# Flow–Solid Coupling Analysis of Ice–Concrete Collision Nonlinear Problems in the Yellow River Basin

^{1}

^{2}

^{*}

## Abstract

**:**

**X**-directional displacement of the flow ice on the water transfer tunnel have the same trend as the simulated values, both of which show an increasing trend with an increase in flow ice velocity. It is shown that the ice material model parameters, ALE algorithm, and grid size used in this paper are able to simulate the impact of drift ice on the water transfer tunnel more accurately. With an increase in drift ice collision angle and drift ice size, the fitted curves of equivalent stress and peak displacement in

**X**-direction all show relationships of exponential function. The peak value of displacement in the

**X**-direction and maximum equivalent stress decrease with an increase in the curvature of the tunnel structure. It is also shown that the influence of change in drift ice size on the tunnel lining is greater than that of a change in tunnel section form. It is found that a high-pressure field will be formed due to extrusion of flowing ice, which should be fully considered in the numerical simulation. The research method and results can provide technical reference and theoretical support for prevention and control of ice jam disasters in the Yellow River Basin.

## 1. Introduction

## 2. Numerical Simulation

#### 2.1. Display Time Integration Principle

#### 2.2. Fluid–Solid Coupling Method

## 3. Modelling of Tunnel Impact by Drift Ice

#### 3.1. Engineering Examples

^{3}/s, and the tunnel has a clear height of 4.40 m, a clear width of 4.20 m, and a vault radius of 2.10 m.

#### 3.2. Selection of Model Material Parameters

- (1)
- Drift ice material model: At present, the research on ice materials is mainly focused on sea ice, while the research on river ice materials is relatively scarce. The drift ice in the river shows different mechanical properties under conditions of different temperature and strain rate. In this paper, the ice material is combined with the uniaxial compression test of river ice carried out in [23]. The plastic material related to strain rate (* MAT_STRAIN_RATE_DEPENDENT_PLASTICITY) is selected to simulate the characteristics of drift ice. The parameters of ice material model are shown in Table 1.

- (2)
- Tunnel material model: In the process of ice–tunnel collision, the tunnel lining will be damaged and deformed, so selection of tunnel lining materials should undergo plastic deformation. The concrete material model is CSCM-CONCRETE model developed by FHA Company [24]. The parameters used in the concrete material model are shown in Table 2.

- (3)
- Water and air media models: The constitutive model and equation of state are often used in LS-DYNA software to describe fluid materials (water, air). Therefore, the blank material Null was chosen to simulate water and air [25] with the parameters shown in Table 3. The equation of state is defined by the polynomial equation and the Gruneisen equation [26] for water and air media, respectively, and the parameters of both equations of state are shown in Table 3.

#### 3.3. Model Building

_{o}, only accounting for 2–7% of the hull mass, as, the longer the impact time is, the larger the additional mass is, while the ice–tunnel impact time is very short. Therefore, in this paper, the additional mass coefficient was set at 0.02 to carry out the additional mass of drift ice Δm calculation in the simulation by adjusting the drift ice density parameters to change the unit volume of drift ice mass [30], as well as through Equation (10) to complete additional mass of the conversion calculation. The fluid–solid coupling model, i.e., considering the dynamic load effect of the water medium on the tunnel lining structure in the actual project, was used to establish the water and air domains, respectively. Second, in order to improve the solution efficiency, only the influence of the water medium on the collision in the collision region was considered. Therefore, a 4 × 1.6 × 0.75 m

^{3}air domain model and a 4 × 1.6 × 2.25 m

^{3}water domain model were selected. The air domain is not selected as the full domain, but a layer of 0.75 m thickness is selected for calculation and analysis because, by observing the natural drift ice, most of the floating drift ice is immersed in the water medium, only a small part is in the air medium, and this paper studies the impact of small- and medium-sized drift ice on the collision of the water transfer tunnel, so the impact of the air domain on the drift ice is small, and, based on the flow–solid coupling method used in this paper for calculation and analysis, if the Eulerian grid is too much, the computation time of fluid–solid coupling will be greatly increased. Therefore, this paper selects part of the air domain for calculation and analysis of the whole model under the condition of ensuring the accuracy of the collision results.

