Experimental Study of Flow Kinematics and Impacting Pressures on a Suspended Horizontal Plate by Extreme Waves

Abstract

The flow kinematics and impacting pressures on a suspended horizontal plate under extreme waves were investigated experimentally. Three different stages of extreme waves, unbreaking, incipient breaking, and broken, were separately generated using a dispersive focusing method. The flow field kinematics around the plate during the slamming process was measured using a combination of particle image velocimetry and bubble image velocimetry techniques. We found that for aerated areas, there are significant differences in flow patterns under different conditions. The velocity distribution in aeration areas is more discrete. The slamming peak on the upper surface is influenced greatly by the aeration effect, resulting in the maximum slamming peak of the unbreaking case being 3.8 kPa, which is 0.41 times larger than that of the incipient-breaking case and 1.12 times larger than that of the broken case. However, for the area below the plate, the slamming force and flow evolution under different types of breaking exhibit similarity.

1. Introduction

Marine structures are subjected to significant design challenges due to extreme waves. In rough seas, when large waves hit ships or platforms, many violent flow phenomena occur, such as the upward wave of water along the frontal deck, a large amount of air entrainment in the fluid, and the erosion of deck top surface by water bodies. There is a potential risk that ships and platforms may be damaged as a result of this type of event, which was called “green water” by many researchers.

Over the past thirty years, an increasing number of scholars have studied the interaction between extreme waves and marine structures. Fixed structures are common in coastal and marine engineering, including coastal bridge decks, cylindrical columns, marine jackets, seawalls, and box-shape decks. Matemu et al. [1] proposed a computational method to explore tsunami forcing on the bridge problem. In the method, using a finite difference algorithm, the governing equations were solved, water forces were computed from the results of the diffraction/trapped air calculation, and an excellent agreement between these models was observed. Moideen et al. [2] studied the forces of extreme waves impacting a coastal bridge deck numerically. A reduction in the maximum vertical impact force can be achieved by enlarging the bridge crossbeam interval and the depth of the bridge crossbeam. Qin et al. [3] built a two-dimensional numerical water tank where the Peregrine breather was generated as a freak wave impacting on a deck. Yan et al. [4] investigated the impact of a breaking wave on a rectangular cuboid structure experimentally and numerically.

Floating structures such as marine vessels, ocean platforms, floating breakwaters, and oscillating wave energy converters are severely prone to risks due to extreme waves. Ransley et al. [5] simulated the impact between focused extreme waves and two taut-moored, floating structures: a cylinder with a hemispherical bottom and a cylinder with a moon pool. Ren et al. [6] employed a smoothed particle hydrodynamics model to investigate the nonlinear interaction between ocean waves with a series of moored floating breakwaters. To have a better understanding of the porous breakwater in weakening wave effects on the tunnel, Sharma et al. [7] developed a coupled numerical model of wave interaction with an underwater marine tunnel and a bottom anchor rope-fixed porous breakwater. Rudman and Cleary [8] simulated ocean extreme waves impacted on a semisubmersible platform employing a very commonly used mesh-free numerical model, and the effectiveness of the mooring system was investigated. Banks and Abdussamie [9] experimentally studied the heave and pitch motion of a semi-submersible marine platform with the impact of ocean extreme wave groups of differing wave steepness, highlighting the intense effects of waves. Chuang et al. [10] studied extreme wave action on a TLP installed with a box-shaped deck. Their results show that the transient wave peak on a TLP with a box-shaped deck was much more significant than that of the fixed marine platform at the initial stage. Using a scaled TLP, Luo et al. [11] have studied ocean rogue wave actions and examined how impact conditions affect impact pressures, platform motions, and tether forces.

