CRANFIELD UNIVERSITY

Martin Agar

Simulation of Hydrodynamic Ram during impact of fluid-filled tank using SPH

SCHOOL OF ENGINEERING ADVANCED LIGHTWEIGHT STRUCTURES AND IMPACT

MSc Thesis Academic Year: 2010 - 2011

Supervisor: Rade Vignjevic September 2011

CRANFIELD UNIVERSITY

SCHOOL OF ENGINEERING Advanced Lightweight Structures And Impact

MSc Thesis

Academic Year 2010 - 2011

MARTIN AGAR

Simulation of Hydrodynamic Ram during impact of fluid-filled tank using SPH

Supervisor: Rade Vignjevic

September 2011

This thesis is submitted in partial fulfilment of the requirements for the degree of MSc

Cranfield University 2011. All rights reserved. No part of this publication may be reproduced without the written permission of the

copyright owner.

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ABSTRACT

Hydrodynamic ram (HRAM) is a phenomenon occurring when a high-kinetic energy projectile penetrates a fluid-filled tank. During HRAM, when the projectile comes through the tank, it transfers its kinetic energy to the fluid creating an extra pressure. This pressure can produce catastrophic effects on the tank. This is of important issue in the design of wing fuel tanks for aircraft since it has been recognized as one of the important factors in aircraft vulnerability. In this thesis, the LSDYNA FE/SPH code has been used to simulate an HRAM event created by a spherical projectile impacting a water filled tank. The SPH formulation is employed to modelling the fluid during this. The simulation is based on an experimental test done previously (consisting in a massive tank impacted by a spherical projectile at 341m/s). In this experiment, pressure transducers located at different points of the fluid has record the pressure evolution for understanding pressure transfer mechanisms during the HRAM. Those transfer mechanisms are reproduced numerically and compared with experimental results in order to assess accuracy of the SPH technique in reproducing such a complex phenomenon. Results show that SPH has the potential for simulating this event. However some important limitations exist.

Keywords:

Hydrodynamic Ram; Impact; Tank; Fluid-Structure Interaction; SPH; Cavitation; Super-Cavitation

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ACKNOWLEDGEMENTS

I would like first to offer thanks to my supervisor, Pr. Rade Vignjevic for his guidance and his advises for defining clearly my project and for the autonomy given during the thesis.

I would like to express my gratitude to Dr. Kevin Hughes, Dr. Tom de Vuyst and Dr. James Campbell, who always managed to find time for helping me and providing me precious advice.

I would also like to thanks the Caf Comet and its lovely ladies for the marvellous chocolate muffins provided.

I would at last like to thank the ALSI students, especially Laura Garnier, for the great year spent here and all the DOD games played during those months.

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TABLE OF CONTENTS

ABSTRACT ......................................................................................................... i ACKNOWLEDGEMENTS.................................................................................... ii LIST OF FIGURES ............................................................................................. v LIST OF TABLES ............................................................................................... vi 1 INTRODUCTION ......................................................................................... 7

1.1 Project background ............................................................................... 7 1.2 Thesis objectives ................................................................................... 8

2 THE HYDRODYNAMIC RAM PHENOMENON ........................................... 9 2.1 Description of the phenomenon ............................................................ 9

2.1.1 The shock phase ............................................................................ 9 2.1.2 The drag phase............................................................................. 11 2.1.3 The cavitation phase ..................................................................... 12 2.1.4 The exit phase .............................................................................. 14

2.2 Consequences .................................................................................... 14 3 RESEARCH DONE PREVIOUSLY ........................................................... 16

3.1 Thesis-based Experiment .................................................................... 16 3.2 History of the simulations of hydrodynamic ram .................................. 18 3.3 Recent simulations done ..................................................................... 20 3.4 Mitigation System ................................................................................ 22

4 METHODOLOGY BACKGROUND ........................................................... 24 4.1 Lagrangian Method ............................................................................. 24 4.2 Eulerian Method .................................................................................. 25 4.3 ALE Method ........................................................................................ 26 4.4 SPH Method ........................................................................................ 26

4.4.1 Integral interpolants ...................................................................... 27 4.4.2 Kernel function .............................................................................. 29 4.4.3 Equations of motion ...................................................................... 30

5 MODEL DEVELOPMENT ......................................................................... 32 5.1 Model size reduction ........................................................................... 32

5.1.1 Symmetry conditions .................................................................... 33 5.1.2 Ghost particles .............................................................................. 34

5.2 Projectile modelling ............................................................................. 35 5.3 Tank modelling .................................................................................... 35 5.4 Fluid modelling .................................................................................... 40

5.4.1 Particle density sensitivity analysis ............................................... 41 5.4.2 Water equation of state ................................................................. 43 5.4.3 Silent boundaries .......................................................................... 45

5.5 Parts interaction .................................................................................. 46 5.5.1 Projectile/Tank interface ............................................................... 47 5.5.2 Projectile/Fluid interface ............................................................... 49 5.5.3 Tank/Fluid interface ...................................................................... 50

5.6 Pressure initialisation .......................................................................... 52 5.7 Pressure measurement ....................................................................... 53 5.8 Final model .......................................................................................... 54

6 RESULTS .................................................................................................. 57

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6.1 Initial problems .................................................................................... 57 6.1.1 Distributed memory computer ....................................................... 57 6.1.2 Double precision solver ................................................................ 58

6.2 Shock phase ....................................................................................... 59 6.3 Drag phase .......................................................................................... 66 6.4 Cavitation phase ................................................................................. 69 6.5 Exit phase ........................................................................................... 73

7 CONCLUSION .......................................................................................... 75 8 FURTHER WORKS................................................................................... 76 REFERENCES ................................................................................................. 77 APPENDICES .................................................................................................. 83

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LIST OF FIGURES

Figure 1 - Phases of the HRAM [1] ..................................................................... 9 Figure 2 - Pressure transducers [7] .................................................................. 16 Figure 3 - Experimental Arrangement [7] ......................................................... 16 Figure 4 - Pressure History [7] .......................................................................... 17 Figure 5 - Cavity collapse steps [7] .................................................................. 18 Figure 6 - Cavity creation in D. Varas et al. [41] ............................................... 21 Figure 7 - Mitigation system for HRAM [10] ...................................................... 23 Figure 8 - Example of a Lagrangian mesh [24]................................................. 24 Figure 9 - Example of an Eulerian mesh [24] ................................................... 25 Figure 10 - Neighbouring particles including by the kernel function [44] .......... 28 Figure 11 - Model dimension justification ......................................................... 33 Figure 12 - Symmetry Conditions ..................................................................... 33 Figure 13 - Ghost particles reflecting real particles [22] ................................... 34 Figure 14 - Projectile Mesh............................................................................... 35 Figure 15 - Two possible tank configurations ................................................... 36 Figure 16 - Different mesh sizes tested ............................................................ 38 Figure 17 - Impact Holes .................................................................................. 39 Figure 18 - Resulting velocity after impact ....................................................... 39 Figure 19 - Boundary conditions of the tank walls ............................................ 40 Figure 20 - Model with 3mm particle density .................................................... 41 Figure 21 - Projectile velocity in function of time for different particle density .. 42 Figure 22 Example of shock front velocity measurement .............................. 44 Figure 23 - Silent boundary creation ................................................................ 46 Figure 24 - Projectile/tank interface study ........................................................ 47 Figure 25 - Contact penetration ........................................................................ 48 Figure 26 - Good contact behaviour ................................................................. 48 Figure 27 - Projectile/fluid interface study ........................................................ 49 Figure 28 - Particle penetration problem .......................................................... 50 Figure 29 - Projectile/fluid interface study ........................................................ 51 Figure 30 - Offset between the tank and the fluid particles .............................. 51 Figure 31 - Pressure initialisation ..................................................................... 53 Figure 32 - Coordinate system ......................................................................... 54 Figure 33 - Small model ................................................................................... 55 Figure 34 - Big model ....................................................................................... 55 Figure 35 - Post-processing problem ............................................................... 58 Figure 36 - Projectile impact with the tank at t=0.03ms .................................... 59 Figure 37 - Von Mises stress generated by the impact on an empty tank at

0.023ms (MPa) .......................................................................................... 60 Figure 38 - Pressure wave generated at 0.06ms .............................................. 60 Figure 39 - Von Mises stress generated by the impact on a filled tank at

