Identification and simulation mathematical model of thermo and physical loading of a small-caliber artillery barrel

The growth of requirements for the intensification of firing modes (increase in the number of shots in the queue and the number of bursts of shots, reducing the time intervals between bursts of shots), the consistency of developments to create new rapid-fire artillery guns determine the exceptional importance of adjusting the methods and ways for describing the barrel heating process, which limits the combat properties of aviation artillerary weapons [1]. Therefore, the scientific and technical task of formalizing the temperature field of the barrel, which is cyclically subjected to high thermomechanical loads, seems to be a priority task for the design and study of aviation artillerary weapons.

When solving the problem under consideration, it is firstly necessary to determine to what extent traditional methods for solving the generalized problem of heating engineering structures will be effective and, secondly, what new approaches can be proposed to achieve the goal set in the work. Thus, in articles [2– 4] schemes of experimental studies and methods for processing output data are proposed, which provide an increase in the accuracy of determining body temperature and an expansion of the measurement range; publication [5] presents a unique version of a thermal model developed on the basis of the apparatus of probability theory; in [6] the temperature fields of finned walls of various configurations were determined by numerical solutions of the multidimensional problem of heat conduction; work [7] proposes tools for modeling the temperature field in the nodes of gas turbines, which maximally take into account the set of parameters in multifactorial boundary conditions of the boundary layer; in the article [8], correlation regression dependences of the optimal extrema of loading the barrels of small arms and cannon artillery weapons were obtained. Publications [9–13] can be cited as examples of works on similar topics in the field of aviation artillery science.

In the analyzed approaches, both a simple method of enumeration of various options and their combinations, and a group of complex methods of the “predictor-corrector” class, which provide for extrapolating and correcting formulas, are used. Naturally, it is impossible to indicate any specific, strictly regulated technique that gives true results in heating bodies of various designs and operating conditions. Nevertheless, the search for solutions that adequately compare the complexity of the object with the complexity of mathematical or experimental means (the law of adequacy of the object and the controller) is relevant in the short and long term. According to the accepted terminology [14; 15] direct construction of individual blocks of the model implies the simulation process. In the case of using known borrowings and establishing the appropriate organization of parameters, the problem of model identification is apparent in the work.

Synthesis of differential operators for identification and simulation of the barrel temperature field

When solving problems of heat conduction, the basic Fourier differential equation of heat conduction is used, generally describing the temperature distribution in a solid [16]. For a barrel having a complex geometric shape of a compound cylinder, it is advisable to represent the Fourier heat equation in a cylindrical coordinate system, and since the shaft is a body of revolution and is symmetric about the longitudinal axis, consideration of the problem determining the temperature field of the barrel is limited to a two-dimensional form in one of the radial planes [16; 17]:

д T    ( д ² T  1 д T  1 д ² T ^Л

— a       +1

д т     ^ д r ²    r д r   r д О² _?

where T is the barrel temperature; τ is time; a is the temperature conductivity coefficient of the barrel steel; r , θ are the radius vector and the polar angle of the cylindrical coordinate system, respectively.

The temperature conductivity coefficient of the material in equation (1), which characterizes the rate of change in body temperature, is especially important for describing precisely non-stationary thermal processes and is calculated by the formula:

λ

a — —,                                    (2)

сρ where λ, с, ρ are the coefficients of thermal conductivity, specific heat capacity and density of the barrel steel, respectively.

The real phenomenon of the barrel heating and cooling is abstracted by some assumptions that simplify, but do not reduce the quality of the model construction and are based on the generalization of the experience accumulated in the applied studies noted earlier:

– the material of barrel steel OHN3MFA is considered to be isotropic and homogeneous;

– the initial temperature of the barrel is equal to the ambient temperature ( T 0 = T в );

– heat exchange between the chamber part of the barrel and the cartridge during the shot is neglected due to the insignificance of the considered period;

– the cartridge is represented as a model temperature concentrator with constant thermophysical characteristics.

Accepting an extreme assumption, while studying the issue of the safe presence of the fresh round in a barrel heated by firing, causes the objective need to directly take into account in formula (2) the dependence of the thermal conductivity coefficients λ and specific heat c of the barrel steel on the barrel temperature T by procedural queries of the required values from the data array [18]. At the same time, the non-linearity of the barrel material thermophysical coefficients undoubtedly leads to a significant complication of the problem and an increase in the time of computer calculations. Since the cylinder is axisymmetric, it was expediently to consider its longitudinal section. Based on the first assumption, in order to differentiate the calculation procedures, with a known relationship between cylindrical coordinates and rectangular Cartesian coordinates [17], equation (1) is replaced by two one-dimensional differential equations, one of which is written in cylindrical and the other in rectangular Cartesian coordinates. As a result, a system of equations was obtained:

дT   ( д2 T  1 д T ^

— — a —у +---;

д т     ^ д r²   r д r ;

д T    д ² T

— a , д т     д x ²

where x is the x-axis of a rectangular Cartesian coordinate system.

