A GAS DYNAMICS BASED METHOD FOR ARTILLERY INTERIOR
BALLISTICS DEDICATED TO NEW PROPULSIVE CHARGES
DESIGN
LIVIU MATACHE
MARIUS VALERIU CÎRMACI – MATEI
ADRIAN NICOLAE ROTARIU
NISTORAN GEORGETA DIANA
Faculty of Mechatronics and Integrated Armament Systems
Military Technical Academy
Bucharest, 39-49 George Coşbuc Avenue,
ROMANIA
[email protected], [email protected], [email protected], www.mta.ro
Abstract:
New propulsive charges for artillery systems require, very often, an inverse approach. The designers
start with the limitations of the existing gun. Firstly, an interior ballistic code must be developed (if it
doesn’t exist) using, as initial conditions, the existing barrel geometry, the ballistic powder in use, the
specific projectiles, all the loading conditions and igniting systems of the fielded weapons. The main
goals in ballistic design is to fulfil an imposed initial velocity of the projectile without exceeding the
limit pressure value inside the barrel. A good control of the propelland combustion is mandatory in this
achievement. In order to do it, the propellant’s chemical composition and shape play an important role.
Things evolve towards combinations between different propellants, each one with specific geometry and
composition, which lead to different combustion velocities. As a consequence, the combustion gases will
be produced with various rates, according to projectile movement down the barrel and free volume left
behind it. Temperature is an important parameter, as every chemical composition of the ballistic
propellants is associated with a very high combustion temperature. Particular homogenous
compositions, as triple base propellants have been made to accomplish the design requirements but also
to reduce the overall temperature in order to increase the barrel life.
A good start in finding the solution of our concerns may be represented by imagining ballistic powders
with various burn rates inside the powder element. A combination of two or more types of such powders
may give the required charge. In this way it is possible to maintain the high pressure level inside the
gun barrel, in order to transfer more energy to the projectile. As in the work-energy theorem, the net
work applied on an object causes a change in the kinetic energy of the object. In this paper we want to
show a general method for interior ballistic evaluation of artillery gun systems. The method is based on
Fluent software. An application is made around the 76 mm field cannon. For a high fidelity in
simulations, a fine knowledge of the combustion parameters is required. They influence the
thermodynamic behavior of combustion gases and they also are considered entry data for the direct
problem in interior ballistic methods. In our case, these parameters are calculated using a Matlab code.
The code covers a set of thermochemical equations.
Keywords: gas dynamic, equivalent model, Fluent, numerical simulation, interior ballistic, propellant
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1. Overview
It is of utmost importance for the new propulsive charges to be less sensitive, to have
higher energetic performance and to limit barrel erosion [1]. An acceptable solution is
combining several layers of powders (co-layered propellants) [1-2] a method patented by US
Patent 4,581,998 (1986) by A.W. Horst et al. Although it is a time-consuming and difficult
manufacturing method, it turned out to be very useful in improving the performance of the
weapon systems due to the increasing momentum of the gases in the projectile. A relatively
new method is the one proposed by TNO [1-2], called co-extrusion propellants. This method
has the following advantages [1-2]: it increases the weapon system performances, it increases
progressively the burning for propellant elements without interior channel or with one channel,
it decreases barrel erosion and improves propellants initiation (e.g. LOVA-type propellants), it
allows the possibility of achieving a large number of geometries, leading to multiple
applications.
Figure 1 shows two types of propellant elements made using this method
a – Double-based propellant
b – LOVA-type propellant
Fig. 1 Two types of propellant elements obtained by co-extrusion [1-2]
Source: Authors representation based on the research undertaken
In Figure 2 there are shown the curves of pressure and velocity variation versus time for
two types of propellant elements: one – conventional, and one – obtained by the method of
combining several layers of propellants (co-layered propellants) [2].
275
Fig. 2 Variation of pressure and velocity versus time for two types of propellant elements
Source: Authors representation based on the research undertaken
As one may notice, the pressure curve corresponding to the powder obtained by the colayered propellants method, presents the particularity of a plateau at the level of maximum
pressure, which leads to improvements in weapon system performances. Based on the data
above mentioned, in the present paper, a method for solving the fundamental problem of interior
ballistics has been developed using numerical simulation.
2. The method
In order to design and implement a new energetic material to equip an existing weapon
system, an iterative working method has been achieved, that has as input the ballistic and
thermo-gas-dynamic properties of the propellant, its geometry, its size, the type of the powder
(conventional powder obtained by the method of combining multiple powder layers), etc.
The solving method achieved is based on a gas-dynamic equivalent model, the
parameters calculation being carried out using numerical calculation, namely the finite volume
method.
Taking into account the fact that, in case of weapon systems, the work field is quite large
and given that a solution is required in a reasonable time, the issue has been approached by
transposing it into a 2D axial symmetrical model.
Generalizing, the propellant elements from the propelling charge were considered to be
9/7 type, i.e. cylindrical elements with 7 channels. These elements cannot be designed in a realaxially symmetric model, which is why a model of equivalence between the real propellant and
the one used in numerical calculation has been built. Since it has been considered that all the
real elements of powder have the same weight, every element of real powder has been
associated with a toroidal element, equivalent in terms of weight with the real element.
Since the rifled barrel cannot be effectively represented in a 2D axial-symmetric model,
the problem has been formulated for a barrel having an average diameter. Moreover, an
equivalent model has been considered for the muzzle brake of the weapon system, given that
the real brake reveals no axial symmetry.
276
The condition of equivalence for the considered muzzle brake is given by the real brake
shape and size and by the areas through which gases are discharged into the environment in the
real case.
The values of the input data are either determined experimentally, or calculated based
on certain experimental or theoretical calculation, taking into account the recommendations
from the literature [3-4]. In addition, working hypotheses for solving the fundamental problem
by various methods are not completely preserved. Thus, the hypotheses are as follows:
a. plastic deformation of the rotating band in the barrel ridges is considered progressive,
as it occurs in reality, and not instantaneously;
b. the mechanical work due to the projectile rotation is taken into account by a coefficient
of fictitious mass;
c. the composition of the propellant gases does not change, which allows the force and
covolume (f and α) to be considered constant values;
d. mechanical work for the elastic deformation of the barrel and the thermal energy that is
lost due to the gas leakage between the rotating band and the barrel are not taken into
account;
e. losses due to heat transfer to the barrel wall and to the projectile body are taken into
account;
f. each powder element has the same weight;
g. the law of propellant burning rate is expressed by u  u  p ;
1
h. the propellant ignition is performed using a priming charge that has the same weight as
the real one, but which generates only gaseous products;
i. the barrel blow-back is not considered
3. The simulation environment
3.1 Initial data
The finite volume method can be used to solve a wide range of issues: compressible gas
dynamics (Euler equations), applications in aerodynamics, astrophysics, detonics and related
fields where shockwaves are involved. For the proposed issue it is one of the most suitable
approaches. This method is applied in FLUENT software, a powerful tool for fluid dynamics
calculus.
The software provides a comprehensive finite volume network flexibility, including the
ability to solve flow problems using unstructured networks, which can be easily generated in
complex geometries. The types of networks are 2D triangular and quadrilateral, 3D tetrahedral
/ hexagon / pyramid / polyhedral elements and hybrid networks. FLUENT also allows local
modification of the network based on flow solutions.
In order to solve the proposed problem, there were used the following capabilities
provided by this software [5]:
 adoption and use of a "real" gas, taking into account the real powder gases covolume
and that the parameters of gases generated into the barrel are a temperature function
("user-define function");
277