0.023ms (MPa) .......................................................................................... 61 Figure 40 - Drop in pressure measured along the shot line on the big model .. 62 Figure 41 - Initial wave pressure recorded by the transducer P1 in the

experiment ................................................................................................ 63 Figure 42 - Pressure history at P3 .................................................................... 63

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Figure 43 Decrease of pressure of the transducer P4, P5 and P7 compared with the transducer P3 .............................................................................. 64

Figure 44 Angle [11] ................................................................................... 65 Figure 45 - Projectile pressure field .................................................................. 66 Figure 46 - Decrease of the velocity due to drag forces ................................... 67 Figure 47 - Comparison of the drag pressures between the simulation and the

experiment ................................................................................................ 68 Figure 48 - Projectile position relative to the transducer P3 at 1.6ms ............... 69 Figure 49 - Cavity formation in the experiment [11] .......................................... 70 Figure 50 - Cavity formation at 1.6ms .............................................................. 71 Figure 51 - Cavity grow [11] ............................................................................. 71 Figure 52 - Cavity collapse pressure in the small model for a transducer located

at 150mm from the impact......................................................................... 72 Figure 53 - Cavity collapse pressure in the experiment .................................... 73 Figure 54 - Deflection of the tank before the impact ......................................... 74 Figure 55 - Pre-stress in the tank wall before impact ....................................... 74

LIST OF TABLES

Table 1 - Johnson-Cook parameters [8] ........................................................... 37 Table 2 - EOS Comparison .............................................................................. 45 Table 3 - EOS Parameters [41] ........................................................................ 45 Table 4 - Pressure transducer position ............................................................. 54 Table 5 - Model statistics .................................................................................. 56

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1 INTRODUCTION

1.1 Project background

Fuel tank in an aircraft has been identified as one of the important components in aircraft vulnerability [15; 41]. A failure of the fuel tank can result with catastrophic consequences for the aircraft (large fuel leakage, loss of the centre of gravity, fire hazard). This failure can occur due to impacts. During high velocity impacts with fluid-filled tank, it appears the problem of the Hydrodynamic Ram (HRAM) phenomenon. It increases the risk of catastrophic failure and the damages are more serious than impacting an empty tank.

During the HRAM, when the projectile comes through the tank, it transfers its kinetic energy to the fluid creating an extra pressure. This pressure can produce catastrophic effects on the tank.

The HRAM phenomenon was before more related to military aircrafts. Those aircrafts are more exposed to the threats capable to produce the HRAM (ballistic impact, missile fragment) [6]. But now its a serious issue concerning civil aviation. The Federal Aviation Administration established the analysis of the effects of turbine engine fragment impacting fuel tank as one its research area in 1990 [41]. In 2000, the BEA determined that the HRAM has played an important role in the dramatic crash of a Concorde during takeoff from Charles de Gaulle Airport [2].

The HRAM phenomenon is a complex fluid-structure interaction which encloses physical phenomena like high velocity impact, fluid mechanics, large deformation and material failure [43]. Its difficult to characterise this phenomenon analytically so most of the research are experimental results and numerical simulation. Simulations of HRAM using numerical methods have been tried for almost 40 years [15]. With the latest numerical methods, good correlation between the experimental and numerical results starts to appear [41]. However its still an ongoing area of research because the simulations require high computational resources and there are still some limitations for

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modelling failure and cracks [35]. Arbitrary Lagrangian-Eulerian (ALE) and SPH methods have shown the best results.

1.2 Thesis objectives

Simulating numerically the HRAM phenomenon is not an easy thing to do. A lot of physical phenomena are involved during this event. Research to simulate it started 35 years ago, but due to computer limitation, results werent so accurate. Recently with the development of new methods, some studies have shown good results in reproducing experimental tests [41; 43].

The objective of the thesis is to assess the capability of the coupled Finite Element/SPH code implanted in LS-DYNA to represent accurately the HRAM phenomenon. Recently, Disimile et al [11] did an experimental study of the hydrodynamic ram impacting a fuel tank with a high velocity. In their paper, a lot of data is provided about the results obtained. The final aim of the thesis is to be able to reproduce numerically this experiment and to have a good fitting between results in term of pressure history and cavity dimension.

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2 THE HYDRODYNAMIC RAM PHENOMENON

2.1 Description of the phenomenon

As said before, the HRAM effect corresponds to the action of a projectile impacting a tank and transferring kinetic energy to the fluid creating an extra pressure. The HRAM phenomenon is a complex problem, difficult to predict. Indeed, the pressure in the fluid caused by the projectile creates a displacement of the wall. In return this displacement modifies the pressure of the fluid in the tank [3]. This complex interaction is call as the fluid-structure interaction. During an HRAM, 3 interactions are presents and have to be taken into account: the interaction between the projectile and the structure, between the projectile and the fluid and between the fluid and the structure [15]. The HRAM is divided in four phases: a shock phase, a drag phase, a cavitation phase and an exit phase as shown in Figure 1.

Figure 1 - Phases of the HRAM [1]

2.1.1 The shock phase

In his paper [7], J.P. Borg et al. show that the catastrophic failure or not of the tank is highly dependent of the nominal pre-stress. If the difference is very high between the internal and external pressure, a projectile with a low kinetic

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energy will be able to cause a catastrophic failure. But for a fuel tank, the internal pressure is due to the hydrostatic pressure so the nominal pre-stress is very low before the impact (this is not true for the impact with the exit wall as it will be explained later) and so the catastrophic failure will not be caused by the impact with the wall but by the wave created in the fluid.

Indeed, after the projectile goes through the tank, it penetrates the fluid accelerating the fluid at the impact point. The fluid initially with no velocity has now the same velocity than the projectile. This sudden acceleration creates a high pressure peak travelling through a shock wave. This pressure generates a stress in the entry wall. This stress coupled with the dynamic stress can result with a catastrophic failure of the tank because the expansion of the shock wave can produce a petalling of the entrance panel.

The shock wave is hemispherical expending from the impact point and its velocity is greater than the speed of sound in the fluid [23]. Townsend et al. [39] tried to characterize the shock wave dynamic, providing an analytical model using the Hugoniot-Rankine relations. The velocity of the shock front is given by:

= + : Velocity of the shock front : Projectile velocity after water impact : Sound speed in the fluid : Hugoniot slope coefficient of fluid And the shock pressure P is equal to:

= Where is the density of the fluid.

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If the peak of pressure due to shock can be very high, its very short in time because this pressure will be dissipated very quickly in the fluid as shown experimentally by Stepka et al. in [38]. So, during the shock phase, only the entry panel will be affected, the pressure field will be very low close to the exit and lateral panels.

2.1.2 The drag phase

The next phase in the HRAM is the drag phase. Now the projectile has passed the impact point and it is moving into the fluid. This displacement created a drag force exerted by the fluid on the projectile, and so the projectile transfers its momentum to the fluid. This force created is proportional the square of the projectile velocity and it is equal to:

= 12 Where is the fluid density, is the drag coefficient of the projectile, A is the reference area and is the projectile velocity after water impact. Stepka et al. decided in [38] to use the Newtons second law to determine the velocity of the fluid. So:

() = 12 Where is the mass of the projectile. With this equation they assumed that the projectile isnt deformed, the drag coefficient stays constant and the fluid is incompressible. Then they rearranged terms, integrated and solve this equation. They end with the projectile velocity decay ratio for a spherical projectile:

= 11 + 34

This equation allows us to predict the velocity of the projectile during its displacement into the fluid knowing the initial velocity after impact.

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The weak created by hydrodynamic friction will be responsible for the cavitation formation, an important phenomenon of the HRAM.

The pressure field created in front of the projectile can be a source of catastrophic failure especially in thin fuel tank as it has been show in [37].

2.1.3 The cavitation phase

Cavitation is an important property of the liquid. Liquid are not able to resist to a very low pressure. For example water is almost uncompressible and its density doesnt change a lot even with very high pressure. Nevertheless, when the pressure of the fluid drops under the vapour pressure of water (0.0023 MPa at 20 [14]), some bubbles and cavitations filled with vapour appear in the fluid. The liquid changes phase to become a gaz. This phenomenon is involved in the cavitation phase of the hydrodynamic ram.

During the drag phase, the projectile is moving into the fluid. Due to the drag a difference of pressure exist between the front of the projectile (static pressure) and the projectile wake. In the projectile wake the pressure drop under the vapour pressure and so a cavity starts to appear.

