TANG Yuezhao, WANG Yang, WANG Wanhu, WANG Hongbing

1. Hubei Key Laboratory of Marine Geological Resources,China University of Geosciences,Wuhan 430074,Hubei provine,China;

2. Haikou Marine GeologicalSurvey Center, China GeologicalSurvey,Haikou 570100,Hainan province,China

Abstract: Dam breaks are easily triggered by heavy rains due to extreme weather such as typhoons, causing serious economic losses and casualties. Through the investigation of Chaoshan coastal zone, it is found that there have been dam breaks caused by geological disasters. In the design and management of water conservancy project, it is very important to analyze the effect of disastrous flow caused by dam break on the building. In this paper, the effect of the dam break flow on the cylinder is simulated numerically by taking the water body with initial velocity as the dam break flow, and the flow characteristics around the cylinder and the water body are analyzed. Numerical model adopted the Renault Average Navier-Stokes(RANS) model and volume of fluid(VOF) method to analyze the evolution of free water surface. It is found that there are different patterns of water movement in the process of dam break resulting in the creation of several isolated convex hull forms of dam-break waves on the stationary water surface,which causes longer disturbances in the water near the cylinder and makes the cylinder more vulnerable to fatigue damage. The increase of the height of the dam breaking water will lead to the increase of the hydrodynamic force on the pipeline. This study has guiding significance for the study of dam break and dam body design in water conservancy projects.

Keywords: dam break, wave impact, flow characteristics, hydrodynamic load

1 Introduction

The removal and overturning of the gates in extreme cases lead to dam failure,which triggers off rapid unsteady flow. The first task after dam failure is to analyze the hydraulic characteristics, flood path, risk management and hazard analysis of destructive floods. The initial conditions when a dam breaks and the geometry of the water layer play a key role in hydraulic characteristics[1]. Ritter(1892) proposed an analytical solution based on de Saint-Venant system of equations to describe the non-viscous dam-break in a smooth horizontal dry bed. The front end of dam break wave flows downstream at a constant speed. The initial conditions have a serious effect on the dam break flow. In Ritter's solution, the resistance is ignored and the flow rate and free surface profile are considered constant and smooth[2]. The resistance factors mainly include the roughness of the tank bed and wall, which causes the free surface flow in the top region to bend[3].

In the actual dam break, the influence of water viscosity and riverbed topography on water flow evolution is significant. Inertia and shear mechanisms are controlled from the initial stage to the final stage of dam break flow[4]. In addition, by studying the first-order and second-order asymptotic solutions for the instantaneous collapse of the flat bed with resistance of infinite water body, a practical solution is obtained on this basis, which provides a reference for the selection of a reasonable time step[5]. And it is found that the TVD explicit scheme using gas dynamics can be applied to the numerical simulation of dam break surge wave that many difference schemes are not competent for[6]. In addition,a numerical study is carried out on the flow-around problem of 2D dam break wave,revealing the complex motion characteristics of reflection, diffraction and deformation of the wave when it meets obstacles, and providing a scientific basis for disaster analysis and prevention and reduction of dam break[7]. The wet-bed dam break flow characteristics have also been reported in previous studies. In addition, many studies have focused on the application of shallow water equations in solving dam break problems[8]. Meanwhile,the dam break flow is three-dimensional at the initial stage of the near field. Therefore,the distribution of non-hydrostatic pressure and the vertical components of velocity are also very important.

The three-dimensional numerical model based on Navier-Stokes equations can accurately predict the unsteady flow characteristics with high precision[9]. In this regard,standard VOF techniques are used to track free surface propagation and expansion of the water-air interface. This technique provides superior performance in modeling free surface flows where the water-air interface suddenly deforms. In addition, compared with other numerical methods, it requires less computation[10]. Therefore, accurate results can be obtained in the modeling of dam break, and the hydraulic characteristics of dam break flow can be predicted reliably. Similarly, the application of this model effectively reduces the computational burden by eliminating the minimum turbulence length scale[11].

Previous experimental studies have focused on the evolution of the wave front of a dam break, the forward distance and wave velocity, and pressure fluctuations within a given range, rather than the entire flow field. However, the unsteady flow characteristics of dam break have not been reported in depth. Some of these features are the hydrograph of the dam, the hydrograph of the downstream control section, the specific energy and the resultant force in the reservoir. In the preliminary design and efficient operation of DAMS, disaster analysis, flood evolution and zoning, it is necessary to predict the characteristics of dam break flow in detail. In addition, by setting different heights of water body of dam break, this paper compares the characteristics of its velocity, vorticity and force, and obtains rules. Therefore, this paper aims to analyze the water surface curve,flow characteristics and force characteristics acting on the structure after the dam break,and draw corresponding conclusions to provide reference for the project.