possibility of introducing variable boundary conditions through direct access on certain
variables via C program ("user-define function");
possibility of modeling real movements that occur at the muzzle barrel during firing
(combustion of powder elements, movement of the powder elements in the barrel,
projectile motion in and outside the barrel, etc.) using the "dynamic mesh" option;
possibility of collecting data from various points of the computing field and writing
them into external files via C program, in order to validate the theoretical model;
possibility to obtain values mediation for each cell in the computing field via C program,
for the flow parameters, for comparison purposes with certain computing models;
using a heat transfer model for the barrel and the projectile wall;
assessment of certain sizes (mass flow, gas pressure on different surfaces, amount of
heat, etc.), in the eye of a strict control of the formulated model.
3.2 Complementary data
In solving the fundamental problem of interior ballistics, the following physical and chemical
characteristics are required (in addition to data on the barrel weapon system, projectile,
propelling charge weight, priming charge weight):
 reaction temperature of the priming powder;
 reaction temperature of the propelling powder;
 burning rate coefficient for the priming powder;
 exponent and coefficient of burning rate for the propelling powder;
 force and covolume of the propelling powder;
 average molecular weight of the powder gases;
 specific heats of the powder gases versus temperature;
 dynamic viscosity of the powder gases versus temperature;
 thermal conductivity of powder gases versus temperature;
 local speed of the sound in the powder gases versus temperature
Most ballistics calculation models based on fluid dynamics require additional
thermodynamic functions to be obtained from the equation of state. They are intended, in
particular, to the combustion modeling with high-level software, such as FLUENT, but can be
adapted to define the input data into any other code of fluid dynamics computation (CFD Computational Fluid Dynamics) [6].
In order to simulate the interior ballistics processes, the models require the description
of the thermodynamic behavior of gases resulted from powder combustion. In order to assess
the parameters that describe the thermodynamic behavior of the combustion gases, a Matlab
code has been used. It is based on the thermochemical calculation model. Its results are entry
data in the direct problem-solving method of interior ballistics.
The equation of state for real gases describes with enough accuracy the burning gases
state at high temperatures and densities inside the gun and inside the ballistic bomb.
278
3.3 Underlying equations for the proposed method
As noted above, the problem-solving mode is based on the finite volume method using
FLUENT software. The program has the possibility to solve the problem in two different ways:
based on pressure or based on density. We have chosen this working method because the
problem is particularly complex. The pressure-based solver, due to its conception design,
provides a faster resolution of the problem. The major disadvantage of the pressure-based
working method implemented within FLUENT software is that it does not contain an integrated
definition for the real gas. This led to the achievement of a mathematical model. Its means are
intended to give a correct solution for a real gas flow both inside the barrel and in the
environment after the projectile exit.
The equations used in the formulation of the method and that were included in the
FLUENT software [5] are presented below.
The equation of conservation of mass and equation of continuity for a 2D axial-symmetric
geometry, has the following form:
 v
 

