DETERMINATION OF AXIAL PLUME IMPINGEMENT FORCE ON THE
MULTITUBE LAUNCHING DEVICE BY ANALOGY WITH PROCESS
OF HOT TANDEM STAGE ROCKET SEPARATION
SINIŠA JOVANČIĆ
Military Technical Institute, Belgrade, [email protected]
DRAGOSLAV ŽIVANIĆ
Military Technical Institute, Belgrade, [email protected]
MILAN ŽIVANOVIĆ
Military Technical Institute, Belgrade, [email protected]
Abstract: This paper presents a requirement for using the equation of plume impingement force during tandem
separation of multistage rocket to the case of determining the axial force on the launching device of MLRS during
launching. It also presents the calculation of axial plume impingement force in dependence on the position of the tube
which the missile is launched from, and the possibility of applying this model to study the dynamics of MLRS, as well
as to study the stress on the adjacent tubes’ caps.
Key words: : impingement force, surface, fictitious radius, nozzle exit distance.
1. INTRODUCTION
2. SIZE AND SHAPE OF ROCKET PLUME
The problem of determining the plume impingement force
to the launching device turn out to be important from the
aspect of dynamic behavior of the launch platform, but
also in the design of its subsystems. Although this
problem is significant, there are very few available studies
on this subject. One of the widely accepted models for the
case of missile leaving the launching tube, is given in [6],
and it is based on the theoretical and experimental results
applied to determine the plume impingement force during
separation of multistage rocket [1], [2], [3], [7]. In this
case, it is taken that the speed of another object (a
launching device) is equal zero. This model has been
accepted by other authors [8], [9] and [10].
The plume is the moving formation of hot rocket exhaust
gases and small particles outside the rocket nozzle. This
gas formation is not uniform in structure, velocity and
composition. It contains several different flow regions
with supersonic shock waves and Prandtl–Meyer
expansions, where gas dynamic parameters are changed.
The plume characteristics (size, shape, structure, visibility
or smokiness) depend not only on the characteristics of
the particular rocket propulsion system or its propellants,
but also on the flight path, flight velocity, altitude,
weather conditions and the particular rocket
configuration.
In this paper, shape of rocket exhaust gases will be
approximated by two identical truncated cones whose
bases are de and Dm (where larger base Dm is mutual, and
de is diameter of nozzle’s output cross-section) and hight
is 0,5Lm (Fig.1). Also, it will be taken that rocket flies at
subsonic velocity on low altitude, while the pressure on
the nozzle’s output cross-section is higher than
surrounding air pressure (underexpanded flow). Changes
of gas-dynamic parameters will be not taken in
consideration (eg. method of characteristic solution), but
the simplified model exhaust plume flowfield given in [1]
will be accepted. The shape of the missile plume in case
there is no obstacle to their spread is shown in Figure 1.
The equation that links force with distance traveled,
derived in references [1], [2] and [3], is given as a
function of radius (or caliber) of rejected rocket’s stage.
That means that for proper use of this relation gas stream
should act to the circular surface. As in our case the
surfaces are not circular, they can be approximated by a
circular cross-section of an adequate surface. Thus, the
fictitious radius which corresponds to a real surface
strained by the gas stream is obtained. It provides the
axial plume impingement force to be determinated
independently of the launching tube position, and it is a
new contribution to the former model.
321
that at the time t = 0.1925 s from the moment of
launching, radius r1 = 0.5 Dm ≈ 0.529 m reaches a
maximum value, when the output section of nozzle is
approximately X ≈ 4.217 m far from the launching device.
At the moment t = 0.2314 s the last contact between
rocket plume and launching panel occurs, because the
distance of nozzle’s output section at that moment is
equal to length of plume X = Lm ≈ 8.339 m. Due to
rocket acceleration, the flow of the other part of plume
(the other cone) lasts shorter (it is obvious that r1 after
t = 0.1925 s declines faster than that it increased by this
moment).
