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0Russian Journal of Nonlinear Dynamics, 2022, vol. 18, no. 3, pp. 333-348. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd220301
NONLINEAR PHYSICS AND MECHANICS
MSC 2010: 35Q30, 76T25, 76F35, 80A25
Numerical Simulation of the Nonstationary Process of the Shot Based on the Navier - Stokes Equations
This paper gives a spatial mathematical formulation of the problem of internal ballistics based on the Navier-Stokes equations, taking into account the swirl of the flow due to the rotation of the projectile. The k-e model of turbulent viscosity is used. The control volume method is used for the numerical solution of systems of equations. The gas parameters at the boundaries of the control volumes are determined by the method of S.K.Godunov using a self-similar solution to the problem of the decay of an arbitrary discontinuity. The MUSCL scheme is used to increase the order of approximation of the difference method. For equations written in a cylindrical coordinate system, an orthogonal difference grid is constructed using the complex boundary element method. A comparative analysis of the results obtained with different approaches to modeling the process of an artillery shot is given. Quantitative data are presented on the influence of factors not previously taken into account on the characteristics of the process.
Keywords: internal ballistics, mathematical model of a shot, mechanics of heterogeneous media, Navier-Stokes equations, axisymmetric swirling flow, computational algorithms
1. Introduction
In mathematical modeling of the process of firing from an artillery gun, two approaches are currently used: thermodynamic and gas-dynamic. The first approach is based on averaging the intra-ballistic parameters over the projectile space. This approach was finally formulated at the
Received March 30, 2022 Accepted June 08, 2022
The reported study was funded by RFBR, project number 20-01-00072.
Ivan G. Rusyak primat@istu.ru Valentin A. Tenenev tenenev@istu.ru Stanislav A. Korolev stkj@mail.ru
Kalashnikov Izhevsk State Technical University ul. Studencheskaya 7, Izhevsk, 426069 Russia
I. G. Rusyak, V. A. Tenenev, S. A. Korolev
turn of the 19th and 20th centuries by the efforts of French and Russian ballistics scientists: H. Rezal, E.Sarro, I. P. Grave, N.F. Drozdov and others. A generalization of this approach can be found in the book by M. E. Serebryakov [1]. For the first time, the problem of internal ballistics in a one-dimensional gas-dynamic formulation was solved by S. A. Betekhtin in 1947, the description of the solution results of which was published in [2]. The mathematical model used was founded on the Euler equations and was based on the gas-powder mixture hypothesis, according to which the burning powder elements move at the same speed as the surrounding gases. In the 1970s, models appeared based on the principles of the mechanics of heterogeneous media and interpenetrating continuums, taking into account the separate motion and interaction of phases. Subsequently, these models were reflected in publications [3-6]. At present, it is this direction of modeling the firing process that is being intensively developed. The main efforts are aimed at taking into account the geometric (loading chamber) and physical (powder charge design) multidimensionality [7-9]. The models used for the behavior of burning powder elements in the bore during firing are based on the equations of motion of a heterogeneous system, which take into account the processes of heat and mass transfer and friction between the phases. In this case, the motion of the carrier gas phase is described by the gas-dynamic Euler equations with source terms responsible for the exchange of momentum and energy with the solid phase.
The firing process is characterized by a high level pressure. Therefore, the equation of state for an ideal gas is unacceptable and is replaced by the equation of state for powder combustion gases in the form of Dupre [10]. This equation is a special case of the state equation for a real van der Waals gas without taking into account the force of attraction between molecules, but taking into account the total volume of gas molecules (covolume). The use of the state equation for a real gas in various forms leads to additional computational difficulties in the implementation of difference schemes and methods. Models of the equation of state for estimating the thermodynamic properties of a continuous medium and transport characteristics are considered in [11]. For the HLLC [12], AUSM+ [13], Kurganov and Tadmor [14] schemes, a general approach is proposed that allows them to be related to exact thermophysical models for calculating real gas flows. Numerical experiments demonstrate the applicability of the proposed methods in comparison with the exact solution of the one-dimensional Riemann problem. The work [15] presents the flow splitting method for the one-dimensional system of Euler equations, according to which the initial vector is divided into the flow vector and the pressure vector, which makes it possible to use the general equation of state. The energy of the gas is determined on the basis of the internal specific energy, expressed in terms of the equation of state, written in various forms. The method of flow splitting for the spatial Euler equations is also described.
