Understanding spray cloud formation by wave impact on marine objects

Cold Regions Science and Technology 129 (2016) 114–136

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Cold Regions Science and Technology journal homepage: www.elsevier.com/locate/coldregions

Understanding spray cloud formation by wave impact on marine objects Armin Bodaghkhani a,⁎, Saeed-Reza Dehghani a, Yuri S. Muzychka a, Bruce Colbourne b a b

Department of Mechanical Engineering, Memorial University of Newfoundland (MUN), St. John's, Canada Department of Ocean and Naval Architectural Engineering, Memorial University of Newfoundland (MUN), St. John's, Canada

a r t i c l e

i n f o

Article history: Received 31 August 2015 Received in revised form 10 June 2016 Accepted 24 June 2016 Available online 30 June 2016 Keywords: Spray cloud formation Wave spray Wave impact process Sheet and droplet breakup Wave slamming Air entrainment and bubble generation

a b s t r a c t Wave impacts on vessels and offshore structures can induce significant spray. This process leads to topside icing in sufficiently cold and windy conditions. This paper establishes the current state of the art understanding of the physical behaviour of wave impact and the process of spray cloud formation upstream of a ship or marine structure. Previous work on the behaviour of spray at the bow is extensively reviewed. The process of spray formation is related to several complicated phenomena including wave slamming, jet formation after impact, sheet and droplet breakup, and production of the spray cloud on the top surface of the ship bow. Progress has been made in modeling some of these phenomena, including numerical methods for modeling the free surface, the phenomena of slamming, air entrainment, and water breakup. Field observation methods for measuring characteristic parameters of the spray are also reviewed. Related phenomena, such as wave slamming on a wall and bow waves, are followed from the numerical and experimental point of view. Although direct numerical simulations of spray formation following by wave impacts are not yet a practical option, constituent models for each separate part of this problem (e.g., free surface modeling, slamming on walls, air entrainment, etc.) show promising progress. This work is a guide for researchers of off-deck phenomena in the field of marine icing. The puzzle pieces and the gaps are examined to help design a new research strategy in this field. © 2016 Elsevier B.V. All rights reserved.

Contents 1. 2.

3.

4.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid water content based on field observations . . . . . . . . . . . . . . . . . . . . 2.1. Spray cloud formulation for vessels . . . . . . . . . . . . . . . . . . . . . . . 2.1.1. Liquid water content formulation . . . . . . . . . . . . . . . . . . . . 2.1.2. Maximum height of spray cloud . . . . . . . . . . . . . . . . . . . . 2.1.3. Spray cloud duration . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. LWC formula for offshore structure . . . . . . . . . . . . . . . . . . . . . . . 2.3. Usage of LWC formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . Interaction of sea waves and marine objects . . . . . . . . . . . . . . . . . . . . . . 3.1. Wave-body interaction and slamming in deep-water . . . . . . . . . . . . . . . 3.1.1. Wave slamming on a wall . . . . . . . . . . . . . . . . . . . . . . . 3.1.2. Wave-ship slamming modeling . . . . . . . . . . . . . . . . . . . . . 3.2. Bow wave and shoulder wave breaking . . . . . . . . . . . . . . . . . . . . . Computational fluid dynamics (CFD) analysis of wave-vessel interactions and spray formation 4.1. Non-linear free surface modeling . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1. Marker and cell (MAC) . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2. Volume of fluid (VOF) . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3. Level set method . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4. Constrained interpolation profile (CIP) . . . . . . . . . . . . . . . . . 4.1.5. Inviscid computation of waves . . . . . . . . . . . . . . . . . . . . . 4.1.6. Smooth particle hydrodynamics . . . . . . . . . . . . . . . . . . . . 4.1.7. Other methods . . . . . . . . . . . . . . . . . . . . . . . . . . . .

⁎ Corresponding author. E-mail address: [email protected] (A. Bodaghkhani).

http://dx.doi.org/10.1016/j.coldregions.2016.06.008 0165-232X/© 2016 Elsevier B.V. All rights reserved.

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4.2. Air entrainment and bubble generation 4.3. Turbulence model . . . . . . . . . . 5. Water breakup and atomization . . . . . . . 5.1. Splash plate atomizer . . . . . . . . 5.2. Sprinkler atomizer . . . . . . . . . 6. Conclusion . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . .

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1. Introduction Offshore oil exploration and development in cold ocean regions are becoming more common. The safety and security of fixed and floating platforms and vessels in these regions are receiving increased industry attention. Understanding the problem of marine icing, one of the environmental hazards, is important. The effect of icing on shipping and offshore exploration and production has been well documented (Lozowski and Zakrzewski, 1993; Ryerson, 2013; Roebber and Mitten, 1987). Studies show that icing in Arctic conditions on offshore structures and vessels arises from two major causes, categorized as atmospheric conditions and sea-water spray (Brown and Roebber, 1985). Sea-water spray is produced in two different ways, from wind and waves. Wind spray is a spray cloud drawn from the sea surface by wind, and wave spray is the upstream spray cloud caused by a wave impact on a vessel's bow or marine structure. N 3000 observations from ships in different sea conditions show that N 90% of icing on vessels results from wave spray (Borisenkov and Pchelko, 1975). Wave spray is produced after the impact of vessels with waves and this impact produces a cloud of water, which leads to impingement of droplets on the ship's deck and bow. In cold conditions, these droplets turn into ice, which is a hazard to crew operations and to vessel stability. Similar effects are observed on offshore structures. Although wave spray is the major cause of icing on the vessels, the physical behaviour of wave impact is still poorly understood (Hendrickson et al., 2003). Besides a few observational studies, which provide a rough understanding of Liquid Water Content (LWC) and spray heights, little research on this subject is available in the literature (Borisenkov and Pchelko, 1975; Ryerson, 2013). In recent years, efforts to model marine icing phenomena using numerical methods have shown progress, but the multi-scale nature and complexity of the problem necessitate separation of the problem into smaller singlephenomenon steps. The generation of spray is divided into several stages, including free surface modeling, wave slamming on the bow, the air entrainment process during impact, water sheet and jet formation on a wall after wave slamming, water sheet and droplet breakup caused by wind, and droplet trajectories when they meet the surface of the ship's deck (Hendrickson et al., 2003; Dommermuth et al., 2007). The present discussion is focused on direct and related work on spray cloud formation after wave impact. Field observations that have considered the spray flux, spray heights, and other specific characteristics of spray clouds are covered in the first section. Numerical and experimental studies on wave-vessel interactions are addressed in the next section. A relatively small number of direct studies lead to consideration of related topics with features that have the same behaviour as spray clouds. These related topics, such as spray production after wave slamming and bow waves, are similar phenomena to the proposed problem. Techniques for numerical and experimental modeling of wave slamming and bow waves are adaptable to the study of spray cloud formation. The remaining sections of the paper cover numerical methods for modeling the free surface, wave slamming, air entrainment, and turbulence related to the proposed problem. Finally, the most similar situations in which sheet and droplet breakup can occur are reviewed. Splash plates and sprinklers are two pieces of equipment that work by employing the breakup phenomena on a rigid surface.

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Although this paper looks at a number of individual subjects, it is the potential for the combination that provides a broader contribution and understanding of the marine icing problem; especially each part that is a distinctly relevant and complementary element. Linking these subjects and covering the gaps are crucial for progress in this field. 2. Liquid water content based on field observations Water droplets are dispersed into the atmosphere and carried by the wind after waves hit a structure. Observations of the phenomena establish the importance of this aspect, but unfortunately there is a relatively small body of literature that contributes to physical understanding or provides measurements of wave spray phenomena. Field data available from vessels are limited and thus numerical and analytical formulations are based on a few field observations (Borisenkov and Pchelko, 1975; Ryerson, 1990). These observations mostly report a vertical distribution of the Liquid Water Content (LWC) above the ship's bow. The LWC is defined as the volume of liquid water per unit volume of air. Generally these observations do not include information about the velocity of the water droplets or the size distribution of droplets. Information about the position of the measurement instruments on the ship's deck and the geometry of ship's bow are also unknown in most of the cases. Effective measurements of a spray cloud from field observations should contain the following factors which are categorized in five groups and are shown in Table 1. Most of the parameters in Table 1 are well-recognized, except for droplet information, such as their size and velocity distribution in front of the vessels or offshore structures. These parameters are crucial for understanding the amount of spray flux and are addressed briefly in the paper by Lozowski and Zakrzewski (1990) as significant parameters. In addition, information about droplets such as size distribution, velocity distribution and distribution of droplet concentration guide Table 1 Effective parameters in the study of spray cloud formation for field observation. Effective parameters

Parameter name

Ship parameters

Ship speed Ship bow geometry Ship movement (acceleration, pitch, and heave) Wind speed Wind direction Wave height Wave length Frequency distribution (wave spectra) Wave propagation direction Wave speed Wave profile relative to bow Droplet size distribution Velocity distribution of droplet Distribution of droplet concentration Frequency of spray cloud generation Duration of spray cloud Air and ocean surface temperature Relative humidity Barometric pressure Water salinity

Wind parameters Wave parameters

Spray cloud information

Time Meteorological factors

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the estimation of the spray flux, vertical distribution and maximum height of the spray cloud. A few previous studies focused on defining a relationship for measuring the vertical distribution of the LWC and the amount of water delivered to topside from the spray cloud. Zakrzewski (1986), who used the data from the Borisenkov and Pchelko (1975) observations, reports some of the parameters listed in Table 1, such as ship speed, ship heading angle, wind parameters, wave parameters and time. Information about droplet size and droplet velocity distribution which lead to calculation of the LWC and maximum height of the spray cloud are not reported. The LWC and maximum height of the spray cloud are calculated based on the empirical formulation by Zakrzewski et al. (1988). The following two sections review the previous research for understanding the LWC formulations for both vessels and offshore structures.

approximations of the wind speed and fetch values in the Sea of Japan, which cannot be found in the Borisenkov and Pchelko (1975) paper. Based on these approximations, the values of the two parameters are calculated as H0 = 3.09 m and V0 = 11.01 ms−1. Vr in Eq. (3), which is the ship speed relative to the surface of the wave, suggested by Zakrzewski (1986) as: V r ¼ 1:559 P w −0:514 V s cosα

m s

ð4Þ

where Vs, Pw and α are ship speed in knots, period of the wave, and heading angle, respectively. The Horjen and Vefsnmo (1987) model, which is based on the spray measurement on a Japanese ship, is defined by the following formula:


kg m3

 ð5Þ

2.1. Spray cloud formulation for vessels

w ¼ 0:1 H expðH−2zÞ

The most significant parameters for modeling spray cloud formation are thought to be the LWC, maximum vertical height of spray cloud, and duration time. This section reviews studies that have proposed formulations for the LWC, maximum vertical height of a spray cloud and the spray cloud duration.

where H is the wave height and z is the elevation above sea level. Another model is proposed by Brown and Roebber (1985) based on the Borisenkov and Pchelko (1975) work. This model, incorporates the statistical behaviour of the wave height distribution, and defines the LWC using the following formula:

2.1.1. Liquid water content formulation The first and simplest LWC equation for the water cloud after wave impact on a ship is defined by the formula (Katchurin et al., 1974)
w ¼ ξH

kg m3

 ð1Þ

where w is LWC, H is the wave height, and ξ is a constant equal to 10−3 kgm− 4. This formula was proposed based on measurements from a Medium-sized Fishing Vessel (MFV), where the ship moves relative to waves with the heading angle of α ≥ 140°, at speeds of 6–8 kn. Unfortunately, information about the techniques and locations of the measuring facilities on the ship was not reported. Comparison of the data from Katchurin et al. (1974) with the listed factors and parameters in Table 1 shows that some of the essential parameters in the study of the LWC or spray cloud are not reported. Another empirical formulation for the vertical distribution of the LWC based on field data for an MFV (Narva) in the Sea of Japan was introduced by Borisenkov and Pchelko (1975) and calculated by the formula: w ¼ 2:36  10−5 expð−0:55hÞ


