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## State of the art of Hydrodynamic Ram simulations

This section offers an overview of the state of the art of hydrodynamic ram simulations presented in the literature. This review is based on Varas (2009) and Abrate (2011) with addition of the most recent works.

### Analytical models

Analytical models have been developed to deal with the water entry of projectile which is a phenomenon that is highly similar to the hydrodynamic ram phenomenon. Numerous analytical studies have been done on the penetration of projectile at low speed ( afew m.s−1) as in (Scolan and Korobkin, 2003; Korobkin, 1994). However extension of these models to higher impact velocities and to hydrodynamic ram has not been done. Projectile movement The only law used for the motion of rigid projectile for instance in (Varas et al., 2009a) is based on Newton’s second law. mp¨zp = − 1 2 ρlACdz˙p 2(1.1) with zp the depth of penetration, ρl the liquid density, A the projected area of the projectile and Cd its drag coefficient. Assuming a constant drag coefficient for a spherical projectile, the velocity of the projectile is found to decay exponentially. z˙p = Ce−zp.

#### Rayleigh-Plesset modelling

A large amount of studies on bubble dynamics can be found in the literature. These studies could be useful to understand the physics of the cavity evolution process that is observed during HRAM events generated by the impact of projectiles in liquid filled containers. These studies began with Lord Rayleigh (1917) on the pressure prediction during the collapse of a spherical bubble, assuming that the surrounding liquid is incompressible and inviscid, and that surface tension forces are negligible. His work was extended by Plesset (1949), who derived the secondorder non-linear ordinary differential equation for the time-dependent bubble radius evolution, which became the well known Rayleigh-Plesset equation of bubble dynamics. Improvements of this equation have been proposed by numerous authors, like by Keller and Miksis (1980), or extensions by Fujikawa and Akamatsu (1980), Prosperetti and Lezzi (1986) and Hauke et al. (2007). These improvements mainly concern effects that are important at the end of the collapse stage: liquid compressibility, thermal effects, and the effect of the non-equilibrium of vapour condensation.

The classical Rayleigh-Plesset equation for a single bubble dynamics has proven its efficiency in the physical analysis of bubble dynamics in various applications and for different bubble dimensions (cavitating flow, underwater explosion, …). This equation is a simple and efficient model for the first approach of bubble expansion phenomena. The Rayleigh-Plesset equation is used here to discuss the influence of several physical parameters on HRAM bubble dynamics in the case of the ballistic impacts in the water filled tank and ONERA pool.

**Derivation of Rayleigh-Plesset equation**

To obtain this equation and use it to predict HRAM bubble dynamics, a spherical gas bubble in an infinite domain of liquid is considered (confinement effect is not investigated in this first part), and the following assumptions are made:

• incompressibility of the liquid.

• spherical deformation of the bubble interface.

• instantaneous energy transfer of kinetic energy of the bullet to the liquid.

• gravity effects are negligible.

• idealised case of zero mass transport across the bubble interface: as the transferred mass is small, the influence is negligible on large radius bubbles, which is the case here (0.2 m in pool, 0.12 m in tank)

• the dynamic viscosity and the surface tension effects are negligible due to the large dimensions of the bubbles. Viscous effects are supposed to be important for bubbles with a radius smaller than 10−3 m (Chapman and Plesset, 1971), but they are usually negligible when the bubble’s radius is 10−1 m, or more.

• the pressure in the liquid at the bubble boundary is equal to the pressure inside the bubble;

• assumption of isothermal process since thermal effects are negligible for cavitation in water (Brennen, 1995)1.

• in addition to the hypothesis classically used to obtain Rayleigh-plesset equation the presence of an initial amount of non-condensable gas (here air) is added, in a first approach its behaviour is assumed to be adiabatic. The equation of mass conservation for a radial movement is expressed in spherical coordinates. It reduces due to the previous assumptions to (2.1).

**Application of Rayleigh-Plesset equation for HRAM events**

In the case of bubbles created by HRAM events, it is proposed here to use the Rayleigh- Plesset equation in a non classical manner. To solve this differential equation, some initial conditions have to be set. Generally for classic cases of Rayleigh-Plesset applications, these initial conditions are Rb0 6= 0, Pb0 > P∞, and ˙Rb0 = 0. In the present case Rb0 6= 0, Pb0 < P∞,and ˙Rb0 6= 0, which means that the dynamics of the bubble is not created by an initial pressure difference between the bubble gas and the liquid at infinity but by momentum.

The initial conditions of the Rayleigh-Plesset equation determined from the ONERA experiments are linked to the initial time chosen for the analysis (when a bubble cavity reasonably appears). To choose the starting times for Rayleigh-Plesset simulations, energetic considerations are also used e.g. when the liquid initial kinetic energy in Rayleigh-Plesset equation is equal to the theoretical initial kinetic energy of the projectile that created the bubble (approximately 3.5 kJ in the pool and 2.9 kJ in the tank). It has been observed that it corresponded approximately to the beginning of the growth stage of the bubble cavity in the tests. As the energetic partition between the kinetic energy transferred to the liquid and the energy dissipated by the deformation of the projectile is not known, no dissipative phenomena are considered here: the whole projectile kinetic energy is assumed to be transferred to the liquid. The amount of kinetic energy of the liquid is calculated using the assumption of incompressibility of the liquid (2.4).

