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Experimental study on flow kinematics and pressure distribution of green water on a rectangular structure
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Experimental study on flow kinematics and pressure distribution of green water on a rectangular structure Gang Nam Lee a, Kwang Hyo Jung a, *, Sime Malenica b, Yun Suk Chung c, Sung Bu Suh d, Mun Sung Kim e, Yong Ho Choi f a
Department of Naval Architecture and Ocean Engineering, Pusan National University, Busan, Republic of Korea Bureau Veritas, Paris, France Bureau Veritas, Busan, Republic of Korea d Department of Naval Architecture and Ocean Engineering, Dong-Eui University, Busan, Republic of Korea e Offshore Engineering Team, Samsung Heavy Industries Co., Ltd., Gyeonggi-do, Republic of Korea f Ship and Offshore Research Center, Samsung Heavy Industries Co., Ltd., Dae-jeon, Republic of Korea b c
A R T I C L E I N F O
A B S T R A C T
Keywords: Green water Flow kinematics Bubbly flow Pressure distribution Impulsiveness
In this study, flow kinematics and pressure distribution of green water on a fixed rectangular structure were investigated with a series of experiments performed in a two-dimensional wave flume. The experiments were conducted under regular wave conditions which were scaled down by a ratio of 1:125 from the BW Pioneer Floating Production Storage and Offloading (FPSO) operated in the Gulf of Mexico. PIV and BIV measurements were applied to obtain the water and bubbly flow in order to study the flow kinematics of multi-phase flow induced by green water. The mass flow rate and the kinetic energy, which were calculated from the water ve locity profiles obtained by the PIV measurements were compared with and without the structure to investigate the quantitative flow kinematics of the water flow at the weather side of the structure. The characteristics of the bubbly flow in the water shipping phase on the deck were investigated based on the velocities of the bubbly flow obtained by the BIV measurements, and the velocities of the bubbly flow were compared with the solutions of dam break theory. Pressure measurements were taken along the structure to estimate the local loading of the green water phenomena and they provided the maximum peak pressure, rise time, impulsiveness of the pres sures, and the peak frequencies of the pressure oscillation due to the air pocket generation on the wall of the structure during the green water.
1. Introduction
For decades, several studies on green water have been conducted with experimental and numerical methods in efforts to understand the phenomena better and to predict the damage they cause. Greco (2001) conducted a series of experiments with a fixed rectangular model in a 2-D wave flume and observed the characteristics of the water that overtopped on the deck in regular waves. Various numerical approaches have been validated with the results of Greco’s experimental study using the volume of fluid (VOF) model (Nielsen and Mayer, 2004), the MPS method (Zhang et al., 2013), and the immersed boundary method (Yan et al., 2018). Ariyarathne et al. (2012) investigated the impact of green water due to plunging breaking waves on a simplified 3-D model while measuring the impact pressures on the structure and the void fractions of green water flow. Lee et al. (2012) provided a series of measured data that include wave elevations and pressure distributions on an FPSO with
Green water impact is one of the major concerns in the design of ships and offshore structures in harsh environments because it can cause tremendous damage to the topside equipment and injuries to crew on deck. Mitigation of green water loading has been an important consid eration in the design of the shapes of hulls, especially for offshore structures that must operate for a long time in survival conditions. Despite the fact that numerous efforts have been made to predict the structural loads and characteristics of the green water phenomena, these phenomena are still challenging because of the high-nonlinear charac teristics of the wave-structure interaction and the violent, multi-phase flow, which make it difficult to estimate damage with analytical theories.
* Corresponding author. E-mail addresses: [email protected] (G.N. Lee), [email protected] (K.H. Jung). https://doi.org/10.1016/j.oceaneng.2019.106649 Received 14 May 2019; Received in revised form 2 September 2019; Accepted 26 October 2019 0029-8018/© 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: Gang Nam Lee, Ocean Engineering, https://doi.org/10.1016/j.oceaneng.2019.106649
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water assumption. Ryu and Chang (2005) compared the solution of the dam-break theory with experimental results and suggested a method to determine the initial water depth of the dam-break model. The dam-break model has also been used in various numerical simulations, e.g., the Navier-Stokes solver (Kleefsman et al., 2005), the smooth par ticle hydrodynamics (SPH) method (Khayyer et al., 2009), and the semi-analytical solution using motion equations in Lagrangian form (Yilmaz et al., 2003). To understand the flow kinematics and the multi-phase character istics of the green water phenomena, Ryu and Chang (2007) measured the velocity fields of the green water with a plunging wave breaker using the particle image velocimetry (PIV) technique and the bubble image velocimetry (BIV) technique. They provided the dominant flow veloc ities for different green water phases around a model structure. Lee et al. (2016) used the PIV and BIV techniques to perform model tests to investigate the behavior and flow kinematics of green water on a rect angular structure. In Lee et al. (2016)’s study, the generation of green water was categorized into five steps, the flow characteristics of the multi-phase flow of green water around a structure were studied, and the experimental data were compared with the results of their CFD simulations. Even though they investigated the flow kinematics of the multi-phase flow due to green water, there was no description of the relationship between its highly-aerated flow and the impact loading of green water. In order to investigate the pressures due to green water, some re searchers have focused on the features of the air pockets that are generated when waves impact structures. The air pockets might be captured on the wall when a steep wave approaches a vertical structure, and they start to be compressed and oscillate as the wave propagates. It has been shown that the entrapped air reduces the impact pressure on a structure caused by the incoming waves and also distributes the pressure over a wide surface of the structure, so the total impulse on the wall might not be decreased (Hattori et al., 1994). For the characteristics of the air pocket in the wave impact on a vertical wall, Topliss et al. (1993) suggested a formula for the frequency of pressure oscillations derived from a boundary value problem and the natural frequency of pressure oscillations provided by the formula was compared with the
Table 1 Principal dimensions of rectangular structure model. Length (m) Breadth (m) Depth (m) Freeboard (m) Draft (m)
BW Pioneer
Model
241.0 42.0 20.4 6.5 13.9
N/A 0.336 0.163 0.052 0.111
BW Pioneer
Model
241.0 42.0 20.4 6.5 13.9
N/A 0.336 0.163 0.052 0.111
Table 2 Wave conditions of experiments. Length (m) Breadth (m) Depth (m) Freeboard (m) Draft (m)
changing bow models, and Gatin et al. (2018) conducted a validation of the experimental results using Naval Hydro pack software, which is an extension of the collocated FV-based CFD open course software foam-extend (Weller et al., 1998; Jasak, 2009). Silva et al. (2017a) performed extensive model tests for an FPSO exposed to beam and quartering seas, and the results were compared with CFD simulations (Silva et al., 2017b). Rosetti et al. (2019) studied water shipping and its impacts on a fixed FPSO in beam waves with experimental analysis and CFD simulations. The results showed that the CFD simulations could be used to predict the force impulses of green water accurately. It is well known that the green water flow from the sides of a ship significantly resembles the behavior of dam-break flow (Goda et al., 1979). Because of the difficulty in using experimental techniques to investigate the height or the velocities of the water on deck, researchers have used dam-break theory to predict the flow characteristics of green water on deck. Buchner (2002) applied Stoker’s solution (Stoker, 1957) of the dam-break model on the green water phenomena with the shallow
Fig. 1. Experimental set-up in a 2-D wave tank. 2
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Fig. 4. Field of View for the PIV measurement.
Fig. 2. Comparison of (a) wave elevations and (b) horizontal water particle velocity profiles with wave theory (Stokes’ 3rd order).
Fig. 5. Field of View for BIV measurement.
obtained by Lugni et al. (2010). Even so, more studies are needed to get an accurate understanding of the role of air pockets in green water impacts. In the present study, the green water phenomena on a fixed rectan gular structure was investigated with a series of experiments in a 2-D wave flume by measuring the velocity profiles of the water, and the bubbly flow, and pressure distributions on the structure. The velocity fields of water flow were obtained by PIV, and bubbly flow was assessed by the BIV measurement technique. This paper provides the flow kine matics of the green water phenomena, i.e., the mass flow rate and kinetic
Fig. 3. Wave elevation and the duration of the measurements in the experiments.
experimental results of Hattori and Arami (1993) and Witte (1988). Abrahamsen and Faltinsen (2012) proposed an improved formula that considered the shape of the air pocket in estimating the natural fre quency, and the formula was validated by the experimental results 3
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Fig. 6. Locations of the pressure measurements.
Fig. 7. Spectrum of the background noise of the pressure sensor. Fig. 9. Velocity fields of multi-phase flow for each phase of the green water generation ((a) flip-through; (b) Air-entrapment; (c) Wave run-up; (d) Wave overturning; (e) Water shipping). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
2. Experimental setup and techniques 2.1. Experimental setup A series of experiments was performed in a 2-D wave tank (32 m long � 0.6 m wide � 1 m deep) to investigate the green water phenom ena on a fixed rectangular structure. The wave tank had a piston-type wave maker and a sloped beach (1:3) at the end of the wave tank to absorb energy of the incoming wave with a reflection ratio less than 5%. The depth of the water in the wave tank was maintained at 0.60 m. The acrylic, rectangular shaped model was installed wall-to-wall in the beam sea condition and fixed with an alloy frame at the position 15 m from the wave maker. The dimensions of the model were deter mined based on the BW Pioneer FPSO, which has been operating in the west central of the Gulf of Mexico since 2008, and it was scaled down with the ratio of 1:125 (Table 1). The length of the model was the same as the dimension of the wave tank (0.60 m) in order to avoid having to consider the 3-D effects of the green water phenomena, and the length was not followed by the scale ratio. Table 2 shows that the regular wave conditions were chosen as the significant wave height (H1/3) and the peak period (TP) of the 100-yrs return period in the Gulf of Mexico (API-2INT-MET, 2007). In addi tion, two more wave conditions (H1/10 and H1/100) with larger steepness than the significant wave height were estimated (Lee et al., 2016), and used in the experiments to account for extreme ocean environments. All of the equipment in this experiment was synchronized, and all of the
Fig. 8. Ratio of zeroth moment of the spectrum for the order of the FIR filter.
energy at the weather side of the structure, the velocity profile of bubbly flow on the deck due to green water and comparison with dam-break theory, and the pressure distributions on the wall and the deck with analysis of the impact pressures. The air pocket generated in the green water around the structure was studied, and the peak frequency of the air pocket was compared with the analytical solutions provided by Topliss et al. (1993) and Abrahamsen and Faltinsen (2012).
