHeinecke (1981) [20]. This 47.3 result N/ mdemonstrates under flowour experimental velocity apparatus U of 0.4 is reliable. m/s. The corresponding drag coefficient CD is calculated by, FXM CD = (1) 1/2ρDLU2 Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 of 18
Transition Transition Initial Zero Stable stage Final 100 Zero
F =47.3 N 50 XM ) N ( F 0
-50 0 10 20 30 40 50 60 70 Time(s)
Figure 3. Time historyFigure 3. Time of the history hydrodynamic of the hydrodynamic forces forces on on a fully fully submerged submerged cylinder cylinderunder a steady under flow a steady flow velocity of U = 0.4 m/s. velocity of U = 0.4 m/s. 2.2. Test Arrangement To generate the equivalent horizontal oscillatory flow derived from wave action, the semi- submerged cylinder is forced to oscillate in harmonic motions in the IL direction (Chakrabarti 1987) [21]. The instantaneous displacement X(t) and corresponding velocity of the forced motion U(t) can be respectively indicated as what follows: 2 X( t ) Am sin t (2) T
22 (3) U( t ) Am cos t TT
where Am and T are respectively oscillation amplitude and period. To describe the hydrodynamic features of the semi-submerged cylinder, the key parameters including KC number, Reynolds number (Re) and Stokes number (β) [3,7,18,22] should be considered, which can be respectively expressed as,
2 Am KC (4) D
2 AD Re m (5) T
Re D2 (6) KC T
where υ = 1.0 × 10−6 m2 s−1 is the kinematic viscosity coefficient of water. However, the free surface effects are the distinctive characteristics for flow past a semi- submerged cylinder. The Froude number (Fr) is also an important parameter for this hydrodynamic load on the cylinder [19] and can be defined as,
Um Fr (7) gh
where Um = Am·(2π/T), h = 0.5D, h is the submerged depth. From the definition of the four parameters in Equations (4)–(7), conflicts are clear between the Froude number and Stokes number. In order to investigate the hydrodynamic force on semi- Appl. Sci. 2020, 10, 6404 5 of 17
The CD is about 1.18, which is consistent with the typical value of 1.2 reported by Achenbach and Heinecke (1981) [20]. This result demonstrates our experimental apparatus is reliable.
2.2. Test Arrangement To generate the equivalent horizontal oscillatory flow derived from wave action, the semi-submerged cylinder is forced to oscillate in harmonic motions in the IL direction (Chakrabarti 1987) [21]. The instantaneous displacement X(t) and corresponding velocity of the forced motion U(t) can be respectively indicated as what follows:
2π X(t) = A sin t (2) m T
2π 2π U(t) = A cos t (3) m T T where Am and T are respectively oscillation amplitude and period. To describe the hydrodynamic features of the semi-submerged cylinder, the key parameters including KC number, Reynolds number (Re) and Stokes number (β)[3,7,18,22] should be considered, which can be respectively expressed as, 2πA KC = m (4) D 2πA D Re = m (5) T υ · Re D2 β = = (6) KC υT where υ = 1.0 10 6 m2 s 1 is the kinematic viscosity coefficient of water. × − − However, the free surface effects are the distinctive characteristics for flow past a semi-submerged cylinder. The Froude number (Fr) is also an important parameter for this hydrodynamic load on the cylinder [19] and can be defined as, Fr = Um √gh (7) where Um = Am (2π/T), h = 0.5D, h is the submerged depth. · From the definition of the four parameters in Equations (4)–(7), conflicts are clear between the Froude number and Stokes number. In order to investigate the hydrodynamic force on semi- submerged cylinder in an oscillatory flow, the test cases were divided into four groups according to their KC numbers, as summarized in Table1. Each test group maintained the same KC number with di fferent Stokes number ranging from 5435 to 11,364. We should note that the test cases are designed to reveal preliminary hydrodynamic force features and the cases at larger KC number will be a promising direction for the future work.
Table 1. Test Matrix.
Case No. KC Am (m) T(s) β Fr Re 1–3 6.3 0.25 5.5:3:11.5 5435–11,364 0.12–0.26 3.4 104–7.1 104 × × 4–6 18.8 0.75 5.5:3:11.5 5435–11,364 0.37–0.77 1.0 105–2.1 105 × × 7–9 31.4 1.25 5.5:3:11.5 5435–11,364 0.62–1.29 1.7 105–3.6 105 × × 10–12 37.7 1.5 5.5:3:11.5 5435–11,364 0.74–1.55 2.0 105–4.3 105 × ×
3. Identification Method of Hydrodynamic Coefficients According to the Morison equation [1,2], the hydrodynamic force in the IL direction can be decomposed into two components; that is, the drag force and the inertial force that are respectively proportional to the square of the velocity and the acceleration. The similar equation was also employed Appl. Sci. 2020, 10, 6404 6 of 17 to describe the hydrodynamic force on floating cylinder in the horizontal direction [18]. Thus, the total force can be expressed as,
2 . 1 πD FXP = CDρDLU(t) U(t) + Cmρ LU(t) (8) 2 4 For convenience, it is assumed that,
1 C1(t) = ρDLU(t) U(t) 2 (9) πD2L . C (t) = ρ U(t) 2 4 Then, Equation (8) can be simplified as,
FXP = C (t) CD + C (t) Cm (10) 1 · 2 ·
The least squares method is used to estimate the drag coefficient CD and added mass coefficients Cm in the IL direction, which minimizes the sum of the squared difference J(CD,Cm) between the predicted and measured force, which can be written as,
n X 2 J(CD, Cm) = [FX(t) FXP(t)] (11) − i=1 where FX(t) is the measured force and n is the sample number. In order to minimize the difference between the identified and predicted values of the hydrodynamic force, Equation (11) should be satisfied,
∂J(CD, Cm) ∂J(CD, Cm) = 0, = 0 (12) ∂CD ∂Cm
Transforming Equation (12) into matrix form, we have " # T CD ΛΛ = ΛFX (13) Cm where, FX(t1) " # F (t ) C1(t1) C1(t2) C1(tn) X 2 Λ = ··· , FX = . (14) C (t ) C (t ) C (tn) . 2 1 2 2 2 . ··· FX(tn) The drag and added mass coefficients can be expressed as, " # CD T 1 = (ΛΛ )− ΛFX (15) Cm
However, there is no mature empirical equation to describe the hydrodynamic force in the CF direction. The empirical prediction of the lift force will be further studied in Section 4.3 according to the features observed by our experiments in this paper.
