Calculation of the parameters and characteristics of a rotating lunar jet penetrator

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Introduction

Theoretical and experimental studies on embedding solid bodies into the ground at the expense of kinetic energy accumulated outside the ground section of the trajectory show that the section of motion in the ground sometimes has a niticeable curvilinear character, in which a significant departure from rectilinear motion is possible, up to a complete turn of the penetrating body and movement of its bottom part forward. The character of motion is significantly influenced by forces, which in turn depend both on the shape of the body and on the initial conditions of penetration, determined by the presence of the angle between the velocity vector and the axis of symmetry, as well as the angular velocities of precession, nutation and proper rotation.

When a jet penetrator with a running engine is embedded in the ground, its stability is affected (in addition to the above-mentioned factors) by such factors as the thrust magnitude, its eccentricity and the possibility of swirling motion.

The purpose of this paper is to determine the internal ballistics parameters of a solid propellant jet engine mounted on a jet penetrator entering the ground with a high rotational velocity around its own axis.

To determine the pressure value in the chamber of a rotating engine, the well-known equations of the balance of gas inflow and outflow are usually used, as in the case of a non-rotating solid propellant jet engine. The difference between the internal ballistics of a rotating solid propellant jet engine is that the effect of rotation on the working process is taken into account [1]:

– by the flow coefficient of gases from the rotating engine chamber

$A_{rot}=A_{\text{0}}{\left(\frac{\text{1}}{\text{1}+\frac{k}{k+\text{1}}{\text{α}}_{cr}^{\text{2}}}\right)}^{\frac{\text{1}}{\text{1}-\text{υ}}};$   (1)

– by change in the rate of erosive combustion of solid propellant during the rotation of a solid propellant jet engine

${\text{ε}}_{rot}=\text{1}+B{n}^{\text{0.5}};$   (2)

– by the heat loss coefficient

${\text{χ}}_{rot}=\frac{\text{1}-\text{0.16}{\left(\text{1}+{{\text{tanα}}_{cr}}^{\text{2}}\right)}^{\text{0.4}}}{\text{1}+\text{2ψ}},$   (3)

where ${А}_{\text{0}}$ is a flow coefficient of gases from the combustion chamber of a non-rotating solid propellant jet engine.

The value of the gas flow coefficient is determined according to the folloing dependence

$A_{\text{0}}=\frac{{\dot{M}}_{\text{0}}}{{\dot{M}}_T}\le \mathrm{1,}$   (4)

where ${\dot{М}}_{\text{0}}$ is a real (experimental) mass flow rate, taking into account all possible types of losses that reduce the gas flow rate through the nozzle; ${\dot{M}}_{т}=\frac{p_{cr}{f}_{cr}}{\sqrt{\text{χ}R{T}_{\text{0}}}}$  is theoretical gas flow rate through the nozzle; $p_{\text{кр}}$ is braking pressure at the nozzle inlet; $f_{\text{кр}}$ is a critical cross-sectional area of the nozzle; $\text{χ}$ is a coefficient of heat loss; $R{T}_{\text{0}}$ is a reduced force of solid propellant; B = $\text{3.7}\cdot {\text{10}}^{-\text{6}}$ at n $\le {\text{10}}^{\text{3}}$ $\frac{rot}{\min }$; k is an adiabatic value; ${\text{α}}_{cr}$ is angle of swirl of gas flow in the critical section of the engine nozzle; n is a number of revolutions of the rotating ground jet penetrator; $\text{υ}$ is a degree exponent in the propellant combustion rate law; $\text{ψ}$ is a relative fraction of burnt charge.

The algorithm for determining the pressure in the combustion chamber of a rotating solid propellant engine

1. Steady-state pressure at the stationary operation section of the solid propellant jet engine.

Fig. 1 graphically depicts the principle of stationarity of the operation of the rotating solid propellant jet engine.

Here ${\dot{m}}_{+}$ is gas supply into the combustion chamber of the solid propellant jet engine; ${\dot{m}}_{-\text{0}}$ and ${\dot{m}}_{-rot}$ are gas flow rate of the non-rotating and rotating engine, respectively.