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corresponded to the special resolutions of 2.72 mm between every ve locity vector in the interrogation area of 64ⅹ64 pixels with 50% overlap (Fig. 4). The continuous laser (Max. 8 Watt, wave length of 532 nm) was used to illuminate the PIV particles, which had a mean diameter of 57 μm and a specific gravity of 1.02. The adaptive multi-pass algorithm was used to get a stronger cross-correlation, as shown in Equation (1) (Raffel et al., 1998) CðΔx; ΔyÞ ¼
M 1 X N 1 1 X f ðm; nÞgðm þ Δx; n þ ΔyÞ� MN i¼0 j¼0
(1)
where the C cross-correlation function, f and g are the distribution of intensity in continuous images, i and j are the positions in the horizontal and vertical directions, respectively, and M and N are the number of pixels in each direction, respectively. The false vectors in the vector fields that were calculated were eliminated by using the median test, i.e., Equation (2) (Westerweel, 1994), jU2D ðmedianÞ
U2D ði; jÞj < εthresh
(2)
where U2D(n) is all neighboring velocity vectors, U2D(i, j) is the velocity vector that is being inspected, and εthresh is the threshold value for in spection (1.1 was used in this study). 2.2.2. Bubble image velocimetry (BIV) technique The bubble image velocimetry (BIV) technique (Mori and Chang, 2003) was used to obtain the velocity fields of bubbly flow at the weather side and on the deck of the rectangular model structure. BIV images were obtained with the a CCD camera that had 50 mm f/1.8 macro focal lens (Sigma 50 mm lens) set of f/4.0. The shadow graphy image technique (Settles, 2012) was used for the BIV images with the acquisition rate of 500 Hz, FOV size was 360 mmⅹ264 mm (Fig. 5), corresponding to the spatial resolution of 2.44 mm between each vector in the interrogation area of 32 � 32 pixels with 50% overlap. The ve locity vectors of bubbly flow in the BIV images were calculated as shown in Equation (3) using the minimum quadratic difference (MQD) method, which is more accurate than the cross-correlation method (Mori and Chang, 2003).
Fig. 10. Mass flow rate and kinetic energy within FOVs over one wave period (H1/100) without structure.
measuring procedures were completed before the reflected waves reached the model structure from the wave absorber. The wave heights and water particle velocity profiles of incoming waves were obtained by a wave gauge and the PIV system, respectively (see Fig. 1). The measured data were validated with Stokes’ 3rd order wave theory in which the incoming waves belong and were in good agreement with the theoretical values of the wave heights and the horizontal water particle velocity profiles (Fig. 2). All of the measurements were taken with one wave for each test, and the test was repeated fifteen times as shown in Fig. 3. To avoid the re flected waves, the time duration of the measurements was chosen at the transient wave passed and before the arrival of the reflected wave from the wave absorber.
Dðm; nÞ ¼
M X N 1 X ½g1 ði; jÞ MN i¼1 j¼1
g2 ði þ m; j þ nÞ�
(3)
2.3. Pressure measurements Pressure distributions along the structure of the model were obtained to estimate the local loading caused by the green water phenomena on the rectangular structure. The pressures due to green water were measured at three locations on the weather side and at two locations on the deck of the structure (Fig. 6). The pressure data were acquired by piezo-resistive type pressure sensors (Kistler 4043A2) which could measure the static and dynamic pressures covering the range from 0 to 2 bars. The sampling rate for the measurements was maintained at 5 kHz, which was determined by conducting convergence test. All measurements were repeated 15 times with the same wave conditions. The finite impulse response (FIR) low-pass filter, which has less phase and delay distortion than the infinite impulse response (IIR) filter (McClellan and Parks, 1973) was used to remove the background noise in the pressure data. The cut-off frequency for the low-pass filter was determined to be 150 Hz considering the frequency component of the background noise (Fig. 7), and the 67th order of the FIR filter was used to maintain the ratio of zeroth moment of the spectrum for filtered and raw signals up to 90% for the components that were lower than the cut-off frequency (Equation (4)) and to maintain a ratio lower than 10% for the higher components than the cut-off frequency (Equation (5)) as shown in Fig. 8,
2.2. Experimental techniques 2.2.1. Particle image velocimetry (PIV) technique The particle image velocimetry (PIV) technique was used to obtain the velocity fields of the green water phenomena at the weather side of the structure with exception of the bubbly flow region. The PIV images were obtained with an acquisition rate of 500 Hz using a high-speed CCD camera (Redlake Y5) mounted with a 105-mm, f/1.8 macro focal lens (Nikkon 105 mm lens) set at f/2.8. The lens had 2352ⅹ1728 pixels, a pixel size of 7μm, 8-bit dynamic range. The two different fields of view (FOV) were used to cover the entire region of the weather side of the rectangular structure with each size being 147 mmⅹ200 mm which 5
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Fig. 11. Comparison of mass flow rates with and without the structure ((a) H1/100, (b) H1/10, (c) H1/3).
R FC Sf ðω; nÞdω RA ðnÞ ¼ 0R FC � 0:9 Sr ðωÞdω 0
� Flip-through: Incoming wave is deformed vertically with a concave face before the wave front impacts on the weather side of the structure � Air-entrapment: Air is entrapped on the weather side of the structure � Wave run-up: The water level increases along the weather side and splashes up into air � Wave overturning: The wave overturns like a plunging wave breaker and impacts on the deck � Water shipping: The body of bubbly water is moved forward along the deck
(4)
R∞
Sf ðω; nÞdω F RB ðnÞ ¼ RC∞ � 0:1 S ðωÞdω FC r
(5)
where RA and RB are the ratios of the zeroth moment of the two spectra, FC is the cut-off frequency, Sf and Sr are the spectra of the filtered signal and the raw signal, respectively, ω is the frequency, and n is the order of the FIR filter.
In this study, the PIV and BIV measurements were used to obtain the velocity fields of water and bubbly flow near the rectangular structure in order to investigate the flow kinematics of the multi-phase flow due to the green water phenomena. Fig. 9 shows the vector fields that were acquired at each step of the green water generation. The blue and yellow vectors in the images represent the velocity of water around the struc ture obtained by PIV measurements and the velocity of bubbly flow by BIV measurements, respectively. To investigate the quantitative flow kinematics on the weather side of the structure during the generation of green water, the mass flow rate and the kinetic energy (per unit length in the horizontal direction) were calculated with the water velocity profiles obtained by the PIV measurements. The mass flow rate and the kinetic energy passing through a vertical
3. Results 3.1. Flow kinematics at the weather side of structure For the generation of the green water phenomena on a rectangular structure, Greco (2001) classified the green water generation into two-stages, i.e., 1) the initial stage of the water shipping when the water started to be shipped on deck and 2) the later stage of the water shipping that formed a dam-break type on deck. Lee et al. (2016) suggested that the generation of green water could be categorized into the five phases presented below.
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Fig. 12. Comparison of kinetic energy with and without the structure ((a) H1/100, (b) H1/10, (c) H1/3).
cross section in FOV were calculated using the following equations, Z η _ tÞjx¼C:S: ¼ Mðx; ρbuðx; z; tÞdz (6)
with the structure for the three wave heights over the entire wave period. The total area under the curves in Fig. 11 is the mass of water that went through the cross section at x=B ¼ 0:12 during one wave period; this mass, was less than it would have been due to the existence of the structure. The green water phenomena on the deck and the re flected wave from the structure would be the main reasons for the decrease in the mass flow rate of the water. And, due to the super position of the incoming and reflected waves, a standing wave might develop, and the vertical velocity component obviously dominant near the structure. The structure resulted in the decrease of 49%, 34%, 32% of the water mass at the condition of H1/100, H1/10 and H1/3, respectively. Fig. 12 shows the variations of the kinetic energy on the cross-section of x=B ¼ 0:12 without and with the structure. The kinetic energy without the structure was the maximum magnitude at the wave crest phase, and it was distributed in a symmetrical shape with respect to the phase π. For the case with the structure, the distribution of the kinetic energy changed to an asymmetric pattern that had two peaks. The magnitude of vertical velocity component was increased dramatically due to the green water phenomena, of which the first and second peaks matched with the wave run-up phase and the water shipping phase, as shown in Fig. 9(c) and (e), respectively.
s
Z KEðx; tÞjx¼C:S: ¼
η s
1 1 ρbu2 ðx; z; tÞ þ ρbw2 ðx; z; tÞ dz 2 2 |{z} |{z} Horizontal
(7)
Vertical
where M_ is the mass flow rate, η is the free surface, ρ is the density of water, b is the width of the wave flume, KE is the kinetic energy u and w are the horizontal and vertical velocities, respectively, and s is the bot tom of the FOV. The values were normalized with ρvp db and ρvp 2 db for each mass flow rate and kinetic energy, where vp denotes the phase velocity of the incoming waves (vp ¼ 1.72 m/s). To investigate the spatial differences in the mass flow rate and the kinetic energy, the values were obtained at seven fixed and equallyspaced locations in the FOV, and the averaged values are shown in Fig. 10 with the error bars which indicate the corresponding 95% con fidence interval. The errors in both the mass flow rate and the kinetic energy were less than 10% of each maximum quantity over one wave period. In this study, all of the values of mass flow rate and kinetic en ergy were shown with the results obtained on the cross section at x= B ¼ 0:12, where B is breadth of the model that was out of the bubbly flow region. Fig. 11 shows the comparison of the mass flow rates without and
3.2. Flow kinematics of water shipping on deck When the green water phenomena occurred, air pockets were generated on the wall and on the deck of the structure due to the 7
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Fig. 13. Snapshot of wave overturning phase ((a) H1/100, (b) H1/10, (c) H1/3).