4. Results and Discussion This section first describes the main features of hydrodynamic forces on the semi-submerged cylinder under oscillatory flow. Based on observed features, we then proposed a novel empirical Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 18
FtX ()1 C()()()() t C t C t F t 1 1 1 2 1nX 2 Λ , FX (14) C2()()() t 1 C 2 t 2 C 2 tn FtXn()
The drag and added mass coefficients can be expressed as,
CD T 1 ()ΛΛ ΛFX (15) Cm
However, there is no mature empirical equation to describe the hydrodynamic force in the CF Appl. Sci. 2020direction., 10, 6404 The empirical prediction of the lift force will be further studied in Section 4.3 according to 7 of 17 the features observed by our experiments in this paper. formula that4. Results reduces and the Discussion parameters required to determine lift force but is demonstrated to well predict the lift force.This section Furthermore, first describes we the perform main features a comprehensive of hydrodynamic study forces on on the the influence semi-submerged of KC number, Stokes number,cylinder Froude under numberoscillatory on flow. the Based drag on coe observedfficient features, and added we then mass proposed coefficient a novel for empirical the IL direction formula that reduces the parameters required to determine lift force but is demonstrated to well as well as liftpredict coe thefficients lift force. and Furthermore, phase di ff weerence perform for a the comprehensive prediction study of lift on forcethe influence in the of CF KC direction. The phenomenanumber, including Stokes number, slamming Froude andnumber overtopping on the drag coefficient are also observedand added andmass discussed.coefficient for the IL direction as well as lift coefficients and phase difference for the prediction of lift force in the CF 4.1. Hydrodynamicdirection. Forces The phenomena on the Semi-Submerged including slamming Cylinder and overtopping are also observed and discussed.
As shown4.1. Hydrodynamic in Figure4, Forces the time on the history Semi-Submerged of the hydrodynamicCylinder loads on a semi-submerged cylinder under the steadyAs flowshown velocity in Figure of4, the U =time0.8 history m/s was of the presented. hydrodynamic The loads blue on anda semi red-submerged solid lines cylinder respectively represent theunder hydrodynamic the steady flow forces velocity in of the U = IL0.8 andm/s was CF presented. directions. The Theblue meanand red force solid lines in the respectively IL direction FXM represent the hydrodynamic forces in the IL and CF directions. The mean force in the IL direction FXM and mean lift force in the CF direction FYM are 27.6 N and 178.9 N, respectively. A novel phenomenon and mean lift force in the CF direction FYM are 27.6 N and 178.9 N, respectively. A novel phenomenon of an obvious mean downward lift force is witnessed for semi-submerged cylinder and reported by of an obvious mean downward lift force is witnessed for semi-submerged cylinder and reported by Ren et al. (2019)Ren et [al.19 ].(2019) This [19]. mean This liftmean force lift force is greater is greater than than the the drag drag forceforce and and should should be considered be considered in in the real engineeringthe real projects. engineering projects.
400
Steady Flow: U=0.8m/s FX FY
F =178.9 N ) 200 YM N (
F FXM=27.6 N 0
-200 10 20 30 40 50 60 70 Time(s) Figure 4. TimeFigure history 4. Time ofhistory the hydrodynamicof the hydrodynamic forces forces for for semi-submergedsemi-submerged cylinder cylinder underunder a flow velocity a flow velocity of U = 0.8 mof/s. U = 0.8 m/s.
For the hydrodynamic force on semi-submerged cylinder in an oscillatory flow, the time history For the hydrodynamic force on semi-submerged cylinder in an oscillatory flow, the time history of of these forces in both IL and CF directions for KC = 18.8 and β = 111,364 is shown in Figure 5. Similar these forcesto in hydrodynamic both IL and force CF directionson semi-submerged for KC cylinder= 18.8 in and steadyβ = flow,111,364 the mean is shown downward in Figure force in5 .the Similar to hydrodynamicCF direction force on FYM semi-submerged can be obviously witnessed. cylinder The in steadyvalues of flow, FYM were the mean162.3 N downward for KC = 18.8 force at β =in the CF 111,364, while no mean values of the force in the IL direction can be observed. The mean lift force in direction FYM can be obviously witnessed. The values of FYM were 162.3 N for KC = 18.8 at β = 111,364, while no meanAppl. Sci. values 2020, 10 of, x FOR the PEER force REVIEW in the IL direction can be observed. The mean lift force8 of 18 in the CF direction can reach and even exceed the fluctuating force amplitude of hydrodynamic force in the IL the CF direction can reach and even exceed the fluctuating force amplitude of hydrodynamic force F direction ( Xin). the The IL direction fluctuating (FX). The components fluctuating components of two forces of two are forces in the are samein the same magnitudes. magnitudes.