The graph shows that a decrease in the gas flow rate of a rotating engine leads to an increase in the steady-state pressure in its combustion chamber, i.e. $P_{strot}\ge P_0.$

 

Рис. 1. График, иллюстрирующий принцип стационарности

Fig. 1. Graph illustrating the principle of stationarity

 

In this case the following equation is used for the calculation of $P_{\text{st}rot}$

$P_{strot}={\left(\frac{\text{1}}{N_{\text{1}}}\right)}^{\frac{\text{1}}{\text{1}-\nu }},$   (5)

where $N_{\text{1}}=\frac{N}{\text{ε}}$; $\epsilon =\frac{P}{{\rho }_{т}{\chi }_{rot}R{T}_0}$; ${\chi }_{rot}\left(\alpha \right)$ – is from $\left(\text{3}\right)$; $N=\frac{{\phi }_2{A}_{rot}{p}_k{f}_{cr}}{S_g{U}_{т}{\rho }_{т}\sqrt{{\chi }_{rot}R{T}_0}}$; $U_{т}=f_1\left(T_3\right)f_2\left(p_c\right)f_3\left({\alpha }_{cr}\right)f_{\text{4}}\left({\chi }_{\text{0}}\right)$ – $U_{т}$=$f_{\text{1}}{f}_{\text{2}}{f}_{\text{3}}{f}_{\text{4}}$ is solid propellant burning speed depending on the charging temperature $\left(f\left({Т}_{\text{3}}\right)\right)$, pressure in the combustion chamber $f_{\text{2}}\left(p_{\text{к}}\right)$, degree of swirl $f_{\text{3}}\left({\text{α}}_{cr}\right)$ of gas flow and the Pobedonostsev criterion $f_{\text{4}}\left({\chi }_{\text{0}}\right)$ [2; 3].

Fig. 2 shows the dependence of the steady-state pressure in the chamber of a rotating solid propellant jet engine on the degree of swirl of the gas flow.

 

Рис. 2. Зависимость величины установившегося давления в камере сгорания от степени закрутки газового потока

Fig. 2. Dependence of the steady-state pressure in the combustion chamber on the degree of swirl of the gas flow

 

The calculations $P_{strot}$ were performed for a real engine of a 40 mm diameter model ground jet. Here ∆ marks are used to indicate experimental values of steady-state pressure. A good agreement between the calculated and experimental data can be seen.

Thus, the steady-state pressure in the chamber of solid propellant jet engine varies depending on the speed of its rotation around its own axis. In this case, with the increase in the degree of swirl of the gas flow, the value of the steady-state pressure increases, the rate of pressure build-up in the process of the engine entering the steady-state mode of operation decreases, and at a given propellant mass, the engine operation time decreases (Fig. 3).

 

Рис. 3. Типовые зависимости давления в камере сгорания для вращающихся РДТТ: 1 – для вращающегося РДТТ; 2 – для невращающегося РДТТ; 3 – отмечается некоторое увеличение установившегося давления в камере для вращающихся двигателей при $\mathit{n}\mathbf{<}{\mathbf{\text{10}}}^{\mathbf{\text{3}}}\frac{\mathbf{\text{об}}}{\mathbf{\text{мин}}}$; 4 – показана возможность появления второго максимума, величина которого больше первого

Fig. 3. Typical pressure dependences in the combustion chamber for rotating solid propellant rocket engines: 1 – for a rotating solid propellant rocket engine; 2 – for a non-rotating solid fuel jet engine; 3 – there is a slight increase in the steady-state pressure in the chamber for rotating engines at $\mathit{n}\mathbf{<}{\mathbf{\text{10}}}^{\mathbf{\text{3}}}$ rpm/min; 4 – shows the possibility of the appearance of a second maximum, the value of which is greater than the first

 

It should be noted that the pressure in the combustion chamber of a rotating engine can be corrected either by using an afterburning volume in its design, which increases the free volume of the combustion chamber, or by changing the thermal and hydraulic loss coefficients. The hydraulic loss coefficient can be calculated using the following formula

$\xi ={\xi }_0{\left(1+\text{tg}{{\alpha }_{cr}}^2\right)}^{1.375},$   (6)

where ${\xi }_0$ is a hydraulic loss coefficient at one-dimensional gas flow through the pipe at ${\alpha }_{cr}=\text{0}$.