The initial shape of this cavity is cylindrical but the cavity expends during the event to a spherical shape (Shi et al. [36] realised some optical observation of the cavitation phenomenon). The cavity will reach its maximum radius and then collapse. In this case, when the cavity dimensions exceed the projectile ones, some authors prefer to use the term of supercavitation.

Bachelor [5] explains the process in his book and defines a non-dimensional number to characterise the cavity:

= 12 Where: is the static pressure and is the cavity pressure.

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He says that the smaller the K coefficient is, the bigger the cavity will be. So if the projectile is travelling at a high velocity, the cavity will be very big. Also, deeper the projectile is travelling smaller will be the cavity (the static pressure is increasing with the depth).

According to him, a supercavitation (so a small K parameter) has an influence on the drag coefficient of the projectile. He found this linear relationship for the new drag coefficient:

() = (1 + ) Where is the drag coefficient without cavitation. As said before, if the projectile has a high velocity, the cavity size grows. K. Reichardt, cited by N. Lecysyn [23] found a relation to calculate the maximal cavity diameter in function of the cavity parameter K:

!"# = $ ()(1 0.132.') Where is the projectile diameter. Some authors have try to determinate the radial growth velocity of the cavity. N. Lecysyn et al. [23] rearranging relations found by M. Held in [18], gives us a analytical relation of the radial growth velocity of the cavity without knowing the K parameter. Using the Bernouilli relation, assuming that the radial growth velocity is proportional to the projectile velocity, we have:

) * = $

) (1 + +) 2,

= 4 1 + . + = 34

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Where ) is the cavity diameter, t is the time after impact, h is the depth at the shot level axis, andis the density of the water and the projectile respectively.

All those analytical models presented dont take into account the fact that the pressure in the cavity is inferior to the atmospheric pressure. So the air present outside will be draw in the cavity through the hole created by the impact.

However it can be interesting to assess the results which will be obtained with the numerical simulation with those equations.

Once the cavity reaches its maximal radius, it collapses. In this case of a supercavitation, the collapse is particularly violent. P.J. Disimile et al. [11] describe this as: when the ends of the cavity collapse and meet with the collapsing interface, a shock wave radiates outwards. This pressure emitted by the collapse is very high, more the one emit during the shock phase but very short in time [17]. During the collapse of the cavity, the air inside will be compress and several oscillations will occur but with a decreasing intensity.

2.1.4 The exit phase

This phase correspond to the exit of the projectile from the tank. This phase can occur before or after the cavity collapse, it depends of the tank size. This phase may also not happen if the projectile looses too much kinetic energy due to the drag force. This phase is different from the shock phase. When the projectile impact the entry wall in the shock phase, there is no pre-stress due to the fluid, here the exit wall is pre-stress due to the fluid [15]. Due to the fluid, the high pressure area on the wall is much larger than the projectile profile and so loads the structure over a large region.

2.2 Consequences

According to J.P. Borg [7], NASA and a lot of authors agreed on a damage criterion for fuel tank stated as follows: "The criterion for catastrophic failure of

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a pressure vessel subjected to hypervelocity impact is met if at least one axial crack exists that is equal or larger than the vessel's axial length."

One of the most dramatic consequences of the hydrodynamic ram effect is the catastrophic failure of the fuel tank. Without the hydrodynamic ram effect, a small projectile impacting a fuel tank would have created a small hole resulting with a small leakage. But due to the HRAM, even a small projectile can create a very important crack. This crack creates then an important leakage and the fuel flowing out of the tank can arrive on an ignition surface [33]. Moreover the loss of fuel can also create a loss of centre of gravity.

This catastrophic failure can occurs during any phases of the HRAM. I can be due to an impact with the entry or exit panel or during the cavity collapse.

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3 RESEARCH DONE PREVIOUSLY

3.1 Thesis-based Experiment

The simulations which will be carried out during the thesis are based on an experimental study made by Disimile et al. [11]. This experiment had for goal to show and give a clear explanation of the pressure transfer mechanism during the HRAM.

For this experiment, they created a generic box representing a fuel tank into which spherical projectile was fired using a gas gun. The tank was able to contain 3785.4 L of water with dimensions of 1.168m from the front wall to the back wall and 1.829m from side to side and in height. Those dimensions are not representative of a real fuel tank but they are made in order to observe more easily the HRAM. The walls are made with 9.53 mm thick ASTM A-36 steel plates and 1.587 mm thick 2024-T3 aluminium at the impact location. The experimental arrangement is showed in Figure 3. Seven pressure transducers were implanted inside the tank to measure the pressure during the impact. The location of those transducers is shown in Figure 2.

Figure 3 - Experimental Arrangement [7] Figure 2 - Pressure transducers [7]

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In the study, three projectiles were fired through the tank, one in steel, one in aluminium and one in tungsten. Regarding the data provided in the paper, I will reproduce the one realized using a tungsten projectile. It was fired at a velocity of 341 17 m/s. The data provided are the pressure history as shown in Figure 4 and also the cavity collapse video as shown in Figure 5.

Figure 4 - Pressure History [7]

In Figure 4, we can clearly show the different phases of the hydrodynamic ram describe in a previous part. Here the cavity collapse represents the highest pressure in the tank but only during a very short time. The aim of the numerical model is to be able to reproduce those results.

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Figure 5 - Cavity collapse steps [7]

3.2 History of the simulations of hydrodynamic ram

The hydrodynamic ram has been studied for a long time. Since computational method appeared some researchers have try to represent it numerically. In [15], C.J. Freitas give us a good review of the different computational simulation of HRAM used in the past:

The first HRAM simulation was made by Ball in 1972 [4]. The idea was to incorporate the piston theory method into a structural analysis code. But this code had some limits since he didnt simulate the plastic response. Then another code was developed based also on the Balls idea, the BR-1. It allowed the plastic deformation but when results were compared with experimental data, it didnt fit at all. Strain and deflection was greatly inferior to experimental data. So the code has been abandoned.

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Then the variable image methodology was developed in 1977 by Lundstrom [29]. In this method the pressure field is determined by a potential function where the effect of the projectile and the cavitations are described by a series of sources placed along the trajectory of the projectile. But C.J. Freitas states that this method wasnt a success because it can only take into account limited types of geometries, linear and 2D problems.

With the fail of the variable image methodology, some other codes have been developed. There was the Unimodal Hydraulic Ram Structural Response (UHRSR) program, developed in 1977. Then this code was coupled with the BR-1 to give the Hydraulic Ram Structural Response (HRSR) code. In addition of theses codes, a derivative of the UHRSR was also developed, the ERAM. It was the first one to take into account the cavitation in the fluid [15].

All those code werent very good for representing the fluid-structure coupling [15]. So, even if the HRAM phenomenon was well represented, the codes werent able to predict accurately the damage on the structure due to the HR event. Moreover the geometry had to be very simple. Its with the development of the finite element methods in the 80s that good simulation results started to appear.

In 1980, the EPIC-2 code (a Lagrangian finite element method) was used by Kimsey [20]. He tried to simulate the impact of a steel rod with a cylindrical fuel tank. The results were qualitatively quite good but no experimental data was available for assessing them quantitatively. The use of a Lagrangian method was suitable for this case because no big distortions occurred during the simulation. But this method is not the best for impact simulation.

Recently, the ALE and SPH methods shown that they can produce good results for impact problem and fluid-structure interaction. Thats why, nowadays, the community is always using those method for simulating the impact on a fuel tank.

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3.3 Recent simulations done

As said before, some simulations were done recently using ALE of SPH methods. Probably, the first one to use the SPH method for this problem was P.W Randles et al. in 1998 [34]. He produced some interesting results but without experimental data, it was difficult to determine the accuracy of the method.

However in 2002 and 2008, two simulations made were quite interesting, because the quality of the results was evaluated using experimental data. The first one was made by R. Vignjevic et al. in 2002 [43]. The purpose of this experiment was to the show that the SPH method was suitable for impact on a fuel tank. The tank and the fluid were made using SPH particles. Only the pressure in the fluid was measured during the simulation but the results were similar with the equivalent experiment. So this paper was a good start for showing that the SPH can be a useful tool for HRAM impacts.

Then a more complete study was made in 2008 by D. Varas et al [41]. They impacted liquid filled aluminium tube with a steel spherical sphere. They simulated it with two different formulations (SPH and ALE).Then experiments tests were made to study the pressure at different points of the fluid, also for studying the displacement of the wall and the cavity development. Then those results were compared to the numerical results. The purpose of this study was to assess the validity of ALE and SPH method for reproducing complex HRAM phenomenon.

