Modelling of internal ballistics of gun systems: A review

Abstract

A deep understanding of the internal ballistic process and the factors affecting it is of primary importance to efficiently design a gun system and ensure its safe management. One of the main goals of internal ballistics is to estimate the gas pressure into the combustion chamber and the projectile muzzle velocity in order to use the propellant to its higher efficiency while avoiding over-pressure phenomena. Dealing with the internal ballistic problem is a complex undertaking since it requires handling the interaction between different constituents during a transient time lapse with very steep rise of pressure and temperature. Several approaches have been proposed in the literature, based on different assumptions and techniques. Generally, depending on the used mathematical framework, they can be classified into two categories: computational fluid dynamics-based models and lumped-parameter ones. By focusing on gun systems, this paper offers a review of the main contributions in the field by mentioning their advantages and drawbacks. An insight into the limitations of the currently available modelling strategies is provided, as well as some considerations on the choice of one model over another. Lumped-parameter models, for example, are a good candidate for performing parametric analysis and optimisation processes of gun systems, given their minimum requirements of computer resources. Conversely, CFD-based models have a better capacity to address more sophisticated phenomena like pressure waves and turbulent flow effects. The performed review also reveals that too little attention has been given to small calibre guns since the majority of currently available models are conceived for medium and large calibre gun systems. Similarly, aspects like wear phenomena, bore deformations or projectile-barrel interactions still need to be adequately addressed and our suggestion is to dedicate more effort on it.

Headnote

ABSTRACT

A deep understanding of the internal ballistic process and the factors affecting it is of primary importance to efficiently design a gun system and ensure its safe management. One of the main goals of internal ballistics is to estimate the gas pressure into the combustion chamber and the projectile muzzle velocity in order to use the propellant to its higher efficiency while avoiding over-pressure phenomena. Dealing with the internal ballistic problem is a complex undertaking since it requires handling the interaction between different constituents during a transient time lapse with very steep rise of pressure and temperature. Several approaches have been proposed in the literature, based on different assumptions and techniques. Generally, depending on the used mathematical framework, they can be classified into two categories: computational fluid dynamics-based models and lumped-parameter ones. By focusing on gun systems, this paper offers a review of the main contributions in the field by mentioning their advantages and drawbacks. An insight into the limitations of the currently available modelling strategies is provided, as well as some considerations on the choice of one model over another. Lumped-parameter models, for example, are a good candidate for performing parametric analysis and optimisation processes of gun systems, given their minimum requirements of computer resources. Conversely, CFD-based models have a better capacity to address more sophisticated phenomena like pressure waves and turbulent flow effects. The performed review also reveals that too little attention has been given to small calibre guns since the majority of currently available models are conceived for medium and large calibre gun systems. Similarly, aspects like wear phenomena, bore deformations or projectile-barrel interactions still need to be adequately addressed and our suggestion is to dedicate more effort on it.

Keywords:

Internal ballistics

Numerical modelling

Lumped-parameter model

Computational fluid dynamics

Gun systems

(ProQuest: ... denotes formulae omitted.)

1. Introduction

Internal ballistics is the part of mechanics that studies the movement of a projectile inside a gun system (Fig. 1). According to the definition in Ref. [1], it deals with the interactions of the gun, projectile and propelling charge spanning from striking the firing pin on the primer up to the projectile leaving the muzzle of the barrel.

From a physical point of view, internal ballistics can be described as a process where the chemical potential energy stored into the propellant, which is a strong and metastable energetic material with the ability to produce hot gases containing a large amount of energy [2], is converted into kinetic energy driving the projectile along the barrel. The way the propellant converts its potential energy to kinetic energy is by combustion, a phenomenon leading to the transition of the propellant from a solid state to a gaseous one. This causes a rapid pressure rise in the combustion chamber, i.e., a closed volume where the propellant is contained, that exerts a force on the base of the projectile. Once a sufficient level of pressure is achieved, the projectile starts to travel down the barrel until it exits the gun system [3,4].

Even if some details may differ from one system to another, a general IB cycle, which begins with the ignition of the propellant and ends when the projectile leaves the weapon's barrel, can be schematically represented as in Fig. 2. By referring the interested reader to Refs. [5,6] for a more detailed description, 3 main phases can be identified in Fig. 2. The first is the ignition of the propellant grains, which start burning once their ignition temperature (adiabatic flame temperature) is reached. It thus begins the combustion process of the propellant, leading to the spread of hot gases and burning particles into the combustion chamber. This rapidly raises the pressure and temperature into the combustion chamber and exerts a force on the base of the projectile. Because of the chamber pressurisation, the propellant burning rate increases, as typical of gun powders. More and more hot gases are thus produced and, consequently, higher values of pressure and temperature are reached into the combustion chamber. Higher forces are also exerted on the projectile's base. When a sufficient level of pressure is achieved, denoted with Ps (shot pressure) in Fig. 2, the projectile begins to move through the barrel and, at the same time, the effective volume available for the combustion gases starts to increase. However, even if a higher volume is available, the gas production is still very intense and, as illustrated in Fig. 2, the chamber pressure continues to increase up to the maximum value Pmax. Starting from this point, the chamber pressurisation starts to decrease since the work done by the combustion gases to accelerate the projectile and the volume increase compensate the mass flow of gases produced by the burning propellant grains. This final phase is also characterised by two events: the end of the combustion process, denoted with E in Fig. 2, and the moment in which the projectile leaves the muzzle of the gun with a certain velocity vm. At this point, to which the pressure value Pm (muzzle pressure) corresponds, the internal ballistics problem is over.

With this regard, a deep understanding of the entire process and, in particular, the factors affecting it, is of primary importance in order to efficiently design a gun system and ensure its safe management [7,8]. Specifically, one of the main goals of internal ballistics is to estimate the gas pressure into the combustion chamber, in order to avoid dangerous over-pressure phenomena, and evaluate the muzzle velocity of the projectile, in order to use the propellant to the highest efficiency.

However, analyzing the shot event from a theoretical point of view is not a trivial undertaking. In fact, this triggering mechanism involves complex phenomena that are often overlooked or poorly controlled, like the propellant combustion law, the mass and energy transfer within different phases, i.e., unburnt propellant grains and combustion gases, the chemical reactions of the different species and the gas-particle interactions during a transient timelapse with a very steep rise of pressure and temperature, typical of internal ballistics [9,10]. Similarly, from the experimental point of view, the quantification of such phenomena suffers from many difficulties [9]. Firstly, because the ballistics process takes place in extreme conditions of pressure and temperature within a very short period of time. Secondly, because of the risk in manipulating solid propellants. Thirdly, because of the shortcoming of the experimental tests, which are often unable to measure all the involved parameters or, to do so, substantial hardware and expensive sensors are required [9]. Consequently, even though analytical approaches and experiments are fundamental to shed light on the physical phenomena involved, nowadays they are more and more supported by computer simulations, providing a fast, safe, economical and highly flexible way for investigating the ballistics problem and evaluating the performance of a gun system. This is confirmed in the literature where, mainly because of the exponential growth in terms of computational power and the development of new mathematical models, computational simulations have thrived in the last decade and have become a catchy tool for all people interested in the field.

As usual, considering the complexity of the internal ballistics problem that requires properly adapted laws of thermodynamics and fluid mechanics [11], a balance of modelling accuracy and simplicity is required. In fact, over-simplified models may provide unsatisfactory results if compared to experimental data while, on the other hand, including a high level of detail leads to an increase in the model complexity and computational cost, as well as possible difficulties with the parameters availability could emerge. As a result, a variety of computational codes can be found in Refs. [12-18].

Generally, depending on their modelling approach, they can be broadly classified into two categories: lumped-parameter (or globalparameter) codes and computational fluid dynamics (CFD)based (or local-parameter) codes.

Also known as 0.5-dimensional or thermodynamics codes, lumped-parameter models are based on the thermodynamic behavior of the combustion gases behind the projectile. In particular, a defined pressure distribution between the gun breech and the projectile base is considered, together with the assumption of a uniform and instantaneous ignition of the entire propellant charge, which then combusts in a well-stirred mixture evenly distributed throughout the barrel [19-23]. Lumped-parameter codes have been successfully used to simulate a variety of gun systems with relatively simple geometries. Because of their simplicity and minimum requirement of computer resources, they have also been employed to perform parametric analysis and optimisation processes. However, lumped-parameter codes are nowadays mainly adopted to obtain quick design evaluations or a quick aid during experimental tests [19]. The reason lies in their inability to address the hydrodynamics of the ballistics problem, so that important phenomena causing catastrophic failures of gun systems, like the ignition-induced pressure waves, can not be captured [24]. Similarly, propellant grains fracture, different modes of ignition or complex geometries can not be included. These limitations are overcome by the more sophisticated CFD-based codes that have a better capability to treat transient multiphase systems where more advanced aspects are included, as complex gun geometries, different ignition processes, non-uniformities in flame spreading and the formation of pressure waves, among others. Generally, CFD-based codes are formulated in terms of conserved variables and by applying the mass, momentum and energy conservation equations over control volumes occupied by the propellant grains and the surrounding combustion gases. A key modelling aspect of CFD-based codes, which can be one-, two- or three-dimensional depending on whether axial and radial changes inside the gun are considered, consists in how the dispersed phase is represented. Two approaches exist: the Lagrangian approach, where each particle is followed along its trajectories, and the Eulerian approach, where the ensemble of particles is treated as a continuum governed by the mass and energy conservation equations. Thus, considering that the gas phase is always treated from an Eulerian point of view, the available CFD-based codes lay between two different approximations: Eulerian-Lagrangian and Eulerian-Eulerian. Regarding the latter, a further classification can be done on the basis of the modelling philosophy adopted: separated flow theory and continuum mixture theory. In the first, formal averaging techniques are applied over the two separated continuum flows of gas and solid phases [25] while, in the second, the gas and solid phases are treated as a one continuum mixture flow [26].

In order to homogenise the used mathematical frameworks and help making a comparison among the large set of computational models proposed in the literature, the aim of this review is to present and classify the main (not-restricted) contributions by mentioning their advantages and drawbacks. Possible links among the proposed lumped-parameter and CFD-based models are suggested, aiming at highlighting common features and analogies. Some comments on the choice of one model over another are also provided, as well as an outlook on the limitations of the currently available modelling strategies and some suggestions on possible future developments. A comprehensive discussion on the current issues and future directions of work development, however, goes beyond the scope of the present paper. This will be the topic of future work.

In particular, following the aforementioned modelling features, the plan of the review is as follows. Initially, a short description of the standard test AGARD gun, very important for validating internal ballistic codes is provided in Section 2. Lumped-parameter codes are then presented in Section 3, where the basic theoretical tools to understand this modelling approach are illustrated, together with a survey of its main applications available in the literature. CFDbased codes are then presented in Section 4, where the large set of models reported in the literature are organised in terms of the used mathematical framework. The Eulerian-Eulerian and Eulerian-Lagrange approaches are separately described, with a special attention on how the physical processes involved are modelled in the two cases. Finally, Section 5 concludes the paper. In particular, possible links among the proposed This is followed by some comments on the choice of one model over another, with an insight into the limitations of the currently available modelling strategies and an outlook on possible future developments.

Considering the extreme large amount of literature flourished in the field of internal ballistics, which makes impossible to quote without omissions all the currently available contributions, it should be noted that here the analysis is restricted to gun systems. However, an extended list of references can also be found throughout the text, which provide the interested reader additional details and/or modelling approaches, such as analytical and empirical models, not presented in this paper.

2. The standart test AGARD gun

Before specifically focusing on the lumped-parameter and CFDbased models, it is worth mentioning the AGARD (Advisory Group for Aerospace Research and Development) gun test, very important for verifying the internal ballistic codes. As it will be seen in the following, in the literature this test has been often adopted for validation purposes in the case of both lumped-parameter and CFDbased models.

he AGARD gun test, specifically established as a standard test in the development of internal ballistic codes [27], is completely fictitious and does not physically represent any particular gun configuration. Specifically, it involves a smooth bore 132 mm diameter gun having a chamber length of 762 mm and a total length of 5080 mm, cylindrical seven-hole propellant grains with a total mass of 9.5255 kg and a projectile with a mass of 45.359 kg. Finally, a constant projectile resistance pressure of 13.79 MPa is considered [28,29]. As specified in the AGARD Advisory Report 172 [29], two versions of the test exist, one-dimensional and twodimensional, depending on how the igniter material is injected into the domain. A detailed description of the AGARD test, which goes beyond the scope of the present paper, can be found in Refs. [27-30] and references therein.

