Qingtong Chen, Baoyu Ni, Shuping Chenand Jiangguang Tang

Numerical Simulation of the Water Entry of a Structure in Free Fall Motion

Qingtong Chen1, Baoyu Ni2＊, Shuping Chen1and Jiangguang Tang1

1. China Helicopter Research and Development Institute, Jingdezhen 333001, China
2. College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China

To solve the problems concerning water entry of a structure, the RANS equations and volume of fluid (VOF) method are used. Combining the user-defined function (UDF) procedure with dynamic grids, the water impact on a structure in free fall is simulated, and the velocity, displacement and the pressure distribution on the structure are investigated. The results of the numerical simulation were compared with the experimental data, and solidly consistent results have been achieved, which validates the numerical model. Therefore, this method can be used to study the water impact problems of a structure.

structure; free fall motion; water impact; RANS equations; VOF; water entry

1 Introduction1

The phenomenon of water entry of a wedge in free falling widely exists in aerospace, underwater weapons, the shipping field, etc. Well known examples include the landing on water with air planes, helicopters, spaceships in the aerospace field, the water entry of aerial torpedos in the weapons field, the sailing of planing boats at high speeds in the shipping field, etc. In the process of a structure entering into water, the water slams the structure. Not only does the fluid have a great slamming force on the structure, but it also significantly impacts the flow field. The water surface produces serious fluid splashing, which is prone to damage the structures.

To attempt to solve the problems of a structure entering into water, scholars have done extensive research, both with experimental studies and theoretical calculations. With the experimental aspect, El-Mahdi et al. (2006) had an experimental study done on the water impact on a symmetrical wedge. Sun et al. (2003) had experimental research done on the fluid-structure interaction in water entry of a two-dimensional (2D) elastic wedge. Based on the hydroelasticity theory, Chen et al. (2012) used a numerical simulation and model test to investigate the water impact on a ship’s cabin in waves. With the aspect of theoretical calculation, the water entry problem of the 2D cross-planes was solved by using a boundary element method developed by Sun (2010). Zhang et al. (2010) utilized the software FLUENT to simulate numerically the water entry of the wedge-shaped sections with different deadrise angles at a constant velocity. Zhang (2003) calculated and analyzed the initial flow of the wedge entering into water using the VOF method numerically, and the numerical simulation was compared with the PIV experiments and high-speed photographic visualization. Based on the potential flow theory, Ni (2012) researched the additive mass variation of a missile during its entry into water by using the semi-analytic and semi-numerical methods. Duan and Wu (2011, 2012) have also done some theoretical research on the water entry of the structures in recent years.

In this paper, the water entry of helicopters is taken as the research background and fundamental prediction method for water impact of a simple structure entering into water. For the problem of water entry of a structure, a large number of domestic researches regards falling velocity as constant, which does not reflect truly the water entry state of a structure. So, to simulate the true state of a structure entering into water, the water entry of a 2D wedge in free-fall motion is simulated numerically based on the software FLUENT with UDF by using VOF to capture the free water surface. The calculation results are consistent with the experimental results with El-Mahdi et al. (2006), which validates this numerical procedure presented. The motion state and surface pressure distribution of the wedge are obtained.

2 Numerical calculation method

2.1 Governing equation

Basic governing equations include continuity, momentum conservation (Navier-Stokes), and energy conservation equations. Energy equations are not needed because conversion of energy is not related. The equations are expressed as follows (2004):

Continuity equation:

Constant viscous Navier-Stokes equations:

where u, v, w are the components of the fluid veocity in the x, y, z directions respectively; fx, fy, fzare the unit components of the volume force in the x, y, z direction respectively; p is the fluid pressure force; ρ is the fluid density; μ is the fluid motion viscosity.

