HUA Cheng, FANG Chao

Department of Mechanics and Engineering Science, Fudan University, Shanghai 200433, China, E-mai: huacheng@fudan.edu.cn

CHENG Jin

School of Mathematical Sciences, Fudan University, Shanghai 200433, China

SIMULATION OF FLUID-SOLID INTERACTION ON WATER DITCHING OF AN AIRPLANE BY ALE METHOD*

HUA Cheng, FANG Chao

Department of Mechanics and Engineering Science, Fudan University, Shanghai 200433, China, E-mai: huacheng@fudan.edu.cn

CHENG Jin

School of Mathematical Sciences, Fudan University, Shanghai 200433, China

Ditching is considered as one of the important aspects of safety performances of airplanes. It is related primarily with the fluid-solid interaction, whose studiesmainly depend on experiments at the present time. Numerical and analyticalmethods for fluid-solid interaction by using 3-D full scale airplane’smodel w ill reduce the dependence on the expensivemodel tests. Numerical studies can be used to estimate the safety of ditching and provide a reference for the crashworthiness design. This article proposes a 3-D dynamical structuralmodel after the real shape of an airplane and an Arbitrary Lagrange-Euler (ALE) fluid-fieldmodel, to simulate the fluid-solid interactions caused by low speed ditching. The simulation is based on interaction computationalmethods, within LS-DYNA nonlinear finite-element code. The results of pressure distributions and accelerating time histories of the airplane’s subfloor are discussed in the context of the safety of ditching, and the simulation results and the analyticalmethods are verified.

ditching, fluid structure interaction, Arbitrary Lagrange-Euler (ALE), finite elementmethod

Introduction

Ditchingmeans that a plane has no alternative but to land on water surface such as sea or lake, in view of the safety of crews and passengers. There are some cases of successful ditching, also some cases of crashes. Airline companies have strict rules for ditching[1]. According to structural dynam ics, an airplane should be in very low speed when ditching, the deformation of an airplane can be approximately considered as elastic-plastic, and it is feasible to simulate an airplane’s ditching by using numerical analysismethods such as finite element analysis.

Unlike a ground impact, a large area of the airplane’s outer skin contacts with water in ditching. Thus, the fluid-solid interaction and some othermechanical problems should be primarily addressed. The problem of water entry and impact can be studied theoretically, experimentally or numerically. Von Karman (1929) developed the first theory for this problem. This pioneering work uses the concept of addedmass for investigation of impact loads on a seaplane during ditching. Themajority of the subsequent theoretical work in the early period was based on Von Karman’s concept. The experimental techniques related with the problem weremainly developed by NASA since the late 1950s. Boeing airplanes were tested and the operating specifications of ditching were developed at that time. Recently, the numerical analysis is w idely used in this respect.

Seddon and Moatamedi[2]reviewed the studies of water entry and impact between 1929 and 2003, and it is pointed out that the numerical analysis should be considered as amain choice, and accurate numericalmodels of fluid-solid interactionmay replace expensive and time-consuming full-scaled tests. Brooks and Anderson[3]first used the finite element software, LSDYNA, to investigate the water entry of spacecraft. The simulation results were validated by a comparison with the full-scaled test data, but the response after 30ms-40ms is not very satisfactory because the field of fluid was calculated as a kind of solids in this soft-ware. Later, with the development of Arbitrary Lagrange-Euler (ALE)[4]finite elementmethod, Tutt and Taylor[5]used an updated software to simulate the water landing characteristics of space vehicles, where the fluid-field was defined by ALE elements. A comparison between test data and numerical analysis shows the value of using the finite elementmethods for simulating the water entry and impact. Furthermore, numerical analysis of water impact was used for optimal designs. Li et al.[6]used a cylindermodel and a similarmethod, to simulate the shell structure dropping into water, and a comparison between simulation and experiment wasmade. Based on a numericalmodel, they optim ized thematerial and the structure to enhance the safety of the structure and to save the experiment time. Sun et al.[7]used a V-shaped plate in the experiment to simulate a 2-D elastic wedge, the response of the structures in the water entry process was dynamicallymeasured, and the influences of some parameters were analyzed. Gong et al.[8]studied the water entry of a wedge by the SPHmethod, and with an improved non-reflection boundary treatment. Wei et al.[9]studied the high-speed water entry impact of an underwater vehicle by the ALEmethod using MSC Dytran software, where the underwater vehicle was treated as a rigid body, and different impact conditions were considered.

Ditching problems were not well studied by using the 3-D full-scaled numericalmodel up to now. However, it is important to study the airplane’s crashworthiness by using actual and accuratemodels. In this article, a 3-D numericalmodel with the shape of a real-scaled airplane is built. Simulations of the ditching under related conditions are conducted using the finite elementmethod, including the ALE and interactive computational approaches. The pressure applied to the subfloor, and the acceleration of the passenger cabin are obtained in the simulations. Based on these results, the safety of the low speed ditching is discussed, and the validity of thismodel and themethods are also verified.