The dualism of conditions in formula (3) is explained by the need to match the accuracy characteristics of the physical and mathematical methods applied with the available characteristics of computer time, memory and bit grid of the calculator. This technique involves the sequential connection of procedure subprograms containing calls to more simplified mathematical models, compared to the complete ones that describe local changes in the barrel temperature field in the transverse r and longitudinal x directions.

As applied to modeling the barrel thermophysical loading, the first equation of system (3) determines the unsteady barrel temperature T exclusively in the radial direction r . The second equation of system (3) determines the same value in the longitudinal direction along the x axis. On the whole, the system of equations (3) characterizes the spatiotemporal change in the cylinder temperature at any point, uniting all types of the heat conduction phenomenon, regardless of the body physical properties and conditions of interaction with the environment. To isolate the phenomenon under study, the system of equations (3) is supplemented with singlevalued conditions specific to the object and subject of the study. The single-valued conditions include the required interpretations of the geometric, boundary (set of initial and boundary) physical single-valued condition. Fig. 1 shows a diagram of the axial symmetry of the most common GSH-301 aircraft gun barrel.

r, м 0,04

0,03

0,02

0,01

0,3              0,6              0,9              1,2              1,5

Рис. 1. Схема осевой симметрии ствола авиационной пушки ГШ-301

Fig. 1. Scheme of axial symmetry of the GSh-301 aircraft gun barrel

With the highest 30-mm cannons rate of fire per barrel, this aviation artillerary weapon with a typical OFZ-30 high-explosive fragmentation projectile provides firing modes with the maximum barrel thermal load. The x axis in fig. 1 is aligned with the barral longitudinal axis, thus consideration of the area located only on one side of the barrel longitudinal is sufficient.

The initial single-valued conditions are determined by the given distribution of temperatures in the barrel for a fixed time preceding the considered one and taken as the initial time τ = 0. The equation for the barrel temperature field for this time τ is written as:

T ( r , x , 0) = T 0 ( r , x ). (4)

Since the initial barrel temperature T 0 at any point ( r, x ) at time τ = 0 is equal to the ambient temperature, equation (4) has the form:

T ( r , x , 0) = T в = const. (5)

The boundary conditions determine the values of the heat transfer parameters at the barrel boundaries. The GSh-301 aircraft gun barrel is not thermally insulated, so the boundary conditions are set in the form of ambient temperatures and the laws of heat transfer between the medium and barrel surfaces, depending on the design parameters and conditions for the use of aviation artillerary weapons (conditions of the third kind) [16].

The intensity of heat exchange between powder gases (hereinafter referred to as gases) formed in the process of burning gunpowder, barrel and air depends on the physical and mechanical processes occurring at the barrel boundaries. These processes are quite accurately described by the well-known equations of the main problem of internal ballistics for automatic artillery pieces [1; 19; 20] and the heat balance equation [16].

For each point of the internal boundary of the barrel, the boundary condition in which the incoming values depend on time τ and coordinate x , written through the heat balance equation:

а г [ Т г (т) - T (x , т) ] = -Х— , д r

where α г is the coefficient of heat transfer from gases to the bore; T г is the temperature of gases in the bore.

Equation (6) determines the convective heat flux from gases directed deep into the barrel wall to its outer surface at a given point.

The impact of a moving projectile on the channel leads to additional heating of the barrel. When using copper leading belts in automatic shooting, the average friction temperature is calculated according to an empirical formula that takes into account the gas velocity v [20] current in coordinates and time in the bore.

The boundary condition for each point of the outer barrel boundary, on which the convective heat flow from air is formed similarly to equation (6)

ав [Тв - Т(x, т)] = -к—, дr where αв is the coefficient of heat transfer from the barrel outer surface to the air.

A convective heat flow acts on the muzzle of the barrel, which is formed by the gas temperatures at the muzzle Tд. In this case, equations (6) and (7) are written for the end of the firing process а г [Тд (т) - Т (r, .)] = -! дТ.

дх

For the time intervals between bursts of shots, equations (7) and (8) have the form:

а в [ Т в - T ( r , т) ] = -Х — . д х

Since the bore is closed, i.e., there is no direct propagation of thermal radiation into the external environment, and the value of the flux of thermal radiation from the barrel outer wall into the environment is insignificant and, according to preliminary calculations, is less than 1% of the barrel temperature T on its outer surface, then calculations of the heat flux of radiant energy are not carried out in the work.