^{3}; m

_{x}is the additional mass factor; and ${\rho}_{o}$ is the initial drift ice density in kg/m

^{3}.

**X**-direction in practical engineering, the initial velocity of ice was in the

**X**-direction in this simulation to provide power for the ice. In order to reduce the calculation time, the model was simplified, thus ignoring wind and air temperature. Second, in order to avoid the penetration phenomenon at the beginning of the collision and ensure small energy loss during movement of the drift ice, the distance between the drift ice and the tunnel in the

**X**-direction was set to 0.005 m in the simulation. The additional mass diagram and flow−solid coupling model diagram of the ice-tunnel collision are shown in Figure 1 and Figure 2.

#### 3.4. Mesh Sensitivity Study

^{3}, and the peak equivalent stress and peak displacement in the

**X**-direction derived from the numerical calculations at different grid scales are given in Table 4.

**X**-direction obtained from the simulations of A1, A2 and A3 meshes and the experimental values are 0%, −13% and −21%, respectively. Through the above mesh size sensitivity analysis, the A1 (0.05 × 0.05 × 0.05 m

^{3}) mesh is used in this paper for the following numerical calculations.

#### 3.5. Test Verification

**X**-direction under different flow ice velocity conditions, as shown in Figure 4.

**X**-direction under different flow ice velocity is +7.1%, the minimum error is +0.75%, and the average error is +2.88%. The error may be due to the fact that the ice material model chosen for the numerical simulations was an idealized ice model. Second, there were errors in the strain measurement process, the source of which was mainly due to the residual stresses in the concrete lining during the casting of the structure itself and the accuracy of the strain gauges. However, overall, the numerical simulation results and the results obtained from the test values were basically consistent. The data match well, and the error is within the permissible range, indicating that the numerical simulation model meets the accuracy requirements. It can also be seen from Figure 4 that, with an increase in flow ice velocity, both the equivalent stress and peak displacement in

**X**-direction show an overall increasing trend. According to the kinetic energy theorem, when the mass of drift ice remains unchanged, as the velocity of drift ice increases, the kinetic energy of drift ice increases and the energy transformed to the tunnel lining structure by the collision energy also increases. This, in turn, causes an increase in equivalent stress and an increase in displacement of the tunnel lining in the

**X**-direction. In addition, when the ice speed is greater than 3.0 m/s, the impact of drift ice on the concrete lining of the tunnel is obvious, so engineers should fully consider the impact of drift ice speed on a tunnel when designing the tunnel.

## 4. Numerical Simulation Results and Analysis

#### 4.1. Simulation Results under Typical Operating Conditions

^{3}was selected for the analysis. At the same time, in order to clarify the influence of the water medium on the collisions, the additional mass model without considering the water medium and the fluid–solid coupling model considering the water medium were established for typical working conditions. The finite element numerical simulation results show that, under the combined working conditions of a drift ice velocity of 3.0 m/s and a drift ice size of 0.7 × 0.7 × 0.3 m

^{3}, the maximum equivalent stress cloud map and

**X**-direction displacement cloud map of the tunnel lining under the time–history curves is shown.

^{3}. The maximum equivalent stress in the tunnel lining collision zone and the maximum displacement in the

**X**-direction are 2.475 × 10

^{6}Pa and 5.126 × 10

^{−5}m, respectively. (2) Additional mass model, the maximum equivalent stress in the tunnel lining collision zone and the maximum displacement in the

**X**-direction are 3.960 × 10

^{6}Pa and 9.191 × 10

^{−5}m, respectively, under the above combined conditions.