Oscillatory Flow: A=0.75 m, T=5.5 s, KC=18.8, β=111364 1 ) m ( (a) 0 X
-1 10 20 30 40 50 60 Time(s) 500 FYM=162.3 N ) N (
(b) 0 F F =0 N XM FX FY -500 10 20 30 40 50 60 Time(s) 200 F F 150 X Y ) f=0.18Hz N
(c) ( 100 f=0.36Hz F 50 0 0 0.5 1 1.5 2 Frequency (Hz)
Figure 5. FigureTime 5. Time history history of of thethe hydrodynamic hydrodynamic forces forcesfor semi- forsubmerged semi-submerged cylinder under Keulegan cylinder- under Keulegan-CarpenterCarpenter (KC) == 18.818.8 at atβ =β 111,364.= 111,364. (a) Time (a) Time history history of forced of motion; forced ( motion;b) Time history (b) Time of the history of hydrodynamic forces on semi-submerged cylinder; (c) The spectrum of the hydrodynamic loads. the hydrodynamic forces on semi-submerged cylinder; (c) The spectrum of the hydrodynamic loads. From the spectrum of the hydrodynamic forces in both IL and CF directions as shown in Figure 5c, the dominant frequency of force in the IL direction is the same as expected for the forced motion frequency (fo) of 0.18 Hz and the response frequency in the CF direction is found to be always dominated by twice the value of fo (f = 0.36 Hz) under KC = 18.8 and β = 111,364. It indicates that this lift force with higher frequency could cause more severe damage to related structures and cannot be neglected. To understand the background mechanics of the mean lift downward force in an oscillatory flow, a camera is placed on the apparatus to record the surface waveforms. Figure 6a clearly shows the recorded local surface distortions. According to the phenomenon recorded by the camera, we infer that the mean downward lift force can be attributed to the asymmetric flow field and the pressure difference between the top and bottom of the cylinder [19,23] as shown in Figure 7c. Furthermore, Figure 7 presents the time history of hydrodynamic force and sketches of flow field around cylinder in one period to give more detailed features. Combining with forced motion velocity, we observed that the lift force reaches its amplitude value at around time points I, III, V. From Figure 7c, the top-sided flow velocity is more significant when the forced motion velocity reach its maximum (Umax). Thus, the maximum lift force value appears near the time points of Umax. Appl. Sci. 2020, 10, 6404 8 of 17
From the spectrum of the hydrodynamic forces in both IL and CF directions as shown in Figure5c, the dominant frequency of force in the IL direction is the same as expected for the forced motion frequency (f o) of 0.18 Hz and the response frequency in the CF direction is found to be always dominated by twice the value of f o (f = 0.36 Hz) under KC = 18.8 and β = 111,364. It indicates that this lift force with higher frequency could cause more severe damage to related structures and cannot be neglected. To understand the background mechanics of the mean lift downward force in an oscillatory flow, a camera is placed on the apparatus to record the surface waveforms. Figure6a clearly shows the recorded local surface distortions. According to the phenomenon recorded by the camera, we infer that the mean downward lift force can be attributed to the asymmetric flow field and the pressure difference between the top and bottom of the cylinder [19,23] as shown in Figure7c. Furthermore, Figure7 presents the time history of hydrodynamic force and sketches of flow field around cylinder in one period to give more detailed features. Combining with forced motion velocity, we observed that the lift force reaches its amplitude value at around time points I, III, V. From Figure7c, the top-sided flow velocity is more significant when the forced motion velocity reach its maximum (Umax). Thus, the maximum lift force value appears near the time points of Umax. Appl.Appl. Sci. Sci. 20 202020, ,10 10, ,x x FOR FOR PEER PEER REVIEW REVIEW 99 of of 18 18 Forward ((aa)) ((bb)) Forward
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FFYY II IIII IIIIII IIVV VV
Figure 7. TimeFigureFigure history 77. . TimeTime of history history forced of of motion, forced forced motion, motion, hydrodynamic hydrodynamic hydrodynamic force force force and and and sketches sketches sketches of of of flow flow flow field field field around around cylinder cylindercylinder in in one one period period under under KC KC = = 18.8 18.8 at at β β = = 111,364. 111,364. ( (aa)) Time Time histories histories of of forced forced motion motion displacement displacement in one period under KC = 18.8 at β = 111,364. (a) Time histories of forced motion displacement and andand velocity;velocity; ((bb)) TheThe hydrodynamichydrodynamic loadsloads onon semisemi--submergedsubmerged cylinder; cylinder; ((cc)) SketchesSketches ofof flowflow fliedflied velocity; (change.bchange.) The hydrodynamic loads on semi-submerged cylinder; (c) Sketches of flow flied change.
However,However, there there is is a a phase phase difference difference between between F FYmaxYmax and and U Umaxmax. .These These two two maximum maximum values values do do not not reachreach atat thethe samesame time.time. TheThe maximummaximum ofof forceforce FFYmaxYmax isis reachedreached beforebefore thethe UUmaxmax. . ItIt maymay bebe relatedrelated toto thethe effectseffects ofof thethe excitinexcitingg wave.wave. TheThe velocityvelocity ofof thethe excitingexciting wavewave willwill bebe superimposedsuperimposed withwith thethe forcedforced motion motion velocity. velocity. The The time time of of the the “real “real velocity” velocity” superimposed superimposed by by the the velocity velocity of of exciting exciting wave wave andand the the forced forced motion motion reachreach the the maximum maximum value value isis not not locatedlocated at at that that of of forcedforced mot motionion velocityvelocity amplitude.amplitude. MoreMore insightfulinsightful experimentsexperiments andand CFDCFD simulationssimulations willwill bebe carriedcarried outout inin thethe nearnear futurefuture toto furtherfurther investigateinvestigate thisthis phenomenon.phenomenon. ThisThis phasephase differencedifference wouldwould bebe anan importantimportant parameterparameter toto describedescribe the the relationship relationship between between the the lift lift force force and and flow flow velocity velocity generated generated by by forced forced motion. motion. BeyondBeyond that,that, aa shockshock signalsignal ofof thethe forceforce inin thethe ILIL directiondirection appearsappears whenwhen thethe cylindercylinder reversesreverses itsits motionmotion directiondirection (II,(II, IV).IV). TheThe phenomenonphenomenon isis causedcaused byby thethe “slamming”“slamming” phenomenonphenomenon asas shownshown inin FigureFigure 66a.a. FigureFigure 66bb showsshows thethe developingdeveloping processprocess sketchsketch ofof thethe slamming.slamming. WhenWhen thethe cylindercylinder movemove forward, forward, the the water water attachedattached on on the the cylinder cylinder andand exciting exciting wavewave will will be be moved moved in in the the same same Appl. Sci. 2020, 10, 6404 9 of 17
However, there is a phase difference between FYmax and Umax. These two maximum values do not reach at the same time. The maximum of force FYmax is reached before the Umax. It may be related to the effects of the exciting wave. The velocity of the exciting wave will be superimposed with the forced motion velocity. The time of the “real velocity” superimposed by the velocity of exciting wave and the forced motion reach the maximum value is not located at that of forced motion velocity amplitude. More insightful experiments and CFD simulations will be carried out in the near future to further investigate this phenomenon. This phase difference would be an important parameter to describe the relationship between the lift force and flow velocity generated by forced motion. Beyond that, a shock signal of the force in the IL direction appears when the cylinder reverses its motion direction (II, IV). The phenomenon is caused by the “slamming” phenomenon as shown in Figure6a. Figure6b shows the developing process sketch of the slamming. When the cylinder move forward, the water attached on the cylinder and exciting wave will be moved in the same direction as the cylinder. As the cylinder reverse its moving direction, the attaching water and exciting wave cannot change their movement direction immediately. The cylinder will strike the exciting wave and attaching water. The water climbs significantly in an instant and results in an impulse force in the IL direction. Slamming will result in high pressure and introduce damage on local structure and subsequent failure of the entire system. Thus, it is necessary to take some considerations to the local structure designs.