2.2 VOF method

In computational fluid dynamics, the volume of the fluid method is a numerical technique used for tracking and locating the free surface (or fluid-fluid interface). The method is based on the idea of the so-called fraction function F. It is defined as the integral of the fluid′s characteristic function in the control volume (namely the volume of a computational grid cell). Basically, when the cell is empty (there is no traced fluid inside), the value of F is zero; if the cell is full, then F = 1. And if the interphasal interface cuts the cell, then 0 ＜ F ＜ 1. F is a discontinuous function, its value jumps from 0 to 1 when the argument moves into the interior of the traced phase. The F function satisfies:

2.3 The equations in the UDF program

2.3.1 Motion equations

When the wedge falls freely, the relationship between the force and speed is expressed as follows:

where v0is the velocity in a last time step, f the external force on the structure, m the mass, and tΔ the time step.

2.3.2 Calculation of fluid density

Water compressibility should be considered when it comes to high-speed impact problems, such as underwater explosion shocks and water hammer problems.Bulk modulus is known as follows:

The mass of a body is always a constant, so dm is expressed as follows:

where ρ can be calculated by combining Eqs. (5) and (6).

where B is the bulk modulus, V is volume, P is pressure, ρ is the changed density of the liquid, and0ρ is the initial density.

2.4 The numerical simulation process

The real-time motion state of a structure is predicted numerically with UDF. Motion in y direction is just considered. The forces and moments of the wedge at this moment can be obtained by solving the RANSE equation, and the forces and moments are then put into the six-degrees-of-freedom equation of motion. By solving the accelerations of the motion, velocities and displacements (translations and rotations) are obtained by integrating in time. The new position of the wedge is updated and the fluid flow is computed again for the new position. By iterating this procedure over the time, the wedge trajectory is obtained. The flow chart is shown in Fig.1.

Fig.1 The flow chart of the numerical simulation

3 Experimental set-up of of El-Mahdi et al. (2006)

In order to study the water entry of the structure, the wedge was released at a height of 1.3 m from the freesurface, and fell freely, as shown in Fig.2. The wedge velocity, displacement and stress distribution were measured by El-Mahdi et al. (2006).

Fig. 2 Schematic diagram of test equipment (El-Mahdi et al., 2006)

The wedge had a square dimension of 1.2 m × 1.2 m that could support additional steel pieces with masses of 94 Kg. To measure the time-varying pressure distribution on one side of the 2D wedge, twelve pressure transducers were used. The pressure range of these transducers is 0-3 Mpa and each one has a diameter of 19 mm. They were distributed along the median of one side of the wedges. The distance between each transducer was 50 mm. They were numbered 1-12, with number 1 being located near the apex of the wedge, as shown in Fig.3. The first natural frequency of the transducer is 10 kHz.

Fig.3 Distribution of the pressure transducers

To measure the instantaneous position and velocity of the wedge, a potentiometric cable extension transducer was used. The transducer position raw data were low-pass filtered using a cut-off 45 Hz to remove spurious noise generated by slight vibrations of the cable. The velocity was calculated by a numerical differentiation of the position signal.

4 Numerical simulation

4.1 Geometry and computational domain

In order to compare with the experimental results, the calculation geometry and conditions are the same as the experimental ones. The 2D wedge entering into water is simulated, whose computational model is shown in Fig.4. The size of the computational domain is 7 m × 8 m, as shown in Fig. 5.

Fig.4 Computation model

Fig.5 Computational domain

4.2 Mesh and boundary conditions

Meshing is one of the most difficult parts in building the computation models. The quality of the grids can determine the accuracy of the solution and it can also determine the consumption of the solving time.

The computational domain is discretized by a structured grid, which greatly reduces the number of grids and computational cost. The entire computational domain is divided into two parts: the inner part and the external part. The inner part corresponds to the domain near the wedge with a greater grid density, while the external parts are the domain outside the inner part, with a relatively sparse grid density. The inner domain is moving with the wedge in the same falling velocity. As a result, the grids around the wedge will not change and have good quality, which ensures the computational accuracy during falling. The computational grids are shown in Fig. 6.

Fig.6 Meshes of the flow field

The boundary conditions are: rigid wall surface on thewedge; pressure outlet on the top surface of the computational domain; rigid wall surfaces on the other surfaces of the computational domain.

4.3 Numerical simulation results

The simulation was performed over 21 000 time steps with a time-step size Δt = 0.000 1 s and 50 iterations per time step. The CPU time was 40 hours for this simulation with four-AMD opteron 2 350 processors.