During bursts of shots, the channel and the barrel outer surface have sufficiently high temperatures, so it is necessary to take into account the design features of the aviation artillery weapons reference sample. The simulation of the process of the GSh-301 aircraft gun standard cooling system functioning is implemented by introducing a local heat transfer coefficient.

It is necessary to take into account the influence of the cartridge case located in the breech during the shot on the distribution of the barrel temperature field. Since the cartridge located at a distance x = 0,175 m of the barrel length, under the influence of gas pressure it is tightly pressed against the wall of the chamber, it can be assumed that such contact is ideal, i.e. heat transfer is carried out only through heat conduction. Since the wall thickness of the cartridge is relatively small, it is assumed that it is instantly heated to the gas temperatures in the channel barrel T г . Based on the above, a boundary condition was formulated on the surface of the chamber at the characteristic points of the barrel, where there is a direct contact between the cartridge and the wall (condition of the first kind) [16]

Т ( r = 0, x = 0...0,175) = Т г (т), (10)

The physical processes of convective heat transfer during automatic firing and in between bursts of shots are quite complex. A perfect mathematical account of heat transfer on the barrel walls by means of heat transfer coefficients from gases to the barrel channel α 1 and from the outer surface of the barrel to air α 2 seems to be an independent task, based, as a rule, on the application of the similarity theory of heat transfer processes in artillery pieces [19; 20].

Thus, under given differential equations of the process (3), the geometric single-valued condition shown in Fig. 1, the physical single-valued condition, the two-dimensional, non-stationary, nonlinear problem of determining the barrel temperature field is represented by a system of equations that completely determines the boundary value problem (4)–(10):

д Т    f 5 2 Т  1 д Т ^

— = a —у +--- дт    ^ д r2   r д r ;

д Т    д ² Т

— = a —т;

д т     д x ²

а_г [ Т _г (т) - Т ( x , т) ] = -Х— ;

д r

а_в [ Т в - Т ( x , т) ] = - ^;

д r аг [ Тд (т) - Т (r, т) ] = -Х   ;

дх ав [Тв - Т(r, т)] = -^;

д х

Т ( r = 0, x = 0...0,175) = Т г (т);

Т ( r , x , 0) = Т _в = const.

The advantages of the equation system (11) lie in the simplicity of the initial data grouping and visibility of their distribution by hierarchy levels with a constantly increasing degree of detail. Nevertheless, the system of equations (11) cannot be directly applied to solve the problem stated. The fact is that the analytical solution of the heat conduction and heat transfer problems is possible for simple shaped bodies and under simple boundary conditions [21; 22].

Analytical methods for solving heat transfer problems are in depth described in [23]. A complex two-dimensional, non-stationary, nonlinear applied problem of heat transfer in the region with a complex configuration of boundaries in such a formulation does not have an exact analytical solution. Obtaining a result close to a probably existing analytical solution is possible only by selection of numerical methods, the essence of which is reduced to finite-difference approximations of the equation system (11).

Formation of a difference scheme and selection of methods for calculating the barrel temperature field

An approximate solution of the problem of heat transfer in the barrel will be carried out using the finite difference method as the most approved and acceptable for bodies with sections variable along the length [21–24]. The zone of continuous variation of the argument is replaced by a difference grid (hereinafter referred to as the grid) – a discrete set of points intersections of which form nodes. Instead of functions of continuous arguments, functions of discrete arguments are introduced - grid functions defined at the nodes of the grid. The partial derivatives entering into the non-differential equations and boundary conditions are replaced (approximated) by difference relations.

As a result of such replacements, the system of partial differential equations (11) must be reduced to a system of finite-difference algebraic (hereinafter referred to as difference) equations - difference schemes. If a solution to the system of difference equations exists and tends to solve the problem as the degree of mesh detail increases, then this solution will be the desired approximate solution of the analytical problem. Despite the fact that the number of unknowns in the system of difference equations will be very significant, from the point of mathematical difficulties, its solution is simpler than the solution of the original system of differential equations (11). Therefore, when solving system (11) it is necessary: to select the configuration and size of the grid; build a difference scheme; determine the stability of the difference scheme; perform a multi-step approximation of the original differential problem; find out the convergence of the difference scheme.