3  m ðwÞ 3 m ðaÞ

ð2Þ

where h is the elevation of an object above the deck of the MFV, and the freeboard height of the MFV is 2.5 m. The LWC is based on the volume of water (w) in a unit volume of air (a). The MFV moves into the waves with an angle of α = 90° − 110° with a speed of 5 − 6 kn. The wind speed reported is 10 − 12 ms−1. This formulation is limited to the specific type of ship under certain sea conditions, and cannot be used to calculate the vertical distribution of LWC for different type of ships and different sea conditions. Zakrzewski (1986) extended this formulation to other types of ship's geometry and other sea conditions leading to the following formulation for the vertical distribution of the LWC:

2
 H Vr kg ∙ ∙ expð−0:55hÞ w ¼ 2:032  10−2 ∙ V0 H0 m3


2 2z M ¼ 4:6∙ exp Hs




kg m3



where z is the elevation above mean sea level and Hs is the mean wave height. Ryerson's (1995) observations of spray events on a U.S. Coast Guard cutter show new results, compared with previous observations, which were for small trawlers or Medium-sized Fishing Vessels (MFV). The spray flux was measured on six different parts of the ship, various distances behind the bow, such as the starboard, port main deck, first level bulkhead surfaces, second level bulkhead and flying bridge deck surfaces, but is reported for only one location in the paper. Frequency, location, height, duration, and size distribution of spray clouds were recorded by camera. The weather conditions, ship position, speed, and heading angle were recorded every hour. Several events of spray on the ship were observed and the results of the LWC, spray duration, droplet size, and droplet number concentration were reported. Mean droplet concentration is about 4 × 105 droplets per m3 and average cloud droplet concentration is 1.05× 107 droplets per m3. These observation data show that the spray droplet size varies from 14 to 7700 μm. The average spray event duration was reported as 2.73 s which is longer than the results measured from the Soviet MFV, which was 2 s (Borisenkov and Panov, 1972). Ryerson (1995) tried to formulate the LWC but his comparison between his reported measurements and previous LWC formulations was not satisfactory. All the models described above are compared with each other in the following graph. The constants in the equations are set as V s ¼ 10m =s ; P w ¼ 15; α ¼ 0, and z − h =5 (See Fig. 1).

ð3Þ

where H and Vr are the wave height and the ship speed relative to a wave, respectively. H0 and V0 are the wave height, and ship speed relative to the wave from the Borisenkov and Pchelko (1975). Zakrzewski (1986) attempted to calculate the values of H0 and V0 based on

ð6Þ

Fig. 1. Comparison between different LWC models as a function of wave heights.

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117

2.1.2. Maximum height of spray cloud The maximum height of the spray cloud is the highest elevation that spray droplets can reach above the ship deck. It was determined with a simple geometric model by Zakrzewski et al. (1988). This calculation was based on field data from Kuzniecov et al. (1971) which was reported for an MFV. This model was tested for several moving ships and sea conditions. The formula for the calculation of the maximum height of spray above the ship's deck is 0

Z max ¼ 0:65 V r þ h

ð7Þ

where h′ is the height of the bulwark (0.75 m) and Vr is the ship speed relative to the wave speed Vw. This formulation is based on the data observation from Jan. Turlejski, which is an MFV, although the size of this vessel is larger than the typical Soviet MFV. The spray height is influenced by the ship size and bow geometry but because of the complexity of calculation, these factors were neglected in most of the studies. It was suggested by Zakrzewski (1987) that there is a need to have a relation between the air-sea and ship motion parameters and the maximum height of spray flight for calculating the vertical extent of the collisiongenerated spray Zakrzewski et al. (1988). The authors used the results from Kuzniecov et al. (1971), who published the relation between the maximum height of ice accretion on the MFV and the relative wind speed, for defining the relation between the maximum height of spray and the ship motion parameters. Subsequently, equations of droplet motion were used in their paper to calculate the trajectory of droplets which hit the vessel's foremast. Zakrzewski (1987) reported a formula for the relationship between the maximum height of the spray jet and the air-sea and ship motion which is defined as: Z 0 ¼ aV r

ð8Þ

where Z0 is the maximum height of the spray with respect to the ship deck, and a is an empirical constant, which is calculated from the data in Kuzniecov et al. (1971) and is equal to 0.535. They calculated the heights of the upper limit of ice accretion on the foremast of a MFV and reported parameters such as the ship heading angle, ship speed and the velocity of wind for 13 different cases of icing. Sharapov (1971) reported several observations for the maximum height of spray for different parts of the MFV, such as the foremast, rigging, front side of the superstructure, and roof of the superstructure. The results show a maximum height of spray for different wind speeds from 5 to 24.5 Beaufort, which is equivalent to 8–32 ms−1. The ship heading angle for most of the values is α = 180° but for some of the values is α = 135°. The original data from the Sharapov (1971) observations are shown in Table 2. Lozowski et al. (2000) proposed a new formula for the calculation of the maximum height of spray jet, H as: H ¼ Hs þ

v2r 2g

ð9Þ

where Hs is significant wave height, vr is the relative ship velocity compared to wave velocity, and g is the acceleration due to gravity. Air drag force is neglected and it was assumed that the spray jet was driven vertically into the air with the vsw speed.

Fig. 2. Comparison between different maximum heights of spray models as a function of wave speed.

The equations for different wave heights as a function of relative velocity are compared in the following graph to illustrate differences. Hs in the Lozowski et al. (2000) equation is equal to 3.5 m (See Fig. 2). 2.1.3. Spray cloud duration Another significant parameter is the duration of the spray cloud formation from the wave-vessel impact through to the fully expanded spray above the ship's bow. Zakrzewski (1987) defines the total duration of the spray cloud as: Δt ¼ c

HV r

ð10Þ

U 210

where c is an empirical constant based on the shape and size of the ship hull, H is the wave height, Vr is the ship velocity relative to wave speed, and U10 is the wind speed. The empirical constant c is equal to 20.62, under specific conditions of a wave length equal to 3.09 m, wind speed of 11 ms−1 and ship velocity of 2.83 ms−1. The duration of the spray event from the Lozowski et al. (2000) model was computed with the empirical constant c equal to 10. This empirical constant value was changed for the United States Coast Guard Cutter (USCGC) based on the Ryerson (1995) observations, which indicated that the spray duration is about 3 to 5 s. Ryerson (1995) computed the spray event duration by subtracting the time frame of the first event from the time frame of the last event. It was shown that the minimum spray duration was 0.47 s and the maximum was 5.57 s with the mean cloud duration of 2.73 s over a total of 39 events. Horjen (2013) used the same formulation and suggested that the mean duration of each spray cloud is 2.9 s based on his personal communication with Zakrzewski and his measurements from the MFV Zandberg. 2.2. LWC formula for offshore structure Different field observations and empirical formulations have been reported for calculation of the spray mass flux and icing on offshore rigs and structures. Several computer models have been produced to model icing on these structures. The major differences between vessels and offshore structures are their geometries (impact and droplet

Table 2 Observed height of spray over the ship moving through waves (Sharapov, 1971). Wind force in Beaufort

Wind speed (m s−1)

Foremast rigging

Front side of super-structure

Roof of super-structure

Boat deck

5B 6B 7B 8B ≥10 B

8–10 11–13 14–17 18–20 25–32

Up to 5.5 m above deck Up to 7.9 m above deck Up to 10.5 m above deck 10.5 m above deck 10.5 m above deck

No spray Spray hits object Spray hits object Objects entirely sprayed Objects entirely sprayed

No spray No spray Spray hits object Objects entirely sprayed Objects entirely sprayed

No spray No spray No spray No spray Boat deck hit by spray

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impingement areas), heights, and their relative velocities. Offshore structures are generally higher in height compared with vessels. The shape, impact areas, and droplet impingement areas vary widely between structures and vessels and shape factor should be take into account. Because the geometry and height differences the duration and amount of spray are also different. Forest et al. (2005) worked on the RIGICE code and upgraded a previous version of this code to RIGICE04, which is a program for simulating ice accretion on offshore structures due to spray generation from wave impact. A new Liquid Water Content formula is derived for a height of 10 m, which is in agreement with the data from the Tarsiut Island field data. The LWC is based on the work of Borisenkov and Pchelko (1975) and Zakrzewski (1987). The data from the Tarsiut Island (Muzik and Kirby, 1992) field observations were used and the original formula changed to a new model for offshore structures, which is proposed as: w ¼ K 1  H 21=3  expð−K 2 hÞ

kgm

−3

ð11Þ

where K1 and K2 are the empirical constants. H1/3 is the significant wave height. This formula represent the LWC above the top of the wave-wash zone. The horizontal spray distribution was reported and derived analytically by Forest et al. (2005). They reported the LWC for one field observation on Tarsiut Island and for one single spray event which can be calculated as the formula: w ¼ 1:35  H 21=3 expð−0:53hÞ

−3

kgm

ð12Þ

These LWC formulas are based on one specific field observation and the results and formulations of these observations only match their own data, and cannot be used for other conditions (Roebber and Mitten, 1987; Forest et al., 2005). Forest et al. (2005) compared different LWC formulas from different authors, such as Horjen and Vefsnmo (1984), Brown and Roebber (1985), and Zakrzewski (1986) with the data from the semisubmersible drilling platform, Ocean Bounty, which experienced several icing events. Spray flux calculated for the amount of icing on this structure was reported as 5 to 10 kg m−2 h−1 (Roebber and Mitten, 1987). This calculation was based on the average wind speed of 45 ms−1, the mean significant wave height of 3.8 m, and the deck level of 10–15 m. The results conclude that the LWC calculation from the Zakrzewski (1986) and Forest et al. (2005) models are in agreement with the field observations from the Ocean Bounty, but the other models do not show comparably accurate results. Roebber and Mitten (1987) reported the number of significant waves which produced significant spray clouds as 24 waves per hour for the Tarsiut Island, and the vertical height of these spray clouds was estimated as 10 m. As noted, these data were calculated for an artificial island and assumed to be the same as for offshore rigs. However, other researchers expressed uncertainty about this correlation. Horjen and Vefsnmo (1987) introduced an empirical formula which was used by Kulyakhtin and Tsarau (2014). This formula models the spray on a semi-submersible oil rig, to model the spray flux for further calculation of icing. The compact formula for spray flux is:
   2  1− 1−10−2 U 10 exp − 4zHV9 þ2 M HV U 10 τper   cos α ð13Þ F HV ðzÞ ¼ τ dur 2 exp kHV U 0:667 10 zHV where MHV =6.28× 10−4 kgm−3, kHV =0.0588 s0.667m−0.667, zHV =(2z/ Hs)− 1, Hs is the significant wave height, z is the height above mean sea level, α is the angle between the normal vector to a surface and the wind direction, U10 is the wind speed at z = 10 m, τper is the wave period, and τdur is the duration of spray events.