plunging wave breaking which overturned on the deck (Fig. 13). In the wave overturning phase (Fig. 9 (d)), the air pockets start to rotate in clock wise (CW) and fluctuate, and the bubbly flow with air pockets start to run along the surface of the structure with the motion of the bubbles in the water shipping phase (Fig. 9 (e)). This type of overtopping could be in the green water regimes of Plunging plus Dam Break (PDB) as classified by Greco et al. (2007) and Zhang et al. (2019). Due to the limitation of the PIV technique for measuring the flow kinematics of bubbly flow, due to the reflection of the laser sheet on the bubbles, the BIV measurement was used to study the flow kinematics of the water shipping phase on the deck. The time the water stayed on the deck during the generation of the green water was less than 0.1s, which was shorter than the relaxation time for the rising bubble that corresponded to the bubble size (1–2 cm) of the water shipping flow (Hysing et al., 2009). This indicated that the inertial force of the water around the bubbles had a more dominant effect on the motion of bubbles than the buoyancy force, and the vertical displacement of the bubbles in the water shipping phase on the deck was much smaller than the horizontal motion. To investigate the characteristics of bubbly flow in the water ship ping phase on the deck, the spatial mean velocities in the horizontal direction were obtained for every 20ⅹ2.5 mm2 using the velocity profiles obtained by BIV measurements, as shown in Fig. 14. The horizontal mean velocities were normalized by the phase velocity of the incoming waves (vp ¼ 1.72 m/s), of which magnitude presented in the distance of each symbol from the dashed line of its left-hand side. Fig. 14 shows the mean velocity profiles every 0.03 s after the wave overturning phase for each wave height. The distribution of the symbols presented the velocity profiles of bubbly flow erratically, because they could only be measured in the bubble region by the BIV technique. The error bar indicates the 95% confidence interval of each mean velocity. More bubbles were generated with the higher wave height condition, and those velocities were, for the most part, faster than the velocities with the lower wave condition. The magnitude of the velocity of the bubbly flow increased gradually along the deck, and the maximum values occurred in the range
of 120–160 mm away from the corner of the structure. The horizontal velocities of the bubbly flow were compared with the analytical solution of the dam breaking phenomena (Stoker, 1957), as shown in Equation (8), 2 �pffiffiffiffiffiffi x� gh’ þ (8) uðx; tÞ ¼ 3 t where u is horizontal velocity, g is the gravitational acceleration, and h’ is the initial water depth of the dam. Buchner (2002) suggested the use of the freeboard exceedance instead of the initial water depth to predict the horizontal velocity of the water shipping phase due to the green water phenomena (Equation (9)). 9 h’ ¼ h 4
(9)
where h is the freeboard exceedance. Ryu et al. (2007) tested the green water phenomena on a fixed rectangular structure due to the plunging breaking wave and suggested the use of Equation (10) to estimate the horizontal velocity of the water on the deck. h’ ¼ 0:36
vp 2 g
(10)
Fig. 15 compares the horizontal velocities of the bubbly flow spatially averaged in the z-direction and the solutions of a dam break problem for every 0.03 s after the wave overturning phase. The symbols and the error bars are the averaged bubbly flow velocities and its cor responding 95% confidence interval on the deck, respectively, which were obtained by BIV measurements. In the study, it was suggested that, rather than using initial water depth in Equation (8) to predict the horizontal velocity of the water on the deck, the water level from the corner of the structure at each time should be used, which was obtained in the instantaneous images taken by the high speed CCD camera (Fig. 16), and are shown by the blue lines in Fig. 15. This method could 8
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Fig. 14. Velocity distributions of bubbly flow on the deck using the BIV technique ((a) H1/100, (b) H1/10, (c) H1/3).
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Fig. 15. Comparison of bubbly flow on the deck with dam breaking theories (Wave condition of H1/100).
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increased up to the maximum water level at the corner of the structure which was defined as the wave overturning phase (Fig. 9 (d)). Fig. 17 (c), (d), and (e) show that the pressures on the deck (PG3, PG4, PG5) increased when the water overtopped the deck after the wave was overturned at the corner of the structure, and the pressures gradu ally diminished due to the drainage of water during the water shipping phase along the deck. Bubbly flow occurred at the corner of the structure in the overturning phase, and it ran along the deck during the water shipping phase. But, unlike what occurred on the weather side wall, the bubbly flow was not found to have an effect on the increase in the pressure and the oscillation on the deck. Fig. 18 shows that the pressure on the deck was induced mainly by the hydrostatic pressure over each pressure sensor. Fig. 19 shows that the pressures that were measured were nondimensionalized by 0:5ρu2max, where umax is the maximum horizontal velocity of the water of the incoming wave using linear wave theory. At the weather side wall (PG1 and PG2), the non-dimensionalized pressures at the smaller wave height condition were larger than those at the larger wave height condition. This occurred because, except for the first peak pressure, the pressure on the wall did not increase as much as the incoming wave increased. This showed that the largest non-dimensionalized pressures on the deck (PG3, PG4, and PG5) were related to each incoming wave height condition, and there was an almost linear relationship with the square of the maximum horizontal water velocity of the incoming wave. Fig. 20 shows the mean values and the corresponding 95% confi dence interval of the maximum peak pressure and rise time at the first pressure peak obtained by repeating the tests at each wave condition fifteen times. The rise time was calculated as twice the time duration from the half peak to the peak value, as suggested by DNV-GL (DNV-GL, 2016). In Fig. 20 (a), because of the bubble entrapped on the weather-side wall at the wave conditions of H1/100 and H1/10, the pres sures measured at PG1 and PG2 had larger mean magnitudes and standard deviations than the pressures on the deck. Also, the rise times at the weather-side wall, i.e., less than 0.01 s, were shorter than those on the deck (Fig. 20 (b)). The magnitude of the peak pressure was inversely proportional to the duration of the rise time in the air-entrapment phase, which showed similar patterns with previous studies of the wave impact €z, phenomena on the vertical wall (Bagnold, 1939; Denny, 1951; Kırkgo 1995; Hattori et al., 1994; Cuomo et al., 2011). Fig. 21 shows that the oscillation of the pressure occurred at the weather side wall (PG1, PG2) when the air pocket was entrapped at wave conditions of H1/100 and H1/10. The symbols indicates the raw pressures that were measured with the sampling rate of 5 kHz, and the solid line is the signal filtered by an FIR low-pass filter. The specifica tions of the low-pass filter were presented in section 2.3. Fig. 22 shows the results when the pressure spectra were plotted for fifteen repeated tests at the same wave conditions. Note that the peak magnitude of the spectrum and its corresponding frequency at the wave condition of H1/ 100 were larger and lower, respectively, than that of H1/10. The pressure spectra were different for each repeated test, but the peak frequencies of the spectra were coincident in all of the tests. The peak frequencies of pressure oscillation were matched for the same wave condition at PG1 and PG2, even though the air pocket was rotated at the location of PG1 and turned to the corner of the structure. Fig. 21 and Equations (9) and (10) show that the peak frequency of the incoming wave height, H1/100, was lower than that of H1/10 because it depended on the size of air bubble and its location with respect to the free surface. Topliss et al. (1993) and Abrahamsen and Faltinsen (2012) suggested a formula for estimating the peak frequencies of the wave impact on the vertical wall and the sloshing in the rectangular tank based on 2-D po tential flow theory assuming an adiabatic pressure-density relationship. Equation (11) is the analytical solution provided by Topliss et al. (1993),
Fig. 16. Measurement of water level on the deck.
estimate the measured velocities better than the solutions of Equations (9) and (10) for every instance. The solutions of Equations (9) and (10) overestimated the velocities because of the fixed y-intercepts that did not vary with time. The method suggested in this study includes the effect of variations in the free surface during the generation of the green water, which was not provided by the solutions with Equations (9) and (10). 3.3. Pressure measurements Pressures were measured on the wall and on the deck of the structure during the generation of green water, and Fig. 17 shows the results for one wave period for three different wave heights. Fig. 17 (a) and (b) show that the pressures on the wall of the structure (PG1 and Pg2) increased suddenly, the first pressure peak was reached at the moment the air pocket was entrapped when the wave face hit the wall, and the pressure was oscillated and decreased during the airentrapment phase (Fig. 9 (b)). The first pressure peak at PG1 was larger than the pressure peaks at PG2 even though the wave face hit directly near PG2 after the flip-through phase. When the wave height was H1/3, the pressure peak and oscillation did not occur because no air pocket was entrapped on the wall (Lee et al., 2016). Thus, the magnitude of the first pressure peak and the pressure oscillation phenomena occurred due to the bubble effect of the entrapped air on the wall. The pressure increased gradually during the wave run-up along the wall (Fig. 9 (c)), and the second peak was reached as the free surface 11
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Fig. 17. Time histories of pressure as the incoming wave heights changed.
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Fig. 18. Comparison of the pressures at P1 and P3 during the different phases of the generation of green water. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
ω2 ¼
2γpatm
ρr2 logðr=2hÞ
defined as the radius of a semi-circle that corresponded to the area of the ellipsoidal air pocket on the structure. Both methods could predict the peak frequencies with reasonable agreement with test results. This may be the result of the differences between the assumptions of the 2-D theories and the uncertainty of the experimental sizing of the area of air bubbles near the wall of the wave tank. The wave load acting on a vertical wall was suggested by two pat terns, i.e., “impact” (impulsive or dynamic) and “quasi-static” (pulsat ing) wave loads (Allsop et al., 1996) as follows;
(11)
where γ is the ratio of specific heats, which is 1.4 for air, r is the equivalent radius of the air pocket, and h is the distance from the free surface to the center of the air pocket (Fig. 23). Abrahamsen and Faltinsen (2012) suggested a formula (Equation (12)) that considers the shape of the air pocket which might not be a semi-circle, � � �� 2γpatm K sin π2ab � � �� ω2 ¼ (12) Ω0 K sin π2ab
� An impact (impulsive or dynamic) wave load could be caused by the special conditions that arise when a wave breaks onto the structure with greater intensity than the quasi-static load in a shorter duration. � A quasi-static (pulsating) wave load varies with the order of magnitude of the free surface elevation for a relatively long duration.
where a and b are the distances from free-surface and the upper part of air pocket (Fig. 23), respectively, Ω0 is the area of the air pocket, and K is obtained by Equation (13). Z 1 1 pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx KðkÞ ¼ (13) 1 x2 1 k 2 x2 0
A few studies have provided quantitative definitions of the impact criteria. Hayashi and Tanaka (1988) classified the dynamic mode of loading versus the strain rate using the five criteria of creep, static or quasi-static, dynamic, impact, and hypervelocity impact. Oumeraci et al. (2001) suggested defining the impact phenomena as a peak loading 2.5 times larger than the subsequent quasi-static loading due to the water
Fig. 24 shows the peak frequencies measured in tests compared with the results of Equations (9) and (10). The equivalent radius (r) was 13
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Fig. 18. (continued).
waves on the vertical wall. Ariyarathne et al. (2012) suggested a specific value of impulsiveness using the non-dimensional pressure rate to the rise time with the wave celerity for the green water phenomena due to a plunging breaking wave. In this study, the impulsiveness (Ariyarathne et al., 2012) was esti mated using the pressure rate to the rise time and the maximum water velocity under the regular wave. This was done because the dominant factors would be the severity of pressures and the horizontal water ve locity acting on the vertical wall instead of the wave celerity. The impulsiveness of Ariyarathne et al. (2012) was modified as shown in Equation (14), Pimp
IP ¼ 0:5ρtur
max
2
> 1500
where Pimp is the maximum pressure, tr is the rise time, umax is the maximum horizontal velocity of the water of the incoming wave using the linear wave theory. The impulsiveness (Ip) was calculated for each test of fifteen repeated trials at the same wave condition for three wave heights i.e., H1/100, H1/ 10, and H1/3, as shown in Fig. 25. The impulsiveness was larger at the location on the weather side where an air pocket was generated. The impulsiveness at PG1 was noticeable at the conditions of H1/100 and H1/ 10 at which the movement of the air pocket was very active on the weather side. At the location of PG2, the impulsiveness was smaller than it was at PG1, because its location was out of the air pocket region. But, the pressure rate to the rise time would be more significant for the higher wave height condition, i.e., H1/100, than the pressure rate to rise time would be for the wave height of H1/10 and H1/3 due to increased
(14)
T
14
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Fig. 19. Non-dimensionalized pressure time histories with changing incoming wave height.