3. The lumped-parameter model

3.1. Overview

As previously mentioned, even if less sophisticated than the CFD-based model, the lumped-parameter approach is internationally accepted as an essential tool to perform fast and accurate internal ballistics calculations. Being very simple and less computationally expensive, it plays a vital role during the design process of propellants, ammunitions and guns [31-33], from the conceptual phase to the optimisation one. It is also very useful for the prediction of a given gun performance, by computing the pressure, projectile velocity and projectile position inside the gun, or to determine whether or not it is safe to fire such weapon with a particular mass of propellant.

Generally, the development of a lumped-parameter model for describing the physiochemical processes during the shot, proceeds through consideration of the following areas.

- gas equation of state,

- propellant burning rate equation and form function analysis,

- projectile motion and variable volume considerations,

- pressure considerations,

- heat-loss effects.

From a mathematical point of view, these aspect leads to a set of ordinary differential equations that, lumped together and solved by numerical integration, simulate the internal ballistics cycle of a gun system. These equations are the equation of energy, expressing the energy conservation for the gas behind the projectile and directly arising from the 1st principle of thermodynamics, the gas equation of state, the burning rate equation, which gives the dependency between the propellant burning rate and the gas pressure, the equation of motion for the projectile, written on the basis of the Newton's second law, and the pressure gradient equation, which takes into account the gas pressure gradient between the chamber breech and the projectile base.

Generally, in the numerical computational routine for the simultaneous solution of the aforementioned equations, the internal ballistics problem is split into three phases described by a different set of governing equations that, if solved at different time steps, provide predictions of quantities like maximum gas pressure, temperature or projectile muzzle velocity [20]. A schematic of the solution algorithm for each phase is represented in Fig. 3. As it can be seen, in phase I, which begins with the ignition of the propellant and ends when the projectile starts to move, the first step consists in solving the burning rate equation, from which the fraction of propellant burnt is obtained. The gas temperature, the spaceaveraged pressure in the chamber and the projectile base pressure can then be obtained by respectively solving the energy equation, gas state equation and pressure gradient equation. From these results, which are used as initial conditions for the calculations during the ensuing time interval, a new fraction of burnt propellant is obtained, leading to new values of gas temperature, chamber space-mean pressure and projectile base pressure. As illustrated in Fig. 3, this calculation loop continues until the pressure exerted on the projectile base exceeds a preselected shot-start pressure and the projectile begins to move.

The second phase thus begins. Its governing equations coincide with those of the first phase, i.e., burning rate equation, energy equation, gas state equation and pressure gradient equation, with the addition of the projectile motion equation. If solved, the latter provides the acceleration, velocity and displacement of the projectile, from which the gas volume enlargement due to the projectile motion can be calculated. Again, calculations are performed at different time steps and, as before, results obtained at the end of each time interval define the initial conditions for the following one. This solution process continues until the propellant is burnt out, event that marks the end of the second phase and the beginning of the third one (Fig. 3). Being the propellant completely consumed, in the third phase the fraction of burnt propellant becomes unitary, so that the corresponding burning rate equation can be eliminated from the solution loop. As illustrated in Fig. 3, also in the third phase the gas temperature, the space-mean pressure in the chamber and the projectile base pressure are evaluated at each time step by solving, in turn, the energy equation, the gas state equation and the pressure gradient equation, while the new gas volume and projectile acceleration, velocity and displacement are given by the projectile motion equation. At the end of each time interval, a check of the projectile displacement is also made to determine whether or not it has reached the muzzle of the gun. When the projectile passes the muzzle, the third phase terminates and the computational routine is stopped.

Numerous lumped-parameter models can be found in the literature but most of them are just a variation of one or more of the following aspects: gas equation of state, pressure gradient equation, propellant form function and propellant burning rate equation. Regarding the latter, considerations on determining the burning rate of fine-grained propellants based on closed-vessels tests are reported in Ref. [34] while, in Ref. [35], the results of a preliminary application correctness assessment of the physical burning law in interior ballistics modelling are presented. For a detailed discussion, which goes beyond the scope of the present paper, the reader is referred to Refs. [34,35] and references therein. Additional details on the various approaches to the abovementioned areas can also be found in the contributions presented in the following where, for the interested reader, an extended list of references in also provided.

For sake of clarity, it should be noted that the analysis is restricted to gun systems since, as previously mentioned, an extreme large amount of contributions are currently available in the field of internal ballistics (e.g. rocket motor systems, ...).

3.2. The lumped-parameter model of Baer and Frankle

The first lumped-parameter model considered is the model developed by Baer and Frankle at the US Ballistic Research Laboratory [20]. Being the baseline for many of the currently available lumped-parameter models, the description of its main characteristics is of order.

3.2.1. Model assumptions

The Baer and Frankle model aims at simulating the onedimensional motion of a spin stabilised artillery projectile inside a conventional gun tube. The model, general enough to include a propelling charge composed by n different species of propellants, is based on a number of simplifying assumptions.

The first consists in assuming that, for each propellant specie, all grains are identical in terms of physical and chemical properties and that, at the beginning of the simulation, all propellant inside the chamber is ignited instantaneously and simultaneously. A linear burning rate equation expressed as a function of the gas space-mean pressure is also considered. In addition, before the simulation commences, the primer is completely consumed, so that the initial gas within the chamber only consists of primer combustion products (ambient air within the chamber is neglected). The second simplification concerns the combustion process, assumed to be an adiabatic phenomenon (heat loss to the gun and related components is ignored) taking place in a closed-volume with a constant speed on all propellant grains. As typical of lumped-parameter models, from a geometrical point of view, any chambrage in the gun system is neglected, i.e., both the chamber and the bore have the same diameter. The projectile, whose mass is constant for assumption, has a flat base on which the gas pressure is uniformly distributed. Finally, no gas leakage between the moving projectile and the bore is considered, being prevented by the projectile rotational band for assumption.

Obviously, not all these simplifications hold true, so that, to provide reasonably good predictions in terms of pressure and velocity, an extended experimental campaign has been performed to verify/calibrate data prone to errors.

3.2.2. Governing equations

3.2.2.1. Energy equation. Direct application of the 1st law of Thermodynamics to the combustion gases behind the projectile leads to the energy conservation equation of the Baer and Frankle model [20].

...(1)

...(2)

the energy consumed to move the projectile, assumed to be the sum of the energies released by the individual n-th propellant specie,

...(3)

the internal energy of the gas,

...(4)

the energy consumed to move the projectile, which coincides with the projectile's gain in kinetic energy (translational movement),

...(5)

the sum of the energy losses of the system: Epr due to the projectile rotation, Epk to the motion of the combustion gas and unburned propellant grains, Ebr the energy lost in engraving the rotating band and in overcoming the friction down the bore, Ed due to the air resistance, Erc to the recoiling parts of the gun, Eh the heat transfer to the environment because of non-isolated gun walls.

In Eqs. (2) and (3), T0i, mi and Zi stand, respectively, for the adiabatic flame temperature, initial mass and fraction of mass burnt of the i-th propellant specie, ... for its specific heat capacity at constant volume, Fi its force per unit mass [36] and ... the ratio between its specific heat capacity at constant pressure , cpi, and at constant volume, cvi. The corresponding properties of the igniter are denoted with ... I. Note that, according to the Baer and Frankle model, cvi (and cvI) is assumed to not vary over the temperature range form Tg, the gas temperature, to T0i (and T0I), being Tg <T0i (and T0I) because of the gas expansion and external work performed in the gun system. Accordingly, a constant value of cvi (and cvI) can be considered. The same applies to cpi (and cpI). Finally, in Eq. (4), Pb is the pressure acting on the projectile base having area Ab, xp the projectile travel and mp and vpr the projectile mass and velocity, on order.

Substituting Eqs. (2)e(5) into Eq. (1) and solving for Tg lead to [37]

...(6)

where ZI≡1, i.e., igniter assumed to be completely burned at zerotime. For sake of conciseness, we refer the interested reader to Refs. [20,37] for the extended expression of El.

3.2.2.2. Equation of state. Considering that in interior ballistics the ideal gas equation of state is not suitable because of the extreme values of pressure and temperature reached at a very short time step, in the Baer and Frankle model the Noble-Abel equation of Ref. [38] is used. Differently from the ideal gas law, to which is very similar at a first look, the Noble-Abel equation takes into account the real gas behavior and, in particular, the volume taken up by the gas molecules themselves.

In its general form, the Noble-Abel equation can be expressed as [38].

...(7))

with P, T and v, respectively, the pressure, temperature and specific volume of the gas, R the gas constant associated with the specific gas component (... R, with b R the universal gas constant and nmol the number of moles/kg of the gas) and b, known as covolume, a sort of correction related to the molecular volume per unit mass of gas.

By directly applying Eq. (7) to the internal ballistics case and specialising it for n propellant species, in Ref. [20] the space-mean gas pressure between the chamber breech and the projectile base, Pg, is derived

...(8)

with

...(9)

the volume available for the combustion gas. This quantity is influenced by the initial chamber volume V0, the volume of nonenergetic charge components associated with the i-th propellant Vpi (second term), the volume increase resulting from the projectile motion (third term), being Ap the projectile surface, and by the volume occupied by the unburnt propellant (fourth term) and the gas molecules of the burnt propellant and igniter (fifth and sixth term). The terms bi, bI and ri introduced in Eq. (9) are, respectively, the co-volume of the i-th propellant and igniter, both of them assumed to be constant over the temperature range from ... and the i-th propellant density. Again, in Eqs. (8) and (9) ... 1. It should be noted that, in calculating Eq. (8), Baer and Frankle simply assume that the total gas pressure is the sum of the pressures resulting from the n burned propellants.

3.2.2.3. Pressure gradient model. Due to the pressure gradient effect [39], produced by the continuous reflection of expansion and rarefaction waves against the gun breech and the base of the moving projectile, the gas pressure acting at the gun breech, P0, and at projectile base, Pb, are different from Pg, with Pb < Pg.

Different models exist to take into account this phenomenon. In Ref. [20], an improved version of the Pidduck-Kent gradient model [8] is used in order to include the possibility of treating composite charges. This leads to

...(10)

...(11)

with wt the total weight of the propellant and igniter, wp the weight of the projectile,

...(12)

n ... 0 the effective ratio of specific heats, ... the Pidduck-Kent constant, expressed as a fifth-order polynomial in propellant charge-to-projectile mass ratio and scaled with the propellant specific heat ratio [40]. For more details, we refer the interested reader to Refs. [20,40].

3.2.2.4. Burning rate equation. In Eqs. (2), (3), (6) and (8) we have introduced the term Zi, representing the fraction of mass burnt of the i-th propellant. This quantity can be evaluated at a given time t according to the following relation

...(13)

where ... dt is the i-th propellant mass fraction burning rate, expressing the rate of consumption of propellant and thus the rate of evolution of combustion gas. In the Baer and Frankle model, _ Zi is calculated based on Piobert's law [37]

...(14)

with Si and Vi, respectively, the surface area and volume of the unburned i-th propellant grain. An adjusted version of the linear burning rate equation of Saint Robert's (also called Vieille's law) is also used, given by

...(15)

being bi and ai experimental coefficients depending on the propellant constituents and burning conditions.

3.2.2.5. Equation of projectile motion. Governed by the Newton's second law, the equation of motion provides the translational acceleration, velocity and displacement of the projectile as a function of the forces acting on it during the internal ballistic cycle.

As illustrated in Fig. 4, three types of forces are considered in Ref. [20]: the propulsive force Fb : = Ab,Pb, resulting from the pressure of the propellant gas Pb, and the two resistive forces Fr := Pr,Ab and Fa := Pa,Ab due to the resistive pressure of the bore Pr and to the pressure of air ahead of the projectile Pa. As a result, the acceleration of the projectile centre of mass takes the form [20]

...(16)

relation from which the projectile velocity vpr and travel xp (with respect to the ground reference frame) at a given time t can be evaluated

...(17)

3.3. IBHVG2

Another well validated lumped-parameter model is IBHVG2 (Interior Ballistics of High Velocity Guns, version 2), developed by the US Army Ballistic Research Laboratory [41]. Based on the work of Baer and Frankle [20], IBHVG2 aims at simulating the interior ballistic cycle of large calibre guns by predicting the gross ballistic variables: the space-mean gas pressure between the chamber breech and the projectile base, the gas temperature of the gun chamber, the acceleration and velocity of the projectile, the mass fraction of each unburned propellant.