Thus, let us replace the region ΩT of continuous variation of the arguments of the required quantity T with some finite set of points lying in this zone. The grid points of the formation of the finite difference of the function of the integer argument Tkj along the x axis will be denoted by k, and the same points along the r axis – by j. In accordance with the specifics of the problem being solved, the region ΩT is transformed into Ω for calculating the temperature Tkj at kj-points of the barrel sections under the value τ. The solution of the original problem is reduced to finding a table of numerical values of the function Tkj at the grid nodes of the region Ω .

In accordance with the chosen coordinate system (0, x, r, ), we divide the barrel in the direction of the x axis into ϑ equal parts ϑ = l / Δ x , and in the direction of the r axis – into ν equal parts ν = r y / Δ r , where Δ x , Δ r are the grid steps along the corresponding coordinates; l – barrel length; r y is the maximum thickness of the barrel. To do this, we draw ϑ – 1 rays in the direction perpendicular to the x axis and ν – 1 rays in the direction perpendicular to the r axis, as shown in Fig. 2. As a result of such partitioning, we will have a grid consisting of a set of internal (marked • in Fig. 2) and boundary (marked • in Fig. 2) nodes. Since in the case under consideration Δ x = l / ϑ = const and Δ r = r y / ν = const, then the set of nodes x k defined by points with numbers k = 0, 1, 2, …, K ϑ , and the set of nodes r j defined by points with numbers j = 0, 1, 2, …, J ν , is a uniform spatial grid in the region Ω .

A r,

T
1
1__

A x

- QHh

Тkj

Рис. 2. Сеточная схема ствола авиационной пушки ГШ-301

Fig. 2. Grid diagram of the GSh-301 aircraft gun barrel

Here we should dwell on one of the features of the grid construction. This feature lies in the fact that the x and r axes are divided into segments not from the origin, but from the surface of the barrel, taking into account the configuration of its longitudinal section and rifling. Such a partition is done so that in the process of compiling the difference equations, the boundary nodes coincide as much as possible with the position of the region surface Q

^Тkj .

In this case, clearly not all boundary nodes will

be on the line that defines the boundary of the barrel surface, and the distance between r j –1 and r j nodes at all ν rays may not be equal to Δ r . However, these circumstances are less significant in comparison with the meeting of the boundary conditions of the third kind, since they mainly determine the nature of the process of its heating and cooling description.

When constructing difference analogs of differential system operators (11), we use the method of derivatives formal replacement by finite-difference relations. The most natural way to replace the derivative is based on the definition of the derivative (for example, with respect to the coordinate r of the 1st system equation (11)) as a limit [17; 24]

^ - = lim [ T ( r + A r ) - T ( r ) ]— . dr  A r A0                 a r

If we fix the step Δ r in equation (12), then we obtain an approximate formula for the first derivative expressed in terms of finite differences.

For the so-called right-hand difference relation or "forward" difference ат .

— - [ T ( r + A r ) - T ( r ) ]— .                           (13)

dr                  A r

Similarly, the left difference relation (the “backward” difference) is introduced, which is written as:

ат .

— . [ T ( r ) - T ( r -A r ) ]— .                          (14)

dr                  A r

When solving heat conduction problems, it is also necessary to approximate the second derivative. For the second derivative, a linear combination of relations (13) and (14) is considered

^^ « [ T ( r + A r ) - 2 T ( r ) + T ( r -A r ) ]— .                   (15)

5 r ²                                  A r ²

Each transition by 1 step "forward" is conditionally indicated by "+1", and "backward" - by "-1". Then, for the j point of the grid of the value Tkj finite difference formation along the r axis, the right difference relation (13) is transformed to dT ar

The left difference relation (14) is similarly transformed

2- = ( T - T )—.                           (17)

д r      j    j ^{- 1} A r

The difference analogue of the second derivative corresponding to formula (15) is represented by the relation:

• -2-    = (- +i - 2 - + - _i )—.                          ⁽¹8)

• - r                   A r

Formulas (12) – (18) and their justifications are also valid when the derivative ∂² T /∂ x ² in the second equation of system (11) is replaced by difference relations. In this case, the variable x will be present in the analogue equations instead of the variable r , and the index j will be replaced by the index k . The shown analogies will be kept in mind further, sometimes without resorting to direct detailing of the problem with respect to the second spatial variable x .

To construct relations approximating the time derivative ∂ T /∂τ in the first and second equations of system (11), theoretically one can use the temperature values at kj -points of the barrel sections at different moments of time: T k , j , i , T k , j , i –1 , T k , j , i –2 , … , where i is the grid point of the finite temperature difference T kj formation at kj -points barrel sections in time τ. However, in the practice of solving most problems of heat conduction, in the overwhelming majority of cases, exclusively two-layer (in time τ) difference schemes are used, approximating the values of the desired temperatures at the current i and previous ( i – 1) time point. Temperature values at the ( i – 2) time point are taken into account much less often by obtaining three-layer difference schemes [21–24].