15
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Fig. 22. Spectra of the pressure oscillations at PG1 ((a) H1/100, (b) H1/10).
Fig. 20. Mean and Standard deviation of the maximum pressure and rise time of the first peak.
Fig. 23. Snapshots of air pockets in the air-entrapment phase ((a) H1/100, (b) H1/10).
Fig. 21. Oscillation of the pressure at the PG1 (wave condition of H1/100).
which was defined as the quasi-static loading by Allsop et al. (1996). The impulsiveness values at the locations of PG3, PG4, and PG5 on the deck were smaller than the suggested threshold, and the wave loading on the deck could be represented by the quasi-static criteria.
horizontal velocity of the water acting on the vertical wall. This suggests an impulsiveness threshold of (Ip > 1500) to define the wave impact criteria, which was estimated to be 1500 at PG1 in the condition of H1/3 without the generation of an air pocket due to the wave action on the vertical wall. The pressures along the deck were similar to the hydrostatic pressure estimated by the water elevation over each of the pressure sensors,
4. Conclusions Green water phenomena on the fixed rectangular structure were 16
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Fig. 24. Comparison of the peak frequencies of pressure oscillation with the analysis formula.
studied with a series of experiments conducted in 2-D wave flume. The model was simplified and scaled down with the ratio of 1:125 from FPSO BW Pioneer which has been operating in the west central of the Gulf of Mexico. Three regular wave conditions were tested with the wave heights of H1/100, H1/10, and H1/3, and the peak period of 100-yrs return period was used at the operating site. PIV and BIV measurements were used to obtain the water and bubbly flow to study the flow kinematics of the multi-phase flow induced by green water. The mass flow rate and the kinetic energy, which were calculated from the water velocity profiles obtained by PIV measure ments, were compared with and without the structure to investigate the quantitative flow kinematics of water flow at the weather side of the structure. The characteristics of bubbly flow in the water shipping phase on the deck were investigated based on the velocities of the bubbly flow obtained by BIV measurements, and the velocities of the bubbly flow were compared with the solutions of dam break theory. Pressure measurements were conducted along the structure of the model to estimate the local loading due to green water, and the maximum peak pressure and rise time were provided. The peak fre quencies of pressure oscillation due to the air pocket entrapped on the weather side wall were analyzed and compared with the results of the formulas suggested by Topliss et al. (1993) and Abrahamsen and Fal tinsen (2012). The impulsiveness suggested by Ariyarathne et al. (2012) was modified, and a new equation using the horizontal velocity of the water was suggested to classify the impact criteria. From the results obtained in this study, the findings are summarized as below: - It was observed that the horizontal mass flow rate in FOV at the weather side of the structure was decreased due to the existence of the structure. The kinetic energy in the same FOV was increased drastically due to the existence of structure having two peaks at the wave run-up and the water shipping phases. The main reason was that the horizontal velocity component of the incoming wave became the vertical direction velocity, which might have been caused by the local standing wave that was generated by the superposition of the incoming and reflected waves near the structure. - At the higher wave height condition, the velocity of bubbly flow had more bubbles and became faster on the deck. Those velocities increased gradually along the deck and had the maximum values at 120–160 mm away from the corner of the structure. The horizontal bubbly flow velocities were compared with the solutions of the dam breaking problem. It was suggested that the dam break theory to be used with the water level from the corner of the structure for the initial water depth. In this study, the suggested method showed
Fig. 25. Impulsiveness results of pressure measurements ((a) H1/100, (b) H1/10, (c) H1/3).
better agreement with the measured bubbly flow velocities than those of Buchner (2002) and Ryu et al. (2007), because it was able to utilize the free surface variation during the green water generation instead of the initial water depth, which was constant (Buchner, 2002; Ryu et al., 2007). 17
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- The pressure at the weather side wall increased the first pressure peak at the air-entrapment phase and, it oscillated due to the air pocket that was entrapped on the wall. Following the first peak and oscillation, the pressure increased gradually, with the free surface rising, and the second peak was reached at the wave overturning phase. The pressures on the deck increased when the water over topped the deck in the wave overturning phase, and it was reduced gradually during the water drainage along the deck. The pressure peaks on the deck, which were non-dimensionalized with the square of the maximum horizontal water velocity of the incoming wave, had similar magnitudes for each incoming wave height. - Pressure oscillations were observed at the weather side wall when the air pocket was entrapped at the wave conditions of H1/100 and H1/10. The peak frequencies of the pressure oscillation were compared with the analytical formulas suggested by Topliss et al. (1993) and Abrahamsen and Faltinsen (2012). The formulas could predict the peak frequencies reasonably with the differences due to the limitations of the equations limited in 2-D and the experimental uncertainty associated with sizing the area of the air pocket. - The impulsiveness (Ariyarathne et al., 2012) was applied to estimate the severity of the pressures using the maximum horizontal water velocity instead of the wave celerity. The impulsiveness due to the dynamic pressure was more dominant at the location of the air pockets generated at the weather side wall. Smaller impulsiveness was estimated on the deck because the hydrostatic pressure was dominant. A threshold was suggested to define the wave impact criteria as 1500, because it was the smallest impulsiveness due to the air pocket on the structure.
Greco, M., 2001. A Two-Dimensional Study of Green-Water Loading. Ph.D Thesis. University of Trondheim. Greco, M., Colicchio, G., Faltinsen, O.M., et al., 2007. Shipping of water on a twodimensional structure, Part 2. Journal of Fluid Mechanics 581, 371–399. Hattori, M., Arami, A., 1993. Impact breaking wave pressures on vertical walls. Coast Eng. 1785–1798, 1992. Hattori, M., Arami, A., Yui, T., 1994. Wave impact pressure on vertical walls under breaking waves of various types. Coast Eng. 22 (1–2), 79–114. Hayashi, T., Tanaka, Y., 1988. Impact Engineering. Nikkan Kogyo Simbunsha (Daily Engineering Newspaper Company), Tokyo (in Japanese). Hysing, S.-R., Turek, S., Kuzmin, D., Parolini, N., Burman, E., Ganesan, S., Tobiska, L., 2009. Quantitative benchmark computations of two-dimensional bubble dynamics. Int. J. Numer. Methods Fluids 60 (11), 1259–1288. Jasak, H., 2009. OpenFOAM: open source CFD in research and industry. Int. J. Nav. Archit. Ocean Eng. 1 (2), 89–94. Khayyer, A., Gotoh, H., Shao, S., 2009. Enhanced predictions of wave impact pressure by improved incompressible SPH methods. Appl. Ocean Res. 31 (2), 111–131. Kırkg€ oz, M.S., 1995. Breaking wave impact on vertical and sloping coastal structures. Ocean. Eng. 22 (1), 35–48. Kleefsman, K., Fekken, G., Veldman, A., Iwanowski, B., Buchner, B., 2005. A volume-offluid based simulation method for wave impact problems. J. Comput. Phys. 206 (1), 363–393. Lee, G.N., Jung, K.H., Chae, Y.J., Park, I.R., Malenica, S., Chung, Y.S., 2016. Experimental and numerical study of the behaviour and flow kinematics of the formation of green water on a rectangular structure. Brodogradnja: Teorija i praksa brodogradnje i pomorske tehnike 67 (3), 133–145. Lee, H.H., Lim, H.J., Rhee, S.H., 2012. Experimental investigation of green water on deck for a CFD validation database. Ocean. Eng. 42, 47–60. Lugni, C., Brocchini, M., Faltinsen, O., 2010. Evolution of the air cavity during a depressurized wave impact. II. The dynamic field. Phys. Fluids 22 (5), 056–102. McClellan, J., Parks, T., 1973. A united approach to the design of optimum FIR linearphase digital filters. IEEE Trans. Circuit Theory 20 (6), 697–701. Mori, N., Chang, K.-A., 2003. Introduction to MPIV. Review of Reviewed Item. Nielsen, K.B., Mayer, S., 2004. Numerical prediction of green water incidents. Ocean. Eng. 31 (3–4), 363–399. Oumeraci, H., Kortenhaus, A., Allsop, N.W.H., De Groot, M.B., Crouch, R.S., Vrijling, J. K., Voortman, H.G., 2001. Probabilistic Design Tools for Vertical Breakwaters. Balkema, Rotterdam, p. 392. Raffel, M., Willert, C.E., Kompenhans, J., 1998. Mathematical Background of Statistical PIV Evaluation, Particle Image Velocimetry. Springer, pp. 61–78. Rosetti, G.F., Pinto, M.L., de Mello, P.C., Sampaio, C.M., Simos, A.N., Silva, D.F., 2019. CFD and experimental assessment of green water events on an FPSO hull section in beam waves. Mar. Struct. 65, 154–180. Ryu, Y.U., Chang, K.A., 2005. Breaking wave impinging and green water on a twodimensional offshore structure. Proceedings of the Fifteenth International Offshore and Polar Engineering Conference; 2005. Int. Soc. Offshore Polar Eng. 660–665. Ryu, Y.U., Chang, K.A., Mercier, R., 2007. Application of dam-break flow to green water prediction. Appl. Ocean Res. 29, 128–136. Settles, G.S., 2012. Schlieren and Shadowgraph Techniques: Visualizing Phenomena in Transparent Media. Springer Science & Business Media. Silva, D.F., Coutinho, A.L., Esperança, P.T., 2017. Green water loads on FPSOs exposed to beam and quartering seas, part I: experimental tests. Ocean. Eng. 140, 419–433. Silva, D.F., Esperança, P.T., Coutinho, A.L., 2017. Green water loads on FPSOs exposed to beam and quartering seas, Part II: CFD simulations. Ocean. Eng. 140, 434–452. Stoker, J.J., 1957. Water Waves: the Mathematical Theory with Applications. Topliss, M., Cooker, M., Peregrine, D., 1993. Pressure oscillations during wave impact on vertical walls. Coast Eng. 1639–1650, 1992. Weller, H.G., Tabor, G., Jasak, H., Fureby, C., 1998. A tensorial approach to computational continuum mechanics using object-oriented techniques. Comput. Phys. 12, 620–631. Westerweel, J., 1994. Efficient detection of spurious vectors in particle image velocimetry data. Exp. Fluid 16 (3–4), 236–247. Witte, H., 1988. Wave Induced Impact Loading in Deterministic and Stochastic Reflection. Mitt Leichtweiss Inst. Wasserbau, vol. 102. Yan, B., Bai, W., Qian, L., Ma, Z., 2018. Study on hydro-kinematic characteristics of green water over different fixed decks using immersed boundary method. Ocean. Eng. 164, 74–86. Yilmaz, O., Incecik, A., Han, J., 2003. Simulation of green water flow on deck using nonlinear dam breaking theory. Ocean. Eng. 30 (5), 601–610. Zhang, X., Draper, S., Wolgamot, H., Zhao, W., Cheng, L., 2019. Eliciting features of 2D greenwater overtopping of a fixed box using modified dam break models. Appl. Ocean Res. 84, 74–91. Zhang, Y., Wang, X., Tang, Z., Wan, D., 2013. Numerical simulation of green water incidents based on parallel MPS method, the Twenty-third International Offshore and Polar Engineering Conference. Int. Soc. Offshore Polar Eng. 931–938.