As with the Baer and Frankle model, IBHVG2 assumes that the primer is completely burned at the beginning of the simulation and that the propellant grains, all identical in terms of physical and chemical properties, burn uniformly and instantaneously after the ignition.

Again, the interior ballistic event is described as a set of ordinary differential equations solved by marching forward in time. The first equation is the energy equation, obtained by applying the 1st principle of Thermodynamics to the closed ballistic system and providing the same expression of Eq. (6) for the gas mean temperature. Note that, also in this case, the same type of energy losses listed in Eq. (5) are considered, as well as the possibility of considering n propellant species is included.

It follows the gas equation of state, based on the Noble-Abel equation and leading to the same gas space-mean pressure of Eq. 8), and the propellant burning rate equation, having the same form of Eq. (15), with the addition of the propellant burning rate acceleration factor xi (dimensionless empirical factor) [41].

...(18)

bi and ai are the same experimental coefficients defined in Eq. (15), Pg is the gas space-mean pressure and vp is the projectile velocity. Two possibilities are offered by IBHVG2 for considering the pressure gradient effect of the system. The first is an improved version of the Lagrangian pressure distribution [8] in order to include the resistive pressure that the projectile experiences while accelerating along the barrel, aspect neglected in the classical Lagrange model. From this, the derived gas pressure at the gun breech and at projectile base take the form

...(19)

...(20)

with wt, wp, Pa and Pr the quantities previously defined.

The second is the Pidduck-Kent gradient model [8], where the gas pressure at the breech and at projectile base are a function of the Pidduck-Kent constant as in Eqs. (10) and (20).

The proposed lumped-parameter model is completed by the equation of projectile motion written, also in this case, in terms of Newton's second law. This provides the acceleration (and corresponding velocity and travel) of the projectile, which have the same form of Eq. (16), being the same type of resistance forces considered.

Finally, all the input data in IBHVG2 is based on closed-vessel experiments [42,43] (for propellant data) and physical measurements (for the resistance profile encountered by the projectile), obtained from a series of 155 mm howitzer firing with short, slotted propellant sticks (25 mm). These values, according to the authors, are representative of the results that one would expect for granular charges in large calibre weapons [41].

Additional subroutines can also be added to the basic version of IBHVG2. These include, among others, a number of propellant form-functions for calculating at any instant the surface area, volume and/or fraction burned for a burning propellant grain based on its initial dimensions and depth burned [44].

3.4. The lumped-parameter model of STANAG 4367

STANAG 4367 [45] is the lumped-parameter model standardised by the North Atlantic Treaty Organisation (NATO) and is considered one of the most accurate lumped-parameter models for internal ballistics simulations [46]. However, the STANAG 4367 documentation has a limited distribution only so that, because of this confidentiality restriction, for more details we refer the interested reader to Ref. [45].

3.5. The Mayer-Hart lumped-parameter model

Another lumped-parameter model that is worth mentioning has been developed by Mayer and Hart in 1945 at the US Ballistic Research Laboratory [47,48]. The authors, based on a number of simplifying assumptions like propellant burning rate proportional to pressure, constant burning surface and no heat losses through gas or projectile frictions, propose a set of simple equations for describing the internal ballistic process of artillery weapons.

Being of the greatest importance from a practical point of view, the expressions for the maximum pressure Pmax;MH and muzzle energy Em;MH are more fully discussed and considerations on how they are affected by a change in the weapon's parameters are reported. This has been done by focusing on the approximated pressure and energy relations given, respectively, by [48]

...(21)

...(22)

with ... and r analytical constants calculated from the weapon and charge characteristics, e = 2.718 the base of the natural logarithm system, g0 the ratio of specific heats, C the total charge, ... the specific propellant force.

Interestingly, denoted with wp the projectile weight, the projectile muzzle velocity Em;MH can be easily derived from the relation [48]

...(23)

A more detailed discussion can be found in Refs. [47,48].

3.6. Practical applications and possible improvements of the lumped-parameter method

Because of its simplicity and reduced implementation time, the lumped-parameter approach is an efficient tool to obtain quick and accurate results in terms of projectile motion and pressure levels reached in a given gun system. Quoting without omissions the extreme large amount of literature currently available in the field would be impossible. The main (not-restricted) contributions are presented in the following, categorised according to the type of application. For the interested reader, a list of references is also provided, describing more special applications of the lumpedparameter model.

3.6.1. Optimisation framework

The reduced implementation time of the lumped-parameter model is particularly useful for the designer trying to optimise the interior ballistic performance, since the ideal internal ballistic model should be simple and fast in order to save computation time when coupled with the optimisation code.

One example is the work in Ref. [49], where the lumpedparameter model is coupled with the particle swarm optimisation technique (PSO) to optimise the propelling charge design of a 76 mm naval gun with guided projectile. The propelling charge considered is a mixture of seven-perforated granular propellant and one-hole tubular propellant, so that a properly defined form shape function needs to be used in the burning rate equation of the lumped-parameter model. Two scenarios are investigated: a singleobjective problem, aiming at maximising the projectile muzzle velocity without violating the gun pressure constraints, and a multi-objective problem, where the muzzle velocity, muzzle pressure and energy efficiency of the charge are selected as objective functions. In both cases, from the optimisation results obtained, it emerges not only a very good agreement with the experimental data, but also a better performance in terms of convergence speed and solution accuracy [49].

A further example of internal ballistic performance optimisation based on the lumped-parameter approach is described in Ref. [50]. Also in this case, the authors couple an optimisation code with the lumped-parameter model in order to find the optimum charge design for a 76 mm naval medium calibre gun. Two types of propelling charge are investigated: a single propellant charge consisting of granular seven-perforated propellant, and a mixed propellant charge consisting of granular seven-perforated propellant and tubular one-hole propellant. By focusing on the projectile muzzle velocity, peak pressure and muzzle pressure, whose values at each optimisation step are provided by the lumped-parameter model, as objective functions, just few seconds of computations on a personal computer lead to the optimum charge configuration that simultaneously decreases the muzzle pressure and peak pressure and increases the muzzle velocity. A very good accordance between the experimental and computational results also emerges.

Similar results are proposed in Ref. [33], where the lumpedparameter model is used in conjunction with the PSO technique to optimise the charge design of a real 130 mm guided projectile. Two specific applications are discussed: a single-objective optimisation problem for conventional seven-perforated propellants, and a multi-objective optimisation problem for deterred sevenperforated propellants. As in Ref. [49], the maximisation of the muzzle velocity is considered in the first case while, in the second, the maximisation of the muzzle velocity, the minimisation of the muzzle pressure and the maximisation of the propellant efficiency are analysed. Again, the values of the objective functions at each optimisation step are provided by the lumped-parameter model.

The application of the lumped-parameter model for exploring novel grain geometries produced by non-conventional means like additive manufacturing is reported in Ref. [30]. As in Refs. [33,49,50], the lumped-parameter model is selected because of its accuracy and very fast run time, making it the best choice for the parametric optimisation of different grain geometries. For validation purposes, the theoretical AGARD gun is considered. The analysis reveals a good consistency of the obtained results, as well as a running time of 0.74 s to perform 1000 simulations (on a Intel Xeon CPU E5-2687W@ 3.40 GHz), confirming the suitability of the lumped-parameter model for optimisation purposes.

An additional application of the lumped-parameter model for performing optimisation studies can be found in Ref. [51], where the authors investigate the potential performance of new propellant formulations. The developed code, called CONPRESS (Constant Pressure Interior Ballistics Code), is based on a constant breech pressure assumption that, according to the authors, has the advantage of eliminating the dependency of the velocity predictions upon the grain geometry. This feature is essential for propulsion concepts involving liquid, gel, emulsified or slurry propellants due to the lack of fixed propellant geometry.

Same conclusions as in Ref. [30] are reported in Refs. [31,32], where the lumped-parameter model is employed in optimisation problems to compute the pressure and the projectile velocity and position inside the barrel. Again, the lumped-parameter model is found to be a relevant tool for the optimum design of propellants, ammunitions and guns.

3.6.2. Study of the internal ballistics of rifle systems

Outside of the optimisation framework, Ref. [52] documents the study of the internal ballistic process of a rifle 7.62 performed with the lumped-parameter model described in Ref. [53], which is based on the work of Baer and Frankle [20]. The maximum pressure, muzzle pressure, and projectile velocity are predicted and compared with the experimental data. A discrepancy of approximately 1% emerges, indicating that, despite its simplicity, the lumped-parameter model is still a relevant tool for gaining engineering information. Note that it is not specified if this very good comparison is obtained by using fitting parameters. However, discussing the validity of the proposed strategy goes beyond the aim of the present paper, which simply consists in describing the status of the art.

The influence of the characteristics of the barrel and powder charge on the gas space-mean pressure and projectile travel and velocity is discussed in Ref. [54] based on the lumped-parameter approach. The authors, in particular, simulate a number of different shot processes involving a 7.62 mm SVD sniper rifle with 7.62 52 rifle cartridge and a 122 mm D-30 howitzer with 122 mm shot at nominal values of charging parameters [54]. Models of influence are constructed, where typical defects like swelling, wear of the barrel, fall of the powder force, increase in the combustion rate, are also included. From a practical point of view, the collected data allow the creation of a database of ballistic elements useful in the diagnosis of weapons and ammunitions.

3.6.3. Lumped-parameter model coupled with the finite element method

A method of coupling the lumped-parameter model with the finite element method (FEM) is proposed in Refs. [55,56]. In Ref. [55], in particular, the lumped-parameter model is utilised to predict the propellant combustion of a gun launch system driven by high-pressure reactive gases, while the FEM approximates the system's mechanical interactions between the projectile and the barrel. The coupling effects are performed by exchanging parameters between the two methods: the FEM calculations provide the kinetic parameters of the moving projectile, i.e., the travel and velocity, for the combustion calculations, the lumped-parameter model gives the loading conditions, i.e., projectile base pressure, for the FEM approximations. Based on the coupled computational framework proposed, validated through experimental tests, the influence of multiple structural parameters determining the geometry of the system main components on the overall performances is also investigated.

A similar strategy is described in Ref. [56] to study the influence of the projectile-barrel interaction on the internal ballistic performance of low-velocity launchers. Again, the lumped-parameter model governs the propellant combustion while the mechanical aspects including position and velocity of the projectile are predicted by FEM. The obtained results, verified by comparison with experimental data, reveal a very good agreement in terms of projectile muzzle velocity.

3.6.4. Rarefaction wave guns

If compared to traditional guns, rarefaction wave guns can significantly reduce the recoil moment and barrel heat while having a minimal effect on the projectile velocity during the launching process. This topic, which goes beyond the aim of the present paper, has been considered in Refs. [57e59], where the lumpedparameter model is coupled with a two-dimensional gas-solid two-phase model to gain a better understanding of the muzzle flow characteristics during a rarefaction wave gun firing.

However, in the case of rarefaction wave guns and, in general, in cases involving special gun designs, an adapted version of the lumped-parameter model including additional terms should be used. More details can be found in Refs. [57e59] and references therein.

3.6.5. Composite charges

As seen in the previous sections, a number of simplified assumptions are adopted by the lumped-parameter model in its classical form (cf. Subsections 3.2, 3.3). These include the same diameter for the gun chamber and bore, propellant grains ignited instantaneously and simultaneously, homogeneous distribution of the gas thermodynamic parameters behind the projectile, Lagrangian or Pidduck-Kent pressure gradient distribution, among others. Despite good comparisons between the predicted results and test data are usually found, in the literature much effort has been devoted to develop an improved version of the lumpedparameter model.

Examples are the works in Refs. [8,60], where a lumpedparameter model considering the case of composite propellant charges is presented. The authors suggest to replace the composite charge with an equivalent one, whose ballistic properties are the sum of the ballistic properties of the single components.

Composite charges are also considered in Ref. [61], where the authors suggest a modification to the IBHVG2 code so that, at each step of the ballistic cycle, the gas composition and pressure of each individual propellant are calculated. Equilibrium and pressure calculations for the overall gas composition are then performed.

Evolvements of this pioneering approach can be found in Refs. [62e65], where the process of combustion for the individual charge components is introduced, and in Ref. [66], where not coincidental instants of ignition for both the different charge components and the propellant grains pertaining to a given component are taken into account. The thermodynamic properties of the combustion gases are then modified accordingly. In Ref. [66], in particular, the accuracy of the resulting model is given on the basis of experimental data involving a 125 mm 2A46 tank gun firing a hard core projectile. By considering the maximum pressure and projectile muzzle velocity as verification criteria, a very good agreement emerges, with a discrepancy of 3.1% and 0.8%, on order. As in Ref. [52], it is not clear if this very good comparison is the result of the application of fitting parameters. However, as previously stated, discussing the validity of the described strategy goes beyond the aim of the present paper.