**Table of contents :**

Acknowledgement

Nomenclature

Bibliographie et synth`ese des travaux en fran¸cais

**I Introduction **

**II Utilisation d’´equations de type Rayleigh Plesset pour l’analyse des bulles cr´e´ees lors d’un coup de b´elier hydrodynamique**

II.1 Pr´esentation des cas d’´etude

II.2 Rappels g´en´eraux sur l’´equation de Rayleigh-Plesset

II.3 Estimation analytique du param`etre de confinement dans l’´equa-tion de Rayleigh-Plesset confin´ee

III ´Etude de l’influence de la compressibilit´e du liquide sur la dynamique de bulles confin´ees

III.1 Simulations ´el´ements-finis compressibles de dynamiques de bulles confin´ees

III.2 R´esultats des simulations compressibles de dynamique de bulles confin´ees

**IV D´eveloppement et validation d’un mod`ele compressible de type Keller-Miksis confin´e **

IV.1 Formulation du mod`ele de Keller-Miksis

IV.2 Application des ´equations dans le cas d’un conteneur sph´erique ´elastique

**V Conclusions et perspectives **

**1 Context of the research and state of the art **

1.1 Hydrodynamic Ram

1.2 State of the art of Hydrodynamic Ram simulations

1.2.1 Analytical models

1.2.2 Numerical models

**2 Analysis of bubbles dynamics created by hydrodynamic ram in confined geometries using the Rayleigh-Plesset equation **

2.1 Introduction

2.2 Studied cases

2.2.1 Description of the ballistic experiments

2.2.2 Exploitation of the test results for Rayleigh-Plesset simulation

2.3 Rayleigh-Plesset modelling

2.3.1 Derivation of Rayleigh-Plesset equation

2.3.2 Application of Rayleigh-Plesset equation for HRAM events

2.4 Confined Rayleigh-Plesset equation

2.4.1 Modification of RP equation to take confinement effect into account.

2.4.2 Confined Rayleigh-Plesset equation for bubble created by HRAM events

2.4.3 Influence of gas modelling in numerical simulations

2.5 Discussion

2.6 Conclusion

**3 Confined Rayleigh-Plesset equation for Hydrodynamic Ram analysis in thin-walled containers under ballistic impacts **

3.1 Introduction

3.2 Studied cases

3.2.1 Description of the ballistic experiment

3.2.2 Use of the test results for Rayleigh-Plesset simulation

3.3 Confined Rayleigh-Plesset equation

3.4 Application of the confined Rayleigh-Plesset equation for bubbles created by HRAM events

3.4.1 Experimental calibration of α

3.4.2 Use of elasticity formula for the structure response

3.5 Conclusion

**4 Cross validation of analytical and finite element models for Hydrodynamic Ram loads prediction in thin walled liquid filled containers **

4.1 Introduction

4.2 Studied cases

4.3 Confined Rayleigh-Plesset equation

4.4 Quasi-incompressible confined bubble dynamics finite element simulations .

4.4.1 Description of the quasi-incompressible confined bubble problem .

4.4.2 Material Laws used in quasi-incompressible confined bubble dynamics simulations

4.4.3 Convergence study for the quasi-incompressible simulations

4.4.4 Simulation results of quasi-incompressible confined bubble dynamics

4.5 Compressible confined bubble dynamics finite element simulations

4.5.1 Description of the compressible confined bubble problem

4.5.2 Convergence study for the compressible simulations

4.5.3 Simulation results of compressible confined bubble dynamics

4.6 Discussion

4.7 Conclusion

**5 Development and validation of a confined Keller-Miksis model for Hydro dynamic Ram loads prediction in liquid-filled containers. **

5.1 Introduction

5.2 Formulation of the Keller-Miksis model

5.2.1 Basic fluid mechanics equations

5.2.2 Simplifications introduced in the Keller-Miksis model

5.2.3 Derivation of the Keller-Miksis differential equation

5.2.4 Determination of g

5.3 Application to a rigid wall case

5.3.1 Verification of the consistency of analytical models for rigid containers

5.3.2 Finite element modelling

5.3.3 Comparison of the confined Keller-Miksis and finite element simulations for bubbles in rigid containers

5.4 Application to an elastic wall case

5.4.1 Verification of the consistency of analytical models for elastic containers .

5.4.2 Comparison of the confined Keller-Miksis and finite element simulations for bubbles in elastic containers

5.4.3 Evaluation of the improvement in prediction using the confined Keller- Miksis model

5.5 Discussion

5.6 Conclusion and outlooks

General conclusion and outlooks

**A Study of the capabilities of an ALE bi-material fluid simulation for solving cavity expansion and collapse during an Hydrodynamic Ram event **

A.1 State of the art of HRAM modelling

A.2 Importance of the cavitation phase simulation

A.3 Numerical simulation of the water entry of the projectile

A.4 Finite Element model

A.4.1 Description

A.4.2 Material laws

A.4.3 Fluid structure interaction

A.5 Results

A.5.1 May’s experiment (water entry 10.6 m.s−1)

A.5.2 Water entry with confinement tank

A.6 Conclusion

A.7 Acknowledgement

**B Thermal effects in cavitation **

B.1 Characteristics of usual aeronautic fuel materials

B.2 Indicator of thermal effects

**C Confinement effects of a spherical container on the dynamic of a single bubble created by optic cavitation **

C.1 Experimental setup

C.1.1 Laser source

C.1.2 Optical system

C.1.3 Containers description

C.1.4 High-speed camera and lighting system

C.1.5 Millimetric grid

C.2 Experimental results

C.2.1 Time history of bubble radius

C.3 Conclusion

**D Demonstration of the relation between κ and α**