The modelling of the box and the projectile was done using a Lagrangian FE model. Only the fluid was made with a SPH or ALE formulation. In the results it appears that the deformation of the tube was good represented by the two methods. The pressure was well predicted only for the shock phase, for the cavity collapse phase the data doesnt appear. For the cavity creation, it seems to have good results as shown in Figure 6.

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Figure 6 - Cavity creation in D. Varas et al. [41] When they compared the two methods, the results are slightly better with the SPH method. But this method is much more time consuming than the ALE method. In conclusion, it appears that the two methods seem suitable for representing the HRAM even if the cavity collapse is not yet perfectly represented.

In 2011, they published a similar study using partially fluid-filled tube [40].

More recently, two papers were published about the impact of composite fuel tank. In the first one [9], the authors simulated the impact on a composite and steel fuel tank and compared the results. It appears that the composite tank is easier to destroy than the steel tank. In [42], D. Varas et al. studied again the HRAM but this time in composite tube. The study was only experimental.

A Master thesis was done in Cranfield University by J. Foll in 2009 about this subject [13]. He used the Cranfield Universitys coupled FE-SPH code to simulate the impact. In his study the impacted wall wasnt represented. The

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results obtained were qualitatively good but it was difficult to assess them quantitatively.

3.4 Mitigation System

As the HRAM can have some catastrophic consequences for an airplane, some systems were developed to mitigate its effect. Moussa et al. presented in [33], two possible solutions:

The first one is a solution already used on military helicopter. I consist in the use of foam against the structural panels of the tank. Using this, the HRAM effect will be reduced by foam compression. However, the authors stated this protection is not enough. Nevertheless with recent research in foam, this technology can be better improved.

The second solution is the Nitrogen-Inerted Bladder developed by Boeing. This system consists in porous bladders disposed along the tank walls. In case of impact, nitrogen is pumped, inflating the bladder. Because the bladder is porous, nitrogen can ooze out and fill the ullage. This system is efficient against the HRAM since the bladder operates as a cushion during the impact. Its also a good way for preventing from fire or explosion since the nitrogen inerts the ullage.

Recently, an article has been published by Disimile et al. [10]. Its about the mitigation of the shock waves during an HRAM event. They tried to mitigate the pressure in the fuel tank by installing triangular bars inside as shown in Figure 7.

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Figure 7 - Mitigation system for HRAM [10]

The role of those bars is to reduce the HRAM pressure by reducing the severity of the wave front. The idea is when the wave arrives on a triangular face, a weak reflection of the shock wave should occurs. This system cannot mitigate the shock wave on the front wall during the shock phase. However the results show a great reduction of the pressure due to cavity collapse. The initial pressure wave measured in the back wall is reduced by 60% compared with the pressure obtain in tank without mitigation member. For the cavity collapse pressure, the reduction is between 25% and 75%. Now the disposition and the shape of the bars have to be optimized.

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4 METHODOLOGY BACKGROUND

For representing the fluid different methods are available. In this chapter, the most-used methods are briefly presented. As we are using SPH during the simulation for representing the fluid, this section is more focused on this method.

4.1 Lagrangian Method

For solving mechanical problem, this method is most popular. The major characteristic of this method is that the mesh is fixed to the material during the simulation. As the mesh is fixed, a large deformation of the material creates a large distortion of the mesh.

Figure 8 - Example of a Lagrangian mesh [24] This is why this method has some limitations for large deformation. Indeed a large distortion of the elements results with an inaccuracy in the results. Moreover this distortion also decreases the time step resulting with an increase of the computational time.

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However this method has some advantages. First, as the mesh stay fixed to the material, its easy to track the material history for calculating the constitutive equations. Furthermore, treating the boundaries is easy as the mesh edges stay attached to the material surface.

4.2 Eulerian Method

This method is the opposite of the previous one; the mesh is relative to the spacial domain. The mesh is fixed in space and the material can flow through the mesh.

Figure 9 - Example of an Eulerian mesh [24] The advantage of this method is the mesh stays fixed and doesnt undergo distortion. So this method is suitable for large deformation problems like fluid. However this formulation has some disadvantages: as the material move through the grid, the model required a large mesh representing the area where the material is likely to flow. The treatment of the boundaries is more complex as they are depending of the material and not of the mesh. Moreover the results are difficult to extract as the data history is defined in spacial coordinate and not in material.

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4.3 ALE Method

Its a hybrid formulation between the Lagrangian and the Eulerian. This method tries to combine the advantages of each method. It starts as the lagrangian method but during the calculation the mesh is allowed to move in the material. There is distortion criterion for controlling the mesh and restoring a good element shape if necessary. Moreover with this method the material is also allowed to flow through the mesh.

This method is quite efficient for fluid problem and could have been also used for modelling Hydrodynamic Ram [24; 41].

4.4 SPH Method

In this section, we briefly describe the SPH method, explaining the theory used in LS-DYNA. This description is based on two reviews explaining in more detail this method [19; 44]. Smooth particles Hydrodynamics was initially created for simulating astrophysical gas dynamics problems. It has been developed by Monaghan and Gingold [16] and Lucy [28]. At this time, the available computational mesh methods were not suitable for this kind of problem. The SPH method looks like a Lagrangian method but the points are never connected together creating a mesh. SPH is a mesh-free method. This means it can undergo large deformation. The particularity of this method is the fluid dynamic variables are calculated with an integral interpolant using a smoothing function and the integral is then approximated by a summation over the interpolation points.

So the SPH was initially made for astrophysical problem but due to all the advantages that represent the absence of a mesh, this formulation has been adapted for many others applications like including high velocity impact and fluid-structure interaction. Thats why this method is particularly suited for simulating the HRAM.

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4.4.1 Integral interpolants

The base of the method is to express each variable as an integral interpolant.

Any function A(r) can be expressed in the form:

(/) = 0(/1)2(/ /1)3/ Where 2 is the Dirac function. This form is exact but not very useful. The idea of the SPH is to replace the Dirac function by a continuous function. This function is called the kernel as the integral interpolant of any function is expressed as:

(/) = 0(/1)7(/ /1, )3/ The choice of this function is very important for the success of the method. This function W has also to respect those two properties:

07(/ /1, ) 3/1 = 1 lim< 7(/ /1, ) =2(/ /1)

The variable h is half-width of the kernel function and it represents the spacial area where W smoothes A. This variable is often called the smoothing length.

The integral interpolant needs to be discretised for numerical simulations. In SPH problem, the fluid is defined by a set of points /> distributed regularly through the fluid. The fluid has a density defined by the variable (/). So the fluid is discretised into several point each of those points having a mass m>. So (/>)3 = m> and the integral interpolant can be expressed as:

(/) = 0 (/1)(/)7(/ /1, )(/)3/

28

(/) @m> >> 7(/ /A, )B>CD

Where > , > is the respectively the value of A and the density at /A. This equation is the basis of SPH method. With this equation the value of A at a point /A can be express as the summing of the contributions from the neighbouring point including in the kernel function:

> @mE EE 7F/A /G, HB

ECD

Figure 10 - Neighbouring particles including by the kernel function [44] The major advantage of using such an integral interpolant is for calculating the expression for the gradient of a function. Indeed (/) can be interpoled as:

(/) = 0(/1)7(/ /1, )3/ By integrating by part, we have:

(/) = 0(/1)7(/ /1, )3/ 0((/1)7(/ /1, ))3/ Using the gradient theorem:

(/) = 0(/1)7(/ /1, )3/ 0F(/1)7(/ /1, )HJ/

29

The second term of the right side on the equation is only over the boundary of the domain. So, most of the time, this integral is neglected because the kernel function is equal to zero around the domain boundary. Therefore if this term is equal to zero, we have:

(/) = 0(/1)7(/ /1, )3/ (/) @m> >> 7(/ /A, )

B>CD

The last equation demonstrate one of the most important advantages of the SPH: the gradient of any function can be determined by differentiating the kernel.

4.4.2 Kernel function

As said before the choice of the kernel function is very important. Originally, the kernel functions used were Gaussian functions or bell-shaped function. But, by far, the most popular kernel equation currently in use is the one developed by Monaghan and Lattanzio [32]. Its a function based on cubic spline functions:

7(q, ) = LM NOPOQ1 32R + 34RSTU0 |R| < 114 (2 R)STU1 < |R| < 20TU|R| > 2

Z

Where [ is the number of dimension of the problem and L a constant equal to S, D\] and D] for respectively 1D, 2D, 3D problems. The advantage of this function is for r > 2h the value are strictly equal to zero (the algorithm has only to summarize the particles include in a radius of 2h from the point of interest. Also it has a continuous second derivative. This function is the one used in LS-DYNA [26].