1. Computation scheme

1.1 ALE formulation

The ALE description allows an arbitrarymovement of the reference domain, as compared with thematerial description or the spatial description. The large deformation can thus easily described, together with themoving boundary of fluid[10]. So ALE description is w idely used for fluid-solid interaction problems. Hughes et al.[11]developed the ALE description for the finite elementmethod, and later, Souli et al.[10,12]added an ALE solver for the finite element software, LS-DYNA, with smoothing algorithms and advection processes. They also presented some numerical examples to show that the program worked well. Karen and Yvonne[13]used this program to analyze the vertical drop of a fuselage section, and the simulation results were compared with experimental tests, with a good agreement.

In this article, we use this ALE solver in LSDYNA for the part of the fluid-field.

In the ALE description, thematerial derivative can be described as[14]

where χ is the ALE coordinate, X is the Lagrangian coordinate, x is the Eulerian coordinate.

The relative velocity ci= ui?viis introduced to simplify the equation, where uiis the velocity of thematerial and viis the velocity of the referential coordinate. Thus when χi= Xior χi= xi, the ALE description becomes the Lagrange description or the Euler description.

The governing equations are as follows:

(1) The equation for conservation ofmass

where ρ is the density of fluid

(2) Themotion equation or the Navier-Stokes equation in hydrodynam ics

whereib is the unit body force.

The stress tensor σijin a New tonian fluid is related with velocities as

where μ is the coefficient of kinematic viscosity, p is the pressure.

The equations are to be solved with the follow ing boundary conditions:where niis the outward unit normal vector on the traction free boundary1Γ, and u0iis the velocity on the constrained boundary2Γ.

The ALE equations are implemented by two phases, using the alternative approach:

(1) First is a Lagrangian phase, in which themeshmoves with thematerial, to calculate the velocity changes due to the internal and external forces. The equilibrium equation is

(2) Second, the advection phase, to transport themass and themomentum:

(a) to decide which nodes tomove,

(b) tomove the boundary nodes,

(c) tomove the interior nodes,

(d) to calculate the transport of the element-centered variables,

(e) to calculate themomentum transport and update the velocity.

1.2 Coupling algorithm

In this article, we use a penalty-based coupling algorithm, in the LS-DYNA ALE package. The interaction of the fluid and the airplane’s structure involves three key aspects[15]:

(1) The construction of the coupling interface on the Lagrange structure side.

(2) The construction of the coupling interface on the ALE fluid side.

(3) The detection of penetrations between two interfaces and the separation by using penalty forces.

The program defines the coupling segments in the Lagrange coupling surface, and if the penetrations between the structure and the fluid aremeasured, then sampling points are generated in the coupling surface. The ALE fluid coupling surface is determ ined by tracking the volume fraction interface of thematerial, the interface is reconstructed after each advection step of the ALE. A fter the coupling interfaces are located, the relative displacement dibetween two coupling interfaces ismeasured at each coupling point, the penalty forces are determ ined by the penalty spring stiffness kiof the Lagrange structure side

Penalty forces are applied on coupled points tomake themmove apart. It can be visualized as a spring added between the sampling points of the structure and the fluid to avoid their penetration.

The whole algorithm flow chart is shown in Fig.1.

Fig.1 A lgorithm flow chart

2. Modeling of ditching

2.1 The finite elementmodel

Our numericalmodel involves the follow ing considerations:

(1) A full-scaled airplane is used in the numerical simulationmodel, based on the Boeing plane with some important simplifications, ignoring the effects of the shape of engines. The dimensions of the plane’smodel are shown in Table 1.

Table 1 Dimensions of airp lanemodel

Fig.2 A finite-element-model of one half of the airplane

A finite elementmodel of one half of the airplane is established as in Fig.2. It consists of the airframe,the w ing, the horizontal stabilizer and the vertical stabilizer.

(2) Structure parts of the airplane such as the skin and the frame aremodeled with shell elements.

Due to the fact that the ditching is in a low speed, we ignore the plastic deformation. In the elastic range, the alum inum alloy’smaterial parameters are shown in Table 2.

Table 2 Material of airp lane

Themass of the inner structure, the airborne goods and persons are converted into themass of the airplane’s outer skin, the thickness of shell elements is assumed to be 0.001m and severalmass elements are added in the subfloor. Alsomass elements are added for engine’s weight where the engine is located, 5 t on both w ings.

In view of the effects of the water wave in ditching, we use a static watermodel. The region of ALE elements consists of a rectangle tank of water with a rectangle volume of air above the water. Gruneisen state equation is defined for the watermaterial governed by the relation p=ρ0C2λ (where λ=ρ/ ρ0?1, ρ0is the initial density, ρ is the density and C is the speed of sound in water), as is detailed in Table 3.