For complex gas-dynamic calculations, the S. K. Godunov difference scheme [16], which uses the physical behavior of the medium at contact discontinuities in a wide range of parameter variations, has received wide application. The work [17] presents the results of the numerical solution of the equations of multicomponent gas dynamics in problems with a high energy density in matter. The gas-dynamic part is based on the S. K. Godunov scheme and the solution of the discontinuity decay problem using the approximate local equation of state. The conservative method of S. K. Godunov of a high order of accuracy in the Euler form was used for a multiphase medium modeled by constructing an effective phase [18].
When using the S. K. Godunov scheme, to calculate the flows through the faces of the control volume, it is required to solve the Riemann problem on the decay of an arbitrary discontinuity. The exact solution of the problem is laborious, even for a simple state equation for an ideal gas. The complexity increases when using the equations of state for a real gas. Therefore, much attention is paid to the development of various solvers of the Riemann problem for the
general equations of state for a real gas. The authors of [19] described a modification of the Roe linearization method for real gases and its application to a particular case of a van der Waals gas. By reducing all the physical parameters that depend on the thermodynamic equations of state to a single intermediate density equation, this method can be extended to work with more complex physical systems, such as, for example, accurate thermodynamic models for dense gases or chemically reacting gases in a local thermodynamic balance. The article [20] presents a modified S.K.Godunov scheme WFRoe, based on the approximate solution of the Riemann problem. To evaluate the method, such special problems as the motion of a cloud of particles in a gas vacuum, two-phase gas flows with rigid smooth particles are considered. The article [21] presents an analysis of averaging procedures for the Riemann scheme, which can be used to model or resolve compressible real gas flows described using complex equations of state. The works [22, 23] offer algorithms for solving the Riemann problem when implementing the method of S. K. Godunov for gas-dynamic reacting flows. It is shown that their application can significantly reduce the amount of calculations and allows expanding the range of equations of state used. The work [24] considers an approach to solving the problem of the decay of an arbitrary discontinuity in media obeying normal equations of state, based on the Newton method for solving problems with two-term equations of state that locally approximate the original ones. The work [25] describes a number of practical recipes for overcoming the computational difficulties that arise when using complex equations of state. As examples of the use of high-resolution numerical methods, problems of wave propagation in media with complex equations of state are considered. The use of local approximation made it possible to calculate the spatial dynamics of a real gas under near-critical conditions close using the difference scheme of S. K. Godunov [26].
In well-known shot models, as a rule, an assumption is made that there is no viscous interaction of combustion products with the surface of the barrel and projectile. In this case, the rotation of the projectile, taking into account the viscosity of the gas, can lead to swirling of the flow when the gas moves along the barrel. In this study, in the numerical simulation of the movement of the carrier gas phase, the Navier-Stokes equations are solved together with the equations describing the movement of burning powder elements and the movement of the projectile. The aim of the work is to obtain quantitative results of the influence of factors not taken into account previously on the characteristics of the shot process.
2. Mathematical model of the shot
We will consider a two-phase flow consisting of powder gases and a set of burning particles of granulated powder, the characteristic size and concentration of which is such that the assumption is made that the distances at which the flow parameters change significantly are much greater than the distances between the particles and the size of the particles themselves. Different phases are present simultaneously at all points in space, at the same time, each phase occupies a part of the volume of the mixture. Particles are on average the same size, collisions, i.e., the interaction between them can be neglected. We assume that the particle material is incompressible and the gas parameters inside and outside the powder elements in each section are the same. The system of equations for the internal ballistics of an artillery shot, which describes the flow of a heterogeneous reacting mixture, taking into account intergranular interaction in an axisym-metric formulation based on the Euler equations, is described in the work of the authors [27]. In addition to this model, instead of the Euler equations for the carrier phase, we will consider
the Navier-Stokes equations. The computational domain and grid in the combustion chamber of the barrel are shown in Fig. 1.