30

4.4.3 Equations of motion

The equation interpolant can be written for any equation so it can be written for the conservation equations. Here if given the form of the momentum equation written in a lagrangian framework:

d^t = 1 P So if the equation of state of the fluid relates the pressure P with only the density , we can express P in the SPH form as:

P = @mc cc 7(/ /d, )B

cCD

So the momentum equation can be written as:

d^t = 1e@mc cc e7fcB

cCD

Where 7fc = 7(/e /d, ) And we can also express the density at any point as:

e = @mc7fcBcCD The form presented above was the one use used in the first version of the method. However this form had a problem. It didnt conserve the linear and angular momentum: the force on particle a due to a particle b wasnt equal to the force on particle b due to a particle a.

For making this forces symmetrical, gh has been written using the chain rule:

P = P P P = P+ P

31

We can then rewrite the momentum equation in the SPH form:

d^t = @mc(cc + Pee)e7fcB

cCD

With this form, the equation converses the momentum.

Then two important modifications have been implanted to this method especially important for fluid simulation. First, instead of having a particles motion described normally by:

/edt = ^e The motion is described by:

/edt = ^e + 12@mc(^deed)7fcB

cCD

Where ed = hjkhl and ^de = ^d ^e. This expression ensures there is no big difference of velocity between particles in a same neighbourhood. Its important for treating nearly incompressible fluid as water due to the absence of viscosity. This method is called the XSPH method.

The second modification is the way how is calculated the density. For a lot of application this way is admissible. But, for fluid problem with a free surface (as in our case), using this method results with problem of discontinuity at the surface.

For overcoming this problem, the initial density is set at the beginning. And during the calculation, it only recalculate when particles move relative to each other. This can be calculated using the continuity equation in the SPH form:

edt = @mc^de e7fcB

cCD

32

5 MODEL DEVELOPMENT

In this chapter, the methodology used for modelling the experiment is explained. The model is developed using Hypermesh and LS-PrePost and the simulation are launched using LS-DYNA.

The unit system used was the following: tonne, mm, s, N, MPa.

5.1 Model size reduction

The first thing to do was to determinate the domain which will be represented during the simulation in order to start the modelling of the different part. In their experiment [11] Disimile et al. used a 1168x1829x1829mm tank. For an equivalent projectile size Varas et al. [41] used a 2.5mm particle density. So with this density, representing the entire tank would have resulted with a model of 250 million particles. Such a model would have been too big to compute even with distributed memory computer like Astral. A reduction of the model size was needed. For doing that, two actions were performed:

A reduction of the size of the tank

A representation of only a quarter of the experiment using symmetry conditions

For the reduction in the size of the tank, two models have been made:

A small one: a 400x200x200mm tank is modelled. The dimensions have been chosen for being able to launch the simulation on a simple computer and to acquire the pressure data from the first transducer.

A bigger one: a 1168x500x500mm tank is modelled. The same distance has been kept from the original experiment in the lengthways in order to recreate all the pressure transducer. For the transversal dimensions, I decided to keep 3 times the space need for the cavity formations as shown in Figure 11.

Figure

5.1.1 Symmetry conditions

For representing only a quarter osymmetry planes. This is madethe tank), by creating nodal constrains

For the symmetry plane the Z translation and in the X and Y rotation. For the symmetry plane Xnodes located on this plane are constrained in the Y translation and in the X and Y rotation.

Figure

500mm

TY, RX, RZ constrained

33

Figure 11 - Model dimension justification

Symmetry conditions

For representing only a quarter of the tank, its necessary to symmetry planes. This is made for the finite element parts (the projectile and

nodal constrains on the boundaries.

For the symmetry plane X-Y, the nodes located on this plane are constrained in the Z translation and in the X and Y rotation. For the symmetry plane X

s located on this plane are constrained in the Y translation and in the X

Figure 12 - Symmetry Conditions

the tank, its necessary to create two element parts (the projectile and

Y, the nodes located on this plane are constrained in the Z translation and in the X and Y rotation. For the symmetry plane X-Z, the

s located on this plane are constrained in the Y translation and in the X

TZ, RX, RY constrained

34

5.1.2 Ghost particles

The way the symmetry is done with the finite elements cant be done with SPH for two reasons: the first reason is that SPH is a mesh-free method and particles can mix together. If a layer of particle is constrained in displacement, there isnt any reason for the neighbouring particles not to pass through these particles and penetrate the boundary. The second reason is that for a particle which is closed to the boundary, that particle will miss some particles to have equilibrium of pressure. This will lead to a decreasing pressure near that boundary. Thats why a new method for symmetry with SPH has been developed.

The way its done in LS-DYNA is by creating a set of ghost particles which is an image of the particles close (within a distance of 2h) to the boundary. For each particle close to the boundary, a ghost particle is automatically created by reflecting the particle itself. The ghost particle has the same mass, pressure, and absolute velocity than the real particle. The ghost particle is then in the list of neighbours of that particle and contributes to the particle approximation.

Figure 13 - Ghost particles reflecting real particles [22]

In LS-DYNA, the ghost particles are created using the card *BOUNDARY_SPH_SYMMETRY_PLANE symmetry plane has to be specified.

5.2 Projectile modelling

Since during the event large deformation doesnt appear on the projectile, itbeen modelled using Lagrangian finite element. 3D elements have befor representing the projectile. In where only one quarter is represented.

It appears in the literature during the impact. So it has been decided to make the projectile as a rigid body part using a rigid material model (MAT_020 in LScomputational time of the model.

An initial velocity is applied to allset to 341 000mm/s in the X

5.3 Tank modelling

Since the tank dimensions are reduced, only the entry and exit wall arrepresented.

35

DYNA, the ghost particles are created using the card OUNDARY_SPH_SYMMETRY_PLANE [25] where only the normal of the

symmetry plane has to be specified.

modelling

Since during the event large deformation doesnt appear on the projectile, itbeen modelled using Lagrangian finite element. 3D elements have befor representing the projectile. In Figure 14are showed the projectile mesh where only one quarter is represented.

Figure 14 - Projectile Mesh appears in the literature [11; 41], that the projectile is not really deformed

during the impact. So it has been decided to make the projectile as a rigid body part using a rigid material model (MAT_020 in LS-DYNA). This wi

time of the model.

An initial velocity is applied to all the node of the projectile. The initial velocity is set to 341 000mm/s in the X-axis.

Tank modelling

Since the tank dimensions are reduced, only the entry and exit wall ar

DYNA, the ghost particles are created using the card where only the normal of the

Since during the event large deformation doesnt appear on the projectile, it has been modelled using Lagrangian finite element. 3D elements have been used

are showed the projectile mesh

deformed during the impact. So it has been decided to make the projectile as a rigid body

DYNA). This will reduce the

the node of the projectile. The initial velocity is

Since the tank dimensions are reduced, only the entry and exit wall are

36

For representing a plate, two choices are possible. It could be modelled using solid element or shell element. Solid element would have been better for representing the damage as the petalling of the tank but would have resulted with more computational time. Since we are more focused on the HRAM event than on the tank damage, shell elements are still good for modelling the tank. The only important think is the resulting velocity of the projectile after the impact. This one has to fit with the experiment for reproducing it properly.

The tank walls are made with steel and aluminium as shown in Figure 15.

Figure 15 - Two possible tank configurations

Another possible configuration was a tank only made by aluminium. A study between those two configurations has shown they give the same resulting velocity. So the fully aluminium configuration has been chosen as the aluminium has a bigger time step due to its properties. However the different thicknesses of the wall have been respected.

During the impact, high strain-rates are involved so It has been decided to use a Johnson-Cook material model (MAT_015 in LS-DYNA) [25]. The Johnson-Cook

Aluminium

Steel

Experiment configuration Only Aluminium configuration

37

model is empirically based. It allows us to determine the flow-stress y in

function of the plastic strainp , the normalised strain rate *& and the

temperature T. This model is widely used because is quite accurate, simple to use and requires a small number of constants. Those constants can be so easily found in the literature.