3.6.6. Small calibre gun systems

Starting from the consideration that lumped-parameter models are mainly conceived for medium and large calibre gun systems, the authors in Ref. [67] propose an adapted version of the lumped model suitable for small calibre guns, in which deterred propellants with multivariate size distribution of grains are often used. This distinguishing feature, incorporated into the lumped-parameter model described in STANAG 4367 [45], leads to the so-called ICTSCIBS code (Small Calibre Interior BallisticS). By addressing the shortcomings of the existing lumped models, this code allows the user to incorporate a non-uniform multivariate size distribution of the propellant grains, a depth-dependent burning rate and a depthand pressure-dependent thermochemistry of the propellant.

Small calibre guns and deterred propellants are also considered in Refs. [68,69] where, again, a modified version of the lumpedparameter model in Ref. [45] is presented. Several amendments are introduced in order to deal with the variability of the propellant characteristics, including the force, flame temperature, covolume and burning rate coefficients [69] (cf. Subsection 3.2.2). Because of these variable ballistic properties, a modification of Eqs. (6) and (8) is foreseen, together with an adjustment in the calculation of the heat transferred to the barrel from the combustion gases (cf. Eq. (5)).

3.6.7. High-low (HiLo) pressure systems

Investigations into the use of the lumped-parameter model for HiLo systems [70], i.e., weapons built on high-low pressure principle involving the existence of two chambers between which the gas flow occurs, is described in Ref. [71]. For these systems, the lumped-parameter approach is preferred since the more complex multi-phase flow models would require numerous input data and considerable computational efforts to achieve high fidelity. However, an improved version of the lumped-parameter model needs to be developed, based on more realistic assumptions like an adiabatic transformation of the gases into the high pressure chamber, an isentropic flow of real gases through the nozzles, a variable burning surface of the propellant grains and a more accurate definition of the heat losses [71]. As reported in Ref. [71], notwithstanding the complex physical processes involved in HiLo systems, these modifications lead to a lumped-parameter model giving results consistent with the experimental measures, obtained through real firings on a 40 mm grenade launcher, i.e., the most popular application of HiLo principle, and focusing on the muzzle velocity.

High-low pressure systems are also investigated in Ref. [72], where the two-fluid model (TFM) and the lumped-parameter model are combined to describe the multi-dimensional behavior of the combustion gas and propellant grains during the launching process. In the proposed coupled approach, which takes into account the gas production, interphase drag, intergranular stress and heat transfer between the two phases, the lumped-parameter model is used in the first stage of the simulation, i.e., before the opening of the vent-holes connecting the two chambers. Then, to characterise the multi-dimensionality of the flow, the TFM model is adopted due to its ability to capture the radial effects, which are very important for high-low systems [72]. After a validation process against experimental results, different types of serial and parallel structures are compared in terms of pressure and temperature distribution, projectile velocity, energy conversion and particle distribution.

3.6.8. Gun chambrage

In the Army context, Refs. [37,73,74] report a new closed-form pressure gradient equation that, differently from the Lagrange and Pidduck-Kent approximations, takes into account the chambrage of the gun, i.e., a non-uniform cross-sectional area of the chamber, and the two-phase structure of the flow inside the barrel, i.e., propellant grains and combustion gases. The so-called propellant velocity lag, a phenomenon in which propellant grains and surrounding combustion gases, because of their different mass, does not move with the same velocity, is also represented. This aspect is neglected in the Lagrange analysis, according to which either the propellant is all burnt at the initial instant of the simulation, or gas-phase and burning propellant grains have the same velocity [74], so that the two-phase flow can be considered as a uniform mixture moving along the barrel. Embedded into Eqs. (10) and (20) and programmed into a lumped-parameter code, the modifications in Refs. [73,74] provide an improved tool that, even if not sophisticated as a fully two-phase analysis, is able to threat, among others, the occurrence of pressure waves and rarefaction waves, while retaining the computational economy of the lumpedparameter approach [74]. If compared to the traditional Lagrange formula, significant enhancements are observed when high performance charges are concerned.

An extension of the Lagrange gradient equation to include the chambrage of the gun is also proposed in Ref. [75] to evaluate how the intergranular stresses and flame-spreading process are influence by the boattail intrusion.

3.6.9. Hypervelocity and electrothermal gun system

Additional considerations about the range of applicability of the traditional Lagrange and Pidduck-Kent pressure gradient models in hypervelocity and electrothermal gun systems can be found in Ref. [39]. By referring the interested reader to Ref. [39] for more details, the authors generally suggest that traditional pressure gradient models are not suitable when low molecular weight combustion gases are involved.

4. Multiphase computational fluid dynamic (CFD) method

4.1. Overview

From Section 3 it clearly emerges that lumped-parameter models are an efficient tool to estimate the pressure levels and the motion of the projectile in a given gun system. However, because of their simplicity, these models are not able to treat the hydrodynamics of the ballistic problem so that the unsteady, multidimensional flow of propellant grains and combustion gases behind the projectile can not be accurately addressed. Similarly, any localised pressure field effects resulting because of the pressure waves and propellant interactions [41,76] can not be captured. Consequently, as emphasised by Goldstain [77] and Klingenberg [78], more complex modelling techniques capable of describing non-steady, turbulent two-phase flow effects are required. This paves the way to the development of the more sophisticated twophase (i.e., combustion gas and propellant grains) computational fluid dynamic (CFD) approach where, according to Ref. [78], heat and mass transfer effects, chambrage, erosive burning and fouling can be included. The importance of not neglecting the propellant grains interactions is also emphasised. However, this requires large computational efforts and computer abilities, so that many of the currently available CFD-based models presented in the following employ simplifying assumptions that reduce the complexity of the real phenomena while still retaining the core dynamics of the original problem.

As illustrated in Fig. 5, existing CFD-based methods can be classified into different categories, according to their degree of detail and complexity. The first distinction is based on how the particulate solid phase is considered. Two approaches are possible [79,80]: the Eulerian-Eulerian one, also called two-fluid model, where both the gas and the solid phase are represented as a continuum with its mass, momentum and energy conservation equations, and the Eulerian-Lagrangian one, which treats the gas phase as a continuum and the solid phase as individual particles traced in time and space. Depending on weather axial, radial and circumferential changes inside the gun are considered, both the EulerianEulerian and Eulerian-Lagrangian models can be one-dimensional, two-dimensional axial-symmetric or fully three-dimensional (Fig. 5). In the first case, only axial changes are involved while, in the second and third, axial * radial and axial * radial * circumferential changes are respectively taken into account. Obviously, given their complexity, fully three-dimensional models require huge computer resources and a larger running time if compared to the other two approaches. Additional computer resources are also required because of the CFD adaptive mesh techniques, i.e., mesh movement and dynamic remeshing, which allow an improved representation of the projectile movement and gas flow field [81]. A comprehensive discussion of these aspects goes beyond our scope. For the interested reader, more details can be found in Refs. [81,82] and the references therein.

Finally, CFD-based models can be grouped according to the degree of coupling between the solid and gas phase (Fig. 5). Three different situations can be distinguished, depending on the volume fraction of the solid phase. Specifically, referring to Fig. 6, we have models with 1-way coupling, which treat the solid phase as a passive scalar with no influence on the gas flow, 2-way coupling, where the mutual effect of the solid phase and gas flow is considered, and 4-way coupling, which also take into account the collisions between solid particles.

Specific to gun systems, military research laboratories played a leading role in the development of multiphase CFD interior ballistic codes to meet varying design needs [83]. These codes are the result of many years of elaboration and validation and, even if most of them are not commercially available, provided the theory behind many of the two-phase CFD models developed outside of the Army Research Laboratories. For example, the U.S. Army Research Laboratory (ARL) popularised the Eulerian-Eulerian one-dimensional, two-phase NOVA code (also known as XKTC) [74,84], the EulerianLagrangian multi-dimensional, two-phase codes of the family NGEN [85e89], and the multi-dimensional CRAFT code for simulating multi-dimensional processes in ETC guns [90]. In Europe, we have the two-dimensional French code MOBIDIC [91,92], the German multi-dimensional code AMI [93,94], the English onedimensional code CTA1 [95] and two-dimensional codes FHIBS [96] and QIMIBS/FNGUN2D [97]. For the interested reader, a comparison of the European codes AMI, MOBIDIC, CTA1 and FHIBS is available in Refs. [98,99], where the authors tested how well the predicted maximum pressure, muzzle velocities and time of shot exit agree with the measured 40 mm gun firings data. A comparison of the lumped-parameter code IBHVG2 and the CTA1 code is also proposed in Ref. [100]. The study involves high velocity ETC guns using several advanced charge concepts with high loading density propellant geometries (19-perforated cylinders, full chamber diameter multi-perforated discs, full chamber diameter layered discs). Pros and cons of the two methods are highlighted, with a special attention to their ability in predicting the projectile muzzle velocity and maximum projectile base pressure.

The most recent advances in the field of internal ballistics CFD modelling of gun systems are presented in the following sections, according to the modelling approach adopted: Eulerian-Eulerian in Subsection 4.2, Eulerian-Lagrangian in Subsection 4.3. The cited references can be complemented with the reviews in Refs. [11,68,78], where the internal ballistic modelling methods up to 1991 are illustrated. An historical review in the area of computational internal ballistics can also be found in Ref. [12].

To summarise, as it will be seen in the following, developing a CFD-based model for describing the physical and chemical processes during the shot, goes through consideration of a number of different aspects. Similarly to the lumped-parameter model (cf. Subsection 3.1), these include.

- gas equation of state,

- propellant burning rate equation and form function analysis,

- projectile motion and variable volume considerations,

- considerations on how the solid phase is modelled (discrete or continuum),

- interactions between the solid and gas phase.

As stated, various formulations are currently available in the literature for modelling, for example, the interactions between gassolid phase and/or between solid particles, or for taking into account the variable volume of the domain. Regarding these aspects, a more detailed discussion is provided in the following sections, where a list of references can also be found.

4.2. Eulerian-Eulerian approach

If compared to the Eulerian-Lagrangian, the main advantage of the Eulerian-Eulerian approach consists in requiring less computational resources. Very first attempts of describing the internal ballistic process via CFD-based modelling are thus based on the Eulerian-Eulerian approach since, at that time, Lagrangian simulations were not affordable.

As anticipated in Subsection 4.1, two possibilities exist when analysing the two-phase flow of solid particles and combustion gas with the Eulerian-Eulerian approach [9]. The first is the Two-phase Flow model, where the two flows of gas and particles are treated separately as two complementary homogeneous continua described by appropriately defined volume-averaged properties. The second is the One Mixture Flow Model, according to which the two flows of gas and solid are considered as a mixture of one continuum flow whose properties are obtained by applying the mixture theory.

For sake of clarity, in the following the two approaches are presented separately even if, from a mathematical point of view, whatever is the strategy adopted, the type of governing equations considered is the same. Namely, the mass, momentum and energy conservation equations, plus some closure relations accounting for the thermodynamic behavior of the two phases and their interactions. Notwithstanding additional inertial coupling terms are present in the continuum mixture approach, the work in Ref. [101] demonstrates that the two formulations have no practical influence on the outcome of interior ballistic calculations.

4.2.1. Two-phase flow model

Various two-phase flow models are available in the literature, each of them with their own advantages and drawbacks [102,103].

Very popular in the ballistic community is the model proposed by Gough and Zwarts [104,105], aiming at simulating the twophase flow of combustion gas and solid propellant grains in a conventional medium calibre gun. Developed in the late seventies, the theoretical background of the model has already been extensively elaborated [105e107], so that only its crucial points are addressed in this section.

As usual, to reduce the complexity of the problem to treat, a number of simplifying assumptions are made, including the solid phase considered as an incompressible granular pseudo-fluid and the ignition assumed to be perfect, i.e., all propellant grains ignited simultaneously and uniformly and burning in parallel layers according to the Piobert's law. A regression rate of the burning propellant related to average gas pressure and propellant surface temperature via the Vieille's exponential law is also used. In addition, when the propellant combustion occurs, the energy deposition is taken to be in the gas only, with a loss of mass for the solid phase and an equivalent gain by the gas phase. The gas, inviscid except for its action on the particles through the interphase drag, is modelled as a compressible flow described by the Noble-Abel equation of state with specific heats independent of the temperature. Finally, adiabatic gun walls are considered.