When obtaining variants of two-layer difference schemes, the time derivative τ is approximated by the “back” time difference dT  ,         1

— = (2 - T i -i )-,                             (19)

дт          At where Δτ is the grid step in time τ.

Spatial differential operators for a two-layer difference scheme are also approximated on the basis of temperature values T kj at kj -points of the barrel sections at i and ( i – 1) moments of time τ. In this respect, two limiting cases are possible.

In the first case, the approximation involves only the temperature values T kj at kj -points of the barrel sections for the current i moment of time τ. Therefore, for the spatial variable r , the onedimensional space-time approximation of the corresponding differential operator of equation (11) will have the form:

-r = (- +1, i - 22 , i + - -1, i )- 2 .                           (20)

д r                     A r

In the second case, only the temperature values T kj are used in the approximation at kj -points of the barrel sections for the previous moment of time ( i – 1)

T ,■- Tn, j,i л j,i 1 = - (Tj-1,i-I - 2Tj,i-I + Tj-1,i-1) —т. (23) Ат с p Ar

The difference equation of the form (23) allows us to express the solution of the heat conduction problem in the barrel explicitly on i time layer through the known solutions on the previous ( i – 1) layer. Together with the approximation of the single-valued conditions (4)–(10), the difference equation (23) forms an explicit difference scheme. Algorithms for numerical calculation using an explicit difference scheme are quite simple in programming; however, they impose requirements on computer time.

The difference scheme given by the difference equation of the form (22) is more complicated, since each difference equation of the form (22) contains, in addition to the unknown solution for the j spatial point, two more desired solutions for the neighboring ( j – 1) and ( j + 1) spatial points. All the desired solutions turn out to be “tied” with each other into a common non-decomposing system of difference equations. Thus, in this case, at each i time layer, the solutions are determined not by explicit formulas of the form (23), but from the solution of a system of difference equations, so the difference scheme given by the difference equation of the form (22) is implicit.

Efficient algorithms for solving the equation system (11) using an implicit difference scheme are much more complicated than algorithms for numerical calculation using an explicit difference scheme, but the time for solving the problem can be significantly reduced by a rational choice of steps Δ x , Δ r and Δτ.

To construct a finite-difference analogue of the equation system (11), an exclusively implicit difference scheme was used in the work due to its unconditional stability, i.e., the ability to ensure the accuracy of the solution at any degree of grid detail.

When solving a two-dimensional problem in spatial coordinates, the locally one-dimensional method, which belongs to the group of splitting methods, turned out to be quite effective [21; 23]. A complex boundary value problem of mathematical physics is reduced to a consistent solution of onedimensional problems. When solving a two-dimensional problem of determining the barrel temperature field, the locally one-dimensional method made it possible in two stages to calculate the temperature T at a set of wellbore points at a fixed time τ by sequentially solving two one-dimensional problems. The grid step in time t is divided in half at each stage of the solution – 0.5 Δτ.

In the development of the above justifications, taking into account expression (22), the system of equations (11) will take the form:

T k , j , i

—

T k , j' , i —1/2 _

^к    T k , j +1, i

0,5 А т

с Р

^2T k , j , i + T k , j —1, i

А r ²

k , j' , i      T k , j' , i —1

А r

;

T k , j , i

T k , j , i —1/2

0,5 А т

A ( T k +1, j , i — 2T k , j , i + T k —1, j , i ' с p к          A x ²          ^

^а г [ T r (^{T) —} T k , j —1, i ⁽ x , ^т) ] =

—

. T k , j , i ⁽ x , ^{т) —} T k , j —1, i ⁽ x , ^т)

■к-----------------;------------------;

A r

a

в

[ T b — T k , j , i ( x , т) ] =

, T k , j , i ⁽ x , ^{т) —} T k , j —1, i ⁽ x , ^т)

■к-----------------------;-------------------------;

T      r             1 T k , j , i ⁽ r , ^{T) —} T k —1, j , i (r , ^T)

^a r [ T g ^{(T) —} T k —1, j , i ⁽ r , ^т) ] = ^—к —-------------------

A X

,              z хП у T k , j , i ⁽ r , ^{т) —} T k —1, j , i (r , ^T)

^a e [ T B^— T k —i, j , i ⁽ r , ^т) ] = ^—к—------г— ------ ;

A X

T ( r = 0, x = 0...0,175) = T _r (т);

T ( r , x , 0) = T _B = const.