Acknowledgement This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSP) through GCRC-SOP (No. 2011-0030013), and the Ministry of Trade, Industry & Energy (MOTIE, Korea) under Industrial Technology Innovation Pro gram (No. 10063405) and PNU Korea-UK Global Graduate Program in Offshore Engineering (N0001288). References Abrahamsen, B., Faltinsen, O., 2012. The natural frequency of the pressure oscillations inside a water-wave entrapped air pocket on a rigid wall. J. Fluids Struct. 35, 200–212. Allsop, N.W.H., Vicinanza, D., McKenna, J.E., 1996. Wave Forces on Vertical and Composite Breakwaters. Report SR 443. Hydraulics Research, Wallingford, United Kingdom. API BULLETIN 2INT-MET, 2007. Interim Guidance on Hurricane Conditions in the Gulf of Mexico, first ed. (Washington, D.C). Ariyarathne, K., Chang, K.-A., Mercier, R., 2012. Green water impact pressure on a threedimensional model structure. Exp. Fluid 53 (6), 1879–1894. Bagnold, R.A., 1939. Interim report on wave pressure research. J. Inst. Civ. Eng. 12, 202–225. Buchner, B., 2002. Green Water on Ship-type Offshore Structures. Delft University of Technology. Cuomo, G., Piscopia, R., Allsop, W., 2011. Evaluation of wave impact loads on caisson breakwaters based on joint probability of impact maxima and rise times. Coast. Eng. Procee. 1 (33) structures. 62. Denny, D.F., 1951. Further experiments on wave pressures. J. Inst. Civ. Eng. 35, 330–345. DNV-GL, 2016. Sloshing analysis of LNG membrane tanks. DNV Classification Notes 30 (9). Gatin, I., Vuk�cevi�c, V., Jasak, H., Seo, J., Rhee, S.H., 2018. CFD verification and validation of green sea loads. Ocean. Eng. 148, 500–515. Goda, K., Miyamoto, T., Yamamoto, Y., 1979. A study of shipping water pressure on deck by two-dimensional ship model test. Naval Arch. Ocean Eng. 17.
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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Experimental study on flow kinematics and pressure distribution of green water on a rectangular structure Gang Nam Lee a, Kwang Hyo Jung a, *, Sime Malenica b, Yun Suk Chung c, Sung Bu Suh d, Mun Sung Kim e, Yong Ho Choi f a
Department of Naval Architecture and Ocean Engineering, Pusan National University, Busan, Republic of Korea Bureau Veritas, Paris, France Bureau Veritas, Busan, Republic of Korea d Department of Naval Architecture and Ocean Engineering, Dong-Eui University, Busan, Republic of Korea e Offshore Engineering Team, Samsung Heavy Industries Co., Ltd., Gyeonggi-do, Republic of Korea f Ship and Offshore Research Center, Samsung Heavy Industries Co., Ltd., Dae-jeon, Republic of Korea b c
A R T I C L E I N F O
A B S T R A C T
Keywords: Green water Flow kinematics Bubbly flow Pressure distribution Impulsiveness
In this study, flow kinematics and pressure distribution of green water on a fixed rectangular structure were investigated with a series of experiments performed in a two-dimensional wave flume. The experiments were conducted under regular wave conditions which were scaled down by a ratio of 1:125 from the BW Pioneer Floating Production Storage and Offloading (FPSO) operated in the Gulf of Mexico. PIV and BIV measurements were applied to obtain the water and bubbly flow in order to study the flow kinematics of multi-phase flow induced by green water. The mass flow rate and the kinetic energy, which were calculated from the water ve locity profiles obtained by the PIV measurements were compared with and without the structure to investigate the quantitative flow kinematics of the water flow at the weather side of the structure. The characteristics of the bubbly flow in the water shipping phase on the deck were investigated based on the velocities of the bubbly flow obtained by the BIV measurements, and the velocities of the bubbly flow were compared with the solutions of dam break theory. Pressure measurements were taken along the structure to estimate the local loading of the green water phenomena and they provided the maximum peak pressure, rise time, impulsiveness of the pres sures, and the peak frequencies of the pressure oscillation due to the air pocket generation on the wall of the structure during the green water.
1. Introduction
For decades, several studies on green water have been conducted with experimental and numerical methods in efforts to understand the phenomena better and to predict the damage they cause. Greco (2001) conducted a series of experiments with a fixed rectangular model in a 2-D wave flume and observed the characteristics of the water that overtopped on the deck in regular waves. Various numerical approaches have been validated with the results of Greco’s experimental study using the volume of fluid (VOF) model (Nielsen and Mayer, 2004), the MPS method (Zhang et al., 2013), and the immersed boundary method (Yan et al., 2018). Ariyarathne et al. (2012) investigated the impact of green water due to plunging breaking waves on a simplified 3-D model while measuring the impact pressures on the structure and the void fractions of green water flow. Lee et al. (2012) provided a series of measured data that include wave elevations and pressure distributions on an FPSO with
Green water impact is one of the major concerns in the design of ships and offshore structures in harsh environments because it can cause tremendous damage to the topside equipment and injuries to crew on deck. Mitigation of green water loading has been an important consid eration in the design of the shapes of hulls, especially for offshore structures that must operate for a long time in survival conditions. Despite the fact that numerous efforts have been made to predict the structural loads and characteristics of the green water phenomena, these phenomena are still challenging because of the high-nonlinear charac teristics of the wave-structure interaction and the violent, multi-phase flow, which make it difficult to estimate damage with analytical theories.
* Corresponding author. E-mail addresses: [email protected] (G.N. Lee), [email protected] (K.H. Jung). https://doi.org/10.1016/j.oceaneng.2019.106649 Received 14 May 2019; Received in revised form 2 September 2019; Accepted 26 October 2019 0029-8018/© 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: Gang Nam Lee, Ocean Engineering, https://doi.org/10.1016/j.oceaneng.2019.106649
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water assumption. Ryu and Chang (2005) compared the solution of the dam-break theory with experimental results and suggested a method to determine the initial water depth of the dam-break model. The dam-break model has also been used in various numerical simulations, e.g., the Navier-Stokes solver (Kleefsman et al., 2005), the smooth par ticle hydrodynamics (SPH) method (Khayyer et al., 2009), and the semi-analytical solution using motion equations in Lagrangian form (Yilmaz et al., 2003). To understand the flow kinematics and the multi-phase character istics of the green water phenomena, Ryu and Chang (2007) measured the velocity fields of the green water with a plunging wave breaker using the particle image velocimetry (PIV) technique and the bubble image velocimetry (BIV) technique. They provided the dominant flow veloc ities for different green water phases around a model structure. Lee et al. (2016) used the PIV and BIV techniques to perform model tests to investigate the behavior and flow kinematics of green water on a rect angular structure. In Lee et al. (2016)’s study, the generation of green water was categorized into five steps, the flow characteristics of the multi-phase flow of green water around a structure were studied, and the experimental data were compared with the results of their CFD simulations. Even though they investigated the flow kinematics of the multi-phase flow due to green water, there was no description of the relationship between its highly-aerated flow and the impact loading of green water. In order to investigate the pressures due to green water, some re searchers have focused on the features of the air pockets that are generated when waves impact structures. The air pockets might be captured on the wall when a steep wave approaches a vertical structure, and they start to be compressed and oscillate as the wave propagates. It has been shown that the entrapped air reduces the impact pressure on a structure caused by the incoming waves and also distributes the pressure over a wide surface of the structure, so the total impulse on the wall might not be decreased (Hattori et al., 1994). For the characteristics of the air pocket in the wave impact on a vertical wall, Topliss et al. (1993) suggested a formula for the frequency of pressure oscillations derived from a boundary value problem and the natural frequency of pressure oscillations provided by the formula was compared with the
Table 1 Principal dimensions of rectangular structure model. Length (m) Breadth (m) Depth (m) Freeboard (m) Draft (m)
BW Pioneer
Model
241.0 42.0 20.4 6.5 13.9
N/A 0.336 0.163 0.052 0.111
BW Pioneer
Model
241.0 42.0 20.4 6.5 13.9
N/A 0.336 0.163 0.052 0.111
Table 2 Wave conditions of experiments. Length (m) Breadth (m) Depth (m) Freeboard (m) Draft (m)
changing bow models, and Gatin et al. (2018) conducted a validation of the experimental results using Naval Hydro pack software, which is an extension of the collocated FV-based CFD open course software foam-extend (Weller et al., 1998; Jasak, 2009). Silva et al. (2017a) performed extensive model tests for an FPSO exposed to beam and quartering seas, and the results were compared with CFD simulations (Silva et al., 2017b). Rosetti et al. (2019) studied water shipping and its impacts on a fixed FPSO in beam waves with experimental analysis and CFD simulations. The results showed that the CFD simulations could be used to predict the force impulses of green water accurately. It is well known that the green water flow from the sides of a ship significantly resembles the behavior of dam-break flow (Goda et al., 1979). Because of the difficulty in using experimental techniques to investigate the height or the velocities of the water on deck, researchers have used dam-break theory to predict the flow characteristics of green water on deck. Buchner (2002) applied Stoker’s solution (Stoker, 1957) of the dam-break model on the green water phenomena with the shallow
Fig. 1. Experimental set-up in a 2-D wave tank. 2
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Fig. 4. Field of View for the PIV measurement.
Fig. 2. Comparison of (a) wave elevations and (b) horizontal water particle velocity profiles with wave theory (Stokes’ 3rd order).