* *1 ln 1

n m

y pA B C T = + + &

With *T defined as ( )* 298 / ( 298)meltT T T= . A B C n and m are constant which have to be determined through experiments.

The material model used in LS-DYNA has also a damage model where the strain at failure is given as:

n = opD + pqr spS L

38

Melt temperature TM 775 K Reference temperature TR 294 K

Effective strain rate EPSO 1 s-1

Specific Heat CP 8.75e8 N-mm/t-K D1 parameter D1 0.13 D2 parameter D2 0.13 D3 parameter D3 -1.5 D4 parameter D4 0.011 D5 parameter D5 0

The last thing to determine was the size of the tank mesh. A quite refined mesh was wanted in order to predict quite accurately the decrease of velocity due to the impact. But the problem was a very refined mesh leads to a too big time step reduction. So the idea was to have a mesh size small enough with a time step close to the SPH particles time step.

Different sizes of mesh have been tested during an impact with the projectile:

Figure 16 - Different mesh sizes tested

0.625x0.625mm mesh

1.25x1.25mm mesh

2.5x2.5mm mesh 5x5mm mesh

39

Those three meshes have impacted at 341m/s by the projectile. The resulting velocities have been compared.

Figure 17 - Impact Holes

Figure 18 - Resulting velocity after impact

0

50

100

150

200

250

300

350

5x5mm mesh 2.5x2.5mm mesh 1.25x1.25mm mesh0.625x0.625mm mesh

Ve

loci

ty (

m/s

)

2.5x2.5mm mesh 5x5mm mesh

0.625x0.625mm 1.25x1.25mm mesh

40

As we can see there, with a coarse mesh there is a big decrease in velocity and the impact hole is too big. However with a mesh of 1.25x1.25mm the convergence of the resulting velocity starts to appear (as we can see in Figure 18) and the shape of the impact hole seems good (Figure 17). Moreover with this mesh size, the time step remains acceptable. So finally, the tank has been mesh with fully-integrated shell element of 1.25x1.25mm size.

Figure 19 - Boundary conditions of the tank walls

5.4 Fluid modelling

The last part to create was the fluid. This is the critical part of the simulation because the HRAM occurs in the fluid. The fluid has been made using SPH.

In LS-DYNA, for modelling water with SPH, you need to use the material card *MAT_NULL (*MAT_009). This card permits to consider an equation of state and only computes the hydrostatic stress which is suited because there is no deviatoric stress in water. So this card had to be use with an equation of state.

In the modelling process, one of the first things to do was to determine the particle density of the fluid.

Symmetry Conditions

Fully constrained

41

5.4.1 Particle density sensitivity analysis

The particle density is the spacing between each particle. It can be compare with the mesh size for finite elements. So the particle density has an important influence on the quality of the element and has to be chosen carefully. Thats why a sensitivity analysis has been performed for this parameter.

A simple box of water (dimensions: 150x90x90mm) has been realised. The fluid is impacted by the spherical projectile as shown in Figure 20.

Figure 20 - Model with 3mm particle density

Different density has been studied and what is assessed for this study is the velocity decrease due to the drag forces acting on the sphere. For assessing that the theory presented in page 11 was used:

= 11 + 34

It has been found in the literature [12] that the drag coefficient (CD) of a sphere is equal to 0.47.

42

So the theoretical velocity of the projectile in function of the time has been compared to the results obtained from the simulations. This comparison is presented in Figure 21.

Figure 21 - Projectile velocity in function of time for different particle density

As we can see, for a particle density of 1.5mm, the velocity curve fits perfectly with the theoretical values. The problem is with such a density, the model would have been too big to compute due to a high number of particles. So its been decided to use 1.5mm as particle density to reduce the computational time. With this value, the error in the velocity decrease is still acceptable (7%).

0

50000

100000

150000

200000

250000

300000

350000

400000

0 0,0001 0,0002 0,0003 0,0004 0,0005 0,0006

Ve

loci

ty m

m/s

Time (s)

Theoretical

1.5mm

3mm

4.5mm

6mm

43

5.4.2 Water equation of state

As said before the material null needs an Equation of State (EOS). In physics an EOS is a relation between state variables. Here in our problem, the EOS used relates the pressure to the density.

For modelling fluid, two equations of state are mainly used: the linear polynomial EOS and the Gruneisen EOS.

The linear polynomial EOS is stated as presented below:

= + D| + | + S|S + (w + '| + }|)~ Where | = 1 , E is the internal energy per initial volume and C0, C1, C2, C3, C4, C5, C6 are EOS coefficients.

The Gruneisen EOS is stated as presented below:

= | 1 + 1 2 | 2 |1 (D 1)| || + 1 S |S(| + 1) + ( + |)~ Where C, S1, S2, S3, 0, are EOS coefficients.

Whatever the type of EOS used, accurate parameters are needed for getting the right behaviour of the water.

Available EOS in the literature have been searched and 3 EOS have been found from articles [1; 13; 41]: 2 Gruneisen EOS and 1 Linear Polynomial EOS.

Those 3 EOS have been tested using a simple model equivalent to the one shown in Figure 20. For determining which one is the best for modelling water during an HRAM, it has been decided to assess them through 2 criteria:

The same criterion as used before, the decrease of velocity due to the drag forces. The velocity of the projectile is measured at 1.7ms and compared with the theoretical velocity at the same time.

44

The shock front velocity. The velocity of the wave generated due to the impact is measured and compared with the theory. The theory used is the one explained in page 9. The theoretical velocity is calculated using the Hugoniot-Rankine relations:

= + Where the Hugoniot slope coefficient of water has been choose at a value of 1.79 [45].

For measuring the velocity of the shock front, the time where the wave peak appears has been measured at two different points located at 10mm and 20mm away from the wall as shown in Figure 22. The difference between those two times divided by the distance between the two points gives us the shock front velocity.

Figure 22 Example of shock front velocity measurement

The results obtained with the different EOS are presented in the table below:

0,00E+00

2,00E+01

4,00E+01

6,00E+01

8,00E+01

1,00E+02

1,20E+02

0,00E+00 2,00E-05 4,00E-05 6,00E-05 8,00E-05 1,00E-04

Pre

ssu

re (

MP

a)

Time (s)

10mm

20mm

t

45

Table 2 - EOS Comparison

EOS type: Projectile Vel. at 1.7ms (error) Front wave velocity (error)

Gruneisen 1 140 m/s (8%) 1568m/s (31%) Gruneisen 2 140.5 m/s (7.5%) 1612m/s (27%)

Linear Polynomial 141 m/s (7%) 1851m/s (11%) Theory 151 m/s 2061m/s

As we can see, the values obtained for the first criterion are approximately the same. So the second criterion has been used for determining the good EOS. However, according Korobkin [21] the initial impact wave is moving at a supersonic velocity only for a short duration. After, the wave returns to the acoustic velocity. As it was quite difficult to evaluate the right theoretical velocity, it has been decide to the Gruneisen 1 EOS from [41] because it the only one which has been used before for the representing a HRAM.

Table 3 - EOS Parameters [41] Parameters name: Value Units

C 1.448e6 mm/s S1 1.979 S2 0 S3 0 0.11 a 3.0

5.4.3 Silent boundaries

As we are not representing the entire model, its important to create silent boundaries. Indeed by modelling only a small portion of the experimental tank, we are creating free surfaces which shouldnt be there. Those free surfaces

46

reflect the pressure waves contaminating the results. In the experiment, because the lateral walls are very far from the shot line, there is no wave reflection. The waves are dissipated before. So thats why its important to add silent boundaries on the lateral side of the fluid, the wave wont be reflected.

In the current version of LS-DYNA, silent boundary for SPH has not been implanted yet. Its currently available only for solid element. So the free surface has been represented using 3D finite elements and SPH and solid elements are linked together.

Figure 23 - Silent boundary creation

SPH particle forming the free surface are connected to solid elements using the card *CONTACT_TIED_NODES_TO_SURFACE. This card permits to link to 2 parts together. So in this type of contact, the slave nodes are constrained to move with the master surface [27]. Then a non reflective boundary is applied on the free surface made with solids elements using the card *BOUNDARY_NON_REFLECTING.

5.5 Parts interaction

Now each part is created, the interfaces between each part had to be created.

Non reflective boundaries

Solid elements

SPH

Tied contact

47

5.5.1 Projectile/Tank interface

First the interaction between the projectile and tank has been studied. This is a contact between two finite element parts. There are a lot of contact algorithms available in LS-DYNA for modelling this type of contact.