By resolving the difference equations of system (24) with respect to T k , j , i –1/2 , T k , j –1, i ( x ,τ), T k , j , i ( x ,τ), T k –1, j , i ( r ,τ), T k , j , i ( r ,τ) the system received has the form:

^T k , j , i —1/2

—

0,5 к Aт сpAr2 ;

T i i +

к Ат ксpAr 2

—

0,5 кAт с p(r0 Ar + Ar 2 Jv)

+1 Tk и k,j,i

—

—

0,5Х А т

—

0,5 к A т

_к с p A r ²    с p( r ₀ A r + A r ² J _v) _;

T k , j —1, i ;

^T k , j , i —1/2

—

0,5k A r к с p A x J

С

^T k +1, j , i ^{+   1 +}

к

к A т । с p A x ² ^

T k , j , i

—

0,5к A т сp A x ² _;

^T k —1, j , i ;

с

T k , j —1, i ^{( x} , ^т) = T k , j , i ^{( x} , ^т)

к

' a_r T _r (т) A r I

к

к + a_r A r

к

к + а г A r J ;

T k , j , i ^{( x} , ^т) = T k , j —1, i ^{( x} , ^т)

T k —1, j , i ⁽ r , ^т) = T k , j , i ⁽ x , ^т)

T k , j , i ⁽ r , ^т) = Tk —1, j , i ⁽ r , ^т)

^

к — a„ A r ^в

к к + ar Ax

Г к

, к — a„ A x ^в

^a B ^T B ^{(т) А} r I.

к — а„ Ar в г аг Tg(т)Ax к к + аг Ax

;

;

" ^а в T B ^{(т) A x} \

₍ к — a_R A x /

в

T (r = 0, x = 0...0,175) = Tr (т);  T (r, x ,0) = TB = const, where r0 is the distance from the longitudinal axis of the barrel to its inner wall.

To solve the system of difference equations (25), one can use the usual methods of linear algebra or iteration methods. However, the implementation of the locally one-dimensional splitting method makes it possible to organize calculations by the sweep method or the factorization method [21–23], which is the most profitable and economical in terms of the amount of calculations.

By denoting the expressions in brackets through the coefficients of the sweep method at the desired temperature T kj on the current i time layer, the system of difference equations (25) is transformed into a compact form

T k , j , i -1/2 =   A j T k , j +1, i + C j T k , j , i    B j T k , j -1, i ;
T k , j , i -1/2 = - A k T k +1, j , i + C k T k , j , i - B k T k -1, j , i ;
T k , j -1, i ( x , t) = T k , j , i ( x , тК + ф Г ;
T k , j , i ( x , t) = Tk , j -1, i ( x, tKB - ф В ;
,,                   ,    ,	f .                               (26)
T k -1, j , i (r , t) = T k , j , i ( x, t)^ x + ф ? ;
T k , j , i (r , t) = T k -1, j , i ( r, T) ^ x -ф ? ;
T (r = 0, x = 0...0,175) = Т г (t);
T ( r , x , 0) = Т в = const.

The identity in the difference equation system (26) of groupings of parameters at the coefficients of the sweep method greatly simplifies the compilation of a computer program (hereinafter referred to as the program) for calculating the barrel temperature field, taking into account the special type of sweep matrices of the first two difference equations of system (26) - their tridiagonality. Thus, for example, for the first difference equation of system (26), the sweep matrix has the form

1		0         .	..         0	0	0	...         0	0	0
A 1	C 1	B 1        .	..         0	0	0	...         0	0	0
... 0	... 0	...             . 0.	..           ... ..       A	... C j	... B j	...           ... ...         0	... 0	... 0	.    (27)
... 0	... 0	...             . 0.	..           ... ..         0	... 0	... 0	...           ... ..      A r j -1	... C j -	... B r j - 1
0	0	0.	..         0	0	0	...         0	r в	1

Under given boundary single-valued conditions of the third kind, the order of matrix (27) is equal to r j – 1, and only the sweep coefficients located on three diagonals, the main one and two adjacent ones, are nonzero. The tridiagonal form of a matrix of the form (27) makes it possible to organize calculations according to the Gauss method [17; 23; 24] so as not to carry out operations with zero elements. Thus, the amount of calculations can be significantly reduced. The advantage of the scheme (26), (27) is that it allows implementing a rectangular grid - the most tested for solving heat conduction problems, as well as significantly simplify the processes of obtaining objective results. In addition, the system of equations (26) makes it possible to apply an implicit representation of the finite-difference analogs of the differential equation (11), which describes the heat transfer in the barrel, ensuring the absolute stability of the difference scheme. In this case, the time for solving the problem is significantly reduced by choosing an arbitrarily large value Δτ of the grid step in time τ without the risk of violating the difference scheme stability.