Fig. 5. Field of View for BIV measurement.
obtained by Lugni et al. (2010). Even so, more studies are needed to get an accurate understanding of the role of air pockets in green water impacts. In the present study, the green water phenomena on a fixed rectan gular structure was investigated with a series of experiments in a 2-D wave flume by measuring the velocity profiles of the water, and the bubbly flow, and pressure distributions on the structure. The velocity fields of water flow were obtained by PIV, and bubbly flow was assessed by the BIV measurement technique. This paper provides the flow kine matics of the green water phenomena, i.e., the mass flow rate and kinetic
Fig. 3. Wave elevation and the duration of the measurements in the experiments.
experimental results of Hattori and Arami (1993) and Witte (1988). Abrahamsen and Faltinsen (2012) proposed an improved formula that considered the shape of the air pocket in estimating the natural fre quency, and the formula was validated by the experimental results 3
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Fig. 6. Locations of the pressure measurements.
Fig. 7. Spectrum of the background noise of the pressure sensor. Fig. 9. Velocity fields of multi-phase flow for each phase of the green water generation ((a) flip-through; (b) Air-entrapment; (c) Wave run-up; (d) Wave overturning; (e) Water shipping). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
2. Experimental setup and techniques 2.1. Experimental setup A series of experiments was performed in a 2-D wave tank (32 m long � 0.6 m wide � 1 m deep) to investigate the green water phenom ena on a fixed rectangular structure. The wave tank had a piston-type wave maker and a sloped beach (1:3) at the end of the wave tank to absorb energy of the incoming wave with a reflection ratio less than 5%. The depth of the water in the wave tank was maintained at 0.60 m. The acrylic, rectangular shaped model was installed wall-to-wall in the beam sea condition and fixed with an alloy frame at the position 15 m from the wave maker. The dimensions of the model were deter mined based on the BW Pioneer FPSO, which has been operating in the west central of the Gulf of Mexico since 2008, and it was scaled down with the ratio of 1:125 (Table 1). The length of the model was the same as the dimension of the wave tank (0.60 m) in order to avoid having to consider the 3-D effects of the green water phenomena, and the length was not followed by the scale ratio. Table 2 shows that the regular wave conditions were chosen as the significant wave height (H1/3) and the peak period (TP) of the 100-yrs return period in the Gulf of Mexico (API-2INT-MET, 2007). In addi tion, two more wave conditions (H1/10 and H1/100) with larger steepness than the significant wave height were estimated (Lee et al., 2016), and used in the experiments to account for extreme ocean environments. All of the equipment in this experiment was synchronized, and all of the
Fig. 8. Ratio of zeroth moment of the spectrum for the order of the FIR filter.
energy at the weather side of the structure, the velocity profile of bubbly flow on the deck due to green water and comparison with dam-break theory, and the pressure distributions on the wall and the deck with analysis of the impact pressures. The air pocket generated in the green water around the structure was studied, and the peak frequency of the air pocket was compared with the analytical solutions provided by Topliss et al. (1993) and Abrahamsen and Faltinsen (2012).
4
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corresponded to the special resolutions of 2.72 mm between every ve locity vector in the interrogation area of 64ⅹ64 pixels with 50% overlap (Fig. 4). The continuous laser (Max. 8 Watt, wave length of 532 nm) was used to illuminate the PIV particles, which had a mean diameter of 57 μm and a specific gravity of 1.02. The adaptive multi-pass algorithm was used to get a stronger cross-correlation, as shown in Equation (1) (Raffel et al., 1998) CðΔx; ΔyÞ ¼
M 1 X N 1 1 X f ðm; nÞgðm þ Δx; n þ ΔyÞ� MN i¼0 j¼0
(1)
where the C cross-correlation function, f and g are the distribution of intensity in continuous images, i and j are the positions in the horizontal and vertical directions, respectively, and M and N are the number of pixels in each direction, respectively. The false vectors in the vector fields that were calculated were eliminated by using the median test, i.e., Equation (2) (Westerweel, 1994), jU2D ðmedianÞ
U2D ði; jÞj < εthresh
(2)
where U2D(n) is all neighboring velocity vectors, U2D(i, j) is the velocity vector that is being inspected, and εthresh is the threshold value for in spection (1.1 was used in this study). 2.2.2. Bubble image velocimetry (BIV) technique The bubble image velocimetry (BIV) technique (Mori and Chang, 2003) was used to obtain the velocity fields of bubbly flow at the weather side and on the deck of the rectangular model structure. BIV images were obtained with the a CCD camera that had 50 mm f/1.8 macro focal lens (Sigma 50 mm lens) set of f/4.0. The shadow graphy image technique (Settles, 2012) was used for the BIV images with the acquisition rate of 500 Hz, FOV size was 360 mmⅹ264 mm (Fig. 5), corresponding to the spatial resolution of 2.44 mm between each vector in the interrogation area of 32 � 32 pixels with 50% overlap. The ve locity vectors of bubbly flow in the BIV images were calculated as shown in Equation (3) using the minimum quadratic difference (MQD) method, which is more accurate than the cross-correlation method (Mori and Chang, 2003).
Fig. 10. Mass flow rate and kinetic energy within FOVs over one wave period (H1/100) without structure.
measuring procedures were completed before the reflected waves reached the model structure from the wave absorber. The wave heights and water particle velocity profiles of incoming waves were obtained by a wave gauge and the PIV system, respectively (see Fig. 1). The measured data were validated with Stokes’ 3rd order wave theory in which the incoming waves belong and were in good agreement with the theoretical values of the wave heights and the horizontal water particle velocity profiles (Fig. 2). All of the measurements were taken with one wave for each test, and the test was repeated fifteen times as shown in Fig. 3. To avoid the re flected waves, the time duration of the measurements was chosen at the transient wave passed and before the arrival of the reflected wave from the wave absorber.
Dðm; nÞ ¼
M X N 1 X ½g1 ði; jÞ MN i¼1 j¼1
g2 ði þ m; j þ nÞ�
(3)
2.3. Pressure measurements Pressure distributions along the structure of the model were obtained to estimate the local loading caused by the green water phenomena on the rectangular structure. The pressures due to green water were measured at three locations on the weather side and at two locations on the deck of the structure (Fig. 6). The pressure data were acquired by piezo-resistive type pressure sensors (Kistler 4043A2) which could measure the static and dynamic pressures covering the range from 0 to 2 bars. The sampling rate for the measurements was maintained at 5 kHz, which was determined by conducting convergence test. All measurements were repeated 15 times with the same wave conditions. The finite impulse response (FIR) low-pass filter, which has less phase and delay distortion than the infinite impulse response (IIR) filter (McClellan and Parks, 1973) was used to remove the background noise in the pressure data. The cut-off frequency for the low-pass filter was determined to be 150 Hz considering the frequency component of the background noise (Fig. 7), and the 67th order of the FIR filter was used to maintain the ratio of zeroth moment of the spectrum for filtered and raw signals up to 90% for the components that were lower than the cut-off frequency (Equation (4)) and to maintain a ratio lower than 10% for the higher components than the cut-off frequency (Equation (5)) as shown in Fig. 8,
2.2. Experimental techniques 2.2.1. Particle image velocimetry (PIV) technique The particle image velocimetry (PIV) technique was used to obtain the velocity fields of the green water phenomena at the weather side of the structure with exception of the bubbly flow region. The PIV images were obtained with an acquisition rate of 500 Hz using a high-speed CCD camera (Redlake Y5) mounted with a 105-mm, f/1.8 macro focal lens (Nikkon 105 mm lens) set at f/2.8. The lens had 2352ⅹ1728 pixels, a pixel size of 7μm, 8-bit dynamic range. The two different fields of view (FOV) were used to cover the entire region of the weather side of the rectangular structure with each size being 147 mmⅹ200 mm which 5
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Fig. 11. Comparison of mass flow rates with and without the structure ((a) H1/100, (b) H1/10, (c) H1/3).
R FC Sf ðω; nÞdω RA ðnÞ ¼ 0R FC � 0:9 Sr ðωÞdω 0
� Flip-through: Incoming wave is deformed vertically with a concave face before the wave front impacts on the weather side of the structure � Air-entrapment: Air is entrapped on the weather side of the structure � Wave run-up: The water level increases along the weather side and splashes up into air � Wave overturning: The wave overturns like a plunging wave breaker and impacts on the deck � Water shipping: The body of bubbly water is moved forward along the deck
(4)
R∞
Sf ðω; nÞdω F RB ðnÞ ¼ RC∞ � 0:1 S ðωÞdω FC r
(5)
where RA and RB are the ratios of the zeroth moment of the two spectra, FC is the cut-off frequency, Sf and Sr are the spectra of the filtered signal and the raw signal, respectively, ω is the frequency, and n is the order of the FIR filter.
In this study, the PIV and BIV measurements were used to obtain the velocity fields of water and bubbly flow near the rectangular structure in order to investigate the flow kinematics of the multi-phase flow due to the green water phenomena. Fig. 9 shows the vector fields that were acquired at each step of the green water generation. The blue and yellow vectors in the images represent the velocity of water around the struc ture obtained by PIV measurements and the velocity of bubbly flow by BIV measurements, respectively. To investigate the quantitative flow kinematics on the weather side of the structure during the generation of green water, the mass flow rate and the kinetic energy (per unit length in the horizontal direction) were calculated with the water velocity profiles obtained by the PIV measurements. The mass flow rate and the kinetic energy passing through a vertical
3. Results 3.1. Flow kinematics at the weather side of structure For the generation of the green water phenomena on a rectangular structure, Greco (2001) classified the green water generation into two-stages, i.e., 1) the initial stage of the water shipping when the water started to be shipped on deck and 2) the later stage of the water shipping that formed a dam-break type on deck. Lee et al. (2016) suggested that the generation of green water could be categorized into the five phases presented below.