A simple model has been made (Figure 24) and several contacts have been tested for finding the best contact algorithm.

Figure 24 - Projectile/tank interface study The first algorithms which have been tested were the ones defined by the cards:

*CONTACT_AUTOMATIC_SURFACE_TO_SURFACE *CONTACT_SURFACE_TO_SURFACE

Those contacts are two-way treatment contact where a master and slave surface is defined and LS-DYNA checks the slave nodes for penetration through master segments and then a second time the master nodes for penetration through slave segments [27].

However whatever the penalty scale factor used, some penetrations appear during the contact as shown in Figure 25.

48

Figure 25 - Contact penetration

This can be explained by a time step too important compared with the projectile velocity and also by the presence of sharp edges due to the projectile geometry. A reduction of the time step can solve this but its not acceptable.

The solution was to the change the contact algorithms. I use the contact card *CONTACT_AUTOMATIC_GENERAL [25]. In this card the contact is defined wholly by the slave part. With this algorithm, contact is considered between all the surfaces in the slave list. This contact is quite efficient for sharp edges but more time consuming than other contact.

Figure 26 - Good contact behaviour

49

As we can in Figure 26, this contact gives good result. So it has been kept.

5.5.2 Projectile/Fluid interface

After, it was the interaction between the projectile and fluid to be studied. Once again, a small model has been made as shown in Figure 27.

Figure 27 - Projectile/fluid interface study For a contact between an SPH part and a finite element part, only contact types NODE TO SURFACE are permitted. So for this case two contacts were permitted:

*CONTACT_AUTOMATIC_NODES_TO_SURFACE *CONTACT _NODES_TO_SURFACE

In [22], for coupling SPH with FE, the author recommend to use the card *CONTACT_ AUTOMATIC_NODES_TO_SURFACE. So I decided to use this card instead of *CONTACT _NODES_TO_SURFACE. With this contact, some penetration problems occurred initially as shown in Figure 28. They were removed by using the soft constrain formulation (flag SOFT in the LS-DYNA card [25]). But some particle were still penetrating, this was due to the sharp edges of the projectile. Increasing the Maximum parametric coordinate in

50

segment search parameter has solved the problem (set to 1.20 instead of 1.025).

Figure 28 - Particle penetration problem

5.5.3 Tank/Fluid interface

The last interface to study was the tank/fluid one. This contact is very similar to the previous one. The difference is here the relative velocity between the particles and the wall is not very fast during the simulation and the shape on the tank wall is simple without sharp edges.

For defining the appropriate contact, the simple model presented in Figure 29 has been made.

51

Figure 29 - Projectile/fluid interface study As the card *CONTACT_AUTOMATIC_NODES_TO_SURFACE was working with the previous contact, it has been used again for this interface.

Contrary to the previous contact, this one is working fine with the default card parameters.

For representing the radius of influence of the SPH particles, an offset has been set between the particle and the tank as shown in Figure 30.

Figure 30 - Offset between the tank and the fluid particles

Tank Fluid

3mm 1.5mm

52

For creating the contact properly, a virtual thickness of 1.5mm (half distance between each particle) has been applied to the SPH. This can be done through the flag SST (Optional thickness for slave surface) in the contact card.

The same has been done for the contact between the projectile and the fluid.

5.6 Pressure initialisation

During the Hydrodynamic Ram, the cavitation appears when the water pressure drops below the vapour pressure of water. So initially the pressure in the water cant be set to zero. Thats why an atmospheric pressure had to be recreated in the water.

So the pressure in the fluid had to be initialised to 0.1013MPa at the beginning of the simulation. Since the pressure due to gravity is very small compared with the atmospheric pressure, the gravity has not been set in the model.

Adding pressure initially in the fluid has been done through the EOS. The Gruneisen EOS is stated like that:

= | 1 + 1 2 | 2 |1 (D 1)| || + 1 S |S(| + 1) + ( + |)~ The initial pressure when | = 0 is equal to = ~. So by specifying a ~ = 0.921, the pressure is initialised to 0.1013MPa. Adding a pressure in the fluid, create a pressure on the tank wall. So for having a model at equilibrium at the initial state is important to create a counter pressure on the tank walls equal to the atmospheric pressure. This pressure has been made using the card *LOAD_SHELL_SET.

53

Figure 31 - Pressure initialisation

As we can see in Figure 31, the pressure is correctly initialised and no pressure appears on the wall.

5.7 Pressure measurement

The last important thing in the model was the way the pressure is measured. From the experiment used for assessing the capability of LS-DYNA to represent the HRAM, the most important data is the pressure measurement.

A way for doing that was first tracking the pressure from an SPH particle located close from a transducer. This can easily done with the card *DATABASE_ HISTORY_SPH. But this method has a major problem, the SPH particles are moving during the simulation and the pressure transducer is supposed to stay at the same place.

So the way which has been chosen for doing it was using the card *DATABASE _TRACER. This card enables to track the history of either a material point or a spatial point into an ASCII file. As the transducers are not moving with the fluid, some special points have been defined in this card representing the position of the transducers.

In the experiment 7 transducers are mounted. In the numerical models, they are all represented for the big model and only two for the small model. Their positions are summarized in Table 4.

54

Table 4 - Pressure transducer position

Transducer: X position (mm) Y position (mm) Z position (mm) 1 0 301 37 2 0 215 215 3 37 0 301 4 74 0 602 5 111 0 903 6 0 0 1167 7 0 143 1167

Figure 32 - Coordinate system

5.8 Final model

As say before two models has been build. The small one (dimensions: 400x200x200mm) is presented in Figure 33. The bigger one (dimensions: 1168x500x500mm) is presented in Figure 34.

(0, 0, 0)

55

Figure 33 - Small model

Figure 34 - Big model

56

Table 5 - Model statistics

Small model Big model Number of SPH particles 127813 2608912 Number of shell elements 12800 80000

The termination time has been set to 30ms which is the necessary time in the experiment for the cavity collapse. Pressure was recorded each s.

57

6 RESULTS

In this section, the results obtained from the two models are presented and discussed. First the problems which occurred during the initial simulations are presented. Then, as the HRAM is divided in 4 four phases, the result discussion is divided in four parts: one for each phase.

6.1 Initial problems

6.1.1 Distributed memory computer

Primarily, the big model was made for being use on the distributed memory computer of Cranfield: Astral. This computer allows the use of a great number of processors and the calculation is very fast. However some problems occurred and it was impossible to use it properly.

Indeed the first problem was due to contact. As the contact was defined for all the particles, processors had some communication problem between themselves. This results with some penetration problems which induce bad results. For solving that I had to create to reduce the interfaces. For instance for the fluid/projectile contact, I reduce the contact to only the particles located on the shot line. This, coupling with the card *CONTROL_MPP_DECOMPOSITION _CONTACT_DISTRIBUTE (which force the distribution of the contact over all the processors used), solved the problem.

However some problems were still present. Due probably to a communication problem between processors, during the post-processing of the results, only pressure for small groups of particles were plotted on the screen as shown in Figure 35.

58

Figure 35 - Post-processing problem

Moreover, the version of LS-DYNA for distributed memory is not able to handle the card *DATABASE_TRACER used for representing the pressure transducer [25]. So finally, it has been decided to not use the ASTRAL computer. A shared memory computer (the GRID) has been used instead. However the use of this computer allows uniquely 4 processors instead of the 24 used on ASTRAL. The calculation for the big model was much longer and only one simulation has been launched for the big model.

6.1.2 Double precision solver

Other problem important problem was the noise in the pressure history. Indeed the first curves obtained were suffering from a lot noise. This problem could have been overcome by filtering but this would have reduced the quality of the result.

The problem was because water is a quasi-uncompressible fluid, the density varies very slightly and so only in the decimals of the value. That is why using only the single precision solver induces some imprecision in the calculation. The use of the double precision solver has permitted to solve this problem and to have smoother curves.

59

6.2 Shock phase

In this section, we will talk about the results obtain during the shock phase. We are mainly focused on the pressure wave created during the shock.

First we are looking at the impact of projectile with the tank as shown in Figure 36.

Figure 36 - Projectile impact with the tank at t=0.03ms

This impact will create a shock wave travelling through the tank wall generating dynamic stress. The Figure 37 shows the stress generated by the impact with an empty tank. This will permit to compare later on with the additional stress generated by the pressure wave.