However, the fulfillment of the stability requirement for the difference scheme does not at all mean meeting the approximation condition for the original differential problem. Therefore, it is essential to solve the problem of choosing the degree of mesh detail which affects the accuracy of the results. At the same time, it should be taken into account that an excessive decrease in grid steps Δ x and Δ r along the corresponding x and r axes leads to an increase in the number of unknowns in the difference scheme (26), (27), and consequently to an increase in computation time. Enlargement of grid steps Δ x and Δ r along the corresponding axes x and r does not improve the accuracy of the required result. The conjugation of the value Δτ of the grid step in time τ with a similar value specified in the procedures for solving the main problem of internal ballistics and the aftereffect period [1; 19; 20] determines the variability of time intervals τ. When debugging the model, the choice of the values of the grid steps

Δ x , Δ r according to the corresponding coordinates x, r and the value of the time step Δτ was carried out based on the conditions for the highest solution accuracy, the smallest number of unknowns in the system of difference equations, taking into account the real dimensions of the region of discrete change in the arguments of the T kj value of temperature in kj -points of the barrel sections. The priority in solving the problem seemed to take into account the configuration of the rifling, since their presence leads to uneven temperature distribution along the perimeter of the threaded part of the bore [8; 9; 11]. The initial requirement of incomparable smallness, the value Δr of the grid step along the r axis in relation to the height of the groove field is obvious.

Mathematical analysis of the difference schemes degree of detail is a rather difficult task. The results of research on the influence of the grid step values Δx, Δr along the corresponding coordinates x, r and the time step size Δτ on the accuracy of the problem solution are given in this paper without proof. The presence of approximation was easily established in practice by means of numerical experiments in the course of trial calculations of elementary problems of heat conduction in cylindrical walls and comparison of the obtained results with the known ones [25].

When the spatiotemporal grid is refined, the accuracy of solving problems increases, which indicates that the approximation error tends to zero, but at the same time it is clear that the computational time increases. Acceptable grid steps should be considered along the x coordinate - Δ x = 0.001 m, along the r coordinate – Δ r = 4 · 10^–4 m, since the averaged root-mean-square error over all points in this case does not exceed 10%, and the simulation time of a single loading is no more than 17 s. A further increase in the degree of grid detail, for example, by a factor of 2 leads to a significant increase in the solving problems time with an increase in the calculation accuracy by only 3%.

Since the velocity of gases in the bore v in time τ and along the barrel length l gradually increases, this feature does not allow us to build a uniform grid in time Δτ, since along the barrel length l the grid step Δ x along the x axis will also increase. It, in turn, can lead to the fact that the accuracy of the solution results obtained at various points in the region of discrete change in the temperature arguments at kj -points of the barrel sections T kj will differ significantly from each other, which is unacceptable. Considering also the fact that calculations at each i time layer are based on the values of the previous ( i – 1) time layer, the error will accumulate rather quickly. In order to exclude this, it is advisable to use a variable time step, determined on the basis of the gas velocity in the bore v , obtained by solving the main problem of internal ballistics [1; 19; 20]

Δx Δτ =

.

v ( l )

Unlike a spatial grid, the set of nodes τ i defined by points i = 0, 1, 2, …, I ο is a non-uniform time grid in the region Ω .

Formula (28) shows a rigid relationship between the value Δτ of the grid step in time τ and the values Δ x of the grid step along the x axis, since the accuracy of the solution of the problem directly depends on the correct choice of the latter.

The convergence of all types of difference schemes in the presence of conditions for their stability and approximation was proved in [21–24].

The calculation of the barrel temperature field when using aviation artillerary weapons is reduced to multiple (by the number of shots in the queue, the number of bursts of shots and the time intervals between the bursts of shots) solving the system of equations (26) with the initial distribution of the barrel temperature T , which is set by the beginning of the next shot and is determined when solving the same system of equations (26) for the previous shot. The completed form of model development was a program that allows calculating the temperature field of the GSh-301 aircraft gun barrel.

Due to the adoption of certain assumptions, the model has a certain level of abstraction and, due to the inevitable loss of information does not give a complete picture that characterizes the physical processes under study. Justification of particular formulations and further discussions of the consequences seem possible after checking the adequacy of the model to real heat transfer processes in the barrel.

Verification the adequacy and resource intensity of the model

The reliability of the developed model was established by checking the spatial distribution of non-stationary values of the barrel temperature T in various firing modes. The values of the main thermophysical parameter under consideration for the OHN3MFA tool steel are well known and make it possible to compare the simulation results with the experimental data classified, for example, in [8; 10; 11] and simulation results that are closest to the experimental ones registered in [8; 9; 12; 13].