4.2. Empirical Prediction of the Lift Force for Semi-Submerged Cylinder in Oscillatory Flow As previously mentioned in Section 4.1, the lift force fluctuates at a frequency twice the forced motion frequency and an obvious phase difference between the force and velocity motion can be observed. Moreover, the mean value is found to be the main feature of the lift force in the CF direction. However, there is no empirical formula for lift force on semi-submerged cylinder in an oscillatory flow. In the real engineering project design, it is very difficult to efficiently determine the hydrodynamic force on these relevant marine structures. To reasonably describe the relationship between the lift force and flow velocity, above main features should be satisfied and thus we further propose the application of this empirical prediction of the lift force to partially submerged cylinder under an oscillatory flow:
1 1 2π 2π 2 F = ρC DL[U(t + τ)]2 = ρC DLU2 cos t + τ (16) YP 2 L 2 L m T T where FYP is the predicted lift force in the CF direction. CL is lift force coefficient and τ is a time delay representing a phase difference. For convenience of description, the phase difference ϕ is expressed as,
2π 180 ϕ = τ (◦) (17) T × π The Equation (16) can be simplified as,
1 2π F = ρC DLU2 [cos(ωt + ϕ)]2, ω = (18) YP 2 L m T Expanding Equation (18) out, we obtain
1 1 F = ρC DLU2 + ρC DLU2 cos(2ωt + 2ϕ) (19) YP 4 L m 4 L m From Equation (19), the predicted force can satisfy the main features observed in Section 4.1 and can be divided into two parts—the mean lift force FYPM and twice the forced motion frequency fluctuating lift force FYPo.
FYP = FYPM + FYP◦ (20) Appl. Sci. 2020, 10, 6404 10 of 17
Through Equation (20), the lift coefficient can be calculated by,
2 CL = FYM/(1/4ρDLUm) (21) where FYM is the measured mean lift force in the CF direction. Here, Equation (18) only has two parameters that remain to be determined, the lift coefficient CL and the phase difference ϕ. However, if we attempt to directly describe the lift force as the mean and fluctuating component, we need to determine four parameters including the mean value, fluctuating frequency, fluctuating amplitude and phase difference. This novel empirical formula in Equation (18) directly assigns the fluctuating frequency as twice of the frequency of oscillatory flow and the fluctuating amplitude the same as the mean value, which reduced undetermined parameters from fourAppl. to Sci. two. 2020 This, 10, x reduction FOR PEER REVIEW is directly based on our observation of the main feature of the measured11 liftof 18 force; for example, red line in Figure6b. In the following, we demonstrate that such a reduction of parametersof parameters still still well well predicts predicts the fluctuatingthe fluctuating lift force lift force in a rangein a range of KC of number, KC number, Froude Froude number number and Stokesand Stokes number. number. ToTo determine determine the the value value of ofϕ φ, the, the sum sum of of the the squared squared di diffefferencerenceJ1 J(1ϕ(φ)) between between the the predicted predicted and and measuredmeasured fluctuating fluctuating force, force, which which can can be be expressed expressed as, as,
X 2 o JFFFF( ) [( ) ( )]2 , 0 180◦ J1(ϕ1 ) = [(FY YFYM YM) (FYP YPFYPM YPM)] , 0 ϕ 180 (22)(22) − − − ≤ ≤ Figure 8 presents the value of J1(φ) versus phase difference angle φ under KC = 18.8 at β = 11,364. Figure8 presents the value of J1(ϕ) versus phase difference angle ϕ under KC = 18.8 at β = 11,364. It can be seen that J1(φ) minimizes at φ = 21°. Thus, the phase difference angle at this case is adopted It can be seen that J1(ϕ) minimizes at ϕ = 21◦. Thus, the phase difference angle at this case is adopted as as 21°. Using the same method, the value of φ for each test case can be obtained. To validate the 21◦. Using the same method, the value of ϕ for each test case can be obtained. To validate the predicted method,predicted the method, identified the lift identified coefficient lift and coefficient phase angle and phase are substituted angle are intosubstit theuted predicted into the formula predicted of Equationformula (18).of Equation The recalculated (18). The recalculated lift force and lift measured force and force measured in an oscillatoryforce in an flowoscillatory are shown flow inare Figureshown9. Thein Figure blue dashed 9. The andblue red dashed solid lineand respectively red solid line represents respectively the measured represents and the calculated measured force. and Thecalculated calculated force. forces The are calculated in a good forces agreement are in a with good the agreement measured with forces. the Therefore,measured itforces. is feasible Therefore, and promisingit is feasible to useand thepromising prediction to use formula the prediction described formula in Equation described (18) toin calculateEquation the(18) hydrodynamic to calculate the forcehydrodynamic in the CF direction force in the in an CF oscillatory direction flow.in an Suchoscillatory a good flow. agreement Such a usinggood thisagreement novel empiricalusing this formulanovel empirical in Equation formula (18) also in Eq providesuation a(18) potential also provides explanation a potential of the mechanismexplanation of of the the fluctuating mechanism lift of forcethe fluctuating with frequency lift force twice with of the frequency oscillatory twice flow, of whichthe oscillatory is worthwhile flow, which for further is worthwhile study through for further flow visualizationstudy through and flow CFD. visualization and CFD.
7 x 10 4.5
4
3.5
3 ) 2
N 2.5 (
) φ
( 2
1 J 1.5
1
0.5
0 0 20 40 60 80 100 120 140 160 180 φ（°）
Figure 8. The sum of the squared difference J1(ϕ) between the predicted and measured fluctuating Figure 8. The sum of the squared difference J1(φ) between the predicted and measured fluctuating force versus phase difference under KC = 18.8 at β = 11,364 (A = 0.75 m, T = 5.5 s). force versus phase difference under KC = 18.8 at β = 11,364 (A = 0.75 m, T = 5.5 s).
Measured force Calculated force 300
200
Force(N) 100
0 20 25 30 35 40 45 50 55 60 Time (s) Figure 9. Time history of the measured and calculated hydrodynamic forces for semi-submerged cylinder in the cross-flow (CF) direction under KC = 18.8 at β = 11,364 (A = 0.75 m, T = 5.5 s).
Moreover, for computational fluid dynamics, this problem is a typical two-phase flow problem, which is not simple to solve. However, this empirical formula can greatly avoid the complicated and Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 18
of parameters still well predicts the fluctuating lift force in a range of KC number, Froude number and Stokes number. To determine the value of φ, the sum of the squared difference J1(φ) between the predicted and measured fluctuating force, which can be expressed as,
2 o JFFFF1( ) [(Y YM ) ( YP YPM )] , 0 180 (22)
Figure 8 presents the value of J1(φ) versus phase difference angle φ under KC = 18.8 at β = 11,364. It can be seen that J1(φ) minimizes at φ = 21°. Thus, the phase difference angle at this case is adopted as 21°. Using the same method, the value of φ for each test case can be obtained. To validate the predicted method, the identified lift coefficient and phase angle are substituted into the predicted formula of Equation (18). The recalculated lift force and measured force in an oscillatory flow are shown in Figure 9. The blue dashed and red solid line respectively represents the measured and calculated force. The calculated forces are in a good agreement with the measured forces. Therefore, it is feasible and promising to use the prediction formula described in Equation (18) to calculate the hydrodynamic force in the CF direction in an oscillatory flow. Such a good agreement using this novel empirical formula in Equation (18) also provides a potential explanation of the mechanism of the fluctuating lift force with frequency twice of the oscillatory flow, which is worthwhile for further study through flow visualization and CFD.