**X**-direction displacement of the collision zone shows loading characteristics. The maximum peak occurs at t = 0.00250 s, after which the drift ice bounces back, the ice load gradually decreases, and the

**X**-direction displacement in the collision zone shows unloading characteristics. At t = 0.00250–0.0330 s, the displacement in the collision zone shows multiple peaks, indicating that the tunnel lining structure continuously fails after being damaged by the impact. After t = 0.0330 s, the

**X**-directional displacement of the tunnel lining stabilizes at around 1.65 × 10

^{−5}m. (2) Additional mass model, the trend of the curve is similar to that of the stress–time curve without considering the water medium, and the reasons for this are basically the same. The difference is that, compared to the displacement–time curve under the water medium, the displacement of the tunnel lining in the

**X**-direction is stable at around 1 × 10

^{−5}m after the collision has been completed.

**X**-direction at different locations in the impact zone in the flow–solid coupled model are provided in the figure below.

**X**-direction displacement time–history curves at different locations are basically similar. In addition, it can also be seen from the figure that the

**X**-directional displacement of the tunnel lining at different locations is stable at about 1.65 × 10

^{−5}m after the collision is completed.

#### 4.2. Simulation Results for Different Ice Flow Angles

^{3}. The effect of shear collision between drift ice and tunnel on tunnel lining is investigated. In the finite element simulation, the collision angle

**θ**between drift ice and tunnel is 0° (positive collision), 15°, 30°, 45°, 60°, and 75°, and the collision angle is shown in Figure 10 (

**θ**= 0°, θ = 45° for example) without considering secondary collision and other problems.

**X**-direction under different drift ice collision angles. It can also be seen from Figure 11 (right) that, when the collision angle is in the range of 0°–15°, the damage deformation is stable at about 1.6 × 10

^{−5}m after the impact is completed; when the collision angle is in the range of 30°–60°, the damage deformation is stable at about 8 × 10

^{−6}m; when the collision angle is equal to 75 degrees, the damage displacement fluctuates at 0. It shows that the influence of drift ice on tunnel lining becomes weaker with an increase in collision angle. The influence is most obvious when the collision angle is 0°, mainly because the impact of drift ice on the tunnel lining is greater at the moment of collision and most of the energy of drift ice acts directly on the tunnel lining and only a small part of the energy is absorbed by water medium. With an increase in collision angle, the influence time of water medium on drift ice increases and then consumes more kinetic energy of flow ice.

**X**-direction under different drift–ice collision angles are plotted and the curves are fitted. The graphs are shown below.

**X**-direction at different drift ice angles show an exponential function (y = y

_{0}+ Ae

^{(−x/t)}). The peak equivalent stress and the peak displacement in

**X**-direction are decreasing with an increase in drift ice angle. This is similar to the findings of Liu et al. [34]. This indicates that the influence of drift ice angle on the collision of tunnel lining is obvious, so the influence of drift ice angle should be fully considered in the engineering design.

#### 4.3. Simulation Results for Different Drift Ice Sizes

^{3}(0.252 m

^{3}), and six other conditions to study the “size effect” of drift ice on the ice–tunnel impact. The drift ice velocity was taken to be 3.5 m/s. The following figure provides the equivalent stress and

**X**-direction displacement time–history curves for different flow ice size working conditions.

**X**-direction for different sizes of flow ice conditions also show nonlinear characteristics for the same reasons as those analyzed in Section 4.1. It can also be seen from the figure that equivalent stress and peak displacement in the

**X**-direction do not occur simultaneously for different size conditions, which is caused by the “water cushion effect” and the “size effect” of the flow ice. Due to the different sizes of drift ice in the process of hitting the tunnel lining, the size of the drift ice increases, the corresponding collision area also increases, and the resistance of the water medium to the drift ice also increases, which, in turn, causes a time difference in the contact between the drift ice and the tunnel lining, resulting in the curve peak not appearing at the same time.