Table 3 Material of water

Fig.3 Computationalmodel

The whole finite elementmodel is shown in Fig.3. The airplane is represented by 1 671 elements and 1 161 nodes, and the fluid part is described by 75 000 elements and 81 932 nodes.

2.2 Initial and simulation settings

For low speed ditching (when the plane’s weight approaches to the lowest, the attack angle is between 9o-12o, the descending speed should as slow as possible, along the longitudinal wave[1]), we set the initial conditions as shown in Table 4. W ith the attack angle and the speed indicated there, the plane starts at the initial height and ditches into the static water. In the structure domain, the gravity is considered; and in the water domain, the viscosity is considered.

Table 4 Initial conditions

Table 5 Parameters for solution

Boundary conditions and some simulation parameters are as shown in Table 5. Themodel is solved by LS-DYNA.

3. The results

3.1 Landing scenario

Our simulation predicts a particular landing scenario, as shown in Fig.4. It is shown that the scenario agrees with the conditions of the actual low speed ditching, whichmeans that during the simulation time, the airplane does not jump or roll. Fig.4(d) shows that after 1.3 s, the airplane starts to contact with water globally, and in Fig.4 (f),most parts of the airplane are still beyond water.

3.2 Pressure time history

We trace three points on the airplane, in the front of passenger cabin, in them iddle of the plane near the center ofmass, and in the rear which contacts with water firstly. The pressure and the velocity and acceleration time histories are recorded to show the conditions of the whole airplane through these three points. The exact locations are shown in Fig.5.

To find out if the outer skin can suffer the impact force, we present the pressure time history of the three points, as shown in Fig.6

From this figure, it can be seen that the point ofthe peak pressure starts at the rear thenmoves to the front. It can also be seen that the pressure reaches a peak when this area impacts with water. Moreover, as the w ing joins the frame, there are stress concentrations in them iddle of the plane, and a great vibration would occur in them iddle point. A ll the values of pressures on the subfloor are within themargin of safety.

Fig.4 Ditching history

Fig.5 Location of sampling points on plane

Fig.6 Pressure time history graph

3.3 Acceleration time history

For protecting passengers and testing if people can endure the overload, we trace the vertical accelerations of the three same points.

Fig.7 Velocity time history graph

Fig.8 Acceleration time history graph

Because we solve the equations with the displacement of nodes as the basic variable, the acceleration obtained by directly differentiating the velocity can not be very accurate. Because of the continuity of the velocity, we can smooth the velocity as shown in Fig.7 first before the differentiation. Thus we obtainthe three acceleration time histories in Fig.8.

In overall, the rear part impacts water first, then them iddle part and the front part. When the rear part impacts water, the head of the airplane is still 8m above the water. The head of the airplane drops very fast because of the gravity and the torsionmoment, and it impacts harder than the rear part. Whenmost areas of the airplane impact water, the acceleration reaches a peak. Consequently, themaximum value of subfloor’s acceleration is not too high, and people can endure it in a short time.

The real ditching can also be affected by the attack angle, the velocity attitude, etc.. When the velocity increases, the responsemay be similar as before, but with a higher peak. In that situation, the accelerationmay be beyond the tolerance of human being, and the pressures can be beyond the strength lim it of thematerial to cause a large deformation, or even a breakdown. The cabin doormay not be open, the cracks develop, and the whole airplanemay even be teared apart.

4. Concluding remarks

A 3-D full-scaled airplane’s numericalmodel is built in this study, and then the fluid-solid interaction simulation of low speed ditching is carried out using the ALE finite element analysis. Themajor results of this study are:

(1) Dangerous situations like jumping and secondary collision do not occur during the ditching simulation. The results of the numerical simulation agree with the floating features of the actual ditching. It is shown that themodel andmethods in this paper are valid.

(2) From the pressure time history on the airplane’s subfloor, the high pressures appear first at the rear, and then at the front. The pressures reach a peak when these areas impact with water. Themaximum pressures is about 20 MPa,much lower than the strength limit of the alum inum alloy.

(3) Accelerations of different passenger cabins are discussed. The front part’s acceleration is greater than the rear part’s. Themaximum peak acceleration is below 6 g in 200ms, a very short period.

(4) From the acceleration time history, it is shown that once all areas of the airplane’s subfloor impact water, the acceleration of the whole plane reaches a peak.

As a real airplane has a complicated structure, it is somehow simplified by themodel, and it should be further refined, by adding some structure units like boards, beams, frames, etc., to get better results Also enginemodels can be taken into the simulation. Besides, the wave of water is important to the ditching. These improvements w ill be implemented in a future study.

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March 21, 2011, Revised May 8, 2011)

10.1016/S1001-6058(10)60159-X

* Project supported by the Shanghai Key Basic Research Program of China (Grant No. 07JC14001).

Biography: HUA Cheng (1963-), Male, Ph. D., Associate Professor