Under these assumptions, the application of mass, momentum and energy conservation equations to the system composed by the solid propellant and surrounding combustion gas in a given control volume leads to the following set of differential equations for the gas phase (compressible flow equations) [19,105].

...(24)

...(25)

...(26)

and for the solid phase (incompressible flow equations) [19,105]

...(27)

...(28)

with ... and ..., ... the volume fraction, density and velocity of the gas phase, the first, and of the solid phase, the second, Pg and Egas, respectively, the pressure and total energy of the gas phase. The source terms ...in Eqs. (24)e(28) are linked to the coupling between phases due to the mass, momentum and energy exchange during the reaction process. They are given, respectively, by [19,105]

...(29)

...(30)

...(31)

where m_ is the mass rate per unit volume generated by the propellant combustion, Eprop the internal energy of the propellant, f d the interphase drag, fig the inter-granular stress, i.e., a measure of the particle-particle force that arises in the packed propellant bed to keep the grains apart [10,19], qp a quantity accounting for the heat transfer per unit area and per unit time between the two phases .

It should be noted that in the mass, momentum and energy conservation Eqs. (24)e(28), directly taken from Refs. [19,105], the contribution of the igniter is not considered. The corresponding ignition mass, momentum and energy are thus not included in the system. Alternative models exist in Refs. [108e110], where igniter elements are taken into account with a level of detail depending on the complexity of the chosen approach. For a detailed analysis, which goes beyond the aim of the present paper, the interested reader is referred to Refs. [108e110].

As anticipated, closure relations need to be added to the system of Eqs. (24)e(28), so that the source terms can be determined, the number of independent variables reduced and the system resolution becomes feasible. For that purpose, constitutive equations, geometrical correlations and empirical formulas are used. Specifically, the first set of equations coincide with the equations of state for the system components: the gas phase, for which the NobleAbel equation is adopted, and the solid phase, for which, in this case, a thermodynamic law is not required due to the incompressibility assumption [10]. It follows the propellant form functions, which provide the grains surface Sp and volume Vp according to their geometry, and allow the calculation of the rate of gas generation [19,105]

...(32)

with r_ the rate of surface regression, assumed to be related to the average gas pressure according to the Vieille's exponential law. A constitutive law is also required to define the inter-granular stress fig. Several proposals are available in the literature, where this macroscopic stress is considered to be isotropic and, when the Mach number based on the relative flow velocity is small, a function of the propellant bed porosity [104,111].

In Ref. [105], fig is assumed to be a function of the intergranular stress tensor Fig

...(33)

with a an empirical constant and 4c the critical porosity above which the inter-granular stress vanishes. Usually, 4c is taken equal to the porosity at which the propellant bed is initially packed [10].

An additional closure relation coincides with the definition of the interphase heat flux, given by Refs. [10,107].

...(34)

with h the convective heat transfer coefficient depending on the grain geometry, Nusselt number and thermal conductivity of the gas [10], Tg and Tps, respectively, the temperature of the gas and propellant surface. Regarding the latter, as there is no energy equation for the solid phase, an assumed temperature distribution within the solid grains is considered [112,113]. Usually, for spherical grains, a parabolic temperature profile is suggested [10,107].

The final closure relation is an empirical law for modelling the drag force f d since, for two-phase flows with high particles concentrations, theoretical expressions are not suitable. An equation often adopted for spherical grains is provided in Ref. [105], based on the Ergun work [114]

...(35)

where 4 is an empirical coefficient to include the effect of the propellant bed fluidisation and Dpe = 6Vpp=Sp the effective particles diameter. In evaluating Dpe with this relation, Ref. [105] implicitly assumes that, at each instant, propellant grains have the same size and are uniformly distributed into the space. However, as observed in Refs. [9,115], this is far from accurate since the pressure gradient and projectile motion induce a spatial distribution of particle size inside the gun. For medium and large caliber guns and for twostage projectile accelerators, some authors use the alternative 'representative particles method' [116], where the spatial characteristics of the propellant grains are derived by focusing on a limited number of representative particles inside the domain. The Spalding's 'shadow method' [117] and its variants [118] can also be adopted [9,115], according to which the particles information in a given location are provided by considering a 'shadow' volume fraction of the solid phase, i.e., the volume fraction that the solid phase would have possessed in absence of combustion, and by solving an additional transport equation expressing the conservation of the number of grains per unit volume.

A further observation regards Eq. (35) that, as discussed in Ref. [101], must be used with some caution. The reason lies in the fact that this relation is based on a limited number of experimental data involving a packed propellant bed, steady-state conditions and a low Reynolds number. In reality, during the internal ballistic cycle, the propellant bed becomes fluidised, Reynolds number is high and the flow is transient. This aspect is partially solved with the factor 4, even if the complexity of the drag force calculation still remains [101].

Finally, the Newton's second law to simulate the motion of the projectile concludes the described two-phase flow model. As previously mentioned, because of the projectile movement, the computational domain moves so that the dynamic remeshing techniques are required. Different approaches to tackle this problem are possible and a number of them are presented in the following. A detailed description, however, goes beyond the aim of this paper, which coincides with the presentation of the more significant (to the authors' opinion) contributions in the field of CFD modelling of internal ballistics.

4.2.1.1. Model applications. In the literature, this approach is chosen by various authors due to its versatility and rapid implementation time.

A first example is the work in Ref. [10], where the EulerianEulerian two-phase flow model is used to develop a onedimensional numerical code for internal ballistic problems involving solid propellant combustion. The code, named UXGun, is based on a conservative numerical method and uses the finite volume approach in conjunction with the splitting technique [119] to separately handle the convective part and source terms of Eqs. (24)e(28). To simulate the movement of the projectile inside the gun barrel, a dynamic meshing algorithm is adopted, while ignition is modelled by adding in the system an amount of energy corresponding to that of the igniter. Numerical results are verified by comparison with experimental test performed in a M114 155-mm howitzer, i.e., a 155 mm/23 calibre howitzer, with a standard single-base cylindrical propellant having 7 holes. A very good agreement emerges in terms of pressure time evolution, maximum pressure in the chamber and projectile muzzle velocity. The capability of the code to simulate pressure waves during the combustion process is also highlighted.

In the same spirit, Ref. [19] present the CFD numerical code Casbar (Collaborative Australian Ballistics Research code) to analyse two-phase flow problems in two-dimensional axisymmetric geometries. The governing Eqs. (24)e(28) are solved by adopting the finite volume approach together with a timestep-splitting technique to separate the various physical processes involved. The suitability of the code is validate through comparison with experimental tests involving, among others, the standard test case AGARD gun.

A two-dimensional axisymmetric model based on the twophase Eulerian-Eulerian approach is also reported in Refs. [9,96,120e122]. By focusing on the pressure profiles in the chamber and projectile muzzle velocity, Ref. [121] claims a quantitatively good agreement with experimental results performed with triple-base smokeless gunpowder in a domain composed by a 781 mm-long chamber and 4740 mm-long barrel having the same diameter of 105 mm. The movement of the solid propellant grains in the chamber and the propagation of pressure waves can also be simulated with the developed procedure. In Ref. [9], the governing Eqs. (24)e(28) are implemented into the CFD software Fluent to investigate the combustion process of solid propellant grains in small-caliber gun systems and predict the interior ballistic performance in terms of gas pressure inside the chamber and projectile velocity. Even if an extensive user customisation is required, simulation results adequately reproduce the experimental pressure-time curves and projectile velocity history of a 5.56 caliber gun utilising spherical single-component propellant.

Regarding the work in Ref. [96], the time-splitting technique in conjunction with the Weighted Average Flux Scheme are adopted to develop the internal ballistic code named FHIBS (Fort Halstead Internal Ballistics Software). The capability of the code in simulating the ignition and combustion of solid propellant charges used in conventional and electrothermal-chemical (ETC) guns is demonstrated via experimental tests with a 40 mm conventional gun charge comprising stick propellant and a 155 mm ETC gun having a capillary plasma generator mounted in the breech.

In a slightly different way, the CFD code described in Ref. [122] has the ability to handle modular artillery charges with granular propellant encased in combustible cases. The latter, in particular, are considered as a secondary propellant having its own ignition and combustion criteria, as well as an appropriately defined material strength and rupture behavior.

A sophisticated algorithm for the simulation of the two-phase, multi-dimensional viscous flow of internal ballistics can be found in Ref. [120]. The algorithm, applied to an idealised Lagrange gun, i.e., a smooth tube of constant radius closed at the end by the breech, has been reported to facilitate the understanding of the mechanisms present in real guns. If compared to the analytical solutions derived with the lumped-parameter model, the obtained numerical results in terms of pressure histories at the breech centre and at the projectile base provide a discrepancy of 2.8% and 3.4% respectively. Several submodels such as heat transfer and turbulence can be included, allowing the determination of their effects on the flow field.

The TWOPIB (TWO-Phase Interior Ballistics) computer code is proposed in Refs. [118,123,124] for the solution of different types of interior ballistic problems involving a two-phase flow of solid granular propellants and combustion gases. The developed code includes a stable fast-converging numerical scheme for the discretisation and solution of the balance Eqs. (24)e(28), as well as a novel procedure of numerical grid adaptation to simulate the flow field increase caused by the projectile motion and predict shock wave creation and/or irregularities inside the gun system. Firings of 100 mm APFSDS (Armour-Piercing Fin Stabilised Discarding Sabot) ammunition with 19-perforated propellant in a experimental 100 mm gun served for the TWOPIB verification. Measured basic parameters like projectile muzzle velocity and maximum gas pressure show excellent compatibility with computational results [118].

Additional contributions are discussed in Refs. [32,80,125,126], where the second-order McCormack technique [127] is used to solve the system of conservative Eqs. (24)e(28). The abovementioned technique is an explicit predictor-corrector approach based on the finite difference method with second-order accuracy in both space and time. To follow the projectile motion and the consequent expansion of the computational domain, a moving mesh is selected in Ref. [125] while the more sophisticated moving control volume conservation (MCVC) technique in conjunction with a grid adaptation algorithm is applied in Ref. [80]. Both [80,125] consider a one-dimensional domain even if, differently from the former where a specific application of the model is not envisaged, in the latter the authors aim at developing a two-phase numerical code for large caliber guided projectile-naval gun systems (NGGPS).

Regarding Ref. [32], the established two-phase flow model is the foundation of the charge design optimisation process. Projectile muzzle velocity and gas pressure, which are in conflict with each other in physical meaning, are set as objective functions, meaning that the first should be as higher as possible without exceeding the maximum pressure allowing a safe firing. Taking into account the large amount of design possibilities, a sort of filter is applied to ensure a better computational stability and a faster running time. The filter, based on a lumped-parameter model [32], is a preprocess that picks out the unqualified combinations of design variables that may induce calculation instabilities or abnormal simulation results, like peak pressures much higher than the design demand. By considering the example of a 30 mm caliber gun with seven-perforated propellant grains as a validation test, it emerges that numerical and experimental results are in good agreement with each other.

Finally, a study on the propellant dynamic burning rate equation is proposed in Ref. [126], based on a previously developed twophase internal ballistic numerical code. Two simulation scenarios are considered, characterised by two different propellant burning rate equation. In the first, the static equation obtained by closedbomb tests is used while, in the second, the dynamic equation derived according to the erosive burning theory [128,129] is adopted. By comparing the experimental and numerical results of the two cases, the analysis reveals a significant improvement provided by the dynamic rate equation in terms of simulation precision of projectile muzzle velocity and maximum gas pressure. Specifically, a decrease of the maximum computation error from 2.97% to 0.75% and from 6.68% to 0.38% respectively emerges for tests involving 220 g, 200 g, and 180 g of propellant.

By following the idea presented in Ref. [130], a non-conservative finite volume strategy is proposed in Ref. [107] to simulate not only the combustion process of the propellant but also that of the primer. After an academic validation test in a virtual 132 mm AGARD gun, simulations performed with the developed two-phase flow model on a real 60 mm gun are compared with experimental measurements. By considering a cylindrical perforated primer and three different initial masses of the propellant (1.17 kg, 1.365 kg, 1.4625 kg), it emerges that numerical results are in agreement with the expected ones.

