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Combining nonlinear boundary conditions of free liquid surface, the impact force of wedge shape profile and the change of free liquid surface were calculated and analyzed based on the solving ideas of Von Karman T. The results were compared with the results of the existing literature, and the results were in good agreement with the existing results. In addition, the impact forces of U profile and the external drift profile were calculated, and the impact load of two kinds of profiles were analyzed through time evolution.

The water entry problem of structure was the strong nonlinear interaction between the structure surface and the free liquid surface. The water entry problem of structure was widely used in the field of vessel and marine engineering. From the end of the last century, the problem of landing on water was studied by von Karman [

In the experimental studies, wedge shape was experimented from the bottom lift angle 1˚ to 15˚ by Chuang S L, the peak value of impact pressure was measured in the water entry process. And the regression formula between the peak value of impact pressure and bottom lift angle was given [

Numerical method was widely used in the calculation of structure water entry. Aimed at the problem of aircraft landing on the water, the conservation of momentum theorem was applied to calculate the impact force and the average pressure of structure by von Karman T. Due to neglect many important factors, the calculation results were relatively small, and there were many limitations in the application. Subsequently, the free liquid surface motion was considered to improve the calculation of wet half width on the basis of von Karman T, and the distribution of impact pressure on the surface was calculated [

In this paper, the characteristics of movement, pressure distribution and impact force on free liquid surface were studied in the water entry process. First, the impact force solution idea of Karman T von was introduced. Secondly, the calculation theory of free liquid surface movement based on the Lagrange grid was described. The last, the free liquid surface movement, pressure distribution and the variation of the impact force for wedge profile, U profile and external drift profile were calculated and compared between each other.

As shown in

To reduce the number of units and improve the calculation efficiency, the calculation tank and water entry profile were divided into two parts along Y. The model was gotten as

Three typical water entry profiles were calculated in this paper. For

In this paper, according to the impulse theory of von Karman and combing with the solution method of the attached water quality, the water entry problem of the two dimensional structure was solved. From

impact force calculation idea of von Karman T was as follows: assuming the water entry body was a wedge profile which enters the water at a constant speed and the bottom lift angle was

In Formula (1), F is impact force, V was velocity of body water entry, y was body water entry part which was relative static water depth,

Joined the Formula (2) to Formula (1), the dimensionless form of impact force was:

The basic ideas of Boundary Element Method is that the boundary discrete into several elements. In this method, boundary condition can be transformed into the boundary of each element. Then, the integral equations can be calculated by the summation of each element

where

In this work, the linear boundary element is used to disperse the field. Finally, the governing equations can be regarded as

where

In our work, the integral equations can be calculated by transforming the coordinate of elements. Firstly, the midpoint of elements can be regarded as the origin of local coordinate. The x axis is the direction along the border. And the y axis is the vertical direction on elements, pointing to the internal area, as shown in

In the coefficient matrix,

where A, B, C, D can be calculated as

Specially,

Boundary conditions were shown in

Free surface was given the first kind of boundary condition, its velocity potential could be calculated according to the water entry process about the time iteration by Formula (8).

Free surface satisfied kinematic equation

In addition, the free liquid surface also satisfied the Lagrange integral:

Sun pointed out that the gravity was small proportion in the impact process [

In this paper, the free surface used Lagrange grid. For

Free surface deformation

Taken the wedge shape with the bottom lift angle as 10 degrees as an example, All time free surface motion was calculated and the results were shown in

The free liquid surface motion results of non-dimensionless were given by Zhao R [

In

Pressure distribution

The surface pressure could be calculated by Formula (9) [

Taken 10 degree wedge shape as an example, the pressure distribution on the surface in the water entry process was shown in

In _{0 }represented atmosphere pressure. The pressure was the largest near intersection of body surface and free liquid surface at a certain moment. For a point on the body surface, when the free liquid surface reached this point, this pressure quickly became large. Then as time goes by, the pressure of this point gradually became small and gone stable finally. And all the points on the wedge body were all the final approximation

Impact force

Taken the wedge shape with the bottom lift angle as 10 degrees as an example, the change of attached water quality in water entry process and the impact force was calculated by Formula (1). The results were shown as

From

Form

The curve of the U-shaped section was selected as

Similar to the wedge shape, the free surface movement, pressure distribution and the impact force of the U-shaped section in the water entry process were shown as

Pressure distribution curves of different time were shown as

As was shown in

The curve of flared section surface was shown in

The free liquid surface movement, pressure distribution and the impact force of external drift surface were as

As was shown in

entry. The pressure peak and pressure steady value of external drift profile gradually became large. It indicated that the impact load increased with the process passage of the water entry. For

In this paper, three typical profiles are selected. The free liquid surface movement, the impact pressure distribution and the impact force are calculated in the water entry process. Based on the results of the three surfaces, the main conclusions are as follows:

1) When three typical profiles go into water, the free liquid surface happen movement, and the pressure near the intersection of the free liquid surface and body surface is far greater than the pressure value of other body surfaces.

2) For impact pressure, the pressure peak value of wedge shape doesn’t change much at different times, and steady value is basically unchanged; the pressure peak value of U shaped section becomes small, and the stationary value also becomes small; the pressure peak value of external drift profile becomes large and the stationary value also becomes large.

3) The impact force of three surfaces becomes larger with the increasing water entry depth. The relationship between the impact of wedge shape and water entry depth is approximately linear. The impact force of U shaped profile becomes smaller with the increasing time. The impact force of external drift profile increases quickly with the increasing time.

This research is funded by National Natural Science Foundation of China (51379033; 51379030; 51239002; 51221961), Special funds for basic scientific research business of Central University (DUT15LK43, DUT15LK32) and 973 Project (2013CB036101).

L. Zou,Z. Zong,S. Feng,G. X. Zhu,G. P. Yu, (2016) Boundary Element Analysis of Two Dimensional Structure Water Entry Based on Attached Water Quality. Journal of Applied Mathematics and Physics,04,1533-1545. doi: 10.4236/jamp.2016.48163