2.3. Usage of LWC formulation Several models were made to calculate the icing rate on different kinds of ships and most of them use the Zakrzewski (1987) LWC model to calculate the spray rate after wave-ship impact. Lozowski and Zakrzewski (1993) developed the icing model for ships. A larger sized ship, the United States Coast Guard Cutter (USCGC) Midgett model, rather than an MFV, was considered in their simulation, and the spray model of Zakrzewski (1987) was modified. Chung and Lozowski (1999) improved this methodology and also developed the new model based on data from the Canadian fishing trawler, MV Zandberg. Lozowski et al. (2000) reviewed the computer models for icing on vessels and offshore rigs. The four computer models simulating topside icing included RIGICE, ICEMOD, Ashcroft (1985) and Romagnoli (1988). These models compute ice accretion for each forecasting time-step caused by a single spraying event. Lozowski et al. (2000) reported the formulation of the wave-ship spray model based on the Zakrzewski (1987) model. It was reported that an increase in fetch causes increasing wave heights, which increases spray flux on the ship. An increase in ship speed into the waves causes more spray flux on ships. The consequent influence of fetch and ship speed on the ice load were reported as well. The ICEMOD model was produced at the Norwegian Hydrodynamic Laboratories (NHL) and calculates icing on structural segmentations of offshore rigs. This model was introduced by Horjen and Vefsnmo (1986a,b, 1987). ICEMOD can compute ice weight and thickness along horizontal and vertical planes and cylinders in one dimension for two modes, categorized as ship mode and rig mode. The recently published paper by Horjen (2013) described the improved model of ICEMOD2, which is a two-dimensional code. The mass flux of sea spray is calculated based on the following formula: _ c ¼ f ðU 10 ; H s ; T s ; W; α Þ  ðAZ þ BÞ M



−2 −1

kgm

s



ð14Þ

where Z is the height above the mean sea level and Hs and Ts are the significant wave height and period, respectively. Hs and Ts are introduced as empirical relations for Norwegian waters and calculated based on Eqs. (15) and (16) respectively. H s ¼ 0:752  U 0:723 10 T s ¼ 6:16  H 0:252 s

ðmÞ

ð15Þ

ðsÞ

ð16Þ

A and B are non-dimensional constants, dependent on the relative wind and spray heading angle α, and W is the vessel speed. These constants were reported for different heading angles based on the field data from the vessel Endre-Dyroy for the specific wave height and period of a specific wave spectrum.f in Eq. (14), is defined by the formula:

f ðU 10 ; Hs ; T s ; W; α Þ ¼

!
 gT s þ W cos α 2 2 4π T s U 10 ρw H2s

  −2 kgm s−1

ð17Þ

where U10 is the wind speed at Z = 10 m, which is the height above the mean sea level, ρw is the water density, and α is the heading angle. A similar formula was introduced by Horjen and Carstens (1990), which was presented based on field observation from various test objects that were attached to the front mast of the Endre Dyroy vessel, operating in the Barents Sea. The formula is presented as: _ c ¼ A  f ðU 10 ; H s ; T s ; W; α Þ  ðZ  ÞB M

  −2 kgm s−1

ð18Þ

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where Z ¼

2Z −1 Hs

ð19Þ

Comparing the parameters' reported in the field observations of Horjen and Carstens (1990), with the parameters listed in Table 1, shows that the only oceanographic parameter was the significant wave height. The spray flux was measured by spray collectors at the front mast, which were located 14 m from the front of the bow and placed at elevations of 1.10, 2, 3.60 and 5.35 m above the base of the mast. The height of the bow from the waterline was reported as 5.5 m. The spray frequency and the duration of spray were not evaluated but, it was stated that duration of spray and median droplet diameter can be calculated based on the data from the Zandberg, which were equal to 2.9 s and 1.8 mm, respectively. RIGICE is another computer model for the calculation of icing, produced by Roebber and Mitten (1987). They reported that spray flux varies strongly as a function of wave steepness, wave energy flux, and structural shape of impact surface. Wave-generated LWC was calculated as: w ¼ 1:31715  10−3  H2:5  expð−0:55hÞ

  3 kgm

ð20Þ

where H is wave height, and h is elevation above the deck. The LWC vertical distribution is assumed to be equal to the Borisenkov and Pchelko (1975) model. The model was compared with the Zakrzewski (1987) model with several wind and wave conditions and shows an agreement with that model. However, models were compared with results from the Treasure Scout field program, but an adequate agreement was not reached, most likely due to scaling issues. Lozowski et al. (2000) described the spray formulation improvement of the RIGICE code, and discussed the simulation of spray generation as a result of wave impact for both vessels and offshore structures. Lozowski et al. (2000) worked on improving the spray model and the LWC formulation. In the RIGICE code it was assumed that the vertical distribution of the LWC does not vary exponentially when a point of interest, close to the ejection point of droplets, is considered. The vertical distribution of the LWC is assumed to be a thin liquid sheet (spray jet) that is already broken up due to wave-vessel impact. Droplet impingement strongly depends on the distance between the ejection point and the point of interest on the structure. The new spray algorithm, which was used in RIGICE, calculates the flux distribution based on droplet trajectories. It depends on wave period, wave height, wave spectrum, wave run-up on the structure, and consideration of wave force for analysis of the droplet trajectory. The new spray algorithm defines horizontal and vertical velocity components of the droplet for the calculation of droplet trajectory as the following formulas, respectively: u¼

πH kz e sin θ þ C τ

ð21Þ

w¼

πH kz zr −z e cos θ þ τ tr

ð22Þ

where u and w are the horizontal (x) and vertical (z) components, respectively. H is the height of the wave at zero force, τ is the wave period, k is the wave number, z is the amplitude of the wave at zero force, θ is the phase angle of the wave at zero force, C is the wave celerity, zr is the wave run-up height at maximum force, and tr is the rise-time of the force. In this model wave diffraction theory is used to calculate wave run-up on the structure and the deep water linear wave theory is used to find the component of droplet velocity with the wave at the zero-force point. This situation happens at π/2 rad before the maximum force point. The vertical and horizontal velocity components are

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calculated based on the wave run-up on the structure and the wave celerity, respectively. The spray jet is assumed to start at the tip of the bow and extend symmetrically along both sides of the ship to a distance x, which is given by: x ¼ 2 H þ 0:04 v2sw −10

ð23Þ

where H is the significant wave height and vsw is the ship velocity relative to wind velocity and is calculated as: vsw ¼ 1:56 τw þ vs cosðπ−α Þ

ð24Þ

where τw is the significant wave period, vs is the ship speed and α is the ship heading angle, which is equal to zero when the vessel is heading directly into the waves or wind. Lozowski et al. (2000) used the LWC formulation that is reported in Lozowski and Zakrzewski (1993). Another computer simulation code for ice accumulation on offshore structures, called MARICE, was described by Kulyakhtin and Tsarau (2014). The code is a 3-dimensional, time-dependent model and uses Computational Fluid Dynamic (CFD) techniques to estimate icing formation on an offshore structure caused by sea spray. The spray part of this code can solve the spray flux on the surface in two ways: first, introduce spray flux formulation on the structure, and second, calculate spray flux based on the droplet trajectories using the Discrete Phase Model (DPM). DPM is a Lagrangian method that allows the simulation of a discrete second phase consisting of spherical particles dispersed in the continuous phase. The spray model in this simulation was introduced by Horjen and Vefsnmo (1985), previously was reported in this paper as Eq. (13). The spray flux duration and wave period in this study were assumed as 2 s and two wave periods, respectively. However, they reported limitations due to uncertainty in the spray generation because of the unknown droplet-size distribution and droplet trajectories. Shipilova et al. (2012) used the LWC model of Zakrzewski (1987) to generate droplet clouds in front of two types of ships in order to calculate the icing rate. Mean Volume Diameter (MVD) is used to characterize the spray cloud in front of the ships and was set to 250 and 2000 μm. It was assumed in this computational model that the droplet cloud was generated in the form of a square, and the spray cloud widths in front of the two types of ships, the Skandi Mongstad and Geosund, were assumed to be equal to 21 and 19 m, respectively. Kulyakhtin et al. (2012) used a mixed Eulerian-Lagrangian approach with the commercial software, FLUENT, to model the distribution of spray flow rate per unit area on the vessel surface. Similarly to Shipilova et al. (2012), they used the Zakrzewski (1987) LWC formulation for defining a spray distribution in front of the vessel. Shipilova et al. (2012) validated their results, by conducting a series of simple on ground spray measurements. They modeled the simple experiment with CFD and compared the results of the computational and experimental models of ice accretion. The spray period and duration of spray were set according to the values from the Lozowski et al. (2000) model with the assumption of ship speed was 5 ms−1 and wind speeds were 10 and 20 ms−1. The spray period values for two different wind speeds are assumed to be 17.7 and 25.4 s, respectively, and spray durations are chosen as 4.06 and 3.73 s, respectively. The authors modeled two different vessels, the Geosund and Skandi Mongstad. The amounts of ice accretion for different wind velocities, droplet sizes, and temperatures was predicted for these vessels, which showed higher ice accretion with conditions of higher wind velocities and smaller droplet sizes. The authors indicated that the results show that the total amount of icing from smaller droplets is higher than that from larger droplets. All of the main field observation studies are compared with the ideal requirements outlined in Table 1 and the differences are depicted in Table 3, to show the need for more detailed observation of more parameters for better comprehension of the phenomenon.

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Table 3 Available field observation data compared with the categorized factors for successful field observation. Field observation

Ship parameters

Wind

Wave

Droplet Information

Time

Borisenkov and Pchelko (1975) (Russian MFV) Horjen and Vefsnmo (1986a) (Endre Dyroy trawler) Ryerson (1995) (U.S. Coast Guard Cutter)

Ship speed and ship heading angle

Wind velocity

Only wave velocity

Ship speed and ship heading angle

Wind velocity

Only wave velocity

Spray duration for only one event. (No Frequency) –

Ship speed and ship heading angle—Bow geometry is available

Wind velocity

Only wave velocity

No information related to droplets (Empirical Formula for LWC) No information related to droplets (Empirical Formula for LWC) Droplet size, No information related to droplet velocity, calculation of LWC for each event

3. Interaction of sea waves and marine objects Sea water, has been determined to be the primary source of icing in cold and harsh conditions. Therefore, the behaviour of wave-ship and wave-structure interactions needs to be considered as part of the icing process. Aside from the field observation approach, several researchers have explored different techniques, such as analytical, numerical, and experimental methods, for better understanding of the marine icing. There is limited information in the literature related particularly to wave spray formation. In the absence of directly related studies, the topics covered in this section are related to the fundamental spray formation problem, such as waves slamming on walls, spray production after a bow waves, and ship-wave interaction. In this section, phenomena relevant to the wave-vessel or wave-offshore structure interaction are presented and work to simulate these subjects is reviewed.