In order to accurately display the characteristics of aerated flows and to gain a deeper understanding of the kinematics of flow resulting from wave–structure interactions, experimental methods are necessary. Cox and Scott [12] performed experiments and measured the free surface, velocities, and overtopping rates on a deck induced by extreme waves. According to their findings, the probability of free surface overtopping exceeding the volumetric overtopping exceedance follows an exponential distribution. Furthermore, more detailed wave surface evolution and velocity measurements were given by Cox and Ortega [13]. Based on their research, they showed that the maximum crest velocity in the case without the thin deck is 2.4 times larger than the flow peak speed in the case with the thin deck. Ryu et al. [14] investigated the velocity distribution of green water events caused by an example of a plunging breaking wave generated by using a wave-focusing technique. The flow fields were measured using the particle image velocimetry (PIV) technique and the bubble image velocimetry (BIV) technique. As a result, they were able to provide the complete flow velocities in the cases of different phases of aerated water around structures. Chang et al. [15] experimentally studied green waves impinging on a three-dimensional structure by the BIV method. In their experiment, the wave impinging on the vertical wall of the model and the horizontal deck surface was tested. The maximum horizontal velocity in the structural impinging case shows around 1.44 times the incident wave speed. Chuang et al. [16] experimentally studied the dynamics of aerated flow caused by waves impinging on a marine platform in an extensive size, deep-water trial basin. A combination of ocean plunging waves as well as irregular waves was adopted for the generation of incident-aerated water. In addition to the experimental method, a variety of numerical simulation methods have been developed to simulate the flow process of green water, including the smoothed particle hydrodynamics method [17,18], finite element method [19,20], finite volume method [21], and boundary element method [22].

Previous studies have mainly investigated the specific effects of extreme waves impacting flat plates. However, in extreme circumstances, the evolution process of extreme waves is crucial for their interaction with structures; different stages of evolution can lead to significant differences in flow characteristics and slamming processes. In the present study, a plunging breaking wave with different stages, before breaking, incipient breaking, and broken, impacting on a 2D suspended plate model is studied experimentally. Flow structures, velocity field, and slamming pressure were measured. PIV and BIV techniques are combined to measure the flow field around the plate with high fidelity.

2. Experimental Methodology

2.1. Experimental Apparatus and Model Structure

An experimental wave flume with a glass wall was used to conduct the experiments in the State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology. This flume measures a length of 20 m, a width of 45 cm, and a height of 60 cm. To simulate the ocean’s extreme waves, a piston-type wave maker was emplaced at one end of the flume. At the other end of the flume, a wave-absorbing porous structure was installed to minimize the reflection of waves.

As shown in Figure 1a, the entire experimental setup was sketched. In the present study, the depth of still water was kept at 0.30 m. A horizontal plate model was fixed at 9.2 m of the flume, the dimension of which is a length of 0.25 m (L), a thickness of 10 mm, and a width of 0.45 m. The plate was made of Plexiglas and suspended at 6 cm height above the mean water level (MWL) in the absence of any supporting structure inside the flume. There is a high level of rigidity and a tight attachment between the plate model and the wall of the flume. Recorded by the high-speed camera, the displacement of the plate is too low to be observed by the recorded images. The definition of Cartesian coordinates x and z is presented in Figure 1a, in which the coordinate starting point x = 0 is defined at the front side of the plate model and z = 0 at the flume bottom. In addition, the y-axis represents the horizontal direction perpendicular to the x-z plane. Additionally, the time starting point t = 0 refers to when the plunging wave just touches the plate’s front edge.

The surface elevations were recorded by four wave gauges placed along the flume central line. They are capacitance type with a fixed sampling frequency of 50 Hz. The location of the wave gauges is marked in Figure 1a. A total of ten analog pressure sensors were employed to measure the slamming pressures. The accuracy of the analog pressure sensors was 0.1% full scale, and the response time was less than one millisecond. These sensors have a circular measuring diaphragm of 5 mm in diameter and an interval of 2.2 cm between them. A measurement speed of 1000 Hz was used to record wave pressures, while Figure 1b shows the respective positions of the employed pressure sensors. This study used piezoresistive sensors to detect pressure changes caused by a single-phase water body, pure air, or even a two-phase flow. The sensors capture pressure changes whether it is single-phase flow or mixed fluid. The experiments were carefully calibrated and measured to minimize errors caused by measurement errors and to ensure repeatability and reliability.