According to [5], maximum possible value of plume
diameter Dm should be determined on the basis of the
known parameters of environment in which plume
spreads (Mach number M ∞ , pressure P∞ , and ratio of
specific heats κ a ), and of the nozzle’s output crosssection parameters (Mach number M e , pressure Pe ,
diameter d e , and ratio of exhausted gases specific heats
κ ):
Dm = d e
M e κ Pe
M ∞ κ a P∞
(1)
The maximum possible length of the rocket’s plume
which will be considered in this paper can be
approximately determined by empirical equation:
Lm ≈ 8 ⋅ Dm
9
8
7
(2)
X [m]
D m [m]
6
The semi-angle of the plume cone could be defined too:
ψ = arctg
Dm − d e
Lm
L m [m]
r 1 [m]
5
4
(3)
3
2
1
0
0.14
0.15
0.16
0.17
0.18
0.19
t [s]
0.2
0.21
0.22
0.23
Figure 2. Geometric plume parameters change in function
of time
3. PLUME IMPINGEMENT FORCE TO
LAUNCHING DEVICE
Immediately after left the launching tube, hot gases and
unburnt particles of combustion dynamicaly and
thermally strain the launching device. The force that
occurs when gases flow over the launching device is of
spatial character, primarily due to nonuniformity of gas
stream, but also due to misalignment between firing line
and rocket axis. Missiles in the initial phase of flight is
oscillating around its center of mass, and a launching
platform around its supports. This paper ignores these
phenomena, and assumes that the firing line and rocket
axis are collinear. The forces in directions perpendicular
to the firing line will be ignored too, as well as the
resistance and contraction of the plume due to its inflow
into the lunching tube.
Figure 1. Plume geometry without obstacle
For accepted model of rocket’s plume formed of two
identical truncated cones, a variable r1 is introduced which
represents radius of the circle which acts onto the
launching device surface. In general radius r1 is a function
of changes in the distance between the nozzle’s output
section and launching device X and of time t (because the
parameters on the output section are Me, and Pe is a in
function of time). Approximate value of the radius r1 is:
r1 = X ⋅ tgψ + 0.5 d e
In order to determine dependence of the plume
impingement force and launching tube position, the
change of area which is affected by gas stream plume
should be determined firstly. Figure 3 shows the
launching device consisting of 24 tubes. The basic
geometric parameters for the launching tube No. 12,
required for calculation are shown. A whole area of the
launching device is divided into four areas S1, S2, S3 and
S4, as shown, and then the calculations for each area will
be carried out separately. S1 is used as an example to
present a method for calculating the changes in area as the
function of r1 (Fig. 4).
(4)
Figure 2 shows the change in distance of nozzle's output
section X, the maximum possible values of diameter Dm,
length Lm and radius r1 in function of time, for the rocket
which will be considerde in this paper. It could be noticed
322
In the same way, the other surfaces S2, S3 and S4, should
be determined as a function of change in plume radius r1.
S ( r1 ) = S1 ( r1 ) + S 2 ( r1 ) + S3 ( r1 ) + S4 ( r1 ) −
d 2π
(5)
4
Equation (5) shows the change of the area which plume
acts on, where the last term of the equation represents a
reduction in area for characteristic surface of rocket being
launched.
Figure 3. Schematic image of launching tube
The surface being flown over S (r1) corresponds to a
fictitious circular surface with radius R1. For each new
value of the plume radius r1 , fictitious radius is given by
the relation:
Regardless of the launching tube position, there are four
possible phases in flowing over. Table 1 shows the
corresponding equations for each phase, as well as the
necessary conditions to determine the current stage.
S ( r1 )
R1 =
(6)
π
After radius R1 is determined, it is possible to calculate
plume impingement force using the formula derived in
[1], [2], [3], [6] and [7]:
{
}
F = κ M e 2 Pe Ae C1 ⎡⎣1 − f1a ( x ) ⎤⎦ + C2 ⎡⎣1 − f1b ( x ) ⎤⎦
(7)
Terms C1, C2, a and b in equation (7) represent function
of gas characteristics, while term f1 ( x ) is function of
distance traveled and fictitious radius. It should be
mentioned that term C1 in fact represents a gas pressure
between launching device and the shock wave. Term C2
represents an increase of pressure due to return flow from
the cavity of loaded tubes. These terms could be solved
using following equations:
Figure 4. Possible plume positions on the part of
lunching device
Table 1: Calculation of area under the plume
Period
Condition
Area
I
r1 ≤ h
S1 = P1
II
h < r1 ≤ b
S1 = P1 − P2
III
b < r1 < c
S1 = P1 − P2 − P3
IV
r1 ≥ c
S1 = bh
C1 = 1 +
C2 =
( ) ⋅⎛⎜⎝ κ 2−1 + M2
κ (κ −1) Me2
2
⋅
κ (κ −1) Me2 − 2 κ +1
f1 ( x ) =
Parameters used in table are:
h = min ( b1 ,h2 )
b = max ( b1 ,h2 )
a=
c = b2 + h 2
b=
P1 =
P2 =
P3 =
2
1
r
4
2
+
2
⎞
⎟
(κ −1) Me4 ⎠
1
2
R
1 + ⎛⎜ 1 ⎞⎟
⎝X⎠
2
κ (κ − 1) M e 2
2
Ae =
( 2δ1 − sin 2δ1 ) , δ1 = arccos rh
e
κ (κ − 1) M e 2
r12 π
4
1
κ M e2
de2
4
π
+1
−1
(8)
In the example of a launching device in Figure 3, and for
the rocket of which the plume characteristics are shown in
diagram 1, the plume impingement forces are calculated
at the individual launchings from four different tubes T1,
T12, T15 and T21, as shown in the Figure 5.
