WANG Yang, HUANG Cheng, YI Jiaji, FENG Kecong, XIA Xiaoyu

1. Haikou Marine Geological Survey Center, China Geological Survey, Haikou 570100, Hainan province,China;

2. Hubei Key Laboratory of Marine Geological Resources,China University of Geosciences,Wuhan 430074,Hubei province, China

Abstract: Tsunamis have a severe impact on marine coastal structures. Tsunami is generally simplified as solitary wave as they propagate, and the presence of the aftermath of Tsunami is similar to a second solitary wave. Waveform evolution occurs as solitary wave propagate down a gentle slope. This paper reveals the propagation of double solitary waves and slope climbing by numerical simulation where the prototype of the embankment is around Shantou city in the Guangdong Province, China. It not only enriches the theory of solitary wave, but also has important implications for the analysis of tsunami disaster mechanism and the hydrodynamic load characteristics of structures.Based on the average Navier-Stokes equation and the VOF approach, numerical simulation results are given, including changes in the velocity field of the climbing and falling process species. The results show that the double solitary waves produce a strong reflux effect, which results in the second solitary wave climbing significantly less than the height of the first solitary wave without the influence of the current. At the same time,double solitary waves can have a much stronger effect on the embankment.

Keywords: double solitary waves, embankment, run-up height, hydrodynamic characteristics

1 Introduction

Tsunamis are generated when extreme ocean conditions such as earthquakes,volcanic eruptions, and submarine landslides cause water bodies to shake. Tsunami wavelengths are extremely long, reaching hundreds of kilometers, and tsunami waves travel with great energy and speed. However, in deep water, tsunami waves are less amplitude and harder to detect. As the tsunami propagates from the deep ocean zone to the offshore zone, the tsunami waves will increase dramatically in wave height, creating huge lapping waves tens of meters high, combined with the extremely high speed, which can be very destructive. In recent years, the Indian Ocean tsunami of 2004 has caused the most devastation. A 9.3 magnitude earthquake under the sea in northern Sumatra triggered a tsunami that killed nearly 300 000 people. The tsunami triggered by the Great East Japan Earthquake in 2011 took away nearly 20 000 lives. When waves from the deep ocean areas are transmitted to offshore areas, due to the influence of topography,islands and environmental flows, the waves undergo changes such as shallowing,refraction, bypassing, deformation and fragmentation, which not only involves the modeling theory and numerical calculation methods of nonlinear water waves, but also needs to take into account the influence of complex factors such as seafloor topography and currents. Coastal zone disaster prevention and mitigation and deep-sea resource development have created new and urgent needs for research on extreme ocean dynamics such as tsunamis and distortion waves, respectively[1-3].

The waveform and motion characteristics of tsunami wave before breaking are close to those of solitary wave, and the waveform is stable with little energy loss when propagating in deep water. Solitary wave is characteristically found in shallow waters, has one peak, is waveform stable, and can propagate steadily over long distances. A complete model of tsunami wave generation, propagation and climb has been developed based on linear shallow water equations. However, as the tsunami wave propagates into the offshore region, the wave amplitude increases and the nonlinear term of the tsunami wave cannot be ignored[4,5].

Previously, the studies mentioned were all based on the single solitary wave slope climbing problem. Although there are many similarities between tsunami waves and solitary waves, it is still not possible to directly replace tsunami wave with single solitary wave. Offshore tsunami waves can be considered as a superposition of wave trains consisting of multiple solitary waves. So it is necessary to conduct multiple solitary wave climbing studies based on a single solitary wave slope climbing. In the case of a tsunami,the tsunami wave is not strictly solitary wave, but more like a series of waves added together[6]. Therefore, in order to derive the theoretical equations for tsunami wave, the Korteweg-de Vries equation for solitary wave is commonly used as the fundamental wave for the superposition of tsunami waves. At the surface of the sea at water depth h, the profile of a solitary wave can be represented as:

where H0is the wave height of solitary wave, k0= （3h0/4h3）1/2denotes the effective wave number, x0is the location of wave crest at t = 0, c0= [g（H0+ h）]1/2is the wave velocity with the gravitational acceleration value g. According to the N wave theory, tsunami waves can be approximated as a se ries of positive and negative solitary waves superimposed on each other to form a series of waves called tsunami-like waves. The parametric control for the superposition of the solitary wave determines whether the tsunami-like wave is in good agreement with the real-world tsunami waves[7]. The formula for the wave superposition is:

where n is the number of superimposed waves, εiand αiare scale factors reflecting the wave height, frequency and period of each underlying solitary wave, tiis the start time of different waves. Real tsunami waves can be approximated using equation (2). However,in practical calculations, it is difficult to obtain all the control parameters of the tsunami waves. Subsequently, the concept of double solitary waves emerged and experiments were conducted. The correlation experiments show that during double solitary wave climbing, the first wave has a large influence on the maximum climbing height value of the second wave, and the position relationship between the two waves also has an influence on wave climbing[8]. Numerical simulations further give the flow field and energy conversion laws of double solitary waves along the straight wall and steep slope climbing process.

In this paper, the numerical simulation of wave climbing process on a slope under the action of double solitary waves is carried out to study the flow of double solitary waves and their climbing characteristics, so as to provide a reference for the design of engineering embankments.

2 Calculation method

Numerical simulation is used to study the climbing characteristics of double solitary waves. It is assumed that the fluid is incompressible, and the basic equations of motion of the fluid and the RNGκ-ε turbulence equation are used to close the group of equations[9,10].

2.1 Continuity equation

The motion of the fluid satisfies the continuity equation, the differential formula for the continuity equation:

where r is the fluid density, t is the time , isthe fluid velocity vector, and div is the differential operator.

2.2 Momentum equation

The momentum equation is derived according to Newton's second law. The sum of the external combined forces exerted per unit time within a specified fluid micro-element should be consistent with the change in the momentum of the fluid within that microdollar[11,12]. Based on Newton's second law, the equation of momentum in the three directions x, y, and z can be derived:

where: p is the pressure on the fluid micromotion, Pa; fx, fy, fzare unit mass force in three directions, m/s2; txx, tyy, tzz are the components of the viscous stress t acting on the surface of the micromere due to molecular adhesion, Pa.

2.3 Turbulence equation

where kTis the turbulent kinetic energy, the turbulent kinetic energy dissipation rate, and Cu is a constant, which is 0.085 in the RNG κ - ε turbulence model.

3 Model setup

Using the similarity principle, the entrance boundary in the model employs a numerical wave-making plate technique to produce double solitary waves. The wave height of the first wave is 0.06 m, the wave height of the second wave is 0.12 m, and the waveform is shown in Fig. 1. The length of the calculation area is 30 m, the height is 0.6 m and the slope is 10 degrees.

Fig. 1 Double solitary waveform

4 Analysis of results

4.1 Double solitary wave velocity analysis

The temporal curves of the horizontal and vertical flow velocity at the location of the flow velocity monitoring point (2 m, 0.2 m) are shown in Fig. 2.

Fig. 2 Monitoring point flow velocity time course curve

Prior to the arrival of the first wave of the solitary wave, the monitoring point is almost stationary when the wave is far away. When the first wave approaches, the water particle at the monitoring site produces velocity vectors along the positive direction of the wave propagation and upward direction. Thereafter the velocity of the current is controlled by the first wave, with the horizontal velocity propagating in a positive direction and the vertical velocity oscillating between positive and negative, which appears as multiple peaks on the graph. After the first wave, the second wave arrives and the transition is controlled by both, showing a faster change. Thereafter, the second wave dominates, and the overall situation is similar to that of the first wave. After the second wave, under the combined effect of inertia, gravity and resistance, the velocity of the water shows oscillating changes, and then gradually returns to the initial calm in a relatively short period of time, and the whole process ends.

From Fig. 2(a), we can see that there are two peaks in the horizontal velocity of the monitoring point during the propagation of double solitary wave, and the velocity vector in the propagation direction is the same as that of the solitary wave, and the movement is obviously influenced by the wave from t = 1.5 s onwards. After the first wave, the velocity starts to decrease. After t = 5.5 s, the solitary wave is far away, and the horizontal velocity at the monitoring site shows a simple harmonic motion similar to the existence of damping under the inertia and viscous resistance, and finally comes to a standstill. The maximum value of the horizontal velocity is proportional to the amplitude of the solitary wave.