7 x 10 4.5
4
3.5
3 ) 2
N 2.5 (
) φ
( 2
1 J 1.5
1
0.5
0 0 20 40 60 80 100 120 140 160 180 φ（°）
Figure 8. The sum of the squared difference J1(φ) between the predicted and measured fluctuating Appl. Sci. 2020, 10, 6404 11 of 17 force versus phase difference under KC = 18.8 at β = 11,364 (A = 0.75 m, T = 5.5 s).
Measured force Calculated force 300
200
Force(N) 100
0 20 25 30 35 40 45 50 55 60 Time (s) FigureFigure 9. 9.Time Time history history of of the the measured measured and and calculated calculated hydrodynamic hydrodynamic forces forces for for semi-submerged semi-submerged cylindercylinder in in the the cross-flow cross-flow (CF) (CF) direction direction under under KC KC= =18.8 18.8 at atβ β= =11,364 11,364 (A (A= =0.75 0.75 m, m, T T= = 5.55.5 s).s). Appl. Sci. 2020, 10, x FOR PEER REVIEW 12 of 18 Moreover,Moreover, for for computational computational fluid fluid dynamics, dynamics, this this problem problem is is a a typical typical two-phase two-phase flow flow problem, problem, time-consuming hydrodynamic calculation problem and will be very efficient and convenient for whichwhich is is not not simple simple to to solve. solve. However, However, this this empirical empirical formula formula can can greatly greatly avoid avoid the the complicated complicated and and time-consumingengineer to calculate hydrodynamic the force on calculation semi-subm problemerged cylinder and will, such be as very the efloatingfficient collar and convenient of fish farming for . engineerHowever, to calculatein real engineering the force on project, semi-submerged the semi-submerged cylinder, such cylinder as the will floating not have collar a ofconstant fish farming. vertical However,displacement, in real which engineering is likely project, to result the in semi-submerged a different lift force. cylinder This willwork not mainly have focused a constant on verticalthe static displacement,semi-submerged which status is likely of the to resultcylinder, in a which different will lift also force. provide This worka preliminary mainly focused estimation on the of staticthe lift semi-submergedforce and basic references status of the for cylinder, further investigation. which will also More provide systematic a preliminary experiments estimation will be ofcarried the lift out forceto determine and basic the references influence for of further vertical investigation. and horizontal More vibration systematic and experiments even the submerged will be carried depths out on tohy determinedrodynamic the load influence in near of future. vertical and horizontal vibration and even the submerged depths on hydrodynamic load in near future. 4.3. Drag and Added Mass Coefficients for Semi-Submerged Cylinder 4.3. DragCombining and Added measured Mass Coeffi hydrodynamiccients for Semi-Submerged force and Cylinder forced motion in the IL direction, we can calculateCombining the drag measured coefficient hydrodynamic (CD) and added force and mass forced coefficient motion in(Cm the) using IL direction, the method we can proposed calculate in theSection drag coe3. Toffi cientverify (C theD) andidentified added method mass coe offfi hydrodynamiccient (Cm) using coefficients, the method the proposed hydrodynamic in Section force3. is Torecalculated verify the identified through methodEquation of (8) hydrodynamic by identified coedragffi cients,and added the hydrodynamic mass coefficient force as shown is recalculated in Figure through10. The Equation blue dashed (8) by line identified and red drag solid and line added represent mass coe theffi cient measured as shown force in and Figure calculated 10. The blue force, dashedrespectively. line and The red solid good line consistency represent the is witnessedmeasured force between and calculated measured force, and respectively. calculated value. The good This consistencydemonstrates is witnessed the validity between of the measured identification and calculated method value. for hydrodynamics This demonstrates of semi the validity-submerged of thecylinder identification under an method oscillatory for hydrodynamics flow. of semi-submerged cylinder under an oscillatory flow.
200 Measured force Calculated force 100 0
Force(N) -100
-200 0 5 10 15 20 25 Time(s) FigureFigure 10. 10.Time Time history history of of the the measured measured and and calculated calculated hydrodynamic hydrodynamic forces forces for for semi-submerged semi-submerged cylindercylinder in in the the in-line in-line (IL) (IL) direction direction under under KC KC= 18.8= 18.8 at atβ =β =11,364 11,364 (A (A= =0.75 0.75 m, m, T T= =5.5 5.5 s). s).