1
r12
( 2δ 2 − sin 2δ 2 ) , δ 2 = arccos rb
4
1
323
Previous diagram clearly shows the influence of the tube
disposition not only on maximum value of axial force, but
on total impulse that is transferred to the structure. Thus,
in the case of launching from the farthest point from the
launching device center (tube T1) maximum force FT1 is
approximately 20% less than total maximum force in case
of launching from central tubes (T12, T15), while the
difference in total impulse delivered is about 60% higher
for the central tube.
1800
FT1
1600
FT12
FT15
1400
FT21
F [daN]
1200
1000
800
Figure 6. Possible plume positions on cap
600
400
Table 2: Area of the cap under plume calculation
200
0
0.14
0.15
0.16
0.17
0.18
0.19
t [s]
0.2
0.21
0.22
0.23
Figure 5. Change of axial plume impingement force on
launching device front panel
Maximum value of plume impingement force will be
achieved before the plume radius reaches its maximum
value. It can also be noticed that from the moment of
reaching the maximum value of force to the moment
t = 0.1925 s (when it reaches the maximum value of the
plume radius r1 ≈ 0.529 m), curves have a mild decrease,
because even though that rocket runs away from a
launching device, covered surface of launcher's front
panel increases. From that point the curves decreases
rapidly as a result of missile removal and due to the
reduction of stream surface on the launching device.
Period
Condition
Area
I
Z ≤ q−R
S =0
II
q−R< Z ≤ q
S = I1 + I 2 − I 3
III
q<Z <q+R
S = R 2 π + I1 − I 2 − I 3
IV
Z ≥ q+R
S = R2π
Parameters used in table are:
Y=
(
4 qr12 − r12 + q 2 − R 2
2
4 q2
Z=
4. PLUME IMPINGEMENT FORCE TO CAPS
OF LAUNCHING DEVICE
)
r12 + q 2 − R 2
2q
⎛Y ⎞
I1 = r12 arcsin ⎜ ⎟ + Y r12 − Y 2
⎝ r1 ⎠
This model for calculating the axial plume impingement
force to launching device can also be applied to the
problem of calculating the mechanical load on sealing
tube caps (if the launching device has ones). The only
difference in this case is that the equation (7) term C2=0,
because there is no return flow from the cavity in the
tube-rocket system to the shock wave (now the tube
cavities are closed with sealing caps).
⎛Y ⎞
I 2 = R 2 arcsin ⎜ ⎟ + Y R 2 − Y 2
⎝R⎠
I 3 = 2 qY
For the same example as in the previous discussion, the
plume impingement force delivered to tube T1 cap is
calculated. Launching from four tubes T2, T3, T10 and T12
is simulated. The results are shown in the Figure 7.
Figure 6 shows the possible plume positions r1. To
calculate the current plume surface on the cap, both the
radius R of the cap and inter-axial distance between cap
and tube from which rocket is to launch, q, must be
known. Table 2 gives the appropriate equations for each
phase, as well as the necessary conditions to determine
the current position of the rocket plume.
All four curves show the same character with a tendency
to decrease in force intensity with inter-axial distance
increase between cap and launching tube.
324
150
T2
T3
125
T10
T12
F [daN]
100
75
50
25
0
0.14
0.15
0.16
0.17
0.18
0.19
t [s]
0.2
0.21
0.22
0.23
Figure 7: Change of axial plume impingement force on
the tube T1 cap depending on what tube is launched from
This model of calculations can be useful in designing
launching tube caps, because it determine mechanical
loads to cap, which are part of the overall complex
stresses (thermal-mechanical-abrasive).
5. CONCLUSION
The objective of this paper is to consider the complex
phenomena at the initial stage of rocket flight and their
impact on the launcher.
By introducing a fictitious circular surface, ie. fictitious
radius, the application of algebraic equations for the axial
force of multi-stage rocket separation is enabled, in case of
the combustion shock to front panel of launching device.
The paper presented the dependance of the launching tube
position to the intensity of the axial force maximum
value, as well as to total impulse delivered to the
launching platform. Differences which occur in delivered
impulses to the launching device have their effects on the
overall dynamics of the system, and on the differences in
changes of its momentum. Therefore, this analysis is
needed when determining the stability of the launching
platform. It can also be used as a criterion for determining
the sequence of launching rockets.
As an example of the influence of axial plume
impingement force to the subsystems of launching
platform, the analytical model of the mechanical effect to
launching tubes is presented too.
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