Focusing on the ignition and flame-spreading dynamics in the gun chamber, in Ref. [131] the XKTC code is employed to model the interior ballistics cycle of a 5.56-mm small calibre gun. The effects of such phenomena on the gun performance are investigated, together with the influence of the gas formation and intergranular stress waves on the gas peak pressure and projectile muzzle velocity. Results reveal, among others, that the large intergranular stresses that can result during the flame-spreading process may be responsible for the debulleting of the projectile from the cartridge case and, in some cartridges, for its early motion. If this happens before the complete ignition of the propellant, the gun performance in terms of projectile base peak pressure and muzzle velocity could be significantly reduced.

Additional applications of the XKTC code can be found in Refs. [132,133] for predicting the thermochemical erosion in gun barrels, in Ref. [134] for performing ignition studies for the 25-mm cannon system mounted on the Bradley Fighting vehicle and in Ref. [135] for investigating the bore surface temperature profiles in the US Army M256 120-mm cannon firing M865, M829, and DM13 cartridges.

Further applications of the CFD codes developed by the military research laboratories are illustrated in Refs. [136e138]. In Ref. [136], the QIMIBS code is used to calculate the stresses exerted on a mortar bomb tail unit during the launch process while, in Ref. [137], the NOVA code describes the transient heating flux at the inner wall of the 155-mm M199 cannon tube over a long period of continuous firings. Regarding Ref. [138], the NGEN3 code is coupled with the DYNA3D software to perform coupled gun-projectile dynamics/CFD simulations for predicting the implications of the combustion of densely packed advanced charges on the projectile structure. A practical application involving a telescoping projectile with large intrusion into the chamber region normally occupied by the propellant is proposed. Pressure-time curves and pressure field contours during the flame-spreading process are obtained, revealing the formation of significant pressure waves during burning. For the projectile structure, these pressure oscillations translate into a significant challenge which could result in either a loss of cargo volume or a non-survival of the projectile [138].

A three-dimensional theoretical/numerical code for the prediction of the ignition, combustion, flame-spreading and pressurisation process in various parts of a 120-mm mortar system is described in Ref. [139]. Due to the complexity of the internal ballistic process in mortar propulsion systems, the developed code, named 3DMIB, solves the problem in a modular fashion. Namely, by separately simulating the different components of the propulsion system and interfacing them with appropriate boundary conditions. Applied to a 120-mm mortar system using two different pyrotechnic materials, Black Powder (BP) and Moisture Resistant Black Powder Substitute (MRBPS), it generally emerges a good agreement between numerical and experimental data in terms of pressure-time curves. The reliability of the code in simulating pressure wave phenomena in the mortar tube is also reported.

The identifying characteristic of the previously presented models is the incompressibility assumption for the solid phase, so that no equation of state is required for it. As mentioned, this is a simplifying hypothesis but, in light of the cited references, it has proven to be valid for various internal ballistic applications. A somewhat different approach can be found in Refs. [92,140e143], where not only the gas phase, but also the solid one are assumed compressible. Consequently, both phases are now modelled by their respective conservation equations of mass, momentum and energy, as well as closure laws and appropriate relations for intergranular stress, drag force and burning distance.

In Refs. [92,140], to which we refer the interested reader for the complete system of equations, attention is given to develop a twophase numerical code able to address interior ballistic and safety problems for complex propulsive systems like consolidated charges, dual grain and stick charges, ETC ignition. Delivered to military organisations of different European countries, the capabilities of the code, called MOBIDIC-NG, are presented in Ref. [92] through different examples of applications. Small calibre guns, for which the propellant combustion rate, burning temperature and energy are higher in the central part of the grains, can also be handled by using step functions allowing properly adapted source terms in the conservative equations.

Based on the MOBIDIC-NG code, a two-dimensional CFD code named M2DNGETC is presented in Ref. [144] for the simulation of gun systems based on the ETC concept. Appropriately modified propellant characteristics like burn rate coefficient, energy equation, thermodynamic properties, are included, leading to a very good agreement between the numerical and experimental results. The latter, in particular, concern 45-mm gun firing with 600 g of double base propellant grains and 750 cm3 combustion chamber, and 120-mm gun firing with 8 kg of double propellant grains and 9600 cm3 combustion chamber.

In Ref. [145], the internal ballistics code MOBIDIC-NG is coupled with the erosion code COPED for the numerical modelling of the chemical erosion phenomena in gun tubes, with particular attention to the thermal and chemical attack of iron barrels by the oxidising gases. MOBIDIC-NG, in particular, provides the timedependent history of the aerodynamic field at any location within the gun. COPED, starting from these data and using some empirical correlations, calculates the heat and mass transfer to the gun tube, as well as the surface temperature and rate of removal of the tube steel. 20 experimental tests including 13 propellants, namely, one single-base (SB1), five double-base (DB1 to DB5), six multi-base (MB1 to MB6) and one lova-propellant (LOVA1), with a different loading density of 0.12 and 0.19 have been performed for validation purposes. Results reveal the ability of the proposed model to predict a good order of magnitude of the erosion level, especially at low loading densities.

With reference to Ref. [141], a two-phase numerical model for pressure and velocity calculations in internal ballistic is deployed. The model is based on the classical Drozdov model [146] that expresses the internal ballistic parameters as a function of one independent variable coinciding, in Ref. [141], with the relative thickness of the burned propellant grains during the combustion process and with the position of the projectile during the projectile movement. The finite difference scheme for the solution of the system of conservative equations is also selected, leading to a better stability and convergence of the solution. By considering a rifle 12.7 mm and two different types of propellants, i.e., seven-channel perforated cylindrical propellant and spherical propellant, it emerges that simulation results give a good estimate of the experimental measures of gas pressure and projectile muzzle velocity.

Moreover, maintaining the two-phase Eulerian-Eulerian approximation, Ref. [142] implements a three-dimensional and multi-component numerical model capable of describing internal ballistic problems with a number of different energetic materials, as in the case of modular artillery charge systems (MACS). Contrary to conventional artillery charges, the main challenge of MACS is the presence of a combustible cartridge case, characterised by its own mechanical and chemical behavior. To overcome this, Ref. [142] adopts a finite volume conservative flux scheme coupled with a dynamic mesh generator tackling the problem of a variable computational domain. The versatility of the developed code in simulating artillery firing with conventional and modular charges is demonstrated by experimental tests involving a 155 mm howitzer.

Three-dimensional simulations based on the Eulerian-Eulerian approach are also reported in Ref. [147] to investigate the different pressure distributions between the gun breech and projectile base obtained by varying the ignition system. Different lengths of the primer and igniter masses are considered, from which it emerges that shortening the primer and increasing the igniter mass leads to an increase in the negative differential pressure fluctuations.

A further two-phase numerical code based on the AMI (Axisymmetric Model of Interior ballistics) code is described in Ref. [143]. The code adopts a multi-dimensional finite volume scheme based on a non-conservative two-velocity one-pressure model enabling the computation of the pressure distribution, projectile velocity and flow field of the muzzle flesh in a 40 mm weapon system. The impact of novel propellant grain geometries on the pressure evolution and projectile velocity can also be explored. To validate the proposed code, a comparison with the experimental results and the prediction of the lumped-parameter model in STANAG 4367 is also reported.

After performing a number of two-dimensional flow field calculations and focusing on the radial and axial gradients of the physical quantities involved, the authors in Ref. [148] observe that the radial effect is very obvious at the ignition stage but, after the flow reaches a relative stability, it basically disappear. Consequently, only the axial distribution of the flow characteristic quantities, i.e., pressure, density and velocity, can be reasonable considered. This paved the way to the development of a hybrid numerical method that links the two previously established onedimensional and two-dimensional two-phase flow models by simultaneously creating a dimension transformation in the radial direction and a grid mergence in the axial one. Based on an appropriately defined conversion standard, the proposed model is able to switch from a 2D to a 1D solver at various stages of calculation, resulting in an improved computational efficiency by maintaining the level of trust of classical codes. Specifically, if compared to a traditional two-dimensional model, the total running time is estimated to be reduced by 86.5%.

A computational study of the flow characteristics around a 155mm projectile moving through a gun barrel is undertaken in Ref. [149] using a phenomenological approach implemented into the commercial code Fluent. The flow field, whose analysis is very complex due to the presence of turbulence, stress-field anisotropy and time-variation of the computational domain among others, is considered to be unsteady, compressible, turbulent and nonisothermal by incorporating these characteristics in Fluent via User Defined Functions. A modified Kolmogorov energy spectrum for compressible turbulence is used, together with a compressible constitutive relation for the Reynolds stresses. In both cases, model coefficients are based on empirical data.

4.2.2. The arbitrary Lagrangian-Eulerian approach (ALE)

An additional class of Two-phase flow models are those based on the arbitrary Lagrangian-Eulerian approach (ALE). This method, specifically developed for moving boundary problems like those involving the process of a projectile motion, combines the Lagrangian and Eulerian algorithms in order to reduce the computational cost and improve the solution accuracy. The Lagrangian description, in particular, is used for tracking the moving mesh while the Eulerian one is used for the stationary areas where a moving mesh is not required. The ALE method has a number of advantages, including the fact that no special equations need to be established for maintaining the solution accuracy at the boundaries of the two descriptions, e.g. special equations to obtain a continuous rezoning capability. For a detailed description of the ALE approach, which goes beyond the scope of the present paper, the interested reader is referred to Refs. [108,150,151].

4.2.2.1. Model applications. In the literature, the ALE technique has been used to model a variety of phenomena like sloshing in water tanks, bending beams, high velocity impacts, soil-structure interactions, fracture and blood flow, among other [152]. In the ballistic community, the ALE methodology has been applied in Refs. [109,110,153,154] to numerically investigate the internal ballistic process of high-low pressure chamber systems [153], twostage combustion weapons [109], multi-projectile launching processes [154] and low-recoil guns [110]. In all cases, numerical predictions are verified against experimental pressure-time profiles and projectile velocity histories, as well as against the standard test AGARD gun [109,110,154]. A very good agreement is generally reported.

It should be noted that in Refs. [109,110,153,154] the ALE method has been formulated for a two-dimensional axisymmetric domain. A three-dimensional version based on the parallel technology also exists, having the advantage of further reducing the computational cost of calculations. More details can be found in Ref. [155].

4.2.3. One Mixture Flow Model

The second modelling strategy when adopting the EulerianEulerian approach is the One Mixture Flow Model. As anticipated, this method consists in representing the two-phase flow of gas and solid particles as a one continuum mixture whose characteristics are provided by the application of the mixture theory. Accordingly, the mass, momentum and energy conservation relations listed in Eqs. (24)e(28) are now substituted by [115].

...(36)

...(37)

...(38)

with nph the number of phases present in the flow ... i and Ei, in turn, the volume fraction, velocity and internal energy of the i-th phase, Pg the gas pressure, rm and vmix the mixture density and mass-averaged mixture velocity, given respectively by

...(39)

Similarly to Eqs. (24)e(28),

...(40)

stand for the source terms that, again, take into account the coupling between phases in terms of momentum and energy exchange. Note that, in this case, they are obtained by summing the individual contribution of all the phases in the system, as reflected in Eq. (40), where f d;i and fig;i are, respectively, the interphase drag and inter-granular stress of the i-th phase, Km and Em, in turn, the mass-averaged heat conductivity of the mixture and the effective energy exchange between phases. For sake of conciseness, for more details we refer the interested reader to Ref. [115] and the references therein.

4.2.3.1. Model applications. Despite representing a simplified and economical alternative to the previously presented Two Phase Flow Model, in the literature the One Mixture Flow Model is scarcely used.

One example of application can be found in Ref. [115], where the authors adopt this method to elaborate a two-dimensional axisymmetric ballistic model for small caliber gun systems. Implemented into the commercial CFD software Fluent, the obtained numerical code provides the characteristic curves of pressure, projectile velocity and projectile travel. If compared with firing test data for a 9 mm caliber gun with a propellant weight of 0.245 g, numerical results are found to be in a very good agreement in terms of average maximum pressure of the gas and projectile muzzle velocity and exit time. A comparison with the lumpedparameter model in STANAG 4367 is also illustrated and, again, a very good agreement emerges. However, a significant computational effort is required to properly implement into Fluent the governing Eqs. (36)-(38) and the corresponding source terms in Eq. (40) via User Defined Functions (UDF) and User Defined Scalars (UDS). The Fluent's Dynamic mesh module is also required to update the computational domain during the motion of the projectile represented, for simplicity, by a moving wall.

