numerical simulation of cushioning problem for blunt ...

POLISH MARITIME RESEARCH Special Issue 2018 S1 (97) 2018 Vol. 25; pp. 85-93 10.2478/pomr-2018-0028

NUMERICAL SIMULATION OF CUSHIONING PROBLEM FOR BLUNT BODIES USING BOUNDARY ELEMENT METHOD

Mojtaba Barjasteh Hamid Zeraatgar Amirkabir University of Technology, Tehran, Iran

ABSTRACT Induced air pressure and resulting free surface profile due to air cushioning layer is studied. The study is mainly focused on 2D blunt circular bodies with constant downward speed. The problem is first solved for the air flow between the body and the free surface of the water. Then the results are employed to solve the problem for the water problem, numerically. Both air and water problem are assumed to be governed by Laplace potential equation. Depending on the induced pressure and velocity of the escaping air flow from cushioning layer, compressibility of the air is also included in the modeling. Gravitational acceleration is also included in the model. An iterative boundary element method is used for numerical solution of both air and water problems. Instantaneous pressure distribution and free surface profile are evaluated for different bodies. The results of calculation for large blunt bodies show that inviscid potential method can fairly approximate the problem for large blunt bodies. Additionally, the behavior of the air pressure for the very blunt body is impulsive and the magnitude of the peak pressure is in order of impact pressure of water entry. The obtained results are compared with analytical method. The comparison shows that as the bluntness of a body increases, the better agreement is concluded.

Keywords: Air cushion, compressible potential flow, water entry, slamming

INTRODUCTION Hydrodynamic evaluation of several typesof marine vehicles are based onsolution of water entry problem. It is common to study a 3D geometry of a vessel by 2D sections along ship length. A 2D section of a ship is a classical water entry problem. The water entry problem may be solved analytically, numerically as well as by experiment. The water entry problem is correlated to some other phenomena which complicates the evaluation of the problem. One of these phenomena is the cushioning layer right before impact. As a body drops and get close to free surface of the water, air between body and the water surface may not scape and compressed as body closes to the surface. An air layer is

formed between the body and free surface of the water. Based on the mass continuity, the velocity of escaping air flow from the layer exponentially increases as the body closes the surface. Since the gap layer is usually very narrow and the body momentum is generally high, the induced air pressure along the cushioning layeris very high. Physically, for the force balancein vertical direction, the free surface is deformed to balance the induced air pressure at the air-water interface. This deformation causes a multipoint impact rather than asingle point impact as it is assumed in most of water entry theories. Depending on the shape of impacting body and deformed free surface profile, some volume of air may be entrapped during entry. The cushioning layer reduces the impact speed as well as the hydrodynamic impact pressure POLISH MARITIME RESEARCH, No S1/2018 Unauthenticated Download Date | 6/10/18 3:42 AM

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due to air drag and compression of entrapped air, respectively. Thus accurate prediction of maximum impact pressure requires considering the cushioning layer particularly for high speed impact of blunt bodies. There are several researches on the topic mainly focused on the evaluation of impact pressure and free surface deformation in bubble dynamics and droplet impacts. The survey was started by Verhagen[1] who modeled the air layer using a channel flow problem with constant geometry of moving walls. He concluded that the lubrication equation[2] may properly simulate the channel flow instead of traditional equations. The proposed model was just a case study and cannot be further developed for different geometries. The next significant try was made by Asryan[3]. He recommended two different formulation including inviscid incompressible and viscous compressible. Both models are available for initial stage of the body motion when the surface deformation isnegligible. Thus, the free surface deformation was not investigated. The obtained results overestimate the pressure and underestimate the air speed particularly at throat of the air gap layer. Wilson [4] suggested a non-dimensional asymptotic model to simulate the problem based on two dimensional Navier-Stokes equation. The model is initially stable but as the body gets closer to the free surface the results are unstable. This instability also occurred for blunter bodies. Hicks and Purvis [5] conducted a great series of experiments and analytical research on the problem. They studied the problem in all details primarily for a bubble impact onto liquid or solids and its resulting deformation due to presence of cushioning layer. Different parameters such as viscosity dissipation, compressibility, topography and the validity of the proposed models are fully discussed in their works. They employed different forms of lubrication equation for the air problem and potential flow for the bubble deformation. They validated the obtained results with some experimental research [8] and with different available theories. All studies show that the dimensional analysis generally controls the problem and its corresponding governing equation. Different research were also made on this analysis to provide some better understanding of the problem such as those carried out by Mandre and Brenner [9] and Smith et. all[10]. They introduced different time and length scales based on the different configuration of the problem. On that basis, they concluded which governing equations such as lubrication equation, incompressible potential flow and compressible flow are applicable. Some experimental studies also investigated different applications of cushioning problem. Two distinctive experimental researches are those carried out by Marston et. all[11] and Tran et. all[12]. Both studies were concentrated on cushioning layer and the resulting air entrapment for small bodies. Some unique visualization and precise measurement were reported. One of the interesting tests results is to determine a scaling law for non-dimensional volume of air entrapment. This scaling simply implies that the volume of air entrapment depends to the Stokes number of the flow. This relation is changed for different regimes of the flows. There are also several research to investigate the 86

problem such as those conducted by Lewison and Maclean [13], Thoroddsen et. all[14] and Bouwhuis et. all[15] which have different contribution to the problem consideration. This study, as a relatively simple approach, represents a fully potential flow for both air and water problem and disregards the lubrication equation for very narrow air layer. Therefore the validity of the formulation is limited to high speed impact of blunt bodies. A fast and practical boundary element method is employed to evaluate all desired parameters of the problem. The proposed method can be efficiently used for developing a commercial software in full prediction of water impact including air cushioning effects.

FORMULATION The present model investigates the cushioning problem in water entry of blunt body such as depicted in Figure 1. A blunt body with downward speed of v0 moves toward to the water free surface which is initially at the rest. The downward speed is assumed to be constant and vertical, for the sake of simplicity. Since the body is essentially blunt and its speed is assumed to be high enough, the dominant phenomenonforfree surface deformation is normal pressure balance. Therefore the viscosity is neglected for both air and water problem. Two length scales are defined here to check the model validity as depicted in Figure 1. The first one in horizontal direction and shows the maximum length of disturbed free surface which is called L. The second length scale indicates the maximum depth of free surface, H, due to air pressure. The present formulation is not valid when H