The calculations show that ${\alpha }_{cr}$ value due to hydraulic losses to the values ${\alpha }_{cr}\approx \mathrm{0,}\text{2}$ is almost unchanged, Therefore, its reduction should be taken into account at ${\alpha }_{cr}>0.3-\text{0.4}$, when  is reduced by 13–35 %.

2. Switching of a rotating solid propellant jet engine to steady-state mode

When calculating the pressure-time dependence of the rotating solid-propellant engine on the steady-state mode of operation, as in the case of the flow rates of a solid propellant jet engine [3; 4], the following parameter is determined

$a=\frac{{\phi }_2{A}_{rot}b{f}_{\text{cr}}\sqrt{{\chi }_{rot}R{T}_0}\left(1-\upsilon \right)}{W_g}$,   (7)

where rotation is taken into account by introducing the coefficients $A_{rot}$ and ${\chi }_{rot}$; b and $\nu$ are coefficients in the propellant combustion law; $W_g=\rho u{S}_g$ is gas supply to the combustion chamber; u is a combustion rate; $S_g$ is a combustion surface of the propellant charge.

After that, the total time for the solid propellant jet engine to reach steady-state is calculated

${\tau }_{р}=\frac{1}{a}\ln \frac{1-p_b^{1-\upsilon }}{1-{\overline{p}}^{1-\upsilon }}$,   (8)

where $\overline{p}$ = 0.99 is limit relative pressures in the combustion chamber in the process of the solid-propellant jet engine reaching the steady-state mode of operation; $p_b$ is pressure in the chamber when the charge is ignited.

 

Рис. 4. Зависимость давления в камере сгорания от времени при выходе двигателя на установившийся режим

Fig. 4. Dependence of pressure in the combustion chamber on time when the engine reaches steady state

 

The calculations given for a 240 mm diameter rotating ground jet penetrator at swirl angles ${\alpha }_{cr}$ = 0.1; 0.2; 0.3 showed that: 1) engine steady-state time increases by 23 % with increasing rotational speed at ${\alpha }_{cr}$ = 0.1, by 46 % at ${\alpha }_{cr}$ = 0.2 and by 130 % at ${\alpha }_{cr}$ = 0.3, i.e. from 0.13 s to 0.3 s; 2) increases the steady-state pressure compared to a non-rotating engine.

In order to obtain the dependence (Fig. 4), ${\tau }_{р}$ was first defined using the fomula (8), then three following values were chosen ${\tau }_1$, ${\tau }_2$, ${\tau }_3$, which are in the interval between ${\tau }_{р}$ and 0, and according to the value of these times the relative pressures ${\overline{p}}_1$, ${\overline{p}}_2$, ${\overline{p}}_3$ were determined by the formula

${\overline{p}}_i={\left[1-{\left(1-p_b\right)}^{1-\upsilon }{e}^{-a{\tau }_i}\right]}^{{}^{\frac{1}{1-\upsilon }}}$,   (9)

Then $\overline{pi}$ were recalculated into real design pressures according to the dependence:

${\overline{p}}_i=p_{strot}{\overline{p}}_i$,   (10)

where $p_i$ is calculated up to $\overline{p}$ = 0.99.

3. Calculation of pressure during the period of free flow of gases from the chamber of a solid propellant jet engine

As in the case of calculation of the after-effect period for a non-rotating engine, the end of charge combustion time is determined by the formula [3–5]

${\tau }_k=\frac{e}{u}$,   (11)

where e is a burning vault thickness; for a tubular charge burning on the outer (D) and inner (d) surfaces it is, in particular, equal to

$е=\frac{D-d}{4}$.   (12)

Taking into account the dependence of the charge burning rate on the pressure in the combustion chamber, it is evident that the end of combustion time for the rotating engine will be less than the end of combustion time for the charge of the non-rotating engine, because the steady-state pressure of the rotating engine is greater than the steady-state pressure of the non-rotating engine.

The time of full flow of gases from the combustion chamber after combustion of solid propellant is calculated by the following formula

${\tau }_{fr}=\frac{1}{В}\left[{\left(\frac{p_{krot}}{1.8}\right)}^{0.1}-1\right]$,   (13)

where $В=\frac{K-1}{2}\frac{{\phi }_2{A}_{rot}{f}_{cr}b\sqrt{X_{rot}R{T}_0}}{W_{km}}$; $p_k$ = 1.8 bar is the pressure in the combustion chamber up to which the supercritical flow formula is valid.