2.2. Wave Generation

Both linear and nonlinear simulation methods can be used in the laboratory to simulate extreme waves [23,24,25,26]. Among those methods, the dispersive focusing method is wildly employed to generate a plunging wave in a flat-bottom experimental flume [27]. After a wave group is generated by the programmed wave maker, as the waves propagate the phases of the wave component gradually focus, forming a steep wave peak and generating a phenomenon of plunging breaker. In the framework of the present approach for plunging wave generation, the mathematical formulation of the wave surface η with propagation distance x and evolution time t can be expressed as

$$\eta (x,t)={\displaystyle \sum _{i=1}^N{a}_i\cos \left[k_i\left(x-x_f\right)-2\pi f_i(t-t_f)\right]}$$

(1)

where N is the total amount of the wave-packet component; f_i, k_i, and a_i are the wave frequency, wave number, and wave amplitude of the i-th wave-packet component, respectively; x_f and t_f are the predefined focusing position and focusing time, respectively.

In the present experiments, the wave frequencies of the wave components were uniformly distributed over the frequency band [28,29], and the amplitudes of corresponding components were determined by the constant-steepness approach. In this method, the global wave steepness is given in order to obtain a steady focus wave group and prevent it from breaking prematurely [30,31]. The amplitude in the spectrum can be expressed as

$$a_i=\frac{S_p}{N·k_i}$$

(2)

in which S_p is the global wave steepness of the focused wave group.

The stage of breaking development is critical to the slamming process of the plunging wave against the structure. As described by Chan and Melville [32], the slamming process of breaking waves and structures is categorized into three typical types. Stage 1 corresponds to the wave being unbreaking; the breaking of the water occurs after wave–structure contact. Stage 2 is the transition stage where the breaking of the wave is happening. Stage 3 corresponds to those in which the front surface of the wave has already plunged back into the fluid before interaction with the structure. In the present study, those three typical plunging waves were realized by adjusting the wave-focusing position x_f. The schematic diagram Figure 1c illustrates different types of breaking waves. The main parameters of the experimental cases are shown in

Table 1. Wave cases.

Case	Sp	f	T	C0	Focal Time	Plate Location	Focal Location	Plunging Phases
A	4.2	0.4~1.8 Hz	1.5 s	1.56 m/s	20 s	9.2 m	9.4 m	Unbreaking
B							9.2 m	Incipient breaking
C							9.0 m	Broken wave

. C₀ is the phase speed of the incident wave group. It is calculated from the linear dispersion relation of the spectrally weighted period [33]. T is the spectrally weighted period as defined by

$$T=\frac{{\displaystyle \sum _{i=1}^N{a}_i{}^2}}{{\displaystyle \sum _{i=1}^N{f}_i{a}_i{}^2}}$$

(3)

The incident wave in the experiment is an extreme breaking wave, which includes unstable phenomena such as gas–liquid mixing and turbulence. The experiment tested the repeatability of the generated wave. The standard deviation of the incident wave surface on the experimental surface is 1.2%, and the standard deviation of the velocity field is about 3%.

2.3. Velocity Measurement

In the present experiments, the 2D velocity field around the structure is measured using an image-based method. For the region of the non-aerated liquid region, a high-performance PIV (particle image velocimetry) system was employed to measure flow velocity. The hardware devices of the PIV measurement system consist of a high frame rate camera, a continuous laser, and special tracer particles for fluid experiments.

We used a continuous laser with a wavelength of 532 nm, which produced a light output of 8 W. Scattering through cylindrical lenses, light sheets 1.5 mm thick were generated and spread into the water flume centerline vertically from the bottom of the flume. Borosilicate glass particles of 10 μm in mean diameter were employed as the tracer material, which was introduced into the water in a uniform manner before the experiments. The particles have a density of 1.1 g/cm³ and maintain a well-suspended and fluid-following suspension in water. The experimental images were captured by a high-speed PCO.dmaxS4 camera. The sensor of the camera is a complementary metal–oxide semiconductor (CMOS) equipped with a 12-bit dynamic range. The pixel range used in the experiment is 2016 × 2016 pixels, in which the collection speed of the image is set to a 1000 Hz framing rate. The camera is equipped with a 32 G internal hard drive and mounted with a 65 mm focal lens.