3.1. Wave-body interaction and slamming in deep-water 3.1.1. Wave slamming on a wall A principal step for analyzing the wave-body interaction is to understand the physical behaviour of a wave slamming on a wall or body in deep water. The wave-body impact process produces high impulsive pressure on the wall and some air that is trapped in this process creates higher pressure, which is called an air-pocket or air-entrainment. Chan and Melville (1988) reported that breaking-wave impacts created higher pressure on the object. Chan and Melville (1988) experimentally investigated the dynamics of trapped air at the moment of impact on a surface-piercing plane wall by making an accurate simulation of the plunging wave. The deep-water wave plunging in this experiment was similar to the work of Longuet-Higgins (1974) and Greenhow et al. (1982). In this experiment, impact pressure, fluid velocity, and surface elevation were monitored simultaneously with a pressure gauge, a laser anemometer and a laser wave gauge. A high speed camera was used to provide a qualitative description of the impact process. Chan and Melville (1988) reported that spray occurred immediately after the impact in all the cases of the experiment. This research also showed that trapped air and the shape of the breaking crest at the moment of impact with the wall is a significant factor in changing the pressure and velocity of the wave impact. A comprehensive review of wave slamming on walls can be found in Peregrine (2003). An important factor in the study of wave-body interaction is the shape of the wave crest as it meets the wall, which has a significant influence on the quality of impacts (Bogaert et al., 2010). The relative angle between the body and the fluid surface was reported as a factor in analyzing impact pressure and velocity by Kapsenberg (2011). Different wave-body interaction angles led to dissimilar and complex impact phenomena with different air entrainment mechanisms. Three different types of wave-wall impacts with different air entrainment mechanisms are shown in Fig. 3 from Bogaert et al. (2010). The difference between a plunging wave and a deep water wave in wave-body impact is another area of study, which has not received as much attention. The majority of researchers have focused on analyzing and calculating the pressure, whereas the wave impact on a ship, the

Spray duration, No frequency

spray formation, and the fluid flow behaviour after the impact have not been a concern (Fig. 4). The water impact problem has received considerable attention. Wagner's theory (1932) was introduced as the basic applied theory for the water impact problem. The Wagner theory is based on a flatdisc approximation and assumes potential flow theory for an incompressible liquid. The boundary conditions and Bernoulli equations are linearized to develop a solution for a wedge impacting on a flat water surface. However, this model assumes a local small dead-rise angle, and that no air pocket is entrapped, which are not applicable for large water-vessel impact. However, it is practical for small water impacts because it provides simple analytical solutions. A schematic view of the Wagner theory, which shows the interaction of the flow between the free surface and body surface is shown in Fig. 5. In this figure, the interaction between fluid and body produces a jet flow, which in practice, ends up as spray. No details or research directly related to the inner domains, which are the jet formation and spray cloud formation, have been found. Later studies worked on improving the Wagner theory by reducing the complexity of the model. These led to solving the analytical solution as an axisymmetric, two-dimensional model (Korobkin, 1996; Korobkin and Scolan, 2003), as well as the solution for the three-dimensional problem (Scolan and Korobkin, 2001). However, the point of interest for this theory and other studies related to slamming was to analyze the impact pressure on an object, and not to track fluid flow and capture spray formation. Detailed review of Wagner's theory can be found in Korobkin (2004) and Howison et al. (1991). Another method associated with the Wagner theory is the BoundaryElement Method (BEM). This method was used by Greenhow and Lin (1985), Zhao and Faltinsen (1993) and Zhao et al. (1996) to solve the boundary-value problem. The distribution of velocity on the free surface, the shape of the free surface and the splash-up height of the domain are known at any time step. The pressure distribution and slamming force results from this method are in agreement with the experimental data and other numerical methods (Mei et al., 1999; Zhao et al., 1996). However, the effect of compressibility and other specific effects must be considered to understand the slamming phenomenon (Ogilvie, 1963). Because of the complexity of the slamming problem, researchers tend to examine this problem from an experimental point of view. Bogaert et al. (2010) and Brosset et al. (2009) experimentally investigated the slamming problem in full scale and large scale by using two types of waves with different parameters. The reported reason that two types of waves were chosen was the difficulty of producing repetitive waves at full scale. Different types of wave impact occurred in the full scale experiment, such as a flip-through and wave pocket impact. The cause of these differences was reported to be the effect of wind. Wind clearly has an influence on the shape of the wave crest before and at the moment of impact, which changes the shape of the wave crest and makes repeatability uncertain. The effect of wind was not a concern for the large scale (1:6) experiment. The aims of the described experiment were to find a deterministic comparison between the full scale and large scale for defining the wave crest shape at impact, the air pocket behaviour, and the pressure associated with the wave crest impact and air pocket. Other similar experimental attempts for

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Fig. 3. Water slamming on a wall. Left: broken wave impacting on wall. Center: wave impact with air entrainment at large scale. Right: Wave impact at full scale with air entrainment. The moment that a wave crest impacts the wall, spray is produced (Bogaert et al., 2010).

modeling the wave pressure on the wall were reported by Chan (1994), Ding et al. (2008), Rudman and Cleary (2013) and Colagrossi et al. (2010). Greco (2001) investigated the behaviour of a dam breaking, which was considered in relevant literature as the same as the behaviour of a vertical wall of water around the bow generated from wave-ship impact (Faltinsen, 2005). A 2-D case of dam breaking, which was defined as the water impact with the wall, was extensively studied numerically and experimentally by many researchers (Greco, 2001; Marsooli and Wu, 2014; Ran et al., 2015). In most of these studies, the pressure of impact on a wall, water overturning, free surface level, and velocity distribution along the free surface were analyzed and discussed. Fullerton et al. (2009, 2010) numerically and experimentally investigated the wave impact pressure on a cube. The cube was located in three positions, completely under the free surface, at half-height above the free surface, and completely above the free surface. The angles of the cube relative to the incoming wave are 0° , + 45° , and − 45°, shown schematically in Fig. 6. Two types of waves, breaking and non-breaking, with different wave heights and wave lengths were used to study the physics of wave impact, and the trends of wave slap loads. The experiment was performed in a high speed tow basin.

Fig. 4. Behaviour of spray formation and spray cloud at different heights was not a concern. The impact pressure was studied. (Sloshel Project, Brosset et al., 2009).

Pressure gauges were used on the side and top of the cube to measure the impact pressure of the wave. Numerical Flow Analysis (NFA) code was used to simulate the described model for understanding the behaviour of flow around the cube and the pressure on the side and top faces of the cube. NFA solves the Navier-Stokes equations and can directly model air entrainment and the generation of droplets. NFA uses the Volume of Fluid (VOF) technique to capture an interface. A detailed description of the NFA code can be found in Dommermuth et al. (2007), O'Shea et al. (2008), Fu et al. (2008) and Brucker et al. (2010). Along with the results of pressure on the cube, the graphical images in this research show a well-formed jet and spray after the impact, but unfortunately the characteristics of the spray cloud, such as spray height or droplet velocity distribution, are not considered as part of the study. 3.1.2. Wave-ship slamming modeling The effect of wave bottom-slamming on ships and structures is an element of spray formation from wave-vessel interaction. The bottomslamming phenomenon is dependent on different variables and physical features and understanding these features are essential for studying the resultant spray formation. Greco et al. (2004, 2012) experimentally modeled water moving to the ship's deck (green water) and the wave bottom-slamming of a ship. Further, the model was investigated numerically in Greco and Lugni (2012). The experiment was performed in a towing tank equipped with a flap wave maker. A 1:20 scaled selfpropelled patrol ship was used to separately investigate the effects of the ship motion on the water on deck using quantity measurement and optimal visualization. The bottom pressure, local forces, wave elevation, rigid ship motion, and water on deck were recorded by different instruments. These and the flare pressure were discussed extensively for both types of modeling. The bottom slamming problem was modeled numerically using the Wagner theory to predict the maximum impact pressure at the time of impact. Overall, the experimental measurements and numerical solutions are in agreement with each other. However, the aim of this study was not directly related to spray formation. Greco et al. (2004, 2012) reported that wave-body interactions producing water jets along the hull and the spray cloud led to higher vertical motion. These two non-linear features of wave-ship interaction are not taken into account in their formulation. Underestimation of the experiment measurements was probably due to this neglect. The amount of water that exceeds the freeboard and turns into spray does not totally return to the deck, and overtopping events depends on the movement between incoming waves and the ship, which can be substantially different in a model compared with a full scale ship. Greco and Lugni (2012) investigated the water on deck problem using a 3-D weakly non-linear seakeeping method coupled with the water-on-deck method based on shallow-water theory. Different parts

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Fig. 5. Wagner problem, no details and research directly relative to the inner and jet domain can be found in literature (left). Experimental view of wedge impact with free surface by Qian et al. (2006) (Right).

of the problem were divided into simplified models. A 2-D in-deck problem was split into a series of 1-D problems, which can be solved by the Godunov method for the main flux variables. The bottom-slamming problem was modeled with the Wagner theory, and the free surface was captured by the Level Set (LS) technique, which will be discussed in the next section. The transient Green function was used to linearize the radiation and scattering waves around the incident waves and solve the problem in the time domain. Voogt and Buchner (2004) experimentally investigated wave impact pressure and probability of wave impact on an FPSO. In this test, the simplified bow wave at the test scale of 1:60 in deep water was considered. The incident wave data, vessel motions, and bow pressure were calculated using data from a large array of pressure transducers and force panels. A video recording system which was capable of determining the correlation between the wave shapes and the impact pressure time traces was also used. The experiment shows that the linear theory for waves under-predicts wave steepness, but the second order wave theory provides a reasonable water surface model. The wave impact with the FPSO bow is shown in Fig. 7, which depicts the jet and spray after the wave impact. Correlation between wave impact and wave steepness was found to be highly correlated. It was observed in this experiment that up to a certain vertical free surface velocity no impact occurred between the wave and bow, but after this level, the probability of impact increases linearly up to a probability of 100% (Guedes Soares et al., 2004). This is illustrated in the left diagram of Fig. 7. Hu et al. (2006) and Hu and Kashiwagi (2009) numerically simulated the strongly non-linear behaviour of a ship-wave interaction with the Constrained Interpolation Profile (CIP) method, which is based on a Cartesian grid method, to capture the interface of a two dimensional

Fig. 6. The cube, which is depicted as a plate, was placed at different submergence levels and angles related to the incoming wave (Fullerton et al., 2010).

wave impact with a floating body. An experiment was performed in a two dimensional wave channel. A box type floating body was used instead of a ship scale model, shown in Fig. 8. A wave maker and a wave absorbing device were used at the two ends of the tank. Heave, roll and sway of the floating body, wave elevation and a forced oscillation test in heave were performed. The results from the numerical method were compared to the experimental results for the case of fixed sway. The results show adequate agreement for different case studies, both for green water events and after its occurrence (Fig. 8). Hu et al. (2010) used the same method and numerically investigated the behaviour of extreme wave-body interaction to measure both ship motions and wave impact loads. This research modeled the wave impact problem in three dimensions and virtual particles were used to define the body. An experiment was performed in a towing tank with the Wigley ship model. The pressure of green water on the deck, as well as the heave motion of the ship, were calculated. For the numerical part of the solution, three grid sizes were used to predict the pressure and height of the free surface. All the models under-predicted the peak values of pressure, especially the low-resolution grid model. The free surface for the low grid resolution was not clearly modeled, but for two other high resolution models, the results of the free surface level is in agreement with the experimental data. A container ship (S175) was modeled and the snapshots that were presented in this research paper clearly showed water-on-deck and wave slamming features. Study of the spray cloud and jet formation after impact were not an objective of the presented paper.

3.2. Bow wave and shoulder wave breaking In recent years, substantial observations and modeling of fluid motion at the ship bow in waves has been undertaken. The behaviour of a liquid sheet around the bow, the bow wave, is related to the study of wave-ship impact behaviour and the material associated with this phenomenon can be useful for understanding the behaviour of the water after impact with the bodies. Baba (1969) simulated a bow wave and concluded from the observation that there is a significant component of wave resistance related to breaking the bow wave. Dagan and Tulin (1972) considered the steady, inviscid, irrotational flow past the bow of a ship which moves with constant velocity in infinite depth. They solved this flow for a small Froude number using a perturbation expansion method. They observed that the free surface rises smoothly to a stagnation point on the bow. It was shown later by Vanden-Broeck et al. (1978) and VandenBroeck (1985) that waves are always present at the bow in infinitely deep waters and that there is no continuous free surface profile for a bow. They suggested that the form of solution for the free surface at the bow is an overturning jet. This assumed that the free surface was attached to the stagnation point for small Froude number, which is based on the draught of the ship as the important physical length scale for bow flows in infinitely deep water.