**X**-direction for different flow ice sizes were plotted and the curves were fitted. The graphs are shown below.

_{0}+ Ae

^{(−x/t)}) with respect to both the equivalent stress and the maximum displacement in the

**X**-direction. It can also be seen from the figure that the peak equivalent stress and the peak

**X**-directional displacement decrease in the range of 0.189–0.210 m

^{3}, which is due to the combined effect of the water medium and the “size effect” of the drift ice size. In this paper, the impact of small- and medium-sized drift ice on the tunnel lining is studied, and the increase in drift ice size increases the mass of drift ice relatively little, so the differences in peak equivalent stress and

**X**-direction displacement of different sizes of drift ice are correspondingly small. In addition, with an increase in drift ice size, the resistance of water medium to drift ice increases accordingly, the slowing down effect on drift ice velocity increases, and the impact area between drift ice and tunnel lining also increases accordingly, so the equivalent stress and peak

**X**-direction displacement of tunnel lining will show a slight downward trend instead of a linear growth trend. When the drift ice size is larger than 0.7 × 1.0 × 0.3 (0.210 m

^{3}), the peak equivalent stress and peak

**X**-direction displacement both increase with an increase in drift ice size. The above phenomenon shows that, when the drift ice size is smaller than 0.7 × 1.0 × 0.3 (0.210 m

^{3}), the “size effect” of water medium and drift ice size has obvious influence on it, while, when the drift ice size is larger than 0.7 × 1.0 × 0.3 (0.210 m

^{3}), the “size effect” of the water medium and size of flow ice diminish.

#### 4.4. Simulation Results for Different Tunnel Cross-Sections

^{3}during the process. Using the combined working conditions of the above two factors, the

**X**-direction displacement and maximum equivalent stress of the tunnel lining under different cross-sectional forms are shown in the following figure.

^{6}Pa, and the maximum displacement in the

**X**-direction is 6.89 × 10

^{−5}m. Similarly, according to Figure 16 and Figure 17, the maximum equivalent stress values of the tunnel lining under the horseshoe shape and circular section are 2.34 × 10

^{6}Pa and 1.53 × 10

^{6}Pa, respectively, and the maximum displacement values in the

**X**-direction are 4.24 × 10

^{−5}m and 3.12 × 10

^{−5}m, respectively. Further, when drift ice hits the tunnel lining, the city gate type has the greatest influence on the tunnel lining, with the horseshoe shape having the second greatest influence, and the circular section has the least influence. The equivalent stress time–history curve and the

**X**-direction displacement time–history curve of drift ice on the tunnel lining under different cross-sectional forms are shown in the figure.

**X**-direction displacement curves for the horseshoe and circular cross-sectional forms are similar, both showing obvious nonlinear characteristics. The nonlinear characteristics of the equivalent stress and

**X**-directional displacement time curves are weaker in the Shing Mun form. The maximum equivalent stress and maximum displacement in the

**X**-direction decrease as the curvature of the tunnel structure increases, indicating that, the greater the curvature of the tunnel structure, the better its crashworthiness. It can also be seen from Figure 18 (right) that the drift ice causes some damage to the tunnel lining in all three different section forms, with the damage deformation in the Shing Mun section form stabilizing at around 2.18 × 10

^{−5}m after impact and the horseshoe and circle showing fluctuations around 0. The reason for this phenomenon is that, because the curvature of the city gate tunnel structure is 0, with deepening of the drift ice impact, the water medium is completely squeezed out, whereas the horseshoe and circular tunnel structures have a certain curvature, so the water medium does not completely escape when the drift ice impacts, thus causing the curve form to fluctuate. The water medium that does not escape plays a certain role in blocking the drift ice, resulting in a significantly smaller impact on the tunnel lining than in the Shing Mun Tunnel section. To sum up, it is particularly important to select a reasonable tunnel cross-section according to the characteristics of regional drift ice hazards during engineering design.

**X**-directional displacement, indicating that the effect of the change in drift ice size on the tunnel lining is greater than the effect of the tunnel section forms.