The pressure dependence on the free gas flow time is determined in the following sequence:

1) time ${\tau }_{fr}$ is divided into three intervals, where ${\tau }_1$, ${\tau }_2$ and ${\tau }_3$ are less than ${\tau }_{fr}$;

2)   $p_{\text{1}}$, $p_{\text{2}}$ and $p_{\text{3}}$ are calculated by the formula $p_i$=$\frac{p_{krot}}{{(1+B{\tau }_i)}^{\frac{2k}{k-1}}}$.

The curve passing through the calculation points describes the period of free gas flow from the rotating solid propellant jet engine.

Fig. 5 shows the graph of dependence of the free flow time from the rotating engine chamber on the degree of swirl of a 240 mm diameter ground jet penetrator.

It was obtaned at ${\alpha }_{cr}$ > 0, ${\tau }_{fr}=0.173\text{ s};$ at ${\text{α}}_{cr}$ = 0.1, ${\text{α}}_{cr}$ = 0.2 and ${\text{α}}_{cr}$= 0.3, ${\text{τ}}_{\text{fr1}}$= 0.22 s, ${\tau }_{\text{fr2}}=0.32\text{ s}$ and ${\tau }_{\text{fr3}}=0.55\text{ s}$, respectively. 

It can be seen from the graph (Fig. 5) that the time of free flow of gases from the combustion chamber after the end of propellant combustion increases with the increment of swirl parameters and, consequently, the number of revolutions of the jet penetrator.

 

Рис. 5. Расчётная зависимость времени истечения от угла закрутки газового потока РДТТ

Fig. 5. Calculated dependence of the exhaust time on the swirl angle of the gas flow of a solid fuel jet engine

 

Selection of linear and angular dimensions of the rotating engine nozzle

The dimensions of a single nozzle or nozzles of the nozzle block of a rotating solid propellant jet engine are selected according to the same dependences as for a transforming engine, but taking into account the previously established dependences and coefficients.

Using dependences (5) for calculations of steady-state pressure in the chamber of a rotating engine, it is possible to find the area of the critical cross-section of the engine nozzle using the formula [1]

$f_{cr}=\frac{s_r{U}_{\tau }{\rho }_{\tau }\sqrt{X_{rot}R{T}_0}}{{\phi }_2{A}_{rot}b{p}_{rot}^{1-v}},$   (14)

$d_{cr}=\sqrt{\frac{\text{4}{f}_{cr}}{\text{π}n}}$,   (15)

where n is the number of nozzles; $A_{rot}\left({\alpha }_{cr}\right)$, $X_{rot}\left({\alpha }_{cr}\right)$ are coefficients; $p_{rot}$ is a design pressure at the engine chamber wall.

The comparative analysis of the calculations of the supersonic nozzle part of rotating and non-rotating engines showed that the optimum angle of the supersonic part of the rotating engine corresponds to the optimum angle of the nozzle of a non-rotating solid propellant jet engine and is equal to 20°. The experimental data presented in [1] confirm this conclusion and also show that it is necessary to choose a larger nozzle entrance angle in the presence of flow rotation than for a nozzle with one-dimensional flow.

Fig. 6 shows the experimental dependence of the single impulse J_un on half of the nozzle entry angle α. The graph shows that J_un reaches a maximum at 2α = 180°, i.e., at a flat wall of the nozzle block. This effect is explained by the fact that the flat wall completely dampens the axial component of the gas flow velocity and increases its radial component, which increases the gas flow rate through the nozzle.

 

Рис. 6. Зависимость величины единичного импульса от половины угла входа в сопло двигателя

Fig. 6. Dependence of the magnitude of a single impulse on half the angle of entry into the engine nozzle

 

For a single nozzle, the thrust formula can be written as follows

$P_{rot}=K_d{p}_{rot}{f}_{cr}{\phi }_1{\phi }_2{A}_{rot}$,   (16)

where $K_d$ is a thrust coefficient; $f_{cr}$ is a nozzle critical cross-sectional area; ${\phi }_1$ = 0.95–0.98 is a velocity coefficient; ${\phi }_2$ is a nozzle flow coefficient at gas flow without swirling; $A_{rot}$= f $\left({\alpha }_{cr}\right)$ is a flow coefficient for rotating gas flow.