FigureFigure 11 11shows shows the dragthe drag and added and added mass coe massfficients coefficients distribution distribution with KC withnumber KC under number diff erent under Stokes number β. The relationship between C and β is indistinct as shown in Figure 11a. When the different Stokes number β. The relationshipD between CD and β is indistinct as shown in Figure 11a. KCWhen number the KC is lessnumber than is 20, less the than discrete 20, the distribution discrete distribution features is features very obvious. is very obvious.Therefore, Therefore, the Stokes the numberStokes number would more would greatly more greatly affect theaffect drag the coedragffi cientcoefficient at this at conditionthis condition (KC (K
(a) C m 0.6 D 0.4 (b)
C 0.4
(a) C 0.40.3 0.4 0.3 0.40.2 0.3 0.2 0.2 0.20 10 20 30 40 0.20 10 20 30 40 0.20 10 KC N20umber 30 40 0 10 KC 20Number 30 40 0 10 KC N20umber 30 40 0 10 KC 20Number 30 40 KC Number KC Number Figure 11. Drag and added mass coefficients changing with KC number under different Stokes Figure 11. Drag and added mass coefficients changing with KC number under different Stokes Figure 11. Dragnumber and ((a)added drag coefficients; mass coe (bffi) cientsadded mass changing coefficient). with KC number under different Stokes number Figurenumber 11. ((a )Drag drag andcoefficients; added mass(b) added coefficients mass coefficient). changing with KC number under different Stokes ((a) drag coenumberfficients; ((a) drag (b) addedcoefficients; mass (b) coe addedfficient). mass coefficient). 0.7 1.2 0.7 (a) β=5435 1.2 (b) β=5435 1.2 0.70.6 (a) β=7353 1 (b) 0.6 (a) βββ===1751343536354 1 (b) ββ==171335634 1 0.60.5 0.8 β=11364 m 0.8 D 0.5 C m C
D 0.8 0.5 C
m 0.6 C 0.4 D 0.4 C 0.6 C β=5435 0.6 0.40.3 0.4 ββ==75345335 0.3 0.4 βββ===1751343536354 0.3 0.4 ββ==171335634 0.2 0.2 0.20 0.5 1 1.5 2 0.20 0.5 1 1.5 β=11364 2 0 0.5 1 1.5 2 0.20 0.5 1 1.5 2 0.2 Fr Fr 0 0.5 1 1.5 2 0 0.5 F1r 1.5 2 Fr Fr Fr Figure 12. Drag and added mass coefficients changing with Froude number under different Stokes Figure 12. DragFigure and12. Drag added and added mass mass coeffi coefficientscients changing changing with Froude Froude number number under under different di Stokesfferent Stokes Figurenumber 12. ((a Drag) drag and coefficients; added mass (b) addedcoefficients mass changingcoefficient). with Froude number under different Stokes number ((a)number drag coe((a) ffidragcients; coefficients; (b) added (b) added mass mass coe coefficient).fficient). number ((a) drag coefficients; (b) added mass coefficient). 0.7 1.2 0.7 (a) KC=6.3 1.2 (b) 0.7 (a) 0.6 KC=61.83.8 1.21 (b) (a) 0.6 KC=613.813.84 1 (b) 0.6 KC=1381.84 0.5 KC=37.7 0.81
D 0.5 KC=317.47 m 0.8 C C D 0.5 KC=37.7 m 0.8 C 0.4 C 0.6 D m
C 0.4 0.6 C KC=6.3 0.4 KC=6.3 0.3 0.60.4 KC=18.8 0.3 0.4 KC=613.813.84 0.3 KC=1381.84 0.2 0.4 KC=37.7 0.2 KC=317.47 0.20 0.5 1 1.5 2 0.20 0.5 1 1.5 2 0 0.5 1 1.5 2 KC=37.7 0.2 Fr 0.20 0.5 F1r 1.5 2 0 0.5 F1r 1.5 2 0 0.5 F1r 1.5 2 Fr Fr Figure 13. DragFigure and 13. addedDrag and mass added coe massfficients coefficients changing changing with with Froude Froude number number under under di differentfferent KC number Figure 13. Drag and added mass coefficients changing with Froude number under different KC ((a) drag coeFigurenumberfficients; 13. ((a )Drag drag (b) added andcoefficients; added mass mass(b) coe added coefficientsfficient). mass coefficient). changing with Froude number under different KC number ((a) drag coefficients; (b) added mass coefficient). number ((a) drag coefficients; (b) added mass coefficient). 4.4. Lift Coefficients for Semi-Submerged Cylinder 4.4. Lift Coefficients for Semi-Submerged Cylinder 4.4. LiftThrough Coefficients the abovefor Semi identified-Submerged method Cylinder described in Section 4.2, the lift coefficients and phase Through the above identified method described in Section 4.2, the lift coefficients and phase angle are determined and summarized in Figures 14—16. Figure 14 shows the lift coefficients and angleThrough are determined the above and identified summarized method in Figure describeds 14— in16 Section. Figure 4.2, 14 showsthe lift thecoefficients lift coefficients and phase and phase angle changing with KC number under the same Stokes number. The maximum lift coefficient anglephase areangle determined changing withand summarizedKC number under in Figure the s ame14— Stokes16. Figure number. 14 shows The maximum the lift coefficients lift coefficient and can reach 1.5, which is even larger than the typical value of drag coefficient for fully submerged phasecan reach angle 1.5, changing which iswith even KC larger number than under the thetypical same value Stokes of number.drag coefficient The maximum for fully lift submerged coefficient cancylinder reach (C 1.5,D = 1which.2) as shown is even in largerFigure than 14a. Therefore,the typical such value large of liftdrag force coefficient for semi -forsubmerged fully submerged cylinder cylinder (CD = 1.2) as shown in Figure 14a. Therefore, such large lift force for semi-submerged cylinder cannot be ignored in the related structure design and engineering projects. cylindercannot be (C ignoredD = 1.2) as in shown the related in Figure structure 14a. Therefore, design and such engineering large lift forceprojects. for semi-submerged cylinder cannot be ignored in the related structure design and engineering projects. Appl. Sci. 2020, 10, x FOR PEER REVIEW 14 of 18
1.6 60 (a) (b) β=5435 50 1.5 β=7353 40 β=11364
1.4 ） ° 30 （ L
C 1.3 φ 20 β=5435 1.2 β=7353 10 β=11364 1.1 0 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 KC Number KC Number
Appl. Sci. 2020, 10, 6404 13 of 17 Figure 14. Lift coefficient and phase difference angle changing with KC number under different Stokes number ((a) lift coefficients; (b) phase difference angle). 4.4. Lift Coefficients for Semi-Submerged Cylinder Similar to the drag coefficient, the discrete distribution can be also observed in lift coefficients Through the above identified method described in Section 4.2, the lift coefficients and phase angle when KC < 15. The lift coefficient varies from 1.19 to 1.51. At this region, the lift coefficients are are determined and summarized in Figures 14–16. Figure 14 shows the lift coefficients and phase angle significantly affected by the Stokes number and seem to increase with the value of the parameter β. changing with KC number under the same Stokes number. The maximum lift coefficient can reach 1.5, When KC > 15 and β < 11,364, the lift coefficients are stable at approximately 1.40 with KC number. which is even larger than the typical value of drag coefficient for fully submerged cylinder (C = 1.2) The effects of Stokes number and KC number seem minor. Under higher Stokes number (β =D 11,364), as shown in Figure 14a. Therefore, such large lift force for semi-submerged cylinder cannot be ignored an unexpected drop of CL occurs. CL decreases from 1.45 to 1.25. This phenomenon will be discussed inAppl. the Sci. related 2020, 10 structure, x FOR PEER design REVIEW and engineering projects. 14 of 18 in next section. As for phase difference angle, the angles are stable with KC number under the same