2 Calculation method

In this paper, the numerical simulation method is mainly used to study the fluid characteristics and the structural response characteristics after dam break. The fluid is assumed to be incompressible, and the basic equations of motion and κ - ε turbulence are used in the model.

2.1 Continuity equation

The motion of the fluid satisfies the continuity equation. The differential equation of the continuity equation is:

where ρ is fluid density, t is time,is fluid velocity vector, div is differential operator.

2.2 Momentum equation

The momentum equation is derived from Newton's second law. The sum of external resultant forces received per unit time in a specified fluid microelement shall be consistent with the momentum change of the fluid in the microelement. Based on Newton's second law, the momentum equation in x, y and z directions can be deduced:

where p (Pa) is the pressure on the fluid microelement body; fx, fyand fzare the unit mass force in three different directions, m/s2; τxx, τyy, τzz, …, are the components of the viscous stress τ, Pa; τ is the viscous stress on the surface of the microelement caused by the molecular viscosity.

3 Model setup

In the model, the dam break landslide is set as a circular arc with a diameter of 1 m,and the center of the arc is at the position of(0 m,0 m)in the rectangular coordinate system.The cylindrical coordinate center is(2 m, 0.1 m), and the radius is 0.075 m. The calculated area is 10 m long, 1 m wide and 1 m high, and the grid resolution is 0.01 m. The height and width of the water body of dam break are 0.8 m and 0.3 m respectively, and the water surface height of the initial water body is 0.3 m. Additionally, the different water heights of dam break(h=0.7 m,0.75 m,0.85 m and 0.9 m)are also taken into account.The locale is shown in Fig. 1.

Fig. 1 Schematic diagram of model setting

4 Results and discussion

4.1 Velocity characteristics

This paper mainly simulates instantaneous dam break, one of the break dam types where the whole dam body disappears. As shown in Fig. 2, the velocity contours of flow fields at different moments after dam break are given. Fig. 2a shows that immediately after the dam break. It can be seen that after the dam break, a large amount of water body of dam break will discharge within a short time. When the water body of dam break slides into the wetland, it will cause the oscillation of the boundary water body of the wetland and form the dam break wave. The manifestation of dam break wave is that it forms obvious surface bulge similar to isolated wave at the free surface of water. This surface bulge propagates downstream in the form of isolated wave, and has obvious influence on the water level downstream of the weir site. There are some velocity distributions characteristics inside the dam break wave. The dam break water which has not left the dam foundation has a higher velocity. The velocity of water body near the dam foundation and the top of the water body of dam break wave uplift are higher. The rest of the more downstream flow more sharply reduced, at the distance of 1 m in dam flow velocity decreases to zero.

As can be seen from Fig. 2b, at this time, the dam break wave has been propagated directly above the cylinder, and the part where the flow rate is higher is still mainly distributed at the top of the convex hull, while the flow velocity of other parts is centered on the apex of the convex hull and decreases sharply to both sides. Compared with Fig. 2a,it can be seen that at this time, the velocity distribution near the cylinder changes significantly, and boundary layer separation occurs in the upper and lower reaches near the cylinder. In the range of 0.5 D and 1 D downstream of the cylinder, the velocity is 0;in other places near the cylinder, the velocity in the upper and lower reaches increases to 0.4-0.5 m/s.Under the influence ofdam-break wave,although the dam-break wave increases the area of the circulation section at the cylinder, the velocity of the convex hull of the dam-break wave is too large, which results in a relatively high velocity distribution at the place directly above the cylinder and near the cylinder.

It is worth noting that at the dam foundation in Fig. 2b, the dam break water in Fig. 2a,which does not leave the dam foundation, has slipped into the wetland and caused the oscillation of the wetland boundary water, forming a secondary dam break wave on a smaller scale, and continue to spread downstream. Different from the first dam-break wave, the first dam-break wave is a large amount of dam-break water that pours into the wetland, while the second dam-break wave is a small amount of water that slides into the wetland along the dam foundation. Therefore, as can be seen from Fig. 2b, with the sliding of dam-break water, a part of the water flow with a velocity of about 0.7 m/s is generated first, and then the second dam-break wave is formed.

In Fig. 2c, after leaving the cylinder, the dam-break wave continues to spread downstream in the form of isolated wave, and the velocity distribution in the dam-break wave is completely consistent with that of the isolated wave. The velocity near the cylinder drops sharply, almost to zero. The secondary dam break wave in Fig. 2b has completely left the dam foundation and is also traveling as an isolated wave. In Fig. 2b, the water flow with a velocity of about 0.7 m/s in the range of x = 0.5 - 1 m propagates downstream in the manner of decreasing velocity. When it propagates to Fig. 2c, the flow velocity decreases to about 0.4 m/s. In addition, similar to the formation of the secondary dam break wave, a small part of the dam break water at the dam foundation is sliding towards the wetland. However, due to the small volume of this part of the dam break water, it cannot arouse the oscillation of the boundary water of the wetland and cannot form new dam break wave, but only slightly changes the flow field distribution near the dam foundation.