The analysis of the output data showed that the discrepancy between the results of calculations and the experiment does not exceed 10%, and the difference between the simulation results and other theoretical calculations is not significant. A comparison of the joint calculated and experimental data shows a slight (no more than 1.3%) increase in the accuracy of obtaining the total non-stationary values of the barrel temperature T . At the same time, the assessment of the resources required to carry out the claimed calculations revealed some advantages of the model. Time efficiency [26], which depends on the list, type and structure of the interaction of program operators and, of course, on the computer speed (clock frequency of processors, amount of RAM), is considered as an estimated resource. The fact that the same program in the same cases on different computers is executed in different times was taken into account.

In order to compare the temporal efficiency of the model, the versions of programs previously used in [9; 12; 13] were imported to the same modern calculator. Based on the results of direct timekeeping and absolute quantitative comparisons, a 1.6-fold reduction in machine calculating time using the model was observed, which is considered quite acceptable for problems of this class. These conclusions make it possible to simulate the working processes of a shot and barrel loading with an accuracy that at least does not reduce the previous one, but with an essential gain in time.

Saving time spent on modeling the thermal state of the barrel was facilitated, apparently, by the following activities:

• –    rational choice of the grid steps Δx, Δr values for the corresponding coordinates x, r and the value of the time step Δτ, systematized based on the results of evaluations of the final data of numerical experiments carried out in the work;

• –    adaptation of the time step Δτ, which first of all affects temporary resources to the identified parameters of coolants in the bore;

• –    successful combination of advantages (unconditional stability) and compensation for disadvantages (increased costs of computer time) of implicit difference schemes.

Recommendations for the solutions extension in the study of different types of artillery pieces

As a reference sample in the work, an aviation artillerary weapon of the GSh-301 type was chosen which has a basic single-barreled automation scheme, being in service with most of the modern ones, and is also planned to be equipped with advanced aviation weapon systems. However, this research does not exclude the fact of switching to research of other aviation artillerary weapon models, other types of land and sea rapid-firing artillery pieces, as well as large-caliber artillery pieces provided that a number of conditions are met:

• 1.    The solution of the main problem of internal ballistics in relation to an artillery gun of a specific mechanism and design. Thus, solutions are known for recoilless artillery guns, howitzers, mortars, etc. [19; 20].

• 2.    Construction of an axial symmetry scheme and a grid scheme for a region with a specific geometry. Giving preference to one or another variant is currently a question that can be resolved, but not closed [21–25]. In many ways, preferences are conditioned by the characteristics of the problem being solved, and largely depend on the researcher’s choice.

• 3.    Multifactorial composition of differential and difference operators, which is two sides of a single modeling process. For example, the incomparably greater length and relatively small wall thickness in similar units of tank and anti-tank guns make it necessary to additionally take into account the heat from solar radiation. As regards the approximation of the system of equations (11), we note that usually after comparing the explicit and implicit difference schemes, it is concluded that the use of the first one is inexpedient. However, the practice of solving real problems does not confirm the absolute correctness of such a recommendation. The following considerations can be given in favor of the explicit difference scheme. First, when analyzing fast processes, the advantage of the implicit difference scheme which consists in a freer choice of the time step may not manifest itself. Secondly, explicit difference schemes are convenient for implementation on computers with multiple parallel processors, which are widely used currently.

• 4.    Finding a solution to the most difficult problem of heat transfer between the barrel and the surrounding gases. The complete system of differential heat transfer equations includes the equations of heat transfer, heat loss, motion, continuity [16], and the solution should be relatively simple and acceptable for engineering practice.

• 5.    The indispensable consideration of the mutual influence of heated barrels in aviation artillerary weapons, high-temperature land and naval artillery guns of a multi-barrel scheme serves as a criterion for the usefulness of the conclusions and increases the practical significance of the model.

Conclusion

Methods and ways of differential-difference description of fast thermal processes formed one of the possible versions of the model, which allows quite objectively, but with less time to calculate the non-stationary temperature field of a multi-walled composite cylinder. The directions for modifying the model when including a diverse typology of artillery pieces into the research tools are determined.

Thus, the proposed model can be used in research organizations of the Aerospace Forces of Russia to determine the optimal conditions for the use of aviation artillerary weapons as part of aircraft weapon systems for strike carriers; in design organizations when carrying out calculations related to determining the thermal state of the barrel; in educational activities for the development of the scientific foundations of academic disciplines.