Stokes number.1.6 φ stabilizes at approximately 16°,60 19° and 26° for β = 11,364, 7353 and 5435, (a) (b) β=5435 respectively. Different from lift coefficient, the angle decreases50 with Stokes number. Slight fluctuation can be found1.5 in the angle distribution under smaller Stokes number (β = 5435). β=7353 40 β=11364
To further1.4 investigate the effects of Froude number,） Figure 15 illustrates the above two values ° 30 （ L versus Froude number under the same Stokes number. The stable trend of CL can be seen in the region C 1.3 φ 20 of Fr ranging from 0.3 to 1.3 as presentedβ=543 5in Figure 15a. Relatively dispersed distribution is also found 1.2 β=7353 10 under lower Froude number. Sharpβ =downward11364 trend happens when Fr is around 1.5. In addition, there is slightly1.1 obvious phenomenon that the lift coefficient0 rises with Stokes number when the 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 Froude number is greater than 0.5. Figure 15b shows the phase angle with Froude number under the KC Number KC Number same Stokes number. More discrete features of the phase distribution can be observed under Fr < 0.5, Figure 14. Lift coefficient and phase difference angle changing with KC number under different Stokes whileFigure stable 14. and Lift slight coefficient downtrend and phase appears difference under angleFr > 0.5. changing with KC number under different number ((a) lift coefficients; (b) phase difference angle). Stokes number ((a) lift coefficients; (b) phase difference angle). 1.6 60 (a) (b) β=5435 Similar to the drag coefficient, the discrete distribution50 can be also observed in lift coefficients 1.5 β=7353 β=11364 when KC < 15. The lift coefficient varies from 1.19 to40 1.51. At this region, the lift coefficients are 1.4 ） °
significantlyL affected by the Stokes number and seem to increase with the value of the parameter β. 30 C （
When KC > 11.35 and β < 11,364, the lift coefficients areφ stable at approximately 1.40 with KC number. 20 The effects of Stokes number and KC numberβ=54 3seem5 minor. Under higher Stokes number (β = 11,364), 1.2 β=7353 10 an unexpected drop of CL occurs. CL decreases from 1.45 to 1.25. This phenomenon will be discussed β=11364 Appl. Sci. 2020, 1.110, x FOR PEER REVIEW 0 15 of 18 in next section.0 As for 0.5phase difference1 angle,1.5 the2 angles0 are stable0.5 with 1KC number1.5 under2 the same Stokes number. φ stabilizes atF r approximately 16°, 19° and 26° for βFr = 11,364, 7353 and 5435, respectively.the lift coefficient Different and fromphase lift angle coefficient, when the the Stokesangle decreases number and with Froude Stokes numbernumber. areSlight in thefluctuation confl ict Figure 15. Lift coefficient and phase difference angle changing with Froude number under different KC for theFigure semi 15.-submerged Lift coefficient cylinder. and phase difference angle changing with Froude number under different can benumber found (( ain) lift the coe anglefficients; distribution (b) phase diunderfference smaller angle). Stokes number (β = 5435). ToKC furthernumber investigate((a) lift coefficients; the effects (b) phase of Froude difference number, angle). Figure 15 illustrates the above two values 1.6 versus Froude number under the same Stokes number.60 The stable trend of CL can be seen in the region (a) (b) KC=6.3 Under the same KC number, the distribution of CL versus Froude number is presented in Figure of Fr ranging1.5 from 0.3 to 1.3 as presented in Figure 15a.50 Relatively dispersed distributionKC=18.8 is also found 16. CL increases with Froude number when KC ≤ 31.4, while decreases with Fr forK CKC=31 .>4 31.4. Similar under lower Froude number. Sharp downward trend40 happens when Fr is around 1.5. In addition, 1.4 KC=37.7 evolutionary phenomenon with drag and added mass） coefficient can also be found in lift coefficient. there is slightly obvious phenomenon that the lift ° coefficient rises with Stokes number when the L
（ 30 C However, there is no linear relationship between CL and Fr. The trend of the CL changes from rapid Froude number1.3 is greater than 0.5. Figure 15b showsφ the phase angle with Froude number under the KC=6.3 20 samerise to Stokes flat increase number. andK MoreC= 1finally8.8 discrete shows features a downward of the phase trend. distribution As for the canphase be observedangle, φ decreasesunder Fr
1.3 φ Similar to the drag coefficient, the discrete distribution20 can be also observed in lift coefficients 4.5. Effects of Instantaneous Overtopping β=5435 when KC < 15.1.2 The lift coefficient variesβ from=7353 1.19 to10 1.51. At this region, the lift coefficients are significantly affected by the Stokes numberβ=1 and1364 seem to increase with the value of the parameter β. Besides1.1 the slamming phenomenon, the overtopping0 is another tough problem for semi- Whensubmerged KC > 15cylinder0 and β
4.5. Effects of Instantaneous Overtopping Besides the slamming phenomenon, the overtopping is another tough problem for semi-submerged cylinder in an oscillatory flow [18,19,24]. In Section 4.3, an unexpected drop of the lift coefficient was observed in Figures 14 and 15. To further investigate the hydrodynamic mechanism behind this phenomenon, the water surface waveforms, the corresponding water surface sketches at the maximum forced motion velocity, the time histories of forced motion and hydrodynamic force in both CF and IL directions are compared and shown in Figure 17. Under the same Stokes number, the upstream water level of the cylinder keeps rising with KC number increasing as shown in Figure 17a,e,i,m. To clearly illustrate the water level change, the corresponding sketches are shown in Figure 17b,f,g,n. The symbol “H” represents the upstream water level. When KC = 37.70, the upstream water was over the top of the semi-submerged cylinder, which is termed as “overtopping” phenomenon [18,19]. Different from that in steady flow, the overtopping only happens at around the time of forced motion velocity amplitude. We redefine the phenomenon as “instantaneous overtopping.” Combining with the lift coefficient distribution, we observe that the drop phenomenon happens as the instantaneous overtopping occurs. The reason can be attributed to the changes of the flow field asymmetry. When the overtopping occurs, this asymmetry weakens, which results in a reduction in the lift coefficient. The same phenomenon was also found in the results under the steady flow reported by Ren et al. (2019) [19]. However, the effects of overtopping in an oscillatory flow were not seen in drag and added mass coefficients. Moreover, the drag force and lift force appear relatively high frequency fluctuation in the peak area as shown in red rectangle part of Figure 17c,d,g,h,k,l,o,p. The fluctuation of the hydrodynamic force is also attributed to the instantaneous “overtopping” phenomenon at forced motion velocity amplitude. The extreme surface distortions and water fragmentation caused by the instantaneous “overtopping” phenomenon result in this higher frequency hydrodynamic force. Limited by the quantity of experimental data, more systematic experiments will be conducted to study these effects of instantaneous “overtopping” phenomenon. Appl. Sci. 2020, 10, 6404 15 of 17 Appl. Sci. 2020, 10, x FOR PEER REVIEW 16 of 18