From Fig. 2(b), it can be seen that the vertical flow velocity at the monitoring site oscillates between positive and negative, and at t = 1.5 s, the flow velocity is positive when influenced by the first wave, and with the movement of the wave, the velocity increases to an extreme value and then begins to decrease. When the wave peak passes, the vertical velocity is 0, after the wave peak, the vertical velocity is negative, the water level begins to fall, and tends to calm. t = 3.5 s onwards, the second wave arrives,the flow velocity is controlled by the second wave, the whole process is basically the same as the first wave. After the wave peak, the water level falls back. The velocity is negative and tends to be stable. When the wave is far away, also under the inertia and resistance, the vertical velocity falls to 0 as the center of the damping of the simple harmonic motion of the change law, and the horizontal velocity in the same time with the restoration to a standstill. The extremes of the vertical velocity are also proportional to the amplitude of the solitary wave.

Combining Fig. 2(a) and Fig. 2(b), the wave speed is controlled by the first wave in 1.5 s -3 s,controlled by both of the two waves in 3 s -4 s, by the second wave in 4 s -5.5 s,and after 5.5 s it is controlled by gravity and various resistances. The horizontal velocity reaches maximum when the wave crest arrives, and the vertical velocity is 0 but the vertical acceleration is maximum; the horizontal velocity acceleration is maximum when the vertical velocity is maximum. It can be determined that the trajectory of the particle is close to the pendulum, but there is resistance and gradually tends to lose energy.

4.2 Waveform analysis

Solitary waves are prone to fragmentation during climbing, and the state of solitary waves after climbing can be determined for different slopes and relative wave heights according to the solitary wave fragmentation index defined by Grilli[13]:

where s is the slope of the slope, the relative wave height of a solitary wave is determined by the wave height and water depth, H being the wave height and d the water depth.When S0＞0.37, the solitary wave does not break; when 0.3 ＜S0＜0.37, the solitary wave breaks to surge wave; when S0＜0.3, the solitary wave breaks to plunging wave.

The waveform during the simulation of the climbing height of double solitary waves towards the bank in this paper is shown in Fig. 3.

Fig. 3 Double solitary wave climbing process

According to equation(6), the first wave S0= 0.839 and no fragmentation occurs. The simulated waveform shows that the first wave arrives at the shore slope, the water depth decreases, the bottom friction increases, and the waveform shows a steeper front end,but the amplitude is limited, the waveform remains intact, and there is no breakage. The first wave height is small, there is no return current at the front end, and the kinetic energy consumption is mainly through friction and overcoming gravity, so the overall consumption is stable. Therefore, under the control of the first wave, the current is impacted upward along the coastal slope, and due to the steady energy consumption, the current has a strong ability to climb high but slow energy loss under the control of the first wave, and the overall current moves along the coastal slope with low destructive power.When the motion reaches a certain height, the velocity returns to zero, and the first wave by gravity causes the climbing water to fall back down.

The calculation shows that the secondary wave S0= 0.593, which also does not break if propagated alone as a solitary wave. The process shows that when there is a first wave, the second wave arrives, and under the joint action of bottom friction and flowback water, the force at the bottom is much larger than the first wave. The waveform changes greatly, and it breaks into surge wave. The whole current strikes the shoreline as a whole and releases a large amount of energy instantaneously. Therefore, the height of the current caused by the second wave is lower than that of the first wave, but it has a greater destructive effect on the shore slope than the first wave.

In general, the current caused by the first wave climbs higher, but the current caused by the first wave is relatively flat and less destructive; the current caused by the second wave climbs lower than the first wave, but the energy release is fast, and the impact on the shore slope is larger and more destructive, so it is necessary to focus on preventing it.

4.3 Water level profile analysis

The four images (Fig.4)display the change in water level between the first wave-induced flow climb back down and the arrival of the second wave and the start of the flow climb back down after the second wave-induced flow climb.