In Fig. 1, the boundary AD is the axis of symmetry, BC is the cylindrical generatrix of the combustion chamber, AB is the bottom of the chamber, CD is the outlet section of the chamber. A cylindrical area of the barrel with a movable mesh is attached to the CD section, bounded on the right by the bottom of the projectile.
The dimensionless equations describing the motion of a viscous heat-conducting gas in the case of an axisymmetric swirling flow have the following form [27]:
9(rq) | d[r( A-A,)] | d[r( B-BJ] = g , H dt dx dr
where t, x, r are time, longitudinal and radial coordinates, respectively,
q
p pu pv
pu puu + p puv
pv pvu pvv + p
pw , A = pwu , B = pwv
e u(e + p) v(e + p)
pK puK pvK
pE puE \ pvE y
(2.1)
0 nii n21
31
. f +
f + -M o^
je t (tk j p dx
O—
P dx
0 0
(:v + pw2 -(-pvw + if) 0
rp(2xD2 — E) \r§p(2ClXD2-c2E)J
H
0
n12 n22
32
w i (fs+x)~ de
P2 * ' pr Pi dr
f + -M o^
Je ^ ctk I P dr
rG
0—
P dr
P_ drm 1 n _ rT m dx Tl^Up I'u
¿drm 1 Q _ m dr >Tv
rGwp — rrw rGQ 0 0
(
\
\
X
a
E
(
\
S
Here p = mp, p = mp, e = mev, ev = pe + pu"+v^+w"; p is density, p is the pressure of powder gases, m is the porosity of the gas-powder mixture, e is the internal energy of a unit mass of powder gases, K is the kinetic energy of turbulence, E is the rate of turbulence energy dissipation, are the components of the viscous stress tensor, Pr is the Prandtl number, Re is the Reynolds number, f£ is the dimensionless function of the dependence of viscosity on internal energy, % is the dimensionless turbulent viscosity, Cl, c2, aK, aE are the parameters of the turbulence model, G is the gas income of combustion products from the surface of the powder charge per unit volume per second, Q is the calorific value of gunpowder, and up, vp, wp, tu, tv, tw are the parameters of interfacial interaction.
The velocity vector V = (u, v, w)T has components in a cylindrical coordinate system (x, r, y), where y is the angular coordinate. The vectors q, A, B, Av, Bv, S, H contain gas-dynamic complexes, where A, B are convective terms, Av, Bv are viscous terms, and S, H are source terms.
We write the equation of state using the local approximation [25]:
t = -, k = --7 =
/ i \ —-« rvn ' '
(k~l)p 1-f
c
v
where a is the covolume of powder gases, y is the adiabatic exponent, and
Cp, Cv are isobaric
and isochoric heat capacities of combustion products, respectively.
Variables p, V are referred to critical values p*, a*, pressure p and energy ev are referred
to where a* = ./7^ is the speed of sound. The spatial scale is the barrel radius Rb, and
p r
the time scale is r = -Jl. The critical values of the variables are related to the braking param-
/ \ —y/' (Y—1) / \ —I/(Y—1)
eters p0 and p0 by the known relationships p* = p0 () and P* = Po ()
As the scales of the turbulence energy K and the rate of its dissipation E, ^ and ^ are chosen, respectively. Turbulent viscosity, referred to v0, is x = The Reynolds number is defined
R a
as Re = The coefficient of kinematic viscosity v0 corresponds to the stagnation temper-
ature T0. The dependence of viscosity on temperature is taken into account by the Sutherland formula:
v (e) = V0 L, fe =
Y + 1\ —y/(y—1) ( e V/2 eo + es
-0/ c ~ cs
cv - _ cyTs
2 J \e0J e + es
where tn = e, = ^,T = 122 K.