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Fig. 7. Left: Wave impact with FPSO Bow (Voogt and Buchner, 2004), Right: Probability of wave impact with FPSO bow related to the free surface vertical velocity for three groups of experiments.

Grosenbaugh and Yeung (1989) investigated the non-linear behaviour of the free surface around the ship bow using a mixed EulerianLagrangian boundary integral equation method. Vanden-Broeck (1989) extended the solution to all values of Froude numbers between pffiffiffi 1.22 and 2, when the Froude number is based on the depth of the fluid upstream. Hocking (1993) reported a wave-less solution for Froude numbers N1 when the fluid was attached to the front of the bow. Mccue and Forbes (1999) solved a boundary integral method for a rotational flow at a finite depth. Their solutions show the same results as the irrotational flow passing a rectangular body. Dong et al. (1997) visually investigated the free surface around a ship model with the attached liquid sheet and the behaviour of the bow wave near the ship bow. Particle Image Velocimetry (PIV) was used to measure the velocity and vorticity distribution. Karion et al. (2004) performed an experiment in a towing tank to investigate the effect of scaling on the bow wave. The quality of spray generated by the bow wave was analyzed and the droplet size and velocity distribution were measured with a high speed camera. The number of droplets in each of the images was counted as a function of time. It was shown that even though the bow wave is a steady breaking wave, the local spray droplet population can be extremely unsteady. Most of the runs were done in calm water, but some of those were reported for rough water, and the amount of bow spray in the rough water increased significantly. However, the depth of the focal plane is important and was not measured in these experiments. Most of the droplet sizes in this experiment were smaller than 0.2 cm, and the droplet velocity was about 10–60 cm−1. Observations from this experiment show that in some cases the bow wave, which is a sheet of water on the hull, was curled

over and breaking, but spray (white water) was not generated. Researchers called this event a smooth breaking case. Based on this observation, critical values of the Froude and Weber numbers for the formation of spray droplets were identified. The values of these two numbers should exceed a limit for spray to form. From the figure that was reported in this report, it was shown that the critical values of Froude and Weber numbers were estimated as N 0.75 and 100, respectively. Qualitative observation in this field leads to understanding some characteristics of bow wave spray, and the same ideas can be used to model spray formation due to wave impact with the bow. Some types of spray generation mechanisms were observed and categorized as ligament formation and droplet pinch-off, bow sheet thinning and disintegration, and secondary droplet break-up. These were studied in detail by Sarpkaya and Merrill (2001). Another significant factor is air entrainment by breaking bow waves. The white water that is produced from the movement of a ship through the water rises from the bow wave breaking, and the air entrainment of the overturning water sheet attaches to the hull. Waniewski et al. (2002) experimentally used a plate with an angle θ to the oncoming flow in a tow tank and in a flume as a wedge-shaped hull to analyze the behaviour of the bow waves and the air entrainment process for different flow conditions. Depending on the Froude number, three different flow regimes (subcritical, critical and supercritical) were introduced for the flume experiment (Miyata and Inui, 1984). From three different flow regimes, different behaviours of flow were observed, and only the supercritical flow regimes produced a wave similar to the bow wave. Fig. 9 from Waniewski et al. (2002) shows a schematic view of a bow waves generated from supercritical flow. The waves near the plate

Fig. 8. Left: The computational domain and floating body shape of Hu et al. (2006) experiment. Right: the computational and experimental results before and after the green water event. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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Fig. 9. Left: Schematic view of bow wave. Right: Cross section at A and B of the Plan-view, showing the procedure and different characteristics of bow wave breaking near the hull wall. Splash point, water jet break point, and secondary liquid sheet are shown.

produce a secondary liquid sheet, which is attached to the plate. Gravity causes the rest of water, when it has reached its maximum height, to curl over and impact on the free surface. This impact produced spray and some amount of splash. The contact line, splash area and wave crest tip are schematically shown in this Fig. 9. Shakeri (2005), Shakeri et al. (2009a,b) and Maxeiner (2009) experimentally studied a large variation of wave crest shapes using a two dimensional plus time approximation for the wave maker. The maximum bow wave height and plunging wave geometry were measured. The bow waves' heights and the contact point heights along the hull were measured with a Laser-Induced Fluorescence (LIF) method. An image, which captured the wave crest and its features with this method is shown in Fig. 10. The laser sheet that interacts with the bow wave clearly shows the contact point of bow, wave crest and jet formation. Different types of breakers in this study were analyzed and categorized as spillers and plungers. The main goal of this study was to characterize bow wave geometry for different bow types under different wave conditions. The spray production after the wave crest and impact of the jet with the free surface were not studied. Some simple analytical studies can be found in the literature studying bow waves. Noblesse et al. (2008) analytically defined a simple expression in terms of ship speed, draught, and waterline entrance angle to define the height of a ship bow wave and the bow wave profile. This expression provides a relationship between the main features of a ship bow wave such as wave height, flow location, and flow profile with basic ship parameters such as speed, draught, and waterline entrance angle. This analytical expression for the height of an overturning

bow wave was compared with experimental data from measurements of Noblesse et al. (2006), and good agreement was achieved. Other analytical expressions for distance between a ship stem and bow wave crest, rise of water at the ship stem, and ship bow wave profile were introduced in this research and compared with experimental data. Noblesse et al. (2014) introduced analytical expressions for measuring the size, shape and thickness of the overturning bow wave. These expressions are explicitly related to ship design parameters and are compared with experimental and CFD results. Correct trends were achieved, but the results from the analytical expressions do not predict accurate results. The methods and expressions of these analytical methods are not reported in this paper and readers can refer directly to the cited articles. 4. Computational fluid dynamics (CFD) analysis of wave-vessel interactions and spray formation Numerical studies of wave-vessel interaction and spray production are complex problems in CFD (Brucker et al., 2010; Dommermuth et al., 2006). The non-linear interaction between an extreme wave and a floating body needs to include the behaviour of free surface turbulence, wave impact, wave breakup, air entrainment and bubble generation, spray sheet formation, and droplet trajectory (Hendrickson et al., 2003). The stages of this complicated phenomena are shown schematically in Fig. 11. Each part of this process is discussed in this section. 4.1. Non-linear free surface modeling Different numerical methods have been developed to address discontinuities. Lagrangian and Eulerian approaches have been used in successful numerical approaches to approximate free boundaries in fluid flow. Both methods have advantages and disadvantages and selection of technique is based on the nature of the problem. Several reviews of free surface flow are available by Mei (1978), for linear free surface flow using integral methods; Floryan and Rasmussen (1989), for numerical methods for viscous flows with moving interfaces; and Tsai and Yue (1996) for computational methods of non-linear free surface flows.

Fig. 10. Typical view of bow wave using the LIF measurement method (Maxeiner et al., 2011). Contact line, wave crest and jet tip are clearly captured. Locations of Contact point, crest and jet tip are tracked over time and space and shown visually in contrast with schematic view in Fig. 9.

4.1.1. Marker and cell (MAC) Harlow and Welch (1965) introduced the MAC method for simulating free surface flow which is a development on the previous Particlein-cell (PIC) code. PIC used mass particles that carried position, mass, and species information (Evans and Harlow, 1957; Harlow, 1965). The MAC method is a finite difference technique on a staggered grid for solving incompressible viscous free surface flow and employs variables of pressure and velocity as the primary dependent variables. The MAC method defines whether a cell contains fluid or not (McKee et al.,

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Fig. 11. Modeling procedure for solving the wave-vessel impact problem.

2008). The cell contains a particle it is labelled “contains fluid”, and if not, it is labelled “empty” (Harlow et al., 1976). This means that marker particles can cover the whole fluid area while each specific particle moves with the fluid velocity in its own coordinates. Storage requirements for this method increase significantly because of the number of point locations that must be stored. Amsden and Harlow (1970) introduced a new simplified version of MAC which is called SMAC. It reduces the difficulties of the first version by splitting the calculation into two parts in the ZUNI code. Current MAC methods incorporate the ability to follow the free surface, and they are able to calculate separation. The MAC method was the base of the creation of other advanced methods which are particle-less and based on the implicit scheme such as the implicit-fluid Eulerian (ICE) method (Harlow and Welch, 1981). Before the appearance of the MAC method, Francis Harlow in the 1960s created a particle-in-cell (PIC) method (Evans and Harlow, 1957; Harlow, 1965), which uses mass particles and includes material position and mass information. This method has a special capability to numerically model flow, which has large distortion and compressions in several spatial dimensions. In this method a single fluid tracked in the continuous phase uses the Lagrangian approach, while densities and currents are calculated with the Eulerian approach (Evans et al., 1957). This method does not give an accurate solution because of numerical diffusion and is noisy. The difficulties of this method are welldocumented. To make the PIC method as accurate as other methods, the viscosity and heat conduction calculation must be reduced. Brackbill (1988, 1991) introduced a new model entitled Fluid Implicit Particle method (FLIP) which overcame the problem of numerical diffusion using fully Lagrangian particles. 4.1.2. Volume of fluid (VOF) Hirt and Nichols (1981) first introduced the VOF method which was simple and efficient compared to other models for the calculation of free boundaries. This Eulerian technique was used to numerically simulate deformation of a free surface. VOF defines a fluid fraction function F(x,y,t), which has a value between unity and zero which represents the volume fraction of the cell occupied by fluid. Cells with F value between zero and one represent a free surface and they must have at least one neighbouring cell for which the F value equals zero. The VOF method requires less storage capacity for each cell. The derivative of F can be used to calculate the boundary normal. By knowing the normal direction and value of F, a line representing the free surface can be shown. The VOF technique is a simple and economical way to track the interface and free boundaries. The first VOF code for calculating the free surface was introduced by Nichols and Hirt (1975) which they titled SOLA-VOF. Subsequent improvements were made by Chorin (1980) and Lafaurie et al. (1994). More accurate and advanced techniques were introduced and used, such as the Piecewise Linear Interface Construction (PLIC) method

(Ashgriz and Poo, 1991; Rider and Kothe, 1998). In this method, the free surface is approximated by segments fitted to the boundary of every two neighbouring cells. The velocity field fluxes are computed at cell faces and the fluid is moved from a donor cell to an acceptor cell (Youngs, 1987). One of the advantages of the PLIC method is that it does not attempt to reconstruct the interface as a continuous chain of joined segments. Rather, it reconstructs the interface as a discontinuous chain with asymptotically small discontinuities. Whenever the curvature is small (i.e. the radius of curvature is large with respect to the grid size) this method will be accurate (Scardovelli and Zaleski, 1999). Use of the original VOF method leads to the appearance of small air-pockets called ‘flotsam’ which is due to the piecewise constraint (Rider and Kothe, 1998). Refer to the original VOF method, fluid can be lost or gained due to rounding in the F function (Harvie and Fletcher, 2000). The VOF method was applied to the simulation of the turbulent free surface and wave impact problem by Kleefsman et al. (2005) and Ganjun et al. (2013). Kleefsman et al. (2005) used the VOF method together with a local height function to overcome the problem of flotsam. The combination of these two methods was first introduced by Gerrits and Veldman (2000) for simulating sloshing on board of a spacecraft. Ganjun et al. (2013) used the VOF model coupled with the turbulence model for modeling the tank numerically to increase the accuracy. Wang and Ren (1999); Ren and Wang (2004) and Xuelin et al. (2009) used the VOF method to simulate waves slamming into an object. The VOF method does not resolve the interface at high Reynolds number with a large density difference between the two flow phases. Velocity jumps occur right at the free surface interface in the VOF method, thus artificial tearing occurs. In order to mitigate these occurrences, smoothing and filtering are required to reduce the velocity jump (Fu et al., 2010; Brucker et al., 2010). Fu et al. (2010) used a density weighted velocity smoother which is effective for simulating the free surface with the VOF method in the applications to modeling bow waves and stern waves. Dommermuth et al. (2006) used the VOF method with piece-wise linear polynomials to simulate ship waves with good results reported. 4.1.3. Level set method Recently, the Level Set method has become popular for problems of tracking fluid boundaries and was first introduced by Sussman et al. (1994) and Osher and Sethian (1988). In this method the interface can be modeled sharply. Osher and Sethian (1988) introduced a new algorithm called the PSC scheme for representing a moving surface with free boundaries. The PSC scheme numerically solves the HamiltonJacobi equations with a viscous term. This scheme uses the approximation technique from hyperbolic conservation laws. In general, the level set function is a method of capturing the interface represented by the zero contour of a signed distance function (Olsson and Kreiss, 2005). The movement of the interface is governed