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Fig. 12. Comparison of kinetic energy with and without the structure ((a) H1/100, (b) H1/10, (c) H1/3).
cross section in FOV were calculated using the following equations, Z η _ tÞjx¼C:S: ¼ Mðx; ρbuðx; z; tÞdz (6)
with the structure for the three wave heights over the entire wave period. The total area under the curves in Fig. 11 is the mass of water that went through the cross section at x=B ¼ 0:12 during one wave period; this mass, was less than it would have been due to the existence of the structure. The green water phenomena on the deck and the re flected wave from the structure would be the main reasons for the decrease in the mass flow rate of the water. And, due to the super position of the incoming and reflected waves, a standing wave might develop, and the vertical velocity component obviously dominant near the structure. The structure resulted in the decrease of 49%, 34%, 32% of the water mass at the condition of H1/100, H1/10 and H1/3, respectively. Fig. 12 shows the variations of the kinetic energy on the cross-section of x=B ¼ 0:12 without and with the structure. The kinetic energy without the structure was the maximum magnitude at the wave crest phase, and it was distributed in a symmetrical shape with respect to the phase π. For the case with the structure, the distribution of the kinetic energy changed to an asymmetric pattern that had two peaks. The magnitude of vertical velocity component was increased dramatically due to the green water phenomena, of which the first and second peaks matched with the wave run-up phase and the water shipping phase, as shown in Fig. 9(c) and (e), respectively.
s
Z KEðx; tÞjx¼C:S: ¼
η s
1 1 ρbu2 ðx; z; tÞ þ ρbw2 ðx; z; tÞ dz 2 2 |{z} |{z} Horizontal
(7)
Vertical
where M_ is the mass flow rate, η is the free surface, ρ is the density of water, b is the width of the wave flume, KE is the kinetic energy u and w are the horizontal and vertical velocities, respectively, and s is the bot tom of the FOV. The values were normalized with ρvp db and ρvp 2 db for each mass flow rate and kinetic energy, where vp denotes the phase velocity of the incoming waves (vp ¼ 1.72 m/s). To investigate the spatial differences in the mass flow rate and the kinetic energy, the values were obtained at seven fixed and equallyspaced locations in the FOV, and the averaged values are shown in Fig. 10 with the error bars which indicate the corresponding 95% con fidence interval. The errors in both the mass flow rate and the kinetic energy were less than 10% of each maximum quantity over one wave period. In this study, all of the values of mass flow rate and kinetic en ergy were shown with the results obtained on the cross section at x= B ¼ 0:12, where B is breadth of the model that was out of the bubbly flow region. Fig. 11 shows the comparison of the mass flow rates without and
3.2. Flow kinematics of water shipping on deck When the green water phenomena occurred, air pockets were generated on the wall and on the deck of the structure due to the 7
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Fig. 13. Snapshot of wave overturning phase ((a) H1/100, (b) H1/10, (c) H1/3).
plunging wave breaking which overturned on the deck (Fig. 13). In the wave overturning phase (Fig. 9 (d)), the air pockets start to rotate in clock wise (CW) and fluctuate, and the bubbly flow with air pockets start to run along the surface of the structure with the motion of the bubbles in the water shipping phase (Fig. 9 (e)). This type of overtopping could be in the green water regimes of Plunging plus Dam Break (PDB) as classified by Greco et al. (2007) and Zhang et al. (2019). Due to the limitation of the PIV technique for measuring the flow kinematics of bubbly flow, due to the reflection of the laser sheet on the bubbles, the BIV measurement was used to study the flow kinematics of the water shipping phase on the deck. The time the water stayed on the deck during the generation of the green water was less than 0.1s, which was shorter than the relaxation time for the rising bubble that corresponded to the bubble size (1–2 cm) of the water shipping flow (Hysing et al., 2009). This indicated that the inertial force of the water around the bubbles had a more dominant effect on the motion of bubbles than the buoyancy force, and the vertical displacement of the bubbles in the water shipping phase on the deck was much smaller than the horizontal motion. To investigate the characteristics of bubbly flow in the water ship ping phase on the deck, the spatial mean velocities in the horizontal direction were obtained for every 20ⅹ2.5 mm2 using the velocity profiles obtained by BIV measurements, as shown in Fig. 14. The horizontal mean velocities were normalized by the phase velocity of the incoming waves (vp ¼ 1.72 m/s), of which magnitude presented in the distance of each symbol from the dashed line of its left-hand side. Fig. 14 shows the mean velocity profiles every 0.03 s after the wave overturning phase for each wave height. The distribution of the symbols presented the velocity profiles of bubbly flow erratically, because they could only be measured in the bubble region by the BIV technique. The error bar indicates the 95% confidence interval of each mean velocity. More bubbles were generated with the higher wave height condition, and those velocities were, for the most part, faster than the velocities with the lower wave condition. The magnitude of the velocity of the bubbly flow increased gradually along the deck, and the maximum values occurred in the range
of 120–160 mm away from the corner of the structure. The horizontal velocities of the bubbly flow were compared with the analytical solution of the dam breaking phenomena (Stoker, 1957), as shown in Equation (8), 2 �pffiffiffiffiffiffi x� gh’ þ (8) uðx; tÞ ¼ 3 t where u is horizontal velocity, g is the gravitational acceleration, and h’ is the initial water depth of the dam. Buchner (2002) suggested the use of the freeboard exceedance instead of the initial water depth to predict the horizontal velocity of the water shipping phase due to the green water phenomena (Equation (9)). 9 h’ ¼ h 4
(9)
where h is the freeboard exceedance. Ryu et al. (2007) tested the green water phenomena on a fixed rectangular structure due to the plunging breaking wave and suggested the use of Equation (10) to estimate the horizontal velocity of the water on the deck. h’ ¼ 0:36
vp 2 g
(10)
Fig. 15 compares the horizontal velocities of the bubbly flow spatially averaged in the z-direction and the solutions of a dam break problem for every 0.03 s after the wave overturning phase. The symbols and the error bars are the averaged bubbly flow velocities and its cor responding 95% confidence interval on the deck, respectively, which were obtained by BIV measurements. In the study, it was suggested that, rather than using initial water depth in Equation (8) to predict the horizontal velocity of the water on the deck, the water level from the corner of the structure at each time should be used, which was obtained in the instantaneous images taken by the high speed CCD camera (Fig. 16), and are shown by the blue lines in Fig. 15. This method could 8
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Fig. 14. Velocity distributions of bubbly flow on the deck using the BIV technique ((a) H1/100, (b) H1/10, (c) H1/3).
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Fig. 15. Comparison of bubbly flow on the deck with dam breaking theories (Wave condition of H1/100).
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increased up to the maximum water level at the corner of the structure which was defined as the wave overturning phase (Fig. 9 (d)). Fig. 17 (c), (d), and (e) show that the pressures on the deck (PG3, PG4, PG5) increased when the water overtopped the deck after the wave was overturned at the corner of the structure, and the pressures gradu ally diminished due to the drainage of water during the water shipping phase along the deck. Bubbly flow occurred at the corner of the structure in the overturning phase, and it ran along the deck during the water shipping phase. But, unlike what occurred on the weather side wall, the bubbly flow was not found to have an effect on the increase in the pressure and the oscillation on the deck. Fig. 18 shows that the pressure on the deck was induced mainly by the hydrostatic pressure over each pressure sensor. Fig. 19 shows that the pressures that were measured were nondimensionalized by 0:5ρu2max, where umax is the maximum horizontal velocity of the water of the incoming wave using linear wave theory. At the weather side wall (PG1 and PG2), the non-dimensionalized pressures at the smaller wave height condition were larger than those at the larger wave height condition. This occurred because, except for the first peak pressure, the pressure on the wall did not increase as much as the incoming wave increased. This showed that the largest non-dimensionalized pressures on the deck (PG3, PG4, and PG5) were related to each incoming wave height condition, and there was an almost linear relationship with the square of the maximum horizontal water velocity of the incoming wave. Fig. 20 shows the mean values and the corresponding 95% confi dence interval of the maximum peak pressure and rise time at the first pressure peak obtained by repeating the tests at each wave condition fifteen times. The rise time was calculated as twice the time duration from the half peak to the peak value, as suggested by DNV-GL (DNV-GL, 2016). In Fig. 20 (a), because of the bubble entrapped on the weather-side wall at the wave conditions of H1/100 and H1/10, the pres sures measured at PG1 and PG2 had larger mean magnitudes and standard deviations than the pressures on the deck. Also, the rise times at the weather-side wall, i.e., less than 0.01 s, were shorter than those on the deck (Fig. 20 (b)). The magnitude of the peak pressure was inversely proportional to the duration of the rise time in the air-entrapment phase, which showed similar patterns with previous studies of the wave impact €z, phenomena on the vertical wall (Bagnold, 1939; Denny, 1951; Kırkgo 1995; Hattori et al., 1994; Cuomo et al., 2011). Fig. 21 shows that the oscillation of the pressure occurred at the weather side wall (PG1, PG2) when the air pocket was entrapped at wave conditions of H1/100 and H1/10. The symbols indicates the raw pressures that were measured with the sampling rate of 5 kHz, and the solid line is the signal filtered by an FIR low-pass filter. The specifica tions of the low-pass filter were presented in section 2.3. Fig. 22 shows the results when the pressure spectra were plotted for fifteen repeated tests at the same wave conditions. Note that the peak magnitude of the spectrum and its corresponding frequency at the wave condition of H1/ 100 were larger and lower, respectively, than that of H1/10. The pressure spectra were different for each repeated test, but the peak frequencies of the spectra were coincident in all of the tests. The peak frequencies of pressure oscillation were matched for the same wave condition at PG1 and PG2, even though the air pocket was rotated at the location of PG1 and turned to the corner of the structure. Fig. 21 and Equations (9) and (10) show that the peak frequency of the incoming wave height, H1/100, was lower than that of H1/10 because it depended on the size of air bubble and its location with respect to the free surface. Topliss et al. (1993) and Abrahamsen and Faltinsen (2012) suggested a formula for estimating the peak frequencies of the wave impact on the vertical wall and the sloshing in the rectangular tank based on 2-D po tential flow theory assuming an adiabatic pressure-density relationship. Equation (11) is the analytical solution provided by Topliss et al. (1993),
Fig. 16. Measurement of water level on the deck.
estimate the measured velocities better than the solutions of Equations (9) and (10) for every instance. The solutions of Equations (9) and (10) overestimated the velocities because of the fixed y-intercepts that did not vary with time. The method suggested in this study includes the effect of variations in the free surface during the generation of the green water, which was not provided by the solutions with Equations (9) and (10). 3.3. Pressure measurements Pressures were measured on the wall and on the deck of the structure during the generation of green water, and Fig. 17 shows the results for one wave period for three different wave heights. Fig. 17 (a) and (b) show that the pressures on the wall of the structure (PG1 and Pg2) increased suddenly, the first pressure peak was reached at the moment the air pocket was entrapped when the wave face hit the wall, and the pressure was oscillated and decreased during the airentrapment phase (Fig. 9 (b)). The first pressure peak at PG1 was larger than the pressure peaks at PG2 even though the wave face hit directly near PG2 after the flip-through phase. When the wave height was H1/3, the pressure peak and oscillation did not occur because no air pocket was entrapped on the wall (Lee et al., 2016). Thus, the magnitude of the first pressure peak and the pressure oscillation phenomena occurred due to the bubble effect of the entrapped air on the wall. The pressure increased gradually during the wave run-up along the wall (Fig. 9 (c)), and the second peak was reached as the free surface 11
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Fig. 17. Time histories of pressure as the incoming wave heights changed.