KC=6.28 A=0.25 m T=5.5 s β=11364 0.5 0.5 (c) X(t) U(t) Um ) ) s /
(a) (b) m ( 0 m H ( X U U -0.5 -0.5 m 50 (d) FX FY ) N
( 0 F -50 40 45 50 55 60 Time(s) KC=18.84 A=0.75 m T=5.5 s β=11364
Um 1 (g) X(t) U(t) 1 ) s / ) H m (e) (f) m ( ( 0 U X -1 -1
400 (h) FX FY Um ) 200 N (
F 0 -200 40 45 50 55 60 Time(s) KC=31.42 A=1.25 m T=5.5 s β=11364 2 2 Um (k) X(t) U(t) ) ) s / m H (i) (j) ( 0 m ( X U
Um -2 -2 1000 (l) FX FY
) 500 N (
F 0 -500 40 45 50 55 60 Time(s) KC=37.70 A=1.50 m T=5.5 s β=11364
Um 2 (o) X(t) U(t) 2 ) s H / )
(m) (n) m ( m 0 ( U X -2 -2 F F Um 1000 (p) X Y
) 500 N (
F 0 -500 40 45 50 55 60 Time(s)
Figure 17. The water surface waveforms and the sketches at the maximum forced motion velocity, Figure 17. The water surface waveforms and the sketches at the maximum forced motion velocity, the the time histories of forced motion and hydrodynamic force in both CF and IL directions changing time histories of forced motion and hydrodynamic force in both CF and IL directions changing with with KC number under the same Stokes number. (a,e,i,m) The pictures of water surface waveforms at KC number under the same Stokes number. (a,e,i,m) The pictures of water surface waveforms at the the maximum forced motion velocity; (b,f,j,n) Corresponding sketches of water surface waveforms; maximum forced motion velocity; (b,f,j,n) Corresponding sketches of water surface waveforms; (c,d,g,h,k,l,o,p) time histories of forced motion and hydrodynamic forces. (c,d,g,h,k,l,o,p) time histories of forced motion and hydrodynamic forces. 5. Conclusions 5. Conclusions In this paper, the hydrodynamic loads on semi-submerged cylinder in an oscillatory flow are experimentallyIn this paper, studied. the hydrodynamic To generate the loads equivalent on semi oscillatory-submerged flow, cylinder the semi-submerged in an oscillatory cylinder flow are is forcedexperimentally to oscillate studied. in various To combinations generate the ofequivalent different amplitudesoscillatory flow, and periods. the semi The-submerged KC number, cylinder Stokes is numberforced to and oscillate Froude in number various respectively combinations ranges of fromdifferent 6.3 to amplitudes 37.7, from 5434and toperiods. 11,364 The and fromKC number, 0.12 to 1.55.Stokes The number hydrodynamic and Froude loads number in both respectively IL and CF ranges directions from are 6.3 measured. to 37.7, from The 5434 main to hydrodynamic11,364 and from features0.12 to are 1.55. observed The hydrodynamic and summarized. loads The in empirical both IL and prediction CF directions formula are of the measured. lift force inThe the main CF directionhydrodynamic and the features relationships are observed among theand hydrodynamic summarized. coeTheffi empiricalcient, KC prediction number, Stokes formula number of the and lift Froudeforce in number the CF are direction further and investigated. the relationships The main among conclusions the hydrodynamic are as follows: coefficient, KC number, Stokes number and Froude number are further investigated. The main conclusions are as follows: (1) Similar to the results in steady flow, a significant mean downward lift force of semi-submerged (1) cylinderSimilar can to the also results be witnessed. in steady The flow, dominant a significant frequencies mean of fluctuating downward lift lift forces force are of always semi- twicesubmerged the forced cylinder oscillation can frequency. also be witnessed. The dominant frequencies of fluctuating lift (2) Basedforceson are the always observed twice load the features,forced oscillation a novel empirical frequency. prediction formula of the lift force on (2) semi-submergedBased on the observ cylindered underload features, oscillatory a novel flow was empirical first proposed, prediction which formula expresses of thethe lift lift forces force ason proportional semi-submerged to the square cylinder of the under oscillatory oscillatory flow velocityflow was in first the IL proposed, direction. which This novel expresses empirical the lift forces as proportional to the square of the oscillatory flow velocity in the IL direction. This novel empirical formula directly explains the underlying mechanism of twice Appl. Sci. 2020, 10, 6404 16 of 17
formula directly explains the underlying mechanism of twice frequency oscillation observed in lift forces, which also provides a more accurate prediction of dynamic load. This formula can well describe the main lift force features including mean value, oscillation that with frequency twice of the oscillatory flow frequency and the leading phase difference. (3) The maximum lift coefficient can be found to be 1.5, which is larger than the typical values of drag coefficient (CD = 1.2) for a fully submerged cylinder with infinite depth. Such a large lift force may cause related structure damage and should be considered in real engineering projects. (4) With KC number increasing, the significant evolution trends between drag coefficients, added mass coefficients, lift coefficient and phase difference angle with Froude number under the same KC number were found. It suggests that using the Froude number and KC number to predict the hydrodynamic coefficient is more reasonable when the Stokes number and Froude number are in the conflict for the semi-submerged cylinder.
Author Contributions: Conceptualization, H.R., S.F. and C.L.; methodology, H.R.; validation, M.Z., Y.X., S.D. and S.F.; formal analysis, H.R.; investigation, H.R., S.F.; resources, S.F.; data curation, H.R.; writing—original draft preparation, H.R.; writing—review and editing, H.R.; visualization, H.R.; supervision, S.F.; project administration, S.F.; funding acquisition, S.F. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the National Natural Science Foundation of China (No. 51825903, 51909159), the Shanghai Science and Technology Program (No. 19XD1402000, 19JC1412800, 19JC1412801), the Joint Funds of the National Natural Science Foundation of China (No. U19B2013), the Key Projects for Intergovernmental Cooperation in International Science (No. 2018YFE0125100). Acknowledgments: The authors gratefully acknowledge the support from the National Natural Science Foundation of China (No. 51825903, 51909159), the Shanghai Science and Technology Program (No. 19XD1402000, 19JC1412800, 19JC1412801), the Joint Funds of the National Natural Science Foundation of China (No. U19B2013), the Key Projects for Intergovernmental Cooperation in International Science (No. 2018YFE0125100). Conflicts of Interest: The authors declare no conflict of interest.
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