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by a differential equation, and the advection is calculated based on the Weighted and Non-Weighted Essentially Non-Oscillatory methods (WENO, ENO). The level set method represents the two-phase flow approximately using the flow of a single fluid, the physical properties (density and viscosity) vary across the interface. In this method, the fluid density and viscosity are defined as smooth functions of the distance from the interface. The function is continuous across the interface (Iafrati et al., 2001). Vogt and Larsson (1999) applied this method to the two dimensional wave problem and achieved good agreement with experimental data. Sussman et al. (1994) re-initialized the distance function to keep the thickness of the transition zone constant. They solved the bubble generation function, denoted by the level set function as ϕ, with ϕ taken to be positive outside the bubble, negative inside the bubble and zero at the bubble interface. Application of the single and two-phase Level Set methods to study flow around a container ship can be found in the Cura Hochbaum and Shumann (1999) and Carrica et al. (2005). The advantage of the Level Set method is that using the advection equation and computing the free surface curvature is more accurate than computing these equations with a non-smooth function. Another advantage of the level set method is that a smooth function distance gives the interface a thickness fixed in time, and density and surface tension both depend on the level set function. The drawback of the Level Set method is that the method is not conservative (Tornberg and Enhquist, 2000). In problems of incompressible two phase flow, loss or gain of mass can occur. Several attempts have been made to improve the poor mass conservation of the Level Set method by combining the method with other methods such as VOF (Sussman and Puckett, 2000), the marker particles method (Enright et al., 2002) and by exchanging the level set function with a new advection scheme (Olsson and Kreiss, 2005). Olsson et al., 2007 introduced a new level set model which is easy to implement for three dimensional problems. For this model, mass conservation is significantly better than the original level set model. Our primary interest in this method is for applications of wave interaction with vessels and offshore structures. These applications require robust interface capturing that is accurate and scalable on a high performance computing platform. The Level Set method is robust over large deformation, and applicable on a large scale with the high Reynolds number flow. Kees et al. (2011) introduced a new conservative level set method for modeling wave breaking. The method maintains a sharp and stable air-water interface, and shows that the method is accurate for large scale and complex free surface problems. Note that, the interface can be defined as a single-phase level set method for which only the water phase of the problem would be solved. This kind of approach has several advantages and disadvantages. A comparison of this approach with the wave–vessel interaction problem needs shows that the continuity condition will not be satisfied in the air phase. Thus the method is not suitable for problems such as when air is trapped or bubbles are formed inside the liquid, or when the air phase gets pressurized (Di Mascio et al., 2004). 4.1.4. Constrained interpolation profile (CIP) A new CFD simulation approach for the non-linear free surface problem is the constrained interpolation profile (CIP) method. The method is an Eulerian approach on a regular Cartesian grid with multiphase computation, and is suitable for multiphase flow problems. This method treats compressible and incompressible fluids with a large density ratio and simulates the interaction of gas with a liquid or solid (Yabe and Wang, 1991). Yabe et al. (2001) reviewed the CIP method for different problems. The conservative pressure-based algorithm of CIP, which is robust and stable, is presented by Tanaka et al. (2000). In general, the CIP method can be defined as an upwind scheme with sub-cell resolution for the advection process, and a pressure based algorithm that treats liquid, gas and solid phases with good robustness and stability. This method can handle the complicated free surface geometry

in extreme deformation. Because no re-meshing is required, the computation time is reduced. Hu and Kashiwagi (2004) used the CIP method to simulate extreme wave-body interaction problems. The numerical results were compared with experimental data and showed satisfactory agreement. Hu and Kashiwagi (2009) and Zhao and Hu (2012) used the CIP based Cartesian grid method, in combination with the Tangent of Hyperbola for Interface Capturing (THINC) scheme, and a virtual particle method, for simulating floating body interaction with waves. The results show that implementation of THINC in conjunction with the original CIP shows much better results in terms of mass conservation and capturing the interface. 4.1.5. Inviscid computation of waves In addition to the discussed methods for modeling the non-linear free surface problem for viscous flows, another method called the Mixed-Eulerian-Lagrangian (MEL) method can be used for simulation of inviscid free surface problems. The MEL method is based on a boundary integral formulation and is in the form of a potential flow problem. Longuet-Higgins and Cokelet (1976) introduced this method for a simulation of 2-D breaking waves. Another early work in this field is the simulation of a floating body on a non-linear free surface by Faltinsen (1977). The detailed numerical procedure for this method can be found in a review paper by Tsai and Yue (1996). Other researchers, used this method for modeling wave and wave breaking in threedimensional problems, such as Dommermuth and Yue (1987), Tian and Choi, 2013, and Romate (1990). Dommermuth et al. (1988) compared a numerical model using the MEL method with measurements on a breaking focusing wave. The researchers indicated that the surface elevation, as well as vertical and horizontal particles velocities, agreed with the experimental data. Song and Banner (2002) developed a wave criterion for focusing wave groups, which was compared with the experimental research of Andonowati et al. (2006). A higher order MEL method was used by Fochesato et al. (2007) to numerically model three-dimensional breaking of focusing waves. Another approach for modeling non-linear waves is the pseudo-spectral method combined with a Fourier Transform, introduced by West et al. (1987) and Dommermuth and Yue (1987). The linear theory of rogue waves indicates that an extreme wave can be produced by superimposing several wave components with different phases and directions over a small region of space and time (Dalrymple, 1989). She et al. (1994) used Particle Image Velocimetry (PIV) to study the kinematics of a breaking rogue wave. Brandini and Grilli (2001) used a snake wave-maker to model extreme overturning waves and numerically modeled this type of wave by solving the three-dimensional potential flow. Fochesato et al. (2007) introduced a three-dimensional Numerical Wave Tank (NWT) to model a rogue wave by solving a fully non-linear potential flow equation for a free surface using a mixed Eulerian-Lagrangian method and a high order boundary element method. A complete review of inviscid and viscous waves with both numerical and experimental simulations for deep and intermediate waters is provided in Perlin et al. (2012) and Dysthe et al. (2008). Potential flow equations for the free surface are as follows. Laplace's equation for mass conservation is: ∇2 ϕðx; y; z; t Þ ¼ 0

ð25Þ

Green's second identity transform equation for the boundary integral equation is: Z α ðxl Þϕðxl Þ ¼

  ∂ϕ ∂G ðx; xl Þ dΓ ðxÞGðx; xl Þ−ϕðxÞ ∂n Γðt Þ ∂n

ð26Þ

where G(x, xl) = 1/4π | x − xl | is the three-dimensional free surface Green's function. Γ indicates the whole domain boundary, ϕ is the velocity potential, where velocity u= (u, v, w) is given by ∇ϕ. The vector n is

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the normal vector exterior to the boundary, and α(xl) is proportional to the exterior solid angle of the boundary at point xl. Along with the boundary conditions and numerical methods for the fully non-linear potential flow model for water waves, Fochesato et al. (2007) and Grilli et al. (2001) used the non-linear kinematics and dynamic boundary conditions for a free surface boundary. The inlet boundary condition was presented with a specific velocity and a wave-maker was introduced at this boundary to generate a wave. The bottom of the domain satisfied the no-flow conditions and an absorbing wave-maker was used for the outlet boundary conditions. The boundary integral equation technique was used to solve the Green's function, which defined as the internal velocity u, and local acceleration a inside the domain as (Fochesato and Dias, 2006): Z uðxl Þ ¼ ∇ϕðxl Þ ¼

aðxl Þ ¼ ∇

∂ϕ ðxl Þ ¼ ∂t

  ∂ϕ ∂Q ðxÞQ ðx; xl Þ−ϕðxÞ ðx; xl Þ dΓ ∂n Γ ðt Þ ∂n (

Z Γ ðt Þ

ð27Þ )

2

∂ ϕ ∂ϕ ∂Q ðxÞQ ðx; xl Þ− ðxÞ ðx; xl Þ dΓ ∂t∂n ∂t ∂n

ð28Þ

where Q ðx; xl Þ ¼

1 4πjr j3

! r

n   o ∂Q 1 ! ! ! ! ðx; xl Þ ¼ r n −3 er  n er ; 3 ∂n 4π jrj

ð29Þ

and

r ! er ¼ jr j

ð30Þ

where r indicates the distance from the boundary point x to the interior point xl. Dias and Bridges (2006) reviewed the extensive details of the numerical computations and methods for modeling time-dependent inviscid water waves. Hedges (1995) showed a better boundary between Stokes and cnoidal theories, which is called U, its Ursell number, and defined as: U¼

Hλ2 3

d

ð31Þ

which can be used to characterize waves. A larger Ursell number defines long high waves where the cnoidal theory can be applied. A small Ursell number (deeper water) is applicable for Stokes theory. Different wave theories and their boundary conditions are extensively discussed in Fenton (1990). 4.1.6. Smooth particle hydrodynamics Besides the Eulerian methods, Lagrangian methods are an appropriate solution for modeling wave-vessel impact and large free surface deformation. Most of the Lagrangian approaches cannot be used for large and sharp deformation of free boundaries at large scales because of computational costs (Altomare et al., 2015). Smooth Particle Hydrodynamics (SPH) is a meshless Lagrangian method which was introduced by Gingold and Monaghan (1977). The meshless character of SPH makes this method very flexible for simulating complex physical problems that are difficult to simulate with grid based techniques. In general, SPH computes trajectories of the particles of a fluid, which interact according to the Navier-Stocks equation. The fluid domain is represented by a number of points as particles and each point carries scalar information, pressure, velocity, and density among other factors. Complete review of this method can be found in Monaghan (1992). Monaghan and Kos (1999) simulated run up and return of waves travelling over shallowing water with SPH and their results were validated with an corresponding experiment. Monaghan (1994) used SPH to model the free surface. Dalrymple and Rogers (2006) used the SPH method to model wave breaking. These studies indicate