For the highly air–water mixture region, the typical PIV technique fails due to overexposure to the laser light. In the present experiment, the BIV method (bubble image velocimetry) proposed by Ryu et al. [34] is employed here to conduct measurements of the aerated flow velocities. In this method, the bubble texture is illuminated by uniform white light (e.g., LED panel light). Many researchers have employed and verified this technique with success to measure the air–water mixture flow, including structural upwelling flows and wave breaking [15,35]. Due to the influence of three-dimensional effects, the depth of field of the camera can lead to measurement errors. The measurement error calculation in BIV is as follows: ε = D_c/2L_c, in which D_c is the camera’s depth of field, and L_c is the distance between the measurement position and the aggregation location. The D_c in the experiment is about 0.067 m and L_c is about 2.1 m; therefore, approximately 1.6% of the error is attributed to the thickness of the depth of field.

For the aerated and non-aerated regions, the field of view (FOV) of them are the same as 0.7 m × 0.5 m, as shown in Figure 1a. A cross-correlation analysis was conducted to calculate velocity fields. For the analysis, we used a position search window of 32 × 32 pixels, while 50% overlapping space between adjacent windows is used in order to achieve maximum quality. To reduce measurement uncertainty, the experimental case entailed 20 runs being repeated, and an ensemble average was calculated for both the speed measurement and the surface elevation measurement [36,37]. Figure 2 shows the source images of BIV and PIV captured in the experiments. Due to the different light sources, the PIV and BIV measurements are conducted separately.

3. Results and Discussion

3.1. Evolution of Flow Structures

The global behavior of flow evolution during plunging breakers slamming on the plate structures is described in this section. Through a high-speed camera, the two-phase flow evolution around the horizontal plate was recorded. Figure 3 and Figure 4 covering the time from t/T = 0 to t/T = 0.7 present the origin images of aerated flow to illustrate the physical process. The aerated flow was lighted by a white LED plane light projected from the back of the tank. The aerated flow around the plate may not meet the gravity similarity criterion due to the influence of surface tension. Therefore, the aerated breaking flow in our experiments needs to consider the scale effect for the prototype size.

For the case of unbreaking wave slamming, marked as in Figure 3a and Figure 4a, before contact with the plate structure, the wavefront surface has not broken up and keep smooth. After the impact on the plate corner (t/T = 0.033), a significant water tongue turned over from the wavefront. In the front region of the water tongue, the water surface has broken and presented an aerated state. Part of the water body ruptures into independent droplets. At t/T = 0.067, as shown in Figure 3(a3), in response to gravity, the aerated water tongue fell to the surface of the structure on the upper side. On the lower surface of the horizontal plate, there was a noticeable water jet phenomenon, leading toward the wave propagation direction. As the flow progresses, the aerated water on the plate propagates downstream quickly, and the jet flow under the structure extrudes from the end of the plate. As a result of this extrusion phenomenon, the water body and structural surface may come into rapid contact, thereby causing significant damage to the plate’s bottom. As shown in Figure 3(a5), the aerated flow on the plate moves downstream and collides with the aerated flow in the rear region of the plate. The water bodies on both sides mixed with each other during the impact process, causing a more obvious phenomenon of aeration. At t/T = 0.167, it is observed that a small portion of the water body water is beginning to separate from the structure on the weather side, and an aerated flow of multiple flipping occurred in the region behind the plate and moved downstream, as shown in Figure 4(a1). In the following stages, the wave-crest water gradually separates from the structure, causing a strong air–water admixture under the plate. In the process of detaching from the structure surface, aerated water flows downward in the form of a jet flow from the rear end of the suspended structure; see Figure 4(a4,a5).

