Small Deadrise Angles Research Articles

Numerical simulation of water impact for 2D and 3D bodies

Slamming forces on 2D and 3D bodies have been computed based on a CIP method. The highly nonlinear water entry problems governed by the Navier–Stokes equations were solved by a CIP-based finite difference method on a fixed Cartesian grid. In the computation, a compact upwind scheme was employed for the advection calculations and a pressure-based algorithm was applied to treat the multiple phases. The free surface and the body boundaries were captured using density functions. For the pressure calculation, a Poisson-type equation was solved at each time step by the conjugate gradient iterative method.Validation studies were carried out for 2D wedges with various deadrise angles ranging from 0 to 60° at constant vertical velocity. In the cases of wedges with small deadrise angles, the compressibility of air between the bottom of the wedge and the free surface was modeled. Studies were also extended to 3D bodies, such as a cylinder and a catamaran, entering calm water. Computed pressures, free surface elevations and hydrodynamic forces were compared with experimental data and the numerical solutions by other methods.

Experimental investigation of water slamming loads on panels

Rigid-body slamming has become increasingly important as ships travel at higher speeds experiencing larger loads during hull impacts against surface water which can result in structural damage and crew injury. It is necessary to characterise the hydrodynamic loading during water impacts. We present a series of experiments conducted in order to study slamming force events seen by flat plates during free surface impacts. The experiments focus on the characterisation of the loads experienced by flat plates during the first phase of the slamming event, the water entry. They have been conducted in an especially designed test apparatus, the Slingshot Impact Testing System (SITS), which allows us launching objects against the free surface of an open channel, with the possibility of setting up different speeds and deadrise angles. We can study slamming with trapped air between the plate and the water free surface, at high impact speeds and small deadrise angles, allowing us to quantify the resulting cushioning effect. High velocity impacts up to 5 m/s were conducted at angles between 0.3° and 25°. It was found that the trapped air phenomenon significantly cushions flat plate impacts with angles less than 5° and impacts with larger angles adhere to Von Kármán's equations.

Second-order Wagner theory for two-dimensional water-entry problems at small deadrise angles

The theory of Wagner from 1932 for the normal symmetric impact of a two-dimensional body of small deadrise angle on a half-space of ideal and incompressible liquid is extended to derive the second-order corrections for the locations of the higher-pressure jet-root regions and for the upward force on the impactor using a systematic matched-asymptotic analysis. The second-order predictions for the upward force on an entering wedge and parabola are compared with numerical and experimental data, respectively, and it is concluded that a significant improvement in the predictive capability of Wagner's theory is afforded by proceeding to second order.

Hydroelastic coupling of beam finite element model with Wagner theory of water impact

The purpose of the paper is to demonstrate the feasibility of the direct coupling of the finite element method for the structural part with a Wagner representation of the hydrodynamic loads during the impact of an elastic body onto the water surface. An efficient and very general method is developed and validated in two dimensions. Advantages of the present method are outlined for the elastic wedge impact problem; however, the method is applicable to any elastic body with small deadrise angle entering water vertically at moderate velocity. Strategy for coupling of this method with commercial finite element codes is discussed.

A free boundary problem arising in a model for shallow water entry at small deadrise angles

A free boundary problem arising in a model for inviscid, incompressible shallow water entry at small deadrise angles is derived and analysed. The relationship between this novel free boundary problem and the well-known viscous squeeze film problem is described. An inverse method is used to construct explicit solutions for certain body profiles and to find criteria under which the splash sheet can `split'. A variational inequality formulation, conservation of certain generalized moments and the Schwarz function formulation are introduced.

Study on Water Impact of Elastic Plate with Small Deadrise Angles

It is well known that in the case of the collision between a flat body with very small deadrise angle and a water surface, Wagner's theory is not applicable because of the trapped air effect. This paper describes the study on this trapped air affected region of the water impact problem. In order to investigate the transient situation from the air trapped impact to the Wagner type impact, we have carried out the pressure and the stain measurement at the water impact by dropping an aluminum flat plate of 200 mm breadth and 5 mm thick in the condition of attack angles from zero to 4.0 degrees at intervals of 0.5 degrees.Based on the experimental results, the following conclusions have been derived.(1) Any pressure pattern in time space is clearly identified with the Wagner type or the air trapped type.(2) The impact situation is classified to the following three regions according to the attack angle β. (a) Air trapped type impact region : 0≤5β< 1°The pressure pattern is the air trapped type all over the impact surface. Flat pressure distribution rises and falls smoothly.(b) Transient region : 1°≤β<3°The Wagner pattern around the keel of the plate and the air trapped pattern at the edge side are observed. The Wagner pressure pattern expands toward the edge with increasing β. Sharp peak pressure is observed.(c) Wagner type impact region : β≥3°The pressure pattern is the well-known Wagner type all over the plate. (3) The maximum strain that has the maximum value around β=0 is moderately depend on β from zero to 4 degrees. The sharp peak pressure hardly affects the strain.(4) The strain calculated statically by loading the measured average pressure is in good agreement with the measured strain.The structural response at impact depends on not only the load but its own structural characteristics. Therefore, further research is needed to evaluate the strength of more complicated structures in practice.