0 a% ' s a% ' s
The components form a tensor
n = p{f£ + x) def (V) — ^ div(V)^ , D = def(V), Pi=(irij).
The parameters of the turbulence model have the following values [29]: C1 = c1-
i1 ~ ^ = c» = °-085; Cl = L42; c2 = L68; Pe = °-012; VE0 = 4-38;
aK = 0.72; aE = 0.72.
The equations for the solid phase with velocities Vp = (up, vp, wp)T in a cylindrical coordinate system (x, r, y) are presented in the form [27]
dra draup dravp --1--- H--- = 0,
dt dx dr
dr6(l - m)up dr (p + 8up) (1 - m) dr§( 1 - m)upvp dr{ 1 - m) ( , dt dx dr ^ dx 1 l'p
drS(l — m)vp drS(l — m)upvp dr (p + 5vp) (1 — m) dr(l — m)
dt dx dr ^ dr 7 lp 7Txn
dr5(1 — m)wp dr5(1 — m)up wp dr5(1 — m)vpwp
dt + di. + d~r = ~7'GWp + TTw'
The combustion equation for grained powder elements is written as:
> before the phase of disintegration of powder elements z ^ l or ^ ^ ^r = k(1 + A + j):
dz , dz , dz u
o ^
and
% + Upfx + = = 1 + 2A* + (2-3)
dt +Updx +Vpdr -h<7Wei'
after the disintegration of powder elements ^ > = k(1 + A + j):
d^ d^ d^ . ,. uk . I , . , . / 1 — ^
+ = = (2.5)
< ^ < 1, a($r) = 1 + 2A +
The porosity of the gas-powder mixture (the volume of voids per unit volume) is determined by the formula
m = 1 — aA0(1 —
and the current geometric dimensions of the powder element, according to the formulas
d = d0 + 2ze1, D = D0 — 2ze1, L = L0 — 2ze1.
In the above equations, tu, tv, tw are the projections of the force of resistance to the movement of combustion products in the layer of granular powder elements on the axes of the cylindrical coordinate system (x, r, y), respectively, G is the gas income of combustion products from the surface of the powder charge per unit volume per second, Q = / = RTV is
gunpowder power, R is the specific gas constant of combustion products, Tv is the temperature of gunpowder combustion products in a closed volume, e = cvT, T is the temperature of powder gases, 5 is powder material density, a is the countable concentration of grained powder elements, uk = uk(p) is gunpowder burning rate, z = f- is the relative thickness of the burnt vault, 2e1 is the initial thickness of the burnt vault, ^ is the relative proportion of burnt gunpowder, a(^) is the ratio of the current combustion surface to the initial one, k, A, j are the shape factors of the powder element determined by the initial dimensions of the powder grain, d0, D0, L0 are the initial diameter of the inner channel, the outer diameter and the length of the 7-channel powder
d2 — 7d2
element, respectively, and A0 is the initial volume of the powder element: A0 = it °4 °L0. The functions of mass, force, energy interaction between the phases have the form
G=
aS0a(z)5uk(p), if z ^ 1 or ^ ^ ^r, aS0a(^)5uk(p), if z > 1 or ^ > ^r, _ , p(u-up)\V -Vp\ S^
Tu ~ \ 2 a 4 '
. P(v-vp)|V-Vp| tv = Xv---a—, (2.6)
_ > P(w-wp) |V - Vp| ¿v Tw — \ 2 a 4 '
where Xv = 0.5 is the drag coefficient of the powder grain in the layer, Sa the current limiting
surface of the powder element: Sa = + ttDL = irD (-j + L), and S0 is the initial surface
D2— 7d2
of the powder element: S0 = 2tt—^—21 + it(D0 + 7d0)L0.
The equations for the velocity upr and displacement xpr of the projectile have the form
dUpr
~dT
>pr uiujjiuuviiiv™ Mpr
I R \
2n p(t, xpr, r)r dr
17pr !
0
dx
n(Ppr - Pf), (2.7)
/
where ppr is the average pressure on the bottom of the projectile, Pf is the forcing pressure, and n(£) is the Heaviside function. The projectile begins to move after reaching the forcing pressure on its bottom ppr ^ Pf.