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Fig. 18. Comparison of the pressures at P1 and P3 during the different phases of the generation of green water. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
ω2 ¼
2γpatm
ρr2 logðr=2hÞ
defined as the radius of a semi-circle that corresponded to the area of the ellipsoidal air pocket on the structure. Both methods could predict the peak frequencies with reasonable agreement with test results. This may be the result of the differences between the assumptions of the 2-D theories and the uncertainty of the experimental sizing of the area of air bubbles near the wall of the wave tank. The wave load acting on a vertical wall was suggested by two pat terns, i.e., “impact” (impulsive or dynamic) and “quasi-static” (pulsat ing) wave loads (Allsop et al., 1996) as follows;
(11)
where γ is the ratio of specific heats, which is 1.4 for air, r is the equivalent radius of the air pocket, and h is the distance from the free surface to the center of the air pocket (Fig. 23). Abrahamsen and Faltinsen (2012) suggested a formula (Equation (12)) that considers the shape of the air pocket which might not be a semi-circle, � � �� 2γpatm K sin π2ab � � �� ω2 ¼ (12) Ω0 K sin π2ab
� An impact (impulsive or dynamic) wave load could be caused by the special conditions that arise when a wave breaks onto the structure with greater intensity than the quasi-static load in a shorter duration. � A quasi-static (pulsating) wave load varies with the order of magnitude of the free surface elevation for a relatively long duration.
where a and b are the distances from free-surface and the upper part of air pocket (Fig. 23), respectively, Ω0 is the area of the air pocket, and K is obtained by Equation (13). Z 1 1 pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx KðkÞ ¼ (13) 1 x2 1 k 2 x2 0
A few studies have provided quantitative definitions of the impact criteria. Hayashi and Tanaka (1988) classified the dynamic mode of loading versus the strain rate using the five criteria of creep, static or quasi-static, dynamic, impact, and hypervelocity impact. Oumeraci et al. (2001) suggested defining the impact phenomena as a peak loading 2.5 times larger than the subsequent quasi-static loading due to the water
Fig. 24 shows the peak frequencies measured in tests compared with the results of Equations (9) and (10). The equivalent radius (r) was 13
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Fig. 18. (continued).
waves on the vertical wall. Ariyarathne et al. (2012) suggested a specific value of impulsiveness using the non-dimensional pressure rate to the rise time with the wave celerity for the green water phenomena due to a plunging breaking wave. In this study, the impulsiveness (Ariyarathne et al., 2012) was esti mated using the pressure rate to the rise time and the maximum water velocity under the regular wave. This was done because the dominant factors would be the severity of pressures and the horizontal water ve locity acting on the vertical wall instead of the wave celerity. The impulsiveness of Ariyarathne et al. (2012) was modified as shown in Equation (14), Pimp
IP ¼ 0:5ρtur
max
2
> 1500
where Pimp is the maximum pressure, tr is the rise time, umax is the maximum horizontal velocity of the water of the incoming wave using the linear wave theory. The impulsiveness (Ip) was calculated for each test of fifteen repeated trials at the same wave condition for three wave heights i.e., H1/100, H1/ 10, and H1/3, as shown in Fig. 25. The impulsiveness was larger at the location on the weather side where an air pocket was generated. The impulsiveness at PG1 was noticeable at the conditions of H1/100 and H1/ 10 at which the movement of the air pocket was very active on the weather side. At the location of PG2, the impulsiveness was smaller than it was at PG1, because its location was out of the air pocket region. But, the pressure rate to the rise time would be more significant for the higher wave height condition, i.e., H1/100, than the pressure rate to rise time would be for the wave height of H1/10 and H1/3 due to increased
(14)
T
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Fig. 19. Non-dimensionalized pressure time histories with changing incoming wave height.
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Fig. 22. Spectra of the pressure oscillations at PG1 ((a) H1/100, (b) H1/10).
Fig. 20. Mean and Standard deviation of the maximum pressure and rise time of the first peak.
Fig. 23. Snapshots of air pockets in the air-entrapment phase ((a) H1/100, (b) H1/10).
Fig. 21. Oscillation of the pressure at the PG1 (wave condition of H1/100).
which was defined as the quasi-static loading by Allsop et al. (1996). The impulsiveness values at the locations of PG3, PG4, and PG5 on the deck were smaller than the suggested threshold, and the wave loading on the deck could be represented by the quasi-static criteria.
horizontal velocity of the water acting on the vertical wall. This suggests an impulsiveness threshold of (Ip > 1500) to define the wave impact criteria, which was estimated to be 1500 at PG1 in the condition of H1/3 without the generation of an air pocket due to the wave action on the vertical wall. The pressures along the deck were similar to the hydrostatic pressure estimated by the water elevation over each of the pressure sensors,
4. Conclusions Green water phenomena on the fixed rectangular structure were 16
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Fig. 24. Comparison of the peak frequencies of pressure oscillation with the analysis formula.
studied with a series of experiments conducted in 2-D wave flume. The model was simplified and scaled down with the ratio of 1:125 from FPSO BW Pioneer which has been operating in the west central of the Gulf of Mexico. Three regular wave conditions were tested with the wave heights of H1/100, H1/10, and H1/3, and the peak period of 100-yrs return period was used at the operating site. PIV and BIV measurements were used to obtain the water and bubbly flow to study the flow kinematics of the multi-phase flow induced by green water. The mass flow rate and the kinetic energy, which were calculated from the water velocity profiles obtained by PIV measure ments, were compared with and without the structure to investigate the quantitative flow kinematics of water flow at the weather side of the structure. The characteristics of bubbly flow in the water shipping phase on the deck were investigated based on the velocities of the bubbly flow obtained by BIV measurements, and the velocities of the bubbly flow were compared with the solutions of dam break theory. Pressure measurements were conducted along the structure of the model to estimate the local loading due to green water, and the maximum peak pressure and rise time were provided. The peak fre quencies of pressure oscillation due to the air pocket entrapped on the weather side wall were analyzed and compared with the results of the formulas suggested by Topliss et al. (1993) and Abrahamsen and Fal tinsen (2012). The impulsiveness suggested by Ariyarathne et al. (2012) was modified, and a new equation using the horizontal velocity of the water was suggested to classify the impact criteria. From the results obtained in this study, the findings are summarized as below: - It was observed that the horizontal mass flow rate in FOV at the weather side of the structure was decreased due to the existence of the structure. The kinetic energy in the same FOV was increased drastically due to the existence of structure having two peaks at the wave run-up and the water shipping phases. The main reason was that the horizontal velocity component of the incoming wave became the vertical direction velocity, which might have been caused by the local standing wave that was generated by the superposition of the incoming and reflected waves near the structure. - At the higher wave height condition, the velocity of bubbly flow had more bubbles and became faster on the deck. Those velocities increased gradually along the deck and had the maximum values at 120–160 mm away from the corner of the structure. The horizontal bubbly flow velocities were compared with the solutions of the dam breaking problem. It was suggested that the dam break theory to be used with the water level from the corner of the structure for the initial water depth. In this study, the suggested method showed
Fig. 25. Impulsiveness results of pressure measurements ((a) H1/100, (b) H1/10, (c) H1/3).
better agreement with the measured bubbly flow velocities than those of Buchner (2002) and Ryu et al. (2007), because it was able to utilize the free surface variation during the green water generation instead of the initial water depth, which was constant (Buchner, 2002; Ryu et al., 2007). 17
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- The pressure at the weather side wall increased the first pressure peak at the air-entrapment phase and, it oscillated due to the air pocket that was entrapped on the wall. Following the first peak and oscillation, the pressure increased gradually, with the free surface rising, and the second peak was reached at the wave overturning phase. The pressures on the deck increased when the water over topped the deck in the wave overturning phase, and it was reduced gradually during the water drainage along the deck. The pressure peaks on the deck, which were non-dimensionalized with the square of the maximum horizontal water velocity of the incoming wave, had similar magnitudes for each incoming wave height. - Pressure oscillations were observed at the weather side wall when the air pocket was entrapped at the wave conditions of H1/100 and H1/10. The peak frequencies of the pressure oscillation were compared with the analytical formulas suggested by Topliss et al. (1993) and Abrahamsen and Faltinsen (2012). The formulas could predict the peak frequencies reasonably with the differences due to the limitations of the equations limited in 2-D and the experimental uncertainty associated with sizing the area of the air pocket. - The impulsiveness (Ariyarathne et al., 2012) was applied to estimate the severity of the pressures using the maximum horizontal water velocity instead of the wave celerity. The impulsiveness due to the dynamic pressure was more dominant at the location of the air pockets generated at the weather side wall. Smaller impulsiveness was estimated on the deck because the hydrostatic pressure was dominant. A threshold was suggested to define the wave impact criteria as 1500, because it was the smallest impulsiveness due to the air pocket on the structure.
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Acknowledgement This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSP) through GCRC-SOP (No. 2011-0030013), and the Ministry of Trade, Industry & Energy (MOTIE, Korea) under Industrial Technology Innovation Pro gram (No. 10063405) and PNU Korea-UK Global Graduate Program in Offshore Engineering (N0001288). References Abrahamsen, B., Faltinsen, O., 2012. The natural frequency of the pressure oscillations inside a water-wave entrapped air pocket on a rigid wall. J. Fluids Struct. 35, 200–212. Allsop, N.W.H., Vicinanza, D., McKenna, J.E., 1996. Wave Forces on Vertical and Composite Breakwaters. Report SR 443. Hydraulics Research, Wallingford, United Kingdom. API BULLETIN 2INT-MET, 2007. Interim Guidance on Hurricane Conditions in the Gulf of Mexico, first ed. (Washington, D.C). Ariyarathne, K., Chang, K.-A., Mercier, R., 2012. Green water impact pressure on a threedimensional model structure. Exp. Fluid 53 (6), 1879–1894. Bagnold, R.A., 1939. Interim report on wave pressure research. J. Inst. Civ. Eng. 12, 202–225. Buchner, B., 2002. Green Water on Ship-type Offshore Structures. Delft University of Technology. Cuomo, G., Piscopia, R., Allsop, W., 2011. Evaluation of wave impact loads on caisson breakwaters based on joint probability of impact maxima and rise times. Coast. Eng. Procee. 1 (33) structures. 62. Denny, D.F., 1951. Further experiments on wave pressures. J. Inst. Civ. Eng. 35, 330–345. DNV-GL, 2016. Sloshing analysis of LNG membrane tanks. DNV Classification Notes 30 (9). Gatin, I., Vuk�cevi�c, V., Jasak, H., Seo, J., Rhee, S.H., 2018. CFD verification and validation of green sea loads. Ocean. Eng. 148, 500–515. Goda, K., Miyamoto, T., Yamamoto, Y., 1979. A study of shipping water pressure on deck by two-dimensional ship model test. Naval Arch. Ocean Eng. 17.
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