4.3. Eulerian-Lagrangian approach

From the examples reported in the previous section, it can be said that the Eulerian-Eulerian model is very popular in the ballistic community. From the point of view of computational cost, this approach is advantageous in those cases where the density of the solid particles is high and the volume fraction of the solid phase is thus a dominating flow parameter. On the other side, by only providing a macroscopic description of the two-phase flow, this model does not recognise the discrete character of the solid phase and local phenomena like a different size of the burning particles or an imperfect grain geometry can not be captured. As reported in Ref. [156], including these aspects in the computational model, provides a more realistic scenario producing numerical predictions much closer to the experimental measurements. However, of the different features simulated, i.e., propellant grain size variation, propellant movement, refined mesh, the analysis in Ref. [156] reveals that the greatest effect on the accuracy of pressure-time curves is due to the propellant movement within the breech and the barrel. An accurate modelling of the burning propellant along the barrel can be also useful in erosion studies.

Considering this, the more sophisticated Eulerian-Lagrangian approach is proposed to numerically solve two-phase flow models in internal ballistics. Differently from the Eulerian-Eulerian, the Eulerian-Lagrangian method allows focusing on the behavior of single propellant grains and getting those grain-size information, hardly obtained experimentally, that the fully Eulerian-Eulerian model can not predict.

Basically, the method consists in modelling the gas phase as a continuum described, at the macroscopic scale, by the same set of mass, momentum and energy conservation Eqs. (24)e(26) introduced in subSection 4.2.1.

Conversely, the solid phase is treated in a discrete way as an ensemble of individual particles, whose state and trajectory are tracked in space and time. Even if, starting from Ref. [157], different discrete particle methods have been proposed in the literature, almost all Eulerian-Lagrangian models refer to the Discrete Element Method (DEM) and Discrete Particle Method (DPM). They consist in predicting the particles movement by solving, for each particle, the equation of motion taking into account collision dynamics and including all the resulting forces acting on the particles, i.e., contact forces between particles or between a particle and the wall, drag forces, gravity, etc. Indeed, as pointed out in Ref. [158], the particles movement in the gas flow is caused not only by the particle-particle or particle-wall collision, but also by gas-solid interphase forces like the drag force, Magnus force or Basset force. However, in the literature, the majority of works only consider the drag and particle-particle contact forces. Accordingly, by applying the Newton's second law, the translational and rotational (in some cases rotational degrees of freedom are considered as well) equations of motion are described, for each particle, by [158].

...(41)

...(42)

with ... and ve : = dxe =dt, respectively, the mass and velocity of the e-th particle, xe its displacement, f ei the contact force due to the collision with the ne contacting particles, f de the gas-particle interphase drag. With reference to Eq. (42), Ie and ue denote, in turn, the e-th particle momentum of inertia and angular velocity, tei the torque due to the particle-particle contact.

As in Subsection 4.2, to close the above calculation equations for the particle elements, a series of auxiliary relations need to be provided.

The first deals with the definition of the contact force f ei, which can be modelled with different levels of accuracy. Usually, in DEM approaches, the multi-contact interactions among particles are described by means of mechanical elements like springs, dampers and sliders [76,158-160] according to hard-sphere or soft-sphere models, as illustrated in Ref. [161]. In Ref. [158], for instance, particle-particle interactions are simulated by a soft collision model, where a set of linear springs and dampers are applied in the normal and tangential direction of the collision. By splitting the total force in its normal, f n ei, and tangential, f t ei, components, this approach leads to Ref. [158]

...(43)

...(44)

...(45)

... and ... respectively, the spring and damper coefficients in the normal and tangential directions, ... the displacement vector between the particles e and i split in the normal and tangential components, vei the relative velocity (or slip velocity) between the considered couple of particles, nei the unit vector from the centre of particle e to particle i. The torque tei is also given by [158]

...(46)

being rei the distance between the centers of the particles e and i.

The second auxiliary relation concerns the drag force f de acting on the e-th particle. Based on the Anderssen law for a fluidised bed, in Ref. [158] the following expression is proposed

...(47)

with Ve and de, in turn, the volume and equivalent diameter of the considered particle, 4p the volume fraction of the solid phase, rg the gas density and Cd an empirical coefficient. Being the drag force obeying the Newton's third law, the total gas-solid interphase drag force can be obtained by summing up the contribution of all the np particles in a given domain:

...(48)

Inserted into Eqs. (24)e(26), the derived equation, in conjunction with the gas-phase closure relations introduced in Section 4.2, completes the definition of the examined problem.

Note that, for completeness, the described method is based on a 4-way coupling, being included not only the gas-particles mutual interactions, but also the particle-particle ones (Fig. 6). This model provides an high level of accuracy but large computational resources are required. Alternative approaches also exist, in which the more simple 2-way coupling [162,163] or 1-way coupling [164] are used. However, in all cases, a sufficient large set of particles need to be considered in order to be statistically representative of the real ensemble of propellant grains. This makes EulerianLagrangian approaches computationally expensive and, for this reason, few works in the literature are based on this method.

4.4. Model applications

The first example is Ref. [158], where the Eulerian-Lagrangian strategy is used to simulate the gas-solid reacting flow in the interior ballistic process, with the objective of more clearly understanding the phenomena involved. The finite volume scheme is selected to discretize the gas phase while, for the solid phase, the direct mapping contact detection algorithm [165] is adopted to simulate the particle-particle contact forces. In the proposed procedure, to take into account the simultaneous presence of the two phases, the multigrid technique is included: the DEM-grid, for the detection of the contact forces, the CFD-grid, for the gas phase, and the porosity-grid, to compute the system porosity. The accuracy and reliability of the proposed method is demonstrated through several verification tests, among which the simulation of an experimental igniter device in open air. Computational and experimental pressure-time curves are compared during the entire combustion process and a good agreement is found. The particles position and temperature distributions at different times is also reported, allowing a detailed representation of the flame propagation process.

A further contribution dealing with the Eulerian-Lagrangian approach is Ref. [76], where the method is applied to perform parametric studies on a simplified two-dimensional planar gun system. The author, in particular, aims at determining the sensitivity of the gas, propellant and projectile conditions to the variation of the parameters related to the propellant and projectile. All work is done with the computational fluid dynamics software Fluent through extensive user code subroutines to implement the adopted modelling techniques. Regarding the latter, the novelty of Ref. [76] consists in directly incorporating the individual propellant grains into the computational domain so that, even for densely packed particle conditions, the particle-particle interaction effects can be simulated. Thus, differently from the standard Lagrangian models which are typically appropriate for diluited particle concentrations of under 10%, in Ref. [76] there are no limitations on the allowable particle concentrations. Particle-particle interactions are described by the soft collision model while the influence of the propellant grains presence and motion on the gas are included through standard boundary conditions not requiring correlation equations for the drag coefficient. In addition, the size of individual particles can be independently altered based on local burn rate calculations formulated from the conditions at individual particle location. Local pressure waves moving from the gun breech to the projectile base can also be captured, demonstrating the ability of the proposed methodology in exploring localised phenomena developing during the propellant burn.

A two-dimensional two-phase flow method based on the Eulerian-Lagrangian technique is also applied in Ref. [166] to study the ignition performance of composite charge structures. The mathematical model of various particle types is firstly established, followed by a numerical study of the multi-scale reaction flow by the two-phase flow method coupled with an adapted version of the particle element methodology. This allows obtaining a detailed description of the flow field corresponding to different charge structures and performing a sensitivity analysis of the gun internal ballistic performance to an adjustment of the charge parameters. After validating the proposed strategy with the standard test AGARD 132 mm gun in terms of projectile velocity and maximum gas pressure in the chamber and breech, four different charge configurations are investigated, based on the artillery charging requirements: fully tubular and fully granular propellant charges, bottom granular propellant charge suitable for fin-stabilised projectile, center tubular propellant charge suitable to high-density charging. Various parameters describing the internal ballistic performance are considered, including pressure curves, axial and radial flame propagation, projectile velocity. If compared to the granular one, a better ignition performance emerges for the tubular propellant, independently of the charge configuration. Results also reveal that the charging proportion of tubular and granular propellants does not affect the ballistic performance, being more tubular propellant ineffective in significantly improving ignition parameters as flame propagation speed and flame propagation time.

A less computationally expensive procedure is presented in Refs. [162,163] where, differently from Refs. [76,158] that adopts a 4-way coupling, a 2-way coupling approach is selected.

Based on this strategy, a three-dimensional model is described in Ref. [162] for simulating the gun firing phenomena. The combustion and flow calculations are separated into two parts so that, unlike conventional methods, simplicity and efficiency is improved. The flow part, in particular, is performed by using the commercial program STAR-CCM * while the combustion part is conducted through a previously developed numerical code [167,168]. A data exchange between the two parts is foreseen and, only at the end of the whole computations, both calculation outcomes are combined. To verify the proposed model, the standard test AGARD gun is considered, as well as 40 mm gun firing with 7-perforated cylindrical propellant. In both cases, simulation results provide a good estimate of the breech pressure, projectile base pressure and projectile muzzle velocity.

Regarding Ref. [163], the authors employ the EulerianLagrangian approach to investigate the internal ballistic performance variation as a function of the solid propellant position in the chamber. Three cases are explored, all of them involving a large caliber gun and 7-perforated cylindrical propellant: propellant located in the region near the breech, propellant in the center of the chamber, propellant in the region near the projectile base. Despite not founding difference in terms of mean pressure, the authors claim much higher values of negative differential pressure and a much higher difference between breech pressure and projectile base pressure with the propellant located in the breech and projectile base regions. From this point of view, propellant in the centre of the chamber is the most profitable configuration.

An extension of the Eulerian-Lagrangian method to polydisperse systems is conducted in Ref. [169] to investigate the effects of the propellant movement on the maximum breech pressure and projectile muzzle velocity. A one-dimensional numerical code is developed by using the semi-implicit method to deal with the pressure-linked relations and the unsteady compressible flow [170-172], the moving boundary technique to simulate the projectile movement and the finite volume method to discretize the analysed domain. The proposed numerical scheme is verified through the free piston motion problem, as well as by comparing the simulation results of 40 mm high-velocity gun firings with the IBHVG2 lumped-parameter code. No quantitative difference is found in terms of breech pressure and projectile base pressure. An error of approximately 2% emerges for the gas mean pressure, mainly due to the pressure gradient assumption of the lumpedparameter model.

In the framework of high-speed combustion systems, a mixed two-phase flow method is established in Ref. [164] to simulate the evolution of the flow field in the chamber. As in the previous works, the gas and solid phases are respectively described by the Eulerian and Lagrangian model. The 1-way coupling approach is adopted, meaning that the solid phase is treated as a transported passive element (Fig. 6). The main advantage of the suggested calculation model is its ability to efficiently describe the propellant state during the combustion process and to obtain accurate temporal and spatial distributions of the two-phase flow parameters. The main idea of Ref. [164] consists in modelling the initial charging state, i.e., size of the propellant grains much larger than that of the fluid microclusters, and the fluidised state, i.e., the particles shrink and become smaller than the system characteristic size, by respectively using the particle element method [173] and the two-fluid method. The coupling and transformation strategy of the two models is finally realised by the boundary mapping method. Similarly to Ref. [162], a 132 mm AGARD gun is the object for carrying out a verification study which reveals a good coherence in terms of pressure curves and a good continuity at the transition point, i.e., from the particle element method to the two-fluid one. The 1-way coupling method is also adopted in Ref. [174] for the development of a gas dynamic computer model capable predicting the pressure and heat transfer distribution in gun systems having a chamber diameter greater than that of the bore. Acceleration loadings on saboted projectiles launched from high velocity guns can be explored, as well as the feasibility of new weapon concepts. Numerical predictions are verified against a total of 68 rounds field in a high velocity 90e37 mm smooth bore gun. By using the projectile velocity- and pressure-time curves as a base for comparison, it emerges a maximum error of 10% in terms of velocity and of 25% in terms of acceleration, whose origin is not understood by the authors. Conversely, in terms of projectile muzzle velocity, the maximum error is 5%.

5. Conclusions

This paper provides a survey of the computational models proposed in the literature for investigating the internal ballistic problem and evaluating the performance of a given gun system in terms of projectile motion and pressure levels reached. To homogenise the different modelling approaches and help making a comparison among them, the large set of contributions are classified according to the used mathematical framework. Lumpedparameter models and computational fluid dynamics (CFD)-based codes are analysed separately by presenting their basic theoretical tools and mentioning their main advantages and drawbacks, summarised in a condensed form in Table 1.