The system of equations (2.1)-(2.8) must be supplemented with initial and boundary conditions. The conditions of rest and the conditions corresponding to the state of the combustion products at the moment of complete combustion of the igniter mass wig in a closed volume are set as the initial conditions.
The initial conditions at t = 0, 0 ^ x ^ Lc, 0 ^ r ^ Rc(x) are
V = 0, Vp = 0, p = pig, p = Aig, T = Tv, a = A , m = l-j,
0 = 0, A = V = 0' xpr = Lc■ (2-9)
Here Lc is the chamber length, Rc(x) is the variable radius of the chamber, Wc is the volume of
the chamber, pig = 1_^t9A is the pressure developed by the igniter, Aig = ^Tj?^ is the igniter
zg c 5
loading density, and u is the charge mass.
On the impermeable boundaries of the volume, the no-slip conditions for the gas and the reflection conditions for the solid phase near the wall are set. Symmetry conditions for the desired functions are set on the channel axis.
The control volume method described in [28] is used for the numerical solution of systems of gas-dynamic equations (2.1). For each face, the corresponding discontinuity decay problem is solved with two sets of gas-dynamic parameters in control volumes separated by this face. The gas parameters at the boundaries of the control volumes are determined by the method of S.K.Godunov using a self-similar solution to the problem of the decay of an arbitrary discontinuity. To solve the hyperbolic equations of motion of the condensed phase (2.2) and the combustion of gunpowder (2.3)-(2.5), a difference scheme of S.K.Godunov type is also used with the definition of flows on the faces of the control volume according to the scheme presented in [30]. The MUSCL scheme is used to increase the order of approximation of the S. K. Godunov difference method. The implemented difference scheme has the second order of approximation in space and time, which is shown in [9, 28].
For equations written in a cylindrical coordinate system, in the general case, an orthogonal curvilinear difference grid is constructed using the complex boundary element method [31], which
turns into a rectangular one in the bore's region with a constant outer radius. The dimension of the grid along the length and the radius is N = Nx x Nr = 250 x 50 cells with condensation near the solid boundary. The minimum cell size in the radial direction was chosen to ensure
that, two boundary cells get into a viscous sublayer of thickness = ^ pa 11, where =
is the dynamic velocity and tr = pu R is the friction stress on the wall. A movable grid is constructed in the barrel, which stretches as the projectile moves; the calculation algorithm taking into account the movable grid is given in [32].
The gradients of the variables included in the components are calculated in the middle of each face via the values of the variables in the surrounding control volumes. Since the mesh is orthogonal, the task of calculating gradients on the faces of the control volume is not difficult.
The transition to the next time step is carried out according to a two-step scheme with the second order of accuracy:
n+l/2 wra+1/2 _ n nn I nn Q™ f~m I
qj1G7+1 = qlj Gij - (£ Qn+1/2 - si;;1/2 c^ дл
where i, j is the number of the control volume, Сij is the value of the control volume at the time step n, and the summation of flows is carried out over all faces of the control volume. The control volumes for the barrel are stretched when the projectile moves, in the combustion chamber they do not change over time.
3. Calculation results
Initial data for calculations: the diameter of the combustion chamber is 0.2 m, the barrel diameter is 0.1 m, the chamber length is 1 m, the barrel length is 5 m (the geometry of the combustion chamber is shown in Fig. 1), the projectile mass is 15 kg, and the charge mass is 17 kg (the charge consists of 7 channel powder elements).
The calculations were carried out before the departure of the projectile from the bore for three options:
1) the components of the viscous tensor are equal to 0;
2) viscous formulation of the problem with the initial value of turbulent viscosity % = v0;
3) viscous formulation of the problem with the initial value of turbulent viscosity % = 5v0 (combustion in a layer of powder grains can lead to additional turbulence of combustion products).