## 5. Conclusions

- (1)
- The experimental values of maximum equivalent stress and
**X**-directional displacement show the same trend as the simulated values by comparing with the experimental results of the model of impact of drift ice on the water transfer tunnel, both of which show an increasing trend with an increase in drift ice velocity. This also indicates that the ice material model parameters, ALE algorithm, and grid size used in this paper can simulate the impact of drift ice on the water transfer tunnel more accurately. In addition, it is found that, when the ice velocity is greater than 3.0 m/s, the impact of drift ice on the water transfer tunnel is obvious, so engineers should fully consider the impact of drift ice velocity on a tunnel when designing the tunnel. - (2)
- Comprehensive analysis of the different drift ice conditions simulated in this study shows that, as drift ice collision angle and drift ice size increase, the fitted curves of equivalent stress and peak displacement in
**X**-direction all show an exponential function. The difference is that equivalent stress and peak displacement in**X**-direction keep decreasing as drift ice collision angle increases, while the most obvious effect of positive drift ice collision on the tunnel lining is found. However, as size of drift ice increases, peak equivalent stress and**X**-directional displacement show different trends, and peak equivalent stress and**X**-directional displacement decrease in the range of 0.189–0.210 m^{3}, indicating that the “size effect” of drift ice and the influence of water medium on the ice–tunnel collision process are obvious. When the drift ice size is larger than 0.210 m^{3}, the peaks all increase with an increase in drift ice size, indicating that “size effect” and water medium have a significantly lower influence at this time. In the case of changing only the tunnel section form, both the peak equivalent stress and peak displacement in the**X**-direction in the tunnel lining impact zone decrease as the curvature of the tunnel structure increases, indicating that, the greater the curvature of the tunnel structure, the better its crashworthiness. In addition, a comparative analysis of the effect on tunnel lining under different drift ice sizes and tunnel section forms was carried out and it was found that the average peak equivalent stress when the drift ice size changes is 1.53 times higher than the average peak value of different tunnel section forms and 1.41 times higher than the average peak**X**-directional displacement. Therefore, regional drift ice disaster characteristics in the engineering design should fully consider drift ice collision angle, drift ice size, and tunnel cross-section form on the impact of a water transmission tunnel. - (3)
- Through comparative analysis of the water medium under typical working conditions by the additional mass method and fluid–solid coupling method, it was found that a high-pressure field will be formed by extrusion of the water medium during drift ice movement. This will have an impact on the tunnel lining during the impact process, and, through analysis of drift ice velocity under different tunnel sections, it was found that water medium has an obvious influence on drift ice movement. This should be fully considered in numerical simulations.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**(

**a**) Test setup diagram. 1. Instrument mounting cabinet. 2. Dynamic strain tester mounting table. 3. Computer mounting table. 4. Power system mounting table. 5. Cabinet doors. 6. Pulse transmitter. 7. Stepper motor driver. 8. Stepper motors. 9. Dynamic strain tester. 10. Computers. 11. Main wiring hole. 12. Concrete lining. 13. Strain gauges. 14. Drift ice. 15. Sub-wiring hole. 16. Wiring holes. 17. Open mouth. (

**b**) Test stands and data acquisition and analysis systems.

**Figure 4.**Comparison of simulation and experimental results of different drift ice speeds. (

**a**) Maximum equivalent stress; (

**b**) maximum displacement in

**X**-direction.

**Figure 5.**Equivalent stress clouds under different collision models, where: Local enlargement of the ice−tunnel impact zone is performed.