Thus, knowing the laws of pressure change in the combustion chamber of a rotating a solid propellant jet engine and using the above formulae for thrust force, it is possible to graphically construct $P_{rot}\left(\tau \right)$ dependences for any type of a rotating engine [6–8].

The analysis of dependences for the thrust force of rotating ground jet vehicles allows us to state that the thrust force value of such engines will be less than that of non-rotating ones, all other conditions being equal.

The difference in thrust forces will be determined by the following ratio

$\frac{{А}_0}{{А}_{rot}}={\left(\frac{P_{rotcr}}{P_{\text{0cr}}}\right)}^{\frac{\text{1}}{\text{1}-v}}={\left(\text{1}+\frac{k}{k-\text{1}}{\text{α}}_{cr}^{\text{2}}\right)}^{\frac{\text{1}}{\text{1}-v}}$,   (17)

then

$\frac{P_0}{P_{rot}}\approx {\left(1+\frac{k}{k-1}{\alpha }_{cr}\right)}^{\frac{1}{1-v}}$.   (18)

For real solid propellants υ = 0.5–0.67 at ${\alpha }_{cr}$= 0.1–0.15 the value of thrust relations is within  $\frac{P_{\text{0}}}{P_{rot}}$ = 1.1–1.36, i.e. the thrust of a non-rotating engine is 10–36 % greater than that of a rotating engine [9–11].

The experimental studies of rotating solid propellant jet engines equipped with multi-ball solid propellant charges have shown that (unlike solid propellant jet engines with single-ball charges) pressure nonuniformity in the combustion chamber is observed only in the pre-nozzle chamber. Herewith, the more draughts in the charge, the less is the degree of swirl both in the channel of a single draughts and in the pre-nozzle block as a whole [12–15].

Conclusion

Within the framework of the conducted research the following tasks have been solved:

1. It has been established that the internal ballistics parameters of rotating solid propellant jet engines are mainly influenced by the coefficient of gas flow rate from the rotating engine chamber, the effect of erosive combustion of solid propellant, and the change in the heat loss coefficient.
2. The basic calculation dependences for determining the pressure in the combustion chamber of a rotating solid propellant engine are given for the periods of pressure release on the stationary mode of engine operation, engine operation on the stationary mode and during the period of free flow of gases from the chamber of a solid propellant jet engine.
3. The methodology for selecting linear and angular dimensions of the nozzle of a rotating engine is presented, which allowed a comparative analysis of the calculations of the supersonic part of the rotating and non-rotating engines.
4. An estimate of the thrust force for a single nozzle of a rotating solid propellant jet engine is given. It is found that the thrust force of rotating engines (with other identical conditions in the combustion chamber) is 1.1–1.36 times less than that of non-rotating solid propellant jet engines.
5. The conducted experiments showed a decrease in the degree of swirl of the gas flow of rotating solid propellant engines with increasing the number of propellant draughts in the engine charge.

About the authors

Evgeniy V. Gusev

Moscow Aviation Institute (National Research University)

Author for correspondence.
Email: ccg-gus@mail.ru

Cand. Sc., Associate Professor of Department 610 “Operation Management of Rocket and Space Systems”

Russian Federation, 4, Volokolamskoe highway, Moscow, 125993

Vladimir A. Zagovorchev

Moscow Aviation Institute (National Research University)

Email: zagovorchev@mai.ru

Cand. Sc., Associate Professor, Associate Professor of Department 610 “Operation Management of Rocket and Space Systems”

Russian Federation, 4, Volokolamskoe highway, Moscow, 125993

Vladimir V. Rodchenko

Moscow Aviation Institute (National Research University)

Email: rodchenko47@mail.ru

Dr. Sc., Professor, Professor of Department 610 “Operation Management of Rocket and Space Systems”

Russian Federation, 4, Volokolamskoe highway, Moscow, 125993

Elnara R. Sadretdinova

Moscow Aviation Institute (National Research University)

Email: elnara-5@mail.ru

Cand. Sc., Associate Professor, Deputy Director of the Aerospace Institute

Russian Federation, 4, Volokolamskoe highway, Moscow, 125993

Elizaveta A. Shipnevskaya

Moscow Aviation Institute (National Research University)

Email: Shipnevskaya.E@gmail.com

Master

Russian Federation, 4, Volokolamskoe highway, Moscow, 125993

References

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