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that the SPH method can be used to simulate wave-vessel interaction problems and spray formations (Liu et al., 2013; Lo and Shao, 2002). A modified particle method called the Moving-Particle Semi-implicit method (MPS) was presented by Koshizuka and Oka (1996) and Koshizuka et al. (1998) for simulating a water collapse experiment. The new method was based on a modified kernel function. Both numerical stability and computational speed are improved in this method. The model represents fluid by moving particles, and the convection term is calculated by the motion and the interaction of these particles. Numerical diffusion does not appear in this method because of the convection calculation procedure, in which convection is directly calculated by the motion of the particles. Several researchers used this type of numerical model to simulate water collapse experiments and free surface problems with different kernel equations (Ataie-Ashtiani and Farhadi, 2006; Hu and Kashiwagi, 2004; Khayyer et al., 2008). 4.1.7. Other methods The front tracking method is used to model a multi-phase fluid in each phase, the interface is represented by a system of curves, and moves through a fixed grid in time. The main idea, reported by Richtmyer and Morton (1967) was implemented for the first time by Glimm et al. (2001) which leads to the introduction of the front tracking method. Bifurcation arises from a system of curves when it moves dynamically and intersects on the topology of the system of curves. Three main concerns arising from the study of the front tracking method are reported by Glimm et al. (2001) in their paper: structure data, reconstruction topology and theory behind the location and type of discontinuities. Tryggvason et al. (2001) present a new method which is a type of front tracking and uses some aspects of the front capturing technique. Fluid flow is simulated on a stationary regular grid, but the interface is modeled by a separate grid of lower dimension. All the phases are modeled together with a single set of equations. The front tracking method reduces the resolution needed to keep the front sharp, and eliminates numerical diffusion (Unverdi and Tryggvason, 1992). In general, the interface is considered as a set of points forming an unstructured mesh. This mesh moves through a staggered Eulerian grid where the fluid pressure and velocity are calculated (Sousa et al., 2004). The main drawbacks and problems of the front tracking method arise due to the complexity of this method. Another problem results from the interaction of the front with another front or another part of the same front. The computational analysis cannot recognize two fronts in one single cell. The main application of this method is to simulate and analyze bubbly flow problems (Unverdi and Tryggvason, 1992; Ervin and Tryggvason, 1997). Another application of this method is for the study of spray, and can be implemented for solving and analyzing spray formation after a wave-vessel impact. Several numerical simulations in this field such as the break-up of a liquid jet (Homma et al., 2000), and the secondary breakup of drops (Han and Tryggvason, 1999a,b) show satisfactory results in the modeling of break up. One attempt using the front tracking method for simulating the wave can be found in Yang and Tryggvason (1998). Another approach in simulating the viscous free surface flow is, instead of using the methods which are based on the interface reconstruction steps, the advection equation could be solved directly (Vincent and Caltagirone, 1999). In this method, a colour function C is defined corresponding to the volume of the fluid in the cells. Solving the problems by this method requires three steps; firstly, the function C should be known and an interface able to be constructed; secondly, the NavierStokes equation should be solved for the velocity field; thirdly, the interface should be moved at time n, to calculate a new C function at time n + 1. The colour function C is equal to one for cells that fully contain fluid, and this procedure can be used for several fluids. Classic schemes such as Lax-Wendroff (LW), QUICK and others are not sufficient to treat the hyperbolic character of the mass transfer

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equation and the advection of shocks (Hirsch, 1990 and LeVeque, 1992). This leads to oscillation in fluids. The Total Variation Diminishing (TVD) theory by Harten (1997), Leveque (1990), Sandham and Yee (1989) and Yee (1987) shows good agreement when compared with the classical scheme and shows no oscillations, controlled diffusion and mass conservation (Vincent and Caltagirone, 2000). The TVD-LW scheme, coupled with the Large Eddy Simulation (LES) turbulent model was used by Lubin et al. (2006) for simulating three dimensional plunging breaking waves. This model shows improvement in the numerical modeling of breaking waves. Several studies validate the TVD-LW scheme for multiphase flow configurations and coastal applications (Lubin et al., 2003; Helluy et al., 2005). Finally, hybrid models have been introduced to overcome the drawbacks of the methods previously described. The main drawback of the Lagrangian or Eulerian approaches is their inability to capture large deformation of the free surface for two phase flow problems. Sussman and Puckett (2000) developed a coupled Level Set (LS) and VOF method to overcome the mass conservation problem of the LS method when a sharp interface was produced. The results of this method also accurately model surface tension driven flows, for which the VOF method is not accurate. Another hybrid approach is the coupled level set with the front tracking method. This hybrid method for the simulation of two phase flows leads to a more accurate solution and better mass conservation. The front tracking method has a problem with complex global topological operations on the front. In order to overcome this difficulty. Shin and Juric (2002) and Ceniceros et al. (2010) applied a new isocontour reconstruction to the Front Tracking Method. This was further developed by Shin and Juric (2007, 2009). Maric et al. (2015) introduced a new hybrid LS method coupled with the front tracking method, which is very similar to the work done by Shin and Juric (2002) but with a different definition of front. These newer hybrid methods have application in the simulation of bubbly problems and track the interface with a Lagrangian approach. Further review did not find any usage of hybrid methods for large scale free surface deformation problems in the literature. 4.2. Air entrainment and bubble generation The simulations of flows with air entrainment and air bubble generation are important in the study of problems involving wave-vessel impacts (Hendrickson et al., 2003; Ding et al., 2008). During wave-vessel interaction, a large number of bubbles forms when the air-water interface is highly unstable. Analysis is complex because of the multi-scale nature of the problem. The surface tension and the turbulence of air entrainment are at micron scales, while, analyses of the fluid flow, modeled by Navier-Stokes equation, are on the scale of meters (Moraga et al., 2008). Analyses when conducting Direct Numerical Simulation (DNS) are complicated and computationally expensive because of the number of cells and storage capacity required for simulation of multi-scale problems. For a review of gas entrainment by plunging liquid jets, readers could refer to Bin (1993). Iafrati et al. (2004) used the level set approach to simulate air entrainment of a plunging liquid jet. This study was unable to capture the formation of small bubbles around the main air cavity because of the lack of spatial resolution. Moraga et al. (2008) presented a physically based model to predict the location of air bubble entrainment. Due to the lack of spatial resolution, this model cannot capture the fine details of air entrainment regions. Waniewski (1999) designed an experiment to measure air entrainment caused by breaking waves. Some photographic studies of air entrainment and bubble generation have been reported by Loewen et al. (1996) and Bonmarin (1989). The simulation procedure for following air entrainment and bubble generation is complicated. The two phase flow, the dispersed phase and the continuous phase, must be followed by the system of equations which cover mass conservation and momentum conservation. A

turbulence model must be introduced to the system of equations and interfacial force density should be modeled. Finally, free surface methods for capturing an interface should be introduced as well. The system of equations for the dispersed phase are introduced here based on the work of Martinez-Bazan et al. (1999a, 1999b, 2002) and Larreteguy et al. (2002). For details, the reader should refer to the references. The Boltzmann-type transport equation for simulation of the mass conservation of polydispersed flow is: ∂N‴g ∂t

  − þ ∇  ug N‴g ¼ E g þ Bþ g −B g

ð32Þ

where N‴g is the bubble number density for a characteristic diameter of Dg, moving with the velocity ug, E g is the source of bubbles due to air in− gestion, and Bþ g and Bg are the source of bubbles due to bubble breakup. The E g (bubble source) is modeled by Moraga et al. (2008) as: X

δ x−xg E g ðxÞ ¼ S0 f E Dg ΔDg

ð33Þ

g

where S0 is a constant that determines the bubble source intensity, fE(Dg) is a bubble size probability density function, ΔDg is the width of the bin, δ(x − xg) is the sum over the Dirac delta function and is used to activate the source in the numerical grid (xs) of the liquid region. The other source term in Eq. (31) is introduced by Martinez-Bazan et al. (1999a, 1999b, 2002) as: Bþ g ¼

Z

∞ Dg



mðD0 Þf Dg ; D0 g ðϵ; D0 ÞN‴0 dD0

‴ B− g ¼ g ðϵ; D0 ÞN g

ð34Þ ð35Þ

where ϵ is the turbulent dissipation, m(D0) is the mean number of bubbles resulting from the breakup of the mother bubble, of which the diameter is D0. f(Dg, D0) and g(ϵ, D0) are the size distributions of the daughter bubble formed from the breakup of the mother bubble and the bubble breakup frequency, respectively. These two parameters are represented by the following equations: g ðϵ; D0 Þ ¼

K g D−1 0

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 12σ βðϵD0 Þ3 − ρD0

ð36Þ

i 2 2 9 5 1−D3 −Λ 3

 f Dg ; D0 ¼ f ðD Þ ¼ Z D h i 2 max 2 5 2 9 5 D3 −Λ 3 1−D3 −Λ 3 dD h

D3 −Λ 3 2

5

ð37Þ

Dmin

where the coefficient Kg and β are found experimentally, σ is the surface tension of the coupled air-water interface and the critical bubble diameter, and the daughter bubble diameter is calculated based on the following equations, respectively: Λ¼

Dc ¼ D0

D ¼


3 2 12σ 5ϵ −5 −1 D0 βρ

Dg D0

ð38Þ

ð39Þ

The momentum conservation equation for the dispersed phase with certain assumptions, such as neglecting viscous shear stress and Reynolds stress of the dispersed phase, can be calculated as

! ∂υg N‴g ρd ug ∂t

! ! þ ∇:υg N‴g ρd ug ug ¼ υg N‴g ∇P c þ υg N‴g ρd g þ M 0g

ð40Þ

A. Bodaghkhani et al. / Cold Regions Science and Technology 129 (2016) 114–136

where υg is the average volume of a bubble, Mg′ is the interfacial force gravity related to the fluctuation values, Pc is the pressure of one side which is approximately equal to the other side, and g is the gravity vector. Much research has been completed in this area of study including bubble size distribution and population of bubbles (Carrica et al., 1999; Moraga et al., 2008). These papers simulated the location of air entrainment and the size distribution of the air entrainment around naval ship hulls, and at the breaking bow wave when specific conditions, such as a zero Froude number condition (flat surface) are met. However, no simulations of the air entrainment and bubble generation are available for wave-vessel impact. The probability that the air entrainment and bubble generation have a significant impact in spray production and force a water surface to eject water into the air is very high.

4.3. Turbulence model Most parts of any simulation of wave-vessel interaction such as free surface simulation, wave and wave breakdown, air entrainment and bubble generation, and spray formation require turbulence models. The first use of turbulence modeling for wave-vessel interaction problems is the simulation of a turbulent free surface. Many studies related to turbulent free surface have been published, such as Melville (1996), Sarpkaya (1986) and Tsai and Yue (1996). Most non-linear and timedependent free surface models, are based on numerical solutions. Borue et al. (1995), used Direct Numerical simulation (DNS) to solve the time-dependent Navier-Stokes equation and continuity equation for open channel flow problems capturing turbulence at the free surface. Komori et al. (1993) analyzed the fully non-linear free surface coupled with turbulent flow, but only a small deformation of free surface was considered. Tsai (1998) simulated the interaction between free surface and turbulence with the DNS method. Brocchini and Peregrine (2001a) reported several features of turbulence at the free surface and described the deformation and breakup of the free surface, based on different types of turbulence. Turbulence modeling with new boundary conditions, which is introduced for the strong turbulence event, is reported in Brocchini and Peregrine (2001b). Several approaches for capturing large deformations, turbulence production, and dissipation at the free surface are available. The twoequation k − ε eddy viscosity model, Large Eddy Simulation (LES), and Reynolds Stress Model (RSM) can all be used. Advantages and disadvantages of some of these model are reported by Ferreira et al. (2004). The results of implementing the two-equation k − ε eddy viscosity model for high-Reynolds number, for two cases of turbulent boundary layers on a flat plate and jet impingement flow, were examined and reported by Ferreira et al. (2004) and were showed satisfactory in comparison with the experimental data. Yue et al. (2003) used the LES method with the Level Set (LS) method to numerically simulate turbulent free surface flow, and calculated the effect of surface tension. The LES method attempts to resolve motions accounting for bulk of the turbulent kinetic energy and can be applied at relatively high Reynold number in even complex flows, which cots significantly less than the DNS method. The DNS method should only be applied for low Reynolds numbers with simple geometries, the method attempts to resolve the energy dissipation scale (Kolmogorov) of turbulent motions, which makes DNS an extremely expensive method. Besides the free surface simulation, waves and wave breakdown under the influence of turbulence are another challenge in simulating the wave-vessel impact problem. Lin and Liu (1998a,b), Tian et al. (2012) and Bradford (2000) modeled a wave train and wave breaking by solving RANS equations and the k − ε equations. The non-linear Reynold stress model (RSM) was used in this simulation and agreement was achieved between the numerical simulation and the experimental data. Bradford (2000) compared the performance of the k model, k − ε

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model and the Renormalized Group model (RNG) with experimental data from Ting and Kirby (1995, 1996) and Lin and Liu (1998a,b). All of the models generally show satisfactory agreement; however, the wave breaking incident occurred earlier than usual in the experiment, which might not be a significant factor in modeling the wave-vessel impact. Zhao et al. (2004) used the VOF method to track the interface and applied a log-law profile for the mean velocity. They compared their results with the RANS model and they show an improved outcome compared with the experimental data. The three-dimensional LES method and the DNS method were used to simulate breaking waves by Christensen and Deigaard (2001) and Wijayaratna and Okayasu (2000), respectively. The results show a satisfactory agreement with experimental data, but usage of these two methods in three dimensional modeling is computationally time consuming. Another factor involved with turbulence is air entrainment. High void fraction and strong turbulence breakup occur in the air entrainment and bubble generation process. Moraga et al. (2008) employed RANS and Reynolds stress closure models which are used in the case of low spatial resolution and the results were compared with the experiments of Waniewski et al. (2001) for measuring the location of a bubble source. Carrica et al. (1999) show that all processes involving bubble generation, such as bubble collision, bubble coalescence and bubble breakup should be considered and modeled with a turbulence model.