At the time presented in Fig. 2d, the water near the dam foundation has basically returned to the initial state, and there is no water slide on the dam foundation. The cylinder has been in contact with the water flow with decreasing velocity mentioned above, and the distribution of the flow field near the cylinder has changed, with a slight increase in the velocity at the top and bottom of the cylinder. The secondary dam-break wave is propagating to the cylinder as a complete isolated wave and is about to contact the cylinder.

Fig. 2 The velocity contours of flow fields at different time of dam break

4.2 Vorticity characteristics

Fig. 3 shows vorticity maps at different times of dam break. As can be seen from Fig. 3a, when the dam-breaking water slides into the wetland, it causes the oscillation of the boundary water body of the wetland. There are two different vortices produced at the wall surface and the water surface. The two spiral mainly concentrated in the dam break wave on the surface, in the majority with blue, which is in the majority with the negative counterclockwise vortex. This is because the dam water influx wetland basic only in one direction when the velocity, thus has a direction of water flow velocity of the dam and originally the stillness of the wetland water formed a counterclockwise vortex[12]. There are some positive vorticity of clockwise rotation on the surface of dam foundation, which is because a small amount of dam break water is sensitive to the friction in the opposite direction of the slide when dam break water slides along the foundation surface, resulting in the water body of this part of dam break sliding in the form of clockwise rotation.

When the isolated wave reaches the top of the cylinder, it causes the velocity around the cylinder to change, and two vortices in opposite directions are generated behind the cylinder(Fig. 3b). The first dam-break wave is still dominated by negative vortices, while the second dam-break wave is still dominated by positive vortices due to the clockwise rotation generated during the rolling dam foundation, and the water surface between the second dam-break wave and the first dam-break wave is dominated by positive vortices.

After the dam-break wave leaves the cylinder (Fig. 3c), the vortex near the cylinder cannot maintain the shape as shown in Fig. 3b due to the absence of continuous water flow, and it begins to fall off, and the vorticity also decreases. Different from Fig. 3b, at this time, the vorticity at the top of the second dam break wave has become negative,and the vorticity distribution inside the first dam break wave is similar. This is because with the propagation of the secondary dam break wave, the form of the wave evolves into a complete isolated wave, so that the vorticity distribution is not consistent with that of the initial dam break wave when it just left the dam foundation, but the first burst wave remains consistent.

It can be seen from Fig. 3d that when the secondary dam break wave approaches the cylinder, the vorticity near the cylinder has little change, which is far less obvious than the first dam break wave. This is due to the small size of the secondary dam break in the examples in this paper. In the actual dam break accident, the secondary dam break wave has a large effect on the cylinder, which cannot be ignored.

Fig. 3 The vorticity contours of flow fields at different time of dam break

4.3 Mechanics characteristic

Fig. 4 shows the force diagram of horiz ontal and vertical hydrodynamic forces of a cylinder changing with time starting from dam break. It can be seen that the horizontal and vertical hydrodynamic forces on the cylinder are the largest for about 1 s after the dam break, and then weaken and maintain a significant fluctuation. Different from the case in which a single isolated wave ACTS on the cylinder, the cylinder will first experience a large fluctuation of hydrodynamic force under the action of isolated wave,and then the fluctuation of hydrodynamic force will become very small[13]. Under the action of dam-break wave, even though the form of dam-break wave is very similar to the isolated wave, the formation of secondary dam-break wave causes the cylinder to be subjected to the hydrodynamic force with large fluctuation many times. In Fig. 4a, t = 0～2 s is the first large fluctuation of the hydrodynamic force, and t = 2～3 s is the second large fluctuation of the hydrodynamic force. In the case of general isolated waves, the cylinder will only be subjected once to a large fluctuation of the hydrodynamic force[14]. This phenomenon can also be seen in Fig. 4b. Before t = 3 s, the hydrodynamic fluctuation of the cylinder is relatively significant, and only after t = 3 s does the hydrodynamic fluctuation decrease to a weak degree.

In addition, it can be found that the fluctuation amplitude of horizontal hydrodynamic force subjected to a cylinder is within the range of ±18 N, while the fluctuation amplitude of vertical hydrodynamic force is within the range of±6 N. On the whole, the fluctuation of horizontal hydrodynamic force subjected to a cylinder is larger and more drastic. However,within t = 0～3 s, the horizontal hydrodynamic fluctuation is only 2 cycles, while the vertical hydrodynamic fluctuation is about 3.5 cycles. Therefore, although the fluctuation amplitude of the vertical hydrodynamic is less than that of the horizontal hydrodynamic, it is faster in frequency than that of the horizontal hydrodynamic. Therefore, in practical engineering,the strength and stiffness in the horizontal direction of the cylinder and the fatigue failure in the vertical direction should be given more consideration.