Figure 3b and Figure 4b demonstrate the flow evolution of the incipient-breaking type wave impact on the plate. At the moment of contact between the wave and the plate, the breaking of the plunging wave has already occurred, as shown in Figure 3(b1). Differing from the unbreaking case, in case b, the water tongue on the upper side of the plate does not appear, with aerated water moving forward against the plate surface. At t/T = 0.067, as shown in Figure 3(b3), the jet flow under the plate shows a more obvious aeration phenomenon and moves toward the wave propagation direction. For the region on the plate, more bubbles are involved in the water body; those bubbles are caused by the air drawn in within the process of water breaking. In the water–structure separation stage (Figure 4(b1–b5)), the flow phenomenon during the separation process is similar for cases A and B; flow separation occurs at the front end of the suspended plate and ends at the rear end.

For case C, the waves have evolved into completely broken waves; the flow evolution around the plate is present in Figure 3c and Figure 4c. The wavefront surface has already returned to the fluid before the wave impacts on the plate, and a large amount of air is drawn into the fluid before the wave is impacting on the plate. Compared to the former cases, a significant aerated water tongue is formed on the upper part of the plate, as shown in Figure 3(c2). The water tongue evolved into the fluid splash phenomenon as waves propagate, and the water gradually breaks up, forming a series of water droplets. The aerated splash collides with the water behind the plate, forming more complex aerated flow phenomena. There is a large amount of splashing caused by aerated water on the plate, which is partly due to a decrease in density due to water aeration as a result of a greater flow velocity. Conversely, it is caused by the upwelling of the water body as a result of the wave breaking, resulting in a greater amount of visible water splashing. In addition, during the flow separation stage, this broken-wave case remains similar to the previous two types of cases.

3.2. Characteristics of Flow Velocity Fields

Figure 5, Figure 6 and Figure 7 show the measured velocity field in the region surrounding the plate. They were obtained using PIV and BIV methods for the unbreaking, incipient-breaking, and broken wave cases, respectively. Before the wave hits the plate, the wave surface is not broken, and the measured velocity is the wave velocity field of the single-phase water body. Figure 5a illustrates a wave velocity field before collision with the structure. The velocity field is continuous, and there is no aeration phenomenon. The entire velocity field is dominated by a horizontal velocity of 0.87 C₀. The vertical speed is relatively small; the maximum value among them is around 0.41 C₀. In our experiment, C₀ corresponds to the phase speed of the incident focusing wave. Throughout the entire process, the velocity of the water body is mainly dominated by horizontal velocity. The flow velocity in the aerated area is much higher than that in the non-aerated area. This is mainly because the air entrainment in the aerated area reduces the density, resulting in a higher flow velocity. In addition, the flow velocity distribution in the aeration area is more discrete, which means that the flow in this area is more complex. On the contrary, for single-phase water bodies, the flow velocity is more uniform. In these three different wave conditions, the initial aeration rate before the wave and structural contact also lead to different aeration amounts during the flow process.

For the broken wave case, the dispersion of flow velocity caused by the amount of air entrainment is also more pronounced. The process of water spray movement brings more significant vertical velocity. The velocity in the upper region of the horizontal plate is mainly dominated by aerated flow, while the typical single-phase flow is mainly below the plate structure. In the stage of water–structure separation, the aerated water is dominated by the fluid–structure separation phenomenon, and the global velocity magnitude gradually decreases.

Figure 8 shows the time variation of maximum velocity under different cases and is standardized using C₀. The speed of the water body increases significantly after contact with the structure. The maximum speed occurs within a time range of 0.5 to 1 s. A maximum speed of 2.09 v_m/C₀ is achieved in the case of the broken wave. Following this, the flow velocity gradually decreases over time. The maximum velocity of the unbreaking case is larger than that of the incipient breaking case; their velocities are 1.72 v_m/C₀ and 1.41 v_m/C₀, respectively. This is mainly due to the formation of the water tongue phenomenon in the case of the unbreaking wave. There is a consistency in the rate of decrease in velocity across different cases.