In Fig. 4(a), t = 16.8 s, the pattern of the secondary wave is still relatively complete,and the first wave has reached the shore slope and caused the current to climb higher with a complete waveform. This corresponds to stages 3 to 4 in the flow field diagram.The water body before 22 m in the x direction of the calculation field is no longer affected by isolated waves, and the velocity has returned to 0, and the water surface height is 0.The water level of the water body after 25 m in the x direction of the calculation field is clearly controlled by the solitary wave, the secondary wave has just started to deform,while the first wave has been converted to the water level climbing.

In Fig. 4(b), t = 17.6 s. The second wave begins to interact with the bank slope, and the climbing part of the first wave ends with the climbing current, corresponding to the period between stages 4 and 5 in the flow field diagram. When the first wave returns, the second wave quickly deforms and breaks up into a surge breaking wave, and the water disperses. As can be seen in the figure, the majority of the current caused by the second wave climbs upward, and the water gathered at the crest of the wave shoots to the shoreline slope, pushing the current to climb higher for a second time, which is shown as a rapid increase in the height of the water level after 26 m of the calculation field. At 28 m of the calculation field, the water level map of its trend has been vertical, indicating that the front end of the water, and the secondary wave by the return current and friction, the peak of the water broken at the peak of the wave, after beating the shore slope of the water body in the air rupture. In addition to the upward flow, Fig. 4(b) shows that the starting water level at 20 m - 25 m of the calculation field is higher than in Fig. 4(a),indicating the presence of some water return to this point, including the return flow of the first wave and some of the secondary bottom flow reversed by the first wave return.

Fig. 4 Water level curves at different times

In Fig. 4(c), t = 18.2 s. The climbing process of the second wave is over, all the currents start to fall back, and the echoes start to form, which corresponds to the 6th stage in the flow field diagram. In the water level diagram, the water level after 26 m of the calculation field shows a straight line. At this time, the water caught in the air in Fig. 4 (b) falls down, and the water as a whole is close to the bank slope and starts to fall back. At 26 m of the calculation field there is a groove in the water level, indicating that the water began to fall back, here is the return flow into the water surface of the "sink",the return flow of water hit the water surface, where the water level is lowered. At the same time, due to the conservation of mass, the collected water begins to be transported in the opposite direction x, gradually forming a reverse echo.

In Fig. 4(d), t = 20.4 s. The return flow kinetic energy is greater than that in Fig. 4(c),which corresponds to after stage 6 in the flow field diagram. At this point, the climbing currents caused by the double solitary waves had all ended, and the water level was lowered above the bank slope than in Fig.4(c).The return flow of water is rapidly returning to the surface, still converging at 26m of the calculation field. At this time, the water level here is already below 0m due to the stronger backflow, and the water begins to move in the opposite direction, forming an echo that propagates negatively in the x direction. As a large amount of energy has been consumed in the impact slope, the wave heights of the echoes are smaller than two solitary waves, and the individual wavelengths are longer.Thereafter, any remaining return flow is returned to the surface and causes reverse transport of the water column in the form of a return wave.

Combining the changes in the water level profiles in the 4 plots, the first wave causes the current to climb higher under the action of double solitary waves, followed by a return flow. The second wave, under the influence of the return flow, climbs higher and lower,but with greater energy, and impacts a large number of banks on the slope. Thereafter,the whole current returns to the water surface along the bank slope and impacts the water surface, forming an echo and reverse motion.

5 Conclusions

(a) The water particles flow trajectory caused by the solitary wave resembles a shrinking pendulum, and the overall water displacement is in the same direction as the propagation of the solitary wave.

(b) The maximum value of the water flow velocity is proportional to the amplitude of the solitary wave, and the effect time is positively correlated with the wave propagation speed and the individual wavelength.

(c) The first wave causes the current to climb higher, and the second wave causes the current to climb relatively lower due to the return flow of the first wave.

(d) The waveform of the first wave is basically intact, with slow energy release, gentle water flow and relatively low destructive force. The second wave in the form of a broken surge wave releases energy rapidly. The current is more rapid with more destructive force. The overall damage caused by the double solitary waves is greater than a single solitary wave, which need to be considered as the focus of disaster prevention.

Acknowledgements

This work is supported by Comprehensive Geological Survey of Chaoshan Coastal Zone (No. DD20208013).