Fig. 4 Stress diagram of cylinder after dam break

4.4 Effect of height of dam break water

Fig. 5 describes the velocity distribution in the flow field at the same time under different dam break water heights. With the increase of the height of the break water, the propagation distance of the dam break wave increases in the same time. This character can be seen from Fig. 5. In Fig. 5a, the peak of the dam break wave reaches the upstream of the pipeline when h = 0.7 m. With the continuous increase of h, the peak gradually moves downstream of the pipeline, and finally moves to the downstream of the pipeline when h = 0.9 m. This is because when the height of the water body of the dam break increases, the gravitational potential energy of the whole water body of the dam break increases, and the dam break wave generated during the dam break has a higher propagation velocity. Therefore, the propagation distance of dam break wave increases with the increase of the height of water body. It can be seen from Fig. 5 that the high-speed flow area inside the dam break wave increases with the increase of the height of the dam break water body. This is because, on one hand, the influence of the height of the water body of dam break mentioned above on the velocity of the dam break wave; on the other hand, the increase of the height of the water body of dam break leads to the increase of the total amount of water body sliding into the wetland, thus resulting in a larger volume of the dam break wave and a larger high-speed flow area. Another notable phenomenon is that the flow velocity near the dam foundation increases with the height of the dam break water body. This is because the high velocity area near the dam foundation in Fig. 5c, and Fig. 5d is caused by the water sliding along the dam foundation not the water pouring into the wetland. As the height of the dam break water increases,the amount of water sliding into the wetland also increases, leading to an increase in the high-speed flow area near the dam foundation.

Fig. 5 The velocity contours of flow fields at different height of dam break water

Fig. 6 shows the variation trend of horizontal and vertical hydrodynamic forces of pipelines with time under different heights of dam break water. It can be seen from Fig.6a that the peak value of the first wave of the curve of the horizontal hydrodynamic force increases with the increase of the height of the dam break water. However, the peak value of the second wave of the curve is not positively correlated with the height of the dam break water, but suddenly increases when h exceeds 0.8 m. This is because the first wave peak of horizontal hydrodynamic force is formed by the action of the first dam break wave on the pipeline. As the height of the water body of the first dam break increases,the first dam break wave has higher height, higher propagation speed and greater kinetic energy, which causes the increase of the horizontal hydrodynamic force on pipeline. The second crest of the horizontal hydrodynamic curve corresponds to the second dam-break wave. The second dam-break wave is caused by a small amount of water sliding down the dam foundation into the wetland. When the height of the water body of dam break is below 0.8 m, most of the water body of dam break pours into the wetland and a small part of the remaining water body slides into the wetland along the dam foundation.However, when the height of water body of dam break is greater than 0.8 m, the volume of water body sliding into the wetland along the dam foundation is too large, which causes the movement pattern of this part of water body change from sliding into pouring, and thus arouses a larger second wave of dam break. The increase and abrupt change of hydrodynamic force can also be seen in Fig. 6b. In general, the increase of the height of the dam breaking water will lead to the increase of the hydrodynamic force on the pipeline. And excessively height of dam break water will lead to abrupt change of hydrodynamic action of the second dam break wave on the pipeline.

Fig. 6 Stress diagram of cylinder after dam break at different height of dam break water

5 Conclusion

In this paper, the cha racteristics of cylinder flow around the instantaneous full dam break are analyzed. Three aspects of the characteristics of cylinder flow around the instantaneous full dam break are studied: pressure field, flow field and vorticity field.According to the analysis, under the condition of instantaneous total failure, most of the water body of dam break will flood into the wetland and a small part of the water body of dam break will slide down along the dam foundation to the wetland, making the boundary water body of the wetland subject to multiple oscillations and forming multiple dam-break waves of different scales. Under the influence of multiple dam-break waves, the hydrodynamic wave duration of the cylinder is longer than that under the action of a single isolated wave, which makes the cylinder vulnerable to fatigue damage. With the increase of the height of the break water, the propagation distance of the dam break wave increases, the high-speed flow area inside the dam break wave increases and the hydrodynamic force on the pipeline increases. In addition, excessively height of dam break water will lead to abrupt change of hydrodynamic action of the second dam break wave on the pipeline. In a word, in the real marine environment, due to the different forms of dam break, different dam foundation forms and the influence of different water volume of dam break, the flow around characteristics of pipelines downstream of dam foundation change correspondingly. According to the different influences, it is helpful to adjust the specifications and layout of pipelines accordingly to protect pipelines