3.3. Wave Impact Pressure

The wave impact pressure on the upper and lower surfaces of the flat plate was measured separately in the experiment. The ensemble-averaged pressure (P) of the upper surface at the points P1 (a1–c1), P2 (a2–c2), P5 (a3–c3), and P7 (a4–c4) are shown in Figure 9. These measuring points represent the typical slamming curves of the front, middle, and rear ends of the plate structure, respectively. The three wave cases mentioned above, unbreaking wave, incipient-breaking wave, and broken wave, are shown separately for comparison. For the unbreaking case, a water tongue is formed on the upper side of the plate and impacts the middle of the plate. The changes in pressure are consistent with the phenomena observed in the experiment. The peak pressure on the front side of the flat plate is smaller than 1.4 kPa at sensor P1, while the pressure in the middle and back of the plate is very significant. At sensors P5 and P7, the peak pressure is 3.8 kPa and 3.4 kPa, respectively. Furthermore, the time when the pressure peaks appear corresponds to their position on the plate.

In the case of the incipient-breaking case, the waves have already broken before interacting with the plate, and there is a certain amount of energy consumption during this process. Due to the absence of tongue curling above the plate, the slamming phenomenon on the middle side and back of the plate is no longer obvious. The slamming peak at the front end of the flat panel is more pronounced, and due to the addition of gas, there is a significant oscillation phenomenon. The maximum peak pressure is 2.7 kPa, which is significantly lower than the unbreaking case. This phenomenon indicates that the aeration during the wave-breaking process reduces the slamming pressure. Air acts as a buffer layer during the slamming process, leading to longer slamming time and reduced slamming force.

The effect of aeration on slamming is more significant in the broken wave case. Compared to the first two wave-impact conditions, there is a significant reduction in the slamming peak. Although aeration has an impact on the magnitude of velocity in flow velocity analysis, a greater impact is reflected in the measurement of slamming force. On the one hand, this is due to the greater energy dissipation during the wave-breaking process. On the other hand, due to the entry of more gas, a more pronounced air cushion appears during the slamming process.

For the pressure on the lower surface of the plate, measured points P1 (a1–c1), P2 (a2–c2), P5 (a3–c3), and P7 (a4–c4) are shown in Figure 10. In the description of flow in the previous section, it can be seen that the aeration in the lower region of the flat plate is much smaller than that in the upper region. For the three types of waves, the difference in slamming force on the lower surface of the flat plate is not particularly significant. The peak slamming occurs gradually over time from the front to the back of the plate. During the flow separation stage, varying degrees of negative pressure phenomena occur. The pressure changes on the lower surface of the flat plate also indicate that the similarity of flow will be reflected in the similarity of slamming pressure.

4. Conclusions

Flow structure, velocity field, and slamming pressure measurements were conducted on a two-dimensional suspended plate structure under extreme waves impact. The plate-type structure is two-dimensional, rigid, and fixed at 6 cm above the still water level. Three types of plunging phases—unbreaking wave, incipient-breaking wave, and broken wave—are generated to mimic different stages of wave–structure interaction. The particle image velocimetry (PIV) technique and bubble image velocimetry (BIV) have been employed to measure the flow field around the plate. In addition, the correlation between slamming force and flow characteristics was discussed. As a result of this study, the following conclusions can be drawn:

(1)

Multiple flow types are involved in wave and plate slamming processes: water tongue generation and turnover, wave breakup into independent droplets, water jet phenomenon, water extrusion, water collision, and water–structure separation. For aerated areas, there are significant differences in flow under different operating conditions, while for non-aerated areas, the flow exhibits obvious similarity.

(2)

The velocity of the flow is mainly dominated by horizontal velocity. The flow velocity in the aerated area is much higher than that in the non-aerated area. The flow velocity distribution in the aeration area is more discrete. The initial aeration rate before the wave and structural contact also leads to different aeration amounts during the flow process. The maximum speed occurs within a time range of 0.5 to 1 s. Across different cases, the rate of velocity decrease is consistent.

(3)

The effect of aeration on slamming is more significant in the broken wave case, causing a significant reduction in slamming peak, owing to the stronger energy dissipation during the wave-breaking process and more pronounced air cushion during the slamming process. The maximum slamming peak of the unbreaking case is 3.8 kPa, which is 0.41 times larger than that of the incipient-breaking case and 1.12 times larger than that of the broken case, appearing on the upper surface of the plate. For the area below the plate, the slamming force and flow evolution under different types of breaking exhibit similarity.