^{1,2}and Bum-Sang Yoon

^{1}

### Abstract

For the oblique impact of a wedge on a liquid half space, the limiting angles of the entry velocity and the wedge orientation corresponding to flow separation from the wedge vertex during the initial stage of the impact are investigated on the basis of an analytical solution of the problem. The liquid is assumed to be ideal and incompressible; gravity, surface tension, and air cushioning effects are ignored. The flow generated by the impact is two dimensional and potential. The solution is presented in terms of two governing expressions, which are the complex velocity and the derivative of the complex potential defined in a parameter region. These expressions are obtained using generalized integral formulas for solving mixed and uniform boundary-value problems for the first quadrant. They include two unknown functions, which are the velocity magnitude and angle to the free surface determined from the dynamic and kinematic boundary conditions. The obtained system of integral equations is solved by using the method of successive approximations. The effect of the horizontal component of the entry velocity is studied for various wedge orientations. The analysis of the computations revealed configurations of the impact such that the pressure along the whole length of one side of the wedge becomes less than the pressure on the free surface. Although air effects are not included in the analysis, such a pressure distribution provides conditions for the ventilation of the wedge side, which, in the presence of the air, starts from the contact point on the free surface and extends suddenly along the whole length of the wedge side, thus leading to flow separation from the wedge vertex. The theoretical predictions of flow separation and the experimental data on flow separation by Judge *et al.* [“Initial water impact of a wedge at vertical and oblique angles,” J. Eng. Math.48, 279 (2004)] are remarkably close to each other.

Both authors gratefully acknowledge the University of Ulsan and Hyundai Heavy Industries Co. Ltd. for their support of the present study.

I. INTRODUCTION

II. THEORETICAL FORMULATION OF THE PROBLEM

A. Analytical derivation of the flow potential

B. Evaluation of the hydrodynamic loads

C. System of integral equations

III. ANALYSIS OF THE FLOW PARAMETERS AT THE ONSET OF FLOW SEPARATION

A. Numerical approach

B. Validation of the numerical approach

C. Oblique entry of a wedge

D. Flow separation and convergence of the solution

IV. CONCLUSIONS

### Key Topics

- Free surface
- 48.0
- Cavitation
- 16.0
- Boundary value problems
- 14.0
- Integral equations
- 10.0
- Bubble dynamics
- 8.0

## Figures

Sketch of the physical plane (a) and the parameter plane (b).

Sketch of the physical plane (a) and the parameter plane (b).

The variation of the function along the boundary of the fluid region. The continuous changes in the function are shown by solid lines while its step changes are shown by dashed lines.

The variation of the function along the boundary of the fluid region. The continuous changes in the function are shown by solid lines while its step changes are shown by dashed lines.

Streamline patterns for wedge angle and entry angles , , and .

Streamline patterns for wedge angle and entry angles , , and .

Pressure distributions along the wedge sides for cases (a) (dotted line), (b) (dashed line), and (c) (solid line) in Fig. 3.

Pressure distributions along the wedge sides for cases (a) (dotted line), (b) (dashed line), and (c) (solid line) in Fig. 3.

Effect of the entry angle on the nondimensional coefficients of normal forces on the right (a) and the left (b) sides of a wedge. The heel angle varies providing wedge orientation (solid line), (dashed line), (dotted line), and (dashed-dotted line). The lower part of graph (b) shows the force coefficients for zero pressure along the wedge side, and corresponding to the experiments (Ref. 1).

Effect of the entry angle on the nondimensional coefficients of normal forces on the right (a) and the left (b) sides of a wedge. The heel angle varies providing wedge orientation (solid line), (dashed line), (dotted line), and (dashed-dotted line). The lower part of graph (b) shows the force coefficients for zero pressure along the wedge side, and corresponding to the experiments (Ref. 1).

Contact angles and on the right (decreasing curves) and left (increasing curves) wedge sides vs the entry angle for wedge rotation (solid line), (dashed line), (dotted line), and (dashed-dotted line).

Contact angles and on the right (decreasing curves) and left (increasing curves) wedge sides vs the entry angle for wedge rotation (solid line), (dashed line), (dotted line), and (dashed-dotted line).

Streamline patterns corresponding to the onset of the ventilation of the left side of a wedge of angle : wedge rotation , (a); , (b).

Streamline patterns corresponding to the onset of the ventilation of the left side of a wedge of angle : wedge rotation , (a); , (b).

Pressure distributions along the wedge sides ( for the left side and for the right side) corresponding to the onset of the ventilation of the left side of the wedge for three angles of rotation: , (solid line); , (dashed line); , (dotted line). The pressure coefficient corresponding to the vapor pressure in the experiments (Ref. 1) is shown by the horizontal dashed-dotted line.

Pressure distributions along the wedge sides ( for the left side and for the right side) corresponding to the onset of the ventilation of the left side of the wedge for three angles of rotation: , (solid line); , (dashed line); , (dotted line). The pressure coefficient corresponding to the vapor pressure in the experiments (Ref. 1) is shown by the horizontal dashed-dotted line.

Onset of flow separation from the wedge vertex in terms of the wedge rotation and the entry velocity angle : the calculated data (solid line), the experimental data (Ref. 1) (solid squares), and the criterion (dashed line). The separation-free region lies below the lines.

Onset of flow separation from the wedge vertex in terms of the wedge rotation and the entry velocity angle : the calculated data (solid line), the experimental data (Ref. 1) (solid squares), and the criterion (dashed line). The separation-free region lies below the lines.

Effect of the wedge half angle for the wedge bisector perpendicular to the initially flat free surface on the critical entry angle shown as the ratio .

Effect of the wedge half angle for the wedge bisector perpendicular to the initially flat free surface on the critical entry angle shown as the ratio .

## Tables

Tip jet angles and for oblique wedge water entry. The results in the columns correspond to different distances between the singular point and the nearest node. The results in the rows correspond to different numbers of nodes. The column marked “CH” shows the result of Chekin’s (Ref. 9) nonlinear theory.

Tip jet angles and for oblique wedge water entry. The results in the columns correspond to different distances between the singular point and the nearest node. The results in the rows correspond to different numbers of nodes. The column marked “CH” shows the result of Chekin’s (Ref. 9) nonlinear theory.

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