A good comparison with experimental data in different contexts is obtained by both approaches, so that the choice of one model over another is mainly related to the application of interest. Lumped-parameter models, for example, can be successfully used to simulate a variety of gun systems with relatively simple geometries and, because of their minimum requirement of computer resources, to perform parametric analysis and optimisation processes. On the other hand, important phenomena causing catastrophic failures of gun systems like pressure waves or nonuniformities in flame-spreading can not be captured.

Much effort has been devoted to overcome these limitations and various improved versions of the lumped-parameter model are currently available. Several amendments are embedded into the model governing equations in order to deal with more advanced applications including high performance charges and weapon systems. More advanced features can be addressed, like a depthand pressure-dependent thermochemistry of the propellant or a non-uniform multi-variate size distribution of the grains. The occurrence of pressure waves and rarefaction waves can also be described, while retaining the computational economy of the original lumped-parameter approach. These improved tools, however, are not sophisticated as the two-phase CFD-based models, which have a better capability of treating nonsteady, turbulent two-phase flow effects and locally studying particular pressure phenomena occurring in internal ballistics.

According to their degree of detail and complexity, existing CFDbased methods can be classified into different categories. The main distinction is based on how the particulate solid phase is represented. Two approaches are possible: the Eulerian-Eulerian one, where the gas and solid phases are considered as a continuum, and the Eulerian-Lagrangian one, which treats the gas phase as a continuum and the solid phase as individual particles traced in time and space along their trajectories.

Fully Eulerian models, which have a much faster implementation and running time, are advantageous in those cases where the density of the solid particles is high or when grain-size information are not required. The latter can be predicted by the EulerianLagrangian model even if a bigger modelling effort is required to reproduce the dispersed particle behavior. The large number of particles needed to get a statistically reliable description of the real ensemble of propellant grains limits the size of the domains that can be investigated and, eventually, the range of applicability to ballistic applications. It can be thus said that continuum approaches, i.e., Eulerian-Eulerian models, are suitable for analysing medium-large scale phenomena, while discrete ones, i.e., EulerianLagrangian models, are a precious tool for investigating local particle transportation effects. Results of one approach can also be used to improve the lacks of the other. Specifically, based on the outputs from the sophisticated Eulerian-Lagrangian methods, some quantities characterising the fully Eulerian or the lumpedparameter models can be identified, e.g., a better definition of the drag force, inter-granular stress, ...

In this respect, it is worth noting that the majority of currently available models, both lumped-parameter and CFD-based ones, are conceived for medium and large calibre systems. These models are the result of many years of elaboration and validation but, being often developed by military research laboratories, many of them are not commercially available. Despite this, they offer a very good starting point for the development of internal ballistic numerical models suitable for small calibre systems, in which deterred propellants with a depth-dependent burning rate and/or a multivariate size distribution of the grains are often used.

Based on the performed review, it can be said that too little attention has been given to small calibre guns. Indeed, very few studies focus on this topic and our recommendation, whenever possible, is to dedicate more effort on it. Developing a numerical predictive model combining, for example, the Finite Element Method and Computational Fluid Dynamics specifically suited for small calibre weapons, would be highly beneficial for the ballistic community. Similarly, notwithstanding a myriad of methods are now available for the internal ballistic numerical modelling, a number of aspects still need to be addressed. Among others, the wear phenomenon, which manifests itself through damage inside the barrel and deformation of the bore diameter.

Acknowledgement

The authors gratefully acknowledge the support provided by the Royal Higher Institute for Defence (RHID) of the Belgian Defence, which has contributed to the progress of this ongoing research.

List of symbols

Ab area of the projectile base/in2

ap projectile acceleration/(in,s 2)

Ap projectile surface/in2

a0 multiplying parameter (Pidduck-Kent model)/

b co-volume/(m3 ,kg1)

bi co-volume of the i-th propellant specie/(in3 ,lb1)

bI co-volume of the igniter/(in3 ,lb1)

C total charge/lbs

Cd empirical coefficient

cn damper coefficient in the normal direction/(N$s$m1)

cpi specific heat capacity at constant pressure of the i-th propellant specie/(in-lb$(lb-K)1)

cpI specific heat capacity at constant pressure of the igniter/ (in-lb/lb-K)

ct damper coefficient in the tangential direction/ (N$s$m1)

cvi specific heat capacity at constant volume of the i-th propellant specie/(in-lb/lb-K)

cvI specific heat capacity at constant volume of the igniter/ (in-lb/lb-K)

de equivalent diameter of the e-th particle/m

Dpe effective particles diameter/m

Ebr energy lost in engraving the rotating band and in

overcoming the friction down the bore/(in-lb)

Ed energy lost due to the air resistance/(in-lb)

Eg internal energy of the gas/(in-lb)

Egas internal energy of the gas (CFD model)/(J,kg1)

Eh energy lost due to the heat transfer to the environment/ (in-lb)

Ei internal energy of the i-th phase (CFD, One Mixture Flow model)/(J,kg1)

El energy losses/(in-lb)

Em effective energy exchange between phases (CFD, One Mixture Flow model)/(in-lb)

Em;MH muzzle energy (Meyer-Hart lumped-parameter model)/ (Pa,s1)

Ep energy consumed to move the projectile/(in-lb)

Epk energy loss due to the motion of the combustion gas and unburned propellant grains/(in-lb)

Epr energy loss due to the projectile rotation/(in-lb)

Eprop internal energy of the propellant/(J,kg1)

Er energy released by the igniter and the burning propellant/(in-lb)

Erc energy lost due to the recoiling parts of the gun/(in-lb)

Fa resistive force of the air ahead of the projectile/lb

Fb propulsive force of the projectile base/lb f d interphase drag/(Pa,m1)

f d;Er interphase drag according to Ergun/N f de gas-particle interphase drag force acting on the e-th propellant particle/(Pa$m2)

f d;i interphase drag force of the i-th phase (CFD, One Mixture Flow model)/Pa

f ei contact force acting on the e-th propellant particle/ (Pa$m2)

f n ei normal component of the contact force acting on the eth propellant particle/Pa

f t ei tangential component of the contact force acting on the e-th propellant particle/Pa

Fi force per unit mass of the i-th propellant specie/(in-lb/ lb)

FI force per unit mass of the igniter (in-lb/lb) fig inter-granular stress/(Pa,m1)

Fig inter-granular stress tensor/Pa

fig;i inter-granular stress of the i-th phase (CFD, One Mixture

Flow model)/Pa

Fr frictional force on the projectile/lb

h convective heat transfer coefficient/(W$m2 $K1 )

Ie e-th propellant particle momentum of inertia/(kg$m2 )

Km mass-averaged heat conductivity of the mixture/ (J$kg1 )

kn spring coefficient in the normal direction/(N$m1 )

kt spring coefficient in the tangential direction/(N$m1 )

m_ rate of gas generation/(kg$m3 $s 1)

me mass of the e-th propellant particle/kg

mi initial mass of the i-th propellant specie/(lb-mols/mol)

mI initial mass of the igniter/(lb-mols/mol) mp projectile mass/(slugs/12)

n number of different species of propellants (composite charges)

n0 exponential parameter (Pidduck-Kent model)

ne number of contacting propellant particles

nei unit vector from the centre of particle e to particle i

nmol number of moles/kg of the gas/(mol$kg1 )

np number of propellant particles in a given domain

nph number of phases present in the flow (CFD, One Mixture Flow model) P gas pressure (Noble-Abel equation)/Pa

P0 gas pressure acting at the gun breech/psi

Pa pressure of air ahead of the projectile/psi

Pb pressure acting on the projectile base/psi

Pg gas pressure (CFD model)/Pa

Pg space-mean gas pressure/psi

Pm muzzle pressure/Pa

Pmax maximum chamber pressure/Pa

Pmax;MH maximum pressure (Meyer-Hart lumped-parameter model)/(lbs$in2 )

Pq analytical constant related to pressure (Meyer-Hart lumped-parameter model)/(lbs$in2 )

Pr resistive pressure of the bore/psi

Ps shot pressure/Pa

qp heat transfer between the solid and gas phases/ (W$m2 )

r analytical constant (Meyer-Hart lumped-parameter model)/

r_ rate of surface regression/(m$s 1 )

R gas constant/(J$kg1 $K1 )

b R universal gas constant/(J$kg1 $K1 )

rei distance between the centers of the particles e and i/m

ri i-th propellant burning rate/(in$s 1 )

Si surface area of the unburned i-th propellant grain/in2

Sp propellant grains surface/m2

T gas temperature (Noble-Abel equation)/K tei torque due to the e and i particles contact/(N$K1 )

Tg gas temperature/K

Tps temperature of the propellant surface/K

T0i adiabatic flame temperature of the i-th propellant specie/K

T0I adiabatic flame temperature of the igniter/K

v specific volume of the gas (Noble-Abel equation)/ (m3 $kg1 )

ve velocity vector of the e-th propellant particle/(m$s 1 )

Ve volume of the e-th particle/m3

vei relative velocity (or slip velocity) between the particles e and i/(m$s 1 )

vi velocity vector of the i-th phase (CFD, One Mixture Flow model)/(m$s 1 )

vg velocity vector of the gas phase/(m$s 1 )

Vg volume available for the combustion gas/in3

Vi volume of the unburned i-th propellant grain/in3

vm projectile muzzle velocity/(m$s 1 )

vmix mass-averaged mixture velocity vector (CFD, One Mixture Flow model)/(m$s 1 )

vm;MH projectile muzzle velocity (Meyer-Hart lumpedparameter model)/(ft$s 1 )

vpr projectile velocity/(in$s 1 )

vp velocity vector of the solid phase/(m$s 1 )

Vp propellant grains volume/m3

Vpi volume of non-energetic charge components associated

with the i-th propellant/in3

V0 initial chamber volume/in3

wp projectile weight/lb

wt total weight of the propellant and igniter/lb

xe displacement of the e-th propellant particle/m

xei displacement vector between the particles e and i/m

xn ei normal component of the displacement vector between the particles e and i/m

xt ei tangential component of the displacement vector between the particles e and i/m

xp projectile travel with respect to the ground reference frame/in

Zi fraction of mass burnt of the i-th propellant specie

z_ i i-th propellant mass fraction burning rate/s1

ZI fraction of mass burnt of the igniter

a empirical constant (CFD, Two Phase Flow model)/ (m$s 1 )

ai experimental coefficient (adjusted version of the Saint Robert's equation)

bi experimental coefficient (adjusted version of the Saint Robert's equation)

g0 effective ratio of specific heats

gi ratio between the specific heat capacity at constant pressure and at constant volume of the i-th propellant specie

gI ratio between the specific heat capacity at constant pressure and at constant volume of the igniter

dpk Pidduck-Kent constant

l analytical constant (Meyer-Hart lumped-parameter model)

xg source term, energy exchange (gas)/(Pa$s 1 )

xi i-th propellant burning rate acceleration factor

xm source term, energy exchange (mixture)/(Pa$s 1 )

rg density of the gas phase/(kg$m3 )

ri i-th propellant density/(lb$in3 )

rm mixture density (CFD, One Mixture Flow model)/ (kg$m3 )

gI ratio between the specific heat capacity at constant pressure and at constant volume of the igniter

dpk Pidduck-Kent constant

l analytical constant (Meyer-Hart lumped-parameter model)

xg source term, energy exchange (gas)/(Pa$s 1 )

xi i-th propellant burning rate acceleration factor

xm source term, energy exchange (mixture)/(Pa$s 1 )

rg density of the gas phase/(kg$m3 )

ri i-th propellant density/(lb$in3 )

rm mixture density (CFD, One Mixture Flow model)/ (kg$m3 )

rp density of the solid phase/(kg$m3 )

4 empirical coefficient (propellant bed fluidisation)

4 analytical constant (Meyer-Hart lumped-parameter model)

4c critical porosity

4g volume fraction of the gas phase

4i volume fraction of the i-th phase (CFD, One Mixture Flow model)

4p volume fraction of the solid phase

cg source term, mass exchange (gas)/(kg$m3 $s 1 )

cp source term, mass exchange (solid)/(kg$m3 $s 1 )

jg source term, momentum exchange (gas)/(Pa$m1 )

jm source term, momentum exchange (mixture)/(Pa$m1 )

jp source term, momentum exchange (solid)/(Pa$m1 )

ue e-th propellant particle angular velocity/(rad$s 1 )

Sidebar

ARTICLE INFO

Article history:

Received 21 March 2024

Received in revised form

18 April 2024

Accepted 10 May 2024

Available online 15 May 2024

Footnote

* Corresponding author.

E-mail address: [email protected] (F. Ongaro).

Peer review under responsibility of China Ordnance Society

References

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