The changes of the grids at different times are shown in Fig.7, while Fig.8 shows the contours of the free surface when the wedge falls freely.

Fig.7 Changes of the grids at different times

Fig.8 Contours of the free-surface at different times

Fig.9 shows the time history of the velocity of the wedge, while Fig.10 shows the time history of the displacement of the wedge. In Fig.9 and Fig.10, it can be seen that the difference between the calculated and experimental results is small. From these figures, the velocity and the displacement of the wedge increase along with the time before in contact with the water surface. And then the velocity decreases along with the time. Because of the water damping effect, the velocity will fall to zero and the displacement of the wedge will remain unchanged at last.

Fig.11 shows the time history of the velocity of the wedge 40 ms after the water impact. In Fig.11, the computated value is compared with the experimental data and the numerical results from the other different numerical models. It can be seen that the Von kanman curve coincides quite well with the computated value in this paper.

Fig.9 Time history of the velocity of the wedge

Fig.10 Time history of the displacement of the wedge

Fig.11 Time history of the velocity of the wedge in 40 ms after the water impact

Fig.12 shows the pressure time variation of the transducers. Refering to Fig.12, it can be seen that the numerical results match the experimental pressure measured by the No.3 and the No.5 transducers well, but there is a small error between the calculated results and the experimental pressure measured by the No.7 and the No.9 transducers. The reason for this is because there exists splash when the wedge impacts the water face in the actual flow, which cannot be well simulated by our numerical model. As a result, the difference between the calculated results and the experimental results gets larger for the latter sensors. This problem may be alleviated by refining the meshes on the free surface.

Fig.13 shows the predicted pressure distribution along the wedge face at different times. As seen in Fig.13, the peak of the pressure decreases along with the time. The peak pressures are located near the water jet tip edge apex, and the peak pressure positions are changed as the water jet travels up the wedge.

Fig.12 Predicted pressure time histories for tranducers

Fig.13 Predicted pressure distribution along the wedge face at different times

5 Conclusions

The water entry of a free falling wedge was researched based on the FLUENT software using the VOF method and the moving grid model. The calculation results were compared with the experimental results. The comparison shows that there is a very small error rate between the calculation and experiment values for the structure velocity and displacement. The predicted values of the surface pressure distribution are in good agreement with the experimental data. In order to improve the accuracy of the predictions further, improving the computational grid and boundary conditions or selecting other turbulence models may be taken into account. The method presented can be applied to the numerical simulations of structures entering into water with various variable speeds. Predicted results may provide references for engineering applications.

The wedge impact model can be extended to calculate the forces on a number of wedges, which can be linked together with the strip theory to create a 3D model for the landing on water of helicopters. Furthermore,the present study can be extended to simulate the landing water state of 3D helicopters directly. Impact loads, pressure distribution and movement can also be obtained.

The method presented provides a solid foundation for the 3D simulation of the landing water state of 3D helicopters. On the other hand, this method can provide a theoretical basis and technical support for landing water experiments of helicopters.

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Author biographies

Baoyu Ni was born in 1986. He is a doctor and lecturer at the College of Shipbuilding Engineering, Harbin Engineering University. He was a visiting research scholar at University College London (UCL). His current research interests are underwater bubble dynamics, and coupled fluid-structure analysis.

1671-9433(2014)02-0173-05

Chen was born in 1987. He

his master’s degree of engineering in the Design and Construction of Naval Architecture from Harbin Engineering University in 2012. He is an assistant engineer at China Helicopter Research and Development Institute. His research interests include Computational Fluid Dynamics and static tests.

Received date: 2013-09-14.

Accepted date: 2013-10-23.

Supported by the National Natural Science Foundation of China (11302056), China Postdoctoral Science Foundation (2013M540272), Heilongjiang Postdoctoral Fund (LBH-Z13051), the Fundamental Research Funds for the Central Universities (HEUCF140116) and Research Fund of State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University (1309).

＊Corresponding author Email: nibaoyu@hrbeu.edu.cn

? Harbin Engineering University and Springer-Verlag Berlin Heidelberg 2014