r S

 v 
 v 
x
r
m , where
t x
r
r
(1)
x - axial coordinate;
r - radial coordinate;
vx - velocity in the axial direction;

 

vr
- velocity in the radial direction;
ρ – density of the combustion products;
Sm
- source providing the continuous phase mass to be dispersed in the secondary phase, if
necessary (e.g., phase changes).
The equations of motion and momentum conservation, axially and radially, are:

1 
1 
(  vx ) 
( r  vx vx ) 
(r  vr vx ) 
t
r x
r r
p 1    vx 2
  1    vx vr  


r  2
 (  v )   

  Fx ;

r  
x r x   dx 3
dx  
  r r   dr
(2)

1 
1 
p 1    vr vx  
(  vr ) 
(r  vx vr ) 
(r  vr vr )  

r 

 
t
r x
r r
r r x   dx
dr  
1    vr 2
vr 2 
vz2


r

2



v

2




v


 Fr .






r r   dr 3
r2 3 r
r

,
Fx, Fr – axial and radial volume forces;
 - kinematic viscosity of the combustion products.
The equation of energy conservation is represented as:
279



E    vx    E  p   1   r  vr    E  p  
t
x
r r



c  
c 
 
1   
p t  T
p t
k

 u  ( )  
r k



ij
eff
 r r 
x 
Pr
x
Pr
t 
t


 
 T

 ru  ( )
 x
ij eff




,
(3)
where: E is the total energy;
( )
ij eff is the viscous heating;
Pr
t is the number of Prandtl.
Due to the fact that the generated powder gases show high pressures and temperatures, the ideal
gas law does not describe the situation with accuracy. As a consequence, there will be used the
real gases law of state:
1

p     RT


, where
(4)
p - powder gas pressure;
ρ - gas density;
α – gas covolume;
R - gas constant;
T - gas temperature.
For the law of the burning rate, the equation used is
ua  u1 p , where
(5)
ν - burning rate exponent;
u1 - burning rate coefficient
For the projectile motion, there has been considered the following equation:


m a  F
(6)
The system of equations presented has introduced a number of 6 equations with 6 unknowns:
v x vr p T
, , , , ρ and a.
4. Model simulation. Equivalent model and discretization
The equivalent physical model of a cannon system designed in the view of addressing
the interior ballistic cycle modeling is illustrated in Figure 3.
280
Fig. 3 The cannon equivalent physical model used in numerical modeling – 2 D case
Source: Authors representation based on the research undertaken
The equivalent physical model developed is based on the following considerations (equivalence
conditions):
 the volume of the loading chamber of the equivalent model is the same as the real one;
 the volume of the barrel for the equivalent model is the same as the real one, i.e. the
equivalent barrel section area is equal to the real-section area of the rifled barrel and the
barrel length of the model is the same with the barrel real length;
 the gases exhaust surfaces areas of the muzzle brake must have the same values;
 physical and chemical properties of the equivalent powder and the generated gas are the
same as in the real case;
 propelling powder mass of the proposed model is the same with real propelling powder
mass;
 the density of the equivalent powder is the same with the real one;
 the mass of the equivalent powder element must be equal to the average real powder
elements mass (real powder element-mediated), thus leading to the same number of
powder elements in both equivalent and in real case;
 the combustion products flow inside the powder element is the same, both in equivalent
and in real powder;
 The fraction of burnt powder of an element has to be the same in both cases.
According to the working hypothesis no. 6, the law of burning rate for the propelling powder is
u  u  p
1
expressed as
. Based on the equivalence conditions mentioned above and making
u
the assumption that the exponent of burning rate remains constant, 1e has been determined.
The system of equations previously defined, which includes continuous field variables
such as velocities, pressure, temperature, density, is submitted to meshing transformation. In
this process, the continuous fields are replaced with discrete fields of values defined in the
volume control centers. The FLUENT software operates with the discrete set of these measures.
The solutions accuracy is conditioned by the meshing network of the field. A sophisticated
network computing leads to higher accuracy, instead of involving a greater computational
effort.
The analyzed model has been meshed using triangular elements as shown in Figure 4.
281
Fig. 4 Meshing the analyzed equivalent model - detail in the projectile
Source: Authors representation based on the research undertaken
In the case of the weapon system under analysis, there has been used a dynamic mesh
network. Taking into account the fact that the weapon internal configuration is variable, the
mesh allows the domain transformation following the manner in which the powder blast
evolves.
The transport movement and interactions (elastic type collisions) of the powder
elements as a consequence of the projectile movement allow the area occupied by the real
combustion products to be filled with finite volumes with controlled dimensions.
5. Results
Numerical simulation of interior ballistics was performed for a 76 mm cannon, equipped
with two types of propelling charge: a 9/7 conventional one, and a co-extruded one which
contains both 9/7 powder and a powder with higher burning vivacity, arranged as a circular
crown inside the conventional powder while maintaining the same charge density.
By solving the problem of interior ballistics, the determination of flow variation for
relevant parameters is allowed, as shown further. Variations in status and kinematic parameters
of powder gases at a subsequent time to the achievement of the maximum pressure in the barrel
for conventional powder are given in Figure 5.
282
Fig. 5 Distribution of the pressure and the velocity at t = 3,2 ms
Pressure distribution
Velocity distribution
a – conventional powder
b – co-extruded powder
Source: Authors representation based on the research undertaken
In Figure 6 the plotted curves of the barrel pressure and the projectile velocity versus time are
shown in the two cases considered.
Figure 6. Variation of the pressure at the projectile base and the projectile velocity versus time
a – Pressure variation versus time
b – Velocity variation versus time
Source: Authors representation based on the research undertaken
283
6. Conclusions
The work presented in this paper is based on how to build equivalence between the real
weapon system (geometry, shape, muzzle brake characteristics, projectile shape, type of powder
etc.) and the one used in numerical simulation. Further, we analyze the general method for
solving the fundamental problem of interior ballistics for any type of medium, namely for a
large caliber weapon system whose powder elements show simple or multi-channel features.
An equivalent real gas model has been defined in relation to the gas generated by the
deflagration of the powder when firing. The way to achieve a product is greatly shortened if
experimental data are correlated to the data obtained by numerical simulation in order to
optimize the product. At this point the method presented in this paper shows its advantage. It
allows obtaining design data in short time, at no additional cost, in order to optimize the
propelling charge and the entire system.
References
[1] SCHOLTES, G. Novel techniques for improved munitions development, 44th annual Gun and
Missile System Conference, April 2009
[2] ZEBREGS, M, van DRIEL,C, Experimental set-up and results of the process of co-extruded
perforated gun propellants, IM/EM Symposium, Tucson, AZ, May 2009
[3] FITT, A.D., CROWLEY, A.B., ASTON, J, TORO, E, Contrasting Numerical Methods for TwoDimensional Two-Phase Internal Ballistics Test Problems, Proceedings of the 11th International
Symposium on Ballistics, Vol.1, Royal Military Academy, Brussels, Belgium, May 1989
[4] VASILE, T, Balistica Interioară a gurilor de foc, Vol I, Ed. ATM, 1993
[5] ***FLUENT User Guide – Fluent Inc. Centerra Resource Park 10 Cavendish Court Lebanon, NH
03766 – FLUENT 6.3 User’s Guide – PathScale Corporation 2003-2004
[6] P.G. BAER, J.M. FRANKLE, The Simulation of Interior Ballistic Performance of Guns by Digital
Computer Program, U.S. Army Ballistic Research Lab., Aberdeen Proving Ground, MD, Dec.
1962
Acknowledgement:
This paper is financially supported within the project entitled “Horizon 2020 - Doctoral
and Postdoctoral Studies: Promoting the National Interest through Excellence,
Competitiveness and Responsibility in the Field of Romanian Fundamental and Applied
Scientific Research”, contract number POSDRU/159/1.5/S/140106. This project is cofinanced by European Social Fund through Sectorial Operational Program for Human
Resources Development 2007-2013. Investing in people!
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