The main characteristics of the shot are the muzzle velocity of the projectile (at the exit from the barrel) and the distribution of pressure over the combustion chamber and the barrel. Figure 2 shows the change in the speed of the projectile along the barrel in time. Figure 3 shows the change in pressure at the bottom of the combustion chamber and at the bottom of the projectile over time. The distribution of the longitudinal components of the velocity of gas and powder particles at the moment of departure of the projectile from the bore is shown in Fig. 4, and the change in pressure along the axis of symmetry is shown in Fig. 5. The curves are
upr, m/s 1200
1000
800
600
400
200
0 -, — 0 2 4 6 8 10 12 14 16
Fig. 2. The dependence of the speed of the projectile along the barrel in time
p, MPa 450
400
350
300
250
200
150
100
50
0 -, — 0 2 4 6 8 10 12 14 16
Fig. 3. Change in pressure at the bottom of the combustion chamber (1) and at the bottom of the projectile (2) in time
presented for the first version of the problem statement — the components of the viscous tensor are equal to 0.
For the case of viscous turbulent flow, a boundary layer is formed on the wall of the bore. Figure 6 shows the relative value of the turbulent viscosity at the final moment of the shot
(t = 15.5 ms) for the second calculation option (the initial value of the turbulent viscosity % = = v0). The figure shows that, with an increase in the gas flow rate towards the end of the barrel, the intensity of turbulence increases.
The radial distribution of the value -f- at the moment of time t = 13 ms is shown in Fig. 7.
v0
The ratio of the value of the viscous friction stress to the velocity head averaged over the borehole section ^ at various points in time is shown in Fig. 8. For the flow area in the bore,
u, m/s
Fig. 4. Distribution of longitudinal velocities of gas (1) and solid (2) phases p, MPa
Fig. 5. Change in pressure along the axis of symmetry
r, mi
L 0.0e+00100 200 300 400 500 600 700 8.4e+02
Fig. 6. Distribution of the relative value of turbulent viscosity in different areas along the barrel
x ______
"o
200 --------
20 25 30 35 40 45 r, mm
Fig. 7. Radial distribution of the relative value of turbulent viscosity in various sections: 1) x = 1.0 m (barrel entrance); 2) x =1.8 m; 3) x = 2.6 m; 4) x = 3.4 m
4
I
•A
2
J. 1 \
Fig. 8. The distribution of the relative value of the viscous friction stress to the velocity head at the instants of time: 1) t = 12.0 ms; 2) t = 13.5 ms; 3) t = 14.5 ms; 4) t = 15.5 ms
the value ^¡jj is approximately 0.18%, which has little effect on the integral indicator of the shot process — the muzzle velocity of the projectile.
To assess the effect of the initial flow turbulence in the combustion chamber, consider the results of calculations for options 2 and 3, which differ in the initial value of turbulent viscosity. Note that option 3 (the initial value of turbulent viscosity % = 5v0) is a hypothetical option based on the fact that the intense combustion of gunpowder itself is a turbulence generator. Figure 9 shows the change in the maximum value of turbulent viscosity in the cross section of the barrel along the barrel (1.0 ^ x ^ 6.0 m) for calculation options 2 and 3.
As can be seen from the graphs shown in Fig. 9, the greatest difference in the level of turbulent viscosity for the 2nd and 3rd calculation options is observed in the initial sections of the barrel, as the projectile speed increases, the influence of the initial value of the turbulence intensity decreases.
x.
VQ 1200
1000
800
600
400
200
0 -1 2 3 4 5 6
Fig. 9. The change in the maximum turbulent viscosity over the cross section of the barrel along the barrel for various options of the problem formulation: viscous setting (dot line); viscous setting, additional turbulence (dash line)
Figure 10 shows the friction stress on the barrel wall tr for calculation options 2 and 3. The friction stress on the barrel wall in option 3 is approximately two times higher than that in option 2. This explains a slight decrease in the muzzle velocity of the projectile by 0.4%.
tr, MPa 0.6
0.5
0.4
0.3
0.2
0.1
0 -1 2 3 4 5 6
Fig. 10. Variation of the friction stress on the barrel wall for various options of the problem formulation: viscous setting (dot line); viscous setting, additional turbulence (dash line)
Figure 11 shows the dependence of the tangential velocity of rotation of the gas caused by the rotation of the projectile in the vicinity of the wall (at a distance of the thickness of the boundary layer) on the distance to the bottom of the projectile at the time of its departure from the bore for calculation options 2 and 3.