5. Water breakup and atomization The phenomena of breakup and atomization caused by wave impact on a vessel's bow have not been investigated. There are a few reports reflecting guesses and ideas about these phenomena. Zakrzewski et al. (1988) believe that the spray cloud upstream of a ship is an upward water jet formed by wave impact on a hull or bow. They assumed that the water is lifted vertically and dispersed by the wind and therefore, the final droplet distribution arises from impact and wind effects. The details of the creation of the upward sheet of water and the mechanism of droplet creation were not reported. Ryerson (2013) stated that water spray arises from the encounter of a wave or swell with the bow. This causes a jet or a sheet of water to rise above the ocean surface along the hull of the ship. Air and wind split the water sheet into fine droplets. It seems the overall phenomenon has been considered and that general understanding of the process has been approximately reported in similar ways. However, there are not any direct investigations of water breakup on a ship's bow. Knowledge of the quantity and quality of spray clouds, the most important inputs of wave-impact marine icing, is crucial for a good estimation of ice accretion on marine objects. Obviously the breakup phenomena on the bow, or other surfaces, obeys the principles of the physics of breakup. There is a high relative velocity between a water stream and a rigid surface and the expectation is the creation of unstable water sheets, sheet breakup and finally droplet breakup (Sazhin, 2014). The understanding of these phenomena can be gained from similar situations which included sheet breakup and droplet breakup. The most similar cases, which are dependent on the breakup of an input water stream, are splash plate atomizers and sprinkler sprays. Both work based on sheet breakup and droplet breakup to create atomized water from an input jet of water. In both cases a vertical or inclined rigid surface forces the water jet to change its path and create a water sheet. A high velocity water sheet results in instability and sheet breakup. If the situation is designed for droplet breakup, fine droplets will be produced. In this section these two mechanisms and the related investigations will be reviewed. The result of these studies can clarify the governed atmosphere of the breakup on a rigid surface. The analytical works are useful and may be customized by some manipulation for the study of

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bow breakup in future. The experimental attempts show the real behaviour of sheet breakup and droplet breakup. 5.1. Splash plate atomizer Sheet breakup on a rigid surface has been investigated in splash plates by several researchers. A splash plate creates splash and spray droplets using the impact of water on an inclined surface plate (Ahmed et al., 2009; Bussmann et al., 2000; Sarchami et al., 2010; Ashgriz et al., 1996). This procedure includes sheet water breakup and droplet breakup, which are fundamentally very similar to the breakup mechanisms on a ship bow. Fig. 12 shows a schematic of a splash plate, which illustrates water flow and breakup procedure. On splash plates, the droplet velocity at the end of the plate depends on the initial velocity of the water impact on the plate. In addition, there is a correlation between the velocity and size of the produced droplets. Ashgriz et al. (1996) conducted an experimental investigation on a splash plate atomizer. They used six surface-to-jet diameter ratios with the jet velocity reaching 31 m/s. They concluded that at low jet velocities and larger surface-to-jet diameter ratios, which are the ratios of the diameters of the impingement surfaces to the jet diameters, the cone angles will be smaller. Fig. 13 shows droplet size distribution at various distances from the nozzle. The X-axis is in the direction of the jet and the origin is at the intersection point of the spray and on the impingement surface. It shows that as X increases the droplets are more split up, and therefore the majority of droplets are finer. Fard et al. (2007) focused on a numerical model for sheet and droplet breakup on the surface of a splash plate. They used three dimensional CFD and reported correlation between liquid film characteristics and mean drop sizes. The weak point of the model is in the secondary breakup in the air flow. Fig. 14 shows a top view of the numerical solution. The sheet formation, disturbance growth, and sheet breakup are modeled numerically. Ahmed et al. (2008, 2009) focused on sheet break up by splash plates. The core of their study is focused on the experimental tests. They reported rim and thin sheet patterns on splash plates. The rim occurs around the sheet and is a thicker area of fluid. According to the Reynolds number, rim instability can either be laminar or turbulent. The various behaviours of breakup for different inlet velocities and inlet diameters are shown in Fig. 15. Karami et al. (2010) also reported an experimental study of splash plates. The temperature and pressure effects were investigated in this study, and the results show that the droplet size is affected by these variables. Sarchami et al. (2010) developed a model for atomization on splash plates. The model was KIVA-based, which is a sophisticated CFD model (Lee and Reitz, 2012), and can determine sheet thickness and velocity distribution on the plate. Fig. 16 shows the numerical result of the model for a special case. It shows that the droplet size around the centerline are bigger than around the sides of the splash sheet. Although the proposed models and experimental efforts are not sufficient for a deep understanding of sheet and droplet breakup on a rigid surface, they are a good starting point for developing and customizing models for wave breakup on a vessel bow.

Fig. 12. Schematic view of a splash plate (Dehghani et al., 2016).

Fig. 13. Drop size distribution for different X (mm) locations (Ashgriz et al., 1996).

5.2. Sprinkler atomizer A sprinkler jet, which is particularly used for distributing water droplets works based on sheet and droplet breakup. Wu et al. (2007) developed a physics-based atomization model for predicting the spray using a free surface boundary layer and dispersion theories. Geometry, pressure, and ambient environment were the main variables. They reported that the atomizer model predicts a realistic drop size distribution. They divided the sheet formation into four individual regions: the stagnation region, which is close to the center of the plate, the boundary layer region, which is a region including the growing stage of the boundary layer, the transition region, which is the region with influence of perturbation, and the last region, which is a region of fast decreasing velocity. The sheet breakup was investigated and the governing equations were developed. The ligament breakup—an important breakup stage after sheet breakup—was also considered. The schematic view and the experimental picture of water breakup in a sprinkler are shown in Fig. 17. After the sheet breakup, the ligaments are created. The ligaments are unstable and dilatational waves grow along their axes; this procedure converts the ligaments into droplets.

Fig. 14. Top view of numerical results modeling the sheet breakup on splash plate (Fard et al., 2007).

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Fig. 15. Effect of splash plate nozzle flow velocities on the breakup. (a) Nozzle diameter = 2.0 mm, (b) nozzle diameter = 1.0 mm (Ahmed et al., 2009).

Ren and Marshall (2014) expanded on the study with more experiments. They investigated water breakup in sprinklers using four impinging jet nozzles. The Weber number for these nozzles was between 103 and 105. Photographs taken with a short exposure time were used to capture the sheet breakup and drop sizes. A − 1/3 power law with the Weber number was confirmed for the breakup distance. They also reported that the final drop sizes are not just dependent on the Weber number, the other ambient parameters are important as well. Three

Fig. 16. Drop size distribution across the spray. Nozzle diameter = 2 mm and velocity = 21.2 m/s (Sarchami et al., 2010).

sheet breakup modes were captured: rim, hole, and ligament. Fig. 18 shows the result of shadowgraph imaging. Using this technique, makes measuring droplet sizes possible. The cumulative volume fraction is shown in Fig. 19. It shows the greatest population of the droplets to be around 1 mm in size, with decreasing droplet population as the droplet diameter gets larger or smaller. 6. Conclusion The focus of this review is on the direct and related topics to spray cloud covering the complex process, such as wave slamming, free surface modeling, air entrainment, and sheet breakup. In the first section, existing field observations involved in the study of the LWC and spray height are reviewed. This shows a need for more observations with better and accurate measurements of the variables of spray clouds, such as droplet velocities, sizes, and locations. Full scale measurements are useful in studying the process. Early measurements illustrated the importance of spray cloud formation, but these measured data were not sufficiently detailed to be used to validate the computational results or lab scale experimental data. Section two covers numerical and experimental studies of waveship impact. Studies directly related to modeling the spray cloud that occurs upstream of a ship bow have not been found in the literature, and most of the studies were focused on the impact pressure of the wave. Instead, similar phenomena, which have an almost identical behaviour as spray cloud formation, have been reported and analyzed. It is clear that the full direct numerical simulation of wave-ship interaction for the generation of spray clouds is not a viable option

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Fig. 17. (a) Schematic view of atomization process in sprinklers (b) Photograph of breakup and spreading the droplets (Wu et al., 2007).

at this time, but the stages of this process can be modeled separately. However, even these separate process involved in spray cloud formation need to be further developed. Even predicting and modeling wave slamming behaviour with the bow is a complex problem. Phenomena, such as air entrainment, compressibility effects, and predicting full scale values from scaled experiments remain to be better developed. The experimental and/or numerical methods associated with the particular parts of the process, such as wave-slamming, free surface capturing, air entrainment, turbulence models, and droplet breakup are covered in the remaining sections of the review. Some of these individual phenomena have specific solutions and corresponding lab experiments, but some were reviewed because of the similarity of the solutions. These similar methods and studies are suggested for adaptation to the present problem. The future of numerical prediction in this field is in CFD. However, the multi-scale nature of this problem makes calculation computationally expensive. The power and memory capability of new multi-processor computers makes modeling the phenomena possible; however, it takes days to simulate the prediction of wave slamming incidents, droplet breakup, and a turbulence

model with small computational cells in the domain. The recent numerical prediction method, SPH is becoming more popular and is a robust method for prediction of the body-fluid interface, which can be used together with classic CFD methods to produce cost-efficient and accurate modeling for the interaction and spray generation phenomenon. This study shows the current trends in modeling and analyzing spray cloud formation arising from wave impact with objects at the sea surface in cold regions, which leads to icing on vessels and offshore structures. In order to better understand these phenomena, the best approach at present is numerical and experimental studies of the separate stages of the processes, which should be verified and compared with field observation data. The influence of all the identified parameters on spray cloud formation are still poorly understood. More precise considerations will be needed in studying each of these parts, which can then be built into an overall model. The present gaps in this field have been highlighted, and appropriate methods and solutions for future study were suggested for each stage of the process. Undoubtedly there will be new challenges as individual component models are assembled into a comprehensive model.

Fig. 18. A result of shadowgraph imaging for the Weber number 4500 (Ren and Marshall, 2014).

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Fig. 19. Drop size distribution for the Weber number 49096 (Ren and Marshall, 2014).

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