60

Figure 37 - Von Mises stress generated by the impact on an empty tank at 0.023ms (MPa)

Indeed, after that the projectile is accelerating the fluid particles situated around the impact area. This suddenly acceleration results with a pressure wave created, as we can see in Figure 38.

Figure 38 - Pressure wave generated at 0.06ms

61

As say before this pressure wave is creating an extra stress on the tank wall as shown in Figure 39.

Figure 39 - Von Mises stress generated by the impact on a filled tank at 0.023ms (MPa)

If we compare the Figure 37 with the Figure 39, we can find that:

In magnitude, the stress generated by the impact in the filled tank is more important. The additional stress created by the pressure wave represent 50% more in comparison with the dynamic stress due to the impact on only the plate.

High values of stress are concentred around the entry hole for the empty tank whereas for the filled tank the area is larger.

Those pictures explain why the HRAM can be a cause of major failure of the tank. Instead of having a small hole created by the penetration of the projectile, the high pressure generated on the wall can create a petalling of the tank increasing largely the size of the hole.

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The size of the holes is the same within or without tank in this simulation but this is due to small impact velocity used (341m/s). In really impact of fuel tank can occurs at much higher velocity.

If this pressure wave increases the risk of important failure of the tank, its concerning only the impact zone. Indeed, the pressure wave is significantly reduced when its travelling through water. As shown in Figure 40.

Figure 40 - Drop in pressure measured along the shot line on the big model

We can see that after 50mm from the tank the pressure drops below 10 MPa which is not anymore significant regarding the stress induced on the tank.

It seems that from a qualitative point of view, LS-DYNA is able to reproduce this phase. Now its important to assess those results quantitatively.

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From the experiment made by Disimile et al.[11], two types of data are available: the pressure history for the transducer P3 shown in Figure 41 and the pressure decrease between the transducer P3, P4, P5 and P7 compared with the simulation results in Figure 43.

Figure 41 - Initial wave pressure recorded by the transducer P1 in the experiment

If we are looking at the Figure 41, for transducer, the shock phase start with a sharp pressure rise of 1.2Mpa and then followed by secondary waves of 0.3 MPa. The data obtained for the shock phase during the simulation are the following:

Figure 42 - Pressure history at P3

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For the wave arrival time, its difficult to assess it from the experiment as we dont know the impact time. However we can calculate the wave velocity, this one is equal to 1403m/s approximately the acoustic velocity in water which agrees with impact dynamic theory.

From the data we have in Figure 41 (which has to be taking into account carefully has, it has been filtered), its quite difficult to analyse it precisely. However it seems that the pressure rises for the first pike agree with the value of the experiment (3% of difference). Then the secondary waves fit also correctly in term of amplitude.

The other data provided for this phase was the decrease of pressure of the transducer P4, P5 and P7 compared with the transducer P3. The comparison with simulation results is shown in Figure 43.

Figure 43 Decrease of pressure of the transducer P4, P5 and P7 compared with the transducer P3

The curves seem to fit correctly, except for the pressure transducer P7 located on the tank wall where no pressure wave has been recorded. Two possibilities can explain that: The pressure record is not correctly working or the wave has

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been completely dissipated. But from the others transducer, we can assume that LS-DYNA predict correctly the dissipation of the pressure wave within the fluid.

The last thing to look at for this phase is the pressure decrease of the wave when its going away from the shoot line. Indeed, McMillen et al. in [31] has shown that the magnitude of the pressure along the arc of the wave decreases with the angle from the shot line. In their experiment, Disimile et al. establish an empirical law which state that P=P90SIN(+7), where P90 is the pressure in the shoot line and the angle away from the shoot line as shown in Figure 44.

Figure 44 Angle [11] This empirical law has been tested in the simulation with the pressure transducer P2 which is at an angle of 45 and along the same arc than P3. So

we should found the relation S = sin( + 7) ( + )

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This agrees with the empirical relation from the experiment.

So, globally, we can conclude that for this phase, the SPH method implanted in LS-DYNA is able to produce correct result from a qualitative and quantitative point of view.

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6.3 Drag phase

Now the next phase to study is the drag phase. During this phase the projectile is moving through the fluid transferring to it, its momentum.

Figure 45 - Projectile pressure field In Figure 45, we can observe the pressure field in front of the projectile. When the projectile is approaching from a transducer a gradually pressure increase occurs has shown in Figure 43.

The first thing to look is the velocity decrease due to the drag forces acting on the sphere, if we are comparing this with the theory (where the velocity reduction due to impact has been taken into account) we have:

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Figure 46 - Decrease of the velocity due to drag forces

In the methodology part, we have been able to validate the behaviour of the projectile moving through a fluid. Here, a difference between the two curves appears, the velocity of the projectile is decreasing faster than the theory predicts it. This means that the drags force applied on the projectile are more important in the simulation. The explanation of this can be the fact that, as it explained in page 2.1.312, the cavity created in the wake of the projectile increase the drag coefficient. Moreover, in the methodology the fluid is free to move instead of here, where the fluid is constrained at the free surface. So as water is quasi-uncompressible, the drag forces are greater. For a better fitting with the experiment, the entire tank should have been represented and the lateral tank wall also.

Then, we can also look at the projectile pressure field. From the data we have within the Figure 43, pressure history at the transducer P3 has been extracted and compared with the pressure recorded in the simulation (NB: the extraction of the data has been made approximately from a curve in a paper, so the accuracy is not perfect). We obtain the following:

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Figure 47 - Comparison of the drag pressures between the simulation and the experiment

We can see that the curve obtained in the simulation is very noisy even after having removed high frequency oscillation with a filter (this is why pressure peak due to shock phase is reduced due to the filtering). In the experiment, oscillations also appears and has been removed my filtering. According to the authors those oscillations are due to the projectile oscillations.

The two curves obtained have some differences. Globally, if we removed the oscillations, they have the same shape. After the shock phase, the pressure stays stable around 0.4 MPa. And then, when the projectile is approaching from the transducer the pressure field is gradually increasing. However in the simulation the gradually increase of the pressure occurs later and the amplitude is lower than the experiment. This is due to what it has been explained before. The projectile velocity history is not the one expected. So the projectile is travelling slower than it should. That is why the increase occurs later and the intensity is less (the intensity is related with velocity of the projectile).

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To check if this pressure increase correspond to the passage of the projectile, its position relative to the transducer at t=1.6ms is show in Figure 48.

Figure 48 - Projectile position relative to the transducer P3 at 1.6ms

SPH used in LSDYNA has the capability to reproduce correctly this phase. However the model made is not totally accurate and a remaining task is to investigate further why the velocity decrease is not what it should be and correct it. A possible hint could be the boundaries condition used around the fluid.

6.4 Cavitation phase

An important phase of the HRAM is the cavitation phase. During this phase, the collapse of the cavity can create an entire failure of the tank. So its very important to be able to reproduce correctly this phase.

The cavitation appears when the pressure in the weak of the projectile drops below the vapour pressure. At this moment the water initially a liquid changes of

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phase and become vapour. In LS-DYNA, an equation of state able to reproduce this doesnt exist. So a mean for representing this had to be found. A simple way for doing that is to define a pressure cut-off. According to the LS-DYNA manual [25], it allows for a material to numerically cavitate. When the pressure drops below the value of the pressure cut-off, the material doesnt resist any more to this dilatation. This method is supposed to allow the material to cavitate. Here the pressure cut-off has been set to 0.0023 MPa, the value of the vapour pressure.

The cavitation starts to form itself in the trail of the projectile. Indeed, in its trail the pressure decrease and goes below the vapour pressure. From the experiment a picture of the formation of this cavity is given. We can compare the size of those cavities with the simulation.

Figure 49 - Cavity formation in the experiment [11]

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Figure 50 - Cavity formation at 1.6ms

If we compare the Figure 49 and the Figure 50, we can see that the shape and the size of the cavity behind the projectile are the same. This shows that LS-DYNA until this step is able to reproduce this phase.

The problem is when the cavity will grow. Indeed once the cavity is formed, this mixture of gas and vapour will grow as a bubble until reaching its maximum radius as shown in Figure 51 - Cavity grow.

Figure 51 - Cavity grow [11] In all the simulation carried out, the cavity has never expended itself. This is the major limitation of the SPH method in LS-DYNA for reproducing the HRAM event.

But this limitation is not due to the SPH formulation. Its due to the method used for representing the cavitation. The pressure cut-off method is not accurate