Bow Flow Field with Spray Phenomena around a Planing Plate

A perturbational study is made of the bow flow field with spray phenomena around a planing plate with a small dead rise angle and a small trim angle in order to obtain the basis on the configuration of the wetted surface area. A hypothesis is proposed whereby the plate planes on a water surface deformed as a parabolic cylinder with transverse curvature near the bow, the deformation seems to be caused by the displacement effect. The hypothesis allows the assumption that a similarity flow changing the scale of transverse section as a square root of longitudinal length. The flow field is divided into three regions for the analysis. By analysing the bow near field, the spray root line is obtained as forming a parabola on the plate, so that good coincidence with an experimental result of pressure contour is shown with the help of the same flow near the spray root line as Wagner's local flow. The spray flow developing on the plate outside the spray root line is analyzed. The flow rate and thickness of the spray and the spray drag show good coincidence with an experimental result. The spray edge line is shown by assuming a lower bound of spray thickness.

Water entry of two-dimensional bodies

A numerical method for studying water entry of a two-dimensional body of arbitrary cross-section is presented. It is a nonlinear boundary element method with a jet flow approximation. The method has been verified by comparisons with new similarity solution results for wedges with deadrise angles varying from 4° to 81°. A simple asymptotic solution for small deadrise angles α based on Wagner (1932) agrees with the similarity solution for small α.

Incompressible water-entry problems at small deadrise angles

This paper summarizes and extends some mathematical results for a model for a class of water-entry problems characterized by the geometrical property that the impacting body is nearly parallel to the undisturbed water surface and that the impact is so rapid that gravity can be neglected. Explicit solutions for the pressure distributions are given in the case of two-dimensional flow and a variational formulation is described which provides a simple numerical algorithm for three-dimensional flows. We also pose some open questions concerning the well-posedness and physical relevance of the model for exit problems or when there is an air gap between the impacting body and the water.

Water Impact of Wedge model

Impact of a wedge body dropping on a water surface is analyzed by means of the boundary element method. Wagner's theory predicts maximum pressure in good agreement with experimental results in case of larger deadrise angle β>15°. In case of small deadrise angle, however, there is a large discrepancy between experiments and the theory, which gives an infinitely large pressure in the limiting case β→0°. In reality, pressure is reduced due to air cushioning effects. The present paper concerns with the behavior of the wedge of an intermediate deadrise angle, 1°≤β≤5°, and air cushioning effects are shown to be neglected for β>3°.In the present calculation, the water is idealized by boundary elements, and the instantaneous configuration of free surface and the wedge body are determined by considering the fluidsolid intersection. Numerical and experimental results show the variation of maximum pressure along the width depending on the dropping conditions, which is in contrast to Wagner's theory predicting uniform distribution.The relation between the maximum pressure caused by dropping at constant speed and that by dropping freely is also shown as a function of wetted width and impact velocity.In order to improve accuracy of numerical results, the splash condition should be included in the analysis, which will be left to further study.

Experiments on Slamming of Wedge-Shaped Bodies

A series of experimental investigations of rigid-body slamming was performed at the Naval Ship Research and Development Center by dropping one flat-bottom steel model and five wedge-shaped steel models with small deadrise angles (up to 15 deg) from various elevated positions above a calm water surface. The test results were used to provide a set of charts for estimating the maximum impact pressure due to rigid-body slamming of wedges.

Hydroelastic Effects in Ship Slamming

A theoretical investigation is presented on hydroelastic effects during the hydrodynamic impact of an idealized structure model of a ship. This is a summary of the author's paper, that will be published soon in Schiffstechnik. Based on H. Wagner's theory of hydrodynamic impact, the equations are formulated for the impact, including elastic deformations of the structure and their influence on the pressure field. This formulation holds for the 2-dimensional, vertical, symmetric impact of a structural model of small but nonzero deadrise angle. Compressibility effects are neglected. The general equations are then specified and evaluated numerically for a certain type of model. A normal-mode series expansion is used to calculate the motions of the bottom plates loaded by transient hydrodynamic pressures. The normal modes are those of the plates in vacuum. In order to obtain the pressures. the influence of deformations as well as of velocities and accelerations is considered. The pressures are obtained from a Fourier series involving Chebyshev polynomials. Special attention is given to convergence criteria of both series expansions. The original paper contains numerical results for five parameter combinations. Principal results of one of these examples are discussed.

THE MAXIMUM PRESSURES AND TOTAL FORCE ON STRAIGHT‐SIDED WEDGES WITH SMALL DEADRISE

SUMMARYA new theory for solving wedge‐entry problems is utilized to give an approximate solution for the pressures and forces acting on blunt (small deadrise angle) straight sided wedges. Closed form analytical expressions are given for the maximum pressures, pressure at any point and total vertical force acting on a blunt symmetrical wedge entering an infinite ocean at constant vertical velocity.