**Figure 6.**Displacement clouds in

**X**-direction under different collision models, where: Local enlargement of the ice−tunnel impact zone is performed.

**Figure 8.**Time–history curves of equivalent stresses (

**left**) and

**X**-displacements (

**right**) at different positions.

**Figure 11.**Time–history curves of equivalent stresses (

**left**) and

**X**-displacements (

**right**) for different collision angles.

**Figure 12.**Peak graph of Equivalent stress (

**left**) and

**X**-direction displacement (

**right**) for different collision angles.

**Figure 13.**Time–history curves of equivalent stresses (

**left**) and

**X**-displacements (

**right**) for different drift ice sizes.

**Figure 14.**Peak graph of Equivalent stress (

**left**) and

**X**-direction displacement (

**right**) for different drift ice sizes.

**Figure 15.**Cross section of city gate tunnel. (

**a**) Equivalent stress peak value. (

**b**) Peak displacement in

**X**-direction.

**Figure 16.**Horseshoe tunnel section. (

**a**) Equivalent stress peak value. (

**b**) Peak displacement in

**X**-direction.

**Figure 17.**Circular tunnel section. (

**a**) Equivalent stress peak value. (

**b**) Peak displacement in

**X**-direction.

**Figure 18.**Time–history curves of equivalent stresses (

**left**) and

**X**-displacements (

**right**) for different cross-sections.

Parameters | Values |
---|---|

Density (kg·m^{−3}) | 910 |

Modulus of elasticity/GPa | Related to strain rate |

Poisson’s ratio | 0.3 |

Parameters | Values |
---|---|

Mass density/(kg·m^{−3}) | 2500 |

Maximum strain increment | 0 |

Rate effect switch | 1 |

Pre-existing damage | 0 |

Erosion coefficient | 1.1 |

Coefficient recovery parameter | 10 |

Blocking options | 0 |

Compression strength/MPa | 29 |

Aggregate size/m | 0.02 |

Parameters | Water | Air |
---|---|---|

Density/(kg/m^{3}) | 1.1845 | 998.21 |

Cutting stress/(Pa) | −10 | −1 × 10^{5} |

Viscosity coefficient | 1.7456 × 10^{−5} | 8.684 × 10^{−4} |

Constant C | 1647 | |

Constant S_{1} | 1.921 | |

Constant S_{2} | −0.096 | |

Constant C_{4} | 0.4 | |

Constant C_{5} | 0.4 | |

Constant | 0.35 | |

Initial internal energy(E_{0}/J) | 2.53 × 10^{5} | 2.895 × 10^{5} |

Initial internal energy/V_{0} | 1.0 | 1.0 |

**Table 4.**Comparison of peak equivalent stress and peak displacement in

**X**-direction at different mesh scales.

Mesh Serial Number | A1 | A2 | A3 | Experimental Value |
---|---|---|---|---|

mesh size | 0.05 × 0.05 × 0.05 m^{3} | 0.08 × 0.08 × 0.08 m^{3} | 0.10 × 0.10 × 0.10 m^{3} | |

Peak equivalent stress (×10^{6} Pa) | 2.475 (2.1%) | 2.916 (−15%) | 3.211 (−27%) | 2.528 |

X-direction displacement peak (×10^{−5} m) | 5.216 (0%) | 5.913(−13%) | 6.295 (−21%) | 5.216 |

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## Share and Cite

**MDPI and ACS Style**

Gong, L.; Dong, Z.; Jin, C.; Jia, Z.; Yang, T.
Flow–Solid Coupling Analysis of Ice–Concrete Collision Nonlinear Problems in the Yellow River Basin. *Water* **2023**, *15*, 643.
https://doi.org/10.3390/w15040643

**AMA Style**

Gong L, Dong Z, Jin C, Jia Z, Yang T.
Flow–Solid Coupling Analysis of Ice–Concrete Collision Nonlinear Problems in the Yellow River Basin. *Water*. 2023; 15(4):643.
https://doi.org/10.3390/w15040643

**Chicago/Turabian Style**

Gong, Li, Zhouquan Dong, Chunling Jin, Zhiyuan Jia, and Tengteng Yang.
2023. "Flow–Solid Coupling Analysis of Ice–Concrete Collision Nonlinear Problems in the Yellow River Basin" *Water* 15, no. 4: 643.
https://doi.org/10.3390/w15040643