As can be seen from the figure, the velocity of rotation of the gas phase decreases quickly as it moves away from the rotating projectile. The effect of swirling of the flow by a rotating projectile, as follows from a comparison of options 1 and 2, is included in the total reduction in muzzle velocity by 0.07%. An increase in the initial turbulence does not lead to a significant change in the distribution of the angular velocity of the flow rotation (see Fig. 11).
/
___X r / \ v.-"" " ~
/ /
/ /---
/
y / /
/ /
__/
w, m 2500
2000
1500
1000
500
0 -, 0 2 4 6 8 10 12 14
Fig. 11. Distribution of the angular velocity of the gas phase for various options of the problem formulation: viscous setting (dot line); viscous setting, additional turbulence (dash line)
/в
\ \
4 \
\
4 ^
Figure 12 shows the change in the speed of the projectile (a) and pressure (b) on the bottom of the combustion chamber and on the bottom of the projectile in the final section of the barrel in time for three calculation options.
upr> 1 1100
1000
900
800
700
600 -, — -, — 12 13 14 15 16 8 10 12 14 16
(a) (b)
Fig. 12. Dependence of the speed of the projectile (a) and pressure (b) on the bottom of the combustion chamber (1) and on the bottom of the projectile (2) in the final section of the barrel in time for different options of the problem setting: inviscid setting (solid line); viscous setting (dot line); viscous setting, additional turbulence (dash line)
As can be seen from the graphs presented in Fig. 12, the dependences for the velocity of the projectile and the pressure on the bottom of the combustion chamber and on the bottom of the projectile differ insignificantly for all calculation options.
A comparison of the main parameters of the shot for the three options is given in the table. Here u™ is the muzzle velocity of the projectile, pmax is the maximum pressure on the bottom of the chamber during the shot, and pmrax is the maximum pressure on the bottom of the projectile.
Table 1. Calculation results of the parameters of the shot
Option 1 2 3
u™, m/s 1068.4 1067.7 1063.7
p™, MPa 393.42 393.49 399.45
MPa 244.19 243.93 241.41
As calculations show, an increase in flow turbulence leads to some decrease in the speed of the projectile due to the additional cost of charge energy for swirling and flow turbulization. It follows from the table that taking into account viscosity leads to a decrease in the calculated muzzle velocity by 0.7 m/s. Additional turbulence of combustion products in the projectile volume reduces the velocity by more than 4.7 m/s (0.5%). Also, additional turbulence of the combustion products leads to an increase in pressure on the bottom of the chamber by 1.5%, and the pressure on the bottom of the projectile decreases by 1.1%. However, in general, the results indicate an insignificant effect of viscosity on the gas-dynamic characteristics of the shot.
4. Conclusion
A mathematical model of internal ballistics using the Navier-Stokes equations made it possible to take into account the influence of viscosity and projectile rotation on the main characteristics of an artillery shot. The calculations showed that the upper estimate of the decrease in the muzzle velocity of the projectile due to viscous friction is less than 0.5 %, the increase in the maximum pressure on the bottom of the channel is 1.5% and the decrease in the maximum pressure on the bottom of the projectile is 1.1%. The estimates made in [27] showed that one-dimensional and two-dimensional gas-dynamic models of internal ballistics based on the Euler equations differ in muzzle velocity by 1.0%, in maximum pressure on the channel bottom by 3.0%, and in maximum pressure on the bottom of the projectile by 2.4%. The level of differences between the studied parameters due to viscosity is less than these values. For practical calculations of gas-dynamic characteristics, the assumptions about the insignificant effect of gas friction on the barrel walls and projectile rotation on the dynamics of the shot process are justified.
Conflict of interest
The authors declare that they have no conflicts of interest.
References
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